Stochastic kinetic treatment of protein aggregation and the effects of macromolecular crowding
John Bridstrup, John S Schreck, Jesse L Jorgenson, Jian-Min Yuan
SStochastic kinetic treatment of proteinaggregation and the effects of macromolecularcrowding
John Bridstrup, * , † John S. Schreck, * , ‡ Jesse L. Jorgenson, S and Jian-Min Yuan * , † † Department of Physics, Drexel University, Philadelphia, PA 19104 ‡ National Center for Atmospheric Research, Boulder, CO 80305 ¶ Department of Chemistry, Drexel University, Philadelphia, PA 19104 S The Evernote Corporation, Austin, TX 78705
E-mail: [email protected]; [email protected]; [email protected]
Abstract
Investigation of protein self-assembly processes is important for the understanding ofthe growth processes of functional proteins as well as disease-causing amyloids. Insidecells, intrinsic molecular fluctuations are so high that they cast doubt on the validity ofthe deterministic rate equation approach. Furthermore, the protein environments insidecells are often crowded with other macromolecules, with volume fractions of the crowdersas high as . We study protein self-aggregation at the cellular level using Gillespie’sstochastic algorithm and investigate the effects of macromolecular crowding using modelsbuilt on scaled-particle and transition-state theories. The stochastic kinetic methodcan be formulated to provide information on the dominating aggregation mechanismsin a method called reaction frequency (or propensity) analysis. This method revealsthat the change of scaling laws related to the lag time can be directly related to thechange in the frequencies of reaction mechanisms. Further examination of the time a r X i v : . [ phy s i c s . b i o - ph ] F e b volution of the fibril mass and length quantities unveils that maximal fluctuationsoccur in the periods of rapid fibril growth and the fluctuations of both quantities can besensitive functions of rate constants. The presence of crowders often amplifies the rolesof primary and secondary nucleation and causes shifting in the relative importance ofelongation, shrinking, fragmentation and coagulation of linear aggregates. Comparisonof the results of stochastic simulations with those of rate equations gives us informationon the convergence relation between them and how the roles of reaction mechanismschange as the system volume is varied. TOC GraphicIntroduction
Protein self-assembly is an important process for the formation of natural polymers likeactin filaments and microtubules, but also for the formation of amyloids, culprits formany neurodegenerative diseases including Alzheimer’s and Parkinson’s diseases. Decadesof studies of self-assembly of proteins reveal that the molecular reactions involved can becomplicated, for example, in the processes of amyloid formaton.
Among the many theoretical methods to investigate the protein aggregation problem,the rate equation approaches based on mass-action laws often provide direct fits toexperimental data and interpretation of the aggregation processes. Processes considered in2hese studies often include primary nucleation, monomer addition and subtraction, fibrilfragmentation, merging of oligomers, heterogeneous (or surface-catalyzed) nucleation, etc.The reaction rates associated with the reaction steps considered can be independent ordependent on the oligomer/fibril size.
Almost all of our current knowledge comes from the studies of systems in vitro , and littlehas been done in understanding processes in vivo . In the latter case, the volume of thecompartment or inside confining boundaries is usually small and the numbers of copies ofcertain protein species can be low, resulting in large number fluctuations.
Furthermore,experiments carried out on protein aggregation often starts out with monomers of amyloidpeptides or proteins. This may be what happens in the brains also, starting with monomers.Above the critical concentration, monomers then aggregate into dimers, trimers, tetramers,. . . , by steps. The system necessarily goes through the stages in which the numbers of dimers,trimers, . . . , oligomers are very small. On the other hand, deterministic rate equations basedon mass-action laws are often used to simulate the growth of the oligomer species. Rigorouslyspeaking, rate equations, that is, concentrations, are well-defined only when large numbers ofrelevant molecules exist, for the fluctuations are inversely proportional to the square root ofthe number of molecules. Therefore, application of the rate equations to the cases involvingsmall numbers of oligomers cannot fully be justified. For these cases, one should use stochasticdynamics methods, such as the Gillespie algorithm.
From another point of view, it wouldbe important to evaluate the accuracy of the rate equation results, especially in the earlystage of aggregation, or at any times when any numbers of the chemical species involved aresmall, by comparing them to those obtained using a stochastic kinetic method.Gillespie’s method of carrying out a stochastic chemical kinetics study is to solve themaster equations defined on the probabilities, P ( 𝑁 , 𝑁 , 𝑁 , . . . , 𝑁 𝑘 , . . . , 𝑁 𝑀 ) , where 𝑁 𝑘 isthe number of 𝑘 -mer, 𝑀 𝑘 , denoting an oligomer containing 𝑘 monomers. We can estimate thedimension, 𝐷 , of the vector space of P . Consider an experiment for which only 𝑀 monomersexist initially at 𝑡 = 0, that is, 𝑁 𝑘 (0) = 0 , for 𝑘 ≥ . Since 𝑁 is bound above by 𝑀 , 𝑁 by3 / , 𝑁 𝑘 by 𝑀/𝑘 , and 𝑁 𝑀 by 1, the dimension of the vector P has an upper bound given by 𝑀 𝑀 / ( 𝑀 !) . Thus for a large 𝑀 , 𝐷 is bounded above by 𝑒 𝑀 , using the Sterling approximation.This implies that the dimension of the state space grows very fast as 𝑀 increases. We cannotrealistically integrate the master equations numerically for an 𝑀 larger than a few hundreds.In reality, for amyloid fibrils, 𝑀 can go as high as several thousands. Thus currently thestandard stochastic kinetic method is restricted to investigating the early stage of the proteinself-assembly processes or, alternatively, to investigating systems of small volume, in whichthe total number of proteins in the system is relatively small.The fact that the stochastic kinetic methods can be used to study chemical reactionsin small volumes was pointed out early on by Gillespie. Applications of the stochastickinetic methods to study protein self-assembly processes in small volumes have been carriedout recently by several groups. Szavits-Nossan, et al. have derived an analytic expressionfor the lag-time distribution based on a simple stochastic model, including in it primarynucleation, monomer addition, and fragmentation as possible reaction mechanisms. Tiwariand van der Schoot have carried out an extensive investigation of nucleated reversible proteinself-aggregation using the kinetic Monte Carlo method. They focused on the stochasticcontribution to the lag time before polymerization sets in and found that in the leadingorder the lag time is inversely proportional to system volume for all nine different reactionpathways considered. Michaels, et al. on the other hand, have carried out a study of proteinfilament formation under spatial confinement using stochastic calculus, focusing on statisticalproperties of stochastic aggregation curves and the distribution of reaction lag time. At the cellular level, besides the reaction being contained in a small volume, the environ-ments of proteins are crowded with other biomolecules, such as DNA, lipids, other proteins,etc. The fraction of volume occupied by these “crowders” can be as high as - , whichcan affect the reaction rates of proteins as well as other biopolymers in the cell in significantways. In this article, we extend the earlier stochastic works on protein self-assembly by in-4estigating beyond the early stage of aggregation to include the polymerization phase andpresent a general study of fluctuations for systems in small volumes (i.e., low total number ofproteins). Furthermore, we examine the role of macromolecular crowding in changing themicroscopic behavior of an aggregating system, and present a new method for extracting thedominant aggregation mechanisms that is more explicit than scaling law analysis.
