How domain growth is implemented determines the long-term behavior of a cell population through its effect on spatial correlations

Domain growth plays an important role in many biological systems, and so the inclusion of domain growth in models of these biological systems is important to understanding how these systems function. In this work we present methods to include the effects of domain growth on the evolution of spatial correlations in a continuum approximation of a lattice-based model of cell motility and proliferation. We show that, depending on the way in which domain growth is implemented, different steady-state densities are predicted for an agent population. Furthermore, we demonstrate that the way in which domain growth is implemented can result in the evolution of the agent density depending on the size of the domain. Continuum approximations that ignore spatial correlations cannot capture these behaviors, while those that account for spatial correlations do. These results will be of interest to researchers in developmental biology, as they suggest that the nature of domain growth can determine the characteristics of cell populations.

Figure 1

Before and after the growth events for both (a) GM1 and (b) GM2, in which growth is along the $x$ axis (horizontal direction) for a two-dimensional lattice. In each row the yellow (light gray) site has been chosen to undergo a growth event. Following this the yellow (light gray) site is moved to the right with its contents, for instance an agent (represented by a black cell). The blue (dark gray) sites are the new lattice sites and are always initially empty. The contents of all the other sites remain unaffected, although in some cases their neighboring sites will change.

Figure 2

The two types of configuration of lattice sites: (a) colinear and (b) diagonal. The lattice sites in question are labeled m and n and bordered by blue (thick border). In (a), two colinear lattice sites share the same row but not the same column (or vice versa). In (b), two lattice sites are diagonal, meaning they do not share the same row or column. In both (a) and (b), $r_x$ is the distance between two lattice sites in the horizontal direction, and $r_y$ is the distance between two lattice sites in the vertical direction. In (b), this means $r_x=3$ and $r_y=1$.

Figure 3

The evolution of the colinear and diagonal components of the pairwise density functions for (a) GM1 and (b) GM2. $P_{g_x}=P_{g_y}=0.1$.

Figure 4

For GM1 including pairwise correlations in a correlation ODE model provides a more accurate approximation of the evolution of the macroscopic density in the IBM than the MFA. (a) $P_m=1$, $P_p=1$, $P_{g_x}=0.05$, $P_{g_y}=0.05$, (b) $P_m=1$, $P_p=1$, $P_{g_x}=0.1$, $P_{g_y}=0.1$.

Figure 5

Increasing the growth rates, $P_{g_x}$ and $P_{g_y}$, causes the pairwise correlations to augment for GM1 in the correlation ODE model [compare (a) with (b)], and in the IBM [compare (c) and (d)]. In the case of GM1 the pairwise correlations do not return to $F=1$. In (a)–(d) the distance increases from $\mathrm{\Delta }$ to $6\mathrm{\Delta }$ in steps of $\mathrm{\Delta }$. Increasing distance is from top to bottom as indicated by the arrows. The parameters for (a) and (c) are $P_m=1$, $P_p=1$, $P_{g_x}=0.05$, $P_{g_y}=0.05$, and for (b) and (d) are $P_m=1$, $P_p=1$, $P_{g_x}=0.1$, $P_{g_y}=0.1$.

Figure 6

For GM2 including pairwise correlations in a correlation ODE model provides a more accurate approximation of the results of the evolution of the macroscopic density in the IBM. (a) $P_m=1$, $P_p=1$, $P_{g_x}=0.05$, $P_{g_y}=0.05$, (b) $P_m=1$, $P_p=1$, $P_{g_x}=0.1$, $P_{g_y}=0.1$.

Figure 7

For GM2 increasing the growth rate, $P_{g_x}$ and $P_{g_y}$, causes the pairwise correlations to shrink in the correlation ODE model [compare (a) with (b)], and in the IBM [compare (c) and (d)]. In the case of GM2 the pairwise correlations return to $F=1$. The distance increases from $\mathrm{\Delta }$ to $6\mathrm{\Delta }$ in steps of $\mathrm{\Delta }$. Increasing distance is from top to bottom as indicated by the arrows. The parameters for (a) and (c) are $P_m=1$, $P_p=1$, $P_{g_x}=0.05$, $P_{g_y}=0.05$, and in (b) and (d) are $P_m=1$, $P_p=1$, $P_{g_x}=0.1$, $P_{g_y}=0.1$.

Figure 8

Altering the initial size of the domain does not cause the evolution of the agent density to change for GM1 but does for GM2. As the initial size of the domain is increased, from 50 by 50 to 200 by 200 lattice sites, the evolution of the agent density to steady-state remains the same in GM1, as can be seen in (a) IBM and (b) correlation ODE model. In the case of GM2 altering the initial domain size does cause the evolution of the agent density to change. As the initial size of the domain is increased, from 50 by 50 to 200 by 200 lattice sites, the evolution of the agent density accelerates in GM2 as can be seen in (c) IBM and (d) correlation ODE model. $P_m=1$, $P_p=1$, $P_{g_x}=0.1$, $P_{g_y}=0.1$ for all.

Figure 9

Altering the initial size of the domain changes the evolution of the pairwise correlations in the case of GM2. In the case of GM2 the correlations are reduced as the initial domain size becomes larger. IBM: (a) 50 by 50, (b) 100 by 100, (c) 200 by 200. Correlation ODE model: (d) 50 by 50, (e) 100 by 100, (f) 200 by 200. Increasing distance is from top to bottom as indicated by the arrows. $P_m=1$, $P_p=1$, $P_{g_x}=0.1$, $P_{g_y}=0.1$ for all.

Figure 10

Altering the initial size of the domain does not change the evolution of the pairwise correlations in the case of GM1. IBM: (a) 50 by 50, (b) 100 by 100, (c) 200 by 200. Correlation ODE model: (d) 50 by 50, (e) 100 by 100, (f) 200 by 200. Increasing distance is from top to bottom as indicated by the arrows. $P_m=1$, $P_p=1$, $P_{g_x}=0.1$, $P_{g_y}=0.1$ for all panels.