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

  • 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.

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Metadaten
Author:Jakob WitzigORCiD, Isabel Beckenbach, Leon Eifler, Konstantin Fackeldey, Ambros Gleixner, Andreas Grever, Marcus Weber
Document Type:Article
Parent Title (English):Multiscale Modeling and Simulation
Volume:16
Issue:1
First Page:248
Last Page:265
Tag:Markov State Models; Mixed-Integer Programming; NESS; Non-reversible Markov Processes
MSC-Classification:60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX)
82-XX STATISTICAL MECHANICS, STRUCTURE OF MATTER
90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING
Year of first publication:2018
ISSN:1438-0064
Published in:Multiscale Modeling and Simulation
Preprint:urn:nbn:de:0297-zib-60353
DOI:https://doi.org/10.1137/16M1091162