Developmental Biology: Mathematical Modelling of Development

Abstract

Understanding how structures (e.g. hair, teeth, feathers, limbs and pigmentation patterns) arise from the initially unstructured fertilised egg is one of the key challenges in developmental biology. Mathematical models enable us to investigate how certain biochemical and/or biophysical processes interact to produce pattern and form. They provide a unifying theme for spatio‐temporal patterning across a vast range of biological applications by suggesting a set of underlying principles for pattern formation. Such models suggest that patterns and structures must have certain properties and these predictions motivate experiments. The results of such experiments help refine models and lead to more precise predictions. In this way, modelling, combined with experiment, can be a powerful investigative tool in helping unravel the complexity of morphogenesis (the formation of structure) in biology.

Key Concepts:

  • Patterning arises due to short‐range activation, long‐range inhibition.

  • Mathematical models suggest constraints on development.

  • Instabilities emerge from stabilising processes.

  • Pattern properties can be mechanism‐independent.

  • It is the integration of biochemical and biophysical processes that lead to structure formation.

Keywords: morphogenesis; pattern formation; emergent behaviour; self‐organisation; developmental constraints; mathematical modelling

Figure 1.

The model of Weliky and Oster reproduces the essential features of Fundulus epiboly showing that the forces they hypothesise to act on the cell vertices are sufficient to produce results of tissue movement and rearrangement consistent with experimental observations. (a) Early stage of epiboly. (b) Middle stage of epiboly. (c) Late stage of epiboly. (d) End of epiboly. Reproduced with permission from Weliky and Oster . Copyright Palgrave Macmillan.

Figure 2.

An illustration of the vast variety of spatial patterns arising in the Turing model (Chang et al., ). Reproduced with permission from Chang et al.. Copyright UBC Press. (a), (b) A typical pattern in chemical concentration. (c)–(e) The skin patterns that may arise from the chemical pattern in (a), (b). (f)–(h) More complex examples of patterns that may arise from the Turing model. (i), (j) The interaction of two Turing patterns. Colours indicate different concentration levels.

Figure 3.

Patterns for feather buds forming on the skin of a chick embryo (Lin et al., ). This sequence of experimental snapshots illustrates the sequential formation of feather buds (top row) and more detailed views (bottom row). Reproduced from Lin et al.. Copyright (2009) with permission from Elsevier.

Figure 4.

An example of the patterns possible on a long thin domain (Chang et al., ). (a)–(e) Illustrate the range of patterns possible and the idea of a developmental constraint can be seen by understanding that the pattern in (b) is much more likely to occur in a Turing system than the pattern in (a). Furthermore, if the domain was tapering then the pattern would change from essentially a two‐dimensional pattern (i.e. pattern that varies in both horizontal and vertical directions, such as that in (c)) to one that is essentially one‐dimensional, such as that shown in (b) (which only varies in the vertical direction). (f)–(h) Illustrate the skin patterns possible as a result of (e). Reproduced from Chang et al.. Copyright (2009) with permission from Elsevier.

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Maini, Philip K, and Baker, Ruth E(Apr 2012) Developmental Biology: Mathematical Modelling of Development. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net [doi: 10.1002/9780470015902.a0001067]