Geometric stability considerations of the ribosome flow model with pool

TL;DR

This paper applies a geometric approach to analyze the stability of the ribosome flow model with pool, extending previous monotone systems theory results and enabling analysis of more complex gene translation models.

Contribution

It introduces a geometric method to study equilibrium stability in the RFM with pool, broadening applicability beyond monotone systems.

Findings

Equilibria form a normally hyperbolic invariant submanifold.

Equilibria are asymptotically stable relative to certain affine subspaces.

Method applicable to complex systems like bi-directional flows.

Abstract

In order to better understand the process of gene translation, the ribosome flow model (RFM) with pool was introduced recently. This model describes the movement of several ribosomes along an mRNA template and simultaneously captures the dynamics of the finite pool of ribosomes. Studying this system with respect to the number and stability of its equilibria was so far based on monotone systems theory (Margaliot and Tuller, 2012). We extend the results obtained therein by using a geometric approach, showing that the equilibria of the system constitute a normally hyperbolic invariant submanifold. Subsequently, we analyze the Jacobi linearization of the system evaluated at the equilibria in order to show that the equilibria are asymptotically stable relative to certain affine subspaces. As this approach does not require any monotonicity features of the system, it may also be applied for more complex systems of the same kind such as bi-directional ribosome flows or time-varying template numbers.