In this paper, we introduce a stochastic model for the dynamics of actin polymers and their interactions with other proteins in the cellular envelop. Each polymer elongates and shortens, and can switch between several modes depending on whether it is bound to accessory proteins that modulate its behaviour as, for example, elongation-promoting factors. Our main aim is to understand the dynamics of a large population of polymers, assuming that the only limiting quantity is the total amount of monomers, set to be constant to some large N. We first focus on the evolution of a very long polymer, of size \({\mathcal {O}}(N)\), with a rapid switch between modes (compared to the timescale over which the macroscopic fluctuations in the polymer size appear). Letting N tend to infinity, we obtain a fluid limit in which the effect of the switching appears only through the fraction of time spent in each mode at equilibrium. We show in particular that, in our situation where the number of monomers is limiting, a rapid binding–unbinding dynamics may lead to an increased elongation rate compared to the case where the polymer is trapped in any of the modes. Next, we consider a large population of polymers and complexes, represented by a random measure on some appropriate type space. We show that as N tends to infinity, the stochastic system converges to a deterministic limit in which the switching appears as a flow between two categories of polymers. We exhibit some numerical examples in which the limiting behaviour of a single polymer differs from that of a population of competing (shorter) polymers for equivalent model parameters. Taken together, our results demonstrate that under conditions where the total number of monomers is limiting, the study of a single polymer is not sufficient to understand the behaviour of an ensemble of competing polymers.

在本文中，我们介绍了肌动蛋白聚合物动力学及其与细胞包膜中其他蛋白质的相互作用的随机模型。每种聚合物都会伸长和缩短，并且可以在几种模式之间切换，这取决于它是否与调节其行为的辅助蛋白质结合，例如，作为伸长促进因子。我们的主要目标是了解大量聚合物的动力学，假设唯一的限制数量是单体的总量，设置为常数，以保持一些大的N。我们首先关注一个非常长的聚合物的演化，尺寸为\({\mathcal {O}}(N)\)，在模式之间快速切换（与聚合物尺寸出现宏观波动的时间尺度相比）。让N趋向于无穷大，我们得到了一个流体极限，其中切换的效果仅通过在平衡时在每个模式中花费的时间部分出现。我们特别表明，在单体数量有限的情况下，与聚合物被困在任何模式中的情况相比，快速的结合-解结合动力学可能会导致伸长率增加。接下来，我们考虑大量的聚合物和复合物，由一些适当类型空间上的随机测度表示。我们证明作为N趋于无穷大，随机系统收敛到一个确定性极限，其中切换表现为两类聚合物之间的流动。我们展示了一些数值示例，其中单个聚合物的限制行为与等效模型参数的竞争（较短）聚合物群体的限制行为不同。总之，我们的结果表明，在单体总数有限的情况下，对单一聚合物的研究不足以理解竞争聚合物集合的行为。