Abstract
Birth–death–movement processes, modulated by interactions between individuals, are fundamental to many cell biology processes. A key feature of the movement of cells within in vivo environments is the interactions between motile cells and stationary obstacles. Here we propose a multi-species model of individual-level motility, proliferation and death. This model is a spatial birth–death–movement stochastic process, a class of individual-based model (IBM) that is amenable to mathematical analysis. We present the IBM in a general multi-species framework and then focus on the case of a population of motile, proliferative agents in an environment populated by stationary, non-proliferative obstacles. To analyse the IBM, we derive a system of spatial moment equations governing the evolution of the density of agents and the density of pairs of agents. This approach avoids making the usual mean-field assumption so that our models can be used to study the formation of spatial structure, such as clustering and aggregation, and to understand how spatial structure influences population-level outcomes. Overall the spatial moment model provides a reasonably accurate prediction of the system dynamics, including important effects such as how varying the properties of the obstacles leads to different spatial patterns in the population of agents.
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References
Agnew DJG, Green JEF, Brown TM, Simpson MJ, Binder BJ (2014) Distinguishing between mechanisms of cell aggregation using pair-correlation functions. J Theor Biol 352:16–23
Bajenoff M, Egen JG, Koo LY, Laugier JP, Brau F, Glaichenhaus N, Germain RN (2006) Stromal cell networks regulate lymphocyte entry, migration, and territoriality in lymph nodes. Immunity 25(6):989–1001
Baker RE, Simpson MJ (2010) Correcting mean-field approximations for birth–death–movement processes. Phys Rev E 82:041905
Barraquand F, Murrell DJ (2013) Scaling up predator–prey dynamics using spatial moment equations. Methods Ecol Evol 4(3):276–289
Binny RN, Plank MJ, James A (2015) Spatial moment dynamics for collective cell movement incorporating a neighbour-dependent directional bias. J R Soc Interface 12(106):20150228
Binny RN, Haridas P, James A, Law R, Simpson MJ, Plank MJ (2016a) Spatial structure arising from neighbour-dependent bias in collective cell movement. PeerJ 4:e1689
Binny RN, James A, Plank MJ (2016b) Collective cell behaviour with neighbour-dependent proliferation, death and directional bias. Bull Math Biol 78(11):2277–2301
Bolker B, Pacala SW (1997) Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor Popul Biol 52(3):179–197
Browning AP, McCue SW, Binny RN, Plank MJ, Shah ET, Simpson MJ (2018) Inferring parameters for a lattice-free model of cell migration and proliferation using experimental data. J Theor Biol 437:251–260
Condeelis J, Segail JE (2003) Intravital imaging of cell movement in tumours. Nat Rev Cancer 3(12):921–930
Dini S, Binder BJ, Green JEF (2018) Understanding interactions between populations: individual based modelling and quantification using pair correlation functions. J Theor Biol 439:50–64
Dyson L, Baker RE (2015) The importance of volume exclusion in modelling cellular migration. J Math Biol 71(3):691–711
Edelstein-Keshet L (2005) Mathematical models in biology (classics in applied mathematics). Society for Industrial and Applied Mathematics, New York
Ellery AJ, Simpson MJ, McCue SW, Baker RE (2014) Characterising transport through a crowded environment with different obstacle sizes. J Chem Phys 140:054108
Ellery AJ, Baker RE, McCue SW, Simpson MJ (2016) Modelling transport through an environment crowded by a mixture of obstacles of different shapes and sizes. Phys A 449:74–84
Finkelshtein D, Kondratiev Y, Kutoviy O (2009) Individual based model with competition in spatial ecology. SIAM J Math Anal 41(1):297–317
Friedl P, Wolf K (2003) Tumour-cell invasion and migration: diversity and escape mechanisms. Nat Rev Cancer 3(5):362–374
Ghosh SK, Cherstvy AG, Grebenkov DS, Metzler R (2016) Anomalous, non-Gaussian tracer diffusion in crowded two-dimensional environments. New J Phys 18:013027
Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361
Hansen MM, Meijer LH, Spruijt E, Maas RJ, Rosquelles MV, Groen J, Heus HA, Huck WT (2016) Macromolecular crowding creates heterogeneous environments of gene expression in picolitre droplets. Nat Nanotechnol 11:191–197
Harley BA, Kim HD, Zaman MH, Yannas IV, Lauffenburger DA, Gibson LJ (2008) Microarchitecture of three-dimensional scaffolds influences cell migration behaviour via junction interactions. Biophys J 95(8):4013–4024
Hu K, Ji L, Applegate KT, Danuser G, Waterman-Storer CM (2007) Differential transmission of actin motion within focal adhesions. Science 315:111–115
Jin W, McCue SW, Simpson MJ (2018) Extended logistic growth models for heterogeneous populations. J Theor Biol 445:51–61
