A diffusional bimolecular propensity function

Abstract

We derive an explicit formula for the propensity function ͑stochastic reaction rate͒ of a generic bimolecular chemical reaction in which the reactant molecules move about by diffusion, as solute molecules in a bath of much smaller and more numerous solvent molecules. Our derivation assumes that the solution is macroscopically well stirred and dilute in the solute molecules. It effectively extends the physical rationale for the chemical master equation and the stochastic simulation algorithm from well-stirred dilute gases to well-stirred dilute solutions, with the former becoming a limiting case of the latter. This extension is important for cellular systems, where the solvent molecules are typically water and the solute ͑reactant͒ molecules are much larger organic structures, whose relatively low populations often require a discrete-stochastic formalism. In the course of our derivation, we illuminate some limitations on the ability of the classical diffusion equation to accurately describe how a diffusing molecule moves on spatial and temporal scales that are relevant to collision-induced chemical reactions.

Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators

“…Tikhonov's first theorem [16,17] states that a simplification of the above equations under timescale separation conditions is possible whenever certain requirements are met: (1) the solutions of both the degenerate and adjoined systems [Eqs. (15)] are unique and their right-hand sides are continuous functions; (2) the root x f = h( x s ,τ ) is the stable solution of the adjoined system; and (3) the initial values x f (τ = 0) are in the domain of influence of the solution as in (2). Whenever these prerequisites are met, the solution of the full system (15) for x s tends to the solution of the reduced system…”

“…We consider four general types of elementary reactions depending on the order j of the reaction, for which the microscopic rate functions have been rigorously derived from microscopic considerations [14,15]: (1)f j ( n, ) = k j , for a zeroth-order reaction by which a species is input into a compartment; (2)f j ( n, ) = k j n u −1 , for a first-order unimolecular reaction describing the decay of some species u;…”

Discreteness-induced concentration inversion in mesoscopic chemical systems

“…Intrinsic noise is taken into account by chemical master equations (CMEs), which are exact mesoscopic descriptions of well-stirred and thermally equilibrated gas-phase chemical systems 11 and chemical reactions occurring in well-stirred dilute solutions 12 . Unfortunately, CMEs are generally analytically intractable, and many studies have therefore resorted to the linear-noise approximation (LNA) of the CME (see, for example, refs [13][14][15][16][17][18][19].…”

Communication: Limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks

“…where is the compartment volume in which the reactions are occurring, n S is the absolute number of substrate molecules, P(n S , t) is the probability that the system has n S substrate molecules at time t and E m S is the step operator which upon acting on a function of n S changes it into a function of n S + m. 14 We note and emphasize that the physical basis of this master equation is not clear because such equations have been derived from first principles for elementary reactions 16,17 while Eq. (3) involves a non-elementary reaction.…”