- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Phenomenological modeling of the nucleated polymerization...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Phenomenological modeling of the nucleated polymerization of human islet amyloid polypeptide : a combined.. 2008
pdf
Page Metadata
Item Metadata
Title | Phenomenological modeling of the nucleated polymerization of human islet amyloid polypeptide : a combined experimental and theoretical approach |
Creator |
Bailey, James |
Publisher | University of British Columbia |
Date Created | 2008-10-02 |
Date Issued | 2008-10-02 |
Date | 2008 |
Description | The inverse scattering problem is based on the scattering theory in physics, where measured data such as radiation from an object is used to determine the unique structure of the object in question. This approach has been widely successful in fields ranging from geophysics and medical imaging, to quantum field theory. In 1996 Henrik Flyvbjerg suggested that a similar approach could be used to study a reaction far from equilibrium of the self-assembly of a nucleation dependent biopolymer and, under certain conditions, uniquely determine the kinetics of the assembly. Here we use this approach to elucidate the unique structure of human islet amyloid polypeptide, also known as amylin, in-vitro. We use a systematic phenomenological analysis of the amount of monomer in fibril, of amylin, for various initial concentrations from an unstructured monomer pool. Using the assumption that nucleation is the rate-limiting step in fibril formation, we invoke mass action to develop our model. We find that the fibrillogenesis of amylin is well described by a nucleation dependent polymerization event that is characteristic of the sigmoidal shape of the reaction profile generated by our data. Furthermore, we find a second nucleation event is needed to accurately match model predictions to the observed data for the kinetic profiles of fibril formation, and the experimental length distributions of mature fibrils from in-vitro assays. This analysis allows for the theoretical determination of each step of assembly in the nucleation process. Specifically, we find the number of steps to nucleation, the size of each oligomer formed in the nucleation process, the nucleus size, and the elongation kinetics of fibrils. The secondary nucleation process is found to be a fibril dependent surface mediated nucleation event and is similar in reaction order to the primary nucleation step. Model predictions are found to be congruent with experimental assay results of oligomer populations and monomer concentration. We demonstrate that, a persistent oligomer formation is a natural and necessary consequence of nucleated fibril formation, given certain qualitative features of the kinetic profile of fibril formation. Furthermore, the modeling assumptions about monomer and fibril mass are in agreement with experiment. |
Extent | 3559458 bytes |
Subject |
Nucleated Polymerization Islet Amyloid Polypeptide |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | Eng |
Collection |
Electronic Theses and Dissertations (ETDs) 2008+ |
Date Available | 2008-10-02 |
DOI | 10.14288/1.0066659 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2008-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/2441 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/24/items/1.0066659/source |
Download
- Media
- ubc_2008_spring_bailey_james.pdf [ 3.39MB ]
- ubc_2008_spring_bailey_james.pdf
- Metadata
- JSON: 1.0066659.json
- JSON-LD: 1.0066659+ld.json
- RDF/XML (Pretty): 1.0066659.xml
- RDF/JSON: 1.0066659+rdf.json
- Turtle: 1.0066659+rdf-turtle.txt
- N-Triples: 1.0066659+rdf-ntriples.txt
- Citation
- 1.0066659.ris
Full Text
Phenomenological Modeling of The Nucleated Polymerization of Human Islet Amyloid Polypeptide; a Combined Experimental and Theoretical Approach by James Bailey B.Sc., Appalachian State University, 2005 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2008 ©James Bailey Abstract The inverse scattering problem is based on the scattering theory in physics, where measured data such as radiation from an object is used to determine the unique structure of the object in question. This approach has been widely successful in fields ranging from geophysics and medical imaging, to quantum field theory. In 1996 Henrik Flyvbjerg suggested that a similar approach could be used to study a reaction far from equilibrium of the self-assembly of a nucleation dependent biopolymer and, under certain conditions, uniquely determine the kinetics of the assembly. Here we use this approach to elucidate the unique structure of human islet amyloid polypeptide, also known as amylin, in - vitro. We use a systematic phenomenological analysis of the amount of monomer in fibril, of amylin, for various initial concentrations from an unstructured monomer pool. Using the assumption that nucleation is the rate-limiting step in fibril formation, we invoke mass action to develop our model. We find that the fibrillogenesis of amylin is well described by a nucleation dependent polymerization event that is characteristic of the sigmoidal shape of the reaction profile generated by our data. Furthermore, we find a second nucleation event is needed to accurately match model predictions to the observed data for the kinetic profiles of fibril formation, and the experimental length distributions of mature fibrils from in- vitro assays. This analysis allows for the theoretical determination of each step of assembly in the nucleation process. Specifically, we find the number of steps to nucleation, the size of each oligomer formed in the nucleation process, the nucleus size, and the elongation kinetics of fibrils. The secondary nucleation process is found to be a fibril dependent surface me- diated nucleation event and is similar in reaction order to the primary nucleation step. Model predictions are found to be congruent with experimental assay results of oligomer populations and monomer concentration. We demonstrates that, a persistent oligomer for- mation is a natural and necessary consequence of nucleated fibril formation, given certain qualitative features of the kinetic profile of fibril formation. Furthermore, the modeling assumptions about monomer and fibril mass are in agreement with experiment. ii Contents Abstract ^ Table of Contents ^ List of Figures Acknowledgments ii iii vi xi 1 Introduction 1 1.1 Phenomenological Modeling of The Nucleated Polymerization of Human Islet Amyloid Polypeptide; a Combined Experimental and Theoretical Ap- proach ^ 1 1.2 Amyloidosis 2 1.3 Islet Amyloid Polypeptide ^ 3 1.3.1^Secretion of IAPP Mirrors That of Insulin ^ 3 1.3.2^Physiological Function of Islet Amyloid Polypeptide ^ 4 2 Cytotoxicity of Islet Amyloid ^ 5 2.1 Fibril Toxicity ^ 5 2.2 Oligomer Toxicity 5 2.3 Need For a Unified Model ^ 6 3 Experimental Methods 7 3.1 Thioflavin T ^ 7 3.2 Dot Blots 8 3.3 Cell Toxicity 8 3.4 Solubility Assay^ 8 3.5 Fibril Length Distributions ^ 8 4 Materials and Methods 10 4.1 Protein Preparation ^ 10 4.2 Fibril Elongation Kinetics 10 4.3 Soluble Oligomer Filtration Assay ^ 10 4.4 Oligomer Time Course; Dot Blots 11 5 Experimental Results ^ 12 5.1 Kinetic Profiles for Fibrillogenesis ^ 12 5.1.1^Scaled Data Collapse 12 5.2 Elongation Kinetics ^ 13 5.3 Oligomer Time Course 13 5.4 Soluble Monomer and Small Oligomer Time Course ^ 13 5.5 Cytotoxicity ^ 13 iii 6 Models of IAPP Polymerization ^ 15 6.1 Simple Nucleation Dependent Polymerization ^15 6.1.1 Application of The Characteristic Scale to The Model ^17 6.2 The Three-Stage Kinetic Model of Amyloid Fibrillation ^18 6.3 Nucleation Dependent Model with Off-Pathway Kinetics for Aggregate For- mation ^ 21 6.4 A Generic Nucleation Dependent Model with Off-Pathway Kinetics for Ag- gregate Formation ^ 24 7 Generic Nucleation Model with Fibril -Dependent Secondary Nucleation 28 7.1 Equation Number and Initial Formation Kinetics ^ 28 7.2 Elongation Kinetics ^ 30 7.3 Fibril Length Distribution ^30 7.4 Impact of Parameter Variation on Fibril and Oligomer Populations^32 7.4.1 Impact of Heparin on The Formation Kinetics of hIAPP 32 7.5 Relevance to Current Biological Hypotheses ^ 33 8 Conclusions ^ 34 8.1 Self Consistency of The Assumption of Trivial Monomer Mass in Oligomer Populations 35 8.2 Room for Improvement ^ 36 8.3 Qualitative Failure of Ordinary Differential Equations to Explain All Ob- served Phenomenon 36 9 Figures ^ 38 9.1 Experimental Fluorescence Data ^ 38 9.2 Power Law ^ 40 9.3 Fluorescence of Seeded Reactions 41 9.4 Experimental Oligomer Concentrations ^ 42 9.5 Monomer and Monomer in Fibril Comparison ^ 43 9.6 Nucleation Dependent Model ^ 44 9.7 The Three Stage Model of Amyloid Fibril Formation ^ 48 9.8 Nucleation Dependent Model With Off-Pathway Aggregates ^ 51 9.9 Nucleation Dependent Model With Fibril Dependant Secondary Nucleation 54 9.10 Elongation Kinetics ^61 9.11 Length Distribution ^62 9.12 Parameter Variations ^ 63 9.13 Scaling Violation ^66 10 Bibliography ^ 67 iv 11 Appendix ^ 73 11.1 Scaled Data. Collapse ^73 11.2 NDP Model reduction ^ 74 11.3 The Three Stage Model of Amyloid Fibrillation ^76 11.4 Scaling of the Nucleated Polymerization Model with Competing Off-Pathway Aggregat ion ^ 80 11.5 Scaling of the Generic Nucleated Polymerization Model with Competing Off-Pathway Aggregation ^ 83 List of Figures 1 Characteristic sample of the fluorescence of Th-T, A(t) (vertical axis), for initial monomer concentrations of hIAPP from 25 to 100 ,uM, right to left respectively. Each data set has been scaled in fluorescence and shifted along the t axis's (horizontal axis) incrementally by bt -= 2.5, for ease of viewing individual plots. This data shows three distinct phases of fibril formation. 1) A pronounced lag phase of constant fluorescence where nu- clei are formed. 2) A rapid rise in fluorescence corresponding to an abrupt elongation of polymers. 3) A plateau in fluorescence that is approached ex- ponentially, corresponding to an exponential decrease in polymer formation. Furthermore, the maximum fluorescence, A„, monotonically increases with increasing initial concentration of hIAPP. Conversely, the time required to reach the plateau in fluorescence monotonically decreases with increasing initial hIAPP concentration. 2 Scaled fluorescence data of Th-T, A(t)1A.,0 (vertical axis) versus scaled timet/to (A„) (horizontal axis) . Each data set has been divided by its maximum recorded florescence, A„, for various initial monomer concen- trations of hIAPP from 25 to 100,u,M. Likewise, each data set has been scaled by to(A„,), the time recorded for that data set to reach 20% maxi- mum fluorescence, A. This approximate collapse of data for all assayed initial concentrations of hIAPP allows us to form a model for the mass of monomers in polymer form that is independent of the initial concentration of monomers. Furthermore, this data collapse, given certain assumptions, allows us to work with scaled variables. We have effectively removed any uncertainty in initial monomer concentrations from our data and subse- quent model calibrations and fits 3 Log(A„) (vertical axis) plotted against Log(to(A„)) (horizontal axis) for several data series between 20 and 100pM of initial hIAPP monomer. Here we see the slope of the best fit line is —2.09, auspiciously close to an integer. Thus, we note y ti 2 in the relationship to (Ao0 ) oc A„-7 and we may now impose this power law, a strict mathematical condition, on any model we use to describe the accumulation of monomers into fibrils 4 Fluorescence for seeded reactions of hIAPP (vertical axis) versus time (hori- zontal axis). Initial concentrations of 25, 35, 50 and 100AM monomers were added to "seeds" generated from vortexed 50/2M initial monomer concen- tration mature reactions of hIAPP. The rate of changes in fluorescence, corresponding to the slopes of the best-fit lines, are used in determining the kinetics of elongation. vi 5 Relative oligomer abundance for initial monomer concentrations of 25, 30, 35, 40, 45, and 50,uM hIAPP, from bottom to top respectively. Each data set is plotted on a normalized y axis, 0 to 1, with each data set being shifted vertically 1 unit for visibility. The data suggest that oligomers form quickly and are found at their maximum concentrations within the first 15 minutes in most cases and by 30 min in all cases. Furthermore, all concentrations exhibit a persistent oligomer population for at least 24 hours of approximately 50-60% of the maximal oligomer concentration. . 42 6 Characteristic experimental results for relative monomer concentration, c(t)/co (diamond), mass in fibril, A(t)12400 (squares), and the sum of the two data sets, c(t)/co + A(t)/A„„ (triangles) versus time (horizontal axis). The sum suggests that most monomer mass is present in either fibril or monomer form throughout fibrillogenesis of hIAPP. This graph was ex- trapolated from results presented by Miranker and Ruschak (6). 43 7 Generic nucleation dependent model, the basic model that describes the nucleated polymerization of a self assembling polymer. Monomers are as- sumed to quickly associate and dissociate to form short lived oligomers. These oligomers are assumed to quickly come into steady state with the monomer population. Stable nuclei are then formed on a slower time scale and quickly elongate to form polymers. 44 8 Schematic of the scaled generic nucleation-dependent model as determined for hIAPP fibril formation. This figure represents the only possible subset of the nucleation dependent polymerization model that is consistent with the observation that the fluorescence profiles generated from various initial monomer concentrations of hIAPP scale systematically to produce a single curve, implying that the initial monomer concentration is immaterial to the formation kinetics of hIAPP fibrils. 45 9 Characteristic fit for the time-dependent behavior of scaled fluorescence, A(t) Ao„ (vertical axis) versus time (horizontal axis), of hIAPP to the scaled single nucleation dependent polymerization model. This fit exempli- fies the inability of the NDP model to capture the rapid rise-time seen in the fluorescence, after the pronounced lag phase. Thus we have elucidated the fact that no version of the nucleation dependent polymerization model will accurately describe the fibrillogenesis of hIAPP, given the observation that initial monomer concentration is not relevant to the fluorescence profile generated for the formation of hIAPP fibrils under systematic scaling. . . 46 10 Representative residuals of the best fits of scaled experimental fluorescence A(t)/A„„, Figure (9), to the single nucleation dependent polymerization model. This result implies a systematic failure of the generic NDP model to accurately capture the fluorescence data for hIAPP. 47 vii 11 The three stage model of amyloid fibril formation, as proposed by Chaung- Chung Lee (2007). This model allows unstable oligomers to assemble by monomer addition at each stage until an unstable nucleus is created. Un- stable fibrils then form by monomer addition to nuclei and may grow by addition of monomers or oligomers of any size. 48 12 Typical best fit of monomer in fibril and corresponding total oligomer pop- ulation (vertical axis) for the three stage model of amyloid fibril formation, versus time (horizontal axis). In this model fibrils are allowed to elongate through both monomer and oligomer addition to fibril ends. While this model obeys both our scaling and the artificial power law given by 7 = 1, it does not support the observation of persistent oligomers for non trivial oligomer-fibril elongation rates. Furthermore, elongation kinetics do not support a theory of fibril elongation through oligomer addition for hIAPP. 49 13 Typical best fit of monomer in fibril and corresponding total oligomer pop- ulation (vertical axis) for the three stage model of amyloid fibrillation with- out fibril elongation through oligomer addition, versus time (horizontal axis). While this model obeys both a scaling and power law, and sup- ports the observation of persistent oligomers, it reduces to the NDP model with the imposed scaling 'y = 1 50 14 Schematic for the NDP model with off-pathway kinetics of aggregation, as proposed by Powers (30). Oligomers, as well as aggregates, are allowed to form by addition of a monomer at each step in the formation process. How- ever, aggregate and oligomer formation are only allowed to "communicate" information through the monomer pool. This model has been proposed to account for the persistent oligomer / aggregate population observed along- side growing and mature fibrils. 