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Energetic and conformational studies of nonspecific adsorption of simple protein-like chain molecules… Liu, Susan Marisa 2004

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ENERGETIC AND CONFORMATIONAL STUDIES OF NONSPECIFIC ADSORPTION OF SIMPLE PROTEIN-LIKE CHAIN MOLECULES USING DYNAMIC MONTE CARLO SIMULATIONS by Susan Marisa Liu B.Eng., McGill University, 1993 MA.Sc, The University of British Columbia, 1997 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES The Department of Chemical and Biological Engineering and The Michael Smith Laboratories We accept this thesis as conforming to thereouired standard  THE UNIVERSITY OF BRITISH COLUMBIA October 2004 © Susan Marisa Liu, 2004  Abstract Dynamic Monte Carlo simulations of short HP (hydrophobic-polar) protein-like chains to solid-liquid surfaces are used to probe thermodynamic and dynamic aspects of protein adsorption. The HP model enables the enumeration of all chain conformations, thereby aiding understanding of the relation between adsorption thermodynamics and changes in accessible chain conformations resulting from the sorption process.  Simulation results  indicate that HP chains having a single conformation at their lowest energy in solution adsorb such that the new lowest energy state of the system is conformationally degenerate.  As a result, adsorption can lead to an increase in chain entropy.  Entropically-driven adsorption is found to be likely when the interaction energy between the hydrophobic segments of the chain and the sorbent is weak and equals the contact energy between two hydrophobic units within the chain. Chain sequence and sorbent properties are shown to profoundly influence adsorption thermodynamics.  Simulations are carried out where  intra- and intermolecular  hydrophobic interaction energies are varied to examine the influence of the stability of the native-state conformation on adsorption thermodynamics over a range of sorbent hydrophobicities. Lower stability chains tend to adsorb more readily on hydrophilic sorbents and experience greater average changes in conformation, usually accompanied by a loss in entropy. Adsorption to more hydrophobic sorbents leads to a loss in chain conformational entropy, irrespective of the stability of the native state. Lateral confinement on the sorbent surface is shown to greatly reduce the degrees of freedom in the chain, thereby resulting in a strong stabilization of the native-state conformation of the chain in its adsorbed state. This effect is compared to experimental  ii  data for nonspecific adsorption of hen egg-white lysozyme to silica to explain the increase in adsorbed enzyme activity as a function of surface loading and geometry. Studies of run-averaged energy trajectories for chain adsorption indicate that the process follows a basic energy path characterized by well-defined energy levels, suggesting the presence of natural kinetic barriers. This thesis demonstrates the value of simple mesoscopic protein-like chain models and dynamic Monte Carlo simulations of their adsorption behavior in understanding better the mechanisms and forces driving nonspecific protein adsorption.  iii  Table of contents  Abstract  ii  Table of contents  iv  List of tables  vii  List of figures  viii  List of nomenclature  xi  Main symbols  xi  Superscripts  xii  Subscripts  xiii  Acknowledgements 1  xiv  Introduction, background and objectives 1.1  Literature review . 1.1.1 General aspects of proteins 1.1.2 Driving forces for nonspecific protein adsorption 1.1.3 Kinetics of nonspecific protein adsorption 1.1.4 Simulating and modeling protein adsorption  1 5 5 12 16 17  1.2  Thesis objectives  21  1.3  References  23  1.4  Tables  1.5  Figures  ;  31 32  2 Mesoscopic analysis of conformational and entropic contributions to nonspecific adsorption of HP copolymer chains using dynamic Monte Carlo simulations 36 2.1  Introduction  36  2.2 Theory 2.2.1 Dynamic Monte Carlo simulations of HP-model polymers 2.2.2 The model system 2.2.3 Calculation of thermodynamic parameters  42 42 45 47  2.3 Results 2.3.1 Dynamics and thermal unfolding behaviour of model sequences 2.3.2 Chain adsorption to the surface 2.3.3 Thermodynamics of chain adsorption  49 49 51 54 iv  2.3.4 2.3.5  Influence of sorbent surface geometry Dependence on the total hydrophobicity  56 57  2.4 Discussion 58 2.4.1 Protein adsorption kinetics are linked to energetic barriers that frustrate conformational trajectories of the peptide chain 62 2.5  Summary  65  2.6  References  67  2.7  Tables  72  2.8  Figures  77  3 Mesoscopic dynamic Monte Carlo simulations of the adsorption of protein-like HP chains within laterally constricted spaces 85 3.1  Introduction  85  3.2  Dynamic Monte Carlo simulations of chain adsorption  88  3.3 Materials and Methods 3.3.1 Reagents 3.3.2 Measurement of Adsorption Isotherms 3.3.3 Lysozyme activity measurements 3.3.4 Isothermal Titration Calorimetry  92 92 93 95 96  3.4 Results and discussion 97 3.4.1 dMC simulations of HP chain adsorption within confined spaces 97 3.4.2 Influence of lateral confinement on the thermodynamics of HP chain adsorption 103 3.5  Summary  107  3.6  References  109  3.7  Tables  112  3.8  Figures  113  4 Energy landscapes for adsorption of protein-like HP chains as a function of nativestate stability 122 4.1  Introduction  122  4.2  Protein-like HP chain and dMC simulation algorithm  125  4.3  Results and discussion  128  4.3.1 4.3.2 4.3.3  .•  HP chain adsorption thermodynamics 128 Thermally-averaged energy landscape analysis of HP chain adsorption 133 The deformation entropy for HP chain adsorption 137  4.4  Summary  142  4.5  References  144  4.6  Tables  4.7  Figures  5  :  Conclusion 5.1  References  Appendix  147 152 157 161 162  Description of dynamic Monte Carlo Simulations and related program code The lattice grid The HP chain The algorithm Ending the program  162 . 162 163 164 165  Program files Main program <*.for> Input file <*.daO .• Output files <*.ow/>, <Screen>, etc  166 166 167 167  Editing, compiling and running the program Running a single program Running a collection of programs in series  168 168 169  Example program for dynamic Monte Carlo simulation, CONTACT9 Main program, <contact9.for> Input data, <test.dat> : Output files  170 171 219 228  vi  List of tables Table 1.1: Interactions governing the native-state structural stability of globular proteins. AN-DG refers to the Gibbs energy of denaturation for the protein 31 Table 2.1: Thermodynamic changes for the native to denatured state transition of sequence I at reduced temperatures T* = 0.44 and T* = 0.71 72 Table 2.2: Thermodynamic changes for the adsorption of sequence I to a relatively hydrophilic surface (XHW -1) as a function of chain stability (%HH) 73 =  Table 2.3: Thermodynamic changes for the adsorption of denatured sequence I (%HH = 0.25) on a hydrophobic surface (XHW -4) 74 =  Table 2.4: Thermodynamic changes for the adsorption of sequence I on surfaces of varying geometry with XHH = XHW = -4 75 Table 2.5: Thermodynamic changes for the adsorption of sequence I to a planar surface when XHH = XHW-  76  Table 3.1: Molar enthalpy change AH d as a function of percent sorbent surface coverage for the adsorption of H E W L to particulate silica in 50-mM KCI (pH 7) at 37°C. AH ds is expressed on a per mole of H E W L adsorbed basis 112 a  s  a  Table 4.1: Adsorption thermodynamics for HP chain sequence I as a function of nativestate stability and sorbent surface affinity 147 Table 4.2: The average number of favorable intramolecular and intermolecular contacts formed by sequence I in solution and when adsorbed onto a planar surface 148 Table 4.3: Adsorption thermodynamics for sequence II as a function of native-state stability and sorbent surface affinity 149 Table 4.4: The average number of favorable intramolecular and intermolecular contacts formed by sequence II in solution and when adsorbed onto a planar surface 150 Table 4.5:  The density of unique conformational states for the adsorbed chain at the  global energy minimum energy state as a function of  XHHIXHW  151  vii  List of figures Figure 1.1: The basic chemical structure of an amino acid and schematic diagrams of the 20 naturally-occuring amino acids 32 Figure 1.2: acids  The condensation reaction forming the peptide link between two amino : 34  Figure 1.3: Isotherm for the adsorption of hen egg-white lysozyme (pl~l 1) to particulate silica at pH 7 and 37°C. Shown is the concentration of protein on the surface as a function of protein in bulk solution. Data points collected from two separate trials are indicated. The solid line represents the ascending isotherm, while dotted lines represent descending isotherms. The vertical dashed line extended from zero indicates that to the lowest possible detectable levels, no desorbed protein was detected. This data is the author's own from reference (43) 35 Figure 2.1: Examples of Verdier-Stockmeyer moves used to manipulate the protein-like chains during simulations. Shown are a 2-bead crankshaft move, a 1-bead flip and an example of an end-bead turn 77 Figure 2.2: Schematic diagrams of the two model chains: sequence I and sequence II. Filled circles represent hydrophobic (H) units while open circles represent polar (P) units .' 78 Figure 2.3: Simulation results demonstrating the conformational dependence of sequence I on temperature. Shown is the ensemble-averaged number of intramolecular hydrophobic contacts as a function of reduced temperature, T*. The line drawn indicates the trend of the data 79 Figure 2.4: System energy probability histograms for sequence I: filled grey bars - chain in solution when XHH -4 (T* - 0.71); filled black bars - chain adsorbed on a relatively hydrophobic surface (XHW ~ -4) when XHH —4- The error bars refer to the standard deviation of 5 runs 80 =  =  Figure 2.5: Energy trajectory for the adsorption of sequence I to a planar hydrophobic surface when XHH ' XHW  =  -4  81  Figure 2.6: Representative conformational states of sequence I shortly after adsorbing to a planar hydrophobic surface when XHH = XHW = -4- A l l conformations correspond to system energies E, > -ASkT. 82 Figure 2.7: Representative examples of lowest energy states of sequence I adsorbed to a hydrophobic surface when XHH XHW -4- At the global minimum, E = -5 6AT.. 83 =  =  T  Figure 2.8: The 6 lowest energy states for sequence I when XHH -3 and XHW = -1. The structure in the centre is the adsorbed chain in its native state 84 =  viii  Figure 3.1: Schematic diagram of the model HP chain. Filled circles represent hydrophobic (H) chain segments while open circles represent polar (P) chain segments 113 Figure 3.2: Energy trajectory for adsorption of the HP chain to a planar sorbent surface when XHH -1 and XHW -4- All other segment-segment interaction energies set equal to zero 114 =  =  Figure 3.3: Dependence of the degeneracy of the lowest energy state {i.e., the total number of unique chain conformations) on accessible sorbent surface area and volumetric space above it. The x-axis indicates the number of lattice sites on the sorbent surface available for binding. Non-zero segment-segment interaction energies are XHH = XHW = -1 115 Figure 3.4: Energy probability histograms for the HP chain adsorbed to sorbent surfaces of two different widths: etched, grey bars - width of sorbent surface, xs, is 4 lattice units; solid black bars - xs is 18 lattice units. The error bars refer to the standard deviation of 5 runs. Non-zero segment-segment interaction energies are XHH ~ XHW = -\ 116 Figure 3.5: Ratio of the probability of the HP chain being in its native-state conformation when adsorbed, P , to that in solution, P s° ", as a function ofthe width of the available adsorption site, xs. Non-zero segment-segment interaction energies are XHH = XHW = -4 (open squares), and XHH = -4, XHW = -1 (open circles). The line drawn indicates the trend of the data 117 lutio  ADSORBED  NS  N  Figure 3.6: HEWL-catalyzed pNP-C5 hydrolysis kinetics when H E W L is dissolved in aqueous solution (pH 6, 22 °C) and when HEWL is nonspecifically adsorbed to particulate silica at different levels of surface coverage. In all experiments, the amount of H E W L and the initial concentration of pNP-C5 are held constant at 50 u.g and 0.20 mM, respectively. Shown are measurements taken for H E W L in solution with no surface (squares), H E W L adsorbed on silica at monolayer coverage (circles), and H E W L adsorbed at 20% monolayer coverage (triangles) 118 Figure 3.7: Thermodynamics of adsorption of the HP chain to a planar sorbent surface as a function of accessible sorbent surface area and volumetric space above it. The xaxis indicates the number of lattice sites on the sorbent surface available for binding. Non-zero segment-segment interaction energies are XHH ~ XHW = -1. Trends indicated are AUIkT (solid line), AA/k (dashed line) and AS/k (dash-dotted line). To avoid excessive clutter, simulation points are shown for AUIkT only. Error bars are calculated from the standard deviation of 5 runs 119 Figure 3.8: Thermodynamics of adsorption of the HP chain to a planar sorbent surface as a function of accessible sorbent surface area and volumetric space above it. Nonzero segment-segment interaction energies are XHH = XHW = -4. Symbology is the same as in Figure 3.7 120  ix  Figure 3.9: Thermodynamics of adsorption of the HP chain to a planar sorbent surface as a function of accessible sorbent surface area and volumetric space above it. Nonzero segment-segment interaction energies are XHH -1, XHW ~ -2. Symbology is the same as in Figure 3.7 121 =  Figure 4.1: Schematic of HP chain sequences used in these simulations: sequence I and sequence II. Hydrophobic (H) segments are filled and polar segments (P) units are unfilled 152 Figure 4.2: Helmholtz energy (AadsA/kT, squares) and entropy (AadsS/k, circles) of adsorption for the HP chain sequence I adsorbing on a weakly attractive surface (ZHW = -1) as a function of Ao-N^/kT, the stability of the native-state of the chain relative to its fully denatured state. Lines drawn indicate data trends. Error is within 11% and 22% for values of AadsA/kT and AadsS/k, respectively 153 Figure 4.3: Thermally-averaged energy landscapes for the HP chain sequence I in solution (squares) and adsorbed (circles) to a weakly attractive sorbent surface (%HW = -1) under conditions where the stability of the native-state conformation is low (ZHH = -1)- Lines drawn indicate data trends 154 Figure 4.4: The average length, l , in the x dimension of each bond vector in HP chain x  sequence I as a function of XHHIXHW, the ratio of the average intramolecular to intermolecular contact energies. The dotted line represents the average length of each bond vector in the x dimension when the chain is in its native-state conformation. The line drawn indicates the data trend 155 Figure 4.5: Comparison of Ad fi/k values calculated with Eq. [4.12] to dMC simulation e  data values over a range of chain deformations, l I l : dashed line a = 2, solid line x  xo  a = 2.8. The dMC data (open squares) shown correspond to adsorption conditions where  HHIXHW  X  respectively  =  0.25  = 2.2), 1 (T /J ~ = 1), and 4 (F /J2= 0.85), x  x  0  x  156  List of nomenclature Main symbols  a  x  P =  = ratio of projection of bond vectors = y,—=  /pnl  l ll x  xo  , relation from Eq.[4.7]  2 x  Xy  - Flory energy parameter between components/residues i and j  coij  = interaction energy between components / and j  Q(x)  = density of states having value x  a  = length of one lattice unit  a  = dimensionality of system [Eq. 4.12]  A  = Helmholtz energy  E  = energy  E  - energy of conformation /  SE  = energy difference between start and end of Monte Carlo move  h  = number of intramolecular hydrophobic contacts  hu  = number of intramolecular hydrophobic contacts in the native state conformation  k  = Boltzmann constant  l  = projection of bond vector on x-axis  t  x  l  = mean projection of bond vector on x-axis  L  = chain length, number of residues in a chain  n  = number of bonds within the chain Eq. [4.7]  n  = random number, chosen for comparison in Monte Carlo algorithm  x  n  = number within sample having end-to-end distances within Xj and Xj+dx Eq.[4.11]  N  = total number of end-to-end distances sampled (Chapter 4)  N  = number of lattice sites in the simulation grid  t  Pi(EI) = probability that the energy of the lattice is £ , P = calculated probability for comparison to randomly chosen number in the Monte Carlo algorithm Q  = partition function  r,  = coordinates of lattice site i  c\r rj) = function based on relation .of positions of lattice sites / and j r  S  = entropy  T  = temperature  U  = internal energy  T  = "melting" temperature  T*  = reduced temperature  V  - volume  W(x)  = distribution of chain end-to-end distances in x-dimension  m  Ay<A> = difference of ensemble averaged values of X for transition Y <X>  - ensemble average of value X  (xt, yi) = coordinates of lattice site /'  Superscripts  o  = in maximum entropy conditions (Chapter 4)  xii  Subscripts  A  = wall (athermal)  ads  = adsorption (non-adsorbed to adsorbed transition)  conf  = conformational  D  = denatured state  def  = deformation  free  = in solution, non-adsorbed  H  - hydrophobic residue of chain  N  = native state  o  = at maximum entropy conditions (Chapter 4)  P  = polar residue of chain  S  = solvent  W  - wall (active)  x  = with respect to the x-axis  Acknowledgements  I would first like to express my heartfelt gratitude to my research supervisor, Chip Haynes. His scientific guidance and unwavering support during my time in U B C will always be appreciated and remembered.  I am grateful to the Institute of Applied Mathematics for the generous use of their computer facilities. Grants from the National Science and Engineering Research Council and the Protein Engineering Network of Centres of Excellence enabled this work.  I would also like to acknowledge fellow members of the Michael Smith Laboratories (formerly the Biotechnology Laboratory) past and present, who contributed to my work and learning there. I'd especially like to thank Louise Creagh, for her tutelage and sound advice regarding all matters practical.  My appreciation goes out to members of the  Haynes Lab and also to the labs of Robin Turner, Jamie Piret, Douglas Kilburn and Donald Brooks for frequent sharing of their resources.  The richness of my graduate student experience can be attributed in large part to Green College, U B C . I feel fortunate to be a part of this special and inspiring community.  Finally, I am indebted to many kind friends, my father, Dirkson, and dear sisters, Dora and Jean for their encouragement and support.  It is with great sadness that my late  mother, Esther Liu, is not able to share this personal milestone with me.  xiv  1  Introduction, background and objectives  The need to understand and control protein adsorption at solid-liquid interfaces is driven in large part by current limitations in synthetic biomaterials for human implantation and in bioprocessing equipment, particularly chromatography resins (1-3).  A substantial effort is currently being made to tailor properties of artificial materials to minimize or eliminate a negative immunological host response, such as the formation of thrombi at or near the blood-implant interface (1,4). As adsorption of plasma proteins to a poorly designed foreign surface is known to trigger a biochemical reaction cascade leading to the formation of thrombi (5, 6), much of this work is devoted to identifying surface chemistries which properly control the density, composition and conformation of the adsorbed protein layer (1, 7).  Similarly, a myriad of empirical strategies are now  being tested to improve the performance of other synthetic body-fluid-contacting materials, including contact lenses (8, 9), dental fixtures (10, 11), and hemodialysis equipment (7).  An increased dependence on biomolecule-based pharmaceuticals has further intensified our need to better understand and control protein adsorption.  FDA-approved  recombinant-protein, D N A and viral products have increased by more than 1000% over the past decade, and are now a preferred strategy to treat a number of life-threatening cancers and auto-immune diseases (12-14). Production costs for protein and DNA-based therapeutics are substantive, and the need to deliver a product free of contaminants is  1  increasing due to growing concerns over product safety (i.e. the avoidance of transmission of Creutzfeldt-Jakob disease, hepatitis, etc.). Identifying ways to reduce costs and increase safety by effectively streamlining the production of these valuable materials is therefore crucial. The separation of complex protein mixtures is typically carried out using large-scale chromatographic processes where the proteins are purified by preferential adsorption or partitioning from a mobile liquid carrier to a solid matrix. In general, the protein's specific affinity for and interaction with the stationary phase dictates the quality of the separation. Improvements in column performance are therefore intimately linked with the ability to carefully control the sorbent chemistry and geometry.  Over the past half century, a significant body of literature has been devoted to adsorption of chain molecules at solid-liquid interfaces. Adsorption phenomena in such systems are complex due to the unique properties of chains.  Monodisperse linear homopolymer  chains (possibly the simplest member of this adsorption group) can fluctuate among a large number of chain configurations at the solid-solution interface. Characterizing the adsorption behavior of the chain is therefore difficult, as it generally takes several parameters to describe the state o f the polymer at the interface.  Representative  parameters include the average number of points of attachment, the horizontal spread  (often defined in terms of the average chain radius Ar.  ), and the average chain thickness  There are in fact more parameters than can be uniquely specified from fitting of  adsorption data, and this has made it difficult to confirm or deny proposed models for homopolymer adsorption. Advances in our understanding of homopolymer adsorption have therefore come largely through experiments that specifically probe conformations of  2  adsorbed chains (x-ray diffraction, neutron scattering, etc.) or, more commonly, through adsorption data generated for model systems using molecular simulation techniques (15, 16).  On and off-lattice Monte Carlo simulations, other random-walk approaches, and  various molecular dynamics and statistical approaches have all been used to generate useful model adsorption data for understanding homopolymer adsorption phenomena (17-23).  Although it shares many of the classic features associated with flexible homopolymer adsorption, globular protein adsorption has proven considerably more difficult to understand at the molecular and thermodynamic levels.  For example, in aqueous  solutions, both linear flexible homopolymer adsorption and globular protein adsorption can be endothermic, indicating that adsorption is entropy rather than energy favored. However, as flexible polymers are known to lose conformational entropy upon adsorption, the gain in system entropy can be precisely ascribed to a gain in solvent entropy. The origin of the entropy gain in globular protein adsorption systems is more complex and therefore far less well understood.  Flexible polymer adsorption and  globular protein adsorption also share a tendency for surprisingly slow adsorption kinetics (24).  Adsorption that appears to have reached equilibrium can often display a  slow but continuous drift for days (25, 26). For the case of flexible polymer adsorption, Cohen-Stuart et al. (27, 28) have argued that the slow adsorption kinetics are due to polymer polydispersity effects.  The same argument cannot hold for globular proteins,  which are uniform in size and chemistry.  3  This thesis is founded on the hypothesis that simulation of the adsorption of simple protein-like chains using a dynamic Monte Carlo (dMC) method will serve to improve our fundamental understanding of nonspecific protein adsorption. The argument is supported by the successful use of molecular simulations to greatly improve our understanding of the adsorption of flexible linear polymers to solid-liquid interfaces (18, 21, 23), and by recent dMC simulations performed by Anderson et al. (29) on the adsorption of simple mesoscopic protein-like chains to a liquid-liquid interface which gave important insights into the dynamics and thermodynamics of the process.  Currently, full atomistic simulations of a related problem, protein folding, can only be carried out over a millisecond time period (30). This duration limit is not practical for the simulation of protein adsorption processes, since they are known to occur over periods of minutes to days (26).  We therefore choose to restrict our studies to the simulation of  simple heteropolymer chains composed of two types of units, hydrophobic (H) and polar (P), known commonly as the HP model. By proper sequencing of the residues, HP chains will fold into unique protein-like conformations.  Despite the obvious simplicity of the  HP model, Onuchic (31), Dill (32, 33), and many others (34, 35) have shown it to be useful in understanding the mechanism of protein folding, the forces that stabilize the native-state conformation, and the sensitivity  of native-state stability on system  conditions.  4  1.1  Literature review  1.1.1 General aspects of proteins  Proteins are biological macromolecules. Diverse in structure and function, they are abundant in all living beings. Proteins often function as biological catalysts, but are also known to be important in ligand transport and cell signaling and as structural materials within biological systems.  The building blocks of proteins are L-amino acids. The  general structure of an amino acid is RCH(NH3 )COi r  , where R represents  its  characteristic side chain (see FigureT.l). Twenty naturally-occurring amino acids exist, each having characteristics defined by the chemical properties of their respective side chain.  Within a protein, amino acids are joined through a condensation reaction between the carbonyl and amino groups (Figure 1.2). The peptide link formed is highly rigid due to its partial double bond character. The sequence in which amino acids connect, referred to as the primary structure of the protein, is significant in that it is the interactions of the peptide units and their respective side chains which impart the principal characteristics and eventual function to the molecule. The primary sequence determines the formation of highly organized substructures such as a-helices or P-sheets, whose content define the secondary structure of the protein.  Similarly, its composition, as determined by how  secondary structures and other less ordered forms are placed, determines its overall threedimensional arrangement (its tertiary structure).  The chain's noncovalent association  5  with like or unlike chains (its quaternary structure) is also ultimately defined by its amino acid sequence.  Protein structures are varied and complex.  Unlike many other types of linear chain  molecules, proteins in an aqueous solution of moderate temperature and pH usually display a specific conformation or a limited set of similar conformations. Maintaining this state, known as the native-state conformation, is essential to the function of the protein. The native state is stabilized relative to the large ensemble of denatured states through a closely-balanced combination of energies.  Perturbations, whether originating  from changes in the protein's solvent environment (alterations in temperature, ionic strength, or pH for instance), or the presence of external factors (e.g. a solid-liquid interface nearby) can easily shift the balance of these stabilizing energies and cause the protein to unfold and therefore cease to function.  Proteins comprised of a single polypeptide chain can be divided into three broad categories according to their tertiary structure: a) b)  expanded coil structures: flexible and highly solvated, fibrillar  proteins: mainly consisting of regular secondary structures such as a-  helices and p-sheets, and c)  globular proteins:  compact proteins that are made up of both random and  structured parts, all folded into a roughly spherical configuration. Most proteins of interest, such as enzymes and antibodies, are globular, and they are the type of proteins of interest in this work.  6  1.1.1.1  Globular  proteins  Globular proteins in an aqueous solution have a number of general characteristics (36), some of which are described here. 1)  Globular proteins are roughly spherical in shape with average diameters of the  order of angstroms to nanometers. 2)  Hydrophobic side groups have a tendency to reside in the interior of globular  proteins to avoid contact with water. This does not necessarily mean that all hydrophobic residues are sheltered from the solvent or that the interior is composed entirely of hydrophobic groups. Internal hydrophobicity is limited, for instance, by the presence of the relatively hydrophilic polypeptide backbone. 3)  Charged groups are found predominantly on the exterior of the protein, while the  very few charged groups in the interior are almost always found in ion pairs. 4)  Globular proteins are very densely packed, i.e. comparable to densities for  polymer glasses. The atomic packing fraction of a protein is about 75%, significantly higher than the packing fraction of liquid water which is 58% at 25°C and 1 atm.  1.1.1.2 Factors  affecting  native-state  stability in aqueous  solution  A folded protein is a densely packed molecule stabilized by an intricate heterogeneous network of intra- and intermolecular forces. Thermodynamic investigations carried out by Privalov (37) indicate the Gibbs energy stabilizing the native state of a globular  protein at physiological conditions typically lies between 30 and 70 kJ/mol, roughly equivalent to the energy of 4 to 12 hydrogen bonds.  Table 1.1 summarizes favorable and unfavorable interactions and forces known or at least thought to affect the stabilities of proteins dissolved in aqueous solutions.  The table  shows that hydrophobic dehydration, dispersion forces and hydrogen bonding can drive folding of the protein into its native state. Compensating those forces are the associated loss of conformational entropy and distortions of bond lengths and bond angles. Coulombic forces can be favorable or unfavorable to the stability of the native structure, depending on the overall pH of the system relative to the isoelectric point of the protein.  Hydrophobic dehydration is generally believed to be a major driving force for protein folding in aqueous solutions (38-40).  It refers to the change in solvation of hydrophobic  amino acid side chains of a protein in its folded (native) state relative to its unfolded (denatured) state. When a protein is fully denatured, most if not all of its side chains are exposed to the aqueous solvent. This results in a high degree of solvation which requires the water molecules to locally arrange themselves around the apolar solute (in this case, the denatured protein) in a relatively ordered shell which seeks to maximize solvent hydrogen bonding at the solvent-solute interface.  In general, the native-state structure  has considerably fewer hydrophobic side chains exposed to the solvent since a majority of the apolar groups fold into the interior of the protein. Solvent entropy is therefore gained from dehydration of these apolar groups.  8  Intramolecular hydrogen bonding is also thought to make a substantial contribution to the stability of the native state. Creighton (36) has argued that hydrogen bonding contributes ca. 30-45% of the energy driving folding, with less significant contributions from dehydration, van der Waals and electrostatic interactions.  The significance of electrostatic forces to the state of the protein can be estimated by observing the dependence of protein stability on changes in pH and ionic strength. For example, at extreme pH relative to the isoelectric point, the charge density of a folded protein becomes high, and there is a tendency for the protein to unfold.  Specific  electrostatic interactions such as ion pairing within a protein have an opposite effect in that they usually lead to the stabilization of the native protein. A rough gauge of the degree of electrostatic interaction occurring at the surface of the protein is given by the local water density adjacent to the protein surface: an ion of significant charge density increases the molar density of water directly surrounding it (41).  Dispersion forces, which are highly dependent on the distance between atoms (r a r" ), 6  are likely important for local protein structure due to the dense atomic packing in a typical protein.  The total magnitude of stabilization energy from dispersion effects,  however, is thought to be less than that due to the effects of hydrophobic dehydration and hydrogen bonding (39).  Counteracting all the positive stabilization forces is one main destabilizing force: the loss of conformational entropy resulting from the folding of the polypeptide chain (39, 40).  9  Creighton has estimated that for a polypeptide chain in a random coil, approximately 4 distinct backbone conformations exist per peptide unit (36). Assuming that a peptide unit has only one backbone conformation when involved in an a-helix or p-sheet, the loss of entropy per peptide unit is R ln 4 = 11.53 J mol" K" . For a protein consisting of 100 1  1  residues, the loss in entropy is approximately 1200 J/K per mole of protein due to freezing of the backbone structure. This results in a Gibbs free energy gain of 350 kJ/mol at 300 K. Additional entropy losses occur from reductions in conformational freedom of side chains within the interior of the folded protein.  A somewhat less significant force opposing the native state is the distortion of covalent bond lengths and bond angles as determined by energy minimization calculations (36, 42). These distortions, which add approximately 4 to 8 kJ per distorted bond to the native state energy, are believed to be necessary to optimize the various interactions (hydrophobic, dispersive and peptide-peptide hydrogen bonding) required for a tightly packed, compact molecule.  1.1.1.3 Macroscopic properties ofprotein adsorption systems  Steady-state behavior of a protein adsorption system is most often illustrated by an adsorption isotherm, where the surface concentration of protein is measured against the concentration of free protein in bulk solution. Figure 1.3 shows an example isotherm for the adsorption of hen egg-white lysozyme to silica at pH 7 and 37°C (43). The direction of the arrows within the figure indicate the type of isotherm being represented. Arrows  10  up and to the right indicate the ascending isotherm, where the total protein concentration is progressively increased. Arrows to the left indicate the descending isotherm where the free protein concentration is diluted at otherwise constant conditions.  The isotherm shown in Figure 1.3 demonstrates some general features typical of many globular protein adsorption systems. First, the behavior of proteins on solid surfaces is complex.  Initial slopes of ascending protein adsorption isotherms are usually steep,  indicating a strong affinity of the protein for the sorbent surface, but not infinite, suggesting establishment of a quasi-equilibrium between the sorbate and protein in the bulk phase.  Second, ascending and descending isotherms differ in an apparent time-  independent manner, indicating that at a given free protein concentration, the system can exist in more than one state. As a result, a protein adsorption process shows some classic features of irreversible thermodynamics. What is also true for this particular system and many others is that for a given ascending isotherm there are an infinite number of descending isotherms, each of which is defined by the departure point from the ascending isotherm. In most cases, there is no evidence that the descending isotherms rejoin the ascending isotherm.  Despite the apparent irreversible nature of their binding interaction, adsorbed proteins are dynamic on a sorbent surface. For example, fluorescence recovery after photobleaching (FRAP) experiments demonstrate that nonspecifically adsorbed proteins are mobile on the sorbent surface and therefore are able to reposition themselves in response to a concentration gradient (44). Limited exchange of the protein on and off the surface has  11  also been shown in radiotracer experiments, where labelled proteins are displaced from the surface by non-labelled proteins (45, 46). nonspecific  protein  adsorption  appears  These results suggest that although  macroscopically  irreversible,  atomic  intermolecular contacts between sorbent and protein are in a constant state of flux, indicating some level of reversibility at the microscopic level.  1.1.2 Driving forces for nonspecific protein adsorption  Isothermal titration calorimetry has allowed for the direct quantification of the heat associated with protein adsorption. Often, as one might expect, enthalpy is found to drive adsorption. But in many cases, the process is endothermic, indicating that adsorption occurs through an increase in system entropy (47, 48). This supports growing evidence that protein adsorption is not the result of a simple single-step reaction process, but a combination of multiple, perhaps synergistic subprocesses occurring at different rates (47). O f those known to occur, three subprocesses are routinely observed: 1) changes in protein conformation, 2) dehydration of parts of the sorbent and protein surfaces, and 3) redistribution of charged groups at the protein-sorbent interface (40,49).  1.1.2.4 Protein structural changes  The introduction of a sorbent interface can influence the stability of the native-state conformation of a protein by interacting directly with the protein (40).  Structural  rearrangement of adsorbed proteins has been observed using a wide range of analytical 12  techniques, including many spectroscopic techniques such as total internal reflection fluorescence (TIRF) (50), microscopic techniques such as atomic force microscopy (51, 52), probing of the charge profile of the protein using proton titrations (53, 54), and thermodynamically, using calorimetry (47, 55).  The severity of the conformational  change varies. In some instances, the structural perturbation away from the native state is small. Ellipsometry and reflectometry experiments, for instance, have shown thicknesses of adsorbed layers which are comparable to the native protein adsorbed in a side-on or end-on orientation  (56, 57).  Activity measurements for many structurally stable  enzymes adsorbed onto hydrophilic surfaces have also indicated minimal structural changes in the adsorbed state (58). In most cases, however, the conformational changes are more severe, and several analytical studies have suggested multiple conformational states of adsorbed proteins (59-61).  Structural denaturation has been shown to increase with certain properties of the system. In particular, high sorbent hydrophobicity, low surface coverage, and low protein stability are correlated with increases in conformational change (43, 50, 62).  These aspects will  be discussed in detail in upcoming chapters, and therefore will not be discussed further here.  1.1.2.5 Hydrophobic interactions  The major contributing effect to a positive entropy of adsorption is thought to be the hydrophobic effect (40),  where, in an aqueous environment, the presence of a  13  hydrophobic solute causes the solvating water molecules to arrange themselves in order to minimize unfavorable polar-nonpolar contacts with the surface while maximizing preferred hydrogen bonding amongst themselves (38, 39). This local enhancement in solvent structure is characterized by a significant decrease in entropy and a large positive change in heat capacity, as demonstrated by the AS and AC for transfer of non-polar P  solutes into dilute aqueous solutions (63), and by differential scanning calorimetry (DSC) studies of protein unfolding (38).  The significance of the hydrophobic effect has been investigated from the perspective of both the protein and the sorbent surface. Greater hydrophobicity, either at the sorbent or protein surface, often lead to increased adsorption and energies of adhesion (50, 64-66). Some studies however show that moderate degrees of hydrophobicity maximize adsorption affinity (67, 68), indicating that although hydrophobic interactions are significant, they do not necessarily dominate the driving force for protein adsorption.  1.1.2.6 Electrostatics  Protein adsorption electrostatics are complex, involving the overlap of the electric double layers of the protein and the sorbent surface, charge interactions between the protein macromolecules, and, especially in the case of repelling charge, involvement of lowmolecular-weight ions present in solution (40).  As a result, protein adsorption is often  strongly dependent on the pH of the system. For example, when electrostatic interactions between neighbouring proteins dominate, adsorption levels are typically maximized when  14  the solution pH is near the isoelectric point of the protein, a condition where the protein surface charge is neutralized, thus allowing proteins to pack closely together on the surface (40, 64, 69).  Alternatively, for cases when electrostatic forces between the protein and the interface dominate, proteins tend to show maximum affinity for the sorbent at the condition of charge complementarity given by the pH where the protein charge is equal and opposite to that of the sorbent surface (40).  In the case where charge complementarity does not  occur, ions from solution must be incorporated into the protein-sorbent interface to achieve a neutral situation. Changes in the pH of the system prior to and after adsorption have therefore been monitored to determine the net transfer of charge to the interface (53).  1.1.2.7 Other driving forces  Other subprocesses can also contribute to the driving force for adsorption, including formation of specific  intermolecular ion pairs or hydrogen bonds, intra- and  intermolecular van der Waals interactions, and those forces which scale with 1/(distance) such as dipole-dipole and dipole-induced dipole interactions (70, 71).  15  1.1.3  Kinetics of nonspecific protein adsorption  The apparent irreversible nature of nonspecific protein adsorption has motivated a large number of fundamental investigations of the kinetics of the process, both for singleprotein and competitive adsorption systems (45, 46, 72, 73). Some of the central goals of these kinetic studies are 1) to identify time constants for the various steps (subprocesses) along the reaction pathway, 2) to identify that step, and associated energy barrier, which allows limited protein exchange (i.e. the desorbed protein is replaced by a protein of the same or different kind) but prevents spontaneous desorption upon dilution, and 3) to obtain the necessary database to test proposed kinetic models for the adsorption process.  The protein adsorption reaction involves a number of time-dependent steps, including 1) diffusion of the protein to the surface or corresponding boundary layer area, 2) attachment (and detachment) of the protein to (from) the surface, and 3) reorientation, rearrangement and conformational changes to the protein once adsorbed. Due to the lack of sensitivity in available analytical techniques, these reaction steps cannot in general be directly probed, but only inferred from the nature of the total signal.  Additional subprocesses have been considered in kinetic theories for nonspecific protein adsorption. For example, Lundstrom and Elwing (74) include the possibility of protein exchange in their simple theoretical model. They hypothesize that the state of the protein on the surface determines its exchange rate with protein in solution; specific rates exist for self-exchange reactions, when the adsorbed protein remains in a native conformation,  16  and for alternate-exchange conformational change.  reactions, when the adsorbed protein has undergone a  Competitive and multiple layer adsorption have also been  considered (45, 46, 75).  In general, model parameters are not determined directly, but are simultaneously regressed from global kinetic data obtained from ellipsometry (59, 76) or surface pressure experiments (72, 77-79).  As a result, development of kinetic models for nonspecific  protein adsorption remains a largely empirical science.  1.1.4 Simulating and modeling protein adsorption  Although the body of knowledge regarding protein structure is substantial, practical limitations (in computational power, for example) have so far restricted the development of detailed molecular simulations of protein adsorption.  Simulations that have been  carried out generally treat the protein as a hard sphere or conformationally static macromolecule and focus primarily on describing electrostatic  interactions  (80),  dispersive effects (81, 82) and solvent interactions (83, 84). However, a small number of more advanced and realistic simulations have recently been reported, and results from these studies are discussed in detail in subsequent chapters of this thesis.  In general, analytical models describing protein adsorption isotherms and kinetics are largely based on comparison with experimental data and tend to be highly empirical in nature. The simplest models are those which assume the protein to be a hard particle  17  which adsorbs reversibly to the sorbent surface (85-87).  For example, the Langmuir  equation has often been used to describe protein adsorption isotherms (87)  At first  glance, this approach seems reasonable as proteins often adsorb at monolayer coverage (47, 85, 88), and as a consequence result in isotherms exhibiting a shape similar to that of a Langmuir curve.  However, mechanistic inferences drawn from the good fit of the  Langmuir equation are clearly misleading (40, 89), as protein structural changes and adsorption irreversibilities are not accounted for in the model.  The random sequential adsorption (RSA) model represents proteins as hard particles, while acknowledging the irreversible nature of protein adsorption (90-93). In a classic RSA-type model, hard particles (most commonly a round disk or sphere) adsorb randomly and are not allowed to overlap. Once adsorbed, the particles are immobile and eventually reach a "jamming limit" where surface coverage cannot increase. Tarjus et al. (90) modified the RSA model to investigate the more realistic situation where desorption and surface diffusion take place.  They presented a generalized description of how a  process can reach an equilibrium-like steady state characterized by the relative rates of adsorption, desorption and diffusion. Van Tassel et al. (91) introduced conformational change into the RSA model by including an expansion factor for the sorbate.  Upon  contact with the surface, the particles (in this case, circular disks representing proteins) are allowed to symmetrically expand to a greater diameter, representing the tendency for a globular protein to flatten out when adsorbed. A later paper by the same group which incorporate partial reversibility in the model (92) showed that greater saturation occurs  18  when sorbate expansion is allowed and that conformational change is dependent on bulk concentration.  In an attempt to advance the RSA model beyond steric effects, electrostatic forces were added by Adamczyk et al. (94) and Lenhoff et al. (95, 96).  Both groups incorporated  charged particles and solution electrolytes into the model, resulting in particle-particle and particle-surface energy considerations.  Oberholtzer et al. (95) also incorporated a  perpendicular force imposed on approaching particles to represent a barrier to adsorption taking into account the opposing forces of surface attraction and electrostatic repulsion of already-adsorbed particles.  The latter 3D model was more successful in describing  experimental results and demonstrated the importance of including an energy barrier prior to contact in the process. Changes in protein conformation were not addressed in these models.  Electrostatic and dispersion forces were incorporated in a protein adsorption model in the case of charged proteins (represented by spheres) adsorbing to a similarly charged surface. (97, 98). In a later study, Stahlberg et al. (99) solved for the electrostatic and van der Waals interaction energies between two parallel plates to determine a capacity factor for ion-exchange chromatography. Despite the coarse geometry of the model, their results correlated well with experimental data over a large range of ionic strengths. Roth and Lenhoff (100) calculated equilibrium constants for proteins adsorbing to oppositely charged surfaces at low coverage where proteins were represented as a low dielectric spheres each containing a single central charge.  Interactions with the surface were  19  determined to be functions of the size and charge of the protein, the charge density on the surface and solvent characteristics. Later papers by the same group indicated that success of the model is highly dependent on proper estimation of particle shape and the charge distribution on the surface of the protein (71,101).  Models which attempt to take into account protein conformational change are few in number. Combining experimental and theoretical estimates, Haynes et al. (43, 47) developed a model based on the enthalpic and entropic contributions of six adsorption subprocesses:  changes in the protein structure, hydration effects, protonation or  deprotonation of titratable residues on the protein and sorbent surfaces, and three electrostatic effects originating from overlapping of the protein and sorbent electric fields, i) coulombic interactions, ii) specific ion pairing between oppositely charged residues, and iii) ion co-adsorption from the solvent into the interfacial layer. Estimates of protein structural changes in the model are derived from thermodynamic parameters determined using microcalorimetry.  A mean-field approach was taken by Fang and Szleifer (102)  who incorporated  conformational change and competitive adsorption to their model. A general diffusion equation was used to define the movement of the protein to the surface while the protein's chemical potential gradient provided a measure of the overall driving force for adsorption.  The study of both kinetic and thermodynamic aspects allowed for the  examination of the initial adsorption sequence as well as transition of the adsorbed layer to its eventual equilibrium state. They investigated the situation where particles undergo  20  a surface-induced conformational change upon adsorption. Their results revealed that the composition of the adsorbed layer is dependent on the bulk concentration of the protein and the degree of intermolecular interaction at the surface.  In their study of the  adsorption of protein mixtures, they were able to mimic some aspects of the Vroman effect, where, due to differences in diffusion rates and degrees of attraction to the surface, larger particles eventually displace smaller particles from the sorbent surface.  1.2  Thesis objectives  The objective of this thesis is to study the adsorption behavior of simple protein-like HP chains using dynamic Monte Carlo simulations with the aim of understanding how the tendency of such chains to form unique low-energy conformations in solution alters their adsorption thermodynamics relative to the adsorption behavior of a random-coil homopolymer. A two-dimensional lattice is used to simplify the problem such that all chain conformations can be observed. The model is used to:  1)  Determine how system variables such as sequence order, structural stability and  surface hydrophobicity affect adsorption thermodynamics, particularly their effect on the change in the conformational entropy of the chain.  2) Study average adsorption trajectories of protein-like chains to better understand the dynamic behavior of nonspecific protein adsorption systems.  21  3) Determine the effects of sorbent geometry and available surface area on adsorption thermodynamics through simulations involving spatial restrictions of an adsorbing protein-like chain.  4) Identify the driving forces and mechanisms leading chains of varying conformational stability to adsorb onto hydrophobic and hydrophilic surfaces.  5) Identify conditions where adsorption is entropically favored and then use the model to obtain a clear understanding of the origin of the gain in entropy.  6) Evaluate and compare energy landscapes for the HP chains in solution and adsorbed to a solid-liquid interface to assess how the multiplicity of states are distributed in both systems and how these differences determine the accessible conformation(s) (and energies) of the chain.  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Xu, S.Q. and Damodaran, S., Kinetics of adsorption of proteins at the air-waterinterface from a binary mixture. Langmuir, 10 (1994) 472-480.  76.  Elwing, H., Protein adsorption and ellipsometry Biomaterials, 19(1998) 397-406.  77.  Suttiprasit, P., Krisdhasima, V. and McGuire, J., The surface-activity of alphalactalbumin, beta-lactoglobulio, and bovine serum-albumin .1. surface-tension measurements with single-component and mixed-solutions. J. Colloid Interface Sci., 154(1992)316-326.  in biomaterial research.  28  78.  Razumas, V . , Nylander, T. and MacRitchie, F., Surface pressure study of hemin, microperoxidase-8, -11, and cytochrome c adsorption at the air-water interface. J. Colloid Interface Sci., 178 (1996) 303-308.  79.  Ybert, C. and di Meglio, J.M., Study of protein adsorption by dynamic surface tension measurements: Diffusive regime. Langmuir, 14 (1998) 471-475.  80.  Juffer, A . H . , Argos, P. and DeVlieg, J., Adsorption of proteins onto charged surfaces: A Monte Carlo approach with explicit ions. J. Comput. Chem., 17 (1996) 1783-1803.  81. Lu, D.R. and Park, K., J. Biomater. Sci. Polymer Edn., 1 (1990) 243-. 82. Noinville, V . , Vidalmadjar, C. and Sebille, B., Modeling of protein adsorption on polymer surfaces - Computation of adsorption potential. J. Phys. Chem., 99 (1995) 1516-1522. 83.  Lu, D.R., Lee, S.J. and Park, K., Calculation of solvation interaction energies for protein adsorption on polymer surfaces. J. Biomater. Sci. Polymer Edn., 3 (1991) 127-147.  84. Latour, R.A.J. and Rini, C.J., Theoretical analysis of adsorption thermodynamics for hydrophobic peptide residues on S A M surfaces of varying functionality. (2002) 56477. 85.  Almalah, K., McGuire, J. and Sproull, R., A macroscopic model for the singlecomponent protein adsorption-isotherm. J. Colloid Interface Sci., 170 (1995) 261268.  86. Johnson, R.D., Wang, Z.G. and Arnold, F.H., Surface site heterogeneity and lateral interactions in multipoint protein adsorption. Journal of Physical Chemistry, 100 (1996)5134-5139. 87. Perrin, A . , Elaissari, A . , Theretz, A . and Chapot, A . , Atomic force microscopy as a quantitative technique: correlation between network model approach and experimental study. Colloids and Surfaces B-Biointerfaces, 11 (1998) 103-112. 88.  Arai, T. and Norde, W., The behavior of some model proteins at solid-liquid interfaces 1. Adsorption from single protein solutions. Colloid. Surface., 51 (1990) 1-15.  89.  Martensson, J., Arwin, H . , Lundstrom, I. and Ericson, T., Adsorption of lactoperoxidase on hydrophilic and hydrophobic silicon dioxide surfaces - an ellipsometric study. J. Colloid Interface Sci., 155 (1993) 30-36.  90. Tarjus, G., Schaaf, P. and Talbot, J., Generalized random sequential adsorption. J. Chem. Phys., 93 (1990) 8352-8360.  29  91.  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Yuan, Y . , Oberholzer, M.R. and Lenhoff, A . M . , Size does matter: electrostatically determined surface coverage trends in protein and colloid adsorption. Colloid. Surface. A, 165 (2000) 125-14L  97. Ruckenstein, E. and Prieve, D.C., Adsorption and desorption of particles and their chromatographic-separation. AIChE J., 22 (1976) 276-283. 98.  Prieve, D.C. and Ruckenstein, E.Jn "Colloid and Interface Science" (M. Kerker, Eds.), p.73. Academic Press, New York, NY, 1976.  99. Stahlberg, J., Jonsson, B. and Horvath, C , Combined effect of coulombic and van der Waals interactions in the chromatography of proteins. Anal. Chem., 64 (1992) 3118-3124. 100.  Roth, C M . and Lenhoff, A . M . , Electrostatic and van der Waals contributions to protein adsorption: Computation of equilibrium constants. Langmuir, 9 (1993) 962972.  101.  Asthagiri, D. and Lenhoff, A . M . , Influence of structural details in modeling electrostatically driven protein adsorption. Langmuir, 13 (1997) 6761-6768.  102.  Fang, F. and Szleifer, I., Kinetics and thermodynamics of protein adsorption: A generalized molecular theoretical approach. Biophys. J., 80 (2001) 2568-2589.  30  1.4 Tables  Table 1.1: Interactions governing the native-state structural stability of globular proteins. AN-DG  refers to the Gibbs energy of denaturation for the protein.  Type of Interaction  Contribution to  Comments  AN-DG  Hydrophobic dehydration  « 0  An increase in entropy results from the release of water molecules contacting hydrophobic residues.  Hydrogen bonding  < 0 (?)  Intramolecular hydrogen bonding, especially in ordered secondary structures, may contribute to stability.  Dehydration of polar groups  <0  Dehydration of polar groups may contribute up to 5 kJ/(mol amino acid).  Electrostatic forces  > or < 0  Contribution is dependent on the pH of the system relative to the pi of the protein.  Dispersion forces  <0  Favorable due to the dense packing of the atoms in a protein structure.  Conformational entropy  » 0  A substantial loss of conformational freedom from folding and the formation of highly ordered secondary structures.  Distortion of covalent bond lengths and bond angles  >0  Unfavorable strains existing to accommodate other, more dominant interactions.  31  1.5 Figures Figure 1.1: The basic chemical structure of an amino acid and schematic diagrams of the 20 naturally-occuring amino acids.  R I +  H  3  N - C - C O ;  I H  32  33  Figure 1.2: The condensation reaction forming the peptide link between two amino acids.  +  R-i  R2  R1  O  R2  I  I  I  I  I  H N-C-C0 " 3  I  H  2  +  +  H N-C-C0 " 3  I  H  2  •  H N— C— C — N — C— C 0 " + H 0 3  H  III  H  2  2  H  34  Figure 1.3: Isotherm for the adsorption of hen egg-white lysozyme (pl~l 1) to particulate silica at pH 7 and 37°C.  Shown is the concentration of protein on the surface as a  function of protein in bulk solution. Data points collected from two separate trials are indicated. The solid line represents the ascending isotherm, while dotted lines represent descending isotherms. The vertical dashed line extended from zero indicates that to the lowest possible detectable levels, no desorbed protein was detected.  This data is the  author's own from reference (43).  35  2  Mesoscopic  analysis  of  conformational  and  entropic  contributions to nonspecific adsorption of HP copolymer chains using dynamic Monte Carlo simulations* 2.1  Introduction  The success of invasive and extracorporeal medical devices is typically limited by the incompatibility of the materials of construction of the device with the tissue and blood with which they come into contact. Despite improvements over the past several decades in the physical and chemical properties of artificial biomaterials, inflammatory reactions against the foreign substances of the devices have not been eliminated.  In critical  therapeutic interventions such as major surgery, the activation of the immune and inflammatory system within a patient, as well as the coagulation process, contribute to a slower  recovery  and  increased  susceptibility  to  infections  and  post-operative  complications. Moreover, when the use of a device is required in long-term care (e.g. implantable devices and hemodialysis), the body's chronic response against the foreign materials of construction requires lifelong medication.  * A version of this chapter is published in the Journal of Colloid and Interface Science. [Reference:  Liu, S.M. and Haynes, C.A., Mesoscopic analysis of conformational and  entropic contributions to nonspecific adsorption of HP copolymer chains using dynamic Monte Carlo simulations. J. Colloid Interface Sci. 275 (2004) 458-469.]  36  Regulating the behaviour of cells and tissues at a biomaterial interface requires strict control over the surface properties of the material and an ability to impart to the material a defined biological response.  Among the greatest challenges for meeting these  requirements are controlling protein adsorption, retaining protein activity following adsorption, and tailoring protein distributions on the artificial surface to elicit a desired cellular response. The hydrophobicity, charge, and chemical makeup of the surface and the contacting protein have all been shown to impact the energetics and kinetics of the protein adsorption process, and can affect both the stability and orientation of a protein at a surface (1-3). Most protein-material interactions therefore result in decreased protein activity.  Improving the biocompatibility of synthetic materials for traditional device-based therapies will require a better fundamental understanding of the kinetics and energetics of protein adsorption.  Considerable attention has therefore been given to understanding  how and why proteins adsorb at interfaces (1, 3). Regrettably, the problem has proven recalcitrant, due in part to the inherent complexity of the process and to the lack of experimental methods capable of accurately measuring molecular contributions to overall adsorption energetics and of visualizing the protein adsorption process at the molecular level.  Numerous experimental studies have shown that electrostatics, dehydration forces, interactions between neighboring adsorbed proteins, and protein conformation all contribute to the adsorption process (1, 2).  Protein and sorbent surface geometry,  37  including surface roughness, are also thought to be important. Globular proteins appear to prefer surfaces of greater hydrophobicity and greater roughness (3).  For example,  Kondo et al. (4) showed that for identical total protein loads, the surface density of adsorbed a-amylase was lowest on silica particles, the least hydrophobic (most polar) surface tested, and increased with increasing hydrophobicity of the sorbent surface, indicating a strong correlation between the affinity of the protein for the surface and sorbent surface polarity.  A dependence of adsorption rates on the sorbent surface  hydrophobicity has also been shown, with increasing sorbent hydrophobicity leading to an increase in the forward rate constant for adsorption and a decrease in the off-rate (5). Kull et al. (6), for example, showed that P-casein adsorbs quickly onto hydrophobic silica, while the adsorption kinetics appear to be much slower on hydrophilic silica under similar conditions.  Sorbent hydrophobicity has also been shown to influence the extent of differences between a protein's adsorbed and native-state conformations. Circular dichroism (7, 8), ellipsometry (9), N M R (10), TIRF (11, 12), and calorimetry (7, 13) data have all shown evidence of significant conformational changes in proteins adsorbed at hydrophobic surfaces.  Activity assays, an indirect measurement of the extent of protein structural  change, have also shown that the tendency for a protein to change conformation during adsorption can be correlated to sorbent surface hydrophobicity (4, 14)  The extent of conformational change in proteins upon adsorption is influenced by other factors, most notably by protein stability (1). McGuire and coworkers (15-17) conducted  38  a number of studies using wild-type T4 bacteriophage lysozyme and a set of mutants of varying thermal stability.  They concluded that T4 variants of greater thermodynamic  stability retain greater amounts of both secondary and tertiary structure upon adsorption, and that the less stable mutants adhere more strongly to the surface.  The rate of  adsorption was also affected, with less stable proteins adsorbing more quickly. Similar conclusions were made by Billsten et al. (18, 19), who studied adsorption of variants of human carbonic anhydrase II on silica particles.  Based on analysis of differential  scanning calorimetry, circular dichroism, and fluorescence  spectroscopy data, they  concluded that adsorption of the more-stable pseudo-wild-type protein resulted in the smallest perturbation in protein conformation relative to the native state.  Efforts have been made to model the thermodynamics and kinetics of globular protein adsorption (5, 20) by assuming that the protein first adsorbs to the surface in its native state, then undergoes structural rearrangements.  Van Tassel et al. (21), for instance,  represented proteins as disk-shaped particles that symmetrically spread on the surface once adsorbed. While validation of these rather coarse models is generally restricted to successful  correlation with adsorption isotherm and binding kinetics data, their  predictions suggest a relation between protein conformational change and adsorbed protein concentration that has been observed experimentally. Thus, despite their rather simple nature, the models indicate that an accurate description of protein conformational changes during adsorption is essential to understanding the thermodynamics and kinetics of protein adsorption.  39  More detailed descriptions of conformational changes in a protein during adsorption have been elucidated from thermodynamic studies. Haynes and Norde utilized a combination of calorimetry, proton titrations and adsorption isotherm measurements to quantify the contribution of changes in protein structure to the thermodynamics of the adsorption process (1,  13).  investigations.  A number of general conclusions  can be drawn from these  First, the degree of protein structural change during adsorption is often  significant, and consequently can have a large effect on the overall energy change accompanying the adsorption process. For example, in a study of the adsorption of ctlactalbumin to negatively-charge polystyrene surfaces at pH 10, protein structural changes were estimated to make the dominant contribution to the overall enthalpy of adsorption (13).  Conformational changes in the protein must also result in a change in entropy, but data defining the magnitude and sign of entropy changes accompanying protein adsorption are limited and generally indirect. In their seminal early work, Norde and Lyklema (22, 23) argued that protein secondary structure losses upon adsorption led to a net increase in the rotational mobility of the backbone and side chains of the protein. As a consequence, the conformational entropy of the protein was predicted to increase.  They argued that the  resulting entropy gain would be large enough to compensate for the unfavorable decrease in enthalpy associated with adsorption. adsorption have been observed.  Losses in protein secondary structure upon  In particular, circular dichroism has been used in several  instances to measure decreases in a-helix content during nonspecific protein adsorption to relatively hydrophilic sorbents such as silica (7, 17). While these studies provide no  40  direct evidence that losses in protein secondary structure lead to an increase in the conformational entropy of the adsorbed protein relative to its native state structure in solution, the overall entropy change for the adsorption process is often favorable.  In  contrast, Giacomelli et al. (24) and Sane et al. (25) and others have recently shown that changes in protein conformation upon adsorption at hydrophobic surfaces often involves an increase in ordered structure, and therefore a decrease in conformational entropy. This is consistent with recent modeling results by Ben-Tal et al. (26) which predict a net loss in chain entropy of approximately 1.7 kT for a short peptide, pentalysine, during adsorption to a hydrophobic lipid membrane.  The aim of this paper is to examine more closely the dependence of the energy and entropy of adsorption on associated changes in the conformation of a chain molecule that folds into a unique compact lowest-energy state, such as the native state of a globular protein. Regrettably, experimental methods capable of directly quantifying changes in chain conformation and entropy and then relating them to system energy and entropy are not available.  However, such connections can be made within much simpler model  systems, such as the well-known HP chain model of Dill, that involve linear copolymer chains that fold in solution into a unique lowest-energy conformation and therefore share with single-domain globular proteins important conformational and energetic properties. Here, we ask what can be learned from mesoscopic dynamic Monte Carlo simulations of the adsorption of protein-like HP model chains about the dependence of adsorbed-state chain conformations and entropy on both the stability of the lowest-energy ("native state") conformation of the chain in solution and the topography and hydrophobicity of  41  the sorbent surface?  Throughout the paper, connections are made with general  experimental observations of protein adsorption processes to identify where results from our simple model are in qualitative agreement and may therefore provide insight into the contributions of chain energy and entropy to the overall thermodynamics of protein adsorption.  The complexity of the folding behaviour of proteins, combined with limitations in computing capabilities, set limits oti the class of protein simulation problems that can currently be solved at the atomistic level. In general, molecular dynamics simulations of protein folding can cover only a portion (e.g., microseconds) of the estimated 10 to 100 milliseconds required for a small protein to fully fold (27). Such an approach is therefore not applicable to the study of protein adsorption, which is a process that typically takes seconds to hours to reach an energetic steady state (28).  Thus, we feel that simplified  mesoscopic chain-on-a-lattice models, such as those developed by Dill (29) and Socci and Onuchic (30), remain the most practical way of investigating at a fundamental level the dependence of chain conformational entropy on system properties.  2.2  Theory  2.2.1 Dynamic Monte Carlo simulations of HP-model polymers The HP model, first proposed by Dill (29) and used extensively by others (see, for example, 31, 32), assumes that the major contribution to the free energy of the native conformation of a protein is due to interactions between hydrophobic (H) amino acids. A hydrophobic core in the folded structure tends to form that is shielded from the 42  surrounding solvent by polar (P) amino acids in the polypeptide sequence.  In the HP  model, the amino acid sequence of a peptide-like chain is represented mesoscopically as a binary string of H and P monomers. The HP model therefore ignores the fact that some amino acids cannot be unambiguously classified as being either hydrophobic or hydrophilic. Despite the simplicity of the HP model, folding processes for HP-model chains simulated by the dynamic Monte Carlo (dMC) method or the molecular dynamics method appear to show important similarities with the protein folding process (33), allowing a number of research groups to successfully evaluate new hypotheses for protein structure formation and driving forces for protein folding (30, 34, 35). For example, the HP model has been successfully used to understand better the formation of compact secondary structures and hydrophobic cores in proteins (36).  It has also been used to  specify energy pathways and intermediates of protein folding which in turn have led to a probable explanation for the Levinthal paradox (37). The success of the model stems in part from the fact that the simple HP representation of the peptide makes it possible to enumerate and consider all possible chain conformations and associated  system  energetics. Given their ability to document changes in chain conformation resulting from perturbations to system environment, dMC simulations of HP-model chains may also provide a useful means of evaluating putative models for protein adsorption, including those that postulate that conformational changes in the peptide during adsorption result in an increase in the entropy of the chain that drives the adsorption process and its apparent irreversibility.  43  Recently, dMC simulations have been used to describe adsorption of mesoscopic proteinlike chains at a liquid-liquid interface.  Anderson et al. (38) simulated the adsorption  behaviour at an oil-water interface of a uniquely folding linear copolymer in which intraand intermolecular interactions between chain segments were defined by the interaction energies of Miyazawa and Jernigan (39). Solvent-chain interactions were considered, as well as the mixing energies of the oil and water phases. They observed that due to the unfavorable energetics between the oil and water, the chain is overwhelmingly (>99%) likely to adhere to the interface in an unfolded state. The transition to this denatured state requires overcoming significant energy barriers, but once attained, is apparently irreversible. A key result from this important study is that the conformational entropy of the denatured adsorbed chain is greater than that of the native-state protein in solution.  Zhdanov and Kasemo (40, 41) used dMC simulations to investigate the relative rates of denaturation of HP chains in the presence of a solid-liquid interface.  Their results  showed that at relatively high temperatures, unfolding of the chains follows an apparent first-order rate equation, similar to what was observed without a surface present. At lower temperatures however, the denaturation pathway at the surface differed in that metastable states were formed. More recently, Castells et al. (42) used a dMC approach to demonstrate that mesoscopic protein-like chains attach to a surface in an unfolded state, and that the degree of unfolding is dependent on the degree of attraction of the residues to the surface.  44  Here we apply dMC simulations to Dill's HP model in two dimensions to further investigate conformations of protein-like chains adsorbed at a solid-liquid interface and the contribution of changes in chain conformation to adsorption energetics and system entropy. Results from these model calculations are then interpreted in the context of real protein adsorption systems and previous experimental results of Norde et al. (13, 43) and others (44, 45) that suggest, albeit indirectly, that nonspecific protein adsorption to hydrophilic surfaces is often driven by an increase in the conformational entropy of the polypeptide.  2.2.2  T h e model system  The 2D protein-like HP chains we used are taken from the work of Dill et al. (29, 33) and contain L residues, connected through L-l vectors, all one lattice unit a in length. The chain is placed on a Cartesian coordinate grid in a self-avoiding configuration so that at any instance, no more than one residue occupies a given lattice site. Each residue is therefore uniquely positioned on a coordinate point (lattice site) of the grid (i.e. the first residue lies on coordinates (x\, yi), the second on (xi, y\) etc.) and all connecting vectors t  run parallel to either the x or y-axis.  Spaces unfilled by the chain are assumed to be  solvent units. The grid size is defined in each simulation as grid height = grid width = lOx the fully stretched chain length, thereby creating a lattice of sufficient size to allow the HP chain to freely sample all conformational space.  For adsorption studies, a  potential is applied at selected walls to represent a model surface, while in all other cases, walls remain athermal with respect to interactions with the remaining components of the system.  The model chain is initially placed away from the adsorbing interface and  45  allowed to undergo Verdier-Stockmeyer moves as shown in Figure 2.1.  The model is  ergodic, as the set of three allowable Verdier-Stockmeyer moves (crankshaft, flip, and turn) enable the chain to adopt all possible conformations within the lattice.  Our dMC simulations are based on the Metropolis algorithm where chain movements are allowed or disallowed depending on the change in energy of the system (46). The total energy of the lattice of N sites is defined as the sum of the interaction energies between all contacting elements of the system, excluding interactions between directly connecting chain residues.  In all simulations reported here, four components are considered:  hydrophobic (H) and polar (P) residues of the chain, the solvent (S), and the active wall (W). Interaction energies between each of the components and residues are defined by an associated set of Flory energy parameters, Xy ~ (OylkT, where / and j index the four components/residues in the system. The overall lattice (system) energy can therefore be calculated as follows for each sequence in a particular conformation and position in the grid,  i j>i  I  The first term in Eq. [2.1] sums over all nearest-neighbor interactions in the lattice, where r, and r, are the coordinates of lattice sites i and j, respectively, and S is a function based on the positioning of the involved sites. For neighboring residues, 6\a) - 1, while for nonneighboring residues, d\\r rj\>a) = 0. Interactions between connected chain residues are r  not considered. The second term in Eq. [2.1] therefore subtracts energies discounted due to connected residues within the chain of length L.  46  The dMC simulation algorithm used for both HP chain annealing studies in solution and all adsorption studies is based on randomly selecting a segment of the HP chain and evaluating all Verdier-Stockmeyer moves consistent with the position of the segment. Where multiple move options are allowed for a particular chosen segment, a single move is randomly selected and the new system energy is evaluated. If the new system energy is found to be equal or less than that of the original conformation, the move is accepted and a new cycle started. If the new conformation results in an increased system energy, then the difference in energy SE is weighted using a Boltzmann relation, P = exp(-SE/kT)  [2.2]  and the calculated probability P is compared to a random number, n, where 0 < n < 1. Moves with a calculated probability higher than n are accepted while moves of lower probability are rejected.  Overall, the weighted method allows for both favorable and  unfavorable moves to take place in the simulation. The frequency of acceptance of unfavorable moves, however, is significantly lower, but sufficient to allow the chain to escape local energy minima and sample all conformational space. The time coordinate of the simulations is presented in units of attempted moves.  2.2.3 Calculation of thermodynamic parameters The on-lattice d M C simulations reported here represent a canonical ensemble.  The  energy of the lattice, E, is therefore equivalent to the internal energy, U, and the natural free energy of the system is the Helmholtz energy, A. The direct observable from each simulation is the energy histogram ptfEj), generated through Monte Carlo sampling at  47  system temperature, T, from which one can calculate the ensemble-averaged internal energy of the system, U  U^<E>=Y PXE )E J  i  [2.3]  i  where p is the probability that the energy of the lattice is E . t  t  The Helmholtz energy of the lattice, A, is calculated directly from the partition function,  Q A=<A>=-kTlnQ  [2.4]  In the simplest representation of the HP model, only contacts between adjacent' hydrophobic segments are considered, with all other interactions assumed athermal. In this case, folding is solely driven by net favorable interactions between hydrophobic residues within the protein-like HP chain (i.e.  OHH^T  <  0, with all other interaction  energies set equal to zero). Thus, hydrophobic effects associated with a repulsive energy of interaction between water and other components are not considered here.  Each model sequence folds into a finite number of conformations, each having an energy value based on h, the number of formed hydrophobic contacts. For this simple model, the partition coefficient, Q, is given by  Q = f Q(h)exp(-^L) j  [2.5]  where Ci(h) is the density of states for which the number of hydrophobic contacts is h, COHH  is the interaction energy between hydrophobic residues,  k is Boltzmann's constant, T  48  is the temperature, and /?# is the number of hydrophobic contacts formed in the most stable conformation(s).  Extension of Eq. [2.5] to include contributions from other  segment / - segment j interactions is straightforward.  Finally, the system entropy, S, is given by the standard thermodynamic relation  For the simple model described by Eq. [2.1], S represents the chain entropy at infinite dilution.  2.3 Results 2.3.1 Dynamics and thermal unfolding behaviour of model sequences Two HP model chains that fold into unique "native-state" conformations at low temperatures when driven by intramolecular interactions between hydrophobic residues (33) were selected for our simulation studies (Figure 2.2).  The first model sequence,  referred to here as sequence I, consists of 18 units and folds into a globular native-state (lowest energy) conformation defined by 9 pair-wise contacts between two hydrophobic residues not directly connected on the chain. It is predominantly hydrophobic, with an entirely hydrophobic interior and partially hydrophobic exterior.  Sequence II is  comprised of 20 units and folds into a globular native-state conformation having 8 hydrophobic contacts.  In this lowest energy state, sequence II has an entirely polar  exterior.  49  In the HP model, the temperature  dependence of the ensemble-average chain  conformation is captured by the Flory interaction parameter XHH (=o)HH(T)lkT) between hydrophobic residues of the chain.  The conformational state of the chain at a given  temperature can therefore be specified by the average number of contacts formed between these residues.  A low number of hydrophobic contacts indicates an open  "denatured" conformation, whereas a high number of contacts indicates a more compactly arranged chain, with the maximum hydrophobic contact number indicating the unique lowest energy conformation.  For model sequence I, Figure 2.3 shows dMC simulation results for the ensemble average number of intramolecular hydrophobic contacts formed as a function of the reduced temperature T* (= T/T ). m  The simulation data for sequence I resemble a thermal  denaturation curve for a single domain globular protein, showing a transition from a highly structured chain to a largely unstructured chain at a defined 'melting' temperature T. m  At very low temperatures (far below T ), the protein-like chain forms an average m  number of hydrophobic H H contacts of 9, indicating that the chain is essentially always found in its lowest energy native-sta.te conformation when the temperature is sufficiently low (i.e., near 0 K).  However, at temperatures 10°C to 80°C below T , corresponding to XHH values between m  -1 and -5, the conformation of the chain in solution is not static.  Conformational  fluctuations of the folded chain in solution when XHH = -4 are shown in Figure 2.4 in the form of a probability histogram for the system energy. The dynamic nature of the chain  50  conformation in solution is apparent, in that the chain is found in its unique lowestenergy conformation only a fraction of the time. The remainder of the time, the chain is in one of a relatively large number of conformations of slightly higher energy.  It is well known that protein denaturation results in a large increase in chain entropy that is compensated in part by an associated increase in the internal energy of the chain due to the loss of favorable intramolecular contacts  between chain segments.  These  compensating effects are captured in the HP model. For sequence I, Table 2.1 lists the ensemble-average energy  AN-D E> <  and entropy  TAN-D <S>  of the denatured state relative  to that of the native state at reduced temperatures T* of 0.44 and 0.71 assuming zero excess heat capacity for the denatured state. energetically unfavorable (i.e.  AN.D<A>  As expected, chain denaturation is  is greater than 0) at both temperatures due to an  unfavorable increase in chain internal energy.  Denaturation, however, is favored by the  increase in chain entropy that would accompany the process at either temperature.  2.3.2 Chain adsorption to the surface The presence of a sorbent surface was introduced into the simulations by setting a favorable interaction energy between a hydrophobic segment of the chain and an adjacent lattice segment of the simulation boundary (i.e., the wall). At the start of each simulation, the chain was positioned in close proximity to the wall to ensure that contact was made early in the simulation. In cases where adsorption to a planar surface was studied, the length of the active dimension of the grid was set to be greater than the fully-stretched length of the chain.  Consequently, the chain could interact with the surface without  51  experiencing physical constraint from the opposite and two adjacent walls, which were all assigned to be energetically neutral.  The approach of the chain to the wall was  simulated by Monte Carlo moves and random diffusion, while movement along the wall following initial chain contact resulted solely from Monte Carlo moves.  Figure 2.5 shows the system energy over the initial 3 x 10 cycles of a dMC simulation of 8  the adsorption of sequence I, initially in its folded lowest energy conformation in solution, to a planar hydrophobic surface. Relatively strong interaction energies are considered here by setting XHH = XHW - -4-  Initially, the energy £ , of the system is > -  2>6kT, indicating that the chain is not in contact with the surface. After a relatively short number of cycles, the system energy decreases by ca. -\2kT\o  a new energy minimum of  —48AT, consistent with favorable side-on adsorption of the native-state chain to the hydrophobic wall (sequence I in its lowest energy native-state conformation has a maximum of 3 exposed hydrophobic residues per solvent-exposed side). For the initial ca. 2.5x107 simulation cycles following chain adsorption, E > -A%kT and the chain t  fluctuates among a large number of conformations with energy similar to that of the native-state conformation. Examples of these conformational states are shown in Figure 2.6.  The chain then adopts a new set of adsorbed conformations characterized by a system energy of -52kT and in a relatively small number of additional cycles begins to assume one or more conformations characterized by an E equal to -56kT, which represents the t  global energy minimum for the system.  The probability of finding the system in this  52  lowest energy state remains high for the remainder of the simulation, but the dynamics of the system are such that the chain frequently adopts higher energy conformations, including conformations close to its native-state conformation on occasion.  