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

  • * Corresponding author: Marcello Delitala

    * Corresponding author: Marcello Delitala 
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  • 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.

    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 ($n^0_2=0.08$)]

    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 ($\mu=0.01$)]

    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

    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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