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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 and Continuous Dynamical Systems, 2014, 34 (12) : 5123-5133. doi: 10.3934/dcds.2014.34.5123
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.

[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.

[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.

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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.

[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.

[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.

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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.

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

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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.

[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.

[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.

[12]

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

[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.

[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.

[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.

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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.

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

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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.

[19]

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

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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.

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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.

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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.

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[39]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[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.

[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.

[12]

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

[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.

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[39]

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

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