Snapshots of conformations of the chain when the system is at its global energy minimum are shown in Figure 2.7. Unlike the chain in solution, which is characterized by a single lowest energy conformation, the adsorbed chain can assume at least 84 unique conformations on the surface to reach the global energy minimum of -56kT.  The chain  conformations corresponding to this lowest energy state vary in dimensions and aspect ratio, but are generally flat, taking on the landscape view of a pancake or mound lying on a plate. Each such conformation differs noticeably from the chain in its native state.  A similar set of dMC simulations with XHH = XHW = -4 was completed for the adsorption of sequence II. As with sequence I, a severe perturbation of the chain conformation away from the native state is required to access the new ensemble-average system energy of <E> - ca. -40kT.  Conformational degeneracy at the global energy minimum is again  observed. In this case, 5 unique adsorbed chain conformations are observed at -44kT, the global energy minimum of the system.  The total change in entropy requires solution of Eq. [2.6] for the adsorption process. It is, however, instructive to first evaluate the contribution to the overall change in chain entropy of the degeneracy in the conformation of the adsorbed chain at the global energy minimum for the system. This analysis, we believe, is analogous to popular models for  53  nonspecific protein adsorption which view the protein in solution as having a single native-state conformation, but capable of accessing multiple conformational states on the surface (5, 20, 21, 47).  For sequence I, for example, the change in the entropy of the  chain for the hypothetical case where it is restricted to its global energy minimum in the absence and presence of the sorbent surface is +4AkT, or an average of +0.25&T per chain residue. This per residue entropy change is similar to those estimated for nonspecific protein adsorption based on measurement of the total entropy change for adsorption and model estimates of the chain entropy. If it represented the true change in chain entropy, it would certainly provide a strong driving force for adsorption. However, as shown in the energy histograms provided in Figure 2.4, the HP chain in our model system accesses a range of energy states and a large number of associated conformations when the sorbent surface is both absent and present. Thus, as will be shown below, the actual change in chain entropy for the adsorption process can differ both in magnitude and in sign from the crude estimate provided above.  2.3.3 Thermodynamics of chain adsorption For sequence I adsorbing to a planar sorbent, Table 2.2 reports changes in the ensembleaverage internal energy A b E>, <  ac  Helmholtz energy Aads<A>, and chain entropy  TA s S> <  aa  under conditions where the sorbent surface is relatively hydrophilic (%HW — -1)- When the stability of the native-state conformation of the chain is low (XHH change in system entropy TA s S> <  aa  =  -1), a positive  of 3.6kT is found to make the dominant contribution  to the overall Helmholtz energy change driving the adsorption process.  In this case,  intramolecular and intermolecular energies of interaction are symmetric and weak, 54  allowing the adsorbed chain to sample a very large number of conformations of similar energy on the sorbent surface. An increase in chain entropy continues to make the dominant contribution to A d <A> a  s  when the stability of the folded chain in solution is increased two-fold {XHH ~ -2). However, a further increase in the stability of the native-state conformation in solution results in an adsorption process characterized by either a relatively small increase (%HH -3) or a slight decrease (XHH ~ -4) in the chain entropy. A ds E> = ca. -3kT.  For the XHH - -3 system,  Side-on adsorption of the chain in its native-state conformation  <  a  would yield a A d E <  a  =  >  s  =  -3kT,  and indeed one of the observed lowest energy  conformations of the adsorbed chain is the native state. However, as shown in Figure 2.8, five additional non-native conformations are observed at the global energy minimum. In each of these conformations, the adsorbed chain accesses the lowest energy state by reducing its total number of intramolecular contacts to form a larger number of new intermolecular contacts.  To achieve a A ds<E> of -3kT, 6 intermolecular HW contacts a  must form in place of a unit reduction in the total number of intramolecular H H contacts. This requirement severely limits the conformational space that is effectively available to the adsorbed chain. The adsorbed chain entropy is further limited by the impenetrable nature of the sorbent surface, which removes a degree of freedom for any chain segment in the lattice layer adjacent to it due to the fact that the segment is not free to step into the sorbent surface. In addition, like for the chain in solution, the interaction energy between the solvent and hydrophobic segments of the adsorbed chain is net unfavorable and therefore favors adsorbed state conformations that limit such contacts.  55  Finally, Table 2.3 reports thermodynamic values for adsorption of sequence I on a hydrophobic surface (JHW = -4) at conditions where the chain in solution is denatured and therefore fluctuates among a very large set of random coil conformations (achieved by setting XHH = -0.25). As one would expect, the random-coil chain in solution loses conformational entropy upon adsorption due to the required localization of hydrophobic chain segments on the sorbent surface.  2.3.4 Influence of sorbent surface geometry Although the effect is not easily studied through experiment, the topography (roughness, porosity, surface area to volume ratio) of the sorbent surface is thought to influence adsorption behaviour, including adsorption thermodynamics. The effect of a non-planar sorbent surface on chain adsorption was investigated by dMC simulations by activating two connected walls of the simulation cell, thereby allowing the chain to interact with a corner of the grid.  When XHH XHW = -4, sequence I preferentially adsorbs to the corner of the grid in order =  to maximize solvent-free intermolecular contact area. As shown in Table 2.4, adsorption of sequence I to the grid corner results in a lower ensemble-average energy A d <E> a  S  and  a lower global energy minimum E „ than observed for adsorption of the same sequence mi  to a planar surface. When adsorbed to the corner, sequence I forms a maximum of 17 favorable hydrophobic contacts (sum of intramolecular and intermolecular contacts), 3 more than when adsorbed to the planar surface. This results in adsorbed chain conformations that are non-native, but which retain a globular structure similar in aspect 56  ratio to that of the native-state in solution. In comparison, sequence I adopts pancake or mound-like conformations to access the global energy minimum on a planar surface at otherwise identical conditions.  The preferential adsorption of the chain to the corner region of the sorbent surface is reflected in the more favorable A d <A> for this process (Table 2.4). a  s  The change in  &ads A> is not as great as the change in A d <E> due to the lower entropy of the chain <  a  adsorbed to the corner.  s  The formation of a larger average number of hydrophobic  contacts effectively restricts the conformational freedom of the chain adsorbed to the corner. This effect is reflected in the conformational degeneracy of the adsorbed chain at the global energy minimum. Sequence I forms a total of 15 unique lowest energy conformations when adsorbed at the hydrophobic corner, much less than the 84 conformations found for adsorption on the planar sorbent.  When XHH  =  XHW ~ -4, sequence II. also preferentially adsorbs to the corner, forming a  total of 12 favorable contacts, or 1 more than formed on the planar surface. In this case, however, chain conformations at the global energy minimum are more pancake in nature and similar to those observed when sequence II adsorbs to the equivalent planar surface.  2.3.5 Dependence on the total hydrophobicity The influence of the total hydrophobicity of the system was also investigated. Table 2.5 reports values Aads<E>, Aads<A>, and TAads <S> for dMC simulations of adsorption of sequence I to a planar sorbent surface under conditions where the value of XHH  =  XHW is 57  varied from -1 to -4. narrows as  COHH  In the absence of the sorbent, the distribution of chain energies  becomes more favorable, resulting in a higher probability for the chain to  be in its lowest energy native-state conformation and an overall lower chain entropy.  Despite the lower conformational entropy of the chain in solution when %HH ~ —4, adsorption of the chain to the hydrophobic planar surface results in a net decrease in chain entropy. Under these conditions, the chain adopts non-native conformations on the surface to achieve a total of two additional favorable contacts (HH and HW) relative to adsorption in its native-state conformation. The observed loss in conformational entropy of the chain during this transition is due to the additional constraints imposed by the impenetrable, inflexible sorbent surface and the connected nature of the chain, which make it possible to reach the lowest energy state (and fluctuate around it) through only a limited number of conformational trajectories.  2.4  Discussion  Despite the simplicity of the model, dynamic Monte Carlo simulations of model HP chains capture many of the phenomenological events that are frequently associated with nonspecific  protein adsorption at a solid-liquid interface.  adsorption, particularly to hydrophobic sorbents,  often results in  observable changes in protein conformation (7, 13, 48). mesoscopic picture of this phenomenon.  For example, protein experimentally  Our results provide a  The dMC data show that the conformation of  the adsorbed chain is constantly changing, so that the system fluctuates in the total number of favorable contacts (HH and HW) formed. As a result, the system energy also  58  fluctuates in time between energy states at or relatively near the global energy minimum. Therefore, adsorbed proteins are not frozen into a fixed, energetically most favorable conformation or energetically degenerate set of lowest energy conformations. Instead, as shown in the adsorbed-state energy probability distribution (Figure 2.4), the system is not always in its lowest-energy state due to the low conformational degeneracy of that state, and is often at a slighter higher energy E because the conformational degeneracy of those t  energy states is orders of magnitude larger than that of the global energy minimum. Thus, the adsorbed HP chains exist as an ensemble of energies and conformations, a conclusion that has recently been proposed by others to describe the adsorbed protein state (49, 50).  Based on this concept, many researchers have argued that an increase in conformational entropy of the peptide chain contributes to the driving force for protein adsorption. Direct (isothermal titration calorimetry) and indirect (e.g., adsorption isotherms measured at several temperatures) measures of heats of protein adsorption indicate that the process is often endothermic, so that the driving force for adsorption is provided by a positive change in system entropy (13, 51). However, whether this entropy change is due, at least in part, to an increase in the conformational entropy of the protein remains unclear. For adsorption to hydrophilic surfaces, indirect evidence suggests that the conformational entropy of the protein often increases upon adsorption (13).  In contrast, adsorption to  hydrophobic surfaces has recently been shown to result in an increase in the secondary structure of a protein, suggesting a loss in chain conformational entropy (24, 25, 49, 50, 52).  59  Our d M C simulation results suggest that changes in the conformational entropy of a protein-like chain during adsorption to a solid-liquid interface can provide a substantial, even dominant driving force for adsorption under certain conditions.  Consistent with  inferences made from experiment, substantial increases in the conformational entropy of HP sequence I are observed when the stability of the native-state fold is low (XHH  =  -1)  and the surface is relatively hydrophilic (XHW — - 1 ) - However, if the stability of the native state or the hydrophobicity of the sorbent is increased, any favorable contribution of chain conformational entropy to A d <A> is quickly lost. a  s  For instance, for adsorption to a more hydrophobic surface {%HW - -4), sequence I loses conformational entropy upon adsorption, a result that agrees with the experimental studies cited above. Careful comparison of the simulation data for the model HP chain in solution and adsorbed to the surface provides an understanding of why this is the case. Two phenomena are responsible. First, although the adsorbed chain does sample a large number of conformational states, these accessible conformations are those corresponding to system energies near the global energy minimum. To reach energies near this global energy minimum, the adsorbed chain must maintain a well-defined number of favorable intramolecular (HH) and intermolecular (HW) contacts with the rigid impenetrable sorbent surface, effectively restricting the number of accessible chain conformations. As a result, although the adsorbed chain does indeed adopt many conformations, these accessible  conformations  represent  a relatively  small  fraction of  all  possible  conformations of the linear chain. This point is illustrated in the dMC results for  60  adsorption of sequence I in its denatured state in solution to the same hydrophobic surface {XHW = —4)- In this case, the chain in solution samples all conformational space with near equal probability, so the conformational entropy of the chain is a maximum. The conformational entropy change TA ds<S> for adsorption of this denatured chain is a  l2kT (see Table 2.3), nearly identical to the loss in the conformational entropy of the chain in the denatured to native-state transition (TAD-N<S> = -\3kT). Thus, relative to the denatured chain in solution, the adsorbed chain and the folded chain in solution have similar conformational entropy.  The second phenomemon of importance to understanding changes in chain entropy upon adsorption is the realization that the native-state chain in solution also fluctuates between a large number of conformational states of energy near that of the unique native-state conformation (see Figure 2.4).  That is, while each HP chain may assume only one  conformation to access its lowest energy state in solution, energy states very near the lowest energy are populated by a large number of conformations.  As a result, the  ensemble-average conformational entropy <S>chain of the chain in solution is significantly larger than one would anticipate by treating the structure in solution as that corresponding to its unique lowest-energy solution state. For proteins, the possible existence of this phenomenon is supported by proton-exchange and other N M R relaxation measurements by Hwang et al. (53) and others (54) which reveal considerable mobility in the peptide backbone, and even more in the side chains of proteins in solution.  61  Finally, it is important to note that in their dMC simulations of mesoscopic protein-like chains adsorbing to a liquid-liquid interface, Anderson et al. (38) observed larger increases in the conformational entropy of the folded chain upon adsorption. The rigidity of the sorbent interface therefore appears to influence adsorption thermodynamics. In our system, the surface is rigid and impenetrable, so that the chain must adopt very specific conformations and points of contact to access low energy states.  In the liquid-liquid  system, the fluidity of the interface provides the adsorbed chain additional degrees of freedom, such that the chain can adopt a large number of conformations involving partial chain penetration into the sorbent (oil) phase. Nevertheless, Anderson et al. (38) found that the large positive changes in total entropy A d S> <  a  s  often observed in protein  adsorption to oil-water interfaces are likely to be largely due to interfacial dehydration effects and not changes in chain conformational entropy.  2.4.1 Protein adsorption kinetics are linked to energetic barriers that frustrate conformational trajectories of the peptide chain The dMC simulation algorithm is, in general, restricted to determination of equilibrium thermodynamic properties. However, several studies (see for example references 38, 40) have shown that the dMC approach can yield insights into dynamic and/or kinetic events if a sufficiently large number of configurational states are sampled at each step number. In this context, it is important to note that the trend in the energy trajectory shown in Figure 2.5 was observed for many independent simulations at the same conditions, indicating that the adsorption path is general in nature and not an artifact of the dMC simulation algorithm. That is, although the chain conformation at any given simulation  62  step is quite different in each independent simulation, the energy trajectory remains the same, showing E values > -A%kT iox the initial 3 x 10 cycles following chain contact, 7  t  followed by a relatively rapid drop in Ef to -56kT.  In such cases, several groups have  shown that dMC simulations can provide insights into the reaction pathway (38, 40-42) in addition to their classic use in determining equilibrium properties. Here, we use this approach to provide a few qualitative observations concerning the adsorption pathway for sequence I.  The energy <E> trajectory (a typical one is shown in Figure 2.5) for adsorption of  • ••  7  sequence I shows that the folded chain fluctuates during the initial 1 x 1 0  R to 1 x 1 0  simulation steps following contact with the sorbent surface between a large number of adsorbed conformations that are close to the native-state in structure and energy. All of these conformations are of energy 8kT or higher than the global energy minimum of the system (-56AT). Thus, the simulations identify a barrier to relaxation of the chain to those conformations associated with or close to the global energy minimum.  Analysis of our d M C data suggests that this barrier occurs because trajectories for conformational change are limited both by the compact segment density of the nativestate chain when it contacts the surface and by the additional constraints placed on chain conformational freedom imposed by favorable HW contacts formed during the initial phase of adsorption. These two constraints serve to frustrate the chain's attempts to access the global energy minimum through conformational change.  As a result, any  63  attempted move by the chain generally leads to an increase in system energy, such that the adsorbed native state and sequences close to it sit in a local energy minimum.  Eventually, the chain randomly samples a conformation on the surface that allows a new segment move to be selected that results in a decrease in system energy. As shown in Figure 2.5, the energy of the system then falls toward the global energy minimum through a series of previously inaccessible conformational trajectories. For 2D proteinlike HP chains such as sequence I, our dMC simulations therefore predict that a significant number of simulation steps are required for the protein to find and follow an adsorbed-state conformational trajectory that enables it to reach the global energy minimum.  This result is qualitatively analogous to the typical time-course of a  nonspecific protein adsorption event.  Experimental observations of desorption of  nonspecifically bound proteins during the initial stages (short times) of protein contact are numerous (20, 48), and several groups have proposed models for the kinetics of nonspecific protein adsorption that segregate the process into a fast reversible adsorption step, followed by a slow irreversible adsorption process (5).  Protein exchange  experiments by Balasubramanian et al. (55) and Bentaleb et al. (56) show that direct exchange between solution and surface-bound proteins occurs during short contact times with the sorbent and then rapidly diminishes with longer adsorption times.  In our simulations, once the system, first accesses its global energy minimum, the chain appears to be irreversibly adsorbed to the surface, at least with respect to the longest simulation interval we could achieve (a series of ten 1 x 10 step dMC simulations of the 10  64  adsorbed protein in which the starting conformation of the new simulation corresponded to the final conformation of the last). Therefore, a striking feature of our dMC results is the observation that a series of reversible actions (all d M C moves are intrinsically reversible) can result in effectively irreversible adsorption behaviour, at least with respect to the duration of the simulation.  This same observation was made by Anderson et  a/.(38) in their dMC studies of the adsorption of protein-like chains to a liquid-liquid interface. More importantly, slow (or non-existent) desorption kinetics following longtime exposure of a protein to a solid surface have been reported in most experimental studies of nonspecific protein adsorption (1).  For example, studies by Suttiprasit et al.  (57) and Giacomelli et al. (28) show that conformational changes originating from adsorption occur slowly and can be detected for hours to days after an initial adsorption event.  2.5 Summary There is now ample experimental evidence showing that protein conformational changes occur during the nonspecific adsorption of proteins to solid-liquid interfaces. However, it remains unclear under what conditions these structural changes contribute to adsorption energetics through a concomitant gain in chain conformational entropy. Dynamic Monte Carlo simulations of short protein-like HP chains on a solvent-filled two-dimensional lattice were used to show that adsorption to a solid-liquid interface of chains having a global energy minimum in solution' characterized by a single native-state conformation results in a new lowest energy state populated by multiple adsorbed-chain conformations. While the HP chains used are not proteins, we have shown that their simulated adsorption  65  properties qualitatively mirror those of real proteins.  Thus, dMC simulations of HP  chains may provide a simple but useful means of understanding molecular aspects of nonspecific protein adsorption.  In simulations of low-stability HP chains adsorbing to relatively hydrophilic sorbents, the conformational entropy of the chain increases  significantly, in part due to the  conformational degeneracy of the global energy minimum.  However, gains in the  conformational entropy of the chain quickly disappear when either the stability of the native-state fold or the hydrophobicity of the sorbent is increased.  Under these  conditions, energetically accessible' conformations of the adsorbed chain are severely restricted due both to the need to maximize the total number of favorable intramolecular (HH) and intermolecular bonds formed within the chain and with the surface, respectively, and to the physical restrictions imposed by the impermeable nature of the sorbent surface.  The adsorption trajectory of our model protein-like chains exhibit features consistent with experimental studies of nonspecific protein adsorption. Initially, the model chain adsorbs reversibly at energies well above the global energy minimum. The dMC simulations indicate that this is due to local energy barriers that frustrate attempts by the chain to access conformational trajectories leading to the global energy minimum. 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Colloid Interface Sci., 112(1986)447-456. 44.  Welzel, P.B., Investigation of adsorption-induced structural changes of proteins at solid/liquid interfaces by differential scanning calorimetry. Thermochim. Acta, 382 (2002) 175-188.  45.  Jackler, G., Steitz, R. and Czeslik, C , Effect of temperature on the adsorption of lysozyme at the silica/water interface studied by optical and neutron reflectometry. Langmuir, 18 (2002) 6565-6570.  46.  Metropolis, N . , Rosenbluth, A.W., Rosenbluth, M.N., Teller, A . H . and Teller, E . , Equation of state calculations by fast computing machines. J. Chem. Phys., 21 (1953) 1087-92.  47. Norde, W., Adsorption of proteins from solution at the solid-liquid interface. Adv. Colloid Interface Sci., 25 (1986) 267-340. 48. Wertz, C.F. and Santore, M . M . , Effect of surface hydrophobicity on adsorption and relaxation kinetics of albumin and fibrinogen: Single-species and competitive behaviour. Langmuir, 17 (2001) 3006-3016. 49.  Przybycien, T . M . A phase diagram for the structural response of proteins to hydrophobic chromatography. (2003).  50. Fernandez, E.J. Toward modeling of hydrophobic interaction chromatography (hie): Multiple tools to connect thermodynamics, adsorption and conformation. (2003). 51.  Omanovic, S. and Roscoe, S.G., Interfacial behaviour of beta-lactoglobulin at a stainless steel surface: An electrochemical impedance spectroscopy study. J. Colloid Interface Sci., Ill (2000) 452-460.  52. Zoungrana, T., Findenegg, G.H. and Norde, W., Structure, stability, and activity of adsorbed enzymes. J. Colloid Interface Sci., 190 (1997) 437-48. 53.  Hwang, P.M., Choy, W.Y., Lo, E.I., Chen, L . , Forman-Kay, J.D., Raetz, C.R.H., Prive, G.G., Bishop, R.E. and Kay, L . E . , Solution structure and dynamics of the outer membrane enzyme PagP by NMR. PNAS, 99 (2002) 13560-13565.  54.  Prompers, J.J. and Bruschweiler, R., Reorientational eigenmode dynamics: A combined M D / N M R relaxation analysis method for flexible parts in globular proteins. J. Am. Chem. Soc, 123 (2001) 7305-7313.  70  55.  Balasubramanian, V . , Grusin, N.K., Bucher, R.W., Turitto, V.T. and Slack, S.M., Residence-time dependent changes in fibrinogen adsorbed to polymeric biomaterials. J. Biomed. Mater. Res., 44 (1999) 253-260.  56.  Bentaleb, A., Abele, A., Haikel, Y., Schaaf, P. and Voegel, J.C., FTIR-ATR and radiolabeling study of the adsorption of ribonuclease A onto hydrophilic surfaces: Correlation between the exchange rate and he interfacial denaturation. Langmuir, 14 (1998) 6493-6500.  57.  Suttiprasit, P., Krisdhasima, V . and McGuire, J., The surface-activity of alphalactalbumin, beta-lactoglobulin, and bovine serum-albumin .1. surface-tension measurements with single-component and mixed-solutions. J. Colloid Interface Sci., 154(1992)316-326.  71  2.7  Tables  Table 2.1:  Thermodynamic changes for the native to denatured state transition of  sequence I at reduced temperatures T* = 0.44 and 7* = 0.71.  Reduced  Internal energy  Helmholtz energy  Entropy  temperature A . <E>. N D  A . <A> N D  TA . <S> N D  (kT)  (kT)  (kT)  T* = 0.44  +90.4  +76.9  +13.5  7* = 0.71  +31.4 •  +25.5  +5.8  72  Table 2.2:  Thermodynamic changes for the adsorption of sequence I to a relatively  hydrophilic surface  Energy ratio  XHH/XHW  (XHW  = -1) as a function of chain stability  Internal energy  A <E> ads  (%HH)-  Helmholtz energy  Entropy  Aads<A>  TAa <S> ds  (kT)  (kT)  (kT)  (kT)  I  -2.1  -5.7  +3.6  2  -1.9  -5.6  +3.7  3  -3.3  -4.4  +1.1  4  -4.1  -3.8  -0.4  73  Table 2.3: Thermodynamic changes for the adsorptiion of denatured sequence I (ZHH 0.25) on a hydrophobic surface (XHW  =  -4)  Entropy  Internal energy  Helmholtz energy  Kds<E>  &ads<A>  (kT)  (kT)  (kT)  -53.9  -41.8  -12.1  TA <S> ads  Table 2.4:  Thermodynamic changes for the adsorption of sequence I on surfaces of  varying geometry with %  HH  = XHW = -4.  Global energy Chain state  Plane  Internal energy  Helmholtz  Entropy  energy  minimum  TA <S>  Emin  Kds<E>  (kT)  (kT)  (kT)  (kT)  -56.0  -23.8  -22.1  -1.8  -68.0  -35.7  -32.4  -3.3  ads  adsorption  Corner adsorption  75  Table 2.5: Thermodynamic changes for the adsorption of sequence I to a planar surface when XHH = XHW-  Interaction energy  Internal energy  Helmholtz energy  Entropy  XHH XHW  Kds<E>  &ads<A>  TKds<S>  (kT)  (kT)  (kT)  (kT)  -1  -2.1  -5.7  +3.6  -2  -11.3  -10.7  -0.6  -3  -17.7  -16.4  -1.4  -A  -23.8  -22.1  -1.8  =  76  2.8 Figures  Figure 2.1: Examples of Verdier-Stockmeyer moves used to manipulate the protein-like chains during simulations. Shown are a 2-bead crankshaft move, a 1-bead flip and an example of an end-bead turn.  O^KX)*"  1  • OOQO  crankshaft  OOOO * H  77  Figure 2.2: Schematic diagrams of the two model chains: sequence I and sequence II. Filled circles represent hydrophobic (H) units while open circles represent polar (P) units.  Sequence I  Sequence II  78  Figure 2.3: Simulation results demonstrating the conformational dependence of sequence I on temperature.  Shown is the ensemble-averaged number of intramolecular  hydrophobic contacts as a function of reduced temperature, T*. The line drawn indicates the trend of the data.  T*  79  Figure 2.4: System energy probability histograms for sequence I: filled grey bars - chain in solution when XHH = -A (T* = 0.71); filled black bars - chain adsorbed on a relatively hydrophobic surface {% w = -4) when XHH = -4H  The error bars refer to the standard  deviation of 5 runs.  80  Figure 2.5: Energy trajectory for the adsorption of sequence I to a planar hydrophobic surface when XHH = XHW = -4.  Figure 2.6: Representative conformational states of sequence I shortly after adsorbing to a planar hydrophobic surface when XHH = XHW = -4.  All conformations correspond to  system energies E > -4%kT. t  82  Figure 2.7: Representative examples of lowest energy states of sequence I adsorbed to a hydrophobic surface when XHH = XHW = -A. At the global minimum, E - -56kT. t  83  Figure 2.8: The 6 lowest energy states for sequence I when %HH =-3 and XHW = -1. The structure in the centre is the adsorbed chain in its native state.  84  3  Mesoscopic adsorption  dynamic of  Monte  protein-like  Carlo  simulations  HP chains  of the  within  laterally  constricted spaces* 3.1  Introduction  The process and consequences of protein adsorption to solid-liquid interfaces have received significant attention over the past half century, due in part to the importance of control of protein adsorption to the design and performance of biomedical implants (e.g., artificial hip and knee joints, contact lenses, vascular grafts), chromatography columns, food processing equipment, etc., and to the inherent complexity of nonspecific protein adsorption that effectively limits our ability to understand and model the process using classic adsorption theories. Much of what we know about nonspecific protein adsorption has been derived from experimental studies, beginning with the now classic works of Vroman on blood proteins (1, 2), and the early seminal contributions of Norde and Lyklema (3-8). Consistent with the chemical heterogeneity of proteins, experiments have shown that intra- and intermolecular Coulombic, hydration, hydrogen-bonding, and shortrange van der Waals forces can each contribute, either favorably or unfavorably, to the kinetics and energetics of protein adsorption (9).  Intermolecular contacts between the  sorbent and the adsorbed protein are often formed at the expense of intramolecular contacts that are stable in the native-state conformation of the protein.  As a result,  changes in protein conformation are often observed during adsorption (9,10).  * A version of this chapter is currently in press in the Journal of Colloid and Interface Science.  85  Conformational changes in proteins during adsorption have been shown to depend on a number of factors, including protein and sorbent surface hydrophobicity (11, 12), and the thermodynamic stability of the native-state conformation of the protein (13, 14). Percent occupation of total binding sites on the sorbent surface also affects the extent of conformational change, with the magnitude of the change typically being largest at low surface coverage (10, 15, 16).  For example, Norde and Favier (17) used circular  dichroism spectra of proteins displaced from the surfaces of various sorbents to show that proteins adsorbed at high surface coverage retain more secondary structure than do proteins adsorbed at lower surface concentrations. Norde et al. (18) also used differential scanning calorimetry (DSC) to show that denaturation temperatures of adsorbed proteins tend to decrease with decreasing surface coverage. These DSC studies also show that denaturation enthalpies (per mole of protein) are often smaller for adsorbed proteins and tend toward zero as the surface coverage decreases.  Enzyme activity assays (16, 18),  N M R and Raman spectroscopy (19, 20), total internal reflectance fluorescence (21, 22), atomic force microscopy (AFM) (23, 24), and ellipsometry (25, 26) are but some of the many other experimental techniques that have confirmed changes in protein conformation upon adsorption and the dependence of these changes on the adsorption conditions.  One consequence o f increased surface coverage is increased excluded volume effects between proximally adsorbed protein molecules that prohibit one protein molecule from accessing the volume occupied by that of a second. As a result, at high surface coverage, the peptide chain o f an adsorbed protein molecule cannot easily 'spread out' to sample low-energy extended conformations. Ellipsometry (26, 27) data support this model by  86  showing that the thickness of adsorbed protein layers tends to increase with increasing surface coverage.  However, little is known about how changes in accessible sorbent  surface area and restrictions in the volumetric space above it affect the energy and entropy of an adsorbed protein macromolecule.  Recently, we have shown that mesoscopic 2D dynamic Monte Carlo simulations of the adsorption to a planar surface of a uniquely folding linear copolymer comprised of a specified sequence of hydrophobic (H) and polar (P) segments (i.e., the HP chain model of Dill (28)) share many properties characteristic of nonspecific protein adsorption (29). Although they are based on an idealized representation of protein-like chains, the simulations predict energy and chain entropy changes consistent with those observed in experiments of protein adsorption and therefore add to our general knowledge of the adsorption process by providing molecular-level insights and the ability to precisely quantify contributions of the various adsorption sub-processes (e.g., conformational changes in the chain and the associated change in chain entropy, etc.). Here, mesoscopic dynamic Monte Carlo simulations of simple protein-like HP chains are used to explore the dependence of adsorbed-chain conformations and energetics on the amount of sorbent surface area and proximal volumetric space available to the native-state sequence for adsorption. In the dynamic Monte Carlo method, the conformational trajectory of the adsorbing chain molecule is determined by the change in system energy that results from a change in chain conformation and any associated change in the number of intra- and intermolecular contacts. The method therefore allows one to calculate changes in chain conformational space and entropy, as well as changes in chain and system energy and  87  their dependence on adsorption conditions, including the amount of sorbent surface area and associated volume available to the adsorbing chain. Results from our simulations are compared with experimental data for the adsorption of hen egg-white lysozyme to silica with the aim of qualitatively connecting observations drawn from the simple mesoscopic simulations to relevant macroscopic data for a real protein adsorption process.  3.2 Dynamic Monte Carlo simulations of chain adsorption Dynamic Monte Carlo (dMC) simulations have been used by our group (29) and others (30-32) to study adsorption of simple protein-like chains to planar liquid-liquid (oilwater) (31) and solid-liquid interfaces.  The details of the simulation method and its  application to the adsorption of flexible copolymer chains can be found in those references. However, application of the dMC technique to analysis of the dependence of adsorbed-state conformations and energetics on the available sorbent surface area and interfacial volume, the purpose of this study, required modification of the model and our previously reported dMC simulation code.  A brief description of the basic d M C  simulation method and necessary modifications made for this study are therefore presented below.  Chain movements are carried out by standard Verdier-Stockmeyer moves on a 2D lattice where the axial distance between adjacent lattice sites is x (or y) = a.  The model is  ergodic, as the set of Verdier-Stockmeyer moves allow the chain to adopt all possible conformations within the lattice. Chain dynamics and associated system energetics are determined using the Metropolis algorithm, which uses Boltzmann-weighted statistics to  88  determine allowed moves. The energy,  of the lattice is defined by the sum of pair-  wise interaction energies between all non-connecting neighboring units of the system:  N-\  L-\  N  i=ZEvfc - j) - IX/ i  E  r  +  t j>,  where  [ j]  i  3  represents the total number of lattice sites in the grid, % is the interaction  energy between the components occupying lattice sites / and j, r and /} are the respective t  coordinates of lattice sites /' and j, and S is a delta function based on the relative positioning of the two interacting species. For neighboring residues, b\a) = 1, while for non-neighboring residues, <5(|r, - r \ > a) = 0. The second term on the right-hand side of }  Eq. [3.1]  subtracts all contributions from pair-wise energies between all directly  connected residues on the polymer chain. The interaction energy coy is related to the Flory interaction parameter %i/(T) by Xy  =  (OylkT. Finally, / indexes the chain bond  number and therefore counts from 1 to 1-1, where L is the number of segments in the chain.  In the dMC simulations reported here, each lattice site is occupied by one of five different components: a hydrophobic (H) chain segment, a polar (P) chain segment, solvent (S), a sorbent wall (W) segment, or an athermal wall (A) segment.  Reflective boundary  conditions are used at each wall. The positions of wall segments are fixed in the lattice and solvent and chain segments are not allowed to occupy or pass through wall segments. The Cartesian lattice is allowed to vary in width in one lattice unit increments, with the  89  top and bottom surfaces of the lattice always presenting a planar sorbent wall (S) and the two side surfaces serving as athermal walls (A) which act only to limit the sorbent surface area and volumetric space available to the chain for adsorption. Changes in the available surface area and associated volumetric space above it for an individual proteinlike chain to adsorb are therefore achieved in the dMC simulations by moving the parallel side walls of the lattice closer or further apart.  All simulations involve the adsorption of individual protein-like chains to the planar sorbent surface at the bottom of the lattice. At the start of each run, the chain is placed directly on the surface and centered equidistantly from the parallel athermal side walls (A) which are impenetrable to the chain. As a result, the sorbent surface and the two side walls act to confine the chain by restricting chain movement in three of four directions. In cases when the initial chain dimensions match the grid width, contact with the adjacent walls at the start of the simulation is permitted. The starting chain conformation is dependent on the width of the sorbent surface. In most simulations, the chain is initially in its lowest energy "native state" conformation; however, simulations in which the chain is initially in a random denatured conformation were also performed to ensure that the simulation results are independent of initial chain conformation.  Non-native chain  conformations were also used to initiate dMC simulations in which the distance of separation between the athermal side walls of the lattice is less than the width of the chain in its native state.  90  Adsorption thermodynamics were calculated from probability histograms generated from Monte Carlo sampling according to,standard statistical mechanical methods. Details of the sampling method and calculation procedures can be found in Liu and Haynes (29). All simulation runs were repeated at least 5 times and averaged to obtain thermodynamic values.  The HP model of Dill and coworkers (28, 33) was used to specify chain composition in the simulations. In the HP model, a linear chain is comprised of two types of segments: hydrophobic (H) and polar (P). The sequence of H and P segments is specified such that the chain adopts a single lowest-energy conformation in the solvent. chain exhibits protein-like properties.  As a result, the  At very low temperatures, the chain is almost  exclusively in its lowest-energy "native state" conformation.  As the temperature is  raised, higher entropy conformations are increasingly observed.  Analogous to real  proteins, the model chain also experiences a melting transition over a limited temperature range, such that below T (the melting temperature) the chain is most often observed in a m  compact folded conformation, while above T the chain conformation is that of a random m  coil (denatured state). In making this connection, we are not implying that the HP chain captures all physical and thermodynamic properties of real proteins. It clearly does not. The value of the model lies in the fact that the conformation, entropy and energy of the HP chain are sensitive to solution environment in a manner that is qualitatively similar to the observed behavior of globular proteins (see (29) and references therein for a more detailed discussion of the protein-like properties of the HP chain model).  However,  unlike for a complex peptide chain, the simplicity of the mesoscopic HP chain allows one  91  to consider all possible conformations and energetics accessible to the chain in a given environment. connection  It therefore provides a simple but useful model for exploring the  between  conformational  states  of  adsorbed  chains  and adsorption  thermodynamics that is relevant to nonspecific protein adsorption.  The chain is placed on a Cartesian coordinate lattice in a self-avoiding configuration and all lattice sites not occupied by the chain are by definition occupied by solvent.  In all  cases, the sorbent surface at the bottom of the lattice is specified to be hydrophobic, and therefore interacts favorably with the H-residues of the chain. Simulations are run for a sufficient number of steps to allow for sampling of the entire conformational space (usually ca. 10 to 10 9  10  cycles).  The chain used in the simulations (shown in Figure 3.1) is one used in our previous study (29).  The sequence, an 18-mer, is predominantly hydrophobic and folds into a lowest-  energy (native) state having 9 intramolecular H H bonds.  3.3  Materials and Methods  3.3.1 Reagents Hen egg-white lysozyme (HEWL) and P-N-acetylglucosaminidase (NAHase), extracted from jack bean, were purchased from Sigma Chemicals (St. Louis, MO) and used without further purification.  The substrate, p-nitrophenyl penta-N-acetyl-P-chito-pentaoside  92  (pNP-C5) was purchased from Seikagaku Co. (Japan). A l l buffer and background salts were purchased from Fisher Scientific (Nepean, Canada).  All water used in the experiments was distilled and filtered through a Sybron/Barnstead NANOpure II system.  