Johnston ST, Shah ET, Chopin LK, McElwain DLS, Simpson MJ (2015) Estimating cell diffusivity and cell proliferation rate by interpreting IncuCyte \(\text{ ZOOM }^{\rm TM}\) assay data using the Fisher–Kolmogorov model. BMC Syst Biol 9:38
Keller EF, Segel LA (1971) Model for chemotaxis. J Theor Biol 30:225–234
Kurosaka S, Kashina A (2008) Cell biology of embryonic migration. Birth Defects Res Part C Embryo Today 84(2):102–122
Law R, Dieckmann U (2000) A dynamical system for neighbourhoods in plant communities. Ecology 81:2137–2148
Law R, Murrell DJ, Dieckmann U (2003) Population growth in space and time: spatial logistic equations. Ecology 84:252–262
Le Clainche C, Carlier M (2008) Regulation of actin assembly associated with protrusion and adhesion in cell migration. Physiol Rev 88(2):489–513
Lewis MA (2000) Spread rate for a nonlinear stochastic invasion. J Math Biol 41:430–454
Martin P (1997) Wound healing-aiming for perfect skin regeneration. Science 276:75–81
Middleton AM, Fleck C, Grima R (2014) A continuum approximation to an off-lattice individual-cell based model of cell migration and adhesion. J Theor Biol 359:220–232
Murray JD (1989) Mathematical biology. Springer, New York
Murrell DJ, Dieckmann U, Law R (2004) On moment closures for population dynamics in continuous space. J Theor Biol 229:421–432
Murrell DJ (2005) Local spatial structure and predator-prey dynamics: counterintuitive effects of prey enrichment. Am Nat 166:354–367
North A, Ovaskainen O (2007) Interactions between dispersal, competition, and landscape heterogeneity. Oikos 116(7):1106–1119
North A, Cornell SJ, Ovaskainen O (2011) Evolutionary responses of dispersal distance to landscape structure and habitat loss. Evolution 65(6):1739–1751
Ovaskainen O, Cornell SJ (2006) Asymptotically exact analysis of stochastic metapopulation dynamics with explicit spatial structure. Theor Popul Biol 69(1):13–33
Ovaskainen O, Finkelshtein D, Kutoviy O, Cornell SJ, Bolker B, Kondratiev Y (2014) A general mathematical framework for the analysis of spatiotemporal point processes. Theor Ecol 7(1):101–113
Plank MJ, Law R (2015) Spatial point processes and moment dynamics in the life sciences: a parsimonious derivation and some extensions. Bull Math Biol 77:586–613
Plank MJ, Simpson MJ (2012) Models of collective cell behaviour with crowding effects: comparing lattice based and lattice-free approaches. J R Soc Interface 9:2983–2996
Raghib M, Hill NA, Dieckmann U (2011) A multiscale maximum entropy moment closure for locally regulated space-time point process models of population dynamics. J Math Biol 62:605–653
Simpson MJ, Towne C, McElwain DLS, Upton Z (2010) Migration of breast cancer cells: understanding the roles of volume exclusion and cell-to-cell adhesion. Phys Rev E 82:041901
Simpson MJ, Binder BJ, Haridas P, Wood BK, Treloar KK, McElwain DLS, Baker RE (2013) Experimental and modelling investigation of monolayer development with clustering. Bull Math Biol 75:871–889
Simpson MJ, Plank MJ (2017) Simplified calculation of diffusivity for a lattice-based random walk with a single obstacle. Results Phys 7:3346–3348
Smith S, Cianci C, Grima R (2017) Macromolecular crowding directs the motion of small molecules inside cells. J R Soc Interface 14:20170047
Sun M, Zaman MH (2017) Modelling, signaling and cytoskeleton dynamics: integrated modelling-experimental frameworks in cell migration. WIREs Syst Biol Med 9:e1365
Tan C, Saurabh S, Bruchez MP, Schwartz R, LeDuc P (2013) Molecular crowding shapes gene expression in synthetic cellular nanosystems. Nat Nanotechnol 8:602–608
Tobin P, Bjornstad ON (2003) Spatial dynamics and cross-correlation in a transient predator-prey system. J Anim Ecol 72:460–467
Wedemeier A, Merlitz H, Langowski J (2009) Anomalous diffusion in the presence of mobile obstacles. Europhys Lett 88:38004
Welch MD (2015) Cell migration, freshly squeezed. Cell 160:581–582
Zaman MH, Trapani LM, Sieminski AL, Mackellar D, Gong H, Kamm RD, Wells A, Lauffenburger DA, Matsudaira P (2006) Migration of tumor cells in 3D matrices is governed by matrix stiffness along with cell-matrix adhesion and proteolysis. Proc Natl Acad Sci USA 103:10889–10894
Acknowledgements
This work is supported by the Australian Research Council (DP170100474). MJP is partly supported by Te Pūnaha Matatini, a New Zealand Centre of Research Excellence. We thank the anonymous referee for their helpful suggestions.
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Surendran, A., Plank, M.J. & Simpson, M.J. Spatial Moment Description of Birth–Death–Movement Processes Incorporating the Effects of Crowding and Obstacles. Bull Math Biol 80, 2828–2855 (2018). https://doi.org/10.1007/s11538-018-0488-1
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DOI: https://doi.org/10.1007/s11538-018-0488-1