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December  2014, 34(12): 5123-5133. doi: 10.3934/dcds.2014.34.5123

Modelling collective cell behaviour

Deborah C. Markham 1, , Ruth E. Baker 1, and  Philip K. Maini 1,

1. 

Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom, United Kingdom, United Kingdom

Received  February 2014 Revised  May 2014 Published  June 2014

The classical mean-field approach to modelling biological systems makes a number of simplifying assumptions which typically lead to coupled systems of reaction-diffusion partial differential equations. While these models have been very useful in allowing us to gain important insights into the behaviour of many biological systems, recent experimental advances in our ability to track and quantify cell behaviour now allow us to build more realistic models which relax some of the assumptions previously made. This brief review aims to illustrate the type of models obtained using this approach.
Keywords: volume exclusion, cell biology., moment dynamics, continuum models, Discrete models.
Mathematics Subject Classification: 35K57, 92B9.
Citation: Deborah C. Markham, Ruth E. Baker, Philip K. Maini. Modelling collective cell behaviour. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5123-5133. doi: 10.3934/dcds.2014.34.5123
References:
[1]

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B. M. Bolker, S. W. Pacala and C. Neuhauser, Spatial dynamics in model plant communities: What do we really know?, Am. Nat., 162 (2003), 135-148. doi: 10.1086/376575.  Google Scholar

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U. Dieckmann and R. Law, Relaxation projections and the method of moments, Cambridge University Press, 21 (2000), 412-457. Google Scholar

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R. Durrett and S. Levin, The importance of being discrete (and spatial), Theor. Pop. Biol., 46 (1994), 363-394. doi: 10.1006/tpbi.1994.1032.  Google Scholar

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L. Dyson, P. K. Maini and R. E. Baker, Macroscopic limits of individual-based models for motile cell populations with volume exclusion, Phys. Rev. E, 86 (2012), 031903. doi: 10.1103/PhysRevE.86.031903.  Google Scholar

[11]

A. Fasano, M. A. Herrero and M. R. Rodrigo, Slow and fast invasion waves in a model of acid-mediated tumour growth, Math. Biosci., 220 (2009), 45-56. doi: 10.1016/j.mbs.2009.04.001.  Google Scholar

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R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753. Google Scholar

[13]

F. Graner and J. A. Glazier, Simulation of biological cell sorting using a two-dimensional extended Potts model, Phys. Rev. Lett., 69 (1992), 2013-2016. doi: 10.1103/PhysRevLett.69.2013.  Google Scholar

[14]

S. T. Johnston, M. J. Simpson and R. E. Baker, Mean-field descriptions of collective migration with strong adhesion, Phys. Rev. E, 85 (2012), 051922. doi: 10.1103/PhysRevE.85.051922.  Google Scholar

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[16]

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R. Law, D. J. Murrell and U. Dieckmann, Population growth in space and time: Spatial logistic equations, Ecology, 84 (2003), 252-262. Google Scholar

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M. A. Lewis and S. Pacala, Modeling and analysis of stochastic invasion processes, Theor. Pop. Biol., 41 (2000), 387-429. doi: 10.1007/s002850000050.  Google Scholar

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P. M. Lushnikov, N. Chen and M. Alber, Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact, Phys. Rev. E, 78 (2008), 061904. doi: 10.1103/PhysRevE.78.061904.  Google Scholar

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J. Mai, V. N. Kuzovkov and W. von Niessen, A general stochastic model for the description of surface reaction systems, Physica A, 203 (1994), 298-315. doi: 10.1016/0378-4371(94)90158-9.  Google Scholar

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D. C. Markham, M. J. Simpson and R. E. Baker, Simplified method for including spatial correlations in mean-field approximations, Phys. Rev. E, 87 (2013), 062702. doi: 10.1103/PhysRevE.87.062702.  Google Scholar

[25]

D. C. Markham, M. J. Simpson, P. K. Maini, E. A. Gaffney and R. E. Baker, Incorporating spatial correlations into multispecies mean-field models, Phys. Rev. E, 88 (2013), 052713. doi: 10.1103/PhysRevE.88.052713.  Google Scholar

[26]

