Page 1

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

tissue-level effects.

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

patterning.

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

California Berkeley,

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.

e-mail: tomlin@stanford.edu

doi:10.1038/nrg2098

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.

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

REVIEWS

© 2007 Nature Publishing Group

Page 2

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.

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

emergent properties.

A classification of modelling approaches

Mathematical models in development can be classified

along two axes: model architecture and mathematical

implementation.

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).

REVIEWS

332 | MAY 2007 | VOLUME 8

www.nature.com/reviews/genetics

© 2007 Nature Publishing Group

Page 3

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.

biologically feasible interactions were added to the

model. These act to inhibit en in cells that lie anterior

to the stripe of cells expressing wg, and provide a spatial

bias in the response of wg-expressing cells to HH. Once

these were encoded in a modified mathematical model,

a large number of parameter sets for which the model

reproduced the observed expression were identified. Of

course, this does not prove that the modified mathemati-

cal model is correct and, indeed, experimental evidence

has yet to emerge, but it supports the model and can

be used to guide experimental research. Understanding

the mechanisms behind patterning in the D. melangaster

embryo remains a grand challenge in development, with

new data and new data-visualization techniques com-

ing online8. Computational experiments like that of

von Dassow and colleagues are extremely valuable in

highlighting which experiments will be most valuable

to guide the collection of such data.

A range of models

An appealing finding of von Dassow and colleagues’

study is its conclusion that the behaviour of the segment

polarity network is largely dependent on the types of

influences between components, whether positive or

negative, rather than on the details of their rates of react-

ion and strengths of cooperativity. If this is indeed the

case, then a much simpler mathematical model could

be used to simulate the system and symbolically analyse

its behaviour. This point was eloquently reinforced by

Albert and Othmer9, who used von Dassow and col-

leagues’ results to design a discrete-state model of the

segment polarity network, in the form of a Boolean rep-

resentation, which only requires knowledge of whether

components are absent or present rather than their actual

concentrations. Because the rules by which signals are

updated are so simple in this model, it can be mathemat-

ically analysed to discover its invariants, or properties

that are always true. By analytically determining all of

the steady state patterns that could be produced by the

model, Albert found only six distinct ones, and showed

that the system converges to these correct patterns for a

large range of incorrect initial patterns. She concluded

that the network is remarkably robust.

As discussed above, von Dassow and colleagues

changed their original model in response to the dif-

ficulty of finding a corresponding parameter set. In

general, determining the necessary changes can be a

time-consuming and frustrating exercise, even with spe-

cific qualitative knowledge of how the system functions.

Taking a step back from the details of the interactions, it

is therefore informative to ask whether there are general

guiding principles about the behaviour of different net-

work topologies that explain what it is about the change

that made von Dassow and colleagues’ new model that

much more robust. In a later paper, von Dassow and

Odell showed that an important difference in the new

model is the existence of two positive feedback loops, one

for each of en and wg10. They also showed that this model

could be represented as an interaction of simple dynamic

elements, including a number of multistable switches,

so named for their ability to achieve multiple stable

expression states in response to a single external input.

Ingolia took this analysis one step further, showing that

multistability is a necessary property for correct func-

tioning of segment polarity11. He used this to constrain

the steady-state values of the solutions to von Dassow

Box 2 | Model architectures

The first axis in model classification for developmental systems is model architecture. Noble’s breakdown of the

different strategies for formulating model architectures includes ‘bottom-up’ and ‘top-down’ approaches 5. The bottom-

up approach starts with the properties of the component parts and their interactions, and higher-level processes are

constructed by assembling these detailed components. This approach suggests that, to understand the function of a

system, it is first necessary to understand the function of its components and their interactions61–65. It is therefore the

approach that is advocated by the exciting new field of synthetic biology66. In his review of this field, which he helped to

create, Endy advocates an ordered, hierarchical design procedure that relies on standard biological parts67; he and

others have used such a design procedure to successfully engineer biological circuits68–70. Although this principled

approach is probably the only way to build a foundational systems science for systems biology, the challenge is to

manage the computational complexity of a bottom-up approach for most biological systems; to answer questions about

the function of biological systems we often require a model that describes higher-level interactions than those that

occur between the component parts that have been categorized today.

According to the top-down approach, one starts with a high-level functional model of the entire system and then

successively replaces each functional block with a model of the mechanism that implements it. Because a high-level

system description might be implemented with many lower-level mechanisms5, the challenge here is to determine the

mechanisms involved and to understand how close they are to those that the biological system of interest actually uses.

It is not surprising that recent top-down approaches that use, for example, network or control theory to suggest an

overall system model, also couple this with experimentally obtained details of the underlying components21,71,72. For

example, Alon and colleagues73 have advocated the use of network motifs in top-down modelling as a way to

characterize the basic building blocks that make up a biological network. Each block has a single function, such as a

feedforward, in which a first transcription factor regulates a second, and then they both regulate one or more operons,

or a single input module, in which a transcription factor regulates itself as well as a set of operons.

