Research article Special Issues

A hybrid discrete-continuum approach to model Turing pattern formation

  • Received: 05 July 2020 Accepted: 13 October 2020 Published: 29 October 2020
  • Since its introduction in 1952, with a further refinement in 1972 by Gierer and Meinhardt, Turing's (pre-)pattern theory (the chemical basis of morphogenesis) has been widely applied to a number of areas in developmental biology, where evolving cell and tissue structures are naturally observed. The related pattern formation models normally comprise a system of reaction-diffusion equations for interacting chemical species (morphogens), whose heterogeneous distribution in some spatial domain acts as a template for cells to form some kind of pattern or structure through, for example, differentiation or proliferation induced by the chemical pre-pattern. Here we develop a hybrid discrete-continuum modelling framework for the formation of cellular patterns via the Turing mechanism. In this framework, a stochastic individual-based model of cell movement and proliferation is combined with a reaction-diffusion system for the concentrations of some morphogens. As an illustrative example, we focus on a model in which the dynamics of the morphogens are governed by an activator-inhibitor system that gives rise to Turing pre-patterns. The cells then interact with the morphogens in their local area through either of two forms of chemically-dependent cell action: Chemotaxis and chemically-controlled proliferation. We begin by considering such a hybrid model posed on static spatial domains, and then turn to the case of growing domains. In both cases, we formally derive the corresponding deterministic continuum limit and show that that there is an excellent quantitative match between the spatial patterns produced by the stochastic individual-based model and its deterministic continuum counterpart, when sufficiently large numbers of cells are considered. This paper is intended to present a proof of concept for the ideas underlying the modelling framework, with the aim to then apply the related methods to the study of specific patterning and morphogenetic processes in the future.

    Citation: Fiona R. Macfarlane, Mark A. J. Chaplain, Tommaso Lorenzi. A hybrid discrete-continuum approach to model Turing pattern formation[J]. Mathematical Biosciences and Engineering, 2020, 17(6): 7442-7479. doi: 10.3934/mbe.2020381

    Related Papers:

  • Since its introduction in 1952, with a further refinement in 1972 by Gierer and Meinhardt, Turing's (pre-)pattern theory (the chemical basis of morphogenesis) has been widely applied to a number of areas in developmental biology, where evolving cell and tissue structures are naturally observed. The related pattern formation models normally comprise a system of reaction-diffusion equations for interacting chemical species (morphogens), whose heterogeneous distribution in some spatial domain acts as a template for cells to form some kind of pattern or structure through, for example, differentiation or proliferation induced by the chemical pre-pattern. Here we develop a hybrid discrete-continuum modelling framework for the formation of cellular patterns via the Turing mechanism. In this framework, a stochastic individual-based model of cell movement and proliferation is combined with a reaction-diffusion system for the concentrations of some morphogens. As an illustrative example, we focus on a model in which the dynamics of the morphogens are governed by an activator-inhibitor system that gives rise to Turing pre-patterns. The cells then interact with the morphogens in their local area through either of two forms of chemically-dependent cell action: Chemotaxis and chemically-controlled proliferation. We begin by considering such a hybrid model posed on static spatial domains, and then turn to the case of growing domains. In both cases, we formally derive the corresponding deterministic continuum limit and show that that there is an excellent quantitative match between the spatial patterns produced by the stochastic individual-based model and its deterministic continuum counterpart, when sufficiently large numbers of cells are considered. This paper is intended to present a proof of concept for the ideas underlying the modelling framework, with the aim to then apply the related methods to the study of specific patterning and morphogenetic processes in the future.


    加载中


    [1] A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B, 237 (1952), 37-72. doi: 10.1098/rstb.1952.0012
    [2] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39. doi: 10.1007/BF00289234
    [3] H. Meinhardt, Models of Biological Pattern Formation, Academic Press, London, 1982.
    [4] J. D. Murray, A pre-pattern formation mechanism for animal coat markings, J. Theor. Biol., 88 (1981), 161-199. doi: 10.1016/0022-5193(81)90334-9
    [5] P. K. Maini and T. E. Woolley, The Turing model for biological pattern formation, in The Dynamics of Biological Systems, Springer, 2019,189-204.
    [6] P. Arcuri and J. D. Murray, Pattern sensitivity to boundary and initial conditions in reaction-diffusion models, J. Math. Biol., 24 (1986), 141-165.
    [7] M. A. J. Chaplain, M. Ganesh, and I. G. Graham, Spatio-temporal pattern formation on spherical surfaces: Numerical simulation and application to solid tumour growth, J. Math. Biol., 42 (2001), 387-423. doi: 10.1007/s002850000067
    [8] E. J. Crampin, E. A. Gaffney, and P. K. Maini, Reaction and diffusion on growing domains: Scenarios for robust pattern formation, Bull. Math. Biol., 61 (1999), 1093-1120. doi: 10.1006/bulm.1999.0131
    [9] E. J. Crampin, W. W. Hackborn, and P. K. Maini, Pattern formation in reaction-diffusion models with nonuniform domain growth, Bull. Math. Biol., 64 (2002), 747-769. doi: 10.1006/bulm.2002.0295
    [10] S. Kondo and R. Asai, A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus, Nature, 376 (1995), 765-768. doi: 10.1038/376765a0
    [11] A. L. Krause, M. A. Ellis, and R. A. Van Gorder. Influence of curvature, growth, and anisotropy on the evolution of Turing patterns on growing manifolds, Bull. Math. Biol., 81 (2019), 759-799. doi: 10.1007/s11538-018-0535-y
    [12] A. L. Krause, V. Klika, T. E. Woolley, and E. A. Gaffney, From one pattern into another: Analysis of Turing patterns in heterogeneous domains via WKBJ, J. R. Soc. Interface, 17 (2020), 20190621. doi: 10.1098/rsif.2019.0621
    [13] P. K. Maini, T. E. Woolley, R. E. Baker, E. A. Gaffney, and S. S. Lee, Turing's model for biological pattern formation and the robustness problem, Interface Focus, 2 (2012), 487-496. doi: 10.1098/rsfs.2011.0113
    [14] A. Madzvamuse, A. H. W. Chung, and C. Venkataraman, Stability analysis and simulations of coupled bulk-surface reaction-diffusion systems, Proc. R. Soc. A, 471 (2015), 20140546. doi: 10.1098/rspa.2014.0546
    [15] A. Madzvamuse, A. J. Wathen, and P. K. Maini, A moving grid finite element method applied to a model biological pattern generator, J. Comp. Phys., 190 (2003), 478-500. doi: 10.1016/S0021-9991(03)00294-8
    [16] K. J. Painter, P. K. Maini, and H. G. Othmer, Stripe formation in juvenile Pomacanthus explained by a generalized Turing mechanism with chemotaxis, Proc. Nat. Acad. Sci., 96 (1999), 5549-5554. doi: 10.1073/pnas.96.10.5549
    [17] S. S. Lee, E. A. Gaffney, and R. E. Baker, The dynamics of Turing patterns for morphogenregulated growing domains with cellular response delays, Bull. Math. Biol., 73 (2011), 2527-2551. doi: 10.1007/s11538-011-9634-8
    [18] J. D. Murray, Mathematical Biology: I. An Introduction, volume 17. Springer Science & Business Media, 2007.
    [19] B. Ermentrout, Stripes or spots? Nonlinear effects in bifurcation of reaction-diffusion equations on the square, Proc. R. Soc. A, 434 (1991), 413-417.
    [20] H. Shoji, Y. Iwasa, and S. Kondo, Stripes, spots, or reversed spots in two-dimensional Turing systems, J. Theor. Biol., 224 (2003), 339-350. doi: 10.1016/S0022-5193(03)00170-X
    [21] H. Meinhardt, Tailoring and coupling of reaction-diffusion systems to obtain reproducible complex pattern formation during development of the higher organisms, Appl. Math. Comput., 32 (1989), 103-135.
    [22] R. Chaturvedi, C. Huang, B. Kazmierczak, T. Schneider, J. A. Izaguirre, T. Glimm, H. G. E. Hentschel, J. A. Glazier, S. A. Newman and M. S. Alber, On multiscale approaches to three-dimensional modelling of morphogenesis, J. R. Soc. Interface, 2 (2005), 237-253. doi: 10.1098/rsif.2005.0033