51 15 Generic NDP model with off-pathway kinetics. This is a generalization of the model proposed by Powers (30) where oligomers, as well as aggregates, are allowed to form by addition of any number of monomers at any step in the formation process. However, aggregation and oligomer formation are only allowed to "communicate" information through the monomer pool. 52 16 Generic NDP model with off-pathway aggregates as restricted by the scal- ing and the power law observed from fluorescence data of hIAPP fibril formation. Here we see that aggregates are restricted in the same way as oligomers and grow by the same number of monomers at each step of as- sembly, except for perhaps the first aggregate species. Furthermore, we see that aggregate populations are only allowed to grow at a non trivial rate, implying that aggregates cannot act as a buffer to the monomer concentra- tion at "large" initial concentrations. 53 viii 17 Generic nucleation dependent model with fibril-dependent secondary nucle- ation. This figure represents a nucleation dependent polymerization pro- cess that is facilitated by a second nucleation step which is fibril dependent. This additional nucleation step has been proposed to address the inability of the nucleation dependent polymerization model to describe the fibrillo- genesis of hIAPP as observed with a Th-T assay. The secondary nucleation process could include branching, breaking, or surface mediated nucleation. We allow the data to "suggest" the most probable candidate for secondary nucleation. 18 Assembly kinetics of hIAPP as discerned from phenomenological modeling of nucleated polymer formation of the generic nucleation dependent model with fibril-dependent secondary nucleation. This figure shows that nuclei are formed in five steps. First, four monomers combine to form the first oligomer cl, then two monomers bind to ci forming c2. This addition of monomer pairs continues until c4, after which a stable nucleus of twelve monomers is formed from c4 by addition of two monomers. Fibrils then grow by unit addition. As fibrils form, a secondary fibril dependent nu- cleation occurs by addition of a single monomer to the c4 species. These secondary nuclei then form fibrils by unit addition 19 Characteristic fit for the nucleation dependent polymerization model with secondary fibril surface mediated nucleation, as derived from the phenomeno- logical model of the fluorescence profile corresponding to monomer of hu- man islet polypeptide in fibril mass over time. The predictions from the unique model, as determined by systematic scaling and application of the corresponding power law, suggest our phenomenological theory is in excel- lent agreement with the data up until maximum fluorescence is reached for all observed initial concentrations of monomers observed. 20 A closer view of the lag and elongation phases of fibril formation fluores- cence. Here we see our phenomenological model of nucleation dependent polymerization with secondary fibril surface mediated nucleation captures both the rapid transition from lag phase to fibril elongation, as well as, the rapid exponential decrease in fluorescence, corresponding to an exponential decrease in fibril formation. 21 Theoretical oligomer and nuclei populations formed during the fibrillogene- sis of human islet amyloid polypeptide as predicted by the scaled nucleation dependent model of fibril formation with secondary surface mediated nu- cleation. ci , c2, c3, c4, and nuclei top to bottom, respectively ix 22 Theoretical formation rates of oligomer and nuclei populations formed dur- ing the fibriogenesis of human islet amyloid polypeptied as predictied by the scaled nucleation dependant model of fibril formation with secondary surface mediated nucleation. ci , bold dsahed, c2, c3, c4 , solid line, top to bottom respectively, and nuclei, dashed. 23 Theoretical values of the time to 50% completion t t516 (vertical axis) vs experimental time to 50% completion r5Igt. (horizontal axis). 24 Log Log plot of the slope of elongation fluorescence (vertical axis) as read from Figure (4) vs. initial monomer concentration of hIAPP added to seeded reactions (vertical axis). Here we see the slope of the best fit line is approximately 1, suggesting that elongation of fibrils is due to monomer ddition. 25 Predictions for normalized fibril length distribution of the nucleation de- pendent polymerization model with secondary fibril dependent surface me- diated nucleation, based on scaled fluorescence data, solid line, plotted with experimental fibril length distribution, dots 26 Effects of altering fibril elongation rates on number of fibrils (dashed) and total oligomer populations (solid). This chart shows as elongation rates increase total number of fibrils present decreases. Furthermore, the to- tal oligomer populations also monotonically decrease with increasing elon- gation rates. Here increased elongation rates correspond to thicker plot lines. Thus, according to our model, acceleration of fibril formation kinet- ics will result in fewer, longer fibrils and a lower total oligomer population throughout fibrillogenesis. This implies less cytotoxicity for accelerated fibril elongation and a dramatic increase in potentially toxic oligomer and fibril densities for retarded fibril elongation kinetics. 27 Effects of altering fibril nuclei formation rates on number of fibrils (dashed) and total oligomer populations (solid). This chart shows as nucleation rates increase total number of fibrils present increase. However, the total oligomer populations are predicted to monotonically decrease with increas- ing nucleation rates. Here increased nucleation rates correspond to thicker plot lines. Thus, according to our model, acceleration of fibril nuclei forma- tion kinetics will result in increasing fibril numbers of shorter lengths, but a lower total oligomer population throughout fibrillogenesis. This implies greater cytotoxicity for accelerated fibril nucleation, should fibril density increase cytotoxicity. Conversely, a dramatic increase in potentially toxic oligomer is seen for retarded fibril nucleation kinetics. 28 Fluorescence data for heparin. Note the drop in Fluorescence after max- imum fluorescence has been reached. This drop implies that fibril may catastrophe, a feature beyond the scope of the model presented here. Fur- thermore, various initial concentrations of hIAPP do not scale for a given heparin concentration. Thus, we conclude that heparin amplifies the hith- erto trivially small reverse rates, which we have scaled out of our model. 65 29 Plot of tmax /to vs. A. This plot reveals that out scaling is not perfect. For perfectly scalable data tmax/to would be constant and not dependent on A. Thus we see, for later times that our approximation is inconsistent with the observed data. 66 xi Acknowledgments The procedure used here to elucidate the kinetic steps in the formation of human islet amyloid polypeptide fibrils was first suggested by Henrik Flyvbjerg in 1996 in a case study of microtubule dynamics. While the model of amyloid fibrillogenes presented here is an extension of this work, it finds its foundation in the work preformed by Flyvbjerg. I am indebted to Dr. Leah Keshet for the suggestion to model amyloid in this manner, and to Dr. Daniel Coombs for first exposing me to the fascinating procedure outlined by Flyvbjerg. Furthermore, Dr. Keshet has proven invaluable in the formation of this thesis. I would also like to acknowledge Dr. Bruce Verchere and Kate Potter, a MD/PHD student in Vercher's lab at the Childrens' Hospital, for giving me my introduction to the assays I have used to produce the data outlined in this thesis. Funding for this research has been provided by MITACS, the BC Childrens' and Women's Hospital, and the University of British Columbia. I would also like to acknowledge Dr. Das Raibatak for his insights and my introduction to Mathmatica. xi i 1 Introduction 1.1 Phenomenological Modeling of The Nucleated Polymerization of Human Islet Amyloid Polypeptide; a Combined Experimental and Theoretical Approach The presence of amyloid deposits is indicative of the diseased state in many degenerative disorders. Normally soluble precursor proteins form deposits as oligomers and amyloid fibrils which amalgamate to form plaques. These fibrilary tangles and associated oligomers are thought to be responsible for decreased cellular function and cytotoxicity (toxic cell death). Kinetic profiles of amyloid fibril formation, in- vitro, reveal a stereotypical sig- moid shape. This kinetic profile is characteristic of nucleated polymerization. However, the profile generated by amyloid fibrils shows a pronounced lag phase, followed by an abrupt formation of fibrils which does not readily fit current nucleation polymerization models. Here we review several models proposed for the nucleated polymerization of amy- loid fibril formation and demonstrate their inability to accurately describe the formation of fibrillogenesis for human islet amyloid polypeptide (hIAPP) in- vitro. We then outline a generic procedure for determining the kinetic steps of amyloid fibril assembly, which exploits the observation that kinetic profiles for various initial concentrations of hIAPP may be resealed by their initial precursor protein concentration and characteristic time to completion, resulting in approximant collapse of data to a single kinetic profile. This property of hIAPP, and likely other amyloid fibril formation kinetics, allows for the com- plete theoretical determination of the kinetics of assembly of hIAPP in- vitro. Here we have determined that hIAPP fibrils are assembled through a primary nucleation event and facilitated by a rapid secondary nucleation event that is fibril dependent. As a bonus, the analysis predicts the formation of oligomers as a natural consequence of fibril formation and describes their lifetimes. Thus, we have elucidated not only the complete kinetics of formation of fibrils but also shown how this process leads to a persistent oligomer pop- ulation. The model presented here may serves as a predictive tool allowing researchers to "see" the effects of altering formation kinetics at specific steps of fibril and oligomers populations. The procedure we demonstrate is readily applicable to any nucleated poly- merization event that exhibits collapsible kinetic data under systematic scaling. Models of nucleated polymerization have been widely successful in describing the as- sembly kinetics of many self-assembling polymers such as actin (1,2) and microtubules (3,4). The kinetic profiles of these polymerization events exhibit a distinct lag phase where little to no polymer is formed, followed by an abrupt growth phase where polymer- ization occurs rapidly. This process results in a sigmoidal curve representing the mass of monomers in polymer form over time. However, single nucleated polymerization alone is inadequate to describe the kinetics of amyloid fibril formation due to the rapid rise time observed in in- vitro experiments. Specifically, in hIAPP, the rise time for polymerization 1 is considerably shorter than the lag phase. The observed time scale of the rise time relative to the lag phase in hIAPP and other amyloid fibrils suggest a secondary means of fibril tip formation, on which fibrils may grow and elongate. Several models have been sug- gested to describe this process. Notably, branching, fibril breaking and fibril dependant nucleation have all been proposed as mechanisms for secondary nucleation of fibril growth (5,6,7). Despite these modeling attempts, the nature of fibril formation in amyloid disease is poorly understood. The necessity for understanding the kinetics of assembly is relevant not only to the physical chemistry of amyloid fibril formation but also to the pathology of amyloidosis as well. Nucleation rates, specifically secondary nucleation process, may be drastically altered by environmental factors and may contribute to the pathology of the disease (6). Recent cell toxicity assays suggest that the oligomeric species found alongside grow- ing and mature fibrils may be the primary cause of cell toxicity in hIAPP fibrillogenesis. Much work has been done to describe an alternate pathway for oligomer construction which competes and is distinct from fibrillogenesis. Here we show that persistent oligomer populations are a natural consequence of fibril formation in hIAPP and we give a global analysis for their time course, size in monomeric units and relative abundance. In this analysis we use scaling properties of hIAPP kinetic data to elucidate the exact steps in- volved in primary and secondary nucleation. The modeling procedure utilized here allows the data to determine the type of secondary nucleation that most naturally fits the ki- netic profile observed and makes definite predictions about length distributions of hIAPP fibrils. These predictions are in agreement with observation. Furthermore, the analysis of the data reveals a natural source of potentially toxic oligomer species and predicts time courses for their assembly and lifetimes. These predicted time courses are shown to be congruent with experimental observation. 1.2 Amyloidosis Amyloidosis is the diseased state associated with abnormal, insoluble, amyloid deposits in organs and tissue. Insoluble amyloid deposits from normally soluble precursor proteins in the form of oligomers, fibrils, and amyloid plaques are pathogenic features in many known diseases. Numerous species of amyloid deposits are known to exist, stemming from over 20 categorical precursor proteins (8,9,10). The presence of these deposits may be classified into two groups of amyloidosis, systematic and localized (8). Systematic amyloidosis encompasses diseases where precursor proteins circulate through an organ or system. For example, chronic inflammation associated with N-terminal frag- ment deposits in rheumatoid arthritis and tuberculosis is driven by acute phase serum amyloid proteins, a class of apolipoproteins. These apolipoproteins are secreted during 2 acute inflammation by the liver and circulate to the site of inflammation. Other com- mon examples of systemic amyloid precursor producers include the heart and kidneys. Localized amyloidosis, describes the deposition of amyloid in a target organ where the production of the precursor protein is created on-site in the organ. Examples include Alzheimer's disease and type II diabetes where amalgamations of fibrous deposits are indicative of the diseased state. In Alzheimers disease, amyloid-/3 (A/3) is known to ac- cumulate in the cerebrovascular and cortical regions (11,12,13). In type II diabetes, islet amyloid polypeptide (IAPP) fibrils aggregate in plaques in the extracellular regions sur- rounding pancreatic islets. Both animal studies which show, amyloid deposits in the islets before the onset of hypoglycemia and abnormally low blood sugar levels, and in- vitro studies, which show the formation and presence of IAPP fibrils are cytotoxic, imply that amyloid deposition plays a causative role in type II diabetes. Despite this evidence of causality and the well documented nature of amyloid deposits, the kinetics of formation of these deposits is largely unknown. 1.3 Islet Amyloid Polypeptide 1.3.1 Secretion of IAPP Mirrors That of Insulin IAPP is a precursor protein that is co-secreted with insulin by the pancreatic 0-cell islets. The accumulation of islet amyloid in the extracellular space between islets is indicative of type II diabetes and is present in nearly all instances of the disease. Though discovered over a century ago (14), the insoluble nature of islet amyloid posed serious impairments to the isolation and analysis of its components until the late 1980's when Westermark and others (15,16,17) successfully isolated and identified the unique amino acid sequence for the precursor protein IAPP, also known as amylin. Secretion levels of hIAPP have been correlated with insulin secretion levels as re- lated to nutritional stimuli (18-22). Fasting mice assays report IAPP/insulin ratios of 1/10 — 1/6 (23,24). However, the clearance rates of insulin and IAPP are dissimilar in the plasma. IAPP, like C Peptide, the peptide created when pro insulin is split into insulin and C-peptide, is cleared by the kidneys. In contrast, insulin clearance is a complex pro- cess involving the liver, kidney, and muscle as major participants (25) This variation in clearance rates, as well as recorded IAPP plasma levels following stimulation of 0-cells for peptide release with glucose or non glucose response suggest a physiological ratio of IAPP to insulin closer to 1/100 (23,26,27,28). This level is in good agreement with in- vitro experimental observations (29). Furthermore, IAPP levels have been shown to mirror the secretion of insulin in humans, and factors regulating insulin levels have also been shown to regulate IAPP levels (8). Specifically, insulin sensitivity as associated with type II dia- betes has been shown to affect secretion of IAPP from islets (26,27). Notable experimental evidence indicates increased levels of IAPP in populations of obese individuals (21,23,24). 