A stochastic model of the production of multiple proteins in cells

TL;DR

This paper introduces a stochastic Markovian model for multiple protein production in cells, accounting for resource limitations and showing that ribosome allocation converges to a Poisson distribution, revealing insights into cellular resource management.

Contribution

It develops a novel stochastic model for multiple protein production considering resource constraints, providing asymptotic analysis and new insights into ribosome allocation dynamics.

Findings

Ribosome allocation converges to a Poisson distribution at equilibrium.

Different protein production processes are asymptotically independent with modified parameters.

The model captures the impact of resource saturation on protein synthesis.

Abstract

The production processes of proteins in prokaryotic cells are investigated. Most of the mathematical models in the literature study the production of {\em one} fixed type of proteins. When several classes of proteins are considered, an important additional aspect has to be taken into account, the limited common resources of the cell (polymerases and ribosomes) used by the production process. Understanding the impact of this limitation is a key issue in this domain. In this paper we focus on the allocation of ribosomes in the case of the production of multiple proteins. The cytoplasm of the cell being a disorganized medium subject to thermal noise, the protein production process has an important stochastic component. For this reason, a Markovian model of this process is introduced. Asymptotic results of the equilibrium are obtained under a scaling procedure and a realistic biological assumption of saturation of the ribosomes available in the cell. It is shown in particular that, in the limit, the number of non-allocated ribosomes at equilibrium converges in distribution to a Poisson distribution whose parameter satisfies a fixed point equation. It is also shown that the production process of different types of proteins can be seen as independent production processes but with modified parameters.