Separation of time scales in one-dimensional directed nucleation-growth processes

Proteins involved in homologous recombination such as RecA and hRad51 polymerize on single- and double-stranded DNA according to a nucleation-growth kinetics, which can be monitored by single-molecule in vitro assays. The basic models currently used to extract biochemical rates rely on ensemble averages and are typically based on an underlying process of bidirectional polymerization, in contrast with the often observed anisotropic polymerization of similar proteins. For these reasons, if one considers single-molecule experiments, the available models are useful to understand observations only in some regimes. In particular, recent experiments have highlighted a steplike polymerization kinetics. The classical model of one-dimensional nucleation growth, the Kolmogorov-Avrami-Mehl-Johnson (KAMJ) model, predicts the correct polymerization kinetics only in some regimes and fails to predict the steplike behavior. This work illustrates by simulations and analytical arguments the limitation of applicability of the KAMJ description and proposes a minimal model for the statistics of the steps based on the so-called stick-breaking stochastic process. We argue that this insight might be useful to extract information on the time and length scales involved in the polymerization kinetics.