Wealso compare the stochastic and rate-equation approaches for a model case to gain insight onwhen and why the methods begin to diverge.The organization of the present article is as follows: In Section II we describe the kineticmodels used in the study. In Section III we introduce the master equation, present Gillespie’sstochastic simulation algorithm (SSA), discuss how it connects to a rate equation treatment,and finally introduce our reaction frequency method of analysis. In Section IV we give a briefoverview of how the effects of macromolecular crowding are included in the models throughscaled-particle and transition-state theories. Our results are presented in Section V. Finally,we conclude with discussion and comments in Section VI.
Kinetic Models Studied
The Oosawa Model
Figure 1: Mechanisms of protein aggregation included in this study. (i) Primary nucleation,(ii) one-step secondary nucleation, (iii) monomer addition and subtraction, (iv) coagulationand fragmentation. 5he Oosawa model is a simple and widely studied model of protein filamentformation. Protein aggregates of size 𝑛 𝑐 , the critical nucleus size, form initially by primarynucleation, as is illustrated schematically in Fig. 1(i), and grow in one possible way by simplemonomer addition and subtraction, shown in Fig. 1(iii). It is assumed based on the classicalnucleation theory that the concentrations of aggregates smaller than 𝑛 𝑐 are zero. The modelis described by the kinetic equations 𝑑𝑐 𝑛 𝑐 𝑑𝑡 = 𝑘 𝑛 𝑐 𝑛 𝑐 − ( 𝑎𝑐 + 𝑘 * 𝑛 ) 𝑐 𝑛 𝑐 + 𝑏𝑐 𝑛 𝑐 +1 𝑑𝑐 𝑟>𝑛 𝑐 𝑑𝑡 = 𝑎𝑐 ( 𝑐 𝑟 − − 𝑐 𝑟 ) + 𝑏 ( 𝑐 𝑟 +1 − 𝑐 𝑟 ) (1) 𝑑𝑐 𝑑𝑡 = 𝑛 𝑐 ( 𝑘 * 𝑛 𝑐 𝑛 𝑐 − 𝑘 𝑛 𝑐 𝑛 𝑐 ) − ∞ ∑︁ 𝑟 = 𝑛 𝑐 +1 𝑟 𝑑𝑐 𝑟 𝑑𝑡 where 𝑎 , 𝑏 and 𝑘 𝑛 are, respectively, the monomer addition, monomer subtraction and primarynucleation rate constants, 𝑘 * 𝑛 is the rate constant for a nucleus dissolving and 𝑐 𝑖 is theconcentration of an oligomer containing i monomers, denoted by 𝑀 𝑖 . There are severalsystems, such as the growth of actin, for which this model describes experimental dataquite well. Generalized Smoluchowski Model
The generalized Smoluchowski model extends the Oosawa model by allowing for proteinaggregates to break and merge with each other. This is a more general and thus more widelyapplicable model of protein aggregation behavior. To account for breaking and merging in6he kinetic equations, Eq. (1) is modified in the following way 𝑑𝑐 𝑟 𝑑𝑡 = ... + 𝑘 𝑎 [︃ 𝑟 − 𝑛 𝑐 ∑︁ 𝑠 = 𝑛 𝑐 𝑐 𝑠 𝑐 𝑟 − 𝑠 − ∞ ∑︁ 𝑠 = 𝑛 𝑐 [1 + 𝛿 𝑟𝑠 ] 𝑐 𝑟 𝑐 𝑠 ]︃ + 𝑘 𝑏 [︃ ∞ ∑︁ 𝑠 =2 [1 + 𝛿 𝑟𝑠 ] 𝑐 𝑟 + 𝑠 − ( 𝑟 − 𝑐 𝑟 ]︃ , (2) 𝑑𝑐 𝑑𝑡 = 𝑛 𝑐 ( 𝑘 * 𝑛 𝑐 𝑛 𝑐 − 𝑘 𝑛 𝑐 𝑛 𝑐 ) − 𝑎𝑐 ∞ ∑︁ 𝑟 = 𝑛 𝑐 𝑐 𝑟 + 𝑏 ∞ ∑︁ 𝑟 = 𝑛 𝑐 +1 𝑐 𝑟 + 𝑘 𝑏 [︃ ∞ ∑︁ 𝑠 = 𝑛 𝑐 𝑛 𝑐 − ∑︁ 𝑟 =2 (1 + 𝛿 𝑟𝑠 ) 𝑟𝑐 𝑟 + 𝑠 ]︃ where 𝑘 𝑎 and 𝑘 𝑏 are the coagulation and fragmentation (Fig. 1(iv)) rate constants, respectively, 𝛿 𝑟𝑠 is the Kronecker delta function, and the factors of and 𝑟 − take into account doublecounting and aggregates having multiple points at which they can break. The last term onthe RHS of the monomer concentration equation is due to the assumption that any polymerssmaller than 𝑛 𝑐 which form via fragmentation immediately dissolve into monomers. Secondary Nucleation
In some cases, an additional mechanism is needed to accurately describe aggregation: sec-ondary or heterogeneous nucleation. Secondary nucleation is the process of existing polymerscatalyzing the formation of new aggregates on their surface and has been shown to playan important role in many aggregating systems. We investigate a simple model of secondarynucleation, where the process is modelled as a one-step process, shown in Fig. 1(ii). Inreality, secondary nucleation can be more generally represented as a two-step process, whichreduces to a one-step process in the low monomer concentration limit.
But for the presentpurposes, we will focus on the one-step secondary nucleation process.For one-step secondary nucleation, the rate equations in 1 and 2 are modifed by 𝑑𝑐 𝑛 𝑑𝑡 = ... + 𝑘 𝑐 𝑛 𝑀 𝑓𝑟𝑒𝑒 , (3)7here 𝑛 is the secondary nucleus size, and 𝑀 𝑓𝑟𝑒𝑒 is the total mass of the polymers whichcontain 𝑛 or more monomers. Stochastic Chemical Kinetics
In classical chemical kinetics, it is assumed that the concentrations of chemical species varycontinuously over time, the so-called mean-field approach. While this is often appropriate forbulk systems, where volume can be assumed to be macroscopic and populations are roughlyof the order 𝑁 𝐴 , Avogadro’s number, it does not take into account that species populationsare discrete and change in integer numbers. Further, chemically reacting systems displayrandom (stochastic) behavior due to their contact with a reservoir. Stochastic fluctuationscan become increasingly relevant as system size or overall concentration decrease. Thus,the assumption that populations can be represented as continuous concentrations becomesless valid as the number of molecules becomes smaller. Stochastic chemical kinetics aims todescribe how a well-stirred reacting system evolves in time, taking into account fluctuationsand the discreteness of populations. The primary goal of stochastic kinetics is to solve thechemical master equation (CME) 𝜕𝑃 ( x , 𝑡 | x , 𝑡 ) 𝜕𝑡 = 𝑆 ∑︁ 𝑗 =1 [ 𝑎 𝑗 ( x − v 𝑗 ) 𝑃 ( x − v 𝑗 , 𝑡 | x , 𝑡 ) − 𝑎 𝑗 ( x ) 𝑃 ( x , 𝑡 | x , 𝑡 )] , (4)where x is the state of the system, v 𝑗 is change in system state due to a single reaction 𝑅 𝑗 , 𝑃 ( x , 𝑡 | x , 𝑡 ) is the probability of the system being in state x at time 𝑡 given an initial state x at time 𝑡 , and 𝑎 𝑗 ( x ) is the propensity function, or transition rate, of a given reaction 𝑅 𝑗 𝑎 𝑗 ( x ) 𝑑𝑡 (cid:44) the probability, given x , that one (5) 𝑅 𝑗 reaction occurs in the time interval 𝑑𝑡. We will show later that the propensity function can be directly related to the reaction ratesfrom classical chemical kinetics.In principle, the probability distribution, P ( x , 𝑡 | x , 𝑡 ) is entirely described by eq. 4. Inpractice, however, analytical solutions are often impossible due to the CME being, in general,a very large system of coupled ODEs. Other complications with a direct analysis of the CMEare described by Gillespie. Thus, it is necessary to use computational methods to solvefor the evolution of the probability distribution function. Several methods of simulating exactand approximate solutions to the CME have been proposed.
In our study we use theGillespie stochastic simulation algorithm (SSA), which allows for a highly detailed, albeitsomewhat computationally expensive, look at how a stochastic system evolves over time.
