Spatial segregation across travelling fronts in individual-based and continuum models for the growth of heterogeneous cell populations

TL;DR

This work develops a phenotype-structured model for growing cell populations in which cells move down a pressure gradient and contribute to a phenotype-weighted pressure $p(t,x)=\sum_i \omega_i n_i(t,x)$. It derives the continuum PDE system $\partial_t n_i - \mu_i \partial_x(n_i \partial_x p)=G_i(p) n_i$, with $p=\sum_i \omega_i n_i$, as the limit of an on-lattice, branching IB model, enabling analysis of travelling waves that produce spatial segregation across fronts. The authors prove existence of segregated travelling wave solutions and provide interface conditions, supported by numerical simulations that show excellent agreement between the IB and PDE models and confirm the predicted wave speeds. The results demonstrate that inter-cellular mobility differences can sustain spatial segregation during invasion and suggest broad biological relevance to tumor invasion and tissue organization, while outlining future directions such as phenotypic switching and continuous-phenotype extensions.

Abstract

We consider a partial differential equation model for the growth of heterogeneous cell populations subdivided into multiple distinct discrete phenotypes. In this model, cells preferentially move towards regions where they feel less compressed, and thus their movement occurs down the gradient of the cellular pressure, which is defined as a weighted sum of the densities (i.e. the volume fractions) of cells with different phenotypes. To translate into mathematical terms the idea that cells with distinct phenotypes have different morphological and mechanical properties, both the cell mobility and the weighted amount the cells contribute to the cellular pressure vary with their phenotype. We formally derive this model as the continuum limit of an on-lattice individual-based model, where cells are represented as single agents undergoing a branching biased random walk corresponding to phenotype-dependent and pressure-regulated cell division, death, and movement. Then, we study travelling wave solutions whereby cells with different phenotypes are spatially segregated across the invading front. Finally, we report on numerical simulations of the two models, demonstrating excellent agreement between them and the travelling wave analysis. The results presented here indicate that inter-cellular variability in mobility can provide the substrate for the emergence of spatial segregation across invading cell fronts.

Figures (7)

• Figure 1: Schematic overview of the mechanisms incorporated in the individual-based model. Between time-steps $k$ and $k+1$, each cell with phenotype $i = 1,\ldots,I$ at spatial position $x_j$ may: divide with probability $\tau \, G_i(p_j^k)_+$, die with probability $\tau \, G_i(p_j^k)_-$, and remain quiescent with probability $1-\tau \, \left(G_i(p_j^k)_+ + G_i(p_j^k)_- \right) = 1 - \tau \, |G_i(p_j^k)|$ (left panel); move to spatial positions $x_{j-1}$ or $x_{j+1}$ with probabilities $M_{i,L}(p_j^k-p_{j-1}^k)$ or $M_{i,R}(p_j^k-p_{j+1}^k)$, respectively, or remain stationary with probability $1-M_{i,L}(p_j^k-p_{j-1}^k)-M_{i,R}(p_j^k-p_{j+1}^k)$ (right panel)
• Figure 2: Schematic overview of spatial segregation across travelling waves. Schematic of how, under assumptions \ref{['eq:G_conditions']}-\ref{['eq:alpha_conditions']} and \ref{['ass:modpar3']}, cells with phenotypes labelled by different values of the index $i$ are spatially segregated across invading fronts, which are represented by travelling wave solutions of the continuum model \ref{['eq:PDEmodel']}, i.e. solutions of the system of differential equations \ref{['eq:PDEmodelTW']}, subject to conditions \ref{['ass:SuppTW1']}-\ref{['ass:SuppTW2']}
• Figure 3: Main results under the baseline parameter setting for $I=3$. Comparing numerical solutions of the continuum model (top panels) with the averaged results of 10 simulations of the individual-based model (bottom panels), when $I=3$ and the values of the parameters $\alpha_i$, $\mu_i$, and $\omega_i$ are set according to \ref{['eq:parametersI=3']}. Plots display the cell pressure $p(t,x)$ (left panels) and the cell densities $n_i(t,x)$ (right panels) at three successive time instants -- i.e. $t=50$(dotted lines), $t=100$(dashed lines), and $t=150$(solid lines). The insets of the left panels display the plots of $x_1(t)$ (cyan), $x_2(t)$ (magenta), and $x_3(t)$ (yellow) defined via \ref{['eq:xplots']}. The coloured markers in the plot of $p(t,x)$ highlight the values of $p(t,x_1(t))$ (cyan), $p(t,x_2(t))$ (magenta), and $p(t,x_3(t))$ (yellow) at $t=50$, $t=100$, and $t=150$. The numerically estimated wave speeds are $c_{\text{PDEn}}= 0.42$ and $c_{\text{IBn}}=0.42$, and the analytically predicted wave speed is $c_{\text{a}}=0.42$
• Figure 4: Main results under the baseline parameter setting for $I=4$. Comparing numerical solutions of the continuum model (top panels) with the averaged results of 10 simulations of the individual-based model (bottom panels), when $I=4$ and the values of the parameters $\alpha_i$, $\mu_i$, and $\omega_i$ are set according to \ref{['eq:parametersI=3']}-\ref{['eq:parametersI=4']}. Plots display the cell pressure $p(t,x)$ (left panels) and the cell densities $n_i(t,x)$ (right panels) at three successive time instants -- i.e. $t=50$(dotted lines), $t=100$(dashed lines), and $t=150$(solid lines). The insets of the left panels display the plots of $x_1(t)$ (cyan), $x_2(t)$ (magenta), and $x_3(t)$ (yellow) defined via \ref{['eq:xplots']}. The coloured markers in the plot of $p(t,x)$ highlight the values of $p(t,x_1(t))$ (cyan), $p(t,x_2(t))$ (magenta), and $p(t,x_3(t))$ (yellow) at $t=50$, $t=100$, and $t=150$. The numerically estimated wave speeds are $c_{\text{PDEn}}= 0.35$ and $c_{\text{IBn}}=0.35$, and the analytically predicted wave speed is $c_{\text{a}}=0.35$
• Figure 5: Quantitative comparison between the individual-based and the continuum models. Plot of the quantity $\dfrac{|p_{\text{P}DE}(t,x) - p_{\text{I}B}(t,x)|}{\overline{p}}$ at $t=150$, where $p_{\text{P}DE}$ is the cell pressure computed from numerical solutions of the continuum model displayed in Figure \ref{['fig:Fig3']} (left panel) and Figure \ref{['fig:Fig4']} (right panel), while $p_{\text{I}B}$ is the cell pressure computed from the averaged results of 10 simulations of the individual-based model displayed in the same figures. The supports of the cell densities $n_i$ for $i=1,\ldots, I$ with $I=3$ (left panel) or $I=4$ (right panel) are highlighted in the same colours as those of the curves of the cell densities displayed in Figures \ref{['fig:Fig3']} and \ref{['fig:Fig4']}

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Theorem 1
• proof
• Remark 3
• Remark 4