3.3.2 Measurement of Adsorption Isotherms Nonporous microcrystalline silica particles (Sigma, St. Louis, MO) with a size distribution of 0.5 um to 10 um (over 80% of the particles between 1 to 5 microns) were used as the sorbent. The surface of the silica particles used is hydrophilic and has a point of zero charge of ca. pH 3. The silica was washed overnight in a stirring solution of 0.7% sodium persulfate (BDH Ltd., Poole, England) in concentrated sulphuric acid (Fisher Scientific, Nepean, ON). Once cleaned, the mixture was transferred to pyrex glass tubes, spun down, and the acid discarded. The silica was then repeatedly rinsed with water to remove all residual acid and placed overnight in a Precision vacuum oven at 100 ° C . Cleaned and dried silica was stored in a dessicator at room temperature until ready for use.  Stock solutions of H E W L in 50 mM sodium phosphate/sodium citrate buffer (1:1 parts), pH 6.0 were made prior to each experiment. Concentrations of lysozyme were measured by spectral absorption using a Carey IE UV/VIS Spectrometer.  The extinction  coefficient used was 24.826 mL mg' cm" (10 mg/mL, A,=280 nm). A stock suspension 1  1  of silica in sodium phosphate/sodium citrate buffer was also prepared. The mixture was  93  sonicated for 5 minutes. The sonication step effectively disperses the particles in solution with no measurable change in specific surface area. The specific surface area of the silica particles, A , was determined by multipoint BET measurements using nitrogen gas and a s  Quantisorb B E T apparatus (Quantachrome Corporation). The specific surface area, A , s  was found to be 5.6 ± 0.45 m /g silica.  Isotherms for adsorption of H E W L on silica particles were measured by the depletion method.  Adsorption experiments  microcentrifuge tubes.  were  carried  out  in  1.5  mL polypropylene  To each tube, specific amounts of buffer, HEWL and silica stock  solutions were added in proper proportions to achieve a predetermined total protein concentration and sorbent surface area. The total volume of solution in each tube was 1 mL.  Samples were left to turn end-over-end at room temperature (22 °C) for at least 12  hours, giving the system sufficient time to reach steady state.  Samples were then  centrifuged at 14000 rpm for 1 minute, and the supernatant recovered, filtered to remove any residual silica, and analyzed by absorbance at 280 nm to determine the free protein concentration.  A total H E W L  mass balance was then used to determine the  corresponding concentration of adsorbed HEWL.  The centrifuged HEWL-loaded sorbent was recovered and then washed three times with a five-fold excess volume of 50 m M sodium phosphate/sodium citrate buffer (1:1 parts), pH 6.0. The protein bearing sorbent was then resuspended in the same buffer to a total volume of 0.5 mL and allowed to equilibrate for 24 to 48 hours. The solution phase was  94  then assayed for desorbed protein by adsorbance at 280 nm. In all cases, adsorption of H E W L to the silica particles was found to be irreversible.  3.3.3 Lysozyme activity measurements H E W L activity, either in solution or adsorbed to silica particles was determined by measuring the rate of hydrolysis of the soluble substrate pNP-C5 according to a modified version of the assay originally described by Nanjo et al. (34).  HEWL-catalyzed  hydrolysis of pNP-C5 yields short chitooligosaccahrides that are susceptible to further hydrolysis by NAHase to release /?-nitrophenol (pNP), a colorimetric compound that can be detected at 400 nm.  H E W L was adsorbed to silica at either 20% or full surface saturation according to the method used to determine the adsorption isotherm.  Steady-state adsorbed H E W L  samples (see above) were centrifuged at 14000 rpm for 1 minute and the supernatant discarded. Fresh buffer was added and the silica pellet resuspended. The centrifugation, decanting and resuspension steps were then repeated 5 to 7 times in order to eliminate protein that might still be in solution. Following the final rinse, an appropriate amount of buffer was added to adjust the adsorbed protein concentration to 0.0035 mM. To each lysozyme/silica/buffer mixture, appropriate volumes of NAHase and pNP-C5 stock solution were added. The resulting concentrations of the enzyme (NAHase) and substrate were 3.8 xlO" m M and 0.2 mM, respectively. Samples were left to rotate end-over-end 5  at room temperature for a given incubation period (either 20, 40 ot 60 minutes), after which, the solutions were centrifuged and the supernatant removed.  95  Membrane syringe filters (0.2 um Gelman PVDF membrane, Pall Corporation, Ann Arbor, MI) were used to remove any residual silica particulates from the solutions. The absorbance of each sample was then measured at 400 nm to quantify released pNP. Readings from the spectophotometer were adjusted to compensate for any losses of the product during the filtration step.  Measurements were also made of the corresponding activity of the buffer solution and H E W L in buffer solutions containing no silica.  In these samples, absorbance was  measured continuously by absorption spectrophotometry at 400 nm.  3.3.4 Isothermal Titration Calorimetry A n aqueous titrate solution containing 3.5 mg mL" HEWL was prepared in 50 mM KCI, 1  adjusted to pH 7.0 through appropriate addition of HC1 or K O H .  Clean dry silica was  also prepared as a 50-mM KCI titrahd solution adjusted to pH 7.0. Isothermal titration calorimetry (ITC) experiments were conducted in a Calorimetry Sciences Corp. Model 4200 Isothermal Titration Calorimeter. The titrand, reference, and titrate solutions were thoroughly degassed prior to loading. Once the system reached thermal equilibrium (ca. 2 hrs.), 25 10-uL aliquots of protein solution were sequentially injected into the sample cell, which contained 1 mL of well-mixed, thermally equilibrated silica suspension (0.005 g/mL), with a thermally equilibrated silica suspension of identical volume and composition serving as reference. The time between injections was set at 2400 s to allow a return to baseline signal after each thermal peak. The titrations were carried out so that  96  the surface coverage reached 95% of the adsorption plateau after 10 to 15 injections of titrate solution. Samples were continuously mixed with an internal Rushton turbine-type blade rotating at 100 rpm, which was sufficient to fully suspend the silica and eliminate mass-transfer effects that might broaden the thermal peak.  3.4 Results and discussion 3.4.1 d M C simulations of H P chain adsorption within confined spaces  Figure 3.2 reports the energy £ , trajectory for the first 1.5 x 10 steps of a typical dMC s  simulation of the adsorption of the HP chain to a planar sorbent surface (bottom wall of the lattice) when the length of the sorbent surface is large (i.e. lattice length is greater than the fully extended chain length). The chain is initially in its lowest-energy folded state in solution. Prior to chain contact with the sorbent surface £ , > -9kT. Following chain contact, E rapidly decreases to ca. -AQkT, an energy level corresponding to t  adsorption of the chain to the sorbent surface in an ensemble of relatively compact nonnative conformations. In this simulation, the stability of the native-state conformation of the chain is low (XHH  =  - 1 ) and formation of intermolecular contacts between a  hydrophobic (H) residue of the chain and a sorbent surface site (W) is energetically highly favorable since the wall is hydrophobic (%HW  =  -4). A relatively small number of  simulation steps are therefore required to allow the chain to adopt new more surface associated conformations that further reduce E toward the global energy minimum E „ t  mi  of the system, -56kT. The probability of finding the system at or near the global energy minimum then remains high for the remainder of the simulation. The conformation of  97  the chain is dynamic however, such that the chain frequently adopts higher energy conformations, including conformations close to the native-state conformation. Thus, even under conditions where the native state of the chain is only marginally stable and the hydrophobic segments of the chain have a strong preference for the interface, the adsorbed chain will adopt the native-state conformation, albeit very infrequently. While the results in Figure 3.2 represent a single dMC trajectory, they are consistent with results obtained from a large set of independent dMC trajectories for the same system, indicating that the reported results are not unique to a single simulated adsorption trajectory but are reflective of the general dynamic properties ofthe chain at equilibrium.  Unlike in solution, where the lowest-energy state is occupied by a single conformation (the native state), the global energy- minimum for the adsorbed HP chain can include a rather large number of distinct conformations. Figure 3.3 plots the conformational degeneracy of the global energy minimum, E , for the adsorbed HP chain as a function min  of the width, xs, of the sorbent surface available for chain contact. When xs is equal to or greater than the length of the fully extended chain (18 lattice units), 84 unique conformations of the adsorbed HP chain are observed at E „. mi  decreased, the conformational degeneracy of E  min  However, as xs is  falls rapidly, such that when xs equals 6  lattice units (i.e., 2 lattice units wider than the smallest dimension of the native state of the HP chain), E „ is occupied by only two distinct conformations. At all sorbent widths, mi  the adsorbed chain also adopts a large number of higher energy conformations, as suggested in Figure 3.2. However, our simulation results indicate that the more than one order of magnitude loss in conformational degeneracy observed at E„m is consistent with  98  that observed at all other accessible energy levels, indicating significantly lower chain entropy when adsorption occurs within a confined volume.  The energies accessible to the adsorbed sequence are shown for two different xs values in Figure 3.4 in the form of probability histograms, which report the probability at any given step number of the system being at energy E and the chain in a conformation unique to h  that energy. Reducing the sorbent surface area (and the volumetric space directly above it) reduces the number of accessible energies in addition to reducing the conformational degeneracy of the chain at each E . However, the native-state conformation of the chain t  remains accessible at all xs > 4.  Figure 3.5 reports the probability of finding the adsorbed HP chain in its native-state conformation at any step number following system equilibration. When the native-state conformation of the HP chain is relatively stable in solution {% H = -4), reducing xs H  enhances the probability of finding the adsorbed chain in its native-state conformation, irrespective of the hydrophobicity of the sorbent surface. Confinement eliminates many expanded conformations of the chain (see above) while still allowing the chain to access its native-state conformation. In addition, side-on adsorption of the chain in its nativestate conformation becomes one of the lowest energy conformations when xs = 4. As a result, restricting the volumetric space available to the adsorbed chain significantly increases the stability of its native state conformation. Previously, Zhou and Dill (35) developed an elegant theory to show that confining a model protein obeying two-state unfolding thermodynamics to a small inert space increases the stability of the native state  99  by as much as 15 kcal/mol. For our adsorbed HP chain, this same stabilizing effect is somewhat weaker due to the fact that one of the walls of the confining space is no longer inert, but instead serves as a sorbent surface offering a favorable energy of interaction with hydrophobic segments of the chain.  When XHH  =  -4 and XHW  =  -1, chain  confinement on three sides (by setting xs = 4) increases the probability of finding the adsorbed HP chain in its native state conformation by over an order of magnitude relative to that observed in solution. This corresponds to an adsorption process that results in a net increase in the stability of the native state of AA4 = -2 kcal/mol despite the energetic driving force to denature the chain provided by the favorable energy of interaction between hydrophobic segments of the chain and the sorbent surface.  Hyperstabilization of the native-state conformation in the adsorbed chain relative to the same chain in solution is no longer observed when the energy of interaction between the sorbent surface and hydrophobic chain becomes more favorable (XHW ~ -4)- In this case, the probability of finding the HP chain in its native-state conformation is reduced by nearly three orders of magnitude following adsorption to the sorbent surface when xs = 18. A decrease in xs again results in a dramatically higher probability that the adsorbed chain will adopt its native-state conformation, such that the probability of finding the adsorbed chain in its native-state conformation becomes equal to (but not greater than) that observed for the chain free in solution.  The ability to stabilize native-state conformations of adsorbed proteins through volumetric confinement has been observed experimentally. Eggers and Valentine (36)  100  showed that the melting temperature of a-lactalbumin could be increased by as much as 32 C by confinement in the pores of silica glass.  An alternate means of restricting the volumetric space available to an adsorbed protein is to increase the concentration of protein on the sorbent surface to near monolayer coverage.  Excluded volume forces with neighboring adsorbed protein macromolecules  then effectively eliminate adsorbed state conformations requiring chain extension in lateral directions.  Figure 3.6 reports, in the form of initial reaction rates, the molar  activity of H E W L in solution and when adsorbed to nonporous particulate silica at two different surface concentrations.  All H E W L in the solution phase was removed by  washing prior to analyzing the activity of the adsorbed protein. In all three experiments, the total mass of H E W L in the reaction cell was held constant. Based on activity being a sensitive measure of protein conformation, the results show that the native-state conformational stability of adsorbed HEWL increases with increasing surface coverage. At 20% monolayer coverage, adsorbed HEWL catalyzes the hydrolysis of pNP-C5 at an initial rate approximately 6% of that observed for an equivalent loading of H E W L in solution. At monolayer coverage, trie initial rate of pNP-C5 hydrolysis increases to 34% of that observed in solution, indicating, as predicted in our simulations for simple proteinlike HP chains, that the stability of the native-state of an adsorbed protein can be increased by reducing the lateral volume (i.e. along the sorbent surface) available to the adsorbed protein to adopt extended conformations.  The generality of our model  prediction is supported by a number of previous studies (10, 15-26) on the adsorption of a  101  number of globular proteins to different sorbents which show that structural changes in adsorbed proteins tend to decrease with increasing surface coverage.  When qualitatively linked to the experimental results of Eggers and Valentine (36) and those reported in Figure 3.6, our dMC simulations on protein-like HP chains suggest that any physically meaningful isotherm model for globular protein adsorption to solid-liquid interfaces must account for the dependence of the adsorbed-protein partition function on surface concentration. Confinement reduces both the number of realistically accessible energies, E , and, to an even greater degree, the conformational degeneracy of each t  accessible E . (  As a result, when an adsorbed protein becomes confined, its partition  function, which represents the weighted sum of all possible chain conformations, decreases substantially.  Our simulation results also suggest strategies for improving the performance of technologies and processes based on protein adsorption. selective  adsorptive  chromatography  columns  under  For example, operation of shock-wave,  high-feed-  concentration conditions may serve to reduce irreversible denaturation and inactivation of desired protein products during their purification, particularly in the case of proteins of relatively low native-state stability in solution. Similarly, entrapment of purified proteins through adsorption into volumetrically defined matrices may serve as a powerful formulation tool for long-term storage of protein products in their functionally active form.  102  Recent experimental data indicate that nonspecific protein adsorption to hydrophobic sorbents can result in an observed increase in secondary structures (20). Our HP model is not capable of defining chain secondary structure at a level where one can observe distinct changes in either ct-helix or P-sheet content. measure of segment packing densities.  Instead, the model provides a  Our simulations show that under certain  adsorption conditions, particularly at higher surface coverages, the average segment packing density within the adsorbed chain is similar to or greater than that in solution. Thus, although denaturation of the chain occurs, it does not necessarily lead to a more open average chain conformation on the sorbent surface. Instead it leads to an ensemble of non-native, high segment density conformers of low energy and a concomitant reduction in the conformational entropy of the chain.  3.4.2 Influence of lateral confinement on the thermodynamics  of HP chain  adsorption As reported above, our dMC simulations of simple protein-like HP chains agree with the previous work of Zhou and Dill by predicting that adsorption of chains possessing highly stable native-state conformations in solution to a restricted volume for which the confining walls are either inert or offer only a very weak attraction for the chain results in stabilization of the native-state conformation by shifting the equilibrium away from the denatured state. For the HP chain under adsorption conditions where XHH < -4 and -1 < XHW < 0, the ensemble averaged energy of adsorption AUIkT approaches the energy difference for side-on adsorption of the native-state as xs nears 4, the edge-length of the chain in its native state. Thus, the chain has little tendency to change conformation  103  during adsorption and adsorption isotherm models that ignore conformational changes in the chain are adequate.  However, our dMC simulation results also suggest that adsorption thermodynamics will quickly become intimately linked with perturbations in chain conformation when the stability of the native state is reduced, the attraction for the sorbent surface is increased, or average sorbent surface area available per protein macromolecule is increased. Here, we report adsorption thermodynamics for these more interesting cases.  For a weakly stable (%HH - -1) native state of the HP chain adsorbing to a weakly attractive sorbent surface (%HW  =  -1), ensemble-averaged thermodynamic properties  calculated from our dMC simulation data indicate that binding is both energetically (AU/kT = A<£>/£7/= -2.1±0.1 when x =\S) s  and entropically (AS/* = A<S>/k = 3.7±0.1)  favored, particularly when xs is large (Figure 3.7). The favorable entropy of adsorption arises in this case because the adsorbed chain accesses a significantly larger number of conformations and associated energies E than does the native-state chain in solution. t  Energy indexing of all chain conformations shows that laterally-stretched conformations of the adsorbed chain preferentially populate energy states (E values) near the global t  energy minimum E  min  of the system. When xs is reduced sufficiently, these stretched  conformations are no longer accessible and the probability of finding the system in the associated low-energy state is either substantially decreased or zero. As a result, AUIkT is a function of surface coverage.  The entropic contribution to adsorption AS/k also  104  decreases with decreasing xs due to the inherent loss in conformational degrees of freedom that accompanies a reduction in free volume.  Evidence of a change in adsorption energy with increasing surface coverage is provided in Table 3.1, which shows the dependence of the molar enthalpy of adsorption (AH ds), a  measured by isothermal titration calorimetry, for binding of H E W L to silica as a function of percent surface coverage.  At low surface coverage, &H ds (per mol of adsorbed a  protein) is large and endothermic, indicative of an adsorption process which results in significant changes in protein conformation and the disruption of a large number of favorable intramolecular contacts. At high surface coverage, AH ds becomes exothermic, a  in qualitative agreement with our simulation results which predict that adsorption thermodynamics (and the ensemble of conformations accessible to the adsorbed protein) will vary with percent surface coverage.  Adsorption thermodynamics of the HP chain and the influence of confinement are quite sensitive to the stability of the native state of the chain in solution, and the energy of attraction between hydrophobic segments of the chain and the sorbent surface. system reported in Figure 3.7,  In the  AU/kT and AS/k are both favorable and their sensitivities to  available sorbent surface area both influence the dependence of the overall driving force for adsorption AAlkT on xs. In contrast, Figure 3.8 reports adsorption thermodynamics regressed from dMC simulation data for the HP chain when XHH  =  XHW = -4. In this case,  the formation of intramolecular H H and intermolecular HW contacts are both highly favorable and the dominant contribution to AAlkT is provided by the energy change  105  AU/kT. As a result, adsorbed chain conformations that maximize the total sum of H H and H W contacts are favored and the probability of finding the system at or near E  min  high.  is  Since the sorbent surface is rigid and impenetrable, the chain must adopt very  specific conformations and points of contact to access these low energy states. At xs = 18, the simulation results in Figure 3.8 therefore show the entropy of adsorption AS/k to be unfavorable; that is, the coriformational entropy of the HP chain is lower in the adsorbed state than in solution. Although the conformation of the adsorbed chain is far from its native state, the chain entropy is lower than in solution due to the conformational restrictions imposed by the need to maximize energetically favorable contacts.  When xs is reduced, AS/k increases linearly, as opposed to the highly nonlinear dependence observed when both XHH and XHW are set to -1. Our dMC results indicate that this linear trend arises because, although it does decrease as £ , moves sufficiently far away from E , the weighted contribution of laterally elongated conformations of the HP min  chain at each E remains fairly constant for energies near E . t  min  landscape for the system is effectively restricted to these low  As the accessible energy ASIk exhibits a linear  dependence on xs-  Finally, Figure 3.9 reports ensemble-averaged adsorption thermodynamics for the HP chain when the attraction of the sorbent surface for hydrophobic (H) segments of the HP chain is stronger than that between two H segments {XHW = -2 and XHH = -1)-  In this  asymmetric system, the adsorption energy, AU/kT, favors breakage of n H H contacts to form 0.5/z + 1 or a greater number of HW contacts. At equilibrium, the average number  106  of H W contacts therefore increases relative to that observed in the two symmetric adsorption systems described above, and the weighted contribution of laterally stretched conformations of the chain also increases at all E since these conformations favor t  formation of HW contacts. As a result, AU/kT and AAlkT depend more strongly on xs, increasing with decreasing sorbent surface area at all xs less than 18, the length of the fully stretched chain.  3.5  Summary  Dynamic Monte Carlo simulations of a protein-like HP chain were used to investigate the influence of lateral chain confinement on adsorption thermodynamics and adsorbed chain conformational space. Adsorption results in a net loss (generally substantial) of nativestate conformation when the volume available for adsorption, defined by the volumetric space proximal to and directly above the available sorbent surface area per adsorbing chain, is large compared to the fully stretched chain length.  Confinement of the  adsorbed chain is shown to dramatically stabilize the native-state conformation due to selective removal of denatured (particularly elongated) chain conformations from the ensemble of accessible states. For the case where the native-state conformation of the chain is relatively stable in solution and its energy of attraction to the sorbent surface is relatively weak, adsorption to an expansive sorbent surface results in destabilization of the native-state. However, increasing lateral confinement of the adsorbed chain leads to hyperstabilization of the native-state conformation of the chain in the adsorbed state relative to free in solution.  This stabilization effect correlates with a loss in the  107  conformational degeneracy of favorable low energy states that results in an increased probability of finding the adsorbed chain in its native-state conformation.  Lateral confinement of the adsorbed chain has a more complicated effect on overall adsorption thermodynamics, with AU/kT and AS/k typically showing significantly different dependences on xs. When the sorbent surface is relatively hydrophilic, AS/k becomes progressively more unfavorable as xs is decreased, consistent with the decreased free volume conformation.  and the For  associated adsorption  hyperstabilization to  hydrophobic  of  the  surfaces,  unique  native-state  however,  AS/k  is  thermodynamically unfavorable at all xs, indicating that the number of conformations accessible to the chain in the adsorbed state is largely dictated by the energetic penalty associated with conformations that lead to a net reduction in the total number of favorable intermolecular H W contacts.  108  3.6  References  1. Vroman, L., Blood. Natural History Press, Garden City, NY, 1967. 2. Vroman, L . and Adams, A.L., Adsorption of proteins out of plasma and solutions in narrow spaces. J. Colloid Interface Sci., I l l (1986) 391-402. 3. Norde, W. and Lyklema, J., The adsorption of human plasma albumin and bovine pancreas ribonuclease at negatively changed polystyrene surfaces: I. Adsorption isotherms, effects of charge, ionic strength and temperature. J. Colloid Interface Sci., 66(1978) 257-265. 4. Norde, W. and Lyklema, J., The adsorption of human plasma albumin and bovine pancreas ribonuclease at negatively changed polystyrene surfaces: II. Hydrogen ion titrations. J. Colloid Interface Sci., 66 (1978) 266-276. 5. Norde, W. and Lyklema, J., The adsorption of human plasma albumin and bovine pancreas ribonuclease at negatively changed polystyrene surfaces: III. Electrophoresis. J. Colloid Interface Sci., 66 (1978) 277-284. 6. Norde, W. and Lyklema, J., The adsorption of human plasma albumin and bovine pancreas ribonuclease at negatively changed polystyrene surfaces: IV. The charge distribution in the adsorbed state. J. Colloid Interface Sci., 66 (1978) 285-294. 7. Norde, W. and Lyklema, J., The adsorption of human plasma albumin and bovine pancreas ribonuclease at negatively changed polystyrene surfaces: V. Microcalorimetry. J. Colloid Interface Sci., 66 (1978) 295-302. 8. Norde, W. and Lyklema, J., Why proteins prefer interfaces. J. Biomater. Sci. Polymer Edn., 2 (1991) 183-202. 9.  Haynes, C A . and Norde, W., Globular protein at solid/liquid interfaces. Colloids Surfaces B, 2 (1994) 517-566.  10. Malmsten, M . , Formation of adsorbed protein layers. J. Colloid Interface Sci., 207 (1998) 186-99. 11. Wertz, C.F. and Santore, M . M . , Effect of surface hydrophobicity on adsorption and relaxation kinetics of albumin and fibrinogen: Single-species and competitive behaviour. Langmuir, 17 (2001) 3006-3016. 12.  Vermeer, A.W.P., Giacomelli, C E . and Norde, W., Adsorption of IgG onto hydrophobic teflon. Differences between the Fab and Fc domains. Biochim. Biophys. Acta, 1526 (2001) 61-69.  109  13. Tian, M . , Lee, W.-K., Bothwell, M.K. and McGuire, J., Structural stability effects on adsorption of bacteriophage T4 lysozyme to colloidal silica. J. Colloid Interface Sci., 200(1998) 146-154. 14. Billsten, P., Carlsson, U., Jonsson, B.-H., Olofsson, G . , Hook, F. and Elwing, H . , Conformation of human carbonic anhydrase II variants adsorbed to silica nanoparticles. Langmuir, 1 5 (1999) 6395-6399. 15.  Galisteo, F. and Norde, W., Adsorption of lysoyzme and alpha-lactalbumin on poly(styrenesulphonate) latices 2. Proton titrations. Colloid. Surface. B, 4 (1995) 389-400.  16. Kondo, A., Urabe, T. and Yoshinaga, K., Adsorption activity and conformation of alpha-amylase on various ultrafine silica particles modified with polymer silane coupling agents. Colloids Surfaces A, 1 0 9 (1996) 129-36. 17. Norde, W. and Favier, J.P., Structure of adsorbed and desorbed proteins. Colloids Surfaces A, 6 4 (1992) 87-93. 18.  Zoungrana, T. and Norde, W., Thermal stability and enzymatic activity of alphachymotrypsin adsorbed on polystyrene surfaces. Colloids Surfaces B, 9 (1997) 157167.  19. Sane, S.U., Cramer, S.M. and Przybycien, T . M . , Protein structure perturbations on chromatographic surfaces. J. Chromatogr. A, 8 4 9 (1999) 149-159. 20.  McNay, J.L. and Fernandez, E.J., Protein unfolding during reversed-phase chromatography: I. Effect of surface properties and duration of adsorption. Biotech. Bioeng, 7 6 (2001) 225-32.  21. Robeson, J.L. and Tilton, R.D.,' Spontaneous reconfiguration of adsorbed lysozyme layers observed by total internal reflection fluorescence with a pH-sensitive fluorophore. Langmuir, 1 2 (1996) 6104-6113. 22.  Buijs, J. and Hlady, V . , Adsorption kinetics, conformation, and mobility of the growth hormone and lysozyme on solid surfaces, studied with TIRF. J. Colloid Interface Sci., 1 9 0 (1997) 171-181.  23. Cullen, D.C. and Lowe, C.R., A F M studies of protein adsorption .1. Time-resolved protein adsorption to highly oriented Pyrolytic-Graphite. J. Colloid Interface Sci., 166(1994) 102-108. 24. Sagvolden, G., Giaever, I. and Feder, J., Characteristic protein adhesion forces on glass and polystyrene substrates by atomic force microscopy. Langmuir, 1 4 (1998) 5984-5987. 25.  Elwing, H . , Protein adsorption and ellipsometry Biomaterials, 1 9 (1998) 397-406.  in biomaterial  research.  110  26.  Giacomelli, C.E., Esplandiu, M.J., Ortiz, P.I., Avena, M.J. and De Pauli, C P . , Ellipsometric study of bovine serum albumin adsorbed onto electrodes. J. Colloid Interface Sci., 218 (1999) 404-411.  27.  Malmsten, M . and Lassen, B.,Ellipsometry studies of protein adsorption at hydrophobic surfaces. In "Proteins at Interfaces II: Fundamentals and Applications" (T.A. Horbett and J.L. Brash, Eds.), p.228-238. American Chemical Society, Washington, DC, 1995.  Ti/Ti02  28. Lau, K.F. and Dill, K.A., A lattice statistical mechanics model of the conformational and sequence spaces of proteins. Macromolecules, 22 (1989) 3986-3997. 29.  Liu, S.M. and Haynes, C.A., Mesoscopic analysis of conformational and entropic contributions to nonspecific adsorption of HP copolymer chains using dynamic Monte Carlo simulations. J. Colloid Interface Sci., In press (2004)  30.  Zhdanov, V.P. and Kasemo, B., Simulations of denaturation of adsorbed proteins. Phys. Rev. E, 56 (1997) 2306-9.  31.  Anderson, R.E., Pande, V.S. and Radke, C.J., Dynamic lattice Monte Carlo simulation of a model protein at an oil/water interface. J. Chem. Phys., 112 (2000) 9167-85.  32. Castells, V., Yang, S. and Van Tassel, P.R., Surface-induced conformational changes in lattice model proteins by Monte Carlo simulation. Phys. Rev. E, 65 (2002) 33. Dill, K.A., Bromberg, S., Yue, K., Fiebig, K . M . , Yee, D.P., Thomas, P.D. and Chan, H.S., Principles of protein folding - A perspective from simple exact models. Protein Sci., 4 (1995) 561-602. 34. Nanjo, F., Sakai, K. and Usui, T., p-nitrophenyl penta-N-acetyl-beta-chitopentaoside as a novel synthetic substrate for the colorimetric assay of lysozyme. J. Biochemistry, 104 (1988) 255-258. 35.  Zhou, H.-X. and Dill, K . A . , Stabilization of proteins in confined spaces. Biochemistry, 40 (2001) 11289-93.  36. Eggers, D.K. and Valentine, J.S., Molecular confinement influences protein structure and enhances thermal protein stability. Protein Sci., 10 (2001) 250-261.  Ill  3.7  Tables  Table 3.1:  Molar enthalpy change AH  ads  as a function of percent sorbent surface  coverage for the adsorption of H E W L to particulate silica in 50-mM KCI (pH 7) at 37°C. AH  ads  is expressed on a per mole of H E W L adsorbed basis.  AHads  100  xT/T  max  (kcal mol" ) 1  4.5  8.1 ± 6  9.2  8.0 ± 5  18.2  .  7.7 ± 6  32.1  4.8 ± 4  63.8  0.2 ± 4  95.0  -3.4 ± 5  112  3.8  Figures  Figure 3.1:  Schematic diagram of the model HP chain.  Filled circles represent  hydrophobic (H) chain segments while open circles represent polar (P) chain segments.  113  Figure 3.2: Energy trajectory for adsorption of the HP chain to a planar sorbent surface when XHH = -1 and XHW = -4. A l l other segment-segment interaction energies set equal to zero.  114  Figure 3.3:  Dependence of the degeneracy of the lowest energy state (i.e., the total  number of unique chain conformations) on accessible sorbent surface area and volumetric space above it. The x-axis indicates the number of lattice sites on the sorbent surface available for binding. Non-zero segment-segment interaction energies are XHH  =  XHW = -  1.  115  Figure 3.4: Energy probability histograms for the HP chain adsorbed to sorbent surfaces of two different widths: etched grey bars - width of sorbent surface, xs, is 4 lattice units; solid black bars - xs is 18 lattice units. The error bars refer to the standard deviation of 5 runs. Non-zero segment-segment interaction energies are XHH - XHW - \ =  0.25 4  i  1 1 1 1 1  r  0.20 4  0.154  i  0.104  0.05 A  0.00 -14-13-12-11-10 -9  -8  -7  -6  -5  -4  -3  -2  L,  -1  0  116  Figure 3.5: Ratio of the probability of the HP chain being in its native-state conformation when adsorbed, P  ,  adsorbed NS  to that in solution, P  ,  SOLUUON NS  as a function ofthe width ofthe  available adsorption site, xs. Non-zero segment-segment interaction energies are %HH ~ XHW  = -4 (open squares), and XHH - -4, XHW = -1 (open circles).  The lines drawn  indicate the trends of the data.  117  Figure 3.6: HEWL-catalyzed pNP-C5 hydrolysis kinetics when H E W L is dissolved in aqueous solution (pH 6, 22 °C) and when HEWL is nonspecifically adsorbed to particulate silica at different levels of surface coverage. In all experiments, the amount of H E W L and the initial concentration of pNP-C5 are held constant at 50 p.g and 0.20 mM, respectively.  Shown are measurements taken for HEWL in solution with no surface  (squares), HEWL adsorbed on silica at monolayer coverage (circles), and H E W L adsorbed at 20% monolayer coverage (triangles).  0.120.10-  E  o o  CD O  0.08 H 0.06-j  £ i_ o  0.04-I  §  0.02-  CO  0.00-  0  1  20  1  1  '  40  r—  60  80  Reaction time (minutes)  118  Figure 3.7: Thermodynamics of adsorption of the HP chain to a planar sorbent surface as a function of accessible sorbent surface area and volumetric space above it. The x- axis indicates the number of lattice sites on the sorbent surface available for binding. Nonzero segment-segment interaction energies are %HH = XHW = -1 • Trends indicated are AU/kT (solid line), AA/k (dashed line) and AS/k (dash-dotted line). To avoid excessive clutter, simulation points are shown for AU/kT only. Error bars are calculated from the standard deviation of 5 runs.  119  Figure 3.8: Thermodynamics of adsorption of the HP chain to a planar sorbent surface as a function of accessible sorbent surface area and volumetric space above it. Non-zero segment-segment interaction energies are XHH = XHW = -4- Symbology is the same as in Figure 3.7.  -16J  -M.0  -18 J  -1.5  AU/kT -20 H  J-2.0 AS/k  AAlkT -22 A  1-2.5  -241  -1-3.0  120  Figure 3.9: Thermodynamics of adsorption of the HP chain to a planar sorbent surface as a function of accessible sorbent surface area and volumetric space above it. Non-zero segment-segment interaction energies are XHH -1, XHW = -2. Symbology is the same as =  in Figure 3.7.  121  4  Energy landscapes for adsorption of protein-like HP chains as a function of native-state stability*  4.1  Introduction  More than a decade ago, Arai and Norde (1) published their landmark paper describing the dependence of globular-protein adsorption on the thermodynamic stability of the protein in solution at adsorption temperature and pH. Their experimental data indicate that although changes in protein conformation are likely to occur in all adsorption events, the perturbation of a protein's conformation away from the native state on adsorption tends to increase with decreasing thermodynamic stability of the native state in solution. This effect correlated well with their adsorption isotherm data, which revealed a general tendency for less stable proteins to adsorb with higher affinity, particularly in systems where the sorbent surface is weakly attractive. To specifically capture the contribution of native-state thermal stability, Arai and Norde coined the term "soft" and "hard" proteins, with the degree of softness reflecting the susceptibility to denaturing changes in protein conformation upon adsorption. While this concept is perhaps a bit too simplistic, it has remained a part of protein adsorption dogma, in part because it provides a concise and qualitatively useful measure of the importance of protein conformation and stability to the overall adsorption process.  * A version of this chapter is currently in press in the Journal of Colloid and Interface Science.  122  More recent studies by Malmsten and others (2, 3) have contributed further to our understanding  of  thermodynamics.  the  connection  between  structural  and  and  adsorption  For example, Haynes and Norde (4) used differential scanning  microcalorimetry to compare denaturation enthalpies lysozyme  stability  bovine  milk  cc-lactalbumin.  Their  of adsorbed hen egg-white results  revealed  that  the  thermodynamically less stable protein, a-lactalbumin, adsorbed more strongly and denatured more extensively on hematite, a weakly attractive hydrophilic surface.  These findings are supported by a number of adsorption studies involving variations in solution conditions which indicate that proteins generally adsorb with higher affinity under conditions where the temperature (5, 6) or solution pH (7) render the protein less stable. As well, the addition of an ion known to bind specifically and thereby stabilize (through mass action effects) the native state of a protein has been shown to reduce the net force of adhesion between a protein and a sorbent surface (8, 9).  Although comparisons of different proteins over a range of adsorption conditions are useful, interpretation of the results is limited by the fact that factors other than protein stability are altered as well.  Perhaps a more specific approach to understanding the  connection between protein structural stability and adsorption thermodynamics is to study the adsorption of a family of site-directed variants of a protein designed to alter the native-state stability of that protein. McGuire and coworkers investigated the adsorption of wild-type T4 bacteriophage lysozyme and a series of single-site and multi-site mutants of lysozyme to silica and mica surfaces (10-14).  Their circular dichroism data indicate  123  that proteins of lower native-state stability lose larger amounts of a-helix content during adsorption (10, 12, 13).  Interferometric surface-force measurements on the same  adsorption system reinforce this conclusion by showing that the tertiary structures of the lower stability mutants are more severely compromised during adsorption to the negatively charged mica surface, resulting in significantly stronger forces of adhesion (12). Surface-force measurements on T4 lysozyme variants have also been used to show that lower stability variants displace proteins of higher stability during adsorption, indicating a larger force of adhesion for low stability proteins (14).  Billsten et al. (15, 16) observed similar trends for the adsorption of site-directed variants of human carbonic anhydrase II to silica. Using a combination of circular dichroism, fluorescence  spectroscopy and differential scanning calorimetry, they showed that  adsorption resulted in much larger changes in the conformation ofthe less stable variants.  Precisely how protein adhesion forces are strengthened by a decrease in native-state stability remains unclear, in part because the contributions of the various reaction subprocesses (e.g., changes in chain entropy, dehydration effects, etc.) cannot be precisely defined.  Previously, we have shown that two-dimensional (2D) dynamic  Monte Carlo (dMC) simulations of uniquely folding copolymers composed of a linear sequence of hydrophobic (H) and polar (P) segments (i.e., the HP chain model of Dill (17)) allow one to unambiguously compute changes in system entropy and energy accompanying nonspecific adsorption of the chain (18). Although they are based on an idealized protein-like chain, simulation results for this simple model add to our general  124  knowledge of the adsorption process by providing molecular-level insights and the ability to enumerate all conformations and energies available to the chain in solution and adsorbed to the sorbent surface. This is something that cannot at present be done with real proteins.  Here, we  use dMC simulations to improve our general understanding of the  thermodynamics of nonspecific adsorption of a protein-like HP chain and its dependence on the stability of the native-state conformation of the adsorbing chain. Of particular concern to us is how the stability of the native-state conformation limits changes in chain conformation and chain conformational entropy upon adsorption. Calorimetric studies of nonspecific protein adsorption reveal that the process often results in a net increase in system entropy (4).  However, the source of this entropy gain and its dependence on  chain stability remain unclear, as a number of subprocesses, including dehydration of ordered water molecules and an increase in chain conformational entropy, could be at least in part responsible (2, 19).  4.2  Protein-like HP chain and dMC simulation algorithm  Simulation results reported here are' for two linear copolymers (Figure 4.1), sequence I and II, both having specific sequences of hydrophobic (H) and polar (P) residues. The basic chain architectures are drawn from the HP chain model of Dill and coworkers (17, 20), and the sequences are designed so that the chains fold at their global energy minimum into unique compact conformations (hereafter referred to as their native states, respectively) (18). HP chains like that shown in Figure 4.1 have been studied extensively  125  by Dill (17, 20) and many others (see for instance, 21-23) using Monte Carlo and molecular dynamics simulation techniques. Despite their obvious simplicity, the folding dynamics and solution thermodynamics of these model chains have significantly enriched our fundamental understanding of protein folding (20-23) and other macromolecular association events (24, 25).  The details of the algorithm and fundamental equations used in our dynamic Monte Carlo simulations of HP chain adsorption to a planar solid-liquid interface are described in a previous paper (18).  We therefore restrict ourselves to a brief description of the  simulation method and its specific application to the interrogation of the influence of the chain's native-state stability on adsorption thermodynamics.  A single protein-like HP  chain is placed on a Cartesian 2D lattice of sufficient size to allow all possible chain conformations and to prevent the chain from interacting with more than one lattice boundary (which serves as the sorbent surface) during the simulation. The four walls of the lattice, including the one selected as the sorbent surface, are impenetrable and reflective boundary conditions are employed.  The system energy is defined by Flory-  type interaction energies between adjacent components within the lattice and its boundaries. These components include hydrophobic (H) and polar (P) chain segments, the sorbent surface (W) and solvent (S).  In all simulations reported, folding of the HP chain into its native state in solution is exclusively driven by the value of XHH, the Flory parameter between two H residues not directly connected within the chain. Increasingly negative values of XHH shift equilibrium  126  toward stabilization of the native-state conformation. Solvent interactions are assumed to be athermal (i.e., XHS  =  0) and hydration effects in the model are therefore represented  only indirectly, since the favourable H H interaction (and HW interaction when the sorbent wall is active) implicitly makes the HS interaction (and the SW interaction) net unfavourable.  Unless otherwise stated, all dMC simulations were run for lxlO or more cycles, with 9  o system equilibration typically observed within the first 5x10 cycles. Samples were taken every 5000 cycles, giving l x l 0 or more data points for each run. Each model condition 5  was simulated 5 or more times and average values are reported. The change in system energy upon chain adsorption is computed from the resulting energy distribution functions for two well-defined conditions: the chain is initially placed within the lattice in its lowest energy conformation and all boundaries are athermal (initial state), and a previously athermal wall is made attractive to H-segments of the chain (final state). The free energy change AadsA and entropy change A fcS for this process are then computed a<  using standard thermodynamic integration algorithms described by Allen and Tildesley (26) and previously used by Socci and Onuchic (22).  127  4.3  Results and discussion  4.3.1 HP chain adsorption thermodynamics 4.3.1.1 Adsorption of sequence I Table 4.1 reports the calculated change in Helmholtz free energy, A dsA, internal energy, a  AadsU, and entropy, A dsS, resulting from adsorption of sequence I to planar sorbent a  surfaces with different degrees of attraction for H segments of the chain. Consistent with the soft and hard protein model originally proposed by Arai and Norde based on their experimental studies of nonspecific protein adsorption (1), the mesoscopic adsorption thermodynamics for our simple protein-like HP chain reveal a strong correlation between adsorption affinity and thermal stability of the native-state conformation of the chain in solution.  The value of AadsA, which provides a measure of the overall affinity of the  chain for the surface, moves toward positive values with increasing stability of the nativestate conformation, irrespective of the hydrophobicity of the sorbent surface.  For  adsorption on the weakly attractive sorbent (jfrw = -1), AidA increases from -5.67ATto 3.77kT, and therefore becomes thermodynamically less favourable, with a change in XHH from -1 to -4.  On the higher affinity surface (XHW = -4), the change in AadsA is even  more pronounced, with the affinity reduced by nearly half when XHH is changed from -1 to -4.  In contrast to the more complex process of nonspecific protein adsorption, simple thermodynamic integration algorithms (26) can be used to compute zW4 for the adsorption of our simple HP chain, allowing one to identify the molecular basis for the  128  observed dependence of both A dsA and A S on the thermal stability of the native-state a  ads  conformation. In our model, the overall change in system entropy, AadsS, reflects both the change in chain conformational entropy upon adsorption and the increase in the total degrees of freedom that accompanies introduction of the sorbent surface into the system. Figure 4.2 reports both AadsAlkT and AadsS/k as a function of Ao-^A/kT, the Helmholtz energy difference between the native and fully denatured (zero energy) conformations of sequence I for the case where XHW ~ -1-  The dMC data reveal a transition in both  AadsA/kT and AadsS/k centered near Ap^AVkT = -12. At low absolute values of Ao^AVkT (i.e. low native-state stability), the chain in solution trades weakly favourable intramolecular H H interactions for conformational entropy.  As a result, the chain in  solution forms an average of only ca. 4 H H contacts, significantly less than the 9 H H contacts that define its lowest-energy  (native state) conformation.  Along with  hydrophobic residues on the surface of the native state, additional unpaired hydrophobic residues are therefore present, and each may form an intermolecular H W contact to reduce the energy of the system.  The system energy may also be lowered by reducing the average number of H H contacts to form a set of lower energy HW contacts, and this concept has been applied frequently in the protein adsorption literature.  In particular, many have correctly argued that  disruption of specific intramolecular interactions to form more favourable intermolecular contacts can perturb the average chain conformation away from the native state towards a larger density of (denatured) conformational states on the sorbent surface (4,27). In such cases, adsorption may be expected to result in a net increase in entropy due to a net  129  decrease in H H contacts.  Our dMC simulations identify certain adsorption conditions  where a net decrease in intramolecular (HH) contacts is observed, but rarely does this decrease lead to an increase in chain entropy. Moreover, as shown in Table 4.2, a net decrease in H H contacts often does not occur upon adsorption of sequence I, even in cases where an increase in entropy is observed. For instance, adsorption of HP chain sequence I when XHH  =  XHW  =  -1 results in relatively little change in the average number  of H H contacts within the chain, but favourable changes in both Aad U/kT and A dsS/k are S  observed.  a  The resulting decrease in AadsA/kT that drives adsorption is not due to an  increase in chain entropy resulting from disruption of specific intramolecular contacts. Rather, it results from the energy decrease and entropy increase generated by the large density of chain conformations that allow unpaired hydrophobic residues (including solvent-exposed residues in the native state structure) to form contacts with the sorbent.  In contrast, at high absolute values of Ao-NAVkT, such as when XHH  =  -A, sequence I in  solution forms an average of 8 H H contacts and is often observed in its lowest energy conformation. Conformations of the chain in solution which expose additional (relative to those exposed on the surface of the native state) unpaired H residues to the sorbent surface are therefore greatly reduced. Moreover, for adsorption of this more stable chain to the XHW - -1 surface, breakage of an intramolecular H H contact is energetically unfavourable unless a significantly greater number of intermolecular HW contacts can be formed as a result. Thus, as shown in Table 4.2, we again observe no net reduction in H H contacts upon chain adsorption. Conformations the adsorbed chain can adopt to lower the system energy are therefore limited by the general need to form specific  130  intermolecular contacts with the sorbent while retaining highly favourable intramolecular H H contacts. When the sorbent surface is weakly attractive (XHW - -1), adsorption then results in a small loss in entropy (Aad S < 0) and is driven entirely by the decrease in s  internal energy that accompanies chain adhesion.  In the transition region, the system (both in the presence and absence of the sorbent surface) responds to %HH taking on more negative values by trading access to highenergy/high-entropy states to increase the number of energetically favourable H H contacts. The derivatives  8LJJIKT  m  d  <?A S/k ads  HH  carry the same sign; both are positive and thereby confer on the adsorption process a weak form of energy-entropy compensation.  A further examination of the statistical mechanical definition of S and U in terms of the partition function shows that they depend in the same qualitative way on the distribution of the system among different energy levels. If the system is closed and the perturbation adiabatic (e.g., a change in system temperature), the entropy must decrease as the mean energy of the system decreases since no new degrees of freedom have been added into the system (28). In our system, however, the perturbation is isothermal and involves the introduction of the sorbent surface.  As a result, the total degrees of freedom are  131  significantly increased and positive values of A S can be observed despite a decrease in aas  the mean energy.  4.3.1.2 Adsorption of sequence II  Sequence II differs significantly from sequence I.  While sequence I is overall quite  hydrophobic with an amphipolar surface in its native state, sequence II is more hydrophilic and symmetric in nature, displaying a completely hydrophilic surface and hydrophobic core in its native state.  Nevertheless, adsorption thermodynamics for  sequence II are qualitatively similar to those reported for sequence I (Tables 4.3 and 4.4). For example, adsorption affinity again weakens with increasing thermodynamic stability of the native-state conformation, irrespective of the attractiveness of the sorbent surface (Table 4.3). This effect is sufficiently strong to disfavor adsorption of sequence II when the native-state conformation is highly stable (%HH for FI segments of the chain is weak (XHW  =  =  -4) and the attraction of the sorbent  -!)•  In this case, non-native chain  conformations that allow contact between hydrophobic residues of the chain and the sorbent surface are disfavoured relative to the native-state conformation.  Adsorption of sequence II is entropically favored under several conditions (Table 4.3). For example, when XHH  =  XHW  =  -4, AdsS is positive, due in large part to the associated  net decrease in H H contacts (Table 4.4) that allows the chain to access a significantly larger density of conformational states on the sorbent surface.  This mechanism for  increasing the system entropy is well known, primarily through the work of Norde and  132  coworkers [1,2,4]. However, as with sequence I, adsorption of sequence II may lead to a favourable A dsS through other mechanisms. When XHH = XHW a  =  -1, AidsS/k  =  +5.62 and  adsorption is driven purely by entropy (i.e., A - U/kT~ 0), despite a small to insignificant aa  s  increase in the average number of H H contacts. The origin of this favourable AadsS is less obvious, but can be understood through more careful analysis of the energy landscapes for the system when the sorbent wall is first athermal (initial state) and then attractive (final adsorbed state).  4.3.2 Thermally-averaged energy landscape analysis of HP chain adsorption  Energy landscape analysis has become a central tool in understanding the folding of HP and other protein-like chains (20, 29, 30), allowing one, for instance, to visualize the ensemble of parallel pathways a chain may follow to fold into its native-state conformation.  For relatively simple protein-like chains possessing a global energy  minimum occupied by a single chain conformation, the energy landscape for the chain in solution has been shown to resemble a funnel, with the lowest energy state, the native state, occupied with a large Boltzmann weight at temperatures well below the native-todenatured state transition temperature, T , but still high enough that the chain folding m  kinetics are not limited by the inability of the chain to escape from conformations that represent local energy minima.  133  In its simplest thermally-averaged form, energy landscape analysis of chain folding yields a symmetric funnel whose shape is specified by the density of conformational states of the chain as a function of the system energy. Due to its symmetry, the energy landscape can be displayed by plotting the contour of the funnel wall without loss of information content.  Figure 4.3 compares the thermally-averaged energy landscape for HP chain  sequence I in solution to that for the same chain adsorbed to the sorbent surface when XHH XHW =  -1 • Comparing the volumes of the two funnels shows that although the total  =  number of possible chain conformations is the same in the two systems, introduction of the sorbent surface increases the total density of unique states to ca. 4.6 times that for the same chain in solution. This increase in total degrees of freedom is observed because, while a given chain conformation has a single energy in solution, it can be found at different system energies in the presence of a sorbent surface depending on its position and orientation relative to the surface. In a 2D lattice with a planar sorbent surface, a given chain conformation can reside either off the surface with an energy identical to that in the sorbent-free system (initial state), or adsorbed to the surface in one of four possible orientations. Thus, when the sorbent surface is present, any given chain conformation can appear in at most five different energy levels, with the average for all chain conformations being 4.6 different energies due to conformational symmetry effects. The adsorbed-state system (ads) thereby has a higher or equal density of states (Q,) at every energy level Ej. This is the dominant reason why, under certain conditions, A S aas  can be  positive in an adsorption process that lowers the mean energy of the system.  Because Q. ds(N,V,E,) > Q^ (JV,F,£,) at all E iia  ee  h  134  E<  [4.1]  where Q is the partition function of state i and the sum covers all energy levels i t  available to the adsorbed state. The sign and magnitude of A dsS is then determined by a  the fundamental statistical thermodynamic relation,  A S/k ads  =  A U/kT+\rl ads  a  Cads  [4.2]  Q  which states that a positive AadsS can be observed in an isothermal adsorption process in which the mean energy of the system (AdsU) is reduced when Q ds is sufficiently larger a  than Q/ . Because QadJQfree is attenuated by the natural logarithm, Q d must in general ree  a  s  be substantially larger than Qfr for an adsorption process to provide a net increase in ee  entropy when AadsU < 0.  Our M C simulations show that  X ^ = 4.6]TQ  /)/ree  E,  [4.3]  E,  for adsorption of the HP chain to the planar sorbent surface. Eq. [4.3] further constrains Eq. [4.1] such that the value of Qads - Qfree can only be increased by altering how the  135  excess density of states in the adsorbed system is distributed among the energy levels. Taken together with the requirement that Q ds(N,V,Ei) iia  easy to prove that Q  ads  > Q.ij (N,V,Ei) ree  at all  it is then  - Qfr is maximized when the 3.6-fold new states in the adsorbed ee  system are preferentially distributed among the lowest energy levels for that system; that is, at adsorbed-state energies with the largest Boltzmann weightings. Thus, widening the energy landscape funnel for the adsorbed state system at the lowest energy levels (i.e., the funnel tip) increases / W S toward more positive, thermodynamically favourable values.  To fix ideas, consider again the thermally-averaged energy landscapes for sequence I shown in Figure 4.3 for the case where XHH  XHW = -1. In solution, the chain must trade  =  considerable conformational entropy to find its lowest-energy state, which is occupied by a single chain conformation, the native-state conformation.  The thermally-averaged  energy landscape for this process is therefore a "closed-tip" funnel that intersects the abscissa at EjIkT = -9, the minimum energy level for the chain in solution.  In contrast,  our M C simulation data show that sequence I may adopt any one of 84 unique conformations at its lowest energy level E  in the adsorbed state.  min  (number of unique conformational states) at E  min  The degeneracy  and energy levels just above E „  is  mi  significant and a positive AadsS is observed.  Table 4.5 reports Q. in,ads(Emin) for sequence I as a function of the ratio XHHIXHW- There is m  a coarseness to the data because of the short length of our HP chain.  However, our  results show that the degeneracy of the lowest energy state is maximized when XHHIXHW  =  1, irrespective of the value of XHH and thus, the stability of the native state. Likewise, on  136  any given sorbent surface the second term on the right side of Eq. [4.2] is maximized when the two contact energies are equal.  Our results further show that Q  aas  - Q/  ree  decreases quickly as the value of  XHHIXHW  diverges away from unity in either direction because the lowest energy levels for the adsorbed chain become significantly less populated with unique conformational states. At  XHHIXHW  =  4, which includes the case of adsorption of a stable HP chain (%HH = —4) on  the weakly attractive surface  (XHW -  -1), only one chain conformation is observed at  E . MIN  The adsorbed-state energy landscape is therefore a closed-tip funnel, and a negative AadsS is observed due to a dramatic reduction in Q ads  Our results therefore suggest that the  distinct differences noted by Arai and Norde (1) in the adsorption behaviour of proteins having relatively  stable ("hard") versus  relatively  unstable  ("soft") native-state  conformations may be due to the manner in which the two systems distribute the excess states generated by the introduction of the sorbent surface.  4.3.3 The deformation entropy for HP chain adsorption  In response to a change in solvent quality, a linear polymer chain in solution will adopt new conformations in order to decrease repulsive and increase attractive contacts within the chain and between the chain and solvent. As a result, the total number of accessible conformations of the polymer molecule will change and will reach a maximum under socalled theta solvent conditions (31), where the chain assumes an "undeformed" randomflight configuration with overall dimensions solely determined by the bond lengths and 137  angles within the chain. In its theta solvent, the chain will therefore have maximum conformational entropy S  conf  = S° . conf  If the quality of the solvent is made poorer, net  repulsive interactions with the solvent will cause the polymer chain to collapse. Conversely, the volume of the polymer chain will expand in response to net attractive interactions with a good solvent. In either case, S j will decrease as a result of the lower con  number of accessible chain conformations.  Flory called this entropy loss the chain  deformation entropy, A.d JS, where e  A S def  = S -S° conf  [4.4].  conf  Analogous with the elastic properties of a linear chain in solution, our dMC simulations show that the conformational entropy of an adsorbed HP chain is a strong function of the quality of the sorbent surface, such that S / shows a maximum at XH^XHW con  solution, when  XHHIXHW >  1. Like in  1 the sorbent quality is relatively poor and the chain will  collapse on itself to increase the density of more favourable H H contacts. XHHIXHW <  =  When 0 <  1, the chain will expand along the sorbent surface to maximize contact area.  Our objective is to correlate &defi with a measurable property of an adsorbed chain. One such property is the average end-to-end distance of the chain at the sorbent surface, which is proportional to the average diameter of the adsorbed chain and can therefore be estimated at monolayer coverage with knowledge of the available sorbent surface area and an appropriate adsorption isotherm model such as the random sequential adsorption model (32, 33).  138  Flory has shown for a linear chain on a one-dimensional lattice where one end of the chain is fixed at the origin that the position of the remaining chain end is described by a Gaussian-type distribution, W(x), of the form  V*  2  v 2  [4.5]  .  where W(x) is the probability of finding the chain end between x and x + dx. To simplify our analysis in the case of a multidimensional lattice, we invoke the approximation that the effect of the deformation on Aj jS in the y dimension is equal to that in the x e  dimension, so that  W(x,y) * (W(x))  2  = ^-exp(-/?V )  [4.6]  where  [4.7]  4  2/i/;  n is the number of bonds within the chain and l] is the average square of the projection of each bond vector on the x axis. The average end-to-end distance of the adsorbed chain is then n^t*. A S(l ) def  x  We seek to use this result to develop a simple analytical model for  that captures the dependence of AadsS on XHHIXHW observed in our dMC 139  simulations. Let W (xy), given by Eq. [4.6], represent the distribution of chain end-to0  end-distances at the  XHHIXHW  value where S  conf  = S°  conf  and l = l .  will then shift the average length of each bond vector from l  x  Xo  xo  A change in  XHHIXHW  to a new value l , giving a x  new distribution of end-to-end distances  W(x,y) = V^exp(-(/?)V)= ^ - e x p |  2  [4.8]  where  L  [4.9]  a.  The entropy change associated with the deformation in adsorbed chain structure can be estimated from the change in the total density of states through the fundamental relation  [4.10]  where,  Q=  M  N] YlnW and Q = —UnW Yin, 1LH„.  [4.11]  In Eq. [4.11], Q is the density of states at maximum conformational entropy, N is the 0  total number of end-to-end distances sampled and «, is the number within that sample  140  with end-to-end distances in the x dimension between x, and x, + dx. Substitution of Eq. [4.11]  into  Eq. [4.10],  [X"i X" , =N =  0  followed  by  X ^,=X^o/  and  =  application l]  m  e  n  gi  y e s  of o  u  the r  summation  relations  desired result after some  algebraic manipulations  Y T - V  A S/k def  =  a\2lncc -(a -\)]=  21rJ  2  x  x  -1  [4.12]  where a is the dimensionality of the system (a = 2 in our lattice).  Use of Eq. [4.12]  requires values for a , which can be determined from our dMC simulation data. Figure x  4.4 reports the dependence of l on XHHIXHW for our HP chain. As expected, l increases x  x  with increasing quality of the sorbent surface (i.e., with decreasing asymptote observed at high values of  XHHIXHW  corresponds to l  x  XHHIXHW).  The  for native-state  conformation of the chain.  Based on the data in Figure 4.4, Figure 4.5 plots values of A ^ predicted using Eq. [4.12] as a function of l for sequence I. Despite the rather crude assumptions embodied x  in Eq. [4.6], our simple model (Eq. [4.12]) shows qualitative agreement with dMC simulation data. A^jS is predicted to be a maximum at equal to unity.  When  XHHIXHW  XH^XHW  = 1, where a is also x  < 1, the adsorbed chain is observed to spread on the  sorbent surface (ax > 1) and a decrease in AdejS and therefore A dsS is predicted as a a  increases. A decrease in entropy is also predicted and observed when  x  XHHIXHW  >  1 due  141  to a collapse of the adsorbed chain onto itself that leads to  values less than unity.  Here, however, the dependence of A jS on a is predicted to be stronger, indicating that de  x  the dependence of the conformational entropy on chain segment density is enhanced at high segment densities. As a result, small deformations of the adsorbed chain can lead to relatively large changes in entropy.  As shown in Figure 4.5, dMC simulation results for the adsorption of sequence I show a stronger dependence of A jS on l than predicted by Eq. [4.12]. This is due to additional de  x  limitations to chain expansion or compression in the y dimension arising from the impenetrable nature of the sorbent that are not accounted for in our simple model. As a result, the contribution to A jS of deformations in the dimension normal to the sorbent de  surface are somewhat higher than predicted. To empirically include this effect, we have treated a in Eq. [4.12] as an adjustable parameter. With a = 2.8, Eq. [4.12] shows good agreement with our dMC simulation data under conditions in which the sorbent surface is relatively attractive and chain expansion is observed. However, the model under-predicts the loss in entropy due to chain collapse since it does not account for the unique property of a protein-like chain to energetically favour a single conformation upon collapse.  4.4 Summary  In this paper, we have used dynamic Monte Carlo simulations to explore the relationship between the native-state stability of protein-like chains and the thermodynamics of adsorption of the chains onto a solid-liquid interface.  Our results provide molecular  142  insights that help explain the well-known differences in the adsorption behaviour of proteins of low and high native-state stability. Increases in entropy have been observed in protein adsorption to a solid-liquid interface, particularly when the native-state stability of the protein is low. Such increases are most often attributed to a combination of solvent dehydration effects and conformational changes in the protein upon adsorption that increase chain entropy through a net loss in intramolecular interactions stabilizing the native state of the protein.  Our dMC simulations directly probe the latter effect for  simple protein-like chains and show that a net loss in intramolecular H H contacts leading to an increase in chain conformational entropy can be observed under certain conditions. However, the effect is far from general. Instead, positive A S  are more directly related  aas  to the ability of the system to preferentially distribute new states generated by the sorbent surface into adsorbed-state energy levels with the largest Boltzmann weightings.  This  situation is favoured when the average intramolecular contact energy {XHH) equals the average intermolecular contact energy  (XHW),  and maxima in A sS are always observed at aa  this condition. This result therefore provides a possible new explanation for why positive and negative values of  are observed for proteins of low and high native-state  stability, respectively, adsorbing on a weakly attractive (e.g., hydrophilic) sorbent.  Finally, a simple analytical model based on Flory's theory of polymer elasticity was derived and used to correlate changes in adsorption entropy with a measurable physical parameter, the average diameter of an adsorbed chain. Analogous to the behaviour of a polymer in solution, the model predicts that the conformational entropy of an adsorbed chain will be a maximum when XHHIXHW - 1, which can be loosely thought of as the theta  143  condition for the sorbent. A dsS is predicted to decrease when the sorbent becomes more a  attractive due to expansion of the chain along the sorbent surface to maximum more favourable H W contacts.  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Billsten, P., Freskgard, P.-O., Carlsson, U . , Jonsson, B.-H. and Elwing, H., Adsorption to silica nanoparticles of human carbonic anhydrase II and truncated forms induce a molten-globule-like structure. FEBS Letters, 402 (1997) 67-72.  16. Billsten, P., Carlsson, U . , Jonsson, B.-H., Olofsson, G., Hook, F. and Elwing, H., Conformation of human carbonic anhydrase II variants adsorbed to silica nanoparticles. Langmuir, 15 (1999) 6395-6399. 17. Lau, K.F. and Dill, K.A., A lattice statistical mechanics model of the conformational and sequence spaces of proteins. Macromolecules, 22 (1989) 3986-3997. 18.  Liu, S.M. and Haynes, C.A., Mesoscopic analysis of conformational and entropic contributions to nonspecific adsorption of HP copolymer chains using dynamic Monte Carlo simulations. J. Colloid Interface Sci., In press (2004).  19. Norde, W., Energy and entropy of protein adsorption. J. Disper. Sci. Technol, 13 (1992) 363-377. 20. Dill, K.A., Bromberg, S., Yue, K., Fiebig, K . M . , Yee, D.P., Thomas, P.D. and Chan, H.S., Principles of protein folding - A perspective from simple exact models. Protein Sci., 4 (1995) 561-602. 21. Camacho, C.J. and Thirumalai, D., Kinetics and thermodynamics of folding in model proteins. PNAS, 90 (1993) 6369-6372.  145  22.  Socci, N.D. and Onuchic, J.N., Kinetic and thermodynamic analysis of proteinlike heteropolymers: Monte Carlo histogram technique. J. Chem. Phys., 103 (1995) 4732-44.  23. Konig, R. and Dandekar, T., Solvent entropy-driven searching for protein modeling examined and tested in simplified models. Protein Eng., 14 (2001) 329-35. 24.  Gupta, P., Hall, C K . and Voegler, A . C , Effect of denaturant and protein concentrations upon protein refolding and aggregation: A simple lattice model. Protein Sci., 7 (1998) 2642-2652.  25. Nguyen, H.D. and Hall, C.K., Effect of rate of chemical or thermal renaturation on refolding and aggregation of a simple lattice protein. Biotechnology and Bioengineering, 80 (2002) 823-834. 26. Allen, M.P. and Tildesley, D.J., Computer simulations of liquids. Oxford University Press, New York, NY, 1987. 27.  Norde, W. and Lyklema, J., Why proteins prefer interfaces. J. Biomater. Sci. Polymer Edn., 2 (1991) 183-202.  28. Hill, T.L., A n introduction to statistical thermodynamics. Dover Books, New York, N Y , 1986. 29. Dill, K . A . and Chan, H.S., From Levinthal to pathways to funnels. Nature Structural Biol, 4 (1997) 10-19. 30.  Anderson, R.E., Pande, V.S. and Radke, C.J., Dynamic lattice Monte Carlo simulation of a model protein at an oil/water interface. J. Chem. Phys., 112 (2000) 9167-85.  31. Flory, P.J., Principles of polymer chemistry. Cornell University Press, Ithaca, N Y , 1953. 32. Widom, B., Random sequential addition of hard spheres to a volume. J. Chem. Phys., 44 (1966) 3888. 33. Tarjus, G., Schaaf, P. and Talbot, J., Generalized random sequential adsorption. J. Chem. Phys., 93 (1990) 8352-8360.  146  4.6  Tables  Table 4.1: Adsorption thermodynamics for HP chain sequence I as a function of nativestate stability and sorbent surface affinity.  XHW  XHH  AadsA/kT  AadsU/kT  -1  -1  -5.67  -2.07  3.60  -4  -3.77  -4.13  -0.36  -1  -41.87.  -49.61  -7.75  -A  -22.10  -23.85  -1.75  -A  A^/k  147  Table 4.2: The average number of favorable intramolecular and intermolecular contacts formed by sequence I in solution and when adsorbed onto a planar surface.  Average number of Average number of H H contacts XHW  -1  -A  H H + H W contacts  XHH  Solution state  Adsorbed state  Adsorbed state  -1  4.3  4.4  6.3  -A  7.7  7.8  12.0  -1  4.3  0.3  13.5  -4  7.7  5.8  13.7  148  Table 4.3: Adsorption thermodynamics for sequence II as a function of native-state stability and sorbent surface affinity.  AadsU/kT  AdsS/k  -5.79  -0.17  +5.62  -A  +0.03 '  -0.19  -0.22  -1  -18.93  -30.88  -11.95  -4  -14.76  -9.58  +5.18  XHW  XHH  -1  -1  -A  AadsAJkT  149  Table 4.4: The average number of favorable intramolecular and intermolecular contacts formed by sequence II in solution and when adsorbed onto a planar surface.  Average number of Average number of HH contacts XHW  -1  -A  H H + HW contacts  XHH  Solution state.  Adsorbed state  Adsorbed state  -1  1.5  1.6  1.7  -4  7.5  7.6  7.6  -1  1.5  2.4  9.9  -A  7.5  3.0  10.6  150  Table 4.5: The density of unique conformational states for the adsorbed chain at the global energy minimum energy state as a function of XHHIXHW-  XHH/XHW  Qmin,ads  0.25  4  0.50  4  1.0  84  2.0  5  3.0  6  4.0  1  151  4.7  Figures  Figure 4.1: Schematic of HP chain sequences used in these simulations: sequence I and sequence II.  Hydrophobic (H) segments are filled and polar segments (P) units are  unfilled.  Sequence I  Sequence II  152  Figure 4.2:  Helmholtz energy (AadA/kT, squares) and entropy (A dsS/k, circles) of a  adsorption for the HP chain sequence I adsorbing on a weakly attractive surface {XHW - 1) as a function of Ao-NAVkT, the stability of the native-state of the chain relative to its fully denatured state. Lines drawn indicate data trends. Error is within 11% and 22% for values of A dsA/kT and AadsS/k, respectively. a  153  Figure 4.3:  Thermally-averaged energy landscapes for the HP chain sequence I in  solution (squares) and adsorbed (circles) to a weakly attractive sorbent surface (%HW = -1) under conditions where the stability of the native-state conformation is low (XHH  =  -1).  Lines drawn indicate data trends.  LU  •16  '|  10°  I IIIIIH|  I I m  10  1  m  1  10  2  ' " " ' 1  1 I IIIW|  1  10  3  I i n u n  10  4  1  '"""I  10  5  1  '  " " ' 1  10  6  10  7  Multiplicity of states  154  Figure 4.4: The average length, l , in the x dimension of each bond vector in HP chain x  sequence I as a function of intermolecular contact energies.  XH^XHW,  the ratio of the average intramolecular to  The dotted line represents the average length of each  bond vector in the x dimension when the chain is in its native-state conformation. The line drawn indicates the data trend.  0.3-1  155  Figure 4.5: Comparison of Ad fi/k values calculated with Eq. [4.12] to dMC simulation e  data values over a range of chain deformations, l  x  ll : dashed line a = 2, solid line a = xo  2.8. The d M C data (open squares) shown correspond to adsorption conditions where XHHIXHW  = 0.25  (TjH = 2-2),  1  (JJL =  1), and 4  (TjT2 = 0.85), respectively.  156  5  Conclusion  The objective of designing surfaces to control protein adsorption is certainly not new and, as noted in Chapter 1, a great deal of both experimental and theoretical work has been conducted over the past half century in an effort to understand and control protein adsorption. While much has been learned, very little is yet understood regarding how to actually control protein adsorption behaviour, and even less is understood regarding the submolecular events involved in protein adsorption processes; these interactions must be understood before protein adsorption can be predicted and controlled. New approaches to understand protein adsorption behaviour are thus needed. One of the most powerful techniques to study complex molecular behaviour today is computational chemistry. Very significant advancements have been made in this field over the past decade to improve both the size of the systems that can be modelled and the accuracy of simulations. These ever increasing capabilities have enormous potential for helping us to understand protein adsorption at a submolecular level and to provide a path toward the goal of proactively designing biomaterial surfaces to control biological response.  In general, the initial protein adsorption process must be governed by a balance of the intermolecular interactions between the residues presented by the protein's surface with the functional groups presented by the sorbent surface as a function of separation distance (i.e., residue-surface  interactions) and the intramolecular interactions between the  residues within the protein itself (i.e., residue-residue interactions). While numerous computational chemistry studies have been conducted in the area of protein folding to understand the energetics of residue-residue  interactions, very little is currently  157  understood regarding the energetics of protein residue-surface interactions.  If these  interactions can be quantified, then theoretically they should be able to be combined with an  understanding  of  intramolecular residue-residue  interactions  to  provide  a  thermodynamic basis for the prediction of protein-surface adsorption behaviour. This concept is similar to the approach used in numerous other biomolecular simulations, such as those used to predict ligand-protein, protein-protein, protein folding and RNA folding interactions (see for example, 1-3)  In each of these approaches, contributions of the  enthalpy, entropy, and/or free energy associated  with specific  functional group  interactions, and an overall accounting of these interactions are used to predict the free energy of binding and structural organization of the system.  This thesis is among the first attempts to use advanced computational chemistry, in particular, dynamic Monte Carlo simulations of a simple coarse-grained protein-like chain, to gain insights into the underlying molecular physics of the adsorption to solidliquid interfaces of chain molecules that preferentially adopt specific compact low-energy conformations in solution. Globular,proteins are the most obvious example of such chain molecules, and it is hoped that results from the model developed in this thesis have improved our understanding of the complexities of protein adsorption.  The results of this work give a unique perspective on the mechanisms driving adsorption of protein-like chains and the factors that influence them. Because they allow the direct connection of adsorption thermodynamics to adsorbed-chain conformational space, they also present a richer view of the process that establishes some unique features of the  158  adsorption of a protein-like chain as compared to the well-known behaviour of a randomcoil polymer adsorbing to a solid-liquid interface. Due to anchoring effects associated with multiple points of contact between the sorbent and the chain, unstructured polymer adsorption necessarily results in a decrease in the conformational entropy of the polymer, a fact confirmed by both experiment (4) and theoretical considerations (5). Nonspecific protein adsorption, however, is often an endothermic process, so that an increase in entropy must drive the adsorption process.  While dehydration effects almost certainly  contribute to the entropy increase, our results confirm that chain conformational entropy can also increase during adsorption. This is due in part to the restricted number of low energy conformations accessible to the HP chain in solution. The results of this thesis reveal that an increase in chain conformational entropy arises through an ability of the system to preferentially distribute new states generated by the sorbent surface into adsorbed-state energy levels with the largest Boltzmann weightings.  If the new ground  state energy level is highly degenerate, an increase in chain conformational entropy will generally be observed.  The simulation results reported here indicate that a highly  degenerate ground-state energy is favoured when the average intramolecular contact energy (XHH) equals or is near the average intermolecular contact energy (XHW)Similarly, results from the simulations carried in this work provide insights into the affect of a number of important systems variables (e.g., sorbent hydrophobicity, protein-like chain  sequence,  sorbent  geometry  and  macromolecular  confinement)  on  the  conformational freedom of the adsorbed chain and adsorption thermodynamics.  159  The results of this work therefore indicate that a simple mesoscopic (coarse-grain) model can be useful in helping to further understand adsorption phenomenon unique to proteins and solid/liquid systems. The simulations have successfully demonstrated that adsorption behaviour and thermodynamic properties of HP model chains on a simulated surface can resemble those of real protein adsorption systems and therefore provide a useful simple model for testing and understanding certain fundamental concepts related to nonspecific protein adsorption.  This is not to say that our model is without weaknesses. It is clearly a highly simplified view of the adsorption of proteins and protein-like chain molecules.  As discussed in  several places in this thesis, the approach developed here has several clear limitations that must be kept in mind in order to properly interpret and appreciate the results from the molecular simulations.  First of all, due to computational limitations, relatively short  protein-like HP chains were used and were modeled in only two dimensions. Secondly, more complex events, such as protein-protein lateral interactions and entropic effects emanating from the solvent have not been considered. It also must be recognized, however, that one must walk before learning to run, and this work must therefore be recognized as a humble beginning toward understanding a very complex problem and not the final analysis.  160  5.1  References  1.  Zhou, Y . Q . and Karplus, M . , Folding of a model three-helix bundle protein: A thermodynamic and kinetic analysis. J. Mol. Biol., 293 (1999) 917-951.  2.  Nakajima, N . , Higo, J., Kidera, A . and Nakamura, H., Free energy landscapes of peptides by enhanced conformational sampling. J. Mol. Biol., 296 (2000) 197-216.  3. Kellogg, G.E., Ligand docking and scoring: New techniques and applications in drug discovery. Med. Chem. Res., 9 (1999) 439-442. 4.  Denoyel, R., Durand, G., Lafuma, F. and Audebert, R., Adsorption of cationic polyelectrolytes onto montmorillonite and silica - microcalorimetric study of their conformation. J. Colloid Interface Sci., 139 (1990) 281-290.  5. Cohen-Stuart, M.A., Fleer, G.J. and Bijsterbosch, B.H., The adsorption of polyvinyl pyrrolidone onto silica. 1. Adsorbed amount. J. Colloid Interface Sci., 90 (1982) 310-320.  161  Appendix Description of dynamic Monte Carlo Simulations and related program code  The investigations reported in this thesis describe the behaviour of a protein-like HP chain within and adsorbed to the surface boundary of a two-dimensional lattice. The movements of the chain are carried out using dynamic Monte Carlo algorithms. The purpose of using this method is not necessarily to mimic the exact movements of a protein in solution or on the surface, but to be able to sample the system within a reasonable period of time in order to compute all energetic and conformational states of the system.  Computer programs for the dynamic Monte Carlo simulations were developed specifically for this work. Below are brief explanations of the major program elements and an example of a program used.  The lattice grid  The program shown here is written for a 2D simulation only, although simulations on 3D lattices were also performed. The lattice space used in the simulations is defined as a Cartesian coordinate grid having only positive x and y coordinates ranging from (and including) the lowest value at 0 to a highest value of GRIDLIM.  The values of  GRIDLIM are given in the starting input file and may differ in each dimension. The  162  lattice is therefore bounded by x = 0 and GRIDLIM(x) (left and right boundaries, respectively) and y = 0 and GRJDLIM(y) (bottom and top boundaries, respectively). Lattice site centers are assumed to be a distance, a, apart. All boundaries are reflective.  The input parameter, WALL_SWITCH, indicates whether a boundary is assigned a potential. A lattice boundary designated as a sorbent surface is referred to as an active wall (W), whose interaction energies with the other simulation components are defined. Non-active walls are assumed to be athermal. Four possibilities are allowed by the W A L L S WITCH function:  horizontal active walls (choice 1), vertical active walls  (choice 2), one wall active (choice 3), and all walls active (choice 4).  The HP chain  The specific sequence of the protein-like HP chain is entered into the input file by the user. The chain length, L, is specified, as are the sequence of H and P residues. The chain conformation is entered in the input file as a series of L coordinates (i.e. (x\, y\), (xj, yi)...  (XL, y£)). The chain is then placed in the program with each residue occupying a  unique coordinate position within the allotted grid space. Throughout the simulation, the chain's position and configuration are defined by the coordinates of its first residue ( S T A R T P T ) and a series of unit-length vectors (VECTOR).  