Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells

Marcello Delitala; Tommaso Lorenzi

doi:10.3934/mbe.2017006

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2017, Volume 14, Issue 1: 79-93. Doi: 10.3934/mbe.2017006

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Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells

Marcello Delitala^1, , and
Tommaso Lorenzi^2,

1.
Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24,10129 Torino, Italy
2.
School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, United Kingdom

* Corresponding author: Marcello Delitala
* Corresponding author: Marcello Delitala

Received: October 28, 2015

Accepted: May 22, 2016

Published: September 2017

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Abstract

Accumulating evidence indicates that the interaction between epithelial and mesenchymal cells plays a pivotal role in cancer development and metastasis formation. Here we propose an integro-differential model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Our model takes into account the effects of chemotaxis, adhesive interactions between epithelial-like cells, proliferation and competition for nutrients. We present a sample of numerical results which display the emergence of spots, stripes and honeycomb patterns, depending on parameters and initial data. These simulations also suggest that epithelial-like and mesenchymal-like cells can segregate when there is little competition for nutrients. Furthermore, our computational results provide a possible explanation for how the concerted action between epithelial-cell adhesion and mesenchymal-cell spreading can precipitate the formation of ring-like structures, which resemble the fibrous capsules frequently observed in hepatic tumours.

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Mathematics Subject Classification: Primary: 92B05; Secondary: 92C17.

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Figure 1. Emergence of spots, stripes and hole patterns. Plots of $n_2(t,\mathbf{x})$ at four time instants $t \in [0,400]$. We consider a sample composed of mesenchymal-like cells and chemotactic cytokines only. The cells are initially characterised by a uniform space distribution parametrised by $n^0_2 \in \mathbb{R}_+$, and their velocities are homogeneously distributed. The initial distribution of cytokines is a small positive random perturbation of the zero level. The effects of non-conservative phenomena are neglected (i.e., $\kappa=\mu=0$). Increasing values of the parameter $n^0_2$ pave the way for the emergence of different spatial patterns, such as spots [vid. Panels A and B ($n^0_2=0.02$ and $n^0_2=0.04$)], stripes [vid. Panel C ($n^0_2=0.05$)], and hole structures [vid. Panel D ($n^0_2=0.08$)]

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Figure 2. Emergence of spots, stripes and honeycomb patterns. We consider a sample initially composed of chemotactic cytokines, mesenchymal-like cells and epithelial-like cells in motion. The distribution of cytokines is a small positive random perturbation of the zero level. Cells are uniformly distributed in space and their distributions are parametrised by $n^0_{1,2}=0.04$. Cellular velocities are homogeneously distributed. We set $\kappa=1$ and tune the value of $\mu$. Decreasing values of parameter $\mu$ pave the way for the emergence of different spatial patterns, such as spots [vid. Panels A and B ($\mu=0.1$)], stripes [vid. Panels C and D ($\mu=0.02$)], and segregation patterns with a honeycomb structure [vid. Panels E and F ($\mu=0.01$)]

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Figure 3. Formation of ring-like patterns. Plots at four time instants $t \in [0, 40]$ of of $n_1(t,\mathbf{x}) + n_3(t,\mathbf{x})$ (Panel A), $n_2(t,\mathbf{x})$ (Panel B) and $n_4(t,\mathbf{x})$ (Panel C). We consider a sample initially composed of mesenchymal-like cells and epithelial-like cells in motion only, whose distributions are modelled by (19). The effects of non-conservative phenomena are neglected (i.e., $\kappa=\mu=0$). While epithelial-like cells rapidly stop moving because of adhesive interactions (vid. Panel A), mesenchymal-like cells diffuse throughout the sample (vid. Panel B), and follow the chemotactic path defined by the diffusing cytokines (vid. Panel C). The resulting pattern is an expanding ring-like structure made of mesenchymal-like cells, which surrounds a cluster of epithelial-like cells kept at rest by homotypic adhesion

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Table 1. Summary of the model parameters

Biological Phenomena	Parameters
Secretion from cells of chemotactic cytokines	$\nu$
Consumption of chemotactic cytokines	$\alpha$
Random motion vs chemotactic reorientation	$\beta$
Homotypic adhesion	$\gamma$
Cell proliferation	$\kappa$
Competition for nutrients	$\mu$

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