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Novel heuristic search methods for protein folding and identification of folding pathways Shmygelska, Alena 2006

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Novel Heuristic Search Methods for Protein Folding and Identification of Folding Pathways by Alena Shmygelska B.Sc. Computer Science, Slippery Rock University of Pennsylvania, 2001 B . A . Biology, Slippery Rock University of Pennsylvania, 2001 Minors in Physics and Mathematics, Slippery Rock University of Pennsylvania, U S A , 2001 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S FOR T H E D E G R E E O F Doctor of Philosophy in The Faculty of Graduate Studies (Computer Science) The University of British Columbia September 2006 © Alena Shmygelska 2006 Abstract Proteins form the very basis of life. If we were to open up any l i v ing ce l l , we wou ld find, apart from D N A and R N A molecules whose primary role is to store genetic information, a large number of different proteins that comprise the cel l itself (for example the cell membrane and organelles), as wel l as a diverse set o f enzymes that catalyze various metabolic reactions. If enzymes were absent, the ce l l wou ld not be able to function, since a number of metabolic reactions would not be possible. Funct ions of proteins are the consequences of their functional 3 D shape. Therefore, to control these versatile properties, we need to be able to predict the 3 D shape of proteins; in other words, solve the protein folding problem. The prediction of a protein's conformation from its amino-acid sequence is currently one of the most prominent problems in molecular biology, biochemistry and bioinformatics. In this thesis, we address the protein folding problem and the closely-related problem of identifying folding pathways. The leading research objective for this work was to design efficient heuristic search algorithms for these problems, to em-pir ica l ly study these new methods and to compare them with exist ing algorithms. This thesis makes the fo l lowing contributions: (1) we show that b io log ica l ly inspired approaches based on the notion of stigmergy - where a col lec t ion of agents modifies the environment, and those changes in turn affect the decision process of each agent (particularly artificial colonies of ants that give rise to such properties as self-organization and cooperation also observed in proteins) is a promis ing field of study for the protein folding problem; (2) we develop a novel adaptive search framework that is used to identify and to bin promis ing candidate solutions and to adaptively retrieve solutions when the search progress is unsatisfactory; (3) we develop a new method that efficiently explores large search neighbourhoods by performing biased iterated solution construction for identifying fo ld ing pathways; and (4) we show that our algorithms efficiently search the vast search landscapes encountered and are able to capture important aspects of the process of protein folding for some wide ly accepted computational models. i i i Contents Abstract i i Contents i i i List of Tables . v i List of Figures x i i Abbreviations x x i i i Acknowledgements x x v Dedications x x v i 1 Introduction 1 1.1 Structure and Funct ion of Proteins 1 1.1.1 Three Levels of Protein Structure 3 1.1.2 Protein Functions 4 1.2 The Protein Fo ld ing Problem and its Importance 5 1.2.1 Protein Fo ld ing Paradoxes 5 1.2.2 The Thermodynamic Hypothesis 6 1.2.3 The Central D o g m a of Protein Fo ld ing 6 1.2.4 Forces in Protein Fo ld ing 7 1.2.5 Motivat ions for Studying Protein Fo ld ing Problems . . . . 8 1.3 A n Overv iew of Computat ional Approaches to Protein F o l d i n g . . 9 1.3.1 H o m o l o g y M o d e l i n g 10 1.3.2 Threading 10 1.3.3 N o v e l F o l d Recogni t ion Methods 10 1.3.4 Ab initio Methods 11 1.4. Thesis Statement 11 1.5 Organizat ion o f the Thesis 12 Contents iv 2 Description of Problems, Models, and Notations 14 2.1 Protein Folding on the Lattice 16 2.1.1 Square and Cubic Lattices 16 2.1.2 Hydrophobic Polar Energy Potential 16 2.1.3 Face-Centered Cubic Lattice Model 17 2.1.4 Beta Sheet Energy Potential Used with the F C C Lattice . . 18 2.2 Off-Lattice Protein Folding 20 2.2.1 Models for Off-Lattice Protein Folding 20 2.2.2 Energy Potentials for Off-Lattice Protein Tertiary Structure Prediction 22 2.3 The Problem of Folding Pathway Identification 23 2.3.1 Models Used for the Identification of Folding Pathways 24 2.3.2 Objective Functions Used for the Identification of Folding Pathways 24 3 Background and Related Work 26 3.1 The Protein Folding Problem 26 3.1.1 Search Algorithms Used for Protein Folding 28 3.1.2 Summary and Classification of Search Methods 36 3.1.3 An Overview of Existing Research in 2D and 3D Hydropho-bic Polar Folding . . .' 39 3.1.4 An Overview of Existing Research in F C C /3-Sheet Protein Folding 41 3.2 The Problem ofldentifying Folding Pathways 43 3.2.1 Theoretical and Experimental Work on Protein Folding Path-ways ' 44 3.2.2 An Overview of Computational Approaches for Identify-ing Folding Nuclei from the Native Conformation 45 4 An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Pro-tein Folding 47 4.1 Description of the Algorithm 48 4.1.1 Construction Phase, Pheromone, and Heuristic Values . . 51 4.1.2 Local Search 52 4.1.3 Update of the Pheromone Values 53 4.2 Empirical Results and Discussion 54 4.2.1 Results for Standard Benchmark Instances 55 4.2.2 Result for New Biological and Random Data Sets 57 4.2.3 Characteristic Performance Differences between A C O and P E R M 61 Contents v 4.2.4 Discussion of ACO Results 71 4.3 Summary 76 5 Adaptive Bin Framework Search, Introduced for the F C C /3-Sheet Protein Folding Problem 77 5.1 The Bin Framework and the Bin Framework Monte Carlo Algorithm 78 5.1.1 Storing Conformations in the Bin Framework 82 5.1.2 Retrieving Conformations from the Bin Framework . . . . 87 5.2 Empirical Results and Discussion 88 5.2.1 Comparison of Results with the Literature 89 5.2.2 Further Comparison for Homopolymers of Length 12, 24, 32, and 64 90 5.2.3 Discussion of the Bin Framework 99 5.3 Summary 106 6 Construction Search for Identifying Folding Pathways 108 6.1 Description of the Problem 108 6.2 Description of the Algorithm 109 6.2.1 Generation of the Polymer Graph 110 6.2.2 Sampling of Folding Pathways 110 6.2.3 Collective Analysis of Low Effective Contact Order Path-ways 113 6.3 Empirical Results and Discussion 114 6.4 Summary 121 7 Conclusions and Future Directions 127 7.1 ACO for 2D and 3D Hydrophobic Polar Folding .- 127 7.2 Adaptive Bin Framework for the FCC /? - Sheet Model 128 7.3 Construction Search for Identification of Folding Pathways . . . . 130 7.4 Summary 130 Bibliography 134 A Ant Colony Optimization 145 B Adaptive Bin Framework Monte Carlo Search 152 Index 168 List of Tables 4.1 Benchmark instances for the 2 D and 3 D H P Protein F o l d i n g Prob-lem used in this study with optimal or best known energy va l -ues E*. M o s t instances for 2 D and 3 D H P can also be found at http://www.cs.sandia.gov web site; Sequence S l - 9 (2D) is taken from [67], and the last two instances (2D) are from [110]. Hi and Pi indicate a string of i consecutive H ' s and P 's , respectively; l ike-wise, (s)i indicates an -i-fold repetition o f string s 4.2 Compar i son of the solution quality obtained in 2 D by the evolu-tionary algorithm of Unger and M o u l t ( E A ) [135], the evolutionary Mon te Ca r lo algorithm of L i a n g and W o n g ( E M C ) [77], the M u l t i -Self -Overlap Ensemble algorithm of Chickenj i etal. ( M S O E ) [24], the pruned-enriched Rosenbluth method ( P E R M ) and A C O . For E A and E M C , the reported energy values are the lowest among five independent runs, and the values in parentheses are the numbers of val id conformations scanned before the lowest energy values were found. M i s s i n g entries indicate cases where the respective method has not been tested on a given instance. The C P U times reported in parentheses for M S O E were determined on a 500 M H z C P U , and those for P E R M and A C O are based on 100 - 200 runs per instance on our reference 2.4 G H z Pentium I V machine. The en-ergy values shown in bo ld face correspond to currently best-known solution qualities List of Tables v i i 4.3 Compar i son of the solution quality obtained in 3 D by the hydropho-bic zipper ( H Z ) algori thm [33], the constraint-based hydrophobic core construction method ( C H C C ) [150], the core-directed chain growth algori thm ( C G ) [13], the contact interactions (CI) algo-ri thm [132], the pruned-enriched Rosenbluth method ( P E R M ) and A C O . For C I , only the best energies obtained are shown. For H Z , C H C C and C G , the reported C P U times are taken from [13]; these are the expected times for finding optimal solutions on a Sparc 1 workstation. In the case of H Z , the reported C P U times are based on an extrapolation from the measured times required for finding suboptimal conformations with the energy values listed here. The C P U times for P E R M and A C O were determined on our reference 2.4 G H z Pent ium I V machine based on 50 — 100 runs per instance. The energy values shown in bold face correspond to currently best-known solution qualities 59 List of Tables viii 5.1 Comparison of the solution quality obtained for the homopoly-mer of length N = 64 by the Monte Carlo Simulated Annealing (MCSA) [52], the Replica Exchange Monte Carlo (REMC) with a linear set of temperatures [52] and the Parallel-hat Tempering algorithm (PHAT) [154] with our implementation of Monte Carlo (MC), R E M C and PHAT and our new Bin Framework Monte Carlo (BINMC). The time reported for M C S A and R E M C from [52] is the estimated time required to run on 2.4 GHz machines (in the original paper authors used 500 MHz processor, therefore, re-ported times in [52] are conservatively divided by a factor of 4.8). The time reported for Parallel-hat tempering [154] is the estimated time to run on 2.4 GHz as well (in the original paper [154] C P U of 750 MHz was used, therefore we conservatively applied a fac-tor of 3.2). The authors in [52] also implemented R E M C with an exponential set of temperatures. The exact temperatures were not specified in the paper [52], however, and the authors could not re-call it in personal communication (the lowest energy observed in this case was —374). The number of runs used for comparison was 10 for all algorithms. In the last column of the table, we re-port p-values indicating the probability that the null hypothesis of no difference between the mean energies reached over 10 runs for B I N M C and a particular algorithm listed (within the same C P U cut-off time) is true; we used the Mann-Whitney U test to calculate p-values [59]; * indicates that p-values are below the significance level of 0.05 of wrongly rejecting the null hypothesis 91 5.2 Comparison of the average solution quality obtained and the aver-age time required for the homopolymers of lengths N = 12,24, 32 for the re-implemented M C , R E M C , PHAT, and B I N M C . The time cut-off used was 1 hr on 2.4 GHz reference machine, and the av-erages were calculated from 10 independent runs. Temperature sets used for the re-implemented algorithms are the same as in Table 5.1. In the last column of the table we report p-values in-dicating the probability that the null hypothesis of no difference between the mean C P U run-time (for the homopolymer of length 24) or the mean energies reached over 10 runs (for the homopoly-mer of length 32) for B I N M C and a particular algorithm listed is true; we used the Mann-Whitney U test to calculate p-values [59]; * indicates that p-values are below the significance level of 0.05 of wrongly rejecting the null hypothesis 96 L i s t of Tables ix 5.3 Comparison of the solution quality obtained for the homopolymers of length N = 64 and N — 32 by re-implemented M C , R E M C with the linear set of temperatures, PHAT, and our new B I N M C on 2.4 GHz reference machine with a time cut-off of 10 hrs over 10 independent runs. In the last column of the table, we report p-values indicating the probability that the null hypothesis of no difference between the mean energies reached over 10 runs for B I N M C and a particular algorithm listed (within the same C P U cut-off time) is true; we used the Mann-Whitney U test to calculate p-values [59]; * indicates that p-values are below the significance level of 0.05 of wrongly rejecting the null hypothesis 97 6.1 Set of proteins used in this study. The kinetic data about folding pathways of these proteins is available in [76] and [107] 126 A . l Biological sequences of length « 30 145 A.2 Performance comparison of P E R M and A C O on biological sequences of length ss 30 in 2D 146 A.3 Performance comparison of P E R M and A C O on biological sequences of length « 30 in 3D 146 A.4 Biological sequences of length « 50 147 A.5 Performance comparison of P E R M and A C O on biological sequences of length w 50 in 2D 147 A.6 Performance comparison of P E R M and A C O on biological sequences of length « 50 in 3D 148 A.7 Random sequences of length 30 148 A.8 Performance comparison of P E R M and A C O on random sequences of length 30 in 2D 149 A.9 Performance comparison of P E R M and A C O on random sequences of length 30 in 3D 149 A . 10 Random sequences of length 50 150 A . l 1 Performance comparison of P E R M and A C O on random sequences-of length 50 in 2D 150 A . 12 Performance comparison of P E R M and A C O on random sequences of length 50 in 3D 151 B. l Vectors for the best found conformation of 64 amino acids (total energy = —391, short-range energy = —212, long-range energy = -179) 153 List of Tables x B.2 Continued, vectors for the best found conformation of 64 amino acids (total energy = —391, short-range energy = —212, long-range energy = —179) 154 B.3 Triplets of vectors, angles: 6\ (between vectors v ; _ i and v;), 6*2 (between vectors v; and v ; + i ) , #3 (between vectors v ; _ i and v ; + 1 ) , short-range energy contributions for the best found conformation of 64 amino acids (total energy = —391, short-range energy = -212, long-range energy = -179) 155 B.4 Continued, triplets of vectors, angles: 9\ (between vectors V j _ ! and Vj) , 62 (between vectors v; and V i + 1 ) , #3 (between vectors V j _ ! and v ; + 1 ) , short-range energy contributions for the best found conformation of 64 amino acids (total energy — —391, short-range energy = —212, long-range energy = —179) 156 B.5 Long-range interactions for the best found conformation of 64 amino acids (total energy = —391, short-range energy = —212, long-range energy = —179) 157 B.6 Continued, long-range interactions for the best found conformation of 64 amino acids (total energy — —391, short-range energy = — 212, long-range energy = —179) 158 B.7 Continued, long-range interactions for the best found conformation of 64 amino acids (total energy = —391, short-range energy = -212, long-range energy = -179) 159 B.8 Continued, long-range interactions for the best found conformation of 64 amino acids (total energy = —391, short-range energy — —212, long-range energy = -179) 160 B.9 Vectors for the best found conformation of 32 amino acids (total energy = —161, short-range energy = —112, long-range energy = -49) 161 B.10 Triplets of vectors, angles: 6\ (between vectors v ; _ i and v0, #2 (between vectors v; and v ; + i ) , #3 (between vectors v ; _ i and V i + 1 ) , short-range energy contributions for the best found conformation of 32 amino acids (total energy — —161, short-range energy = — 112, long-range energy = —49) 162 B . l 1 Long-range interactions for the best found conformation of 32 amino acids (total energy = —161, short-range energy = —112, long-range energy = —49) 163 B.12 Vectors for the best found conformation of 24 amino acids (total energy = —109, short-range energy = —68, long-range energy = -41) 164 List of Tables x i B .13 Triplets o f vectors, angles: 6\ (between vectors ^ \ - \ and v ; ) , 82 (between vectors v ; and V j + 1 ) , f9.j (between vectors v ; _ x ' and v;_|_i), short-range energy Contributions for the best found conformation of 24 amino acids (total energy = —109, short-range energy = . —68, long-range energy = —41) 165 B . l 4 Long-range interactions for the best found conformation of 24 amino acids (total energy = —109, short-range energy = —68, long-range energy = —41) 166 B .15 Vectors for the best found conformation of 12 amino acids (total energy — —39, short-range energy = —28, long-range energy = - 1 1 ) 167 B .16 Triplets o f vectors, angles: 0\ (between vectors v ; _ x and v ; ) , t92 (between vectors v ; and v ; + i ) , 0 3 (between vectors V j _ i and v ; + i ) , short-range energy contributions for the best found conformation of 12 amino acids (total energy = - 3 9 , short-range energy = - 2 8 , long-range energy = —11) 167 B . 17 Long-range interactions for the best found conformation of 12 amino acids (total energy = —39, short-range energy = —28, long-range energy = —11) 167 X l l List of Figures 1.1 (a) Structure of an amino acid (electrical charges are not indicated); (b) peptide bond between neighbouring amino acids 2 1.2 (a) Dihedra l angles (4>,ip); (b) Ramachandran plot o f a l lowed val -ues for the dihedral angles. A l l regions shown in white are dis-a l lowed 3 1.3 (a) Majo r elements of secondary structure: a -he l ix (left), /3-sheet (right); (b) tertiary structure of M y o g l o b i n 4 2.1 (a) A n example of a protein conformation on the 2 D square lattice; (b) an example of a protein conformation on the 3 D cubic lattice. Two types of monomers are considered: hydrophobic amino acids are colored in red, polar are green, ye l low dotted lines represent energy interactions 17 2.2 (a) Depic t ion of the 12 base vectors for the F C C lattice model ; (b) an example of several F C C lattice cells 19 2.3 The effective contact order ( E C O ) between i and j is equal to 5, compared to the contact order ( C O ) of 7 between the same residues. 25 3.1 A n example of a protein's funneled energy landscape 28 4.1 Generic outline of A n t C o l o n y Opt imiza t ion (for static combinato-rial problems) 49 4.2 The local structural motifs that form the solution components un-der ly ing the construction and local search phases of our A C O a l -gor i thm in 3 D 49 4.3 The underlying protein sequence (Sequence S l - 1 from Table 4.1) is H P H P P H H P H P P H P H H P P H P H ; black circles represent hy-drophobic amino acids, whi le white circles symbol ize polar amino acids. The dotted lines represent the H - H contacts underlying the energy calculat ion. The energy of this conformation is -9, wh ich is optimal for the given sequence 50 List of Figures xiii 4.4 The iterative first improvement local search procedure that is per-formed by selected ants after the construction phase 53 4.5 The native conformation of Sequence Sl-8 from Table 4.1 (64 amino acids; energy —42), found by A C O in an average C P U time of 1.5 hours and by P E R M in ti — t2 — texp = 78 hours 60 4.7 Mean C P U time (natural log transformed) required by A C O P E R M for reaching the best solution quality, as observed over 10 runs with a cut-off time of 1 C P U hour for sequences of length 30 and 50 in 3D. The left and right plots show the results for the biological and random test-sets, respectively. Performance results for instances of size 30 are indicated by circles, while stars mark results for instances of size 50. The dashed lines indicate the band within which performance differences are not statistically signifi-cant (significance was determined using the Mann-Whitney U test and setting the significance level at 0.05 [59]). Mean run-times were obtained from 10 runs per instance and algorithm. We only show data points for the runs where the best known solution qual-ity was reached at least in some runs out of 10 by both algorithms. When unsuccessful runs were present, the expected time was cal-culated as in [101] . For biological sequences of length 50, there are four instances and for random instances of length 50, there are two instances for which this was the case. Detailed results for both successful and unsuccessful runs are given in Appendix A 63 4.8 Left side: Lowest energy conformation of a biological sequence (B50-7, 45 amino acids, energy —17) that is harder for P E R M (*i = 271, t2 = 299, texp = 284 C P U seconds) than for A C O (texp = 130 C P U seconds; cut-off time 1 C P U hour). Right side: Lowest energy conformation of a biological sequence (B50-5, 53 amino acids, energy —22) that is much harder for A C O than for P E R M ; within a cut-off time of 1 C P U hour, both A C O and P E R M reached this energy in 10 out of 10 runs in tavg = 820 and t\ = 5, t2 — 118, texp = 9 C P U seconds on average, respectively 64 List of Figures xiv 4.9 Left side: Unique minimal energy conformation of a designed se-quence, D - l (length 50, energy -19); A C O reaches this confor-mation much faster than P E R M when folding from the left end (mean run-time over 100 successful runs for A C O : 236 C P U sec-onds, compared to t\ = 3 795, t2 = 1, t'exp — 2 C P U seconds for PERM). Right side: Unique native conformation of another de-signed sequence, D-2 (length 60, energy -17). A C O finds this con-formation much faster than P E R M folding from either end (mean run-time over 100 successful runs for A C O : 951 C P U seconds, compared to tx = 9 257, t2 = 19 356, texp = 12 524 C P U sec-onds for P E R M ) 65 4.10 Left side: Lowest energy conformation of random sequence R50-9 (50 amino acids, energy —30), which is harder for P E R M when folding from the left end than for A C O . With a cut-off time of 1 C P U hour, A C O reached this energy in 10 out of 10 runs with texp = 1000 C P U seconds, while P E R M failed to find a confor-mation with this energy in 7 out of 10 runs when folding from the left end (*i = 9 892, t2 = 2, texp = 3 C P U seconds). The confor-mation displayed here has relative directions: D S R U U R L R R L D D -L U R R U L R S D U R U R L R R D D U S R R S U L R R D U U L S L L U R . Right side: Lowest energy conformation of random sequence R50-7 (50 amino acids, energy —38), which is much harder for A C O than for P E R M . With a cut-off time of 1 C P U hour, P E R M reached this energy in two out of 10 runs when folding from the left and in 10 of 10 runs when folding from the right end in t\ = 15 322, t2 — 46, texp = 92 C P U seconds, while the lowest energy reached by A C O over ten runs was —37. The conformation displayed here has relative directions: S U U R U D U L U U D S S U U S R U D D L D S S U -U R R U U L D L D R D D U R L L D L R U S U R 66 4.11 Distributions of H - H contact order for 500 conformations of Se-quence Sl-7 from Table 4.1 (60 amino acids) in 2D (left side) and Sequence Sl-5 from Table 4.1 (48 amino acids) in 3D (right side) found by A C O and P E R M 67 4.12 Mean hydrophobic solvent accessible area as a function of prefix length for a biological sequence (B50-4, 50 amino acids) in 2D (left side) and Sequence S2-6 from Table 4.1 (48 amino acids) in 3D. Crosses and circles represent mean values for an ensemble of 100 native structures found by A C O and P E R M , respectively. . . . 67 List of Figures xv 4.13 Mean number of H - H contacts as a function of prefix length for a biological sequence (B50-4, 50 amino acids) in 2D (left side) and Sequence S2-6 from Table 4.1 (48 amino acids) in 3D. Crosses and circles represent mean values for an ensemble of 100 native structures found by A C O and P E R M , respectively 68 4.14 Mean H - H contact order as a function as a function of prefix length for a biological sequence (B50-4, 50 amino acids) in 2D (left side) and Sequence S2-6 from Table 4.1 (48 amino acids) in 3D. Crosses and circles represent mean values for an ensemble of 100 native structures found by A C O and P E R M , respectively 68 4.15 Effect of the relative weights of pheromone information, a, and heuristic information, B, on the average C P U time required for obtaining minimal energy conformations of Sequence SI-8 in 2D (length 64, left side) and Sequence S2-5 in 3D (length 48, right side) 73 4.16 Effect of the pheromone persistence parameter,' p, on the average C P U time required for obtaining minimal energy conformations of Sequence S1-8 in 2D (length 64, left side) and Sequence S2-5 in 3D (length 48, right side) 73 4.17 Mean C P U time required for finding minimum energy conforma-tions of Sequence S1-7 in 2D (length 60, left side) and Sequence S2-5 in 3D (length 48, right side), as a function of ant colony size and the maximum number of non-improving local search steps. . . 74 4.18 Mean C P U time required for finding minimum energy conforma-tions of Sequence S1-8 in 2D (length 64, left side) and Sequence S2-5 in 3D (length 48, right side), as a function of the probabil-ity of retaining previous directions (p) during long-range mutation moves 75 5.1 A n illustration of how candidate solutions at a given state of the bin framework relate to the search space of a given problem instance. Eo is the best solution quality found so far and serves as an estimate of the ground state energy, L\E is the energy range of interest, and conformations within this range are binned. Each bin i has energy threshold diversity threshold HDi, and energy window AEi. 81 List of Figures xvi 5.2 High- leve l outline of the main body of the B i n Framework M o n t e Ca r lo algori thm. Eo is the current best solution quali ty; nolm-prRetrieve is a parameter of the algorithm that specifies number of non- improving steps over the best energy that are tolerated before a new conformation is retrieved from the bin system; HDMAX and HDMIN a r e parameters defining the H a m m i n g distance diversity criteria for high-energy and low-energy conformations correspond-ingly; (3MC and are parameters that refer to the inverse tem-peratures used for the Monte Car lo run and for the bin framework correspondingly; AEi is a parameter that represents the w indow of energies by each bin i; AE is a parameter defining the range of energies of interest. Conformations whose energy falls wi th in this range are attempted to be stored in the bin framework. The functions PlacelntoBin and CheckPlacelntoBin are defined in F i g -ures 5.3 and 5.4 correspondingly 83 5.3 Out l ine for the function used to place the conformation with the lowest energy encountered so far into the bin system, where c is the conformation with energy E(c) to be placed into a b in , Eo is the estimated ground state energy (the lowest energy seen in the simulat ion so far), HDMAX and HDMIN are parameters defin-ing the H a m m i n g distance diversity criteria for high-energy and low-energy conformations correspondingly; AEi is a parameter that represents the window of energies considered by each b in i, AE is a parameter defining the range of energies of interest - con-formations whose energy falls wi thin this range are attempted to be stored in the bin framework. '. . 84 5.4 Out l ine for the function used to place the conformation c wi th a low energy E(c) encountered into the bin system, where Eo is the estimated ground state energy (the lowest energy seen in the simulat ion so far), HDMAX and HDMIN are parameters defin-ing the H a m m i n g distance diversity criteria for high-energy and low-energy conformations correspondingly, AEi is a parameter that represents the window of energies considered by each bin i, AE is a parameter defining the range of energies of interest - con-formations whose energy falls wi th in this range are attempted to be stored in the bin framework 85 List of Figures x v i i 5.5 Placement of the low-energy conformation c with energy E(c) into an appropriate bin i wi th the energy threshold Ef and the H a m -ming distance criterion HDi. In order for the conformation c to be stored in the bin framework the fo l lowing conditions need to be satisfied: (1) E(c) < Ef, and (2) the H a m m i n g distance (HD) be-tween conformation c and conformations c' with the same energy already stored in the bin ( if there are any) is larger or equal to HDi. 86 5.6 Outl ine for the function used to retrieve a conformation from the bin system. Eo is the current best solution quality; /?;,;„ is the in -verse temperature used for the b in framework during the retrieval of conformations 88 5.7 (a) The lowest energy conformation of the F C C 64 amino acids homopolymer found by our B i n Framework Mon te Ca r lo method (energy —391, short-range energy is —212, long-range energy is —179). The detailed description of this conformation is also found in Append ix B ; (b) same conformation, view from above; (c) a low-energy conformation of the F C C 64 amino acids homopoly-mer found by B I N M C (energy - 3 8 7 , short-range energy is —212, long-range energy is —175); this was found in the same run that lead to the conformation with the best energy of —391; (d) another low-energy conformation of the F C C 64 amino acids homopoly-mer found in the same run of B I N M C (energy —388, short-range energy is —212, long-range energy is —176) 92 5.8 (a) A low-energy conformation of the F C C 64 amino acids ho-mopolymer found by B I N M C (energy —387, short-range energy is — 208, long-range energy is —179); (b) another low-energy confor-mation of the F C C 64 amino acids homopolymer found in the same run of B I N M C (energy —389, short-range energy is —212, long-range energy is —177); (c) a low-energy conformation of the F C C 64 amino acids homopolymer found by B I N M C (energy —387, short-range energy is —208, long-range energy is —179); (d) an-other low-energy conformation of the F C C 64 amino acids ho-mopolymer found in the same run o f B I N M C (energy —389, short-range energy is —220, long-range energy is —169) 93 List of Figures XVlll 5.9 (a) The low-energy conformation of the F C C 64 amino acids ho-mopolymer found by our bin framework (energy —387, short-range energy is —220, long-range energy is —167); (b) another low-energy conformation of the F C C 64 amino acids homopolymer found by our bin framework (energy —387, short-range energy is —220, long-range energy is —167); (c) the same conformation as in part (a), view from above; (d) the same conformation as in part (b), view from above 94 5.10 The lowest energy conformation of the F C C homopolymers of 12,24, and 32 (same conformation from different point of view) amino acids (from left to right) found by all of the algorithms (corre-sponding energies are: -39 , short-range energy is —28, long-range energy is —11 for the 12 amino acid polymer; —109, short-range energy is —68, long-range energy is —41 for the 24 amino acid polymer; and —161, short-range energy is —112, long-range en-ergy is —49 for the 32 amino acid polymer). The detailed descrip-tion of these conformations is also found in Appendix B 95 5.11 Run-time distributions of C P U times on our 2.4 GHz reference machine to obtain a sub-optimal solution quality of —158 for the homopolymer of length 32 (part (a)) and to obtain sub-optimal so-lution quality of —370 for the homopolymer of length 64 (part (b)) using Monte Carlo (MC), Replica Exchange Monte Carlo (REMC), Parallel-hat Tempering (PHAT), and the Bin Framework Monte Carlo (BINMC). We fit the RTD of B I N M C for the homopolymer of length 64 with exponential distribution, to show by example that RTDs for all algorithms are exponential. For all algorithms, 100 in-dependent runs were performed. In all of them, the target energy was reached 99 5.12 Distributions of energies visited by different replicas in a represen-tative run of the Replica Exchange Monte Carlo (REMC) (part (a)) and in a representative run of the Parallel-hat Tempering Monte Carlo (PHAT) (part (b)) for the homopolymer of length 64. The time cut-off used was 28 min on our reference machine 100 5.13 Distributions of energies visited by the Monte Carlo (MC) and our Bin Framework Monte Carlo (BINMC) for the homopolymer of length 64. The time cut-off used was 28 min on our reference machine 100 List of Figures x ix 5.14 M e a n C P U time required for finding m i n i m u m energy conforma-tions (—161) of the 32 amino acid homopolymer over 20 indepen-dent runs on our reference machine as a function of the number of non- improving steps (noImprRetrieve) before retrieving a confor-mation from bins. Error bars indicate standard deviation observed. Dashed line indicates that energy of —161 was not found after 2 weeks ' C P U time on our reference machine 102 5.15 M e a n C P U time required for finding m i n i m u m energy conforma-tions (—161) of the 32 amino acid homopolymer over 20 indepen-dent runs on our reference machine as a function of combinat ion of parameters: the energy range of interest ( A i ? ) and the bin 's temperature (Tb; n ) , p = e " A f i / T w " . Error bars indicate standard deviation observed 102 5.16 M e a n C P U time required for finding m i n i m u m energy conforma-tions (—161) of the 32 amino acid homopolymer over 20 indepen-dent runs on our reference machine as a function of the H a m m i n g distance criterion (HDMAX is varied, HDMIN = 0-1 is kept con-stant, part (a) and HDMIN is varied, HDMAX = 0-6 is kept con-stant, part (b)). Error bars indicate standard deviation observed. . . 104 5.17 M e a n C P U time required for finding m i n i m u m energy conforma-tions ( - 1 6 1 ) of the 32 amino acid homopolymer over 20 indepen-dent runs on our reference machine as a function of the energy window width AEi. Er ror bars indicate standard deviation observed. 105 5.18 M e a n C P U time required for finding m i n i m u m energy conforma-tions (—161) of the 32 amino acid homopolymer over 20 indepen-dent runs on our reference machine as a function of the bin 's ca-pacity. Error bars indicate standard deviation observed 106 6.1 Po lymer graph generated in phase one of our algori thm for the chy-motrypsin inhibitor 2 (CI2) protein containing 65 nodes (n) and 82 edges (m) 110 List of Figures xx 6.2 Illustration of the sampling phase of the algorithm. Edges that are added to the polymer graph are highlighted in bold, while edges that still have to be added are represented in non-bold. Initially, we start with a fully extended polymer, and edges that are to be added are weighted by their chain separation. At each step one edge is probabilistically added based on its E C O weight, and the E C O weights of the edges that are still to be added have to be revised at each step. The process of addition of edges stops when all of the edges are added to the polymer graph and their respective E C O s shrink to 1 112 6.3 Outline for the probabilistic constructive local search used in the sampling of the folding pathways stage of our algorithm, where k is the number of non-covalent edges already added to the poly-mer graph and m is the total number of non-covalent edges in the polymer graph 113 6.4 Outline of the analysis phase of our algorithm that identifies fold-ing nuclei contacts in the low effective contact order pathway en-semble. Parameter Al is the length threshold used to identify con-tacts whose formation proceeds rapidly 115 6.5 Band representation of the experimental and predicted folding nu-clei for our test set of 27 proteins, Part I. The upper band represents H / X experimental data (darker-shaded residues represent those that gain protection first during folding, lighter-shaded residues repre-sent residues that have slow exchange in the native state H / X [76]). The middle band (if available) represents experimental ^-values (the darker the shade, the higher the $-value (0 < <p < 1) [93]. The lower band represents our folding nuclei predictions, shaded according to percentage of involvement in low-ECO events (the ex-act scale in percent is given on the right side; proteins are grouped according to the maximum percentage of edge usage) 117 List of Figures x x i 6.6 B a n d representation of the experimental and predicted folding nu-clei for our test set o f 27 proteins, Part II. The upper band rep-resents H / X experimental data (darker-shaded residues represent those that gain protection first during folding, lighter-shaded residues represent residues that have slow exchange in the native state H / X [76]). The middle band (if available) represents experimental ^ -va lues (the darker the shade, the higher the value (0 < cp < 1) [93]. The lower band represents our folding nuclei predictions, shaded according to percentage of involvement in l o w - E C O events (the ex-act scale in percent is given on the right side; proteins are grouped according to the m a x i m u m percentage of edge usage) 118 6.7 B a n d representation of the experimental and predicted fo ld ing nu-clei for our test set o f 27 proteins, Part III. The upper band rep-resents H / X experimental data (darker-shaded residues represent those that gain protection first during folding, lighter-shaded residues represent residues that have slow exchange in the native state H / X [76]). The middle band ( if available) represents experimental ^ -va lues (the darker the shade, the higher the <3>-value (0 < qb < 1) [93]. The lower band represents our folding nuclei predictions, shaded according to percentage of involvement in l o w - E C O events (the ex-act scale in percent is given on the right side; proteins are grouped according to the max imum percentage of edge usage) 119 6.8 Predicted order of contact formation for the chymotrypsin inhibi tor 2 protein (pdb id 2ci2) . Contact matrix and three-dimensional structure representations are gray-scale-coded according to percent contact usage (the scale is given on the right). Indexes n \ and r i 2 represent indexes of the residues along the backbone. Darker-shaded edges (contacts) are predicted to form first 120 6.9 Predicted order o f contact formation for the B I immunoglobu l in -binding domain of protein G (left) and protein L (right), (pdb ids l p g a and 2ptl). Darker shaded edges (contacts) are predicted to form first ' 121 6.10 Predicted order o f contact formation for the the barnase protein (pdb id la2p) . Darker shaded edges (contacts) are predicted to form first. . . 122 6.11 Predicted order of contact formation for the chicken src S H 3 do-main (pdb id l s rm) . Darker shaded edges (contacts) are predicted to form first 123 6.12 Predicted order of contact formation for the C h e Y protein (pdb id 3chy). Darker shaded edges (contacts) are predicted to form first. . 123 List of Figures xxii 6.13 Predicted order of contact formation for the ubiquitin protein (pdb id lubi). Darker shaded edges (contacts) are predicted to form first. 124 6.14 Predicted order of contact formation for the T4 lysozyme protein (pdb id 31zm). Darker shaded edges (contacts) are predicted to form first 124 6.15 Predicted order of contact formation for the bovine pancreatic trypsin inhibitor protein (pdb id lbpi). Darker shaded edges (contacts) are predicted to form first 125 6.16 Predicted order of contact formation for the B domain of protein A (pdb id lbdd). Darker shaded edges (contacts) are predicted to form first 125 XX111 Abbreviations 2D Two-Dimensional 3D Three-Dimensional A C O Ant Colony Optimization ATP Adenosine Triphosphate B I N M C Bin Framework Monte Carlo Algorithm C A S P Critical Assessment of Techniques for Protein Structure Prediction C G Core-directed Chain Growth Method C H C C Constraint-based Hydrophobic Core Construction Method CI Contact Interactions Method C O Contact Order C P U Central Processing Unit D N A Deoxyribonucleic Acid DSSP Database of Secondary Structure Assignments for Proteins E A Evolutionary Algorithm E C O Effective Contact Order E L P Energy Landscape Paving E M C Evolutionary Monte Carlo Algorithm F C C Face Centered Cubic FIRST Freely Rotating Rodes Method G A Genetic Algorithm G C P Graph Coloring Problem G N M Gaussian Network Model H-bonding Hydrogen Bonding H-contact Hydrophobic Contact H - H Hydrophobic-Hydrophobic H / X Hydrogen-Deuterium Exchange H D Hamming Distance HP Hydrophobic Polar H Z Hydrophobic Zipper JSP Job-Shop Scheduling Problem M B S Model-Based Search M C Monte Carlo Method Abbreviations xxiv M C M C Markov Chain Monte Carlo MCSA Monte Carlo Simulated Annealing MD Molecular Dynamics MSOE Multi-Self'-Overlap Ensemble M U C A Multicanonical Algorithm NMR Nuclear Magnetic Resonance NP-hard Non-deterministic Polynomial-Time Hard REMC Replica Exchange Monte Carlo RMSD Root Mean Square Deviation RNA Ribonucleic Acid PDB Protein Data Bank PERM Pruned-Enriched Rosenbluth Method PHAT Parallel-Hat Tempering QAP Quadratic Assignment Problem R A M Random-Access Memory RTD Run-Time Distribution SA Simulated Annealing SAT Propositional Satisfiability Problem SLS Stochastic Local Search TS Tabu Search TSP Traveling Salesman Problem VRP Vehicle Routing Problem X X V Acknowledgements I would like to thank my supervisor Holger Hoos for his guidance, thoroughness, and keen insight. I would also like to thank Steven Plotkin for providing point-ers throughout my research and for bringing up and discussing relevant questions. M y respectful and profound gratitude to my supervisory committee: Holger Hoos, Steven Plotkin, Alan Mackworth, Nando de Freitas, Anne Condon, for discussing research ideas, for your keen observations, for helping with different aspects of the research, giving valuable suggestions, providing constructive criticism, and proof-reading the thesis. Thanks to members of the B E T A lab who were always there to discuss any research-related and unrelated issues. M y gratitude and profound ap-preciation to the National Science and Engineering Council of Canada (NSERC) for funding my doctoral studies. Special thanks to Sam Thangiah, my undergrad-uate research supervisor who taught me how gratifying research curiosity can be when one discovers answers to intriguing scientific questions. M y utmost gratitude and admiration to my family (my mom Klara and my brother Valik) for all the remarkable and little things they do in life, who inspire me, give me greatest warmth and happiness, who prove each day that "to live without roots takes a stout heart" [E. M. Remarque]. M y sincere gratitude to my loyal and incomparable friends for guarding my solitude during the trying time of doctoral research, sharing your time, your thoughts, your aspirations, creating the treasure of common memories and emotions. Thank you for sharing the apple-green sky, snow covered mountains, deserts and crystal lakes springing to life and taking on a meaning beyond their natural beauty in your company. Our time to-gether and our travel adventures helped me to find new meaning in old appearances during my time here. "There are no reasons for our closeness... one is in need of a friend in order for one to be oneself" [R. Rolland]. I am incredibly grateful that I completed this scientific journey accompanied and supported by all of you, and now, "...embarked upon my illicit exploration of the world" [A. de S.-Exupery], here is my thesis. xxvi With my love to my M o m : I know myself through you. In memory of my Grandfather, Volf Buhman. x x v i i IF If you can keep your head when all about you Are losing theirs and blaming it oil you, If you can trust yourself when all men doubt you, But make allowance for their doubting too; If you can wait and not be tired by waiting, Or being lied about, don't deal in lies, Or being hated, don't give way to hating, And yet don't look too good, nor talk too wise: If you can dream - and not make dreams your master, If you can think - and not make thoughts your aim; If you can meet with Triumph and Disaster And treat those two impostors just the same; If you can bear to hear the truth you've spoken Twisted by knaves to make a trap for fools, Or watch the things you gave your life to, broken, And stoop and build 'em up with worn-out tools: If you can make one heap of all your winnings And risk it all on one turn of pitch-and-toss, And lose, and start again at your beginnings And never breath a word about your loss; If you can force your heart and nerve and sinew To serve your turn long after they are gone, And so hold on when there is nothing in you Except the W i l l which says to them: "Hold on!" If you can talk with crowds and keep your virtue, Or walk with kings - nor lose the common touch, If neither foes nor loving friends can hurt you, If all men count with you, but none too much; If you can fill the unforgiving minute With sixty seconds' worth of distance run, Yours is the Earth and everything that's in it, And - which is more - you' l l be a Man, my son! Rudyard Kipling xxviii ECJIM ECJIM c y i v i e e i i i B coxpaHHTb CBOH p a 3 y M , Kor.ua Bee 6 y / i y T HBITB, BO BCSM BMHMTL Te6a, M Bepy coxpaHHmb, XOTB noTycKHeeniL pa30M B rna3ax nfoj,eM ( n y c T L , T H MM He cy/ita); ECJIM cyMeeuiL -MR&Th H u,enn ^oSnBaTbCH H , n o B C T p e q a B oSMaH, TBI He n o r p . 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Francis Crick In the entire realm of science, there is probably no class of molecule currently known that can compete with proteins when it comes to diversity of functions per-formed. Proteins serve as structural units, participate in the catalysis or inhibition of various chemical reactions, perform a transport role, carry out energy transduc-tion, control metabolic pathways, stabilize the architecture within the cell, perform a protective function and.many others. If there is a job to be done in the biological organism, there almost always exists a protein to perforin the required task. The protein folding problem has played a prominent role in the fields of biomolec-ular physics and algorithm design for over 50 years. The importance of this prob-lem increases continually with the exponential growth in the number of known protein sequences and only linear increase in the number of known protein struc-tures [12]. 