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Phenomenological modeling of the nucleated polymerization of human islet amyloid polypeptide : a combined.. Bailey, James 2008

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Phenomenological Modeling of The Nucleated Polymerization ofHuman Islet Amyloid Polypeptide; a Combined Experimental andTheoretical ApproachbyJames BaileyB.Sc., Appalachian State University, 2005A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIES(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2008©James BaileyAbstractThe inverse scattering problem is based on the scattering theory in physics, wheremeasured data such as radiation from an object is used to determine the unique structureof the object in question. This approach has been widely successful in fields ranging fromgeophysics and medical imaging, to quantum field theory.In 1996 Henrik Flyvbjerg suggested that a similar approach could be used to study areaction far from equilibrium of the self-assembly of a nucleation dependent biopolymerand, under certain conditions, uniquely determine the kinetics of the assembly. Here weuse 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, ofamylin, for various initial concentrations from an unstructured monomer pool. Using theassumption that nucleation is the rate-limiting step in fibril formation, we invoke massaction to develop our model. We find that the fibrillogenesis of amylin is well described bya nucleation dependent polymerization event that is characteristic of the sigmoidal shapeof the reaction profile generated by our data. Furthermore, we find a second nucleationevent is needed to accurately match model predictions to the observed data for the kineticprofiles of fibril formation, and the experimental length distributions of mature fibrils fromin- vitro assays.This analysis allows for the theoretical determination of each step of assembly in thenucleation process. Specifically, we find the number of steps to nucleation, the size of eacholigomer formed in the nucleation process, the nucleus size, and the elongation kineticsof 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 oligomerpopulations and monomer concentration. We demonstrates that, a persistent oligomer for-mation is a natural and necessary consequence of nucleated fibril formation, given certainqualitative features of the kinetic profile of fibril formation. Furthermore, the modelingassumptions about monomer and fibril mass are in agreement with experiment.iiContentsAbstract ^Table of Contents ^List of Figures Acknowledgments iiiiivixi1 Introduction 11.1 Phenomenological Modeling of The Nucleated Polymerization of HumanIslet Amyloid Polypeptide; a Combined Experimental and Theoretical Ap-proach ^ 11.2 Amyloidosis 21.3 Islet Amyloid Polypeptide ^ 31.3.1^Secretion of IAPP Mirrors That of Insulin ^ 31.3.2^Physiological Function of Islet Amyloid Polypeptide ^ 42 Cytotoxicity of Islet Amyloid ^ 52.1 Fibril Toxicity ^ 52.2 Oligomer Toxicity 52.3 Need For a Unified Model ^ 63 Experimental Methods 73.1 Thioflavin T ^ 73.2 Dot Blots 83.3 Cell Toxicity 83.4 Solubility Assay ^ 83.5 Fibril Length Distributions ^ 84 Materials and Methods 104.1 Protein Preparation ^ 104.2 Fibril Elongation Kinetics 104.3 Soluble Oligomer Filtration Assay ^ 104.4 Oligomer Time Course; Dot Blots 115 Experimental Results ^ 125.1 Kinetic Profiles for Fibrillogenesis ^ 125.1.1^Scaled Data Collapse 125.2 Elongation Kinetics ^ 135.3 Oligomer Time Course 135.4 Soluble Monomer and Small Oligomer Time Course ^ 135.5 Cytotoxicity ^ 13iii6 Models of IAPP Polymerization ^  156.1 Simple Nucleation Dependent Polymerization  ^156.1.1 Application of The Characteristic Scale to The Model  ^176.2 The Three-Stage Kinetic Model of Amyloid Fibrillation  ^186.3 Nucleation Dependent Model with Off-Pathway Kinetics for Aggregate For-mation ^  216.4 A Generic Nucleation Dependent Model with Off-Pathway Kinetics for Ag-gregate Formation ^  247 Generic Nucleation Model with Fibril -Dependent Secondary Nucleation 287.1 Equation Number and Initial Formation Kinetics ^  287.2 Elongation Kinetics ^ 307.3 Fibril Length Distribution  ^307.4 Impact of Parameter Variation on Fibril and Oligomer Populations^327.4.1 Impact of Heparin on The Formation Kinetics of hIAPP  327.5 Relevance to Current Biological Hypotheses ^  338 Conclusions ^  348.1 Self Consistency of The Assumption of Trivial Monomer Mass in OligomerPopulations  358.2 Room for Improvement ^  368.3 Qualitative Failure of Ordinary Differential Equations to Explain All Ob-served Phenomenon  369 Figures ^  389.1 Experimental Fluorescence Data ^  389.2 Power Law ^  409.3 Fluorescence of Seeded Reactions  419.4 Experimental Oligomer Concentrations ^  429.5 Monomer and Monomer in Fibril Comparison ^  439.6 Nucleation Dependent Model ^  449.7 The Three Stage Model of Amyloid Fibril Formation ^ 489.8 Nucleation Dependent Model With Off-Pathway Aggregates ^ 519.9 Nucleation Dependent Model With Fibril Dependant Secondary Nucleation 549.10 Elongation Kinetics  ^619.11 Length Distribution  ^629.12 Parameter Variations ^  639.13 Scaling Violation  ^6610 Bibliography ^  67iv11 Appendix ^  7311.1 Scaled Data. Collapse  ^7311.2 NDP Model reduction ^  7411.3 The Three Stage Model of Amyloid Fibrillation  ^7611.4 Scaling of the Nucleated Polymerization Model with Competing Off-PathwayAggregat ion ^  8011.5 Scaling of the Generic Nucleated Polymerization Model with CompetingOff-Pathway Aggregation ^  83List of Figures1 Characteristic sample of the fluorescence of Th-T, A(t) (vertical axis), forinitial monomer concentrations of hIAPP from 25 to 100 ,uM, right to leftrespectively. Each data set has been scaled in fluorescence and shiftedalong the t axis's (horizontal axis) incrementally by bt -= 2.5, for ease ofviewing individual plots. This data shows three distinct phases of fibrilformation. 1) A pronounced lag phase of constant fluorescence where nu-clei are formed. 2) A rapid rise in fluorescence corresponding to an abruptelongation 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 withincreasing initial concentration of hIAPP. Conversely, the time required toreach the plateau in fluorescence monotonically decreases with increasinginitial hIAPP concentration.  2 Scaled fluorescence data of Th-T, A(t)1A.,0 (vertical axis) versus scaledtimet/to (A„) (horizontal axis) . Each data set has been divided by itsmaximum recorded florescence, A„, for various initial monomer concen-trations of hIAPP from 25 to 100,u,M. Likewise, each data set has beenscaled by to(A„,), the time recorded for that data set to reach 20% maxi-mum fluorescence, A. This approximate collapse of data for all assayedinitial concentrations of hIAPP allows us to form a model for the mass ofmonomers in polymer form that is independent of the initial concentrationof monomers. Furthermore, this data collapse, given certain assumptions,allows us to work with scaled variables. We have effectively removed anyuncertainty 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) forseveral data series between 20 and 100pM of initial hIAPP monomer. Herewe 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 nowimpose this power law, a strict mathematical condition, on any model weuse 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 wereadded 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 determiningthe kinetics of elongation.  vi5 Relative oligomer abundance for initial monomer concentrations of 25, 30,35, 40, 45, and 50,uM hIAPP, from bottom to top respectively. Each dataset is plotted on a normalized y axis, 0 to 1, with each data set beingshifted vertically 1 unit for visibility. The data suggest that oligomersform quickly and are found at their maximum concentrations within thefirst 15 minutes in most cases and by 30 min in all cases. Furthermore,all concentrations exhibit a persistent oligomer population for at least 24hours of approximately 50-60% of the maximal oligomer concentration. . 426 Characteristic experimental results for relative monomer concentration,c(t)/co (diamond), mass in fibril, A(t)12400 (squares), and the sum of thetwo 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 ormonomer form throughout fibrillogenesis of hIAPP. This graph was ex-trapolated from results presented by Miranker and Ruschak (6).   437 Generic nucleation dependent model, the basic model that describes thenucleated 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 themonomer population. Stable nuclei are then formed on a slower time scaleand quickly elongate to form polymers.   448 Schematic of the scaled generic nucleation-dependent model as determinedfor hIAPP fibril formation. This figure represents the only possible subsetof the nucleation dependent polymerization model that is consistent withthe observation that the fluorescence profiles generated from various initialmonomer concentrations of hIAPP scale systematically to produce a singlecurve, implying that the initial monomer concentration is immaterial to theformation kinetics of hIAPP fibrils. 459 Characteristic fit for the time-dependent behavior of scaled fluorescence,A(t) Ao„ (vertical axis) versus time (horizontal axis), of hIAPP to thescaled single nucleation dependent polymerization model. This fit exempli-fies the inability of the NDP model to capture the rapid rise-time seen inthe fluorescence, after the pronounced lag phase. Thus we have elucidatedthe fact that no version of the nucleation dependent polymerization modelwill accurately describe the fibrillogenesis of hIAPP, given the observationthat initial monomer concentration is not relevant to the fluorescence profilegenerated for the formation of hIAPP fibrils under systematic scaling. . . 4610 Representative residuals of the best fits of scaled experimental fluorescenceA(t)/A„„, Figure (9), to the single nucleation dependent polymerizationmodel. This result implies a systematic failure of the generic NDP modelto accurately capture the fluorescence data for hIAPP.   47vii11 The three stage model of amyloid fibril formation, as proposed by Chaung-Chung Lee (2007). This model allows unstable oligomers to assemble bymonomer addition at each stage until an unstable nucleus is created. Un-stable fibrils then form by monomer addition to nuclei and may grow byaddition of monomers or oligomers of any size.   4812 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 elongatethrough both monomer and oligomer addition to fibril ends. While thismodel obeys both our scaling and the artificial power law given by 7 = 1,it does not support the observation of persistent oligomers for non trivialoligomer-fibril elongation rates. Furthermore, elongation kinetics do notsupport a theory of fibril elongation through oligomer addition for hIAPP. 4913 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 (horizontalaxis). While this model obeys both a scaling and power law, and sup-ports the observation of persistent oligomers, it reduces to the NDP modelwith the imposed scaling 'y = 1  5014 Schematic for the NDP model with off-pathway kinetics of aggregation, asproposed by Powers (30). Oligomers, as well as aggregates, are allowed toform 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 toaccount for the persistent oligomer / aggregate population observed along-side growing and mature fibrils. 5115 Generic NDP model with off-pathway kinetics. This is a generalization ofthe 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 inthe formation process. However, aggregation and oligomer formation areonly allowed to "communicate" information through the monomer pool. 5216 Generic NDP model with off-pathway aggregates as restricted by the scal-ing and the power law observed from fluorescence data of hIAPP fibrilformation. Here we see that aggregates are restricted in the same way asoligomers and grow by the same number of monomers at each step of as-sembly, except for perhaps the first aggregate species. Furthermore, we seethat 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. 53viii17 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 inabilityof the nucleation dependent polymerization model to describe the fibrillo-genesis of hIAPP as observed with a Th-T assay. The secondary nucleationprocess could include branching, breaking, or surface mediated nucleation.We allow the data to "suggest" the most probable candidate for secondarynucleation.  18 Assembly kinetics of hIAPP as discerned from phenomenological modelingof nucleated polymer formation of the generic nucleation dependent modelwith fibril-dependent secondary nucleation. This figure shows that nucleiare formed in five steps. First, four monomers combine to form the firstoligomer cl, then two monomers bind to ci forming c2. This addition ofmonomer pairs continues until c4, after which a stable nucleus of twelvemonomers is formed from c4 by addition of two monomers. Fibrils thengrow by unit addition. As fibrils form, a secondary fibril dependent nu-cleation occurs by addition of a single monomer to the c4 species. Thesesecondary nuclei then form fibrils by unit addition 19 Characteristic fit for the nucleation dependent polymerization model withsecondary 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 theunique model, as determined by systematic scaling and application of thecorresponding power law, suggest our phenomenological theory is in excel-lent agreement with the data up until maximum fluorescence is reached forall 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 dependentpolymerization with secondary fibril surface mediated nucleation capturesboth the rapid transition from lag phase to fibril elongation, as well as, therapid exponential decrease in fluorescence, corresponding to an exponentialdecrease in fibril formation.  21 Theoretical oligomer and nuclei populations formed during the fibrillogene-sis of human islet amyloid polypeptide as predicted by the scaled nucleationdependent model of fibril formation with secondary surface mediated nu-cleation. ci , c2, c3, c4, and nuclei top to bottom, respectively ix22 Theoretical formation rates of oligomer and nuclei populations formed dur-ing the fibriogenesis of human islet amyloid polypeptied as predictied bythe scaled nucleation dependant model of fibril formation with secondarysurface mediated nucleation. ci , bold dsahed, c2, c3, c4 , solid line, top tobottom 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 readfrom Figure (4) vs. initial monomer concentration of hIAPP added toseeded reactions (vertical axis). Here we see the slope of the best fit lineis approximately 1, suggesting that elongation of fibrils is due to monomeraddition.  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 withexperimental fibril length distribution, dots 26 Effects of altering fibril elongation rates on number of fibrils (dashed) andtotal oligomer populations (solid). This chart shows as elongation ratesincrease 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 plotlines. Thus, according to our model, acceleration of fibril formation kinet-ics will result in fewer, longer fibrils and a lower total oligomer populationthroughout fibrillogenesis. This implies less cytotoxicity for acceleratedfibril elongation and a dramatic increase in potentially toxic oligomer andfibril 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 nucleationrates increase total number of fibrils present increase. However, the totaloligomer populations are predicted to monotonically decrease with increas-ing nucleation rates. Here increased nucleation rates correspond to thickerplot lines. Thus, according to our model, acceleration of fibril nuclei forma-tion kinetics will result in increasing fibril numbers of shorter lengths, buta lower total oligomer population throughout fibrillogenesis. This impliesgreater cytotoxicity for accelerated fibril nucleation, should fibril densityincrease cytotoxicity. Conversely, a dramatic increase in potentially toxicoligomer 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 maycatastrophe, a feature beyond the scope of the model presented here. Fur-thermore, various initial concentrations of hIAPP do not scale for a givenheparin concentration. Thus, we conclude that heparin amplifies the hith-erto trivially small reverse rates, which we have scaled out of our model. 6529 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 dependenton A. Thus we see, for later times that our approximation is inconsistentwith the observed data.   66xiAcknowledgmentsThe procedure used here to elucidate the kinetic steps in the formation of humanislet amyloid polypeptide fibrils was first suggested by Henrik Flyvbjerg in 1996 in a casestudy of microtubule dynamics. While the model of amyloid fibrillogenes presented hereis 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 byFlyvbjerg. 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 studentin Vercher's lab at the Childrens' Hospital, for giving me my introduction to the assays Ihave used to produce the data outlined in this thesis. Funding for this research has beenprovided by MITACS, the BC Childrens' and Women's Hospital, and the University ofBritish Columbia. I would also like to acknowledge Dr. Das Raibatak for his insights andmy introduction to Mathmatica.xii1 Introduction1.1 Phenomenological Modeling of The Nucleated Polymerization ofHuman Islet Amyloid Polypeptide; a Combined Experimental andTheoretical ApproachThe presence of amyloid deposits is indicative of the diseased state in many degenerativedisorders. Normally soluble precursor proteins form deposits as oligomers and amyloidfibrils which amalgamate to form plaques. These fibrilary tangles and associated oligomersare thought to be responsible for decreased cellular function and cytotoxicity (toxic celldeath). 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 anabrupt formation of fibrils which does not readily fit current nucleation polymerizationmodels. Here we review several models proposed for the nucleated polymerization of amy-loid fibril formation and demonstrate their inability to accurately describe the formationof fibrillogenesis for human islet amyloid polypeptide (hIAPP) in- vitro. We then outlinea generic procedure for determining the kinetic steps of amyloid fibril assembly, whichexploits the observation that kinetic profiles for various initial concentrations of hIAPPmay be resealed by their initial precursor protein concentration and characteristic timeto completion, resulting in approximant collapse of data to a single kinetic profile. Thisproperty 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 wehave determined that hIAPP fibrils are assembled through a primary nucleation event andfacilitated by a rapid secondary nucleation event that is fibril dependent. As a bonus, theanalysis predicts the formation of oligomers as a natural consequence of fibril formationand describes their lifetimes. Thus, we have elucidated not only the complete kinetics offormation 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 researchersto "see" the effects of altering formation kinetics at specific steps of fibril and oligomerspopulations. 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 phasewhere 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 ofmonomers in polymer form over time. However, single nucleated polymerization alone isinadequate to describe the kinetics of amyloid fibril formation due to the rapid rise timeobserved in in- vitro experiments. Specifically, in hIAPP, the rise time for polymerization1is considerably shorter than the lag phase. The observed time scale of the rise time relativeto the lag phase in hIAPP and other amyloid fibrils suggest a secondary means of fibriltip formation, on which fibrils may grow and elongate. Several models have been sug-gested to describe this process. Notably, branching, fibril breaking and fibril dependantnucleation 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 diseaseis poorly understood. The necessity for understanding the kinetics of assembly is relevantnot only to the physical chemistry of amyloid fibril formation but also to the pathology ofamyloidosis as well. Nucleation rates, specifically secondary nucleation process, may bedrastically altered by environmental factors and may contribute to the pathology of thedisease (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 constructionwhich competes and is distinct from fibrillogenesis. Here we show that persistent oligomerpopulations are a natural consequence of fibril formation in hIAPP and we give a globalanalysis for their time course, size in monomeric units and relative abundance. In thisanalysis 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 allowsthe 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 hIAPPfibrils. These predictions are in agreement with observation. Furthermore, the analysis ofthe data reveals a natural source of potentially toxic oligomer species and predicts timecourses for their assembly and lifetimes. These predicted time courses are shown to becongruent with experimental observation.1.2 AmyloidosisAmyloidosis is the diseased state associated with abnormal, insoluble, amyloid depositsin organs and tissue. Insoluble amyloid deposits from normally soluble precursor proteinsin the form of oligomers, fibrils, and amyloid plaques are pathogenic features in manyknown diseases. Numerous species of amyloid deposits are known to exist, stemming fromover 20 categorical precursor proteins (8,9,10). The presence of these deposits may beclassified into two groups of amyloidosis, systematic and localized (8).Systematic amyloidosis encompasses diseases where precursor proteins circulate throughan organ or system. For example, chronic inflammation associated with N-terminal frag-ment deposits in rheumatoid arthritis and tuberculosis is driven by acute phase serumamyloid proteins, a class of apolipoproteins. These apolipoproteins are secreted during2acute 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 theproduction of the precursor protein is created on-site in the organ. Examples includeAlzheimer's disease and type II diabetes where amalgamations of fibrous deposits areindicative 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, isletamyloid polypeptide (IAPP) fibrils aggregate in plaques in the extracellular regions sur-rounding pancreatic islets. Both animal studies which show, amyloid deposits in the isletsbefore the onset of hypoglycemia and abnormally low blood sugar levels, and in- vitrostudies, which show the formation and presence of IAPP fibrils are cytotoxic, imply thatamyloid deposition plays a causative role in type II diabetes. Despite this evidence ofcausality and the well documented nature of amyloid deposits, the kinetics of formationof these deposits is largely unknown.1.3 Islet Amyloid Polypeptide1.3.1 Secretion of IAPP Mirrors That of InsulinIAPP 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 oftype II diabetes and is present in nearly all instances of the disease. Though discoveredover a century ago (14), the insoluble nature of islet amyloid posed serious impairmentsto the isolation and analysis of its components until the late 1980's when Westermark andothers (15,16,17) successfully isolated and identified the unique amino acid sequence forthe 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 of1/10 — 1/6 (23,24). However, the clearance rates of insulin and IAPP are dissimilar in theplasma. IAPP, like C Peptide, the peptide created when pro insulin is split into insulinand 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 inclearance rates, as well as recorded IAPP plasma levels following stimulation of 0-cells forpeptide release with glucose or non glucose response suggest a physiological ratio of IAPPto insulin closer to 1/100 (23,26,27,28). This level is in