In both the bottom-up and top-down approaches, the model is designed from experimental data and existing

knowledge of the biological system, and the components and their interactions are modelled at the specific level of

abstraction that is appropriate to answer the question being asked about the system. Ideally, a successful model

should suggest new experiments that could be used to extend and refine the model.

REVIEWS

NATURE REVIEWS | GENETICS

VOLUME 8 | MAY 2007 | 333

© 2007 Nature Publishing Group

Page 4

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.

and colleagues’ second model, and in doing so derived

constraints on the free parameters, thereby providing

a simple analytic test of whether a parameter set would

make the segment polarity network function correctly.

This work drew motivation from similar deconstruc-

tion of multistability in other biological systems12–14, and

demonstrates one of several powerful tools from applied

mathematical analysis and control theory that have been

used to help understand the behaviour of ODE models of

protein regulatory networks15–22. For more about models

and intuition, see BOX 4.

Modelling diffusion

Sometimes, a simple numerical model can serve to both

demonstrate the feasibility of a particular mechanism

and help us to understand its properties. In a combi-

nation of modelling and experimentation, Gregor and

colleagues recently studied the action of the maternal

morphogens that control an earlier developmental stage

of the D. melanogaster embryo23. The morphogen Bicoid

(BCD) diffuses along the anterior–posterior axis of the

embryo to form a gradient, and cells along this axis

take on different fates depending on the level of signal

they receive. This morphogen therefore helps set up

the patterning that is necessary for the segment polar-

ity network to function. Although it has generally been

assumed that gradients of morphogens arise through

diffusion, it was not clear how identical, scaled patterns

across different species with embryos of vastly different

size (FIG. 2a) could arise from the relationship between

length and time that is imposed by diffusion. In order

to test the diffusion hypothesis, Gregor and colleagues

injected an inert, fluorescently tagged molecule that

mimicked BCD at the anterior pole of the embryo. They

then measured the ensuing concentrations over time at

different spatial points in the embryo (FIG. 2b). Working

at a level between the biochemistry and its system-level

effect, they designed a simple partial differential equation

(PDE) model to describe the change in protein concen-

tration24 as a result of diffusion over space and time as

well as decay due to protein lifetime. They then solved

this PDE numerically over time on a three-dimensional

grid representing a regular array of points on and inside

the embryo. The only free parameter in this model is the

diffusion constant, which they determined for different

species by fitting the model results to the experimental

data. With the diffusion constant instantiated, the model

agreed exceedingly closely with the data for each species.

Interestingly, there was little relative change in diffusion

constants over the species’ different-sized embryos (the

largest embryo length is more than four times the small-

est), and there is little difference in the developmental

timescales of these species. Guided by their model

and its computational results, Gregor and colleagues

concluded that the most plausible explanation for the

scaled identical patterns is that the only other degree

of freedom, the lifetime of BCD itself, changes from

species to species. This remarkable hypothesis, which

emerged as a result of the modelling, remains to be

tested experimentally.

From model to experiment

Similar classes of simple, intuitive models have been

used to help decode the function of the mechanisms

that control patterning in other organisms. One notable

success has been the use of modelling to understand a

striking social behaviour in Dictyostelium discoideum,

a common slime mould. When nutrients are plentiful,

D. discoideum cells grow and divide as individual amoebae

but, when the cells are deprived of food, they aggregate

into large groups of cells called ‘slugs’. The slugs can move

around as a colony25 and eventually form structures

resembling microscopic water towers, each with a stalk

and a head of hardy spores (FIG. 3a). When food again

becomes available, the spores germinate and form new

amoebae. Dictyostelium discoideum therefore provides

a model system for studying cell to cell communication

and cell differentiation. For differentiation, progress has

been made in understanding how the cells determine

which among them will become stalk cells and which

will germinate to populate the next generation26. At the

same time, great strides have been made in understand-

ing how cells communicate with each other to initiate the

aggregation process. For this, mathematical modelling

has played a key part.

Building on experimental observations of D. discoideum

cell aggregation and growing knowledge of the signal

transduction network by which cells communicate to

Box 3 | Mathematical implementation

The second axis in model classification is mathematical implementation. The most

frequently chosen mathematical models of developmental processes have been those

described by differential equations that represent growth or decay in biochemical

concentration over time as continuously varying signals1. These equations include

parameters, such as decay rates, rates of reaction and diffusion, and activation

thresholds, which are usually unknown and must be identified using the available data.