3 A similar study shows increased levels of IAPP mirror increased levels of insulin in preg- nant women (30). This correlation of IAPP secretion with insulin secretion is also noted in pathologies associated with depleted insulin levels, such as 0-cell function depletion associated with type I and II diabetes(19,20,22,27,31,32,33). Individuals with diminished glucose to,lerance associated with type II diabetes, as well as family members of indi- viduals with the disease show retarded release of IAPP to oral glucose intake(22,27,31). Despite the detailed analysis of the secretion and co-production of IAPP, and the impli- cations of a causative role in type II diabetes, little is known about the normal biological function of the peptide. 1.3.2 Physiological Function of Islet Amyloid Polypeptide IAPP has been implicated in insulin regulation, both in the impediment of insulin se- cretion with glucose stimulation (34,35,36) and suppression of insulin mediated glucose uptake, specifically in skeletal muscles (37,38). Furthermore, in-vitro studies of isolated islets show a decrease in the production of glycogen, a short term sugar storage structure. However, these studies used inordinately high levels of IAPP and were not altogether re- producible. Nevertheless, experiments with IAPP deficient mice exhibit enhanced insulin secretion and glycogen clearance rates. Intriguingly, IAPP has been correlated with food intake and weight control. Injections of an IAPP antagonist into the circulatory system of the brain has been demonstrated to show increases in caloric intake and body mass in rats (39-42). Other physiological func- tions implied for IAPP are regulation of renal function (43,44), and calcium homeostasis (45,46). 4 2 Cytotoxicity of Islet Amyloid In non-diseased individuals, IAPP secreted with insulin by ,3-cells is maintained in the well regulated intracellular environment. In this environment, IAPP is maintained in a monomeric form and processed normally. Islet transplant studies with mice suggest the ini- tiation of fibril formation may occur intracellularly with the mis-processing of (pro)IAPP (47,48). This mis-processed (pro)IAPP remains inert while in the intracellular region. Once secreted into the intercellular fluid, this malformed IAPP would be exposed to ag- gravating levels of pH and calcium. Similar perturbations in pH and calcium levels have been shown to facilitate the self assembly of IAPP fibrils in-vitro (49,50,51). Furthermore, heparin sulfate proteoglycans and other compounds found in mature amyloid deposits are readily accessible in this extracellular fluid. In-vitro assays with thioflavin-T demonstrate heparin accelerates the formation of IAPP fibrils with similar pH levels (52), Section [ 7.4.1]. Taken together, these observations suggest an extracellular environment highly favorable for the production of toxic amyloid deposits of IAPP. 2.1 Fibril Toxicity Many studies support the theory of fibril cytotoxicity. Indeed, the almost universal obser- vation of IAPP fibrils in autopsy studies of chronic diabetics gives superficial evidence of their toxicity. Supporting in-vitro studies have shown IAPP fibrils to be cytotoxic in 0- cell cultures (34,35). Despite this body of evidence for fibril toxicity, several studies have shown that increased fibril formation does not increase cell death. This avenue of research has opened the door to an alternate explanation for the cytotoxicity of IAPP deposits, namely oligomeric species that may represent either early stages in fibril development (8,35) or a distinct pathway of oligomer formation(36). 2.2 Oligomer Toxicity Oligomeric species of IAPP have been noted alongside both growing and mature fibrils (53), Section [5.3]. These oligomers have been implicated as the cytotoxic component of amyloid formation (53,54), Section [5.5]. However, the nature of formation of these species is currently the subject of much debate. Models have been proposed that imply that the oligomers are a disperse phase balanced by the presence of IAPP monomers (55). This theory evolved to account for the lack of concentration-dependence in IAPP fibril formation as measured by anisotropy assays. Other studies have suggested that oligomer formation is a process distinct from and competitive with fibril formation (36). This model suggests that the persistent nature of oligomers, well beyond the formation phase of fibrils, supports a distinct pathway for the formation of oligomers. In this thesis, a theory for nucleated fibrillogenesis of IAPP is presented which demonstrates that persistent oligomer formation is a natural and necessary consequence of nucleated fibril formation, given certain qualitative features of the kinetic profile of fibril formation. 5 2.3 Need For a Unified Model Given the contradictory evidence for fibril toxicity cited above and the growing body of evidence of the role of oligomeric cytotoxicity, an exact understanding of fibrillogenesis of IAPP is needed to predict the potential effects of inhibitors on not only fibril popu- lations, but oligomer populations as well. It is the intent of this thesis to help elucidate the underlying kinetics of IAPP fibrillogenesis and, thereby, help to clarify conflicting observations found in the literature. Furthermore, a unique kinetic description for the self assembly of amyloid fibrils and their associated oligomers may help in the search for potential therapies for the early detection and control of type II diabetes. 6 3 Experimental Methods Historically, the "starch-like" appearance of fibrous deposits found in diseased tissue were termed amyloid by Virchow in 1855, after observing blue staining of acid-treated fibrils by iodine. This staining is now known to be due to the protein present in these fibril deposits. Several compounds have been shown, under appropriate conditions, to selectively stain amyloid deposits. The histological benzothiazole dyes thioflavin S (Th-S) and thioflavin T (Th-T) along with the diazobenzide sulfate dye, and Congo red were found to stain amyloid in a number of pathological settings (56). Furthermore, several other fluorescent and non fluorescent dyes including Phorwhite BBU and Sirus Red share this property. Th-S, a mentholated, sulfonated polymerized primulin preparation with an unchar- acterized structure, is commonly used for histological demonstrations of amyloid fibrils where long wavelength UV excitation can be used. The enhanced emission intensity ob- served by the binding of Th-S to amyloid fibrils exhibits a magnification of several fold, while the emission spectra is unaltered. This characteristic excitation of TH-S results in high background levels of fluorescence in solution and renders TH-S unsuited for quanti- tative analysis in solution(56). Th-T however, exhibits a notable spectral shift, without enhanced emission intensity, when bound to amyloid fibrils but not monomers or 0-sheet conformations (56). The speci- ficity with which Th-T exhibits fluorescence shift when bound to amyloid fibrils forms the basis for assays using Th-T fluorescence shift as a measure of monomer mass in fibril form. 3.1 Thioflavin T The characteristic behavior of Th-T of exhibiting a fluorescence spectral shift without altering emissions spectrum implies Th-T's 115-nm red shift in fluorescence is associated with a change in the ground state energy upon binding to amyloid fibrils, and does not result from changes in the excitation state energies. The later is implied by a shift in the emission spectrum (56). Regardless of these observations, the nature of the observed behavior of the Th-T amyloid fibril complex remains to be elucidated. The unknown nature of the amyloid fibril Th-T complex makes exact quantification of the relationship between fluorescence and amyloid fibril all but impossible to obtain. Nevertheless, the consistency in fluorescence shift associated with the complex is useful for determining relative amounts of amyloid fibril formed over time, as well as relative rates of fibril formation for various initial concentrations of IAPP monomer. Fortunately, relative rates of fibril formation for various initial monomer concentrations is all that is required, in the case of hIAPP, to determine the unique kinetics of assembly in fibrillo- 7 genesis. Given these considerations, this thesis utilizes a Th-T assay as described in the Materials and Methods section as a basis for modeling hIAPP fibrillogenesis. 3.2 Dot Blots Western Blot assays have been traditionally used to determine the presence of specific proteins in a sample. Gel electrophoresis is used to segregate proteins in the sample; these proteins are then exposed to antibodies specific to the protein under scrutiny and examined for interactions. Recently, Dot Blot assays have replaced Western, Southern, and Northern Blot assays in a number of studies. The Dot Blot assay has the advantage of not requiring gel electrophoresis or the complex blotting procedures associated with the other assays. However, Dot Blots give no information about the size of the protein present. This thesis reports results from Dot Blot assays used to establish the time order of formation and corresponding lifetimes of oligomers during and after fibril formation of IAPP, as described in the Materials and Methods section. 3.3 Cell Toxicity Cultured 0-cells were exposed to various initial concentrations of hIAPP and were screened for viability over time by measuring cellular metabolism. These assays give a rough estimate of when toxicity began and help elucidate any dependence of cytotoxicity on initial IAPP monomer concentration. 3.4 Solubility Assay To test for monomer and soluble oligomer over time a filtration assay was utilized. This assay tracks the soluble monomer and small soluble oligomer present in solution over time. The assay was used as confirmation of the assumption presented in this thesis model that all monomers are consumed as fibril formation ceases, corresponding with a plateau in Th-T fluorescence. Furthermore, this assay allows for the quantitative tracking of soluble monomers and small soluble oligomers over time. It can be matched to model predictions to help support the theory presented here. 3.5 Fibril Length Distributions Fibril length distributions for hIAPP were measured using atomic force microscope (AFM) by Peter Marek, State University of New York, Stony Brook, New York. The distribution used was from mature fibrils formed under conditions similar to the ones used in fibril kinetics assays in this thesis, supplementary material (53). These fibril length distributions for mature hIAPP reactions in - vitro were compared to model predictions of fibril length distributions at t oc. Specifically, this data allows for an alternate means of determining 8 two key model parameters. One of these parameters is found in a way that relates directly to the kinetic Th-T data. The other relates to a specific stage of nucleation This is information that is not available from the Th-T data and gives validation to our theory. 9 4 Materials and Methods To determine the kinetics of self assembly of hIAPP in - vitro a standard Th-T assay was used to produce kinetic profiles of fibril formation for various initial concentrations of the monomeric precursor protein. This data was systematically scaled and analyzed in the context of a generic nucleation-dependent polymerization model, based on mass action assumptions, which allows for secondary nucleation to occur through one of several path- ways. Specifically, kinetic data for hIAPP fibril formation was collected from in - vitro experiments in Dr. Bruce Verchere's laboratory (Childrens' Hospital, Vancouver BC.). Fluorescence shift associated with the binding of Th-T to amyloid fibrils, but not beta sheet, monomer, or oligomeric species (56) was observed to capture monomer of hIAPP in fibril over time, for concentrations of monomers ranging from 10,uM to 100AM. 4.1 Protein Preparation Samples of lmg hIAPP, were obtained in powdered monomeric form from (Bachem.com ). The monomeric hIAPP was dissolved in 1000/2L Hexafluoroisopropanol (HFIP) and aliquoted into 100,aL units. The solubilized hIAPP was frozen at —20°C for 2 hours, then moved to —80°C overnight. Lyophilization was carried out the next day and the hIAPP was stored at room temperature until needed. Aliquots of hIAPP were dissolved in Dimethyl sul- foxide (DMSO) and mixed with filtered buffer, pH 7.4, in 500,aL 96-well plates with 44, Th-T, to the desired concentration of hIAPP. Buffer was prepared from filtered (.2pM) X10 stock diluted in distilled, filtered water. Rat IAPP, which does not form fibrils, along with "Blanks" (DMSO, Th-T, and buffer) were used as controls. Measurements were taken over a 20 hour period using a fluorescanner. 4.2 Fibril Elongation Kinetics Fibril elongation kinetics were ascertained with a 50pM mature reactions which was vor- texed for 30 sec. Monomeric hIAPP solutions of 25, 35 and 50 auM were added to the mature vortexed reactions and monitored with a Th-T assay. The first 600 seconds of the elongation kinetics were analyzed, assuming a constant number of fibril ends were present during this time. 4.3 Soluble Oligomer Filtration Assay Filtration assays were conducted with centrifugal filter device from Millipore (ultrafree- MC Durapore PVDF 0.22um). It has been observed that hIAPP binds to this membrane, 10 thereby decreasing the level of hIAPP. In order to accommodate for this binding, mem- brane blocking with 10% skim milk in TBS was carried out. 1004 of this blocking solution was spun-down at 8,000rpm for 10mins. The membrane was then washed with distilled water. Washing was carried out by flushing the membrane surface with water. After washing the filter was spun-down for 4 minutes at 10,000rpm to rid the unit of residual water. The water was then removed by pipetting. This procedure prepared the filter unit for use with IhAPP. 101/L of the desired concentration of hIAPP was then mixed with 50/11 of PBS just prior to filtration. This step compensates for the 104 hold up volume associated with the units. The sample was centrifuged at 10,000rpm for 6 minutes, until all the solution had passed through the membrane. Blots of 2/./I., samples were placed on girded nitrocellulose membrane at 15 minute intervals for the first hour. Data points were then taken every 30 min for 2 hours then every hour, until 4 hours had passed. One final data point was collected at 24 hours. The membrane was allowed to air dry and was blocked with a 10% non-fat milk solution at 4°C overnight. The membrane was then washed 3 times in a 1X TBST solution for 5 minutes. Primary antibody was then diluted in a 5% non-fat milk and 1X TBST solution. The membrane was then incubated under the solution for 1 hour with gentle shaking, and washed 3 times in 1X TBST for 5 min. Secondary antibody solution was made from diluted .1pglmL antibody in a 5% non-fat milk TBST solution. The membrane was then covered in this solution and incubated for 1 hour with gentle shaking. The membrane was then washed thee times in TBST solution for 5 min each washing. Substrate solution was then added and the images were developed and analyzed. The remainder of each sample was frozen at —80°C at the time of sampling for later use. 4.4 Oligomer Time Course; Dot Blots Oligomer time course assays were conducted using Dot Blots as described in the filtration section above, with two distinct differences. First, no filtration was used to separate non-soluble oligomers and fibrils from the solution prior to blotting the nitrocellulose membrane. Secondly, antibodies for oligomer staining as opposed to monomer antibodies were used. Furthermore, time points were taken at 15 and 30 minutes, at 1, 2, 3,6,8,10 hour time points. A final time point was taken at 24 hours. The Dot Blots were washed and incubated as above and developed within 24 hours. 1 1 5 Experimental Results 5.1 Kinetic Profiles for Fibrillogenesis Kinetic profiles generated from Th-T assays, as described in Materials and Methods, re- sulted in characteristic sigmoidal curves as shown in Figure (1). Three distinct phases of the polymerization are noted from these kinetic profiles. 1) A pronounced lag phase of constant fluorescence where oligomers are formed but little monomer mass is in polymer form. 2) A rapid rise in fluorescence corresponding to an abrupt elongation of polymers that quickly accelerates. This elongation phase is notably shorter than the preceding lag phase over all concentrations. 3) A plateau in fluorescence that is approached expo- nentially, corresponding to a cessation of polymer formation. Furthermore, the maxima fluorescence ./ 61,, monotonically increases with increasing initial concentration of hIAPP. Conversely, the time required to reach the plateau in fluorescence monotonically decreases with increasing initial hIAPP concentration. 