Interplay between morphology and competition in two-dimensional colony expansion

Letter

Interplay between morphology and competition in two-dimensional colony expansion

Daniel W. Swartz, Hyunseok Lee, Mehran Kardar, and Kirill S. Korolev

Phys. Rev. E 108, L032301 – Published 8 September 2023

Abstract

In growing populations, the fate of mutations depends on their competitive ability against the ancestor and their ability to colonize new territory. Here we present a theory that integrates both aspects of mutant fitness by coupling the classic description of one-dimensional competition (Fisher equation) to the minimal model of front shape (Kardar-Parisi-Zhang equation). We solve these equations and find three regimes, which are controlled solely by the expansion rates, solely by the competitive abilities, or by both. Collectively, our results provide a simple framework to study spatial competition.

Received 24 January 2023
Accepted 1 July 2023

DOI:https://doi.org/10.1103/PhysRevE.108.L032301

Physics Subject Headings (PhySH)

Ecological population dynamics

Front propagation Surface growth

Coarse graining

Physics of Living Systems

Authors & Affiliations

Daniel W. Swartz , Hyunseok Lee , and Mehran Kardar

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Kirill S. Korolev

Department of Physics, Graduate Program in Bioinformatics and Biological Design Center, Boston University, Boston, Massachusetts 02215, USA

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Issue

Vol. 108, Iss. 3 — September 2023

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Images

Figure 1
Sample simulation results generated by solving Eqs. (1) and (2) numerically in the pulled wave regime $s (f) = s_{0}$ . (a) Profiles of mutant frequency $f (x, t)$ at five equally spaced time slices labeled by color. (b) Height profiles $h (x, t)$ taken at the same five time points as in (a) with the same color convention. Starting from a flat initial condition, the height field develops a nontrivial morphology through a dependence of the growth rate of $h$ on the mutant frequency $f$ . (c) Spatial distribution of the two competitors visualized by plotting successive solutions of the height field $h$ and coloring each point according to the value of $f$ at the corresponding $x$ and $t$ .
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Figure 2
Invasion dynamics in pulled waves. Shown on top is the invasion velocity showing two regimes with dependence on either $α$ or $s$ . The horizontal dashed lines are the predicted Fisher velocities. The curved dashed line is $u = \sqrt{2 α v_{0}}$ , as predicted by Eq. (3). For each value of $s_{0}$ the closed circle shows the location of the transition point $α_{c}$ between the composite and circular arc morphologies. Shown on the bottom are three distinct colony morphologies, depending on $α$ . When $α < 0$ the front shape is a V-shaped dent. When $0 < α < α_{c}$ the morphology is a composite bulge consisting of a central circular arc transitioning to a constant slope at the bulge edges. When $α > α_{c}$ the front is entirely a circular arc. The red arrow shows the invasion velocity $u$ , which is the speed of the mutant–wild-type boundary along the horizontal axis. The parameters are $v_{0} = 10$ and $D_{f} = D_{h} = 1$ .
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Figure 3
Invasion dynamics in pushed waves. Symbols show the measured invasion velocity from simulations as a function of the expansion velocity difference $α$ . The dashed curves show the result of the perturbation theory discussed in the text and the horizontal dash-dotted lines are the invasion velocity predicted from simple linearization of Eq. (2). When $α$ is large and positive, all invasion velocities match the black solid line, the speed of the invading circular arc. The behavior for $α$ sufficiently negative (corresponding to a mutant growing much slower than the wild type) depends on the sign of $f_{0}$ . When $f_{0} < 0$ , the invasion speed approaches a value predicted by the linearized equations $u = 2 \sqrt{- s_{1} f_{0} D_{f}}$ . For $f_{0} > 0$ , the invasion velocity changes sign at large negative $α$ , reversing the competitive outcome. The negative invasion speed is that of a leftward-moving circular arc ( $u = - \sqrt{2 | α | v_{0}}$ ), which is depicted by the black solid line. The inset is a sample morphology which arises when the mutant is invaded by the wild type ( $u < 0$ ). The simulation shown is initialized as a half space where the left half is occupied by the mutant and the right by the wild type. The parameters are $v_{0} = 20, s_{1} = 4$ , and $D_{f} = D_{h} = 1$ .
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Figure 4
Numerical results showing the dependence of the coefficient $κ$ as defined in Eq. (5) on $f_{0}$ . The numerical values of $κ$ are obtained by fitting measured invasion velocities as functions of $α$ in the limit $α \to 0$ . The best-fit slope is then used to obtain $κ$ in Eq. (5). The red dashed line is the theoretical prediction of our perturbative analysis for the value of $D_{h}$ used in simulation at small $α$ . The yellow dashed line is the theoretical value of $κ$ when $D_{h} = 0$ . The parameters are $v_{0} = 10, D_{h} = D_{f} = 1$ , and $s_{1} = 2$ .
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