Modeling growing confluent tissues using a lattice Boltzmann method: Interface stability and fluctuations

Tissue growth underpins a wide array of biological and developmental processes, and numerical modeling of growing systems has been shown to be a useful tool for understanding these processes. However, the phenomena that can be captured are often limited by the size of systems that can be modeled. Here, we address this limitation by introducing a lattice Boltzmann method (LBM) for a growing system that is able to efficiently model hydrodynamic length scales. The model incorporates a bounce-back approach to describing the growing front of a tissue, which we use to investigate the dynamics of the interface of growing model tissues. We find that the interface grows with scaling in agreement with the Kardar-Parisi-Zhang (KPZ) universality class when growth in the system is bulk driven. Interestingly, we also find the emergence of a previously unreported hydrodynamic instability when proliferation is restricted to the tissue edge. We then develop an analytical theory to show that the instability arises due to a coupling between the number of cells actively proliferating and the position of the interface.

Figure 1

A LBM for a growing biological system. (a) We discretize our system (left) onto a hexagonal lattice (middle), with each lattice point representing a number of cells. There is a growing front beyond which the distribution function $f_i$ is zero in all lattice directions, (white sites in right). (b) Each time step is subdivided into a streaming step and redistribution step. In the streaming step, $f_i$ is advanced by one time step to get an intermediate distribution $f_i^{*}$. A steady-state distribution $f_i^{ss}$ is then calculated using $f_i^{*}$ and $f_i$ is redistributed such that it relaxes towards $f_i^{ss}$. (c) The growing periphery is enforced using a “bounce-back” condition during the redistribution step. When the density at a site is below a critical density ${\rho }_c$, instead of relaxing $f_i$ towards the steady-state distribution $f_i^{ss}$, the directions of the distribution function are reversed such that the mass that would next stream to an empty site (dashed arrow in top schematic) is now bounced back in the opposite direction (dashed arrow in bottom schematic). The mass just streamed from an empty site (dotted arrow in top schematic), which would be zero, is then set to stream back out instead (dotted arrow in bottom schematic).

Figure 2

KPZ scaling. (a) Time evolution of the interface width $w$ for systems of different length $L$. (b) Curves collapse when rescaled by KPZ exponents. The dashed line showing scaling $t^{1/3}$ highlights the corresponding KPZ growth exponent. KPZ scaling of (c) $w_{\infty }$ and (d) $t_{\infty }$ with $L$. The dashed line is a guide showing KPZ scaling for (c) the roughness exponent $\alpha , w_{\infty }\sim L^{1/2}$ and (d) the dynamic exponent $z, t_{\infty }\sim L^{3/2}$.

Figure 3

A system spanning instability causes divergence from KPZ scaling. (a) Interface growth curves for systems with $L<320$ collapse when rescaled by KPZ exponents, but diverge for $L>320$. (b) Examples of steady-state profiles of the interface position $h$ shifted by the average interface position $\overline{h}$ for systems with $L=320$ (light blue), $L=640$ (dark blue), and $L=1280$ (black). (c) Probability distributions for the difference between the number of cell divisions $\mathrm{\Delta }{\rho }_{\mathrm{cell}}$ around $h_{\max }$ and $h_{\min }$ for $L=40$ (light blue) and $L=640$ (dark blue). (Inset) Schematic of the proposed instability mechanism, proliferative regions are shown in dark blue. The more advanced sections of the interface are more proliferative.

Figure 4

Growth rate of perturbations of different wave numbers. Perturbations only become unstable at wave numbers $q<\sqrt{k/\gamma }$, corresponding to lengths $L>\sqrt{\gamma /k}$. Beyond this threshold, the fastest growing mode is always the longest wavelength permitted by the system.

Figure 5

Restricting proliferation to the bulk removes the instability. (a) Time evolution of $w$ when proliferation is (blue) density dependent and (orange) permitted only in the bulk. Stars indicate times at when profiles of the same color are depicted in (b) and (c). Time evolution of the interface position $h$ shifted by the average interface position $\overline{h}$ for a system of size $L=640$ with (a) density-dependent proliferation and (b) proliferation restricted to the bulk, away from the interface. Darker colors indicate later times.