In this array, the structure  of the chain is described by L-l vectors (i.e. (vector \,vector \), (vector^vector i)... x  y  (vector .i,vector -i)), each running parallel to either the x or y axis. xL  yL  y  Vector subscript  163  numbers refer to the chain residue the vector originates from (i.e. vectori refers to the vector originating from (xi,yi) and ends at (X2, yi))-  The algorithm  The program runs as follows. READDATA.  Initial parameters are read into the main program using  The initial chain position and conformation are scanned to ensure that  they are workable within the given simulation conditions. Chain characteristics are also analyzed at this point. Calculations, of the chain energy and conformational likeness to a reference structure (usually the native-state lowest-energy structure) are made. Whether the chain conformation should be stored or not is also considered. In certain simulations, only the lowest energy conformations are stored. In other simulations, conformations at other or all energy levels are collected.  At this point, the main loop of the main subroutine begins. Upon entering the loop, all sampling frequencies are checked to determine whether information should be written to output files. The frequency of the WHIRLING function is also evaluated to determine whether the chain should be translated or rotated at this time.  The program then enters the move algorithm section. selected.  A chain residue is randomly  Depending on the position of the residue in relation to the overall chain  conformation, a successful move may or may not occur. A successful move is one that results in a conformational change without chain overlap and without exceeding the  164  boundaries of the grid space (after having been once reflected off the walls).  A  successful move also has to pass the energy criteria given by the Metropolis algorithm.  For successful moves, conformation and energy parameters are calculated for the newly formed conformation and a transition protocol then takes place to replace old simulation parameters with new ones. Finally, just prior to returning to the beginning of the loop for the next attempted move, a scan is made to decide which data for the new conformation should be saved in the storage files.  If a move is unsuccessful, the program returns to the start of the loop, with only the simulation step number having changed throughout the process.  Ending the program  The program is designed to terminate itself. This can happen in a few ways.  The program can end at its last designated cycle. For instance, if the number of times to run the main subroutine, C H A N G E L I M , is 2, and the number of cycles in each main subroutine, LIMIT, is lxlO cycles, then the program 9  will end in its 2xl0 -th cycle, as long as the assigned storage capacities are not exceeded. 9  The program can end when its storage files are full.  165  For instance, if the size of the storage files, NSTOR, is 50, the simulation will end when the 50th unique conformation having energy lower than the defined maximum, E M A X , is found.  Program files  Running of the simulations involves 3 types of files - the main program, the input file, and output files.  Main program <*.for>  The  dynamic Monte Carlo simulation programs for these investigations  programmed in Fortran 77.  were  All program coding is original work by the author, except  for the random number generators, R A N L U X and GASDEV.  Coded by F. James in  1993, R A N L U X is a well-known subroutine easily found on shareware sites (e.g. http://tonic.phvsics.sunvsb.edu/docs/num meth.html).  GASDEV  is  taken  from  "Numerical Recipes in Fortran77" .and is also found on numerous web sources (e.g. http://lib-www.lanl.gov/numerical/index.htmn.  166  Input file <*.dat>  The input file holds initial simulation parameters for the program. Information such as the chain sequence, grid parameters, interaction energies and sampling frequency are specified here. program.  Also included is designation of the type of output generated by the  The name of the input file is important in that it becomes the name of the  output file.  For example, a program run using the input file <test2.daf> will have the  output files <test2.out>. Also important is that the spacing and format of the input file remain the same during the editing procedure. All numbers in the input file are rightjustified.  Output files <*.out>, <Screen>, etc.  The number of output files varies with each program.  As some simulations run for long periods of time, it is often necessary to have multiple outputs so that files remain reasonably sized.  Having more than one output file also  allows for parallel sampling, and consequently, avoidance of having to store large amounts of information in a single file during the simulation.  167  In all programs, the main output file is <*.out>, where "*" refers to the title of the input file.  <*.out> generally presents detailed sampling data collected at the frequency  indicated by NWRITE.  There is, also a screen output where data sampled at the  frequency of NSIGN is written to the screen.  Editing, compiling and running the program  For most part, the simulation programs used for this work were edited, compiled and run on LINUX systems supported by the Biotechnology Laboratory (www.biotech.ubc.ca) and the Institute of Applied Mathematics (www.iam.ubc.ca), both at the University of British Columbia. A portion of the work was conducted on Windows run PC's, using either Microsoft  Fortran PowerStation  or the shareware program, GNU g77 (found at  http://www.cs.yorku.ca/Courses/1540/ftn.htm).  Instructions for running a program on a LINUX server will be given here.  Running a single program  To compile the program, Type "f77 <program.for> -o <commandname> -03" where, p7\  Calls the Fortran77 compiler.  168  <program.for> or <program.f>: The name of the program. -o <commandname>: The program is compiled to become a command, and the name of the command can be referred to here. -03:  Optimization command. Some brief trials were run, and it was found that level 3  optimization resulted in the most efficient program.  To run the program in background mode, Type 'nice <commandname> <inputfde.dat> > <screenfile> &" where, nice: A command used so that the computationally-intensive program doesn't overtake the entire server. <commandname>: Running the command. <inputfile.dat>:  The input file.  > <screenfile>: Channelling the screen output to a storage file. &: Sets the program to run in the background.  R u n n i n g a collection of programs in series  Compile the program. Set up a command file listing the programs to be run on separate lines. Type in an open editor window: <commandnamel>  <inputfilel.dat> > <screenfilel>  <commandname2> <inputfile2.dat> > <screenfile2>  169  (etc.) Save the window as  <commandfilename>.  Type "chmod +x <commandfilename>" Type "nice <commandfilename>  to activate the command file.  &. h  Example program for dynamic Monte Carlo simulation, CONTACT9  C O N T A C T 9 is a program designed to frequently sample a chain adsorbed onto an active surface.  Simulations using this program begin with the chain sitting on the surface.  C O N T A C T 9 carries out the dynamic Monte Carlo moves, recording unique chain conformations into file <*.out> while regularly sampling energy and structural data into <*.gph>.  170  Main program, <contact9.for> *  *************************  *  *************************  PROGRAM C O N T A C T 9  ********************************************************************** * * *  General program d e s c r i p t i o n :  * * * *  This program surface i n a dynamics are The i n t e r n a l  * *  the  * *  The c h a i n series of  * *  s i m u l a t e d t o b e i n a s p a c e whose d i m e n s i o n s a r e i n d e p e n d e n t o f one another (GRIDLIM(X) v s G R I D L I M ( Y ) ) . The remainder o f the l a t t i c e is  * *  filled  * *  Sorbent  * *  C0NTACT9  method  s i m u l a t e s the a d s o r p t i o n of a p r o t e i n - l i k e c h a i n to a 2D C a r t e s i a n c o o r d i n a t e l a t t i c e . Monte C a r l o u s e d to manoeuver the c h a i n i n t o random c o n f i g u r a t i o n s . energy of the c h a i n i s c a l c u l a t e d and weighted using  outlined  by M e t r o p o l i s .  is of length L, h y d r o p h o b i c (H)  with  solvent  surfaces  and i s composed o f a c o n n e c t e d a n d h y d r o p h i l i c (P) u n i t s . It  units.  are  Spatial  simulated  to  boundaries  be  at  are  specific  is  reflective.  boundary  planes.  ********************************************************************** * program description:  *  This  *  conformations  program is  *  continually  *  first  evaluated  * *  their  isomers.  *  The p r o g r a m can  *  parallel  2-sided  *  of  and y  the  x  a modification  with  runs  energies  without by  run with  *  simulation; and r o t a t i o n )  * *  mechanism  *  Energy  * *  expressed  *  T h e CONTACT v e r s i o n  *  original  *  the  for  a  function off  of  active  It  possible  is  switch is  when  to  terms  native  of of  the  chain  the  T -  lowest  in  an a c t i v e  interactions  counts  energy  the  number  *  The n o n - n e i g h b o u r i n g  contacts  of  the  lowest  conformations  *  by  * *  and the  *  The CONTACT9 v e r s i o n  *  points  for  a  recorded  is  to  (but  not  conformation  size  of  are  *  using  the  Monte C a r l o  *  energy  of  adsorption.  *  to  read  *  There  *  information.  * *  RADIUS_SWITCH.  file  also  an  in  the data  it,  sample  In  minimize  is  designed  simulation.  *  the  compared a g a i n s t  the  are  catalogued.  original  found contacts  compared.  *  the  be  First,  state  are  can  of  conformations.  native  numbers  whirling  PSI.  the  then  sizes  the  of  are  active  surface.  contacts  simulation  and  (transposition  non-neighbouring the  are  this  WHIRLING  and s o l v e n t - w a l l /  or  differing  on a u t o m a t i c ,  contacts  simulation  in  surface  have  store  conformations  mechanism e x i s t s When s e t  CHI = KAPPA  contacts  to  to  maximum. The p r o g r a m  conformations  stored  4-sided  control  added. the  a program designed specified  Acceptable  fully  hydrophobic-solvent in  the  previously  No d i f f u s i o n  however,  shuts  a  surface.  *  DEGEN10, than  restarting.  comparison  axes.  of  less  files).  histogram  in  to  state  version  c a n be  number  and c o n t a c t  coordinates  technique  calculate  this  The c a l c u l a t i o n  a large  energy  The d a t a  The n a t i v e  order  option  the  can  to  to  made  be  calculate state request  to  analyzed the  has  been  added  contacts.  record radius at  data  themselves  then  conformation native  of  information  of  using  gyration the  switch,  * * ********************************************************************** * .  171  L i s t of subroutines: ******************** CENTRE OF MASS  = Determines the c e n t r e o f mass c o o r d i n a t e s f o r a configuration. C o o r d i n a t e s used as a p o i n t o f r o t a t i o n f o r movement o f the c h a i n i n WHIRLING.  CENTRE OF MASS REAL  = Determines the c e n t r e o f mass v a l u e s f o r a c o n f i g u r a t i o n . R e s u l t s are r e a l numbers, used radius of g y r a t i o n c a l c u l a t i o n s  for  COORD MAKER  = caculates s e r i e s of coordinates d e s c r i b i n g p o s i t i o n s and c o n f o r m a t i o n of c h a i n  FILER  = t r a n s f e r s i n f o r m a t i o n from a c t i v e p a r t o f the program i n t o the s t o r a g e f i l e s a s s i g n e d t o r e c o r d unique lowest energy c o n f o r m a t i o n s  FILTER  = screens c o n f o r m a t i o n s which are tagged as h a v i n g the s p e c i f i e d c r i t e r i a ( e . g . h a v i n g energy lower than the g i v e n maximum t h r e s h o l d ) as b e i n g unique. Comparison of the c o n f o r m a t i o n and i t s r e f l e c t i v e and r o t a t i o n a l v e r s i o n s to a l l p r e v i o u s l y s t o r e d s t r u c t u r e s are c a r r i e d o u t .  FIND  = the p o s i t i o n o f a s p e c i f i c a l l y chosen bead r e l a t i v e t o c o n n e c t i n g beads i s a s s e s s e d f o r the purpose o f d e t e r m i n i n g a p o s s i b l e f u t u r e move.  SITUATION  INCONSISTANCY_CHECK  = scans the i n i t i a l c h a i n c o n f o r m a t i o n g i v e n at the s t a r t o f the s i m u l a t i o n f o r e r r o r s , such as d i a g o n a l v e c t o r s or s k i p p e d c o o r d i n a t e s .  ISWITCH  = s w i t c h e s two g i v e n i n t e g e r s . Used t o c o o r d i n a t e s or v e c t o r s i n the x=y l i n e .  reflect  MOVE 2D  = u s i n g v e c t o r d a t a , c a l c u l a t e s new c h a i n c o o r d i n a t e s a f t e r a s e l e c t e d move has taken p l a c e .  NEIGHBOUR_CATALOG_SINGLE = compares the i n t r a m o l e c u l a r c o n t a c t s o f the working c o n f o r m a t i o n w i t h the c o n f o r m a t i o n g i v e n i n the i n p u t f i l e f o r comparison ( u s u a l l y the n a t i v e state conformation). NEIGHBOUR COUNT  = counts the number o f c o n t a c t s s p e c i f i c conformation.  NEIGHBOUR ID  = i d e n t i f i e s the s e r i e s o f i n t r a m o l e c u l a r e x i s t i n g w i t h i n a given conformation.  OVERLAP CHECK  = scans a p r o p o s e d c h a i n c o n f i g u r a t i o n f o r overlap.  PROTEIN ADSORTPION  2  made f o r a  contacts  possible  = the main p r o c e s s s u b r o u t i n e . A s i n g l e run o f t h i s s u b r o u t i n e cannot exceed 2**32 c y c l e s , and t h e r e f o r e had t o be l o o p e d w i t h i n a l a r g e r program f o r the p o s s i b i l i t y of longer s i m u l a t i o n s .  RANDOM INT  = c o n v e r t s a random number determined i n RANLUX from RVEC t o i n t e g e r v a l u e between two s p e c i f i e d limits  RANDOM REAL  = c o n v e r t s a random number determined i n RANLUX from RVEC t o r e a l v a l u e between two s p e c i f i e d limits  RANLUX  READ DATA  = random number g e n e r a t o r , g i v e s an a r r a y o f random numbers between 0 and 1 reads i n i t i a l  i n p u t d a t a from <*.DAT>.  REFRESH  = t a k e s accepted them i n t o a c t i v e  R_G_CALC  = calculates  VECTOR MAP  = t r a n s l a t e s the c o o r d i n a t e s o f the s t a r t i n g c o n f o r m a t i o n read i n t o the program and t r a n s l a t e s i t i n t o s t a r t i n g p o i n t s and a v e c t o r s e r i e s .  WALL BOUNCE  = the boundary f u n c t i o n f o r the s i m u l a t i o n . Given a c o n f o r m a t i o n , i t r e f l e c t s any p a r t o f the c h a i n o u t s i d e the l a t t i c e g r i d back i n t o the g r i d space.  WHIRLING  = c a r r i e s out r o t a t i o n or t r a n s l a t i o n moves a t an assigned frequency. A t r a n s l a t i o n a l distance i s s p e c i f i e d i n the i n p u t f i l e , but the t r a n s l a t i o n d i r e c t i o n or i f degree o f r o t a t i o n i s chosen randomly.  WRITE DATA  = w r i t e s the s i m u l a t i o n parameters  potential variables variables.  and  shifts  radius of g y r a t i o n of a conformation.  from <*.dat>.  ************************************************************** List  of  functions:  DOT  = f i n d s the dot p r o d u c t between two v e c t o r s . Used t o i d e n t i f y p e r p e n d i c u l a r c o n f o r m a t i o n s the c h a i n .  F_PROB  = c a l c u l a t e s p r o b a b i l i t y u s i n g Boltzmann weighted s t a t i s t i c s .  F_ENERGY  = c a l c u l a t e s the c o n f o r m a t i o n a l of the c h a i n .  energy  ******************************************************************** L i s t parameters o f main program: ********************************* PARAMETER (MAXD=2,MAXC=100,MAXSTOR=100000, MAXCONTACTS=1000)  + + + + + + + + + + + + + + + +  INTEGER NDIM, LIMIT, NWRITE, NSTOR, NWHIRL_START, NWHIRL, NTRANS, L , COORD (MAXD, MAXC) , STOR, NCHANGE, CHANGE_LIM, NFRAME, NSTARTREC, XHH, XPP, XHP, XHS, XPS, XHW, XPW, XSW, NHOURS, NMIN, NSEC, WHIRL_SWI,TCH, RADIUS_SWITCH, WALL_SWITCH, MAP_SWITCH, PAST_STORTAG, NNEIGHBOUR_ORIGIN, MAP_FLAG,NNATIVE_ORIGIN, START_ORIGIN (MAXD) , PAST_START (MAXD) , START_PT (MAXD) , GRIDLIM (MAXD) , PAST_VECTOR (MAXD, MAXC) , VECTOR (MAXD, MAXC) , COORD_ORIGIN (MAXD, MAXC) , VECTOR_ORIGIN (MAXD, MAXC) , LOW_STEP (MAXSTOR) , LOW_XHH (MAXSTOR) , LOW_XHP (MAXSTOR) , LOW_XPP (MAXSTOR) , LOW_XHS (MAXSTOR) , LOW_XPS(MAXSTOR), LOW_XHW(MAXSTOR), LOW_XPW(MAXSTOR), LOW_XSW (MAXSTOR) , LOW_START (MAXD, MAXSTOR) , LOW_VECTOR (MAXD, MAXC, MAXSTOR) , LOW_CHANGE (MAXSTOR) , NEIGHBO"R_ORIGIN (2, MAXCONTACTS) INTEGER*2 TIME  + + +  REAL GRIDX, GRIDY, CHI_HH, CHI_PP, CHI_HP, CHI_HS, CHI_PS, CHI_HW, CHI_PW, CHI_SW, KAPPA_HS, PSI_HS, KAPPA_SW, PSI_SW, TEMP, E_MAX, TIME_COUNT, ETIME, TARRAY(2) , LOW_E(MAXSTOR)  CHARACTER*1 TYPE(MAXC) CHARACTER*40 ARG, F_Name  within  COMMON / C H I  /  CHI_HH, CHI_PP, CHI_HP, CHI_HS, CHI_PS, CHI_HW, CHI_PW, CHI_SW, KAPPA H S , P S I H S , KAPPA _SW, P S I _ S W COMMON / G R A P H _ L I M / L , N D I M , NVECTOR " COMMON / TEMP / TEMP COMMON / WHIRL / COMMON / L A T T I C E  NWHIRL, /  NTRANS  GRIDLIM  COMMON / T Y P E / T Y P E COMMON / NEIGHBOUR / NEIGHBOUR +  ORIGIN,  NNEIGHBOUR  ORIGIN  NNATIVE_ORIGIN  Open files ********** IF  (  IArgCO . L T . 1 ) THEN WRITE(*,*) ' S P E C I F Y INPUT STOP END I F Call GetArgd, F_Name) IF  ( Access( WRITE(*,*) STOP  END  F_NAME , ' r ' ) . N E . 0 ) THEN ' F I L E I S NOT A C C E S S I B L E ! ! '  = index(F_Name, .')  -1  1  (  INC  inc END  .It.  0  )  THEN  = index(F_Name,'  ')  -  1  IF  OPEN  (UNIT=10, FILE=F_NAME, IOSTAT=IERROR)  + +  PLEASE  IF  inc IF  FILE  OPEN ( U N I T = 2 0 , IOSTAT=IERROR) OPEN  STATUS='OLD  1  ,  ERR=1000,  FILE=F_Name(1:INC)//('.out'),ERR=1000,  (UNIT=42, F I L E = F _ N a m e ( 1 : I N C ) / / ( ' . g p h ) , ERR=1000, IOSTAT=IERROR) 1  +  TIME_COUNT =  ETIME(TARRAY)  I n i t i a l i z e parameters and e x t e r n a l loop *************************************** NCHANGE =  0  NTOTALSTEP = MAP_FLAG CALL  0  0  READ_DATA  + +  =  GRIDY, '  (NDIM,  CHANGE_LIM,  NWHIRL_START,  NSTARTREC, START_PT,  +  CHI_HH,  CHI_PP,  +  CHI_SW,  KAPPA_HS,  +  TEMP,  WHIRL_SWITCH,  E_MAX,  L,  CHI_HP,  CHI_HS,  PSI_HS,  COORD,  GRIDX,  WALL_SWITCH, CHI_HW,  CHI_PW,  PSI_SW, COORD_ORIGIN)  1 *  REAL(L))  = INT(GRIDY  *  REAL(L))  VECTOR_MAP  (COORD_ORIGIN,  MAP_SWITCH,  VECTOR_ORIGIN,  MAP_FLAG)  (MAP_FLAG.EQ.l) WRITE(20,*)'THE WRITE(*,*)  NSTOR,  N FRAME,  TYPE,START_ORIGIN,  = INT(GRIDX  IF  NWRITE,  CHI_PS,  KAPPA_SW,  GRIDLIM(2)  STARTJDRIGIN,  NSIGN,  RADIUS_SWITCH,  GRIDLIM(1)  CALL +  LIMIT,  MAP_SWITCH,  +  NVECTOR = L -  NTRANS,  THEN GRID  IS  TOO S M A L L FOR ( N A T I V E )  CONFORMATION'  ' T H E GRID  IS  TOO SMALL FOR ( N A T I V E )  CONFORMATION  STOP ENDIF  +  C A L L NEIGHBOUR_COUNT ( V E C T O R , S T A R T _ P T , X H S , X P S , XHW, XPW, XSW, W A L L _ S W I T C H )  XHH, XPP, XHP,  WRITE(20,*)  '********************************************!  WRITE(20,*)  'DATA  WRITE(20,7) +  ',  F_NAME X:',  GRIDLIM(l),  'Y:',GRIDLIM(2) WRITE(42  *)  •********************************************'  WRITE(42,*)  *********************************************'  WRITE(42,*)  'DATA  WRITE(42,7) + 7  FILE:  'GRID DIMENSIONS  FILE:  ',  F_NAME  'GRID DIMENSIONS  X:',  GRIDLIM(1)  ,  'Y:',GRIDLIM(2) FORMAT  (IX,A21,15,8X,A2,15)  C A L L WRITE_DATA  (NDIM,  +  GRIDY,  +  NSTARTREC,  +  CHI_HH,  CHI_PP,  +  CHI_SW,  KAPPA_HS,  +  CHANGE_LIM,  NWHIRL_START,  TEMP,  NTRANS,  LIMIT,  NSIGN,  MAP_SWITCH, S T A R T _ P T ,  E_MAX,  CHI_HP,  L,  CHI_HS,  PSI_HS,'  COORD,  NWRITE,  NSTOR,  GRIDX,  NFRAME,  WHIRL_SWITCH, CHI_PS,  KAPPA_SW,  WALL_SWITCH,  CHI_HW,  CHI_PW,  PSI_SW,  TYPE,START_ORIGIN,  COORD_ORIGIN)  WRITE(20  *)  '********************************************'  WRITE(20  *)  '********************************************t  WRITE(42,*)  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * i  WRITE(42,*)  •********************************************'  DO 20  DIM  =  1,  NDIM  PAST_START(DIM) 20  =  0  CONTINUE DO 25  VNUM = DO 23  1,  NVECTOR  DIM =  1,  NDIM  PAST_VECTOR(DIM,VNUM) 23  =  0  CONTINUE  25  CONTINUE DO 40  STOR  =  1,  NSTOR  LOW_STEP(STOR)  =  LOW_XHH(STOR)  =  0 0  LOW_XPP(STOR)  =  0  LOW_XHP(STOR)  =  0  LOW_XHS(STOR)  =  0  LOW_XPS(STOR)  =  0  LOW_XHW(STOR)  =  0  LOW_XPW(STOR)  =  LOW_XSW(STOR)  =  0 0  LOW_E(STOR)  =  E_MAX +  DO  1,  NDIM  30  DIM =  0.0001  L O W _ S T A R T ( D I M , STOR) 30  =  0  CONTINUE DO 35  VNUM =  DO 34  DIM «= 1,  1,  NVECTOR NDIM  L O W _ V E C T O R ( D I M , VNUM, NSTOR) 34  =  0  CONTINUE  35  CONTINUE LOW_CHANGE (STOR)  40  =  0  CONTINUE  CALL IF  VECTOR_MAP  (COORD,  ( M A P _ F L A G . E Q . 1)  MAP_SWITCH,  VECTOR,  START_PT,  W R I T E ( 2 0 , * ) ' T H E GRID  IS  TOO S M A L L FOR T H E C H A I N  CONFORMATION  WRITE(*,*)  IS  TOO S M A L L FOR T H E C H A I N  CONFORMATION  ' T H E GRID  STOP ENDIF  * Set d i f f u s i o n on or o f f a u t o m a t i c a l l y a c c o r d i n g to s w i t c h *********************************************************** IF  (WHIRL_SWITCH.EQ.l) IF  (XHW.GT.0)  THEN THEN  NWHIRL  =  LIMIT  NWHIRL  =  NWHIRL_START  ELSE ENDIF ELSE  MAP_FLAG)  THEN  NWHIRL  =  NWHIRL_START  ENDIF * C a l c u l a t e contact information for o r i g i n a l conformation ********************************************************* CALL +  NEIGHBOURED  (START_ORIGIN,  VECTOR_ORIGIN,  NEIGHBOUR_ORIGIN,  NNEIGHBOUR_ORIGIN) DO 45  HOOD = IF  1,  NNEIGHBOUR_ORIGIN  (NEIGHBOUR_ORIGIN(2,HOOD).GT.O) NNATIVE_ORIGIN  =  THEN  NNATIVE_ORIGIN  +  1  ENDIF 45  CONTINUE  * S t a r t of e x t e r n a l loop ************************ 50  CONTINUE  * Ending program ***************** IF  ( (NCHANGE.GE . CHANGE_LIM) . O R . ( P A S T _ S T O R T A G . E Q . N S T O R )  )  THEN  WRITE)*,*) 'PROGRAM I S C O M P L E T E WRITE(20,*)'PROGRAM IS COMPLETE' T I M E _ C O U N T = E T I M E (TARRAY) - T I M E _ C O U N T NHOURS = I N T ( T I M E _ C O U N T / 3 6 0 0 ) • 1  + + 60  NMIN = I N T ( T I M E _ C O U N T (NHOURS*3600) ) / NSEC = INT (TIME_COUNT) (NHOURS*3600) WRITE(20,60) 'PROGRAM R U N - T I M E : ' , N H O U R S , ' N M I N , ' MINUTES ' , N S E C , ' SECONDS ' WRITE(42,60) 'PROGRAM R U N - T I M E : N M I N , ' MINUTES ' , N S E C ,  1  ',NHOURS,' SECONDS '  60 (NMIN*60) HOURS ', HOURS  ',  FORMAT(A17,I4,A7,I2,A9,I2,A9) CLOSE(UNIT=42) CLOSE(UNIT=20) CLOSE(UNIT=10) STOP ENDIF  * External loop counter *********************** NCHANGE =  NCHANGE +  1  * Call simulation ***************** CALL  PROTEIN_ADSORPTION_2  +  PAST_STORTAG,  +  LOW_XHS,  +  LOW_START,  LOW_STEP,  LOW_XPS,  (NCHANGE, LOW_XHH,  LOW_XHW,  PAST_START, LOW_XHP,  LOW_XPW,  LOW_XSW,  L O W _ V E C T O R , LOW_CHANGE)  * End e x t e r n a l l o o p ******************* GOTO  50  •Error statements: ****************** 1000  WRITE!*,*)  ' * * * * T R O U B L E OPENING  WRITE(*,*)  'IOSTAT  WRITE(*,*) END  IS',  IERROR  FILE***'  PAST_VECTOR,  LOW_XPP; • LOW_E,  SUBROUTINES:  *************************************************************** SUBROUTINE  PROTEIN_ADSORPTION_2  +  PAST_STORTAG,  +  LOW_XHS,  +  LOW_VECTOR,  LOW_STEP,  LOW_XPS,  (NCHANGE,  LOW_XHH,  LOW_XHW,  PAST_START,  LOW_XHP,  LOW_XPW,  PAST_VECTOR,  LOW_XPP,  LOW_XSW,  LOW_E,  LOW_START,  LOW_CHANGE)  ******************************************************************* T h i s i s the main subroutine that runs the s i m u l a t i o n . Given the data r e a d from <*.dat>, i t e s t a b l i s h e s the l a t t i c e and i t s boundaries, the i n t e r a c t i o n e n e r g i e s and c h a i n c o n f o r m a t i o n and p o s i t i o n . This s u b r o u t i n e g o v e r n s c h a i n movements, s a m p l i n g t o the o u t p u t f i l e s and screen, a n d d e t e r m i n a t i o n o f w h i c h c o n f o r m a t i o n s s h o u l d be s t o r e d .  ******************************************************************* Parameter list: **************** PARAMETER  +  (MAXD=2,MAXC=100,MAXSTOR=100000,MAXCONTACTS=l000)  INTEGER L , POSITION, NDIM, NVECTOR,  C O N F I G , D I M , VNUM, NCHANGE, R A N D _ B E A D ,  +  NSTEP,  +  X H H , X P P , X H P , X H S , X P S , 'XHW, XPW, XSW,  +  MOVE_FLAG,  + +  WHIRL_FLAG, NTOTALSTEP, NSTARTREC, NEW_XHH, N E W _ X P P , NEW_XHP, N E W _ X H S ,  +  NEW_XPW,  +  NSUCCESS,  NWRITE,  +  NFRAME,  LIMIT,  OVER_FLAG,  NEW_XSW,  CHANGE_LIM,  MAP_FLAG,  NWHIRL,  STORTAG,  NSTOR,  WHIRL_SWITCH,  INCON_FLAG,  NTRANS,  NEW_XPS,  STOR,  NNEIGHBOUR_ORIGIN,  RADIUS_SWITCH,  NEW_XHW,  NWHIRL_START, NNATIVE_ORIGIN,  WALL_SWITCH,  S T A R T _ P T (MAXD) ,  + +  GRIDLIM(MAXD), COORD(MAXD,MAXC), COORD_ORIGIN(MAXD,MAXC), V E C T O R (MAXD,MAXC) , NEW_VECTOR (MAXD,MAXC) , T Y P E _ N U M (MAXC) ,  + +  LOW_XHH (MAXSTOR) , LOW_XHS (MAXSTOR) ,  + +  LOW_XPW (MAXSTOR) , LOW_XSW (MAXSTOR) , L O W _ S T E P (MAXSTOR) , LOW_START (MAXD,MAXSTOR) , L O W _ V E C T O R ( M A X D , M A X C , M A X S T O R ) ,  +  NEW_START (MAXD) ,  PAST_STORTAG,  +  LOW_CHANGE(MAXSTOR) ,  S T A R T _ O R I G I N (MAXD) ,  LOW_XPP (MAXSTOR) , LOW_XPS (MAXSTOR) ,  LOW_XHP (MAXSTOR) , LOW_XHW (MAXSTOR) ,  NEIGHBOUR_ORIGIN(2,MAXCONTACTS)  +  P A S T _ S T A R T (MAXD) ,  P A S T _ V E C T O R (MAXD, MAXC) ,  +  NNATIVE_CONTACTS,  NNONNATIVE_CONTACTS,  REAL GRIDX,  GRIDY,  CHI_HH,  CHI_PP,  +  CHI_SW,  KAPPA_HS,  +  ENG, NEW_E,  +  LOW_E(MAXSTOR),  +  R_G_AVG,  COMMON /  NWALL_CONTACTS  TEMP,  +  CHARACTER*1  ,  CHI_HP,  CHI_HS,  PSI_HS,  D_E, P_CALC,  CHI_PS,  KAPPA_SW,  CHI_HW,  CHI_PW,  PSI_SW,  P_RAND,  NATIVE_FRAC,  E_MAX,  SUCCESS,  R_G_DEFORM  TYPE(MAXC) CHI_HH,  CHI_PP,  CHI_HP,  CHI_HS,  +  CHI /  CHI_PS,  CHI_HW,  CHI_PW,  CHI_SW,  +  KAPPA_HS, COMMON /  GRAPH_LIM /  COMMON /  TEMP /  COMMON /  WHIRL  COMMON /  LATTICE  L,  PSI_HS, NDIM,  KAPPA_SW,  PSI_SW  NVECTOR  TEMP /  NWHIRL, /  COMMON /  TYPE /  COMMON /  NEIGHBOUR  NTRANS  GRIDLIM  TYPE /  +  NEIGHBOURJDRIGIN,  NNEIGHBOUR_ORIGIN,  NNATIVE_OSlGIN  * Read data *********** CALL +  READ_DATA GRIDY,  (NDIM,  CHANGE_LIM,  NWHIRL_START,  NTRANS,  LIMIT, NSIGN,  NWRITE, NFRAME,  NSTOR,  GRIDX,  + +  NSTARTREC, MAP_SWITCH, S T A R T _ P T , WHIRL_SWITCH,  +  CHI_HH,  + +  CHI_SW, K A P P A _ H S , P S I _ H S , KAPPA_SW, PSI_SW, T E M P , E _ M A X , L , COORD, TYPE,START_ORIGIN,COORD_ORIGIN)  CHI_PP,  RADIUS_SWITCH,  CHI_HP,  CHI_HS,  WALL_SWITCH,  CHI_PS,  CHI_HW,  CHI_PW,  I n i t i a l i z e i n t e r n a l loop parameters: ************************************** NVECTOR = L -  1  NSTEP = 0 NSUCCESS = 0 INCON_FLAG = 0 OVER_FLAG = 0 MAP_FLAG = 0 WHIRL_FLAG = 0 STORTAG =  0  NTOTALSTEP =  (NCHANGE  -  1)  *  LIMIT  * C r e a t i n g v e c t o r map: ********************** CALL IF  VECTOR_MAP  (COORD,  MAP_SWITCH,  (MAP_FLAG.EQ.1) THEN W R I T E ( 2 0 , *) ' T H E GRID WRITE(*,*)  ' T H E GRID  VECTOR,  START_PT,  MAP_FLAG)  IS  TOO SMALL FOR T H E C H A I N  CONFORMATION'  IS  TOO S M A L L FOR T H E C H A I N  CONFORMATION'  STOP ENDIF IF  (NCHANGE.GT.1)  THEN  STORTAG = DO 50  PAST_STORTAG  DIM =  1,  NDIM  START_PT(DIM) 50  =  PAST_START(DIM)  CONTINUE DO 60  VNUM =  1,  DO 55  NVECTOR  DIM =  1,  NDIM  VECTOR(DIM,VNUM) 55  =  PAST_VECTOR(DIM,VNUM)  CONTINUE  60  CONTINUE ENDIF  * Checking f o r problems i n o r i g i n a l conformation: ************************************************* CALL IF  INCONSISTANCY_CHECK  (INCON_FLAG.EQ.1) WRITE(20,*) WRITE(*,*)  (VECTOR,  INCON_FLAG)  THEN  ' T H E ORIGINAL  CONFORMATION I S  ' T H E O R I G I N A L 'CONFORMATION I S  INCONSISTENT' INCONSISTENT'  STOP ENDIF CALL IF  OVERLAP_CHECK (VECTOR,  (OVER_FLAG.EQ.1) WRITE(20,*) WRITE!*,*)  START_PT,  OVER_FLAG)  THEN ' T H E ORIGINAL  ' T H E ORIGINAL  CONFORMATION HAS O V E R L A P ' CONFORMATION HAS O V E R L A P '  STOP ENDIF * Count c o n t a c t s and c a l c u l a t e e n e r g y o f o r i g i n a l c o n f i g u r a t i o n *************************************************************** CALL +  NEIGHBOUR_COUNT  (VECTOR,  X P S , XHW, XPW, XSW, WALL  START_PT, SWITCH)  XHH, XPP, XHP, XHS,  ENG = F ENERGY(XHH, XPP, XHP, XHS, XPS, XHW, XPW, XSW) CALL NEIGHBOUR_CATALOG_SINGLE (START_PT,VECTOR, + NATIVE_FRAC, NNATIVE_CONTACTS, NNONNATIVE_CONTACTS, + NWALL_CONTACTS) IF + + + +  (ENG.LE.E MAX) THEN CALL FILER (STORTAG, NSTEP, ENG, XHH, XPP, XHP, XHS, XPS, XHW, XPW, XSW, START_PT, VECTOR, NCHANGE, LOW_STEP, LOW_E, LOW_XHH, LOW_XPP, LOW_XHP, LOW_XHS, LOW_XPS, LOW_XHW, LOW_XPW, LOW_XSW, LOW_START, LOW_VECTOR, LOW_CHANGE)  ENDIF * Set bead type to numerical value ********************************** DO 70  BEAD = IF  1,  L  ( T Y P E (BEAD) . E Q . ' P ' )' THEN TYPE_NUM(BEAD)  =  0  TYPE_NUM(BEAD)  =  1  ELSE ENDIF 70  CONTINUE  * Start of i n t e r n a l loop ************************ 100  CONTINUE  * E n d i n g i n t e r n a l loop at at g i v e n l i m i t and s t o r i n g f i n a l r e s u l t s ****************************************************************** IF  ((NSTEP.EQ.LIMIT).OR.(STORTAG.EQ.NSTOR)) SUCCESS  =  100  *  THEN  (REAL(NSUCCESS))/(REAL(NSTEP))  PAST_STORTAG = STORTAG DO 123  DIM =  1,  NDIM  P A S T _ S T A R T (DIM) 123  = S T A R T _ P T (DIM)  CONTINUE DO 125  VNUM = DO 124  1,  NVECTOR  DIM =  1,  NDIM  PAST_VECTOR(DIM,VNUM) 124  =  VECTOR(DIM,VNUM)  CONTINUE  125  CONTINUE IF  ( (STORTAG.EQ.NSTOR)  . O R . (NCHANGE. E Q . C H A N G E _ L I M ) )  WRITE(20,*)'LIMIT WRITE(20,*)  REACHED FOR S I M U L A T I O N  '%AGE OF S U C C E S S F U L  MOVES I S  THEN  #',NCHANGE ',  SUCCESS  W R I T E ( 2 0 , *) WRITE(20,*) WRITE(20,*)  '******  T H E LOWEST ENERGY CONFORMATIONS  'MAXIMUM ENERGY RECORDED =  ',  ******'  E_MAX  W R I T E ( 2 0 , *)  *  *  *Output option  #1:  short  listing  DO 2 1 5  STOR  =  1,  +  conformations  1  LEC',  1 STOR,  'SIM',  LOW_CHANGE(STOR) ,  +  '#' ,  206  LOW_STEP(STOR),  1  E ' ,  LOW_E(STOR)  FORMAT(2X,A3,I7,2X,A3,I4,2X,A1,I10,2X,A1,F9.3)  215  *  unique  STORTAG -  WRITE(20,206)  *  of  *********************************************************  CONTINUE  *Output option  #2:  detailed  listing  of  unique  conformations  ***********************************************************  179  * * * *206 * * * * * * * *208 * * *212 * * * * *210 * *215 * *  *  DO 215 STOR = 1, STORTAG - 1 WRITE(20,*) WRITE(20,206) 'LOW ENERGY CONFORMATION #', STOR FORMAT(A28,14) WRITE(20,*) ' SIMULATION # ' , LOW_CHANGE(STOR), ' STEP COUNT #', LOW_STEP(STOR) W R I T E ( 2 0 , 2 0 8 ) ' H H , LOW_XHH(STOR),'HP', LOW_XHP(STOR),'PP', LOW_XPP(STOR),'HS', LOW_XHS(STOR),'PS', LOW_XPS(STOR),'HW', LOW_XHW(STOR) , 'PW', LOW_XPW(STOR), S W ' , LOW_XSW(STOR) FORMAT(3X,8(A2,13,4X)) WRITE(20,*) ' POSITION', LOW_START(1,STOR), LOW_START(2,STOR) , ' ENERGY ' , LOW_E(STOR) FORMAT(IX,A20,14,A13,14,A8,14,A14,F9.3) WRITE(20,*) ' VECTORS' DO 210 VNUM = 1, NVECTOR WRITE(20,*) LOW_VECTOR(1,VNUM,STOR), LOW_VECTOR(2,VNUM,STOR) CONTINUE WRITE(20,*) ' * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ' CONTINUE  +  1  + + + +  1  +  +  *Output  end  ***********************************************************  ENDIF WRITE(20,*) WRITE ( 2 0 , * ) RETURN ENDIF * Write and sampling data ******************************* IF  ((MOD(NSTEP,NFRAME).EQ.0).AND.(NTOTALSTEP.GT.NSTARTREC)) IF  (RADIUS_SWITCH.EQ.O) WRITE(42,220)  +  NCHANGE,  NNATIVE_CONTACTS,  +  THEN  THEN NSTEP,  ENG, NATIVE_FRAC,  'NNONNATIVE_CONTACTS,  NWALL_CONTACTS  220  FORMAT(14,110,F9.3,F9.3,14,14,14) ELSE CALL R_G_CALC(VECTOR, +  START_PT,  R_G_AVG,  R_G_DEFORM) WRITE(42,222)  +  NCHANGE,  NNATIVE_CONTACTS,  +  NSTEP,  NWALL_CONTACTS, R G_AVG,  222  ENG, NATIVE_FRAC,  NNONNATIVE_CONTACTS, R_G_DEFORM  FORMAT(I4,I10,F7.2,F7.2,I4,I4,I4,F9.3,F9.3) ENDIF ENDIF  * Data sampling to screen ************************* IF  (MOD(NSTEP,NSIGN).EQ.0) WRITE(*,*)  +  '  'SIM', ENERGY  THEN  NCHANGE,  '  STEP#',NSTEP,  ' , ENG  ENDIF * Data sampling to output file ****************************** IF  (MOD(NSTEP,NWRITE).EQ.0) WRITE(20,228)  228  THEN  'SIMULATION  FORMAT(A13,14,5X,A8,110)  #  ',  NCHANGE,  'STEP  #  ',NSTEP  +  WRITE(20,235) 'HH',XHH, 'HP', XHP,'PP ,XPP, X H S , ' P S ' , X P S , ' H W , XHW, ' PW' , XPW  +  WRITE(20,*) ' ENERGY ' , ENG WRITE(20,*) ' POSITION', START_PT(1), START_PT(2)  'HS',  1  235  F O R M A T ( 2 X , 7 ( A 2 , 1 3 , 4X))  WRITE(20,*) ' VECTORS' DO 240 VNUM = 1, NVECTOR WRITE(20,*) CONTINUE  240  WRITE(20,*)  VECTOR(1,VNUM),  VECTOR(2,VNUM)  '***************************************  1  WRITE(20,*) ENDIF • C o u n t e r and f l a g r e s e t f o r i n t e r n a l loop ***************************************** NSTEP  =  NSTEP  NTOTALSTEP  +  1  = NTOTALSTEP +  MOVE_FLAG  =  0  OVER_FLAG  =  0  1  * Translate or rotate polymer ***************************** IF  (MOD(NSTEP,NWHIRL).EQ.0) CALL  WHIRLING(VECTOR, WHIRL_FLAG)  + IF  THEN  (WHIRL_FLAG.EQ.1) GOTO 100  START_PT,  NEW_VECTOR,  NEW_START,  THEN  ENDIF C A L L O V E R L A P _ C H E C K (NEW_VECTOR, I F ( O V E R _ F L A G . E Q . 1 ) THEN GOTO 100  NEW_START,  OVER_FLAG)  ENDIF CALL + + +  NEIGHBOUR_COUNT (NEW_VECTOR, N E W _ S T A R T , NEW_XHH, NEW_XPP, NEW_XHP, NEW_XHS, NEW_XPS, NEW_XHW,  NEW_XPW,  NEW_XSW,  WALL_SWITCH)  NEW_E = F _ E N E R G Y ( N E W _ X H H , N E W _ X P P , NEW_XHP, N E W _ X P S , NEW_XHW, NEW_XPW, NEW_XSW) D_E IF  = NEW_E  - ENG  (D_E.LE.(0.0)) CALL  NEW_XHS,  REFRESH  THEN (VECTOR,  START_PT,  +  XHH, X P P , XHP, XHS, X P S ,  +  NEW_VECTOR,  XHW, XPW, XSW, E N G ,  NEW_START,  +  NEW_XHH,  NEW_XPP,  NEW_XHP,  NEW_XHS,  +  NEW_XHW,  NEW_XPW,  NEW_XSW,  NEW_E)  CALL NEIGHBOUR_CATALOG_SINGLE +  NEW_XPS, (START_PT,VECTOR,  NATIVE_FRAC,  +  NNATIVE_CONTACTS,  +  NWALL_CONTACTS) NSUCCESS  = NSUCCESS  +  NNONNATIVE_CONTACTS,  1  ELSE P_CALC = CALL IF  F_PROB(D_E)  RANDOM_REAL(P_RAND,0.0,1.0)  (P_RAND.LE.P_CALC) C A L L REFRESH  THEN (VECTOR,  START_PT,  +  XHH, XPP,  XHP, XHS, XPS,  +  ENG, NEW_VECTOR,  XHW, XPW, XSW,  +  NEW_XHH,  NEW_XPP,  NEW_XHP,  NEW_XHS,  +  NEW_XPS,  NEW_XHW,  NEW_XPW,  NEW_XSW,  NEW_START, NEW_E)  CALL NEIGHBOUR_CATALOG_SINGLE +  VECTOR,  +  NNATIVE_CONTACTS,  +  • NSUCCESS  = NSUCCESS  (START_PT,  NATIVE_FRAC, NNONNATIVE_CONTACTS,  NWALL_CONTACTS) +  1  ELSE  181  GOTO  100  ENDIF ENDIF IF +  ( E N G . L E . E _ M A X ) THEN C A L L F I L E R (STORTAG, N S T E P , X H S , X P S , XHW, XPW, XSW, S T A R T _ P T ,  ENG, XHH, XPP, XHP, V E C T O R , NCHANGE,  +  LOW_STEP,  LOW_XHP,  +  LOW_XPS,  +  LOW_E,  LOW_XHH,  LOW_XHW,  LOW S T A R T ,  LOW_XPP,  LOW_XPW,  LOW_XHS,  LOW_XSW,  LOW_VECTOR,. LOW_CHANGE)  ENDIF GOTO  100  ENDIF Randomly choose a bead, find- i t s p o s i t i o n and s i t u a t i o n ******************************************************* CALL  RANDOM_INT  (RAND_BEAD,  CALL  FIND_SITUATION  1,  L)  (RAND_BEAD,.  VECTOR,  POSITION,  CONFIG)  C a l c u l a t i n g a p o t e n t i a l move **************************** C A L L MOVE_2D +  (RAND_BEAD,  NEW_VECTOR, IF  VECTOR,  NEW_START,  (MOVE_FLAG.EQ.1)  START_PT,  POSITION,  CONFIG,  MOVE_FLAG)  THEN  T h e f o l l o w i n g s t a t e m e n t ( I F - T H E N l o o p ) i s o n l y n e c e s s a r y when r e c o r d i n g t h e r a t e o f o c c u r a n c e o f l o w e n e r g y c o n f o r m a t i o n s ; t a k e o u t when scanning for unique conformations because i t i s redundant. ********************************** IF  ( E N G . L E . E _ M A X ) THEN C A L L F I L E R (STORTAG,  NSTEP,  ENG, XHH, XPP, XHP,  +  X H S , X P S , XHW, XPW, XSW, S T A R T _ P T ,  VECTOR,  +  LOW_STEP,  LOW_XHP,  +  LOW_XPS,  +  LOW_E, LOW_XHW,  LOW_START,  LOW_XHH,  LOW_XPP,  LOW_XPW,  LOW_VECTOR,  NCHANGE, LOW_XHS,  LOW_XSW,  LOW_CHANGE)  ENDIF ********************************** GOTO  100  ENDIF C h e c k p o t e n t i a l move f o r o v e r l a p , r e s e t new c o n f i g u r a t i o n ********************************************************* CALL IF  OVERLAP_CHECK(NEW_VECTOR, NEW_START,  (OVER_FLAG.EQ.1)  OVER_FLAG)  THEN  T h e f o l l o w i n g s t a t e m e n t ( I F - T H E N l o o p ) i s o n l y n e c e s s a r y when r e c o r d i n g t h e r a t e o f o c c u r r a n c e o f l o w e n e r g y c o n f o r m a t i o n s ; t a k e i t o u t when scanning for unique conformations because i t i s redundant. ********************************.** IF  ( E N G . L E . E _ M A X ) THEN C A L L F I L E R (STORTAG,  NSTEP,  ENG, XHH, XPP, XHP,  +  X H S , X P S , XHW, XPW, XSW, S T A R T _ P T ,  VECTOR,  +  LOW_STEP,  LOW_XHP,  +  LOW_XPS,  +  LOW_E, LOW_XHW,  LOW_START,  LOW_XHH, LOW_XPW,  LOW_VECTOR,  ENDIF ********************************** GOTO ENDIF  100  LOW_XPP, LOW_XSW,  LOW_CHANGE)  NCHANGE, LOW_XHS,  I d e n t i f y c o n t a c t s a n d c a l c u l a t e e n e r g y o f new c o n f o r m a t i o n ********************************************************** CALL + +  +  NEIGHBOUR_COUNT (NEW_VECTOR, NEW_START, NEW_XHH, N E W _ X P P , NEW_XHP, NEW_XHS, NEW_XPS, NEW_XSW, W A L L _ S W I T C H )  NEW_E = F _ E N E R G Y ( N E W _ X H H , N E W _ X P P , NEW_XHW, NEW_XPW, NEW_XSW) D_E IF  + + + + +  NEW_XHP,  NEW_XHW,  NEW_XHS,  NEW_XPW,  NEW_XPS,  = NEW_E - ENG ( D _ E . L E . ( 0 . 0 ) ) THEN C A L L REFRESH (VECTOR, S T A R T _ P T , X H H , X P P , X H P , X H S , X P S , XHW, XPW, XSW, E N G , NEW_VECTOR, NEW_START, NEW_XHH, N E W _ X P P , NEW_XHP, NEW_XHS, N E W _ X P S , NEW_XHW, NEW_XPW, NEW_XSW, NEW_E) CALL NEIGHBOUR_CATALOG_SINGLE (START_PT, VECTOR, N A T I V E _ F R A C , N N A T I V E _ C O N T A C T S , NNONNATIVE_CONTACTS, NWALL_CONTACTS) NSUCCESS = NSUCCESS  + 1  ELSE P_CALC = CALL IF  F_PROB(D_E)  RANDOM_REAL(P_RAND,0.0,1.0)  ( P _ R A N D . L E . P _ C A L C ) THEN C A L L REFRESH (VECTOR,  START_PT,  +  X H H , X P P , X H P , X H S , X P S , XHW, XPW, XSW,  +  NEW_VECTOR,  +  NEW_XHH,  NEW_XPP,  NEW_XHP,  NEW_XHS,  +  NEW_XHW,  NEW_XPW,  NEW_XSW,  NEW_E)  CALL NEIGHBOOR_CATALOG_SINGLE +  NATIVE_FRAC,  +  NNATIVE_CONTACTS,  +  NWALL_CONTACTS) NSUCCESS  ENG,  NEW_START,  = NSUCCESS  NEW_XPS, (START_PT,VECTOR,  NNONNATIVE_CONTACTS,  + 1  ELSE GOTO  100  ENDIF ENDIF Store lowest energy c o n f i g u r a t i o n ********************************* IF  (ENG.LE.E_MAX) CALL  THEN  FILER  (STORTAG,  NSTEP,  ENG, XHH, XPP, XHP,  +  X H S , X P S , XHW, XPW, XSW, S T A R T _ P T ,  VECTOR,  +  LOW_STEP,  LOW_XHP,  +  LOW_XPS,  +  LOW_E, L0W_XHW,  LOW_START,  LOW_XHH, LOW_XPW,  LOW_VECTOR,  LOW_XPP,  NCHANGE, LOW_XHS,  LOW_XSW,  LOW_CHANGE)  ENDIF End l o o p ******** GOTO  100  END  ******************************************************************* SUBROUTINE CENTRE_OF_MASS (VECTOR, S T A R T _ P T , CENTRE) ******************************************************************* Calculates of  the  of  mass  the  chain for  centre  of  mass.  and determines the  the  Takes  the  position  coordinates  and  closest  to  structure the  centre  chain.  Main parameters: VECTOR: the v e c t o r set d e s c r i b i n g the c h a i n conformation START_PT: the c o o r d i n a t e s o f the f i r s t bead e s t a b l i s h i n g the chain position  * *  CENTRE:  the  coordinates  closest  to  the  chain  centre  of  mass  ********************************************************************* * Parameter list **************** PARAMETER  +  (MAXD=2,  MAXC=100)  I N T E G E R L , NDIM, N V E C T O R , VNUM, D I M , F A C T O R , S T A R T _ P T (MAXD) , T C E N T R E (MAXD) , C E N T R E (MAXD) ,  +  V E C T O R (MAXD, MAXC) COMMON /  GRAPH_LIM /  L , NDIM,  NVECTOR  * Initialize parameters *********************** DO 20  D I M = 1,  NDIM  TCENTRE(DIM)  = 0  CENTRE(DIM) 20  = 0  CONTINUE  * C a l c u l a t e c e n t r e o f mass ************************** DO  60  VNUM = 1,  NVECTOR  FACTOR = L DO 40  DIM =  VNUM 1,  NDIM  TCENTRE(DIM) 40  = TCENTRE(DIM)  + FACTOR *  VECTOR(DIM,VNUM)  CONTINUE  60  CONTINUE DO 80  D I M = 1, NDIM CENTRE(DIM) = CENTRE(DIM)  80  NINT(REAL(TCENTRE(DIM))/REAL(L))  = CENTRE(DIM)  +  START_PT(DIM)  CONTINUE RETURN END  ********************************************************************* SUBROUTINE CENTRE_OF_MASS_REAL (VECTOR, S T A R T _ P T , CENTRE_REAL) ********************************************************************* * *  Calculates  *  of  the  * *  of  mass  * *  Main parameters: VECTOR: the vector  *  the  chain for  centre  the  START_PT:  the  *  of  mass.  and determines chain  in  real  set  the  exact  of  position  location  of  and  structure  the  centre  numbers.  describing  coordinates  chain  Takes  the  the  the  first  chain bead  conformation  establishing  the  position  * CENTRE: t h e c h a i n c e n t r e o f mass ( r e a l numbers) ********************************************************************* * Parameter list **************** PARAMETER INTEGER  (MAXD=2,  L , NDIM,  NVECTOR,  +  S T A R T _ P T (MAXD) ,  +  V E C T O R (MAXD, MAXC) R E A L T A B (MAXD) , COMMON /  *  MAXC=100)  Initialize  VNUM,  DIM, FACTOR,  T C E N T R E (MAXD) ,  C E N T R E _ R E A L (MAXD)  GRAPH_LIM /  parameters  L , NDIM,  NVECTOR  DO 20  DIM = 1, NDIM TCENTRE(DIM) =0.0 CENTRE_REAL(DIM) =0.0 CONTINUE  20  * C a l c u l a t e e x a c t c e n t r e o f mass ******************************** DO  VNUM = 1, NVECTOR F A C T O R = L - VNUM DO 40 DIM = 1, NDIM TCENTRE(DIM) = TCENTRE(DIM) CONTINUE CONTINUE  40 60  60  DO 80  80  DIM =  1,  + FACTOR *  VECTOR(DIM,VNUM)  NDIM  CENTRE_REAL(DIM)  =  CENTRE_REAL(DIM)  =  REAL(TCENTRE(DIM))/REAL(L) CENTRE REAL(DIM)  +  REAL(START_PT(DIM))  CONTINUE RETURN END  ********************************************************************* S U B R O U T I N E COORD_MAKER ( V E C T O R , S T A R T _ P T , COORD_OUT) *********************************************************************  * * *  Converts  starting  point  * *  Main parameters: VECTOR: the vector  *  START_PT:  the  *  COORD_OUT:  chain  and v e c t o r  set  data  describing  coordinates  of  coordinates  the on  into  the  first  the  lattice  chain bead  of  coordinates.  conformation the  chain  grid  ********************************************************************* * Parameter l i s t **************** PARAMETER  (MAXD=2,  MAXC=100)  INTEGER L , DIM, NDIM, NVECTOR, BEAD, START_PT(MAXD), VECTOR(MAXD, MAXC),  +  COMMON /  GRAPH_LIM /  L , NDIM,  COORD_OUT(MAXD,MAXC)  NVECTOR  * Calculate coordinates *********************** DO 20  D I M = 1,  NDIM  COORD_OUT ( D I M , 1) 20  = S T A R T _ P T (DIM)  CONTINUE DO  60  BEAD = 2, DO 40  L  DIM = 1, NDIM COORD_OUT(DIM,BEAD)  + 40  = COORD_OUT(DIM,BEAD-1)  +  VECTOR(DIM,BEAD-1) CONTINUE  60  CONTINUE RETURN END  ********************************************************************* SUBROUTINE  FILER  (STORTAG,  NSTEP,  ENG, XHH, XPP, XHP,  +  X H S , X P S , XHW, XPW, XSW, S T A R T _ P T ,  VECTOR,  +  LOW_STEP,  LOW_XHP,  +  LOW_XPS,  +  LOW_START,  LOW_E, LOW_XHW,  LOW_XHH, LOW_XPW,  LOW_VECTOR,  LOW_XPP, LOW_XSW,  LOW_CHANGE)  NCHANGE, LOW_XHS,  Files *  information into  the  storage  files.  Main  parameters: v a r i a b l e used to i d e n t i f y the next u n f i l l e d storage position c y c l e number energy of the a c t i v e conformation number o f c o n t a c t s b e t w e e n t h e two s p e c i f i e d components XHH, etc. f o r t h e a c t i v e c o n f o r m a t i o n ( e . g . XHH i s t h e n u m b e r o f HH c o n t a c t s )  STORTAG: NSTEP: ENG:  *  the c o o r d i n a t e s of the f i r s t bead of START_PT: the v e c t o r set d e s c r i b i n g the a c t i v e VECTOR: c o n t i n u o u s r e p e a t number NCHANGE: LOW_*, e t c : s t o r a g e a r r a y f o r the given v a r i a b l e  *  storage  array  for  the a c t i v e c h a i n chain conformation (e.g.  