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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

resolution.

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

(CISBAN), Newcastle

University, Newcastle upon

Tyne, Tyne and Wear

NE1 7RU, UK.

e‑mail: d.j.wilkinson@ncl.ac.uk

doi:10.1038/nrg2509

Published online

13 January 2009

Continuous deterministic

mathematical model

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

pre-determined mathematical

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

biological systems

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.

Modelling

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Stochastic model

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

the model.

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 1

a

Time (hours)

Fluorescence

Cell 2Cell 3Cell 4

Cell 5Cell 6

1234

Time (hours)

010 203040

Time (hours)

Time (hours)

010203040

Time (hours)

02468

0510 15 2520

Time (hours)

30

Time (hours)

0 515 2535 0510 1525 20

0

100

200

300

Fluorescence

0

20

40

60

80

100

Fluorescence

Fluorescence

0

20

40

60

80

0

20

10

30

50

40

Fluorescence

Fluorescence

0

50

150

100

0

20

40

60

Fluorescence

0

20

60

40

b

Cell 7

p53

MDM2

ATM

p14ARF

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.

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Michaelis–Menten

A simple kinetic law that

modifies the rate of conversion

from substrate to product

based on enzyme

concentration.

Hill kinetics

A more complex enzyme

kinetic law than simple

Michaelis–Menten kinetics.

Ordinary differential

equation

A mathematical equation

involving differential calculus.

In simple cases, explicit

formulas can be derived for

their solution, but typically

they must be numerically

integrated on a computer.

Probability theory

The mathematical theory of

chance, randomness,

uncertainty and stochasticity.

Markov jump process

A class of stochastic processes

that is well studied in

probability theory and that

includes the class of processes

described by stochastic

chemical kinetics.

Stochastic chemical kinetics

A chemical kinetic theory

which recognizes that

molecules are discrete entities,

and that reaction events occur

at random when particular

combinations of molecules

interact.

can be regarded as small well-stirred containers, thus

ignoring spatial effects, and describe the dynamics of the

process of interest using a set of biochemical reactions10.

The differences between stochastic and deterministic

approaches relate to the assumptions made regarding

the nature of the kinetic processes associated with the

reaction network.

deterministic models

The classical approach to chemical kinetics is to assume

that reactants are abundant and have a level measured

on a continuous scale, traditionally in units of concen-

tration. In the p53 example, reactants will be proteins

and complexes such as p53, MDM2, p53–MDM2,

phosphorylated p53, as well as the mrNA molecules

that encode the different proteins. The state of the

system at any particular instant is therefore regarded

as a vector (or list) of amounts or concentrations.

furthermore, the changes in amount or concentration

are assumed to occur by a continuous and determin-

istic process. The velocity of each reaction is specified

using a rate equation that typically assumes mass action

kinetics or is based on an enzyme kinetic law (such as

Michaelis–Menten or Hill kinetics)11. The way in which the

state of the system evolves can be described mathemati-

cally (by using ordinary differential equations (BOX 1)). In

certain simple but usually not biologically realistic

cases, these equations can be solved to give an explicit

formula that describes the time course trajectory. In

more complicated scenarios, such as the p53 example

described above, computational methods are used

that provide only approximate (but typically accurate)

solutions.

Although some deterministic models of the p53–

MDM2 system have been proposed in the litera-

ture6,7,12,13, they are unsatisfactory for several reasons.

The most fundamental limitation of deterministic

models is that they inevitably fail to explain the highly

noisy and heterogeneous observed cellular response to

DNA damage. The obvious lack of agreement between

the model and the data cannot be attributed to genetic

or environmental effects, as these have been largely

eliminated by the experimental design. It is therefore

difficult to make any sensible assessment of the extent

to which such models explain the observed data.

Another limitation of deterministic models is that they

do not span multiple scales. Either the model oscillates

(as suggested by the single-cell assays), or it has a sin-

gle peak in p53 expression (as observed in population

level data). It is difficult to reconcile these two obser-

vations without accepting a heterogeneous cellular

response, and in practice this involves introducing sto-

chasticity into the models. by contrast, an essentially

stochastic model based on the known biochemical

mechanisms has recently been described14. This sim-

ple model shows that the heterogeneity observed in the

experimental data is entirely consistent with intrinsic

stochasticity in the system; it also has the property that

the population average of the p53 levels of many single

cells over time has the observed single peak in p53

expression.

Modellers aim to find simple explanations for a

range of complex and sometimes apparently contradic-

tory experimental observations. Here, a single simple

mechanistic model simultaneously explains how p53

levels can oscillate and why they do not oscillate in

some cells, the origins of stochasticity and heterogene-

ity in the cellular response, and the apparent conflict

between the single-cell data and population level data.

No comparatively simple deterministic model can do

this. furthermore, because the stochastic model exhib-

its a similar range of behaviour to the experimental

data, it becomes meaningful to try and make a seri-

ous assessment of how well such a model matches the

experimental data, and to try and use the experimental

observations to improve our knowledge about the model

parameters15.

Stochastic models

The continuous deterministic approach to modelling

biochemical reaction networks fails to capture many

important details of a biological process and the experi-

mental data that relates to this process. The ‘missing

detail’ manifests itself as a degree of apparent unpredict-

ability of the system. As a result, the single-cell dynam-

ics of biological systems seem noisy, or stochastic, with

these terms being used more or less interchangeably.

Heterogeneity is then a phenotypic consequence for a

cell population given stochastic single-cell dynamics.

Stochasticity and heterogeneity are aspects of model

biological system behaviour that cannot be ignored, and

attempts to refine experimental techniques to eliminate

them are both hopeless and misguided2,16,17.

There are multiple sources of stochasticity and het-

erogeneity in biological systems, and these can, and

often do, have important consequences for understand-

ing overall system behaviour. Stochasticity influences

genetic selection and evolution18,19; biological systems

have also developed strategies for both exploiting20 and

suppressing18 biological noise and heterogeneity21. Any

useful predictive model of the system must therefore

account for a degree of intrinsic unpredictability.

The only satisfactory quantitative modelling frame-

work that takes into account the inherent unpredictabil-

ity of a system is based on probability theory. Statistical

mechanical arguments are used to understand the proba-

bilistic behaviour of the dynamic stochastic process asso-

ciated with the biochemical network. The dynamics of a

biological system can be modelled using a Markov jump

process, whereby any change in the system occurs dis-

cretely after a random time period, with the change and

the time both depending only on the previous state22–25.

This is a well-understood model from the theory of sto-

chastic processes. It has been known for decades that

this framework can be applied to the simulation of sto-

chastic chemical kinetics26, but it did not become a well-

established approach in biology until the late 1990s27,

when experimental techniques became precise enough

to show that experimental findings could be mod-

elled accurately only by incorporating stochasticity28.

Stochastic modelling has a long tradition in other areas of

biological modelling, including population dynamics24,29.

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Box 1 | A simple model for protein production and degradation

Consider the following artificial model for production and degradation of a single protein, X: the protein is produced at a

constant rate α, and each protein molecule is independently degraded at a constant rate µ. This can be written using

chemical reaction notation as:

??→??

?

??→??

µ

α

Let the number of molecules at time t be denoted Xt, and assume that there are initially no protein molecules, so that

X0 = 0. The plots show the case α = 1, μ = 0.1. The parameters are purely illustrative and not intended to model any real

biological system.

The first plot shows the standard reaction rate equation (RRE) model. Although in this case this model captures the

essential ‘shape’ of the discrete stochastic (Markov jump process) model, shown in the second plot, it completely ignores

the substantial variability. The final plot shows the chemical Langevin equation (CLE) model. Although this model

sacrifices discreteness, it effectively captures both the shape and variability of the discrete stochastic solution, despite the

low copy numbers involved. Note that although in the case of this simple model, the equilibrium means (and, indeed,

time-varying means) of the stochastic models match the deterministic model, in general this is not the case. The plots for

the stochastic models show a single realization of the process, based on independent noise processes.

0

5

10

20

15

0204060 80100

Time

X(t)

0

5

10

20

15

02040 6080100

Time

X(t)

0

5

10

20

15

020406080 100

Time

X(t)

Nature Reviews | Genetics

Solution:

??

???? ??? ?

µα

µ

µ

µα

µ

α

µ

α

????????????????? ??

µ

?????? ?

Continuous deterministic model (RRE):

?????????????????????????????? ????

?????????????????????????????? ????

Solution:

Equilibrium:

?????????????????????????? ??

α

?????????????? ? ?

µα

?????????????????? ?µα

µα

Continuous stochastic model (CLE):

Discrete stochastic model:

??????? ??? ????????????? ??? ???????

µαµα

At equilibrium:

Equilibrium distribution:

?????????????????? ?

Probability distribution

A precise mathematical

description of a stochastic

quantity.

As in the deterministic case, some simple network

models are analytically tractable. In these simple situ-

ations, the full probability distribution for the state of the

biological system over time can be calculated explic-

itly. However, as for the deterministic case, the class of

solvable models is small, mainly covering those models

that contain only single-molecule reactions. As almost

all interesting systems involve interactions between

molecules of different types, these simple models do not

cover systems of genuine practical interest. Here, too,

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Stochastic simulation

algorithm

In the context of stochastic

chemical kinetics, this refers to

an exact discrete event

simulation algorithm for

generating time course

trajectories of chemical

reaction network models.

Monte Carlo error

The unavoidable error

associated with estimating

a population quantity from a

finite number of stochastic

samples from the population.

It can often be reduced by

averaging large numbers of

samples.

Intrinsic noise

A crude categorization of

stochasticity in biological

systems that loosely

corresponds to noise that

cannot be controlled for.

numerical simulation of the process on a computer is the

key tool used for understanding the system behaviour.

for Markov jump process models, an algorithm known

in this context as the stochastic simulation algorithm (but

more commonly known as the Gillespie algorithm)

is used to generate exact realizations (or ‘runs’) of the

Markov jump process26. The algorithm generates time

course trajectories of the system state over a given

time window, starting from a given initial system state

(BOX 2). Of course, these realizations are stochastic, and are

therefore different for each run of the simulation model.