5.1.1 Scaled Data Collapse Scaling of the experimentally observed fluorescence, A(t), by its maximum value, A te , and t by a characteristic time, to(A09 ), resulted in approximate data collapse, as shown in Figure (2). Here we have arbitrarily used the time to 20% maximum fluorescence for to(A,). Plots of the log of A, vs. the log of to(A,,) result in a simple linear relationship. We see from Figure (3) that the slope of the best fit line of this plot is —2.09, auspiciously close to an integer. Thus, we note -y^2, in the relationship to(A,,) cx Acc—r , and we may now impose this power law, a strict mathematical condition, on any model we use to describe the accumulation of monomers into fibrils. Furthermore, the collapse of all data sets over the given concentrations suggest the kinetics of hIAPP are largely independent of em the initial monomer concentration. Therefore, all time-series data is approximately described by a single function, A(t; Aco ) = Acof [t/to(A00)], resulting in the power law 12 to oc A o2 , for any model used to describe the data, where again 'y is determined by the data as in Figure(3), see Appendix (11.1) for details. 5.2 Elongation Kinetics Monomeric hIAPP solutions of 25, 35, 50 and 100,uM were added to solutions of mature vortexed reactions made from 50AM initial monomer concentrations hIAPP and were monitored with Th-T assay, as described above in Materials and Methods. The first 600 seconds of fluorescence data show no lag phase and suggest fibril elongation begins swiftly. Furthermore, the rate of change in fluorescence is approximately constant over this period, Figure (4). These results are shown to be in agreement with fibril elongation by monomer addition. 5.3 Oligomer Time Course Dot Blot data from oligomer time courses assays suggests oligomers form quickly in a concentration dependent manner and persist for at least 24 hours Figure (5). This time span is considerably longer than the time of formation for fibrils as fluorescence from the corresponding Th-T assays plateaus within 2 hours for the lowest concentration observed to produce fibrils. This observation is in agreement with similar assays that suggest oligomers of hIAPP persist on time scales longer than that of fibril formation. 5.4 Soluble Monomer and Small Oligomer Time Course Unfortunately, all assays for soluble monomer conducted by the author, were unreadable and produced no discernable data. However, assays carried out by Miranker (6) using light scattering to detect soluble monomer concentration show soluble monomer concentration approaches 0 as fibril formation plateaus, Figure (6). These assay results are of particular importance as they support the key assumption in the model presented in this thesis, which states all monomers are "consumed" in the production of oligomers and fibrils. Furthermore, the mirroring of monomer depletion to that of fibril formation supports the assumption that most monomer mass is contained in monomers or fibrils. Likewise, this observation, especially at higher concentration, see Miranker and Evan (7,B), suggest that off-pathway kinetics which produce aggregate distinct from fibril formation play a minimal role, if any, in the kinetics of hIAPP fibrillation. 5.5 Cytotoxicity Results from cell viability assays show an increase in cell toxicity as a function of initial monomer concentration, as well as time exposed to the fibril formation process, (data not 13 shown). This observation is in agreement with other assay results in the literature. Un- fortunately, the assay is inconclusive as to the toxic component of hIAPP fibril formation. However, it suggests that the toxic affects are felt early in the process of fibril formation and continue to be present during fibril formation and after completion. This observation coupled with the observation that mature fibrils did not induce cytotoxicity in the ab- sence of oligomers suggest that the toxic component in hIAPP fibrillogenesis is either the oligomers/aggregate populations or a result of maturing, (but not mature) fibrils. The last option seems unlikely. 14 6 Models of IAPP Polymerization 6.1 Simple Nucleation Dependent Polymerization The Nucleation Dependent Polymerization model (NDP) is the basic model that describes the nucleated polymerization of a self assembling polymer. Monomers are assumed to quickly associate and dissociate, to form short-lived species, denoted oligomers. These oligomers are assumed to quickly come into steady state with the monomer population. Stable nuclei are then formed on a slower time scale and quickly elongate to form poly- mers. This model of a rate limiting nucleation process has been successful in describing actin(1,2) and microtubule (3,4) polymer formation. Furthermore, the kinetic profile gen- erated by the NDP model is categorically sigmoidal. A brief review of a generic NDP model and the use of scaling, as presented by Flyvberg (3) for microtubule dynamics is given here in the context of IAPP fibrillogenesis. Following Flyvbjerg we let c(t) =IAPP monomer concentration, c(0) = c, the initial IAPP monomer concentration, ci(t) =number concentration of the ith oligomer, ni =number of monomers added to ci to form ci+ i, k =number of different oligomer species, v(t) = number concentration of nuclei, M(t) =monomer mass in fibril, excluding monomers in oligomer and nuclei, L =forward rate constant for the ith oligomer species, fk =forward rate constant for nuclei formation, fk+1 =forward rate constant for polymer elongation, bi =backward rate constant for the ith oligomer species, di =disintegration rate constant for the ith oligomer species, A(t) = the measured fluorescence, assumed to be a measure of M. Initially all mass is assumed to be in monomer form. These definitions, and the diagram shown in Figure(7), result in the following system of differential equations for nucleated polymerization. For the first oligomeric species we have dc 1 = fen° — h^+ b2c2 — d i c1 . Here fen° represents the rate of formation of c 1 as no monomers bind together to form this species. This oligomeric species ci is also formed through the term b2c2, representing the decay of the next oligomeric species c2 into the ci species. The loss of ci oligomers to c2 oligomers where n1 monomers bind to Cl forming c2 is given by —ficnici. Finally, the c 1 oligomers may completely disassociated into monomers at a rate dice. For the ith oligomer we have dt 15 dci =^1ci-1 — ficn' ci — bici + bi+ ici+ i — dici,^for 2 < i < k. dt The formation of the ith oligomer from the i — 1 oligomer occurs by the addition of ni_i monomers to the ca _1 species, represented by the term f2 _10--- Ic2_ 1 . The ith oligomeric species is also created at a rate b z+ ic,+1 through the loss of n i monomers from the ci + i oligomer species. Loss of the ci species proceeds through the loss of ni_ i monomers from the ith species at the rate bi , and is given by the term V,. Furthermore, ci oligomers may be lost by the complete disintegration of the ith oligomer into monomers via the term dici. The equation for the change in concentration of stable nuclei is given by dv dt ^kCnk Ck. This equation represents the formation of stable nuclei v by the binding of nk monomers to the ck oligomeric species, the largest oligomer allowed in the model. Furthermore, the total change in mass of monomers in fibril form, discounting the monomer in oligomer and nuclei, is given by dM dt fk+lcv. Conservation of mass, under our assumptions gives dc dM dt^dt implying c(0) = c(t) + M(t), and M(t) _ 1^c(t)^A(t) Moo^c(0)^Ao„ • Here, we have assumed that the mass of monomers in nuclei and oligomers is minimal compared to the mass in monomer or polymer form. This assumption is supported by soluble oligomer assays Figure (6) and Miranker (6). 16 6.1.1 Application of The Characteristic Scale to The Model The generic model for nucleated polymerization presented above is now rescaled to elimi- nate the explicit dependence on c0 . Any model term left which explicitly contains c, after the scaling must be set to zero, as described in (Experimental Results). We now enforce our scaling and the power law to cx Aco—r , on the system of differential equations requiring that co , the initial monomer concentra- tion, not appear explicitly in the model. We use the following scaling variables t t = -, and^to cc co ry. Therefore, t Ac;' , and „^c ci ' c with^ X — — for 2 < i < k, where A is needed to keep the units correct, and X is a quantity to be determined by the system. Likewise, v Mv —, and /C//^, co where p, is a quantity to be determined by the system. Substituting the above into the model and requiring that c, not appear explicitly in the equations reduces the model to dci dt^f-= oc27 — dc, dt^ — fze cz, dv f=dt^keck, for 2 < < k, dt^fk+lcv, where the - has been dropped, and A has been absorbed into the reaction constants, Figure(8), Appendix (11.2). dM 17 The observation that the kinetic profiles of hIAPP fibril formation follow a systematic scaling and the implication that the initial monomer concentration is immaterial to the kinetics has allowed us to reduce the generic NDP model to one that uniquely describes the nucleated polymerization of hIAPP. However, we have no assurance that a single nu- cleation scheme, like the one presented above, is the correct description. We are, however, assured that of all the possible pathways allowed by the NDP model the one produced by our scaling is the only plausible candidate. Fits of the scaled NDP model show a systematic error between the fits and the data Figures (9,10). Thus, the process of hIAPP fibril formation, as observed by Th-T assay, suggest that the NDP model is inadequate to describe the fibrillogenesis of hIAPP. Sev- eral alternate models have been suggested for the fibrillogenesis of hIAPP. Most are based on the NDP model and many additional off-pathways, which form oligomers through a process distinct from fibril formation. In this thesis we will examine some of the more prominent models in the literature. The power law discerned from the kinetic data will be used in each case to constrain the models. As we will see, the application of scaling and the power law to these proposed models will have a dramatic impact on their ability to describe the fibrillogenesis of hIAPP. Finally, we will present a simple dual nucleation model that allows both a primary nucleation, as described in the NDP model, with the added feature that fibrils promote a secondary nucleation event. 6.2 The Three-Stage Kinetic Model of Amyloid Fibrillation A model similar to the NDP model by Lee et al. (33), has been put forth to describe the formation of amyloid fibrils in various disorders in a unified manner. However, through the "lens" of scaling and the use of elongation kinetics, we will see that the model reduces to the NDP model with a artificial scaling, to cx for -y =1. Figure (11) shows a schematic of the proposed model for the species of interest. Here F denotes the fibril concentration and fibrils may elongate via addition of any oligomer to a fibril by the term E c3 F. 3=1 The concentration of fibrils F is similar to the number of nuclei formed in the NDP model above, and will be treated as such. Therefore, the assumption that negligible monomer mass is contained in oligomers and nuclei is extended to include the shortest of fibrils, 18 those containing a nuclei and 1 additional monomer. Given that the mean fibril length, in monomer units, is several orders of magnitude greater than the monomer units in a nuclei, this assumption seems plausible. The need to include these shortest fibers in the negligible mass pool assumption stems from the constraint that a fibril may only form by the addition of a monomer to a nuclei. Removal of this constraint should remove this new assumption. Here we precede with the convention used in (33). The rates of attachment and detachment for the ith oligomers to fibrils are given by /i, and s i respectively. Rewrit- ing the model in the notation of this thesis, given the assumptions here and in Section [6.1] leads to the differential equations dc dt =- — fk±icF + soF, dc 1 ^ = foc2dt f1C1C b2C2^SiF — dcz dt - =^— fzczc bz+i ci+ i — bici + s z F — liciF, for 2 < i < k — 1. The change in the ck species, and fibril tip populations are dck dt = fk—ick—ic fkckc bk±i F — bkck + skF — IkckF, and dF dt = fkckc — bk+ 1F. Furthermore, we keep track of mass of monomers in fibril over time as well. The re- sulting equation for monomer mass in fibril is given by dM ^-= dt^ fk± icF + k k F — bk+ 1F. j-=-1 ^ j=0 Systematic scaling of this model Appendix 11.3) results in the following system of equa- tions dc = dcir 2^r dt = oc — ficci — 1 1 c1F, dc, dt fz _icca _i — ficcz — 1,c,F, 19 dt^fk-FicF + dM dck dt^ fkck_lc — fkckc — dF dt f̂kckc, and Elongation kinetics as described in section [7.2] suggest that fibril growth is mediated by monomer addition implying liciF =- 0, for 1 < i < k. Furthermore, the persistent population of oligomers observed experimentally cannot be established if the elongation terms for oligomeric species are not 0, or trivially small, Fig- ures (12,13). The model now becomes dc dt — fk+ icF, dci dt^foc2 — heel, dci dt^ — fcci, dt = fk—iek-1c — fkckc, dF -= fkckc, dt and dM dtdtfk-FicF Therefore, the model as proposed in (33) results in a special case of the generic NDP model from the "view" of collapsible data and persistent oligomer populations, or elonga- tion kinetics. Removing the condition that only one monomer may join other monomers or oligomers to form the next oligomeric species, thus allowing the formation of each oligomeric species through any number of additional monomer units results in the NDP model dck 20 dc dt =- — fk+ic-F, dci dt - foc27 — hc7ei, de, dt = fz-1 c2-1 — fic7 dck dt — fk-ick-ic7 — fkckc, dF f=dt^kekC7 and dM^de fk±i cF = dt dt Thus , we observe that the "Three-Stage Kinetic Model of Amyloid Fibrillation" reduces to the NDP model, under the systematic scaling and elongation kinetics implied by our data. Next we examine a NDP model that allows for off-pathway aggregates to form distinct from fibril formation. 6.3 Nucleation Dependent Model with Off-Pathway Kinetics for Aggre- gate Formation We now look at a model for competing off-pathway aggregation by Powers and Powers (30). This type of model was proposed to account for the presences of oligomers found alongside growing and mature fibrils during amyloid fibrillogenesis. This model is sim- ilar to the model proposed by Chaung-Chung above, but has two distinct differences. Oligomers and fibrils are only allowed to grow through single monomer addition Again we see a problem only the scaling 7 = 1 will satisfy this assumption in the model. The model imposes an off-pathway that allows oligomers to form in competition with the oligomers that lead to nucleated fibril formation. These off-pathway oligomers are referred to as aggregates. Powers and Powers noted in (30) that at some critical concentration of initial monomers scaling of data for fibril formation will become impossible. In the words of the authors "We also find this mechanism has an especially striking feature: although increasing pro- tein concentration generally cause simple nucleated polymerization to reach completion faster, they cause nucleated polymerization with off-pathway aggregation to reach com- pletion more slowly when the protein concentration becomes too high." 21 We find this statement to be untrue for a specific subsystem of nucleated polymerization models with off-pathway aggregation. We will demonstrated bellow that this model does allow for the formation of persistent oligomers as both oligomers and aggregates. How- ever, these aggregates are of the same size, in monomer units, as the oligomers that are formed during the nucleation process. Furthermore, the enforcement of the scaling law removes the back rate terms, see Appendix (11.1,11.2). We are left with a population of oligomers and aggregates that are monomericaly the same. Thus, no distinction can be made between the off-pathway aggregates and the on-pathway oligomers that are left after the monomer pool is consumed. Here we write the model proposed by Powers and Powers (30) in the notation of this thesis. The assumptions made for the basic NDP model apply here. However, Powers and Powers have imposed the arbitrary scaling to a A007 ,^for -y =- 1. The model allows nuclei to form through the addition of one monomer at a time to the preceding oligomeric species, for example, the c, species is formed by the binding of a single monomer c to one of the c a_1 species at some rate fi_i. The first oligomer being formed by two monomers binding to form Cl. Simultaneously, the model allows aggregate species z1 to form by the binding of two monomers to form the first aggregate at some rate a l . These aggregates may then grow by the addition of one monomer at a time to form the next aggregate species. Furthermore, aggregates and oligomers are allowed to lose monomers at any stage at some back rate /3, and ba respectively. Using the notation in Section [6.1] and introduce the new variables and parameters z,(t) =number