The process of parameter identification requires solving the model’s differential

equations: in most cases, a guess is made for the numerical values of the parameters,

the equations are solved using numerical algorithms, and the resulting model is

simulated and compared with experimental data. If the results do not match, a new

guess is made and the process continues. The inability to find a suitable parameter set

might indicate that the equations are not a correct representation of the system; in

such cases, the simulation results might help question the original hypothesis from

which the model was derived and inform changes to this model. Thus, this approach

can lead to testable quantitative or qualitative predictions that distinguish between

alternative biological models. In general, this process can be computationally

intensive, especially for large numbers of parameters and variables. However, savings

can be made by using efficient search methods74 or by optimization-based techniques

(R. Raffard, K. Amonlirdviman, J.D.A. and C.J.T., unpublished observations).

Discrete-state models present an alternative mathematical implementation. These

include Boolean models75,76 and Bayesian networks77 that are used to represent

relationships, influences and interactions of cellular components as graphs or

automata. In general, discrete-state models are more abstract system representations

than differential equations, because they ignore kinetic properties that describe how

the system changes over time. Therefore, they cannot be used to identify parameters

that depend on time, such as growth and decay rates. They can be used to help

determine dependencies and causal relationships between variables in the model.

Hybrid systems, which incorporate both continuous and discrete dynamics, have

been used to model both the logic of influences and the dynamics of biochemical

growth and decay. By modelling the interactions between components using

discrete-state models, the continuous dynamics could be represented as simple

differential equations that can be solved analytically for ranges of numerical

parameter values, thereby simplifying the parameter identification problem78,79.

REVIEWS

334 | MAY 2007 | VOLUME 8

www.nature.com/reviews/genetics

© 2007 Nature Publishing Group

Page 5

a

wg

ptc

CID

WG

PTC

PTC-HH

HH

EN

cid

+

hh

en

CN

HH

PTC-HH

PTC

WG

c

A P

wg

ptc

cid

CID

CN

hh

PTC–HH

b

en

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.

initiate this group behaviour27–30, Levine and colleagues

sought to understand how a small, localized group of

cells could communicate over a relatively vast distance

to attract other cells which would eventually join their

construction effort31. It is well known that D. discoideum

cells secrete a diffusible chemical component, cyclic

adenosine monophosphate (cAMP)32,33, and that the

cells have membrane receptors for cAMP which, when

stimulated, activate an internal regulatory network that

both causes the cell to produce cAMP and inactivates the

membrane receptor for a period of time34,35. At the same

time, this triggers another mechanism inside the cell that

induces the extension of pseudopodia used for chemotaxis

up the concentration gradient of cAMP. Within a colony,

this behaviour results in waves of cAMP that propagate

in large spirals, and the cells in the colony begin to move in

the direction of the wave source (FIG. 3b,c).

Levine and colleagues sought to understand how an

initially disordered spatial structure of cAMP, produced

by random emissions from the individual cells, becomes

the characteristic large spirals of cAMP, the centres of

which define the points of aggregration. Building on the

work of Tang and Othmer36, Levine and colleagues con-

structed a model to describe the relationship between

individual cells and the spatial concentration of cAMP.

In this model, the ambient concentration of cAMP at

each point in space is governed by a simple ODE that

models the rate of change of cAMP concentration over

time as affected by its diffusion and constitutive decay, as

well as by a periodic source of cAMP provided by indi-

vidual D. discoideum cells in the environment. When

the ambient cAMP concentration exceeds a threshold,

these cells become excited and emit cAMP for a fixed

time period. Following this, they enter an inactive

Figure 1 | The segment polarity network. a | Segment polarity gene expression in a Drosophila melanogaster

embryo, showing expression of wingless (wg; in green) and Engrailed (EN; in red). wg expression is detected from a

wg–lacZ fusion gene, from which β-galactosidase is expressed under the control of the wg promoter. Both

β-galactosidase and EN were detected with antibodies in the fixed embryo. b | Segment polarity gene expression

model results from von Dassow and colleagues7. A segmented region of the embryo is boxed (above) and expanded

(below; several complete parasegments, each four cells wide, are shown) to depict the striped steady-state

expression patterns of the segment polarity components (lower-case labels indicate mRNA and upper-case labels

indicate proteins): en, wg, patched (ptc), cubitus interruptus (cid), hedgehog (hh), repressor fragment of Cubitus

interruptus (CN), patched–hedgehog complex (PTC–HH). A, anterior; P, posterior. c | Influence model inside and

between cell boundaries, showing interactions among the genes in von Dassow and colleagues’ model of segment

polarity in D. melanogaster. Solid lines indicate their original model; their modified model additionally incorporates

the dashed lines. Arrowheads indicate positive interactions and barred lines indicate negative interactions.

The + beneath cid indicates constitutive expression. Shading indicates the cell interior. Image in part a courtesy of

S. Vincent, Stanford University School of Medicine. Part b reproduced and part c modified with permission from

Nature REF. 7 © (2000) Macmillan Publishers Ltd.

REVIEWS

NATURE REVIEWS | GENETICS

VOLUME 8 | MAY 2007 | 335

© 2007 Nature Publishing Group