1.1 Structure and Function of Proteins Proteins are the final products of most genes. They are chain-like polymers of small subunits {amino acids). Twenty different amino acids are distinguished. Modified forms of these 20 amino acids do exist, but they are less common. The chemical structure common to all amino acids, with the exception of Proline, is given in Fig-ure 1.1. Each amino acid has an amino group (NH^), a carboxyl group (COO~), a hydrogen atom (H), and a side chain (R-group, attached to a central alpha-carbon C°) that is different from one amino acid to another. In Proline, the side-chain bridges to the nitrogen atom of the amino group of the backbone (backbone atoms are N, Ca, C, O, and hydrogen atoms attached to nitrogen and alpha-carbon). The standard amino acids can be loosely grouped into classes based on their chemical properties. Commonly accepted classes are hydrophobic, polar or hydrophilic, and Chapter 1. Introduction 2 charged [17]. W i t h i n these broad classes, further classifications are possible, for example: aromatic or aliphatic, large or small [17]. The arrangement of amino acids with their distinct side chains gives each pro-tein its unique character. The amino acids jo in v ia a peptide bond between the amino group and carboxyl group, as shown in Figure 1.1, part (b). Therefore, the protein (polypeptide) has polarity: a free amino group {amino terminus or N - ter-minus) on its left and a free carboxyl group (carboxyl terminus or C - terminus) on its right end. The peptide bond is planar and quite r igid, see Figure 1.2. Therefore, the polypeptide chain has rotational freedom only about the bonds formed by the alpha-carbon. The angles that denote this rotational freedom are noted as cf> - rota-t ion around the Ca-N axis, and ip - rotation in reference to the Ca-C axis; thus, a protein of length n has 2n torsion angle degrees of freedom for posi t ioning the backbone (main chain). The dihedral angles ip and <p are graphical ly depicted in Figure 1.2, part (a). However, the rotational freedom about these angles is l i m -ited by steric hindrance between the side chains of the residues and the peptide backbone. The possible conformations of a given polypeptide chain are quite l i m -ited and are represented using a Ramachandran plot (a plot o f permissible cf> vs. ip combinations; see Figure 1.2, part (b)). P e p t i d e b o n d A m i n o g r o u p A l p h a c a r b o n C a r b o x y l g r o u p A m i n o a c i d / A m i n o a c i d 1+1 (a) (b) Figure 1.1: (a) Structure of an amino acid (electrical charges are not indicated); (b) peptide bond between neighbouring amino acids. Chapter I. Introduction 3 $ (ph i ) 1 8 0 0 (a) (b) Figure 1.2: (a) Dihedra l angles (<p,ip); (b) Ramachandran plot o f a l lowed values for the dihedral angles. A l l regions shown in white are d is -a l lowed. 1.1.1 Three Levels of Protein Structure The linear sequence of amino acids constitutes the primary structure o f the pro-tein. A m i n o acids in the protein usually influence parts o f the backbone (amide group and carboxyl group) to interact with each other by means of hydrogen bonds to form a secondary structure. The three most c o m m o n secondary structure ele-ments are: a - helix, which results from hydrogen bonding among near-neighbour amino acids (stabilized by hydrogen bonds between the carbonyl oxygen of the amino acid residue at position n with the amide group, NH, of residue n + 4), as shown in F igure 1.3, part (a); 0 - sheet, which involves extended protein chains packed side by side that interact by hydrogen bonding (hydrogen bonds occur be-tween carbonyl oxygen and the amide group of adjacent strands; see F igure 1.3, part (a)); and turns, which usually connect a-helices and /3-sheets (turns involve 180° change in the direction of the chain, and are stabil ized by a hydrogen bond between the carbonyl oxygen of the residue at position n and the amide group, N H , of residue n + 3). Hel ices are the most abundant secondary structure elements in globular proteins (for example, enzymes), fo l lowed by sheets, and then turns [92]. Those regions that cannot be classified as one of the standard three classes o f sec-ondary structure are classified as random coil. The complete three-dimensional Chapter 1. Introduction 4 shape of a polypeptide is its tertiary structure. Figure 1.3, part (b) shows the ter-tiary structure of the protein myoglobin. Most globular proteins (soluble proteins) take this roughly spherical shape. Many proteins are composed of multiple sub-units [143]. They additionally possess a fourth level of structure - quaternary structure, which involves association of two or more polypeptide chains to form larger protein molecules that function as dimers, trimers, tetramers, etc. (for ex-ample, hemoglobin functions as a tetramer). Complexes higher than octamers are rarely observed [143]. Figure 1.3: (a) Major elements of secondary structure: a-helix (left), /3-sheet (right); (b) tertiary structure of Myoglobin. 1.1.2 Protein Functions Proteins are very versatile molecules that assume a number of functions [74], in-cluding the following: 1. a structural role - give cells their integrity and shape (for example, viral coat proteins, molecules of the cytoskeleton such as actin filaments, epidermal 2. an enzymatic role in a number of metabolic pathways - binding and carrying substances, regulating different catalytic reactions (for example, ribonucle-ase - an enzyme that degrades RNA) (a) (b) keratin) Chapter 1. Introduction 5 3. a hormonal role - carrying signals from one cell to another (for example, insulin) 4. gene-regulating activity (for example, D N A polymerase) 5. a transport and storage role (for example, hemoglobin, myoglobin, ferritin) 6. energy transduction - performing electron transfer (for example, ATP syn-thase) 7. and an immune system role (including proteins involved in cell-cell recogni-tion and signaling, for example, immunoglobulins that function as antibod-ies) 1.2 The Protein Folding Problem and its Importance The protein folding problem consists of predicting the functional (tertiary or na-tive) structure of the protein given its linear sequence of amino acids (primary structure). Even though extensive progress has been made in understanding the process of protein folding, as described in Chapter 3, there still remain a number of unan-swered questions about this process, which is guided just by the propensities coded in the sequence itself. For instance, we would like to find out what exactly is coded within the sequence, and how it determines the structure; we would also like to know if proteins evolve to have the best sequence, and consequently the best struc-ture for their function. 1.2.1 Protein Folding Paradoxes The mystery of protein folding provides three challenging paradoxes [147]: 1. The Levinthal paradox: How can a protein find its low-energy state in time less than geological? On average, the number of independent conformations per amino acid residue is about 10. This means that for a protein of length n, there are 10" possible conformations. Therefore, folding cannot proceed via random sampling of the space of conformations. Consequently there must exist folding pathways that allow folding to proceed efficiently. Little is still known about the folding pathways of proteins, and properties of the intermediate states that a protein undergoes during folding. 2. The Hoyle paradox, which asks how foldable proteins evolved within the lifetime of the universe. The number of possible proteins of length 100, Chapter I. Introduction 6 for example, is 2 0 1 0 0 , which exceeds by far the number of protons in the universe times the turnover time of a typical protein. Thus, it is difficult to argue that any particular protein is perfect for its task. 3. The marginal stability or Honig paradox, which deals with the issue of why proteins seem to have such a delicate compensation of entropy and energy during intermediate stages of the folding process. The native state of globular proteins is typically only 5 — 15 kcal/mol more stable than the denatured state [74]. This is the equivalent of about one or two water-water hydrogen bonds. Precisely why proteins have marginal stability is unclear - it could be to facilitate turnover of proteins, or maybe proteins are as stable as they need to be and there is no selective advantage to optimizing the stabilizing interactions further. 1.2.2 The Thermodynamic Hypothesis In order to predict the tertiary structure of a protein, we need to know something about the nature of its native state (functional three-dimensional state). An impor-tant characteristic of the native state is described by the Thermodynamic Hypoth-esis. It states that the native state represents the ground state of lowest Gibbs free energy, and has been proposed by Anfinsen on the basis of denaturalization - renat-uralization experiments of the ribonuclease enzyme [4]: "The native conformation is determined by the totality of inter-atomic interactions and hence by the amino acid sequence, in a given environment." Usually, the native state of a protein is at the minimum free energy. Exceptions to this rule are very rare (for example, in the case of prions, whose folding happens under kinetic control) [27]. 1.2.3 The Central Dogma of Protein Folding Resulting from Anfinsen's experiment, the Central Dogma of protein folding (or the Central Dogma of genomics) states that "sequence determines structure de-termines function". This dogma assumes that the protein sequence and sequence alone in the given environment determines a protein's tertiary structure, and that the function of the protein is primarily determined by its structure [27]. Molecules that can assist in the folding of some proteins (chaperons) do not violate this dogma, be-cause chaperons do not induce a fold in proteins that is different from one adopted when the proteins are allowed to fold on their own in diluted solution; chaper-ons just expedite the folding or prevent unwanted aggregation. The dogma asserts that it should be possible, ultimately, to deduce the function of a protein from its structure. Chapter I. Introduction 7 1.2.4 Forces in Protein Folding , To approach the ab initio protein folding problem, one needs to quantify physical forces that determine the structure of the protein. These include: hydrophobic, hydrogen bonding, electrostatic, and van der Waals forces. These forces, together with covalent interactions, determine conformation along the folding pathway, the shape of folded and misfolded proteins, and also interactions between proteins, proteins and other molecules, as well as proteins and the solvent. Hydrophobic Force It is widely believed that the folding of globular proteins (most enzymes) is pri-marily driven by the hydrophobic force [27]. Non-polar atoms, such as those in hydrocarbons, are trying to reduce their contact with aqueous environments. The folded state of globular proteins is reflected by a balance between the opposing energetics of H-bonding between hydrophilic R-groups and the aqueous environ-ment and repulsion from the aqueous environment by hydrophobic R-groups. The hydrophobicity of certain amino acid R-groups (for example, Leucine, Isoleucine, and Valine) tends to drive them away from the exterior of proteins into the interior. In contrast, hydrophilic side-chains (for example, Arginine, Aspartic acid, and As-paragine) make hydrogen bonds with the aqueous environment, and are preferen-tially found on the surface of folded proteins. This driving force, characterized by the absence of hydrogen bonding between water and non-polar groups (rather than the presence of favourable interactions between non-polar groups themselves), re-stricts the available conformations into which a protein may fold. Most proteins that occur in the aqueous, intracellular environment or in plasma are of a globular nature because their structure is determined primarily by the hydrophobic force: they are approximately spherical in shape, or consist of several different domains (average domain length is 200 - 300 amino acids) [92]. A domain is a "folding unit" of a protein, the part of the protein sequence that folds largely independently of the rest of the sequence. Hydrogen Bonding Force Polypeptides contain numerous proton donors and acceptors both in their back-bone and in the R-groups of the amino acids. The aqueous environment in which proteins are found also contains ample H-bond donors and acceptors. H-bonding, therefore, occurs not only within and between polypeptide chains but also with the surrounding aqueous medium [27]. Chapter 1. Introduction 8 Elec t ros ta t ic Force Electrostatic forces are mainly of three types: charge-charge, charge-dipole, and dipole-dipole [27]. Typical charge-charge interactions that favour protein folding are those between oppositely charged R-groups such as Lysine or Arginine and Aspartic acid or Glutamic acid. A substantial component of the energy involved in protein folding is a result of charge-dipole interactions, the interaction of ion-ized R-groups of amino acids with the dipoles of the surrounding water molecules. The slight dipole moment that exists in the polar R-groups of amino acids also influences their interaction with water. It is, therefore, understandable that the ma-jority of the amino acids found on the exterior surfaces of globular proteins contain charged or polar R-groups. V a n de r Waals Forces There are both attractive and repulsive van der Waals forces that control protein folding. Attractive van der Waals forces involve the interactions among induced dipoles that arise from fluctuations in the charge densities occurring between adja-cent uncharged non-bonded atoms. Repulsive van der Waals forces involve the in-teractions that occur when uncharged non-bonded atoms come very close together but do not induce dipoles. The repulsion is the result of the electron-electron re-pulsion that occurs as two clouds of electrons begin to overlap. Although van der Waals forces are extremely weak relative to other forces governing conformation, the huge number of such interactions that occur in large protein molecules make them significant in the folding of proteins [27]. 1.2.5 Motivations for Studying Protein Folding Problems The protein folding problem represents an optimization problem (continuous or discrete, depending on the model assumed). Even for simple models that discretize the space on a lattice (grid), it is an /VP-hard combinatorial problem [91, 136]. A detailed description of different models used for protein folding is given in Chap-ter 2. Motivations for studying protein folding are directly connected with the ability to deduce protein functions. Some of these motivations are: 1. being able to predict protein function given the primary structure of the pro-tein (via tertiary structure prediction) 2. understanding a number of diseases that are directly caused by protein mis-folding, aggregation and fibrillogenesis (some of which include Alzheimer's, Huntington's, cystic fibrosis, prion disease) [92] Chapter 1. Introduction 9 3. designing drugs that specifically target certain aspects of a given protein's structure 4. designing proteins with desired structure and function In addition to posing an important biological problem with multiple practical ap-plications, the folding problem constitutes a significant mathematical optimiza-tion problem, that is, finding global minima in highly complex objective function (energy-potential) surfaces. Experimental methods available for determining protein structure include X -ray crystallography and nuclear magnetic resonance (NMR). In the case of X-ray crystallography, the structure of proteins is determined by measuring directions and intensities of X-ray beams diffracted from a high-quality crystal of purified protein molecules. In comparison, N M R determines the structure of proteins using high magnetic fields and radio-frequency pulses to manipulate the spin states of nuclei. The position and intensities of the peaks of the resulting spectrum reflect the chemical environment. N M R requires purification of a protein but does not require crystallization. However, it is limited to smaller proteins (less than about 20 - 50 kilo Daltons). Thus, computational methods for predicting the structure of proteins are very attractive, since experimental methods (X-ray crystallography and N M R ) are highly labour intensive and require purification and / or crystallization of proteins. 1.3 An Overview of Computational Approaches to Protein Folding There are four major approaches to predicting protein structure: 1. homology (comparative) modeling 2. fold recognition (or sequence -^structure threading) 3. novel fold recognition 4. ab initio prediction Progress in the field of protein structure prediction is evaluated by the Critical Assessment of Structure Prediction (CASP) experiments (that take place every two years). C A S P measures in a quantitative way (using structural similarity measures such as root mean square deviation, RMSD) the success in predicting the tertiary structure of a given set of proteins whose structure has been determined experi-mentally but not released to the public. Chapter 1. Introduction 10 1.3.1 Homology Modeling Homology modeling is based on finding a sequence with a similarity greater than 25 - 30% with a known structure (stored in the Protein Data Bank, PDB [12]). When this level of similarity is found, it is commonly believed that the two se-quences had a common ancestor. Homology modeling approaches use a known structure as a starting point for the 3D structure of the sequence. They take advan-tage of the fact that closely-related proteins have significant sequence similarity and therefore close structural similarity. The drawback of this method is that for every unknown sequence of interest, there has to be a relatively close homologue present in the database (PDB), which is not always the case. Usually when se-quence identity is greater than 70%, a high prediction accuracy is possible. 1.3.2 Threading Fold recognition (also known as threading or inverse folding) is usually used on se-quences with a sequence identity less than or equal to 30% to sequences of known structure [15]. Proteins displaying this level'of homology are believed to be ei-ther homologous (evolved from the same ancestor) or analogues (similarity is a result of convergent evolution). This approach stems from the observation that proteins with a level of sequence similarity that could not be detected by conven-tional homology searches sometimes adopt a particular three-dimensional fold. By aligning the residues in the unknown structure with the residues in a known fold, the tertiary location of the target residues can specify secondary structure, expo-sure, and spatial interaction with other residues. There are two known ways of threading: dynamic programming and the use of knowledge-based potentials. In the case of knowledge-based potentials, spatial interactions are often modeled by empirically-derived values quantifying the desirability of a residue of type i being a given distance from a residue of type j. The key problem for this approach is poor alignment between residues of interest and target residues [17]. 1.3.3 Novel Fold Recognition Methods This approach was known in earlier CASPs as ab initio fold prediction, but was renamed in CASP4 to better define current methodologies that are being used -particularly, to separate methodologies that use sequence homology from those that do not. Unlike pure ab initio prediction, novel fold recognition uses thread-ing, and fragments from existing protein structures, as well as secondary structure prediction and multiple alignment techniques, to predict some features of the three-dimensional structure. If prediction is done via secondary structure prediction, the-Chapter 1. Introduction 11 method can be sensitive to errors in secondary structure prediction. After obtain-ing a seed structure from sequence homology, local optimization techniques (such as Monte Carlo methods [56, 88, 122] and Genetic .Algorithms [18]) are used to further optimize the conformation. • 1.3.4 Ab initio Methods In cases when there is no significant homology with any sequence of known struc-ture, ab initio methods based on energetic principles perform a search in the set of all possible conformations. But since this search space is usually exponentially large, it is very challenging to search efficiently. To accelerate conformational searching, simplified models employ techniques that permit coarse (large-scale) searching of the energy landscape. A variety of methods are used in conjunc-tion with reduced complexity models and simplified potentials to perform broad searches through low-resolution structures, including Metropolis Monte Carlo meth-ods [125, 127] and Genetic Algorithms [29, 103]. Methods developed in this thesis belong to the class of pure ab initio methods but can be extended to use sequence homology. A more detailed description of different methods used for coarse-grained searching is given in Chapter 3. 1.4 T h e s i s S t a t e m e n t The primary topic of this thesis is the development of efficient local search methods for protein folding and related problems. Within this vast problem area (which has been studied in the literature for over 50 years with limited success!) we focus on three major directions for the development of more efficient search methods: 1. The development of biologically inspired approaches. The protein folding process has been described in the literature as a self-organizing process. Therefore, we ask whether biologically inspired search algorithms based on self-organization phenomena such as stigmergy (where an agent modifies the environment, and the decision process of other agents is influenced by this change in the environment) can be used as an efficient search metaphor. One of the most prominent biologically inspired approaches is Ant Colony Op-timization, which is inspired by self-reshaping phenomena observed in real ant colonies [40]. 2. The development of construction search methods. These methods work on partial candidate solutions, as opposed to complete conformations. We also focus on a combination of a construction-based search and search methods Chapter I. Introduction 12 based on complete conformations. This direction is particularly interesting, since at the present time the field of protein structure prediction is largely dominated by non-construction-based approaches working with complete conformations. Construction-based approaches are promising, since every round of construction that retains some properties of the previously seen candidate solutions is equivalent to switching to a larger search neighbour-hood. This efficient move set can surmount larger energetic barriers than the standard bond moves used in the literature. Additionally, construction-based methods have been shown to perform well for other NP-hcird problems. 3. The development of adaptive search strategies that can react to the amount of progress made during the search and adjust the search strategy accord-ingly. Most search methods proposed in the literature for the protein folding problem are non-adaptive. Standard methods have difficulty exploring low-energy regions efficiently due to the ruggedness of the search landscapes. Instead we propose to extract important features of the search landscape (by storing a promising and diverse set of local optima) and adapt the search strategy (adjust the amount of diversification vs. intensification) based on the search progress made. This thesis makes contributions in the following areas: 1. in the development and evaluation of the first Ant Colony Optimization (ACO), a construction-based stigmergic stochastic local search algorithm based on the self-organizing phenomenon for widely studied lattice models of protein structure prediction 2. in the development and evaluation of an efficient construction-based search method for the identification of folding nuclei (structural regions that are important for driving the folding process) 3. in the development and evaluation of adaptive search strategies (based on a novel bin framework for storing promising local optima) that can be used in conjunction with current state-of-the-art methods for ab initio and de novo folding 1.5 Organization of the Thesis Chapter 2 presents the protein folding problem and the closely-related problem of identifying folding pathways. Furthermore, in this chapter we introduce a set of basic notions and some widely used simplified models of protein structure. Chap-ter 3 describes general classes of search methods used in the literature for solving Chapter I. Introduction 13 protein folding problems, summarizes related work, and outlines the shortcom-ings of existing methods that motivated the algorithms introduced in this thesis. In Chapter 4, we introduce an Ant Colony Optimization algorithm for the 2D and 3D HP protein folding problem, describe major components of the outlined approach, and provide an empirical evaluation. In Chapter 5, we introduce an adaptive bin framework for the F C C /3-sheet protein folding problem and compare its perfor-mance with results from the literature. In Chapter 6, we introduce a construction-based search algorithm for identifying folding pathways, provide theoretical and empirical justification for the proposed approach, and compare results with the ex-perimental data available. Finally, in Chapter 7, we discuss how the approaches introduced in this thesis relate to each other in particular, and to the field of devel-opment of efficient search methods for protein structure prediction in general. We also discuss strengths and weaknesses of the proposed methods and outline some suggested directions for future research. Chapter 2 14 Description of Problems, Models, and Notations We can't solve problems by using the same kind of thinking we used when we created them. Albert Einstein In this chapter, we define the two problems addressed in this thesis: the protein folding problem and the closely-related problem of identifying folding pathways. Additionally, we introduce models and objective functions (energy potentials) that are used for each of the problems. T h e Ab Initio P r o t e i n F o l d i n g P r o b l e m is the problem of prediction of protein tertiary structure from its amino acid sequence for a given energy function. It was proven that this problem and several variations of it are NV-h&rd even when conformations are restricted to a grid, as in the case of lattice models (the proof was conducted for diamond and cubic lattices) [91, 136]. The technical definition of this problem is given in Chapter 4. T h e P r o b l e m of Ident i fy ing Pro te in F o l d i n g N u c l e i a n d F o l d i n g Pa thways is the problem of identifying a set of native contacts that play an important role in folding along with identifying a time-ordered sequence of their formation. The technical definition of this problem is provided in Chapter 6. The problem of identifying folding pathways without relying on the availability of the native state is believed to be generally more difficult than the protein struc-ture prediction problem [66]. The only variant of this problem that has been exten-sively addressed in the literature is the one outlined above, when folding pathways are deduced from either the native state [2, 32, 36, 107, 144, 151] or additional experimental results [97]. There are no complexity results available for the search for folding nuclei, but it is widely believed that this problem is difficult [66, 144]. To solve these computationally difficult search problems, a number of reduced models have been introduced by biochemists and physicists. These models allow more efficient searching of the possible conformations for the protein structure prediction problem and more efficient searching of the possible folding pathways Chapter 2. Description of Problems, Models, and Notations 15 for the folding pathway problem. Methods for reducing protein structure can be divided into two major classes: lattice and off-lattice models. The main advantages of lattice models include the abi l i ty to develop efficient computational techniques for storing conformations, testing their self-avoidance, and calculat ing contact energy terms (long-range potential); the possibi l i ty o f pre-calculat ing and storing in tables entire sets of some conformational transitions (lat-tice moves), and some elements of the energy function (for example, short-range potential); and reduction of the.conformational search space. The main disadvan-tages of lattice models are related to the distortion of local protein geometry and a l imi ted resolution o f the model (obtained by calculat ing the root mean square deviation between the model and the native state). For low-complexi ty models, for example, the Face Centered cubic lattice (12 vectors), the best resolution achieved is approximately 2 A; for high-complexi ty models, for example, Side C h a i n O n l y (646 vectors) and Ca-C'/3-Side Groups (800 vectors), a resolution less than 1 — 2 A is possible. In general, resolution also depends on the protein's size and secondary structure content [66]. The main advantage of discrete off-lattice models is the abil i ty to achieve high geometrical and resolution accuracy (less than 1 — 2 A) whi le st i l l keeping low complexi ty [100]. The primary disadvantage of off-lattice models is loss o f c o m -putational efficiency due to inabil i ty to test self-avoidance easily, s low calculat ion of long-range contact energy, and the impossibi l i ty of efficient pre-computation of a conformational move set. To address the protein folding problem, we need to consider the fo l lowing three issues: 1. the design of the model , i.e. the choice of the protein structure's representa-tion (which has a desired level o f accuracy) 2. the definition of the energy potential (energy potential should be able to dis-criminate between native and non-native states of proteins for the protein folding problem, and between entropically-favourable and non-favourable folding pathways for the folding pathways problem) 3. searching through the space of possible solutions or sample from the model in an efficient way to find the solution as fast as possible In this chapter, we discuss the design of models and the definition of energy potentials chosen, and provide justifications for specific choices made. In the next chapter, we address the issue of searching through the space of possible conforma-tions. Chapter 2. Description of Problems, Models, and Notations 16 2.1 Protein Folding on the Lattice To address the protein folding problem, we chose to concentrate on lattice models of protein representation, primarily due to the unavailability of a single universal energy function for off-lattice folding and a lack of comprehensive comparison results for state-of-the art methods using off-lattice models. Additionally, lattice models have a number of advantages that can be attributed to regular discretization of the space, conceptual simplicity, and wide usage in the protein folding literature. For small proteins, an extensive study of compact conformations on the lattice is possible and the evaluation of energies can be performed quite efficiently. How-ever, lattice methods have a somewhat restricted ability to accurately represent sec-ondary structure and backbone conformation. Recently, it has been shown that the computational advantages of lattice models outweigh the problems associated with their biases [66]. It has also been shown that lattice models can provide some valu-able insights into the protein folding problem, including information about folding stages, comparing energy potentials, and testing optimality of the native state [15]. 2.1.1 Square and Cubic Lattices Historically, the most common and most widely studied lattices in the field of protein folding are square and cubic lattices. The protein conformations of the sequence are restricted to self-avoiding paths on a lattice. For the 2D model, a 2-dimensional square lattice is used; in the case of the 3D model, a 3-dimensional cubic lattice is commonly used. Examples of protein conformations on the 2D and 3D lattices with two types of amino acids (monomers) are shown in Figure 2.1. 2.1.2 Hydrophobic Polar Energy Potential The Hydrophobic Polar (HP) model is based on the observation that the hydropho-bic force is the primary driving force of folding in globular proteins. Most enzymes fall into this class [72]. In the HP model, every amino acid of a given protein is reduced to a single point on a lattice, and classified as either hydrophobic or polar. Hydrophobic residues are those that avoid contact with the aqueous environment, and therefore are more likely to be found in the interior of the protein, forming a hydrophobic core. Polar amino acids, in contrast, usually bear a charge; therefore, they interact with aque-ous environments and can be found on the surface of the protein. The hydrophobic force is the only force considered by the HP model. In the HP model, the energy is usually given by specifying a symmetric matrix of interaction energies between hydrophobic (H) and polar (P) residues that are Chapter 2. Description of Problems, Models, and Notations 17 1 (a) (b) Figure 2.1: (a) A n example of a protein conformation on the 2 D square lattice; (b) an example of a protein conformation on the 3 D cubic lattice. T w o types o f monomers are considered: hydrophobic amino acids are colored in red, polar are green, ye l low dotted lines represent energy interactions. topological neighbours on the grid but not immediate neighbours in the sequence: In the standard H P model , the energy of a conformation is defined as the number of topological contacts between hydrophobic amino acids that are not neighbours in the given sequence, thus, EHH = —1 and £HP = £PH = £pp = 0. M o r e specifically, a conformation c with exactly n such H - H contacts has free energy E{c) = n • (—1); for example, the conformation shown in Figure 2.1, part (a) has energy —14. Other variations of the energy potential for the H P model exist (see, W h i l e the H P model is most intuitively defined in 3 D to match the geometry, self-avoidance, and entropy of real proteins 1105], in fact the 2 D model captures some properties o f real proteins as we l l . Particularly, the perimeter-to-area ratio o f a short 2 D chain is a close approximation of the surface-to-volume ratio o f the native states of proteins [22]. 2.1.3 Face-Centered Cubic Lattice Model Another important representative of lattice models is the Face-Centered C u b i c ( F C C ) lattice. The F C C lattice is the usual lattice structure for the majority of crystall ine metals. Even though the square and cubic lattices were explored the e.g., [21]). Chapter 2. Description of Problems, Models, and Notations 18 most in the literature and both are important for empirical comparison with other existing approaches, the F C C lattice is preferred for predicting protein structure due to its ability to have low complexity (12 vectors) and still model real protein conformation with the best quality (coordinate root mean square deviation below 2 A ) when compared to other simple lattices [100]. It has also been shown that local packing of amino acids in proteins closely resembles a distorted F C C lattice [7], and that the F C C model has a reasonable description of secondary structure ele-ments; furthermore, hydrogen bonding is geometrically accurate as compared to real proteins [66]. As a result, the F C C lattice is considered the best overall choice among the simpler regular lattices [66]. In the F C C model, the polypeptide is restricted to a face-centered cubic lat-tice [106] that has 12 base set vectors (shown in Figure 2.2): where e i = ( l , l , 0 ) , e 2 = ( l , - l , 0 ) , e 3 = ( l , 0 ) l ) , e 4 = ( l , 0 , - l ) , e 5 = ( 0 , l , l ) , e 6 = ( 0 ; l , - l ) , e 7 = ( 0 , - 1 , 1 ) , e 8 = ( o , - l , - l ) , e 9 = ( -1 ,0,1) , e 1 0 = ( - 1 , 0 , - 1 ) , e n = ( - l , 1 , 0 ) , e 1 2 = ( - 1 , - 1 , 0 ) . ' A protein chain of ./V residues is described by N — 1 vectors, where vector V ; connects residues i and + The 12 base vectors allow for the following valence angles between each pair of vectors: 60, 90, 120, and 180 degrees [52]. 2 . 1 . 4 Beta Sheet Energy Potential Used with the F C C Lattice To model /3-sheet proteins and the stiffness of the polymer chain, the following definition of an extended, /3-type chain conformation was defined in [52]: three' vectors are in an extended state if 1. the angles between vectors Vi-i and v* and between v ; and V j + i are greater than 90 degrees 2. the dot product V j _ i • v i + 1 is larger than 0, which means that the angle between vectors V j _ i and VJ+I is less than 90 degrees The energy potential for this model is composed of two terms: the short-range potential j j + i that depends on three consecutive vectors in the chain (VJ_I , V j , V j + i ) and mimics conformational propensity to form an extended set of /3-strands: Vbase = { e i , e 2 , . . . , e 1 2 } (2.1) SB, if the triple ( v . ; _ i , V j , v . ; + i ) is in extended state, otherwise (2.2) Chapter 2. Description of Problems, Models, and Notations 19 (a) (b) Figure 2.2: (a) Depic t ion of the 12 base vectors for the F C C lattice mode l ; (b) an example of several F C C lattice cells. and the long-range potential Vhj for two non-bonded chain residues, defined as: {+oo, for r-ij = 0 and i ^ j, —£A, for rij = 1 (in lattice units) and \i — j\ > 1, (2.3) 0, for rij > 1 (in lattice units) where nj is the lattice distance between residues i and j . For a chain of length N, the total energy is defined as: N-1 N N i=2 t = l j=l where 5jj is the Kronecker delta (5ij = 1 when i = j, and 0 otherwise) and the values of the force field parameters are defined as fol lows £A — 1-0 [£o] and £B = 4.0 [so] (eo is the unit o f interaction energy and £Q — 1.0) as in Gront et al. to model a semi-flexible polymer [52]. This energy potential was chosen (in combinat ion wi th the F C C lattice), since the accuracy of protein structure prediction methods for /3-sheet proteins is the lowest among other structural classes of proteins [1]. Add i t i ona l ly , it exhibits Chapter 2. Description of Problems, Models, and Notations 20 characteristics of more complicated energy potentials used for off-lattice mod-els [52, 106], particularly, cooperative all-or-none folding transition, characteristic interplay between short- and long-interactions, secondary structure propensity. 2.2 Off-Lattice Protein Folding In this section, we briefly summarize available and commonly used off-lattice mod-els and provide an overview of potential energy functions. Even though our ap-proaches are introduced and evaluated for lattices only, they can easily be extended to a discrete off-lattice case. We did not concentrate on discrete off-lattice repre-sentation due to the unavailability of universally used energy functions and the ab-sence of an appropriate data set for which empirical results of the best-performing algorithms in the literature exists. The two major difficulties that ab initio protein folding faces and that are fur-ther amplified in the off-lattice case are the exponentially large number of possible conformations, and the lack of the potential (the energy function) to discriminate between native folds and misfolded proteins. The usual approach for dealing with an exponentially large continuous search space is to simplify the protein structure representation, and to allow these simplified models to adopt only a small discrete number of conformational states [15, 100]. 2.2.1 Models for Off-Lattice Protein Folding The primary reason for choosing off-lattice models over lattice models is to obtain a better geometrical accuracy. Reduced or discrete state off-lattice models usually fix degrees of freedom of all side chains and assume that all bond lengths are constant. Internal (angular) coordinates for the main chain atoms are usually kept according to the "ideal geometry" of Alanine [113]. There are three generally acceptable ways in which side chains are modeled in reduced off-lattice models: 1. to ignore side chains and model only Ca, while side chain properties (such as secondary structure propensities, hydrophobicity values, participation in disulfide bridges etc.) are taken into account in the energy potential [29, 75, 113] 2. to use a single virtual atom to represent each side chain at the center of mass of the heavy atoms in the side chain [131, 146] 3. to find a conformation of the backbone and then add side chains and model them using rotamer libraries, optimizing their positions [43] Chapter 2. Description of Problems, Models, and Notations 21 The backbone conformation is modeled by the (cp, ip) dihedral angle pair for each amino ac id . This backbone representation effectively reduces the a l lowed conformational space of the protein backbone, whi le uniquely defining atomic co-ordinates of the protein. M o s t discretized off-lattice models consider only m al -ternative (<p, ip) states representing various alpha-helices, beta-strands, and loop conformations, where m — 4 to 32 [29, 100, 113]. The set of m a l lowed angle pairs is chosen by fitting the backbone coordinates to the coordinates of represen-tative natural proteins. The choice of model is particularly important in the off-lattice case, since each model is l imi ted in its possible representational accuracy (which determines the predictive accuracy of protein conformations), and since off-lattice models are pr i -mar i ly chosen on the basis o f their accuracy. The two discrete off-lattice models that show good prediction accuracies are the seven-state model used by Rooman et al. [113] and the four-state model by Park and Levi t t [100]. Searching a con-formational space using reduced off-lattice models st i l l remains computat ionally very chal lenging, even for small proteins. After obtaining a reduced representation of the native fold of a protein, reduced models can be accurately reconstructed to al l-atom protein representations [100]. In practice, most methods currently used in novel fold predictions (apart f rom using a reduced off-lattice model and searching in the discrete space) make exten-sive use of available structural information. This is done by developing scoring energy potentials that use structural information accumulated in protein databases, by using secondary structure predictions that employ homology information, and by choosing structural fragments from a database of known proteins to reduce the size of the conformational space during the search phase [18, 63, 122, 126, 131]. For this reason, the ab initio category was renamed "novel fold predict ion" in C A S P 4 [89]. A number of novel fold methods including R O S E T T A , which is currently the most successful method in the novel fold prediction section of C A S P competi-tions [1, 16], enhance the search process by using a large library of short frag-ments. These fragments consist of a small number of residues with a specified three-dimensional structure [122]. Short segments of the sequence of interest are restricted to the local structures adopted by closely related sequences in the protein structure database [18, 63, 122]. M a n y novel fold prediction methods use well-established secondary structure prediction tools. These methods assemble tertiary structure from candidate frag-ments of secondary structure, either by fixing secondary structure as hard con-straints or using it as soft constraints [56, 63, 88]. M o s t successful novel fo ld prediction methods represented in C A S P use a c o m -bination of approaches based on threading, lattice folding, clustering, and structural Chapter 2. Description of Problems, Models, and Notations 22 refinement using more detailed off-lattice models [125]. 2.2.2 Energy Potentials for Off-Lattice Protein Tertiary Structure Prediction Success in solving the protein structure prediction problem relies heavily on the choice of a potential energy function that can accurately discriminate between na-tively folded and misfolded protein conformations. Three types of energy poten-tials have been developed in the literature [25, 98, 99, 123]: 1. Physical potentials: based on empirical measurements of interactions be-tween atoms, these functions are typically parametrized on data from small proteins [64, 141]. 2. Knowledge-based or statistical potentials ("potentials of mean force", database-derived): based on the statistical properties and features of a database of proteins of known native structures, and transformed into various free en-ergy parameters using statistical measures that calculate the ratio between observed and expected values (e.g., the number of contacts, the distance be-tween amino acids, the number of atoms in contact between different amino acids) [87, 99, 124, 137]. The interactions can be either distance-dependent or only contact-dependent. 3. Learning-based potentials: an energy function is derived that maximizes the difference between correct and incorrect structures (stability gap) by training using machine learning methods or linear optimization of parameters [141]. There are advantages and disadvantages to using each of these potentials. Phys-ical potentials are mostly useful for simplified models of protein structure where not all of the forces are considered, since the exact potential energy governing pro-tein folding is not known [141]. Knowledge-based potentials have achieved some degree of success but they suffer from the following limitations [25]: 1. The theoretical basis of these potentials is weak. There is no reason to be-lieve that the interactions in proteins in the respective ground states of an ensemble of biological proteins would obey Boltzmann statistics. 