Reaching Agreement in Competitive Microbial Systems

TL;DR

It is shown 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.

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., $Ω(\sqrt{n\log n})$, 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 $Ω(n)$. To corroborate our analytical results, we use simulations to show that these consensus dynamics occur within practical biological time scales.

Figures (6)

• Figure 1: Stochastic simulation with initial population counts $A = 151000$ and $B = 149000$ over $1$ day. Population counts are per $\mu$L and plotted on a logarithmic scale.
• Figure 2: Fraction of $A$ in the bacterial population after $60$ min and $120$ min. $N=10$ simulations per initial fraction. Error bars indicate maximum and minimum, markers average fractions.
• Figure 3: Zoomed version with initial population difference on the top abscissa.
• Figure 4: Stochastic simulation as in Figure \ref{['fig:run']}, but with $\alpha = 0$ and shown over $1$ day.
• Figure 5: Same setting as in Figure \ref{['fig:s_curve']}, but with $\alpha = 0$ and snapshot after one day instead of $60$ min.

Theorems & Definitions (20)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof