Mean-field Master Equations for Multi-stage Amyloid Formation 1. Mean-field Master Equations for Multi-stage Amyloid Formation. Blake C.

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1 Mean-field Master Equations for Multi-stage myloid Formation 1 Mean-field Master Equations for Multi-stage myloid Formation lake C. ntos Department of Physics, Drexel University, Philadelphia, P 19104, United States Final Project. PHYS 502 Mathematical Physics 2 March 15, 2016

2 MEN-FIELD MSTER EQUTIONS FOR MULTI-STGE MYLOID FORMTION 2 bstract ased on the master equation formalism of Michaels and Knowles [1] and the kinetic study of fibril growth of Schreck and Yuan [2], we construct an amyloid formation model with a wider scope accounting for a larger number of possible transitions as well as size dependent rate constants. The assumption of diffusion-limited reactions allows for the use of the mean-field master equation formalism, leading to a set of coupled first order differential equations to describe the time evolution of the system s composition. y using a hybrid NP-NCC mechanism, we consider a multi-stage fibril growth pathway including a wider range of potential constructive and destructive pathways than previously studied that may lead to interesting insights into the physical nature of observed amyloid fibril formation over time. Keywords: amyloid, fibril, biofilaments, master equation

3 MEN-FIELD MSTER EQUTIONS FOR MULTI-STGE MYLOID FORMTION 3 Mean-field Master Equations for Multi-stage myloid Formation Introduction Of particular interest in the field of iophysics is the study of protein aggregation and amyloid formation. Under certain circumstances, proteins have been shown to aggregate and sometimes form biofilaments. Such structures are known to be associated with lzheimer s disease, though the mechanisms by which these amyloid fibrils are formed are not fully understood. One heavily studied mechanism is nucleated polymerization (NP), in which proteins undergo spontaneous, primary nucleation of monomers followed by elongation, which, over time, leads to the formation of fibrils. This model, like that put forth by Michaels and Knowles, while it reproduces some experimentally verified features, is likely drastically simplified. For instance, this model precludes the possibility of having disordered oligomers, assuming all oligomers are in a fibril conformation. nother mechanism includes disordered oligomers as an intermediate conformation between monomers and roughly linear fibrils: nucleated conformational conversion (NCC). Such a mechanism allows oligomers to form below the monomer concentration required by the NP mechanism, called the critical fibril concentration. These sub-critical oligomers may then undergo a conformational transition to the fibril state. This mechanism was studied by Schreck and Yuan, though some aggregate transitions were disallowed. This paper aims to construct a model including these transitions for future in-depth study. Protein ggregation Systems Like the models on which this paper is based, we consider a limited number of possible transitions the aggregates can undergo.

4 MEN-FIELD MSTER EQUTIONS FOR MULTI-STGE MYLOID FORMTION 4 Primary Nucleation - ssuming our system has an initial state consisting of only N monomers, the most basic transition is that of the aforementioned primary nucleation, in which a collection of n c monomers aggregate to a disordered oligomer. Due to the high solubility of these monomers, the reverse transition, denucleation, may also occur, returning the n c monomers to the solution. Elongation - Once the oligomer is formed, elongation, a term whose origin will be later elucidated, may occur. n oligomer undergoes aggregation when a single monomer collides (this model assumes diffusively) with the oligomer, increasing its aggregate number by one. gain, the reverse process, dissociation, is also possible, whereby a single monomer, due to thermal fluctuations, detaches itself from the aggregate, returning to the solution and reducing the oligomer s aggregate number by one. ggregation Should two oligomers interact with each other (again, diffusively), they may undergo aggregation and merge to create a large aggregate. The reverse process, called fragmentation, occurs when a subset of aggregates separates from the oligomer, forming two smaller oligomers. Conformational Transition Introduced by, and central to, the NCC mechanism, is the process by which disordered oligomers may transition to an ordered, roughly linear fibril. The reverse process is also considered in this paper. The processes of elongation, dissociation, aggregation, and fragmentation may also occur for fibrils, bringing our total number of possible transitions to 12. This model assumes that primary nucleation cannot lead directly to a fibril, necessitating an oligomer intermediate. In the model put forth by Schreck and Yuan, the oligomer phase was bypassed, joining the processes of

5 MEN-FIELD MSTER EQUTIONS FOR MULTI-STGE MYLOID FORMTION 5 nucleation and conformation transition, and eliminating denucleation, elongation, dissociation, aggregation, and fragmentation of oligomers. Fibril fragmentation, elongation, and dissociation, as well as reverse transition back to the oligomer conformation were also disallowed, bring the total allowed transitions down to 3: nucleation/conformational transition, aggregation, and fragmentation. This model also allows for variable n c down to a minimum value n c min. The Mean-field Master Equation With the assumption of diffusion-limited interactions in mind, we can consider the trajectories all monomers and aggregates of the system to be random. orrowing heavily from Michaels and Knowles NP model, this consideration allows us to employ the use of the master equation treating all the 12 aforementioned processes as probabilistically determined Markov transitions between states. These states are monomeric, oligomeric of aggregate size l = n c min to N, and fibrilic of aggregate size l = n c min to N, yielding a total of 2(N n cmin ) + 1 states. Coupled with the size-dependent transitions from generic state x to state y, T x y, we construct the master equation dp(y, t) dt = T x y P(x, t) T y x P(y, t). x That is, the time derivative of the probability of state y relies positively on the transitions from any state x to state y as well as the probability of state x at time t, and negatively on the transitions from state y to any other state x as well as the probability of state y at time t. Since this model is not meant to provide a microscopic, detailed description of protein aggregation, but a bulk, macroscopic view involving a large number of molecules, we may consider average concentrations, c, of each aggregate number, l, of each conformation, x,: x

