Thermodynamic Efficiency Out of Equilibrium

Abstract

Equilibrium thermodynamics satisfactorily explains the efficiency of macroscopic machines, whose operation is posited as a quasi-static, infinite time, zero power process exemplified by the Carnot heat engine. Microscopic biomolecular motors differ markedly from their macroscopic counterparts, as they are subject to large fluctuations, operate far from equilibrium, and by necessity accomplish their tasks in finite time with non-zero power. They thus demand novel non-equilibrium frameworks. We explore thermodynamic length as an analytic framework for understanding the physical limits on biomolecular motors. Thermodynamic length defines the length of a non-equilibrium transformation as the root-mean squared fluctuations of the variables conjugate to the control parameters. It is a natural measure of distance between equilibrium thermodynamic states, but unlike the free energy change explicitly depends on the path taken through thermodynamic state space. Thermodynamic length equips thermodynamic state space with a Riemannian metric and thus facilitates the discovery of minimum thermodynamic length paths, which minimize the dissipation for slow, but finite time, transformations. We derive analytic expressions for Fisher information (related to the derivative of thermodynamic length) in simple bistable energy landscapes, finding that it can vary by several orders of magnitude across a given energy landscape. Our novel dynamic programming approach allows more detailed analysis of these model landscapes, establishing that thermodynamic length analysis accurately predicts the instantaneous dissipation of far-from-equilibrium processes across the entire energy landscape. We also derive thermodynamic length as a special case of linear response theory, a standard non-equilibrium framework. Thermodynamic length analysis should prove useful in the further analysis of molecular motors, as it gives access to non-equilibrium properties (dissipation) through equilibrium properties (Fisher information and relaxation time).