Kinetic Model of the Emergence of Autocatalysis

TL;DR

This work develops a kinetic framework for the emergence of autocatalytic, self-constructing objects in an open, number-conserving system. It derives a nonlinear integral-differential master equation for the productivity distribution $f(x,t)$ and obtains an exact Fourier-based solution for general jump kernels, showing that the average productivity $igl\uparrow xigr angle$ inevitably grows and the dispersion broadens, implying the spontaneous appearance of autocatalytic objects without selecting a single best type. A diffusional (Fokker–Planck) approximation yields closed-form dynamics for $iglngle x igr angle$ and dispersion, while extensions to random diffusion/drift reveal higher-order time-polynomial corrections and enhanced dispersion due to noise. The nonlinear autocatalysis analysis demonstrates that linear kinetics fail to produce selection, but nonlinear (e.g., quadratic) autocatalysis can generate selection-like outcomes depending on initial conditions and kinetic form, highlighting how higher-order autocatalysis could influence self-organization and origins of life.

Abstract

We develop a formal model of the emergence of self-constructing objects (e.g. heteropolymers with autocatalytic capability) in an open system, which don't contain such objects initially. The objects are constructed from subunits (e.g. monomers). Each object is characterized by the difference of self-instructed reproduction and decomposition rate only. This difference, divided by a common dimensional constant, is called ``productivity''. Due to external influence the productivity of each object can randomly change. The system as a whole is subjected to external limitation: the total number of the objects is conserved (e.g., by the controlled influx of monomers). We consider such process as possibly simplest example of self-organization. We obtained exact solutions of our model for several presumed mechanisms of random change of the productivity. We have shown that the probability to find self-constructing objects in the system necessarily increases, even if initially it was equal to zero.