Open Markov processes: A compositional perspective on non-equilibrium steady states in biology

TL;DR

This paper explores how open Markov processes can model biological systems maintained away from equilibrium, showing non-equilibrium steady states minimize dissipation and differ from entropy production principles.

Contribution

It extends the framework of open Markov processes to biological modeling, demonstrating non-equilibrium steady states and their relation to dissipation and entropy.

Findings

Non-equilibrium steady states minimize dissipation.

Dissipation approximates the rate of change of relative entropy.

Prigogine's minimum entropy production principle often fails.

Abstract

In recent work, Baez, Fong and the author introduced a framework for describing Markov processes equipped with a detailed balanced equilibrium as open systems of a certain type. These `open Markov processes' serve as the building blocks for more complicated processes. In this paper, we describe the potential application of this framework in the modeling of biological systems as open systems maintained away from equilibrium. We show that non-equilibrium steady states emerge in open systems of this type, even when the rates of the underlying process are such that a detailed balanced equilibrium is permitted. It is shown that these non-equilibrium steady states minimize a quadratic form which we call `dissipation.' In some circumstances, the dissipation is approximately equal to the rate of change of relative entropy plus a correction term. On the other hand, Prigogine's principle of minimum entropy production generally fails for non-equilibrium steady states. We use a simple model of membrane transport to illustrate these concepts.