Embryonic lateral inhibition as optical modes: an analytical framework for mesoscopic pattern formation

TL;DR

This paper presents an analytical model for embryonic pattern formation via lateral inhibition, showing how negative coupling coefficients lead to checkerboard patterns and drawing analogies with optical phonon modes.

Contribution

It introduces a tractable analytical framework linking lateral inhibition to optical mode analogies, elucidating pattern formation regimes in embryonic cell arrangements.

Findings

Negative coupling coefficient induces checkerboard patterns.

Periodic and anti-correlated patterns occur in specific regimes.

Multiple stable states exist in hexagonal lattice arrangements.

Abstract

Cellular checkerboard patterns are observed at many developmental stages of embryos. We study an analytically tractable model for lateral inhibition and show that a coupling coefficient with a negative value is sufficient to obtain noisy or periodic checkerboard patterns. We solve the case of a linear chain of cells explicitly and show that noisy anti-correlated patterns are available in a post-critical regime $(\epsilon_c < \epsilon < 0)$. In the sub-critical regime $(-\infty < \epsilon \leq \epsilon_c)$ a periodic and alternating steady state is available, where pattern selection is determined by making an analogy with the optical modes of phonons. For cells arranged in a hexagonal lattice, the sub-critical pattern can be driven into three different states: two of those states are periodic checkerboards and a third in which both periodic states coexist.