Variable species densities are induced by volume exclusion interactions upon domain growth

In this work we study the effect of domain growth on spatial correlations in agent populations containing multiple species. This is important as heterogenous cell populations are ubiquitous during the embryonic development of many species. We have previously shown that the long-term behavior of an agent population depends on the way in which domain growth is implemented. We extend this work to show that, depending on the way in which domain growth is implemented, different species dominate in multispecies simulations. Continuum approximations of the lattice-based model that ignore spatial correlations cannot capture this behavior, while those that explicitly account for spatial correlations can. The results presented here show that the precise mechanism of domain growth can determine the long-term behavior of multispecies populations and, in certain circumstances, establish spatially varying species densities.

Figure 1

Before and after the growth events for both (a) GM1 and (b) GM2, in which growth is along the $x$ axis 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 re-main 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 (thicker line). 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. $r_x$ 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) $r_x=3$ and $r_y=1$.

Figure 3

Including the effects of pairwise correlations renders the correlations ODE model [Eq. (27)] able to accurately approximate the averaged results from the IBM, whereas the standard MFA [Eq. (28)] cannot. In (a) the parameters are $P_m^A=20,P_p^A=0.9,P_d^A=0,P_m^B=1,P_p^B=1$, and $P_d^B=0$. In (b) the parameters are $P_m^A=20,P_p^A=0.9,P_d^A=0.4,P_m^B=1,P_p^B=1$, and $P_d^B=0.4$.

Figure 4

Including the effects of pairwise correlations renders the correlations ODE model [Eq. (27)] able to accurately approximate the averaged results from the IBM, whereas the standard MFA [Eq. (28)] cannot. The parameters for both panels (a) GM1 and (b) GM2 are $P_m^A=20,P_p^A=0.9,P_m^B=1,P_p^B=1,P_{g_x}=0.1$, and $P_{g_y}=0.1$.

Figure 5

(a)–(c) the evolution of the correlation functions for $F_{A,A},F_{B,B}$, and $F_{A,B}$ with exponential growth via GM1. The distance plotted increases from $\mathrm{\Delta }$ to $10\mathrm{\Delta }$ in steps of $\mathrm{\Delta }$. Increasing distance is from top to bottom in (a) and (b), i.e., blue line $=\mathrm{\Delta }$, red line = $2\mathrm{\Delta }$, and so on. In (c) distance increases from bottom to top. In (d)–(f) the correlation functions for $F_{A,A},F_{B,B}$, and $F_{A,B}$ are calculated from ensemble averages from the IBM. In (a, b) and (d, e) the autocorrelation functions are greater than or equal to unity, indicating that lattice sites are more likely to be occupied by a particular species if there are others nearby. The lower movement rate of species $B$ leads to it having larger autocorrelations, and these autocorrelations grow as species $A$ begins to dominate. In contrast, the autocorrelations of species $A$ decrease as it begins to dominate. The parameters for all panels are $P_m^A=20,P_p^A=0.9,P_m^B=1,P_p^B=1,P_{g_x}=0.1$, and $P_{g_y}=0.1$.

Figure 6

(a)–(c) the evolution of the correlation functions for $F_{A,A},F_{B,B}$, and $F_{A,B}$ with exponential growth via GM2. In (d)–(f) the correlation functions for $F_{A,A},F_{B,B}$, and $F_{A,B}$ are calculated from ensemble averages from the IBM. See Fig. 5 for further details. The lower movement rate of species $B$ leads to species $B$ having larger autocorrelations, and these autocorrelations decrease as species $B$ begins to dominate. The parameters for all panels are $P_m^A=20,P_p^A=0.9,P_m^B=1,P_p^B=1,P_{g_x}=0.1$, and $P_{g_y}=0.1$

Figure 7

Altering the initial domain size causes the evolution of both species to change in GM2 but not in GM1. As the initial size of the domain is increased the dominance of species $B$ in GM2 occurs earlier. GM1: (a) 50 by 50, (b) 100 by 100, (c) 200 by 200. GM2: (d) 50 by 50, (e) 100 by 100, (f) 200 by 200. The parameters for all panels are $P_m^A=20,P_p^A=0.9,P_m^B=1,P_p^B=1,P_{g_x}=0.1$, and $P_{g_y}=0.1$.

Figure 8

Including the effects of pairwise correlations renders the correlations ODE model [Eqs. (31) and (33)] able to accurately approximate the averaged results from the IBM, whereas the standard MFA cannot. (a) GM1 linear domain growth, (b) GM2 linear domain growth, (c) GM1 logistic domain growth, (d) GM2 logistic domain growth. The parameters for all panels are $P_m^A=20,P_p^A=0.9,P_m^B=1,P_p^B=1,P_{g_x}=0.1$, and $P_{g_y}=0.1$.

Figure 9

Enteric domain growth. (a) GM1 exponential domain growth, (b) GM2 exponential domain growth, (c) GM1 linear domain growth, (d) GM2 linear domain growth. The parameters for all panels are $P_m^A=20,P_p^A=0.9,P_m^B=1,P_p^B=1,P_{g_x}=0.1$, and $P_{g_y}=0.1$.

Figure 10

The average relative error associated with the domain length closure for individual density functions.

Figure 11

Parameter sweep for (a) GM1 and (b) GM2. $P_{g_x}=0.1$ and $P_{g_y}=0.1$ and domain growth is exponential. $P_p^A=0.9$ and $P_p^B=1$, and $c^A\left(t\right)/c^B\left(t\right)$ is plotted. The red contour line in (a) indicates $c^A\left(t\right)/c^B\left(t\right)=1$.

Figure 12

Apical domain growth. (a) GM1 exponential domain growth, (b) GM2 exponential domain growth, (c) GM1 linear domain growth, (d) GM2 linear domain growth. The parameters for all panels are $P_m^A=20,P_p^A=0.9,P_m^B=1,P_p^B=1,P_{g_x}=0.1$, and $P_{g_y}=0.1$.