Genetic interfaces at the frontier of expanding microbial colonies

TL;DR

It is shown that at the frontier of the colony, the genetic interface width saturates at finite values for long times, both for neutral strains and interspecies interactions such as antagonism.

Abstract

We study the genetic interfaces between two species of an expanding colony that consists of individual microorganisms that reproduce and undergo diffusion, both at the frontier and in the interior. Within the bulk of the colony, the genetic interface is controlled in a simple way via interspecies interactions. However, at the frontier of the colony, the genetic interface width saturates at finite values for long times, both for neutral strains and interspecies interactions such as antagonism. This finite width arises from geometric effects: genetic interfaces drift toward local minima at an undulating colony frontier, where a focusing mechanism induced by curvature impedes diffusive mixing. Numerical simulations support a logarithmic dependence of the genetic interface width on the strength of the number fluctuations.

Figures (10)

• Figure 1: Deterministic FKPP dynamics for two interacting species: (a) Concentration profiles at three different times $t$ for neutral/antagonistic/mutualistic interactions, with $\epsilon =0,-0.1,0.1$. Periodic boundary conditions are employed in the $x$-direction. (b) $AB$-interface width $w_{AB}$ along the x-direction at the frontier of the colonies (computed along the value of $y$ such that $c_T=0.5$) for these three interaction values. Units: $\lambda = \sqrt{D/\mu}$ (length) and $\tau=1/\mu$ (time); System-dimension: $L_X=256$, $LY = 1024$; Grid-resolution: $512 \times 2048$ points in the $(x,y)$-plane; with numerical time-step $dt=0.005$.
• Figure 2: Stochastic FKPP-dynamics for two interacting species, with and without interactions, again with periodic boundary conditions in the $x$-direction (a) Lattice configurations of typical simulations for neutral/antagonistic/mutualistic interactions, with $\epsilon =0,-0.1,0.1$ with $t=1000$. (b) Genetic interface width $w_{AB}$ along the x-direction at the frontier of the colonies along a line where $N_T=\bar{N}_T/2$ (indicated by the black dashed lines in (a)) for these three interaction settings; we also show the width at the initialization height for the neutral setting as a function of time (indicated by the white or black dashed line at the bottom of (a)). After a brief transient, this quantity gives the bulk interface width deep in the colony interior. A typical deme size for these simulations is $\bar{N}_T=\mu/\lambda_\text{self}\approx 100$ Parameters: $\mu = 0.1$, $\lambda_\text{self}=0.001$, $\lambda_\text{cross}= \lambda_\text{self}(1-\epsilon)$, $\rho=0.1$; Lattice-dimension: $M_X=64$, $M_Y=128$;
• Figure 3: Effects of noise strength: (a) Lattice configurations of typical simulations for neutral interactions $\epsilon=0$ with $\bar{N}_T=10,100,1000,10000$. (b) Wave velocities for the pulled FKKP waves as a function of $\bar{N}_T$ with standard error (averaged over $20$ independent simulations) for neutral interactions, deterministic limit (gray dashed line), and correction due to fluctuations (black dashed line) as functions of the carrying capacity $\bar{N}_T$ with $K=1.30$ fit to the functional form $v_\text{FKKP} \sim v_\text{FKKP}(\infty)(1-K/\log^2(\bar{N}_T)$. (c) Interface width $w_{AB}$ along the frontier for interaction parameters $\epsilon=0,-0.1,0.1$ as functions of the carrying capacity $\bar{N}_T$, at time $t=500$. Parameters: $\mu = 0.1$, $\lambda_\text{self}=\mu/\bar{N}_T$, $\lambda_\text{cross}= \lambda_\text{self}(1-\epsilon)$, $\rho=0.1$; Lattice-dimension: $N_X=64$, $N_Y=128$;
• Figure 4: $AB$-interfaces localize at height minima: (a) Lattice configurations at an undulating frontier for three typical simulations for neutral interactions with $\bar{N}_T=100$ (top row), with colony height indicated with a black dashed line, and colored with the local genotype fraction at time $t=500$ (bottom row), expanded vertical scale. Our periodic boundary conditions in the $x$-direction ensure that two genetic interfaces can be studied for each simulation. (b) Relative colony height around the $AB$-interface at position $x_0$, averaged over 20 individual runs and each of the two interfaces, given in units of the lattice spacings. Parameters: $\mu = 0.1$, $\lambda_\text{self}=\lambda_\text{cross}=0.001$, $\rho=0.1$; Lattice-dimension: $N_X=64$, $N_Y=128$;
• Figure 5: Frontier undulations alter interface broadening: Concentration profiles at three different times $t$ for neutral interactions $\epsilon =0$. Initially, the colony height has an imposed long wavelength $\cos(x/L_X)$ undulation. In (a) and (b), the $AB$(i.e. blue/red)-interface is initially positioned as a step function at the minimum and maximum of the cosine function. Note the squeezing of the interface width at the frontier in (a), as opposed to the broadening of the frontier interface width in (b). (c) An $AB$-interface initially positioned off center from cosine indentation is attracted to the minimum of the slowly relaxing cosine undulation. Parameters same as in Fig. \ref{['fig:FKPP']}, but no-flux boundary conditions in the $x$-, and $y$-direction, so that we can focus on a single interface.