Reaching Agreement in Competitive Microbial Systems

TL;DR

This paper analyzes how microbial systems reach agreement under competition, showing direct competition enables consensus with small initial differences, unlike non-competitive systems.

Contribution

It demonstrates that direct competition facilitates majority consensus with small initial gaps, providing theoretical and simulation evidence for practical biological timescales.

Findings

Direct competition achieves consensus with initial gap $ ext{Ω}(\sqrt{n ext{log} n})$

Absence of competition requires a large initial gap $ ext{Ω}(n)$ for consensus

Simulations confirm consensus occurs within practical biological timescales

Abstract

We study distributed agreement in microbial distributed systems under stochastic population dynamics and competitive interactions. Motivated by recent applications in synthetic biology, we examine how the presence and absence of direct competition among microbial species influences their ability to reach majority consensus. In this problem, two species are designated as input species, and the goal is to guarantee that eventually only the input species which had the highest initial count prevails. We show that direct competition dynamics reach majority consensus with high probability even when the initial gap between the species is small, i.e., $\Omega(\sqrt{n\log n})$, where $n$ is the initial population size. In contrast, we show that absence of direct competition is not robust: solving majority consensus with constant probability requires a large initial gap of $\Omega(n)$. To corroborate our analytical results, we use simulations to show that these consensus dynamics occur within practical biological time scales.