Gillespie Simulation Algorithm
The Gillespie stochastic simulation algorithm is a method for generating statistically accuratereaction pathways of stochastic equations, and thus statistically correct solutions to the CME(eq. 4). The algorithm proceeds as follows:1. Set the species populations to their initial values and 𝑡 = 0 .2. Calculate the transition rate, 𝑎 𝑖 ( x ) , for each of the 𝑆 possible reactions.3. Set the total transition rate 𝑄 ( x ) = ∑︀ 𝑆𝑗 =1 𝑎 𝑗 ( x ) .4. Generate two uniform random numbers, 𝑢 and 𝑢 .5. Set ∆ 𝑡 = 𝑄 ( x ) 𝑙𝑛 ( 𝑢 ) . 9. Find 𝜇 ∈ [1 , .., 𝑆 ] such that ∑︀ 𝜇 − 𝑗 =1 𝑎 𝑗 ( x ) < 𝑢 𝑄 ( x ) ≤ ∑︀ 𝜇𝑗 =1 𝑎 𝑗 ( x ) .7. Set 𝑡 = 𝑡 + ∆ 𝑡 and update species populations based on reaction 𝜇 .8. Return to step 2 and repeat until an end condition is met.This process generates a single reaction pathway and may be repeated and averaged tocompare with bulk behavior and experimental results. Modifications to this method existto decrease computation time, such as the next-reaction and tau-leaping methods, butwhere they improve efficiency they sacrifice in accuracy. Relation to Chemical Kinetics
In classical chemical kinetics, differential rate equations are solved to study the bulk behaviorof a continuous system. In order to compare with these studies, as well as with experimentalstudies, it is necessary to relate bulk rates, and rate constants, with the stochastic propensityfunctions and the equivalent stochastic rate constants. We give two examples of how this isdone. For a coagulation process 𝑀 𝑟 + 𝑀 𝑠 𝑘 𝑎 → 𝑀 𝑟 + 𝑠 , we have a rate equation of the form 𝑑𝑐 𝑟 + 𝑠 𝑑𝑡 = 𝑘 𝑎 𝑐 𝑟 𝑐 𝑠 . (6)Making use of the relationship between species population and species concentration, 𝑐 𝑖 = 𝑁 𝑖 𝑁 𝐴 𝑉 ,where 𝑁 𝑖 is the species population and 𝑉 is system volume, we find the relation 𝑑𝑁 𝑟 + 𝑠 𝑑𝑡 = 𝑘 𝑎 𝑁 𝐴 𝑉 𝑁 𝑟 𝑁 𝑠 , (7)which is the rate at which an 𝑟 -mer and an 𝑠 -mer transition into an ( 𝑟 + 𝑠 ) -mer. The RHS isthe sum of the propensity functions from eq. 5 for all possible 𝑀 𝑟 + 𝑀 𝑠 → 𝑀 𝑟 + 𝑠 coagulation10eactions. Finally, the stochastic rate constant can be written 𝑘 ′ 𝑎 = 𝑘 𝑎 𝑁 𝐴 𝑉 . (8)For protein aggregation, we can again make use of the definition of concentration for thetotal number, 𝑁 , and concentration, 𝑐 , of monomers to obtain 𝑘 ′ 𝑎 ≡ 𝑐 𝑁 𝑘 𝑎 . (9)For a primary nucleation process, 𝑛 𝑐 𝑀 𝑘 𝑛 → 𝑀 𝑛 𝑐 , we have a rate equation 𝑑𝑐 𝑛 𝑐 𝑑𝑡 = 𝑘 𝑛 𝑐 𝑛 𝑐 . (10)Following the same process, the stochastic nucleation rate constant is given by 𝑘 ′ 𝑛 = 𝑘 𝑛 ( 𝑁 𝐴 𝑉 ) 𝑛 𝑐 − (11) ≡ (︂ 𝑐 𝑁 )︂ 𝑛 𝑐 − 𝑘 𝑛 . The other stochastic rate constants are found similarly. It is worth noting that all stochasticrate constants have units of frequency ( 𝑠 − ) and thus the stochastic rate constants involvedin shrinking or breaking processes ( 𝑏 ′ , 𝑘 ′ 𝑏 , ¯ 𝑘 ′ , etc.) are identical in value to their bulkcounterparts. Reaction Frequency Method
In numerical simulations using Gillespie’s stochastic algorithm, we can obtain informationon what reactions are occurring in certain intervals of time. In particular, we investigatethe frequencies or propensities of particular reaction types as they evolve over time. This11pproach gives unique insight into the behavior of a system and offers direct confirmation ofwhich mechanisms of reaction are dominating at various phases of the aggregation process.For instance, a common scenario, meaning for a set of parameters showing nontrivial dynamicbehaviors, is that the primary nucleation dominates at the very beginning, then monomeraddition and oligomer coagulation become important, balanced by monomer subtraction andfragmentation. These are followed by secondary nucleation, which becomes active, beforemonomers are depleted. At longer time scale, oligomer coagulation and fragmentation persistas the system approaches an equilibrium or steady state. For some aggregation reactions,such as that involving actin, secondary nucleation never plays a significant role, so it canbe neglected. But, for other reactions, especially under the influence of molecular crowders,secondary nucleation dominates, until monomers are depleted. In the results section, wenormalize the reaction frequencies by the total number of reactions occurring at that time( 𝑓 𝑟𝑒𝑙 ). Doing this allows us to investigate the relative importance of any particular reactionas it evolves over time. Macromolecular Crowding
In the presence of molecular crowders, the rate constants of reaction steps may be affectedand the degree of influence varies widely among the different types of reaction steps involved.The effects of crowders on the rate constants has been worked out in our previous study using the transition-state theory (TST) and the scaled-particle theory (SPT). In thissection, we outline the relevant formulas that we have used in the present simulations. Westart with the forward coagulation reaction of the reversible reaction M r + M s 𝑘 𝑎 −− ⇀↽ −− 𝑘 𝑏 M r+s . (12)In TST one assumes that quasi-equilibrium is established between the reactants and thetransition state which allows us to express the rate constant in terms of the free energy12ifference between the reactants and the transition state. Further, expressing the chemicalpotential of a chemical species in terms of the product of its activity coefficient, 𝛾 , andconcentration, we can describe the forward rate constant using the following relationship 𝑘 𝑎 = 𝛾 𝑟 𝛾 𝑠 𝛾 ‡ 𝑟 + 𝑠 𝑘 𝑎 ≈ 𝛾 𝑟 𝛾 𝑠 𝛾 𝑟 + 𝑠 𝑘 𝑎 (13) = 𝛾 𝛼 𝑘 𝑎 , where 𝛾 𝑟 is the activity coefficient for an 𝑟 -mer, ‡ refers to the transition state, and thesuperscript 𝑘 𝑎 refers to the associate rate constant in the case that all relevant activitycoefficients are unity, including the case where the crowders are absent in the solution. Thuswe call it the crowderless rate constant below. We have also made use of the importantresult that the activity coefficient of an 𝑟 -mer can be related