January  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:
[1]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.

[2]

A. R. A. AndersonM. ChaplainE. NewmanR. Steele and E. Thompson, Mathematical modelling of tumour invasion and metastasis, J. Theor. Med., 2 (2000), 129-154.  doi: 10.1080/10273660008833042.

[3]

N. Bellomo, A. Bellouquid and M. Delitala, Methods and tools of the mathematical kinetic theory toward modeling complex biological systems, in Transport Phenomena and Kinetic Theory, Eds. C. Cercignani and E. Gabetta, Birkhäuser (Boston), (2007), 175-193 doi: 10.1007/978-0-8176-4554-0_8.

[4]

N. Bellomo and M. Delitala, From the mathematical kinetic, and stochastic game theory for active particles to modelling mutations, onset, progression and immune competition of cancer cells, Phys. Life Rev., 5 (2008), 183-206. 

[5]

N. BellomoA. BellouquidJ. Nieto and J. Soler, Modelling chemotaxis from L2-closure moments in kinetic theory of active particles, Discrete Contin. Dyn. Systems B, 18 (2013), 847-863.  doi: 10.3934/dcdsb.2013.18.847.

[6]

R. CallardA. J. George and J. Stark, Cytokines, chaos, and complexity, Immunity, 11 (1999), 507-513.  doi: 10.1016/S1074-7613(00)80125-9.

[7]

F. CerretiB. PerthameC. SchmeiserM. Tang and N. Vauchelet, Waves for an hyperbolic Keller-Segel model and branching instabilities, Math. Models and Meth. in Appl. Sci., 21 (2011), 825-842.  doi: 10.1142/S0218202511005386.

[8]

F. A. C. C. ChalubP. A. MarkowichB. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.  doi: 10.1007/s00605-004-0234-7.

[9]

A. ChauviereT. Hillen and L. Preziosi, Modeling cell movement in anisotropic and heterogeneous network tissues, Netw. Heterog. Media, 2 (2007), 333-357.  doi: 10.3934/nhm.2007.2.333.

[10]

R. H. ChisholmB. D. HughesK. A. Landman and M. Zaman, Analytic study of three-dimensional single cell migration with and without proteolytic enzymes, Cell. Mol. Bioeng, 6 (2013), 239-249.  doi: 10.1007/s12195-012-0261-8.

[11]

R. H. ChisholmB. D. Hughes and K. A. Landman, Building a morphogen gradient without diffusion in a growing tissue, PLoS ONE, 5 (2010), e12857.  doi: 10.1371/journal.pone.0012857.

[12]

R. H. ChisholmT. LorenziA. LorzA. K. LarsenL. Neves de AlmeidaA. Escargueil and J. Clairambault, Emergence of drug tolerance in cancer cell populations: An evolutionary outcome of selection, non-genetic instability and stress-induced adaptation, Cancer Res., 75 (2015), 930-939.  doi: 10.1158/0008-5472.CAN-14-2103.

[13]

S. Cui and A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl., 255 (2001), 636-677.  doi: 10.1006/jmaa.2000.7306.

[14]

J. C. DallonJ. A. Sherratt and P. K. Maini, Mathematical modelling of extracellular matrix dynamics using discrete cells: fiber orientation and tissue regeneration, J. Theoret. Biol., 199 (1999), 449-471.  doi: 10.1006/jtbi.1999.0971.

[15]

M. Delitala and T. Lorenzi, A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions, J. Theoret. Biol., 297 (2012), 88-102.  doi: 10.1016/j.jtbi.2011.11.022.

[16]

M. Delitala and T. Lorenzi, A mathematical model for progression and heterogeneity in colorectal cancer dynamics, Theor. Popul. Biol., 79 (2011), 130-138.  doi: 10.1016/j.tpb.2011.01.001.

[17]

R. Dickinson, A generalized transport model for biased cell migration in an anisotropic environment, J. Math. Biol., 40 (2000), 97-135.  doi: 10.1007/s002850050006.

[18]

R. Erban and H. G. Othmer, Taxis equations for amoeboid cells, J. Math. Biol., 54 (2007), 847-885.  doi: 10.1007/s00285-007-0070-1.

