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Modelling Cell Migration and Adhesion During Development

Abstract

Cell–cell adhesion is essential for biological development: cells migrate to their target sites, where cell–cell adhesion enables them to aggregate and form tissues. Here, we extend analysis of the model of cell migration proposed by Anguige and Schmeiser (J. Math. Biol. 58(3):395–427, 2009) that incorporates both cell–cell adhesion and volume filling. The stochastic space-jump model is compared to two deterministic counterparts (a system of stochastic mean equations and a non-linear partial differential equation), and it is shown that the results of the deterministic systems are, in general, qualitatively similar to the mean behaviour of multiple stochastic simulations. However, individual stochastic simulations can give rise to behaviour that varies significantly from that of the mean. In particular, individual simulations might admit cell clustering when the mean behaviour does not. We also investigate the potential of this model to display behaviour predicted by the differential adhesion hypothesis by incorporating a second cell species, and present a novel approach for implementing models of cell migration on a growing domain.

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References

  • Alberts, B., Bray, D., Lewis, J., Raff, M., Roberts, K., & Watson, J. D. (1994). Molecular biology of the cell (3rd ed.). New York: Garland.

    Google Scholar 

  • Anguige, K. (2011). A one-dimensional model for the interaction between cell-to-cell adhesion and chemotactic signalling. Eur. J. Appl. Math., 22(4), 291–316.

    MathSciNet  MATH  Article  Google Scholar 

  • Anguige, K., & Schmeiser, C. (2009). A one-dimensional model of cell diffusion and aggregation, incorporating volume filling and cell-to-cell adhesion. J. Math. Biol., 58(3), 395–427.

    MathSciNet  MATH  Article  Google Scholar 

  • Armstrong, N. J., Painter, K. J., & Sherratt, J. A. (2006). A continuum approach to modelling cell-cell adhesion. J. Theor. Biol., 243(1), 98–113.

    MathSciNet  Article  Google Scholar 

  • Baker, R. E., Yates, C. A., & Erban, R. (2010). From microscopic to macroscopic descriptions of cell migration on growing domains. Bull. Math. Biol., 72(3), 719–762.

    MathSciNet  MATH  Article  Google Scholar 

  • Berg, H. C. (1975). How bacteria swim. Sci. Am., 233(2), 36–44.

    Article  Google Scholar 

  • Berg, H. C. (1993). Random walks in biology. Princeton: Princeton University Press.

    Google Scholar 

  • Bolker, B., & Pacala, S. W. (1997). Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor. Popul. Biol., 52(3), 179–197.

    MATH  Article  Google Scholar 

  • Brenner, M. P., Levitov, L. S., & Budrene, E. O. (1998). Physical mechanisms for chemotactic pattern formation by bacteria. Biophys. J., 74(4), 1677–1693.

    Article  Google Scholar 

  • Crampin, E. J., Gaffney, E. A., & Maini, P. K. (1999). Reaction and diffusion on growing domains: scenarios for robust pattern formation. Bull. Math. Biol., 61(6), 1093–1120.

    Article  Google Scholar 

  • Erban, R., & Othmer, H. G. (2004). From individual to collective behavior in bacterial chemotaxis. SIAM J. Appl. Math., 65(2), 361–391.

    MathSciNet  MATH  Article  Google Scholar 

  • Foty, R. A., & Steinberg, M. S. (2005). The differential adhesion hypothesis: a direct evaluation. Dev. Biol., 278(1), 255–263.

    Article  Google Scholar 

  • Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem., 81(1), 2340–2361.

    Article  Google Scholar 

  • Keller, E. F., & Segel, L. A. (1970). Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol., 26(3), 399–415.

    Article  Google Scholar 

  • Keller, E. F., & Segel, L. A. (1971a). Model for chemotaxis. J. Theor. Biol., 30(2), 225–234.

    Article  Google Scholar 

  • Keller, E. F., & Segel, L. A. (1971b). Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol., 30(2), 235–248.

    Article  Google Scholar 

  • Khain, E., Katakowski, M., Hopkins, S., Szalad, A., Zheng, X., Jiang, F., & Chopp, M. (2011). Collective behavior of brain tumor cells: the role of hypoxia. Phys. Rev. E, 83(3), 031920.

    Article  Google Scholar 

  • Landman, K. A., Pettet, G. J., & Newgreen, D. F. (2003). Mathematical models of cell colonization of uniformly growing domains. Bull. Math. Biol., 65(2), 235–262.

    Article  Google Scholar 

  • Lieberman, M. A., & Glaser, L. (1981). Density dependent regulation of cell growth: an example of a cell-cell recognition phenomenon. J. Membr. Biol., 11, 1–11.

    Google Scholar 

  • Maini, P. K., & Solursh, M. (1991). Cellular mechanisms of pattern formation in the developing limb. Int. Rev. Cytol., 129, 91–133.

    Article  Google Scholar 

  • Mooney, J. R., & Nagorcka, B. N. (1985). Spatial patterns produced by a reaction-diffusion system in primary hair follicles. J. Theor. Biol., 115(2), 299–317.

    MathSciNet  Article  Google Scholar 

  • Morton, K. W., & Mayers, D. F. (2005). Numerical solution of partial differential equations: an introduction. Cambridge: Cambridge University Press.

    MATH  Book  Google Scholar 

  • Murray, J. D. (2002). Mathematical biology (3rd ed.). Berlin: Springer.

    MATH  Google Scholar 

  • Othmer, H. G., & Hillen, T. (2011). The diffusion limit of transport equations II: chemotaxis equations. SIAM J. Appl. Math., 62(4), 1222–1250.