LOW_XHH i s  the  XHH v a l u e s )  ************************************************************** * Parameter l i s t **************** PARAMETER INTEGER  (MAXD=2,  STORTAG,  MAXC=100,  NSTEP,  MAXSTOR=100000)  XHH, XPP, XHP, XHS,  +  X P S , XHW, XPW,  +  L,  +  S T A R T _ P T (MAXD) ,  +  LOW_XHH(MAXSTOR),  LOW_XPP(MAXSTOR) ,  LOW_XHP(MAXSTOR) ,  +  LOW_XHS (MAXSTOR) ,  LOW_XPS (MAXSTOR) ,  LOW_XHW (MAXSTOR) ,  +  LOW_XPW (MAXSTOR) ,  LOW_XSW (MAXSTOR) ,  L O W _ S T E P (MAXSTOR) ,  +  L O W _ S T A R T (MAXD,MAXSTOR) ,  +  D I M , VNUM,  XSW,  NDIM,  NVECTOR,  NCHANGE,  V E C T O R (MAXD, MAXC) ,  LOW_VECTOR(MAXD,MAXC,MAXSTOR) ,  LOW_CHANGE(MAXSTOR) R E A L E N G , LOW_E(MAXSTOR) COMMON /  GRAPH_LIM /  STORTAG =  L , NDIM,  STORTAG + 1  LOW_STEP(STORTAG) LOW_E(STORTAG)  =  NSTEP  = ENG  LOW_XHH(STORTAG)  = XHH  LOW_XPP(STORTAG)  = XPP  LOW_XHP(STORTAG)  = XHP  LOW_XHS(STORTAG)  = XHS  LOW_XPS(STORTAG)  = XPS  LOW_XHW(STORTAG)  = XHW  LOW_XPW(STORTAG)  = XPW  LOW_XSW(STORTAG)  = XSW  DO 10  NVECTOR  DIM =  1,  NDIM  LOW_START(DIM,STORTAG) 10  =  START_PT(DIM)  CONTINUE DO 40  VNUM = DO 20  1,  NVECTOR  DIM =  1,  NDIM  L O W _ V E C T O R ( D I M , VNUM, STORTAG) 20  = V E C T O R ( D I M , VNUM)  CONTINUE  40  CONTINUE LOW_CHANGE(STORTAG)  = NCHANGE  RETURN END  ********************************************************************* SUBROUTINE +  FILTER  (STORTAG,  NSTEP,  ENG,  XHH,  X P P , X H P , X H S , X P S , XHW, XPW, XSW, S T A R T _ P T ,  +  NCHANGE,  LOW_STEP,  +  LOW_XHS,  LOW_XPS,  +  LOW_VECTOR,  LOW_E, LOW_XHW,  LOW_XHH,  LOW_XPP,  LOW_XPW,  LOW_XSW,  VECTOR,  LOW_XHP, LOW_START,  LOW_CHANGE)  *********************************************************************  * *  Screens  conformations  for  those  which  are  o r i g i n a l or  isomers  of  186  past recorded conformations maximum. Files  w i t h e n e r g i e s lower than t h e g i v e n  i n f o r m a t i o n i n t o the s t o r a g e  files.  Main p a r a m e t e r s : v a r i a b l e used t o i d e n t i f y t h e next u n f i l l e d s t o r a g e p o s i t i o n STORTAG: c y c l e number NSTEP: energy o f t h e a c t i v e c o n f o r m a t i o n ENG: XHH, e t c . number o f c o n t a c t s between the two s p e c i f i e d components f o r t h e a c t i v e c o n f o r m a t i o n (e.g. XHH i s the number o f HH c o n t a c t s ) START_PT: the c o o r d i n a t e s o f t h e f i r s t bead o f the a c t i v e c h a i n the v e c t o r s e t d e s c r i b i n g the a c t i v e c h a i n c o n f o r m a t i o n VECTOR: c o n t i n u o u s r e p e a t number NCHANGE: : s t o r a g e a r r a y f o r t h e g i v e n v a r i a b l e (e.g. LOW_XHH i s the LOW *, e t c s t o r a g e a r r ay ay t f oorr XHH v a l u e s ) * * * * * * * * * * * * * * * * * * * * * * * * * * ******************************************* * Parameter l i s t **************** PARAMETER  (MAXD=2, MAXC=100, MAXSTOR=100000)  INTEGER STORTAG, NSTEP, XHH, XPP, XHP, XHS, + XPS, XHW, XPW, XSW, + STOR, L, DIM, VNUM, NDIM, NVECTOR, NCHANGE, + START_PT(MAXD), VECTOR(MAXD,MAXC) , + LOW_XHH (MAXSTOR) , LOW_XPP (MAXSTOR) , LOW_XHP (MAXSTOR) , + LOW_XHS (MAXSTOR) , LOW_XPS (MAXSTOR) , LOW_XHW (MAXSTOR) , + LOW_XPW (MAXSTOR) , LOW_XSW (MAXSTOR) , LOW_STEP (MAXSTOR) , + LOW_START (MAXD, MAXSTOR) , LOW_VECTOR (MAXD, MAXC, MAXSTOR) , + LOW_CHANGE (MAXSTOR) REAL ENG,  LOW_E(MAXSTOR)  COMMON / GRAPH_LIM / L, NDIM, NVECTOR * I n i t i a l i z e parameter ********************** COUNT = 0 * Loop f o r c h e c k i n g p r e v i o u s s t o r e d c o n f o r m a t i o n and t h e i r isomers ****************************************************************** DO 700 STOR = 1, STORTAG * Comparison w i t h p r e v i o u s l y s t o r e d c o n f o r m a t i o n ************************************************  +  40 60 80  DO 60 VNUM = 1, NVECTOR DO 40 DIM = 1, NDIM IF (VECTOR(DIM,VNUM).NE. LOW_VECTOR(DIM,VNUM,STOR)) THEN GOTO 80 ENDIF CONTINUE CONTINUE GOTO 750 CONTINUE  * Comparison w i t h 90 r o t a t i o n o f p r e v i o u s c o n f o r m a t i o n ****************************************************** DO 100 VNUM = 1, NVECTOR IF ((-l*VECTOR(2,VNUM)) .NE.LOW_VECTOR(1,VNUM, STOR))THEN GOTO 120 ELSEIF (VECTORd, VNUM) .NE. LOW_VECTOR(2, VNUM, STOR) ) THEN GOTO 120 ENDIF  100  CONTINUE GOTO  120  750  CONTINUE  * C o m p a r i s o n w i t h 270 r o t a t i o n o f p r e v i o u s conformation ******************************************************* DO 150  VNUM = IF  1,  NVECTOR  (VECTOR(2,VNUM).NE.LOW_VECTOR(1,VNUM,STOR))  THEN  GOTO 170 ELSEIF ((-l*VECTOR(1,VNUM)).NE. LOW_VECTOR(2,VNUM,STOR))THEN  +  GOTO  170  ENDIF CONTINUE GOTO 750 CONTINUE  150 170  * C o m p a r i s o n w i t h 180 r o t a t i o n o f p r e v i o u s conformation ******************************************************** DO 210  VNUM = DO 200  1,  NVECTOR  D I M = 1,  NDIM  IF((-l*VECTOR(DIM,VNUM)).NE. LOW_VECTOR(DIM,VNUM,STOR)(THEN  +  GOTO  220  ENDIF 200 210  CONTINUE CONTINUE GOTO 750 CONTINUE  220  * C o m p a r i s o n w i t h r e f l e c t i o n o f p r e v i o u s c o n f o r m a t i o n i n x=0 ************************************************************ DO 250  VNUM = IF  1,  NVECTOR  ((-l*VECTOR(l,VNUM)).NE.LOW_VECTOR(1,VNUM,STOR))THEN GOTO  ELSEIF  270  (VECTOR(2,VNUM).NE.LOW_VECTOR(2,VNUM,STOR))THEN GOTO  270  ENDIF 250  CONTINUE GOTO  270  750  CONTINUE  * C o m p a r i s o n w i t h r e f l e c t i o n o f p r e v i o u s c o n f o r m a t i o n i n y=0 ************************************************************ DO 300  VNUM = IF  1,  NVECTOR  ( V E C T O R ( 1 , VNUM) . N E . L O W _ V E C T O R ( 1 , V N U M , S T O R ) ) GOTO  ELSEIF +  THEN  320  ( ( - l * V E C T O R ( 2 , VNUM)) . N E . L O W _ V E C T O R ( 2 ,  VNUM,STOR))  THEN GOTO  320  ENDIF 300  CONTINUE GOTO  320  750  CONTINUE  * C o m p a r i s o n w i t h r e f l e c t i o n o f p r e v i o u s c o n f o r m a t i o n i n x=y ************************************************************ DO 350  VNUM = IF  +  1,  NVECTOR  (VECTOR(1,VNUM).NE.LOW_VECTOR(2,VNUM,STOR)) GOTO 370  ELSEIF THEN  ( (VECTOR ( 2 , VNUM) ) . N E . LOW_VECTOR ( 1 , VNUM, STOR) ) GOTO  ENDIF 350  CONTINUE GOTO 750  THEN  370  CONTINUE  370  * Comparison w i t h r e f l e c t i o n o f p r e v i o u s c o n f o r m a t i o n i n x=-y *********************************************************** DO 400 VNUM = 1, NVECTOR IF((-l*VECTOR(l,VNUM)).NE.LOW_VECTOR(2,VNUM,STOR))THEN GOTO 420 ELSEIF((-l*VECTOR(2,VNUM)).NE.LOW_VECTOR(1,VNUM,STOR)) THEN GOTO 420 ENDIF CONTINUE GOTO 750 CONTINUE  +  400 420 700  CONTINUE STORTAG = STORTAG + 1 CALL  FILER  (STORTAG,  +  XHS,  +  LOW_STEP,  +  LOW_XPS,  +  LOW V E C T O R ,  750  XPS,  NSTEP,  ENG, XHH, XPP, XHP,  XHW, XPW, XSW, LOW_E,  START_PT,  LOW_XHH,  LOW_XHW,  LOW_XPW,  LOW_CHANGE)  LOW_XPP, LOW_XSW,  VECTOR, LOW_XHP,  NCHANGE, LOW_XHS,  LOW_START,  •  CONTINUE RETURN END  *********************************************************************  SUBROUTINE FIND_SITUATION (RAND_BEAD, VECTOR, POSITION, CONFIG) *********************************************************************  * * Determine g e n e r a l p o s i t i o n and s i t u a t i o n o f a p a r t i c u l a r bead: * * Position * 1 = * 2 = * 3 = * 4 = *  l a b e l s (POSITION): f i r s t bead second t o L - 2 bead second l a s t bead l a s t bead  * C o n f i g u r a t i o n l a b e l s (CONFIG): * 1 = f i r s t bead a t r i g h t angle * 2 = f i r s t bead i n s t r a i g h t l i n e * 3 = m i d d l e bead p o s i t i o n e d f o r 3 p t f l i p * 4 = m i d d l e bead p o s i t i o n e d f o r 4 pt f l i p * ( i n c l u d e s v e c t o r p r e v i o u s ^and 2 v e c t o r s * 5 = m i d d l e bead i n s t r a i g h t l i n e * 6 = l a s t bead at r i g h t angle * 7 = l a s t bead i n s t r a i g h t l i n e *  after)  * Main p a r a m e t e r s : * RAND_BEAD: the c h a i n r e s i d u e c u r r e n t l y b e i n g e v a l u t e d * VECTOR: the v e c t o r set d e s c r i b i n g the a c t i v e c h a i n c o n f o r m a t i o n * POSITION: l a b e l o f where the r e s i d u e i s r e l a t i v e t o the r e s t o f the c h a i n * CONFIG: c o n f i g u r a t i o n a l s i t u a t i o n o f the r e s i d u e r e l a t i v e t o n e i g h b o u r i n g * residues *  ********************************************************************** * Parameter l i s t **************** PARAMETER (MAXD=2, MAXC=100)  + + +  INTEGER L , DIM, RAND_BEAD, POSITION, CONFIG, N_VECTOR, DEPTH, F_DOT, NDIM, PROD_FLAG, V _ L I N E 1 ( 3 ) , V_LINE2(3), FLAG(2), OPPOS(2), DOT ARRAY (3), V_ARRAY (2,3) , NVECTOR,  189  +  VECTOR(MAXD,MAXC) COMMON  /  GRAPH_LIM /  L,  NDIM,  NVECTOR  * Initialize parameters ***********************  DO  60  DEPTH = DO  40  1,  3  DIM =  1,  NDIM  V_ARRAY(DIM,DEPTH) 40  DOT_ARRAY(DEPTH) 60  0  =  0  CONTINUE DO  80  =  CONTINUE  80  DIM =  1,  NDIM  V_LINE1(DIM)  =  0  V_LINE2(DIM)  =  0  CONTINUE  * D e f i n e POSITION and CONFIG **************************** IF  (RAND_BEAD.EQ.1) POSITION  =  1  N_VECTOR =  2  DO  140  THEN  DEPTH = DO  120  1,  2  DIM =  1,  NDIM  V_ARRAY(DIM,DEPTH) 120  =  VECTOR(DIM,DEPTH)  CONTINUE  140  CONTINUE ELSE  IF  (RAND_BEAD.EQ.(L-l))  POSITION  =  3  N_VECTOR =  2  DO 2 2 0  220  DIM =  1,  THEN  NDIM  V_ARRAY(DIM,1)  =  VECTOR(DIM,L-2)  V_ARRAY(DIM,2)  =  VECTOR(DIM,L-l)  CONTINUE ELSE  IF  (RAND_BEAD.EQ.L)  POSITION  =  4  N_VECTOR  =  2  DO 2 4 0  240  DIM =  1,  THEN  •  NDIM  V _ A R R A Y ( D I M , 1)  = VECTOR(DIM, L - 2 )  V_ARRAY(DIM,2)  =  VECTOR(DIM,L-l)  CONTINUE ELSE POSITION  =  2  N_VECTOR  =  3  DO 2 6 0  DIM =  1,  NDIM  V_ARRAY(DIM,1)  2 60  = VECTOR(DIM,RAND_BEAD)  V_ARRAY(DIM,3)  = VECTOR(DIM,RAND_BEAD+1)  CONTINUE END  IF  DO 2 9 0  DEPTH = DO 2 8 0  280  1,  DIM =  3 1,  NDIM  V_LINE1(DIM)  =  V_LINE2(DIM)  =  V_ARRAY(DIM,DEPTH+1) V_ARRAY(DIM,DEPTH)  CONTINUE DOT_ARRAY(DEPTH)  290  = VECTOR(DIM,RAND_BEAD-1)  V _ A R R A Y ( D I M , 2)  =  F_DOT(V_LINE1,  CONTINUE IF  (POSITION.EQ.l) IF  THEN  (DOT_ARRAY(1).EQ.0) CONFIG =  1  CONFIG  2  ELSE =  THEN  V_LINE2)  END ELSE  IF  IF IF  (POSITION.EQ.2)  THEN  (DOT_ARRAY(1).EQ.1) CONFIG =5  THEN  ELSE DO 3 5 0  DIM = 1, FLAG(DIM)  NDIM = 0  OPPOS(DIM) = - 1 * V_ARRAY(DIM,1) IF (OPPOS(DIM).EQ.V_ARRAY(DIM,3)) FLAG(DIM) = 1 END I F 350  THEN  CONTINUE PROD_FLAG = 1 DO 360 DIM = 1, NDIM PROD_FLAG = PROD_FLAG * CONTINUE IF (PROD_FLAG.EQ.1) THEN CONFIG = 4  360  FLAG(DIM)  ELSE CONFIG = END IF  END ELSEIF  (POSITION.EQ.3) IF  3  IF  THEN  (DOT_ARRAY(1).EQ.1)  THEN  CONFIG = 5 ELSE CONFIG = END  3  IF  ELSE IF  ( D O T _ A R R A Y (1) . E Q . 0) CONFIG = 6  THEN  ELSE CONFIG = IF  END  7  ENDIF RETURN END  ************************************************************ SUBROUTINE INCONSISTANCY_CHECK (VECTOR, INCON_FLAG) ********************************************************************* * * * * *  Scans the i n i t i a l c h a i n conformation given at the s t a r t of s i m u l a t i o n f o r e r r o r s , such as d i a g o n a l v e c t o r s o r s k i p p e d coordinates.  *  Main  the  parameters:  *  VECTOR:  the  *  ICON_FLAG:  flag  vector  set  describing  signalling  whether  the  chain  *  (=0,  chain  is  fine,  continue  * *  (=1,  chain  in  inconsistent,  active is  chain  conformation  acceptable  or  not  program) need  to  end  program)  ********************************************************************* * Parameter list **************** PARAMETER INTEGER +  L,  NDIM,  INCON_FLAG, COMMON /  *  (MAXD=2,  Checking  for  MAXC=100)  NVECTOR,  GRAPH_LIM / skipped  D I M , VNUM,  V E C T O R (MAXD, MAXC) L,  NDIM,  coordinate  NVECTOR  DO 3 2 0  VNUM = DO 3 0 0  1,  NVECTOR  DIM = IF  1,  NDIM  (ABS(VECTOR(DIM,VNUM)).GE.2)  THEN  WRITE(*,*) ' B A D ORIGINAL CONFORMATION' WRITE!*,*) ' O N E V E C T O R I S G R E A T E R THAN INCON_FLAG = 1 GOTO 330  2'  ENDIF CONTINUE  300  * Checking f o r v e c t o r not f o l l o w i n g lattice ******************************************* IF  (ABS(VECTOR(2,VNUM)).EQ. ( A B S ( V E C T O R ( 1 , V N U M ) ) ) ) THEN WRITE!*,*) ' B A D O R I G I N A L CONFORMATION' WRITE(*,*) ' I L L E G A L DIAGONAL S T E P T A K E N ' INCON_FLAG = 1 GOTO 330  +  320  ENDIF CONTINUE  330  CONTINUE RETURN END  ********************************************************************* S U B R O U T I N E I S W I T C H ( A , B) *********************************************************************  * * *  Switches  integers  A and B  ********************************************************************* INTEGER A ,  B,  C  C = A A = B B = C RETURN END  ********************************************************************* SUBROUTINE +  MOVE_2D  CONFIG,  (RAND_BEAD,  NEW_VECTOR,  VECTOR,  NEW_START,  START_PT,  POSITION,  MOVE_FLAG)  ********************************************************************* * * * *  Carries i s used  *  POS  CONFG CODE  MOVE straighten 180° flip  *  o u t a p o t e n t i a l move f o r a g i v e n b e a d . T h e W A L L - B O U N C E here so t h a t the r e s u l t i n g c o n f o r m a t i o n i s  *  1  1  1  *  1  1  2  * *  1 1  2  1  2  2  90° 90°  *  2  3  NA  3 pt  *  2  4  NA  4 pt  flip  *  3  3  NA  3 pt  flip  *  4  6  1  straighten  *  4 4  2 1  180°  * *  6 7  90°  counter-clockwise  4  8  2  90°  clockwise  *  NA  5  NA  no  Main  counter-clockwise clockwise (270°)  function  (90°)  flip  flip (90°)  (270°)  move  parameters:  RAND_BEAD:  the  chain  VECTOR:  the  vector  residue  START_PT:  the  coordinates  POSITION:  label  set  currently being  describing  describing  of  the  where  the  first the  evaluted  active residue  residue  is  chain of  the  conformation active  relative  to  chain the  rest  of  *  the  chain  * * *  CONFIG:  configurational situation residues NEW_VECTOR:the v e c t o r , set d e s c r i b i n g  * * *  NEW_START: MOVE_FLAG:  * *  of  the  the  residue  newly  relative  calculated  to  conformation  the c o o r d i n a t e s o f the newly c a l c u l a t e d c o n f o r m a t i o n f l a g s i g n a l l i n g w h e t h e r t h e move i s p l a u s i b l e (0 = move i s accepted) (1  = move  is  unacceptable)  **************************************************** * Parameter l i s t **************** PARAMETER  (MAXD=2,  INTEGER L ,  MAXC=100)  D I M , NDIM,  +  MOVE_FLAG,  +  START_PT(MAXD),  +  GRIDLIM(MAXD),  +  NVECTOR,  RAND_BEAD,  B O U N C E _ F L A G , VNUM,  CHECK_PT(MAXD),  V E C T O R (MAXD, MAXC) , COMMON /  GRAPH_LIM /  COMMON /  LATTICE /  BEAD,  POSITION,  CONFIG,  RAND_MOVE,  NEW_START(MAXD) ,  NEW_VECTOR (MAXD, MAXC)  L , NDIM,  NVECTOR  GRIDLIM  * Initialize parameters *********************** MOVE_FLAG = 0 RAND_MOVE = 0 BOUNCE F L A G = DO 20  0  VNUM = 1, NVECTOR DO 10 D I M = 1, NDIM  NEW_VECTOR(DIM,VNUM) CONTINUE CONTINUE  10 20  D I M = 1, NDIM NEW_START(DIM) CONTINUE  =  VECTOR(DIM,VNUM)  DO 25 25  =  START_PT(DIM)  * No move f o r m i d d l e b e a d i n s t r a i g h t line ****************************************** IF  (CONFIG.EQ.5)  THEN  MOVE_FLAG = GOTO  1  300  ENDIF * Moves f o r bead 1 ****************** IF  (POSITION.EQ.l) CALL IF  THEN  RANDOM_INT  (RAND_MOVE,  (CONFIG.EQ.l) IF  2)  THEN  (RAND_MOVE . E Q . l ) ' DO 30  1,  D I M = 1,  THEN NDIM  NEW_VECTOR(DIM,1) NEW_START (DIM) +  VECTOR(DIM,1)  30  =  VECTOR(DIM,2)  = S T A R T _ P T (DIM) -  +  VECTOR(DIM,2)  CONTINUE ELSE DO 40  DIM =  1,  NDIM  NEW_VECTOR(DIM,1)  = -1  *  VECTOR(DIM,1)  NEW_START (DIM) = S T A R T _ P T (DIM) +  2  40  * V E C T O R ( D I M , 1)  CONTINUE ENDIF ELSE IF  (RAND M O V E . E Q . l )  THEN  neighbouring  +  N E W _ V E C T O R ( 1 , 1) NEW_VECTOR(2,1)  = =  -1 * VECTOR(2,1) VECTOR(1,1)  N E W _ V E C T O R ( 1 , 1) NEW_VECTOR(2,1)  = =  VECTOR(2,1) -1 * V E C T O R ( l , l )  ELSE  ENDIF DO 60  DIM =  1,  NDIM  NEW_START(DIM) = S T A R T _ P T ( D I M ) + VECTOR(DIM,1) - NEW_VECTOR(DIM,1) CONTINUE  + 60 ENDIF  * Moves f o r beads 2 t o L - 2 ************************** ELSEIF  (POSITION.EQ.2) IF (CONFIG.EQ.3) DO 80  THEN THEN  DIM =  1,  NDIM  N E W _ V E C T O R ( D I M , R A N D _ B E A D - 1 ) >= V E C T O R ( D I M , RAND_BEAD) N E W _ V E C T O R ( D I M , RAND_BEAD) = VECTOR(DIM, RAND_BEAD-1)  + + 80  CONTINUE ELSE DO 100  DIM =  1,  NDIM  +  NEW_VECTOR(DIM, RAND_BEAD-1) = -1 * VECTOR(DIM, RAND_BEAD-1)  +  N E W _ V E C T O R ( D I M , RAND_BEAD+1) = - 1 * V E C T O R ( D I M , RAND_BEAD+1)  100  CONTINUE ENDIF  * Moves f o r beads L - l ********************* ELSEIF  ( P O S I T I O N . E Q . 3 ) THEN DO 1 2 0 D I M = 1, NDIM NEW_VECTOR(DIM, RAND_BEAD-1) = V E C T O R ( D I M , RAND_BEAD) N E W _ V E C T O R ( D I M , RAND_BEAD) = VECTOR(DIM, RAND_BEAD-1) CONTINUE  + + 120  * Moves f o r b e a d L ****************** ELSE C A L L RANDOM_INT IF  (CONFIG.EQ.6) IF  (RAND_MOVE,  1,  2)  THEN  (RAND_MOVE.EQ.1) DO 140  DIM =  THEN 1,  NDIM  NEW_VECTOR(DIM,L-l) 140  = VECTOR ( D I M , L - 2 )  CONTINUE ELSE DO 160  DIM =  1,  NDIM  NEW_VECTOR(DIM,L-l) +  =  -1  * VECTOR(DIM,L-l)  160  CONTINUE ENDIF ELSE IF  (RAND_MOVE.EQ.1)  THEN  NEW_VECTOR(l,L-l)  = -1  NEW_VECTOR(2,L-l)  =  VECTOR(1,L-1)  *  VECTOR(2,L-l)  NEW_VECTOR(l,L-l)  =  VECTOR(2,L-l)  ELSE NEW_VECTOR(2,L-l)  1  *  VECTOR(1,L-l)  ENDIF ENDIF ENDIF  * Check whether newly formed c o n f o r m a t i o n i s  acceptable  DO 200  DIM =  1,  NDIM  C H E C K _ P T ( D I M ) = NEW_START(DIM) IF ((CHECK_PT(DIM).GT.GRIDLIM(DIM)).OR. (CHECK_PT(DIM).LT.O)) THEN  +  C A L L W A L L _ B O U N C E (NEW_VECTOR, BOUNCE_FLAG) I F ( B O O N C E _ F L A G . E Q . 1 ) THEN MOVE_FLAG = 1 GOTO 300  +  NEW_START,  ENDIF ENDIF CONTINUE  200  DO 2 60  BEAD = 2, L DO 240 DIM = 1,  NDIM  CHECK_PT(DIM) = CHECK_PT(DIM) NEW_VECTOR(DIM,BEAD-1)  +  IF  +  ((CHECK_PT(DIM).GT.GRIDLIM(DIM)).OR. (CHECK_PT(DIM).LT.0)) THEN  +  C A L L W A L L _ B O U N C E (NEW_VECTOR, BOUNCE_.FLAG)  +  IF  (BOUNCE_FLAG.EQ.1) MOVE_FLAG = 1 GOTO 300  NEW_START,  THEN  ENDIF 240 260  ENDIF CONTINUE CONTINUE  300  CONTINUE RETURN END  ****************************************************************** +  S U B R O U T I N E NEIGHBOUR_COUNT (VECTOR, S T A R T _ P T , X H S , X P S , XHW, XPW, XSW, WALL_SWITCH)  XHH, XPP, XHP,  ********************************************************************* * *  Counts  the  contacts  *  NEARBY  array  north  (0,  south  (0, -1) ( 1 , 0) (-1 , 0 )  east west *  Main  existing  a  conformation.  1)  parameters:  *  VECTOR:  the  vector  *  START_PT:  the  coordinates  *  XHH, e t c .  number  of  describing of  the  contacts  for  *  HH c o n t a c t s ) WALL_SWITCH:flag  the  set  * *  within  active  the  first  between  the  conformation  indicating  the  *  (1  = vertical  walls  *  (2  = horizontal  * *  (4  = all  walls  walls on,  walls  active bead two  (e.g.  active x=0,  on,  of  y=0,  chain the  specified XHH i s  in  the  conformation  active the  chain  components number  of  simulation  GRIDLIM) GRIDLIM)  active)  ********************************************************************* * Parameter list **************** PARAMETER INTEGER L , +  (MAXD=2, NDIM,  MAXC=100,  NVECTOR,  MAXS=3)  BEAD,  D I M , QUAD,  NSIDES,  SIDE,  X H H , X P P , X H P , X H S , X P S , XHW, XPW, XSW,  +  PROD_FLAG,  +  S T A R T _ P T (MAXD) ,  SCAN_BEAD,  C H E C K _ P T (MAXD) ,  WALL_SWITCH, WALL_AREA,  +  P R E V I O U S (MAXD) ,  G R I D L I M (MAXD) ,  F L A G (MAXD),  S I D E _ P T (MAXD) ,  +  NEARBY (MAXD, 4) ,  +  V E C T O R (MAXD, MAXC)  S I D E S (MAXD, MAXS) ,  COORD_MAP (MAXD, MAXC) ,  CHARACTER*1 TYPE(MAXC) COMMON / COMMON / COMMON /  GRAPH_LIM / L , NDIM, L A T T I C E / GRIDLIM TYPE / TYPE  DATA N E A R B Y / 0 ,  1,  0,  -1,  1,  NVECTOR  0,  -1,  0/  * Initialize parameters *********************** XHH 0 XPP = 0 XHP = 0 XHS = 0 XPS 0 XHW = 0 XPW = 0 XSW = 0 CALL *  COORD_MAKER ( V E C T O R , S T A R T _ E T , COORD_MAP)  Establishing non-connecting  DO 600  BEAD = DO 70  1,  neighbouring  IF  for  selected  bead  L  DIM =  1,  NDIM  CHECK_PT(DIM) CONTINUE  70  sites  (BEAD.EQ.l) NSIDES  =  SIDE  1  =  DO 120  = COORD_MAP(DIM,BEAD)  THEN 3  QUAD = IF  1,  4  ( ( N E A R B Y ( 1 , QUAD)  +  .NE.VECTOR(1,1))  .OR.(NEARBY(2,QUAD).NE.VECTOR(2,1)))THEN DO 80  DIM =  1,  NDIM  SIDES(DIM,SIDE) 80  = NEARBY(DIM,QUAD)  CONTINUE SIDE  = SIDE  + 1  ENDIF 120  CONTINUE ELSEIF  (BEAD.EQ.L) NSIDES SIDE  =  DO 220  =  THEN  3  1 QUAD = 1, DO 140  4  DIM =  1,  NDIM  PREVIOUS(DIM) 140  = -1  * VECTOR(DIM,L-l)  CONTINUE IF  ((NEARBY(1,QUAD).NE.PREVIOUS(1)).OR.  +  (NEARBY(2,QUAD).NE.PREVIOUS(2))) DO 180  DIM =  1,  SIDES(DIM,SIDE) 180  THEN  NDIM = NEARBY(DIM,QUAD)  CONTINUE SIDE  = SIDE  + 1  ENDIF 220  CONTINUE ELSE NSIDES  =  SIDE  1  =  DO 320  2  QUAD = 1, DO 240  4  DIM =  1,  NDIM  PREVIOUS(DIM) 240  = -1  * VECTOR(DIM,BEAD-1)  CONTINUE IF + +  (((NEARBY(1,QUAD).EQ.PREVIOUS(1)).AND. (NEARBY(2,QUAD).EQ.PREVIOUS(2))).OR. ((NEARBY(1,QUAD).EQ.VECTOR(1,BEAD)).AND.  +  (NEARBY(2,QUAD).EQ.VECTOR(2,BEAD)))) CONTINUE  THEN  ELSE DO 280  DIM =  1,  NDIM  SIDES(DIM,SIDE) CONTINUE SIDE = SIDE + 1  280  =  NEARBY(DIM,QUAD)  ENDIF 320  CONTINUE ENDIF  * E s t a b l i s h i n g number o f i n t r a m o l e c u l a r and w a l l c o n t a c t s ********************************************************* DO 500  SIDE  =  DO 340  1,  NSIDES  DIM =  1,  NDIM  SIDE_PT(DIM) 340  =  CHECK_PT(DIM)  +  SIDES(DIM,SIDE)  CONTINUE IF  (WALL_SWITCH.EQ.1)  IF  THEN  ((SIDE_PT(1).GT.GRIDLIM(1)).OR.  +  (SIDE_PT(1).LT.0)) IF  THEN  (TYPE(BEAD).EQ.'H')  XHW = XHW +  1  XPW = XPW +  1  THEN  ELSE ENDIF GOTO  500  ENDIF ELSEIF  (WALL_SWITCH.EQ.2) IF  +  THEN  ((SIDE_PT(2).GT.GRIDLIM(2)).OR.  (SIDE_PT(2).LT.0)) IF  THEN  (TYPE(BEAD).EQ.'H') XHW = XHW +  1  XPW =  1  THEN  ELSE XPW +  ENDIF GOTO  500  ENDIF ELSE DO 360  DIM =  1,  IF +  NDIM  ((SIDE_PT(DIM).GT.GRIDLIM(DIM)).OR.  (SIDE_PT(DIM).LT.0)) IF  THEN  (TYPE(BEAD).EQ.'H ) 1  XHW = XHW +  1  XPW = XPW +  1  THEN  ELSE .ENDIF GOTO  500  ENDIF 3 60  CONTINUE ENDIF DO 400  SCAN_BEAD =  PROD_FLAG = DO 380  1,  L  1  DIM =  1,  NDIM IF  +  (SIDE_PT(DIM).EQ.  COORD_MAP ( D I M , S C A N _ B E A D ) ) FLAG(DIM)  =  1  =  0  THEN  . ELSE FLAG(DIM) ENDIF PROD_FLAG = 380  PROD_FLAG *  F L A G (DIM)  CONTINUE IF  (PROD_FLAG.EQ.1) IF  THEN  ((TYPE(BEAD).EQ.'H').AND.  +  (TYPE(SCAN_BEAD).EQ.'H'))  ELSEIF  XHH =  XHH +  GOTO  500  THEN  1  ((TYPE(BEAD).EQ.'P').AND.  197  (TYPE(SCAN_BEAD).EQ.'P')) . XPP = XPP + 1 GOTO 500  THEN  ELSE XHP = XHP + GOTO  1  500  ENDIF ENDIF 400  CONTINUE IF  (TYPE(BEAD).EQ.'H') XHS = XHS + 1 ELSE XPS  = XPS  THEN  +1  ENDIF 500  CONTINUE  600  CONTINUE XHH = XHH /  2  XPP  = XPP /  2  XHP = X H P /  2  * D e t e r m i n e number o f s o l v e n t - w a l l contacts * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *,****** IF  (WALL_SWITCH.EQ.4)  THEN  WALL_AREA = (2*GRIDLIM(1)) ELSEIF (WALL_SWITCH.EQ.l) THEN WALL_AREA =  2*GRIDLIM(2)  WALL_AREA =  2*GRIDLIM(1)  +  (2*GRIDLIM(2))  ELSE ENDIF XSW = W A L L _ A R E A -  XHW - XPW  RETURN END  ********************************************************************* SUBROUTINE NEIGHBOUR_CATALOG_SINGLE ( S T A R T _ P T , V E C T O R , N A T I V E _ F R A C , + N N A T I V E _ C O N T A C T S , NNONNATIVE_CONTACTS, NWALL_CONTACTS) ********************************************************************* * *  Compares the i n t r a m o l e c u l a r c o n t a c t s o f the working c o n f o r m a t i o n the conformation given i n the input f i l e for comparison (usually  * *  native  *  Main  *  state  conformation).  parameters:  START_PT:  the  * *  VECTOR:  the  NATIVE_FRAC: NNATIVE_CONTACTS:  set  of  over  number  the  first  describing  NNONNATIVE_CONTACTS: NWALL_CONTACTS:  contacts  all  of  bead  of  the  active  the  active  chain  number  of  match  contacts  number o f  *  with  * *  non-active)  matching  those  of  comparison  comparison matching  those  in  given  conformation  case  do n o t  the  in  contacts  comparison  * *  vector  fraction case  * *  of  conformation  * *  coordinates  chain  * *  with the  in  given  those  of  comparison  contacts  of  the  grid  boundary  active  (either  which  conformation  active  or  •  ********************************************************************* * Parameter list **************** PARAMETER INTEGER  (MAXD=2,  L , NDIM,  MAXC=100,  NVECTOR,  MAXSTOR=l00000,  MAXCONTACTS=1000)  +  NNEIGHBOUR_ORIGIN,  +  NNEIGHBOUR,  +  NNATIVE_CONTACTS,  +  S T A R T _ P T (MAXD) ,  +  NEIGHBOUR_ORIGIN(2,MAXCONTACTS), REAL  HOOD,  NNATIVE_ORIGIN, NNONNATI V E _ C O N T A C T S ,  NWALL_CONTACTS,  V E C T O R (MAXD, MAXC) , NEIGHBOUR(2,MAXCONTACTS)  NATIVE_FRAC  COMMON /  GRAPH_LIM /  COMMON /  L , NDIM,  NEIGHBOUR  NVECTOR  / NEIGHBOUR_ORIGIN, NNATIVE ORIGIN  +  NNEIGHBOUR_ORIGIN,  * Initialize parameters *********************** NNATIVE_CONTACTS  =  0  NNONNATIVE_CONTACTS  =  0  N W A L L _ C O N T A C T S •= 0 NNEIGHBOUR CALL +  =  1  NEIGHBOURED  NEIGHBOUR,  (START_PT,  VECTOR,  NNEIGHBOUR)  * D e t e r m i n e number o f m a t c h i n g , n o n - m a t c h i n g and w a l l c o n t a c t s ************************************************************** DO 100  HOOD = 1, NNEIGHBOUR • DO 90 COMPARE = 1, NNEIGHBOUR_ORIGIN IF  +  ( N E I G H B O U R S , HOOD) . L T . 0 ) NWALL_CONTACTS = NWALL_CONTACTS + 1  THEN  +  GOTO 100 ENDIF I F ( ( N E I G H B O U R ( 1 , HOOD) . E Q . N E I G H B O U R _ O R I G I N ( l , COMPARE)) . A N D .  + +  (NEIGHBOUR(2,HOOD).EQ. NEIGHBOUR_ORIGIN(2,COMPARE)))  THEN  NNATIVE_CONTACTS = NNATIVE_CONTACTS + GOTO  1  100  ENDIF 90  CONTINUE NNONNATIVE_CONTACTS CONTINUE  100  = NNONNATIVE_CONTACTS  NAT I V E _ F R A C = R E A L ( N N A T I V E _ C O N T A C T S ) +  +  1  /  REAL"( N N A T I V E _ O R I G I N ) RETURN END  *************************************,******************************** SUBROUTINE +  NEIGHBOURED  (START_PT,  VECTOR,  NEIGHBOUR,  NNEIGHBOUR)  ********************************************************************* * *  Identify  * *  and c o n t a c t s  *  Contact  * * * * * * *  the  intramolecular made  codes  y-axis (x  with  for  a  wall  the  2D s y s t e m  (x=0):  VECTOR:  the  * *  NEIGHBOUR:  array l i s t of conformation  *  NNEIGHBOUR:total  set  number  NEIGHBOUR  conformation  (NEIGHBOUR  array)  -11  *  *  a  -20 -21  coordinates vector  within  boundary.  -10  = gridlim_x):  x - a x i s w a l l (y=0): (y = g r i d l i m _ y ) : Main parameters: START_PT: the  contacts grid  of  the  first  describing  the  intramolecular of  contacts  for  bead  of  active  the  and boundary the  active  chain  chain  conformation contacts  conformation,  size  of  a  of  array  199  ********************************************************* *  Parameter  list:  PARAMETER  (MAXD=2,  MAXC=100,  MAXS=3,  MAXCONTACTS=1000)  INTEGER L , NDIM, BEAD, D I M , SCAN_BEAD, PROD_FLAG, N S I D E S , S I D E , QUAD, NNEIGHBOUR, PREVIOUS(MAXD), FLAG(MAXD), SIDE_PT(MAXD), GRIDLIM(MAXD) , START_PT(MAXD), V E C T O R (MAXD, MAXC) , COORD_MAP (MAXD, MAXC) , NEARBY (MAXD, 4 ) , SIDES(MAXD,MAXS), N E I G H B O U R ( 2 , MAXCONTACTS)  + + + + +  COMMON / G R A P H _ L I M / L , NDIM, COMMON / L A T T I C E / G R I D L I M  NVECTOR  DATA N E A R B Y / 0 ,  -1,  1,  0,  -1,  1,  0,  0/  * Initialize parameters *********************** CALL  C O O R D _ M A K E R ( V E C T O R , S T A R T _ P T , COORD_MAP)  NNEIGHBOUR =  0  * E s t a b l i s h non-connecting neighbouring s i t e s for s e l e c t e d bead *************************************,************************** DO 600  BEAD =  1,  L - l  DO 60  S I D E = 1, 3 DO 50 DIM= 1, NDIM SIDES(DIM,SIDE) CONTINUE CONTINUE  50 60 IF  =  0  ( B E A D . E Q . l ) THEN NSIDES = 3 SIDE = 1 DO 120  +  80  120 ELSEIF  QUAD = 1, 4 I F ( (NEARBY ( 1 , QUAD) . N E . V E C T O R d , 1) ) .OR.(NEARBY(2,QUAD).NE.VECTOR(2,1)))THEN DO 80 DIM = 1, NDIM S I D E S ( D I M , S I D E ) = NEARBY(DIM,QUAD) CONTINUE SIDE = SIDE + 1 ENDIF  CONTINUE (BEAD.EQ.L) NSIDES = 3 SIDE = 1 DO 220  THEN  QUAD =  1,  4  DO 140  DIM = 1, NDIM PREVIOUS(DIM) = -1 CONTINUE  140  IF +  180  * VECTOR(DIM, L - l )  ( ( N E A R B Y f l , QUAD) . N E . P R E V I O U S ( 1 ) ) . O R . ( N E A R B Y ( 2 , Q U A D ) . N E . P R E V I O U S ( 2 ) ) ) THEN DO 180 DIM = 1, NDIM S I D E S ( D I M , S I D E ) = NEARBY(DIM,QUAD) CONTINUE SIDE = SIDE  +  1  ENDIF CONTINUE  220 ELSE  240  NSIDES = 2 SIDE = 1 DO 320 QUAD = 1, 4 DO 240 DIM = 1, NDIM PREVIOUS(DIM) = -1 CONTINUE  *  VECTOR(DIM,BEAD-1)  200  IF +  (((NEARBY(1,QUAD).EQ.PREVIOUS(1)).AND. (NEARBY(2,QUAD).EQ.PREVIOUS(2))).OR.  +  ((NEARBY(1,QUAD) . E Q . V E C T O R ( 1 , BEAD)) . A N D .  +  (NEARBY(2,QUAD) . E Q . V E C T O R ( 2 , B E A D ) ) ) ) CONTINUE  THEN  ELSE DO 280  DIM =  1,  NDIM  SIDES(DIM,SIDE) CONTINUE SIDE = SIDE + 1  280  =  NEARBY(DIM,QUAD)  ENDIF CONTINUE  320 ENDIF  * E s t a b l i s h p o s i t i o n of bead through e v a l u a t i o n of neighbouring ************************************************************* DO 500  SIDE =  1,  NSIDES  DO 340 340  sites  D I M = 1, NDIM S I D E _ P T ( D I M ) = COORD_MAP(DIM,BEAD)  +  SIDES(DIM,SIDE)  CONTINUE  * Check f o r w a l l - c h a i n c o n t a c t s ******************************* DO 350  DIM = IF  1,  NDIM  (SIDE_PT(DIM).LT.0)  THEN  NNEIGHBOUR = NNEIGHBOUR NEIGHBOUR(1,NNEIGHBOUR) NEIGHBOUR(2,NNEIGHBOUR) GOTO 500 ELSEIF  + 1 = BEAD = -10*DIM  ( S I D E _ P T ( D I M ) . G T . G R I D L I M ( D I M ) ) THEN NNEIGHBOUR = NNEIGHBOUR + 1 NEIGHBOUR(1,NNEIGHBOUR) = BEAD N E I G H B O U R ( 2 , NNEIGHBOUR) = (-10*DIM) GOTO 500  -  1  ENDIF 350  CONTINUE DO 450  SCAN_BEAD  = BEAD + 1,  PROD_FLAG = DO 420  DIM = IF  L  1 1,  NDIM  (SIDE_PT(DIM).EQ.COORD_MAP(DIM,SCAN_BEAD)) FLAG(DIM)  =  1  FLAG(DIM)  =  0  THEN  ELSE ENDIF PROD_FLAG = 420  PROD_FLAG *  FLAG(DIM)  CONTINUE IF  (PROD_FLAG.EQ.l) ' NNEIGHBOUR  THEN +  1  NEIGHBOUR(1,NNEIGHBOUR)  =  BEAD  NEIGHBOUR(2,NNEIGHBOUR)  =  SCAN_BEAD  GOTO  = NNEIGHBOUR  500  ENDIF 450  CONTINUE  500 600  CONTINUE CONTINUE RETURN END  ********************************************************************* SUBROUTINE OVERLAP_CHECK (VECTOR, S T A R T _ P T , OVER_FLAG) ********************************************************************* *  Checking  *  evaluation  for of  overlap whether  in or  the not  conformation a potential  of  the  move  is  chain.  The  acceptable.  201  * Main parameters: * VECTOR: the * START_PT: the * *  OVER_FLAG:  v e c t o r set d e s c r i b i n g the a c t i v e c o o r d i n a t e s of the f i r s t bead of  chain conformation the a c t i v e chain  f l a g i n d i c a t i n g whether o v e r l a p of beads o c c u r s (0 = i n i t i a l v a l u e o f f l a g , n o o v e r l a p d e t e c t e d )  * *  (1  = overlap  detected,  present  conformation  not  possible)  **************************************************************** * Parameter list **************** PARAMETER  +  (MAXD=2,  MAXC=100)  INTEGER NDIM, NVECTOR, D I M , BEAD, PROD_FLAG, L , OVER_FLAG,  +  CHECK_PT(MAXD) ,  +  P R E V _ P T (MAXD, MAXC) , COMMON /  GRAPH_L  FLAG(MAXD),  /  I M  L,  PREV_BEAD, START_PT(MAXD) ,  V E C T O R (MAXD, MAXC)  NDIM,  NVECTOR  * Initialize parameters *********************** DO 80  DIM =  1,  NDIM  CHECK_PT(DIM) 80  =  START_PT(DIM)  CONTINUE  * Identify possible overlap *************************** DO 190  BEAD = DO 110  2,  L  D I M = 1, NDIM CHECK_PT(DIM) = CHECK_PT(DIM) PREV_PT(DIM,BEAD)  110  =  +  VECTOR(DIM,BEAD-1)  CHECK_PT(DIM)  CONTINUE DO 140  PREV_BEAD = B E A D - 1 , PROD_FLAG = DO 130  DIM =  1,  -1  NDIM  FLAG(DIM)  =  0  PREV_PT(DIM, +  1,  1  PREV_BEAD)  =  PREV_PT(DIM,PREV_BEAD+1)  +  IF  VECTOR(DIM,PREV_BEAD)  (PREV_PT(DIM,PREV_BEAD).EQ.CHECK_PT(DIM))  +  THEN FLAG(DIM)  =  1  ENDIF PROD_FLAG = 130  PROD_FLAG *  FLAG(DIM)  CONTINUE IF  (PROD_FLAG.EQ.l) OVER_FLAG =  THEN 1  RETURN ENDIF 140 190  CONTINUE CONTINUE RETURN END  ********************************************************************* S U B R O U T I N E RANDOM_INT ( R _ I N T , LBOUND, UBOUND) *********************************************************************  * *  Choosing  random i n t e g e r  * *  Converts  a  *  Main  between  random number b e t w e e n  parameters:  and i n c l u d i n g 0  and  1 to  upper  the  and  required  lower  bounds.  integer.  R_I : LBOUND: UBOUND:  random i n t e g e r lower boundary upper boundary  N T  chosen for integer for integer  ************************************************************* Parameter list *************** INTEGER REAL  DIFF,  LBOUND,  UBOUND,  R_INT  RVEC(l)  C o n v e r t random number t o i n t e g e r w i t h i n range ********************************************** D I F F = UBOUND CALL  R A N L U X ( R V E C , 1)  R_INT +  +  LBOUND  =  INT(  RVEC(l)  *(REAL(DIFF)+  0.9999999))  LBOUND  RETURN END  ********************************************************************* S U B R O U T I N E RANDOM_RFAL ( R _ R E A L , L R E A L , UREAL) *********************************************************************  * *  Choosing  random r e a l  *  bounds.  Converts  * *  needed.  a  number  between  random number  and  including  between  *  Main  * *  R_RFAL: random r e a l number c h o s e n L R E A L : l o w e r b o u n d a r y f o r r e a l number  * *  UREAL:  0 and  the  1 to  upper the  and  real  lower  number  parameters:  upper  boundary  for  real  number  ********************************************************************* * Parameter list **************** REAL R_REAL,  DIFF,  D I F F = UREAL -  LREAL,  UREAL,  RVEC(l)  LREAL  * C o n v e r t random number t o be w i t h i n r e q u i r e d r a n g e *************************************************** CALL  RANLUX(RVEC,1)  R_REAL  =  (RVEC(l)  * DIFF)  +  LREAL  RETURN END  ********************************************************************* SUBROUTINE RANLUX(RVEC,LENV) ********************************************************************* *  Random number  *  zero  and  generator,  giving  C  Subtract-and-borrow  C  M a r s a g l i a and  C  RCARRY i n  C C C C C  an  array of  LENV  random numbers  between  one.  in Fortran  1993 77  to  1991,  references: M. L u s c h e r , Computer  implemented  and l a t e r  produce  coded  random number  Zaman,  by  F.  "Luxury James,  Physics  generator by  F.  James  improved by M a r t i n Pseudorandom  proposed with  the  Luescher  Numbers".  1993  Communications  79  (1994)  100  by name  C  F . James,  C C C  LUXURY  Computer Physics  C o m m u n i c a t i o n s 79  level  0  (p=24) : e q u i v a l e n t  C  a n d Zaman,  C  level  1  (p=48):  C level  2  (p=97):  C  very  luxury levels are:  t o t h e o r i g i n a l RCARRY  long period,  considerable  now p a s s e s  C  111  LEVELS. The a v a i l a b l e  C C  (1994)  improvement  the gap t e s t ,  passes  but f a i l s  fails  Marsaglia tests.  in quality  but s t i l l  a l l known t e s t s . ,  of  many  over  level  spectral  test.  but theoretically  s t i l l  0,  defective.  C  level  3  (p=223):  C  DEFAULT  correlations  C  level  4  (p=389):  VALUE.  have  highest  Any t h e o r e t i c a l l y  very  small  possible  chance  luxury,  possible  of being  observed.  a l l 24 b i t s  chaotic.  C C! ! ! CM! C!!! C!!! C M ! C!!! C!!! C! ! ! C!!! C!!! C! ! !  +++++++++++++++++++++++++++++++++++++++++++++++++++ C a l l i n g s e q u e n c e s f o r RANLUX: ++ C A L L RANLUX ( R V E C , L E N ) r e t u r n s a v e c t o r RVEC o f L E N • ++ 3 2 - b i t random f l o a t i n g p o i n t numbers between ++ z e r o ( n o t i n c l u d e d ) a n d o n e ( a l s o n o t i n c l . ) . ++ CALL RLUXGO(LUX,INT,KI,K2) i n i t i a l i z e s the g e n e r a t o r from ++ o n e 3 2 - b i t i n t e g e r I N T a n d s e t s L u x u r y L e v e l L U X ++ w h i c h i s i n t e g e r b e t w e e n z e r o a n d MAXLEV, o r i f ++ L U X . G T . 2 4 , i t s e t s p=LUX d i r e c t l y . K I a n d K2 ++ s h o u l d be s e t t o z e r o u n l e s s r e s t a r t i n g a t a break++ p o i n t g i v e n b y o u t p u t o f RLUXAT ( s e e R L U X A T ) . ++  C!!! C! ! !  CALL  RLUXAT ( L U X , I N T , K I , K 2 ) w h i c h c a n be u s e d  gets the values of four integers++ t o r e s t a r t t h e RANLUX g e n e r a t o r ++  C!!!  at the current point  CM!  specify  C!!!  i n i t i a l i z a t i o n with  CM!  skips  C!!! CM!  by c a l l i n g  how many n u m b e r s  over  were  RLUXGO.  generated  LUX a n d I N T .  K1+K2*E9  numbers,  K I a n d K2++ since  t h e ++  The r e s t a r t i n g so i t  c a n be  ++  long.++  A m o r e e f f i c i e n t b u t l e s s c o n v e n i e n t way o f r e s t a r t i n g i s b y : ++ CALL RLUXIN(ISVEC) r e s t a r t s the generator from v e c t o r ++  CM!  ISVEC  C! ! ! C M !  CALL  C! ! !  ISVEC must  o f 25 3 2 - b i t  integers  ( s e e RLUXUT)  ++  RLUXUT(ISVEC) o u t p u t s t h e c u r r e n t v a l u e s o f t h e 25 ++ 3 2 - b i t i n t e g e r s e e d s , t o b e u s e d f o r r e s t a r t i n g ++ be dimensioned  25 i n t h e c a l l i n g p r o g r a m  DIMENSION  RVEC(LENV)  DIMENSION  SEEDS(24),  ISEEDS(24),  PARAMETER  (MAXLEV=4,  LXDFLT=3) '  DIMENSION  NDSKIP(0:MAXLEV)  DIMENSION  NEXT(24)  ++  ISDEXT(25)  PARAMETER  (TWOP12=4096.,  IGIGA=1000000000,JSDFLT=314159265)  PARAMETER  (ITW024=2**24,  ICONS=2147483563)  SAVE  NOTYET,  SAVE  NSKIP,  INTEGER  1 2 4 , J 2 4 , CARRY, NDSKIP,  SEEDS,  NEXT,  TWOM24,  KOUNT,  TWOM12,  MKOUNT,  LUXLEV  INSEED  LUXLEV  INTEGER*2 LOGICAL  IN24,  TIME  NOTYET  DATA N O T Y E T ,  LUXLEV,  IN24,  KOUNT,  MKOUNT  /.TRUE.,  LXDFLT,  0,0,0/  DATA 1 2 4 , J 2 4 , C A R R Y / 2 4 , 1 0 , 0 . / C C  Luxury  Level  DATA  NDSKIP/0,  Corresponds C  time  t o p=24 factor  C C C  NOTYET  is  2  default *3*  4  24, 48  73, 97  199, 223  365 389  1  2  3  6  10  on  1.5  2  3  5  on  i f  no i n i t i a l i z a t i o n h a s been  Default IF  (NOTYET) NOTYET  =  /  1 .TRUE,  C  1  0  Initialization  slow w o r k s t a t i o n fast mainframe performed y e t .  by M u l t i p l i c a t i v e C o n g r u e n t i a l  THEN .FALSE.  JSEED =  INT(TIME())  INSEED = JSEED *  WRITE(6,'(A,112)') LUXLEV  ' RANLUX  DEFAULT  INITIALIZATION:  ',JSEED  = LXDFLT  NSKIP = NDSKIP(LUXLEV) LP  = NSKIP  IN24  + 24  = 0  204  KOUNT = 0 MKOUNT = 0 WRITE(6,'(A,12,A,14)')  * *  +  LUXLEV, ' TWOM24 = 1. DO 25 1= 1, 24 K =  p  '  RANLUX  DEFAULT  LUXURY L E V E L  =  ',  =' , L P  TWOM24 = TWOM24 * JSEED/53668  0.5  JSEED = 40014*(JSEED-K*53668) -K*12211 I F ( J S E E D . L T . 0) J S E E D = JSEED+ICONS I S E E D S ( I ) = MOD(JSEED,ITW024) CONTINUE  25  TWOM12 = TWOM24 DO 50 1= 1,24  *  4096.  SEEDS(I) = REAL(ISEEDS(I))*TWOM24 NEXT(I) = 1-1 CONTINUE  50  N E X T ( l ) = 24 124 = 24 J 2 4 = 10 CARRY = 0 . IF ENDIF C C C C  (SEEDS(24)  .EQ.  0.)  CARRY =  TWOM24  The G e n e r a t o r p r o p e r : "Subtract-with-borrow", as p r o p o s e d by M a r s a g l i a and Zaman, F l o r i d a State U n i v e r s i t y , March, 1989  C DO 100 I V E C = 1, LENV UNI = S E E D S ( J 2 4 ) - S E E D S ( 1 2 4 ) I F (UNI . L T . 0 . ) THEN UNI  =  UNI  -  CARRY  +1.0  CARRY = TWOM24 ELSE CARRY = ENDIF  0.  S E E D S ( 1 2 4 ) = UNI 124 = N E X T ( I 2 4 ) J24 = NEXT(J24) R V E C ( I V E C ) = UNI C  small IF  numbers (with l e s s (UNI . L T . TWOM12)  t h a n 12 THEN  "significant"  bits)  RVEC(IVEC) = RVEC(IVEC) + TWOM24*SEEDS(J24) and z e r o i s f o r b i d d e n i n c a s e someone t a k e s  C  IF ENDIF C  (RVEC(IVEC)  Skipping  to  .EQ.  0.)  luxury.  IN24 = IN24 + 1 I F ( I N 2 4 . E Q . 24) IN24 = 0  As  a  are  "padded".  logarithm  R V E C ( I V E C ) = TWOM24*TWOM24 proposed by  Martin  Luscher.  THEN  KOUNT = KOUNT + N S K I P DO 90 I S K = 1, NSKIP UNI = S E E D S ( J 2 4 ) I F (UNI . L T . 0 . ) UNI  =  UNI  +  -  SEEDS(124) THEN  -  CARRY  1.0  CARRY = TWOM24 ELSE CARRY = ENDIF  0.  S E E D S ( 1 2 4 ) = UNI 124 = N E X T ( I 2 4 ) 90 100  J24 = N E X T ( J 2 4 ) CONTINUE ENDIF CONTINUE KOUNT = IF  KOUNT +  (KOUNT  LENV  . G E . IGIGA)  THEN  MKOUNT = MKOUNT + 1 KOUNT = KOUNT - I G I G A ENDIF RETURN  205  Entry to input RLUXIN(ISDEXT)  ENTRY IF IF  b l o c k added by (NOTYET) THEN  and  float  Phillip  integer  Helbig  after  WRITE(6,'(A)') ' PROPER R E S U L T S I N T E G E R S O B T A I N E D WITH R L U X U T '  $25  NOTYET = ENDIF TWOM24 = DO 195  1,  1  IF  with  run James  INITIALISATION  FROM  1.  1=  24 *  0.5  OF RANLUX WITH  1  SEEDS(I) = CONTINUE CARRY = 0 .  correpondence  ONLY WITH  N E X T ( l ) = 24 TWOM12 = TWOM24 * 4 0 9 6 . WRITE(6, (A)') FULL INITIALIZATION WRITE(6,'(5X,5112)') ISDEXT DO 200 1= 1, 24 0  from p r e v i o u s  .FALSE.  NEXT(I) = I - l TWOM24 = TWOM24  5  seeds  25  INTEGERS:'  REAL(ISDEXT(I))*TWOM24  (ISDEXT(25)  . L T . 0)  CARRY = TWOM24  ISD 124  = IABS(ISDEXT(25)) = M O D ( I S D , 100)  ISD J24  = ISD/100 = MOD ( I S D ,  100)  ISD = I S D / 1 0 0 IN24 = M O D ( I S D , 1 0 0 ) ISD  =  ISD/100  L U X L E V •= I S D I F ( L U X L E V . L E . M A X L E V ) THEN NSKIP = NDSKIP(LUXLEV) WRITE ( 6 , ' ( A , 1 2 ) ' ) ' RANLUX LUXURY L E V E L + LUXLEV ELSE I F ( L U X L E V . G E . 24) THEN N S K I P = L U X L E V - 24 WRITE ELSE  (6,'(A,15)')  '  RANLUX  S E T BY RLUXIN  P - V A L U E S E T BY R L U X I N  NSKIP = NDSKIP(MAXLEV) WRITE ( 6 , ' ( A , 1 5 ) ' ) ' RANLUX  TO: ' ,  TO:',LUXLEV  I L L E G A L LUXURY R L U X I N :  ',LUXLEV  LUXLEV = MAXLEV ENDIF INSEED =  -1  RETURN Entry ENTRY  RLUXUT(ISDEXT)  DO 300  1=  1,  ouput  seeds  as  integers  24  ISDEXT(I) 0  to  =  INT(SEEDS(I)*TWOP12*TWOP12)  CONTINUE ISDEXT(25) IF  (CARRY  "  124  + 100*J24  . G T . 0.)  + 10000*IN24  ISDEXT(25)  =  +  1000000*LUXLEV  -ISDEXT(25)  RETURN Entry ENTRY  to  output  RLUXAT(LOUT,INOUT,Kl,K2)  the  "convenient"  restart  point  •  LOUT = L U X L E V INOUT =  INSEED  Kl  =  KOUNT  K2  = MKOUNT  RETURN Entry ENTRY IF  to  initialize  f r o m one  or  three  integers  RLUXGO(LUX,INS,K1,K2) (LUX . L T . 0) LUXLEV  ELSE  IF  (LUX . L E . MAXLEV)  LUXLEV ELSE  IF  THEN  = LUX  ( L U X . L T . 24  LUXLEV WRITE  THEN  = LXDFLT  . O R . LUX . G T . 2000)  THEN  = MAXLEV (6,'(A,17)')  '  RANLUX  I L L E G A L LUXURY R L U X G O : ' , L U X  ELSE LUXLEV = LUX DO 310  ILX=  0,  MAXLEV  I F (LUX . E Q . N D S K I P ( I L X ) + 2 4 ) CONTINUE ENDIF  310 IF  (LUXLEV NSKIP =  . L E . MAXLEV) THEN NDSKIP(LUXLEV)  WRITE(6,'(A,12,A,14)') LUXLEV,' P=\  +  LUXLEV = I L X  ' RANLUX LUXURY L E V E L NSKIP+24  S E T B Y RLUXGO : ' ,  ELSE N S K I P = L U X L E V - 24 WRITE ( 6 , " ( A , 1 5 ) ' ) ' ENDIF IN24 = 0 IF +  (INS ' Ill I F (INS JSEED  . L T . 0) WRITE ( 6 , ' ( A ) ' ) e g a l i n i t i a l i z a t i o n b y RLUXGO, . G T . 0) THEN = INS  WRITE(6,'(A,3112)') JSEED, K1,K2  +  RANLUX P - V A L U E  '  S E T BY RLUXGO T O : ' , L U X L E V  negative  RANLUX I N I T I A L I Z E D  input  seed'  BY RLUXGO FROM  SEEDS',  ELSE JSEED = JSDFLT WRITE(6,'(A)')' ENDIF INSEED NOTYET TWOM24  = = =  RANLUX I N I T I A L I Z E D  BY RLUXGO FROM D E F A U L T  SEED'  JSEED .FALSE. 1.  DO 325 1= 1, 24 TWOM24 = TWOM24  *  0.5  K = JSEED/53668 JSEED = 40014*(JSEED-K*53668) -K*12211 I F ( J S E E D . L T . 0) J S E E D = JSEED+ICONS ISEEDS(I) = MOD(JSEED,ITW024) CONTINUE  325  TWOM12 = TWOM24 * 4 0 9 6 . DO 350 1= 1,24 S E E D S ( I ) = R E A L d S E E D S ( I ) ) *TWOM24 350  NEXT(I) = CONTINUE N E X T ( l ) = 24 124 = 24 J 2 4 = 10 CARRY = 0 . IF  1-1  C  (SEEDS(24) . E Q . 0.) CARRY = 'TWOM24 If r e s t a r t i n g at a break point, skip  C C  Note that t h i s i s t h e u s e r PLUS t h e  KI +  IGIGA*K2  t h e number o f numbers d e l i v e r e d to number s k i p p e d ( i f l u x u r y . G T . 0 ) .  KOUNT = K I MKOUNT = K2 I F (K1+K2 . N E . 0) THEN DO 5 0 0 IOUTER= 1, K2+1 INNER = I G I G A I F ( I O U T E R . E Q . K2+1) DO 450 I S K = 1, INNER  INNER  = KI  UNI = S E E D S ( J 2 4 ) - S E E D S ( 1 2 4 ) I F (UNI . L T . 0 . ) THEN UNI = UNI +1.0 CARRY = TWOM24 ELSE CARRY = ENDIF SEEDS(124)  450 500 C  -  CARRY  0. = UNI  124 = N E X T ( 1 2 4 ) J24 = NEXT(J24) CONTINUE CONTINUE Get  the  right  value  of  IN24  by d i r e c t  calculation  IN24 = MOD(KOUNT, N S K I P + 2 4 ) I F (MKOUNT . G T . 0) THEN IZIP  = MOD(IGIGA,  NSKIP+24)  207  I Z I P 2 = MKOUNT*IZIP + IN24 IN24 = M 0 D ( I Z I P 2 , NSKIP+24) ENDIF C  + +  Now IN24  had b e t t e r  IF  . G T . 23)  (IN24 '  be b e t w e e n  zero  a n d 23  inclusive  THEN  WRITE ( 6 , ' ( A / A , 3 1 1 1 , A , 1 5 ) ' ) E r r o r i n RESTARTING w i t h R L U X G O : ' , • The v a l u e s ' , K l , K 2 , ' c a n n o t o c c u r a t l u x u r y l e v e l ' , LUXLEV  IN24 ENDIF ENDIF RETURN END  INS,  = 0  ********************************************************************* SUBROUTINE  READ_DATA  +  GRIDX,  +  NSTARTREC,  +  RADIUS_S  +  CHI_HS,  +  BCAPPA_SW,  +  GRIDY,  (NDIM,  CHANGE_LlM, LIMIT,  NWHIRL_START,  MAP_SWITCH, '  W I T C H  NTRANS,  START_PT,  WALL_SWITCH,  CHI_PS,  CHI_HW,  PSI_SW,  TEMP,  NSTOR,  NFRAME,  WHIRL_S ITCH, U  CHI_HH,  CHI_PP/  CHI_P '  CHI_S -  E_MAX,  L , COORD,  W  NWRITE,  NSIGN,  W  CHI_HP,  KAPPA_HS,  PSI_HS,  TYPE,  START_ORIGIN,COORD_ORIGlM)  *********************************************************** * * * * *  Read  start  (LINE  data  refers  to  from  file  blank  <*.dat>  i n <*.dat>  file.)  ********************************************************************* PARAMETER  (MAXD=2,  INTEGER NDIM,  MAXC=100)  CHANGE_LLM,  LIMIT,  NWRITE,  +  NWHIRL_START,  NTRANS,  +  WHIRL_SWITCH,  RADIUS_SWITCH,  +  START_PT(MAXD) ,  +  COORD(MAXD,MAXC) ,  NSIGN,  NSTOR,  NFRAME,  L , BEAD,  MAP_SWITCH,  WALL_SWITCH,  NSTARTREC,  START_ORIGIN(MAXD),  C O O R D _ O R I G I N ( M A X D , MAXC)  REAL GRIDX, GRIDY, + CHI_HH, CHI_PP, CHI_HP, CHI_HS, CHI_P f CHI_HW, CHI_PW, + CHI_SW, KAPPA_HS, P S I _ H S , KAPPA_SW, PSI_SW, TEMP, E_MAX S  CHARACTER*1  LINE  CHARACTER*1  TYPE(MAXC)  REWIND  (UNIT=10)  READ  (10,'(Al)')  LINE  READ  (10,'(Al)•)  LINE  READ  (10,'(Al)')  LINE  READ  (10,'(115)')  NDIM  READ  (10,'(115)')  CHANGE_LIM  READ  (10,'(115)')  LIMIT  READ  (10,'(115)')  NWRITE  READ  (10,'(115)')  NSTOR  READ  (10,'(F15.2)')  GRIDX  READ  (10,'(F15.2)')  GRIDY  READ  ( 1 0 , ' ( 1 1 5 ) ')  READ  (10,'(115)')  NWHIRL_START ' NTRANS  READ  (10, ' ( 1 1 5 ) ' )  NSIGN  READ  (10,'(115)')  NFRAME  READ  (10,'(115)')  READ  (10,'(Al)')  READ  (10,'(115)')  READ  (10,'(Al)')  READ  (10,'(115)')  START_PT(1)  READ  (10,'(115)')  START_PT(2)  READ  (10,'(115)')  READ  (10,'(Al)')  LINE  READ  (10,'(Al)')  LINE  READ  (10,'(Al)')  LINE  NSTARTREC LINE MAP_SWITCH LINE  WHIRL_SWITCH  208  READ  (10,'(115)')  READ READ  (10,'(Al)') (10,'(Al)')  READ  (10,'(115)')  READ  (10,'(Al) )  READ  (10, (Al) )  LINE  READ  (10,'(Al)')  LINE  READ READ  (10,'(Al)') LINE (10,40) CHI_HH, CHI_PP, CHI_HP, CHI_HS, CHI_PS, CHI_HW, CHI_PW, CHI_SW, KAPPA_HS, P S I _ H S , KAPPA_SW, PSI_SW,  + + 40  60 80  RADIUS_SWITCH LINE LINE  1  1  1  WALL_SWITCH LINE  T E M P , E_MAX FORMAT (13(F15.2,/),F15.2) READ ( 1 0 , ' ( 1 1 5 ) ' ) L READ(10,'(Al)') LINE READ(10,'(Al)') LINE DO 60 B E A D = 1 , L READ ( 1 0 , 8 0 ) C O O R D ( 1 , B E A D ) CONTINUE FORMAT ( 4 X , 2 1 4 , 3 X , A l )  , COORD(2,BEAD),  READ  (10,'(Al)')  LINE  READ  (10,'(Al)')  LINE  READ  (10,'(Al)')  LINE  READ  (10,'(115)')  START_ORIGIN(l)  READ  (10,'(115)')  START_ORIGIN(2)  DO 120  B E A D = 1, READ  120  CONTINUE  140  FORMAT  TYPE(BEAD)  L  (10,140)  COORD_ORIGIN(1,  BEAD),COORD_ORIGIN(2,  BEAD)  (4X,2I4)  RETURN END  **************************************  ****************************** SUBROUTINE +  REFRESH (VECTOR, S T A R T _ P T , X H H , X P P , X H P , X H S , X P S , XHW, XPW, XSW,  + +  NEW_VECTOR, NEW_START, NEW_XHH, NEW_.XPP, NEW_XHP,  +  NEW_XPW,  NEW_XSW,  NEW_XHS,  ENG,  NEW_XPS,  NEW_XHW,  NEW_E)  ********************************************************************* * *  Replaces  working  variables with newly-calculated variables  * Main parameters: * VECTOR:  the  vector  *  START_PT:  the  coordinates  *  XHH, e t c .  number  of  set  f o r the  *  HH c o n t a c t s )  active  ENG  energy  *  NEW_VECTOR:  the vector  *  NEW_START:  the  * *  of  the  first  contacts' between  * *  d e s c r i b i n g the  value  coordinates  the  conformation  for active  set  active bead of  chain the  conformation  active  two s p e c i f i e d  (e.g.  XHH i s  the  chain  components number  of  chain  d e s c r i b i n g t h e new c o n f o r m a t i o n of  the  first  bead of  t h e new  conformation NEW_XHH,  etc.:  number  of  *  f o r the  *  number  contacts  between  new c o n f o r m a t i o n of  the  (e.g.  two s p e c i f i e d XHH i s  components  the  HH c o n t a c t s )  * NEW_E: e n e r g y v a l u e f o r new c o n f o r m a t i o n ********************************************************************* * Parameter l i s t **************** PARAMETER  +  (MAXD=2,  MAXC=100)  I N T E G E R L , D I M , VNUM, N D I M , N V E C T O R , X H H , X P P , X H P , X H S , X P S , XHW, XPW,  + + +  XSW,  S T A R T _ P T (MAXD) , V E C T O R (MAXD, MAXC) , N E W _ X H H , N E W _ X P P , NEW_XHP, N E W _ X H S , NEW_XPS, NEW_XHW, NEW XPW, NEW_XSW, N E W _ S T A R T ( M A X D ) , N E W _ V E C T O R ( M A X D , M A X C ) REAL  E N G , NEW_E  209  COMMON /  GRAPH_LIM /  L,  NDIM,  NVECTOR  * Replace variables ******************* DO 40  VNUM = DO 20  1,  NVECTOR  DIM =  1,  NDIM  VECTOR(DIM,VNUM) 20  =  NEW_VECTOR(DIM,VNUM)  CONTINUE  40  CONTINUE DO  60  DIM =  1,  START  PT(DIM)  NDIM = NEW S T A R T ( D I M )  CONTINUE XHH = NEW_ XHH XPP = NEW]]XPP XHP = NEW]]XHP XHS = NEW""XHS XPS = NEW]"XPS XHW = NEW]]XHW XPW = NEW]"XPW XSW = NEW""xsw ENG = NEW""E RETURN END  ********************************************************************* SUBROUTINE R _ G _ C A L C (VECTOR, S T A R T _ P T , R _ G _ A V G , R_G_DEFORM) ********************************************************************* * *  Calculates  * *  chain  the  radius  * *  Main parameters: VECTOR:  of  gyration  parameters  for  the  specified  conformation.  the  vector  *  START_PT:  the  coordinates  set  describing  * * *  R_G_AVG: R_G_DEFORM:  averaged radius of deformation ratio,  of  the  the  first  active bead  of  chain the  conformation  active  chain  g y r a t i o n of the conformation R_G(x-dim)/R_G(y-dim)  ********************************************************************* * Parameter list **************** PARAMETER  (MAXD=2,  INTEGER L , +  MAXC=100) .  D I M , VNUM,  S T A R T _ P T (MAXD) , REAL R_G_AVG, COMMON /  NDIM,  R_G_DEFORM,  GRAPH_LIM /  NVECTOR,  BEAD,  COORD (MAXD, MAXC) ,  L,  V E C T O R (MAXD, MAXC)  CENTRE_REAL(MAXD),  NDIM,  R_G(MAXD),  NVECTOR  * Initialize parameters *********************** R_G_AVG DO 10  =0.0  DIM =  1,  R_G(DIM) 10  NDIM =0.0  CONTINUE  * C a l c u l a t e R _ G _ A V G a n d R_G_DEFORM ********************************** CALL CALL DO 40  COORD_MAKER ( V E C T O R , S T A R T _ P T , COORD) CENTRE O F _ M A S S _ R E A L (VECTOR, S T A R T _ P T , C E N T R E _ R E A L ) BEAD =  DO 20  1,  DIM =  L 1,  NDIM  SUM(MAXD)  20 40  R_G(DIM) CONTINUE CONTINUE  =  R_G(DIM)+((CENTRE_REAL(DIM)-COORD(DIM,BEAD))**2)  60  DO 60 DIM = 1, NDIM R_G(DIM) = R_G(DIM) / L R_G_AVG = R _ G _ A V G + R_G(DIM) CONTINUE R_G_AVG = R_G_AVG**0.5 R_G_DEFORM = R_G(1) / R_G(2) RETURN END  ************************************************************ SUBROUTINE VECTOR_MAP +  (COORD,  MAP_S™ITCH, VECTOR,  START_PT,  MAP_FLAG)  ********************************************************************* * Creates * bead i n *  v e c t o r map f r o m o r i g i n a l c o o r d i n a t e s , the centre of the g r i d i f requested.  and s i t u a t e s  the  centre  * Main parameters: * COORD: c o o r d i n a t e s o f t h e c h a i n ( r e a d f r o m <*.DAT>) * MAP_S ITCH: f l a g i n d i c a t i n g w h e t h e r c o n f o r m a t i o n s h o u l d be p l a c e d * i n middle of g r i d , or i f l o c a t i o n is indicated * (0 = s w i t c h o f f , exact location is given) * (1 = s w i t c h o n , a u t o m a t i c a l l y p l a c e c h a i n i n g r i d centre) * VECTOR: the v e c t o r set d e s c r i b i n g the a c t i v e c h a i n c o n f o r m a t i o n W  *  START_PT:  the  * * *  MAP_FLAG:  indicates chain fit in grid (0 = c h a i n s u c e s s f u l l y e n c a s e d i n g r i d ) (1 = c h a i n e x t e n d s o u t s i d e o f g r i d , n e e d  coordinates  * *  of  the  first  bead  of  the  active  to  chain  restart  simulation)  ********************************************************************* * Parameter list **************** PARAMETER  (MAXD=2,  MAXC=100)  INTEGER L , NVECTOR, NDIM, M I D _ L , MAP_S™ITCH,  + + +  VNUM,  D I M , MAP_FLA(3,  S T A R T _ P T ( M A X D ) , M I D _ G R I D ( M A X D ) , GRIDLIM(MAXD) , COORD (MAXD, MAXC) , V E C T O R (MAXD, MAXC) , COORD_MAP (MAXD, MAXC) COMMON /  GRAPH_LIM /  COMMON /  LATTICE  /  L , N D I M , NVECTOR  GRIDLIM  * C r e a t e v e c t o r map ******************* DO 90  + 50 90  VNUM = 1, N V E C T O R DO 50 DIM = 1, NDIM V E C T O R ( D I M , V N U M ) = COORD(DIM,VNUM+1) COORD(DIM,VNUM)  CONTINUE CONTINUE  * L o c a t e c e n t r e of c h a i n and g r i d ********************************* INT((L+l)12)  MID_L = DO 100  DIM = 1,  NDIM  MID_GRID(DIM) 100 *  =  INT((GRIDLIM(DIM)+1)12)  CONTINUE  Place  chain  on g r i d  and check  for  inconsistancies  -  IF  (MAP_SWITCH.EQ.1) DO 115 DIM = 1,  THEN NDIM  START_PT( ) = MID_GRID(DIM) CONTINUE DO 140 VNUM = M I D _ L ~ l f 1, -1 DO 120 DIM = 1, NDIM S T A R T _ P T (DIM) = S T A R T _ P T (DIM) - V E C T O R ( D I M , VNUM) D I M  115  IF +  ((START_PT(DIM).LT.0).OR. (START_PT(DIM).GT.GRIDLIM(DIM))) MAP_FIAG = 1  .  GOTO ENDIF 120 140  THEN  200  CONTINUE CONTINUE ENDIF DO  150  150  D I M = 1, NDIM C O O R D _ M A P ( D I M , 1)  = S T A R T _ P T (DIM)  CONTINUE DO 170  + +  BEAD = 2, L DO 160 DIM = 1, NDIM COORD_MAP(DIM,BEAD) = COORD_MAP(DIM,BEAD-1) VECTOR(DIM,BEAD-1) IF ((COORD_MAP(DIM,BEAD).LT.0).OR. (COORD_MAP(DIM,BEAD).GT.GRIDLIM(DIM))) MAP_FLAG = GOTO 2 0 0  +  THEN  1  ENDIF 160 170  CONTINUE CONTINUE  200  CONTINUE RETURN END  **************************************************************** SUBROUTINE WALL_B°UNCE (VECTOR, S T A R T _ P T , BOUNCE_FLAG) ********************************************************************* *  Checks  for  *  points  back  *  boundary  * * * *  coordinates into  the  outside  grid.  of  boundaries  The t r e s p a s s e d  and r e f l e c t s  boundary i s  outlying  the  surface.  Main parameters: VECTOR: the START_PT: the  v e c t o r set d e s c r i b i n g the a c t i v e coordinates of the f i r s t bead of  BOUNCE_FLAG: i n d i c a t e s  * * * * *  the  s t a t e .of  coordinates  of  the  chain conformation the a c t i v e chain  newly-reflected  the  new  conformation  (0  = all  conformation  (1  the l a t t i c e space) = chain extends o u t s i d e of g r i d , even a f t e r reflections, the proposed c o n f o r m a t i o n i s unacceptable)  are  within  ********************************************************************* * Parameter l i s t **************** PARAMETER INTEGER  (MAXD = 2 ,  L , NDIM,  MAXC =  100)  DIM, NVECTOR,  +  GRIDLIM(MAXD) ,  +  COORD(MAXD,MAXC),  +  V E C T O R (MAXD, MAXC) COMMON /  GRAPH_LIM /  COMMON /  LATTICE  /  BEAD,  VNUM, B O U N C E _ F L A G ,  START_PT(MAXD) , NEW_COORD(MAXD, MAXC) ,  L , NDIM,  NVECTOR  GRIDLIM  * Initialize parameters ***********************  212  BOUNCE  FLAG  =  0  * Check s t a r t i n g p o i n t and r e f l e c t i n t o g r i d i f necessary **************************************************** DO 20  DIM =  1,  NDIM  COORD(DIM,1) = START_PT(DIM) IF (COORD(DIM,1).GT.GRIDLIM(DIM))  THEN  NEW_COORD(DIM,1) = (-1 * C O O R D ( D I M , 1 ) ) (2 * G R I D L I M ( D I M ) )  + ELSEIF  ( C O O R D ( D I M , 1 ) . L T . O ) THEN NEW_COORD(DIM, 1) = (-1 * C O O R D ( D I M ,  +  1))  ELSE NEW_COORD(DIM, 1) = C O O R D ( D I M , 1 ) ENDIF IF ((NEW_COORD(DIM,1).GT.GRIDLIM(DIM)).OR. ( N E W _ C O O R D ( D I M , 1 ) . L T . O ) ) THEN  +  BOUNCE_FLAG = GOTO 100 ENDIF START_PT(DIM) CONTINUE  20  1  = NEW_COORD(DIM,1)  * Check r e m a i n i n g c o o r d i n a t e s and r e f l e c t i n t o g r i d i f necessary **************************************************************** DO  60  BEAD = 2, DO 40  L  DIM <= 1, NDIM COORD(DIM,BEAD) = COORD(DIM,BEAD-1) + VECTOR(DIM,BEAD-1)  +  IF  (COORD(DIM,BEAD).GT.GRIDLIM(DIM)) NEW_COORD(DIM,BEAD) + (2  + ELSEIF  = *  THEN  (-1 * C O O R D ( D I M , B E A D ) ) GRIDLIM(DIM))  (COORD(DIM,BEAD).LT.O)  THEN  NEW_COORD(DIM,BEAD)  =  (-1  * COORD(DIM, BEAD))  NEW_COORD ( D I M , BEAD)  = COORD ( D I M , BEAD)  ELSE ENDIF I F ((NEW_COORD(DIM,BEAD) .GT.GRIDLIM(DIM)) (NEW_COORD(DIM,BEAD).LT.O)) THEN  +  BOUNCE_FLAG = GOTO 100  .OR.  1  ENDIF 40 60  CONTINUE CONTINUE  * R e - e s t a b l i s h v e c t o r map f r o m new c o o r d i n a t e s ********************************************** DO 90  VNUM = 1, DO 80  NVECTOR  DIM =  1,  NDIM  VECTOR(DIM,VNUM) +  = NEW_COORD(DIM,VNUM+1)  -  NEW_COORD(DIM,VNUM)  80  CONTINUE  90  CONTINUE  100  CONTINUE RETURN END  ********************************************************************* SUBROUTINE +  WHIRLING  (VECTOR,  START_PT,  NEW_VECTOR,  NEW_START,  WHIRL_FLAG)  ********************************************************************* * *  Randomly r o t a t e s  * *  The l e n g t h  *  Possible  of  or  translates  translation  maneouvres:  is  the  given  polymer at  a given  initially in  frequency.  <*.dat>  *  CODE  MANOEUVER rotate rotate  90° (counter-clockwise) 180°  rotate  270°  translate  Main  north  translate  south  translate  east  translate  west  parameters:  VECTOR:  the  START_PT: NEW_VECTOR: NEW_START: WHIRL_F G:  the c o o r d i n a t e s o f the f i r s t bead o f the a c t i v e c h a i n n e w l y - c a l c u l a t e d v e c t o r map newly-calculated starting point i n d i c a t e s the s t a t e of the n e w l y - r e f l e c t e d c o n f o r m a t i o n (0 = a l l c o o r d i n a t e s o f t h e new c o n f o r m a t i o n a r e w i t h i n boundaries) (1 = c h a i n e x t e n d s o u t s i d e o f g r i d a f t e r reflections, the proposed conformation i s unacceptable)  L A  vector  set  describing  the  active  chain  conformation  grid  *************************** ***************************************** * Parameter l i s t **************** PARAMETER  +  (MAXD=2,  MAXC=100)  I N T E G E R L , N D I M , N V E C T O R , D I M , W H I R L , B E A D , B O U N C E _ F L A G , VNUM, W H I R L _ F L A G , GRIDLIM(MAXD) , NWHIRL, NTRANS,  +  S T A R T _ P T (MAXD) ,  + +  RdT_DIFF(MAXD) , CHECK_PT(MAXD) , V E C T O R (MAXD, MAXC) , NEW_VECTOR (MAXD, MAXC) COMMON / COMMON / COMMON /  NEW_START (MAXD) ,  D I F F (MAXD) ,  C E N T R E (MAXD) ,  G R A P H _ L L M / L , NDIM, NVECTOR WHIRL / NWHIRL, NTRANS L A T T I C E / GRIDLIM  * I n i t i a l i z e parameters *********************** CALL DO 20  CENTRE_OF_MASS DIM =  1,  (VECTOR,  START_PT,  CENTRE)  NDIM  N E W _ S T A R T (DIM) = S T A R T _ P T (DIM) DIFF(DIM)  = START_PT(DIM)  R O T _ D i E T (DIM) = 20  -  CONTINUE DO 30  VNUM = DO 25  1,  NVECTOR  DIM =  1,  NDIM  N E W _ V E C T O R ( D I M , VNUM) 25 30  CENTRE(DIM)  0  = V E C T O R ( D I M , VNUM)  CONTINUE CONTINUE  * R a n d o m l y c h o s e move ********************* CALL  RANDOM_INT  (WHIRL,  1,  7)  * C a l c u l a t e new p o s i t i o n o f c o n f o r m a t i o n **************************************** IF  (WHIRL.EQ.l)  THEN  ROT_DIFF(l)  = -1  ROT_DIFF(2)  =  DIFF(l)  DO 40  1,  NDIM  DIM =  *  NEW_START(DIM) 40  DIFF(2)  = START_PT(DIM)  +  ROT_DIFF(DIM)  CONTINUE DO 60  VNUM =  1,  NVECTOR  NEW_VECTOR(l,VNUM)  = -1  NEW_VECTOR(2,VNUM)  =  *  VECTOR(2,VNUM)  VECTOR(1,VNUM)  214  60  CONTINUE (WHIRL.EQ.2) THEN DO 120 DIM = 1, NDIM NEW_START(DIM) = CENTRE(DIM) - DIFF(DIM) CONTINUE DO 160 VNUM = 1, NVECTOR DO 140 DIM = 1, NDIM NEW_VECTOR(DIM,VNUM) = -1 * VECTOR(DIM,VNUM) ' CONTINUE CONTINUE ELSEIF (WHIRL.EQ.3) THEN ROT_DIFF(l) = DIFF(2) ROT_DIFF(2) = -1 * D I F F ( l ) DO 220 DIM = 1, NDIM NEW_START(DIM) = START_PT(DIM) + ROT_DIFF(DIM) CONTINUE DO 230 VNUM = 1, NVECTOR NEW_VECTOR(1,VNUM) = VECTOR(2,VNUM) NEW_VECTOR(2,VNUM) = -1 * VECTOR(1,VNUM) CONTINUE ELSEIF (WHIRL.EQ.4) THEN NEW_START(2) = START_PT(2) + NTRANS ELSEIF (WHIRL.EQ.5) THEN NEW_START(2) = START_PT(2) - NTRANS ELSEIF (WHIRL.EQ.6) THEN NEW_START(1) = START_PT(1) + NTRANS ELSE NEW_START(1) = START_PT(1) - NTRANS ENDIF  ELSEIF  120  140 160  220  230  * Check whether n e w l y - c a l c u l a t e d c o n f o r m a t i o n i s w i t h i n g r i d b o u n d a r i e s ************************************************** DO 235  235  DIM = 1, NDIM CHECK_PT(DIM) = NEW_START(DIM) IF ((CHECK_PT(DIM).GT.GRIDLIM(DIM)).OR. + (CHECK_PT(DIM).LT.O)) THEN CALL WALL_BOUNCE (NEW_VECTOR, NEW_START, + BOUNCE_FLAG) I F (BOUNCE_FLAG.EQ..l) THEN WHIRL_FLAG = 1 GOTO 300 ENDIF ENDIF CONTINUE DO 260  240 2 60 300  BEAD = 2, L DO 240 DIM = 1, NDIM CHECK_PT(DIM) = CHECK_PT(DIM) + + NEW_VECTOR(DIM,BEAD-1) IF ((CHECK_PT(DIM).GT.GRIDLIM(DIM)).OR. + (CHECK_PT(DIM).LT.O)) THEN CALL WALL_BOUNCE (NEW_VECTOR, NEW_START, + BOUNCE_FLAG) IF (BOUNCE_FLAG.EQ.1) THEN WHIRL_FLAG = 1 GOTO 300 ENDIF ENDIF CONTINUE CONTINUE CONTINUE RETURN END  ********************************************************************* SUBROUTINE WRITE_DATA (NDIM, CHANGE_LIM, LIMIT, NWRITE, NSTOR, + GRIDX, GRIDY, NWHIRL_START, NTRANS, NSIGN, NFRAME, + NSTARTREC, MAP SWITCH, START PT,  215  +  WHIRL_SWITCH,  WALL_SWITCH,  CHI_HH,  CHI_PP,  CHI_HP,  + C H I _ H S , C H I _ P ' CHI_HW, CHI_PW, C H I _ S KAPPA_HS, + K A P P A _ S W , P S I _ S W , T E M P , E _ M A X , L , COORD, T Y P E , + START_ORIGIN,COORD_ORIGIN) ************************************************************ S  W  PSI_HS,  * * *  Write  starting  data  from f i l e  <*.dat>  to  output  file  ********************************************************************* PARAMETER  (MAXD=2,  MAXC=100)  I N T E G E R N D I M , C H A N G E _ L I M / L I M I T , N W R I T E , NSTOR, L , B E A D , + NWHIRL_START, NTRANS, N S I G N , NFRAME, M A P _ S I T C H , + WHIRL_SWITCH, WALL_SWITCH, NSTARTREC, + START_PT(MAXD) , START_ORIGIN(MAXD) , + COORD(MAXD,MAXC), COORD_ORIGIN(MAXD,MAXC) W  + +  REAL GRIDX, GRIDY, CHI_HH, CHI_PP/ C H I _ H P , C H I _ H S , C H I _ P S , C H I _ H W , CHI_PVJ, C H I _ S W , K A P P A _ H S , P S I _ H S , K A P P A _ S W , P S I _ S W , T E M P , E_MAX CHARACTER*1  TYPE(MAXC)  WRITE WRITE  (20,'(A15,115) ) (20,'(A15,115)')  WRITE WRITE WRITE WRITE WRITE WRITE  ( 2 0 , ' ( A 1 5 , 1 1 5 ) ') ' L I M I T ' , LIMIT (20,'(A15,115)') 'NWRITE', NWRITE ( 2 0 , ' ( A 1 5 , 1 1 5 ) ) ' N S T O R ' , . NSTOR (20,'(A15,F15.2)') 'GRIDX , GRIDX ( 2 0 , ' ( A 1 5 , F 1 5 . 2 ) ' ) 'GRIDY', GRIDY ( 2 0 , ' ( A 1 5 , 1 1 5 ) ') ' N W H I R L _ S T A R T ' , NWHIRL_START  WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE  (20,'(A15,115)') ( 2 0 , ' ( A 1 5 , 1 1 5 ) ') (20,'(A15,115)') (20, (A15,115) ' ) (20,'(A15,115)') (20,'(A15,115)') ( 2 0 , * ( A 1 5 , 1 1 5 ) ') (20,'(A15,I15)')  1  'NDIM , NDIM 'CHANGE_LLM', 1  CHANGE_LIM  1  1  1  WRITE  'NTRANS', NTRANS ' N S I G N ' , NSIGN ' N F R A M E ' , NFRAME ' N S T A R T R E C , NSTARTREC ' M A P _ S I T C H ' , MAP_SWITCH 'START_PT(1)', START_PT(1) 'START_PT(2) ' , START_PT(2) 'WHIRL_SWITCH', WHIRL_SWITCH W  (20,'(A15,115) ) ' W A L L _ S W I T C H ' , WALL_SWITCH ( 2 0 , ' ( A 1 5 , F 1 5 2) ') ' C H I H H ' , CHI HH •CHI P P ' , C H I _ P ( 2 0 , ' ( A 1 5 , F 1 5 2) ' ' C H I _ H P ' , CHI_HP ( 2 0 , ' ( A 1 5 , F 1 5 2) ' ' C H I _ H S ' , CHI_HS W R I T E ( 2 0 , ' ( A 1 5 , F 1 5 2) ' 1  WRITE WRITE WRITE  WRITE WRITE WRITE  ( 2 0 , ' ( A 1 5 , F 1 5 2) ' ( 2 0 , ' ( A 1 5 , F 1 5 2) ' ( 2 0 , ' ( A 1 5 , F 1 5 2) '  •CHI_PS', 'CHI HW', 'CHI PW',  CHI_PS CHI HW CHI PW  WRITE  (20, ' (A15,F15 (20, ' (A15,F15 (20, ' (A15,F15 (20, ' ( A 1 5 , F 1 5  2) '  'CHI_SW',  CHI_SW  2) ' 2) ' '  KAPPA_HS ' KAPPA_HS ' , ' P S I _ H S ' , P S I HS ' K A P P A _ S W ' , KAPPA_SW  ' ' '  • P S I . S W ' , PSI SW ' T E M P ' , TEMP ' E _ M A X ' , E_MAX  WRITE WRITE WRITE WRITE WRITE WRITE  60 80  p  2) ( 2 0 , ' ( A 1 5 , F 1 5 2) ( 2 0 , ' ( A 1 5 , F 1 5 2) ( 2 0 , ' ( A 1 5 , F 1 5 2) (20, ' ( A 1 5 , I 1 5 ) ' )  L' , L WRITE DO 60 BEAD = 1, L WRITE ( 2 0 , 8 0 ) C O O R D ( 1 , B E A D ) , C O O R D ( 2 , B E A D ) CONTINUE FORMAT ( 4 X , 2 1 4 , 3 X , A l )  ,  TYPE(BEAD)  WRITE ( 2 0 , ' ( A 3 0 ) ' ) ' C O M P A R I S O N CONFORMATION' W R I T E ( 2 0 , ( A 1 5 , I 1 5 ) ' ) ' S T A R T _ O R I G I N (1) ' , S T A R T _ O R I G I N (1) WRITE ( 2 0 , ' ( A 1 5 , I 1 5 ) ' ) ' S T A R T _ O R I G I N ( 2 ) ' , START_ORIGIN(2) DO 1 2 0 B E A D = 1, L 1  120 140  WRITE ( 2 0 , 1 4 0 ) CONTINUE FORMAT ( 4 X , 2 I 4 )  COORD_ORIGIN(1,BEAD),COORD_ORIGIN(2,BEAD)  WRITE WRITE WRITE  'NDIM', NDIM ' C H A N G E _ L I M ' , CHANGE_LrM 'LIMIT', LIMIT  (42,'(A15,I15)') (42,'(A15,I15)') (42,'(A15,I15)')  216  WRITE  (42,  '(A15, 115)')  'NWRITE' ,  WRITE  (42,  ' ( A 1 5 , 115)  'NSTOR',  WRITE  (42,  '(A15, F15.2)  )  'GRIDX',  GRIDX  WRITE  (42,  '(A15, F15.2)  )  'GRIDY',  GRIDY  WRITE  (42,  ' ( A 1 5 , 115)  ')  'NWHIRL_ START' ,  WRITE  (42,  ' ( A 1 5 , 115)  •)  'NTRANS'. ,  WRITE  (42,  ' ( A 1 5 , 115)  ')  'NSIGN',  WRITE  (42,  1  ( A 1 5 , 115)  ')  'NFRAME' ,  WRITE  (42,  ' ( A 1 5 , 115)  •)  'NSTARTREC',  WRITE  ')  (42,'(A15,115)')  WRITE  (42,'(A15,I15)')  WRITE WRITE WRITE  (42,'(A15,115)')  NWRITE NSTOR  NWHIRL  NTRANS NSIGN NFRAME NSTARTREC  'MAP_SWITCH',  MAP_SWITCH  START_PT(1)',  START_PT(1)  (42,'(A15,I15)')  'START_PT(2)',  START_PT(2)  (42,'(A15,115)')  'WHIRL_S ITCH',  1  W  WHIRL_SWITCH  'WALL_SWITCH', WALL_S ITCH ( 4 2 , ' ( A 1 5 , F 1 5 . 2) ') ' C H I _ H H / C H I _ H H CHI PP ( 4 2 , ' ( A 1 5 , F 1 5 2) ') ' C H I _ P / 1 ( 4 2 , ' ( A 1 5 , F 1 5 2) ') ' C H I _ H P / C H I _ H P ' C H I _ H S ' / CHI HS ( 4 2 , ' ( A 1 5 , F 1 5 2) ' ' C H I PS 1 / C H I _ P S ( 4 2 , • ( A 1 5 , F 1 5 2) ' CHI_HW •CHI_HW 1 ( 4 2 , ' ( A 1 5 , F 1 5 2) '  WRITE WRITE  W  p  WRITE WRITE WRITE  WRITE W R I T E ( 4 2 , ' ( A 1 5 , F 1 5 . 2 ) ') • C H I _ P ' ' CHI_PW ' C H I SW 1 / CHI SW W R I T E ( 4 2 , ' ( A 1 5 , F 1 5 2) ' • K A P P A HS ' , KAPPA_ W R I T E ( 4 2 , ' ( A 1 5 , F 1 5 2) ' •PSI_HS 1 PSI_HS W R I T E ( 4 2 , ' ( A 1 5 , F 1 5 2) • ' K A P P A _ SW ' , KAPPA_ W R I T E ( 4 2 , ' ( A 1 5 , F 1 5 2) ' W  ( 4 2 , ' ( A 1 5 , F 1 5 2) '  WRITE  1 ' PSI_S PSI_S ' T E M P ' , TEMP ( 4 2 , ' ( A 1 5 , F 1 5 2) ' W R I T E ( 4 2 , ' ( A 1 5 , F 1 5 . 2 ) ') • E M A X ' , E _ MAX L \ L WRITE (42, • ( A 1 5 , I 1 5 ) ' ) DO 160 B E A D = 1, L W R I T E ( 4 2 , 180) C O O R D ( 1 , B E A D ) , C O O R D ( 2 , B E A D ) , W  W  WRITE  160  CONTINUE  180  FORMAT  TYPE(BEAD)  (4X,214,3X,Al)  WRITE  (42,'(A30)')  'COMPARISON  CONFORMATION'  WRITE  (42,'(A15,115)')  'START_ORIGIN(1)',  START_ORIGIN(1)  WRITE  ( 4 2 , ' ( A 1 5 , I 1 5 ) ')  'START_ORIGIN(2) ' ,  START_ORIGIN(2)  DO 2 2 0  BEAD = 1, WRITE  220  CONTINUE  240  FORMAT  L  (42,240)  COORD_ORIGIN(1,BEAD),COORD_ORIGIN(2,BEAD)  (4X,2I4)  RETURN END  ******************************************************************** *********************************************************** *  FUNCTIONS:  *******************************************************************! I N T E G E R F U N C T I O N F_DOT ( V E C T O R _ A , V E C T O R _ B ) *******************************************************************>  * * *  Dot product  function  *************************************.******************************: * Parameter list **************** PARAMETER INTEGER +  (MAXD = 2)  NDIM,  DIM,'NVECTOR,  VECTOR_A(MAXD), COMMON /  GRAPH_LLM  * Initialize parameter ********************** F *  DOT = 0  Calculate  dot  product  /  VECTOR_B(MAXD) L , NDIM,  NVECTOR  DO 20  DIM =  F_DOT  1,  =  NDIM  F_DOT  + V E C T O R _ A (DIM) * V E C T O R _ B (DIM)  CONTINUE  RETURN END  ************************************************************* R E A L F U N C T I O N F _ E N E R G Y ( X H H , X P P , X H P , X H S , X P S , XHW, XPW, XSW) ***************************************************************** Calculates  chain  energy  ***************************************************************** Parameter list ************** INTEGER XHH,  XPP,  REAL  CHI_PP, CHI_HP, CHI_HS, CHI_PS, CHI_HW, C H I _ P W ,  CHI_HH,  + +  XHP,  XHS,  KAPPA_HS, COMMON /  CHI  COMMON /  TEMP  /  CHI_HH, CHI_PS,  + +  XHW,  PSI_HS,  CHI_PP, CHI_HW,  KAPPA_HS, /  XPS,  XPW,  XSW  CHI_SW,  KAPPA_SW,  CHI_HP, CHI_PW,  PSI_HS,  PSI_SW  CHI_HS, CHI_SW,  KAPPA_SW,  PSI_SW  TEMP  Option:  I f HS a n d SW i n t e r a c t i o n e n e r g i e s a r e f u n c t i o n s of t e m p e r a t u r e , t h e n t h e n e x t two l i n e s s h o u l d be activated ***************************************************************** CHI_HS CHI_SW  = =  KAPPA_HS KAPPA_SW  / /  TEMP + TEMP +  PSI_HS PSI~SW  C a l c u l a t e energy ***************** F_ENERGY  =  (CHI_HH*XHH)  +  (CHI_PP*XPP)  +  (CHI_HS*XHS)  +  (CHI_PS*XPS)  +  (CHI_PW*XPW)  +  (CHI_SW*X3W)  +  +  (CHI_HP*XHP)  (CHI_HW*XHW)  +  +  RETURN END  ****************************************************************** REAL FUNCTION F_PROB(D_E) ****************************************************************** Calculates  probability  according  to  Boltzmann  relation  ****************************************************************** Parameter list *************** REAL  D_E,  INTRINSIC  KJbolz,  COMMON / T E M P / Boltzmann K_bolz  =  TEMP  EXP TEMP  constant,K_bolz, 1.381E-23  Calculate probability ********************** FPROB RETURN END  =  EXP(-1.0*D_E)  in  units  of  J/K  Input data, <test.dat>  The input file, <test.dat>, is shown below. The parameters were chosen to give a short demonstration of the possible outputs from CONTACT9.  Explanations of input  parameters are first given.  [NDIM] The dimension number. For example, C O N T A C T 9 is a 2D simulation, and therefore NDIM = 2.  [CHANGE_LIM] The maximum number of times the central subroutine, PROTEIN_ADSORPTION_2 is to be run. The computer has a counting limit of 2 , and therefore it is necessary to run the main subroutine within an added outer loop to exceed this constraint for very long simulations. C H A N G E L I M defines the maximum number of times the subroutine is to be carried out in series.  [LIMIT] The maximum number of cycles to' be run in PROTEIN_ADSORPTION_2.  It cannot  exceed 2 . 32  [NWRITE]  219  Frequency of sampling to output file <*.ou(>. Sampling occurs at multiples of the number listed.  For example, when NWRITE = 10000, sampling occurs at 0, 10000,  20000, etc. Information provided by this function includes the energy and configurational profiles of the chain.  [NSTOR]  Maximum size of storage array of unique low energy conformations.  [GRIDX] The size of the grid in the x-dimension as multiple of chain length. For example, in a simulation of an 18-residue chain when GRIDX = 0.50, the grid is 9 units wide. Since the chain occupies the coordinate sites (and not the spaces in between them), a lattice 9 units wide accommodates 10 chain residues. [GRIDY] The size of the grid in the ^-dimension as multiple of chain length (not necessarily equal to GRIDX).  [NWHIRLSTART]  Frequency of a rotation or translation move. The type of move is chosen randomly.  [NTRANS] Length of each translation move.  220  [NSIGN] Frequency of screen output, This function serves to inform the user on the progress of the simulation.  [NFRAME] Frequency of output to <*.gph> file. Information provided by this function is restricted to single line reports of chain energy and its resemblance to a comparison conformation. This function is designed to accommodate frequent sampling of long simulations.  [NSTARTREC] The number of cycles to pass before sampling takes place.  This function is used when  an equilibrium period is needed.  [MAP_SWITCH] The switch controlling the automatic chain placement function at the start of the simulation. 0 = off, the chain is placed according to the coordinates given. 1 = auto, the chain is automatically placed in the middle of the lattice at the start of the simulation.  [STARTJPT] (startingx-coordinate) [STARTJPT] (starting^-coordinate)  221  Coordinates of the first chain residue at the start of the simulation.  Only valid if  MAP_S WITCH = 1.  [WHIRL_SWITCH] The switch controlling automatic suspension of the WHIRLING mechanism. 0 = off,  the WHIRLING mechanism  continues at the frequency  indicated by  NWHIRL_START 1 = auto, the WHIRLING mechanism shuts off when contacts between the chain and an active wall are detected.  [RADIUS_SWITCH] The switch controlling automatic calculation and presentation of radius of gyration data. Data using this function are generated at the frequency indicated by N F R A M E . 0 = off, the report does not include R data g  1 = on, the calculations includes R data g  [WALL_SWITCH] The switch controlling which boundaries become active surfaces. 4 = all sides are active walls 1 = vertical walls on (i.e. surfaces at x = 0 and GRID_LIM(y)) 2 = horizontal walls on (i.e. surfaces aty = 0 and GRID_LIM(JC))  [CHI_HH]  222  Interaction energy between H H chain residues.  [CHI_PP]  Interaction energy between PP chain.residues.  [CHI_HP]  Interaction energy between HP chain residues.  [CHI_HS]  Interaction energy between H residues and solvent units.  [CHIPS]  Interaction energy between P residues and solvent.  [CHI_HW]  Interaction energy between H residues and an active wall unit. [CHI_SW] Interaction energy between solvent and an active wall unit. [CHI_PW] Interaction energy between P residues and an active wall unit.  [KAPPA_HS] Internal energy portion of the temperature-dependent interaction energy between H residue and solvent units. There is a choice in the F E N E R G Y function to set the HS interaction energy to be temperature dependent. (The program code must be accessed to control this function.)  In this case, the interaction energy, C H I H S , is divided into  internal energy, KAPPA_HS, and entropy (PSIHS) components using the relation: CHI_HS = (KAPPA_HS / TEMP) + PSI_HS  [PSI_HS] The entropy portion of the temperature-dependent interaction energy between H-residue and solvent units. (See KAPPA_HS for further explanation.)  [KAPPA_SW] The internal energy portion of the temperature-dependent interaction energy between solvent and active wall units. There is a choice in the F E N E R G Y function to set the SW interaction energy to be temperature dependent. (The program code must be accessed to control this function.) In this case, the interaction energy, C H I S W , is divided into internal energy, K A P P A S W , and entropy (PSI_SW) components using the relation: CHI_SW = (KAPPA_SW / TEMP) + PSI_SW  [PSI_SW] The entropy portion of the temperature-dependent interaction energy between solvent and active wall units. (See KAPPA_SW for further explanation.)  224  [TEMP] System temperature.  [E_MAX] Maximum energy limit when searching for low energy conformations. In searching for low energy structures, the selected conformations have to have calculated energies equivalent or lower than E _ M A X .  [L]  Chain length.  [COORD] and [TYPE] Parameters describing the chain sequence. The sequence is entered into the input file as a series of coordinates and letters describing the residue type.  The location of the  coordinates with respect to the simulation space is not relevant because the program automatically places the chain either automatically in the centre of the grid or according to the position specified by START_PT. The format for the input is as follows: Columns 1-4: indicator of residue number Columns 5-8: x-coordinate (justify right) Columns 9-12: v-coordinate (justify right) Column 16: residue type, designated as H or P  225  The number of lines allotted for the sequence data is exactly the number of residues in the chain, and therefore no empty lines can be inserted between data sets in <*.dat>.  [STARTORIGIN] jc-coordinate [STARTORIGIN] y-coordinate Coordinates for the first residue of the comparison conformation.  A second chain  conformation is entered into the data file to serve as a comparison for conformations generated by the simulation.  [COORD_ORIGIN] Sequence of the comparison chain. The input format here resembles that of the active chain sequence without the column indicating the residue type. Columns 1-4: indicator of residue number Columns 5-8: x-coordinate (justify right) Columns 9-12: ^-coordinate (justify right)  Input  data  for  2D A D S O R P T I O N  PROGRAM -  CONTACT9  EDITION  ******************************************************** 2  [NDIM]  1  [CHANGE_LIM]  5000000 50000 50  [LIMIT] [NWRITE] [NSTOR]  dimensions  number  number o f frequency size  of  of  simulations  cycles of  for  each  simulation  conformations  storage  array  for  p r i n t e d to  lowest  output  energy  [GRIDX]  grid  x-dimension  (as  multiple  of  chain  length)  5.00  [GRIDY]  grid  y-dimension  (as  multiple  of  chain  length)  5 20000 500 0  [NWHIRL_START] [NTRANS] [NSIGN] [NFRAME]  length  frequency of  frequency steps  rotation  or  tranlation  translation of  frequency  [NSTARTREC]  of  file  confns  1.00 10000  Diffusion  number o f  output  of to  output pass  to  screen  to  before  movie  or  output  other  output  file  recorded  parameters 0  [MAP_SWITCH] (0  10  = off,  [START_PT]  automatic 1 =  placement  of  chain  in  centre  auto)  starting  x-coordinate  226  3 1  [ST7ART_PT]  starting  y-coordinate  [WHIRL_S 1 automatic suspension of d i f f u s i o n d e p e n d e n t on c h a i n c o n t a c t w i t h s u r f a c e , and of contact monitor with surface (0 = o f f , 1 = auto) W I T C H  1  [RADIUS_S I CH] automatic (0 = o f f , 1 = on) W  T  calculation  of  Rg,  mechanism  frequency  of  Boundary  parameters 2 [WALL_SWITCH] a c t i v a t i o n o f w a l l e n e r g y (4 = a l l s i d e s o n , 1 = v e r t i c a l w a l l s on, x = 0 and G R I D _ L 2 = h o r i z o n t a l w a l l s o n , y = 0 and GRID_LIM) I n t e r a c t i o n energy of contacts ( p e r kT) : -4.00 [CHI_HH] h y d r o p h o b i c - h y d r o p h o b i c c o n t a c t 0.00 [CHI_P l polar-polar contact 0.00 [CHI_HP] h y d r o p h o b i c - p o l a r c o n t a c t 0.00 [CHI_HS] h y d r o p h o b i c - s o l v e n t contact 0.00 [CHI_PS] p o l a r - s o l v e n t contact -4.00 [CHI_HW] h y d r o p h o b i c - w a l l c o n t a c t -0.00 [CHI_P 1 p o l a r - w a l l c o n t a c t 0.00 [CHI_SW] s o l v e n t - w a l l contact 2 7 0 0 . 0 0 [KAPPA_HS] e n t h a l p i c c o n t r i b u t i o n t o CHI_HS ( i s kappa / I M  p  W  -7.71 2700.00 -7.71 300.00  [ P S I _ H S ] e n t r o p i c c o n t r i b u t i o n o f C H I _ H S ( i s p s i / R) [KAPPA_SW] e n t h a l p i c c o n t r i b u t i o n t o C H I _ H S ( i s k a p p a / [ P S I _ S W ] e n t r o p i c c o n t r i b u t i o n o f C H I _ H S ( i s p s i / R) [TEMP] t e m p e r a t u r e o f s y s t e m  - 5 2 . 0 0 [ E _ M A X ] maximum e n e r g y v a l u e i n l i s t o f l o w 18 [ L ] n u m b e r o f monomers i n c h a i n S e q u e n c e o f m o n o m e r s i n c h a i n s t a r t i n g f r o m f i r s t monomer ;d w i t h s p a c e s , H o r P m o n o m e r s ) (coordinates sepa <fig_i_ H 1 0 0 n  2  0  3 4 5 6 7  1 2 2 3 3 2 2 3  8 9 10 11 12 13 14  3 2 1  1 1 1 0 0 -1 -1 -2 -2 -3 -3 -3 -2  a  t  i  v  e  energy  >  P P P H P H H H H H H H  H 1 H 15 0 -2 H 16 0 -1 H 17 1 -1 H 18 1 0 N A T I V E CONFORMATION FOR C A L C U L A T I O N S O F CONTACT S e q u e n c e o f m o n o m e r s i n c h a i n s t a r t i n g f r o m f i r s t monomer ( c o o r d i n a t e s s e p a r a t e d w i t h s p a c e s , H o r P monomers) 10 [ S T A R T _ O R I G I N ] s t a r t i n g x - c o o r d i n a t e f o r n a t i v e 10 [ S T A R T _ O R I G I N ] s t a r t i n g y - c o o r d i n a t e f o r n a t i v e <fig_i_ ive> H 0 0 1 n a t  2  0  1  P  3 4  1 2 2 3  1 1 0 0 -1  P  5 6 7 8  3 2  9 10  2 3  11 12 13 14 15 16 17 18  3 2 1 1 0 0 1 1  -1 -2 -2 -3 -3 -3 -2 -2 -1 -1 0  confs  P H P H H H H H H H H H H H H  confn confn  R) R)  NFRAME  Output files <Screen>  The output is a series of short statements printed to the screen (and in this case channelled to file <Screeri>) at the frequency specified by NSIGN. "SIM" refers to which run of the main subroutine is currently being carried out. The number of times the subroutine is carried out is specified by C H A N G E _ L I M . "STEP" refers to the cycle in the simulation (called NSTEP in the program). "ENERGY" refers to the energy of the chain.  SIM SIM SIM SIM SIM SIM  1 1 1 1 1 1  STEP# STEPt STEP# STEPt STEP# STEP#  0 ENERGY -48. 20000 ENERGY - 44. 40000 ENERGY - 48. 60000 ENERGY - 40. 80000 ENERGY - 48. 100000 ENERGY -48.  (etc ) SIM 1 STEP! 3840000 SIM 1 STEP# 3860000 SIM 1 STEP# 3880000 SIM 1 STEP# 3900000 SIM 1 STEP# 3920000 SIM 1 STEP# 3940000 SIM 1 STEP# 3960000 SIM 1 STEP# 3980000 PROGRAM IS COMPLETE  ENERGY ENERGY ENERGY ENERGY ENERGY ENERGY ENERGY ENERGY  -52 -52 -52 '-52 -52 -52 -52 -52  <test.out>  The first lines in <test.out> list simulation parameters given by the input file. Below this, reports of the chain's position and conformational state are given, sampled at the  228  frequency specified by NWRITE.  Also included is a list of the number of contacts made  at the sampling point. "SIMULATION #" refers to which run of the main subroutine is currently being carried out. "STEP #" refers to the cycle in the simulation. "HH", "PP", etc. refer to the number of contacts made between the components indicated. "ENERGY" refers to the energy of the chain. "POSITION" refers to the coordinates of the first residue of the chain. "VECTORS" refers to the series of vectors describing the chain conformation.  The final portion of <*.out> gives information regarding the low energy conformations stored by the program. " % A G E OF SUCCESSFUL M O V E S " refers to the percentage of successful moves over the total number of cycles. move.  One simulation cycle is equivalent to a single attempted  The Monte Carlo algorithm specifies that only energetically favourable moves  and a small weighted number of unfavourable moves are allowed. As a consequence, relatively few attempted moves are actually carried out. The ratio of these successful moves is calculated in the simulation. " M A X I M U M E N E R G Y RECORDED" refers to the upper energy limit for the list of low energy conformations. It is the value specified by E M A X . "LOW E N E R G Y CONFORMATION #" refers to the order in which the configurations were detected in the simulation.  229  "SIMULATION #" refers to which run of the main subroutine is currently being carried out. "STEP COUNT" refers to the cycle in the simulation. "HH", "PP", etc. refer to the number of contacts made between the components indicated. "POSITION" refers to the coordinates of the first residue of the chain. "ENERGY" refers to the energy of the chain. "VECTORS" refers to the series of vectors describing the chain conformation.  The last output line gives the run-time of the simulation.  *********************** ********************* ************ ******************************** DATA GRID  FILE:  test.dat  DIMENSIONS NDIM CHANGE_LIM LIMIT NWRITE NSTOR GRIDX GRIDY  NWHI R L _ S T A R T  X:  18  Y: 1  5000000 50000 50 1.00 5.00 10000  NTRANS  5  NSIGN  20000  NFRAME  500  NSTARTREC MAP_SWITCH  0  START_PT(1) START_PT(2)  0 10 3  WHIRL_SWITCH.  1  WALL_SWITCH  2  CHI_HH CHI_PP  -4.00  C H I HP  0.00  CHI  0.00  HS  CHI_PS C H I HW C H I PW CHI_SW KAPPA_HS PSI  0.00  0.00 -4.00 0.00 0.00 2700.00  HS  K A P P A SW  -7.71  PSI_SW  2700.00 -7.71  TEMP  300.00  E_MAX  -52.00  L 0 0  0 1  90  2  18 H P  230  1  1  P  2 2  1 0  P H  3  0  P  3 2 2  -1 -1 -2  H H H  3  -2  H  3 2  -3  H  -3 -3  H  1  H  1  -2  H  0  -2  H  0 1 1  -1 -1 0  H H H  COMPARISON START_ORIGIN(1) START_ORIGIN(2) 0 0  0 1  1  1  2  1  2  0  3 3  0 -1  2  -1  2  -2  3 3 2  -2 -3  1 1  -3 -3 -2  0  -2  0  -1  CONFORMATION 10 10  1  -1 1 0 ******************************************** ******************************************** SIMULATION HH  9  #  1  HP  ENERGY  -48.  POSITION  10  1  STEP PP  #  0  0 HS  8  PS  7  HW  3  6  HW  4  PW  0  3  VECTORS 0  1  1  0  1  0  0 1 0  -1 0 -1  -1  0  0  -1  1  0  0  -1  -1  0  -1 0  0 1  -1 0  0 1  1  0  0  1  *************************************** SIMULATION HH 8 ENERGY POSITION VECTORS 0 1 1 0 1 0  # HP 2 -48. 9 2  1  STEP PP  0  #  50000 HS  8  PS  PW  0  1 0 1 0 0 -1 0 1 0 0 -1 -1 -1 0 1 1 0 0 1 -1 0 * * * *****************************  (etc.)  SIMULATION # 1 HH 7 HP 2 ENERGY -52. POSITION 15 0 VECTORS 0 1 0 1 0 1 -1 0 -1 0 0 -1 1 0 0 -1 0 -1 -1 0 -1 0 -1 0 -1 0  PP  STEP # 0 HS  4900000 8 PS  6  HW  6  PW  0  6  HW  6  PW  0  0 1 10 ' 1 0 1 0 *************************************** SIMULATION # 1 HH 7 HP 2 ENERGY -52. POSITION 15 0 VECTORS 0 1 0 1 0 1 -1 0 -1 0 0 -1 1 0 0 -1 0 -1 ' -1 0 -1 0 -1 0 -1 0 0 1 1 0 1 0  PP  STEP # 0 HS  4950000 8 PS  LIMIT REACHED FOR SIMULATION # 1 %AGE OF SUCCESSFUL MOVES IS 5.97279978  232  ******  T H E LOWEST ENERGY CONFORMATIONS  MAXIMUM ENERGY RECORDED =  LOW ENERGY CONFORMATION # SIMULATION # 1 S T E P COUNT  #  HH  HS  7  POSITION  HP  2  15  PP  0  0  ENERGY  ******  -52.  1 4048833 8  PS  6  HW  6  PW  0  SW 30  6  HW  6  PW  0  SW 30  6  HW  -52.  VECTORS 0 1 0 1 0 1 -1 0 -1 0 0 1 0 0 -1 -1 -1 -1  -1 0 -1 -1 0 0 0 0  0  1  1 0 1 0 1 0  LOW ENERGY CONFORMATION # SIMULATION HH  7  POSITION  #  1 STEP  HP  2  15  0  COUNT  PP  0  ENERGY  2 #  4108634  HS  8  PS  -52.  VECTORS 0  1  0  1  0  1  -1  0  -1  0  0  -1  1  0  0  -1  0  -1  -1  0  -1  0  -1  0  -1 0  0 1  1  0  1  0  1  0  ********************************* (etc. )  LOW ENERGY CONFORMATION # SIMULATION HH  7  POSITION  #  1 STEP  HP 15  2 0  PP ENERGY  COUNT 0  9 # HS  4874088 8  PS  6  PW  0  SW 30  -52.  VECTORS 0  1  0  1  0  1  -1  0  -1  0  0  -1  1  0  0  -1  0  -1  -1  0  233  1 1 1  0 0 0  PROGRAM  IS  COMPLETE  PROGRAM. R O N - T I M E :  0 HOURS  0 MINUTES  33  SECONDS  <test.gph>  The first part of <test.gph> is the list of simulation parameters.  The second half of the file lists results sampled at the frequency specified by N F R A M E , but starting only after the value defined by NSTARTREC.  The simulation output shown  here includes energy, conformation and radius of gyration data. Columns 1 to 4: Simulation # Columns 5 to 14: Step # Columns 15 to 21: Energy Columns 22 to 28:  Fraction matching comparison configuration.  The intramolecular  contacts within the chain at the time of sampling are matched to those given for the comparison conformation. The fraction given is the number of similar contacts found in the sampled chain over the total number of intramolecular contacts of the comparison configuration. Columns 29 to 32: Number of intramolecular contacts in the sampled chain matching those in the comparison configuration.  234  Columns 33 to 35: Number of intramolecular contacts in the sampled chain not matching those in the comparison configuration. Columns 36 to 39: Number of contact made between the sampled chain and a boundary site (either active or non-active). Columns 40 to 48: The calculated radius of gyration for the sampled chain. Columns 49 to 57:  The deformation ratio of the sampled chain.  This value is  dimensional ratio of the x and y components of the radius of gyration when calculating the averaged value.  ****************************************** ******************************************** DATA F I L E : t e s t . d a t 18 GRID DIMENSIONS X: 2 NDIM 1 CHANGE_LIM LIMIT 5000000 NWRITE 50000 50 NSTOR 1.00 GRIDX GRIDY 5.00 NWHIRL START 10000 5 NTRANS NSIGN 20000 NFRAME 500 NSTARTREC 0 MAP_SWITCH 0 START_PT(1) 10 3 START_PT(2) WHIRL SWITCH 1 WALL_SWITCH 2 CHI HH -4.00 0.00 CHI_PP CHI_HP 0.00 CHI_HS 0.00 CHI_PS 0.00 CHI_HW -4.00 0.00 CHI PW CHI_SW 0.00 2700.00 KAPPA HS -7.71 PSI_HS KAPPA_SW 2700.00 -7.71 PSI_SW 300.00 TEMP -52.00 E MAX 18 L 0 0 H 0 1 P I I P 2 1 P 2 0 H 3 0 P 3 -1 H 2 -1 H  235  2  -2  3 3  -2 -3  2 1  -3  H H H H H  1  -3 -2  0  -2  H  0  -1  1  -1  H H  H  H 1 0 COMPARISON START_ORIGIN(l) START_ORIGIN(2) 0 0 1 2  0 1 1 1  2  0  3 3  0 -1 -1  2 2 3  -2 -2  3  -3  2 1 1  -3 -3 -2  0  -2  0  -1 -1  1 1  CONFORMATION 10 10  0  * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ********** *************************** ************ 1  500  -48  00  0 50  5  5  4  1 722  0 716  1  1000  -48  00  1500  -48  00  5 5  5  1  0 50 0 50  5  4 4  1  2000  -48  00  0 50  5  5  4  1 693 1 693 1 693  0 659 0 659 0 659  1  4997000  -52  00 00  8 8  6 6  1 951 1 951  2  341  1  4998000  -52 -52  1 1  341  4997500  0 10 0 10  2  1  00  0 10  1  8  6  1 951  2  1  4998500  -52  00  0 10  1  8 '  6  1 951  2  341 341  1  4999000  -52  00  0  10  1  8  6  341  1  8  6  1 951 1 951  2  0 10 00 0 HOURS  2  341  c . >•  1 4999500 -52 PROGRAM R U N - T I M E :  0 MINUTES  33  SECONDS  236  

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