to that of a monomerby 𝛾 𝑟 = 𝛾 𝛼 𝑟 − , where the parameter 𝛼 is derived from the change in activity as a monomeris added to a spherocylinder aggregate. Additionally, the approximation has been madethat the transition state has roughly the same structure as the aggregated polymer, thus 𝛾 ‡ 𝑟 + 𝑠 ≈ 𝛾 𝑟 + 𝑠 . This approximation, when applied to the reverse reaction, gives the result 𝑘 𝑏 = 𝑘 𝑏 . (14)13n other words, breaking reactions are unaffected by crowders in this model. This approachcan be applied to the other mechanisms in our model to obtain 𝑎 = 𝛾 𝛼 𝑎 𝑘 𝑛 = (︂ 𝛾 𝛼 )︂ 𝑛 𝑐 − 𝑘 𝑛 (15) 𝑘 = 𝛾 𝑛 Γ 𝑘 𝑏 = 𝑏 , where Γ is a factor related to the change in shape of an aggregate as monomers attachto the surface in a secondary nucleation process. For our study, we assume Γ ≡ ina one-step secondary nucleation model. The activity coefficients, as well as 𝛼 , may becalculated using SPT by treating crowders and monomers as hard spheres and aggregates ashard sphero-cylinders. Expressions for the effects of crowders on the activity coefficientsin this case have been derived by Cotter. Further details on the relevant calculations havebeen presented by Bridstrup and Yuan, and Minton, and a closer look at how crowderscontribute to both the one-step and two-step secondary nucleation has been presented bySchreck et al. Results
Scaling Laws
A powerful tool for extracting the dominant mechanisms of aggregation is by looking athow the half-time of aggregation mass ( 𝑡 / ) scales with increasing initial concentration ofmonomers. We test this method using stochastic simulations by directly examining whichreactions are dominating during the growth phase. In a scaling law analysis, the slope ofa log-log plot of 𝑡 / vs 𝑐 gives the scaling exponent, 𝛾 , which can be related to specific14igure 2: Log-log plot of 𝑡 / vs 𝑐 . At low 𝑐 , nucleation-elongation dominates based on thescaling factor 𝛾 = − . . At higher concentrations, the scaling exponent reaches 𝛾 = − . corresponding to secondary nucleation being the dominant mechanism. These interpretationsare based on the scaling laws of Meisl et al. mechanisms of growth. The reader may refer to Fig. 6 from Meisl et al. for a list of someof these scaling relationships. Fig. 2 shows a scaling relationship plot for the generalizedSmoluchowski model with one-step secondary nucleation. The negative curvature and valuesof the scaling exponent indicate competition between nucleation-elongation (small 𝑐 ) andsecondary nucleation (large 𝑐 ), based on Meisl et al. Fig. 3 shows the relative reactionfrequency of each mechanism of growth as they evolve over time. For low 𝑐 , it is clear thatmonomer addition following an initial phase of primary nucleation is the dominant mechanismof growth. As 𝑐 is increased, the relative frequency of both monomer addition and secondarynucleation increase (competition) before eventually secondary nucleation begins to suppresseven monomer addition during the growth phase. This comparison both supports the validityof the scaling laws and justifies further use of this approach, as it is clear that much can begained from this level of detail. 15igure 3: Relative reaction frequencies vs time. Blue ( ∙ ) corresponds to primary nucleation,orange ( (cid:4) ) to monomer addition, red ( (cid:78) ) to secondary nucleation and green ( x ) to fragmen-tation. As 𝑐 is increased, it is clear that secondary nucleation goes from hardly participatingin the reaction to completely dominating. Relative frequencies are calculated by dividing theindividual reaction rates with the total reaction rate, which includes monomer subtractionand coagulation (not shown in figure). 16 luctuations and System Volume Half-time Fluctuations
Tiwari and van der Schoot showed that nucleation time increases linearly with /𝑉 , theinverse of system volume. We investigate the effects of decreasing the volume, or total numberof monomers, 𝑁 , at fixed concentration, on the the halftime, 𝑡 / , and fluctuations of thehalftime, 𝜎 / (the standard deviation of 𝑡 / ), of polymer mass for two different sets ofparameters, which we call set 1 and set 2 (defined in Fig. 4). We define halftime as thetime it takes for the polymer mass to reach half its equilibrium value. Fig. 4 shows 𝑡 / vs /𝑁 for both sets at fixed 𝑐 . For set 1 , the linear trend is clear at all values of 𝑁 . For set 2 ,however, there is no increase in half-time until a threshold value is reached. This may implythat bulk behavior is recovered at different values of 𝑁 depending on the rate constants.Perhaps unsurprisingly, the relative deviations increase as 𝑁 decreases. More interestingly,the manner in which these fluctuations increases depends on the rate constants themselves.For set 1 , where 𝑡 / increases linearly with /𝑁 , the standard deviation ( 𝜎 / ) also increasesroughly linearly, whereas for set 2 , 𝜎 / appears to increase more like the square root of /𝑁 before abruptly increasing at the same threshold. Moment Fluctuations
System volume has a significant effect on fluctuations of the moments of the distribution(polymer number and polymer mass), but also on the overall rate of mass production aswell as the evolution of the average length of polymers. As mentioned earlier, for certainchoices of rate constants the half-time of the reaction may increase or stay roughly the sameas volume is decreased. In all cases, fluctuations increase with decreasing volume but themagnitude of fluctuations depends strongly on the rate constants themselves. Fig. 5 showsplots of the average polymerized mass as a function of time for set 2 , normalized by the totalmass of the system. For large values of 𝑁 , i.e. larger volumes, deviations from the average17igure 4: 𝑡 / and 𝜎 𝑡 / as functions of /𝑁 for two different sets of rate constants. For set1 , the half-time increases linearly as expected. For set 2 , however, there is no increase inhalf-time until a threshold is reached. Parameters used are, for set 1 : 𝑎 = 5 × µ m − s − , 𝑏 = 𝑘 𝑏 = 1 × − s − , 𝑘 𝑎 = 100 µ m − s − , 𝑘 𝑛 = 1 × − µ m − s − , 𝑛 𝑐 = 2 , and 𝑐 = 10 µ m ;and for set 2 : 𝑎 = 2 µ m − s − , 𝑏 = 𝑘 𝑏 = 1 s − , 𝑘 𝑎 = 0 . µ m − s − , 𝑘 𝑛 = 5 × − µ m − s − , 𝑛 𝑐 = 2 , and 𝑐 = 5 µ m . 