Asymptotics of Stochastic Protein Assembly Models

Self-assembly of proteins is a biological phenomenon which gives rise to spontaneous formation of amyloid fibrils or polymers. The starting point of this phase, called nucleation exhibits an important variability among replicated experiments. To analyse the stochastic nature of this phenomenon, one of the simplest models considers two populations of chemical components: monomers and polymerised monomers. Initially there are only monomers. There are two reactions for the polymerization of a monomer: either two monomers collide to combine into two polymerised monomers or a monomer is polymerised after the encounter of a polymerised monomer. It turns out that this simple model does not explain completely the variability observed in the experiments. This paper investigates extensions of this model to take into account other mechanisms of the polymerization process that may have impact an impact on fluctuations. The first variant consists in introducing a preliminary conformation step to take into account the biological fact that, before being polymerised, a monomer has two states, regular or misfolded. Only misfolded monomers can be polymerised so that the fluctuations of the number of misfolded monomers can be also a source of variability of the number of polymerised monomers. The second variant, based on numerical considerations, represents the reaction rate α of spontaneous formation of a polymer as of the order of N−ν , for some large scaling variable N representing the reaction volume and ν some positive constant. Asymptotic results involving different time scales are obtained for the corresponding Markov processes. First and second order results for the starting instant of nucleation are derived from these limit theorems. The proofs of the results rely on a study of a stochastic averaging principle for a model related to an Ehrenfest urn model, and also on a scaling analysis of a population model.