[19]

P. Friedl and K. Wolf, Tumour-cell invasion and migration: Diversity and escape mechanims, Nat. Rev. Cancer, 3 (2003), 362-374.  doi: 10.1038/nrc1075.

[20]

N. Gavert and A. Ben-Ze'ev, Epithelial-mesenchymal transition and the invasive potential of tumors, Trends Mol. Med., 14 (2008), 199-209.  doi: 10.1016/j.molmed.2008.03.004.

[21]

T. Hillen and K. Painter, A user's guide to PDE Models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[22]

T. Hillen, M5 mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585-616.  doi: 10.1007/s00285-006-0017-y.

[23]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Part Ⅱ, Jahresber. Dtsch. Math.-Ver., 106 (2004), 51-69. 

[24]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Part Ⅰ, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165. 

[25]

M. IshizakiK. AshidaT. HigashiH. NakatsukasaT. KaneyoshiK. FujiwaraK. NousoY. KobayashiM. UemuraS. Nakamura and T. Tsuji, The formation of capsule and septum in human hepatocellular carcinoma, Virchows Arch., 438 (2001), 574-580.  doi: 10.1007/s004280000391.

[26]

A. J. Kabla, Collective cell migration: Leadership, invasion and segregation, Journal of the Royal Society Interface, 77 (2012), 3268-3278.  doi: 10.1098/rsif.2012.0448.

[27]

R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations Philadelphia, SIAM, 2007. doi: 10.1137/1.9780898717839.

[28]

A. LorzT. LorenziJ. ClairambaultA. Escargueil and B. Perthame, Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors, Bulletin of Mathematical Biology, 77 (2015), 1-22.  doi: 10.1007/s11538-014-0046-4.

[29]

A. LorzT. LorenziM. E. HochbergJ. Clairambault and B. Perthame, Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, Math. Model. Numer. Anal., 47 (2013), 377-399.  doi: 10.1051/m2an/2012031.

[30]

E. MahesE. MonesV. Nameth and T. Vicsek, Collective motion of cells mediates segregation and pattern formation in co-cultures, PLoS ONE, 7 (2012), e31711. 

[31] J. Murray, Mathematical Biology Ⅱ: Spatial Models and Biochemical Applications, 3rd edn, Springer, New York, 2003. 
[32]

G. Naldi, L. Pareschi and G. Toscani (Eds. ), Mathematical Modeling of Collective Behavior in Socio-economic and Life Sciences Birkhäuser, Basel, 2010. doi: 10.1007/978-0-8176-4946-3.

[33]

K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147.  doi: 10.1007/s11538-009-9396-8.

[34]

K. J. Painter, Modelling cell migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009), 511-543.  doi: 10.1007/s00285-008-0217-8.

[35]

K. J. Painter and T. Hillen, Volume-filling and quorum sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2003), 501-543. 

[36]

B. Perthame, Transport Equations in Biology Birkhäuser, Basel, 2007.

[37]

Z. SzymanskaC. M. RodrigoM. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models and Meth. in Appl. Sci., 19 (2009), 257-281.  doi: 10.1142/S0218202509003425.

[38]

C. XueH. J. HwangK. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations, Bull. Math. Biol., 73 (2011), 1695-1733.  doi: 10.1007/s11538-010-9586-4.

[39]

M. Yilmaz and G. Christofori, EMT, the cytoskeleton, and cancer cell invasion, Cancer Metastasis Rev., 28 (2009), 15-33.  doi: 10.1007/s10555-008-9169-0.

[40]

F. van ZijlS. MallG. MachatC. PirkerR. ZeillingerA. WeinhäuselM. BilbanW. Berger and W. Mikulits, A human model of epithelial to mesenchymal transition to monitor drug efficacy in hepatocellular carcinoma progression, Mol. Cancer Ther., 10 (2011), 850-860. 

show all references

References:
[1]

W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177.  doi: 10.1007/BF00275919.

[2]

A. R. A. AndersonM. ChaplainE. NewmanR. Steele and E. Thompson, Mathematical modelling of tumour invasion and metastasis, J. Theor. Med., 2 (2000), 129-154.  doi: 10.1080/10273660008833042.