18re small, growing larger as 𝑁 is decreased. As expected, the value of the relative deviationis proportional to / √ 𝑁 . The fluctuations in mass peak when the rate of change of mass isat its maximum, which corresponds with the time period in which the most reactions areoccurring. Later, they reach an equilibrium value in which the total number of reactionsoccurring is smaller and roughly stable. It should be noted that for this set the reactionproceeds at the same time scale for even very small values of 𝑁 . Fig 6 shows the same plotsFigure 5: Average mass as a function of time for various values of 𝑁 using set 2 parameters.It is clear that the relative fluctuations decrease significantly as 𝑁 is increased, proportionalto / √ 𝑁 , and also that the time scale for each value of 𝑁 is the same. Error bars correspondto ± standard deviation.for set 1 . In this case, fluctuations are much larger, and can in fact be larger than theaverage mass. The time-scale of the reaction increases significantly as 𝑁 becomes smaller,corresponding to the half-time of the reaction increasing with /𝑁 . Another interesting resultis presented in Fig. 7. For set 2 , 𝐿 ( 𝑡 ) peaks very low (roughly ) and is almost identicalfor every value of 𝑁 . For set 1 , however, not only does the peak value of 𝐿 decrease, whichwould be expected as for small enough values of 𝑁 it could not possibly reach the same value,but the actual shape of 𝐿 ( 𝑡 ) changes. For large 𝑁 , 𝐿 ( 𝑡 ) peaks sharply early on in the growthphase before rapidly falling back to smaller values. As 𝑁 is decreased, this peak broadens andeventually flattens at small enough values of 𝑁 . This would seem to imply that the reaction19igure 6: For set 1 parameters, 𝑀 ( 𝑡 ) behaves similarly on average for large 𝑁 but hassignificantly larger fluctuations. For smaller N, 𝑀 ( 𝑡 ) takes over 10 times as long to reachequilibrium and fluctuations can be up to of the average value for small 𝑁 . Error barscorrespond to ± standard deviation.mechanisms have very different behavior at different values of 𝑁 . Fig 8 illustrates this clearly.For set 2 (Figs. 8( 𝑎 ) and ( 𝑏 )), the reactions proceed almost identically for both 𝑁 = 40000 and 𝑁 = 600 . For set 1 (Figs. 8( 𝑐 ) and ( 𝑑 )), however, more reactions are involved duringthe growth phase for 𝑁 = 600 than there are for 𝑁 = 40000 , and the reaction rates fluctuatefairly significantly. This relative increase in importance of mechanisms other than monomeraddition, particularly fragmentation, would explain the flattening of the peak in 𝐿 ( 𝑡 ) . Average versus Individual-run Behavior
In addition to giving access to fluctuations, stochastic simulations allow for direct comparisonof individual reaction pathways to the average behavior of a set of simulations. This isanalogous to a comparison of bulk behavior to single-molecule behavior in the field of single-molecule experiments, the latter of which is much more difficult to access experimentally. Forthis section, we ran a sweep of the ratio of parameters ℛ = 𝑐 𝑛 − 𝑘 𝑎 , (16)20igure 7: A comparison of 𝐿 ( 𝑡 ) for the two sets of rate constants. For set 2 shown in (a), 𝐿 ( 𝑡 ) is almost exactly the same even for very small 𝑁 . For set 1 shown in (b), however, 𝐿 ( 𝑡 ) not only peaks lower but it’s overall behavior changes significantly as 𝑁 is decreased.where 𝑐 𝑛 − on the right hand side makes the ratio dimensionless, to gain insight on howindividual reaction pathways may change as parameters are varied. Fig. 9 shows two sets ofsimulations at different values of ℛ by varying 𝑘 and keeping the rest of the rate constantsthe same. It is clear that when secondary nucleation is relatively unimportant, the individualruns of the simulation behave much like their average. However, when this ratio becomeslarge, the individual behavior can deviate quite significantly from the average.Additionally, for low ℛ , it turns out that the distribution of halftime about the meanvalue is close to normal, while for larger values it becomes skewed. Crowders Change Dynamics
A more physiologically relevant example of the present model is to see how the presence ofcrowder molecules can change the local behavior. Fig. 10 shows that as the volume fraction, 𝜑 , of crowders increases, we see similar spread in local behavior compared to the average aswe did when directly changing the rates. This is because the growth rates are directly affectedby 𝜑 as seen in the theory section. From Fig. 11, you can see that secondary nucleationcompletely dominates the growth process as 𝜑 is increased. However, before it can occur, an21igure 8: Relative reaction frequencies for the two different sets of rate constants for bothlarge and small 𝑁 . Orange ( (cid:4) ) corresponds to monomer addition, blue ( ∙ ) to primarynucleation, green ( x ) to fragmentation, red ( (cid:78) ) to coagulation and purple ( (cid:72) ) to monomersubtraction. For set 2 (a and b), there is no change in the reactions even for very small 𝑁 .For set 1 (c and d), not only do more reactions become relevant during the growth phase atsmall 𝑁 , there are significant fluctuations in the reaction rates.22igure 9: Local dynamics changing as ℛ increases. The solid line represents the averagewhile dashed lines represent individual stochastic simulation runs. Simulation were runwith fixed parameters 𝑁 = 2000 , 𝑐 = 10 µ m , 𝑎 = 2 µ m − s − , 𝑏 = 0 . − , 𝑘 𝑎 = 𝑘 𝑏 = 0 , 𝑘 𝑛 = 0 . µ m − s − and 𝑛 = 𝑛 𝑐 = 2 .incubation period exists, as shown in the case of 𝜑 = 0 . of Fig. 11. Hence, in the individualruns at