6 MEN-FIELD MSTER EQUTIONS FOR MULTI-STGE MYLOID FORMTION 6 c x l (t) = N l x P(y, t) V where N l x is the number of aggregates of length l and in state x. We can now write the time y derivative of these concentrations in terms of the transitions rates, k y x l, that map an l -mer in l state y to an l-mer in state x: d dt c l x (t) = k l l y,l y x c l y (t) The transition rates we will use in this model, for which we use to signify oligomers and to signify fibrils, are those for: Nucleation and denucleation: k nc and k nc respectively, mapping n c monomers to an n c -mer oligomer, with the reverse mapping for denucleation. s previously mentioned, the oligomers, once formed are highly soluble and therefore unstable. The nucleation and denucleation rates are thus set to be equal: k nc = k nc. Elongation and dissociation: k + and k +, and k and k respectively, for each corresponding aggregate conformation, mapping a monomer and an l-mer oligomer or fibril to an (l + 1)-mer oligomer or fibril, with the reverse mapping for the dissociation rate constants ggregation and fragmentation: k r,s and k r,s, and k r+s and k r+s respectively, mapping an r-mer and s-mer oligomer to an (r + s)-mer oligomer or an r-mer and s-mer fibril to an (r + s)-mer fibril, with reverse mappings for the fragmentation constants Conformational transition: k r and k r, mapping, respectively, an r-mer from oligomer (state ) to fibril (state ), and an r-mer from fibril (state ) to oligomer (state ).

7 MEN-FIELD MSTER EQUTIONS FOR MULTI-STGE MYLOID FORMTION 7 ll of these transition rate constants, also known as kernels, are considered size-dependent in this model, with the exception of the nucleation rate. Considering all of these transitions and the concentrations on which they depend, including the monomer concentration c 0 (t), we can construct a pair of equations for the time rate of change of an r-mer of each conformation (all concentrations are understood to be timedependent) d dt c r = k (nc =r) (c r 0 c r ) + k r c r k r c r + k (r+1) r n c 2 + [k (r s),s c r s c s k r+s c r ] c r+1 N r k r c r + k (r 1) + [k r,s c r c s ] N c r 1 k r + c r + [k r+s c r+s ] s=n c min s=n c min s=r+n c min and d dt c r = k r c r + k r c r + k (r+1) r n c 2 + [k (r s),s c r s c r+1 k r c r + k (r 1) + c s k (r s)+s c r ] N r c r 1 k r + c r [k r,s c r c s ] s=n c min s=n c min N + [k r+s c r+s ] s=r+n c min as well as the rate of change of the monomer concentration N 1 d dt c 0 = [k (nc=r )( c r 0 + c r ) k r + c 0 c r + (k (r+1) r=n c min + (k (r+1) )c (r+1) ] )c (r+1) k r + c 0 c r It should come as no surprise that the first two equations are nearly identical with a few exceptions: 1) the fibril equation has no nucleation term (the first term in the oligomer equation),

8 MEN-FIELD MSTER EQUTIONS FOR MULTI-STGE MYLOID FORMTION 8 and 2) the signs in front of the conformational transition terms are opposite between the two equations. The summations in the equations are, in order: the sum of all possible transitions to the r-mer by aggregating an s-mer with an (r s)-mer minus all possible fragmentations of the r-mer to an s-mer and an (r s)-mer minus the sum of all possible transitions from the r-mer to an (r + s)-mer by aggregation the sum of all possible transitions to the r-mer by fragmenting an (r + s)-mer to an r-mer and an s-mer. These equations clearly are coupled not only to each other, but to the equations corresponding to every other value of r on the interval [n c, N] as well. Thus we have a system of min 2(N n c ) + 1 first order coupled non-linear differential equations that must be solved min numerically. Elongation & Dissociation Size-Dependent Rate Constants We begin by finding expressions for the elongation kernels. Using the Smoluchowski coagulation kernel [3] for diffusion-limited aggregation of aggregates of size x 1 and x 2 with fractal dimensions y 1 and y 2, respectively β = 2 k T 3 η (x 1 1 y1 1 + x y2 2 ) (x 1 y x 1 y 2 2 ) where k is the oltzmann constant, T is the system temperature, and η is the viscosity of the solution. The fractal dimension, henceforth called D f, for a roughly spherical colloid [4], which the unorganized oligomers likely closest resemble, is D f 2.2,