    [23] S. Christley, M. S. Alber and S. A. Newman, Patterns of mesenchymal condensation in a multiscale, discrete stochastic model, PLoS Comput. Biol., 3 (2007), e76. doi: 10.1371/journal.pcbi.0030076
    [24] B. Duggan and J. Metzcar, Generating Turing-like patterns in an off-lattice agent based model: Handout, Preprint.
    [25] D. Karig, K. M. Martini, T. Lu, N. A. DeLateur, N. Goldenfeld, and R. Weiss, Stochastic Turing patterns in a synthetic bacterial population, Proc. Nat. Acad. Sci., 115 (2018), 6572-6577. doi: 10.1073/pnas.1720770115
    [26] M. A. Kiskowski, M. S. Alber, G. L. Thomas, J. A. Glazier, N. B. Bronstein, J. Pu and S. A. Newman, Interplay between activator-inhibitor coupling and cell-matrix adhesion in a cellular automaton model for chondrogenic patterning, Dev. Biol., 271 (2004), 372-387. doi: 10.1016/j.ydbio.2004.03.038
    [27] S. Kondo, Turing pattern formation without diffusion, in Conference on Computability in Europe, Springer, 2012,416-421.
    [28] J. Moreira and A. Deutsch, Pigment pattern formation in zebrafish during late larval stages: A model based on local interactions, Dev. Dyn., 232 (2005), 33-42. doi: 10.1002/dvdy.20199
    [29] S. Okuda, T. Miura, Y. Inoue, T. Adachi, and M. Eiraku, Combining Turing and 3D vertex models reproduces autonomous multicellular morphogenesis with undulation, tubulation, and branching, Sci. Rep., 8 (2018), 1-15. doi: 10.1038/s41598-017-17765-5
    [30] A. Volkening and B. Sandstede, Modelling stripe formation in zebrafish: An agent-based approach, J. R. Soc. Interface, 12 (2015), 20150812. doi: 10.1098/rsif.2015.0812
    [31] C. M. Glen, M. L. Kemp, and E. O. Voit, Agent-based modeling of morphogenetic systems: Advantages and challenges, PLoS Comp. Biol., 15 (2019), e1006577. doi: 10.1371/journal.pcbi.1006577
    [32] J. A. Izaguirre, R. Chaturvedi, C. Huang, T. Cickovski, J. Coffland, G. Thomas, G. Forgacs, M. Alber, G. Hentschel, S. A. Newman, et al, CompuCell, a multi-model framework for simulation of morphogenesis, Bioinformatics, 20 (2004), 1129-1137. doi: 10.1093/bioinformatics/bth050
    [33] T. M. Cickovski, C. Huang, R. Chaturvedi, T. Glimm, H. G. E. Hentschel, M. S. Alber, J. A. Glazier, S. A. Newman, and J. A. Izaguirre, A framework for three-dimensional simulation of morphogenesis, Trans. Comput. Biol. Bioinform., 2 (2005), 273-288. doi: 10.1109/TCBB.2005.46
    [34] B. D. Hughes, Random walks and random environments: Random walks, vol. 1, Oxford University Press, 1995.
    [35] T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM J. Appl. Math., 61 (2000), 751-775. doi: 10.1137/S0036139999358167
    [36] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3
    [37] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298. doi: 10.1007/BF00277392
    [38] K. J. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart., 10. (2002), 501-543.
    [39] K. J. Painter and J. A. Sherratt, Modelling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339. doi: 10.1016/S0022-5193(03)00258-3
    [40] W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177. doi: 10.1007/BF00275919
    [41] M. Burger, P. Markowich and J.-F. Pietschmann, Continuous limit of a crowd motion and herding model: analysis and numerical simulations, Kinet. Relat. Models, 4 (2011), 1025-1047. doi: 10.3934/krm.2011.4.1025
    [42] A. Stevens, The derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many-particle systems, SIAM J. Appl. Math, 61 (2000), 183-212. doi: 10.1137/S0036139998342065
    [43] A. Stevens and H. G. Othmer, 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
    [44] F. Bubba, T. Lorenzi, and F. R. Macfarlane, From a discrete model of chemotaxis with volume-filling to a generalized patlak-keller-segel model, Proc. R. Soc. A, 476 (2020), 20190871. doi: 10.1098/rspa.2019.0871
    [45] N. Champagnat and S. Méléard, Invasion and adaptive evolution for individual-based spatially structured populations, J. Math. Biol., 55 (2007), 147. doi: 10.1007/s00285-007-0072-z
    [46] M. A. J. Chaplain, T. Lorenzi and F. R. Macfarlane, Bridging the gap between individual-based and continuum models of growing cell populations, J. Math. Biol., 80 (2020), 342-371.
    [47] M. Inoue, Derivation of a porous medium equation from many markovian particles and the propagation of chaos, Hiroshima Math. J., 21 (1991), 85-110. doi: 10.32917/hmj/1206128924
    [48] K. Oelschläger, On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes, Probab. Theory Relat. Fields, 82 (1989), 565-586. doi: 10.1007/BF00341284
    [49] H. G. Othmer and T. Hillen, The diffusion limit of transport equations Ⅱ: Chemotaxis equations, SIAM J. Appl. Math., 62 (2002), 1222-1250. doi: 10.1137/S0036139900382772
    [50] C. J. Penington, B. D. Hughes and K. A. Landman, Building macroscale models from microscale probabilistic models: A general probabilistic approach for nonlinear diffusion and multispecies phenomena, Phys. Rev. E, 84 (2011), 041120. doi: 10.1103/PhysRevE.84.041120
    [51] C. J. Penington, B. D. Hughes and K. A. Landman, Interacting motile agents: Taking a mean-field approach beyond monomers and nearest-neighbor steps, Phys. Rev. E, 89 (2014), 032714.
    [52] R. E. Baker, A. Parker and M. J. Simpson, A free boundary model of epithelial dynamics, J. Theor. Biol., 481 (2019), 61-74. doi: 10.1016/j.jtbi.2018.12.025
    [53] H. M. Byrne and D. Drasdo, Individual-based and continuum models of growing cell populations: A comparison, J. Math. Biol., 58 (2009), 657. doi: 10.1007/s00285-008-0212-0
    [54] T. Lorenzi, P. J. Murray and M. Ptashnyk, From individual-based mechanical models of multicellular systems to free-boundary problems, Interface Free Bound., 22 (2020), 205-244. doi: 10.4171/IFB/439
    [55] S. Motsch and D. Peurichard, From short-range repulsion to hele-shaw problem in a model of tumor growth, J. Math. Biol., 76 (2018), 205-234. doi: 10.1007/s00285-017-1143-4
    [56] 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
    [57] 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
    [58] K. Oelschläger, Large systems of interacting particles and the porous medium equation, J. Diff. equation., 88 (1990), 294-346. doi: 10.1016/0022-0396(90)90101-T
    [59] B. J. Binder and K. A. Landman, Exclusion processes on a growing domain, J. Theor. Biol., 259 (2009), 541-551. doi: 10.1016/j.jtbi.2009.04.025
    [60] 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
    [61] A. E. Fernando, K. A. Landman and M. J. Simpson, Nonlinear diffusion and exclusion processes with contact interactions, Phys. Rev. E, 81 (2010), 011903. doi: 10.1103/PhysRevE.81.011903
    [62] S. T. Johnston, R. E. Baker, D. S. McElwain and M. J. Simpson, Co-operation, competition and crowding: a discrete framework linking allee kinetics, nonlinear diffusion, shocks and sharp-fronted travelling waves, Sci. Rep., 7 (2017), 42134. doi: 10.1038/srep42134
    [63] 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
    [64] K. A. Landman and A. E. Fernando, Myopic random walkers and exclusion processes: Single and multispecies, Phys. A Stat. Mech. Appl., 390 (2011), 3742-3753. doi: 10.1016/j.physa.2011.06.034
    [65] 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
    [66] M. J. Simpson, K. A. Landman and B. D. Hughes, Cell invasion with proliferation mechanisms motivated by time-lapse data, Phys. A Stat. Mech. Appl., 389 (2010), 3779-3790. doi: 10.1016/j.physa.2010.05.020
    [67] C. Deroulers, M. Aubert, M. Badoual and B. Grammaticos, Modeling tumor cell migration: from microscopic to macroscopic models, Phys. Rev. E, 79 (2009), 031917. doi: 10.1103/PhysRevE.79.031917
    [68] D. Drasdo, Coarse graining in simulated cell populations, Adv. Complex Syst., 8 (2005), 319-363. doi: 10.1142/S0219525905000440
    [69] M. J. Simpson, A. Merrifield, K. A. Landman and B. D. Hughes, Simulating invasion with cellular automata: connecting cell-scale and population-scale properties, Phys. Rev. E, 76 (2007), 021918. doi: 10.1103/PhysRevE.76.021918
    [70] A. Buttenschoen, T. Hillen, A. Gerisch and K. J. Painter, A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis, J. Math. Biol., 76 (2018), 429-456. doi: 10.1007/s00285-017-1144-3