good agreement with in- vitroexperimental observations (29). Furthermore, IAPP levels have been shown to mirror thesecretion of insulin in humans, and factors regulating insulin levels have also been shownto 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 experimentalevidence indicates increased levels of IAPP in populations of obese individuals (21,23,24).3A 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 notedin pathologies associated with depleted insulin levels, such as 0-cell function depletionassociated with type I and II diabetes(19,20,22,27,31,32,33). Individuals with diminishedglucose 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 biologicalfunction of the peptide.1.3.2 Physiological Function of Islet Amyloid PolypeptideIAPP 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 glucoseuptake, specifically in skeletal muscles (37,38). Furthermore, in-vitro studies of isolatedislets 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 insulinsecretion and glycogen clearance rates.Intriguingly, IAPP has been correlated with food intake and weight control. Injectionsof an IAPP antagonist into the circulatory system of the brain has been demonstrated toshow 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).42 Cytotoxicity of Islet AmyloidIn non-diseased individuals, IAPP secreted with insulin by ,3-cells is maintained in thewell regulated intracellular environment. In this environment, IAPP is maintained in amonomeric 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 havebeen 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 arereadily accessible in this extracellular fluid. In-vitro assays with thioflavin-T demonstrateheparin accelerates the formation of IAPP fibrils with similar pH levels (52), Section [7.4.1]. Taken together, these observations suggest an extracellular environment highlyfavorable for the production of toxic amyloid deposits of IAPP.2.1 Fibril ToxicityMany 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 oftheir 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 haveshown that increased fibril formation does not increase cell death. This avenue of researchhas 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 ToxicityOligomeric species of IAPP have been noted alongside both growing and mature fibrils(53), Section [5.3]. These oligomers have been implicated as the cytotoxic componentof amyloid formation (53,54), Section [5.5]. However, the nature of formation of thesespecies is currently the subject of much debate. Models have been proposed that implythat 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 fibrilformation as measured by anisotropy assays. Other studies have suggested that oligomerformation is a process distinct from and competitive with fibril formation (36). This modelsuggests 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 fornucleated fibrillogenesis of IAPP is presented which demonstrates that persistent oligomerformation is a natural and necessary consequence of nucleated fibril formation, givencertain qualitative features of the kinetic profile of fibril formation.52.3 Need For a Unified ModelGiven the contradictory evidence for fibril toxicity cited above and the growing body ofevidence of the role of oligomeric cytotoxicity, an exact understanding of fibrillogenesisof 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 elucidatethe underlying kinetics of IAPP fibrillogenesis and, thereby, help to clarify conflictingobservations found in the literature. Furthermore, a unique kinetic description for theself assembly of amyloid fibrils and their associated oligomers may help in the search forpotential therapies for the early detection and control of type II diabetes.63 Experimental MethodsHistorically, the "starch-like" appearance of fibrous deposits found in diseased tissue weretermed amyloid by Virchow in 1855, after observing blue staining of acid-treated fibrils byiodine. 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 stainamyloid deposits. The histological benzothiazole dyes thioflavin S (Th-S) and thioflavinT (Th-T) along with the diazobenzide sulfate dye, and Congo red were found to stainamyloid in a number of pathological settings (56). Furthermore, several other fluorescentand 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 fibrilswhere 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 inhigh 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 thebasis for assays using Th-T fluorescence shift as a measure of monomer mass in fibril form.3.1 Thioflavin TThe characteristic behavior of Th-T of exhibiting a fluorescence spectral shift withoutaltering emissions spectrum implies Th-T's 115-nm red shift in fluorescence is associatedwith a change in the ground state energy upon binding to amyloid fibrils, and does notresult from changes in the excitation state energies. The later is implied by a shift inthe emission spectrum (56). Regardless of these observations, the nature of the observedbehavior of the Th-T amyloid fibril complex remains to be elucidated.The unknown nature of the amyloid fibril Th-T complex makes exact quantificationof the relationship between fluorescence and amyloid fibril all but impossible to obtain.Nevertheless, the consistency in fluorescence shift associated with the complex is usefulfor determining relative amounts of amyloid fibril formed over time, as well as relativerates of fibril formation for various initial concentrations of IAPP monomer. Fortunately,relative rates of fibril formation for various initial monomer concentrations is all that isrequired, in the case of hIAPP, to determine the unique kinetics of assembly in fibrillo-7genesis. Given these considerations, this thesis utilizes a Th-T assay as described in theMaterials and Methods section as a basis for modeling hIAPP fibrillogenesis.3.2 Dot BlotsWestern Blot assays have been traditionally used to determine the presence of specificproteins 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 andexamined 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 advantageof not requiring gel electrophoresis or the complex blotting procedures associated withthe other assays. However, Dot Blots give no information about the size of the proteinpresent. This thesis reports results from Dot Blot assays used to establish the time orderof formation and corresponding lifetimes of oligomers during and after fibril formation ofIAPP, as described in the Materials and Methods section.3.3 Cell ToxicityCultured 0-cells were exposed to various initial concentrations of hIAPP and were screenedfor viability over time by measuring cellular metabolism. These assays give a roughestimate of when toxicity began and help elucidate any dependence of cytotoxicity oninitial IAPP monomer concentration.3.4 Solubility AssayTo test for monomer and soluble oligomer over time a filtration assay was utilized. Thisassay 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 thatall monomers are consumed as fibril formation ceases, corresponding with a plateau inTh-T fluorescence. Furthermore, this assay allows for the quantitative tracking of solublemonomers and small soluble oligomers over time. It can be matched to model predictionsto help support the theory presented here.3.5 Fibril Length DistributionsFibril length distributions for hIAPP were measured using atomic force microscope (AFM)by Peter Marek, State University of New York, Stony Brook, New York. The distributionused was from mature fibrils formed under conditions similar to the ones used in fibrilkinetics assays in this thesis, supplementary material (53). These fibril length distributionsfor mature hIAPP reactions in- vitro were compared to model predictions of fibril lengthdistributions at t oc. Specifically, this data allows for an alternate means of determining8two key model parameters. One of these parameters is found in a way that relates directlyto the kinetic Th-T data. The other relates to a specific stage of nucleation This isinformation that is not available from the Th-T data and gives validation to our theory.94 Materials and MethodsTo determine the kinetics of self assembly of hIAPP in- vitro a standard Th-T assay wasused to produce kinetic profiles of fibril formation for various initial concentrations of themonomeric precursor protein. This data was systematically scaled and analyzed in thecontext of a generic nucleation-dependent polymerization model, based on mass actionassumptions, which allows for secondary nucleation to occur through one of several path-ways. Specifically, kinetic data for hIAPP fibril formation was collected from in- vitroexperiments in Dr. Bruce Verchere's laboratory (Childrens' Hospital, Vancouver BC.).Fluorescence shift associated with the binding of Th-T to amyloid fibrils, but not betasheet, monomer, or oligomeric species (56) was observed to capture monomer of hIAPPin fibril over time, for concentrations of monomers ranging from 10,uM to 100AM.4.1 Protein PreparationSamples of lmg hIAPP, were obtained in powdered monomeric form from (Bachem.com ).The monomeric hIAPP was dissolved in 1000/2L Hexafluoroisopropanol (HFIP) and aliquotedinto 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 storedat 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, alongwith "Blanks" (DMSO, Th-T, and buffer) were used as controls. Measurements weretaken over a 20 hour period using a fluorescanner.4.2 Fibril Elongation KineticsFibril 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 themature vortexed reactions and monitored with a Th-T assay. The first 600 seconds of theelongation kinetics were analyzed, assuming a constant number of fibril ends were presentduring this time.4.3 Soluble Oligomer Filtration AssayFiltration 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,10thereby 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 solutionwas spun-down at 8,000rpm for 10mins. The membrane was then washed with distilledwater. Washing was carried out by flushing the membrane surface with water. Afterwashing the filter was spun-down for 4 minutes at 10,000rpm to rid the unit of residualwater. The water was then removed by pipetting. This procedure prepared the filter unitfor use with IhAPP. 101/L of the desired concentration of hIAPP was then mixed with50/11 of PBS just prior to filtration. This step compensates for the 104 hold up volumeassociated with the units. The sample was centrifuged at 10,000rpm for 6 minutes, untilall the solution had passed through the membrane. Blots of 2/./I., samples were placedon girded nitrocellulose membrane at 15 minute intervals for the first hour. Data pointswere then taken every 30 min for 2 hours then every hour, until 4 hours had passed. Onefinal data point was collected at 24 hours. The membrane was allowed to air dry andwas blocked with a 10% non-fat milk solution at 4°C overnight. The membrane was thenwashed 3 times in a 1X TBST solution for 5 minutes. Primary antibody was then dilutedin a 5% non-fat milk and 1X TBST solution. The membrane was then incubated underthe 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-fatmilk TBST solution. The membrane was then covered in this solution and incubated for1 hour with gentle shaking. The membrane was then washed thee times in TBST solutionfor 5 min each washing. Substrate solution was then added and the images were developedand analyzed. The remainder of each sample was frozen at —80°C at the time of samplingfor later use.4.4 Oligomer Time Course; Dot BlotsOligomer time course assays were conducted using Dot Blots as described in the filtrationsection above, with two distinct differences. First, no filtration was used to separatenon-soluble oligomers and fibrils from the solution prior to blotting the nitrocellulosemembrane. Secondly, antibodies for oligomer staining as opposed to monomer antibodieswere used. Furthermore, time points were taken at 15 and 30 minutes, at 1, 2, 3,6,8,10hour time points. A final time point was taken at 24 hours. The Dot Blots were washedand incubated as above and developed within 24 hours.1 15 Experimental Results5.1 Kinetic Profiles for FibrillogenesisKinetic 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 ofthe polymerization are noted from these kinetic profiles. 