9 MEN-FIELD MSTER EQUTIONS FOR MULTI-STGE MYLOID FORMTION 9 for a polymer chain [5], like our fibrils, is D f 1.37, and for a monomer, which we will consider to be spherical, D f = 3. This gives us our final form for our elongation kernels acting on an r-mer and a monomer k r + = 2 k T 3 η (r ) (r ) = 2 k T 3 η (2 + r r 1 2.2) k r + = 2 k T 3 η (r ) (r ) = 2 k T 3 η (2 + r r ). In order to relate the dissociation kernels to the elongation kernels, we consider the reactions at equilibrium [2][6] when the concentrations c l x (t) settle to ρ l x : k r + ρ (r 1) ρ 0 + k r ρ (r+1) = k r + ρ r ρ 0 + k r ρ r and likewise with the fibrils at equilibrium. Thus the transitions away from the r-mer are equal to those into it. Since we are considering these concentrations at equilibrium, ρ r = Z r where Z r is the partition function of an r-mer oligomer. gain, the same is true for fibrils. For a spherical aggregate, such as the disordered oligomer, the partition function can be written [7] V Z r e β(εr ar2/3 ) where ε is the interaction energy per particle, given as ε 3 [8], β is the thermodynamic 1 k T, and a is related to the surface contribution to the free energy of the aggregate. For approximately linear aggregates [9] however, Z r r n where n is a fit parameter between the values of 4 and 6. Using these expressions, we can write relate the dissociation kernels to the elongation kernels using the size of the aggregate

10 MEN-FIELD MSTER EQUTIONS FOR MULTI-STGE MYLOID FORMTION 10 k r = k eβ(ε(r 1) a(r 1)2/3) e β(εr ar2/3 ) r + e β(εr ar2/3) e β(ε(r+1) a(r+1)2/3 ) k r = k (r 1)n r n r + r n (r + 1) n. ggregation & Fragmentation Utilizing the Smoluchowski coagulation kernel once again, though this time for oligomer-oligomer coagulation and fibril-fibril coagulation, we obtain the aggregation kernels k r,s = 2 k T 3 η (r s 1 2.2) (r s 1 2.2) = 2 k T 3 η (2 + r1 2.2 s r s 1 2.2) k r,s = 2 k T 3 η (r s ) (r s ) = 2 k T 3 η (2 + r s r s ). Through the same process used in the previous step, we find a relationship to the fragmentation kernels based on the aggregate size k r+s = k r,s e βa((r+s)2/3 r 2/3 s 2/3 ) k r+s = k r,s r n s n (r + s) n. Conformational Transition No theoretical model was obtained to predict the r dependence of the conformational transition kernel, so we use in its place a simple two parameter fit k r = γr w.

11 MEN-FIELD MSTER EQUTIONS FOR MULTI-STGE MYLOID FORMTION 11 Once more calling upon the equilibrium argument, we obtain the reverse transition eβ(εr ar2/3 ) k r k r where an unknown proportionality factor remains, since we cannot be sure, and, in fact, have no reason to think, the proportionality constants in the oligomer and fibril partition functions are the same. Conclusion Following in the footsteps of Knowles and Schreck, we have managed to theorize a more inclusive model for multi-stage amyloid formation. ased on the method by which the model was constructed, we expect it to yield results agreeable with those of the aforementioned authors, though the discrepancies between the models could provide insight into how each aspect of the models ultimately affects the behavior of the system over time. We would be interested in seeing this model tested against experimental result to see if helps account for known discrepancies in measurements of amyloid concentration over long incubation times. Unfortunately, the computing power necessary to implement this model was not available to the author, though he spent many hours constructing a Runge-Kutta 4 integrator for the model. Perhaps it may be possible in the near future to test the model on a super-computing cluster such as Drexel s Dirac cluster. References [1] Michaels, T. C., & Knowles, T. P. (201). Mean-field master equation formalism for biofilament growth. merican Journal of Physics. [2] Schreck, J. S., & Yuan, J.-M. (2013). kinetic study of amyloid formation: fibril growth and length distributions. The Journal of Physical Chemistry, r n

12 MEN-FIELD MSTER EQUTIONS FOR MULTI-STGE MYLOID FORMTION 12 [3] Kryven, I., Lazzari, S., & Storti, G. (2014). Population balance modeling of aggregation and coalescence in colloidal systems. Macromolecular Theory and Simulations, [4] Jiang, Q., & Logan,. E. (1991). Fractal dimensions of aggregates determined from steadystate size distributions. Environmental Science & Technology, [5] Havlin, S., & en-vraham, D. (1982). Fractal dimensionality of polymer chains. Journal of Physics, L311-L316. [6] Wattis, J.. (2006). n introduction to mathematical models of coagulation-fragmentation processes: discrete deterministic mean-field approach. Physica D: Nonlinear Phenomena, [7] Sattler, K. D. (2010). Handbook of Nanophysics: Principles and Methods. oca Raton, FL: CRC Press. [8] Volkenstein, M. V. (1977). Molecular iophysics. New York, NY: cademic Press, Inc. [9] Hill, T. L. (1983). Length dependence of rate constants for end-to-end association and dissociation of equilibrium linear aggregates. iophysics J,