    MathSciNet  Article  Google Scholar 

  • Othmer, H. G., & Schaap, P. (1998). Oscillatory cAMP signaling in the development of Dictyostelium discoideum. Comments Theor. Biol., 5, 175–282.

    Google Scholar 

  • Othmer, H. G., & Stevens, A. (1997). Aggregation, blowup, and collapse: the ABC’s of taxis in reinforced random walks. SIAM J. Appl. Math., 57(4), 1044–1081.

    MathSciNet  MATH  Article  Google Scholar 

  • Othmer, H. G., Dunbar, S. R., & Alt, W. (1988). Models of dispersal in biological systems. J. Math. Biol., 26, 263–298.

    MathSciNet  MATH  Article  Google Scholar 

  • Palsson, E., & Othmer, H. G. (2000). A model for individual and collective cell movement in Dictyostelium discoideum. Proc. Natl. Acad. Sci. USA, 97(19), 10448–10453.

    Article  Google Scholar 

  • Patlak, C. S. (1953). Random walk with persistence and external bias. Bull. Math. Biophys., 15(3), 311–338.

    MathSciNet  MATH  Article  Google Scholar 

  • Penington, C. J., Hughes, B. D., & Landman, K. A. (2011). Building macroscale models from microscale probabilistic models: a general probabilistic approach for nonlinear diffusion and multispecies phenomena. Phys. Rev. E, 84(4), 041120.

    Article  Google Scholar 

  • Simpson, M. J., Landman, K. A., Hughes, B. D., & Fernando, A. E. (2010a). A model for mesoscale patterns in motile populations. Physica A, 389(7), 1412–1424.

    Article  Google Scholar 

  • Simpson, M. J., Towne, C., McElwain, D. L. S., & Upton, Z. (2010b). Migration of breast cancer cells: understanding the roles of volume exclusion and cell-to-cell adhesion. Phys. Rev. E, 82(4), 041901.

    Article  Google Scholar 

  • Steinberg, M. S. (1962a). On the mechanism of tissue reconstruction by dissociated cells, I. Population kinetics, differential adhesiveness, and the absence of directed migration. Proc. Natl. Acad. Sci. USA, 48(9), 1577–1582.

    Article  Google Scholar 

  • Steinberg, M. S. (1962b). Mechanism of tissue reconstruction by dissociated cells, II. Time-course of events. Science, 137(3532), 762–763.

    Article  Google Scholar 

  • Steinberg, M. S. (1962c). On the mechanism of tissue reconstruction by dissociated cells, III. Free energy relations and the reorganization of fused, heteronomic tissue fragments. Proc. Natl. Acad. Sci. USA, 48(10), 1769–1776.

    Article  Google Scholar 

  • Woolley, T., Baker, R. E., Gaffney, E. A., & Maini, P. K. (2011). Influence of stochastic domain growth on pattern nucleation for diffusive systems with internal noise. Phys. Rev. E, 84(4), 041905.

    Article  Google Scholar 

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Acknowledgements

RNT would like to thank the Centre for Mathematical Biology, University of Oxford, for the opportunity to carry out this research, the Nuffield Foundation for the bursary that allowed this research to begin, and BBSRC for research funding via the Genes to Organisms doctoral training award. He would also like to thank Endre Suli, Michael Thompson and Nik Cunniffe for helpful discussions and support. CAY would like to thank Christ Church College, Oxford, for a Junior Research Fellowship.

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Correspondence to Robin N. Thompson.

Appendix: Transition Probabilities on a Non-uniform Domain

Appendix: Transition Probabilities on a Non-uniform Domain

When the domain has unequal compartment sizes, the distance that cells jump between neighbouring compartments varies. If this distance is large, then the transition rate between the compartments concerned should be reduced.

We note that, if we consider a stochastic system with transition probabilities of the form

$$T_i^\pm = D/h^2, $$

where diffusivity D is constant, we can derive the PDE

$$ \frac{\partial \rho}{\partial t} = -\frac{\partial}{\partial x} \biggl(-D \frac{\partial \rho}{\partial x} \biggr), $$
(5)

as discussed by Baker et al. (2010).

If we consider approximating a solution to this PDE on a non-uniform domain (Morton and Mayers 2005), we obtain

where the h i s are defined as in Sect. 2.1. Assuming that the number of cells evolves according to

$$\frac{\partial n_i}{\partial t}=T_{i+1}^- n_{i+1} + T_{i-1}^+ n_{i-1} - \bigl(T_i^- + T_i^+\bigr)n_i, $$

we deduce that

$$T_i^+ = \frac{2D}{h_{i+1}(h_i + h_{i+1})}, $$

with a similar expression obtainable for \(T_{i}^{-}\). For ease of notation, we define

$$h_{i+\frac{1}{2}} = \frac{1}{2}(h_i + h_{i+1}). $$

The above consideration motivates the pre-factor in our transition rates on a non-uniform domain, for i=2,3,…,k−1 (where the cell density in compartment i is n i /S i ), given by

with similar expressions derived for i=1 and i=k. Strictly speaking, we have assumed that the compartment edges are halfway between the lattice points.

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Thompson, R.N., Yates, C.A. & Baker, R.E. Modelling Cell Migration and Adhesion During Development. Bull Math Biol 74, 2793–2809 (2012). https://doi.org/10.1007/s11538-012-9779-0

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Keywords

  • Mathematical modelling
  • Cell–cell adhesion
  • Differential adhesion
  • Cell sorting
  • Domain growth