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

A modeling 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 equation. Combining these two equations into one, a comprehensive theory for the Brownian dynamics and statistical thermodynamics of single 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 enzyme catalysis by single fluctuating proteins. As a generalization of the classic polymer theory, the dynamic equation is capable of characterizing a single macromolecule in aqueous solution, in probabilistic terms, (1) its thermodynamic equilibrium with fluctuations, (2) transient relaxation kinetics, and most importantly and novel (3) nonequilibrium steady-state with heat dissipation. A reversibility condition which guarantees an equilibrium solution and its thermodynamic stability is established, an H-theorem like inequality for irreversibility is obtained, and a rule for thermodynamic consistency in chemically pumped nonequilibrium steady-state is given.Comment: 23 pages, 4 figure

Fitness and entropy production in a cell population dynamics with epigenetic phenotype switching

Motivated by recent understandings in the stochastic natures of gene expression, biochemical signaling, and spontaneous reversible epigenetic switchings, we study a simple deterministic cell population dynamics in which subpopulations grow with different rates and individual cells can bi-directionally switch between a small number of different epigenetic phenotypes. Two theories in the past, the population dynamics and thermodynamics of master equations, separatedly defined two important concepts in mathematical terms: the {\em fitness} in the former and the (non-adiabatic) {\em entropy production} in the latter. Both play important roles in the evolution of the cell population dynamics. The switching sustains the variations among the subpopulation growth thus continuous natural selection. As a form of Price's equation, the fitness increases with ($i$) natural selection through variations and $(ii)$ a positive covariance between the per capita growth and switching, which represents a Lamarchian-like behavior. A negative covariance balances the natural selection in a fitness steady state | "the red queen" scenario. At the same time the growth keeps the proportions of subpopulations away from the "intrinsic" switching equilibrium of individual cells, thus leads to a continous entropy production. A covariance, between the per capita growth rate and the "chemical potential" of subpopulation, counter-acts the entropy production. Analytical results are obtained for the limiting cases of growth dominating switching and vice versa.Comment: 16 page

A Hamiltonian-Entropy Production Connection in the Skew-symmetric Part of a Stochastic Dynamics

The infinitesimal transition probability operator for a continuous-time discrete-state Markov process, $\mathcal{Q}$, can be decomposed into a symmetric and a skew-symmetric parts. As recently shown for the case of diffusion processes, while the symmetric part corresponding to a gradient system stands for a reversible Markov process, the skew-symmetric part, \frac{d}{dt}u(t)=\mcA u, is mathematically equivalent to a linear Hamiltonian dynamics with Hamiltonian H=1/2u^T\big(\mcA^T\mcA)^{1/2}u. It can also be transformed into a Schr\"{o}dinger-like equation $\frac{d}{dt}u=i\mathcal{H}u$ where the "Hamiltonian" operator \mathcal{H}=-i\mcA is Hermitian. In fact, these two representations of a skew-symmetric dynamics emerge natually through singular-value and eigen-value decompositions, respectively. The stationary probability of the Markov process can be expressed as $\|u^s_i\|^2$. The motion can be viewed as "harmonic" since $\frac{d}{dt}\|u(t)-\vec{c}\|^2=0$ where $\vec{c}=(c,c,...,c)$ with $c$ being a constant. More interestingly, we discover that \textrm{Tr}(\mcA^T\mcA)=\sum_{j,\ell=1}^n \frac{(q_{j\ell}\pi_\ell-q_{\ell j}\pi_j)^2}{\pi_j\pi_{\ell}}, whose right-hand-side is intimately related to the entropy production rate of the Markov process in a nonequilibrium steady state with stationary distribution $\{\pi_j\}$. The physical implication of this intriguing connection between conservative Hamiltonian dynamics and dissipative entropy production remains to be further explored.Comment: 18 page