An Efficient Kinetic Model for Assemblies of Amyloid Fibrils and Its Application to Polyglutamine Aggregation

Protein polymerization consists in the aggregation of single monomers into polymers that may fragment. Fibrils assembly is a key process in amyloid diseases. Up to now, protein aggregation was commonly mathematically simulated by a polymer size-structured ordinary differential equations (ODE) system, which is infinite by definition and therefore leads to high computational costs. Moreover, this Ordinary Differential Equation-based modeling approach implies biological assumptions that may be difficult to justify in the general case. For example, whereas several ordinary differential equation models use the assumption that polymerization would occur at a constant rate independently of polymer size, it cannot be applied to certain protein aggregation mechanisms. Here, we propose a novel and efficient analytical method, capable of modelling and simulating amyloid aggregation processes. This alternative approach consists of an integro-Partial Differential Equation (PDE) model of coalescence-fragmentation type that was mathematically derived from the infinite differential system by asymptotic analysis. To illustrate the efficiency of our approach, we applied it to aggregation experiments on polyglutamine polymers that are involved in Huntington’s disease. Our model demonstrates the existence of a monomeric structural intermediate acting as a nucleus and deriving from a non polymerizing monomer (). Furthermore, we compared our model to previously published works carried out in different contexts and proved its accuracy to describe other amyloid aggregation processes.


Conversion of the Xue et al ODE model into a PDE model:
example of the best-fit model.
Their best-fit model is given by the following processes: • no conformational exchange, no coalescence and no degradation of polymers or monomers, • the size of the nucleus is i 0 = 2 and denucleation occurs only through depolymerization, • polymerization and depolymerization follow a step function with a step at i = 6, • fragmentation into two smaller polymers occurs.
Thus, using the previously introduced notations, the original ODE system can be simplified as follows: Unlike the formulation of Xue et al., we did not introduce fragmentation into polymers of size 1, which can be included in the depolymerization terms. For the particular choice of fragmentation made in [1], however, fragmentation in polymers of size 1 is close to 0. This ODE system is then formally equivalent to the following PDE system: Note that our notations are slightly different from those presented in the SI of [1]. Due to the particular shape of the polymerization process, with a step at i = 6, it may be preferable to keep all the ODEs (3) occurring for i ≤ 6 and consider the PDE (5) only for i ≥ 6. Though nothing is indicated in [1], we suspect that the polymerization rate is much larger for i ≥ 6 than for i ≤ 6, and on the contrary that depolymerization is much smaller for larger i. If true, this would be interpreted as in some sense an energetic barrier or a kind of 'second nucleus' for i = 6. We then adapt the boundary condition (6) in the case where depolymerization and fragmentation for small polymers can be ignored and state, supposing k 5 on k 6 on and thus an instantaneous equilibrium: c 5 given by Equation (3) taken for i = 5. A more detailed study would require knowing the order of magnitude of the best-fit parameter values.
2 Discussion on the Fragmentation Kernel

Discrete setting
The strategy developed in [1] to analyse the growth of amyloid fibrils consists in fitting transitional parameters of experimental reaction progress curves with 21 mathematical models combining pre-polymerization, elongation and fragmentation processes. In their papers, Xue and co-workers compare two different prepolymerization and three different elongation functions, but only one fragmentation distribution according to the following equation based on statistical mechanical considerations for linear polymers [2]: with k i,j of f the first order fragmentation rate of a species of size i into an aggregate of size j and an aggregate of size j − i, a the overall amplitude and b describing the size and position dependence of the fragmentation rate constant.
In order to generalize this approach to other fragmentation processes, we investigated the effect of the distribution of fragmentation on the transitional parameters, namely the length of the lag phase and the slope of the growth curve at the inflexion point. We simulated the following model (see main text for notations and assumptions, equations [30]-[32]): with two different distributions for the fragmentation : (i) the fragmentation rate defined by (8), as in [1], and (ii) a uniformly distributed fragmentation rate along the aggregates (i.e. k j,i of f = with which a polymer of size i can fragmentate to give smaller aggregates was taken to be equal in both cases to study only the effect of the distribution of the fragmentation rate among smaller polymers: Figure S4 Left represents the typical shape of these two distributions for an aggregate of size i = 20. The other processes were taken as for the best fitting model of [1], i.e. no pre-polymerization and a fragmentation described by a step law. The numerical values used in the simulations were chosen to roughly reproduce the experimental curves of Xue et al [1]: • i 0 = 2 Figure S4 Right represents the normalized reaction progress curves for three different initial concentrations (50µM , 100µM and 150µM ). Table S2 represents the transitional parameters of these curves, automatically extracted as described in [1]. One can observe graphically and numerically that these parameters are not very sensitive to the distribution of fragmentation.

Continuous setting
As stated above, the fragmentation kernel proposed by [1] is given by Equation (8). For a continuous model, a first attempt would be to simply replace j < i by x < y ∈ (0, +∞). However, in such a case ln(x) may become negative, so to avoid this we use the artefact of replacing ln(j) by ln(x + 1), and ln(i − j) by ln(y − x + 1). Indeed, in the original article [2] quoted by [1], polymers' diffusion coefficient are defined using a formula of Riseman and Kirkwood [3]. In this last article, ln appears through ln(1/n), where n being the number of monomers inside a polymer is supposed to be large, so that it is possible to replace it by ln(1/(n + 1)). We thus obtain the following formula: In [1], fragmentation as a secondary process was requested to fit β 2 -microglobulin experimental data. In addition, fragmentation seems to be a critical process in some others fibrillation processes, including prion strains [4]. Thus, developing a strategy that would help to characterize not only the magnitude of the fragmentation but also the distribution of this fragmentation along the aggregate size appears essential. Many studies suggest using the size distribution of aggregates to characterize more fully the distribution of elongation and fragmentation parameters [5][6][7] (see Figure S5), through PDE formalism and inverse problem techniques, for instance.   Transitional parameters extracted from the simulated reaction progress curves represented in Figure S4 Right