# American Institute of Mathematical Sciences

February  2017, 14(1): 79-93. doi: 10.3934/mbe.2017006

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

 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

Received  October 29, 2015 Accepted  May 23, 2016 Published  October 2016

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.

Citation: Marcello Delitala, Tommaso Lorenzi. Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Mathematical Biosciences & Engineering, 2017, 14 (1) : 79-93. doi: 10.3934/mbe.2017006
##### References:

show all references

##### References:
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 ($n^0_2=0.08$)]
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 ($\mu=0.01$)]
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
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 $\mu$
 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 $\mu$
 [1] Catherine Ha Ta, Qing Nie, Tian Hong. Controlling stochasticity in epithelial-mesenchymal transition through multiple intermediate cellular states. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2275-2291. doi: 10.3934/dcdsb.2016047 [2] Julien Barré, Pierre Degond, Diane Peurichard, Ewelina Zatorska. Modelling pattern formation through differential repulsion. Networks & Heterogeneous Media, 2020, 15 (3) : 307-352. doi: 10.3934/nhm.2020021 [3] Alfredo Jose Morales, Werner Creixell, Javier Borondo, Juan Carlos Losada, Rosa Maria Benito. Characterizing ethnic interactions from human communication patterns in Ivory Coast. Networks & Heterogeneous Media, 2015, 10 (1) : 87-99. doi: 10.3934/nhm.2015.10.87 [4] Simone Fagioli, Yahya Jaafra. Multiple patterns formation for an aggregation/diffusion predator-prey system. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021010 [5] Christos V. Nikolopoulos. Mathematical modelling of a mushy region formation during sulphation of calcium carbonate. Networks & Heterogeneous Media, 2014, 9 (4) : 635-654. doi: 10.3934/nhm.2014.9.635 [6] H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433 [7] Yuri B. Gaididei, Carlos Gorria, Rainer Berkemer, Peter L. Christiansen, Atsushi Kawamoto, Mads P. Sørensen, Jens Starke. Stochastic control of traffic patterns. Networks & Heterogeneous Media, 2013, 8 (1) : 261-273. doi: 10.3934/nhm.2013.8.261 [8] Jonathan P. Desi, Evelyn Sander, Thomas Wanner. Complex transient patterns on the disk. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1049-1078. doi: 10.3934/dcds.2006.15.1049 [9] Mark A. Peletier, Marco Veneroni. Stripe patterns and the Eikonal equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 183-189. doi: 10.3934/dcdss.2012.5.183 [10] Elena Izquierdo-Kulich, José Manuel Nieto-Villar. Morphogenesis of the tumor patterns. Mathematical Biosciences & Engineering, 2008, 5 (2) : 299-313. doi: 10.3934/mbe.2008.5.299 [11] Wenbin Yang, Yujing Gao, Xiaojuan Wang. Diffusion modeling of tumor-CD4$^+$-cytokine interactions with treatments: asymptotic behavior and stationary patterns. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021090 [12] Sara Y. Del Valle, J. M. Hyman, Nakul Chitnis. Mathematical models of contact patterns between age groups for predicting the spread of infectious diseases. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1475-1497. doi: 10.3934/mbe.2013.10.1475 [13] Jonathan A. Sherratt, Alexios D. Synodinos. Vegetation patterns and desertification waves in semi-arid environments: Mathematical models based on local facilitation in plants. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2815-2827. doi: 10.3934/dcdsb.2012.17.2815 [14] Bashar Ibrahim. Mathematical analysis and modeling of DNA segregation mechanisms. Mathematical Biosciences & Engineering, 2018, 15 (2) : 429-440. doi: 10.3934/mbe.2018019 [15] A. V. Babin. Preservation of spatial patterns by a hyperbolic equation. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 1-19. doi: 10.3934/dcds.2004.10.1 [16] Luis F. Gordillo, Stephen A. Marion, Priscilla E. Greenwood. The effect of patterns of infectiousness on epidemic size. Mathematical Biosciences & Engineering, 2008, 5 (3) : 429-435. doi: 10.3934/mbe.2008.5.429 [17] Pranay Goel, James Sneyd. Gap junctions and excitation patterns in continuum models of islets. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1969-1990. doi: 10.3934/dcdsb.2012.17.1969 [18] Emiliano Alvarez, Silvia London. Emerging patterns in inflation expectations with multiple agents. Journal of Dynamics & Games, 2020, 7 (3) : 175-184. doi: 10.3934/jdg.2020012 [19] Arno F. Münster. Simulation of stationary chemical patterns and waves in ionic reactions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 35-46. doi: 10.3934/dcdsb.2002.2.35 [20] Julijana Gjorgjieva, Jon Jacobsen. Turing patterns on growing spheres: the exponential case. Conference Publications, 2007, 2007 (Special) : 436-445. doi: 10.3934/proc.2007.2007.436

2018 Impact Factor: 1.313