J. B. McGillen, E. A. Gaffney, N. K. Martin and P. K. Maini, A general reaction-diffusion model of acidity in cancer invasion, J. Math. Biol., 68 (2014), 1199-1224. doi: 10.1007/s00285-013-0665-7.  Google Scholar

[27]

F. A. Meineke, C. S. Potten and M. Loeffler, Cell migration and organization in the intestinal crypt using a lattice-free model, Cell Prolif., 34 (2001), 253-266. doi: 10.1046/j.0960-7722.2001.00216.x.  Google Scholar

[28]

P. J. Murray, C. M. Edwards, M. J. Tindall and P. K. Maini, From a discrete to a continuum model of cell dynamics in one dimension, Phys. Rev. E, 80 (2009), 031912. doi: 10.1103/PhysRevE.80.031912.  Google Scholar

[29]

P. J. Murray, C. M. Edwards, M. J. Tindall and P. K. Maini, Classifying general nonlinear force laws in cell-based models via the continuum limit, Phys. Rev. E, 85 (2012), 021921. doi: 10.1103/PhysRevE.85.021921.  Google Scholar

[30]

D. J. Murrell, U. Dieckmann and R. Law, On moment closures for population dynamics in continuous space, J. Theor. Biol., 229 (2004), 421-432. doi: 10.1016/j.jtbi.2004.04.013.  Google Scholar

[31]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976.  Google Scholar

[32]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canadian Appl. Math. Quarterly, 10 (2002), 501-543.  Google Scholar

[33]

M. Raghib, N. A. Hill and U. Dieckmann, A multiscale maximum entropy moment closure for locally regulated space-time point process models of population dynamics, J. Math. Biol., 62 (2011), 605-653. doi: 10.1007/s00285-010-0345-9.  Google Scholar

[34]

K. J. Sharkey, Deterministic epidemic models on contact networks: Correlations and unbiological terms, Theor. Pop. Biol., 79 (2011), 115-129. doi: 10.1016/j.tpb.2011.01.004.  Google Scholar

[35]

K. J. Sharkey, C. Fernandez, K. L. Morgan, E. Peeler, M. Thrush, J. F. Turnbull and G. B. Bowers, Pair-level approximations to the spatio-temporal dynamics of epidemics on asymmetric contact networks, J. Math. Biol., 53 (2006), 61-85. doi: 10.1007/s00285-006-0377-3.  Google Scholar

[36]

M. J. Simpson and R. E. Baker, Corrected mean-field models for spatially dependent advection-diffusion-reaction phenomena, Phys. Rev. E, 83 (2011), 051922. doi: 10.1103/PhysRevE.83.051922.  Google Scholar

[37]

M. J. Simpson, B. J. Binder, P. Haridas, B. K. Wood, K. K. Treloar, D. L. S. McElwain and R. E. Baker, Experimental and modelling investigation of monolayer development with clustering, Bull. Math. Biol., 75 (2013), 871-889. doi: 10.1007/s11538-013-9839-0.  Google Scholar

[38]

O. Warburg and F. Dickens, The metabolism of tumors, Am. J. Med. Sci., 182 (1931), 123. doi: 10.1097/00000441-193107000-00022.  Google Scholar

[39]

W. R. Young, A. J. Roberts and G. Stuhne, Reproductive pair correlations and the clustering of organisms, Nature, 412 (2001), 328-331. Google Scholar

show all references

References:
[1]

K. Anguige and C. Schmeiser, A one-dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion, J. Math. Biol., 58 (2009), 395-427. doi: 10.1007/s00285-008-0197-8.  Google Scholar

[2]

G. Ascolani, M. Badoual and C. Deroulers, Exclusion processes: Short-range correlations induced by adhesion and contact interactions, Phys. Rev. E, 87 (2013), 012702. doi: 10.1103/PhysRevE.87.012702.  Google Scholar

[3]

R. E. Baker and M. J. Simpson, Correcting mean-field approximations for birth-death-movement processes, Phys. Rev. E., 82 (2010), 041905, 12pp. doi: 10.1103/PhysRevE.82.041905.  Google Scholar

[4]

B. Bolker and S. W. Pacala, Using moment equations to understand stochastically driven spatial pattern formation in ecological systems, Theor. Pop. Biol., 52 (1997), 179-197. doi: 10.1006/tpbi.1997.1331.  Google Scholar