2. These potentials assume statistical independence of various interactions but from theoretical studies of protein folding it has been shown that correlations between interactions may arise to facilitate the folding process. 3. Some contributions can be underestimated due to the selection of the refer-ence state or neglection of chain connectivity. Chapter 2. Description of Problems, Models, and Notations 23 Learning-based potentials define for each protein a Z-score that represents the difference in energy between the correct and random states, divided by the standard deviation of the energy levels of random conformations. The best energy potential would be the one that maximizes Z-scores for all the proteins in the database [25]. To apply this method to a given protein model, one needs knowledge of both the native and non-native states of proteins under this model, which is computationally expensive [55]. One commonly accepted way to test a given energy potential function is to develop a set of decoys (a mixture of native and non-native protein folds) and then study the ability of different potentials to detect native structures [98, 99, 123]. The decoys are generated in different ways: putting the sequence from a particular protein onto the backbone of another protein, lattice models, fragment building, threading, or other alignment methods [99, 141]. Currently, there is no universal energy function that performs consistently well for multiple sequences. Therefore, most approaches used in C A S P use different linear combinations of energy components based on all three types of energy po-tentials described previously; this makes comparison and development of novel search methods increasingly difficult. Therefore, in this work we restrict ourselves to lattice folding, and will consider off-lattice folding in future work. 2.3 The Problem of Folding Pathway Identification The second problem we address, that is, the problem of identifying folding path-ways (or folding nuclei) in native protein structures, is directly related to the prob-lem of protein folding. In the general sense, when no information about the native state is used, it is a more difficult problem. It is now widely accepted that proteins fold by a "nucleation and growth" kinetic mechanism [36, 47]. A folding nucleus is defined as the critical set of interactions (the minimal stable element of struc-ture) that must be present in order for the folding to result in a rapid assembly of the native state [35, 36]. The problem of identifying folding nuclei from the large number of native structures currently available in the Protein Data Bank remains unsolved and of great importance. Accurate predictions of folding nuclei can lead to a detailed de-scription of the hierarchy of the folding process, make it possible to identify con-tacts that have high structural importance, and thus yield possible improvements in solving the ultimate protein problem - the folding problem - by restricting the conformational space that needs to be searched. Insights obtained during the study of folding nuclei can also be useful for rational protein design. Chapter 2. Description of Problems, Models, and Notations 24 2.3.1 Models Used for the Identification of Folding Pathways A variety of models have been introduced for the identification of fo ld ing nuclei . They range from lattice representations o f proteins [65] ( inc luding 3 D cubic lat-tices, as described previously), and reduced continuous Ca - Cp models s imi lar to the off-lattice models described above (which are used in M o l e c u l a r Dynamics simulations [36]), to all-atom models o f protein structure [107]. Lat t ice models are very popular for protein folding. However, studying the structure and formation process of folding nuclei requires models with higher accuracy, since they impose unphysical constraints (distorted geometry) when information about the geometry of the native state is available. In order to search efficiently, a model has to be sufficiently reduced. We used the discrete Ca model in our approach and a graph representation of native contacts. A protein is represented by a polymer graph (a connected undirected weighted graph) with one node per residue. Two residues are connected by an edge, i f they are in contact in the. native state. We used the fo l lowing contact definition: i f any of the heavy atoms of the side-chain or the Ca o f one residue occur wi th in the cut-off radius of 4.55 A from heavy atoms of the side chain or the Ca of the second residue [26], residues are in contact. A s in [26], native contacts between pairs o f residues with j < i + 3 are discarded, as such residues interact due to chain connectivity. Every edge of the polymer graph is augmented by the value of the objective function, discussed in the next section. 2.3.2 Objective Functions Used for the Identification of Folding Pathways The majority of methods described in the literature for finding fo ld ing nuclei [2, 32, 36, 107, 144, 151] only consider topological (entropic) aspects of the protein energy surface, since the precise energetics of the protein folding process is st i l l unknown. Nuclea t ion during the folding process is a kinetic process and there-fore, cannot be precisely determined from just the topological properties o f the native state alone (a single nucleation phase). However, as shown by ^ -va lue anal-ysis, transition states tend to be native-like, that is, have many native contacts [35]. Furthermore, it is known from experimental analysis that the locations of fo ld ing nuclei depends on the topology of the native state [65] and that the mechanism and speed of fo ld ing directly depends on the topology of the fold [104]. Different approaches capture the geometry (topology) of the native state in var-ious ways. In this work, we adapted an objective function based on the notion of the effective contact order ( E C O ) first introduced by D i l l [144]. W h i l e the con-tact order ( C O ) is the sequence separation CO = \i — j\ between two contacting Chapter 2. Description of Problems, Models, and Notations 25 E C O i k I J CO Figure 2.3: The effective contact order (ECO) between i and j is equal to 5, com-pared to the contact order (CO) of 7 between the same residues. residues numbered i and j [104] and does not depend on the order of contact for-mation, the E C O of a contact is dependent upon other contacts already formed in a particular configuration. Thus E C O values are folding pathway dependent and directly relate to the entropy of folding. Effective contact order of a newly-added contact is defined as the effective loop closure size (the number of steps, covalent and non-covalent links taken along the shortest path on the polymer graph), given that other contacts have been formed, as shown in Figure 2.3. It has also been shown that low-ECO pathways often coincide with the lowest free-energy pathways for simple models, such as the hydrophobic polar model. This observation also holds for real protein structures [144]. Thus, the objective function to be minimized in this case is the sum of individ-ual effective contact orders for each contact ECOTOTOI= ECO{i,j,AT) (2.5) },i<j+3 where m is the total number of contacts { e i , e m } or edges in the polymer graph, and i, j iterate only through the set of the non-covalent native contacts (contacts between residues fewer than 3 residues apart are discarded due to chain connec-tivity). At is the set of edges present at the time t when contact is added, t e [0,m - 1], A0 = 0, and At+X = At U Since the ECO(i, j, At) of each contact depends on what contacts have already been formed, this problem is an optimization problem similar to the protein folding problem and maximizes the absolute value of the total energy of contacts formed. 26 Chapter 3 Background and Related Work What makes the desert beautiful is that somewhere it hides a well. Antoine de Saint-Exupery Search methods for hard optimization problems depend on the topology of the search space, which in turn is defined by the two issues discussed in Chapter 2: the choice of the model (more specifically, the resolution of the model - the rep-resentation of a protein - determines the size of the search space) and the choice of the potential energy function (an objective function defines the landscape of the search space). The speed of exploration of the search space for these exponentially large spaces is important. It ranges from Molecular Dynamics Simulations that directly integrate Newton's equation of motion (taking the very small step sizes required for numerical stability) to accelerated conformation or pathway searching that permits coarse sampling of energy landscapes. In this chapter, we provide an overview of the topology of the space, search methods, and particular algorithms that rate among the best for the problems of protein folding and identification of folding pathways from the native state topology. 3.1 The Protein Folding Problem This section focuses on the problem of searching conformational space efficiently, with the goal of finding the global optimum as it applies to systems with complex search landscapes, and particularly to the protein folding problem. First, we will briefly summarize existing optimization methods for protein folding. Then, we will categorize the methods and specify the relationships between the stochastic local search methods employed for this problem. Finally, we will discuss promis-ing directions for the development of efficient search methods for problems with complex energy landscapes, some of which have been explored in this thesis. One important distinction that has to be made in the context of the protein folding problem is the difference between sampling and searching. The goal of Chapter 3. Background and Related Work 27 sampling is to generate a correct ensemble (defined by the Boltzmann probabil-ity) for the calculation of physical quantities; the goal of searching is to find a configuration or a set of configurations with a required property (usually a global energy minimum). In general, since searching involves finding one conformation with specific properties instead of making sure that the ensemble of conformations found conforms to a certain sampling requirement, searching can be more efficient than sampling. Unmodified Molecular Dynamics and Monte Carlo are efficient sampling methods, but inefficient search methods. Here and throughout the thesis we use the term 'efficient algorithm' to describe several desirable properties of a stochastic local search algorithm; besides clean design and reasonable space re-quirements, the main desired property is speed, the time it takes for an algorithm to obtain the global optimum or high-quality solutions. The term "efficient" as applied to stochastic local search algorithms (that may require exponential time to solve certain classes of problem instances) does not imply that algorithms have polynomial time complexity. There are a number of search methods applicable to the protein folding prob-lem that can be used in conjunction with reduced complexity models and sim-plified potentials to perform a broad search through low-resolution structures. The most widely used methods include Metropolis Monte Carlo and Simulated Anneal-ing [63, 95, 122, 125, 153], as well as Genetic Algorithms [18, 29, 103]. The size of the neighbourhood considered in each local move of the search can be quite sig-nificant, and can allow a much more rapid sampling of the conformational space. For example, a simple change in torsion angle space produces large changes in Cartesian coordinates, fragment-based procedures speed up sampling by allowing jumps between different local structures in a single step, secondary structure re-stricts the size of the search space and can also contribute to more coarse sampling, and the usage of templates for search initialization aids the search. Computer simulations suggest that the energy landscape along the folding path-way of the protein is often quite rugged, and usually passes through stable inter-mediate states with low energy [92]. In particular, so-called molten globule inter-mediates have been extensively studied [92]. In protein folding, a rugged land-scape poses a problem since the system has to cross barriers that can be larger than the deterioration of the potential energy function an algorithm searching for low-energy states is willing to accept. These energy barriers arise due to at least three classes of interactions: local barriers separating stable states of the torsion angles (as observed from Ramachandran plots); and close encounters between side-chain atoms and backbone atoms that repel one another once they get too close. Addi-tionally, energetic barriers occur due to the role that non-native interactions play during folding [11]. The most challenging feature of the protein folding problem is the fact that the objective function (energy) has a large number of local minima. Chapter 3. Background and Related Work 28 Figure 3.1: An example of a protein's tunneled energy landscape. Therefore, local optimization is likely to get arbitrarily stuck in one of them, pos-sibly far away from the desired global minimum. Estimates of the number of local minima for the protein folding problem range from 1.4" to 10" for a protein with n residues. However, in most cases a search will not encounter all of them [27]. It is believed that evolution has generally led to tunneled energy landscapes, with the native state at the base of the funnel, by requiring robust folding; this property is also called the principle of minimal frustration [105] . At the same time, the landscape is still rugged, with many local optima resulting from non-native contacts that can lead to specific folding intermediates. A typical example of the tunneled energy landscape of a protein is given in Figure 3 .1. It is generally hypothesized that improvements in energy function can lead to smoother search landscapes, and can result in obtaining more guidance from the potential energy function that will speed up the search [15] . 3.1.1 Search Algorithms Used for Protein Folding Two of the most commonly used algorithms for protein folding are Monte Carlo methods (and their variant, Simulated Annealing) and Genetic Algorithms. Chapter 3. Background and Related Work 29 M o n t e C a r l o a n d S i m u l a t e d A n n e a l i n g Monte Ca r lo ( M C ) methods are the most wide ly used methods for searching and sampling complex energy landscapes in statistical physics. Specif ical ly, general-ized ensemble methods based on the Monte Car lo method represent the state-of-the-art for ab initio protein structure prediction using discrete-state models [86, 94, 130]. A wide ly used set o f Monte Ca r lo methods, known as M a r k o v chain Mon te C a r l o ( M C M C ) , use Markov chains to explore the state space. In this section, I w i l l restrict the discussion of Markov chains to discrete state spaces, but note that the presented definitions and convergence requirements extend naturally to general state spaces. A process is called a Markov process i f the condit ional probabil i ty P(Xin — Sin\Xtn_i = S i n _ 1 , X t l — -S j J is independent o f all states S i 1 , S i n _ 2 but the immediate predecessor [71]. This condit ion results in a search procedure wi th-out memory that is inefficient in searching for a global op t imum but is efficient for sampling, since the probabili ty of vis i t ing a particular conformation depends only on the energy of the proposed conformation and the energy of the current state. The transition probabil i ty from state Si to state Sj is defined as: Wij = WiS, - Sj) = P(Xtn = SjlXt^ = Si), (3.1) where it is required that Wij > 0 and ^ VVij = 1- The total probabil i ty P(Xtn — Sj) that at time tn the system is in state Sj is then equal to: P(Xtn = Sj) = Y , p ( x t n = s3\xtn_l=st)P(xtn_1=sl) i = £ w V ( * t „ - i i A M a r k o v chain is said to be ergodic i f it converges to an equ i l ib r ium distribu-tion in the l imi t when time goes to infinity [3, 59]. This distribution is also known as the stationary or invariant distribution of the process and for physical systems in thermal equi l ib r ium it corresponds to the Bol tzmann distribution. To ensure ergodicity, the chain must satisfy the fo l lowing properties [3]: 1. Irreducibil i ty - for al l states Si and Sj, there exists a time t, such that Wfj >. 0, Le. the transition graph represented by the transition matrix W is connected; thus, the matrix W cannot be reduced to separate smaller ma-trices Chapter 3. Background and Related Work 30 2. Aperiodicity - chain does not get trapped in cycles; thus, ensuring that the stationary distribution is unique The above conditions are hard to verify in practice. To surmount this difficulty, one often adopts an easier to verify alternative condition for convergence known as detailed balance. This is a sufficient, but not necessary condition [3]. The detailed balance condition is defined as [71]: WJtP(Xtn = Sj) = Wl3P(Xtn = SA. (3.2) In practice, when the conformational space to be searched is exponentially large, a simulation becomes quasi ergodic. This means that trajectories between structurally diverse but statistically important energy minima have low (close to zero) transition probabilities that can result in the search or sampling process get-ting trapped in isolated minima; thus, the simulation may not necessarily reach the equilibrium and may not converge. Monte Carlo (MC) and Simulated Annealing (SA) algorithms [82] generate a trajectory of states for a system. Each state is evaluated using an objective function - potential energy. If the value of the objective function (E) decreases, a new state is accepted. If it increases, the new state is accepted with a probability dependent on the difference in the Boltzmann weights between the states, proportional to e-AE/(kBT)^ w | i e r e kB is the Boltzmann constant. While the temperature T is kept constant in M C , it is varied according to some schedule (usually slowly decreasing) in SA. Extremely slow schedules are inefficient, and extremely fast schedules will get the system trapped in the local optima. Thermal annealing can be repeated multiple times, thus producing thermal cycles [37]. Generalized Ensemble Monte Carlo Methods As mentioned above in the canonical simulation, the probability of crossing an energy barrier is proportional to the energy difference and the temperature (em-ploying the Boltzmann probability): PB{E,T)<xe-^ElkBT, (3.3) where e-PAB/kBR-T j s the Boltzmann weight, the energy difference AE = Ej — Ei, and the inverse temperature 8 = 1/ksT. This results in a probability density function over energy levels that is bell-shaped, with a maximum around average energy at temperature T. It decreases exponentially with T. The barrier height is measured in units of kBT. Thus, in canonical Monte Carlo (MC), very high and, more important, very low energy configurations are rarely sampled. To overcome Chapter 3. Background and Related Work 31 these problems, a number of generalized ensemble Monte Carlo methods have been developed [86]. Generalized ensemble methods compute the density of states (or related physical quantity) to perform a uniform sampling rather than sampling the states of the system using the Boltzmann weight (as is the case with Monte Carlo methods). Here we provide a brief overview of the most widely used generalized ensem-ble methods. 1. Multicanonical Algorithm ( M U C A ) (other names for this method are En-tropic Sampling, Flat Histogram method, Adaptive Umbrella Sampling, Uni-form Sampling in energy) [86]. The objective of this method is to overcome energy barriers in the rugged landscape by performing a random walk in the energy landscape (where ide-ally all the states are sampled with the same probability). To achieve this, configurations with energy E are updated with non-Boltzmann (multicanonical) weights [86]: wmu(E) cx e - 0 o E m ^ E ' T ^ = n-l{E), (3.4) where BQ is an arbitrary reference inverse temperature BQ — l/fe^To, the multicanonical potential energy is defined as: Emu(E,T0) = kBT0ln(n(E)) = T0S(E), (3.5) and n(E) is the density of states that quantifies how closely packed energy levels are, S(E) = ln(n(E)) is the multicanonical entropy. Thus, a uniform (flat) distribution of energy is obtained: Pmu(E) oc n(E)wmu(E) — const (3.6) (all states are sampled with comparable frequency). Since the density of states is unknown, multicanonical weights have to be determined iteratively. Simulation is performed using the usual Metropolis criterion. The transition probability from state i with potential energy Ei to state j with energy Ej is given by: f 1 AEmu < 0 where Chapter 3. Background and Related Work 32 AEmu = Emu(Ej,To) — Emu(Ei,To). (3.8) In the first run, a canonical simulation at a sufficiently high temperature To is performed: E£l(E,To) = E, (3.9) and initially wmu is equal to the Boltzmann weight: w£l(E,T0) = e-E/kBTo. (3.10) This run defines the maximum energy value Ernax under which one obtains flat energy distribution, and is equal to the average potential energy at tem-perature To, (Emax —< E >T0) [86]. Above Emax w e have the canonical distribution at T — To-During the simulation, energy (occupancy) histograms H(E) are generated that provide estimates of the probability of finding a configuration having en-ergy E in a multicanonical ensemble. Subsequently, multicanonical weights wmu(E) are "re-weighted" by the estimate of the density of states [86, 94, 130]. The estimated density of states is modified iteratively by recording an occu-pancy histogram, H(E), which counts the number of visits at each energy level [71]. When H(E) is flat in the energy range of the random walk, the density of states has converged with the accuracy proportional to the factor / (the convergence factor) used for updating occupancy histograms The fac-tor / is iteratively reduced in the process and histograms H(E) are reset to get the desired accuracy. This iterative process can take significantly long time [86]. Finally, a single long productive run is conducted. M U C A represents an efficient sampling method, but is an inefficient search method. The product of the estimated density of states and the Boltzmann weight produces an almost flat energy distribution. Sampling this distribu-tion results in a one-dimensional (no temperature T dependence) walk in the energy landscape. It can therefore escape any energy barrier but is unable to search low energy regions efficiently [148, 149], since low-energy regions are not guaranteed to be sampled in great detail because a wide range of en-ergies has to be sampled. Additionally, the weight factors used are derived from previous iterations. Thus, the process has no information concerning unexplored low-energy regions of the landscape, since it does not specifi-cally concentrate on exploring them. Chapter 3. Background and Related Work 33 2. Replica Exchange Monte Carlo (REMC) (also known as the multiple Markov Chain method and Parallel Tempering) [86]. A number of non-interacting copies (replicas) at different temperatures are simulated independently by M C . Every few steps, pairs of replicas are ex-changed with a specified transition probability. The weight factor is a prod-uct of Boltzmann weights (is essentially known). In general, each replica can experience its own temperature, pressure or chemical potential. The acceptance criteria for a trial swap are derived from the product of the elementary moves used to construct it. A swap of conformations i and j is controlled by the following two parts: i —> j is performed with the following acceptance probability: pi=min[l,e-'3i^-Ei)], (3.11) where Ei and Ej are the energies of conformations i and j respectively, and j —> i with probability: Pj = min[l,e-^Ei~E^]. (3.12) A swap move is a double move, whose acceptance probability is the product of pi and p.j: PlJ=w,in[Ce-^-E^-^-E^} ' (3.13) Thus, the transition probability is: Pij = m i n [ l , e ~ ^ - 0 ^ - B i ) ] . (3.14) If the difference (/% — 3j) in temperature is large, the move has a very low acceptance probability. Thus, trial moves have high acceptance probabili-ties only if some degree of overlap exists between probability distribution functions (histograms) corresponding to neighbouring replicas. The probability of exchange Pij = 20% in the literature is believed to pro-vide a good balance between short-range and long-range moves [94]. R E M C is an example of Umbrella Sampling, where to overcome very high barriers, simulations with different umbrellas (biases) are coupled and their umbrellas periodically exchanged. In general we can't construct a perfect umbrella (bias) for efficient sampling or searching without knowledge of the "true" potential energy surface, but we can do this with umbrellas involving temperatures. Chapter 3. Background and Related Work 34 An advantage of this method is that unlike M U C A , it does not require deter-mination of weight factors, since the weights are essentially a priori known (defined by Boltzmann probabilities). The drawback of this method is that as the number of degrees of freedom (AO of the system increases, the required number of replicas also increases (VN), where as only a single replica is simulated in M U C A . Improvements of R E M C introduced in the literature include hybrid approaches between R E M C (for the weight factor determination) and M U C A , or Simulated Tem-pering production runs [86, 94, 130]. R E M C methods are currently the best performing algorithms for ab initio protein structure prediction in the context of the search problem [130]. 3. Energy Landscape Paving (ELP) [53]. This method is an example of a generalized Monte Carlo method that adapts acceptance criteria based on the search history. Hansmann et al. [53] introduced temporary energy sur-face deformation. In this method, the barrier height is decreased proportion-ally to the time the system stays in the minima (configurations are searched with time-dependent weights, similarly to Tabu Search (TS) that uses a time-dependent adaptive memory in the search [50]). Tabu Search does not differentiate between important and non-important re-gions of the landscape. Energy deformation techniques escape from local optima by deforming or smoothing the energy landscape (e.g., lowering bar-riers between relevant parts of the search space similarly to other generalized ensemble approaches). Ideally, we are interested in transforming the origi-nal energy landscape into a funnel landscape (but it is likely impossible to do this in practice). E L P is designed to perform low-temperature M C with a modified energy expression, as follows, to steer the search away from the regions that have already been explored: where f(H(q, t)) is a function of the time-dependent histogram of energies of states q sampled (H(q, t)) updated at each M C step. In ELP, the weight of a local minimum state decreases with the time the system stays in that minimum, thus the probability of escape increases by deforming the energy landscape until the local optimum is no longer favored. (3.15) E = E + f(H(q,t)), (3.16) Chapter 3. Background and Related Work 35 With a short-term memory in the histogram and an infinite cost for 'forbid-den' moves E L P is equivalent to Tabu Search. For f(H(q,t)) — f(H(q)) the method reduces to various generalized ensemble methods. This method has only been applied on short peptides using detailed atomic representa-tion [53]. Our later work presented in Chapter 5 relates to the current ap-proach and builds on improving sampling of low-energy conformations us-ing adaptive strategies based on the search progress. Other Algorithms Employed for Protein Folding To make this high-level overview complete, we mention other methods employed for protein structure prediction, but it should be noted that they are less successful or not as widely used for more realistic models of protein structure representation in the context of ab initio folding. Genetic Algorithms (GAs) [57] optimize the population of solutions by using biologically inspired operations of mutation of the solution and recombination of pairs of solutions (crossover). Possible solutions to a search problem are repre-sented by genes, and a fitness function is used to evaluate the fitness of each gene (candidate solution). GAs have been applied extensively to the problem of protein folding [29, 44, 103, 131]. The advantage of GAs is that a population of solutions can overcome energy barriers by a recombination that is larger than the typical moves employed in M C . The ability of the G A to leave a region of attraction is enhanced by crossing over. However, the G A crossover tends to not work well on systems that are highly coupled and where crossover can be disruptive, which is the case with dense protein configurations. The size of the population that goes onto the next generation can play the role of temperature in G A [138]. Model-based Search (MBS) is a stochastic local search that focuses the search on promising regions of the search space by storing a certain number of conforma-tions L (local minima) in memory and expanding N conformations in every step of the algorithm [20]. During each step of the search, new local optima are stored and all conformations stored are ranked and pruned based on the scoring function that considers their energies and the radius of a local minimum they represent (the ra-dius is estimated by the distance to the nearest neighbours using root mean square deviation). M B S has been shown to outperform in some cases simple M C used in the R O S E T T A algorithm [20]. Al l of the described earlier methods work on complete conformations after the initial creation of the random initial state. The next method is an example of a construction-based search method that works with partial conformations. In the protein folding literature, construction-based methods for protein folding are Chapter 3. Background and Related Work 36 rare due to the problem of attrition when the chain runs into itself. The only construction-based method so far restricted to simplified lattice models of folding is the pruned-enriched Rosenbluth method (PERM) of Grassberger etal. [8, 60]. The pruned-enriched Rosenbluth method (PERM) is based on a biased sam-pling (Sequential Importance Sampling) and uses pruning (termination of con-struction) of partial configurations with low statistical weights and enrichment (creation of multiple copies) of partial configurations with high weights. 3.1.2 Summary and Classification of Search Methods Search methods can generally be classified as being either model-based or non-model-based methods (also called instance-based methods [156]). In this context, model-based methods build a parametric or a non-parametric model of the search space that is updated during the search, and the new candidate solutions are gen-erated using the model. Non-model-based search methods generate new candidate solutions using solely the current solution or the current population of solutions; they do not create a model and do not update it based on the search space encoun-tered. The term 'model' is used here as in [156] to refer to an adaptive stochastic mechanism for generating candidate solutions. Some standard non-model-based methods in their most standard implementa-tion include Monte Carlo, Simulated Annealing, Replica Exchange Monte Carlo, Iterated Local Search, Iterative Improvement, Randomized Iterative Improvement, Variable Neighbourhood Descent, Variable Depth Search, and Evolutionary Algo-rithms (Genetic Algorithms are in this category). For a description of the stochastic local search methods mentioned in this classification and their application to other combinatorial problems, we refer to [59]. Other stochastic local search methods that build a parametric model (either probabilistic or deterministic) that is updated during the search include Importance Sampling (builds a model of the distribution of interest), Ant Colony Optimiza-tion (stores pheromone matrix), Multicanonical Monte Carlo (records histograms of energies), Energy Landscape Paving (accumulates temporary histograms of en-ergies), Tabu Search (records tabu attributes, and adapts the tabu tenure in reactive Tabu Search [9]), and Dynamic Local Search (adapts penalty weights). Examples of model-based methods that build a non-parametric model (record actual candidate solutions) include Model-based Search (MBS) (conformations are stored for future retrieval), the Pruned-Enriched Rosenbluth Method (parametric: weight thresholds for pruning and enrichment, and non-parametric: retrieval of partial conformations from the memory stack), and some population-based elitist search strategies that make use of candidate solutions accumulated over the past iterations of the search (e.g., population-based Iterated Local Search [128]). Chapter 3. Background and Related Work 37 Any of the non-model-based approaches can be turned into a model-based ap-proach. This can be accomplished in two ways: either by extracting some prop-erties of the search landscape and reacting to the search progress made (turning the method into an adaptive search with parametric model) or by storing certain conformations and restarting the search with a new conformation found in previ-ous iterations (turning the method into an adaptive search with a non-parametric model). Model-based methods that build a model which does not concentrate on low energy regions result in good sampling but often inefficient search methods (e.g., M U C A ) . In our work, we introduce an Ant Colony Optimization implementation which is a probabilistic parametric model-based search method (see Chapter 4) and a novel bin framework, which represents a non-parametric model-based search method (see Chapter 5). Our proposed approaches to protein folding are based on the observation that model-based searches seem to perform better for ab ini-tio protein folding problems, for example, P E R M outperforms most methods in Hydrophobic-Polar 2D square and 3D cubic lattice models, and M B S outperforms R O S E T T A in some cases for off-lattice protein folding. This is most likely due to the complexity of the energy landscapes encountered, since for other combinato-rial problems such as the Traveling Salesman Problem (TSP), the Graph Coloring Problem (GCP), and the Propositional Satisfiability Problem (SAT), many non-model-based searches perform well [59]. Now let us direct our attention to the height of the barriers that can be sur-mounted by the search. This is an important property of search methods dealing with rugged landscapes. In general, there are a limited number of ways of in-creasing the height of the barrier that can be surmounted effectively by a given search method. The increase in barrier height is required in order to prevent quasi-ergodicity, and to make a search procedure more efficient in terms of being able to effectively escape from local optima. This achieved in one of the following ways: 1. Change E(x) (for non-construction-based search). Specific examples of this strategy used for protein folding include allowing but penalizing in-feasible conformations (soft-core potentials) [24]; scaling energy function E(x) [145], or approximating it using other simplified functions [140]; adding additional biasing potential (the generally accepted name for this approach is Umbrella Sampling [94]). The addition of the bias can be time or search progress dependent and deformation can be short-lived, as in Energy Land-scape Paving [53]. 2. Change acceptance criteria (transition between samples for non-construction-based search), e.g., Multicanonical Monte Carlo [94], Broad Histogram Chapter 3. Background and Related Work 38 Method [30], Tabu Search (based on number of times visited) [73]. Reactive search strategies such as Energy Landscape Paving [53], which performs lo-cal elevation (modification of the acceptance criterion) based on the amount of time spent in the well also can be classified under this category. Also, modification of the Boltzmann acceptance criterion directly relates to rais-ing the temperature (this strategy is the most commonly explored in the lit-erature, particularly in Simulated Annealing [54], Replica Exchange Monte Carlo [94], and Simulated Tempering [94]). 3. Increase (adapt) the neighbourhood size considered, via local moves (1—, 2—, 3—, 4— residue moves, addition of one residue during construction); non-local moves (macro-mutation - a mutation of multiple solution compo-nents at once, reptation - a snake-like motion of the polymer happening by diffusion of stored length along its own contour, G A cross-over, construction starting from partial conformations, adding multiple residues). Multi-Scale Modeling approaches also belong into this category [45]. A n example of indirect neighbourhood change is an exchange of conformations at different temperatures in the Replica Exchange Monte Carlo method [94]. 4. Specify the region of interest from which samples are generated, e.g., Im-portance Sampling and Sequential Importance Sampling (construction-based search methods) [60]. An example of a non-construction search method in this category is Model-based Search (MBS) [20]. The search methods proposed in this work are closely related to the last cate-gory, which has not been widely studied in the literature. We also use ideas from the other categories described above. An important quality of the search method is how it balances intensification of the search against diversification. Intensification can be performed in the following ways: 1. enrichment - direct (copying) or indirect (reinforcement) of parts for both • construction and non-construction searches 2. replacement of unfit parts (component solutions that have low objective func-tion values) 3. decreasing neighbourhood size 4. making acception criteria more stringent - lowering temperature Chapter 3. Background and Related Work 39 Diversification can be achieved by: 1. random construction (random restart of the search) 2. random mutation of a complete candidate solution that is accepted 3. random walk - accept a number of random mutations 4. increasing neighbourhood size 5. relaxing acceptance criteria - raising temperature, or by adding a search-progress dependent factor To guarantee efficient searching during the search process, one may want to vary the amount of intensification and diversification based on the search progress or landscape features encountered during the search. Currently, very few adaptive, strategies have been tested for the protein folding problem. Our work described in Chapter 5 makes contributions to this area. Apart from choosing an efficient algorithm, it is also important to choose an efficient set of moves used by the algorithm (as defined by the neighbourhood relations used). Moves employed by search algorithms include, mutation moves in Monte Carlo, mutations and crossovers in Genetic Algorithms. Neighbourhood relations describe a set of conformations that can be reached from a given solution. The size of the neighbourhood is important, as is the diver-sity between the original candidate solution and the neighbours that can be reached in a single step. For example, it has been shown that certain local moves work better than others. Elofsson et a/.introduced local moves for protein folding that disrupt only the local region of a protein chain while keeping the rest of the protein fixed [44]. After each change, the dihedral angles at the boundary of the region were changed to compensate for the local change that took place. They showed that M C and G A methods using these local moves perform better than mutations that change the dihedral angles of a protein, and therefore propagate the change along the chain. Similarly, in the case of lattice models, so-called pull moves have been proposed with the same objective [73]. In our work, described in the next chapter, we have designed long-range moves for 2D and 3D HP models to guarantee a more efficient search of the energy land-scape. 3.1.3 An Overview of Existing Research in 2D and 3D Hydrophobic Polar Folding We now direct our attention towards models that were chosen to test newly pro-posed methods. A number of well-known heuristic optimization methods have Chapter 3. Background and Related Work 40 been applied to the 2D and 3D HP Protein Folding Problem, including Evolution-ary Algorithms (EAs) [68, 69, 102, 134, 135] and Monte Carlo (MC) algorithms [8, 24, 60, 77, 96, 110, 115]. The latter have been found to be particularly robust and effective for finding high-quality solutions to the HP Protein Folding Problem [60]. Unger and Moult [134, 135] presented an early application of Evolutionary Algorithms to protein structure prediction. Their non-standard E A incorporates characteristics of the Monte Carlo methods such as reliance on the Boltzmann probability for determining the acceptance ratio. Currently among the best known algorithms for the HP Protein Folding problem are various Monte Carlo algorithms, including the 'pruned-enriched Rosenbluth method' (PERM) of Grassberger et al. [8, 60]. As mentioned previously, P E R M is a biased chain growth algorithm that evaluates partial conformations and employs pruning and enrichment strategies to explore promising partial solutions. Liu et al. [152] introduced a biased sampling algorithm that is very similar to P E R M except that statistical weights for pruning and enrichment are calculated based on the simulation of a number of chains in parallel. Their algorithm seems to be inferior to P E R M in terms of performance. Other Monte Carlo methods (less successful than PERM) developed to ad-dress this problem include the dynamic Monte Carlo algorithm by Ramakrishnan et al. [110] based on a four-cycle change move that disconnects the chain. Liang et al. [77] introduced an evolutionary Monte Carlo (EMC) algorithm that works with a population of individuals, where each individual performs Monte Carlo (MC) optimization. They also implemented a variant of E M C that reinforces certain sec-ondary structures (alpha helices and beta sheets). Chikenji et al. introduced the Multi-Self-Overlap Ensemble (MSOE) Monte Carlo method [24], which considers overlapping chain configurations. Besides general optimization methods, there are other heuristic methods that rely on specific heuristics based on intuitions or assumptions about the folding process. These include methods that consider the co-operativity of folding or the existence of a hydrophobic core. Co-operativity is believed to arise from lo-cal conformational choices that result in a globally optimal state without an ex-haustive search [33]. Among these methods are the hydrophobic zipper method (HZ) [33], the contact interactions method (CI) [132], the core-directed chain growth method (CG) [13], and the constraint-based hydrophobic core construction method (CHCC) [150]. The hydrophobic zipper (HZ) strategy developed by Dill et al. is based on the hypothesis that once a hydrophobic contact is formed it cannot be broken, and that other contacts are formed in accordance with previously folded parts of the chain (co-operativity of folding) [33]. The contact interactions (CI) algorithm developed by Toma and Toma [132] combines the idea of H Z with a Monte Carlo search pro-Chapter 3. Background and Related Work 41 cedure that assigns different conformational freedom to the different residues in the chain. Thus, this allows previously formed contacts to be modified according to their computed mobilities. The core-directed chain growth method (CG) devised by Beutler and Dill [13] biases construction towards finding a good hydrophobic core. This is achieved by using a specifically designed heuristic function and by ap-proximating the hydrophobic core with a square (in 2D) or a cube (in 3D); the result is a restrictive heuristic that finds only certain native states [13]. The constraint-based hydrophobic core construction method (CHCC) by Yue and Dill [150] is complete, i.e., always guaranteed to find a global optimum: It attempts to find the hydrophobic core with the minimal possible surface area by systematically intro-ducing geometric constraints and by pruning branches of a conformational search tree. A similar but more efficient complete constraint satisfaction search method has been proposed by Backofen et cd. [5] for the more complex face-centered cubic lattice. Other Monte Carlo methods that have been particularly useful in off-lattice protein folding, as mentioned in Section 3.1.1, but have not been tested and com-pared on the set of standard benchmarks include generalized ensemble methods, such as Umbrella Sampling [133] (with Replica Exchange Sampling [52, 86] be-ing the most common variant) and Multicanonical Monte Carlo sampling [10, 86]. Replica Exchange Monte Carlo (Parallel Tempering) has also been applied to the off-lattice HP model [62]. Currently, when applied to the square and cubic lattice HP model, none of these algorithms appears to completely dominate the others in terms of solution quality and run-time. Results of comparing the algorithms mentioned above with our proposed algorithm on a set of well-studied instances are provided in Chapter 4. 