1) A pronounced lag phase ofconstant fluorescence where oligomers are formed but little monomer mass is in polymerform. 2) A rapid rise in fluorescence corresponding to an abrupt elongation of polymersthat quickly accelerates. This elongation phase is notably shorter than the precedinglag phase over all concentrations. 3) A plateau in fluorescence that is approached expo-nentially, corresponding to a cessation of polymer formation. Furthermore, the maximafluorescence ./61,, monotonically increases with increasing initial concentration of hIAPP.Conversely, the time required to reach the plateau in fluorescence monotonically decreaseswith increasing initial hIAPP concentration.5.1.1 Scaled Data CollapseScaling of the experimentally observed fluorescence, A(t), by its maximum value, Ate ,and t by a characteristic time, to(A09 ), resulted in approximate data collapse, as shownin Figure (2). Here we have arbitrarily used the time to 20% maximum fluorescence forto(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, auspiciouslyclose to an integer. Thus, we note-y^2,in the relationshipto(A,,) cx Acc—r ,and we may now impose this power law, a strict mathematical condition, on any modelwe use to describe the accumulation of monomers into fibrils. Furthermore, the collapseof all data sets over the given concentrations suggest the kinetics of hIAPP are largelyindependent of em the initial monomer concentration. Therefore, all time-series data isapproximately described by a single function,A(t; Aco ) = Acof [t/to(A00)],resulting in the power law12to oc A o2 ,for any model used to describe the data, where again 'y is determined by the data asin Figure(3), see Appendix (11.1) for details.5.2 Elongation KineticsMonomeric hIAPP solutions of 25, 35, 50 and 100,uM were added to solutions of maturevortexed reactions made from 50AM initial monomer concentrations hIAPP and weremonitored with Th-T assay, as described above in Materials and Methods. The first 600seconds 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 monomeraddition.5.3 Oligomer Time CourseDot Blot data from oligomer time courses assays suggests oligomers form quickly in aconcentration dependent manner and persist for at least 24 hours Figure (5). This timespan is considerably longer than the time of formation for fibrils as fluorescence from thecorresponding Th-T assays plateaus within 2 hours for the lowest concentration observedto produce fibrils. This observation is in agreement with similar assays that suggestoligomers of hIAPP persist on time scales longer than that of fibril formation.5.4 Soluble Monomer and Small Oligomer Time CourseUnfortunately, all assays for soluble monomer conducted by the author, were unreadableand produced no discernable data. However, assays carried out by Miranker (6) using lightscattering to detect soluble monomer concentration show soluble monomer concentrationapproaches 0 as fibril formation plateaus, Figure (6). These assay results are of particularimportance 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 supportsthe assumption that most monomer mass is contained in monomers or fibrils. Likewise,this observation, especially at higher concentration, see Miranker and Evan (7,B), suggestthat off-pathway kinetics which produce aggregate distinct from fibril formation play aminimal role, if any, in the kinetics of hIAPP fibrillation.5.5 CytotoxicityResults from cell viability assays show an increase in cell toxicity as a function of initialmonomer concentration, as well as time exposed to the fibril formation process, (data not13shown). 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 formationand continue to be present during fibril formation and after completion. This observationcoupled 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 theoligomers/aggregate populations or a result of maturing, (but not mature) fibrils. Thelast option seems unlikely.146 Models of IAPP Polymerization6.1 Simple Nucleation Dependent PolymerizationThe Nucleation Dependent Polymerization model (NDP) is the basic model that describesthe nucleated polymerization of a self assembling polymer. Monomers are assumed toquickly associate and dissociate, to form short-lived species, denoted oligomers. Theseoligomers 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 describingactin(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 NDPmodel and the use of scaling, as presented by Flyvberg (3) for microtubule dynamics isgiven here in the context of IAPP fibrillogenesis. Following Flyvbjerg we letc(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 diagramshown in Figure(7), result in the following system of differential equations for nucleatedpolymerization. For the first oligomeric species we havedc1 = fen° — h^+ b2c2 — di c1 .Here fen° represents the rate of formation of c 1 as no monomers bind together to formthis species. This oligomeric species ci is also formed through the term b2c2, representingthe decay of the next oligomeric species c2 into the ci species. The loss of ci oligomersto c2 oligomers where n1 monomers bind to Cl forming c2 is given by —ficnici. Finally,the c1 oligomers may completely disassociated into monomers at a rate dice. For the itholigomer we havedt15dci =^1ci-1 — ficn' ci — bici + bi+ ici+ i — dici,^for 2 < i < k.dtThe formation of the ith oligomer from the i — 1 oligomer occurs by the addition of ni_imonomers to the ca _1 species, represented by the term f2 _10--- Ic2_ 1 . The ith oligomericspecies is also created at a rate b z+ic,+1 through the loss of ni monomers from the ci +ioligomer species. Loss of the ci species proceeds through the loss of ni_ i monomers fromthe ith species at the rate bi , and is given by the term V,. Furthermore, ci oligomers maybe lost by the complete disintegration of the ith oligomer into monomers via the termdici. The equation for the change in concentration of stable nuclei is given bydvdt ^kCnk Ck.This equation represents the formation of stable nuclei v by the binding of nk monomersto the ck oligomeric species, the largest oligomer allowed in the model.Furthermore, the total change in mass of monomers in fibril form, discounting the monomerin oligomer and nuclei, is given bydMdt fk+lcv.Conservation of mass, under our assumptions givesdc dMdt^dtimplyingc(0) = c(t) + M(t),andM(t)_ 1^c(t)^A(t)Moo^c(0)^Ao„ •Here, we have assumed that the mass of monomers in nuclei and oligomers is minimalcompared to the mass in monomer or polymer form. This assumption is supported bysoluble oligomer assays Figure (6) and Miranker (6).166.1.1 Application of The Characteristic Scale to The ModelThe 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, afterthe scaling must be set to zero, as described in (Experimental Results). We now enforceour scaling and the power lawto 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 variablestt = -, and^to cc co ry.Therefore,tAc;' ,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 bythe system. Likewise,v Mv —, and /C//^,cowhere p, is a quantity to be determined by the system. Substituting the above intothe model and requiring that c, not appear explicitly in the equations reduces the modeltodci dt^f-= oc27 —dc,dt^— fze cz,dvf=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).dM17The observation that the kinetic profiles of hIAPP fibril formation follow a systematicscaling and the implication that the initial monomer concentration is immaterial to thekinetics has allowed us to reduce the generic NDP model to one that uniquely describesthe 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 byour scaling is the only plausible candidate.Fits of the scaled NDP model show a systematic error between the fits and the dataFigures (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 basedon the NDP model and many additional off-pathways, which form oligomers through aprocess distinct from fibril formation. In this thesis we will examine some of the moreprominent models in the literature. The power law discerned from the kinetic data willbe used in each case to constrain the models. As we will see, the application of scalingand the power law to these proposed models will have a dramatic impact on their abilityto describe the fibrillogenesis of hIAPP. Finally, we will present a simple dual nucleationmodel that allows both a primary nucleation, as described in the NDP model, with theadded feature that fibrils promote a secondary nucleation event.6.2 The Three-Stage Kinetic Model of Amyloid FibrillationA model similar to the NDP model by Lee et al. (33), has been put forth to describe theformation of amyloid fibrils in various disorders in a unified manner. However, throughthe "lens" of scaling and the use of elongation kinetics, we will see that the model reducesto 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. HereF denotes the fibril concentration and fibrils may elongate via addition of any oligomerto a fibril by the termE c3 F.3=1The concentration of fibrils F is similar to the number of nuclei formed in the NDP modelabove, and will be treated as such. Therefore, the assumption that negligible monomermass is contained in oligomers and nuclei is extended to include the shortest of fibrils,18those 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 anuclei, this assumption seems plausible. The need to include these shortest fibers in thenegligible mass pool assumption stems from the constraint that a fibril may only form bythe addition of a monomer to a nuclei. Removal of this constraint should remove this newassumption. Here we precede with the convention used in (33). The rates of attachmentand detachment for the ith oligomers to fibrils are given by /i, and si respectively. Rewrit-ing the model in the notation of this thesis, given the assumptions here and in Section[6.1] leads to the differential equationsdcdt =- — fk±icF + soF,dc1^ = foc2dt f1C1C b2C2^SiF —dczdt- =^— fzczc bz+i ci+ i — bici + sz F — liciF, for 2 < i < k — 1.The change in the ck species, and fibril tip populations aredckdt = fk—ick—ic fkckc bk±i F — bkck + skF — IkckF,anddFdt = 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 bydM^-=dt^fk± icF +k kF — bk+ 1F.j-=-1^j=0Systematic scaling of this model Appendix 11.3) results in the following system of equa-tionsdc =dcir 2^rdt = oc — ficci — 11 c1F,dc, dt fz _icca _i — fi ccz — 1,c,F,19dt^fk-FicF +dMdckdt^fkck_lc — fkckc —dFdt^fkckc,andElongation kinetics as described in section [7.2] suggest that fibril growth is mediatedby monomer addition implyingliciF =- 0, for 1 < i < k.Furthermore, the persistent population of oligomers observed experimentally cannot beestablished if the elongation terms for oligomeric species are not 0, or trivially small, Fig-ures (12,13). The model now becomesdcdt — fk+ icF,dcidt^foc2 — heel,dci dt^— fcci,dt = fk—iek-1c — fkckc,dF -= fkckc,dtanddMdtdtfk-FicFTherefore, the model as proposed in (33) results in a special case of the generic NDPmodel 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 oligomersto form the next oligomeric species, thus allowing the formation of each oligomeric speciesthrough any number of additional monomer units results in the NDP modeldck20dcdt =- — fk+ic-F,dcidt- foc27 — hc7ei,de,dt = fz-1 c2-1 — fic7dckdt — fk-ick-ic7 — fkckc,dF f=dt^kekC7anddM^defk±i cF =dt dtThus , we observe that the "Three-Stage Kinetic Model of Amyloid Fibrillation" reducesto the NDP model, under the systematic scaling and elongation kinetics implied by ourdata. Next we examine a NDP model that allows for off-pathway aggregates to formdistinct from fibril formation.6.3 Nucleation