Systems biology1 aims to move beyond the study of
single biomolecules and the interaction between
specific pairs of molecules; its goal is to describe, in
quantitative terms, the dynamic systems behaviour of
complex biological systems that involve the interaction
of many components. Traditional reductionist genetic
and molecular biology approaches have yielded huge
amounts of data, but understanding how low-level bio-
logical data translates into functioning cells, tissues and
organisms remains largely elusive. Now that life scien-
tists possess an extensive ‘parts list’ for biology, we can
begin to think about how the function of a biological
system arises from dynamic interactions between its
parts. As even simple dynamic systems can exhibit a
range of complex behaviour, such an approach requires
quantitative mathematical and statistical modelling
of biological system dynamics. At the level of cellular
modelling, this ideally requires time course data on the
abundance of many different biomolecules at single-cell
Traditionally, systems dynamics have been described
by using continuous deterministic mathematical models.
However, it has recently been acknowledged that bio-
chemical kinetics at the single-cell level are intrinsically
stochastic2. It is now generally accepted that stochastic
models are necessary to properly capture the multiple
sources of heterogeneity needed for modelling biosys-
tems in a realistic way. However, such models come at
a price; they are computationally more demanding than
deterministic models, and considerably more difficult to
fit to experimental data.
Statistics is the science concerned with linking models
to data, and as such it is absolutely pivotal to the success
of the systems biology vision. Statistical approaches to
inferring the parameters of deterministic and stochas-
tic biosystems models provide the best way to extract
maximal information from biological data. Effective
methods for statistically estimating stochastic models
by using time course data have only recently appeared
in the systems biology literature; these techniques are the
final piece of the puzzle needed to describe biological
dynamics in a quantitative framework.
This article reviews the key issues that need to be
understood to describe biological heterogeneity prop-
erly, the approaches that have been used and the range of
problems that they solve, together with the most promis-
ing avenues for further development. Many of the exam-
ples in the literature concern single-celled organisms such
as bacteria and yeast; however, heterogeneity is present in
all biological systems, and separating intrinsic stochast-
icity from genetic and environmental sources3 is likely to
become increasingly important in the context of human
genetics and complex diseases in the near future.
Basic modelling concepts: a working example
One of the principal aims of systems biology is to test
whether our understanding of a complex biological
process is consistent with observed experimental data.
As dynamic systems exhibit complex behaviour, our
understanding must be encoded in quantitative mathe-
matical models. A lack of consistency between the model
and the data indicates that further research is required to
School of Mathematics &
Statistics and the Centre for
Integrated Systems Biology
of Ageing and Nutrition
University, Newcastle upon
Tyne, Tyne and Wear
NE1 7RU, UK.
13 January 2009
A model that does not
contain any element of
unpredictability, and that
describes the smooth and
gradual change of model
elements (such as biochemical
substances) according to
rules. The precise behaviour of
the model is entirely
pre-determined (and hence, in
principle, predictable) from the
form of the equations and
the starting conditions.
Stochastic modelling for quantitative
description of heterogeneous
Darren J. Wilkinson
Abstract | Two related developments are currently changing traditional approaches to
computational systems biology modelling. First, stochastic models are being used
increasingly in preference to deterministic models to describe biochemical network
dynamics at the single-cell level. Second, sophisticated statistical methods and
algorithms are being used to fit both deterministic and stochastic models to time
course and other experimental data. Both frameworks are needed to adequately
describe observed noise, variability and heterogeneity of biological systems over a
range of scales of biological organization.
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© 2009 Macmillan Publishers Limited. All rights reserved
A model that contains an
element of unpredictability or
randomness specified in a
precise mathematical way.
Each run of a given model will
produce different results, but
the statistical properties of the
results of many such runs are
pre-determined by the
mathematical formulation of
complete our understanding of the system under study.
Consistent models can be used to make further testable
predictions for more independent validation, and also to
carry out in silico investigation of the system behaviour
that would be difficult or time consuming to do entirely
in the laboratory. These concepts can be illustrated
using the example of oscillations and variability in the
well-characterized p53–MDM2 system.
The human tumour suppressor protein p53 (encoded
by the TP53 gene) is a transcription factor that has an
important role in regulating the cell cycle, tumour sup-
pression and DNA damage response4. Population level
data showed only a single peak in p53 expression, fol-
lowed by decay back to basal levels. More recently, how-
ever, single cell assays in MCf7 breast cancer cell lines
have revealed that levels of p53 sometimes seem to oscil-
late in response to radiation-induced DNA damage5–7.
for example, the Alon laboratory measured p53 and
MDM2 levels in single cells over time using two fluo-
rescent reporters7. FIGURE 1a shows clearly a highly het-
erogeneous cellular response despite some evidence of
p53 and MDM2 oscillations.
Oscillations are indicative of negative feedback in the
system dynamics. We would therefore like to understand
the underlying mechanisms, and to test that understand-
ing by developing quantitative and predictive models
of the system behaviour. The essential feedback feature of
this system is well known: p53 activates transcription
of MDM2, a ubiquitin E3 ligase, which in turn binds
to p53 and thereby enhances its degradation8,9. The sig-
nal for p53 activation can come from more than one
source. In MCf7 cell lines, which do not express the
cyclin-dependent kinase inhibitor p14Arf (also known
as CDKN2A), the strongest signal probably comes from
the kinase ATM (ataxia telangiectasia mutated), which
is activated by DNA damage; ATM phosphorylates both
p53 and MDM2, blocking their binding to each other
and enhancing MDM2 degradation, thereby allowing
accumulation of active p53 (FIG. 1b).
Many systems biology models are concerned with
intracellular processes, and therefore operate (concep-
tually, at least) at the level of a single cell. Most stochastic
and deterministic models for chemical reaction network
kinetics make the assumption that cellular compartments
Cell 2Cell 3Cell 4
Cell 5Cell 6
0510 15 2520
0 515 2535 0510 1525 20
Figure 1 | Fluctuations in p53 and MDM2 levels in single cells. a | Image analysis can be used to extract time courses
of expression levels from time-lapse microscope movies. The plots show the measured fluorescence levels for seven
individual cells from one particular movie (movie 2 in data from REF. 7, provided by the authors); the tumour suppressor
protein p53 is represented by blue circles, and the ubiquitin E3 ligase MdM2 is represented by yellow circles. Although
there is some evidence of p53 and MdM2 oscillations, there is clearly a highly heterogeneous cellular response. b | The
essential interactions between p53, MdM2, and key signalling molecules ataxia telangiectasia mutated (ATM) and
the cyclin-dependent kinase inhibitor p14ARF (also known as CdKN2A). p53 activates transcription of MdM2. MdM2
then binds to p53, thereby enhancing its degradation8,9. p53 can be activated by the kinase ATM, which is activated by
dNA damage; ATM phosphorylates p53 and MdM2, this prevents the binding of p53 to MdM2 and enhances MdM2
degradation, thereby allowing accumulation of active p53. MdM2 can also be inactivated by p14ARF.
NATurE rEVIEWS | Genetics
VOluME 10 | fEbruAry 2009 | 123
© 2009 Macmillan Publishers Limited. All rights reserved
integrated model of this nature that experimental data
at the whole-system level can be used effectively to esti-
mate (that is, calibrate) model parameters and to assess
the adequacy of the model. Therefore, completion of the
iterative cycle of modelling and experimentation that is
central to the systems biology approach actually requires
integrated stochastic models of whole-system behaviour
(see REFs 100,101 for a promising example). The develop-
ment of such integrated multiscale models will require
significant developments in stochastic simulation tech-
nology. The use of fast, approximate stochastic simula-
tors will be necessary, as will the development of new
techniques for simulating multiscale stochastic models.
It is likely that statistically estimated stochastic emula-
tors will be used for some model components in cer-
tain situations to reduce the computational demands of
the algorithms. Techniques for running large stochastic
simulation models on high-performance computing
facilities will also require development.
The second area of this article concerned statistical
estimation of network structure and model parameters.
Here the challenges ahead are similarly formidable.
reliable simultaneous inference for network structure
and kinetic parameters from a combination of high-
throughput time course data and fine-grained time
course data is a clear short-term goal. In the medium
term, the development of techniques for effective cali-
bration of large multiscale integrated stochastic models
of complex biological systems is a key objective. In both
cases sophisticated statistical methods will be required,
and the problem structures make a bayesian approach
the obvious choice. It is therefore likely that we will see
a bayesian ‘revolution’ in computational systems biol-
ogy, similar to that already experienced in genetics102 and
The scientific community must recognize the piv-
otal role of statistics and statisticians in systems biol-
ogy research. No serious genetics laboratory or clinical
trials unit would be considered complete without at least
one expert statistical modeller. The contribution that a
statistician can make to the success of a systems biology
laboratory is every bit as great, but owing to the histori-
cal development of this new discipline, this fact has not
been widely appreciated.
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The author would like to thank three anonymous referees for
numerous suggestions that have helped to improve this arti-
cle. This work was funded by the Biotechnology and Biological
Sciences Research Council through grants BBF0235451,
BBSB16550 and BBC0082001.
ATM | MdM2 | p14ARF | p53
Darren J. Wilkinson’s homepage:
Biology of ageing e‑science integration and simulation
system (BASIS): http://www.basis.ncl.ac.uk
BioModels database: http://www.ebi.ac.uk/biomodels
FERN — A Java Framework for Stochastic Simulation and
Evaluation of Reaction Networks:
Systems Biology Markup Language (SBML):
All links Are Active in the online pDF
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