3.1.4 An Overview of Existing Research in F C C /3-Sheet Protein Folding The following algorithms have been implemented and tested for the F C C lattice of /3-sheet proteins: classical Metropolis Monte Carlo [52], Multicanonical ( M U C A ) Monte Carlo (or Entropy Sampling Monte Carlo) [52], Replica Exchange Monte Carlo (REMC) [52], and Parallel-hat Tempering Monte Carlo (a variant of R E M C that utilizes an additional weight factor based on the histogram of energies sampled by each temperature) [154]. M C , M U C A and R E M C have been described previously in Section 3.1.1. It is worth noting the differences between R E M C and Parallel-hat Tempering [154], given that Parallel-hat Tempering is the best-performing search method for this model among all the other methods tested in the literature. The latter is an exten-sion of the Replica Exchange method that results in more efficient search by the Chapter 3. Background and Related Work 42 introduction of a new hat-like weight factor to each replica. This weight factor results in both low- and high-energy acceptance probabilities being exponentially reinforced, which allows the algorithm to overcome higher barriers and explore a wider range of energies for each replica. The weight factor chosen was: j(E) = exp{-E/kBT + V2\E - (E)\/o), (3.17) where (E) is the average energy of the system at temperature T and a is the root mean square deviation of the energy. Both are updated iteratively, in the mih step of the M C : (E) = a = (3.18) 1 / 2 (3.19) In the initial few steps, the authors set the average energy to (E) = E. Local movements used in each replica are accepted according to the following probabil-ity: P L _ > 2 = exp[-0AE + V2/a(\E2 - (E)\- \Ei - (E)\)]. (3.20) The following observations were made when the move from the state with energy Ei to the state with energy E2 is uphill: 1. If the energies (E\ and E2) are low compared with the average (E), the acceptance rate is decreased (the probability of these low and rare energies is enhanced by not transitioning to a higher energy state). From the search perspective, this helps to hold on to the lower energy values found during the search. 2. If the energies (E\ and E2) are high compared with the average, the ac-ceptance rate is increased and the unusually high energies are sampled with increased probability. This mechanism has the ability to overcome larger energy barriers. The following observations were made when the move from E\ to E2 is downhill: 1. If the energies (E\ and E2) are low compared with the average (E), the move is accepted, as in the case of the canonical simulation. Chapter 3. Background and Related Work 43 2. If the energies {E\ and E2) are high compared with the average, the move may have not been accepted, even though it is downhill, since the acceptance rate is decreased. Exchange moves between replicas at different temperatures (say, i and j ) sim-ilarly follow a modified acceptance probability: Pi^j = exp[(0i - Bj){Ei - Ej) + V2(\Ej - {E^/cn - \E.t -+ \Ei-{Ej)\/aj-\Ej-(Ej)\/aj)]. This results in a similar hat-like modification of the exchange probabilities be-tween replicas as described above for probabilities of moves within a single replica. As of to-date, no comprehensive study of different search methods has been conducted in the protein folding literature for more complex off-lattice discrete models. This could be because researchers are inclined to use different energy functions in addition to using a limited set of algorithms for sampling and a dif-ferent set of proteins for testing. The most comprehensive studies of search and sampling methods are limited to lattice models, particularly the HP square, and cubic lattices and the F C C lattice. The HP model is not very realistic for the study of real proteins due to a parity problem, the inability to represent protein geometry closely, and a significant distortion of the secondary structure elements. The F C C lattice, as mentioned previously, has been shown to be the most accu-rate lattice among elementary lattices. Additionally, the state-of-the-art searching and sampling methods ( M U C A , R E M C , modified R E M C that uses histogram data) have been tested extensively for the F C C lattice of /3-sheet proteins [52, 106, 154]. Therefore, in our own work for the development, testing, and comparison of new adaptive search methods, we have adopted the F C C model of /3-sheet proteins. Our goal is to be able to compare computational results to the results described in the literature. 3.2 The Problem of Identifying Folding Pathways In this section, we describe theoretical, experimental, and computational work available for the problem of identifying protein folding pathways. Our own work builds on theoretical and experimental data from the literature. Our goal is to avoid placing restrictive assumptions on the underlying process of folding and folding nuclei formation while still developing an efficient and reliable approach for the identification of folding pathways. Chapter 3. Background and Related Work 44 3.2.1 Theoretical and Experimental Work on Protein Folding Pathways Experimental results from studying protein folding kinetics and thermodynamics, along with a number of computational studies for a variety of lattice models, sug-gested three viable mechanisms for the process of protein folding [84]: 1. The framework (hierarchical) or diffusion-collision model proposes that local elements of native stable secondary structure form independently of tertiary structure; they collide and adhere to form tertiary interactions. The rate-limiting step is the docking of the secondary structure elements [27]. 2. The nucleation model suggests that proteins have a small set of interactions common to most of the conformations in the transition state ensemble [27]. This small set of interactions is called a folding nucleus, and can bring to-gether distant parts of the chains (largely formed by long-range interactions with stabilizing short-range interactions). The nucleation mechanism does not require secondary structure elements to be -formed before the transition state is reached. In contrast to the diffusion-collision model, secondary struc-ture may be formed simultaneously with the folding of the tertiary structure. 3. The hydrophobic collapse model suggests that a protein rapidly collapses around its hydrophobic side-chains and then rearranges itself from the re-stricted conformational space. In this model, secondary structure is directed by tertiary interactions [27]. This model is plausible for small proteins. One way of studying and testing the nucleation mechanism and diffusion-collision model, stems from the fact that they predict different effects of mutations on the folding rate. The diffusion-collision model predicts that the stabilization of any local secondary structure element (an a-helix or a /3-strand) will always lead to an acceleration of folding. In contrast, the nucleation model predicts that the strength of tertiary contacts formed in the transition state is the primary determi-nant of folding rate. According to the nucleation model, stabilization of local struc-ture accelerates folding rates only if a particular element is present in the transition state ensemble. Through directed site mutagenesis on real proteins it has been ob-served that the nucleation model seems to be correct for small and medium size proteins [84]. However, for certain proteins in simulations the diffusion-collision model seemed to play the role [155]. Another important partly computational study that provided insight into the protein folding process was conducted by Plaxco et al. [104]. They suggested that the average separation between residues interacting in the native state (contact Chapter 3. Background and Related Work 45 order) can be used as a general descriptor of protein topology. The contact order where 6ij = 1 i f residues i and j are in contact and 0 otherwise; C is the total number of contacts, and ./V is the protein length. For a number of two-state fo ld ing proteins, contact order was reported to exhibit a statistically significant correlation with the logari thm o f the folding rate in an aqueous environment. Proteins that had a higher contact order (had more local contacts) folded faster. Dinner and Karp lus later showed that apart from contact order, stability o f the native state (to a much smaller degree) determines the folding rate as wel l [34]. The precise nature of the folding nucleus (the ratio and distribution of long- and short-range interactions, its size, and other topological characteristics) or of mult iple fo ld ing nuclei for longer proteins s t i l l has to be determined. Exper imental methods for identifying folding nuclei include site-directed mu-tagenesis ($-value analysis), which affects the overall fo lding rate and protein stability [46], and identifying folding cores using hydrogen-deuterium exchange ( H / X ) experiments [76]. Bo th experimental techniques are time- and labour-intensive, and can therefore benefit from additional computational analysis. 3.2.2 An Overview of Computational Approaches for Identifying Folding Nuclei from the Native Conformation A number of computational methods for the prediction of fo ld ing nuclei already exist in the literature, but most of them rely on restrictive assumptions about the nature of nuclei or the process of folding. Current approaches in the literature to ident ifying fo ld ing nuclei include time-intensive molecular dynamic unfolding o f the native structures [36], a Gaussian network model ( G N M ) based on an analysis of the highest frequency fluctuations near the native state [32], an elastic network and constraint network models o f freely rotating rods ( F I R S T ) [107], clustering of native contacts and the use of low effective contact order search [144], a motion planning approach (the probabilistic roadmap method) for prediction of folding pathways [2], m i n i m u m cuts unfolding of native structures [151], amino acid con-servation wi th in a family or super-family [112], search for free energy saddle points on networks of protein unfolding pathways [49], and determination of the transi-tion state by restraining Mon te Ca r lo simulations with a pseudo-energy function based on the set o f experimental <J?-values [97]. M o s t of these methods are based on a number of restrictive assumptions about the fo ld ing nuclei or the search process that may not necessarily hold . Fo r exam-of a given conformation is defined as: (3.21) Chapter 3. Background and Related Work 46 pie, these assumptions include an assumption that folding nuclei are evolutionarily conserved [112], that residues participating in folding nuclei are those that are most physically constrained [107], or that low effective contact order clusters of contacts can not be partially formed [144]. The goal of our work, presented in Chapter 6 of this thesis, is to develop a simple and efficient method of identifying folding nuclei that does not rely on restrictive assumptions about the nature of folding nuclei. 47 Chapter 4 An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding Nature uses only the longest threads to weave her patterns, so each small piece of her fabric reveals the organization of the entire tapestry. Richard Feynman Ant Colony Optimization (ACO) is a population-based stochastic search method for solving a wide range of combinatorial optimization problems. A C O is based on the concept of stigmergy - indirect communication between members of a pop-ulation through interaction with the environment. An example of stigmergy is the communication of ants during the foraging process: ants indirectly communicate with each other by depositing pheromone trails on the ground and thereby influenc-ing the decision processes of other ants [14]. This simple form of communication between individual ants gives rise to complex behaviors and capabilities of the colony as a whole. In real life, ants follow higher concentrations of pheromone and find the shortest path to a food source. Our goal is to develop an algorithm that captures the emergence of higher-order intelligence from low-level communication of a population of agents, similar to the functioning of an ant colony. From a computational point of view, A C O is an iterative construction search method in which a population of simple agents ('ants') repeatedly constructs can-didate solutions to a given problem. This construction process is probabilisti-cally guided by heuristic information on the given problem instance as well as by a shared memory of experiences gathered by the ants in previous iterations ('pheromone trails'). Following the seminal work by Dorigo et al. [40, 41], A C O algorithms have been successfully applied to a broad range of hard combinatorial problems: the Traveling Salesman Problem (TSP), the Quadratic Assignment Problem (QAP), Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding 48 the Job-shop Scheduling Problem (JSP), the Graph Coloring Problem (GCP), the Vehicle Routing Problem (VRP), and others (e.g., [37, 38, 42]). In this work, we address the ab initio protein folding problem. It can be for-mally defined as follows: Given an amino acid sequence s — S1S2 . . . sn and an energy function E(c), find an energy-minimizing conformation of s, /. e., find c* e C(s) such that E(c*) = m i n { £ ( c ) | c 6 C(s)}, where C(s) is the set of all valid conformations for s. A C O , which has been very successfully applied to other combinatorial prob-lems [14], appears to be a very attractive computational method for solving the pro-tein folding problem, since it combines aspects of chain growth and permutation-based search with ideas closely related to reinforcement learning. These concepts and ideas apply rather naturally to protein folding: by folding from multiple ini-tial folding points, guided by the energy function and experience from previous iterations of the algorithm, an ensemble of promising, low-energy complete con-formations is obtained. These conformations are further improved by a subsidiary local search procedure and then evaluated to update the accumulated pheromone values that are used to bias the generation of conformations in future iterations of the algorithm.' In this chapter, we introduce an A C O algorithm for the 2D and the 3D HP protein folding problem, and discuss the promising results obtained [120, 121]. This is the first application of the A C O algorithm to this highly relevant problem 1. 4.1 Description of the Algorithm The 'ants' in our A C O algorithm iteratively undergo three phases: (1) the con-struction phase, during which each ant constructs a candidate solution by sequen-tially growing a conformation of the given HP sequence, starting from a folding point that is chosen uniformly.at random among all sequence positions; (2) the local search phase, when ants further optimize protein conformations folded dur-ing the construction phase; and (3) the pheromone update phase, when ants update the pheromone matrix (representing the collective global memory of the colony) based on the energies of the conformations obtained after the construction and lo-cal search phases. A general outline of A C O is shown in Figure 4.1. The solution components used during the construction process, the local search phase, and the pheromone update consists of the following local structural motifs (or relative folding directions): straight (S), left (L), right (R) in 2D, and straight 'A version of this chapter has been published. A. Shmygelska and H.H. Hoos, (2005) An Ant Colony Optimisation Algorithm for the 2D and 3D Hydrophobic Polar Protein Folding Problem, BMC Bioinformatics, 6:30. Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding procedure ACO initialize pheromone trails; while (termination condition not satisfied) do construct candidate conformations; perform local search; update pheromone values; end end Figure 4.1: Generic outline of Ant Colony Optimization (for static combinatorial problems). STRAIGHT LEFT RIGHT UP DOWN Figure 4.2: The local structural motifs that form the solution components underly-ing the construction and local search phases of our A C O algorithm in 3D. (S), left (L), right (R), up (U), down (D) in 3D. These relative directions indicate the position of each amino acid on the 2D or 3D lattice relative to its direct prede-cessors in the given sequence (see Figure 4.2). In 3D, the relative folding directions are defined as in [6]: A local coordinate system is associated with every sequence position, such that S corresponds to the direction of the x axis, L to the direction of the y axis, and U to the direction of the z axis. Each local motif corresponds to a relative rotation of this coordinate system (for the forward construction, S - no rotation, L = 90° counter-clockwise around the z axis, R = 90° clockwise around the z axi's, U = 90° clockwise around the y axis, D = 90° counter-clockwise around the y axis). Since conformations are rotationally invariant, the position of the first two amino acids can be fixed without loss of generality. Hence, we represent candidate conformations for a protein sequence of length n by a sequence of local structural motifs of length n — 2. For example, the conformation of Sequence Sl-1 from Table 4.1 shown in Figure 4.3 corresponds to the motif sequence L S L L R R L R L L -S L R R L L S L . Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding Figure 4.3: The underlying protein sequence (Sequence Sl-1 from Table 4.1) is HPHPPHHPHPPHPHHPPHPH; black circles represent hydrophobic amino acids, while white circles symbolize polar amino acids. The dotted lines represent the H-H contacts underlying the energy calculation. The energy of this conforma-tion is -9, which is optimal for the given sequence. During the construction phase, ants fold a protein from an initial folding point by probabilistically adding one amino acid at a time based on the two follow-ing sources of information: pheromone matrix values r (which represent previous search experience and reinforce certain structural motifs) and heuristic function values rj (which reflect current energy of the considered structural motif); details of this process are provided in the next section. The relative importance of r and rj is determined by the parameters a and 0, respectively. Similar to other ACO algorithms known from the literature, our algorithm for the HP Protein Folding Problem incorporates a local search phase that takes the ini-tially built protein conformation and attempts to optimize its energy further, using probabilistic long-range moves that are described in detail in the next chapter. Finally, the pheromone update procedure is based on two mechanisms: uni-form pheromone evaporation is modeled by decreasing all pheromone levels by a constant factor p (where 0 < p < 1), and pheromone reinforcement is achieved by increasing the pheromone levels associated with the local folding motifs used in a fraction of the best conformations (in terms of energy values) obtained during the preceding construction and localsearch phase. Furthermore, to prevent search stagnation when all of the pheromone is accumulated on very few structural motifs, we introduce an additional renormalization mechanism for the pheromone levels (controlled by a parameter 9, where 0 < 9 < 1; details are given below). Our ACO algorithm iterates construction, local search, and pheromone update phases until a termination condition is satisfied; in the context of this work, we mostly terminated the algorithm upon reaching a given energy threshold. In the following sections, we describe the three search phases in detail. Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding 4.1.1 Construction Phase, Pheromone, and Heuristic Values During the construction phase of ACO, each ant first determines a starting point within the given protein sequence; this is done by uniform random choice. From this starting point, the sequence is folded in both directions, adding one residue at a time. Each ant performs probabilistic chain-growth construction of the protein conformation, by which the structure is extended either to the left or to the right in every step, such that the ratio of unfolded residues at each end of the protein remains (roughly) unchanged. Here, we assume that folding is performed in 3D; the 2D case is handled analo-gously by considering three relative directions {S, L, R} instead of five {S, L, R, U, D} (see also [119]). The relative directions in which the conformation is extended in each construction step are determined probabilistically based on a heuristic func-tion i)itd and pheromone values r ^ , according to the following formula: P i 4 : = M a M ^ ( z U ) 2~2e£{S,L,R,U,D}iTi,e]°'lrli,e}13 The pheromone values indicate the desirability of using the local structural motif with relative direction d € {S, L. R, U, D} at sequence position i. Initially, all Titd are equal, such that local structural motifs are chosen in an unbiased way. Throughout the search process, however, the pheromone values are updated to bias folding towards the use of local motifs that occur in low-energy structures. The updating mechanism will be described in more detail later. The heuristic values rji^d are based on the energy function E. They are defined according to the Boltzmann distribution as 7 7 ^ : = e~lhi<d, where 7 is a parame-ter called the inverse temperature, as in [60], and h^d is the number of new H-H contacts achieved by placing amino acid i at the position specified by direction d. During construction, it may happen that the chain cannot be extended without running into itself. This situation is called attrition, which our algorithm over-comes as follows. First, starting at the end at which attrition occurred, half of the sequence that has been folded up to this point is unfolded. Then, this segment of the chain is refolded; the first residue {i.e., the last one that was unfolded) is placed such that its relative direction differs from what it had been when attrition occurred, while all of the subsequent residues are folded in a feasible direction that is cho-sen uniformly at random. This backtracking mechanism is particularly important for longer protein sequences in 2D, where infeasible conformations are frequently encountered during the construction phase. Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding 4.1.2 Local Search Similar to other A C O algorithms known from the literature, our algorithm for the HP Protein Folding Problem incorporates a local search phase, based on a long-range mutation move that has been designed to avoid infeasible conforma-tions. It also has a number of important advantages over the more commonly used point mutation moves or Monte Carlo moves (i.e., the end, crankshaft, and cor-ner moves [114]). It is easy to implement; it decreases the number of infeasible conformations encountered, even when the protein is very compact (at high densi-ties); it considers a larger neighbourhood that subsumes the single-point mutation neighbourhood; and it has some validity in terms of the physical processes taking place during the protein folding process. Similar attempts have been previously undertaken, but these involved disconnection of the chain [110]. From studies of protein folding dynamics, it is known that proteins display a broad range of motions that range from localized motions to slow large-scale movements [27]. Inspired by this complex process, we designed a long-range mu-tation move that starts by selecting a residue whose relative direction is randomly mutated and then adapts the rest of the chain by probabilistically changing rela-tive directions starting from this initial position [121]. During this adaptation, the previous relative direction for each residue with a probability p (0 < p < 1) is left unchanged, if it is still feasible. Otherwise (i.e., with probability 1 — p, or if the previous direction has become infeasible), a different relative direction is cho-sen, where the probability for each direction d is proportional to the corresponding heuristic value 77^. Formally, this can be written as follows: di at sequence position i. In our initial implementation, we investigated using the following two types of local search: greedy (Iterative Improvement) and non-greedy (Probabilistic Itera-tive Improvement). Consequently, we had two types of ants in the colony: forager ants that use greedy local search while constructing solutions using the pheromone matrix, and improving ants that exploit the best solution found so far by using non-greedy local search. The number of improving ants compared with forager ants was much smaller. Both local searches employed a long range mutational move designed by us to decrease the number of infeasible configurations considered, es-pecially when a protein is quite compact. (4.2) where P[di := d] is the probability of choosing direction d as the relative direction Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding 53 Unlike in our initial implementation [121], the local search phase of the latest version of our ACO algorithm (which has been shown to work better for both the 2D and the 3D HP) is a simple iterative first improvement procedure that is based on the long-range mutation move. The outline of this local search procedure is shown in Figure 4.4. Iterative first improvement accepts a new conformation gen-erated via long-range mutation only if the solution quality of a new conformation c' improves over the current solution quality (energy) of c. This search process is greedy in the sense that it does not allow worsening steps, and it is terminated when no improving steps have been found after a specific number of scans through the chain (this number is a parameter of the algorithm). procedure Iterat,iveImprovementLS(c) input: c a n d i d a t e c o n f o r m a t i o n c output: c a n d i d a t e c o n f o r m a t i o n c' while ( t e r m i n a t i o n c o n d i t i o n n o t sa t i s f i ed) do i := random({l,..., n}); c := longRangeMove(c, i ) ; if E{c') < E{c) then c : = c'; end end return(c) end Figure 4.4: The iterative first improvement local search procedure that is performed by selected ants after the construction phase. Since this local search procedure has a relatively high time-complexity, in each iteration of ACO it is only applied to a certain fraction of the highest-quality con-formations constructed by the ants in the preceding construction phase. 4.1.3 Update of the Pheromone Values After each construction and local search phase, pheromones are updated according to n,d •= p • n,d, (4.3) where 0 < p < 1 is the pheromone persistence, a parameter that determines how much of the information gathered in previous iterations is retained. Subsequently, selected ants with low-energy conformations update the pheromone values accord-ing to n,d •= Ti>d + A i A c , (4.4) Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding where A i ( / C is the relative solution quality of the given ant's candidate conforma-tion c, if that conformation contains local structural motif d at sequence position i, and zero otherwise. We use the relative solution quality, E(c)/E*, where E* is the known minimal energy for the given protein sequence or an approximation based on the number of H residues in the sequence, in order to prevent premature search stagnation for sequences with large energy values. As a further mechanism for preventing search stagnation, we use an addi-tional renormalization of the pheromone values that is conceptually similar to the method used in the MAX-M1N Ant System [129]. After the standard pheromone updates according to Equations 3 and 4, all r values are normalized such that Yld<E{s L RU D} Ti4 = 1 for every residue i; additionally, whenever for a given sequence position i the minimal normalized pheromone value Tidl V ^ de{S,L,R.,U,D} m i n Tid/ 2^ Ti,d d£{S,L,R,U,D} falls below a threshold 9 (which is a parameter of the algorithm), the minimal ri}d value is set to 6, while the maximal value is decreased by d—minde{s,L,R,u,D} Ti, (If there is more than one minimal Titd value, all of these are increased to 9, and if there is more than one maximal T j c / value, one of them is chosen uniformly at ran-dom.) This guarantees that the probability of selecting an arbitrary local structural motif for the corresponding sequence position does not become arbitrarily small, and hence ensures the probabilistic approximate completeness of our algorithm (see [59]). 4.2 Empirical Results and Discussion In the following work, we address the following questions: is A C O a competitive method for solving the ab initio protein folding problem under the 2D and 3D HP models? How does its performance scale with sequence length? What is the role of the parameters of the A C O algorithm for the efficiency of the optimization process? Which classes of structures (if any) are solved more efficiently by A C O than by any other known algorithms? Finally, it should be noted that our A C O algorithm for this problem is based on very simple design choices, in particular with respect to both the solution components reinforced in the pheromone matrix and the subsidiary local search procedure. We discuss which of the many design choices underlying our algorithm should be reconsidered in order to achieve further performance improvements. To compare A C O with algorithms for the 2D and 3D HP Protein Folding Prob-lem described in the literature, we tested it on a number of standard benchmark Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding 55 instances as wel l as on two newly created data sets. We obtained one of these sets by randomly generating amino acid sequences with hydrophobici ty value charac-teristic of globular proteins, whi le the other consists o f b io log ica l sequences that we translated into H P strings using a standard hydrophobici ty scale. These new data sets w i l l be described in more detail later in this section. 4.2.1 Results for Standard Benchmark Instances The 21 standard benchmark instances for 2 D - and 3 D - H P protein fo ld ing shown in Table 4.1 have been wide ly used in the literature [8, 13, 24, 68, 77, 1L9, 134, 135]. Experiments on these standard benchmark instances were conducted by perform-ing a number of independent runs for each problem instance (in 2 D : 500 runs for sequence length n < 50, 100 runs for 50 < ri < 64, and 20 runs for n > 64; in 3 D : 100 runs for each sequence). Unless expl ic i t ly indicated otherwise, we used the fo l lowing parameter settings for all experiments: a :— 1, 0 := 2, p :— 0.8 and 9 : = 0.05. Furthermore, al l pheromone values were in i t ia l ized to 1/3 in 2 D and to 1/5 in 3 D , and a population of 100 ants was used, 5 0 % o f which were a l -lowed to perform local search. The local search procedure was terminated when no improvement in energy had been obtained after between 1 000 (for n < 50) and 10 000 (for n > 50) scans through the protein sequence. We used an elitist pheromone updating scheme in which only the best 1% of a l l ants was a l lowed to perform pheromone updates. The probability p of keeping the previous direct ion when feasible during the long-range mutation move was set to 0.5. These settings were determined in a series o f experiments in which we studied the influence of different parameter settings, and w i l l be discussed further later. A l l experiments were performed on P C s with 2.4 G H z Pentium IV C P U s , 2 5 6 K b cache and 1 M b R A M , running Redhat L i n u x (our reference machine). Run- t ime was measured in terms of C P U time. M o s t studies of E A and M C methods in the literature, inc lud ing [68, 77, 134, 135], report the number of val id conformations scanned during the search. Th i s makes a performance comparison difficult, since run-time spent for backtracking and the checking of partial or infeasible conformations, wh ich may vary substan-t ial ly between different algorithms, is not accounted for. We therefore compared A C O to the best-performing algorithm from the literature for wh ich performance data in terms of C P U time is available - P E R M [60] (we used the most recent implementation, which was k indly provided by P. Grassberger). We note that the most efficient P E R M variant for the H P Protein Fo ld ing Prob lem uses an additional penalty of 0.2 for H - P contacts [61]. Since this corresponds to an energy function different from that o f the standard H P model underlying our A C O algori thm as wel l as other algorithms developed in the literature, we used the best performing variant 9 Seq. No. Length E* Protein Sequence 2D HP 1 20 -9 (HP)2PH2PHP2HPH2P2HPH 2 24 -9 H2(P2H)7H 3 25 -8 P2HP2{H2P4)3H2 4 36 -14 P3 H2 P2 H2 P5 H7 P2 H2 P4 H2 P2 HP2 5 48 -23 P2H(P2H2)2P5H10P6(H2P2)2HP2H5 6 50 -21 H2(PH)3PH4PH(P3H)2P4H(P3H)2PH4(PH)3PH2 7 60 -36 P2H3PHeP3H10PHP3H12P4H6PH2PHP 8 64 -42 Hl2{PH)2(P2H2)2P2H{P2H2)2P2H{P2H2)2P2HPHPHl2 9 85 -53 H4P4H12P6(H12P3)3HP2(H2P2)2HPH 10 100 -50 P3H2P2H4P2H3{PH2)2PH4PaH6P2H6P9HPH2PHnP2H3PH2PHP2HPH3PeH3 11 100 -48 PeHPH2P5H3PH5PH2P4H2P2H2PH5PHwPH2PH7PllH7P2HPH3P6HPH2 3D HP 1 48 -32 HPH2P2H4PH3P2H2P2HPH3PHPH2P2H2P3HPSH2 2 48 -34 . H4PH2PH5P2HP2H2P2HP<iHP2HP3HP2H2P2H3PH 3 48 -34 PHPH2PHeP2HPHP2HPH2(PH)2P3H(P2H2)2P2HpHP2HP 4 48 -33 PHPH2P2HPH3P2H2PH2P3H5P2HPH2{PH)2P4HP2{HP)2 5 48 -32 P2HP3 HP H4 P2 H4 PH2PH3P2 (HP)2HP2HP6 H2PH2PH 6 48 -32 H3P3H2PH{PH2)3PHP7HPHP2HP3HP2H&PH 7 48 -32 PHP4HPH3PHPH4PH2PH2P3HPHP3H3{P2H2)2P3H 8 48 -31 PH2PH3PH4P2H3P6HPH2P2H2PHP3H2{PH)2PH2P3 9 48 -34 (PH)2P4(HP)2HP2HPH6P2H3PHP2HPH2P2HPH3P4H 10 48 -33 PH2PeH2P3H3PHP2HPH2(P2H)2P2 H2 P2 HT P2 H2 Table 4.1: Benchmark instances for the 2D and 3D HP Protein Folding Problem used in this study with optimal or best known energy values E*. Most instances for 2D and 3D HP can also be found at http://www.cs.sandia.gov web site; Sequence S1-9 (2D) is taken from [67], and the last two instances (2D) are from [110]. Hi and Pt indicate a string of i consecutive H's and P's, respectively; likewise, (s)i indicates an i-fold repetition of string s. 3 > 3 n o" 3 N Eo O 3 NO 3 Q. 3-O o-3' 2 3 O N Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding of P E R M [60] based on the standard energy function in our experiments. It may be noted that the chain growth process in P E R M can start from the N- or C-terminus of the given HP sequence; in many cases, this results in substantial differences in the performance of the algorithm. To capture this effect, we always ran P E R M in both directions. In addition to the respective average run-times, t\ and t%, we re-port the expected time for solving a given problem instance when performing both runs concurrently, texp = 2 • (l/t\ + l / ^ ) - 1 - For all runs of P E R M , the following parameter settings were used: inverse temperature 7 := 26 and q := 0.2. The results obtained on standard 2D benchmark instances (see Table 4.2) in-dicate that A C O is competitive with the E A and M C methods described in the literature; it works very well on sequences of sizes up to 64 amino acids and pro-duces high quality suboptimal configurations for the longest sequences considered here (85 and 100 amino acids). On average, A C O requires less C P U time than P E R M for finding best-known conformations for Sequence SI-8. However, P E R M performs better for Sequences S1-6 and Sl-7 as well as for the longer sequences of 85 to 100 residues (Sequences S1-9 to S i l l ) . Sequence Sl-8 has a very symmetrical optimal state (see Figure 4.5), which, as argued in [60], would be difficult to find for any chain-growing algorithm. Al l algorithms from the literature that we are aware of have problems folding this se-quence. A C O , on the other hand, is able to handle this instance quite well, since a number of ants folding from different starting points in conjunction with a local search procedure that involves large-scale mutations originating from different se-quence positions can produce good partial folds for various parts of the chain. In comparison with other algorithms for the 2D HP Protein Folding Problem consid-ered here (EA, E M C , M S O E ) , A C O generally shows very good performance on standard benchmark instances. In the case of the 3D HP Protein Folding Problem (see Table 4.3), the majority of algorithms for which we were able to find performance results in the literature use heuristics that are highly specialized for this problem. Unlike H Z , C G , and CI, A C O finds optimal (or best-known) solution qualities for all sequences. However, P E R M (when folding from the A^-terminus) and C H C C consistently outperform A C O on these standard 3D HP benchmark instances, and C G reaches best-known solution qualities substantially faster in many cases. We note that for Sequences S2-3 and S2-7, PERM'S performance is greatly dependent on the folding direction. 4.2.2 Result for New Biological and Random Data Sets To thoroughly test the performance of A C O , we created two new data sets of ran-dom and biological sequences of length « 30 and ss 50 amino acids (ten sequences for each length; for details, see Appendix A). Random sequences were generated Seq. No. Length Energy GA EMC MSOE PERM ACO 1 20 -9 -9 (30 492) -9 (9374) -9 « 1 sec) -9 « 1 sec) 2 24 -9 -9(30 491) -9 (6 929) -9 « 1 sec) -9 « 1 sec) 3 25 - 8 - 8 (20 400) - 8 (7 202) - 8 (12 sec) - 8 « 1 sec) 4 36 - 1 4 - 1 4 (301339) -14 (12 447) - 1 4 (< 1 sec) - 1 4 (4 sec) 5 48 - 2 3 -23 (126 547) - 2 3 (165 791) - 2 3 (15 min) - 2 3 (1 min) 6 50 - 2 1 - 2 1 (592 887) - 2 1 (74 613) - 2 1 (2 sec) - 2 1 (15 sec) 7 60 - 3 6 -34 (208 781) -35 (203 729) - 3 6 (5 sec) - 3 6 (20 min) 8 64 - 4 2 -37 (187 393) -39 (564 809) -39 -40 (2 days) - 4 2 (1.5 hrs) 9 85 - 5 3 -52(44029) - 5 3 (11 sec) - 5 3 (20 % of runs 1 day) 10 100 - 5 0 - 5 0 (50 hrs) - 5 0 (50% of runs 2 hrs) -49 (12 hrs) 11 100 - 4 8 -47 - 4 8 (2 min) -47 (10 hrs) 9 •a Table 4.2: Compar i son of the solution quality obtained in 2 D by the evolutionary algori thm of Unger and M o u l t ( E A ) [135], the evolutionary Mon te Car lo algori thm o f L i a n g and Wong ( E M C ) [77], the Mul t i -Se l f -Over lap Ensemble a lgori thm of Chicken j i et al. ( M S O E ) [24], the pruned-enriched Rosenbluth method ( P E R M ) and A C O . For E A and E M C , the reported energy values are the lowest among five independent runs, and the values in parentheses are the numbers of val id conforma-tions scanned before the lowest energy values were found. M i s s i n g entries indicate cases where the respective method has not been tested on a given instance. The C P U times reported in parentheses for M S O E were determined on a 500 M H z C P U , and those for P E R M and A C O are based on 100 — 200 runs per instance on our reference 2.4 G H z Pentium I V machine. The energy values shown in bo ld face correspond to currently best-known solution qualities. a oo Seq. No. Length Energy HZ CHCC CG CI PERM ACO 1 48 -32 -31(15 000 min) -32 (30 min) -32 (9.4 min) -32 -32(0.1 min) -32 (30 min) 2 48 -34 -32 (71 000 min) -34 (2.3 min) -34 (35 min) -33 -34 (1 min) -34 (420 min) 3 48 -34 -31 (82 000 min) -34 (30 min) -34 (62 min) -32 -34 (30.5 min) -34 (120 min) 4 48 -33 -30 (1 600000 min) -33 (71 min) -33 (29 min) -32 -33 (2 min) -33 (300 min) 5 48 -32 -30(110 000 min) -32 (32 min) -32 (12 min) -32 -32 (1 min) -32 (15 min) 6 48 -32 -29 (180 000 min) -32 (80 min) -32 (460 min) -30 -32 (1 min) -32 (720 min) 7 48 -32 -29 (220 000 min) -32 (110 min) -32 (64 min) -30 -32 (1 min) -32 (720 min) 8 48 -31 -29(14 000 min) -31 (530 min) -31 (38 min) -30 -31 (0.3 min) -31 (120 min) 9 48 -34 -31 (16 000 min) -34 (8.3 min) -33 -32 -34 (8 min) -34 (450 min) 10 48 -33 -33 (4 100 min) -33 (4.8 min) -33(1.1 min) -32 -33 (0.15 min) -33 (60 min) n i t Table 4.3: Comparison of the solution quality obtained in 3D by the hydrophobic zipper (HZ) algorithm [33], the constraint-based hydrophobic core construction method (CHCC) [150], the core-directed chain growth algorithm (CG) [13], the contact interactions (CI) algorithm [132], the pruned-enriched Rosenbluth method (PERM) and A C O . For CI, only the best energies obtained are shown. For H Z , C H C C and C G , the reported C P U times are taken from [13]; these are the expected times for finding optimal solutions on a Sparc 1 workstation. In the case of H Z , the reported C P U times are based on an extrapolation from the measured times required for finding suboptimal conformations with the energy values listed here. The C P U times for P E R M and A C O were determined on our reference 2.4 GHz Pentium IV machine based on 50 — 100 runs per instance. The energy values shown in bold face correspond to currently best-known solution qualities. 3 Crq Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding Figure 4.5: The native conformation of Sequence Sl-8 from Table 4.1 (64 amino acids; energy —42), found by A C O in an average C P U time of 1.5 hours and by P E R M in = t2 = texp = 78 hours. based on the observation that most globular proteins have a fairly uniform amino acid profile, and that the percentage of hydrophobic residues of the majority of globular proteins falls in the range of 40-50% [90]. Thus, we chose the probability of generating character H at each position of a sequence to be 0.45, and in the remaining cases (i.e., with probability 0.55), we generated a P. For the biological test sets, ten sequences were taken from the P D B S E L E C T data set with homology < 25% from the Protein Data Bank (PDB) in order to obtain a non-redundant representative set of proteins. These protein sequences were translated into HP strings using the hydrophobicity scale classification of R A S M O L [116], according to which the following amino acids were considered hydrophobic: Ala, Leu, Val, He, Pro, Phe, Met, Trp, Gly, and Tyr. Non-standard amino acid symbols, such as X and Z, were skipped in this translation. Figures 4.6 and 4.7 illustrate the performance of A C O vs P E R M in terms of mean C P U time over 10 runs per instance and algorithm. For practical reasons, each run was restricted to 1 C P U hour on our reference machine, and the lowest energies obtained in these runs (listed in Appendix A) are not necessarily optimal. As can be seen from these results, in 2D, A C O performs roughly comparably to P E R M ( P E R M ' S texp was calculated as described in the previous subsection): A C O reaches the same energies as P E R M , but on some instances, particularly of length 50, requires more run-time. In 3D, A C O generally requires a comparable amount of run-time on sequences of length 30 and outperforms P E R M on one ran-dom sequence of length 30. It performs noticeably worse, however, on sequences of length 50 and in some cases does not reach the same energy. We also gener-Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding ated longer sequences of length 75; for these, A C O failed to reach the min imal energy values obtained by P E R M in a number of cases. The run-times for both algorithms are reported in detail in Append ix A ; we note that on some sequences, the performance of P E R M depends significantly on the direction of fo ld ing. Inter-estingly, there is no significant difference in performance between the b io logica l and random test-sets for either P E R M or A C O . In summary, the performance of A C O is comparable wi th that o f P E R M (the best known algori thm for the 2 D and 3 D H P Protein Fo ld ing Problem) on b io log-ical and random sequences of length 30-50 , but worse on longer sequences. Th i s scal ing effect is significantly more pronounced in 3 D than in 2 D . W e note that nei-ther A C O nor P E R M were opt imized for short sequences (n < 30); however, by using parameter settings different from the ones specified earlier, the performance of both algorithms can be significantly improved in this case. 4.2.3 Characteristic Performance Differences between A C O and P E R M To further investigate the conditions under which A C O performs wel l compared to P E R M , we visual ly examined native conformations found by both algorithms, paying special attention to conformations for which one of the two algorithms does not perform wel l (see Figures 4.8 and 4.10). Based on our observations, we hy-pothesized that P E R M usually performs wel l on sequences that have a structural nucleus in the native conformation at one of the ends of the sequence, particularly the end from which P E R M starts folding the sequence. O n the other hand, P E R M has trouble folding sequences whose native conformations have structural nuclei in the middle of the sequence. In comparison, A C O is not significantly affected by the location o f the structural nucleus (or mult iple nuclei) i n the sequence, since it uses construction from different folding points as wel l as the long-range mutation moves in local search, wh ich can initiate re-folding from arbitrary sequence posi-tions. Here, we use the term 'structural nucleus' to refer to a predominantly local ly folded part of the chain that can be folded sequentially relatively easily based on local sequence information [27], For most sequences considered in this study, we observed a single structural nucleus, which is not surprising, given their relatively short length; however, it is generally believed that longer sequences have mult iple folding nuclei [27]. The left side of Figure 4.8 shows an example of a relatively short b io logica l sequence (B50-7 , 45 amino acids) wi th a unique native hydrophobic core in the 2 D H P model . (This is rare for H P sequences, which usually have a high ground state and hydrophobic core degeneracy. A c c o r d i n g to our observations, o f the 11 standard benchmark instances in 2 D , only Sequences S l - 1 , S l - 3 , and S l - 4 have a er 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding . 6 - 4 ? 0 2 4 6 8 - 6 - 4 - ? 