Shape of population interfaces as an indicator of mutational instability in coexisting cell populations

TL;DR

This work addresses how mutational instability within a coexisting invading population shapes the spatial invasion frontier. It develops a three-strain lattice model with a mutating invader and a reinvading bystander, analyzed in $d=1+1$ and $d=2+1$ to reveal how interface roughness encodes internal dynamics. Near the mutational meltdown transition, the invasion front becomes rougher, with a directed-percolation type criticality emerging in $d=1+1$ and front-speed dependent roughening in $d=2+1$, ranging from voter-like coarsening to Fisher-wave behavior. The results suggest that measuring front morphology could provide diagnostic insight into tumor progression or microbial range expansions, complementing molecular sequencing and imaging data and motivating further higher-dimensional and motility-inclusive studies.

Abstract

Cellular populations such as avascular tumors and microbial biofilms may "invade" or grow into surrounding populations. The invading population is often comprised of a heterogeneous mixture of cells with varying growth rates. The population may also exhibit mutational instabilities, such as a heavy deleterious mutation load in a cancerous growth. We study the dynamics of a heterogeneous, mutating population competing with a surrounding homogeneous population, as one might find in a cancerous invasion of healthy tissue. We find that the shape of the population interface serves as an indicator for the evolutionary dynamics within the heterogeneous population. In particular, invasion front undulations become enhanced when the invading population is near a mutational meltdown transition or when the surrounding "bystander" population is barely able to reinvade the mutating population. We characterize these interface undulations and the effective fitness of the heterogeneous population in one- and two-dimensional systems.

Figures (13)

• Figure 1: (a) Update rules for the bystander model for a population in $d=1+1$ dimensions. Each generation is evolved by allowing for two cells from the previous generation to compete for an empty lattice site, as shown by the arrows. The probability of occupation by a cell of a type $i=s,f,b$ is proportional to its growth rate $\Gamma_i$, where $s$ is the slow growing black strain, $f$ is the fast growing red strain, and $b$ is the yellow bystander. If a red (fast-growing) cell is placed in the empty spot, then it in addition has a probability $\mu$ of mutating to the slower-growing black strain. (b) For a two-dimensional population $(d=2+1)$ the generations are evolved on staggered triangular lattices, as shown. This time, three cells compete to divide into empty lattice sites. Otherwise, the dynamics are the same as the $d=1+1$ case.
• Figure 2: Simulated sectors of a black/red mutating population invading a bystander population (surrounding white area), initialized by a single red cell in a (a) one-dimensional and (b) two-dimensional population. The populations are evolved for about 100 generations, with the time direction indicated. In (a), we indicate the motion of the invasion front (which in this case is a point) between the two populations. In (b), the invasion front would be the complicated boundary between the black/red population and the white space at each time slice $t$ along the vertical direction. The phases of the internal dynamics of the invading population (inactive, critical, and active phases) are indicated. In the inactive phase, the red, fast-growing mutant is lost from the population over time. As the invader population transitions from the inactive to the active phase in which the red strain is maintained, the invasion front exhibits enhanced undulations.
• Figure 3: A $d=1+1$ simulation of a red/black mutating population invading a bystander yellow one. Here, the yellow strain grows faster than the black strain but slower than the fit red strain. The invasion front between the black/red population and the bystander strain can be characterized by a random walk with alternating bias. The yellow strain invades the black patches and is invaded by the red patches. The sizes of the red and black patches are controlled by the internal dynamics of the black/red strain. We illustrate the characteristic sizes $\xi_{\perp}$ and lifetime $\xi_{\parallel}$ of the black patches.
• Figure 4: Phase diagrams for (a) the $d=1+1$ case and (b) the $d=2+1$ case, calculated by initializing a well-mixed population of the red and yellow strains and evolving the whole population for $t \approx 10^6$ generations. In (a) we use a one-dimensional population of $L=5000$ cells and average over 256 runs of the evolution. In (b) we have a two-dimensional population with $L^2$ cells, where $L=500$. Here we average over 40 evolution runs. In both cases we set $s=0.3$. After evolving for $10^6$ generations, we calculate the red/black mutating fraction of the total population: $\rho_m=\rho_f+\rho_s$. In each phase diagram, the black dashed line corresponds to the mean field prediction $\mu=s-b$. The green and white dashed lines correspond to the improved predictions [see Eqs. eq:improvedmf1, eq:improvedmf2, eq:improvedmf3d1, eq:improvedmf3d2] for $\mu \ll \mu^*$ and $\mu \approx \mu^*$, respectively, that take into account the spatial structure of the population. We also indicate the line $\mu=\mu^*$ along which we find a mutational meltdown transition within the invading red/black population.
• Figure 5: Long-time survival probabilities $P_{\mathrm{surv}}$ of clusters generated from a single mutating red cell in a yellow bystander population for (a) $d=1+1$-dimensional range expansions for $s=0.3$ (left panel) and $s=0.1$ (right panel), after $t=10^6$ generations on a lattice with size $L=5000$ cells averaged over 128 runs; and (b) $d=2+1$-dimensional range expansions for $s=0.15$ (left panel) and $s=0.1$ (right panel), after $t=2\times 10^3$ generations on a lattice with size $L^2$ cells with $L=500$, averaged over 256 runs. We show the expected transition shape near the $\mu\approx \mu^*$ DP transition in the white dashed line [Eqs. eq:improvedmf2,eq:improvedmf3d2]. The black dashed line is the transition position for a well-mixed population. The green dashed line is an improved mean-field estimate of the transition discussed in the main text [Eqs. eq:improvedmf1,eq:improvedmf3d1].