First passage times in homogeneous nucleation and self-assembly

TL;DR

This paper analyzes the stochastic process of self-assembly of particles, deriving equations for cluster formation times and revealing that increased detachment rates can paradoxically shorten the time to form a maximum-sized cluster.

Contribution

It provides a comprehensive analytical framework for calculating first passage times in particle self-assembly, including the effects of detachment rates, which was not previously understood.

Findings

Faster detachment can shorten the mean cluster formation time.

Derived the Backward Kolmogorov equation for cluster probability distribution.

Developed analytical methods for different assembly rate regimes.

Abstract

Motivated by nucleation and molecular aggregation in physical, chemical and biological settings, we present a thorough analysis of the general problem of stochastic self-assembly of a fixed number of identical particles in a finite volume. We derive the Backward Kolmogorov equation (BKE) for the cluster probability distribution. From the BKE we study the distribution of times it takes for a single maximal cluster to be completed, starting from any initial particle configuration. In the limits of slow and fast self-assembly, we develop analytical approaches to calculate the mean cluster formation time and to estimate the first assembly time distribution. We find, both analytically and numerically, that faster detachment can lead to a shorter mean time to first completion of a maximum-sized cluster. This unexpected effect arises from a redistribution of trajectory weights such that upon increasing the detachment rate, paths that take a shorter time to complete a cluster become more likely.