0 2 4 6 8 P F R M C P U l i m e , ; o g j s e c ] P E R M C P U t i m e , l o g [ s e c | Figure 4.6: M e a n C P U time (natural log transformed) required by A C O vs P E R M for reaching the best solution quality, as observed over 10 runs with a cut-off t ime of 1 C P U hour for sequences of length 30 and 50 in 2 D . The left and right plots show the results for the b io logica l and random test-sets, respectively. Performance results for instances of size 30 are indicated by circles, whi le stars mark results for instances of size 50. The dashed lines indicate the band wi th in wh ich performance differences are not statistically significant (significance was determined using the M a n n - W h i t n e y U test and setting the significance level at 0.05 [59]). M e a n run-times were obtained from 10 runs per instance and algori thm, and we only show data points for the runs where the best known solution quali ty was reached at least in some runs out o f 10 by both algorithms. When unsuccessful runs were present, the expected time was calculated as in [101]. For random sequences of length 50, there are three instances for which this was the case. Deta i led results for both successful and unsuccessful runs are given in Append ix A . Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding - 4 - 2 0 2 4 6 8 - 4 - 2 0 2 * 6 8 PFRM CPU Time tog [Sftc] PERM CPU ItiTtfl. log [soc] Figure 4.7: Mean C P U time (natural log transformed) required by A C O vs P E R M for reaching the best solution quality, as observed over 10 runs with a cut-off time of 1 C P U hour for sequences of length 30 and 50 in 3D. The left and right plots show the results for the biological and random test-sets, respectively. Performance results for instances of size 30 are indicated by circles, while stars mark results for instances of size 50. The dashed lines indicate the band within which performance differences are not statistically significant (significance was determined using the Mann-Whitney U test and setting the significance level at 0.05 [59]). Mean run-times were obtained from 10 runs per instance and algorithm. We only show data points for the runs where the best known solution quality was reached at least in some runs out of 10 by both algorithms. When unsuccessful runs were present, the expected time was calculated as in [101]. For biological sequences of length 50, there are four instances and for random instances of length 50, there are two instances for which this was the case. Detailed results for both successful and unsuccessful runs are given in Appendix A. Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding 50' o o - © G -e—• o <p • • • O—1 <> • • • o Figure 4.8: Left side: Lowest energy conformation of a b io logica l sequence (B50-7, 45 amino acids, energy - 1 7 ) that is harder for P E R M (tx = 271, t2 = 299, 284 C P U seconds) than for A C O (texp = 130 C P U seconds; cut-off t ime vexp 1 C P U hour). R igh t side: Lowest energy conformation of a b io log ica l sequence (B50-5, 53 amino acids, energy - 2 2 ) that is much harder for A C O than for P E R M ; wi th in a cut-off time of 1 C P U hour, both A C O and P E R M reached this energy in 10 out of 10 runs in t, average, respectively. avg 820 and tx = 5, t2 = 118, t, exp 9 C P U seconds on Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding 0 — © Q Q Q Q — e -6 O » © — e — • <b © - • © - ^ > Q O Q O O Q (b 0 © © ©—© Q O 6 o o 60 o—• CP Q - ® (b Q—© C) o O O Q o-f 1 • • • 50 CP O 0 0 6 cb cb f> o o 0 - 0 0 0 0 o -© Figure 4.9: Left side: Unique minimal energy conformation of a designed se-quence, D - l (length 50, energy —19); A C O reaches this conformation much faster than P E R M when folding from the left end (mean run-time over 100 successful runs for A C O : 236 C P U seconds, compared to tx = 3 795, t2 = 1, tt exp 2 C P U seconds for PERM). Right side: Unique native conformation of another designed sequence, D-2 (length 60, energy -17). A C O finds this conformation much faster than P E R M folding from either end (mean run-time over 100 successful runs for A C O : 951 C P U seconds, compared to tY = 9 257, t2 = 19 356, texp = 12 524 C P U seconds for PERM). er 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding Figure 4.10: Left side: Lowest energy conformation of random sequence R 5 0 - 9 (50 amino acids, energy - 3 0 ) , which is harder for P E R M when fo ld ing f rom the left end than for A C O . W i t h a cut-off time of 1 C P U hour, A C O reached this en-ergy in 10 out o f 10 runs wi th texp — 1000 C P U seconds, whi le P E R M failed to find a conformation with this energy in 7 out o f 10 runs when fo ld ing from the left end ( t i = 9 892, t2 = 2, texp = 3 C P U seconds). The conformation displayed here has relative directions: D S R U U R L R R L D D L U R R U L R S D U R U R L -R R D D U S R R S U L R R D U U L S L L U R . Right side: Lowest energy conformat ion o f random sequence R50-7 (50 amino acids, energy —38), wh ich is much harder for A C O than for P E R M . Wi th a cut-off time of 1 C P U hour, P E R M reached this en-ergy in two out o f 10 runs when folding from the left and in 10 o f 10 runs when folding from the right end in t\ = 15 322, t2 = 46, texp = 92 C P U seconds, whi le the lowest energy reached by A C O over ten runs was —37. The conformation displayed here has relative directions: S U U R U D U L U U D S S U U S R U D D L D S S U -U R R U U L D L D R D D U R L L D L R U S U R . Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding 67 Figure 4.11: Distr ibut ions of H - H contact order for 500 conformations o f Sequence S l - 7 from Table 4.1 (60 amino acids) in 2 D (left side) and Sequence S l - 5 from Table 4.1 (48 amino acids) in 3 D (right side) found by A C O and P E R M . 10 20 30 40 50 0 10 20 30 40 length of sequence prefix length ot sequence prefix Figure 4.12: M e a n hydrophobic solvent accessible area as a function o f prefix length for a b io log ica l sequence (B50-4, 50 amino acids) in 2 D (left side) and Se-quence S2-6 from Table 4.1 (48 amino acids) in 3 D . Crosses and circles represent mean values for an ensemble of 100 native structures found by A C O and P E R M , respectively. Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding 68 a j i ixxx °°XXXXXX Ix cm coo / fXCOxxx* 1 I p p a 5«»oool 1D 20 30 A length of sequence prefix 20 30 length of sequence prefix Figure 4.13: M e a n number of H - H contacts as a function of prefix length for a b io log ica l sequence (B50-4 , 50 amino acids) in 2 D (left side) and Sequence S2-6 from Table 4.1 (48 amino acids) in 3 D . Crosses and circles represent mean values for an ensemble of 100 native structures found by A C O and P E R M , respectively. oocccc^a: ! ^ ^ J O O O O O t 3 0 oar* COCO ' COD XXXX ao' ' . wax cpxaxxxxx) jyy,' xxx**** 10 20 length ol sequence prefix 10 20 30 40 length of sequence prefix Figure 4.14: M e a n H - H contact order as a function as a function of prefix length for a b io logica l sequence (B50-4, 50 amino acids) in 2 D (left side) and Sequence S2-6 from Table 4.1 (48 amino acids) in 3 D . Crosses and circles represent mean values for an ensemble o f 100 native structures found by A C O and P E R M , respectively. er 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding unique hydrophobic core; in 3D, none of the sequences studied here have a unique hydrophobic core.) This sequence has no structural nuclei at its ends; instead, the two ends interact with each other. A C O outperforms P E R M by a factor of 2 on this sequence in terms of C P U time. Using a cut-off time of 1 C P U hour per run, P E R M found the optimum with energy —17 in an average run-time of 284 C P U seconds (t\ — 271 sec, t2 = 299 sec), while using the same cut-off time and machine, A C O found the optimum in an average run-time of 130 C P U seconds. We also designed two additional sequences, D - l and D-2, of length 50 and 60, respectively, that have a unique native state in which both ends of the sequence interact with each other (see Figure 4.9). Sequence D - l also has a structural nu-cleus near its C-terminus. When testing the performance of P E R M and A C O on these sequences, we found that on D - l , A C O requires a mean run-time of 236 C P U seconds, compared to tj = 3 795, t2 = 1, texp = 2 C P U seconds for P E R M (val-ues are based on 100 successful runs). When this sequence was reversed, P E R M started folding the sequence from the structural nucleus, and its mean run-time dropped to 1 C P U second. A result similar to that for sequence B50-7 was ob-tained for Sequence D-2, which has no structural nuclei at the ends, but a native state in which the ends interact with each other. Here, A C O was found to require a mean run-time of 951 C P U seconds (again, mean run-times were obtained from 100 successful runs), compared to t\ = 9 257, t2 = 19 356, texp = 12 525 C P U seconds for P E R M . As expected, reversing the folding order of the sequence in this case, did not cause a decrease in P E R M ' S run-time. We also analyzed native conformations of sequences on which P E R M outper-forms A C O and observed that the end from which P E R M starts folding is relatively compact and forms a structural nucleus in the resulting conformation. An exam-ple of a conformation with the structural nucleus at the beginning of the sequence (near the Af-terminus, i.e., residue 1) is shown in the right panel of Figure 4.8. For this biological sequence (B50-5, 53 amino acids), P E R M finds an optimal confor-mation with an energy of —22 in t\ = 5,t2 = 118, texp = 9 C P U seconds, while the average run-time for A C O is 820 C P U seconds. Our A C O algorithm generally performs worse than P E R M on sequences that have structural nuclei at the ends, because it tends to spend substantial amounts of time compacting local regions in the interior of the sequence, while P E R M folds more systematically from one end. These observations also hold in 3D, as seen from two random sequences folded in 3D (see Figure 4.10). To further investigate our hypothesis, we studied differences between the dis-tributions of native conformations found by A C O and P E R M , respectively. For this purpose, we introduced the notion of relative H-H contact order, which captures arrangement of H residues in the core of the folded protein, and thus determines the topology of the conformation (the closely related concept of contact order was Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding 70 first defined in [104]). Relat ive H - H contact order is defined as fo l lows: where I is the number of H - H contacts, n is the number of H residues in the se-quence, and i and j are interacting H residues that are not neighbours in the chain (contact belongs to the set o f H - H contacts HH). Intuitively, COH-H speci-fies the average sequence separation between H - H residues in contact per H in the sequence. Figure 4.11 shows the cumulative frequency distributions of relative H - H con-tact order values for sets of native conformations of a 2 D (left panel) and 3 D (right panel) standard benchmark instance, respectively, found by A C O and P E R M over 500 independent runs. Each run was terminated as soon as a native conforma-tion had been found. These results show that the A C O algori thm finds a set of native conformations with a wider range of H - H contact order values than does P E R M . In particular, A C O finds conformations wi th high relative H - H contact or-der compared with P E R M (more distant parts of the chain interact, for example, relative CO^-H = 0.324 for Sequence S l - 7 in 2 D and relative COH-H = 0.75 for Sequence S2-5 in 3 D are not found by P E R M ; s imi lar results were obtained for other sequences). These results further support our hypothesis that in both, 2 D and 3 D , P E R M is biased toward a more restricted set of native conformations. W e per-formed analogous experiments for the case where P E R M is a l lowed to keep certain statistics from one run to another, that is, runs are no longer independent, as in [60]. We found no significant differences in the set of conformations obtained. To further examine the topological differences between ensembles o f native conformations found by the two algorithms, we also looked at the hydrophobic solvent accessible area (defined as SA^.H '•= Y^h Eh, where Eh is the number of unoccupied lattice sites around H residue h), the number of H - H contacts, and the H - H contact order as a function of the length of the sequence prefix (starting from the A^-terminus of the sequence, where P E R M starts folding). In this analysis, we calculated the properties of interest mentioned above for the native conforma-tions found in 100 independent runs by A C O and P E R M , and plotted the mean values of the respective quantities as functions of the sequence prefix length (see Figures 4.12, 4.13, and 4.14). A s seen in Figure 4.12, A C O is less greedy than P E R M , both in 2 D (left side) and in 3 D (right side). It also tends to leave more lattice sites around H residues accessible for future contacts with other H residues that appear later in the chain. This is also reflected in the mean number of H - H contacts formed when folding pre-fixes o f increasing length; A C O tends to form fewer H - H contacts than P E R M for 1 (4.5) (i,j)€HH,i<j-l Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding short- and medium-size prefixes (see Figure 4.13). By examining the dependence of absolute H - H contact order (defined as } J2(i j)eHH i<j-i N ~~ j\> t n e a v e r a g e sequence separation per H-contact) on prefix length, we furthermore observed that, different from P E R M , A C O realizes the bulk of its local H - H interactions in the middle of the given sequence (see Figure 4.14). This further confirms that A C O is capable of finding native conformations with structural folding nuclei that are not located at or near the end of a given protein sequence. The results illustrated in Figures 4.12, 4.13, and 4.14 are typical for all 2D and 3D HP instances we studied. 4.2.4 Discussion of A C O Results Although conceptually rather simple, our A C O algorithm is based on a number of distinct components and mechanisms. A natural question to ask is whether and to which extent each of these contributes to the performance reported in the previous section. A closely related question concerns the impact of parameter settings on the performance of A C O . To address these questions, we conducted several series of experiments. In this context, we primarily used three standard test sequences: Sequence Sl-7 of length 60 and Sequence SI-8 of length 64 (long sequences) in 2D, as well as Sequence S2-5 of length 48 in 3D (all standard benchmark sequences for 3D are 48 amino acids in length). These sequences were chosen because the C P U time required to find the best-known solutions was sufficiently small to perform a large number of runs (100-200 per instance) needed to obtain reliable distributions. Following the methodology of Hoos and Stiitzle [58], we measured run-time distributions (RTDs) of our A C O algorithm, which represent the (empirical) probr ability distribution over the run-time required to reach (or exceed) a given solution quality. The solution qualities used here are the known optimal or best-known energies for the respective sequences. P h e r o m o n e Values and Heur i s t i c I n f o r m a t i o n Two important components of any A C O algorithm are the heuristic function, which indicates the desirability of using particular solution components during the con-struction phase, and the pheromone values, which represent information learned over multiple iterations of the algorithm. Three parameters control the influence of the pheromone information versus heuristic information on the construction of can-didate solutions: the relative weight of the pheromone information, a; the relative weight of the heuristic information, j3; and the pheromone persistence, p. In the first experiment, we investigated the impact of pheromone (a) and heuris-tic information (/?), and their relative importance for the performance of our A C O er 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding algorithm. As can be seen from the results shown in Figure 4.15, both the pheromone values and the heuristic information are important in 2D and 3D. When either is ignored either (a 0 or 0 := 0, respectively), the algorithm performs worse, par-ticularly for longer 2D sequences (n > 50). For short 2D sequences with n < 50, the pheromone matrix does not appear to play a significant role, since sequences are generally easily solved by the subsidiary local search procedure alone. The optimal settings for a and 0 for most problem instances seem to be around a = 1 and 0 = 2, as shown in Figure 4.15. It should be noted that in 3D, pheromone information appears to be less important than in 2D, which suggests that the spe-cific solution components used in our algorithms are somewhat less meaningful in 3D. The goal of the. next experiment was to further explore the role of experience accumulated over previous iterations in the form of pheromone values. To this end, we varied the pheromone persistence, p, while keeping other parameters constant. The results shown in Figure 4.16 show that in 2D, it is important to utilize past experience (i.e., to choose p > 0), but also to weaken its impact over time (i.e., to use p < 1). At the same time, closer examination revealed that for p > 0, attrition, or the construction of inextensible partial conformations, is a major problem, re-sulting from the accumulation of pheromone from multiple conformations. This is why the backtracking mechanism described earlier is extremely important for the performance of our algorithm in 2D. In 3D, for the previously stated reasons and because of the fact that the attrition problem is much less severe, the impact of the persistence parameter is generally smaller than in 2D. A n t C o l o n y Size a n d L e n g t h of L o c a l S e a r c h Phase During the initial empirical evaluation of our algorithm, we observed that ant colony size, i.e., the number of ants used in each iteration, and the duration of local search, expressed as the number of non-improving search steps we are will-ing to consider before terminating the local search procedure, are correlated and significantly affect its performance. To further investigate this phenomenon, we conducted additional experiments in which we fixed the ant colony size and varied the maximal number of non-improving steps during local search, and vice versa. In this series of experiments, different colony sizes were considered, from a sin-gle ant up to a population of 5 000 ants. The number of non-improving steps in local search was varied, from 100 to 10 000. The results, shown in Figure 4.17, indicate that there is an optimal colony size of about 100 ants for both 2D and 3D. A C O is quite robust with respect to colony size, but performance decreases for very small or very large colony sizes. Intuitively, this is the case because the use of a population of ants provides diversification to the search process, which enables it to explore different regions of the underlying search space. Very small popula-Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding Figure 4.15: Effect o f the relative weights of pheromone information, a , and heuristic information, 0, on the average C P U time required for obtaining m i n i m a l energy conformations of Sequence S l - 8 in 2 D (length 64, left side) and Sequence S2-5 in 3 D (length 48, right side). Figure 4.16: Effect o f the pheromone persistence parameter, p, on the average C P U time required for obtaining minimal energy conformations of Sequence S l - 8 in 2 D (length 64, left side) and Sequence S2-5 in 3 D (length 48, right side). Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding 74 tions provide insufficient diversification, and the search stagnates easily, wh i l e for very large populations, the additional time required for running the search phases for each ant on the same sequential machine is no longer amortized by increased efficiency of the overall search process. Our results also indicate that the performance of A C O is more sensitive to the number of non- improving steps than to ant colony size. The opt imal value for the m a x i m u m number of non-improving steps tolerated (per ant) before the local search phase terminates was found to be around 1 000 for short 2 D sequences (n < 50) and around 10 000 for long 2 D sequences (n > 50). The latter value also appeared to be opt imal for al l 3 D sequences considered here. Th i s observation fol lows the intuit ion that more degrees of freedom, as present for longer sequences and in higher dimensions, require more time for local opt imizat ion, since for any conformation, improv ing neighbours tend to be rarer and hence harder to find. Figure 4.17: M e a n C P U time required for finding m i n i m u m energy conformations of Sequence S I - 7 in 2 D (length 60, left side) and Sequence S2-5 i n 3 D (length 48, right side), as a function of ant colony size and the m a x i m u m number of non-improving local search steps. Selectivity and Persistence of Local Search A s described previously, our A C O algori thm uses selective local search, i.e., local search is performed only on a certain fraction of the lowest energy conformations. W e observed that A C O is fairly robust with respect to the fraction o f conformations to which local search is applied. G o o d performance was obtained for loca l search selectivity values between 5% and 50%, but performance was found to deteriorate Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding 75 when local search is performed by all ants. Intuitively, s imilar to co lony size, local search selectivity has an impact on search diversification. If too few ants perform local search, insufficient diversification is achieved, wh ich typ ica l ly leads to premature stagnation of the search process. O n the other hand, i f local search is performed by too many ants, the resulting substantial overhead in run-time can no longer be amortized by increased search efficiency. S imi l a r ly to selective local search, pheromone update was performed only by the certain fraction of so-called 'elitist ants' whose solution quali ty after the local search phase is highest wi th in the population. A s in the case o f local search selec-tivity, A C O shows robustly high performance for elitist fractions between 1% and 5 0 % (results are not shown here), but performance deteriorates markedly when all ants in the colony are a l lowed to update the pheromone matrix. probability of previous direction probability of previous direction Figure 4.18: M e a n C P U time required for finding m i n i m u m energy conformations of Sequence S l - 8 in 2 D (length 64, left side) and Sequence S2-5 in 3 D (length 48, right side), as a function of the probabili ty of retaining previous directions (p) during long-range mutation moves. In a final experiment, we studied the impact of the persistence o f local search, i.e., o f the probabili ty p o f retaining (feasible) previous relative directions during long-range mutation moves. A s can be seen in Figure 4.18, good performance is generally obtained for p values between 0.3 and 0.7. Bo th extreme cases, i.e., p = 0, wh ich corresponds to an extremely H-contact-greedy mutation operator, and p = 1, in which re-folding always fol lows previous directions when feasible, result in a substantial decrease in performance. W h e n p = 0, the decrease in performance in 3 D is smaller than in 2 D . This result is due to the fact that there is no severe attrition in 3 D as in 2 D , where greedy placement of H residues leads to early Chapter 4. An Ant Colony Optimization for 2D and 3D Hydrophobic Polar Protein Folding formation of very compact partial conformations that often cannot be extended into valid complete conformations. The performance decrease for high p values is due to insufficient ability of the chain to fold into a new conformation accommodating well the local change in structure that triggered the re-folding. 4.3 Summary We have shown that all components of our A C O algorithms contribute to its per-formance. In particular, performance is affected by the following components and parameters, listed in order of decreasing impact: pheromone values, termi-nation criterion for local search, persistence of long-range moves, ant colony size, pheromone persistence, heuristic function, selectivity of local search, and selectiv-ity of pheromone update (i.e., fraction of elitist ants). One of the important results of our study is the observation that the subsidiary local search procedure is crucial for the performance of the algorithm. In particular, to ensure that high-quality conformations are obtained, it is very important to allow the local search procedure to run sufficiently long. In an earlier version of our algorithm [121], we used substantially more stringent termination criteria. These criteria forced us to additionally use non-greedy local search (probabilistic iterative improvement, which accepts worsening steps) in addition to the greedy local search procedure used here. The results presented in this study indicate that by using a new and simpler local search procedure, A C O achieves better performance. This finding is probably due to the fact that the new local search procedure is based on a type of long-range move that leads to a larger effective search neighbourhood. Generally, our empirical results indicate that our new A C O algorithm is able to fold sequences that have structural folding nuclei somewhere within the sequence and not at the ends more efficiently than the best-performing methods for this prob-lem. Additionally, our algorithm finds a more diverse set of optimal states for HP sequences. Considering the simplicity of the underlying local search procedure and the evidence of the importance of all of the parameters of the algorithm, this property is a result of the search diversification provided by the population-based, probabilistically biased, adaptive construction mechanism that is characteristic of A C O algorithms. 77 Chapter 5 Adaptive Bin Framework Search, Introduced for the FCC /5-Sheet Protein Folding Problem Twilight was beginning. Blue veils hung in the trees. It was not any definite object that he recognized -no houses or villages or hills - it was the landscape itself that suddenly spoke. Erich Maria Remarque The performance of stochastic local search algorithms is critically dependent on the properties of the search landscape encountered, such as the number and distribution of local minima, the degree of landscape ruggedness (measured using fitness dis-tance correlation, for example), and detailed information on the plateau and basin structure of a given landscape. Therefore, reactive search strategies that can extract important features of the landscape and adapt the search strategy accordingly are invaluable for solving problems with complex energy landscapes. It is evident from an analysis of the literature describing search methods for protein folding, that adaptive search strategies have not been widely studied for this problem. In this chapter, we describe a general bin framework that can be used adaptively as an efficient search method for this complex problem. We also provide an extensive empirical comparison with existing state-of-the-art methods, which we carried out by re-implementing these methods and carrying out an empirical study. In this part of our work, we chose the F C C lattice over the cubic lattice for the following reasons: (1) the F C C lattice, as mentioned previously, is free from some of the artifacts encountered on the cubic lattice (such as the parity problem, inability to model secondary structure, and others); (2) due to the lack of results in the literature for the state-of-the-art search methods for protein folding (such as Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 78 R E M C ) for the cubic lattice, we are unable to conduct an empir ica l comparison with the best search methods for protein folding on a simpler cubic lattice. A d d i -tionally, the F C C lattice was chosen over more realistic discrete off-lattice models due to the unavailabili ty o f universally used energy functions and the absence o f an appropriate data set for which empir ical results o f the best-performing algorithms (such as R E M C ) in the literature exists. We developed a new stochastic local search that uses a novel b in framework for storing a diverse set of conformations (candidate solutions) in memory. P romis ing conformations, that satisfy the energy and diversity criteria, encountered during the search are stored for future retrieval when a search stagnation is detected (both stor-age and retrieval mechanisms presented here represent an adaptive component o f the search). Solutions are retrieved from a pool of stored conformations according to their energy. 5.1 The Bin Framework and the Bin Framework Monte Carlo Algorithm To be able to search complex energy landscapes efficiently, we devised a flexible framework that stores a population of candidate solutions that are of good quali ty (have low energy) and represent a diverse set o f conformations encountered during the search. This is performed by adapting energy and diversity thresholds during the search for bins that store conformations at different energy ranges. Energy thresholds for each bin are calculated based on the conformations already stored in the b in , and retrieving conformations adaptively once search stagnation is detected. Promis ing conformations whose energy is lower than a bin 's threshold and that satisfy the diversity criterion are placed into the appropriate b in according to their energies (each bin considers a certain window of energies, see Figure 5.1). It should be noted that sorting conformations into bins by a window of different energies is suggested in order to pick a diverse set o f promising conformations that can help to overcome large energy barriers dur ing the search. Thus, a pool of promising conformations encountered during the search and stored for future re-use is subdivided into bins of a certain energy window size AEi. The energy window size represented by bins is a parameter that for s impl ic i ty is kept constant for all of the bins (AEi — AE2 = ... = Ei = ... — AEn). Fo r example, b in 1 can have conformations of energy values ranging from 0 to —10 [eo], bin 2 w i l l ho ld conformations of energies —10 to —20 [eo], and so on. Under the F C C model associated with each conformation c encountered, there are the fo l lowing properties: Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 79 1. its total energy, E(c), subdivided into short-range ESR{C) and long-range energy ELR(C) according to the F C C /3-sheet potential used [52] 2. an array o f residues in extended /3-state, c.BetaEnergy[i], for i = 2 . . . N — 2 (where N is the protein length) 3. its vector representation on the F C C lattice, V = v\vi... v^-i The fo l lowing properties and parameters are associated wi th each b in i: 1. The capacity o f the b in , bins[i].capacity, specifying the number o f con-formations the bin can hold. This is a parameter o f the algori thm that for s impl ic i ty is set to the same fixed value for all bins. 2. The current number of conformations stored, bins[i].currentNumber. The content o f a bin, i.e., conformations, is stored as a sorted list (from the lowest to the highest) according to the objective function (energy) for easy future retrieval, since lower-energy conformations are more desirable. 3. The bin's energy threshold, Ef. This is the highest energy that a conforma-tion can reach and still be placed into the bin. It is adapted during the search based on the energies o f conformations placed in the b in . To be selective when the bin is fu l l , and to be less selective when it st i l l has room, the high-est energy among conformations already in the bin is picked as a threshold i f the capacity has been reached; the threshold is equal to 0 until then. 4. The H a m m i n g distance diversity threshold, HDi, wh ich specifies how dif-ferent a conformation has to be from other conformations o f the same energy already stored in order for the new conformation to get placed into a bin . The H a m m i n g distance between a newly considered conformation c and all conformations c's with the same energy already in the bin is calculated as fol lows: N-2 HD = ^2 \c'.BetaEner gy[i}/number Of (c') — c.BetaEnergy[i]\/eBN i=2 (5.1) where EB — 4.0 [EQ] is the energy contribution of each /3-residue, as in [52]. The sum goes from i = 2 to i = N — 2, since the first residue and the two last residues can never be in the extended /3-state. W e consider different H a m m i n g distance criteria HDi for different bins i. A s has been observed experimentally and is consistent with the tunneled picture o f the energy land-scape, there are more conformations at higher energies, therefore, the crite-rion should be more stringent, and fewer conformations at lower energies, Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 8 0 where the Hamming distance criterion can be relaxed. This is performed by specifying two Hamming distance parameters to the algorithm HDMAX (the highest Hamming distance used for bins with high energy) and HDMIN (the lowest Hamming distance used for low-energy bins). A particular Hamming distance criterion used for bin i is determined by assigning linearly spaced Hamming distance criteria for bins between values HDMAX a n d HDMIN in the following way: HDi = HDMIN + {'1-1){HDMAX-HDMIN)/numBinsTotal, (5.2) where numBinsTotal is the total number of bins in the bin framework, which is not a parameter of the algorithm but instead is defined by the ratio of two other parameters, as described below; i is the number of a bin (i e [1 . . . numBinsTotal]). 5. The width of the energy window represented by each bin, AEi. For simplic-ity, it is kept constant for all bins. The bin framework also has the following properties and parameters associated with it: 1. The total number of bins, numBinsTotal. 2. The energy range of interest, AE, which is a parameter of the algorithm. If conformations encountered during the search have energies within this range, we store them in the bin framework, provided they satisfy the diver-sity criterion. Even though the width of the energy range of interest is kept constant, AE shifts down the funnel as the current estimate of the ground state EQ is updated. 3. The current estimate of the ground state energy Eo, which is the lowest en-ergy found so far. 4. The inverse temperature used for retrieval of conformations from bins Bun — l/kgTbin. This is a parameter of the algorithm. Figure 5.1 depicts some of the properties of bins and conformations in a bin and the overall relationship of conformations stored in the framework to the energy landscape. We keep the number of bins constant during the run of the algorithm. This number is determined by the interval of energies of interest and the energy window width used: numBinsTotal = AE/AEi. (5.3) Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 3-Sheet Protein Folding Problem 81 Hi ) , MI), lin, . . . nn n Figure 5.1: An illustration of how candidate solutions at a given state of the bin framework relate to the search space of a given problem instance. Eo is the best solution quality found so far and serves as an estimate of the ground state energy, AE is the energy range of interest, and conformations within this range are binned. Each bin i has energy threshold Ef, diversity threshold HD{, and energy window AEi. Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 82 The base energy of interest, defined as EQ + AE, is the highest energy a con-formation c can have and still be considered for placement into the bin system. The range AE is the estimate of the maximal barrier height that needs to be surmounted to get to lower energy states, and is a parameter of the algorithm. The bin framework is conceptually a very rich framework that allows for a great amount of flexibility. Design decisions adopted in our implementation of the algorithm proposed here were guided by the principles of simplicity and efficiency. Many other variations are possible and will be studied as part of future work. In this work, the novel bin framework introduced is used to adaptively restart a canonical Monte Carlo search when it stagnates. In general, our bin framework can be used in combination with any stochastic local search algorithm. We chose a combination with the Monte Carlo method since this is the most widely stud-ied search method applied to the protein folding problem. We therefore call this approach the Bin Framework Monte Carlo (BINMC) method. The outline of the algorithm is given in Figure 5.2. We used the same move set as described in [52, 154] which involves n/2 at-tempts at double-bond moves (the location of the move is chosen uniformly at random and two bond vectors are modified) and two attempts at chain-end moves (where we are attempting to modify the location of the first or the last residue). 5.1.1 Storing Conformations in the Bin Framework We run a single Monte Carlo chain search at a low enough constant temperature TMC, which is a parameter of the algorithm; note that TMC — ^-/^B^MC- During a Monte Carlo run, a conformation c is placed into a bin, or attempted to be placed into a bin, if E(c) < EQ + AE and either of the following conditions is satisfied: 1. The energy of a conformation c is the lowest energy observed so far (E(c) < Eo)- The conformation is stored in the bin system and the estimate of the ground state Eo is updated (function PlacelntoBin is called - see Figure 5.3). 2. If a conformation c is accepted according to the regular Metropolis criterion, we check to see if the energy of c, E(c) = E, is lower than the appropriate bin's threshold energy Ef and if the Hamming distance diversity criterion is satisfied - the Hamming distance between the conformation c and other con-formations c' with the same energy E should be larger or equal to HDi. We then add it to the bin (function CheckPlacelntoBin is called - see Figures 5.4 and 5.5). After a new conformation is added to the bin, we drop the conformation with the highest energy from the bin if the capacity of the bin has been reached. Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 83 procedure BINMQConformation c, nolmprRetrieve, BMC, Bbin, HDMAX, HDMIN, AEit AE) input: candidate conformation c, the number of non-improving steps nolmpr Retrieve, the inverse temperature for the Monte Carlo run BMC, the inverse temperature for the bin framework Bun, Hamming distance diversity criteria HDMAX and HDMIN, the window of energies considered by each bin i: AEi, the range of energies of interest AE output: the lowest energy conformation c nolmpr := 0; Eo := 0; while (termination condition not satisfied) do run single chain Monte Carlo at inverse temperature BMC for conformation c //during the run store accepted conformations that satisfy //energy criterion (defined by A E and bins' energy thresholds) //and the Hamming distance criteria (defined by HDMAX and HDMIN)', if (accepted new conformation c) if (E{c) < Eo) PlaceIntoBin(c, Eo, HDMAX, HDMIN, AEi, AE); else CheckPlaceIntoBin(c, AEi, AE); end end if no improvement has bin made over the best energy Eo in the current run nolmpr := nolmpr+l; else nolmpr := 0; end if (nolmpr > nolmpr Retrieve) //retrieve conformation from bins at inverse temperature Bun c := RetrieveFromBin(/3bi„,i5o); nolmpr ;= 0; end end return c; end Figure 5.2: High-level outline of the main body of the Bin Frame-work Monte Carlo algorithm. EQ is the current best solution quality; nolmprRetrieve is a parameter of the algorithm that specifies number of non-improving steps over the best energy that are tolerated before a new conformation is retrieved from the bin system; HDMAX and . HDMIN are parameters defining the Hamming distance diversity cri-teria for high-energy and low-energy conformations correspondingly; 0MC and 0fjin are parameters that refer to the inverse temperatures used for the Monte Carlo run and for the bin framework correspond-ingly; AEi is a parameter that represents the window of energies by each bin i; AE is a parameter defining the range of energies of inter-est. Conformations whose energy falls within this range are attempted to be stored in the bin framework. The functions PlacelntoBin and CheckPlacelntoBin are defined in Figures 5.3 and 5.4 correspondingly. Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 84 procedure PlaceIntoBin(Conformation c, Eo, HDMAX, HDMIN, AEi, AE) input: candidate conformation c, the lowest energy found so far Eo, Hamming distance diversity criteria HDMAX and HDMIN, the window of energies considered by each bin i: AEi, the range of energies of interest AE output: none Eo := E(c); //since all AEiS are equal and are supplied to the algorithm //as parameters, a particular AEi is known and index i of the bin that stores //energy range to which E belongs to is calculated as: i := E/AEi; if (bin[i] is empty) //linearly redistribute FID criteria between //HDMAX and HDMIN to the bins (HDj) HDj := HDMIN+U — 1)(H DM AX-H DM iN)/numBinsTotal; end insert conformation c in bins[i] as the top conformation; shift other conformations already stored (if any) by one entry; if (bins[i].currentNumber > bins[i].capacity) drop highest energy conformation from bin[i] and recalculate Ef; else bins[i].current.Num,ber:=bins[i}.current Number + 1; end end Figure 5.3: Outline for the function used to place the conformation with the lowest energy encountered so far into the bin system, where c is the conformation with energy E(c) to be placed into a bin, Eo is the estimated ground state energy (the lowest energy seen in the sim-ulation so far), HDMAX and HDMIN are parameters defining the Hamming distance diversity criteria for high-energy and low-energy conformations correspondingly, AEi is a parameter that represents the window of energies considered by each bin i, AE is a parame-ter defining the range of energies of interest - conformations whose energy falls within this range are attempted to be stored in the bin framework. Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 85 procedure CheckPlacelntoBinfConformcition c, AEi, AE) input: candidate conformation c, the window of energies considered by each bin i: AEi, the range of energies of interest AE output: boolean variable indicating if c was stored i f ( £ ( c ) > £ 0 + AE) //energy of c is outside the energy range of interest AE return false; end i :=E(c)/AEt; //if bin still has place or if E(c) is lower than bin's threshold; if (bins[i].current,Number < bins[i).capacity or E(c) < Ef) find insertion place for conformation c in bins[i\; //make sure we satisfy diversity criterion; if (there are other conformations c' with the same energy in the bin) for (all residues) do calculate B — H amming Distance between c and average 3 configuration of all c's; end //normalize Hamming Distance to get value between 0 and 1; normHD;=H amming Distance/N; else normHD:=\; end • if (normHD > IIDi) insert conformation c; shift other conformations already stored by one entry if there are any with higher energies; if (bins[i].currentNumber > bins[i].capacity) drop the highest energy conformation; recalculate Ef; else bins[i\. current Number:= bins[i].current Number+1; end return true; else return false; end else return false; end end Figure 5.4: Outl ine for the function used to place the conformation c with a low energy E(c) encountered into the bin system, where Eo is the estimated ground state energy (the lowest energy seen in the s im-ulation so far), HDMAX and HDMIN are parameters denning the H a m m i n g distance diversity criteria for high-energy and low-energy conformations correspondingly, AEi is a parameter that represents the window of energies considered by each bin i, AE is a parame-ter defining the range of energies o f interest - conformations whose energy falls wi th in this range are attempted to be stored in the bin framework. Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 86 HD, HI), 1111 . . . H I ) n Figure 5.5: Placement of the low-energy conformation c with energy E(c) into an appropriate bin i with the energy threshold Ef and the Hamming distance crite-rion HDt. In order for the conformation c to be stored in the bin framework the following conditions need to be satisfied: (1) E(c) < Ef, and (2) the Hamming distance (HD) between conformation c and conformations c' with the same energy already stored in the bin (if there are any) is larger or equal to HDi. Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 87 In general, we need to make sure that conformations in the pool, or the union of all conformations stored in bins (1) represent all energy levels of interest, which is ensured by binning conformations at different levels; (2) represent as many possi-ble 0 and non-0 configurations as possible, which is taken care of by the Hamming distance criterion. 