An exact steady-state solution of the stochastic equations governing the behavior of a gene regulated by a self-generated proteomic atmosphere is presented. The solutions depend on an adiabaticity parameter measuring the relative rate of DNA-protein unbinding and protein degradation. The steady-state solution reveals deviations from the commonly used Ackers et al approximation based on the equilibrium law of mass action, allowing anticooperative behavior in the "nonadiabatic" limit of slow binding and unbinding rates. Noise from binding and unbinding events dominates the shot noise of protein synthesis and degradation up to quite high values of the adiabaticity parameter.

We established a theoretical framework in terms of the curl flux, population landscape and coherence for non-equilibrium quantum systems at steady state, through exploring the energy and charge transport in molecular processes. The curl quantum flux plays the key role in determining transport properties and the system reaches equilibrium when flux vanishes. The novel curl quantum flux reflects the degree of nonequilibriumness and the time-irreversibility. We found an analytical expression for the quantum flux and its relationship to the environmental pumping (non-equilibriumness quantified by the voltage away from the equilibrium) and the quantum tunnelling. Furthermore, we investigated another quantum signature, the coherence, quantitatively measured by the non-zero off diagonal element of the density matrix. Populations of states give the probabilities of individual states and therefore quantify the population landscape. Both curl flux and coherence depend on steady state population landscape. Besides the environment-assistance which can give dramatic enhancement of coherence and quantum flux with high voltage at a fixed tunnelling strength, the quantum flux is promoted by the coherence in the regime of small tunnelling while reduced by the coherence in the regime of large tunneling, due to the non-monotonic relationship between the coherence and tunneling. This is in contrast to the previously found linear relationship. For the systems coupled to bosonic (photonic and phononic) reservoirs the flux is significantly promoted at large voltage while for fermionic (electronic) reservoirs the flux reaches a saturation after a significant enhancement at large voltage due to the Pauli exclusion principle. In view of the system as a quantum heat engine, we studied the nonequilibrium thermodynamics and established the analytical connections of curl quantum flux to the transport quantities such as energy (charge) transfer efficiency (ETE or CTE), chemical reaction efficiency (CRE), energy dissipation, heat and electric currents observed in the experiments. We observed a perfect transfer efficiency in chemical reactions at high voltage (chemical potential difference). Our theoretical predicted behavior of the electric current with respect to the voltage is in good agreements with the recent experiments on electron transfer in single molecules.