Investigating mathematical properties of non-equilibrium signatures in biological information processing systems*

Detecting nonequilibrium behavior in biological information processing systems poses a technical challenge in experimental and theoretical contexts. To address this difficulty, several mathematical signatures of broken detailed balance in Markovian systems have been suggested. While these signatures identify the presence of energy expenditure, little is known about how they relate to underlying thermodynamic forces. Here we use a graph-theoretic approach to Markov processes to probe the relationship between force and the Steinberg signature, which detects nonequilibrium behavior through the inequality of forward and reverse higher-order autocorrelation functions. We have developed software to calculate the Steinberg signature from arbitrary graphs. We find that when a Markovian system is perturbed progressively from equilibrium and its force increases from zero, the Steinberg signature reaches its maximum at an intermediate value of force before decaying asymptotically, potentially to zero. This non-monotonic relationship shows the Steinberg signature's limitations as a tool for detecting energy expenditure. Characterizing the mathematical behavior of such signatures may elucidate new properties of nonequilibrium systems to be tested experimentally in real biological contexts.