First passage times in homogeneous nucleation: Dependence on the total number of particles

Abstract

Motivated by nucleation and molecular aggregation in physical, chemical, and biological settings, we present an extension to a thorough analysis of the stochastic self-assembly of a fixed number of identical particles in a finite volume. We study the statistics of times required for maximal clusters to be completed, starting from a pure-monomeric particle configuration. For finite volumes, we extend previous analytical approaches to the case of arbitrary size-dependent aggregation and fragmentation kinetic rates. For larger volumes, we develop a scaling framework to study the first assembly time behavior as a function of the total quantity of particles. We find that the mean time to first completion of a maximum-sized cluster may have a surprisingly weak dependence on the total number of particles. We highlight how higher statistics (variance, distribution) of the first passage time may nevertheless help to infer key parameters, such as the size of the maximum cluster. Finally, we present a framework to quantify formation of macroscopic sized clusters, which are (asymptotically) very unlikely and occur as a large deviation phenomenon from the mean-field limit. We argue that this framework is suitable to describe phase transition phenomena, as inherent infrequent stochastic processes, in contrast to classical nucleation theory.

Quasi steady state approximation of the small clusters in Becker–Döring equations leads to boundary conditions in the Lifshitz–Slyozov limit

“…Hence, our work shed lights on which appropriate boundary condition should be used for the LS equation (or similar continuous coagulation models) according to specific microscopic hypotheses (unfavorable, balanced or irreversible nucleation). We believe that our procedure could be applied to several related models (for instance, the Lifshitz-Slyozov-Wagner equation mentioned above, or the prion equation [10]) and should help to build reduced structured population models while taking into account of their intrinsic multi-scale nature (see [29,28] for applications). and for all i ≥ 2,ã i = a i /A,b i+1 = b i+1 /B , and the particular scaling at the boundary (we use different letters to emphasize this point):α := a 1 /A 1 ,β := b 2 /B 2 .…”

From Becker-Döring to Lifshitz-Slyozov: deriving the non-local boundary condition of a non-linear transport equation

“…By proving a quasi steady-state result, we were able to prove that each discrete size tend to equilibrate to a value given by the stationary state of an auxiliary system, very similar to the original deterministic BD model (but in a linear version). Finally, second order approximation and large deviation phenomena of the stochastic discrete BD system (14) were observed numerically [10]. In particular, when the formation of large cluster is (asymptotically) very unlikely, the latter appears as a large deviation from the mean-field limit and gives a suitable framework to describe phase transition phenomena, as inherent infrequent stochastic processes, in contrast to classical nucleation theory.…”

“…(18). The importance of deriving such results is both numerical, in order to design fast numerical scheme to approximate large discrete system, and theoretical, to derive steady-state and time-dependent properties of the original discrete system from a (simpler) continuous one [10,11]. In our scaling, each cluster of size initially i ≥ 2 is seen as a cluster of size roughly iε ∈ R + , and our scaling consists in an acceleration of the fluxes (by 1/ε) in Eq.…”

Asymptotics of Stochastic Protein Assembly Models

“…Equation (30) shows that the variance of the lag time is inversely proportional to γ/γ * , a low misfolding rate will thus increase the variability of the polymerisation process.…”

“…As underlined by previous studies Szavits-Nossan et al [28], the nucleation step is intrinsically stochastic, leading to an important variability among replicated experiments, not only in small volumes but even in relatively large ones, see Xue et al [29]. The question of building convenient stochastic models, able to render out the heterogeneity observed, and even to predict it, has recently raised much interest in the biological and biophysical community, see Szavits-Nossan et al [28], Yvinec et al [30], Pigolotti et al [24] and Eden et al [9].…”