[3]

N. Bellomo, A. Bellouquid and M. Delitala, Methods and tools of the mathematical kinetic theory toward modeling complex biological systems, in Transport Phenomena and Kinetic Theory, Eds. C. Cercignani and E. Gabetta, Birkhäuser (Boston), (2007), 175-193 doi: 10.1007/978-0-8176-4554-0_8.

[4]

N. Bellomo and M. Delitala, From the mathematical kinetic, and stochastic game theory for active particles to modelling mutations, onset, progression and immune competition of cancer cells, Phys. Life Rev., 5 (2008), 183-206. 

[5]

N. BellomoA. BellouquidJ. Nieto and J. Soler, Modelling chemotaxis from L2-closure moments in kinetic theory of active particles, Discrete Contin. Dyn. Systems B, 18 (2013), 847-863.  doi: 10.3934/dcdsb.2013.18.847.

[6]

R. CallardA. J. George and J. Stark, Cytokines, chaos, and complexity, Immunity, 11 (1999), 507-513.  doi: 10.1016/S1074-7613(00)80125-9.

[7]

F. CerretiB. PerthameC. SchmeiserM. Tang and N. Vauchelet, Waves for an hyperbolic Keller-Segel model and branching instabilities, Math. Models and Meth. in Appl. Sci., 21 (2011), 825-842.  doi: 10.1142/S0218202511005386.

[8]

F. A. C. C. ChalubP. A. MarkowichB. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123-141.  doi: 10.1007/s00605-004-0234-7.

[9]

A. ChauviereT. Hillen and L. Preziosi, Modeling cell movement in anisotropic and heterogeneous network tissues, Netw. Heterog. Media, 2 (2007), 333-357.  doi: 10.3934/nhm.2007.2.333.

[10]

R. H. ChisholmB. D. HughesK. A. Landman and M. Zaman, Analytic study of three-dimensional single cell migration with and without proteolytic enzymes, Cell. Mol. Bioeng, 6 (2013), 239-249.  doi: 10.1007/s12195-012-0261-8.

[11]

R. H. ChisholmB. D. Hughes and K. A. Landman, Building a morphogen gradient without diffusion in a growing tissue, PLoS ONE, 5 (2010), e12857.  doi: 10.1371/journal.pone.0012857.

[12]

R. H. ChisholmT. LorenziA. LorzA. K. LarsenL. Neves de AlmeidaA. Escargueil and J. Clairambault, Emergence of drug tolerance in cancer cell populations: An evolutionary outcome of selection, non-genetic instability and stress-induced adaptation, Cancer Res., 75 (2015), 930-939.  doi: 10.1158/0008-5472.CAN-14-2103.

[13]

S. Cui and A. Friedman, Analysis of a mathematical model of the growth of necrotic tumors, J. Math. Anal. Appl., 255 (2001), 636-677.  doi: 10.1006/jmaa.2000.7306.

[14]

J. C. DallonJ. A. Sherratt and P. K. Maini, Mathematical modelling of extracellular matrix dynamics using discrete cells: fiber orientation and tissue regeneration, J. Theoret. Biol., 199 (1999), 449-471.  doi: 10.1006/jtbi.1999.0971.

[15]

M. Delitala and T. Lorenzi, A mathematical model for the dynamics of cancer hepatocytes under therapeutic actions, J. Theoret. Biol., 297 (2012), 88-102.  doi: 10.1016/j.jtbi.2011.11.022.

[16]

M. Delitala and T. Lorenzi, A mathematical model for progression and heterogeneity in colorectal cancer dynamics, Theor. Popul. Biol., 79 (2011), 130-138.  doi: 10.1016/j.tpb.2011.01.001.

[17]

R. Dickinson, A generalized transport model for biased cell migration in an anisotropic environment, J. Math. Biol., 40 (2000), 97-135.  doi: 10.1007/s002850050006.

[18]

R. Erban and H. G. Othmer, Taxis equations for amoeboid cells, J. Math. Biol., 54 (2007), 847-885.  doi: 10.1007/s00285-007-0070-1.

[19]

P. Friedl and K. Wolf, Tumour-cell invasion and migration: Diversity and escape mechanims, Nat. Rev. Cancer, 3 (2003), 362-374.  doi: 10.1038/nrc1075.