Dependent Model with Off-Pathway Kinetics for Aggre-gate FormationWe 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 foundalongside 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 wesee a problem only the scaling 7 = 1 will satisfy this assumption in the model. The modelimposes an off-pathway that allows oligomers to form in competition with the oligomersthat lead to nucleated fibril formation. These off-pathway oligomers are referred to asaggregates. Powers and Powers noted in (30) that at some critical concentration of initialmonomers scaling of data for fibril formation will become impossible. In the words of theauthors"We also find this mechanism has an especially striking feature: although increasing pro-tein concentration generally cause simple nucleated polymerization to reach completionfaster, they cause nucleated polymerization with off-pathway aggregation to reach com-pletion more slowly when the protein concentration becomes too high."21We find this statement to be untrue for a specific subsystem of nucleated polymerizationmodels with off-pathway aggregation. We will demonstrated bellow that this model doesallow 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 areformed during the nucleation process. Furthermore, the enforcement of the scaling lawremoves the back rate terms, see Appendix (11.1,11.2). We are left with a population ofoligomers and aggregates that are monomericaly the same. Thus, no distinction can bemade between the off-pathway aggregates and the on-pathway oligomers that are left afterthe 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 applyhere. However, Powers and Powers have imposed the arbitrary scalingto a A007 ,^for -y =- 1.The model allows nuclei to form through the addition of one monomer at a time tothe preceding oligomeric species, for example, the c, species is formed by the binding ofa single monomer c to one of the ca_1 species at some rate fi_i. The first oligomer beingformed by two monomers binding to form Cl. Simultaneously, the model allows aggregatespecies z1 to form by the binding of two monomers to form the first aggregate at somerate a l . These aggregates may then grow by the addition of one monomer at a time toform the next aggregate species. Furthermore, aggregates and oligomers are allowed tolose monomers at any stage at some back rate /3, and ba respectively. Using the notationin Section [6.1] and introduce the new variables and parametersz,(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 diagramshown in Figure(14) result in the following system of differential equations for nucleatedpolymerization of hIAPP with off-pathway aggregates. Here, as in section [6.1], we findthe equations for the oligomeric species to bedeldt- = foc2 f1eC1 b2c2 — —anddcadt =^— ficca — bZci + for 2 < i < k.22Likewise, we see the formation of the aggregate species aredz1^ — ce0C2dt - ai zi c + 132z2 — ,(31 -dzidt =^c — az zic —^+ i3i+i zz+i , for 2 < i < m — 1.Furthermore, the change in the largest aggregate species if found to bedz, —dt^ —We also obtain,dvdt^fkcnk ek — bvv-This equation represents the formation of the unstable nuclei v by the binding of nkmonomers to the ck oligomeric species, the largest oligomer allowed in the model, andallowing decay through the term —bvv. Furthermore, the total mass of monomers in fibrilform, 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 infibrils, which is proportional to the number of fibril ends present. Here we have assumedthat the number of fibril ends present is equal to the number of nuclei formed. Since nucleiform slowly and elongation is rapid this assumption is a reasonable one, as no breakingof fibrils is allowed in the model. We now introduce the additional scaled variableziY for 1 < i < m,here Y is to be determined by the system. Substituting the scaling variables into themodel and dropping the -s givesdci— =dt^A[foc2 - fig ,de,dt ^A [fi_ici_ic — ficicd, for 2 < i < k,dvdt = A[fkckc],23anddM =dtFurthermore, the rates of change for the off-pathway aggregates becomedt^A[aoc2 — ai zic],dzi—dt =^— az zz c], for 2 < i < 7n — 1.The change in the largest aggregate, in monomer units, is now given bydz,^ —dt^A [ani_ z„,_i c] •Therefore, we find, given our data constraints, that the "Nucleation Dependent Modelwith Off-Pathway Kinetics for Aggregate Formation" reduces to the NDP model whereaggregates of the same monomeric size as oligomers are produced. Furthermore, theseaggregate populations are ever increasing, and given that no "bottleneck" of nucleationexist along the aggregate pathway it is likely that such aggregation events would effec-tively compete for monomers, especially during early stages of fibrillogenesis. The resultis a notable mass of monomers being sequestered into aggregates. This would contradictour assumption that all appreciable monomer mass is in either fibril or monomer duringthe 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 sizein monomer units as the persistent oligomer population will hold, except perhaps for thesmallest aggregate species, see Appendix (11.2).6.4 A Generic Nucleation Dependent Model with Off-Pathway Kineticsfor Aggregate FormationWe now rewrite the nucleation dependant model with off-pathway aggregates whereoligomers and aggregates are allowed to form by any number of monomers joining toform the first species. Subsequent species are allowed to form by binding to any numberof monomers to form the next largest species. No constraint is imposed that requiresoligomers and aggregates to be made of the same number of monomer units. However,aggregates may only form from monomers and other aggregates, likewise oligomers mayonly form from monomers and other oligomers. We now introduce the additional param-eter24a, =number of monomers added to zi 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 givenbydt = foe° — f1C1Cn1 + b2c2 —dci- =dt^fi_ici_icTh' - ' — ficicn' — bici bi+ici+i,^for 2 < i < k,anddZi- =dt^ao^+ /32z2 -dzidt^—^— Nizi +13i-ozi+i, for 2 < i < m — 1.The change in the largest aggregate species allowed isam-1 — m _Zrn •dt^rnWe also obtaindvdt- = ikcnk ck — bvv.Furthermore, the total mass of monomers in fibril form, discounting the monomer inoligomer, aggregate and nuclei, is given bydt = 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],dtdci - =—dtdvdt ALAckcl,for 2 < i < k,dc1dzn,dM25and=dt^A Ffk± ivc] .Furthermore, the rates of change for the off-pathway aggregates becomedzl = A [aoc" —^c,),J ,dtdzi -= A^— ai z, ,dt for 2 < i < m — 1.The change in the largest aggregate, in monomer units, is now given bydzm = A [am_iz„,,_]. c-Y].Thus, we see that given fibril formation data that scales systematically, as is in the casein hIAPP fibrillogenesis, off-pathway kinetics are restricted, mathematically, in the sameway as are the on-pathway kinetics of nuclei formation and no distinction can be madebetween such events Figure (16). Furthermore, the time scale of formation of aggregatesis likely faster than the fibril formation time scale due to the "Bottleneck" step at thenucleus. Therefore, aggregate populations can either be assumed to be in equilibrium withthe monomer population on the time scale of fibril formation, implying fibril formation willnot scale for high initial monomer concentrations, or the back-rate terms in the aggregatekinetics must be trivially small, while aggregate sizes in monomer units is restricted tomirror oligomer sizes. We therefore state that off-pathway kinetics for aggregate formationis unlikely a pronounced feature of hIAPP fibril formation and is redundant in this class ofmodels, given the constraints provided by our data. The presence of such terms will onlygo to alter formation rates in data fits, and give no new insight into the structure of theseaggregates. Therefore, models with off-pathway aggregates may fit the kinetics profilesof fibril formation that scale under all initial monomer concentrations, which result infibril formation. However, they introduce an unneeded pathway, from a mathematicalpoint of view. We also note that, under scaling, populations of aggregates will increaseuntil the monomer population is depleted. This competition of aggregates vs. fibrils formonomers could have a notable impact on the total oligomer/aggregate populations, ifaggregate formation rates are not trivially small, resulting in the production of a notableoligomer/aggregate population. However, plots of the sum of normalized monomer andfibril mass populations suggest that oligomer and aggregate populations are trivially smallfor all times, Figure (6). Furthermore, this model, like all single nucleation models, doesnot readily fit the kinetic profile observed for hIAPP and exhibits a systematic error indMdt26these fits, Figures (9,10). Given these observations, we propose a model for the nucleatedpolymerization 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 newnuclei (fibril ends) on which elongation may occur. The scaling law observed from thedata collapse and the corresponding power law are then employed to reduce the rathergeneric model to the unique model, under our assumptions, that describes the data.277 Generic Nucleation Model with Fibril-Dependent Sec-ondary NucleationUsing 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 modelreduces to Figure (18)dci dt = foc2' — fic7c1,dc,- =dt^ci_i — fa c-rez , 2 < i < k.However, for some species class j we havedcidt- foc27 — fic-Y ci — f c(71) M,^if j = 1,ordtdcj^f^ci —^cj — fie(^y-1)^, for some 2 < i = j < k,anddvdt^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, howeverdata fits clearly separate these possibilities. Fits of the model to the data give consistentestimates 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 sequesteredinto 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 existdue to these terms. Typical residual oligomer populations and their rates of formationare shown in Figures(21,22) respectively.7.1 Equation Number and Initial Formation KineticsIn 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-dM28ter. 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 alog log scale, giving the order of initial fibril formation and thus k. Here we have chosento evaluate the model at early times giving a first order solution for the scaled model int. This first order solution is then fit to the data for early times, just as the initial risein Th-T fluorescence is observed and before fibril kinetics accelerate, implying secondaryeffects have become impactful. As a check of our first order solution for the number ofnucleation equations k we then plot both the observed time to 50% completion t50 versusthe predicted time to 50% completion tp50 on a log log plot, Figure (23). Here, as withto our scaling time, t50 is an arbitrary choice and the analysis follows through with anychosen time of completion. We now determine the initial formation kinetics of hIAPP byapproximating the model as a first order system close to t = 0, withdeldt^foc27 ,dcidtdv =dt^fkc7ck,dMdt^fk+icv.Solving the equations successively leads tofoCO2-Yt,c(i + 1) ''ci =I-L=0^°^t ,We then obtain(k+2)7=^fi C(V+i )!andfor 2 < i < k.r_rk.+1^c(!c+2)-y+1^pc 2113=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 thusobtain the nucleation size N = (k + 2)7, where, again, y is determined by the fit of the logof to(A,) vs. the log of (A,), as described in [5.1]. It may be read off as the slope of the29best fit line in Figure(3), Appendix (11.1). Plots of the observed time to 50% completionvs. 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 KineticsMonomeric hIAPP solutions of 25, 35, 50 and 100AM were added to mature vortexedreactions made from a 50pM solution and monitored with Th-T assay as described abovein Materials and Methods. The first 600 seconds of fluorescence data show no lag andsuggest fibril elongation begins swiftly. Furthermore, the rate of change in fluorescenceis approximately constant over this period Figure (4). Assuming elongation occurs byaddition of oligomers of monomer size nf to fibril ends F at the rate fe , and that allappreciable monomer mass is in either monomer or polymer form. We havedc- dt fe Fenf ,anddM de dt^dtGiven the assumption that the fluorescence A(t) is a measure of the mass of monomer infibril M(t), we may writedA= fe Fenf •dtA plot ofdAlog Ti t vs. log(co),gives a slope of nf = 1 as shown in Figure(24). Thus we find, elongation of hIAPPfibrils is well described by monomer addition. The same elongation kinetics were foundmonitoring monomer consumption directly by Miranker for seed concentrations rangingfrom 177M to 100,uM added to 50/2M vortexed reactions (6).7.3 Fibril Length DistributionThe 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 tp ( 1 ;30Given 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 writefo c') p(1; oo) dl = 1.Furthermore, the fibril length our distribution depends on is a relative one and is normal-ized to 1J0°9 p(1; oo)1 dl = 1.Given the rate of formation of fibrils at any time tdvdt = fke7ck + fjc(7-1) ciM,and the corresponding "velocity" of fibril growthfk-Fic(t)•We can determine the length 1, at time t = oo, of a given fibril that formed at timet001 fk+1 J c(e ) de .Using the rate of formation of fibrils, the fibrils of length / to / +dl at time t oo is simplyp(1; 00)d1 —dvdt dt.Therefore, we havep(1; Do )^f kc'Y ck + f 3 c(7- 1 ) e3 mfk-Fic(t)This distribution is shown over data points collected by Peter Marek in figure (25). Theplot of the length distribution at large time was generated by fitting the scaled nucleationmodel with fibril-dependent secondary nucleation to scaled data for an arbitrary initialmonomer 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 timesone simply shifts the horizontal axis to the left.317.4 Impact of Parameter Variation on Fibril and Oligomer PopulationsVariations in the rates of formation of fibrils and nuclei result in predictable changes inpopulations of total oligomers, number of fibrils formed, and, obviously, the time scaleof fibril formation. These observations are summarized in Figures (26, 27). These plotssuggest that blindly slowing or inhibiting fibril elongation rates is not the most prudentcourse for reducing cytotoxicity of hIAPP fibrillogenesis, given that total oligomer andfibril 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 formationrates has a similar effect on oligomer populations. However, fibril densities are impactedin the reverse manner. Therefore, lowering fibril nuclei formation rates will result in anincreased 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 screeningpotential therapeutic drugs for type II diabetes treatment. Specifically, kinetic profilesin the presence of inhibiters may be analyzed in the context of this model to detail theexact mechanism by which the inhibitor affects the fibril formation process. Furthermore,the resulting impact on oligomer population, the relative number of fibrils produced andthe length distributions may be determined readily using the model, given nothing morethan the kinetic profile of fibril formation, as measured by relative mass of monomersin fibril form. We note that, the observation of accelerated fibril kinetics reducing bothfibril density and oligomer populations is in qualitative agreement with assays suggestionaccelerated fibril formation decreases cytotoxicity.7.4.1 Impact of Heparin on The Formation Kinetics of hIAPPHeparin, 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 withfibril-dependent secondary nucleation suggest that both primary and secondary nucleationare drastically accelerated. However, examination of the fluorescence profiles of severalinitial concentrations of hIAPP monomer with constant heparin concentration does notproduce scalable data. Therefore, the theory presented here and its results do not applyto the observed fibrillogenesis of hIAPP in the presence of heparin. Furthermore, thelack of scaling specifically preclude the subclass of models isolated here for hIAPP fibrilformation. Thus, we may conclude that heparin results in the inclusion of non-scalableterms in any NDP model which describes it and implies that heparin acts to increase thetrivially small reverse rates of oligomer and/or fibril decay that we have removed fromour model of hIAPP fibrillogenesis on phenomenological bases. This supposition is further32supported by the observed decrease in fluorescence seen in the profiles in Figure (28) justafter maximum fluorescence is reached.7.5 Relevance to Current Biological HypothesesMany current biological arguments suggest that hIAPP precursor protein may undergo aseparate aggregate pathway that is distinct from fibril formation. This alternate pathwayhas 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 theseoligomeric species is unclear. However, onset of oligomer formation is rapid and beginswell before the first fibrils are observed. Furthermore, experimental evidence suggests alife-time of the oligomeric species of an order of magnitude or greater than the growthkinetics of fibrillogenesis in IAPP, Figure(5). The above model gives a natural candidatefor these oligomeric species, without the need for a separate off-pathway for aggregateformation, namely the oligomers that are "trapped" after the depletion of the monomerpool, unable to grow or decay. It should be noted that we have determined that neitherthe off-rates, bz , nor the disintegration rates, di , are relevant to the kinetics and that nonet 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 thatis too small to be detected by our assay. Simulations suggest that such a small net flux ofmonomers would have a minimal impact on the results presented here, and would allowfor the decay of oligomers as some experiments imply.338 ConclusionsThe observation that various initial concentrations of hIAPP monomer result in fibrilformation kinetics in- vitro, as observed with Th-T assays, that systematically scale andcollapse onto a single profile, gives considerable insight into the kinetics of fibril nucleationand 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 justifiedby the formation kinetics under scaling, but would be considered prudent to include, ingeneral, in the modeling of nucleated polymer kinetics. For example, the back rates ofoligomer formation are often included in such models and are needed, in many cases, todescribe the polymerization event in question. However, inclusion of these terms in amodel that describes scalable data is not only unsupported, but may lead to misleadingresults for model fits. Blind fitting of models to data for hIAPP with out the removal ofnon-scalable terms, as implied by data collapse, has led to suggestions that off-pathwaykinetics play a role. While the use of scaling to reduce fibril formation models does notpreclude the existence of off-pathway kinetics, it does put rather harsh restrictions on theallowed kinetics of the off-pathway aggregates. Specifically, any off-pathway aggregate isconstrained to the same scaling as on-pathway oligomers, (except for perhaps the smallestaggregate), and will have the same size in monomer units, be formed and grow by thesame number of monomer units at each step in assembly, and are mathematically andkinetically indistinguishable.Given these observations the simplest model that accurately fits the kinetic profile forhIAPP fibril formation, and does not violate the observed scaling of these profiles, is a nu-cleation dependent polymerization model with fibril-dependent secondary nucleation andno off-pathway aggregates, as described in section [7]. Thus, phenomenological modelingof fluorescence shift associated with Th-T binding to hIAPP fibrils suggests that fibrilformation is dominated by a primary nucleation event and facilitated by a rapid secondarynucleation event that is fibril dependent. Scaling arguments and first order solutions forearly times to the described model indicate that nuclei are formed in five steps. First, fourmonomers combine to form the first sub-nuclei el, then two monomers bind to el formingc2. This addition of monomer pairs continues until c4, after which a stable nucleus oftwelve monomers is formed from c4 by addition of two monomers. Fibrils then grow byunit addition. As fibrils form, a secondary fibril-dependent nucleation occurs by additionof a single monomer to the c4 species. These secondary nuclei then form fibrils by unitaddition. 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 may34"observe" some otherwise inaccessible details of a biochemical self-assembly process."Through the "lens" of this "microscope" we can observe the kinetics of assembly ofoligomers, 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, whilemathematically rigorous, is not guaranteed to be correct. For example the NDP modelscales in accordance with the data, but does not accurately describe the formation ofhIAPP 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 conditionthat various initial concentrations of hIAPP monomer produce kinetic profiles that willcollapse to one "master" profile, when subjected to a systematic mathematically rigorousscaling. While this scaling is not perfect, it is accurate to a high degree, so that it isdifficult to discern if the small violations in data collapse are due to a systematic erroror 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 inOligomer PopulationsA crucial assumption in our model is that the mass of monomers in oligomers and nuclei istrivially 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) tothe 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 equationc(t) + M(t)^c(0).Inserting the model predictions for oligomer and nuclei population values into the ex-act equation for monomer massk-1c + noel + (n.0 + ni)c2 + +^nick +^niv + M = c(0).i=0^i=0We now observe that these additional terms are trivially small for all time. Thus, weare able to work in the scaled variable c(t)/c(0) and we are justified in assuming thatM(t)/Moo = 1 — c(t)/c(0).35Therefore, 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 ImprovementThe 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 mayobserve this by noting that if the relationshipto(Ao„) oc A0-07 ,were exact then the characteristic time too defined asA00 — 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, fromFigure (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, afeature our system of differential equations cannot describe. It is worth noting however,that this violation of too /to independent of iloo is observed by looking at later times anddoes not imply a grave contradiction to the modeling procedure presented here.8.3 Qualitative Failure of Ordinary Differential Equations to ExplainAll Observed PhenomenonAgain, 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 timeseries 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 ofour model, which says the slope of these plots are described bydt logio(A„„^dt— A(t)) = —d logioc(t) = mco fk± iv(t)•Here we have used the fact thatM(t)^1^c(t)^A(t) Moo^c(0)^Aoc,'and36dMdt = fk+lcv.We observe that fibrils cannot disappear,dvdt  - j f k, k.This observation has allowed us to write the equations for mass in fibril in a finite setof equations. Therefore, no theory based on a finite number of differential equationscan describe the observed phenomenon exactly and a model tracking both growing andshrinking fibrils and their distributions is needed. This observation implies that a moreexacting model of the fibrillogenesis of hIAPP would require partial differential equations.Therefore, we state that the theory presented here is accurate for the maximum numberof fibrils formed which is approximately equal to the number of fibrils at too , and ourmodel is most accurate up until this time tmax , where this time to maximum fibril densityis defined astmax = max[--d logio(A00 — A(t))]•dtTherefore, we note that, despite the seemingly limited amount of information in oursimple assay, we have yet to exhaust all the information about the formation of fibrilsthat is contained in the data.379 Figures9.1 Experimental Fluorescence DataAU)1 teem •^•4p. .4.