concentration of the ith aggregate, m =number of different aggregate species, =backward rate constant for nuclei, bm =backward rate constant for fibrils, a, =forward rate constant for the ith aggregate species, 0, =backward rate constant for the ith aggregate species, Initially all mass is assumed to be in monomer form. These definitions, and the diagram shown in Figure(14) result in the following system of differential equations for nucleated polymerization of hIAPP with off-pathway aggregates. Here, as in section [6.1], we find the equations for the oligomeric species to be del dt - = foc2 f1eC1 b2c2 — — and dca dt =^— ficca — bZci + for 2 < i < k. 22 Likewise, we see the formation of the aggregate species are dz1 ^ — ce0C2 dt - ai zi c + 132z2 — ,(31 - dzi dt =^ c — a z zi c —^+ i3i+ i zz+ i , for 2 < i < m — 1. Furthermore, the change in the largest aggregate species if found to be dz, — dt ^ — We also obtain, dv dt^fkcnk ek — bvv- This equation represents the formation of the unstable nuclei v by the binding of nk monomers to the ck oligomeric species, the largest oligomer allowed in the model, and allowing decay through the term —bvv. Furthermore, the total mass of monomers in fibril form, discounting the monomer mass in oligomer, aggregate and nuclei is given by, dM = dt^ fk±i cv — bMV. The term —bMv allows for the loss of monomers from the total mass of monomers in fibrils, which is proportional to the number of fibril ends present. Here we have assumed that the number of fibril ends present is equal to the number of nuclei formed. Since nuclei form slowly and elongation is rapid this assumption is a reasonable one, as no breaking of fibrils is allowed in the model. We now introduce the additional scaled variable zi Y for 1 < i < m, here Y is to be determined by the system. Substituting the scaling variables into the model and dropping the -s gives dci — = dt^ A[foc2 - fig , de, dt ^A [fi_ici_ic — ficicd, for 2 < i < k, dv dt = A[fkckc], 23 and dM =dt Furthermore, the rates of change for the off-pathway aggregates become dt^ A[ao c2 — ai zic], dzi— dt = ^— az zz c], for 2 < i < 7n — 1. The change in the largest aggregate, in monomer units, is now given by dz,̂ — dt^ A [ani_ z„,_i c] • Therefore, we find, given our data constraints, that the "Nucleation Dependent Model with Off-Pathway Kinetics for Aggregate Formation" reduces to the NDP model where aggregates of the same monomeric size as oligomers are produced. Furthermore, these aggregate populations are ever increasing, and given that no "bottleneck" of nucleation exist along the aggregate pathway it is likely that such aggregation events would effec- tively compete for monomers, especially during early stages of fibrillogenesis. The result is a notable mass of monomers being sequestered into aggregates. This would contradict our assumption that all appreciable monomer mass is in either fibril or monomer during the fibril formation process, a prediction not supported by experimental data Figure (6). This model may be easily modified in such a way that allows for any scaling of 'y, Figure (15). However, the observation that the off-pathway aggregates will be of the same size in monomer units as the persistent oligomer population will hold, except perhaps for the smallest aggregate species, see Appendix (11.2). 6.4 A Generic Nucleation Dependent Model with Off-Pathway Kinetics for Aggregate Formation We now rewrite the nucleation dependant model with off-pathway aggregates where oligomers and aggregates are allowed to form by any number of monomers joining to form the first species. Subsequent species are allowed to form by binding to any number of monomers to form the next largest species. No constraint is imposed that requires oligomers and aggregates to be made of the same number of monomer units. However, aggregates may only form from monomers and other aggregates, likewise oligomers may only form from monomers and other oligomers. We now introduce the additional param- eter 24 a, =number of monomers added to z i to form zi+i . Then the diagram shown in Figure(15) and our standard assumptions result in the fol- lowing system of differential equations for nucleated polymerization with off-pathway ag- gregates. The differential equation for the oligomeric and aggregate species are now given by dt = foe° — f1C1Cn 1 + b2c2 — dci- = dt^ fi_ici_icTh' - ' — ficicn' — bici bi+ ici+ i,^for 2 < i < k, and dZi- = dt^ ao^+ /32z2 - dzi dt^—^— Nizi +13i -ozi+i, for 2 < i < m — 1. The change in the largest aggregate species allowed is am-1 — m _Zrn • dt^ rn We also obtain dv dt - = ikcnk ck — bvv. Furthermore, the total mass of monomers in fibril form, discounting the monomer in oligomer, aggregate and nuclei, is given by dt = fk+icv — bmv. Substituting our scaling variables into the model and dropping the ^s gives, see Ap- pendix(11.2) ci A[foc2 '7 — ficic7 ], dt dci - =— dt dv dt ALAckcl, for 2 < i < k, dc 1 dzn, dM 25 and = dt^ A Ffk± ivc] . Furthermore, the rates of change for the off-pathway aggregates become dzl = A [aoc" —^c,),J , dt dzi -= A^— ai z, , dt for 2 < i < m — 1. The change in the largest aggregate, in monomer units, is now given by dzm = A [am_iz„,,_]. c-Y]. Thus, we see that given fibril formation data that scales systematically, as is in the case in hIAPP fibrillogenesis, off-pathway kinetics are restricted, mathematically, in the same way as are the on-pathway kinetics of nuclei formation and no distinction can be made between such events Figure (16). Furthermore, the time scale of formation of aggregates is likely faster than the fibril formation time scale due to the "Bottleneck" step at the nucleus. Therefore, aggregate populations can either be assumed to be in equilibrium with the monomer population on the time scale of fibril formation, implying fibril formation will not scale for high initial monomer concentrations, or the back-rate terms in the aggregate kinetics must be trivially small, while aggregate sizes in monomer units is restricted to mirror oligomer sizes. We therefore state that off-pathway kinetics for aggregate formation is unlikely a pronounced feature of hIAPP fibril formation and is redundant in this class of models, given the constraints provided by our data. The presence of such terms will only go to alter formation rates in data fits, and give no new insight into the structure of these aggregates. Therefore, models with off-pathway aggregates may fit the kinetics profiles of fibril formation that scale under all initial monomer concentrations, which result in fibril formation. However, they introduce an unneeded pathway, from a mathematical point of view. We also note that, under scaling, populations of aggregates will increase until the monomer population is depleted. This competition of aggregates vs. fibrils for monomers could have a notable impact on the total oligomer/aggregate populations, if aggregate formation rates are not trivially small, resulting in the production of a notable oligomer/aggregate population. However, plots of the sum of normalized monomer and fibril mass populations suggest that oligomer and aggregate populations are trivially small for all times, Figure (6). Furthermore, this model, like all single nucleation models, does not readily fit the kinetic profile observed for hIAPP and exhibits a systematic error in dM dt 26 these fits, Figures (9,10). Given these observations, we propose a model for the nucleated polymerization of hIAPP that does not have an off-pathway for aggregate formation, but does consist of a secondary fibril-mediated nucleation step. This step is left generic, surface mediated nucleation, branching or breaking are all allowed as ways to form new nuclei (fibril ends) on which elongation may occur. The scaling law observed from the data collapse and the corresponding power law are then employed to reduce the rather generic model to the unique model, under our assumptions, that describes the data. 27 7 Generic Nucleation Model with Fibril-Dependent Sec- ondary Nucleation Using the NDP model, as described above, we now allow nuclei to form through a sec- ondary fibril-dependent process Figure(17). After application of the scaling law the model reduces to Figure (18) dci dt = foc2' — fic7c1, dc, - = dt^ ci_i — fa c-rez , 2 < i < k. However, for some species class j we have dci dt - foc27 — fic-Y ci — f c(71) M,^if j = 1, or dt dcj ^f^ci —^cj — fie (^y-1)^, for some 2 < i = j < k, and dv dt^fkcryck 613 (-Y-1) • M , dt^fk+1cv. Here 6 may be interpreted as either a scaled branching, breaking or catalyzing rate. These phenomena are largely indistinguishable in the current model analysis, however data fits clearly separate these possibilities. Fits of the model to the data give consistent estimates for all parameters, and convincing fits to the data Figures(19,20). An interest- ing prediction of this class of scaled models is that as the last monomers are sequestered into polymers they leave behind oligomeric species. Neither the off-rates, bz, nor the dis- integration rates, di , are relevant to the kinetics, within the detection level of our assay, and no observable net flux of monomers into or out of the various oligomer species exist due to these terms. Typical residual oligomer populations and their rates of formation are shown in Figures(21,22) respectively. 7.1 Equation Number and Initial Formation Kinetics In order to determine the equation number, the number of steps required to form a nuclei, we have several options. One could simply fit the scaled model with k as a fitting parame- dM 28 ter. Likewise, one may use k as a slowly varying parameter in data fits of the scaled model. A third option involves extrapolating the slope of the kinetic profile for early times on a log log scale, giving the order of initial fibril formation and thus k. Here we have chosen to evaluate the model at early times giving a first order solution for the scaled model in t. This first order solution is then fit to the data for early times, just as the initial rise in Th-T fluorescence is observed and before fibril kinetics accelerate, implying secondary effects have become impactful. As a check of our first order solution for the number of nucleation equations k we then plot both the observed time to 50% completion t50 versus the predicted time to 50% completion tp50 on a log log plot, Figure (23). Here, as with to our scaling time, t50 is an arbitrary choice and the analysis follows through with any chosen time of completion. We now determine the initial formation kinetics of hIAPP by approximating the model as a first order system close to t = 0, with del dt^foc27 , dci dt dv =dt f̂kc7ck, dM dt^ fk+icv. Solving the equations successively leads to foCO2-Y t, c (i + 1) ''ci =I-L=0^°^t , We then obtain (k+2)7 =^fi C(V+i )! and for 2 < i < k. r_rk.+1^c(!c+2)-y+1^pc 2 113=0 J3 (k+2)! k was determined by fitting A = al tk+2 a2tk+3 to the fluorescence data for hIAPP, such that al >> a2, to assure we have the first order solution to the model. We thus obtain the nucleation size N = (k + 2)7, where, again, y is determined by the fit of the log of to(A,) vs. the log of (A,), as described in [5.1]. It may be read off as the slope of the 29 best fit line in Figure(3), Appendix (11.1). Plots of the observed time to 50% completion vs. theoretical time to 50% completion are shown in Figure (23). We see excellent agree- ment between the theoretical and observed formation rates of fibrils, and the parameter, k, with the theoretical value of k as determined by first order solution of the scaled model. 7.2 Elongation Kinetics Monomeric hIAPP solutions of 25, 35, 50 and 100AM were added to mature vortexed reactions made from a 50pM solution and monitored with Th-T assay as described above in Materials and Methods. The first 600 seconds of fluorescence data show no lag and suggest fibril elongation begins swiftly. Furthermore, the rate of change in fluorescence is approximately constant over this period Figure (4). Assuming elongation occurs by addition of oligomers of monomer size nf to fibril ends F at the rate fe , and that all appreciable monomer mass is in either monomer or polymer form. We have dc - dt f e Fenf , and dM de • dt^dt Given the assumption that the fluorescence A(t) is a measure of the mass of monomer in fibril M(t), we may write dA = fe Fenf • dt A plot of dA log Ti t vs. log(co ), gives a slope of nf = 1 as shown in Figure(24). Thus we find, elongation of hIAPP fibrils is well described by monomer addition. The same elongation kinetics were found monitoring monomer consumption directly by Miranker for seed concentrations ranging from 177M to 100,uM added to 50/2M vortexed reactions (6). 7.3 Fibril Length Distribution The Th-T assay gives no direct information about the number or length of fibrils. Nonethe- less, we are able to develop a relative length distribution for fibrils for any time t p ( 1 ; 30 Given our lack of data about fibril length or number of fibrils, we look for a "scaled" distribution, that is, we normalize our length distribution to 1. Mathematically we write fo c') p(1; oo) dl = 1. Furthermore, the fibril length our distribution depends on is a relative one and is normal- ized to 1 J 0 °9 p(1; oo)1 dl = 1. Given the rate of formation of fibrils at any time t dv dt = fke7ck + fjc (7-1) ciM, and the corresponding "velocity" of fibril growth fk-Fic(t)• We can determine the length 1, at time t = oo, of a given fibril that formed at time t 00 1 fk+1 J c(e ) de . Using the rate of formation of fibrils, the fibrils of length / to / +dl at time t oo is simply p(1; 00)d1 — dv dt dt. Therefore, we have p(1; Do )^f kc'Y ck + f 3 c (7- 1 ) e3 m fk-Fic(t) This distribution is shown over data points collected by Peter Marek in figure (25). The plot of the length distribution at large time was generated by fitting the scaled nucleation model with fibril-dependent secondary nucleation to scaled data for an arbitrary initial monomer concentration. No attempt was made to directly fit the model length distribu- tion to the experimental length distribution data. To view the distribution for early times one simply shifts the horizontal axis to the left. 31 7.4 Impact of Parameter Variation on Fibril and Oligomer Populations Variations in the rates of formation of fibrils and nuclei result in predictable changes in populations of total oligomers, number of fibrils formed, and, obviously, the time scale of fibril formation. These observations are summarized in Figures (26, 27). These plots suggest that blindly slowing or inhibiting fibril elongation rates is not the most prudent course for reducing cytotoxicity of hIAPP fibrillogenesis, given that total oligomer and fibril densities both monotonically increase as elongation kinetics are retarded. Thus, slowing of fibril elongation, without completely preventing it, will result in greater num- bers of short fibrils and an increased number of oligomers. Retarding nuclei formation rates has a similar effect on oligomer populations. However, fibril densities are impacted in the reverse manner. Therefore, lowering fibril nuclei formation rates will result in an increased oligomer population, but will have the effect of reducing total fibril density, revealing a potential avenue of research for determining the cytotoxic component in hI- APP fibrillogenesis. Given these observations, this model may prove useful in screening potential therapeutic drugs for type II diabetes treatment. Specifically, kinetic profiles in the presence of inhibiters may be analyzed in the context of this model to detail the exact mechanism by which the inhibitor affects the fibril formation process. Furthermore, the resulting impact on oligomer population, the relative number of fibrils produced and the length distributions may be determined readily using the model, given nothing more than the kinetic profile of fibril formation, as measured by relative mass of monomers in fibril form. We note that, the observation of accelerated fibril kinetics reducing both fibril density and oligomer populations is in qualitative agreement with assays suggestion accelerated fibril formation decreases cytotoxicity. 7.4.1 Impact of Heparin on The Formation Kinetics of hIAPP Heparin, a widely used anticoagulant with a known molecular structure, has been as- sessed for its ability to impact fibril formation using a Th-T assay as described above in (Materials and Methods). We find the presence of heparin rapidly accelerated the for- mation of amyloid fibrils in-vitro Figure (27). Fits of the scaled nucleation model with fibril-dependent secondary nucleation suggest that both primary and secondary nucleation are drastically accelerated. However, examination of the fluorescence profiles of several initial concentrations of hIAPP monomer with constant heparin concentration does not produce scalable data. Therefore, the theory presented here and its results do not apply to the observed fibrillogenesis of hIAPP in the presence of heparin. Furthermore, the lack of scaling specifically preclude the subclass of models isolated here for hIAPP fibril formation. Thus, we may conclude that heparin results in the inclusion of non-scalable terms in any NDP model which describes it and implies that heparin acts to increase the trivially small reverse rates of oligomer and/or fibril decay that we have removed from our model of hIAPP fibrillogenesis on phenomenological bases. This supposition is further 32 supported by the observed decrease in fluorescence seen in the profiles in Figure (28) just after maximum fluorescence is reached. 