Unbiased on-lattice domain growth

Abstract

Domain growth is a key process in many areas of biology, including embryonic development, the growth of tissue, and limb regeneration. As a result, mechanisms for incorporating it into traditional models for cell movement, interaction, and proliferation are of great importance. A previously wellused method in order to incorporate domain growth into on-lattice reaction-diffusion models causes a build up of particles on the boundaries of the domain, which is particularly evident when diffusion is low in comparison to the rate of domain growth. Here, we present a new method which addresses this unphysical build up of particles at the boundaries, and demonstrate that it is accurate for scenarios in which the previous method fails. Further, we discuss for which parameter regimes it is feasible to continue using the original method due to diffusion dominating the domain growth mechanism.

Incorporating domain growth into hybrid methods for reaction–diffusion systems

“…When a jump occurs from the first compartment into the ghost cell, we reduce n 1 (t) by 1 and add a new particle to the ghost cell by sampling a uniform position within the ghost cell (that is, add a new particle at a position y* ∼ Unif(I(t) − h c (t), I(t))) and subsequently increase n GC (t) by 1. Domain growth is implemented using the stretching method [24] for the mesoscale, and via the SDE for the microscale. The algorithm for the gGCM can be found in algorithm 2, and a schematic for the static case is given in figure 2.…”

“…To extend the domain, we use the stretching method of Smith et al [24]. As the domain grows through a series of discrete fixed size extensions, the compartments grow in length uniformly.…”

“…We briefly present the main steps of the calculation in electronic supplementary material, §S.2.3. For a more complete calculation please see [24] for the mean equations and §S.2.3 in the electronic supplementary material for the small compartment limit.…”

“…The particles in the parent compartment are distributed into the two daughter compartments according to some symmetric probability distribution. This method causes a build-up of particles at the boundaries when growth occurs on a faster time scale than diffusion [24]. The second method (and the method we will employ in this work) is a global method introduced by Smith et al [24].…”

“…This method causes a build-up of particles at the boundaries when growth occurs on a faster time scale than diffusion [24]. The second method (and the method we will employ in this work) is a global method introduced by Smith et al [24]. Compartments grow uniformly until a growth event is due to occur.…”