5.1.2 Retrieving Conformations from the Bin Framework Let us now think about what we search for and what we most fear during the search. We want to reach the best local optimum, or the global optimum, as quickly as pos-sible, and the worst hindrance is to get stuck in a local optimum that is still far from the global optimum. Therefore, our choice of search strategy (for example, the pro-posal mechanism for new candidate solutions) can be guided by the search progress made. During the search process, one can adaptively adjust important parameters of the algorithm to further the search (it may be noted that this is closely related to the concept of reactive search [9]). By varying the search strategy according to a priori defined transition probabilities, the approach results in an algorithm that sacrifices an exact relationship with the canonical ensemble. However, if we are interested in finding global minima, and not in obtaining the canonical ensemble to calculate physical properties of interest, this method can result in reducing the non-convergence, or quasi-ergodicity, in rugged energy landscapes. Therefore, we chose an adaptive strategy for retrieval: retrieval of a conformation from the bin system is performed when the search stagnates. The simplest way to recognize when the search stagnates is to record the time (the number of steps) during which we have not observed any improvement on the lowest energy, (see, e.g., [59]). This criterion relates directly to the objective of the search. Thus, if no improvement on the lowest energy has been seen for a certain number of steps (nolmprRetrieve, which is a parameter of the algorithm) we retrieve a conformation from the bin system. To retrieve a conformation from the bin framework we first need to choose a bin, that is, the range of energies from which we will pick a conformation, and then choose a particular conformation from the selected bin. Since lower energy conformations are preferred, given that the energy landscape of real proteins is be-lieved to be tunneled [105], we choose a bin and conformation according to the corresponding energy threshold and the energy of the conformation, respectively. Conformations can be chosen with or without replacement; here we limited our-selves to choosing conformation with replacement, since the same conformation can yield a different fold each time it is picked. The outline for the described sim-ple retrieval strategy is given in Figure 5.6. Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 3-Sheet Protein Folding Problem 88 procedure RetrieveFromBin(j3bin,Eo) input: the inverse temperature for the bin framework / 3 0 ; n , the lowest energy found so far E0 output: candidate conformation c if (bins.totalStored = 0) //no. conformations are stored return null; end choose bin Num. with probability e~ f3bi"^ Ei among all bins i\ choose conf Num. inside the bin with probability e-h;,,Xbins[binNum}.conm.E^ among all conformations j stored in the bin binNum; c := bins[binNwm}.conf[confNum\; return c; end Figure 5.6: Outl ine for the function used to retrieve a conformation from the bin system. EQ is the current best solution quali ty; Bun's the inverse temperature used for the bin framework during the retrieval of conformations. A s in the stochastic tunneling approach [145], to lessen exponential decay of the probabil i ty function we used Boltzmann-based modified weights proportional to e~@(E~E°\ This weighting preserves the location of all min ima , but maps the entire energy space from Eo to the m a x i m u m energy 0 onto the interval [0,1]. The dynamic process fo l lowing the Bol tzmann distribution can therefore pass through energy barriers o f an arbitrary height. 5.2 Empirical Results and Discussion To evaluate the performance of generalized ensemble methods re-implemented from the literature and the bin framework compared with the available data in the literature, we measured the average solution quality, median solution quality, and 25- and 75-percentiles. We also recorded the best solution quali ty obtained over 10 runs with the specified cut-off time. To further evaluate the performance of our B I N M C algori thm and a number of generalized ensemble methods re-implemented from the literature, we analyzed the run-time distributions ( R T D s ) to obtain the best-known solution quality (or in some cases specified sub-optimal solution qual-ities) for instances of different sizes in 100 independent runs. The fo l l owing lengths for the F C C /3-homopolymers were used: 12, 24, 36, and 64. A l l experiments were performed on P C s with 2.4 G H z Pent ium I V C P U s , 2 5 6 K b cache, and 1 M b R A M , running Redhat L i n u x (our reference machine). Thei r run-time was measured in Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 89 terms of the absolute C P U time required to obtain the specified solution quality or better. 5.2.1 Comparison of Results with the Literature First, we compared our implementation of the simple Monte Carlo (MC) and the Replica Exchange Monte Carlo (REMC) methods with the earlier implementation of Gront etal. [52] described in the literature. In [52], the authors tested algorithms implemented on the homopolymer of length N — 32 and iV = 64, but only pro-vided results for the homopolymer of length 64.1 For the protein of length 64 the authors believed that the lowest energy they reached (-374) is the ground state, but as was shown later [154], lower energies exist for this system (an energy of —387 has been reported in [154]). In Table 5.1 we provide results averaged over 10 independent runs as done in [52] (for M C and R E M C ) , and in [154] (for Parallel-hat Tempering (PHAT)). It should be noted that even though it is not evident from the paper [52], from per-sonal communications with D. Gront, " M C with linear set of temperatures" means that they ran M C and annealed the temperature from 2.75 to 1.25 [eo/ks] (where £o = 1 is the unit of interaction energy [154]). The exact annealing schedule was not provided, therefore, for our M C we chose a constant temperature of 1.25. Thus, results for the M C S A of Gront et al. may not be exactly comparable with our im-plementation of pure M C . We used average C P U times reported in [52] and [154] as the cut-off time for our algorithms, since the number of iterations varies sig-nificantly based on the implementation used. As seen from Table 5.1, our imple-mentations on M C S A and R E M C are comparable to the implementation reported in [52]; as discussed in the table caption, differences in execution environments were accounted for. Our implementation of PHAT performed worse than was described in [154]. We, therefore, contacted Y. Zhang and tried to verify every aspect of the algo-rithm. Unfortunately, however, he could not reproduce the precise details and re-sults, including the coordinates of the best state found, —387, due to data loss. Our novel B I N M C algorithm performs better than M C and R E M C , and than our im-plementation of PHAT. We used the following set of parameters for the bin frame-work for the homopolymer of length 64: AE = 30, AEi = 5, TMC = 1-25, Tbin = 6.521 (since units of T are [eo/ks], the inverse temperature is calcu-'We contacted D. Gront [52] and Y. Zhang [154] but they no longer had the information or the ability to reproduce the data. Both authors commented that all methods were able to reach what they believe is the global minimum quite easily for the polymer of length 32. However, they could not tell what the exact value of the energy was, nor were they able to provide any information on the conformation with the lowest energy found. Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 90 lated as: 0 = 1 / T ) , binCopacity = 100, HDMAX = 0.8, HDMIN = 0.01, nolmpr Retrieve — 2 000 000 steps. These settings were determined in a series of experiments in which we studied the influence of different parameter settings; these w i l l be further discussed in Section 5.2.3. The best solution quality for the homopolymer o f length 64 found by our bin framework is —391 (shown in Figure 5.7), which is lower than the energy o f any conformation previously reported in the literature, as we have seen, the previous best energy was —387 [154]. W e found conformations with energies —391 twice, after 47 hrs and 55 hrs. The two conformations found were mirror reflections of each other, and one of them is shown in Figure 5.7. W e also show examples of other low-energy conformations found by our bin framework for the 64 amino acid homopolymer in Figures 5.8, and 5.9. Conformations wi th energies of —389, — 388, —387, some of which are displayed in these figures, were found mult iple times by B I N M C within the C P U time cut-off o f 10 hrs on our reference machines. 5.2.2 Further Comparison for Homopolymers of Length 12, 24, 32, and 64 To perform further comparison of the re-implemented methods from the litera-ture ( M C , R E M C , P H A T ) with B I N M C , we tested the methods on instances o f length N = 1 2 , 2 4 , 3 2 by performing 10 independent runs on each protein se-quence. We reported the mean solution quality reached and the standard deviation observed wi th in the 1 hr time cut-off. We used the fo l lowing set o f parameters for the bin framework for the homopolymers o f length 12 and 24: AE = 20, AEi = 5, TMc = 1-25, TBIN = 4.344, binCapacity = 100, HDMAX = 0.6, HDMIN — 0.01, nolmpr Retrieve — 100 000 steps. The b in framework pa-rameters for the homopolymer of length 32 were set to: AE = 20, AEi — 5, TMc = 1-25, TBIN = 4.344, binCapacity = 100, HDMAX = 0.6, HDMIN = 0.01, nolmpr Retrieve — 1 000 000 steps. These particular parameter settings are discussed in Section 5.2.3. ' . A s can be seen from our results presented in Table 5.2, al l methods find what is probably the lowest energy (—39) for the homopolymer o f length 12 under 1 sec C P U time on our reference machine. For the homopolymer o f length 24, we are starting to see differences among the methods: B I N M C slightly outperforms all other methods in terms of C P U time needed to reach what is probably also the low-est energy for the homopolymer of length 24 ( - 1 0 9 ) . M C is the next best method in terms of performance, then P H A T , fo l lowed by R E M C . The performance results for R E M C and P H A T are worse than for M C because the polymer is too short, and s imple M C methods that do not invest in exchanges between replicas perform better. For the homopolymer o f length 32, B I N M C outperforms other methods by Method Temperature set Timecltt^0ff Energyavg ± sd Energy p — value MCSA [52] annealed from 2.75 to 1.25 24 min (approx) -349.3 (±2.1) - 3 6 2 REMC [52] linear 1.25 to 2.75 28 min (approx) -368.2 (±0.8) - 3 7 3 PHAT[154] linear 1.25 to 2.75 1 hr 25 min (approx) -380.4 (±1.9) - 3 8 7 our M C 1.25 24 min -367.2 (±1.7) - 3 7 0 our M C 1.25 28 min -367.4 (±2.7) -371 0.1367 our REMC linear 1.25 to 2.75 28 min -368.5 (±2.1) - 3 7 3 0.3425 our PHAT linear 1.3 to 2.75 28 min -367.5 (±3.3) - 3 7 2 0.1599 our BINMC TMC = 1-25, Tbin=6.521 28 min -370.3 (±4.3) - 3 7 9 our M C 1.25 1 hr 25 min -368.2 (±4.6) - 3 7 4 0.0006* our REMC linear 1.25 to 2.75 1 hr 25 min -369.4 (±3.0) - 3 7 6 0.0008* our PHAT linear 1.3 to 2.75 1 hr 25 min -369.5 (±3.2) - 3 7 6 0.0023* our BINMC T M C = 1-25, r f c i n =6.521 1 hr 25 min -375.7 (±3.8) - 3 8 3 Table 5.1: Compar i son of the solution quali ty obtained for the homopolymer of length TV = 64 by the Monte Ca r lo Simulated A n n e a l i n g ( M C S A ) [52], the Rep l i ca Exchange Mon te Car lo ( R E M C ) with a linear set of temperatures [52] and the Paral lel-hat Tempering algorithm ( P H A T ) [154] wi th our implementation of Monte C a r l o ( M C ) , R E M C and P H A T and our new B i n Framework Mon te Car lo ( B I N M C ) . The time reported for M C S A and R E M C from [52] is the estimated time required to run on 2.4 GHz machines (in the or iginal paper authors used 500 MHz processor, therefore, reported times in [52] are conservatively divided by a factor o f 4.8). The time reported for Parallel-hat tempering [154] is the estimated time to run on 2.4 GHz as we l l ( in the or iginal paper [154] C P U of 750 MHz was used, therefore we conservatively applied a factor o f 3.2). The authors in [52] also implemented R E M C with an exponential set o f temperatures. The exact temperatures were not specified in the paper [52], however, and the authors could not recall it in personal communicat ion (the lowest energy observed in this case was —374). The number of runs used for comparison was 10 for all algorithms. In the last co lumn o f the table, we report p-values indicat ing the probability that the null hypothesis of no difference between the mean energies reached over 10 runs for B I N M C and a particular algorithm listed (within the same C P U cut-off time) is true; we used the M a n n - W h i t n e y U test to calculate p-values [59]; * indicates that p-values are below the significance level of 0.05 of wrongly rejecting the nul l hypothesis. 9 "a Cu" 3 Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 92 (c) (d) Figure 5.7: (a) The lowest energy conformation of the F C C 64 amino acids ho-mopolymer found by our Bin Framework Monte Carlo method (energy —391, short-range energy is —212, long-range energy is —179). The detailed description of this conformation is also found in Appendix B; (b) same conformation, view from above; (c) a low-energy conformation of the F C C 64 amino acids homopoly-mer found by B I N M C (energy —387, short-range energy is —212, long-range en-ergy is —175); this was found in the same run that lead to the conformation with the best energy of —391; (d) another low-energy conformation of the F C C 64 amino acids homopolymer found in the same run of B I N M C (energy —388, short-range energy is —212, long-range energy is —176). Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 93 (c) ( d ) Figure 5.8: (a) A low-energy conformation of the F C C 64 amino acids homopoly-mer found by B I N M C (energy -387, short-range energy is —208, long-range en-ergy is —179); (b) another low-energy conformation of the F C C 64 amino acids homopolymer found in the same run of B I N M C (energy —389, short-range energy is —212, long-range energy is —177); (c) a low-energy conformation of the F C C 64 amino acids homopolymer found by B I N M C (energy —387, short-range en-ergy is —208, long-range energy is —179); (d) another low-energy conformation of the F C C 64 amino acids homopolymer found in the same run of B I N M C (energy —389, short-range energy is —220, long-range energy is —169). Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 94 (c) (d) Figure 5.9: (a) The low-energy conformation of the F C C 64 amino acids ho-mopolymer found by our bin framework (energy —387, short-range energy is —220, long-range energy is —167); (b) another low-energy conformation of the F C C 64 amino acids homopolymer found by our b in framework (energy —387, short-range energy is —220, long-range energy is - 1 6 7 ) ; (c) the same conforma-tion as in part (a), v iew from above; (d) the same conformation as in part (b), view from above. Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC [3-Sheet Protein Folding Problem 95 obtaining lower average energy (finding lower energy states such as —161 more often), next is PHAT, then R E M C , followed by M C . We believe that the authors in [52] found only the sub-optimal solution quality of —158 for the 32 amino acid polymer, since energies lower than —158 are more difficult to obtain and methods start to vary in their ability to reach such energies. Unfortunately, they did not re-port the precise energy and could not reproduce it in personal communications. We show the best solution qualities found for polymers of length 12,24, and 32 in Fig-ure 5.10. These solutions seem tobe unique in terms of short-range and long-range energy values, since all of the conformations found by different methods multiple times have the same short- vs. long-range energy interplay. Figure 5.10: The lowest energy conformation of the F C C homopolymers of 12, 24, and 32 (same conformation from different point of view) amino acids (from left to right) found by all of the algorithms (corresponding energies are: —39, short-range energy is —28, long-range energy is —11 for the 12 amino acid polymer; — 109, short-range energy is —68, long-range energy is —41 for the 24 amino acid polymer; and —161, short-range energy is —112, long-range energy is —49 for the 32 amino acid polymer). The detailed description of these conformations is also found in Appendix B. Additionally, we carried out 10 independent long runs for the homopolymer of length 64 and 32 to further investigate behavior of different algorithms over an extended period of time. We used a longer cut-off time of 10 hrs on our reference machine, since the solutions reported in Table 5.2 are sub-optimal. As seen from the results presented in Table 5.3, B I N M C outperforms our implementation of the state-of-the-art R E M C and PHAT methods and the canonical M C method in terms of the solution quality reached. R E M C was second best, then PHAT, then M C . Method Length Energyavg ± sd Energymin CPU Timeavg Timemed Time (775 Time c/25 p — value M C 12 - 3 9 (±0) - 3 9 < 1 sec < 1 sec < 1 sec < 1 sec REMC 12 - 3 9 (±0) - 3 9 < 1 sec < 1 sec < 1 sec < 1 sec PHAT 12 - 3 9 (±0) - 3 9 < 1 sec < 1 sec < 1 sec < 1 sec BINMC 12 - 3 9 (±0) - 3 9 < 1 sec < 1 sec < 1 sec < 1 sec M C 24 - 1 0 9 (±0) - 1 0 9 5.0 m i n (±4.1 m i n ) 5.5 m i n 7.2 m i n 1.5 m i n 0.1230 REMC 24 - 1 0 9 (±0) - 1 0 9 18.3 m i n (±18.0 m i n ) 16.3 m i n 19.4 m i n 4.3 m i n 0.0015* PHAT 24 - 1 0 9 (±0) - 1 0 9 8.7 m i n (±8.2 m i n ) 6.6 m i n 11.7 m i n 2.9 m i n 0.0039* BINMC 24 - 1 0 9 (±0) - 1 0 9 1.7 m i n (±1.2 m i n ) 1.8 m i n 2.7 m i n 0.5 m i n Method Length Energyavg ± sd Lowest Energy CPU TimeaVg Energymed Energy 1J75 Energy g25 p — value MC 32 -158.1 (±0.9) - 1 6 1 4.3 m i n (±8.4 m i n ) - 1 5 8 - 1 5 8 - 1 5 8 0.0155* REMC 32 -158 .2 (±0.7) - 1 6 1 4.5 m i n (±8.6 m i n ) - 1 5 8 - 1 5 8 - 1 5 8 0.0185* PHAT 32 -158 .3 (±0.9) - 1 6 1 5.8 m i n (±8.0 m i n ) - 1 5 8 - 1 5 8 - 1 5 8 0.0214* BINMC 32 -158.9 (±0.6) - 1 6 1 23.0 m i n (±20.1 m i n ) - 1 5 9 - 1 5 9 - 1 5 8 9 Table 5.2: Comparison of the average solution quality obtained and the average time required for the homopolymers of lengths N = 12,24,32 for the re-implemented M C , R E M C , PHAT, and B I N M C . The time cut-off used was 1 hr on 2.4 GHz reference machine, and the averages were calculated from 10 independent runs. Temperature sets used for the re-implemented algorithms are the same as in Table 5.1. In the last column of the table we report p-values indicating the probability that the null hypothesis of no difference between the mean C P U run-time (for the homopolymer of length 24) or the mean energies reached over 10 runs (for the homopolymer of length 32) for B I N M C and a particular algorithm listed is true; we used the Mann-Whitney U test to calculate p-values [59]; * indicates that p-values are below the significance level of 0.05 of wrongly rejecting the null hypothesis. O N Method Temperature set Length Energyavg ± sd Energymsd Energy q75 Energy q25 Energymin p — value our MC 1.25 32 -158.7 (±1.9) -159 -159 -158 -161 0.0271* our REMC linear 1.25 to 2.75 32 -159.6 (±1.3) -160 -161 -158 -161 0.5471 our PHAT linear 1.3 to 2.75 32 -158.9 (±1.4) -159 -159 -158 -161 0.0638 our BINMC TMC = 1-25, T b i „=6.521 32 -160.1 (±0.9) -161 -161 -159 -161 our MC 1.25 64 -372.2 (±2.3) -372 -373 -371 -377 0.0005* our REMC linear 1.25 to 2.75 64 -376.1 (±3.5) -376 -378 -37.3 -382 0.0521 our PHAT linear 1.3 to 2.75 64 -374.1 (±3.8) -374 -377 -371 -383 0.0120* our BINMC T M C = 1-25, T t i „=6 .521 64 -379.5 (±3.3) -381 -382 -376 -389 Table 5.3: Compar i son of the solution quali ty obtained for the homopolymers o f length A r = 64 and N = 32 by re-implemented M C , R E M C with the linear set o f temperatures, P H A T , and our new B I N M C on 2.4 GHz reference machine with a time cut-off o f 10 hrs over 10 independent runs. In the last column of the table, we report p-values indicating the probabil i ty that the null hypothesis o f no difference between the mean energies reached over 10 runs for B I N M C and a particular algori thm listed (within the same C P U cut-off time) is true; we used the Mann-Whi tney U test to calculate p-values [59]; * indicates that p-values are below the significance level o f 0.05 of wrongly rejecting the null hypothesis. Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC (3-Sheet Protein Folding Problem 9 8 Next we conducted a more thorough comparison of the methods by comparing run-time distributions (RTDs) for all of the methods on problem instances of length 32 and 64. For the homopolymer of length 32, we used the sub-optimal solution quality of —158, since, as shown in Table 5.3, the time required to obtain the best solution quality of —161 is computationally prohibitive to conduct 100 runs for some of the methods. For the homopolymer of length 64, we also used the sub-optimal solution quality of —370 in order to make comparison computationally feasible for all of the methods. We conducted 100 independent runs on 2.4 GHz reference machine. Our results are presented in Figure 5.11. For the homopoly-mer of length 32, our B I N M C and M C outperform all of the algorithms in terms of the time required to find the sub-optimal solution quality of —158. The next best algorithm was R E M C , followed by PHAT. For the homopolymer of length 64 our B I N M C outperforms all of the algorithms in terms of the time required to find the sub-optimal solution quality of —370. The next best algorithm was M C , fol-lowed by R E M C and PHAT. It should be noted that for both,the 32 and 64 amino acid homopolymer, the R T D for PHAT has a longer right tail, indicating that the probability of longer runs is higher. A somewhat unexpected result is that M C per-forms better than R E M C and PHAT. However, we have to remember that the RTDs reported in Figure 5.11 are for sub-optimal qualities only. Since M C at a low tem-perature is "greedier" and does not run multiple chains at different temperatures nor attempts exchanges between them, it gets to sub-optimal energies faster. After reaching them, however, it gets stuck. The solution quality does not improve when running M C for a long time (10 hrs or more on our 2.4 GHz reference machine) for homopolymers of length 32 and 64, as shown in Table 5.3. We also looked at the scaling behaviour of R E M C and B I N M C with homopoly-mer length. We measured the median run-time to reach the global minimum over 20 runs for sequences of length 12, 24, and 32 amino acids (the sequence of length 64 was not used since only B I N M C reaches the lowest energy known). When three data points available are fitted with the line on a semi-logarithmic plot, the scal-ing behaviour appears to be exponential for both R E M C and B I N M C ; the median run-time scales as 10° 3 ' 1 * A r " 5 - 2 for R E M C a n d - l O 0 2 8 ^ " 4 ' 7 for B I N M C . Finally, we inspected the distribution of energies sampled by each method for the long homopolymer of length 64, based on approximately 5 x 10 9 conforma-tions each. As seen in Figure 5.12 part (a) and (b), R E M C and P H A T have typical energy distributions for each replica, as reported in [154]. In the case of PHAT, probabilities of low and high energies are enhanced, as expected. Interesting dif-ferences are observed when we examine the distribution of energies visited by M C and B I N M C (see Figure 5.13). The Bin Framework Monte Carlo method samples energies according to the Boltzmann distribution, but this distribution is shifted towards lower energies. Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 3-Sheet Protein Folding Problem 99 (a) (b) Figure 5.11: Run-time distributions of C P U times on our 2.4 GHz reference ma-chine to obtain a sub-optimal solution quality of —158 for the homopolymer of length 32 (part (a)) and to obtain sub-optimal solution quality of -370 for the ho-mopolymer of length 64 (part (b)) using Monte Carlo (MC), Replica Exchange Monte Carlo (REMC) , Parallel-hat Tempering (PHAT), and the Bin Framework Monte Carlo (BINMC). We fit the R T D of B I N M C for the homopolymer of length 64 with exponential distribution, to show by example that RTDs for all algorithms are exponential. For all algorithms, 100 independent runs were performed. In all of them, the target energy was reached. 5.2.3 Discussion of the Bin Framework We now turn our attention to the study of the parameter settings for the Bin Frame-work Monte Carlo method. We are interested in knowing what the reasonable values are for all of the parameters used. In this study, we performed 20 runs for the polymer of length 32, and record the time required to reach the best solu-tion quality of —161. We chose this particular homopolymer to study differences in parameter settings since the solution quality of —161 is most likely the opti-mum, and the instance of length 32 is both challenging and still computationally feasible for performing multiple runs of our algorithm. To study the influence of different parameters, we vary one (or in some cases two parameters when they are closely related), and keep all of the other parameters of the algorithm constant. Unless indicated otherwise, parameters for the homopolymer of length 32 were ChaPter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 100 (a) (b) Figure 5.12: Distr ibutions o f energies visited by different replicas in a representa-tive run o f the R e p l i c a Exchange Monte Car lo ( R E M C ) (part (a)) and in a repre-sentative run of the Parallel-hat Tempering Monte Ca r lo ( P H A T ) (part (b)) for the homopolymer o f length 64. The time cut-off used was 28 min on our reference machine. 0.05 0.045 0.04 Qj 0.035 J? 0.03 I 0.025 •g 0.02 £ 0.015 0.01 0.005 0 \ : BINMC MC — ' I 1 I I A 1 \ u. vv -400 -350 -300 -250 -200 -150 -100 -50 0 E, [<„] Figure 5.13: Distr ibut ions of energies visited by the M o n t e C a r l o ( M C ) and our B i n Framework Mon te Car lo ( B I N M C ) for the homopolymer o f length 64. The time cut-off used was 28 min on our reference machine. Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 101 fixed at the following values: AE = 20, AEi = 5, TMc = 1-25, Tbin = 4.344, binCapacity = 100, HDMAX = 0.6, HDMIN — 0.1, nolmpr Retrieve = 100 000 steps. We plot the average C P U time required to reach the optimum over 20 independent runs on the 2.4 GHz reference machine. The Number of nolmpr Retrieve Steps An important parameter of our Bin Framework Monte Carlo method is the nolmpr Retrieve number of steps. It characterizes how often we reach into a bin and retrieve a conformation based on the indication that our search stagnated. This parameter determines a cut-off for a number steps performed without seeing an improvement over the best energy before reaching into bins to retrieve a new con-formation. To understand the influence of the nolmpr Retrieve step cut-off, we conducted 20 independent runs for the homopolymer of length 32 to reach —161 while varying nolmpr Retrieve from 1000 to 2 000 000 steps. As can be seen from the results shown in Figure 5.14, the nolmpr Retrieve parameter signifi-cantly affects performance of B I N M C . For the homopolymer of length 32, a value close to 1 000 000 steps gives the best performance. If we retrieve conformations too often, we prevent the search from getting to the promising local optima by restarting it with a new conformation too early. If we reach into the bin framework very infrequently, the simulation becomes equivalent to a pure Monte Carlo run at a constant TMC temperature. We also found evidence in informal experiments that the best value of nolmpr Retrieve steps increases with chain length; for the homopolymer of 64 amino acids nolmpr Retrieve = 2 000 000 steps results in good performance. This observation is consistent with the intuition that for longer chains, longer time is required to reach promising local optima. Additionally, we compared B I N M C (performing an adaptive retrieval of di-verse conformations) with a simple restart strategy. If there was no improvement on the best energy observed for nolmpr Retrieve = 1 000 000 steps, instead of retrieving a conformation from bins we restarted the search with a newly con-structed conformation. In this case, the simulation failed to reach the energy of -161 within one week of C P U time on our 2.4 GHz reference machine (only the energy of —159 was reached). Similarly, in the case of the homopolymer of length 64, only an energy of —365 was reached within one week of C P U time. The Energy Range AE and the Bin's Temperature Tun The performance of our bin framework is also dependent on the ratio and the in-dividual settings of the energy range of interest, AE, (conformations with energy within this range are binned) and the temperature that controls the probability of Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 8-Sheet Protein Folding Problem 102 1 e + 0 7 1 e + 0 6 f OJ £ ~> 1 0 0 0 0 0 CL O CD CD E 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 e + 0 6 n o l m p r R e t r i e v e 1 e + 0 7 Figure 5.14: M e a n C P U time required for finding m i n i m u m energy conformations (—161) o f the 32 amino acid homopolymer over 20 independent runs on our refer-ence machine as a function of the number o f non-improving steps {nolmprRetrieve) before retrieving a conformation from bins. Error bars indicate standard deviation observed. Dashed line indicates that energy of - 1 6 1 was not found after 2 weeks ' C P U time on our reference machine. 1 e + 0 6 | 1 0 0 0 0 0 Q-O CD E 1 0 0 0 0 1 0 0 0 I N = 0.01 — • — = 0 .05 — — 10 2 0 3 0 AE 4 0 5 0 Figure 5.15: M e a n C P U time required for finding m i n i m u m energy conformations ( - 1 6 1 ) o f the 32 amino acid homopolymer over 20 independent runs on our ref-erence machine as a function of combination of parameters: the energy range of interest (AE) and the bin's temperature (Tbin), p = e~AE^Tbin. Er ror bars indicate standard deviation observed. Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 6-Sheet Protein Folding Problem 103 retrieving conformations of various energies, Tf,m. Since it is not efficient to place conformations that will later have a very low probability of being retrieved, the relative settings of these two parameters are important. The probability of retrieval of the conformation with the highest energy (EQ + AE) from the bin system is equal to p — e~^bir,AE. Thus, for example, if we would like to have at least a 1% chance of retrieving the highest energy conformation from the bin system (p = e - P b i « A E > 0.01), the following condition must hold: ^ > 4.605. There-fore, if we have fixed the desired probability of retrieval (p) and one other param-eter (either AE or T o i„; we chose AE), we can calculate the value of another parameter (Tbin)- We fixed p at 0.01 and 0.05 and varied AE from 10 to 50 kBT (Tbin was calculated as described previously). Our results presented in Figure 5.15 indicate that B I N M C shows a rather robust performance for various combinations of AE and T^n- The performance worsens when the AE range is too small and we are only storing a few of the best conformations encountered. The search time increases slowly as AE becomes too large and we are storing too many conforma-tions. Each probability value (we tested p = 0.01 and p = 0.05) seems to have its optimal AE range, and as p increases the optimal AE seems to decrease. This ob-servation is consistent with the intuition that if the retrieval probability is increased, the simulation can perform well with a smaller energy range of interest, since the binned conformations have a higher chance of being retrieved. From experiments with the longer homopolymer of length 64 (not reported here) larger AE results in good performance (particularly AE = 30 and Tun — 6.522, which renders p equal to 0.01, works well). As for the relationship between TMC and T o i ( l , here we only investigated the case when M C is run at a low temperature and the bin framework uses a high temperature for conformation retrieval. This scenario seems most promising, given the intuition that the search has to be efficient in minimizing conformational energy and the bin framework can provide diversification and intensification at the same time. We found that running M C at high temperatures, for example, TMC = 2.0 for both 32 and 64 amino acid polymer, does not yield optimal results. Therefore, TMC was kept constant at 1.25, which is below the transition temperature of 1.8 [eo/ks} reported for the homopolymer of length 64 [52, 154]. The Hamming Distance Criterion (HDMAX and HDMIN) The efficiency of our bin framework also depends on the diversity of conforma-tions binned at each energy level. In this experiment, we investigated the role the Hamming distance criterion (controlled by HDMAX and HDMIN) plays in the overall optimization. First, we fixed HDMIN (the fraction of residues that have to be different for low-energy conformations), and varied HDMAX (the fraction of Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 104 residues that have to be different for high-energy conformations) from 0.2 to 0.9. Note that setting the H a m m i n g distance criteria to 0 does not make sense, since the same conformation w i l l be binned over and over. A s seen in Figure 5.16 part (a), HDMAX of about 0.6 seems to result in the best performance. A s expected, i f HDMAX is set too low, the diversity of the set stored decreases and this impairs performance. If HDMAX is set too high, very few conformations may exist to satisfy this stringent criterion. s 1 ooooo 10000 1000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1e+06 100000 3 a. o 10000 1000 0.05 0.1 0.15 0.25 H D M (a) (b) Figure 5.16: M e a n C P U time required for finding m i n i m u m energy conformations (—161) o f the 32 amino acid homopolymer over 20 independent runs on our refer-ence machine as a function of the H a m m i n g distance criterion (HDMAX is varied, HDMIN — 0.1 is kept constant, part (a) and HDMIN is varied, HDMAX = 0.6 is kept constant, part (b)). Error bars indicate standard deviation observed. Next , we fixed HDMAX, and varied HDMIN from 0.01 to 0.9. A s seen from Figure 5.16 part (b), i f HDMIN > 0.1 is set too high, the binning process becomes less efficient, since we are not binning all promising low-energy conformations. Th i s happens because there are fewer conformations at low-energy levels and they are more s imilar to each other, compared with higher-energy level conformations. In our informal experiments, both HDMAX and HDMIN criteria do not dif-fer significantly for a longer polymer of length 64 (HDMAX — 0.8 — 0.6 and HDMIN = 0.01 seem to work wel l ) . Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 105 The Energy Window Width AEi In this experiment, we studied the performance of the B i n Framework Mon te Ca r lo method as a function o f the energy window width AEi. Intuitively, AEi controls the level of coarse-graining during the process o f memorizat ion o f promis ing con-formations. For simplici ty , it is constant for all bins i in our implementation. A s seen in Figure 5.17, we varied AEi from 1 (when every single energy level is stored in its own bin) to 10 £o (when 10 different energy levels are grouped in a single bin). We observed that our bin framework is quite robust wi th respect to the width o f the energy window and that the optimal value appears to be around 5 £ 0 . For the homopolymer of length 64, both AEi = 5 and AEi = 10 seem to perform we l l . 1e+06 jjj 100000 Q. o 10000 1000 10 A E: Figure 5.17: M e a n C P U time required for finding m i n i m u m energy conformations ( - 1 6 1 ) o f the 32 amino acid homopolymer over 20 independent runs on our refer-ence machine as a function o f the energy window width AEi. E r ro r bars indicate standard deviation observed. The Capacity of Bins In our final experiment, we studied the impact of the capacity of bins. The capacity of bins is responsible for determining the number o f conformations stored and used for later retrieval. A s can be seen in Figure 5.18, good performance is generally obtained for a capacity value o f about 100 conformations per b in . Stor ing too few or too many conformations results in less efficient searching, due to the inabi l i ty of the bin system to provide an effective diversification mechanism when too few Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC ft-Sheet Protein Folding Problem 106 3 Q-O c cn tu E 1e+06 | 100000 10000 1000 60 80 100 120 140 160 180 200 Capacity Figure 5.18: Mean C P U time required for finding minimum energy conformations (—161) of the 32 amino acid homopolymer over 20 independent runs on our ref-erence machine as a function of the bin's capacity. Error bars indicate standard deviation observed. conformations are stored, or due to the computational overhead encountered when the search has to memorize too many conformations. This observation is also con-sistent for the homopolymer of length 64, where a capacity of 100 seems to work well. 5.3 S u m m a r y During our study of different parameter settings, we determined that the follow-ing parameters, in the order of their decreasing relevance, play an important role in the performance of the B I N M C algorithm: (1) the number of non-improving steps (over the best energy) that are performed before reaching into a bin and re-placing the current conformation in the Monte Carlo run; (2) the ratio between the width of the energy of interest considered for binning and the bin temperature; (3) the diversity criteria used during binning high- and low-energy conformations (the Hamming distance limits); (4) the width of the energy window considered by bins; and (5) the capacity of bins. This empirical study further shows that the bin framework performs better than the simple restart strategy and than the pure Monte Carlo algorithm, which it is based on. We also determined that all components of our algorithm are important for its efficiency. The bin framework introduced in this chapter belongs to the class of model-Chapter 5. Adaptive Bin Framework Search, Introduced for the FCC 0-Sheet Protein Folding Problem 107 based search ( M B S ) methods, since it uses a non-parametric model represented by conformations stored in the bin framework. The model is updated during the search and influences the choice of a new candidate solution every nolmpr Retrieve steps, when search stagnation is detected. In M B S for the discrete off-lattice model wi th structural fragment insertion as described in [20], the choice o f a new can-didate solution is influenced at every step by the pool o f conformations stored. Thus, M B S exploration of the search space is only dependent on conformations stored [20]. Therefore, regions that are pruned from the model are eliminated and not explored any further. Since the bin framework provides only a subsidiary mech-anism for generating candidate solutions when search stagnation is detected, it does not completely eliminate unexplored regions o f the search space. This is achieved by running a non-model-based search (canonical M C ) for a sufficiently long time to a l low it to explore other regions of the search space. Addi t iona l ly , M B S does not have a mechanism comparable to our diversity criteria between stored confor-mations. In M B S , N best conformations, the quality of which is based on the score described in Chapter 3, are stored. The bin framework sorts conformations into bins representing different energy levels to make sure that the model contains as many energy levels of interest as possible whi le still reducing the search space. This aspect o f the search is some-what conceptually related to histogram-based sampling and search methods such as M U C A and E L P . It should be noted, however, that b in framework is a non-parametric model o f the space composed of the actual subset o f diverse promis ing candidate solutions. M U C A , on the other hand, and to some degree E L P , are based on a parametric model of sampling all energy levels with the same probabil i ty without emphasis on low energies. In addition, our bin framework applies differ-ent H a m m i n g distance criteria based on the energy level o f the bin to ensure that conformations stored are diverse and reflect the overall structure o f the landscape, wh ich is believed to be funnel-like for proteins. F ina l ly , the bin framework proposed here suggests a very general model that can be used adaptively to provide both intensification and diversification during the search. A s has been shown in this work, even the most s imple design choices resulted in very good performance of the algorithm and improved on the best-known solution quality for the homopolymer o f length 64. Development o f other adaptive and possibly more complex mechanisms therefore holds much promise. 108 Chapter 6 Construction Search for Identifying Folding Pathways If the path be beautiful, let us not ask where it leads. Anatole France In this chapter, we propose a novel and simple construction-based approach to address the problem of identifying folding pathways as follows: given the topology of the native state, identify native contacts that form folding nuclei based on a graph theoretical approach that considers effective contact order (effective loop closure) as its objective function. A number of computational methods for the prediction of folding nuclei al-ready exist in the literature, but most of them rely on restrictive assumptions about the nature of nuclei or the process of folding. Our motivation was to develop a simple, efficient and robust algorithm to find an ensemble of pathways with the lowest effective contact order and to identify contacts that are crucial for folding. Our approach is different from previously used methods in that it uses efficient graph algorithrhs and does not rely on restrictive assumptions about the structure, location and process of folding nuclei formation. Our predictions provide more detail concerning the protein folding pathway than most other methods in the lit-erature. We demonstrate the success of our approach by predicting folding nuclei for a data set of proteins for which experimental kinetic data is available. We show that our method compares favorably with other methods in'the literature and that its results agree with experimental results1. 6.1 Description of the Problem The Problem of identifying folding pathways (a set of native contacts that play an important role in folding along with identifying a time-ordered sequence of their 1 A version of this chapter has been published. A. Shmygelska, (2005) Search for Folding Nuclei in Native Protein Structures, Bioinformatics, 2005, 21: i394-i402. Chapter 6. Construction Search for Identifying Folding Pathways 109 formation) of proteins can be formally defined as follows: Given a target backbone conformation (the native state) c represented as the set of pairwise native contacts c = ( e i , 62, • • •, em} (where m is the total number of pairwise contacts present in the native state) and a factor 5 £ (0,1], find an ensemble of folding pathways o — {oi | oi £ O(c)}, where 0(c) is the set of all possible folding pathways for conformation c and each folding pathway o; £ 0(c) consists of a sequence of pairwise contacts in relative order of their formation Oj — o ^ i 0^2 . . • Oi-rn (each o^t is a contact between two amino acids present in the native state that was formed in the folding pathway i as the tth contact, o^t £ {e\,ei,... ,em}, t £ [l,m}). Each pathway Oj is a pathway satisfying the following condition: the value of an objective function E(c, c\) is within a factor 5 of the minimal objective function value E(c,o*) — mm{E(c,0j) | Oj £ 0(c)}, i.e. E(c,Oi) < 5E(c,o*). Usually, the objective function only considers the entropy of contact formation, since the precise energetics of the protein folding process are still unknown. The problem of identifying folding nuclei imposes an additional constraint on folding pathways of. In this case, each folding pathway Oj consists of a sequence of pairwise contacts in relative order of their formation Oi = Oi^Oi^ • • • °i,fc;, such that when all ki contacts have been formed in the pathway Oj, the process of fold-ing results in the downhill rapid assembly of the native state. Thus, the objective function (energy) of adding contacts Oi^i+\ . . . oitkm is smaller than the specified threshold, i. e., E(c, o i ) f c . + 1 . . . o i ] f c v J < E t h r e s h o i d . 6.2 Description of the Algorithm The proposed algorithm for identifying folding nuclei (given the native state, de-termine the order of contact formation and identify those that are critical for the process of folding) is based on the notion of effective contact order (ECO) first introduced by Dill [ 144]. As described in Chapter 2, effective contact order of a newly added contact is defined as the effective loop closure size (the number of steps, i.e., covalent and non-covalent links, taken along the shortest path on the polymer graph) given that other contacts have been formed. Our approach consists of the following three steps, which we describe in detail in subsequent subsections: (1) construction of the polymer graph (a graph of con-tacts present in the native state of a protein); (2) sampling of the space of folding pathways to identify an ensemble of low cumulative effective contact order path-ways; (3) analysis of the ensemble obtained in the previous step to extract contacts that belong to folding nuclei, and analysis of the order of their relative formation. Chapter 6. Construction Search for Identifying Folding Pathways 110 6.2.1 Generation of the Polymer Graph The choice of model for a protein representation, discussed in Chapter 2, was guided by two somewhat opposing objectives: reduction of the search space and accurate geometry of protein structure. A protein is represented as a connected undirected weighted graph with one node per residue. Two residues are connected by an edge, if they are in contact in the native state, as described previously in Chapter 2. Every edge of the polymer graph is augmented by the value of E C O -loop closure size required to form a given contact or edge. Initially, we consider an extended polymer where no non-covalent contacts are present; thus the weight of any edge is set to CO(i,j) — \i — j\. Figure 6.1 shows an example of the polymer graph generated for the chymotrypsin inhibitor 2 (CI2) protein from Table 6 .1 . Generally, such polymer graphs, as has been previously observed [51 ] , are very well connected and exhibit a so-called small-world property, whereby most nodes in the graph are also neighbors of one another, and every node can be reached from every other node by a small number of hops or steps. Figure 6 . 