They are ‘exact’ in the sense that each run is an independ-

ent realization from the true underlying process; proper-

ties deduced about the probabilistic nature of the process

from multiple runs can be made arbitrarily accurate by

averaging over a sufficient number of runs to reduce the

Monte Carlo error associated with the estimates. Accessible

introductions to methods of stochastic simulation

for reaction networks can be found in REFs 25,30,31.

Modelling intrinsic noise. Once stochastic models are

created, they allow a range of investigations that are not

possible using deterministic models. Modelling and

experimental investigation of noise at the single-cell

level in isogenic cell populations is currently the subject

of many active research programmes. Stochastic model-

lers acknowledge the fact that molecules are discrete enti-

ties, and that reactions between molecules are stochastic

events, which typically occur when molecules collide

according to random processes. The state of the system

at a given instant is therefore regarded to be a vector of

counts of molecules, and remains constant until the next

reaction event occurs. for the p53 example, this means

that the state will be the actual number of molecules of

p53, MDM2 and so on present in the cell; and this number

will not change continuously, but will remain constant

until changing abruptly each time a reaction involving

those molecules occurs. for example, a p53–MDM2

binding event might occur when a p53 and MDM2 mol-

ecule collide at random in the cellular environment. The

implications for the system state will be that the number

of molecules of p53 and MDM2 will decrease by one,

and the number of p53–MDM2 complexes will increase

by one. Although it might seem reasonable to view the

molecular dynamics of cellular processes as essentially

deterministic, models concerned only with molecu-

lar counts do not explicitly consider the position and

momentum of every single molecule, and so the timings

of reaction events are essentially unpredictable.

Intrinsic noise in biochemical reactions has many com-

ponents, including randomness of promoter binding

and other DNA binding events, stochasticity in mrNA

transcription and degradation processes, stochasticity of

translation and protein degradation events, and random-

ness of other protein–protein and protein–metabolite

interactions. Stochastic models allow investigation of the

intrinsic variability of the cellular process of interest. for

example, did the system evolve to suppress noisy gene

expression? Such suppression could be achieved using a

variety of techniques, including utilization of a carefully

tuned signalling cascade32. Alternatively, has the sys-

tem evolved to exploit noise? Several examples of noise

exploitation are known for the Gram-positive bacterium

Bacillus subtilis (BOX 3). Stochasticity in gene expression

has been especially well studied in yeast (BOX 4), but

it has also been observed and modelled in mammalian

cells14,33. Experimental technology has developed to

the extent that, in special cases, one can even observe

stochasticity at the single-molecule level34.

A common and well-studied example of noise exploi-

tation is provided by bistability in a reaction network in

conjunction with randomness of expression: this frame-

work allows a single cell to select one of two phenotypic

traits at random, with a probability that is specific to the

network and to its associated initial conditions35. This

allows organisms to express phenotypic heterogene-

ity even in uniform genetic and environmental condi-

tions, and can have selective advantages. It is difficult to

investigate such issues using deterministic models.

Other sources of heterogeneity. There are, of course,

sources of variation in cellular systems other than intrin-

sic stochastic kinetic noise in biochemical reactions that

should be incorporated into the models if they are to

describe cell population behaviour effectively. One is

randomness or uncertainty in the initial state of the

biological system. for example, cells in a particular

experiment might behave differently because they were

different at the start of the experiment — perhaps hav-

ing different rNA and protein levels — even if they are

genetically identical.

One way to incorporate this uncertainty into the

analysis is to construct simulations by first simulating

initial conditions from a specified probability distribu-

tion, and then carrying out the stochastic simulation

Box 2 | outline of the gillespie algorithm

The Gillespie algorithm is used to simulate stochastic time course trajectories of the

state of a chemical reaction network. The essential structure of this discrete event

simulation algorithm is outlined below.

•?Step 1: set the initial number of molecules of each biochemical species in the reaction

network and set the simulation time to zero.

•?Step 2: on the basis of the current molecular abundances, calculate the propensity for

each possible reaction event.

•?Step 3: using the current propensities, simulate the time to the next reaction event,

and update the simulation time accordingly (the larger the reaction propensities, the

shorter the time to the next event).

•?Step 4: pick a reaction event at random, with probabilities determined by the reaction

propensities (higher propensities lead to higher probability of selection), and update

the number of molecules accordingly.

•?Step 5: record the new simulation time and state.

•?Step 6: check the simulation time. If the simulation is not yet finished, return to step 2.

To give an explicit example, consider using the Gillespie algorithm to generate a

realization from the simple model described in the discrete stochastic model of BOX 1,

which considers the production and degradation of a molecule. At each point in the

simulation, the time to the next reaction event is simulated (and the expected time to

wait will be shorter the more molecules there are), and a decision will need to be made

as to whether the reaction should be a synthesis or a degradation event, with the

probability of degradation increasing as the number of molecules in the system

increases. For an accessible introduction to the Gillespie algorithm, and stochastic

modelling for systems biology more generally, see REF. 25.

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