[5]

B. M. Bolker and S. W. Pacala, Spatial moment equations for plant competition: Understanding spatial strategies and the advantages of short dispersal, Theor. Pop. Biol., 153 (1999), 575-602. doi: 10.1086/303199.  Google Scholar

[6]

B. M. Bolker, S. W. Pacala and S. A. Levin, Moment methods for ecological processes in continuous space,, In The Geometry of Ecological Interactions: Simplifying Spatial Complexity, (): 338.  doi: 10.1017/CBO9780511525537.024.  Google Scholar

[7]

B. M. Bolker, S. W. Pacala and C. Neuhauser, Spatial dynamics in model plant communities: What do we really know?, Am. Nat., 162 (2003), 135-148. doi: 10.1086/376575.  Google Scholar

[8]

U. Dieckmann and R. Law, Relaxation projections and the method of moments, Cambridge University Press, 21 (2000), 412-457. Google Scholar

[9]

R. Durrett and S. Levin, The importance of being discrete (and spatial), Theor. Pop. Biol., 46 (1994), 363-394. doi: 10.1006/tpbi.1994.1032.  Google Scholar

[10]

L. Dyson, P. K. Maini and R. E. Baker, Macroscopic limits of individual-based models for motile cell populations with volume exclusion, Phys. Rev. E, 86 (2012), 031903. doi: 10.1103/PhysRevE.86.031903.  Google Scholar

[11]

A. Fasano, M. A. Herrero and M. R. Rodrigo, Slow and fast invasion waves in a model of acid-mediated tumour growth, Math. Biosci., 220 (2009), 45-56. doi: 10.1016/j.mbs.2009.04.001.  Google Scholar

[12]

R. A. Gatenby and E. T. Gawlinski, A reaction-diffusion model of cancer invasion, Cancer Res., 56 (1996), 5745-5753. Google Scholar

[13]

F. Graner and J. A. Glazier, Simulation of biological cell sorting using a two-dimensional extended Potts model, Phys. Rev. Lett., 69 (1992), 2013-2016. doi: 10.1103/PhysRevLett.69.2013.  Google Scholar

[14]

S. T. Johnston, M. J. Simpson and R. E. Baker, Mean-field descriptions of collective migration with strong adhesion, Phys. Rev. E, 85 (2012), 051922. doi: 10.1103/PhysRevE.85.051922.  Google Scholar

[15]

M. J. Keeling, Correlation equations for endemic diseases: Externally imposed and internally generated heterogeneity, Proc. R. Soc. Lond. B, 266 (1999), 953-960. doi: 10.1098/rspb.1999.0729.  Google Scholar

[16]

M. J. Keeling, D. A. Rand and A. J. Morris, Correlation models for childhood epidemics, Proc. R. Soc. Lond. B, 264 (1997), 1149-1156. doi: 10.1098/rspb.1997.0159.  Google Scholar

[17]

J. G. Kirkwood, Statistical mechanics of fluid mixtures, J. Chem. Phys., 3 (1935), 300-314. doi: 10.1063/1.1749657.  Google Scholar

[18]

J. G. Kirkwood and E. M. Boggs, The radial distribution function in liquids, J. Chem. Phys., 10 (1942), 394-403. doi: 10.1063/1.1723737.  Google Scholar

[19]

R. Law, D. J. Murrell and U. Dieckmann, Population growth in space and time: Spatial logistic equations, Ecology, 84 (2003), 252-262. Google Scholar

[20]

M. A. Lewis and S. Pacala, Modeling and analysis of stochastic invasion processes, Theor. Pop. Biol., 41 (2000), 387-429. doi: 10.1007/s002850000050.  Google Scholar

[21]

P. M. Lushnikov, N. Chen and M. Alber, Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact, Phys. Rev. E, 78 (2008), 061904. doi: 10.1103/PhysRevE.78.061904.  Google Scholar

[22]

J. Mai, V. N. Kuzovkov and W. von Niessen, A theoretical stochastic model for the $a + 1/2b_2\to0$ reaction, J. Chem. Phys., 98 (1993), 10017-10025. Google Scholar

[23]