large 𝜑 , there is a short period of no growth before the first primary nucleation eventoccurs, followed by a rapid explosive period of growth once a polymer has formed and theauto-catalytic secondary-nucleation process can occur. These simulations were run with 𝑛 𝑐 = 𝑛 = 2 , and the effect is generally more exaggerated when 𝑛 > 𝑛 𝑐 . This is a purelystochastic phenomenon, as fluctuations in the first-passage time of primary nucleation (aswell as monomer addition in the case of 𝑛 > 𝑛 𝑐 ) leads to the spread of the individual runs,producing an average that does not represent the individual reaction pathways of the system.Moreover, it is clear that the presence of crowders can magnify these differences by increasingcertain reaction propensities (e.g secondary nucleation) more than others (such as merging oraddition).In other words, beyond increasing the overall rate of the reaction, the actual way in whichthe polymers grow is changed significantly. This can be shown in terms of the scaling lawanalysis as well. Fig. 12 shows how the scaling law governing the crowderless reaction issignificantly different from that with even fairly low 𝜑 . The scaling laws here agree with23igure 10: Local dynamics changing as crowder volume fraction is increased. Individualstochastic runs are represented by dashed lines, the average by a solid line. For no crowdersand low crowder concentrations, the individual runs behave similarly to the average. At large 𝜑 , however, the individual runs behave very differently from the average.24igure 11: Relative frequency of the various mechanisms of growth. Curves shown aresecondary nucleation ( (cid:72) ), primary nucleation ( ), fragmentation ( ∙ ) and monomer addition( (cid:4) ). As 𝜑 increases, secondary nucleation begins to supress all other mechanisms. Atintermediate values, it is clear that there is competition between multiple growth mechanisms.25he reaction picture, in that for 𝜑 = 0 the slope is close to (corresponding to nucleation-elongation. 𝛾 = − 𝑛 𝑐 / ) and for 𝜑 = 0 . the slope is . (secondary nucleation dominating. 𝛾 = − ( 𝑛 + 1) / ). One implication of this is that knowledge of the dynamics of an aggregatingsystem in vitro does not necessarily translate to the same protein aggregating in vivo , wheremuch of the system is occupied by molecules which do not participate in the reaction otherthan to exclude volume, resulting in the entropic effects.Figure 12: Scaling law comparison of the same system with different volume fraction ( 𝜑 )of crowders. For the crowder-less case, the slope is close to corresponding to nucleation-elongation. For 𝜑 = 0 . , it is close to . corresponding to secondary nucleation. In between,there is competition between nucleation-elongation and secondary nucleation as the dominantmechanisms shift. Comparison With the PM-model
In order to show more clearly the effects of low particle number, we compared the stochasticapproach to the moment-closure approximation of the reaction-rate-equation approach. Thelatter will be called the PM-model in the present article. This model reduces a large setof rate equations for the concentration of each species to three closed differential equations.One for the monomer concentration, 𝑐 ( 𝑡 ) , and one each for the first two moments of thedistribution of aggregates: the number of polymers, 𝑃 ( 𝑡 ) , and the mass contained in polymers,26 ( 𝑡 ) . Additionally, the average length of polymers, 𝐿 ( 𝑡 ) , is computed as the ratio 𝑀 ( 𝑡 ) /𝑃 ( 𝑡 ) .For a more detailed description, we refer the reader to the references. Figs. 13(a) and(b) show comparisons of 𝑀 ( 𝑡 ) and 𝐿 ( 𝑡 ) for the PM-model (fitted to experimental data inSchreck et al. ) as well as stochastic simulation at different values of 𝑁 . It is clear that forlarge 𝑁 , the stochastic results agree with the bulk behavior predicted by the PM-model, butdiffer significantly as 𝑁 decreases. For lower values of 𝑁 , the overall rate of production ofmass and asymptotic length are much decreased. Of course, for certain small values of 𝑁 ,depending on chosen rate constants, the average length predicted by stochastic dynamicscould not possibly agree with that of the PM-model. It is important to note, however, that 𝐿 converges concurrently with 𝑀 as 𝑁 is increased. An interpretation of this is that, forstochastic dynamics to agree with bulk methods, 𝑁 must be large enough for the lengthprofiles to agree.Figs. 13(c) and (d) show even more interesting behavior. For smaller 𝑁 , the individualstochastic runs actually reach their equilibrium mass values more rapidly , following initialnucleation, than do the individual runs for large 𝑁 . So the reduction in the rate of relativemass production is purely due to the stochastic nature of the system: nucleation events aremore spread out and, on average, take longer to occur. Additionally, at larger 𝑁 , more thanone nucleation event generally occurs as evidenced by kinks in the individual runs. Whenconsidering that the number of molecules in the simulation is analogous to the volume ofthe system at fixed concentration, this implies that the dynamics may change significantlydepending on system volume. Discussion and Conclusions
In summary, we have introduced a powerful stochastic method based on the Gillespie SSAto study the time evolution of the relative frequencies of aggregation reaction mechanisms.We tested this method by comparing with the scaling law treatment of Meisl et al. to27igure 13: The fraction of mass contained in polymers (a) and the average length of polymers(b) compared with the PM-model (dashed). Stochastic simulations were done with N = 1000( ∙ ), 2500 ( ), 10000 ( (cid:4) ), 20000 ( (cid:72) ). In (a) and (b), open circles refer to the mean valueobtained using the PM model. In (c) and (d), comparison of the individual runs (dashed)with the average (solid) of the stochastic model. For large N, the individual runs closelyresemble the average, which fits the PM-model prediction. For smaller N, however, not onlydo the results differ drastically from the continuous model, the individual behavior changesrelative to the average. Simulations were run using the Oosawa model with parameters fromSchreck et al., obtained by fitting ThT data for actin in the presence of dextran from Rosinet al., 𝑎 = 911 µ m − s − , 𝑏 = 50 s − , 𝑘 𝑛 = 1 . × − µ m − s − and 𝑛 𝑐 = 3 .28onfirm that the dominant mechanisms predicted by the scaling exponent, 𝛾 , agreed with therelative reaction frequencies of those mechanisms. In particular, we showed that as the scalingexponent increased, competition between mechanisms was seen as the relative frequency ofmonomer addition was overcome by secondary nucleation as 𝑐 increased. Additionally, detailas to which mechanisms become important and at what time during the reaction can giveinsight into why observable quantities, such as polymer mass, behave as they do. The methodprovides great detail into the behavior of reacting systems and, in principle, can be used forany stochastic system where knowledge of the importance of particular reactions or events asthey change over time is desired.We showed that the halftime of 𝑀 ( 𝑡 ) is not always proportional to /𝑁 (or /𝑉 ), asshown by Tiwari and van der Schoot, and in fact can reach a thermodynamic limit whereincreasing 𝑁 has no effect on 𝑡 / . This effect, along with the behavior of 𝜎 / , is stronglydependent on the values of the rate constants. Accordingly, for a choice of rate constantswhere 𝑡 / does not increase with decreasing volume, the time-scale of 𝑀 ( 𝑡 ) does not changeas volume is decreased, whereas it can increase quite significantly for another set of rateconstants. By looking at the average polymer length over time, we showed that simulationswith rate constants which favored smaller polymers did not show changing results for therange of 𝑁 used in this study, while those using rate constants which favored very longpolymers had dramatic changes in dynamics. This is further confirmed by the reactionfrequencies for longer polymer-forming rate constants fluctuating more significantly, and moremechanisms being important in the reaction at smaller values of 𝑁 . In other words, thedynamics can be very different for small volumes. Physically, this is consistent with smallervolumes being less conducive to very large polymers growing. This implies that, when fittingmodels to experimental data in order to predict behavior within living cells, one shouldalso fit 𝐿 ( 𝑡 ) to data on the average length of polymers, or at the very least bias the fits toachieve a best guess of the actual length profiles, as was done in Schreck et al. That atomicforce microscope (AFM) measurements of polymer length can be used in addition to ThT29easurements to obtain more robust estimates of rate constants was previously pointed outby Schreck and Yuan. We also compared individual stochastic reaction pathways with the average value calculatedfrom many runs for a range of parameter values, as well as in the presence of crowders. Weshowed that for certain parameters, the individual runs can look very different than theaverage. In the case studied, when secondary nucleation was relatively unimportant comparedto monomer addition, the individual runs looked similar to the average. When secondarynucleation was made more important, however, the individual runs differed greatly fromthe average. This reflects both the change in reaction dynamics and the skewness in thedistribution of 𝑡 / . When crowders were included, this change was even more dramatic,and plots of the reaction frequencies indeed confirm that secondary nucleation could becomecompletely dominant.Furthermore, we compared the stochastic approach to the continuous, PM-model. Forlarge values of 𝑁 the two models are in good agreement, but diverge significantly as 𝑁 decreases. Again, we saw that 𝐿 ( 𝑡 ) not being in agreement for the two models was indicativeof this divergence, further confirming the importance of having experimental data on thelengths of polymers. Additionally, analysis of the individual runs shows that, for small valuesof 𝑁 , the local behavior is vastly different than the average behavior. This means that thelocal behavior effect mentioned previously can be caused not only by the presence of crowders,but by decreasing the reaction volume. The specific model compared was the Oosawa modelwithout secondary nucleation, so the effect is present even in the simplest of models providedthey have more than one mechanism of growth.Lastly, to our stochastic kinetic simulator we can add other reaction mechanisms todifferent degrees of sophistication, depending on the system that we are investigating. Animportant next step is to apply our methods to protein aggregation systems that have beenstudied experimentally, especially, in vivo . For such systems, we should first fit the observed 𝑀 ( 𝑡 ) and/or 𝐿 ( 𝑡 ) curves by varying rate constants and other parameters. With the30et of constants determined, the present stochastic scheme can be used to provide valuableinformation on the fluctuations, the reaction dynamics, and pathways involved in the systemand how they vary with the changes of system volume and the amount of crowders.All stochastic simulations were performed using popsim , Acknowledgements
We thank Prof. Frank Ferrone for stimulating discussion, Karsten Chu for useful discussionat the early stage of the work. It is late, but we still want to dedicate this work to Prof.William P. Reinhardt for his influence on our research in general. The authors have nofunding institutions/support to report.
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40 and A 𝛽
42 Peptides.
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