  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.
