Equivalence between two-dimensional cell-sorting and one-dimensional generalized random walk -- spin representations of generating operators

TL;DR

This paper demonstrates a mathematical equivalence between 2D cell-sorting and 1D generalized random walk models, revealing a common structure and enabling analysis of biological stochastic problems via spin system techniques.

Contribution

It establishes a novel equivalence between biological cell-sorting and random walk models using lattice spin representations, broadening analytical tools for biological stochastic systems.

Findings

Mathematical equivalence between 2D cell-sorting and 1D random walk models.

Representation of biological stochastic operators using spin operators.

Potential to analyze biological problems with lattice spin system techniques.

Abstract

The two-dimensional cell-sorting problem is found to be mathematically equivalent to the one-dimensional random walk problem with pair creations and annihilations, i.e. the adhesion probabilities in the cell-sorting model relate analytically to the expectation values in the random walk problem. This is an example demonstrating that two completely different biological systems are governed by a common mathematical structure. This result is obtained through the equivalences of these systems with lattice spin models. It is also shown that arbitrary generation operators can be written by the spin operators, and hence all biological stochastic problems can in principle be analyzed utilizing the techniques and knowledge previously obtained in the study of lattice spin systems.