Lifetime of small systems controlled by autocatalytic reactions

TL;DR

This paper analyzes the stochastic lifetime of small autocatalytic systems using reaction kinetics, revealing asymmetric lifetime distributions and certain extinction.

Contribution

It introduces a point model approach to quantify the lifetime distribution and extinction probability of autocatalytic systems with fixed substrate conditions.

Findings

Lifetime density function is strongly asymmetric.

Extinction probability of such systems is exactly 1.

Lifetime distribution can have a well-defined initial minimum.

Abstract

By using the point model of reaction kinetics we have studied the stochastic properties of the lifetime of small systems controlled by autocatalytic reaction A+X -> X+X -> A+X, X -> B. Assuming that a system is living only when the number of autocatalytic particles is larger than zero but smaller than a positive integer N, we have calculated the probability of the lifetime provided that the number of substrate particles A is kept constant by a suitable reservoir, and the end-products B do not take part in the reaction. We have shown that the density function of the lifetime is strongly asymmetric and in certain cases it has a well-defined minimum at the beginning of the process. It has been also proven that the extinction probability of systems of this type is exactly 1.