[20]

N. Gavert and A. Ben-Ze'ev, Epithelial-mesenchymal transition and the invasive potential of tumors, Trends Mol. Med., 14 (2008), 199-209.  doi: 10.1016/j.molmed.2008.03.004.

[21]

T. Hillen and K. Painter, A user's guide to PDE Models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.  doi: 10.1007/s00285-008-0201-3.

[22]

T. Hillen, M5 mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585-616.  doi: 10.1007/s00285-006-0017-y.

[23]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Part Ⅱ, Jahresber. Dtsch. Math.-Ver., 106 (2004), 51-69. 

[24]

D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Part Ⅰ, Jahresber. Dtsch. Math.-Ver., 105 (2003), 103-165. 

[25]

M. IshizakiK. AshidaT. HigashiH. NakatsukasaT. KaneyoshiK. FujiwaraK. NousoY. KobayashiM. UemuraS. Nakamura and T. Tsuji, The formation of capsule and septum in human hepatocellular carcinoma, Virchows Arch., 438 (2001), 574-580.  doi: 10.1007/s004280000391.

[26]

A. J. Kabla, Collective cell migration: Leadership, invasion and segregation, Journal of the Royal Society Interface, 77 (2012), 3268-3278.  doi: 10.1098/rsif.2012.0448.

[27]

R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations Philadelphia, SIAM, 2007. doi: 10.1137/1.9780898717839.

[28]

A. LorzT. LorenziJ. ClairambaultA. Escargueil and B. Perthame, Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors, Bulletin of Mathematical Biology, 77 (2015), 1-22.  doi: 10.1007/s11538-014-0046-4.

[29]

A. LorzT. LorenziM. E. HochbergJ. Clairambault and B. Perthame, Populational adaptive evolution, chemotherapeutic resistance and multiple anti-cancer therapies, Math. Model. Numer. Anal., 47 (2013), 377-399.  doi: 10.1051/m2an/2012031.

[30]

E. MahesE. MonesV. Nameth and T. Vicsek, Collective motion of cells mediates segregation and pattern formation in co-cultures, PLoS ONE, 7 (2012), e31711. 

[31] J. Murray, Mathematical Biology Ⅱ: Spatial Models and Biochemical Applications, 3rd edn, Springer, New York, 2003. 
[32]

G. Naldi, L. Pareschi and G. Toscani (Eds. ), Mathematical Modeling of Collective Behavior in Socio-economic and Life Sciences Birkhäuser, Basel, 2010. doi: 10.1007/978-0-8176-4946-3.

[33]

K. J. Painter, Continuous models for cell migration in tissues and applications to cell sorting via differential chemotaxis, Bull. Math. Biol., 71 (2009), 1117-1147.  doi: 10.1007/s11538-009-9396-8.

[34]

K. J. Painter, Modelling cell migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009), 511-543.  doi: 10.1007/s00285-008-0217-8.

[35]

K. J. Painter and T. Hillen, Volume-filling and quorum sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2003), 501-543. 

[36]

B. Perthame, Transport Equations in Biology Birkhäuser, Basel, 2007.

[37]

Z. SzymanskaC. M. RodrigoM. Lachowicz and M. A. J. Chaplain, Mathematical modelling of cancer invasion of tissue: The role and effect of nonlocal interactions, Math. Models and Meth. in Appl. Sci., 19 (2009), 257-281.  doi: 10.1142/S0218202509003425.

[38]

C. XueH. J. HwangK. J. Painter and R. Erban, Travelling waves in hyperbolic chemotaxis equations, Bull. Math. Biol., 73 (2011), 1695-1733.  doi: 10.1007/s11538-010-9586-4.

[39]

M. Yilmaz and G. Christofori, EMT, the cytoskeleton, and cancer cell invasion, Cancer Metastasis Rev., 28 (2009), 15-33.  doi: 10.1007/s10555-008-9169-0.

[40]

F. van ZijlS. MallG. MachatC. PirkerR. ZeillingerA. WeinhäuselM. BilbanW. Berger and W. Mikulits, A human model of epithelial to mesenchymal transition to monitor drug efficacy in hepatocellular carcinoma progression, Mol. Cancer Ther., 10 (2011), 850-860. 

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 $\mu$
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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