New sector morphologies emerge from anisotropic colony growth

TL;DR

A model of anisotropic growth, based on coupled Kardar-Parisi-Zhang and Fisher-Kolmogorov-Petrovsky-Piskunov equations, finds that strong anisotropy results in a distinct morphology of spatial invasion with a kink in the displaced strain ahead of the boundary between the strains.

Abstract

Competition during range expansions is of great interest from both practical and theoretical view points. Experimentally, range expansions are often studied in homogeneous Petri dishes, which lack spatial anisotropy that might be present in realistic populations. Here, we analyze a model of anisotropic growth, based on coupled Kardar-Parisi-Zhang and Fisher-Kolmogorov-Petrovsky-Piskunov equations that describe surface growth and lateral competition. Compared to a previous study of isotropic growth, anisotropy relaxes a constraint between parameters of the model. We completely characterize spatial patterns and invasion velocities in this generalized model. In particular, we find that strong anisotropy results in a distinct morphology of spatial invasion with a kink in the displaced strain ahead of the boundary between the strains. This morphology of the out-competed strain is similar to a shock wave and serves as a signature of anisotropic growth.

Figures (15)

• Figure 1: Colony growth can be described by a height function $h(x,t)$ and population composition $f(x,t)$. (Color Online) The top panel shows two neutral strains (blue and yellow) with identical expansion speeds growing under isotropic conditions. The resulting colony shape is circular, and the sector boundaries expand perpendicular to the expansion front. The second panel illustrates neutral expansion with a speed $v(\theta)$ that depends on the growth direction, specified by the angle $\theta$. As an example, an elliptical colony shape is shown. In general, the boundary between the strains is not orthogonal to the growth front but instead forms an angle $\epsilon(\theta)$ with the front normal. The vertical growth $dh$ and horizontal drift $dx_b$ over a time interval $dt$ depend on the slope $\theta$, leading to the nonlinearities in Eqs. \ref{['eqn:FKPP']} and \ref{['eqn:KPZ']}.
• Figure 2: Anisotropy enables an "Escaping Bulge" sector morphology (Color Online). The top panel shows morphologies observed in numerical solutions of deterministic Eqs. \ref{['eqn:FKPP']} and \ref{['eqn:KPZ']} . To the left (negative $\alpha$) is a V-shaped dent with straight edges. Next, at small positive $\alpha$, is the composite bulge morphology, characterized by constant limiting slopes surrounding a circular arc. At larger $\alpha$, the sector shape is a pure circular arc, commonly observed in experiments korolev2012selective. The final shape is the newly identified escaping bulge. This morphology requires large $\alpha$ but, more importantly, can only emerge in the presence of strong anisotropy ($\lambda \geq 2\beta$ for pulled waves). The bottom panel demonstrates that the same morphologies exist in the anisotropic reaction-diffusion system described by Eq. \ref{['eqn:reaction_diffusion']}. For the coupled FKPP-KPZ equations, the parameters used are $D_f = D_h = 1$, $v_0 = 0.15$, $\lambda = 20$, $s_0 = 0.25$, and $\beta = 5$. For the two-dimensional mechanistic model, all panels use $r_1 = r_2 = 1$, $K = 0.1$, $D_{1x} = D_{1y} = D_{2y} = 0.02$, and $D_{2x} = 0.09$ (with type 1 being blue). The interaction strengths $a_{ij}$ used for the different morphologies are, from left to right, $(a_{11}, a_{21}, a_{12}, a_{22}) = (-0.7, -0.7, 0.7, 0), (0.4, 0.4, 0.9, 0), (1, 1, 0.6, 0), (1, 1, 0, 0)$.
• Figure 3: Three regimes for anisotropic pulled waves (Color Online). The mutant can invade with the Fisher velocity $u_0$, the velocity of the circular arc $u_{\text{kpz}} = \sqrt{2\alpha \lambda}$ (dashed black line with green ribbon), or the escaping bulge velocity predicted by Eq. \ref{['eqn:u_escaping']} (dashed lines of different colors) depending on the expansion speed difference $\alpha$ and the degree of anisotropy $\beta/\lambda$. The markers are obtained from numerical solutions of Eqs. \ref{['eqn:FKPP']} and \ref{['eqn:KPZ']} . Regions of the plot are colored based on the morphology, matching the phase diagram in Fig. \ref{['fig:pulled_phase']}. When $\alpha < 0$ the morphology is a V-shaped dent (purple color). When $\alpha > 0$ and $u > u_{\text{kpz}}$ the morphology is composite (orange color). When $\alpha > 0$ and $u = u_{\text{kpz}}$ the morphology is a circular arc (green region with black dashed center line). When $\alpha > 0$ and $u < u_{\text{kpz}}$ we find the escaping bulge morphology (blue color). Parameters are: $v_0 = 1$, $\lambda = 20$, $D_h = 1$, $D_f = 1$, and $s_0 = 0.25$.
• Figure 4: Schematic of the escaping bulge morphology. This diagram illustrates the key geometric features involved in deriving the invasion speed for the escaping bulge morphology (Eq. \ref{['eqn:u_escaping']}). The solid dot marks the instantaneous location of the sector boundary at time $t$, while the square indicates the location of the shock in the invaded strain's morphology, which moves with speed $c$ as given in Eq. \ref{['eqn:shock_speed']}. The dotted line represents the limiting slope of the escaping bulge far ahead of the invasion front, as described by Eq. \ref{['eqn:sigma_escaping']}.
• Figure 5: Predicted phase diagram of morphologies for pulled waves based on Eqs. \ref{['eqn:FKPP']} and \ref{['eqn:KPZ']}(Color Online). The isotropic case corresponds to the line $\beta / \lambda = 1$. Small anisotropy does not produce new morphologies, whereas the escaping bulge can only exist for $\beta / \lambda < 1/2$ (horizontal dashed line). Parameters not shown are $u_0 = 5$ and $\lambda = 20$.