1 : Polymer graph generated in phase one of our algorithm for the chy-motrypsin inhibitor 2 (CI2) protein containing 65 nodes (n) and 82 edges (m). 6.2.2 Sampling of Folding Pathways Since dominant protein folding pathways are those that maximize the formation of favourable native interactions while minimizing the loss of configurational entropy, Chapter 6. Construction Search for Identifying Folding Pathways 111 low effective order contacts are more readily made as compared to high effective contact order contacts during the process of folding. Thus, we are interested in finding an ensemble of folding pathways such that the cumulative effective contact order: ECOtotol= ECO(i,j,At) (6.1) i>JG{ei,...,e„ l},i<j+3 for all of the contacts added is minimized, where rn is the total number of edges { e i , . . . , e m } in the polymer graph, and i, j iterate only through the set of non-covalent native contacts. As previously mentioned, individual values of ECO(i, j, AT) are affected by what other contacts have been formed previously - the set of previously-formed contacts AT, t e [0, m — 1], AQ = 0, and AT+\ = AT U Therefore maximizing the cumulative effective contact order is not trivial. To solve this problem, we use probabilistic constructive local search, as il-lustrated in Figure 6.2. Our algorithm, outlined in Figure 6.3, works as follows: we start with an extended polymer graph that represents a fully extended polymer chain; at each step we add one edge (native contact) to the polymer graph probabilistically (with probability p) based on the effective contact order: y ECO(i,j,At)3'2+k/m' We used a power of 3 /2 based on the theory of the random Gaussian poly-mer, where the probability of a random contact is proportional to the separation along the chain raised to the power of 3 /2 [48]. This local search algorithm is quite "greedy" and will generally tend to pick edges with low E C O , following the intuition that during the process of folding, low-ECO contacts have a higher probability of formation. As non-covalent edges are added to the polymer graph, local search should become even greedier, since the probability that there will be subsequent edges whose E C O will depend on the edges just added will decrease. The goal here is to minimize the sum of E C O of individual contacts during their formation. Thus, we added an additional factor k/m to the power, where k is the number of non-covalent edges already added to the polymer graph, and m is the total number of non-covalent edges in the polymer graph. Thus, this factor reflects the proportion of non-covalent edges already added to the graph. The weights of edges that are still left to be added are updated according to an efficient graph algorithm for the dynamic all pairs shortest path problem by Rama-lingam and Reps [111]. This algorithm maintains information about the shortest paths by keeping a subtree of potentially affected nodes for each source - a des-tination pair containing all edges that belong to at least one shortest path from a source node to a destination node. Distances are updated by running a Dijkstra-like procedure on the affected nodes. The time complexity of this algorithm is Chapter 6. Construction Search for Identifying Folding Pathways 112 Figure 6.2: Illustration of the sampling phase of the algorithm. Edges that are added to the polymer graph are highlighted in bold, while edges that still have to be added are represented in non-bold. Initially, we start with a fully extended polymer, and edges that are to be added are weighted by their chain separation. At each step one edge is probabilistically added based on its E C O weight, and the E C O weights of the edges that are still to be added have to be revised at each step. The process of addition of edges stops when all of the edges are added to the polymer graph and their respective E C O s shrink to 1. Chapter 6. Construction Search for Identifying Folding Pathways 113 procedure ProbabilisticConstructiveLocalSearch input: polymer graph output: folding pathway with low ECO initialize weights of all edges to \i — j\; add all edges with weight equal to 1 to the graph; k:= 0; while (not all edges added i.e., k < m ) do select next edge to add with probability p = e c o ^ j f^/^+k/,,, \ update weights for edges not yet added; k:=k+i; end end Figure 6.3: Outline for the probabilistic constructive local search used in the sampling of the folding pathways stage of our algorithm, where k is the number of non-covalent edges already added to the polymer graph and m is the total number of non-covalent edges in the polymer graph. 0(mn + n2log(n)), where n is the number of nodes in the graph and rn is the number of edges [31]. Samples obtained and added to the ensemble of pathways that have low cumu-lative effective contact order should be independent of each other, thus we used the results of multiple independent runs of the constructive search to construct the low-ECO ensemble. Our simple sampling procedure is repeated multiple times until the number of low cumulative ECO pathways specified by the user is built. A particular low cumulative ECO pathway is added to the ensemble of low-ECO pathways if the cumulative effective contact order obtained is within 10% of the lowest ECO value obtained during the search. 6.2.3 Collective Analysis of Low Effective Contact Order Pathways To identify the subset of contacts that belongs to the folding nuclei and to deter-mine their relative formation order, we analyze a representative ensemble of a large number of low cumulative ECO pathways obtained during the previous sampling step. In order to obtain an upper bound on the subset of native contacts that can potentially participate in folding nuclei, we define a critical point in the folding pathway after which the process of subsequent contact formation becomes down-hill according to our objective function of interest - the effective contact order. This critical point determines a moment in time when all folding nuclei have been Chapter 6. Construction Search for Identifying Folding Pathways 114 assembled. To define this critical point in terms of the effective contact order, we record the subset of native contacts (edges) that, as they are added, result in an effective contact order smaller than the given threshold for all edges still left to be added. The definition of the ECO threshold that determines when edges should no longer be added to the set of potential folding nuclei naturally follows from studies of loop lengths that have high probability of formation without involving interme-diate steps [83]. We defined this threshold to be Al = 15 residues, based on the observations in [83]. After the maximal effective contact order of edges that are left to be added drops below this specified threshold, we stop adding edges to the set of potential folding nuclei contacts. As a result, we end up with the subset of native contacts that potentially belong to the set of folding nuclei contacts. In order to identify which of these edges are true folding nuclei contacts, we analyze relative dependencies among the edges as they were formed, that is, added to the polymer graph. We count the number of times each edge is used in the shortest path (effective contact order) events of the subsequent edges. We do so both directly i.e., the edge belongs to the shortest path of another edge during growth of the polymer graph, and indirectly, the shortest path for an edge that is added during the growth process contains an edge whose shortest path directly or indirectly dependents on the current edge - a recursive step. When this recursive analysis of edge dependencies is completed, we can calculate the proportion of times an edge was used in the shortest-path events of other edges. The edges that are used most often are those that form early and are crucial for reducing the effective contact order (and entropy) for the subsequent edges. Thus, these edges comprise folding nuclei contacts. The outline for the analysis and identification of the folding nuclei phase of our algorithm is given in Figure 6.4. 6.3 Empirical Results and Discussion To test our construction-based algorithm for identification of folding pathways, we used the set of 29 proteins listed in the paper by Rader et al. [107]. Only 27 were used since we had problems processing the pdb files of losp and lhcb. The output of our algorithm consists of the percent usage of each edge in the ensemble of low-ECO folding pathways. Those contacts that are used most often in the shortest path (low-ECO contact formation) represent folding nuclei, since they are crucial for subsequent native contact formation. We display our results in two ways. The first one is more intuitive: we show the actual edges in the three-dimensional protein structures shaded according to usage in the shortest path events. The darker edges represent those contacts predicted to Chapter 6. Construction Search for Identifying Folding Pathways 115 procedure Identify FoldingNuclei input: the set of low-ECO pathways, the length threshold Al output: the percent usage of each edge in low-ECO pathways for (each pathway in the low-ECO ensemble ) do set weights of all edges (i, j) to \i - j\; add all edges with weight equal to 1 to the graph; while (not all edges added) do add edge in order recorded; update ECO for the remaining edges; calculate maximum ECO among the remaining edges; if (maximum ECO > AO then for (each edge added) do get the list of edges that current edge depends on; recursively increment usage of the edges involved; end end end end end Figure 6.4: Outline of the analysis phase of our algorithm that identi-fies folding nuclei contacts in the low effective contact order pathway ensemble. Parameter Al is the length threshold used to identify con-tacts whose formation proceeds rapidly. fold earlier during the folding process - these are important for subsequent folding. Another way to represent edges is using the contact matrix of a protein shaded ac-cording to the order of contact formation. In both cases, contacts that are essential for low-ECO folding are clearly visible, and the overall folding mechanism can be deduced. The second approach is to display results using the "band" representa-tion of a protein and to use shading to represent the usage of residues in the protein during low-ECO contact formation. We also use shading to indicate the order of be-coming structured for individual residues in three-dimensional protein structures. To convert edge usage into residue usage, we sum usage numbers for each edge involving the residue of interest, and normalize it by the total usage of all amino acids in the low-ECO pathways. Using the second representation, we are able to compare our method with other methods in the literature, the majority of which produce the "band" representation of folding nuclei, even though descriptions of a folding nucleus in terms of contacts are more accurate. First we compare our folding nuclei predictions with the experimental results Chapter 6. Construction Search for Identifying Folding Pathways 116 obtained from the hydrogen-deuterium slow exchange ( H / X ) method that was run on the set of 29 proteins by L i et al. and Rader et al., as wel l as from experimental cfr-value analysis [93]. A s mentioned in [107], the experimental folding core was defined in this case as the secondary structural elements (assigned using Database of Secondary Structure Assignments (DSSP) ) containing residues wi th the largest protection factors in the H / X experiments, which indicates that these residues are structured early during the process of folding. Exchange rates are most often mea-sured in the native structures (this does not directly represent fo ld ing events). Fo r some proteins, exchange rates for partially folded species dur ing the actual pro-cess of fo ld ing are available. We indicate these residues using darker shades in the experimental results, since these become structured earlier in the process of folding [76]. In Figures 6.5, 6.6, and 6.7 we show "band" representations of the previously mentioned set of 27 proteins. The top band is for the H / X experimental results [76], the middle band ( if available) are the experimental ^ -va lues that represent the participation of a residue in the transition state based on the mutagenesis exper-iments [93], and the lower band represents our predictions, gray-scale-coded by the percent involvement of the residues in the l o w - E C O events. The results o f our approach generally show good agreement with the experimental results. A l s o using visual comparison of our predictions with Figure 3 from [108] wi th the F I R S T r ig id algori thm, the G N M fast mode peak and the G N M slow mode min ima methods tested by Rader et al., we found that the results from our method are in agreement with the previously mentioned methods and addi t ional ly provide information about the order of contact formation. Nex t we examine case by case a few very well-studied proteins in our data set and compare our predictions with the experimental and computational data avail-able about their fo lding pathways. The chymotrypsin inhibitor 2 protein forms a four-stranded /3-sheet packed against an a-he l ix , as illustrated in Figure 6.8. This is probably the best character-ized protein using <f?-value analysis. Experimental values are high in the a -he l ix , with the highest values for the residues at the N-cap of the a -he l ix . These residues interact with two residues in the /3-sheet to form a core, wh ich is the most h ighly structured region of the protein in the transition state [93]. Our prediction captures the previously described experimental observations. Our gray-scale-coded contact matrix for C I 2 (see Figure 6.8) also agrees wi th computational results o f W e i k l et al. establishing that a-32, a-81, 83-34- form first, and 01-/34 forms after a l l o f the local contacts have been formed. A m o n g the most difficult cases for our algorithm are the folding pathways of proteins G and L . Both protein G and protein L are members of the ubiquit in fold (while they have little sequence identity) and fold via helix-assisted hairpin forma-Chapter 6. Construction Search for Identifying Folding Pathways 117 lbdd lpga 2crs I I I L I l iy5 I lsrm 2ptl lbpi 2ait I I I lbu4 2ci2 lubi 351c Figure 6.5: B a n d representation of the experimental and predicted fo ld ing nuclei for our test set of 27 proteins, Part I. The upper band represents H / X experimen-tal data (darker-shaded residues represent those that gain protection first dur ing folding, lighter-shaded residues represent residues that have slow exchange in the native state H / X [76]). The middle band ( if available) represents experimental <£-values (the darker the shade, the higher the 4>-value (0 < </> < 1) [93]. The lower band represents our folding nuclei predictions, shaded according to percentage of involvement i n l o w - E C O events (the exact scale in percent is given on the right side; proteins are grouped according to the max imum percentage of edge usage). Chapter 6. Construction Search for Identifying Folding Pathways 118 lhrc i n 1. I a 6 m la2p 3c hy l h e l I I I I I lhfx 3lzm 20 40 eo i i i i ! 1 BO 100 120 II 1*0 20 I j 1 1 i 40 63 SO 103 i | i 10 40 1 1 i i i i 1 I I 0 | 0 103 120 » l I I I i 1 20 « eo eo 100 120 2eql 1 i 1 i i i i it 1 i 1 | 1 i 1 20 40 eo ao 100 120 l r b x 1 i i i i ) I i i i 1 1 I  i 1 1 20 40 eo ea 100 120 2rn2 | i 1 1 1 i i i i I 1 I i 1 20 40 60 80 iCO 120 140 l m c p ! 1 i i j > t i l I 1 M l 1 1 1 1 M i l i i i 1 eo *o 00 00 100 I X 140 180 180 200 220 2afg | i i i i i i 1 i 20 40 eo ao 100 ISO l h m l i i 1 1 • l 1 1 I I i l l II 'I I H i l l i i i I L I : 103 120 140 100 Figure 6.6: B a n d representation of the experimental and predicted fo ld ing nuclei for our test set o f 27 proteins, Part II. The upper band represents H / X experimen-tal data (darker-shaded residues represent those that gain protection first dur ing folding, lighter-shaded residues represent residues that have slow exchange in the native state H / X [76]). The middle band (if available) represents experimental values (the darker the shade, the higher the <£-value (0 < <j> < 1) [93]. The lower band represents our folding nuclei predictions, shaded according to percentage of involvement in l o w - E C O events (the exact scale in percent is given on the right side; proteins are grouped according to the max imum percentage o f edge usage). Chapter 6. Construction Search for Identifying Folding Pathways 119 9ilb Istn I ! I Figure 6.7: B a n d representation of the experimental and predicted fo ld ing nuclei for our test set o f 27 proteins, Part III. The upper band represents H / X experimen-tal data (darker-shaded residues represent those that gain protection first dur ing folding, lighter-shaded residues represent residues that have slow exchange in the native state H / X [76]). The middle band ( if available) represents experimental <I>-values (the darker the shade, the higher the <3>-value (0 < cj) < 1) [93]. The lower band represents our folding nuclei predictions, shaded according to percentage of involvement in l o w - E C O events (the exact scale in percent is given on the right side; proteins are grouped according to the m a x i m u m percentage o f edge usage). t ion. Thei r topology consists o f a central a-hel ix packed against a /3-sheet c o m -posed of two anti-parallel /3-hairpins, as shown in Figure 6.9. A c c o r d i n g to H / X experimental data, protein G folds through a transition-state ensemble wi th a w e l l -formed /3-hairpin 2 (formed by /33 - /34), while protein L forms /3-hairpin 1 (formed by /31 - 02) first [76]. Our approach captures helix-assisted fo ld ing, but since the objective function only considers topological effects and ignores energetics, it does not correctly predict the hairpin formation order. The importance o f energetics in the folding of proteins G and L has been shown previously in the literature [23]. The enzyme barnase folds into a five-stranded anti-parallel /3-sheet and two a-helices, see Figure 6.10. The $-value analysis of barnase shows that major sec-ondary structures are formed but the loops are unfolded [93]; this observation is consistent wi th our predicted results. It has also been shown experimental ly (by H / X [139]) and in molecular dynamic simulations [28] that the fo ld ing pathway of barnase is dominated by /3-sheet structures, particularly by the formation of the 03-04 hairpin, and as seen from our predictions this hairpin forms early. The chicken src S H 3 domain folds into a five-stranded orthogonal /3-sheet structure, see Figure 6.11. Both H / X [76] and <£-value [93] experimental results show that the 02-03 hairpin forms first. Our predictions are consistent wi th these results. Chapter 6. Construction Search for Identifying Folding Pathways 120 m Figure 6.8: Predicted order o f contact formation for the chymotryps in inhibi tor 2 protein (pdb i d 2ci2) . Contact matrix and three-dimensional structure representa-tions are gray-scale-coded according to percent contact usage (the scale is given on the right). Indexes n\ and n2 represent indexes of the residues along the backbone. Darker-shaded edges (contacts) are predicted to form first. The C h e Y protein is composed of five parallel /3-sheets and five a-hel ices (with two a-hel ices, A and E , located on one face of the /3-sheet and three other a-helices, B , C , and D , on the other side), see Figure 6.12. H / X protein stabili ty is highest in the hydrophobic core formed by the side chains of the residues in 0-strands 1, 2, and 3 and helices A and E . The a -he l ix A , together wi th /3-strands 1 and 3, is involved in the formation of the folding nucleus [70]. <l>-value analysis shows that only residues in the first sub-domain (1 to 61) exhibit some degree of native-like structure, while the second sub-domain is essentially unstructured (residues 62-129). Our predictions correspond to experimental results: the first sub-domain, particularly /3-strands 1, 2, and 3 as wel l as helices A , B , and C , forms early. The ubiquit in protein (see Figure 6.13) forms a molten globule intermediate that has partially folded 01 and 02 strands of the five-strand /3-sheet, part o f the 83 strand, and a partially structured a-hel ix packed on the hydrophobic side of the sheet ( H / X experiments) [19]. Our predictions are consistent wi th these experi-mental observations. The T 4 lysozyme protein forms a molten globule that involves helices E , H , Chapter 6. Construction Search for Identifying Folding Pathways 121 Figure 6.9: Predicted order of contact formation for the B I immunog lobu l in -binding domain of protein G (left) and protein L (right), (pdb ids l p g a and 2ptl). Darker shaded edges (contacts) are predicted to form first. and A and the C-terminal o f helix C ( H / X analysis) [80], as shown in Figure 6.14. Our predictions also show that interactions between helices E , H and C-terminal o f hel ix C are established early in the process of folding. A c c o r d i n g to experimental H / X data, the bovine pancreatic trypsin inhibi tor protein (see Figure 6.15) forms a-01 and 02 - 03 structures that come together [76]. Our predictions are exactly supported by these experimental observations. The B domain of protein A is composed of three helices, forming a three hel ix-bundle, as shown in Figure 6.16. It has been demonstrated experimentally, using H / X , that Protein A forms a marginally stable intermediate consist ing o f helices al and a3 [76]; according to our approach, these two helices should also come into contact wi th each other during the early stages of the folding process. 6.4 Summary Our graph-theoretical construction method for identifying folding nuclei is in agree-ment with the experimental data obtained using 4>-value analysis and the H / X method for different structural classes of proteins for the data set used. It is able to conduct mult iple runs of the probabilistic constructive search on sequences of significant lengths (sequences of length 54 - 237 amino acids were used here). Since the average length of a protein domain (the part of the protein sequence that folds largely independently of the rest o f the sequence) is between 200 — 300 amino acids [92], our approach is an attractive tool for studying folding of ind iv idua l do-mains in different structural classes of proteins. Our method has a number of advantages compared with the methods used to Chapter 6. Construction Search for Identifying Folding Pathways 122 Figure 6.10: Predicted order of contact formation for the the barnase protein (pdb id la2p). Darker shaded edges (contacts) are predicted to form first. address this problem in the literature (described in Chapter 3). In particular, it is conceptually simple; it has a reasonable and precise definition of a folding nucleus based on the entropy of the chain; and it places no restrictions on structure (as compared with clustering of native contacts and the use of low effective contact order search [144]), location or other properties (such as evolutionary conserva-tion) of folding nuclei. Additionally, it provides information about the folding pathway, unlike other methods (for example, G N M [32], FIRST [107], and amino acid conservation within a family/super-family [112]) that provide only the set of residues predicted to participate in formation of folding nuclei. Unfortunately, performance data for some of the methods mentioned in Chapter 3 (for example, time-intensive molecular dynamic unfolding of the native structures [36], the prob-abilistic roadmap method [2], minimum cuts unfolding of native structures [151], amino acid conservation within a family/super-family [112]) are unavailable for the comprehensive data set [76, 107] chosen for this work. Therefore, we could not directly compare the performance of these methods and our new algorithm. One shortcoming of our method is that, similarly to many other methods in the literature, it considers only native contacts, and even though it scales quite well with the length of the protein, there is a limitation on length. Our method, while still having conceptual simplicity, can be extended to use more realistic energy functions that in addition will consider energetic contribu-tions. It can thus be used as a tool for obtaining further insight into the process of folding. Chapter 6. Construction Search for Identifying Folding Pathways 123 Figure 6.12: Predicted order of contact formation for the C h e Y protein (pdb id 3chy). Darker shaded edges (contacts) are predicted to form first. Chapter 6. Construction Search for Identifying Folding Pathways 124 Figure 6.13: Predicted order of contact formation for the ubiquit in protein (pdb id l u b i ) . Darker shaded edges (contacts) are predicted to form first. F igure 6.14: Predicted order of contact formation for the T 4 lysozyme protein (pdb id 31zm). Darker shaded edges (contacts) are predicted to form first. Chapter 6. Construction Search for Identifying Folding Pathways 125 Figure 6.15: Predicted order o f contact formation for the bovine pancreatic trypsin inhibitor protein (pdb id l bp i ) . Darker shaded edges (contacts) are predicted to form first. Figure 6.16: Predicted order of contact formation for the B domain o f protein A (pdb id l bdd ) . Darker shaded edges (contacts) are predicted to form first. Chapter 6. Construction Search for Identifying Folding Pathways 126 Protein PDB ID length 1 Apo-myoglobin la6m 151' 2 Barnase la2p 108 3 Cytochrome c lhrc 104 4 T4 lysozyme 31zm 164 5 Ribonuclease T l lbu4 104 6 a-Lactalbumin lhml 123 7 Chymotrypsin inhibitor 2 2ci2 64 8 Ubiquitin lubi 76 9 Bovine pancreatic trypsin inhibitor lbpi 58 10 Interleukin-1/3 9ilb 151 11 Hen egg-white lysozyme lhel 129 12 Equine lysozyme 2eql 129 13 Protein A , B-domain lbdd 60 14 Staphylococcccal nuclease Istn 136 15 Ribonuclease A lrbx 124 16 Ribonuclease H 2rn2 155 17 Guinea pig a-lactalbumin lhfx 123 18 BI immunoglobulin-binding domain protein G lpga 56 19 BI immunoglobulin-binding domain protein L 2ptl 78 20 Cardiotoxin analog III 2crs 60 21 Tendamistat 2ait 74 22 Single chain antibody fragment lmcp 237 23 Human acidic fibroblast growth factor-1 2afg 127 24 Cytochrome c551 351c 82 25 Ovomucoid third domain liy5 54 26 Chicken src SH3 domain lsrm 56 27 CheY 3chy 128 Table 6.1: Set of proteins used in this study. The kinetic data about folding path-ways of these proteins is available in [76] and [107]. 127 Chapter 7 Conclusions and Future Directions May I never use my reason against the truth. Hasidic prayer In this chapter, we discuss how the approaches introduced in this thesis relate to each other and outline scientific contributions made towards developing more efficient search methods for protein folding problems. We also draw conclusions based on the observed performance of our new algorithms and outline directions for future research. 7.1 ACO for 2D and 3D Hydrophobic Polar Folding In this work, we have shown that Ant Colony Optimization (ACO) can be applied in a rather straight forward way to the 2D and 3D HP protein folding problems. Even though our A C O algorithm is based on very simple structural components (single relative directions) and a simple subsidiary local search procedure (iterative first improvement), it performs fairly well compared with other algorithms and spe-cialized heuristics on the benchmark instances considered here, particularly in 2D. The only non-specialized algorithm that typically performs better than our A C O algorithm, both in 2D and 3D, is P E R M . We observed that, particularly in 3D, the run-time required by A C O for finding minimum (or best-known) energy con-formations scales worse than P E R M with sequence length. However, our results show that our A C O algorithm finds a different ensemble of native conformations compared to P E R M , and has less difficulty folding sequences whose native states contain structural nuclei located in the middle rather than at the ends of a given sequence, as well as sequences with structures in which the ends interact. Because of its ability to find more balanced ensembles of minimum (or close to minimum) energy conformations, our new A C O algorithm can greatly facilitate investigations of the topology and location of structural nuclei based on the sequence alone (when Chapter 7. Conclusions and Future Directions 128 the native state of a protein is not necessarily known, which is the requirement for the graph-theoretical construction search introduced in Chapter 6). We found that two major components of A C O — the pheromone values, which capture experience accumulated over multiple iterations of the search process from multiple conformations, as well as the heuristic information that provides myopic guidance to the folding process — play a significant role for longer 2D sequences and, to a lesser extent, for 3D sequences. This observation suggests that in 3D, it may be preferable to associate pheromone values with more complex solution components. In future work, it will be interesting to investigate the use of more complex and informative solution components. These can be obtained, for example, based on reinforcement of contacts, similarly to the contact representation used for the graph-theoretical construction search for folding nuclei presented in Chapter 6. Another promising direction is to consider different heuristic functions that take into account certain properties of intermediate and complete conformations, sim-ilar to the more specialized H Z , C H C C , C G , and CI methods. It would also be worthwhile to consider the use of subsidiary local search procedures that are more powerful than the iterative first improvement algorithm used in this work (for ex-ample, the Bin Framework Monte Carlo method introduced in Chapter 5). Further-more, we believe that better techniques for avoiding the attrition problem during the construction process (particularly in 2D) should be developed (for example, a look-ahead strategy could be used). Finally, while HP protein folding problems are of considerable interest because of their conceptual simplicity, ultimately, most applications of protein folding al-gorithms require the use of more realistic models of protein structure. Our A C O algorithm does not rely on heuristics and properties that are specific to the HP model, yet it performs very well on this restrictive, but not entirely unrealistic ab-stract model. We therefore believe that relatively straight forward extension of our A C O algorithm to more complex and realistic models of protein structure, includ-ing other lattice and discrete off-lattice models, holds significant promise. 7.2 Adaptive Bin Framework for the FCC (3- Sheet Model Our bin framework generalizes different adaptive strategies and provides a very rich framework for establishing interactions between the search process and the search landscape. The adaptive component of this framework is very important for the overall efficiency of the search. Here we introduce adaptive mechanisms for both choosing which conformations should be stored, based on the conformations Chapter 7 . Conclusions and Future Directions 129 already stored in memory, and biasing preferences of retrieving particular confor-mations, based on the past history when the search stagnates. This is generally absent from other methods and therefore, even when running other search methods for an extensive time, the best solution quality found may not improve. Since we are not simply performing a memory-less search by keeping only the current conformation or a set of conformations at different temperatures as current state-of-the-art methods for protein folding do, but keep the k best ones encountered at any time during the search (where k is determined by the diversity of a set), we can search multiple regions containing promising conformations and possibly sample compact conformations more efficiently. Similar to the Model-based Search (MBS) method [20], our bin framework is used to store promising candidate solutions for future reuse. However, unlike M B S , we developed and tested an adaptive diversification mechanism that varies based on the energy level considered and takes into account how different a confor-mation is from other conformations of the same energy. Additionally, the energy level of interest, which determines what is the highest energy that conformations are allowed to have and still be memorized in the bin framework, and individual Hamming distance criteria used for each bin, are adapted according to the estimate of the ground state energy. One of the important questions we address in this part of our work is: how do adaptive strategies perform for the complex energy landscape compared with heuristic methods that do not adapt based on the landscape being searched (such as Replica Exchange Monte Carlo and Parallel-hat Tempering) for the protein folding problem? As shown by our analysis of the empirical performance of our novel Bin Framework Monte Carlo method (a combination of the bin framework and a simple Monte Carlo algorithm), we were able to develop more efficient search methods that rely on the bin framework and use an adaptive component. These methods can outperform both canonical Monte Carlo, Replica Exchange Monte Carlo, and its heuristic variant Parallel-hat Tempering when it comes to the search for a global optimum, as shown for the F C C [3-sheet protein folding model. Some of the promising directions for future work that we are planning to con-sider include more advanced adaptive strategies that extract other features of the search landscape and determine what precisely is needed - either diversification or intensification - and adjust the search accordingly. Combination of other stochas-tic local search methods, such as Ant Colony Optimization introduced in Chapter 4 with the bin framework introduced here can be considered. Additionally, we are planning to generalize our bin framework further to work on partial as well as com-plete conformations, producing an efficient generalized framework that combines two distinct search strategies. Finally, we would like to extend our bin framework to address other protein folding models that use more complex energy potentials, Chapter 7. Conclusions and Future Directions 130 such as the F C C model, with a more general energy function and discrete off-lattice models. 7.3 Construction Search for Identification of Folding Pathways Our graph-theoretical construction method for identifying folding pathways, while conceptually very simple, has been shown to produce results that agree with the experimental data available for folding nuclei as well as with a number of more complex computational methods for finding folding nuclei. The presented method has a number of advantages: it represents a natural way to bound and identify the subset of native contacts important for low-ECO folding pathways; it provides additional information on the order of native contact formation and their relative dependencies; it does not require, but can potentially be augmented by, experimen-tal results; and it does not impose restrictions on the complete formation of certain elements (clusters) of structure. Future work will include a more extensive comparison of different computa-tional methods for finding folding nuclei in different structural classes of proteins as well as possible augmentation of the method with physically based energy func-tions, particularly those that consider not only entropic but also energetic contribu-tions. It would also be interesting to incorporate insights gained from the folding nuclei study to design more efficient search methods for protein folding. Specifi-cally, if certain regularities for a specific structural class of proteins are discovered through a computational study of folding pathways (for example, using the graph-theoretical method proposed here), they could be incorporated into search methods for protein folding such as those introduced in Chapters 4 and 5, in the form of initial probabilities during the search. 7.4 Summary The protein folding problem is at the center of this thesis. Within this vast prob-lem we focused on two major sub-problems, with the emphasis on the first: (1) conformational searching for the protein folding problem and (2) the problem of identifying folding nuclei. In the context of search problems associated with the process of protein folding, we explored promising search strategies that result in more efficient searching for problems with complex energy landscapes, such as protein folding and the closely related problem of identifying folding pathways. We designed and evaluated the performance of our new algorithms using em-Chapter, 7. Conclusions and Future Directions 131 pirical evaluation techniques. Additionally, we attempted to understand the be-haviour and properties of our algorithms by studying the influence of various pa-rameters on their empirical performance for problems of interest. Whenever pos-sible, we also determined the theoretical guarantees an algorithm was retaining or giving up. In this thesis, we considered three classes of promising algorithms that have not been developed and studied previously for these problems: 1. Biologically inspired self-organizing algorithms based on the notion of stig-mergy. The phenomenon of stigmergy arises in a multi-agent system, where a collection of agents modifies the environment. These changes in turn affect the decision process of each agent. A well-known algorithm in this category is Ant Colony Optimization that along with other approaches based on stig-mergy, has not been previously implemented and tested for this problem. In this work, we developed and tested the Ant Colony Optimization method for the problem of protein folding under the widely-studied 2D and 3D HP models. 2. The second class of algorithms we have presented in this work explore reac-tive (adaptive) search, which reacts to the progress made and adjusts the search strategy accordingly. We introduced a new framework - the bin framework, which stores a diverse set of promising conformations encoun-tered during the search. The storage and retrieval of solutions is guided by the search progress made, and provides the diversification or intensification of the search. 3. The third class of algorithms studied in this work is based on efficient repet-itive construction. We developed a construction search method that relies on efficient, dynamic, graph-theoretical methods and avoids forming restric-tive assumptions about the process of folding. The Ant Colony Optimization method introduced for the protein folding problem is also a construction-based search, in which partial solution components are reinforced indirectly by means of a pheromone matrix. One of the questions that was not answered explicitly in this thesis is, which of the proposed novel heuristic methods is the most promising for protein fold-ing and closely-related problems? As seen from our empirical evaluations of the proposed methods, usually construction-based methods (such as Ant Colony Op-timization) benefit from further local search optimization on complete candidate solutions. Local search methods relying on complete candidate solutions can ben-efit from occasional restart with a promising initial solution, as we showed using our novel bin framework. This candidate solution can also come from an efficient Chapter 7. Conclusions and Future Directions 132 construction heuristic transferring the search into a different part of the landscape when diversification is required, or from the previously memorized solution that resulted from the search on a complete conformation. The latter was done in our bin framework, and is a more intensification-directed strategy. Thus, the real question is: How can these strategies be combined to result into a more efficient adaptive search method? This is one of the primary directions planned for our future research - when the need for a significant diversification is detected, based on the search progress made, a construction-based search could improve performance; when intensification is preferred, an efficient search based on complete candidate solutions could be more promising. As seen from the individual conclusions and proposed future work described previously for the novel methods introduced in this thesis, all three methods will benefit from future integration of strategies found to work well in the two other parts of this work. Specifically, Ant Colony Optimization (probabilistic construc-tive search) will benefit from incorporation of a more powerful local search on complete candidate solutions, such as the Bin Framework Monte Carlo method. Additionally, A C O can be further improved, particularly when used on more real-istic protein models, incorporating insights that can be derived using an algorithm for identification of folding nuclei. In its turn, our Bin Framework Monte Carlo method can be enhanced by a more powerful diversification mechanism that uses a construction-based search, such as A C O . As in the case of A C O , it can also ben-efit from additional insights derived from the search for folding nuclei. Finally, the graph-theoretical search for folding nuclei can be extended not to rely on the knowledge of the native state, by employing efficient search methods for protein folding, such as A C O and B I N M C . In the course of this work, we have identified several promising general areas for future research. We have shown that biologically inspired, adaptive search mod-els based on the concept of importance sampling, and construction-based search methods provide an alternative to the current algorithms used for these problems that are primarily based on Monte Carlo and generalized Monte Carlo methods. In the previous literature on protein folding, these directions were either not proposed at all (as in the case of the biologically inspired algorithms) or are proposed in a limited setting (as in the case of adaptive strategies and the focus on construction-based search). 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Skolnick, "Parallel-Hat Tempering: A Monte Carlo Search Scheme for the Identification of Low-Energy Structures," Journal of Chemical Physics, Vol. 115, No. 11, 2001, pp. 5027-5032. [155] Y Q . Zhou and M . Karplus, "Folding Thermodynamics of a Model Three-Helix-Bundle Protein," Proceedings of the National Academy of Sciences of the USA, Vol. 94, 1997, pp. 14429-14432. [156] M . Zlochin, M . Birattari, N. Meuleau, and M . Dorigo, "Model-based Search for Combinatorial Optimization," Technical Report, IRIDIA, Brussels, 2001. Appendix A Ant Colony Optimization 145 In this appendix, we include supplemenary information for our Ant Colony Opti-mization (ACO) for the 2D and 3D Hydrophobic Polar (HP) protein folding prob-lems. The following tables provide additional information on our new test sets of biological and randomly generated HP sequences and the results from our compu-tational experiment with A C O and P E R M . For the biological sequences, we show the PDB IDs of the original protein sequences. Note that in our translation from protein sequences into HP strings, non-standard amino acid symbols, such as X and Z , were skipped; consequently, some of our HP strings differ in length from the respective P D B S E L E C T sequences. ID PDB ID HP sequence Length B30-1 1HA9:A phpphhhphphhppppppppphhhphpphphhph 34 B30-2 1CE4:A ppphppppppphphhhphphhhpphhhhphpphpp 35 B30-3 1FW0:A pphpppphpphhhppppphhhhpphphhhhpphhp 35 B30-4 1ZTA hpphpphppphpphhppphphppphhphpphhhpp 35 B30-5 1G1Z:A ppphphhhpphhhhpphhpphhphhhphphp 31 B30-6 1BZG hhpppphhppphpphpphppphhhpphhhphpph 34 B30-7 1BH7 hphhpphhhhhphhphphphhhhpphpphphph 33 B30-8 IB4G hhpphphphphppphpphhpppphppphh 29 B30-9 1BNX:A phhhhhpphpphhhhhhhhhphhhphhhhhhpp 33 B30-10 1FCT hhhhhppphhhphhhphhhphhphhpphpphh 32 Table A . l : Biological sequences of length « 30 Appendix A. Ant Colony Optimization 146 ID Ermn (2D) PERM <2 texp ACO tavg B30-I -12 0.43 0.56 0.49 0.71 B30-2 -12 0.15 4.09 0.29 1.08 B30-3 -14 0.15 0.02 0.04 1.19 B30-4 -10 2.5 0.3 0.54 3.64 B30-5 -13 0.10 0.17 0.12 0.22 B30-6 -13 0.84 10.40 1.55 70.94 B30-7 -16 0.02 0.02 0.02 0.19 B30-8 -8 0.09 0.003 0.006 0.17 B30-9 -18 0.002 0.008 0.003 0.06 B30-10 -16 0.003 12.50 .0.006 4.08 Table A . 2 : Performance comparison o f P E R M and A C O on bio logica l sequences of length « 30 in 2 D ID E m i n (3D) h PERM ti texp ACO tavg B30-1 -17 1.14 1.03 1.08 1.19 B30-2 -16 1.06 1.3 1.17 2.29 B30-3 -18 1.20 1.20 1.20 2.34 B30-4 -16 3.4 1.2 1.77 20.18 B30-5 -20 2.2 3.04 2.55 65.71 B30-6 -19 1.20 1.50 1.33 23.18 B30-7 -25 1.10 2.40 1.51 62.36 B30-8 -13 1.10 1.07 1.09 1.74 B30-9 -26 1.20 1.09 1.14 1.57 B30-10 -24 1.8 3291 3.60 163.23 Table A . 3 : Performance comparison of P E R M and A C O on bio log ica l sequences of length « 30 in 3 D Appendix A. Ant Colony Optimization 147 ID PDB ID HP sequence Length B50-1 1KBF:A hphppphpppphhpphpphpppphhhhhpppppphppppppppphhhpp 49 B50-2 1 KBH:A phppppphhhpphpphhpppphphhpphpphhhhhphhpphphhphp 47 B50-3 1G9P:A hhhphhhphpppppppppphphhppphpphpphhhhhhhhhhhhh 45 B50-4 1YUJ:A hphpphpphhhppphpppppppphhpphhphhhhppppphppphphpphh 50 B50-5 1GAB phpphhhpphppphhhphpphhhppphhhphhpphpphpphphhppphhphph 53 B50-6 2BRZ ppppphhpphhhpppphhpppphppphppphpphpphhppppphpphpphpph 53 B50-7 1VPC pphphhphhhphhpphhhhphphhppppphhhhppppppphhppp 45 B50-8 1VIB hphphhhhhhpppppppphphphppphphhhpphpphhpphhpppppphppppp 54 B50-9 1CEU:A hpphhppphhppphhpphphphhpphppphhpphhphhhpphhpphhpphh 51 B50-10 1CFH hpphphpphhphphpppphppppphpphpphhpppppppphhpphhp 47 Table A . 4 : B i o l o g i c a l sequences of length ss 50 ID E m i n (2D) h PERM *2 ACO tcitig B50-1 -12 5.60 3.90 4.60 39.5 B50-2 -19 650.40 0.90 1.80 283.31 B50-3 -20 3.80 0.04 0.08 2.47 B50-4 -17 3.33 2.24 2.68 184.5 B50-5 -22 4.90 117.80 9.41 820.37 B50-6 -14 0.43 0.74 0.55 2.08 B50-7 -17 270.70 298.80 284.06 130.35 B50-8 -14 0.04 2.16 0.07 0.29 B50-9 -21 0.40 0.23 0.29 55.32 B50-10 -14 0.27 0.3 0.28 3.22 Table A . 5 : Performance comparison of P E R M and A C O on b io log ica l sequences of length « 50 in 2 D Appendix A. Ant Colony Optimization 148 ID E m i n (3D) h PERM <2 texp ACO tav3 B50-1 -18 2.08 1.50 1.74 14.65 B50-2 -29 43.60 18.20 25.68 701.84 B50-3 -28 1.40 1.30 1.35 5.52 B50-4 -28 2.29 1 822.2 117.90 (-27) B50-5 -37 18.2 1 235.9 164.87 (-36) B50-6 -25 2.80 10.7 4.44 1 638.29 B50-7 -26 406.90 2.70 5.36 814.11 B50-8 -25 10.05 (-24) - 1026.12 B50-9 -36 3 492.6 4.03 16.10 (-35) B50-10 -22 2.50 1.40 1.80 200.68 Table A . 6 : Performance comparison of P E R M and A C O on b io log ica l sequences of length « 50 in 3 D ID HP sequence Length H fraction R30-1 pphphpphpphphppppphphhpphhhpph 30 0.40 R30-2 hpppppphhphhhpphppphhhhhhhhpph 30 0.53 R30-3 pphpphphppphhpppphhpphphphhhhh 30 0.47 R30-4 pphhhhppppphphhpphphpppppphphh 30 0.40 R30-5 hhppphhphpphhphhhpphphpphphpph 30 0.50 R30-6 phhpphphpppppphhhphppphppphhhh 30 0.43 R30-7 phhhphhhpppphhphphphhhppppppph 30 0.47 R30-8 ppphpphhphphhpphpphphhhphhphph 30 0.50 R30-9 hhphhpppphppphphpphpphpphhphph 30 0.43 R30-10 hphphpphpphphhppphphhhhphhphhp 30 0.53 Table A . 7 : R a n d o m sequences o f length 30 Appendix A. Ant Colony Optimization 149 ID Emin (2D) t\ PERM ti ACO tavg R30-1 -90 0.4 0.001 0.002 0.27 R30-2 -13 1.60 0.011 0.022 0.58 R30-3 -10 0.14 0.001 0.002 0.17 R30-4 -9 0.04 0.07 0.05 0.26 R30-5 -13 0.01 0.038 0.02 0.44 R30-6 -11 86.3 0.33 0.65 12.85 R30-7 -8 0.007 0.03 0.01 0.10 R30-8 -12 0.033 0.017 0.045 0.38 R30-9 -9 0.002 0.11 0.004 0.14 R30-10 -12 0.007 0.001 0.002 0.34 Table A . 8 : Performance comparison of P E R M and A C O on random sequences of length 30 in 2 D ID E7nin (3D) PERM *2 texp ACO tavg R30-1 -13 0.16 0.16 0.16 0.47 R30-2 -18 0.22 0.44 0.29 3.44 R30-3 -15 1.00 1.85 1.30 17.81 R30-4 -12 0.14 0.26 0.18 0.25 R30-5 -19 0.53 0.51 0.52 28.77 R30-6 -14 0.22 0.10 0.14 1.11 R30-7 -11 0.118 0.067 0.09 0.055 R30-8 -17 0.19 0.28 0.225 0.57 R30-9' -14 0.22 0.24 0.23 1.48 R30-10 -19 2.22 0.27 0.48 16.69 Table A . 9 : Performance comparison of P E R M and A C O on random sequences of length 30 in 3 D Appendix A. Ant Colony Optimization 150 ID HP sequence Length H fraction R50-1 ppphphphppppphphhhphhhpppphphphhppphphphhhhphpphhh 50 0.48 R50-2 hhhhhhphppphpppphhhpphhhhhphpphhhhpphppphppphhppph 50 0.52 R50-3 hhhphhphphhppphpphhhphhhhhhhphppphphhhphppphphpppp 50 0.54 R50-4 hhhhpphpphhpphhhpppppphpphhpppphhhhphhhpppphppphhh 50 0.48 R50-5 ppphhppphpphphpphhphphhhhhhhhpphhhpphpppphpppphphp 50 0.46 R50-6 pphpphphphhhpphpppppphpphhpppphphhhhphhphhhhpphhph 50 0.48 R50-7 phhppphphpphhpppphhhphphhhphhphhphphpphhhphphhhhhp 50 0.56 R50-8 hpppphphphpphphpphhhhhhpphhhhhphppphhpphphhpppphhp 50 0.50 R50-9 pphphhphhphpphpppphphphphphpphhpppppphpphhhhhpphhh 50 0.46 R50-10 hhphphpppphpphphppppphphppphpppppphhhphhhhhphhphph 50 0.44 Table A . 10: Random sequences of length 50 PERM A C O ID Emin (2D) t l i-exp tavg R50 rl -29 575.90 1.05 2.09 367.70 R50-2 -34 1,40 2.40 1.77 1 150.10 R50-3 -32 1.00 5.50 1.69 893.66 R50-4 -32 2.70 2.40 2.54 1 329.01 R50-5 -32 (-31) 5.49 - (-31) R50-6 -32 3.8 1.6 2.25 1 153.81 R50-7 -38 15 322.20 46.20 92.12 (-37) R50-8 -33 1.30 0.91 1.07 837.13 R50-9 -30 9892.00 1.70 3.40 1000.49 R50-10 -27 72.80 1.80 3.51 530.84 Table A . l 1: Performance comparison of P E R M and A C O on random sequences of length 50 in 2 D Appendix A. Ant Colony Optimization 151 PERM ACO ID E m i n (3D) ti R50-1 -29 575.90 1.05 2.09 367.70 R50-2 -34 1.40 2.40 1.77 1 150.10 R50-3 -32 1.00 5.50 1.69 893.66 R50-4 -32 2.70 2.40 2.54 1 329.01 R50-5 -32 (-31) 5.49 - (-31) R50-6 -32 3.8 1.6 2.25 1 153.81 R50-7 -38 15 322.20 46.20 92.12 (-37) R50-8 -33 1.30 0.91 1.07 837.13 R50-9 -30 9892.00 1.70 3.40 1000.49 R50-10 -27 72.80 1.80 3.51 530.84 Table A . 12: Performance comparison of P E R M and A C O on random sequences of length 50 in 3 D Appendix B 152 Adaptive Bin Framework Monte Carlo Search In this appendix, we report the necessary information to reconstruct the lowest energy conformations of homopolymers of length 12,24,32, and 64 obtained by our adaptive bin framework search, introduced for the Face-centered Cubic (FCC) /3-sheet protein folding problem. The homopolymer of length 64 Total energy = -391, short-range energy = -212, long-range energy = -179. The homopolymer of length 32 Total energy = —161, short-range energy = -112, long-range energy = —49. The homopolymer of length 24 Total energy = —109, short-range energy = —68, long-range energy = —41. The homopolymer of length 12 Total energy = -39, short-range energy = -28, long-range energy = -11. Appendix B. Adaptive Bin Framework Monte Carlo Search 153 vector[#] V, z) v[0] (1,1,0) v[l] (0,1,-1) v[2] (0,1,-1) v[3] (-1,0,-1) v[4] (-1, -1,0) v[5] (-1,-1,0) v[6] (0,-1,1) v[7] (0, -1,1) v[8] (1,0,1) v[9] (1,0,1) v[10] (1,1,0) v[ll] (1,1,0) v[12] (0,1,-1) v[13] (0,1, -1) v[14] (0,1,-1) v[15] (-1,0,-1) v[16] (-1,0, -1) v[17] (-1,-1,0) v[18] (-1,-1,0) v[19] (-1,-1,0) v[20] (0,-1,1) v[21] (0, -1,1) v[22] (0, -1,1) v[23] (1,0,1) v[24] (1,0,1) v[25] (0,1,1) v[26] (1,1,0) v[27] (1,1,0) v[28] (0,1,-1) v[29] (0,1,-1) v[30] (0,1,-1) v[31] (-1,0,-1) v[32] (-1,0, -1) v[33] (-1,-1,0) v[34] (-1,-1,0) v[35] (-1, -1,0)' v[36] (0, -1, i) v[37] (0, -1,1) v[38] (1,0,1) v[39] (1,0,1) v[40] (1,1,0) v[41] (1,1,0) v[42] (0,1,-1) v[43] (0,1,-1) v[44] (-1,0,-1) v[45] (-1,-1,0) v[46] (-1,-1,0) v[47] (0, -1,1) v[48] (1,1,0) v[49] (1,1,0) v[50] (1,0,-1) Table B. 1: Vectors for the best found conformation of 64 amino acids (total energy = —391, short-range energy = —212, long-range energy = —179). Appendix B. Adaptive Bin Framework Monte Carlo Search 154 vector[#] v[51] (1 ,0 , - 1 ) v[52] (1,1,0) v[53] (0 , - 1 ,1 ) v[54] ( - 1 , - 1 , 0 ) v[55] ( -1 ,0 ,1 ) v[56] ( -1 ,0 ,1 ) v[57] ( - 1 , - 1 , 0 ) v[58] (1 ,0 , - 1 ) v[59] (0 ,1 , - 1 ) v[60] (1 ,0 , - 1 ) v[61] ' (0 ,1 , - 1 ) v[62] (1,1,0) Table B.2: Continued, vectors for the best found conformation of 64 amino acids (total energy = -391, short-range energy = -212, long-range energy = -179). Appendix B. Adaptive Bin Framework Monte Carlo Search 155 triplet Ol 82 03 Ebeta " 0 , " 1 , " 2 120 180 60 -4 "1 , "2, "3 180 120 60 -4 "2 , "3 , "4 120 120 120 0 "3 , " 4 , V s 120 180 60 -4 " 4 , " 5 , " 6 180 120 60 -4 "5, "6 ,"7 120 180 60 -4 "0, "7 ,"8 180 120 60 -4 V7,Vg, Vc, 120 180 60 -4 Vg,09, "io 180 120 60 -4 "9,"10>"11 120 180 60 -4 "10,"11,"12 180 120 60 -4 "12, "13 120 180 60 -4 "12,"13 ,"14 180 180 0 -4 "13,"14, "15 180 120 . 60 -4 "14,"15,"16 120 • 180 60 -4 "15 ,"16 ,"17 180 120 60 -4 "16,"17,"18 120 180 60 -4 "17,"18 ,"19 180 180 0 -4 "18 ,"19 ,"20 180 120 60 -4 "10, "20, "21 120 180 60 -4 "20,"21,"22 180 180 0 -4 "21 , "22, "23 180 120 60 -4 "22, "23,"24 120 180 60 -4 "23 , "24,"25 180 120 60 -4 "24, "25,"26 120 120 60 -4 "25 ,"26 ,"27 120 180 60 -4 "20, "27,"28 180 120 60 -4 "27 ,"28 ,"29 120 180 60 -4 "28, "29,"30 180 180 0 -4 "29, "30, "31 180 120 60 -4 "30 ,"31 , "32 120 180 60 -4 "31,"32, "33 180 120 60 -4 "32,"33,"34 120 180 60 -4 "33,"34 ,"35 180 180 0 -4 "34,"35,"36 180 120 60 -4 "35,"36, "37 120 180 60 -4 "36,"37, "38 180 120 60 -4 "37,"38 ,"39 120 180 60 -4 "38 ,"39 ,"40 180 120 60 -4 "39, "40, "41 120 180 60 -4 "40, "41, "42 180 120 60 -4 "41 , "42, "43 120 180 60 -4 "42, "43, "44 180 120 60 -4 "43,"44 ,"45 120 120 120 0 "44, "45, "46 120 180 60 -4 "45, "46, "47 180 120 60 -4 "46, "47,"48 120 60 180 0 "47 ,"48 ,"49 60 180 120 0 "48,"49 , "50 180 120 60 -4 "49,"50,"51 120 180 60 -4 "50,"51 , "52 180 120 60 -4 Table B . 3 : Triplets of vectors, angles: 9\ (between vectors v ; _ i and v ; ) , 92 (be-tween vectors v ; and V j + i ) , 9s (between vectors v ; _ i and v i + 1 ) , short-range en-ergy contributions for the best found conformation of 64 amino acids (total energy = - 3 9 1 , short-range energy = - 2 1 2 , long-range energy = —179). Appendix B. Adaptive Bin Framework Monte Carlo Search 156 triplet 01 02 0.1 £be.ta "51,"52 "53 120 60 120 0 "52, "53 "54 60 120 180 0 "5.3, "54 "55 120 120 60 - 4 "54, "55 "56 120 180 60 - 4 "55, "56 "57 180 120 60 - 4 "56, "57 "58 120 60 180 0 "57, "58 "50 60 120 120 0 "58, "59 "60 120 120 0 - 4 "53, "60 "61 120 120 0 - 4 "60, "61 "62 120 120 60 - 4 Table B.4: Continued, triplets of vectors, angles: 6\ (between vectors Vj_x and V;) , 62 (between vectors Vj and V j + i ) , 63 (between vectors V i _ i and v; + 1X short-range energy contributions for the best found conformation of 64 amino acids (total energy = —391, short-range energy = —212, long-range energy = —179). Appendix B. Adaptive Bin Framework Monte Carlo Search 157 # Long-range interactions 1 10 nteracts with 1 2 11 nteracts with 1 3 12 nteracts with 1 4 12 nteracts with 2 5 13 nteracts with 2 6 14 nteracts with 2 7 14 nteracts with 3 8 15 nteracts with 3 9 15 nteracts with 4 10 16 nteracts with 4 11 17 interacts with 5 12 17 nteracts with 4 13 18 nteracts with 5 14 19 interacts with 6 15 19 interacts with 5 16 20 interacts with 7 17 20 interacts with 6 18 21 interacts with 7 19 22 interacts with 8 20 22 interacts with 7 21 23 interacts with 9 22 23 interacts with 8 23 24 interacts with 9 24 25 interacts with 9 25 25 interacts with 10 26 26 interacts with 10 27 26 interacts with 11 28 27 interacts with 11 29 28 interacts with 11 30 28 interacts with 12 31 29 interacts with 12 32 29 interacts with 13 33 30 interacts with 2 34 30 interacts with 13 35 30 interacts with 14 36 31 interacts with 3 37 31 interacts with 14 38 31 interacts with 15 39 32 interacts with 4 40 32 interacts with 15 41 32* interacts with 16 42 33 interacts with 5 43 33 interacts with 4 44 33 interacts with 17 45 34 interacts with 19 46 34 interacts with 5 47 34 interacts with 18 48 35 interacts with 20 49 35 interacts with 6 50 35 interacts with 19 51 36 interacts with 21 52 36 interacts with 7 53 36 interacts with 20 54 37 interacts with 22 55 37 interacts with 21 Table B . 5 : Long-range interactions for the best found conformation of 64 amino acids (total energy = - 3 9 1 , short-range energy = - 2 1 2 , long-range energy = - 1 7 9 ) . Appendix B. Adaptive Bin Framework Monte Carlo Search 158 # Long-range interactions 56 38 nteracts with 23 57 38 nteracts with 22 58 39 nteracts with 24 59 39 nteracts with 23 60 40 nteracts with 24 61 40 nteracts with 25 62 40 nteracts with 9 63 41 nteracts with 25 64 41 nteracts with 26 65 41 nteracts with 10 66 41 nteracts with 27 67 42 nteracts with 10 68 42 nteracts with 27 69 42 nteracts with 11 70 42 nteracts with 1 71 42 nteracts with 28 72 43 nteracts with 1 73 43 nteracts with 28 74 43 interacts with 12 75 43 interacts with 2 76 43 interacts with 29 77 43 interacts with 30 78 44 interacts with 2 79 44 interacts with 3 80 44 interacts with 30 81 44 interacts with 31 82 45 interacts with 3 83 45 interacts with 33 84 45 interacts with 4 85 45 interacts with 31 86 45 interacts with 32 87 46 interacts with 35 88 46 interacts with 6 89 46 interacts with 34 90 46 interacts with 5 91 46 interacts with 33 92 47 interacts with 36 93 47 interacts with 7 94 47 interacts with 35 95 47 interacts with 6 96 48 interacts with 38 97 48 interacts with 37 98 48 interacts with 22 99 48 interacts with 8 100 48 interacts with 36 101 48 interacts with 7 102 49 interacts with 39 103 49 interacts with 38 104 49 interacts with 23 105 49 interacts with 40 106 49 interacts with 9 107 49 interacts with 8 108 50 interacts with 48 109 50 interacts with 8 110 50 interacts with 47 Table B . 6 : Cont inued, long-range interactions for the best found conformation of 64 amino acids (total energy = - 3 9 1 , short-range energy = - 2 1 2 , long-range energy — —179). Appendix B. Adaptive Bin Framework Monte Carlo Search 159 # Long-range interactions 111 51 nteracts w th 47 112 51 nteracts w th 46 113 51 nteracts w th 44 114 51 nteracts w th45 115 52 nteracts w th6 116 52 nteracts w th 46 117 52 nteracts w th 5 118 52 nteracts vv th45 119 52 nteracts w th3 120 52 nteracts w lh4 121 53 nteracts w th 5 122 53 interacts w th4 123 54 interacts w th4 124 54 interacts w th 17 125 54 interacts w th 16 126 55 interacts w th 53 127 55 interacts w th 3 . 128 55 interacts w th4 129 55 interacts w th 15 130 56 interacts w th 52 131 56 interacts w th 53 132 56 interacts w th 3 133 57 interacts w th 1 134 57 interacts w th51 135 57 interacts w th 52 136 57 interacts w th 44 137 57 interacts w th 2 138 57 interacts w th3 139 58 interacts w th 50 140 58 interacts w th 42 141 58 interacts w th 1 142 58 interacts w th 51 143 58 interacts w lh 43 144 58 interacts w th 44 145 59 interacts w th 40 146 59 interacts w th 49 147 59 interacts w th9 148 59 interacts w th 41 149 59 interacts w th 10 150 59 interacts w th 50 151 59 interacts w th 42 152 60 interacts w th 9 153 60 interacts w th 8 154 60 interacts w th 10 155 60 interacts w th 50 156 60 interacts w th 58 157 60 interacts w th 1 158 60 interacts w th57 159 61 interacts w th 8 160 61 interacts w th 7 161 61 interacts w th 50 162 61 interacts w th 47 163 61 interacts w th 6 164 61 interacts w lh 51 165 61 interacts w th 57 Table B.7: Continued, long-range interactions for the best found conformation of 64 amino acids (total energy — -391, short-range energy = —212, long-range energy = —179). Appendix B. Adaptive Bin Framework Monte Carlo Search 160 # Long-range inleractions 166 61 interacts with 52 167 62 interacts with 6 168 62 interacts with 52 169 62 interacts with 56 170 62 interacts with 53 171 63 interacts with 6 172 63 interacts with 19 173 63 interacts with 5 174 63 interacts with 53 175 64 interacts with 5 176 64 interacts with 53 177 64 interacts with 18 178 64 interacts with 17 179 64 interacts with 54 Table B.8: Continued, long-range interactions for the best found conformation of 64 amino acids (total energy = —391, short-range energy = —212, long-range energy = —179). Appendix B. Adaptive Bin Framework Monte Carlo Search 161 vector [#] v[0] (1 ,0 , - 1 ) v[l] (1, - 1 , 0 ) v[2] (1, - 1 , 0 ) v[3] (0, - i , 1) v[4] ( -1 ,0 ,1 ) v[5] ( -1 ,0 ,1 ) v[6] ( -1 ,1 ,0 ) v[7] ( -1 ,1 ,0 ) v[8] (0,1, - 1 ) v[9] ( 0 ,1 , - 1 ) v[10] (1,0, - 1 ) v[ l l ] (1,0, - 1 ) v[12] (1 , - 1 ,0 ) v[13] (1, - i .o) v[14] (1, - 1 , 0 ) v[15] (0, - 1 , 1 ) v[16] (0, - 1 , 1) v[17] ( -1 ,0 ,1 ) v[18] ( -1 ,0 ,1 ) v[19] (0,1,1) v[20] ( -1 ,1 ,0 ) v[21] ( -1 ,1 ,0 ) v[22] (0,1, - 1 ) v[23] (0, 1, - 1 ) v[24] (1 ,0 , - 1 ) v[25] (1 ,0 , - 1 ) v[26] ( 1 , - 1 , 0 ) v[27] ( 1 , - 1 , 0 ) v[28] (0, - 1 , 1 ) v[29] (0, - 1 , 1) v[30] ( -1 ,0 ,1 ) Table B.9: Vectors for the best found conformation of 32 amino acids (total energy — -161, short-range energy = -112, long-range energy = -49). Appendix B. Adaptive Bin Framework Monte Carlo Search 162 triplet Ol 02 03 ^beta " 0 , " 1 , " 2 120 180 60 - 4 " 1 , " 2 , " 3 180 120 60 - 4 " 2 , " 3 , " 4 120 120 120 0 "3 , "4, "5 120 180 60 - 4 V4 , 'U 5 , Vc, 180 120 60 - 4 "5, "6,V7 120 180 60 - 4 V6,V7,VS 180 120 60 - 4 V7,vs,vg 120 180 60 - 4 " 8 , " 9 , " 1 0 180 120 60 - 4 "9,"10,"11 120 180 60 - 4 "10,"11,"12 180 120 60 - 4 "11,"12,"13 120 180 60 - 4 "12,"13,"14 180 180 0 - 4 "13, "14,"15 180 120 60 - 4 "14,"15,"16 120 180 60 - 4 "15 ,"16 ,"17 180 120 60 - 4 " 1 G , " 1 7 , " 1 8 120 .180 60 - 4 "17 ,"18 ,"19 180 120 60 - 4 "18', 'V19, "20 120 120 60 - 4 "19,"20,"21 120 180 60 - 4 "20,"21,"22 180 120 60 - 4 "21,"22 ,"23 120 180 60 - 4 "22,"23 ,"24 180 120 60 - 4 "23,"24 ,"25 120 180 60 - 4 "24, "25,"26 180 120 60 - 4 "25 ,"26 ,"27 120 180 60 - 4 "26,"27,"28 180 120 60 - 4 "27, "28, "29 120 180 60 - 4 "28,"29 , "30 180 120 60 - 4 Table B.10: Triplets of vectors, angles: 9\ (between vectors v;_! and v;), 6>2 (be-tween vectors v; and v ; + i ) , #3 (between vectors v ; _ i and v , + i ) , short-range en-ergy contributions for the best found conformation of 32 amino acids (total energy = -161, short-range energy = —112, long-range energy = —49). Appendix B. Adaptive Bin Framework Monte Carlo Search 163 # Long-range interactions 1 10 nteracts wi th 1 2 11 nteracts wi th 1 3 12 nteracts wi th 1 4 12 nteracts wi th 2 5 13 nteracts wi th 2 6 14 nteracts wi th 2 7 14 nteracts wi th 3 8 15 nteracts wi th 3 9 15 nteracts wi lh 4 10 16 nteracts wi th 4 11 17 nteracts wi th 5 12 17 nteracts wi th 4 13 18 nteracts wi th 5 14 19 nteracts wi th 6 15 19 nteracts wi th 5 16 20 nteracts wi th 7 17 20 nteracts wi th 6 18 21 nteracts wi th 7 19 22 interacts wi th 8 , . 20 22 nteracts wi th 7 21 23 interacts wi th 9 22 23 interacts wi th 8 23 24 interacts wi th 9 24 24 interacts wi th 10 25 25 nteracts wi th 10 26 25 interacts wi th 11 27 25 interacts wi th 1 28 26 interacts wi th 1 29 26 interacts wi th 12 30 26 interacts wi th 2 31 27 interacts wi th 2 32 27 interacts wi th 13 33 27 interacts wi th 14 34 28 interacts wi th 3 35 28 interacts wi th 14 36 28 interacts wi th 15 37 29 interacts WI th 4 38 29 interacts WI th 15 39 29 interacts Wl th 16 40 30 interacts Wl th 5 41 30 interacts Wl th 4 42 30 interacts Wl th 17 43 31 interacts Wl th 19 44 31 interacts Wl th 5 45 31 interacts Wl th 18 46 32 interacts Wl th 20 47 32 interacts Wl th 6 48 32 interacts w th 19 49 32 interacts w th 21 Table B . l 1: Long-range interactions for the best found conformation of 32 amino acids (total energy = —161, short-range energy = —112, long-range energy — - 4 9 ) . Appendix B. Adaptive Bin Framework Monte Carlo Search 164 vector [#] (•x, y, z) v[0] (1 ,0 , -1 ) v[l] ( 0 , - 1 , - 1 ) v[2] ( 0 , - 1 , - 1 ) v[3] (1 , -1 , 0 ) v[4] ( 0 , - 1 , - 1 ) v[5] ( 0 , - 1 , - 1 ) v[6] ( - 1 , - 1 , 0 ) v[7] ( - 1 , 0 , - 1 ) v[8] (0,1,1) v[9] (1,0,1) v[10] (0,1,1) v [ l l ] (0,1,1) v[12] (0,1,1) '[131 (1,1,0) v[14] (0,1,1) v[15] ( - 1 , - 1 , 0 ) v[16] ( - 1 , 0 , - 1 ) v[17] ( 0 , - 1 , - D v[18] ( 0 , - 1 , - D v[19] (1 ,0 , -1 ) v[20] ( 0 , - 1 , - 1 ) v[21] ( 0 , - 1 , - 1 ) v[22] (1 , -1 , 0 ) Table B . 12: Vectors for the best found conformation o f 24 amino acids (total energy = - 1 0 9 , short-range energy = —68, long-range energy = - 4 1 ) . Appendix B. Adaptive Bin Framework Monte Carlo Search 165 triplet 01 02 03 " 0 , " 1 , " 2 120 180 60 - 4 " 1 , " 2 , " 3 180 120 60 - 4 " 2 , " 3 , "4 120 120 0 - 4 "3 , "4 , "5 120 180 60 - 4 V4 j VTj j " 6 180 120 60 - 4 " 5 , " G , " 7 120 120 60 - 4 " 6 , " 7 , " 8 120 60 120 0 V7,vg,Vo 60 120 180 0 " 8 , " 9 , "10 120 120 0 - 4 " 9 , " 1 0 , "11 120 180 60 - 4 " 1 0 , " 1 1 , " 1 2 180 180 0 - 4 "11,"12, "13 180 120 60 - 4 " 1 2 , " 1 3 , " 1 4 120 120 0 - 4 " 1 3 , " 1 4 , " 1 5 120 60 180 0 " 1 4 , " 1 5 , " 1 6 60 120 120 0 "15, "16, "17 120 120 60 - 4 ' " 1 6 , " 1 7 , " 1 8 120 180 60 - 4 " 1 7 , " 1 8 , " 1 9 180 120 60 - 4 "18, " 1 9 , " 2 0 120 120 0 - 4 "19, " 2 0 , "21 120 180 60 - 4 " 2 0 , " 2 1 , " 2 2 180 120 60 - 4 Table B . 1 3 : Triplets of vectors, angles: Q\ (between vectors v;_i and Vj), 92 (be-tween vectors v; and v; + 1), 63 (between vectors v;_i and v; +i), short-range en-ergy contributions for the best found conformation of 24 amino acids (total energy — - 1 0 9 , short-range energy = - 6 8 , long-range energy = - 4 1 ) . Appendix B. Adaptive Bin Framework Monte Carlo Search 166 # Long-range interactions 1 10 nteracts w th8 2 11 nteracts w th 8 3 11 nteracts w th 7 4 11 nteracts w th6 5 12 nteracts w th6 6 12 nteracts w th 5 7 13 nteracts w lh 4 8 13 nteracts w th 5 9 14 nteracts w th3 10 15 nteracts w th 3 11 15 interacts w th2 12 16 interacts w th2 13 17 interacts w th 14 14 17 interacts w th 1 15 17 interacts w th 2 16 17 interacts w th 15 17 18 interacts w th 14 18 18 interacts w th 3 19 18 interacts w th 1 20 18 interacts w lh 2 21 19 interacts w th 13 22 19 interacts w th 4 23 19 interacts w th 14 24 19 interacts w th 3 25 20 interacts w th 12 26 20 interacts w th 13 27 • 20 interacts w th4 28 21 interacts w th 12 29 21 interacts w th 6 30 21 interacts w th 4 31 21 interacts w th 5 .32 22 interacts w th 10 33 22 interacts w th 11 34 22 interacts w th7 35 22 interacts w lh 6 36 23 interacts w th 9 37 23 interacts w th 10 38 23 interacts w th 8 39 23 interacts w th7 40 24 interacts w th8 41 24 interacts w th7 Table B.14: Long-range interactions for the best found conformation of 24 amino acids (total energy = - 109 , short-range energy = -68 , long-range energy = -41). Appendix B. Adaptive Bin Framework Monte Carlo Search 167 vector [if] v[0] (-1,-1,0) v[l] (0,-1,1) v[2] (0,-1,1) v[3] (1,-1,0) v[4] (1,-1,0) v[5] (1,0,-1) v[6] (-1,1,0) v[7] (-1,1,0) v[8] (0 ,1 , -D v[9] (0 ,1 , -D vflO] (-1,1,0) Table B.15: Vectors for the best found conformation of 12 amino acids (total energy = —39, short-range energy = —28, long-range energy = —11). triplet 01 02 03 v0,vi,v2 120 180 60 -4 "1 , V2, l>3 180 120 60 —4 V2,V3,V4 120 180 60 -4 " 3 , " 4 , " 5 180 120 60 -4 " 4 , " 5 , " 6 120 60 180 0 " 5 , " 6 , " 7 60 180 120 0 " 6 , " 7 , " 8 180 120 60 -4 Vj, Vg,VQ 120 180 60 -4 Vg,Vg,Vl0 180 120 60 -4 Table B.16: Triplets of vectors, angles: 6\ (between vectors Vj_! and v;), # 2 (be-tween vectors v; and v; + 1), 83 (between vectors v;_i and v; +i), short-range en-ergy contributions for the best found conformation of 12 amino acids (total energy = —39, short-range energy = -28 , long-range energy = -11). # Long-range interactions 1 8 interacts with 5 2 8 interacts with 6 3 9 interacts with 4 4 9 interacts with 5 5 9 interacts with 3 6 10 interacts with 3 7 10 interacts with 2 8 11 interacts with 2 9 11 interacts with 1 10 12 interacts with 2 11 12 interacts with 1 Table B.17: Long-range interactions for the best found conformation of 12 amino acids (total energy = —39, short-range energy = -28 , long-range energy = —11). 168 Index <D-value analysis an experimental technique that relies on use of a mutation as a reporter of structural change; a mutation in the protein molecule is engi-neered which causes a difference in stability between the mutant and wild-type; structural changes are usually measured indirectly by recording the folding speed, 24 a-helix a common motif in the secondary structure of proteins, a right-handed coiled conformation in which hydrogen bonds are formed between the car-bonyl oxygen of the amino acid residue at position n with the amide group, NH, of residue n + 4, 3 0-sheet a commonly occurring form of regular secondary structure in proteins, it consists of a stretch of amino acids whose peptide backbones are almost fully extended, resulting in an elongated pleat-like structure in which the peptide carbonyls point in alternating directions relative to the plane of the sheet, 3 4> dihedral angle the angle specifying rotation around the Ca-N axis in proteins, 2 •ip dihedral angle the angle specifying rotation around the Ca-C axis in proteins, 2 ab initio (physics) from first principles; a calculation is said to be ab initio (or from first principles) if it relies on basic and established laws of nature without additional assumptions or special models, 7 aliphatic (chemistry) organic compounds in which carbon atoms form open chains (straight or branched), not aromatic rings, 2 amino acid any molecule that contains both amine and carboxylic acid functional groups, 1 amino group functional group composed of a nitrogen and two hydrogen atoms NH2 covalently linked, in the process it gives the free electron pair of the nitrogen atom to a proton, and turns into positively charged NHf, 1 Index 169 A n t Co lony O p t i m i z a t i o n ( A C O ) a population-based stochastic search method inspired by real ant colonies; it is a construction-based search method based on the pheromone information accumulated over previous runs of the search and the heuristic function that provides information about the desirability of the solution component considered, 47 a r o m a t i c (chemistry) having an unsaturated ring, 2 a t t r i t i o n a condition when a chain runs into itself during the chain growth process, 36 backbone (of a p ro te in ) the main chain of proteins, composed of N, C a, C, O atoms, 1 beta sheet energy potent ia l an energy function used to fold /3-sheets; according to this potential residues are classified as either being in an extended beta state or not; for every residue in the extended state there is a favourable en-ergy contribution; additionally, contact (long-range) energy is usually present in the beta sheet potential, 18 C - t e r m i n u s the extremity of the amino acid chain terminated by a free carboxyl group, 2 canon ica l s i m u l a t i o n same as equilibrium simulation, 30 c a r b o x y l g r o u p functional group composed of COOH, in the process it dissoci-ates into COO~ and a proton H +, 1 C e n t r a l D o g m a o f genomics principle believe that asserts "sequence determines structure determines function", 6 c h a p e r o n (biology) proteins whose function is to assist other proteins in achieving proper folding, 6 coarse s a m p l i n g o f energy landscapes broad exploration of the search landscape that includes sampling of all of the "relevant" parts of the search space, 26 complex search landscapes search landscapes arising in complex systems with many degrees of freedom such as spin glass and biomolecular systems such as proteins, 26 c o n f o r m a t i o n a l search space a set of all possible conformations for a molecule, 15 Index 170 construction-based search the search process in which the search starts from an empty candidate solution and solution components are iteratively added until a complete candidate solution is constructed, this process is usually repeated multiple times, 35 contact order (CO) the sequence separation between residues in contact, 24 cubic lattice a three dimensional lattice that has three base vectors perpendicular to each other, 16 decoys (a set of) a mixture of native and non-native protein folds used for testing the ability of energy potentials to discriminate between native and non-native folds, 23 density of states a property in statistical and condensed matter physics that quan-tifies how closely packed energy levels are in a particular physical system, 31 detailed balance condition a condition that ensures that the rate at which the sys-tem transitions into and out of any state are equal; the correct distribution for a physical system in thermal equilibrium is the Boltzmann distribution, 30 diffusion-collision model a model that proposes that local elements of native sta-ble secondary structure form independently of tertiary structure; they collide and adhere to form tertiary interactions, 44 diversification an important property of the search process, that concentrates on further exploration of the search space in order to avoid search stagnation (or entrapment in a local optimum), 39 domain {biochemistry) a "folding unit" of a protein, the part of the protein se-quence that folds largely independently of the rest of the sequence, 7 effective contact order (ECO) effective contact order of a newly-added contact is defined as the effective loop closure size (the number of steps, covalent and non-covalent links taken along the shortest path on the polymer graph), given that other contacts have been formed, 24 energy potential (physics) same as potential energy; energy which depends on mutual positions of bodies, 15 Index 171 energy barriers an increase in the objective function value that search has to over-come in order to move from the current candidate solution to another candi-date solution with the same or lower (in the case o f the min imiza t ion prob-lem) objective function value, 27 Energy Landscape Paving (ELP) adaptive stochastic local search algori thm clas-sified as a generalized ensemble method in which the barrier height is de-creased proportionally to the time the system stays in the min ima (configu-rations are searched with time-dependent weights that take into account both the Bol tzmann Probabi l i ty and time spend in a particular local min ima) , 34 entropy (physics) a measure of randomness or disorder o f a system, 109 ergodicity (or convergence) the process reaches an invariable (stationary) distri-bution in the l imi t , 29 Evolutionary Algorithm (EA) population-based stochastic local search that is in -spired by the process o f evolution according to selection rules (usually has operations o f mutations and recombination, 40 face-centered cubic (FCC) lattice the usual lattice structure for the majority of crystall ine metals, it has 12 base vectors, 17 folding nucleus disjointed structure composed of protein residues that are impor-tant for the process o f folding, 109 funneled energy landscape the view of protein folding that proteins evolved to have the native state at the base o f the search landscape funnel, as a conse-quence o f the min imal frustration; it explains how most proteins fold effi-ciently and robustly to their functional native state and it a l lows robust pre-dict ion of folding kinetics, 28 generalized ensemble Monte Carlo methods simulations in a generalized ensem-ble that strive to perform a random walk in potential energy space; the advan-tage of these methods is that from only one simulat ion run, one can obtain canonical ensemble averages of physical quantities as functions o f temper-ature by the single-histogram and/or multiple-histogram re-weighting tech-niques, 31 Genetic Algorithm (GA) population-based stochastic local search algori thm (a type o f Evolut ionary Algor i thm) in which candidate solutions are encoded as genes (vectors o f integers), and search strategy involves mutation (local Index 111 neighbourhood moves) and cross-over (among two or more candidate solu-tions) moves, 35 global optimum a selection from a given domain which yields either the highest value or the lowest value (depending on the objective: either to minimize or to maximize), when a specific objective function is applied, 26 globular proteins one of the three main protein classes, comprising globe-like proteins that are more or less soluble in aqueous solutions (where they form colloidal solutions); this main characteristic helps distinguishing them from fibrous and membrane proteins (the other classes), which are practically in-soluble, 3 Hamming distance the Hamming distance between two strings of equal length is the number of positions for which the corresponding symbols are different; thus, it measures the number of substitutions required to change one into the other, 79 Hoyle paradox an inquiry asking how foldable proteins evolved within the life-time of the universe, 5 hydrogen bond hydrogen bonding occurs when an atom of hydrogen is attracted by strong forces to two atoms instead of only one, so that it may be consid-ered to be acting as a bond between them; this typically occurs where the partially positively charged hydrogen atom lies between partially negatively charged oxygen and nitrogen atoms; although stronger than most other in-termolecular forces, the typical hydrogen bond is much weaker than both the ionic bond and the covalent bond, 3 hydrogen-deuterium exchange (H/X) a chemical reaction in which a covalently-bonded hydrogen atom is replaced by a deuterium atom, or vice versa; this method gives information about the solvent accessibility of various parts of the molecule, and thus the three-dimensional structure of the protein, 45 hydrophilic having an affinity for water, 1 hydrophobic having an aversion of water, 1 hydrophobic collapse model a model that suggests that a protein rapidly col-lapses around its hydrophobic side-chains and then rearranges from the re-stricted conformational space, 44 Index 173 Hydrophobic Polar (HP) model a model of protein representation that classifies 20 amino acids as ether being hydrophobic or polar, 16 Importance Sampling stochastic local search and sampling technique in which important values are emphasized by sampling more frequently; this estimator achieves reduction of variance; the basic methodology in Importance Sam-pling is to choose a distribution which encourages sampling of the important values more often, 38 intensification an important property of the search process, that concentrates the search on the best candidate solutions (or solution components) found so far, and often uses "greedy" search strategies, 38 kinetic control (chemistry) kinetic reaction control means that the reverse reaction does not occur or is slow; under kinetic control a product may be formed that is less stable but this product is formed faster because the activation energy for this reaction is lower, 6 knowledge-based potential an energy potential that is based on quantities derived from a data base of known structures, 22 lattice (physics) a physical model that is not defined on a continuum, but instead defined on a grid, which is a graph or an n-complex approximating space, 14 lattice moves allowed move set on a lattice, 15 learning-based potential an energy potential derived by maximizing the differ-ence between correct and incorrect structures using machine learning and linear optimization techniques, 22 Levinthal paradox an inquiry asking how can a protein find its low-energy state in time less than geological, since proteins have an astronomical number of possible conformations, 5 local minimum a candidate solution whose objective function value is smaller or equal to the objective function values of any candidate solutions in its neighbourhood, 27 long-range potential an energy contribution resulting from interaction of distant parts of the chain (contact-energy), 19 Index 174 m a r g i n a l s tab i l i t y (Hon ig ) p a r a d o x an inquiry that deals with the issue of why proteins seem to have such a delicate compensation of entropy and energy during intermediate stages of the folding process, 6 M a r k o v process the process in which the probability of transitioning between the current and a new state is only dependent on the current state and the energy difference between the current and the new state, 29 m o d e l (in protein folding) the representation of a protein, 15 model -based search search method that builds a parametric or a non-parametric model (an adaptive stochastic mechanism) of the search space that is updated during the search and the new candidate solutions are generated using the • model, 36 M o l e c u l a r D y n a m i c s ( M D ) molecular modeling technique that addresses numer-ical solutions of Newton's equations of motion on an atomistic or similar model of a molecular system to obtain information about its time-dependent properties, 26 m o l t e n g lobule a stable, partially folded protein state found in mildly denaturing conditions such as low pH (generally pH = 2), mild denaturant, or high tem-perature; they are collapsed and generally have some native-like secondary structure but a dynamic tertiary structure; molten globules often are stable intermediate states during folding, 27 m o n o m e r a small molecule that may become chemically bonded to other monomers to form a polymer, 16 M o n t e C a r l o ( M C ) m e t h o d a widely-used class of stochastic local search algo-rithms for simulating the behavior of various physical systems; they are dis-tinguished from other stochastic local search methods by making transitions between candidate solutions according to the Boltzmann probability and by the fact that the transition probability is only dependent by the energy differ-ence between the current and a new state, 29 move set a set of allowed moves; each move (search step) usually involves the modification, addition or removal of one or more solution components, 15 M u l t i c a n o n i c a l A l g o r i t h m ( M U C A ) stochastic local search that belongs to a class , of generalized ensemble methods where the transition probability between states is determined by the estimated density of states for a particular prob-lem instance, 31 Index 175 N - terminus refers to the extremity of a protein or polypeptide terminated by an amino acid with a free amine group, 2 native state the native state of a protein is its operative or functional form, 6 neighbourhood relation an important component of a local search algorithm, a relationship that defines the direct neighbours of the current candidate solu-tion, 39 non-model-based search search method that generates new candidate solutions using solely the current solution or the current population of solutions, 36 non-native (biochemistry) not a functional state of a protein, opposite to native, 15 NP (non-deterministic polynomial time) is the set of decision problems solvable in polynomial time on a non-deterministic Turing machine; equivalently, it is the set of problems that can be verified by a deterministic Turing machine in polynomial time, 8 NP-hard (non-deterministic polynomial-time hard) refers to the class of decision problems that contains all problems H, such that for every decision problem L in NP there exists a polynomial-time many-one reduction to H, written L <p H; informally, this class can be described as containing the decision problems that are at least as hard as any problem in NP, 8 nuclear magnetic resonance (NMR) spectroscopy the principal techniques used to obtain physical, chemical, electronic and structural information about a molecule in solution; it uses high magnetic fields and radio-frequency pulses to manipulate the spin states of nuclei, 9 nucleation model a model that suggests that proteins have a small set of interac-tions (folding nucleus) common to most of the conformations in the transi-tion state ensemble, 44 objective function an evaluation function that assigns a numerical value to each candidate solution, 24 off-lattice continuous model, or discrete model that does not use lattices, 15 peptide the family of short molecules formed from the chain linking of various amino acids, 2 Index 176 peptide bond a chemical bond formed between two molecules when the carboxyl group of one molecule reacts with the amino group of the other molecule, releasing a molecule of water, 2 physical (empirical) potential an energy potential that is based on empirical mea-surements of interactions between atoms and/or molecules, 22 polar (of a compound) having an electric charge, 1 polypeptide chain of amino acids; proteins are made up of one or more polypep-tide molecules; the amino acids are linked covalently by peptide bonds, 4 primary structure (biochemistry) linear sequence of amino acids in a given pro-tein, 3 prion an abbreviation for proteinaceous infectious particle, it is a unique type of infectious agent, made only of protein, 6 protein chain-like polymer of small subunits (amino acids), 1 protein folding problem problem of prediction of protein tertiary structure from its amino acid sequence for a given energy function, 14 Pruned-Enriched Rosenbluth Method (PERM) stochastic local search that is based on Sequential Importance Sampling; additionally, it performs re-samplin after pruning and enrichment, this is a construction-based search method, 36 quasi ergodicity a situation when trajectories between structurally diverse but sta-tistically important energy minima have low (close to zero) transition prob-abilities that can result in the search or sampling process getting trapped in isolated minima; thus, the simulation may not necessarily reach the equilib-rium and may not converge, 30 quaternary structure (biochemistry) an arrangement of multiple folded protein molecules in a multi-subunit complex, 4 R-group a side-chain; a part of an amino acid that is attached to a central alpha-carbon, 1 Ramachandran plot a plot of permissible dihedral angles <p vs. ip, 2 random walk a process of taking successive steps, each in a random direction, 31 Index 111 random coil a polymer conformation where the monomer subunits are oriented randomly while still being bonded to adjacent units, 3 Replica Exchange Monte Carlo (REMC) stochastic local search algorithm that belongs to a class of generalized ensemble methods; in this method a number of Monte Carlo simulations are performed at different temperatures between Tiow to Thigh, and simulations (sampling processes) with nearby tempera-tures attempt to exchange temperatures every specified number of steps; the exchange between simulations as well as within simulation is performed ac-cording to the Boltzmann probability, 33 reptation a snake-like motion of the polymer happening by diffusion of stored length along its own contour, 38 residue (chemistry) a portion of a large molecule, such as protein; also designates an amino acid in the protein folding literature, 2 root mean square deviation (RMSD) (as applied to biological structures) the mea-sure of the average distance between the backbones of superimposed pro-teins, 9 rugged landscape (ruggedness) a property of a search landscape that describes the fluctuation of the objective function values of candidate solutions that are neighbours according to the neighbourhood relation chosen, 27 run-time distribution probability distribution of the time required by a stochastic algorithm to solve a given problem instance, 71 sampling a process whose goal is to generate a correct ensemble (usually defined by the Boltzmann probability in physical systems), 26 search space a set of all candidate solutions (conformations) for a given problem instance, 20 search landscape a mathematical concept consisting of the search space, the neigh-bourhood relation and the objective function for a given problem instance; a search landscape is characterized by such properties as: local minima distri-bution, global minima distribution, ruggedness, 26 search methods algorithms used to search the space of all possible candidate so-lutions, 26 searching a process whose goal is to find a configuration or a set of configurations with a required property (usually a global energy minimum), 26 Index 178 secondary structure (biochemistry) local structure denned by the hydrogen bonds of the biopolymer; in proteins, the secondary structure is denned by patterns of hydrogen bonds between backbone amide groups; in nucleic acids, the secondary structure is defined by the hydrogen bonding between the nitroge-nous bases, 3 self-avoidance (for a chain) a state of non-collision with itself, 15 Sequential Importance Sampling stochastic local search method and sampling technique similar to Importance Sampling, the only difference being that weights that control transition probabilities between states are computed sequentially as candidate solutions are built (this is a construction-based search), 38 short-range potential an energy contribution resulting from interaction of neigh-bouring amino acids, 18 Simulated Annealing(SA) a stochastic local search algorithm that is inspired by the process of crystallization in physics; it follows the Boltzmann acceptance probability but unlike in Monte Carlo methods the temperature is lowered according to a specified cooling (annealing) schedule, 28 square lattice one of the five two-dimensional lattice types composed of upright squares, 16 state-of-the-art methods best performing methods for a problem of interest, 16 steric hindrance an affect arising from the fact that each atom within a molecule occupies a certain amount of space; if atoms are brought too close together, there is an associated cost in energy due to overlapping electron clouds, and this may affect the molecule's preferred shape (conformation), 2 stigmergy a method of communication in multi-agent systems in which the indi-vidual agents communicate with one another by modifying their local envi-ronment, 47 stochastic local search (SLS) local search algorithm that generates and/or selects candidate solutions (or partial candidate solutions - components of a solu-tion) using randomization techniques, 26 Tabu Search stochastic local search method that exploits memory of the search history for guiding the search, 34 Index 179 tertiary structure (biochemistry) the tertiary structure of a protein is its overall shape, also known as its fold, 4 Thermodynamic Hypothesis a hypothesis that states that the native state repre-sents the ground state of lowest Gibbs free energy, 6 topology of the search space a distribution of various properties for all of the pos-sible candidate solutions of a problem (for example, the distribution of ob-jective function values, their connectedness), 26 transition state the transition state of a chemical reaction is a particular configu-ration along the reaction coordinate; it is defined as the state corresponding to the highest energy along the reaction coordinate, 24 Umbrella Sampling stochastic local search and sampling method in which an ad-ditional biasing potential is added during the simulation to the canonical simulation (usually a particular umbrellas requires knowledge of the system or the exact energy surface in some cases, the only umbrella that does not require this is temperature), 37 X-ray crystallography a technique in crystallography in which the pattern pro-duced by the diffraction of X-rays through the closely spaced lattice of atoms in a crystal is recorded and then analyzed to reveal the three-dimensional shape of a molecule, 9 

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