Adiabatic and non-adiabatic non-equilibrium stochastic dynamics of single regulating genes

We explore the stochastic dynamics of self-regulative genes from fluctuations of molecular numbers and of on and off switching of gene states due to regulatory protein binding/unbinding to the genes. We found when the binding/unbinding is relatively fast (slow) compared with the synthesis/degradation of proteins in adiabatic (nonadiabatic) case the self-regulators can exhibit one or two peak (two peak) distributions in protein concentrations. This phenomena can also be quantified through Fano factors. This shows that even with the same architecture (topology of wiring) networks can have quite different functions (phenotypes), consistent with recent single molecule single gene experiments. We further found the inhibition and activation curves to be consistent with previous results (monomer binding) in adiabatic regime, but, in nonadiabatic regimes, show significantly different behaviors with previous predictions (monomer binding). Such difference is due to the slow (nonadiabatic) dimer binding/unbinding effect, and it has never been reported before. We derived the nonequilibrium phase diagrams of monostability and bistability in adiabatic and nonadiabatic regimes. We studied the dynamical trajectories of the self-regulating genes on the underlying landscapes from nonadiabatic to adiabatic limit, and we provide a global picture of understanding and show an analogy to the electron transfer problem. We studied the stability and robustness of the systems through mean first passage time (MFPT) from one peak (basin of attraction) to another and found both monotonic and nonmonotonic turnover behavior from adiabatic to nonadiabatic regimes. For the first time, we explore global dissipation by entropy production and the relation with binding/unbinding processes. Our theoretical predictions for steady state peaks, fano factos, inhibition/activation curves, and MFPT can be probed and tested from experiments.