Understanding interactions between populations: Individual based modelling and quantification using pair correlation functions

Abstract

Understanding the underlying mechanisms that produce the huge variety of swarming and aggregation patterns in animals and cells is fundamental in ecology, developmental biology, and regenerative medicine, to name but a few examples. Depending upon the nature of the interactions between individuals (cells or animals), a variety of different large-scale spatial patterns can be observed in their distribution; examples include cell aggregates, stripes of different coloured skin cells, etc. For the case where all individuals are of the same type (i.e., all interactions are alike), a considerable literature already exists on how the collective organisation depends on the inter-individual interactions. Here, we focus on the less studied case where there are two different types of individuals present. Whilst a number of continuum models of this scenario exist, it can be difficult to compare these models to experimental data, since real cells and animals are discrete. In order to overcome this problem, we develop an agent-based model to simulate some archetypal mechanisms involving attraction and repulsion. However, with this approach (as with experiments), each realisation of the model is different, due to stochastic effects. In order to make useful comparisons between simulations and experimental data, we need to identify the robust features of the spatial distributions of the two species which persist over many realisations of the model (for example, the size of aggregates, degree of segregation or intermixing of the two species). In some cases, it is possible to do this by simple visual inspection. In others, the features of the pattern are not so clear to the unaided eye. In this paper, we introduce a pair correlation function (PCF), which allows us to analyse multi-species spatial distributions quantitatively. We show how the differing strengths of inter-individual attraction and repulsion between species give rise to different spatial patterns, and how the PCF can be used to quantify these differences, even when it might be impossible to recognise them visually.

A quantitative modelling approach to zebrafish pigment pattern formation

“…If the PCF value is greater or less than unity for a certain distance then this indicates that there is spatial correlation or anticorrelation respectively at that distance. The PCF we employ is specific to on-lattice domains and is called the Square Uniform PCF ( Gavagnin et al, 2018 ) adapted for multiple cell-types (see Appendix 3 and Dini et al, 2018 ). We describe the PCF as homotypic when we are measuring the spatial correlation of one cell-type and heterotypic when we consider the spatial correlation between two different cell-types.…”

“…We implement cell movement so that cells are biased towards cell types they are attracted to and away from cell types they are repelled by. The direction of the cell’s movement is determined using an on-lattice attraction-repulsion mechanism based on a model described by Dini et al, 2018 and is detailed as follows. If a cell is chosen to move, it is able to move in one of eight different possible orientations denoted by .…”

“…In this paper, we use the square uniform PCF ( Gavagnin et al, 2018 ) developed for on-lattice systems of agents where distance is measured using the uniform norm. This PCF was originally developed for determining pair correlation between single agents types on a lattice, however, in a technique similar to Dini et al, 2018 we adapt it here so that it can also be used for identifying correlation between two different types of agents (cell types). First we define the PCF.…”

Neural crest streaming as an emergent property of tissue interactions during morphogenesis

“…Whether neural crest of other species exhibit a chase-and-run-behavior remains to be investigated. A mechanism similar to the chase-and-run has been proposed to drive the patterning of initially mixed populations of motile agents [61]. While such theoretical approaches allow a more complete understanding of the model behavior, simplicity often confounds their applicability and experimental testability.…”

Population dynamics with spatial structure and an Allee effect

“…Short-range interactions can lead to the development of spatial structure that can affect the overall population dynamics [29][30][31]. Spatial structure in biological populations includes both clustering and segregation [32][33][34][35][36][37]. Stochastic individual-based models (IBM) offer a straightforward means of exploring population dynamics without invoking a meanfield approximation [38,39].…”