Bulletin of Mathematical Biology

, Volume 76, Issue 4, pp 895–921 | Cite as

Stochastic Pattern Formation and Spontaneous Polarisation: The Linear Noise Approximation and Beyond

  • Alan J. McKane
  • Tommaso Biancalani
  • Tim Rogers
Original Article


We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics which generalise the more familiar macroscopic equations. We apply this formalism to the analysis of two specific noise-induced phenomena observed in biologically inspired models. In the first example, we show how the stochastic amplification of a Turing instability gives rise to spatial and temporal patterns which may be understood within the linear noise approximation. The second example concerns the spontaneous emergence of cell polarity, where we make analytic progress by exploiting a separation of time-scales.


Stochastic models and the master equation Stochastic patterns and cell polarity 



This work was supported in part under EPSRC Grant No. EP/H02171X/1 (A.J.M. and T.R.). T.B. also wishes to thank the EPSRC for partial support.


  1. Altschuler, S. J., Angenent, S. B., Wang, Y., & Wu, L. F. (2008). On the spontaneous emergence of cell polarity. Nature, 454, 886–889. CrossRefGoogle Scholar
  2. Biancalani, T., Fanelli, D., & Di Patti, F. (2010). Stochastic Turing patterns in the Brusselator model. Phys. Rev. E, 81, 046215. doi: 10.1103/PhysRevE.81.046215. CrossRefGoogle Scholar
  3. Biancalani, T., Galla, T., & McKane, A. J. (2011). Stochastic waves in a Brusselator model with nonlocal interaction. Phys. Rev. E, 84, 026201. doi: 10.1103/PhysRevE.84.026201. CrossRefGoogle Scholar
  4. Biancalani, T., Rogers, T., & McKane, A. J. (2012). Noise-induced metastability in biochemical networks. Phys. Rev. E, 86, 010106(R). doi: 10.1103/PhysRevE.86.010106. CrossRefGoogle Scholar
  5. Black, A. J., & McKane, A. J. (2012). Stochastic formulation of ecological models and their applications. Trends Ecol. Evol., 27, 337–345. doi: 10.1016/j.tree.2012.01.014. CrossRefGoogle Scholar
  6. Boland, R. P., Galla, T., & McKane, A. J. (2009). Limit cycles, complex Floquet multipliers and intrinsic noise. Phys. Rev. E, 79, 051131. MathSciNetCrossRefGoogle Scholar
  7. Bonachela, J. A., Munoz, M. A., & Levin, S. A. (2012). Patchiness and demographic noise in three ecological examples. J. Stat. Phys., 148, 723–739. CrossRefzbMATHGoogle Scholar
  8. Bromwich, T. (1926). An introduction to the theory of infinite series. London: Chelsea. zbMATHGoogle Scholar
  9. Butler, T. C., & Goldenfeld, N. (2009). Robust ecological pattern formation induced by demographic noise. Phys. Rev. E, 80, 030902(R). doi: 10.1103/PhysRevE.80.030902. CrossRefGoogle Scholar
  10. Butler, T. C., & Goldenfeld, N. (2011). Fluctuation-driven Turing patterns. Phys. Rev. E, 84, 011112. doi: 10.1103/PhysRevE.84.011112. CrossRefGoogle Scholar
  11. Butler, T. C., Benayounc, M., Wallace, E., van Drongelenc, W., Goldenfeld, N., & Cowane, J. (2012). Evolutionary constraints on visual cortex architecture from the dynamics of hallucinations. Proc. Natl. Acad. Sci. USA, 109, 606–609. doi: 10.1073/pnas.1118672109. CrossRefGoogle Scholar
  12. Chaikin, P. M., & Lubensky, T. C. (2000). Principles of condensed matter physics (3rd ed.). Cambridge: Cambridge University Press. Google Scholar
  13. Cross, M. C., & Greenside, H. S. (2009). Pattern formation and dynamics in non-equilibrium systems. Cambridge: Cambridge University Press. CrossRefzbMATHGoogle Scholar
