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On the Asymptotic Distribution of Nucleation Times of Polymerization Processes

Philippe Robert 1 Wen Sun 1
1 MAMBA - Modelling and Analysis for Medical and Biological Applications
Inria de Paris, LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
Abstract : In this paper, we investigate a stochastic model describing the time evolution of a polymerization process. A polymer is a macromolecule resulting from the aggregation of several elementary subunits called monomers. Polymers can grow by addition of monomers or can be split into several polymers. The initial state of the system consists of isolated monomers. We study the {\em lag time} of the polymerization process, that is, the first instant when the fraction of monomers used in polymers is above some threshold. The mathematical model includes {\em a nucleation property}: If $n_c$ is defined as the size of the nucleus, polymers with a size smaller than $n_c$ are quickly fragmented into smaller polymers. For polymers of size greater than $n_c$, the fragmentation still occurs but at a smaller rate. A scaling approach is used, by taking the volume $N$ of the system as a scaling parameter. If $n_c{\ge}3$ and under quite general assumptions on the way polymers are fragmented, if $T^N$ is the instant of creation of the first "stable" polymer, i.e. a polymer of size $n_c$, then it is proved that $(T^N/N^{n_c-3})$ converges in distribution. We also show that, if $n_c{\ge}4$, then the lag time has the same order of magnitude as $T^N$ and, if $n_c{=}3$, it is of the order of $\log N$. An original feature proved for this model is the significant variability of $T^N$. This is a well known phenomenon observed in the experiments in biology but the previous mathematical models used up to now did not exhibit this magnitude of variability. The results are proved via a series of technical estimates for occupations measures on fast time scales. Stochastic calculus with Poisson processes, coupling arguments and branching processes are the main ingredients of the proofs.
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Submitted on : Wednesday, December 27, 2017 - 11:34:21 AM
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Philippe Robert, Wen Sun. On the Asymptotic Distribution of Nucleation Times of Polymerization Processes. SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2019, 79 (5), pp.27. ⟨10.1137/19M1237508⟩. ⟨hal-01672800⟩



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