From prelife to life: a bio-inspired toy model

Abstract

We study a one-dimensional lattice of N sites each occupied by a mathematical ‘polymer,’ that is, is a binary random sequence of arbitrary length n, or equivalently, a rooted path of n links on an infinite binary tree. The average polymer length is controlled by the monomer fugacity z. A pair of polymers on adjacent sites carries a weight factor ω for each link on the tree that they have in common. The phase diagram in the zω plane exhibits a critical line z = zc(ω). For z < zc(ω) there exists an equilibrium phase with, in particular, a finite average polymer length. We investigate the equilibrium ensemble by transfer matrix and Monte Carlo methods, paying particular attention to the vicinity of the critical line. For z > zc(ω) the equilibrium is unstable and Monte Carlo time evolution brings about a dynamical symmetry breaking which favors the evolution of a small selection of polymers to ever greater length. While of interest for its own sake, this model may also be relevant to the prelife-to-life transition that has occurred during biological evolution. We compare it to existing models of similar simplicity.