  14. Datta, S., Delius, G. W., & Law, R. (2010). A jump-growth model for predator-prey dynamics: derivation and application to marine ecosystems. Bull. Math. Biol., 72, 1361–1382. doi: 10.1007/s11538-009-9496-5. MathSciNetCrossRefzbMATHGoogle Scholar
  15. Gardiner, C. W. (2009). Handbook of stochastic methods for physics, chemistry and the natural sciences (4th ed.). Berlin: Springer. zbMATHGoogle Scholar
  16. Gillespie, D. T. (1976). A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys., 22, 403–434. MathSciNetCrossRefGoogle Scholar
  17. Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem., 81, 2340–2361. CrossRefGoogle Scholar
  18. Glansdorff, P., & Prigogine, I. (1971). Thermodynamic theory of structure, stability and fluctuations. Chichester: Wiley-Interscience. zbMATHGoogle Scholar
  19. Gupta, A. (2012). Stochastic model for cell polarity. Ann. Appl. Probab., 22, 827–859. MathSciNetCrossRefzbMATHGoogle Scholar
  20. Lawson, M. J., Drawert, B., Khammash, M., Petzold, L., & Yi, T. M. (2012, submitted). Spatial stochastic dynamics enable robust cell polarization. Google Scholar
  21. Lugo, C. A., & McKane, A. J. (2008). Quasi-cycles in a spatial predator-prey model. Phys. Rev. E, 78, 051911. MathSciNetCrossRefGoogle Scholar
  22. Mehta, M. L. (1989). Matrix theory. India: Hindustan Publishing Corporation. Google Scholar
  23. Murray, J. D. (2008). Mathematical biology, Vol. II (3rd ed.). Berlin: Springer. Google Scholar
  24. Ridolfi, L., Camporeale, C., D’Odorico, P., & Laio, F. (2011a). Transient growth induces unexpected deterministic spatial patterns in the Turing process. Europhys. Lett., 95, 18003. doi: 10.1209/0295-5075/95/18003. CrossRefGoogle Scholar
  25. Ridolfi, L., D’Odorico, P., & Laio, F. (2011b). Noise-induced phenomena in the environmental sciences. Cambridge: Cambridge University Press. CrossRefzbMATHGoogle Scholar
  26. Risken, H. (1989). The Fokker–Planck equation—methods of solution and applications (2nd ed.). Berlin: Springer. zbMATHGoogle Scholar
  27. Rogers, T., McKane, A. J., & Rossberg, A. G. (2012a). Demographic noise can lead to the spontaneous formation of species. Europhys. Lett., 97, 40008. doi: 10.1209/0295-5075/97/40008. CrossRefGoogle Scholar
  28. Rogers, T., McKane, A. J., & Rossberg, A. G. (2012b). Spontaneous genetic clustering in populations of competing organisms. Phys. Biol., 9, 066002. CrossRefGoogle Scholar
  29. Scott, M., Poulin, F. J., & Tang, H. (2011). Approximating intrinsic noise in continuous multispecies models. Proc. R. Soc. Lond. A, 467, 718–737. doi: 10.1098/rspa.2010.0275. MathSciNetCrossRefzbMATHGoogle Scholar
  30. Turing, A. M. (1952). The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B, 237, 37–72. doi: 10.1098/rstb.1952.0012. CrossRefGoogle Scholar
  31. Van Kampen, N. G. (2007). Stochastic processes in physics and chemistry (3rd ed.). Amsterdam: Elsevier Science. zbMATHGoogle Scholar
  32. Wiggins, S. (2003). Introduction to applied nonlinear dynamical systems and chaos (2nd ed.). Berlin: Springer. zbMATHGoogle Scholar
  33. Woolley, T. E., Baker, R. E., Gaffney, E. A., & Maini, P. K. (2011). Stochastic reaction and diffusion on growing domains: understanding the breakdown of robust pattern formation. Phys. Rev. E, 84, 046216. doi: 10.1103/PhysRevE.84.046216. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  • Alan J. McKane
    • 1
  • Tommaso Biancalani
    • 1
  • Tim Rogers
    • 1
    • 2
  1. 1.Theoretical Physics Division, School of Physics and AstronomyUniversity of ManchesterManchesterUK
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK

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