Mixed-Integer Programming for Cycle Detection in Non-reversible Markov Processes

TL;DR

This paper introduces a mixed-integer programming approach to detect cyclic behavior in non-reversible stochastic processes, especially useful in biological and chemical systems where small time steps obscure global cycles.

Contribution

The paper presents a novel optimization-based method for cycle detection in non-reversible processes, outperforming classical spectral analysis in synthetic biological networks.

Findings

Successfully detects the most productive cycle in a genetic network

Outperforms spectral analysis methods in synthetic tests

Applicable to non-equilibrium steady state systems

Abstract

In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing non-reversible biological processes, in particular molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring-oscillator with three genes, we show that this approach can detect the most productive overall cycle, outperforming classical spectral analysis methods. Our method applies to general non-equilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst.