Environment heterogeneity creates fast amplifiers of natural selection in graph-structured populations

TL;DR

This work analyzes how environment heterogeneity across demes on graphs reshapes mutant fixation under two regimes: frequent and rare migrations. Using a serial-dilution deme-structured model, a multi-type branching process, and coarse-grained Markov chains, the authors derive first- and second-order fixation probabilities in heterogeneous settings, revealing that circulation graphs obey the circulation theorem to first order but exhibit second-order corrections, while noncirculation graphs (notably star and line) can amplify selection when mutants are more advantageous in high-outflow demes. A key mechanism is that enhanced mutant fitness in demes with stronger migration outflow can boost fixation probability and accelerate both fixation and extinction, sometimes even turning suppressors into amplifiers in the rare-migration regime via refugia effects. The results generalize across diverse graphs, including large and Dirichlet-mixed networks, and extend to deleterious mutants, indicating broad relevance for spatially structured microbial populations and potential applications in directed evolution and microbiome ecology. Overall, the study provides a unifying framework linking graph topology, migration asymmetry, and environment-driven fitness differences to predict when spatial structure will amplify natural selection and hasten evolutionary outcomes.

Abstract

Complex spatial structure, with partially isolated subpopulations, and environment heterogeneity, such as gradients in nutrients, oxygen, and drugs, both shape the evolution of natural populations. We investigate the impact of environment heterogeneity on mutant fixation in spatially structured populations with demes on the nodes of a graph. When migrations between demes are frequent, we find that environment heterogeneity can amplify natural selection and simultaneously accelerate mutant fixation and extinction, thereby fostering the quick fixation of beneficial mutants. We demonstrate this effect in the star graph, and more strongly in the line graph. We show that amplification requires mutants to have a stronger fitness advantage in demes with stronger migration outflow, and that this condition allows amplification in more general graphs. As a baseline, we consider circulation graphs, where migration inflow and outflow are equal in each deme. In this case, environment heterogeneity has no impact to first order, but increases the fixation probability of beneficial mutants to second order. Finally, when migrations between demes are rare, we show that environment heterogeneity can also foster amplification of selection, by allowing demes with sufficient mutant advantage to become refugia for mutants.

Figures (25)

• Figure 1: Serial dilution model for spatially structured populations with environmental heterogeneities. Schematic of an elementary step of the serial dilution model. Starting from a bottleneck, each deme undergoes a deterministic local growth step followed by a dilution and migration step, leading to a new bottleneck. Blue markers represent wild-type individuals and orange markers represent mutants. Gray backgrounds with different darkness represent different environments.
• Figure 2: Impact of environmental heterogeneity on mutant fixation in the star. Panel A: Schematic of the heterogeneous star, with migration asymmetry $\alpha=m_I/m_O$, and deme-dependent mutant advantage prefactor $\delta$ shown in the histogram below. The relative mutant fitness excess in the center is defined as $\sigma_C = (\delta_C - \langle \delta \rangle) / \langle \delta \rangle$. Panel B: Mutant fixation probability versus the baseline mutant fitness advantage $st$, for different relative mutant fitness excess $\sigma_C$ in the center, with migration asymmetry $\alpha=0.2$. Markers: stochastic simulation results; lines: analytical predictions (Eq. \ref{['eq:fo-star']}). Panel C: Heatmap of the first-order coefficient in the expansion of the mutant fixation probability in the baseline effective mutant fitness advantage $st$ (denoted by $a$ in Eq. \ref{['expand']}), versus the migration asymmetry $\alpha$ and the relative mutant fitness excess $\sigma_C$ in the center of the star. Red dashed lines: numerically-determined boundaries of the region where the star amplifies selection; solid black line: analytical prediction for this boundary, given by $f_{\textrm{star}}(\alpha,D)$ in Eq. \ref{['eq:fstar']}. Horizontal dotted line: homogeneous environment case ($\sigma_C=0$). Markers: parameter values considered in panel B. Parameter values: in panels B and C, $D=5$; $K=1000$; $\langle\delta \rangle=0.5$. In Panel B: $m_O=0.6$; $\alpha=0.2$; each marker comes from $5\times 10^5$ stochastic simulation realizations.
• Figure 3: Amplification of selection and accelerated dynamics in the heterogeneous line. Panel A: Schematics of the structures considered: heterogeneous line with step environmental profile and $\alpha=1.5$; homogeneous line with same $\alpha$ and $\langle\delta\rangle$; homogeneous clique with same $\langle\delta\rangle$. Panel B: Condition for amplification of natural selection in the heterogeneous line shown in the top schematic. The functions $S$ and $g$, defined respectively in Eqs. \ref{['eq:cond-line1']} and \ref{['eq:cond-line3']}, are plotted versus migration asymmetry $\alpha$. Pink-shaded region: range of $\alpha$ where $S(\alpha)>g(\alpha)$, i.e. where the heterogeneous line amplifies natural selection. Dotted line: migration asymmetry ($\alpha=1.5$) chosen in panels C-E. Panel C: Mutant fixation probability versus baseline mutant fitness advantage $st$, in the different spatial structures shown in panel A. Predictions from the branching process theory ("Th."), specifically Eq. \ref{['eq:aline']}, and results from stochastic simulations ("Sim."), are shown. Panel D: Mutant fixation time, in number of bottlenecks, in the different structures considered, versus baseline mutant fitness advantage $st$. Panel E: Same as in D, but for mutant extinction time. Theoretical predictions are from Eq. \ref{['eq:text']}. Parameter values for all structures: $D=5$, $K=1000$; for all lines: $\alpha = 1.5$, $m_L = 0.3$; for the clique: $m=0.15$; for the line with step profile: $\delta=1$ in the leftmost deme, $\delta=0$ in other ones. Each marker comes from $5\times 10^5$ stochastic simulation realizations.
• Figure 4: Amplification of selection and accelerated dynamics in structures with a special deme in terms of mutant advantage and migration outflow. Panel A: Mutant fixation probability versus baseline mutant fitness advantage $st$ for population structures with one special deme, characterized by stronger migration outflow ($\tilde{\alpha}=m_1/m_2=2$) and greater mutant advantage than others ($\beta=\delta_1/\delta_2>1$), for various values of $\beta$. Results for a homogeneous clique with the same $\langle\delta\rangle$ are shown for reference. Schematics of these population structures are shown in Panel D. Panel B: Time to mutant fixation, in bottlenecks, versus baseline mutant fitness advantage $st$, for the same structures as in A. Panel C: Same as in B, but for extinction time. "Th." indicates results from the branching process theory (Eq. \ref{['eq:pfix-dirdet']} for panel A, Eq. \ref{['eq:text']} for panel C), "Sim." results from stochastic simulations (panels A-C). Panel D: the structures considered in this figure: a fully-connected structure with a special deme and a homogeneous clique. Panel E: The excess fixation probability in a structure with one special deme, compared to a homogeneous clique with the same $\langle\delta\rangle$, is shown versus the migration asymmetry $\tilde{\alpha}$ and the environment asymmetry $\beta$, in the case where migration probabilities are drawn from Dirichlet distributions (no strong symmetry in migration probabilities, generalizing over the structure considered in panels A-D, see SI Section \ref{['subs:alpha-eta']}). The plotted values represent the difference between the first-order coefficient in $st$ of the fixation probability (denoted by $a$ in Eq. \ref{['expand']}, computed numerically as explained in SI Section \ref{['sec:numerical_pfix']}, and averaged over many sets of migration probabilities), and the value for the homogeneous clique with the same $\langle\delta\rangle$, which is $2\langle\delta\rangle$. Solid and dashed black lines mark theoretical boundaries for amplification in the strongly symmetric structure shown in panels A-D (see Eq. \ref{['eq:alphabelta']} and main text): the solid line corresponds to $\tilde{\alpha} = \beta$ and the dashed line to $\tilde{\alpha} = 1$. Markers show the $(\tilde{\alpha},\beta)$ values considered in panels A-C. Parameter values for all panels: $D=5$, $\langle \delta \rangle = 0.25$; for panels A-C: $K=1000$, $\tilde{\alpha}=2$ and $m_1=0.33$ for the special-deme structure, $m=0.15$ for the homogeneous clique. Each value in the heatmap of panel E is obtained by averaging over $5\times10^3$ structures, whose migration probabilities are each time drawn independently in Dirichlet distributions.
• Figure 5: Amplification of selection across five-node graphs with strong migration outflow from demes where mutants are advantaged. Each panel shows a connected five-node graph, ordered by decreasing first-order coefficient in the expansion of the fixation probability in $st$ (denoted by $a$ in Eq. \ref{['expand']}). These coefficients are indicated in the top left corner of each panel, and shown through the panel frame's color. They were calculated numerically as described in SI Section \ref{['sec:numerical_pfix']}. Migration probabilities and environment heterogeneity follow convention 1 defined in SI Section \ref{['SI:gengraphs']}, ensuring that overall migration flows from demes whith the strongest mutant fitness advantage to those with weaker advantage. For all graphs, $\langle \delta \rangle=0.5$. Hence, circulations have a first-order coefficient $2\langle \delta \rangle=1$, and coefficients larger than one indicate amplification of selection.