Abstract
We consider the first-passage problem for N identical independent particles that are initially released uniformly in a finite domain Ω and then diffuse toward a reactive area Γ, which can be part of the outer boundary of Ω or a reaction centre in the interior of Ω. For both cases of perfect and partial reactions, we obtain the explicit formulas for the first two moments of the fastest first-passage time (fFPT), i.e., the time when the first out of the N particles reacts with Γ. Moreover, we investigate the full probability density of the fFPT. We discuss a significant role of the initial condition in the scaling of the average fFPT with the particle number N, namely, a much stronger dependence (1/N and 1/N2 for partially and perfectly reactive targets, respectively), in contrast to the well known inverse-logarithmic behaviour found when all particles are released from the same fixed point. We combine analytic solutions with scaling arguments and stochastic simulations to rationalise our results, which open new perspectives for studying the relevance of multiple searchers in various situations of molecular reactions, in particular, in living cells.
Highlights
Physical kinetics studies the dynamics of molecular chemical reactions in terms of the time dependence of the reactant concentrations [1]
In the ‘facilitated diffusion picture’ this quantity is based on the typical distance covered by the protein during sliding while it intermittently binds to the DNA [4,5,6,7]
As first noticed in [79] and later explored in [86,87,88], in a finite domain the contribution to the effective rate due to a diffusive search for a target decreases in proportion to 1/N2, while the contribution due to a penetration through a barrier against reaction vanishes as 1/N in the limit N → ∞. This means that (i) placing N searchers at distinct positions gives a substantial increase in the reaction efficiency, which can be orders of magnitude larger as compared to the case when all N searchers start from the same point, (ii) as compared to a standard Collins–Kimball relation, which is valid in the thermodynamic limit and in which both contributions have the same 1/N-dependence on the number of searchers, here the contribution due to a diffusive search acquires an additional power of a concentration of searchers, and (iii) in the limit N 1 reactions become inevitably controlled by chemistry rather than by diffusion
Summary
Physical kinetics studies the dynamics of molecular chemical reactions in terms of the time dependence of the reactant concentrations [1]. As first noticed in [79] and later explored in [86,87,88], in a finite domain the contribution to the effective rate due to a diffusive search for a target decreases in proportion to 1/N2, while the contribution due to a penetration through a barrier against reaction vanishes as 1/N in the limit N → ∞ This means that (i) placing N searchers at distinct positions gives a substantial increase in the reaction efficiency, which can be orders of magnitude larger as compared to the case when all N searchers start from the same point, (ii) as compared to a standard Collins–Kimball relation, which is valid in the thermodynamic limit and in which both contributions have the same 1/N-dependence on the number of searchers, here the contribution due to a diffusive search acquires an additional power of a concentration of searchers, (which is a strong concentration effect), and (iii) in the limit N 1 reactions become inevitably controlled by chemistry (kinetically-controlled reactions) rather than by diffusion (diffusion-controlled reactions). Details of derivations and some additional analyses are presented in the appendices
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