Michaelis-Menten reaction: pump current and other stochastic effects

Stochastic effects, such as the pump current, have been discussed in the context of the Michaelis-Menten mechanism and channel on a cell surface.

References

Thermal Ratchets and Periodic Pumping

1. H Qian. Vector Field Formalism and Analysis for a Class of Thermal Ratchets, PRL 81:3063, 1998. PDF

   Abstract

   To understand the physics of muscle contraction and molecular motor movement, we develop a model for nonequilibrium free energy transduction based on a diffusion in a periodic force field. It is shown that a nonconservative force is sufficient and necessary for a steady state with circular flux, but is not sufficient for a global unidirectional transport synonymous to motor protein movement. A vector potential for the flux is introduced for characterizing the circular flux and global transport. The model provides a natural distinction between the two types of muscle protein movement, namely the mechanical dominant “power-stroke” and the Brownian-motion dominant ratchet.

   Comments

   This is strictly speaking not about pump effect, however, in light of Chen's 1987 article it is relevant. Author considers a system that is known to show a current in steady state, namely particle moving in periodic potential along x and having internal stochastic dynamics along internal coordinate y so that force on particle depends on the position at internal coordinate y. First, author points to the fact that charge is conserved so current must be pure curl of some vector A. He looks at what this vector can be and finds that A is related to part of the evolution Hamiltonian that breaks detailed balance so that system is forced to go around some contour in x-y space. The final effect, i.e. the current in real x-space is proportional to some contour integral of some other vector F over this path. Although it all suggests some relation to nontrivial topology, this is not yet a discovery of the Berry phase. Berry phase is a phase accuired by some eigenvector after adiabatic evolution. In this paper, it is still unclear what this eigenvector is. We can say only that some nonholonomy was found but not Berry phase, although the Berry phase interpretation of the result can be found probably.

2. H Qian and M Qian, Pumped Biochemical Reactions, Nonequilibrium Circulation, and Stochastic Resonance. PRL 84:2271, 2000. PDF.

   Abstract

   Based on a master equation formalism for mesoscopic, unimolecular biochemical reactions, we show the periodic oscillation arising from severe nonequilibrium pumping is intimately related to the periodic motion in recently studied stochastic resonance (SR). The white noise in SR is naturally identified with the temperature in the biochemical reactions; the drift in the SR is associated with the circular flux in nonequilibrium steady state (NESS). As in SR, an optimal temperature for biochemical oscillation is shown to exist. A unifying framework for Hill’s theory of NESS and the SR without periodic forcing is presented. The new formalism provides an analytically solvable model for SR.

   Comments

   Authors play with symple Markov chains of sizes 3 or 4 states. They look at power spectrum. If there is a maximum for some ω they say that there will be a stochastic resonance. They point to the fact that when detailed balance is broken then the chance to find such a maximum is larger.

C. Kwon, P. Ao, D. J. Thouless, PNAS 102, 13029 (2005),  Astumian et al., 2003,  Astumian et al., 1987, Astumian et al., 1989,  Westerhoff et al., 1986,  Robertson and Astumian, 1990,

Astumian and Robertson, 1989, Robertson and Astumian, 1991, R. D. Astumian and B. Robertson J. Chem. Phys. 91, 4891 (1989), B. Robertson, R. D. Astumian, J. Chem. Phys. 94, 7414 (1991), Chen, 1987,Qian-xie 2006