From prelife to life: a bio-inspired toy model

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 $z=z_{\rm c}(ω)$. For $z<z_{\rm c}(ω)$ 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 $z>z_{\rm c}(ω)$ 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 due to Wu and Higgs (2009, 2012) and to Chen and Nowak (2012).

No reviews yet.

No comments yet.