    [71] L. Marcon and J. Sharpe, Turing patterns in development: What about the horse part? Curr. Opin. Genet. Dev., 22 (2012), 578-584. doi: 10.1016/j.gde.2012.11.013
    [72] H. G. Othmer, P. K. Maini and J. D. Murray, Experimental and theoretical advances in biological pattern formation, vol. 259, Springer Science & Business Media, 2012.
    [73] H. G. Othmer, K. J. Painter, D. Umulis, and C. Xue, The intersection of theory and application in elucidating pattern formation in developmental biology, Math. Model. Nat. Phenom., 4 (2009), 3-82. doi: 10.1051/mmnp/20094401
    [74] P. Maini, K. Painter and H. P. Chau, Spatial pattern formation in chemical and biological systems, J. Chem. Soc. Faraday Trans., 93 (1997), 3601-3610. doi: 10.1039/a702602a
    [75] M. R. Myerscough, P. K. Maini, and K. J. Painter, Pattern formation in a generalized chemotactic model, Bull. Math. Biol., 60 (1998), 1-26. doi: 10.1006/bulm.1997.0010
    [76] J. A. Sherratt and J. D. Murray, Mathematical analysis of a basic model for epidermal wound healing, J. Math. Biol., 29 (1991), 389-404. doi: 10.1007/BF00160468
    [77] J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theor. Biol., 81 (1979), 389-400. doi: 10.1016/0022-5193(79)90042-0
    [78] G. Lolas, Spatio-temporal pattern formation and reaction diffusion systems, Master's thesis, University of Dundee Scotland, 1999.
    [79] R. McLennan, L. Dyson, K. W. Prather, J. A. Morrison, R. E. Baker, P. K. Maini, and P. M. Kulesa, Multiscale mechanisms of cell migration during development: Theory and experiment, Development, 139 (2012), 2935-2944. doi: 10.1242/dev.081471
    [80] R. McLennan, L. J. Schumacher, J. A. Morrison, J. M. Teddy, D. A. Ridenour, A. C. Box, C. L. Semerad, H. Li, W. McDowell, D. Kay, et al, Neural crest migration is driven by a few trailblazer cells with a unique molecular signature narrowly confined to the invasive front, Development, 142 (2015), 2014-2025. doi: 10.1242/dev.117507
    [81] R. McLennan, L. J. Schumacher, J. A. Morrison, J. M. Teddy, D. A. Ridenour, A. C. Box, C. L. Semerad, H. Li, W. McDowell, D. Kay, et al, VEGF signals induce trailblazer cell identity that drives neural crest migration, Dev. Biol., 407 (2015), 12-25. doi: 10.1016/j.ydbio.2015.08.011
    [82] J. D. Murray, G. F. Oster, and A. K. Harris, A mechanical model for mesenchymal morphogenesis, J. Math. Biol., 17 (1983), 125-129.
    [83] F. Schweisguth and F. Corson, Self-organization in pattern formation, Develop. Cell, 49 (2019), 659-677. doi: 10.1016/j.devcel.2019.05.019
    [84] L. Tweedy, D. A. Knecht, G. M. Mackay, and R. H. Insall, Self-generated chemoattractant gradients: attractant depletion extends the range and robustness of chemotaxis, PLoS Biol., 14 (2016), e1002404. doi: 10.1371/journal.pbio.1002404
    [85] J. Galle, M. Loeffler, and D. Drasdo, Modeling the effect of deregulated proliferation and apoptosis on the growth dynamics of epithelial cell populations in vitro, Biophys. J., 88 (2005), 62-75. doi: 10.1529/biophysj.104.041459
    [86] J. Galle, G. Aust, G. Schaller, T. Beyer, and D. Drasdo, Individual cell-based models of the spatial-temporal organization of multicellular systems-Achievements and limitations, Cytom. A, 69 (2006), 704-710.
    [87] D. Drasdo, S. Hoehme, and M. Block, On the role of physics in the growth and pattern formation of multi-cellular systems: What can we learn from individual-cell based models? J. Stat. Phys., 128 (2007), 287. doi: 10.1007/s10955-007-9289-x
    [88] M. P. Neilson, D. M. Veltman, P. J. M. van Haastert, S. D. Webb, J. A. Mackenzie, and R. H. Insall, Chemotaxis: A feedback-based computational model robustly predicts multiple aspects of real cell behaviour, PLoS Biol., 9 (2011), e1000618. doi: 10.1371/journal.pbio.1000618
    [89] S. Kondo, An updated kernel-based turing model for studying the mechanisms of biological pattern formation, J. Theor Biol., 414 (2017), 120-127. doi: 10.1016/j.jtbi.2016.11.003
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3475) PDF downloads(181) Cited by(6)

Article outline

Figures and Tables

Figures(17)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog