Stochastic Analysis of Dimerization Systems

TL;DR

This paper introduces a simplified stochastic approach using moment equations to analytically analyze dimerization systems, especially in regimes with small copy numbers where traditional rate equations fail.

Contribution

It provides the first analytical solutions for stochastic dimerization processes using moment equations, covering steady state and time-dependent scenarios.

Findings

Analytical solutions match master equation results in stochastic limit.

Solutions agree with rate equations in deterministic limit.

Applicable to various dimerization processes, including dissociation and hetero-dimer formation.

Abstract

The process of dimerization, in which two monomers bind to each other and form a dimer, is common in nature. This process can be modeled using rate equations, from which the average copy numbers of the reacting monomers and of the product dimers can then be obtained. However, the rate equations apply only when these copy numbers are large. In the limit of small copy numbers the system becomes dominated by fluctuations, which are not accounted for by the rate equations. In this limit one must use stochastic methods such as direct integration of the master equation or Monte Carlo simulations. These methods are computationally intensive and rarely succumb to analytical solutions. Here we use the recently introduced moment equations which provide a highly simplified stochastic treatment of the dimerization process. Using this approach, we obtain an analytical solution for the copy numbers and reaction rates both under steady state conditions and in the time-dependent case. We analyze three different dimerization processes: dimerization without dissociation, dimerization with dissociation and hetero-dimer formation. To validate the results we compare them with the results obtained from the master equation in the stochastic limit and with those obtained from the rate equations in the deterministic limit. Potential applications of the results in different physical contexts are discussed.