Equations for Stochastic Macromolecular Mechanics of Single Proteins: Equilibrium Fluctuations, Transient Kinetics and Nonequilibrium Steady-State

A mathematical framework for the internal conformational dynamics and external mechanical movement of single biological macromolecules in aqueous solution at constant temperature is developed. Both the internal dynamics and external movement are stochastic; the former is represented by a master equation for a set of discrete states, and the latter is described by a continuous Smoluchowski (Fokker-Planck) equation. Combining these two equations into one, a comprehensive theory for the dynamic of macromolecules arises. This approach is shown to have wide applications. It is applied to protein-ligand dissociation under external force, unfolding of polyglobular proteins under extension, movement along linear tracks of motor proteins against load, and catalysis by fluctuating enzymes. The dynamic equation is capable of characterizing thermodynamic equilibrium with fluctuations, transient relaxation kinetics, and nonequilibrium steady-state. A reversibility condition which guarantees the equilibrium solution and its asymptotic stability is introduced, and pumping under irreversible settings is discussed.