J. Mai, V. N. Kuzovkov and W. von Niessen, A general stochastic model for the description of surface reaction systems, Physica A, 203 (1994), 298-315. doi: 10.1016/0378-4371(94)90158-9.  Google Scholar

[24]

D. C. Markham, M. J. Simpson and R. E. Baker, Simplified method for including spatial correlations in mean-field approximations, Phys. Rev. E, 87 (2013), 062702. doi: 10.1103/PhysRevE.87.062702.  Google Scholar

[25]

D. C. Markham, M. J. Simpson, P. K. Maini, E. A. Gaffney and R. E. Baker, Incorporating spatial correlations into multispecies mean-field models, Phys. Rev. E, 88 (2013), 052713. doi: 10.1103/PhysRevE.88.052713.  Google Scholar

[26]

J. B. McGillen, E. A. Gaffney, N. K. Martin and P. K. Maini, A general reaction-diffusion model of acidity in cancer invasion, J. Math. Biol., 68 (2014), 1199-1224. doi: 10.1007/s00285-013-0665-7.  Google Scholar

[27]

F. A. Meineke, C. S. Potten and M. Loeffler, Cell migration and organization in the intestinal crypt using a lattice-free model, Cell Prolif., 34 (2001), 253-266. doi: 10.1046/j.0960-7722.2001.00216.x.  Google Scholar

[28]

P. J. Murray, C. M. Edwards, M. J. Tindall and P. K. Maini, From a discrete to a continuum model of cell dynamics in one dimension, Phys. Rev. E, 80 (2009), 031912. doi: 10.1103/PhysRevE.80.031912.  Google Scholar

[29]

P. J. Murray, C. M. Edwards, M. J. Tindall and P. K. Maini, Classifying general nonlinear force laws in cell-based models via the continuum limit, Phys. Rev. E, 85 (2012), 021921. doi: 10.1103/PhysRevE.85.021921.  Google Scholar

[30]

D. J. Murrell, U. Dieckmann and R. Law, On moment closures for population dynamics in continuous space, J. Theor. Biol., 229 (2004), 421-432. doi: 10.1016/j.jtbi.2004.04.013.  Google Scholar

[31]

H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABC's of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976.  Google Scholar

[32]

K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Canadian Appl. Math. Quarterly, 10 (2002), 501-543.  Google Scholar

[33]

M. Raghib, N. A. Hill and U. Dieckmann, A multiscale maximum entropy moment closure for locally regulated space-time point process models of population dynamics, J. Math. Biol., 62 (2011), 605-653. doi: 10.1007/s00285-010-0345-9.  Google Scholar

[34]

K. J. Sharkey, Deterministic epidemic models on contact networks: Correlations and unbiological terms, Theor. Pop. Biol., 79 (2011), 115-129. doi: 10.1016/j.tpb.2011.01.004.  Google Scholar

[35]

K. J. Sharkey, C. Fernandez, K. L. Morgan, E. Peeler, M. Thrush, J. F. Turnbull and G. B. Bowers, Pair-level approximations to the spatio-temporal dynamics of epidemics on asymmetric contact networks, J. Math. Biol., 53 (2006), 61-85. doi: 10.1007/s00285-006-0377-3.  Google Scholar

[36]

M. J. Simpson and R. E. Baker, Corrected mean-field models for spatially dependent advection-diffusion-reaction phenomena, Phys. Rev. E, 83 (2011), 051922. doi: 10.1103/PhysRevE.83.051922.  Google Scholar

[37]

M. J. Simpson, B. J. Binder, P. Haridas, B. K. Wood, K. K. Treloar, D. L. S. McElwain and R. E. Baker, Experimental and modelling investigation of monolayer development with clustering, Bull. Math. Biol., 75 (2013), 871-889. doi: 10.1007/s11538-013-9839-0.  Google Scholar

[38]

O. Warburg and F. Dickens, The metabolism of tumors, Am. J. Med. Sci., 182 (1931), 123. doi: 10.1097/00000441-193107000-00022.  Google Scholar

[39]

W. R. Young, A. J. Roberts and G. Stuhne, Reproductive pair correlations and the clustering of organisms, Nature, 412 (2001), 328-331. Google Scholar

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