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Biology by numbers: mathematical modelling in developmental biology

Key Points

  • Mathematical models can be useful to developmental biologists, in particular in helping to bridge the gap in understanding between proposed molecular interactions inside and between cells and their tissue-level effects.

  • Differential equation models have been the most popular in development, and there is a growing body of work in the application of more computationally efficient models, such as hybrid systems and discrete state systems. Both 'top-down' and 'bottom-up' model architectures have been used.

  • By focusing on a few developmental systems and the corresponding models that have been developed, this Review illustrates the potential and possibilities in developing an intuition about interactions among biochemical components and their emergent properties.

  • The first mathematical model we discuss represents the segment polarity network in the early Drosophila melanogaster embryo. Modifications to the proposed network of molecular interactions resulting from this model are presented, along with ensuing, simpler models that aid in understanding this system.

  • How diffusion models have helped in forming hypotheses about the properties of a morphogen in different species of embryos that show great variation in size is then discussed.

  • The Review concludes with two sets of mathematical models highlighting cell-to-cell communication in Dictyostelium discoideum and Myxococcus xanthus, describing how these models have helped in postulating molecular mechanisms responsible for cell signalling and aggregation when nutrients are scarce.

Abstract

In recent years, mathematical modelling of developmental processes has earned new respect. Not only have mathematical models been used to validate hypotheses made from experimental data, but designing and testing these models has led to testable experimental predictions. There are now impressive cases in which mathematical models have provided fresh insight into biological systems, by suggesting, for example, how connections between local interactions among system components relate to their wider biological effects. By examining three developmental processes and corresponding mathematical models, this Review addresses the potential of mathematical modelling to help understand development.

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Figure 1: The segment polarity network.
Figure 2: A model for diffusion.
Figure 3: Dictyostelium discoideum cell aggregation.
Figure 4: Social behaviour of Myxococcus xanthus.

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Dictybase

Glossary

Phyllotaxis

The radial organization of the leaves on the shoot of a plant; the basic patterns are spiral, alternating sides, opposite each other and whorled.

Systems biology

There is no single accepted definition for this idea. We define it as the attempt, using mathematical modelling, to examine the structure of a biological pathway or function, and to simultaneously consider all of the relevant component parts and the dynamics of their interactions.

Morphogen

A diffusible protein that forms a concentration gradient to control patterning by inducing at least two distinct threshold dependent responses.

Influence model

A descriptive, non-mathematical model that represents positive (promoting) or negative (inhibitory) influences between components using arrows.

Ordinary differential equation

An equation that contains functions of only one independent variable, and one or more of these functions' derivatives with respect to that variable.

Synthetic biology

The field at the interface of engineering and biology, involving designing and building systems from biological components.

Boolean model

A collection of nodes, each representing a system state, with a set of rules that represent how the system switches between states.

Steady-state pattern

In a tissue-patterning system, this refers to a pattern that is usually reached after some time, and persists indefinitely or until a new tissue-patterning system takes over.

Bayesian network

A collection of nodes connected by edges, in which each node represents a system state and each edge represents an influence or dependency between the two nodes that touch it.

Control theory

A theory built around the behaviour of dynamic systems, in which a controller is designed to automatically manipulate system input variables in order to guide the system output variables to desired values.

Numerical solution

A solution that is represented as a set of numerical values, which are usually obtained using a computer when an analytic solution is difficult or impossible to obtain.

Partial differential equation

An equation that contains functions of many independent variables, and one or more of these functions' partial derivatives with respect to that variable.

Chemotaxis

The movement of cells or organisms in response to chemical stimulation.

Analytic solution

A functional, rather than numerical, representation of a solution.

Flux

The amount of flow per unit time, arising, for example, from diffusion or convection.

Principal component analysis

A method to organize similar data points together in groups, so that an overall data set becomes simpler to analyse.

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Tomlin, C., Axelrod, J. Biology by numbers: mathematical modelling in developmental biology. Nat Rev Genet 8, 331–340 (2007). https://doi.org/10.1038/nrg2098

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