Dissipation, generalized free energy, and a self-consistent nonequilibrium thermodynamics of chemically driven open subsystems

Nonequilibrium thermodynamics of a system situated in a sustained environment with influx and efflux is usually treated as a subsystem in a larger, closed “universe.” A question remains with regard to what the minimally required description for the surrounding of such an open driven system is so that its nonequilibrium thermodynamics can be established solely based on the internal stochastic kinetics. We provide a solution to this problem using insights from studies of molecular motors in a chemical nonequilibrium steady state (NESS) with sustained external drive through a regenerating system or in a quasisteady state (QSS) with an excess amount of adenosine triphosphate (ATP), adenosine diphosphate (ADP), and inorganic phosphate (Pi). We introduce the key notion of minimal work that is needed, $W_{\min }$, for the external regenerating system to sustain a NESS (e.g., maintaining constant concentrations of ATP, ADP and Pi for a molecular motor). Using a Markov (master-equation) description of a motor protein, we illustrate that the NESS and QSS have identical kinetics as well as the second law in terms of the same positive entropy production rate. The heat dissipation of a NESS without mechanical output is exactly the $W_{\min }$. This provides a justification for introducing an ideal external regenerating system and yields a free-energy balance equation between the net free-energy input $F_{\mathrm{in}}$ and total dissipation $F_{\mathrm{dis}}$ in an NESS: $F_{\mathrm{in}}$ consists of chemical input minus mechanical output; $F_{\mathrm{dis}}$ consists of dissipative heat, i.e. the amount of useful energy becoming heat, which also equals the NESS entropy production. Furthermore, we show that for nonstationary systems, the $F_{\mathrm{dis}}$ and $F_{\mathrm{in}}$ correspond to the entropy production rate and housekeeping heat in stochastic thermodynamics and identify a relative entropy $H$ as a generalized free energy. We reach a new formulation of Markovian nonequilibrium thermodynamics based on only the internal kinetic equation without further reference to the intrinsic degree of freedom within each Markov state. It includes an extended free-energy balance and a second law which are valid for driven stochastic dynamics with an ideal external regenerating system. Our result suggests new ingredients for a generalized thermodynamics of self-organization in driven systems.

Figure 1

The thermodynamics of spontaneous ATP hydrolysis and related ATP regenerating process. The entire cycle would have an $\Delta \mu >0$ amount of net free-energy input and the same amount of energy dissipated. $\Delta \mu$ is just the free-energy change of the ATP hydrolysis minus the amount of mechanical energy output. See the main text for details.