•^••• • •re. •• • *we•• • •^• • •^•• •• •^•^•^*• • •• •• ***** oe••• .■^I• :^I^I1.•. 1. ,I ,^1d a Aid 4.11 .44,..."5^1 0^15^20^25^tFigure 1: Characteristic sample of the fluorescence of Th-T, A(t) (vertical axis), for initialmonomer concentrations of hIAPP from 25 to 100 iftIVI, right to left respectively. Eachdata 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 threedistinct phases of fibril formation. 1) A pronounced lag phase of constant fluorescencewhere nuclei are formed. 2) A rapid rise in fluorescence corresponding to an abruptelongation 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 ofhIAPP. Conversely, the time required to reach the plateau in fluorescence monotonicallydecreases with increasing initial hIAPP concentration.• • •38A(t; A„) = A„f[tIto (A„)]•• ..•^•.•^•z.•A(t)A„t43^4^500 (A.)Figure 2: Scaled fluorescence data of Th-T, A(t) Acc (vertical axis) versus scaled timet/to(iloo ) (horizontal axis) . Each data set has been divided by its maximum recordedflorescence, 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 toreach 20% maximum fluorescence, A. This approximate collapse of data for all assayedinitial concentrations of hIAPP allows us to form a model for the mass of monomers inpolymer form that is independent of the initial concentration of monomers. Furthermore,this data collapse, given certain assumptions, allows us to work with scaled variables. Wehave effectively removed any uncertainty in initial monomer concentrations from our dataand subsequent model calibrations and fits.399.2 Power Lawto CC Ac/D21toFigure 3: Log(Ao„) (vertical axis) plotted against Log(to(A 00 )) (horizontal axis) for severaldata series between 20 and 100/tM of initial hIAPP monomer. Here we see the slope ofthe best fit line is —2.09, auspiciously close to an integer. Thus, we note 7 2 in therelationship to (11.09 ) cx Ac,-7 and we may now impose this power law, a strict mathematicalcondition, on any model we use to describe the accumulation of monomers into fibrils40=1.6192x= C.9626Y -0.5622xR - = 0.9102Y =3.5170x`,1 9694Y=C 3081xfZ-6-8269-C 25^0.309.3 Fluorescence of Seeded ReactionsFigure 4: Fluorescence for seeded reactions of hIAPP (vertical axis) versus time (horizontalaxis). 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, areused in determining the kinetics of elongation.41sj--^.....^ -11-• - -•.4_./^-- -0- -I 1'1irri„.1r9.4 Experimental Oligomer ConcentrationsFigure 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 plottedon a normalized y axis, 0 to 1, with each data set being shifted vertically 1 unit forvisibility. The data suggest that oligomers form quickly and are found at their maximumconcentrations 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 24hours of approximately 50-60% of the maximal oligomer concentration.429.5 Monomer and Monomer in Fibril Comparison1.2 -A( t)Af I 10Figure 6: Characteristic experimental results for relative monomer concentration, c(t)/co(diamond), mass in fibril, A(t)/Acx, (squares), and the sum of the two data sets, c(t)/coA(t)I A„ (triangles) versus time (horizontal axis). The sum suggests that most monomermass is present in either fibril or monomer form throughout fibrillogenesis of hIAPP. Thisgraph was extrapolated from results presented by Miranker and Ruschak (6).439.6 Nucleation Dependent ModelFigure 7: Generic nucleation dependent model, the basic model that describes the nu-cleated polymerization of a self assembling polymer. Monomers are assumed to quicklyassociate and dissociate to form short lived oligomers. These oligomers are assumed toquickly come into steady state with the monomer population. Stable nuclei are thenformed on a slower time scale and quickly elongate to form polymers.440 0 0 0° 0 0 0I 0 0/GOOCCO (T)0 0 a./0,Figure 8: Schematic of the scaled generic nucleation-dependent model as determined forhIAPP fibril formation. This figure represents the only possible subset of the nucleationdependent polymerization model that is consistent with the observation that the fluo-rescence profiles generated from various initial monomer concentrations of hIAPP scalesystematically to produce a single curve, implying that the initial monomer concentrationis immaterial to the formation kinetics of hIAPP fibrils.45Figure 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 NDPmodel to capture the rapid rise-time seen in the fluorescence, after the pronounced lagphase. Thus we have elucidated the fact that no version of the nucleation dependentpolymerization model will accurately describe the fibrillogenesis of hIAPP, given the ob-servation that initial monomer concentration is not relevant to the fluorescence profilegenerated for the formation of hIAPP fibrils under systematic scaling.46Residuals•••••..; ■•••./.."••••••••••.•••• •• •^••••S.• ■^••• • •• •• •Figure 10: Representative residuals of the best fits of scaled experimental fluorescenceA(t)/A,„, Figure (9), to the single nucleation dependent polymerization model. Thisresult implies a systematic failure of the generic NDP model to accurately capture thefluorescence data for hIAPP.# • %• • ••s     ^• • •    ^  '• s .•, „:••   , 47o••o•o•••••9.7 The Three Stage Model of Amyloid Fibril FormationMonomers Oligomers FibrilsFigure 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 additionat each stage until an unstable nucleus is created. Unstable fibrils then form by monomeraddition to nuclei and may grow by addition of monomers or oligomers of any size.48A ( 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 andoligomer 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 oligomersfor non trivial oligomer-fibril elongation rates. Furthermore, elongation kinetics do notsupport a theory of fibril elongation through oligomer addition for hIAPP.49tFigure 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 elongationthrough oligomer addition, versus time (horizontal axis). While this model obeys both ascaling and power law, and supports the observation of persistent oligomers, it reduces tothe NDP model with the imposed scaling ry = 1.509.8 Nucleation Dependent Model With Off-Pathway AggregatesFigure 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 additionof 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. Thismodel has been proposed to account for the persistent oligomer / aggregate populationobserved alongside growing and mature fibrils.5100^0 0 0 0°0 0 o o 0 oa^\, 2 b 1 /"o 0° 99COCCCOC0000CCOCOO00CCOb „,1^000Figure 15: Generic NDP model with off-pathway kinetics. This is a generalization of themodel proposed by Powers (30) where oligomers, as well as aggregates, are allowed toform 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" informationthrough the monomer pool.5200 00(i) 0 0 00f / 2 ,pcoocoo at-0°•• r00te'Figure 16: Generic NDP model with off-pathway aggregates as restricted by the scalingand the power law observed from fluorescence data of hIAPP fibril formation. Here we seethat aggregates are restricted in the same way as oligomers and grow by the same numberof 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 trivialrate, implying that aggregates cannot act as a buffer to the monomer concentration at"large" initial concentrations.539.9 Nucleation Dependent Model With Fibril Dependant SecondaryNucleationCDC)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 nucleationstep 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 mediatednucleation. We allow the data to "suggest" the most probable candidate for secondarynucleation.54Figure 18: Assembly kinetics of hIAPP as discerned from phenomenological modelingof 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 toc1 forming c2. This addition of monomer pairs continues until c4, after which a stablenucleus of twelve monomers is formed from c4 by addition of two monomers. Fibrils thengrow by unit addition. As fibrils form, a secondary fibril dependent nucleation occurs byaddition of a single monomer to the c4 species. These secondary nuclei then form fibrilsby unit addition.55A( 0/,4,,Figure 19: Characteristic fit for the nucleation dependent polymerization model withsecondary fibril surface mediated nucleation, as derived from the phenomenological modelof the fluorescence profile corresponding to monomer of human islet polypeptide in fibrilmass over time. The predictions from the unique model, as determined by systematicscaling and application of the corresponding power law, suggest our phenomenologicaltheory is in excellent agreement with the data up until maximum fluorescence is reachedfor all observed initial concentrations of monomers observed.56tFigure 20: A closer view of the lag and elongation phases of fibril formation fluores-cence. Here we see our phenomenological model of nucleation dependent polymerizationwith secondary fibril surface mediated nucleation captures both the rapid transition fromlag phase to fibril elongation, as well as, the rapid exponential decrease in fluorescence,corresponding to an exponential decrease in fibril formation.57c (t)   v(t)tFigure 21: Theoretical oligomer and nuclei populations formed during the fibrillogenesisof human islet amyloid polypeptide as predicted by the scaled nucleation dependent modelof fibril formation with secondary surface mediated nucleation. ci , c2, c3, c4, and nucleitop to bottom, respectively.58dc dvdt dttFigure 22: Theoretical formation rates of oligomer and nuclei populations formed duringthe fibriogenesis of human islet amyloid polypeptied as predictied by the scaled nucleationdependant model of fibril formation with secondary surface mediated nucleation. cl, bolddsahed, c2, c3, c4, solid line, top to bottom respectively, and nuclei, dashed.59f t/i50Data'50Figure 23: Theoretical values of the time to 50% completion tt50 (vertical axis) vs. exper-imental time to 50% completion tsigta (horizontal axis).602.4v - 1 C90GxC.9$492.2dA/dt "1.6 •1.4 •1.2 •9.10 Elongation Kinetics1.3^1.4 1.5^1.bC01./^1.S 1.5Figure 24: Log Log plot of the slope of elongation fluorescence (vertical axis) as readfrom 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, suggestingthat elongation of fibrils is due to monomer addition.619.11 Length Distribution1Figure 25: Predictions for normalized fibril length distribution of the nucleation dependentpolymerization model with secondary fibril dependent surface mediated nucleation, basedon scaled fluorescence data, solid line, plotted with experimental fibril length distribution,dots.629.12 Parameter Variationscr , t (t) v(t)Figure 26: Effects of altering fibril elongation rates on number of fibrils (dashed) andtotal oligomer populations (solid). This chart shows as elongation rates increase totalnumber of fibrils present decreases. Furthermore, the total oligomer populations alsomonotonically decrease with increasing elongation rates. Here increased elongation ratescorrespond to thicker plot lines. Thus, according to our model, acceleration of fibrilformation kinetics will result in fewer, longer fibrils and a lower total oligomer populationthroughout fibrillogenesis. This implies less cytotoxicity for accelerated fibril elongationand a dramatic increase in potentially toxic oligomer and fibril densities for retarded fibrilelongation kinetics.63c r0  (1) v( t)tFigure 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 totalnumber of fibrils present increase. However, the total oligomer populations are predictedto monotonically decrease with increasing nucleation rates. Here increased nucleationrates correspond to thicker plot lines. Thus, according to our model, acceleration offibril nuclei formation kinetics will result in increasing fibril numbers of shorter lengths,but a lower total oligomer population throughout fibrillogenesis. This implies greatercytotoxicity for accelerated fibril nucleation, should fibril density increase cytotoxicity.Conversely, a dramatic increase in potentially toxic oligomer is seen for retarded fibrilnucleation kinetics.64A(t)Figure 28: Fluorescence data for heparin. Note the drop in Fluorescence after maximumfluorescence has been reached. This drop implies that fibril may catastrophe, a featurebeyond the scope of the model presented here. Furthermore, various initial concentrationsof hIAPP do not scale for a given heparin concentration. Thus, we conclude that heparinamplifies the hitherto trivially small reverse rates, which we have scaled out of our model.65•41* •L1.1^1^'9.13 Scaling Violation••1S ••132 5 I-1t maxtotiT1T^_LT^ rL L1. 