7.5 Relevance to Current Biological Hypotheses Many current biological arguments suggest that hIAPP precursor protein may undergo a separate aggregate pathway that is distinct from fibril formation. This alternate pathway has been invoked to account for potentially toxic oligomeric species that are seen along- side growing, and mature fibrils. Experimental evidence about the exact life-times of these oligomeric species is unclear. However, onset of oligomer formation is rapid and begins well before the first fibrils are observed. Furthermore, experimental evidence suggests a life-time of the oligomeric species of an order of magnitude or greater than the growth kinetics of fibrillogenesis in IAPP, Figure(5). The above model gives a natural candidate for these oligomeric species, without the need for a separate off-pathway for aggregate formation, namely the oligomers that are "trapped" after the depletion of the monomer pool, unable to grow or decay. It should be noted that we have determined that neither the off-rates, bz , nor the disintegration rates, di , are relevant to the kinetics and that no net flux of monomers into or out of the various oligomeric species exist due to these terms, within the detection level of the Th-T assay. A small net flux of monomers may exist that is too small to be detected by our assay. Simulations suggest that such a small net flux of monomers would have a minimal impact on the results presented here, and would allow for the decay of oligomers as some experiments imply. 33 8 Conclusions The observation that various initial concentrations of hIAPP monomer result in fibril formation kinetics in- vitro, as observed with Th-T assays, that systematically scale and collapse onto a single profile, gives considerable insight into the kinetics of fibril nucleation and elongation. Specifically, the scaling relationship puts strict constraints on any nucle- ation model used to describe hIAPP fibrillogenesis. These constraints, as detailed above, allow for the systematic removal of superfluous terms in the model which are not justified by the formation kinetics under scaling, but would be considered prudent to include, in general, in the modeling of nucleated polymer kinetics. For example, the back rates of oligomer formation are often included in such models and are needed, in many cases, to describe the polymerization event in question. However, inclusion of these terms in a model that describes scalable data is not only unsupported, but may lead to misleading results for model fits. Blind fitting of models to data for hIAPP with out the removal of non-scalable terms, as implied by data collapse, has led to suggestions that off-pathway kinetics play a role. While the use of scaling to reduce fibril formation models does not preclude the existence of off-pathway kinetics, it does put rather harsh restrictions on the allowed kinetics of the off-pathway aggregates. Specifically, any off-pathway aggregate is constrained to the same scaling as on-pathway oligomers, (except for perhaps the smallest aggregate), and will have the same size in monomer units, be formed and grow by the same number of monomer units at each step in assembly, and are mathematically and kinetically indistinguishable. Given these observations the simplest model that accurately fits the kinetic profile for hIAPP fibril formation, and does not violate the observed scaling of these profiles, is a nu- cleation dependent polymerization model with fibril-dependent secondary nucleation and no off-pathway aggregates, as described in section [7]. Thus, phenomenological modeling of fluorescence shift associated with Th-T binding to hIAPP fibrils suggests that fibril formation is dominated by a primary nucleation event and facilitated by a rapid secondary nucleation event that is fibril dependent. Scaling arguments and first order solutions for early times to the described model indicate that nuclei are formed in five steps. First, four monomers combine to form the first sub-nuclei el, then two monomers bind to el forming c2. This addition of monomer pairs continues until c4, after which a stable nucleus of twelve monomers is formed from c4 by addition of two monomers. Fibrils then grow by unit addition. As fibrils form, a secondary fibril-dependent nucleation occurs by addition of a single monomer to the c4 species. These secondary nuclei then form fibrils by unit addition. The results are summarized in Figure (18). To quote Flyvberg "Thus, by combining a simple, but precise, physical measure with a systematic math- ematical analysis we have created a "microscope" through which one apparently may 34 "observe" some otherwise inaccessible details of a biochemical self-assembly process." Through the "lens" of this "microscope" we can observe the kinetics of assembly of oligomers, nuclei and fibrils on an arbitrarily fine scale, thus allowing for the determi- nation of an exact process of assembly. One should keep in mind that this process, while mathematically rigorous, is not guaranteed to be correct. For example the NDP model scales in accordance with the data, but does not accurately describe the formation of hIAPP fibrils as observed by Th-T assay. We have simply found a detailed model for hI- APP fibril formation which describes all observed data and does not violate the condition that various initial concentrations of hIAPP monomer produce kinetic profiles that will collapse to one "master" profile, when subjected to a systematic mathematically rigorous scaling. While this scaling is not perfect, it is accurate to a high degree, so that it is difficult to discern if the small violations in data collapse are due to a systematic error or experimental error associated with the assays. Nevertheless we can investigate a sys- tematic error in the scaling violation observed by looking at late times in fibril formation, section [8.2]. 8.1 Self Consistency of The Assumption of Trivial Monomer Mass in Oligomer Populations A crucial assumption in our model is that the mass of monomers in oligomers and nuclei is trivially small compared to the mass in fibrils and monomers for all times. This assump- tion has allowed us to relate the experimentally determined value of fluorescence A(t) to the initial monomer concentration c(0). The latter is difficult to determine with a high de- gree of accuracy, while preserving the accuracy with which we measure the fluorescence. We have shown this assumption to be approximately true experimentally, Figure (6). However, we may also look for the self consistency of this assumption by using the theo- retical values of the oligomer populations, as determined by our model, using the equation c(t) + M(t)^c(0 ). Inserting the model predictions for oligomer and nuclei population values into the ex- act equation for monomer mass k-1 c + noel + (n.0 + ni)c2 + +^nick +^niv + M = c(0). i=0 ^ i=0 We now observe that these additional terms are trivially small for all time. Thus, we are able to work in the scaled variable c(t)/c(0) and we are justified in assuming that M(t)/Moo = 1 — c(t )/c (0). 35 Therefore, we preserve the precision with which we measure fluorescence without intro- ducing any uncertainty from the biochemical assay of c(0) into our model. 8.2 Room for Improvement The theory presented above is only approximate. Specifically, if the theory were exact, the observed scaling would be exact and not deviate as observed in Figure (2). We may observe this by noting that if the relationship to(A o„) oc A0-07 , were exact then the characteristic time too defined as A00 — A(t) a exp(—t/t„), would be constant for all A. However, Figure (29) shows that too /to is only approx- imately constant, there is some dependance on A. Furthermore, we may note, from Figure (1), that fluorescence and therefore mass in fibril, M, does not monotonically in- crease for all times but exhibits a slight drop after reaching maximum fluorescence, a feature our system of differential equations cannot describe. It is worth noting however, that this violation of too /to independent of il oo is observed by looking at later times and does not imply a grave contradiction to the modeling procedure presented here. 8.3 Qualitative Failure of Ordinary Differential Equations to Explain All Observed Phenomenon Again, looking at late times in fibril formation reveals a qualitative feature which is be- yond the scope of a finite system of ordinary differential equations to describe. Specifically, plots of t/to vs. 1 — A(t)IA,,, at lower concentrations, suggest that the slope of the time series decreases slightly at later times, thus implying that fibrils are lost. This is in con- tradiction to the assumption that fibrils only grow and is a violation of the prediction of our model, which says the slope of these plots are described by dt logio(A„„ ^dt— A(t)) = — d logioc(t) = mco fk± iv(t)• Here we have used the fact that M(t)^1^c(t)^A(t) Moo^c(0)^Aoc,' and 36 dM dt = fk+lcv. We observe that fibrils cannot disappear, dv dt - j f k, k. This observation has allowed us to write the equations for mass in fibril in a finite set of equations. Therefore, no theory based on a finite number of differential equations can describe the observed phenomenon exactly and a model tracking both growing and shrinking fibrils and their distributions is needed. This observation implies that a more exacting model of the fibrillogenesis of hIAPP would require partial differential equations. Therefore, we state that the theory presented here is accurate for the maximum number of fibrils formed which is approximately equal to the number of fibrils at too , and our model is most accurate up until this time tmax , where this time to maximum fibril density is defined as tmax = max[-- d logio(A00 — A(t))]• dt Therefore, we note that, despite the seemingly limited amount of information in our simple assay, we have yet to exhaust all the information about the formation of fibrils that is contained in the data. 37 9 Figures 9.1 Experimental Fluorescence Data AU) 1 teem •^•4p. .4.•^•• • • •re. •• • *we • • • •^• • •^• • • • •^•^•^* • • • • • • ** *** oe• • • • .■^I • :^I^I 1.•. 1. , I ,^1 d a Aid 4.11 .44,..." 5^1 0^15^20^25^t Figure 1: Characteristic sample of the fluorescence of Th-T, A(t) (vertical axis), for initial monomer concentrations of hIAPP from 25 to 100 iftIVI, right to left respectively. Each data set has been scaled in fluorescence and shifted along the t axis's (horizontal axis) incrementally by St = 2.5, for ease of viewing individual plots. This data shows three distinct phases of fibril formation. 1) A pronounced lag phase of constant fluorescence where nuclei are formed. 2) A rapid rise in fluorescence corresponding to an abrupt elongation of polymers. 3) A plateau in fluorescence that is approached exponentially, corresponding to an exponential decrease in polymer formation. Furthermore, the max- imum fluorescence, Ate , monotonically increases with increasing initial concentration of hIAPP. Conversely, the time required to reach the plateau in fluorescence monotonically decreases with increasing initial hIAPP concentration. • • • 38 A(t; A„) = A„f[tIt o (A„)] •• ..•^•.•^• z. • A(t) A„ t 4 3^4^5 00 (A.) Figure 2: Scaled fluorescence data of Th-T, A(t) Acc (vertical axis) versus scaled time t/to(iloo ) (horizontal axis) . Each data set has been divided by its maximum recorded florescence, Ate , for various initial monomer concentrations of hIAPP from 25 to 100,uM. Likewise, each data set has been scaled by to (A,), the time recorded for that data set to reach 20% maximum fluorescence, A. This approximate collapse of data for all assayed initial concentrations of hIAPP allows us to form a model for the mass of monomers in polymer form that is independent of the initial concentration of monomers. Furthermore, this data collapse, given certain assumptions, allows us to work with scaled variables. We have effectively removed any uncertainty in initial monomer concentrations from our data and subsequent model calibrations and fits. 39 9.2 Power Law to CC A c/D2 1 to Figure 3: Log(Ao„) (vertical axis) plotted against Log(to(A 00 )) (horizontal axis) for several data series between 20 and 100/tM of initial hIAPP monomer. Here we see the slope of the best fit line is —2.09, auspiciously close to an integer. Thus, we note 7 2 in the relationship to (11.09 ) cx A c,-7 and we may now impose this power law, a strict mathematical condition, on any model we use to describe the accumulation of monomers into fibrils 40 = 1.6192x = C.9626 Y - 0.5622x R - = 0.9102 Y = 3.5170x `,1 9694 Y= C 3081x fZ-6-8269- C 25^0.30 9.3 Fluorescence of Seeded Reactions Figure 4: Fluorescence for seeded reactions of hIAPP (vertical axis) versus time (horizontal axis). Initial concentrations of 25, 35, 50 and 100p,M monomers were added to "seeds" generated from vortexed 50pM initial monomer concentration mature reactions of hIAPP. The rate of changes in fluorescence, corresponding to the slopes of the best-fit lines, are used in determining the kinetics of elongation. 41 sj --^.....^ -11-• - -•.4_. /^-- -0- - I 1 '1 irri„ .1r 9.4 Experimental Oligomer Concentrations Figure 5: Relative oligomer abundance for initial monomer concentrations of 25, 30, 35, 40, 45, and 50AM hIAPP, from bottom to top respectively. Each data set is plotted on a normalized y axis, 0 to 1, with each data set being shifted vertically 1 unit for visibility. The data suggest that oligomers form quickly and are found at their maximum concentrations within the first 15 minutes in most cases and by 30 min in all cases. Furthermore, all concentrations exhibit a persistent oligomer population for at least 24 hours of approximately 50-60% of the maximal oligomer concentration. 42 9.5 Monomer and Monomer in Fibril Comparison 1.2 - A( t) A f I 10 Figure 6: Characteristic experimental results for relative monomer concentration, c(t)/co (diamond), mass in fibril, A(t)/A cx, (squares), and the sum of the two data sets, c(t)/co A(t)I A„ (triangles) versus time (horizontal axis). The sum suggests that most monomer mass is present in either fibril or monomer form throughout fibrillogenesis of hIAPP. This graph was extrapolated from results presented by Miranker and Ruschak (6). 43 9.6 Nucleation Dependent Model Figure 7: Generic nucleation dependent model, the basic model that describes the nu- cleated polymerization of a self assembling polymer. Monomers are assumed to quickly associate and dissociate to form short lived oligomers. These oligomers are assumed to quickly come into steady state with the monomer population. Stable nuclei are then formed on a slower time scale and quickly elongate to form polymers. 44 0 0 0 0 ° 0 0 0 I 0 0/ GOO CCO (T) 0 0 a . /0, Figure 8: Schematic of the scaled generic nucleation-dependent model as determined for hIAPP fibril formation. This figure represents the only possible subset of the nucleation dependent polymerization model that is consistent with the observation that the fluo- rescence profiles generated from various initial monomer concentrations of hIAPP scale systematically to produce a single curve, implying that the initial monomer concentration is immaterial to the formation kinetics of hIAPP fibrils. 45 Figure 9: Characteristic fit for the time-dependent behavior of scaled fluorescence, A(t)/A,,, (vertical axis) versus time (horizontal axis), of hIAPP to the scaled single nu- cleation dependent polymerization model. This fit exemplifies the inability of the NDP model to capture the rapid rise-time seen in the fluorescence, after the pronounced lag phase. Thus we have elucidated the fact that no version of the nucleation dependent polymerization model will accurately describe the fibrillogenesis of hIAPP, given the ob- servation that initial monomer concentration is not relevant to the fluorescence profile generated for the formation of hIAPP fibrils under systematic scaling. 46 Residuals • • • • • ..; ■ •••./.."••••••••••.