Mathematical modelling has been used for decades to
help scientists understand the mechanisms and dynamics
behind their experimental observations. In develop-
mental biology, one of the most cited models is Turing’s
reaction-diffusion differential equations. In 1952, Turing
developed these equations as a model for morphogenesis,
and applied the model to simple one-, two-, and
three-dimensional cell networks. He related these
networks to various biological settings, from tentacle
patterns on the radially symmetrical Hydra, to phyllotaxis,
to the gastrulation of animal embryos1. Other pioneering
mathematical models that have a similar basis in differ-
ential equation abstractions of underlying mechanisms
have followed (BOX 1). Like Turing’s equations, these
models specify how the concentrations of biochemical
substances change in time, in and around cells, when
influenced by the presence or absence of other molecules.
If predictions derived from simulating these models
match experimental values, the model could represent a
plausible explanation of how the system works.
Designing such models requires experimental data,
and knowledge or a hypothesis about how the compo-
nents of the system are connected. The usefulness of
the early models was restricted by the limited amount
of experimental data: these models could validate the
plausibility of the experimenters’ hypotheses, but gen-
erally could not provide any information about the
system’s function beyond that which the experimenters
had already proposed to be true. This is changing. Today,
with advances in molecular biology, genetic manipula-
tion and the availability of complete genome sequences,
new models are being developed that incorporate
detailed dynamics of sets of biochemical interactions.
Often, the developmental events of interest, such as pat-
terning or cell-cycle regulation, are large-scale features of
the system, whereas the biochemistry on which current
models are built is of a much finer scale. It is not pos-
sible to simply intuit the fine-scale biochemistry from
its large-scale effects, or vice versa, under a multitude of
experimental conditions. However, computational power
now exists to solve large sets of equations of biochemical
interactions over multicellular networks, to effectively
bridge the gap between cellular-level processes and their
Various modelling methods have been used to suc-
cessfully address specific questions about the systems
of interest. The range of problems that can be tackled
is growing steadily and the modelling methods differ
in their modularity and mathematical implementa-
tion. Although all successful modelling efforts answer
a defined question, we believe that they also succeed to
varying degrees in revealing an intuitive understanding
of why the system behaves as it does. In this Review,
we first present a broad classification of mathematical
models and methods. Then, using several examples
from specific developmental systems, we explore the
gap between successful modelling and intuitive under-
standing, and discuss the extent to which it is reasonable
to expect models to provide a clear understanding
of how fine-scale mechanisms produce large-scale
As many scientists are embarking on collaborative
research in systems biology, we believe that it is essential to
carefully define at the outset the goals and expectations
*Department of Electrical
Engineering and Computer
Sciences, University of
Berkeley, California 94720,
USA, and Department of
Aeronautics and Astronautics,
Stanford University, Stanford,
California 94305-4035, USA.
‡Department of Pathology,
Stanford University School
of Medicine, Stanford,
California 94305-5324, USA.
Correspondence to C.J.T.
The radial organization of the
leaves on the shoot of a plant;
the basic patterns are spiral,
alternating sides, opposite
each other and whorled.
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
Biology by numbers: mathematical
modelling in developmental biology
Claire J. Tomlin* and Jeffrey D. Axelrod‡
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.
NATURE REVIEWS | GENETICS
VOLUME 8 | MAY 2007 | 331
© 2007 Nature Publishing Group
A diffusible protein that forms
a concentration gradient to
control patterning by inducing
at least two distinct threshold
A descriptive, non-
mathematical model that
(promoting) or negative
(inhibitory) influences between
components using arrows.
An equation that contains
functions of only one
independent variable, and one
or more of these functions’
derivatives with respect to
of specific modelling efforts2–4. By focusing on a few
developmental systems and the models that have been
developed for these systems, we show how modelling
has played an integral part in developing intuition about
interactions among biochemical components and their
A classification of modelling approaches
Mathematical models in development can be classified
along two axes: model architecture and mathematical
Model architecture addresses the question of which
components of the system should be modelled. Noble
provides a pragmatic breakdown of the different strate-
gies for formulating model architectures of complex
biological pathways5 (BOX 2). The ‘bottom-up’ approach
builds a model by first representing the details of the
component parts and how they are connected together.
The ‘top-down’ approach starts with a functional repre-
sentation of the entire system, and then breaks this down
into its component parts. In both cases, the components
to model are chosen on the basis of existing experimen-
tal data and the questions being asked about the system.
Both bottom-up and top-down approaches are appeal-
ing because they suggest a modular organization, which
seems to be present in developmental systems and is
useful in providing intuition about the behaviour and
function of the overall system6.
Mathematical implementation addresses the kind of
model that is used to represent the dynamics of each
component. The most popular implementation so far
has been with differential equation models: the initial
success stories in modelling developmental networks
have all used this traditional mathematical form. But
there have been some recent analyses with both discrete-
state and hybrid models that have shown great promise.
We discuss these different mathematical forms and their
applications in BOX 3.
In the following sections, three developmental
processes are examined and framed according to the
mathematical approaches that are outlined here. The
examples we give, of patterning in the early Drosophila
melanogaster embryo and cell aggregation in Dictyostelium
discoideum and Myxococcus xanthus colonies, each show
how modelling has been used to shed light on problems
of current interest to biologists.
Getting the stripes right in Drosophila
von Dassow and colleagues made a splash when they
published their analysis of the segment polarity network
in D. melanogaster embryos7. Mathematical modelling
had been used for some time to validate the proposed
function of protein networks that govern developmental
processes; this work represents one of the first instances of a
model that helped its designers to correct their hypothesis
about the form of the protein network itself.
Patterning in the early D. melanogaster embryo is
controlled by a protein regulatory network that guides
the spatial patterns of gene expression in the embryo
over time. The embryonic patterning develops in several
consecutive stages, progressing towards an increasingly
finely striped pattern: maternal morphogens induce the
activation of gap genes, which initiate the expression of
pair-rule genes, which in turn control the initial pattern
of the segment polarity genes. It is the segment polarity
genes and their products that von Dassow and colleagues
considered in their model, including Engrailed (EN),
Wingless (WG), Hedgehog (HH), Patched (PTC) and
Cubitus interruptus (CID). The characteristic periodic
spatial pattern of segment polarity gene expression along
the anterior–posterior axis of the embryo, shown both
diagrammatically and through experimental results in
FIG. 1a,b, is then maintained throughout development.
This pattern is the precursor to segmental development
and provides positional information for subsequent
developmental events, such as the formation of append-
ages. von Dassow and colleagues asked whether the
known interactions among these segment polarity genes
and their products could result in the observed behav-
iour of the embryo during and after the segment polarity
stage. By extrapolating from experimental results, they
developed an influence model that specifies promotion
or inhibition between these segment polarity proteins
(indicated by solid lines in FIG. 1c). They then designed
a mathematical model that encodes this logic in a set of
13 nonlinear ordinary differential equations (ODEs) with
50 parameters representing, among other terms, reaction
rates and cooperativity coefficients. They asked if a set
of parameters exists for which this model, when simu-
lated with the initial expression pattern of these genes,
maintains this expression stably over time. Choosing
large numbers of parameter sets at random from a con-
strained set, they found no matches. But the results that
came close helped them to reason about the original
influence model and recognize the need for additional
interactions. The best pattern that they achieved
consisted of wg and en expressed alternately in every
other cell along lines of cells parallel to the anterior–
posterior axis. To achieve the desired pattern, two other
Box 1 | Differential equation models in developmental biology
Following Turing’s reaction-diffusion equations to model morphogenesis in
developing plants and animals1, Winfree51 showed that a model of similar structure
could produce spiral waves in a two-dimensional medium. This provided early
evidence for a form of chemical interaction that would be picked up again years later
to help explain cell aggregation patterns in developmental biology31. Also following
was the work of Gierer and Meinhardt on local autocatalysis and long-range
inhibition52, and that of Savageau on the development of a form or pattern by
differential growth of the organism53,54. With advances in molecular biology
and genetic manipulation, more data has become available to use in identifying and
testing models. Focused largely on model organisms, mathematical models have
been designed for spatial pattern formation and network analysis of transcriptional
regulation in Escherichia coli19,55, for cell aggregation and patterning in bacteria and
slime mould colonies31,37, for stress response in Bacillus subtilis56, for patterning in the
early Drosophila melanogaster embryo7,9,57 and for the functioning of different
proteins in chick limb development58. New insights have been gained about
patterning during vulval development in Caenhorhabditis elegans from a new model
that explicitly relates parameter changes to different patterns59. Mathematical
models that predict surprising oscillations in the mitogen-activated protein kinase
(MAPK) cascades used in eukaryotic signal transduction13,60 have recently been
validated using clever experiments with individual cells (D. Ippolito, H. Shankaran,
H. Resat, G. Newton, W. Chrisler, L. Opresko and H. Wiley, personal communication).
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Competing interests statement
The authors declare no competing financial interests.
The following terms in this article are linked online to:
Entrez Gene: http://www.ncbi.nlm.nih.gov/entrez/query.
csgA | cubitus interruptus | engrailed | erkB | hedgehog | regA |
Giant | Hunchback | Runt
Claire Tomlin’s homepage:
Jeffrey Axelrod’s homepage:
Access to this links box is available online.
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