9Figure 29: Plot of t-max Ito vs. A. This plot reveals that out scaling is not perfect. Forperfectly scalable data tmax /to would be constant and not dependent on A. 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Methods in Enzymology 309:274-2847211 Appendix11.1 Scaled Data CollapseThe 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 singlefunction, such asA(t; Ate )^f [t/to (A,,)],^[1]where f is dimensionless. To determine if the data is described by Eq.(1) we letg(e) = log [f (lot)] .We may now write Eq.(l) aslog[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 haveno 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. Valuesof to are readily determined from plots of A/ Ac, vs. t. These values are plotted againstthe corresponding values of A„, on a log-log plot in Figure(3). Fitting a linear functiongives the approximate relationshipto oc^,with-y = 2 over the entire range of initial monomer concentrations assayed here.7311.2 NDP Model reductionFrom section[6.1] we observe that substituting the scaling variables into the model givesd61 no - 7 ^nodt- —[ fOCOX C^hconi 1 enie1 co (b2a2 — died],(rajdt- = A [fi_i^I con-1-7 _1 fiej^"ao—Y(—biâi^diai)],Furthermore we havedi)^AX [fkank croik ak],dt^ = PAL fk+ier 'I)] •dtWe now let/i — CpX =na = -y, for 1 < i < k,andno = 2y.Substituting these values into the equations results indaldt = foê'21" — f 167 el -I- ao—Y (b2e2dOidt =^ei-i — fiery ei + ep ry( — bi6i +^— diai), for 2 < i < k.Likewise we havedv f ak,dtanddt = A +for 2 < i < k.dM74Removal of the remaining explicit terms in co is accomplished by setting(bi+ CCi± i — biai — diôi) = 0,and( b2e2 dial) = 0.Thus we have determined that neither the off-rates, th, nor the disintegration rates, di , arerelevant to the kinetics and no net flux of monomers into or out of the various oligomerspecies exist due to these terms.7511.3 The Three Stage Model of Amyloid FibrillationHere 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 modelreduces to the scaled NDP model where Chaung-Chung Lee have artificially enforced ascaling 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 formationof 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 anysize oligomer from fibril ends. The original model by (33) tracks density of fibrils. Thisterm parallels the density of nuclei in the NDP model, and does not track total monomermass in fibril. However this quantity is easily discerned with the model and is includedin this analysis for easy comparison to observed fluorescence profile of hIAPP. The modelby Chaung-Chung Lee assumes monomers begin as properly folded hexamers which irre-versibly form monomers. Here we have assumed an initial monomer concentration. Thismodification will have no impact on the observations presented here. The original modelof Chaung-Chung Lee, given the assumptions and modifications above, isdcdt - .4+1cF + soF.Here we have again assumed the mass of monomers is predominantly contained in monomersand fibrils. Furthermore, we havedc1^ = foe2dt f1C1C b2c2 — b1 c1 + s i F —dcidt = 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 aredckdt = fk-ick-ic - fkckc +anddt^fkckc - bk+1F.bk+1F bkck SkF 1kCkF,dF76dt = fk.i4cF + F — bk + 1F.dMFurthermore, we keep track of the mass of monomers in fibrils over time as well. Theresulting equation for monomer mass in fibrils is given by^j=1^j=0Here, 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 followingscaling variables,t^ci = ^ c =^Acr coCi= —X '^for 2 < i < k,where A is needed to keep the units correct, and X is a quantity to be determined bythe system. Likewise,F,, and ./1^// = M ,cowhere it is a quantity to be determined by the system. Substituting the above intothe model and dropping the s givesdcdt^co C J k+iccoF + soF1-11,dci^Ac-0-7^2 2dt^X^{f0C Co — f1C1XCCo b2c2X b1c1X SiFIL^F 1-11,dc,^Acodt^X r[ft — 1 C2-1 XCeo — fi caX cco + b,-Fic,±1, X — b,c,X^liciXFk — 1.Furthermore, we havedck = Aco--ydt^X^ {fk — 1 Ck- 1 XeCo — fkckXcco bk±iF — bkckX skF — lkckXFanddF_— ^dt^[f kek X cco — bkH4F it] •for 2 < i <77The mass of monomer in fibrisl iskdt^co [fk+iccoF + iXFp —^Fp, — bk+iF bd•i=odM )coj = 1We now attempt to remove all explicit dependence of the model on co . Inspection ofthe equations for the change in fibril end density and mass imply that the term fkckXccomust survive or no fibrils will ever form. Therefore, we setX = c'O, and a= X.The resulting equations aredM _^ dt^A[fk+icF +k 3^0FC71 — SkFC0-7 — bk±iFC—o 7]=1anddFdt- = A[fkckccr+ 1 — bk±iFc0-7].The equation for the ck species becomesdckdt -= A[fk_ick_icco -Y+1 — fkckcco 7+1 + bk±iFc0 7 — bkCkC0 7 SkFC0 7 — IkCkFC0 71-1 ].Furthermore,dcidt A[fi_ici_iccr+1 — ficicc;7+1 + bi±ici±icr — bic cfor 2 < i < k — 1, + siFcr —dci = A[foc2c0-27+2 — ficiccr+1 + b2c2c7, — bicic0-7 + siFcr —dtand,dcdt=^fk+icF + soc; 1 F].We now set = 1 and absorb A into the reaction rates. Our model reduces todcdt- =78del^r 2dtrJOC^J1Cei — 11c1F,dc,dt = fz _ i cci_ i — fz cc, — lz c,F,dckdt^fkck_lc — fkckc — lk ck F,dF 2dtanddMdt — fk+ickc^kj=1However, elongation kinetics suggest that fibrills grow by monomer addition resultingincjF = 0.3=1Modifying the above model by allowing oligomers to grow by the addition of any numberof 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.7911.4 Scaling of the Nucleated Polymerization Model with CompetingOfd PathwayAggregationHere we write the model proposed by Powers and Powers (30) in the notation of thisthesis. The assumptions made for the basic NDP model apply here. However, Powers andPowers 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 onemonomer at a time to the preceding oligomeric species and fibril elongation by monomeraddition. These definitions, and the diagram shown in Figure(14) result in the followingsystem of differential equations for nucleated polymerization. For the oligomeric and ag-gregate species we havedci^ 2dt- = Joe- — ficci + b2c2 — bici,dcidt fi—icci—i — ficei — bici + bi+ici+i, for 2 < i < k,dz1- aoc2 — aiZle 02Z2 01Z1)dtdzi- =dt^cti_izi_ic — aizic — zi + Oi+i zi+and the change in the largest aggregate species allowed is given bydzmdt =^Zrn_iC —^Zrn  We also obtain,dvdt = fkcnk ck — bvv.Furthermore, the total mass of monomers in fibril form, discounting the monomer inoligomer, aggregate and nuclei, is given by,dt^fk±icv — bmv.We now introduce the scaling variables of section [6.3]. Substituting into the model anddropping the "'s gives the following equations for the change in the c 1 and z1 speciesfor 2 < i < m — 1,dM80dci Acr^2 2dt^X^if0C Co — ficixeco + b2 c2 X — biciX],— anddzi^Ac,T7 [^2dt   ctoc co — aiziYcco 02z2Y —For the ith speciesdci A6-7 ,= ^dt^X Lfz—ici—iXcco — ficiX cc, — biciX + bid-ici+14anddt^Y [ai_izi_iYcco — aiziYcco — OiziY + Oi+izi+1 17] 7The mth aggregate rate of change is nowdzm Aco-7dt^Y^ [am_i^Ycco — i3mzm,Y] .The nuclei and fibril mass rate of changes aredv Acrdt ^ if kekX cco — byvttdzi^Ac0-7for 2 < i < k,for 2 < i < m — 1.anddM = Aco-7 [fk ±i v pcco — bm vdt^coWe now let p = X = co. The fibril mass and nuclei concentration rates of changenow becomedMdt = \[fk+ivc — bm vcn,anddvdt = A[fkckccr+1 — bo vcoThe equations for the ci s are81dci— =-dt^ - ficicco 1H-1 - bicico 7 for 2 < < k.We now write the equation for cidcidt =- A [foc2co-27+2 fleiCC 7+1 b2C2Cr — biCiC0 7].and,We are now in a position to impose the only scaling of y that will allow oligomers andnuclei 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, ^pdt- = - fz czcci,dvdt^A[fkckc],anddM = A [fk± i vc]dtfor 2 < i < k,Furthermore, the rates of change for the off-pathway aggregates becomezi = A [ao c2 — a zi ,dtdzidt =^- aizic], for 2 < < m - 1.The change in the largest aggregate, in monomer units, is now given bydz„t _dt^A [ce,_ z,_Therefore, we see that scaling restricts off-pathway kinetics of aggregate formation, aswell 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 monomerconcentrations as is observed for hIAPP. We now relax the implicit power law with 7 = -1and rewrite the model in a more generic form so as to let our data motivate the powerlaw used.8211.5 Scaling of the Generic Nucleated Polymerization Model with Com-peting Ofd PathwayAggregationWe now rewrite the nucleation dependent polymerization model with off-pathway aggre-gates where oligomers and aggregates are allowed to form by any number of monomersjoining to form the first species. Subsequent species are allowed to form by binding toany number of monomers to form the next largest species. No constraint is imposed thatrequires 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 oligomersmay only form from monomers and other oligomers. The definitions of Section [6.4], andthe diagram shown in Figure(15) result in the following system of differential equationsfor nucleated polymerization.dci =dt^focn° —^+ b2c2 —anddZi^ =dt^CtoCa° —^132Z2 01z1.For the ith aggregate and oligomer we havedcidt = fi lci_lcni-1 bici + bid-ici+i,^for 2 < i < k,anddzidt =^— ceizica' —^+ Oi+izi+i, for 2 < i < m — 1.The change in the largest aggregate species allowed is now given bydz,r,^ —^— Orn Zrn .dtWe also obtaindudt^f ken ck — bvv.Furthermore, the total mass of monomers in fibril form, discounting the monomer inoligomer, aggregate and nuclei, is given bydt = fk±l ev — bmv.We now introduce the scaling variables of Section [6.4]. Substituting into the modeldM83and 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^Xanddz1 Acrdt =^oc—c° — aiziYca' Coal ± /32Z2Y /31Z1/71•For the ith speciesdci Acr— ficiX cni conidt^X^ — bi ci X bi±ici+ 14 for 2 < i < k,anddzi^Aco^ — aiziYcaicoai — AziY /3i+1zi+111, for 2 < i < m — 1.The mth aggregate rate of change is nowdzm = A Co ^— /am z77, Y] .dtThe nuclei and fibril mass rate of changes aredv Acr= ^dt^[fkckX cnk conk —anddM Acrdt^co [fk+ivitcco — bmvti]•We now let= c')O =X.The fibril mass and nuclei concentration rates of change now becomedM^=-dt^fk+ivc — bmvc0 1 ],anddt84dvdt- = A[fkckcnk co-7±nk b,,vcr].The equations for the ci s aredcidt =- A[fi_ici_icni- ' co^— ficicni cr±ni — bicicr b^•^—7 1i_kicz+ ico J ,We now write the equation for c1dcidt A[foc"C2 — ci cm' cr±n' + b2c2c0 7 — bici co-7].We now setni = -y for i < 1 < k,andno = 2'71reducing the equations for oligomers, nuclei and fibril mass todel^ =dt^A[foc27 — ficie7],dcidt = A[.I; lci lc7 — fici c7],^for 2 < i < k,dvdt^A[fkckc,.y],anddt^A[fk+i v c].Furthermore, the equations for ofd pathway aggregates are now given bydzidt^[cei_ zi _ cai --1 co 7±' — aizicai c0-7±a — bizi^+^zi±i co 7] ,anddzi A=-- [aoca°cr±"Y-1 — a ci cal c;'+' + 132 z2c0 1" — 13izic; 7],dtWe now setdMfor 2 < i < k.for 2 < i < k.85ai=ry for 1 < < m,and remove the explicit dependance on co by setting terms explicitly containing co to0. The resulting equations are,dzidt-^[CroCa0C0 7±a° 17-1 alzic7],dzi -- =dt^- ctiz,c7], for 2 < i < m - 1.The change in the largest aggregate is now given bydtdzWe now set Y = c-W. The resulting equation for —dti is given bydzidt = A[aoca°c;"Y -w±a° - alz1c-Y].Thus, we are left with the resultao =- y + co,while the solutioncto = 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° —dtwhereao = -y + W.86

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