A Primer on Reaction–Diffusion Models in Embryonic Development

Abstract

A fundamental problem in developmental biology is understanding how complex patterns and organised tissues develop from a small group of nearly identical cells. A wealth of experimental data has exposed the complexity of the molecular networks guiding cellular decisions of organisation and patterning – networks whose output evolves over space and time as development progresses. Integrating this data into reaction–diffusion (RD) mathematical models that describe the spatiotemporal dynamics of molecular species during development provides a rigorous approach to test the plausibility of hypothesised mechanisms guiding pattern formation, to understand how the complexity is regulated and to optimise experimental design. RD modelling provides a complementary mode of inquiry that both depends on and informs experimental research. RD systems are used in developmental biology to model morphogen‐mediated pattern formation.

Key Concepts

  • Integrating wet lab experiments with spatiotemporal mathematical modelling enables greater hypothesis testing ability than either method alone.
  • Viewing developing embryos as engineered dynamical systems designed to satisfy performance objectives is a helpful mental framework for model‐based analysis in developmental biology.
  • Reaction‐transport equations (simplified to reaction–diffusion here) are indispensable when studying developmental patterning and morphogenesis.
  • Turing networks and positional information are the primary conceptual bases of pattern specification.
  • Model reproducibility is as important to consider as experimental reproducibility and should be the focus during implementation and communication of the model definition.

Keywords: morphogen; reaction–diffusion; pattern formation; dynamical systems; genetic regulatory network; mathematical model; Turing network; positional information; development

Figure 1. The Turing model and the positional information model of morphogen‐mediated pattern formation. The x axes indicate the spatial position along a line of cells. (a) In the Turing model, stochastic fluctuations induce random deviations from a uniform steady state in the concentration of a diffusive signalling molecule (green) which are amplified and propagated due to the reactions between the components (vi) to form a periodic pattern that may collide with other similar patterns with different orientations and starting positions (vii), generating intricate designs (viii). Once a concentration threshold is reached, cells respond by initiating downstream gene transcription and taking on a ‘green’ phenotype. (b) In Wolpert's model of positional information, (i) a signalling molecule concentration gradient emanates from a source, (ii) where multiple phenotypes (red, white or blue) may result based on concentration thresholds. Source: Green, http://dev.biologists.org/content/142/7/1203. Licensed under CC BY 4.0.
Figure 2. The three‐component Bmp–Sox9–Wnt PI‐driven Turing network guiding digit patterning described by the base equations . Positional information provided by Hoxd13–Fgf expression modulates the suppression of Wnt and Bmp by Sox9, initiating Turing instability, which enables the initiation and propagation of a self‐regulating pattern.
Figure 3. The Bicoid morphogen gradient. (a) Confocal image of nuclear localised Bicoid gradient along the Drosophila AP axis. (b) Semiquantitative signal intensity data along the length of the egg following the exponentially decaying profile seen in synthesis–diffusion–degradation models. Adapted with permission from Grimm et al. . © Company of Biologists.
Figure 4. A theoretical morphogen‐modulator two‐component system. Dashed lines indicate autoregulation for either component's own transport, clearance or production not analysed in the cited work.
Figure 5. Bmp signalling in Drosophila. (a) Dpp molecules secreted from a source represented by the green nucleus are prohibited from diffusion through binding with collagen. Upon shuttling complex formation with Dpp–Scw/Sog/Tsg, it can diffuse, creating a concentration gradient along the syncytium. The inhibitory complex is broken by Tld, enabling the Dpp–Scw heterodimer to bind its receptors and initiate downstream signalling. Fading blue nuclei represent decreasing downstream Bmp target gene transcription. (b) Reactions controlling Bmp signalling in Drosophila with terms from equation . In the dashed box, the order of reactions for the Dpp shuttling complex is distinguished. Either diffusive Sog and Tsg bind before complexing with Dpp–Scw, or Tsg binds collagen, then Dpp and then Sog completes the complex formation.
Figure 6. Generalised workflow for modelling morphogen systems using RD equations. The starting step involves asking and answering questions such as: ‘What objective function(s) is/are observed in the system?” “What mechanisms are known, assumed, postulated or missing?’ “What data exist or need to be acquired to proceed?’ ‘Is treating transport as pure diffusion appropriate?’ The choice to pursue a top‐down or bottom‐up approach depends on the level of theoretical abstraction appropriate for the system. Once the system's dynamics have been compared to data of these dynamics, the cycle may continue.
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Further Reading

Hengenius JB, Gribskov M, Rundell AE and Umulis DM (2014) Making models match measurements: model optimization for morphogen patterning networks. Seminars in Cell & Developmental Biology 35: 109–123. DOI: 10.1016/j.semcdb.2014.06.017.

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Umulis DM and Othmer HG (2012) The importance of geometry in mathematical models of developing systems. Current Opinion in Genetics and Development 22 (6): 547–552. DOI: 10.1016/j.gde.2012.09.007.

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Thompson, Matthew J, Othmer, Hans G, and Umulis, David M(Aug 2018) A Primer on Reaction–Diffusion Models in Embryonic Development. In: eLS. John Wiley & Sons Ltd, Chichester. http://www.els.net [doi: 10.1002/9780470015902.a0026599]