•••• • • •^•••• S. • ■^••• • •• • • • Figure 10: Representative residuals of the best fits of scaled experimental fluorescence A(t)/A,„, Figure (9), to the single nucleation dependent polymerization model. This result implies a systematic failure of the generic NDP model to accurately capture the fluorescence data for hIAPP. # • % • • ••s• ••••^• • • • •••^••'•• s . • , „: •••••,• 47 o••o•o• •••• 9.7 The Three Stage Model of Amyloid Fibril Formation Monomers Oligomers Fibrils Figure 11: The three stage model of amyloid fibril formation, as proposed by Chaung- Chung Lee (2007). This model allows unstable oligomers to assemble by monomer addition at each stage until an unstable nucleus is created. Unstable fibrils then form by monomer addition to nuclei and may grow by addition of monomers or oligomers of any size. 48 A ( t )/ Figure 12: Typical best fit of monomer in fibril and corresponding total oligomer pop- ulation (vertical axis) for the three stage model of amyloid fibril formation, versus time (horizontal axis). In this model fibrils are allowed to elongate through both monomer and oligomer addition to fibril ends. While this model obeys both our scaling and the artifi- cial power law given by -y = 1, it does not support the observation of persistent oligomers for non trivial oligomer-fibril elongation rates. Furthermore, elongation kinetics do not support a theory of fibril elongation through oligomer addition for hIAPP. 49 t Figure 13: Typical best fit of monomer in fibril and corresponding total oligomer popula- tion (vertical axis) for the three stage model of amyloid fibrillation without fibril elongation through oligomer addition, versus time (horizontal axis). While this model obeys both a scaling and power law, and supports the observation of persistent oligomers, it reduces to the NDP model with the imposed scaling ry = 1. 50 9.8 Nucleation Dependent Model With Off-Pathway Aggregates Figure 14: Schematic for the NDP model with off-pathway kinetics of aggregation, as pro- posed by Powers (30). Oligomers, as well as aggregates, are allowed to form by addition of a monomer at each step in the formation process. However, aggregate and oligomer for- mation are only allowed to "communicate" information through the monomer pool. This model has been proposed to account for the persistent oligomer / aggregate population observed alongside growing and mature fibrils. 51 00^0 0 0 0 °0 0 o o 0 o a\̂, 2 b 1 /" o 0° 99 COCCCO C0000 CCOCOO 00CCO b „,1^0 0 0 Figure 15: Generic NDP model with off-pathway kinetics. This is a generalization of the model proposed by Powers (30) where oligomers, as well as aggregates, are allowed to form by addition of any number of monomers at any step in the formation process. How- ever, aggregation and oligomer formation are only allowed to "communicate" information through the monomer pool. 52 00 00 (i) 0 0 00 f / 2 , p coocoo at- 0 ° • • r00 te' Figure 16: Generic NDP model with off-pathway aggregates as restricted by the scaling and the power law observed from fluorescence data of hIAPP fibril formation. Here we see that aggregates are restricted in the same way as oligomers and grow by the same number of monomers at each step of assembly, except for perhaps the first aggregate species. Furthermore, we see that aggregate populations are only allowed to grow at a non trivial rate, implying that aggregates cannot act as a buffer to the monomer concentration at "large" initial concentrations. 53 9.9 Nucleation Dependent Model With Fibril Dependant Secondary Nucleation CDC) O, r, Figure 17: Generic nucleation dependent model with fibril-dependent secondary nucle- ation. This figure represents a nucleation dependent polymerization process that is facil- itated by a second nucleation step which is fibril dependent. This additional nucleation step has been proposed to address the inability of the nucleation dependent polymer- ization model to describe the fibrillogenesis of hIAPP as observed with a Th-T assay. The secondary nucleation process could include branching, breaking, or surface mediated nucleation. We allow the data to "suggest" the most probable candidate for secondary nucleation. 54 Figure 18: Assembly kinetics of hIAPP as discerned from phenomenological modeling of nucleated polymer formation of the generic nucleation dependent model with fibril- dependent secondary nucleation. This figure shows that nuclei are formed in five steps. First, four monomers combine to form the first oligomer c1, then two monomers bind to c1 forming c2. This addition of monomer pairs continues until c4, after which a stable nucleus of twelve monomers is formed from c4 by addition of two monomers. Fibrils then grow by unit addition. As fibrils form, a secondary fibril dependent nucleation occurs by addition of a single monomer to the c4 species. These secondary nuclei then form fibrils by unit addition. 55 A( 0/,4,, Figure 19: Characteristic fit for the nucleation dependent polymerization model with secondary fibril surface mediated nucleation, as derived from the phenomenological model of the fluorescence profile corresponding to monomer of human islet polypeptide in fibril mass over time. The predictions from the unique model, as determined by systematic scaling and application of the corresponding power law, suggest our phenomenological theory is in excellent agreement with the data up until maximum fluorescence is reached for all observed initial concentrations of monomers observed. 56 t Figure 20: A closer view of the lag and elongation phases of fibril formation fluores- cence. Here we see our phenomenological model of nucleation dependent polymerization with secondary fibril surface mediated nucleation captures both the rapid transition from lag phase to fibril elongation, as well as, the rapid exponential decrease in fluorescence, corresponding to an exponential decrease in fibril formation. 57 c (t) v(t) t Figure 21: Theoretical oligomer and nuclei populations formed during the fibrillogenesis of human islet amyloid polypeptide as predicted by the scaled nucleation dependent model of fibril formation with secondary surface mediated nucleation. ci , c2, c3, c4, and nuclei top to bottom, respectively. 58 dc dv dt dt t Figure 22: Theoretical formation rates of oligomer and nuclei populations formed during the fibriogenesis of human islet amyloid polypeptied as predictied by the scaled nucleation dependant model of fibril formation with secondary surface mediated nucleation. cl, bold dsahed, c2, c3, c4, solid line, top to bottom respectively, and nuclei, dashed. 59 f t/i 50 Data '50 Figure 23: Theoretical values of the time to 50% completion t t50 (vertical axis) vs. exper- imental time to 50% completion tsigta (horizontal axis). 60 2.4 v - 1 C90Gx C.9$49 2.2 dA/dt " 1.6 • 1.4 • 1.2 • 9.10 Elongation Kinetics 1.3^1.4 1.5^1.b C 0 1./^1.S 1.5 Figure 24: Log Log plot of the slope of elongation fluorescence (vertical axis) as read from Figure (4) vs. initial monomer concentration of hIAPP added to seeded reactions (vertical axis). Here we see the slope of the best fit line is approximately 1, suggesting that elongation of fibrils is due to monomer addition. 61 9.11 Length Distribution 1 Figure 25: Predictions for normalized fibril length distribution of the nucleation dependent polymerization model with secondary fibril dependent surface mediated nucleation, based on scaled fluorescence data, solid line, plotted with experimental fibril length distribution, dots. 62 9.12 Parameter Variations c r , t (t) v(t) Figure 26: Effects of altering fibril elongation rates on number of fibrils (dashed) and total oligomer populations (solid). This chart shows as elongation rates increase total number of fibrils present decreases. Furthermore, the total oligomer populations also monotonically decrease with increasing elongation rates. Here increased elongation rates correspond to thicker plot lines. Thus, according to our model, acceleration of fibril formation kinetics will result in fewer, longer fibrils and a lower total oligomer population throughout fibrillogenesis. This implies less cytotoxicity for accelerated fibril elongation and a dramatic increase in potentially toxic oligomer and fibril densities for retarded fibril elongation kinetics. 63 c r0 (1) v( t) t Figure 27: Effects of altering fibril nuclei formation rates on number of fibrils (dashed) and total oligomer populations (solid). This chart shows as nucleation rates increase total number of fibrils present increase. However, the total oligomer populations are predicted to monotonically decrease with increasing nucleation rates. Here increased nucleation rates correspond to thicker plot lines. Thus, according to our model, acceleration of fibril nuclei formation kinetics will result in increasing fibril numbers of shorter lengths, but a lower total oligomer population throughout fibrillogenesis. This implies greater cytotoxicity for accelerated fibril nucleation, should fibril density increase cytotoxicity. Conversely, a dramatic increase in potentially toxic oligomer is seen for retarded fibril nucleation kinetics. 64 A(t) Figure 28: Fluorescence data for heparin. Note the drop in Fluorescence after maximum fluorescence has been reached. This drop implies that fibril may catastrophe, a feature beyond the scope of the model presented here. Furthermore, various initial concentrations of hIAPP do not scale for a given heparin concentration. Thus, we conclude that heparin amplifies the hitherto trivially small reverse rates, which we have scaled out of our model. 65 •41* • L 1.1^1^' 9.13 Scaling Violation • •1S •• 1 3 2 5 I- 1 t max to ti T 1 T^ _L T^ r L L 1. 9 Figure 29: Plot of t-max Ito vs. A. This plot reveals that out scaling is not perfect. For perfectly scalable data tmax /to would be constant and not dependent on A. Thus we see, for later times that our approximation is inconsistent with the observed data. 66 10 Bibliography 1. Dominique P., Terrell H., Marie C., Edward K. 1985 A model for actin polymerization and the kinetic effects of ATP hydrolysis. Proc. Natl. Acad. Sci. USA 82(21):7207-7211 2. Oosawa, F., and S. Asakura. 1975 Thermodynamics of the polymerization of pro- tein. Academic Press, London. 3. Flyvbjerg H., Jobs E., Leibler, S. 1996 Kinetics of self-assembling microtubules: An inverse problem in biochemistry. Proc. Natl. Acad. Sci. USA 93:5975-5979. 4. Claude B., Nicole B., Benjamin L., Marie-Ccile H., and Christian L., 2007 Kinetic analysis of tubulin assembly in the presence of the microtubule-associated protein TOGp. J Biol Chem. 282(8):5570-5581. 5. Ferrone F. 1999 Analysis of protein aggregation kinetics. Methods Enzymol. 309:256-274. 6. Amy M. R., Andrew D. M. 2007 Fiber-dependent amyloid formation as catalysis of an existing reaction pathway Proc. Natl. Acad. Sci. USA 104:12341-12346 7. Cristian lonescu-Zanetti, Ritu K. 1999 Monitoring the assembly of Ig light-chain amyloid fibrils by atomic force microscopy. Proc. Natl. Acad. Sci. USA 6(23):13175- 13179. 8. Rebecca L. H., Gunilla T. W., Per W., Steven E. K. 2004 Islet Amyloid: A Crtical Entity in the Pathogenesis of Type 2 Diabetes. Clinical Endocrinology and Metabolism 89(8):3629-3643 9. Tan S. Y., Pepys M. B. 1994 Amyloid. Histopathology 25:403-413 10. Westermark P., Benson MD., Buxbaum J. N., Choen A. S., Frangione B., Ikeda S., Masters C. L., Merilin G., Saraiva M. J., Sipe JD. 2002 Amyloid fibril protein nomen- clature. Amyloid 9:197-200 11. Glenner G. G., Wong C. W. 1984 Alzheimer's disease initial report of the initial- tion and purafacation of a novev cyrebrovasculer amyloid protein. Biochem Biophys Res Commun. 120:885-890 12. Glenner G. G., Wong C. W., Quaranta V., Eanes E. D. 1984 The amyloid deposits 67 in Alzheimer's disease: their nature and pathogenesis. Apply Pathology 2:357-369 13. Masters CL., Simms G., Weinman N. A., Multhaup G., McDonald B. L., Beyreuther K. 1985 Amyloid plaque core protein in Alzheimer disease and Down syndrome. Proc Natl Acad Sci USA 82:4245-4249 14. Opie L. 1901 The relationship of diabites Millitus to lessions of the pancrease: hyaline degeneration of the islets of Langerhans. J Exp Med 5:527-540 15. Westermark P., Wernstedt C., Wilander E., Hayden D. W., OBrien T. D., Johnson K. H. 1987 Amyloid fibrils in human insulinoma and islets of Langerhans of the diabetic cat are derived from a neuropeptide-like protein also present in normal islet cells. Proc Natl Acad Sci USA 84:3881-3885 16. Westermark P., Wernstedt C., Wilander E., Sletten K. 1986 A novel peptide in the calcitonin gene related peptide family as an amyloid fibril protein in the endocrine pancreas. Biochem Biophys Res Commun 140:827-831 17. Cooper G. J. S., Willis A. C., Clark A., Turner R. C., Sim R. B., Reid K. M. 1987 Purification and characterization of a peptide from amyloid-rich pancreases of type 2 diabetic patients. Proc Natl Acad Sci USA 84:8628-8632 18. Kahn S. E., D Alessio D. A., Schwartz M. W., Fujimoto W. Y., Ensinck J. W., Taborsky G. J., Porte Jr. D. 1990 Evidence of co-secretion of islet amyloid polypeptide and insulin by. /3cells. Diabetes 39:634 -638 19. Butler P. C., Chou J., Carter W. B., Wang Y. N., Bu B. H., Chang D., Chang J. K., Rizza R. A. 1990 Effects of meal ingestion on plasma amylin concentration in NIDDM and non-diabetic humans. Diabetes 39:752-756 20. Hartter E., Svoboda T., Ludvik B., Schuller M., Lell B., Kuenburg E., Brunn bauer M., Woloszczuk W., Prager R. 1991 Basal and stimulated plasma levels of pancre- atic amylin indicate its co-secretion with insulin in humans. Dia betologia 34:52-54 21. Kahn S. E., Fujimoto W. Y., DAlessio D. A., Ensinck J. W., Porte Jr. D. 1991 Glucose stimulates and potentiates islet amyloid polypeptide secretion bt to /*ells Horm Metab Res 23:577-580 22. Enoki S., Mitsukawa T., Takemura J., Nakazato M., Aburaya J., Toshimori H., Matsukara S. 1992 Plasma islet amyloid polypeptide levels in obesity, impaired glucose- tolerance and non-insulin-dependent diabetes-mellitus. Diabetes Res Clin Pract 15:97-102 68 23. Kautzky-Willer A., Thomaseth K., Pacini G., Clodi M., Ludvik B., Streli C. Wald- hausl W., Prager R. 1994 Role of islet amyloid polypeptide secretion in insulin resistant humans. Diabetologia 37:188-194 24. Thomaseth K., Kautzky-Willer A., Ludvik B., Prager R., Pacini G. 1996 Inte- grated mathematical model to assess /3cell activity during the oral glucose test. Am J Physiol 33:E522-E53 25. Rabkin R., Reaven G. M., and Mondon C. E. 1986 Insulin metabolism by liver, muscle, and kidneys from spontaneously diabetic rats. AJP - Endocrinology and Metabolism 250 (5):530-E537 26. Dechenes CJ., Verchere CB., Andrikopoulos S., Kahn SE. 1998 Human aging is as- sociated with parallel reductions in insulin and amylin release. Am J Physiol 38:E785-E791 27. Kahn S. E., Verchere C. B., Andrikopoulos S., Asberry P. J., Leonetti D. L., Wahl P. W., Boyko E. J., Schwartz R. S., Newell-Morris L., Fujimoto W. Y. 1998 Reduced amylin release is a characteristic of impaired glucose tolerance and type 2 diabetes in Japanese Americans. Diabetes 47:640-645 28. Knowles N. G., Landchild M. A., Fujimoto W. Y., Kahn S. E. 2002 Insulin and amylin release are both diminished in first-degree relatives of subjects with type 2 dia- betes. Diabetes Care 25:292-297 29. Novials A., Sarri Y., Casamitjana R., Rivera F., Gomis R. 1993 Regulation of islet amyloid polypeptide in human pancreatic islets. Diabetes 42:1514-1519 30. Kautzky-Willer A., Thomaseth K., Ludvik B., Nowotny P., Rabensteiner D., Waldhausl W., Pacini G., Prager R. 1997 Elevated islet amyloid pancreatic polypeptide and proinsulin in lean gestational diabetes. Diabetes 46:607-614 31. Hall D., Edskes H. 2004 Silent prions lying in wait: a two-hit model of prion/amyloid formation and infection. J. Mol. Biol. 336:775-786. 32. Tanaka M., Craparo E. F., Martorana V., Bulone D., San Biagio P. L. 2006 The physical basis of how prion conformations determine strain phenotypes. Biophys J 90:4585-4591 33. Chuang-Chung L., Arpan N., Ananthakrishnan S., Georges B., and Gregory J., 2007 A Three-Stage Kinetic Model of Amyloid Fibrillation. Biophysical Journal 92:3448- 69 3458. 34. Lucy M., Christopher J. R., Donald F. S., Leena H., Philippe A. H., Bruce V. 2006 Impaired NH2-Terminal Processing of Human Proislet Amyloid Polypeptide by the Prohormone Convertase PC2 Leads to Amyloid Formation and Cell Death. Diabetes 55, 2192-2201. 35. Porat Y., Kolusheva S., Jelinek R., Gazit E. 2003 The human islet amyloid polypeptide forms transient membrane-active prefibrillar assemblies. Biochemistry 42:10971- 10977 36. Evan T. P., David L. P. 2008 Mechanisms of protein fibril formation: Nucleated polymerization with competing off-pathway aggregation 94:379-391 37. Leighton B., Cooper G. J. S. 1988 Pancreatic amylin and calcitonin gene-related peptide cause resistance to insulin in skeletal-muscle in vitro. Nature 335:632-635 38. Sowa R., Sanke T., Hirayama J., Tabata H., Furuta H., Nishimura S., Nanjo K. 1990 Islet amyloid polypeptide amide causes peripheral insulin resistance in-vivo dogs. Diabetologia 33:118-120 39. Chance W. T., Balasubramaniam A., Zhang F. S., Wimalawansa S. J., Fischer J. E. 1991 Anorexia following the intrahypothalamic administration of amylin Brain Res 539:352-354 40. Chance W. T., Balasubramaniam A., Stallion A., Fischer J. E. 1993 Anorexia following the systemic injection of amylin. Brain Res 607:185-188 41. Lutz T. A., Senn M., Althaus J., DelPrete E., Ehrensperger F., Scharrer E. 1998 Lesion of the area postrema nucleus of the solitary tract (AP/NTS) attenuates the anorec- tic effects of amylin and calcitonin gene-related peptide (CGRP) in rats. Peptides 19:309 -317 42. Arnelo U., Herrington M. K., Theodorsson E., Adrian T. E., Reidelberger R., Larsson J., Marcusson J., Strommer L., Ding X., Permert J. 2000 Effects of long-term infusion of anorexic concentrations of islet amyloid polypeptide on neurotransmitters and neuropeptides in rat brain. Brain Res 887:391-398 43. Wookey P. J., Tikellis C., Du. H. C., Qin H. F., Sexton P. M., Cooper M. E. 1996 Amylin binding in rat renal-cortex, stimulation of adenylyl-cyclase, and activation of plasma-renin. Am J Physiol 39:F289 -F294 70 44. Chai S. Y., Christopoulos G., Cooper M. E., Sexton P. M. 1998 Characteriza- tion of binding sites for amylin, calcitonin, and CGRP in primate kidney. Am J Physiol 43:F51-F62 45. Gilbey S.G., Ghatei M. A., Bretherton-Watt D., Zaidi M., Jones P. M., Perera T., Beacham J., Girgis S., Bloom S. R. 1991 Islet amyloid polypeptide: production by an osteoblast cell-line and possible role as a paracrine regulator of osteoclast function in man. Clin Sci 81:803-808 46. Alam A., Moonga B. S., Bevis P. JR., Huang C. L. H., Zaidi M. 1993 Amylin inhibits bone-resorption by a direct effect on the motility of rat osteoclasts. Exp Physiol 78:183-196 47. Westermark P., Eizirik D. L., Pipeleers D. G., Hellerstrom C., Andersson A. 1995 Rapid deposition of amyloid in human islets transplanted into nude mice. Diabetologia 38:543-549 48. Westermark G., Westermark P., Eizirik D. L., Hellerstrom C., Fox N., Steiner D. F., Andersson A. 1999 Differences in amyloid deposition in islets of transgenic mice ex- pressing human islet amyloid polypeptide versus human islets implanted into nude mice. Metab Clin Exp 48:448-454 49. Porte Jr. D., Kahn S. E. 1989 Hyperproinsulinemia and amyloid in NIDDM: clues to etiology of islet /3-cell dysfunction? Diabetes 38:1333-1336 50. Charge S. B. P., de Koning E. J. P., Clark A. 1995 Effect of pH and insulin on fibrillogenesis of islet amyloid polypeptide in-vitro. Biochemistry 34:14588-14593 51. Westermark P., Li Z. C., Westermark G. T., Leckstro A., Steiner D. F. 1996 Effects of 0-cell granule components on human islet amyloid polypeptide fibril formation. FEBS Lett 379:203-206 52. Hegyi P., Rakonczay Z., Gray M., Argent B., 2004 Measurement of Intracellular pH in Pancreatic Duct Cells: A New Method for Calibrating the Fluorescence Data. Pan- creas. 28(4):427-434. 53. Peter M., Andisheh A. 2007 Aromatic Interactions Are Not Required for Amyloid Fibril Formation by Islet Amyloid Polypeptide but Do Influence the Rate of Fibril For- mation and Fibril Morphology. Biochemistry 46(11):1731-1418. 71 54. Kayed R., Head E., Thompson JL., McIntire TM., Milton SC., Cotman CW., Glabe CG. 2003 Common structure of soluble amyloid oligomers implies common mech- anism of pathogenesis. Science 300:486-489 55. Shae B., Padrik M., Andrew M. 2002 Islet Amyloid: Phase Partitioning and Sec- ondary Nucleation Are Central to the Mechanism of Fibrillogenesis. Bioghemistry 41: 4694-4703 56. Harrey LeVine III 1999 Quantification of 0-Sheet Amyloid Fibril Structure with Thioflavin T. Methods in Enzymology 309:274-284 72 11 Appendix 11.1 Scaled Data Collapse The observed similar shape in the time series fluorescence data motivates the question: Do the observed data sets differ only by a characteristic scale in fluorescence and time? Mathematically we ask whether the observed data sets could be described by a single function, such as A(t; Ate )^f [t/to (A,,)],^[1] where f is dimensionless. To determine if the data is described by Eq.(1) we let g(e) = log [f (lo t)] . We may now write Eq.(l) as log[A] = g[log(t) — log(to)]^log[A,..0 ]. [2] If Eq. (2) is satisfied we may say that the observed fluorescence data only differs by hori- zontal and vertical shifts of log[to] and log[A 00 ], respectively. Therefore, the kinetics have no dependence on co, the initial monomer concentration. Figure(2) shows this to be ap- proximately true for our data. We now look for a relationship between Ao„ and to. Values of to are readily determined from plots of A/ A c, vs. t. These values are plotted against the corresponding values of A„, on a log-log plot in Figure(3). Fitting a linear function gives the approximate relationship to oc^, with -y = 2 over the entire range of initial monomer concentrations assayed here. 73 11.2 NDP Model reduction From section[6.1] we observe that substituting the scaling variables into the model gives d61 no -7 ^no dt - — [ fOCO X C ^hconi 1 enie1 co (b2 a2 — d ied], (raj dt - = A [fi_i^I con-1-7 _1 fiej^"ao—Y(—biâi^diai)], Furthermore we have di)^AX [fkank croik ak], dt ^ = PAL fk+ier 'I) ] •dt We now let /i — Cp X = na = -y, for 1 < i < k, and no = 2y. Substituting these values into the equations results in dal dt = foê'21" — f 167 el -I- ao—Y (b2e2 dOi dt =^ei-i — fiery ei + ep ry( — bi6i +^— diai), for 2 < i < k. Likewise we have dv f ak,dt and dt = A + for 2 < i < k. dM 74 Removal of the remaining explicit terms in co is accomplished by setting (bi+ CCi± i — biai — diôi) = 0, and (b2e2 di al) = 0. Thus we have determined that neither the off-rates, th, nor the disintegration rates, di , are relevant to the kinetics and no net flux of monomers into or out of the various oligomer species exist due to these terms. 75 11.3 The Three Stage Model of Amyloid Fibrillation Here we present a systematic scaling of the model by Chaung-Chung Lee and others (33). We will show under the condition of data collapse as observed for hIAPP that the model reduces to the scaled NDP model where Chaung-Chung Lee have artificially enforced a scaling law of the form, to oc^for 7 = 1. From Figure (11) and section [6.2] we may write differential equations for the forma- tion of fibrils from an unstructured monomer pool. This model allows for the formation of oligomers by the addition of one monomer at each step of oligomer formation. Fur- thermore, the growth and decay of fibrils is allowed by the addition or removal of any size oligomer from fibril ends. The original model by (33) tracks density of fibrils. This term parallels the density of nuclei in the NDP model, and does not track total monomer mass in fibril. However this quantity is easily discerned with the model and is included in this analysis for easy comparison to observed fluorescence profile of hIAPP. The model by Chaung-Chung Lee assumes monomers begin as properly folded hexamers which irre- versibly form monomers. Here we have assumed an initial monomer concentration. This modification will have no impact on the observations presented here. The original model of Chaung-Chung Lee, given the assumptions and modifications above, is dc dt - .4+1cF + s o F. Here we have again assumed the mass of monomers is predominantly contained in monomers and fibrils. Furthermore, we have dc1 ^ = foe2dt f1C1C b2c2 — b 1 c1 + s i F — dci dt = fi-ici-ic - ficic + bi+ici±i - bici + siF - liciF, for 2 < < k - 1. The change in the ck species, the nucleus in this model, and fibril end populations are dck dt = fk-ick-ic - fkckc + and dt^fkckc - bk + 1F. bk+1F bkck SkF 1kCkF, dF 76 dt = fk.i4cF + F — bk + 1F. dM Furthermore, we keep track of the mass of monomers in fibrils over time as well. The resulting equation for monomer mass in fibrils is given by ^j=1^j=0 Here, again, we are tracking fibril density F as opposed to stable nuclei v , as no sta- ble nuclei are assumed in this model. To determine whether our data would support this mechanism, we introduce the following scaling variables, t^c i = ^ c = ^Acr co Ci = —X '^for 2 < i < k, where A is needed to keep the units correct, and X is a quantity to be determined by the system. Likewise, F, and ./1/̂/ = M , co where it is a quantity to be determined by the system. Substituting the above into the model and dropping the s gives dc dt^co C J k+iccoF + soF1-11, dci^Ac-0-7^2 2 dt^X^{f0C Co — f1C1XCCo b2c2X b1c1X SiFIL^F 1-11, dc,^Aco dt^X r [ft — 1 C2-1 XCeo — fi ca X cco + b, -Fic,±1, X — b,c,X^liciXF k — 1. Furthermore, we have dck = Aco--y dt^X̂ {fk — 1 Ck- 1 Xe Co — fkckXcco bk±iF — bkckX skF — lkckXF and dF_ — ^dt^[f kek X cco — bkH4F it] • for 2 < i < 77 The mass of monomer in fibrisl is k dt^co [fk+iccoF + iXFp —^Fp, — bk+iF bd• i=o dM )co j = 1 We now attempt to remove all explicit dependence of the model on co. Inspection of the equations for the change in fibril end density and mass imply that the term fkckXcco must survive or no fibrils will ever form. Therefore, we set X = c'O, and a= X. The resulting equations are dM _ ^ dt^A[fk+icF + k 3^0FC71 — SkFC0-7 — bk±iFC —o 7 ] =1 and dF dt - = A[fkckccr+ 1 — bk± iFc0-7 ]. The equation for the ck species becomes dck dt -= A[fk_ick_icco -Y+ 1 — fkckcco 7+ 1 + bk± iFc0 7 — bkCkC0 7 SkFC0 7 — IkCkFC0 71-1 ]. Furthermore, dci dt A[fi_ici_iccr +1 — ficicc;7+ 1 + bi± ici± icr — bic c for 2 < i < k — 1, + siFcr — dci = A[foc2c0-27+2 — ficiccr+1 + b2c2c7, — bicic0-7 + siFcr —dt and, dc dt =^fk+icF + s oc; 1 F]. We now set = 1 and absorb A into the reaction rates. Our model reduces to dc dt - = 78 del^r 2 dt r JOC^J1Cei — 11c1F, dc, dt = f z _ i cci_ i — fz cc, — l z c,F, dck dt^ fkck_lc — fkckc — lk ck F, dF 2 dt and dM dt — fk+ickc ^k j=1 However, elongation kinetics suggest that fibrills grow by monomer addition resulting in cjF = 0. 3=1 Modifying the above model by allowing oligomers to grow by the addition of any number of monomers at each step and following the scaling procedure results in the NDP model. Thus, no new dynamics or insight have been elucidated with this model. Furthermore, given our data, the model is less general than the basic nucleation dependent model. 79 11.4 Scaling of the Nucleated Polymerization Model with Competing Ofd PathwayAggregation Here we write the model proposed by Powers and Powers (30) in the notation of this thesis. The assumptions made for the basic NDP model apply here. However, Powers and Powers have imposed the artificial scaling, to cx^for^-=- 1. A stated in Section [6.3], the model allows nuclei to form through the addition of one monomer at a time to the preceding oligomeric species and fibril elongation by monomer addition. These definitions, and the diagram shown in Figure(14) result in the following system of differential equations for nucleated polymerization. For the oligomeric and ag- gregate species we have dci^ 2 dt - = Joe- — ficci + b2c2 — bici, dci dt fi—icci—i — ficei — bici + bi+ici+i, for 2 < i < k, dz1- aoc2 — aiZle 02Z2 01Z1)dt dzi - = dt^ cti_izi_ic — aizic — zi + Oi+i zi+ and the change in the largest aggregate species allowed is given by dzm dt =^Zrn_iC —^Zrn • We also obtain, dv dt = fkcnk ck — bv v. Furthermore, the total mass of monomers in fibril form, discounting the monomer in oligomer, aggregate and nuclei, is given by, dt^ fk±icv — bmv. We now introduce the scaling variables of section [6.3]. Substituting into the model and dropping the "'s gives the following equations for the change in the c 1 and z1 species for 2 < i < m — 1, dM 80 dci Acr^2 2 dt^X^if0C Co — ficixeco + b 2 c2 X — biciX], — and dzi^Ac,T7 [^2dt ctoc co — aiziYcco 02z2Y — For the ith species dci A6-7 ,= ^ dt^X Lfz—ici—iXcco — ficiX cc, — biciX + bid-ici+14 and dt^Y [ai_izi_iYcco — aiziYcco — OiziY + Oi+izi+1 17 ] 7 The mth aggregate rate of change is now dzm Aco-7 dt^Y^ [am_i^Ycco — i3mzm,Y] . The nuclei and fibril mass rate of changes are dv Acr dt ^ if kekX cco — byvtt dzi^Ac0-7 for 2 < i < k, for 2 < i < m — 1. and dM = Aco-7 [fk±i v pcco — bm v dt^co We now let p = X = co. The fibril mass and nuclei concentration rates of change now become dM dt = \[fk+ivc — bm vcn, and dv dt = A[fkckccr+1 — bo vco The equations for the cis are 81 dci — =- dt^ - ficicco 1H-1 - bicico 7 for 2 < < k. We now write the equation for ci dci dt =- A [foc2 co-27+2 fleiCC7+1 b2C2Cr — biCiC0 7 ]. and, We are now in a position to impose the only scaling of y that will allow oligomers and nuclei to form in this model. Specifically, 7 = I. Furthermore to allow off-pathway aggre- gates to form we require, Y = c7 . This results in the above equations simplifying to — del = dt^ A [foc2 - cic] , de, ^p dt - = - fz cz cci, dv dt^A[fkckc], and dM = A [fk± i vc] dt for 2 < i < k, Furthermore, the rates of change for the off-pathway aggregates become zi = A [ao c2 — a zi , dt dzi dt =^- aizic], for 2 < < m - 1. The change in the largest aggregate, in monomer units, is now given by dz„t _ dt^ A [ce,_ z,_ Therefore, we see that scaling restricts off-pathway kinetics of aggregate formation, as well as, on-pathway kinetics for oligomer formation. Furthermore the aggregate popula- tion will only grow while monomers are present and not act as an equilibrium "buffer". Thus, the rate of monomer in fibril should continue to scale even with high initial monomer concentrations as is observed for hIAPP. We now relax the implicit power law with 7 = -1 and rewrite the model in a more generic form so as to let our data motivate the power law used. 82 11.5 Scaling of the Generic Nucleated Polymerization Model with Com- peting Ofd PathwayAggregation We now rewrite the nucleation dependent polymerization model with off-pathway aggre- gates where oligomers and aggregates are allowed to form by any number of monomers joining to form the first species. Subsequent species are allowed to form by binding to any number of monomers to form the next largest species. No constraint is imposed that requires oligomers and aggregates to be made of the same number of monomer units. How- ever, aggregates may only form from monomers or other aggregates, likewise oligomers may only form from monomers and other oligomers. The definitions of Section [6.4], and the diagram shown in Figure(15) result in the following system of differential equations for nucleated polymerization. dci =dt^focn ° —^+ b2c2 — and dZi ^ = dt^ CtoCa° —^132Z2 01z1. For the ith aggregate and oligomer we have dci dt = fi lci_lcni-1 bici + bid-ici+i,^for 2 < i < k, and dzi dt = ^— ceizica' —^+ Oi+izi+i, for 2 < i < m — 1. The change in the largest aggregate species allowed is now given by dz,r,^ —^— Orn Zrn . dt We also obtain du dt^f ken ck — bvv. Furthermore, the total mass of monomers in fibril form, discounting the monomer in oligomer, aggregate and nuclei, is given by dt = fk±l ev — bmv. We now introduce the scaling variables of Section [6.4]. Substituting into the model dM 83 and dropping the -s gives the change in the c1 and z1 species as -n1 -0 1 + -2-2— —^j, dc1^Aco- ' [fOCri° —^Xr rn 4-1) r.^hdt^X and dz1 Acr dt =^oc—c° — aiziYca ' Coal ± /32Z2Y /31Z1/71• For the ith species dci Acr — ficiX cni conidt^X^ — bi ciX bi± ici+14 for 2 < i < k, and dzi^Aco^ — aiziYcaicoai — AziY /3i+1zi+111, for 2 < i < m — 1. The mth aggregate rate of change is now dzm = A Co^ — /am z77, Y] .dt The nuclei and fibril mass rate of changes are dv Acr= ^ dt^ [fkckX cnk conk — and dM Acr dt^co [fk+ivitcco — bmvti]• We now let = c')O =X. The fibril mass and nuclei concentration rates of change now become dM^=- dt^ fk+ivc — bmvc0 1 ], and dt 84 dv dt - = A[fkckcnk co-7±nk b,,vcr]. The equations for the cis are dci dt =- A[fi_ici_icni - ' co^— ficicni cr±ni — bicicr b^•^—7 1i_kicz+ ico J , We now write the equation for c1 dci dt A[foc"C2 — ci cm' cr±n' + b2c2c0 7 — bic i co-7]. We now set ni = -y for i < 1 < k, and no = 2 '71 reducing the equations for oligomers, nuclei and fibril mass to del ^ = dt^ A[foc27 — ficie7 ], dci dt = A[.I; lci lc7 — fici c7],^for 2 < i < k, dv dt^ A[fkckc,.y], and dt^ A[fk+i v c]. Furthermore, the equations for ofd pathway aggregates are now given by dzi dt^ [cei_ zi _ cai --1 co 7± ' — aizicai c0-7±a — bizi^+^zi±i co 7 ] , and dzi A=-- [aoca°cr±"Y -1 — a ci cal c;'+' + 132 z2c0 1" — 13izic; 7],dt We now set dM for 2 < i < k. for 2 < i < k. 85 ai=ry for 1 < < m, and remove the explicit dependance on co by setting terms explicitly containing co to 0. The resulting equations are, dzi dt - ^[CroCa0C0 7±a° 17-1 alzic7 ], dzi - = dt^ - ctiz,c7], for 2 < i < m - 1. The change in the largest aggregate is now given by dt dz We now set Y = c-W. The resulting equation for — dt i is given by dzi dt = A[aoca°c;"Y-w±a° - alz1c-Y]. Thus, we are left with the result ao =- y + co, while the solution cto = 2-y, implying = c4) is possible, it is not a unique solution and we are left with an uncertainty in ao . Therefore, dtour equation for —dt may only be reduced to - [aoca° — dt where ao = -y + W. 86
Cite
Citation Scheme:
Usage Statistics
Country | Views | Downloads |
---|---|---|
United States | 5 | 0 |
Japan | 2 | 0 |
China | 1 | 2 |
City | Views | Downloads |
---|---|---|
Unknown | 2 | 1 |
Tokyo | 2 | 0 |
Beijing | 1 | 0 |
Sunnyvale | 1 | 0 |
Redmond | 1 | 0 |
Ashburn | 1 | 0 |
{[{ mDataHeader[type] }]} | {[{ month[type] }]} | {[{ tData[type] }]} |
Share
Share to: