Handbook of the Biology of Aging, Sixth Edition
Copyright © 2001 by Academic Press. All rights of reproduction in any form reserved.
328
I. Introduction
A. The Why, What, and How of
Biological Modeling
Three intersecting processes are making
the application of mathematical and
computer modeling increasingly import-
ant in the biological sciences. First, biol-
ogy itself has become much more of an
informational science, as a result prima-
rily of the development of genomic
(based on advances in gene sequence and
expression data) and post-genomic (based
on advances in proteomic and functional
data) sciences. Our capacity to answer
questions ranging from cell and molecu-
lar function through to evolutionary
genetics requires an increasing ability
to acquire, store, and manipulate large
volumes of raw data. This requirement
has called upon biologists to develop
the necessary computational skills and
understanding.
Second, there is a realization that
complex biological processes cannot be
understood through the application of
ever-more reductionist experimental
programs alone. There needs to be some
integration of the mass of data and
insight from study of the detailed mecha-
nisms at the level of the physiological
“system.”
Third, the sophistication and power of
desktop computer hardware has increased
to a point where the kind of model that
two decades ago might have required an
overnight run on a large mainframe com-
puter can now be done in the individual
scientist’s lab or office with a response
time that makes possible a much more
interactive way of working.
Alongside these changes is the develop-
ment of a different perception of the role
and value of computer modeling in bio-
medical research. To many scientists who
have trained and worked in environments
where modeling has not been a part of the
scientific toolkit, the nature and scope of
computer modeling is still unclear. Many
see models as essentially descriptive, beg-
ging the question “Why bother?” when
the real answer will be revealed in time
by experiment. Others have been indoc-
trinated with the widespread—but largely
Chapter 12
Computer Modeling in the Study of Aging
Thomas B. L. Kirkwood, Richard J. Boys, Colin S. Gillespie, Carole J. Proctor,
Daryl P. Shanley, and Darren J. Wilkinson
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CHAPTER 12 / Computer Modeling in the Study of Aging 329
false—idea that as soon as a model has
more than two or three parameters, it can
“explain” anything, resulting in a suspen-
sion of belief that models, particularly of
complex systems, can be of any real use
at all. Fortunately, the increasing dialog
between modelers and experimentalists is
beginning to break down these barriers of
misunderstanding and giving rise to new
interactions that are likely to change the
way a great deal of science will be done in
the coming decades. This new approach
is commonly being described as systems
biology. This chapter reviews how com-
puter modeling is developing within the
context of the biology of aging.
The distinctive advantages of modeling
a biological process with the rigor that is
needed to build a computer model are as
follows:
1. Model building requires that verbal
hypotheses be made specific and concep-
tually rigorous. Before a mathematical
model can be formulated, the investigator
must specify each element of the model
and how it interacts with other elements.
2. Starting to build a computer model
may help to highlight gaps in current
knowledge. The process of specifying a
mathematical model will highlight any
important unknowns. Sometimes these
can be represented as variables yet to be
estimated or determined.
3. The process of model development
might lead to the recognition of a gap
that needs to be filled by further
experimental investigation, which may
be fundamental to understanding a
complex system. Thus, modeling can be
useful even if the gap means that a
model cannot yet be completed.
4. Computer models yield quantita-
tive as well as qualitative predictions.
A hypothesis can be tested much more
rigorously by a model that permits quan-
titative predictions to be made. In aging,
where multiple mechanisms might be at
work, it often happens that data are
broadly consistent with a hypothesized
mechanism, but modeling can show that
the magnitude of the effect is too small
to explain aging on its own.
5. Modeling can result in improved
experimental design, especially where the
system embodies the potential for
complex interactions. Complexity is very
hard to deal with experimentally but is
relatively straightforward in a computer
model. Models are thus ideal for analyzing
complex interactions prior to experimen-
tal tests. In extreme cases, modeling may
actually reveal that because of interac-
tions within complex systems, a proposed
experiment would be inconclusive.
6. Modeling can provide a low-cost,
rapid test bed for candidate interventions,
thereby enabling a more predictive
approach and effecting significant savings
in time and money.
To the non-modeler, the science of bio-
logical modeling can easily appear to
involve the application of the same set of
skills to a very diverse range of problems,
and in much the same kind of way.
A facility with numbers, knowledge of
computer programming, and some under-
standing of the biological system seem
all that is required. In reality, the range
of approaches and skills in computer
modeling is broad, involving a significant
diversity of skills and research subdisci-
plines. Later sections of this chapter
examine the different kinds of computer
modeling, how they are performed, and
what they can achieve. If the integration
between theoretical and experimental
science is to take place as fast and as
effectively as is needed, researchers from
both communities will need to learn
more about each other’s methods of
working. Curiously, there is more simi-
larity between the methods of working of
experimenters and modelers than is usu-
ally recognized. The experimenter has to
(1) decide which factors to include and
vary in the study in order to address the
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330 T. B. L. Kirkwood et al.
hypothesis most efficiently and directly;
(2) spend a large amount of the total
effort of the study on controls in order to
reduce the possibility of artifact; (3) be
careful in framing the conclusions from
the study so as not to extrapolate beyond
what the results support; and (4) be con-
scious of planning the study within a
constrained budget of time, money, and
human resources. So does the modeler.
A number of texts expand on some of
the issues raised here (e.g., Hilborn &
Mangel, 1997).
In parts of this chapter, we describe the
mathematical, statistical, and computa-
tional approaches that have been brought
to bear on understanding the aging
process. Because these approaches will
be unfamiliar to some readers, we have
explained the terms and basic concepts
as clearly as possible. It is not feasible,
however, to include all of the explana-
tion that would be necessary to equip the
reader new to these approaches with a
complete knowledge base. We have
therefore had to find a balance between
explanation and concision. At all rele-
vant points within the text, we give
references to texts where the reader can
find more detailed explanation.
B. How Computer Models Have Been
Instrumental in Solving Biological
Problems
In many biological domains, it is difficult
to see how a clear understanding of key
processes could be gained without math-
ematical modeling. This includes, for
example, the study of population dynam-
ics in ecology, disease transmission in
epidemiology, and population genetics
and life history theory in evolutionary
biology. In other domains, single exam-
ples exemplify the insights that mathe-
matical investigation can provide, such
as in the study of cardiac fibrillation
in physiology. Modeling is making an
increasingly important contribution in
the relatively new field of systems
biology, which aims, in part, to bridge
molecular biology and physiology by
capitalizing on the large amount of post-
genomic data currently being generated.
Many biochemical networks involve
nonlinear components, which means
that relying on intuition is not reliable.
In a recent essay, Lander (2004) provides
an excellent example of how modeling
has helped reveal the details of a molecu-
lar mechanism that was proving difficult
to understand from a purely experimen-
tal approach: the role of the segmenta-
tion polarity genes in maintaining the
segmentation pattern during Drosophila
development (von Dassow et al., 2000).
Although details differ, a similar process
is shared by all insects, and to a lesser
degree in vertebrates.
Drosophila embryonic development
can be described as a three-stage process.
In the first stage, maternally expressed
mRNA enters the Drosophila oocyte
within the ovary, and following transla-
tion, a polarized protein gradient is
established. In the second stage, gap
and coordinate genes are expressed in
response to the protein gradient, which
in turn govern the periodic expression of
pair-rule genes. The final stage involves
the segment polarity genes. A repetitive
pattern of engrailed (en) and wingless
(wg) expression is established based on
the pair-rule genes. The embryo then
undergoes cellularization, and the pat-
tern of en and wg expression is trans-
ferred to an intracellular context involv-
ing signals between morphologically
distinct bands. The gap and pair-rule
gene products fade, en and wg expression
is maintained via a complex network of
transcription factors and intracellular
signals, and the segmented structure is
retained during substantial morphologi-
cal change.
Von Dassow and colleagues (2000) con-
structed a mathematical model based on
current information about the segment
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CHAPTER 12 / Computer Modeling in the Study of Aging 331
polarity gene network to test whether it
could maintain a stable segmented
structure. The structure was represented as
a series of connected cells, each cell popu-
lated with the focal mRNAs, proteins, and
protein complexes. The known intracellu-
lar and intercellular relationship between
interacting molecules was then repre-
sented as a series of differential equations,
with simulation used to provide informa-
tion on the temporal variation in concen-
tration of all the molecules. Although a
substantial body of information existed to
construct the network, it was insufficient
to fully quantify the model—50 parameters
were unknown. The approach taken was
to run simulations with many randomly
chosen parameter sets, with each para-
meter bounded within realistic limits.
Interestingly, no solution could be found
that satisfactorily reproduced the observed
stable segmented pattern.
Attention was then turned to the
network structure, and it was realized
that modifications were needed to cap-
ture the known biology, namely the
asymmetry in signaling to neighboring
cells anterior and posterior. With appro-
priate modifications in place, the differ-
ential equations were updated and the
process of simulation with randomly
chosen sets of parameters was repeated.
The observed segmented structure was
now reproduced with surprising ease.
The important result was that it was the
fine detail of the network structure that
was key in determining the system
behavior and not exact parameter values.
Interestingly, this may reflect an evolu-
tionary adaptation as minor variations,
such as mutations in components of the
network or environmental fluctuations
affecting levels of signals, would not
unduly affect the segment structure and
subsequent development. The major
message was that the segment polarity
genes represent a “robust developmental
module” that ensures the formation of
an appropriate pattern even across
distantly related insect species in which
earlier stages of development differ.
II. Why Aging Particularly
Needs Models
Recent years have seen rapid progress
in the science of aging. A key factor in
this progress has been the interaction
between evolutionary (why?) and mecha-
nistic (how?) lines of research, which
gives shape to the likely genetic basis of
aging and to the mechanisms that may
be involved (Kirkwood & Austad, 2000).
This has helped overcome a situation
where the field was dominated by a
plethora of rival theories, with little
effective dialog between them. In partic-
ular, the disposable soma theory
(Kirkwood, 1977; Kirkwood & Austad,
2000) suggests that aging is caused
ultimately by evolved limitations in
organisms’ investments in somatic main-
tenance and repair rather than by active
gene programming. This predicts that
aging is due to the gradual accumulation
of unrepaired random molecular faults,
leading to an increasing fraction of dam-
aged cells and eventually to functional
impairment of older tissues and organs.
Genetic effects on the rate of aging are,
in this view, mediated primarily through
genes that influence somatic mainte-
nance and repair.
Although the idea of aging as a buildup
of damage is straightforward in principle
and supported by a growing range of data,
it presents a number of distinctive chal-
lenges (Kirkwood et al., 2003). First, it pre-
dicts that there are multiple mechanisms
that cause aging, instead of just one or a
few. Second, it predicts that aging is inher-
ently stochastic—that is, it is modulated to
an important degree by chance. Extensive
evidence points to an important contri-
bution in aging that arises from chance
variations, which are not explained by
genetic or environmental factors (Finch &
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332 T. B. L. Kirkwood et al.
Kirkwood, 2000). A particularly clear
example of the role of chance in aging
is the threefold range in life span (see
Figure 12.1) and the apparently stochastic
age-related cell degeneration of individ-
ual worms in isogenic populations of
Caenorhabditis elegans reared under
uniform laboratory conditions (Herndon
et al., 2002; Kirkwood & Finch, 2002).
Third, since multiple mechanisms con-
tribute to aging, a high level of complexity
is to be expected. For all of these reasons,
there is exceptional need in aging research
for the use of computer models to help
integrate findings from different lines of
experimental work.
Although the multiplicity of aging
mechanisms is now widely acknowl-
edged, the reductionist nature of
experimental techniques means that, in
practice, most research is still narrowly
focused on single mechanisms. This is
where computer modeling can make a
major contribution. By allowing for
interaction and synergism between dif-
ferent processes, models reveal that
the predicted effects on the system are
often much greater than when mecha-
nisms are considered one at a time.
Furthermore, models can highlight
important differences between the
upstream mechanisms that set a process
in train and the end-stage mechanisms
that dominate the cellular phenotype
at the end of its life. For example, a
gradual accumulation of mitochondrial
(mt)DNA mutations, occurring over
years, might lead to a steady increase in
the production of reactive oxygen
species (ROS) and a gradual decline in
energy production (Kowald & Kirkwood,
1996) . However, although the buildup of
mtDNA mutations initiates the process,
what ultimately destroys the cell is that
eventually a threshold is reached where
homeostatic mechanisms collapse. The
end-stage of the cell’s life span is domi-
nated by dramatic biochemical changes,
such as an accumulation of damaged
protein. Experimental study of the latter
effect, or even of the former cause, in the
absence of a quantitative model to link
the two would find it hard to establish
the connection.
Another benefit of integrative model
building is that it is well suited to take
account of the fact that many of the key
reactions involved in normal cell mainte-
nance and metabolism do not act in
isolation—rather, they belong to a net-
work of activity. When the activity of
one enzyme changes, all connected
metabolite pools and enzyme activities
may be altered. In some cases, there may
be redundancy in pathways, which pro-
vides buffering against damage, whereas
in other cases, the effect of damage may
be propagated.
Another important area for modeling is
to understand the actions of genes that
affect the rate of aging. Over the past
decade, scores of genes have been identi-
fied that affect aging in yeast, nematodes,
fruit flies, and mice, and there is growing
interest in genes affecting human
longevity (Gems & Partridge, 2001;
Jazwinski, 2000; Larsen, 2001; Lithgow,
1998; Tan et al., 2004). Experimental
data are beginning to reveal the interac-
tions of these genes within pathways
0
5
10
15
20
25
30
35
0 10 20 30 40 50
Days
N
o
dy
in
g
Figure 12.1 Life-span distributions for individual
Caenorhabditis elegans nematodes in isogenic pop-
ulations of wildtype (filled bars) and age-1 (open
bars) strains. Redrawn from Kirkwood and Finch
(2002); original data from Johnson (1990).[AU1]
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CHAPTER 12 / Computer Modeling in the Study of Aging 333
that control the aging rate, and there is
evidence that several of the most import-
ant genes are those that affect basic
cellular processes, such as insulin and
insulin-like growth factor (IGF) signaling,
which are strongly conserved across the
species range (Gems & Partridge, 2001;
Rincon et al., 2004). Nevertheless, we are
a long way from understanding the inter-
actions between these effects. These
studies need also to take account of the
intrinsic stochastic nature of gene regula-
tory networks.
III. Different Approaches to
Modeling Biological Systems
A. Descriptive Versus Predictive
A descriptive model describes a process
or behavior that has already been
observed. A predictive model predicts the
behavior of a system not previously
observed. A valid descriptive model is
often easier to develop but it has less
value than a model with predictive
power. However, descriptive models are
useful for highlighting gaps in our cur-
rent knowledge. A model may start out
as being descriptive but can then be used
to predict outcomes when parts of the
system are perturbed. For example, a
descriptive model of a metabolic path-
way of proteins under known conditions
can be used to predict protein functions
under different circumstances or in
different species.
A predictive model provides quantifi-
able as well as qualitative predictions.
The value of quantifiable predictions is
that a hypothesis can be tested much
more rigorously. In aging, where numer-
ous mechanisms might be at work, data
are often broadly consistent with a
hypothesized mechanism, but modeling
can show that the magnitude of the
effect is too small to explain aging on
its own. Another advantage of predic-
tive models is that they can provide a
low-cost, rapid test bed for candidate
interventions.
B. Simple Versus Complex
Biological systems are complex and
involve the interrelationships of many
different “species,” where species can
refer to molecules, cells, tissues, or
organisms. A model is a description of
the system. The “art” in building a good
model is to capture the essential details
of the biology, without burdening the
model with nonessential details. Every
model is to some extent a simplification
of the biology, but it is valuable in taking
an idea that might have been expressed
purely verbally and making it more
explicit. Nevertheless, the question still
remains: what level of complexity should
be incorporated in the model?
At the most basic level, a model must
be able to capture the desired inputs and
outputs of a system. This is where a clear,
prior specification of the problem to be
addressed is as essential in modeling as it
is in experimentation. For instance, if we
wish to investigate how an increase of
ATP will affect the production of ROS by
mitochondria, then obviously these ele-
ments must be included into the model.
Other elements, such as the dynamics of
a cell cycle, are likely to be excluded.
However, it is at this point that the mod-
eler needs to exercise caution and to bear
in mind the opportunities that are avail-
able to include further factors in the
model than could easily be added to an
experiment. The choice of which reac-
tions should be left out of the model—
since the activity of one enzyme may con-
ceivably affect all connected metabolite
pools—has no easy or universal answer.
Some modelers prefer to start simple and
add further detail as required; others pre-
fer to recognize greater complexity from
the outset. Either way, a modeler should
always develop a model with as much
direct biological input as possible.
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334 T. B. L. Kirkwood et al.
Another key factor in the modelers’
decision-making process is the time
scale of processes involved. Molecular
processes often need to be modeled on a
time scale of seconds or less; outcomes
affecting aging develop in months or
years; evolutionary changes occur over
generations. Models that seek to integrate
across levels present particularly challeng-
ing problems that need to be addressed in
defining the aims and scope of the project.
When modeling a process as complex
as aging, an unfortunate side-effect is
that very quickly, the mathematical
representation can become exceedingly
complex. Although the modeler may be
comfortable with each and every detail of
the model, the reader may be presented
with an indecipherable collection of
symbols. Conversely, an oversimplifica-
tion of the systems may lead to the
justified claim that the model does not
represent the structure under considera-
tion. Thankfully, part of this problem is
being overcome with the introduction of
standard methods for describing models
used throughout the biological commu-
nity, such as the Systems Biology Markup
Language (SBML, described in more
detail later). When a standard has been
decided, this enables generic tools to be
developed that aid the understanding of
models. For instance, an SBML-aware
visualization tool should accept any
SBML-encoded model and return a graph-
ical representation of it.
A barrier that limits the amount of
complexity that can be included in a
model is computational power. Put sim-
ply, do we have a computer powerful
enough to calculate a solution to our
model? It is relatively easy to construct a
simple model that when simulated could
take weeks to finish. With the yearly
increase in processor power, models that
would have taken weeks of computa-
tional time 5 years ago can now be solved
in a matter of minutes. Other exciting
avenues include the emergence of the
GRID—the new generation of hardware/
software computer networking that is
designed to facilitate the sharing of data
and compute resources over a network.
A benefit of the GRID is the harnessing
of idle computer power. For instance,
whereas a model may take weeks on a
single 500-MHz processor, if 50 machines,
say in a university computer laboratory,
which are idle for 12 hours per day were
set to the task, a properly formulated
model could take hours.
C. Discrete Versus Continuous
When modeling biological processes, it is
often helpful to treat time as a discrete
quantity divided into a number of inter-
vals. For instance, when dealing with
the cell cycle of the budding yeast
Saccharomyces cerevisiae, it may natu-
ral to deal in terms of generations
(Gillespie et al., 2004; Sinclair, 2002).
Although some types of system natu-
rally lend themselves to discrete-time
modeling, it is important to consider any
distortion that may be introduced. In the
yeast cell cycle example, a mother cell
produces on average 24 daughter cells;
however, the time taken to form a
daughter cell gradually increases. So if
the events being modeled were directly
affected by the interbudding interval, the
model may be of limited validity if only
discrete generations were considered.
D. Deterministic Versus Stochastic
A model can be generally classed as deter-
ministic or stochastic. A deterministic
model is one that takes no account of
random variation and therefore gives a
fixed and precisely reproducible result.
It can be solved by numerical analysis
or computer simulation. Deterministic
models are often mathematically
described by sets of differential equations.
Deterministic models are appropriate
when large numbers of individuals of a
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CHAPTER 12 / Computer Modeling in the Study of Aging 335
species are involved and the importance
of statistical variations in the average
behavior of the system is relatively unim-
portant. However for many biological sys-
tems, this assumption may not be valid.
To illustrate the concepts, let us con-
sider perhaps the simplest of molecular
reactions, spontaneous degradation. The
ordinary differentiation equation for the
degradation of a species X at rate k1 is
given by
(1)
Because this is a simple equation, it
can be solved exactly to give the deter-
ministic solution
(2)
where X(0) is the initial amount of
species X.
A stochastic model should be used
when either the number of a particular
species is small or when there is reason
to expect random events to have an
important influence on the behavior of
the system. Often, a stochastic model
will be more appropriate when we need
to take account of species as discrete
units rather than as continuous variables,
and particularly when the numbers of a
particular species may become small. It
may also be necessary to take account of
events occurring at random times. The
essential difference between a stochastic
and deterministic model is that in a sto-
chastic model, different outcomes can
result from the same initial conditions.
A stochastic model is formulated in
terms of probabilities and is constructed
by considering the probability that an
event occurs during a small time period.
Formulating the model in this manner
enables us to calculate the probability
that the population is of size X at time t,
px(t). Because the model is reasonably
simple, the exact stochastic solution can
be obtained,
X(t)
X(0)ek1t
dx
dt

k1X .
A stochastic model is formulated in
terms of probabilities, so at each time
interval, the degradation of species X has
an associated probability. Again, since
the model is reasonably simple, the exact
stochastic solution can be obtained:
(3)
where X takes the values between 0 and
its initial amount X(0).
In most biological systems, the number
of species involved and the interactions
between them mean that for stochastic
models, an analytical solution—that is,
one that can be obtained by purely alge-
braic formulae without using a com-
puter—will not be feasible. In these cases,
computer simulations of the stochastic
kinetics are used. A simulation keeps
track of the number and state of each
species over time. Therefore, it is neces-
sary to carry out repeated simulations and
then look at the distribution of results to
get a picture of the central tendency, the
dispersion, and outliers. This process is
called Monte Carlo simulation.
Figure 12.2 shows a stochastic real-
ization of a spontaneous degradation
Px(t)



X(0)
X


ek1Xt(1ek1t)X(0)X
0
10
20
30
40
50
60
70
80
90
100
0 10 20 30 40 50 60 70 80 90 100
Time
N
o.
o
f m
ol
ec
ul
es
Deterministic
Stochastic 1
Stochastic 2
Stochastic 3
Figure 12.2 Comparison between a stochastic real-
ization and the deterministic solution for a simple
degradation reaction. In this stochastic realization,
the molecules are degraded quicker than predicted
by the deterministic solution.[AU3]
Chapter 12 1/4/04 10:27 PM Page 335

336 T. B. L. Kirkwood et al.
reaction and its deterministic counter-
part. For a given starting condition, a
single stochastic realization may differ
considerably from its deterministic coun-
terpart. This is particularly important in
models when the number of a particular
species becomes low and the species may
or may not become extinct. Figure 12.2
shows a clear example of a species reach-
ing a very low concentration but never
becoming totally extinct in the deter-
ministic model, whereas in the stochas-
tic model, the species becomes extinct
and the time of extinction varies in
different realizations. In the modeling of
epidemic diseases within a host popula-
tion, where it may matter greatly
whether and when the first or last infec-
tive individual dies or recovers, the
difference between stochastic and deter-
ministic models can be very marked.
Similar considerations can arise in gene
regulatory networks with respect to
the random association of transcription
factor complexes.
Both the deterministic and stochastic
methods have their respective advan-
tages and disadvantages. The modeler
should determine which method is more
suitable to the task at hand (it may
sometimes be both) and use that which
is appropriate.
E. Software Tools
There are many ways to develop a model,
from using traditional programming
languages such as C, Fortran, and Java
to mathematical packages such as
Mathematica, Matlab, and R. Conven-
tional publication of computer models is
generally restricted to the presentation of
a few key predictions, although it is com-
mon to allow the reader to download the
computer program, or source. In order to
replicate the same predictions as were
published, the program must first be
downloaded and any necessary algorithm
libraries installed. To actually use the
model for further investigation generally
requires a significant degree of comput-
ing skill.
As noted above, recently, increasing
effort has been made to agree on a
standard method on representing mathe-
matical models. One standard that has
been widely adopted is the Systems
Biology Markup Language (SBML). SBML
provides a computer-readable format
for representing models of biochemical
reaction networks. Although SBML is
human-readable, it is intended that it
will usually be other software that would
both read and write any models. This is
analogous with the now widespread use
of HTML for Web documents. Although
a human can read HTML source docu-
ments, these are intended primarily for
reading by a browser, such as Internet
Explorer, that transforms the HTML
code into a more easily read document
on screen.
Currently there are over 60 groups
using the SBML standard (see Hucka
et al., 2003). Some tools, such as
CellDesigner, have been created that
enable models to be constructed using a
drag-and-drop approach. Using this
approach, a user creates species that are
assigned to graphical nodes. The nodes
are then connected up using arrows to
denote reactions (see for example
Funahashi et al., 2003).
Other tools, such as JigCell, allow the
user to construct a model using chemical
equations combined with a spreadsheet
approach (Allen et al., 2003). Using this
approach is perhaps more useful when
dealing with large and complex models,
whereas the graphical approach is partic-
ularly useful when constructing a model
for the first time.
SBML is not the only standard that has
emerged. For example, the Petri Net
Markup Language (PNML) and CellML
are similar efforts in creating standards
(see Lloyd et al., 2004; Weber & Kindler,
2003). Although each standard focuses on
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CHAPTER 12 / Computer Modeling in the Study of Aging 337
different aspects of the model, they are
not mutually exclusive. Hence, an effort
is being made to create tools that allow
the transformation from one language to
another.
F. Validation of Models
Once we have constructed a model and
are satisfied with its behavior, we need to
test the model against observations from
the biological system that it represents.
This process is called validation. First, it
is necessary to test the model to see
whether it fits the data that we already
have. Any discrepancies here need to be
addressed. Once the model has been
validated in this way, we should then
test the model against data that were not
used to estimate parameters for our
model. If the model predictions and the
observed data are not in close agreement,
then the modeler needs to study the
model to try and find where the discrep-
ancies arise. This could mean modifying
the model or adding further detail to the
model. This is an important step as it
may highlight that the current knowl-
edge of the system is insufficient and
that further experimental work should be
carried out. Once modifications to the
model have been made, the model is
tested again, as shown in Figure 12.3.
Another aspect of validation is sensitiv-
ity analysis to assess how varying model
parameters affect the model outcomes.
There are two reasons why this is useful.
First, we may be interested in some partic-
ular parameters—for example, the rate of
degradation of a protein and how this
might affect the buildup of damaged pro-
tein. Second, some of the parameter rates
might not have been accurately deter-
mined and so it is important to see how
sensitive the model is to small changes in
these. The size of changes to make for each
parameter depends on how well the para-
meter was determined initially. In the case
of parameters that have been estimated
from data, a multiple of the standard error
is appropriate. For parameter values that
have been guessed, a guess at the percent-
age reliability is also required. Caution
needs to be taken when parameter esti-
mates are correlated because if one para-
meter estimate is changed, some of the
others might have to be changed too.
IV. Currently Available Models
of Aging
Existing work reflects the variety of cur-
rent models in aging research, which
range from detailed modeling of individ-
ual intracellular mechanisms to higher-
level modeling required to address the
fundamental problem of why aging
should occur. We will not attempt to be
wholly comprehensive in this review but
will illustrate the breadth of coverage and
the different methodologies employed.
A. Intracellular Mechanisms
There are a large number of models cur-
rently available that focus on individual
intracellular mechanisms. Currently, the
Model
building
Model
validation
Using the
model
Extending/
refining the
model
Experimental
data
Figure 12.3 Flow diagram to show the steps in
building a model. Once a model has been built, it is
tested against experimental data. If the model does
not agree well, then the modeler goes back to the
model-building stage. Otherwise, the model can be
used and then extended further as more knowledge
becomes available.
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338 T. B. L. Kirkwood et al.
models that relate directly to aging have
been mainly concerned with telomere
shortening, the accumulation of somatic
mutations, and the accumulation of
defective mitochondria.
1. Telomere Models
Telomeres are repetitive DNA sequences
found at both ends of linear chromo-
somes. In telomerase-negative cells,
telomeres shorten with each cell divi-
sion, and this process eventually causes
cells to enter a state of replicative senes-
cence. One cause of telomere shortening
is the end-replication problem caused by
the inability of DNA polymerases to
replicate a linear DNA molecule to its
very end. In the 1990s, models were
developed to try to explain replicative
senescence in human fibroblasts based
solely on the end-replication problem
(Arino et al., 1995; Levy et al., 1992).
Later models included additional mecha-
nisms of telomere shortening. Rubelj and
Vondracek (1999) modeled abrupt telom-
ere shortening due to DNA recombina-
tion or nuclease digestion. It has been
found that an increase in oxidative stress
accelerates the rate of telomere shorten-
ing due to an accumulation of single-
strand breaks in telomeric DNA (von
Zglinicki et al., 1995; von Zglinicki
et al., 2000). More recent models have
included this additional mechanism
and found that the models predict that
oxidative stress plays an important role
(Proctor & Kirkwood, 2002, 2003). For
example, simulations showed that
increasing the level of ROS led to fewer
cell divisions on average. Space does not
permit detail of all of the current models
of telomere shortening, but the inter-
ested reader may refer to the references
for further models (Aviv et al., 2003; den
Buijs et al., 2004; Golubev et al., 2003;
Hao & Tan, 2002; Olofsson & Kimmel,
1999; Sidorov et al., 2004; Tan, 1999a,b,
2001).
2. Somatic Mutations
The role of somatic mutations in aging
is an area of particularly active research,
following new methods for measuring
DNA modification and repair. The
somatic mutation theory was first pro-
posed several decades ago after experi-
ments showed that irradiation shortened
life span in animal models and induced
features of premature aging (Henshaw
et al., 1947; Lindop & Rotblat, 1961).
Szilard (1959) proposed a mathemati-
cal model that assumed that recessive
mutational “hits” in diploid organisms
would accumulate so that a cell could
continue to function until one pair of
genes had both received a “hit.”
Holliday and Kirkwood (1981) developed
a deterministic model of the accumula-
tion of recessive mutations in human
fibroblast populations. A stochastic
model of the same processes was later
developed (Kirkwood & Proctor, 2003),
which also considered the possibility
that there may be synergistic inter-
actions between mutations.
3. Mitochondria Models
The free-radical theory of aging proposes
that ROS, which are constantly gener-
ated through normal cell metabolism in
the mitochondria, cause aging by damag-
ing membranes, proteins, and DNA
(Harman, 1956). The mitochondrial
theory of aging proposes that an accumu-
lation of defective mitochondria is a
major contributor to the cellular deterio-
ration that underlies the aging process
(Harman, 1972). Studies have shown that
defective mitochondria accumulate with
age to a greater extent in post-mitotic tis-
sues (Cortopassi et al., 1992; Lee et al.,
1994), although it has recently been
reported that high levels of mitochondr-
ial defects are observed in aged human
colon (Taylor et al., 2004). In addition,
several studies have shown that muscle
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CHAPTER 12 / Computer Modeling in the Study of Aging 339
fibers are taken over by a single form of
mutant mtDNA (Brierley et al., 1998;
Müller-Höcker et al., 1993).
Hypotheses to explain the apparent
“clonal expansion” of mutant mtDNA are
a replication advantage for the mutant
mtDNA; slower degradation of mutant
mitochondria (de Grey, 1997); and random
intracellular drift. Mathematical models
have been developed to explore quantita-
tive predictions from these ideas. Kowald
and Kirkwood (2000) developed a deter-
ministic model based on de Grey’s
hypothesis. Other models are based on
the idea of random intracellular drift
(Chinnery & Samuels, 1999; Elson et al.,
2001).
4. Chaperone Models
Molecular chaperones have an important
role in helping to maintain protein
homeostasis within cells. It has been
observed that the induction of heat
shock proteins, a major class of chaper-
ones, is impaired with age and that there
is also a decline in chaperone function.
Although there are a few mathematical
models on the role of heat shock pro-
teins in the cell, to date only one model
has looked at the role of chaperones in
the aging process (Proctor et al., 2005).
This model describes how heat shock
proteins are upregulated after an
increase in intracellular stress and can
be used to investigate the effect of stress
on protein homeostasis.
5. Network Models
A few models exist that show how differ-
ent mechanisms interact synergistically
(Kowald & Kirkwood, 1994, 1996; Sozou
& Kirkwood, 2001), examples being the
interactions of defective mitochondria,
aberrant proteins, free radicals, and scav-
engers in the aging process (Kowald &
Kirkwood, 1996); and the interactions of
telomere shortening, oxidative stress,
and somatic mutations in nuclear and
mitochondrial DNA (Sozou & Kirkwood,
2001). We are currently engaged in a
major effort, the Biology of Ageing
e-Science Integration and Simulation
(BASIS) project, to develop interactive
models that can network a variety of
individual processes together in a flexi-
ble, user-friendly manner (Kirkwood et
al., 2003). One of the aims of the BASIS
project is to allow models of individual
mechanisms to be linked together to
form a “Virtual Aging Cell” (Proctor &
Kirkwood, 2003).
B. Tissue Models
The functional properties of an aging
organ or tissue can become compro-
mised, even if most of the cells are in
good working order. Mathematical
models are required to help us try to
understand how a fraction of damaged
cells can lead to altered tissue function.
Early models were motivated by the fact
that cultured human diploid fibroblasts
cannot be grown indefinitely in culture
(Hayflick, 1972). These models were
based on the commitment theory, the
idea that cells become irreversibly com-
mitted to senescence while still out-
wardly healthy (Holliday et al., 1977;
Kirkwood & Holliday, 1975).
Recently, extensive experimental data
has been generated on intrinsic age
changes that affect the function of intes-
tinal stem cells in aging mice (Loeffler
et al., 1993; Martin et al., 1998a,b). This
has led to a number of mathematical
models (e.g., Gerike et al., 1998; Loeffler
et al., 1993; Meineke et al., 2001).
Another model based on data from
muscle-derived stem cells has also been
developed (Deasy et al., 2003).
Another tissue system that has been
extensively studied and modeled is the
population of T cells and their role in
immunosenescence (Luciani et al., 2001;
Romanyukha & Yashin, 2003).
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340 T. B. L. Kirkwood et al.
C. Organism Models
There are only a very limited number of
models dealing with whole-organism
aging, and these are limited to unicellular
organisms. Budding yeast, Saccharomyces
cerevisiae, is commonly used to study
cellular aging. Accumulation of extrachro-
mosomal ribosomal DNA circles (ERCs)
appears to be an important contributor to
aging in yeast, and a mathematical model
has been developed to examine this
process (Gillespie et al., 2004). Another
interesting model contrasts regulatory
with stochastic processes in genetic segre-
gation during division as a mechanism for
the aging observed in the asexually repro-
ducing ciliate Styloncychia (Duerr et al.,
2004).
D. Population Models
An area of research that has contributed to
a fundamental understanding of why aging
occurs is life-history theory—a theory that
essentially deals with schedules of growth,
survival, and reproduction maximizing
Darwinian fitness (Kirkwood & Austad,
2000; Kirkwood & Rose, 1991; Partridge &
Barton, 1993). Many classic life-history
papers included an investigation of senes-
cence (Cole, 1954; Fisher, 1930; Hamilton,
1966; Williams, 1957). Another modeling
approach is represented by the disposable
soma theory (Kirkwood, 1977; Kirkwood
& Rose, 1991), founded on the principles
of optimality theory (Parker & Maynard
Smith, 1990). The disposable soma theory
suggests that aging arises as part of an
optimal life history due to tradeoffs in
resource allocation between investment in
reproduction and maintenance affecting
long-term survival. There is much data in
general support of the existence of such
tradeoffs, including in humans (Lycett
et al., 2000; Westendorp & Kirkwood,
1998). More complex general life-history
models that have incorporated measures
of an organism’s state such as size in addi-
tion to age have been analyzed using tech-
niques such as dynamic programming,
simulated annealing, and Pontryagin’s
maximization principle (Abrams &
Ludwig, 1995; Blarer & Doebeli, 1996;
Cichon, 1997; Clark & Mangel, 2000;
Houston & McNamara 1999; Schaffer,
1983; Teriokhin, 1998; Vaupel et al., 2004).
The following specific issues have
attracted attention using approaches and
modeling techniques drawn from the
domains of life-history theory, demogra-
phy, and population genetics.
1. Dietary Restriction
Dietary restriction is observed to cause
slowing of aging and extension of life in
many species. One hypothesis is that ani-
mals have evolved a response to tempo-
rary fluctuations in resource availability,
in which energy is diverted from reproduc-
tion to maintenance functions in periods
of food shortage, thereby enhancing sur-
vival and retaining reproductive potential
for when conditions improve. A detailed
quantitative development of this hypothe-
sis using a dynamic resource allocation
model revealed that the effect could be
the result of the suggested evolutionary
process provided that the following condi-
tions were satisfied: (1) there is a substan-
tial initial cost to reproduction, and
(2) juveniles are at a disadvantage during
periods of food shortage (Shanley &
Kirkwood, 2000). An alternative approach
is presented using a dynamic energy
budget (van Leeuwen et al., 2002).
Recently, metabolic control analysis has
been used to help identify an increased
proton leak in the mitochondrial inner
membrane as one possible mechanism
whereby ROS production is reduced
(Lambert & Merry, 2004).
2. Negligible Senescence
Species that exhibit negligible senescence
are of particular interest (Finch, 1998) and
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CHAPTER 12 / Computer Modeling in the Study of Aging 341
are well represented in species that con-
tinue to grow after maturation, so-called
indeterminate growth. A dynamic model
that explicitly included size as a state vari-
able predicts that this should be the case,
and indeed predicts that some species
should show negative senescence (Vaupel
et al., 2004). Interestingly, conditions for
non-aging can be found in a general life-
history model in an analysis of vitality: a
term that combines the declining fecun-
dity and increasing mortality characteris-
tic of senescence (Sozou & Seymour,
2004).
3. Gompertz, Mortality Plateaus, and
Heterogeneity
Most species exhibit an exponential
increase in mortality with age that can be
described by the Gompertz model or by
the Gompertz-Makeham model that has
an extension to include extrinsic sources of
mortality (Golubev, 2004). One problem in
the acceptance of this model is the obser-
vation that in large laboratory populations,
mortality rates appear to plateau at later
ages (Vaupel et al., 1998). A number of
models have been proposed to account for
this pattern, such as an evolutionary trade-
off (Mueller & Rose, 1996; Mueller et al.,
2003), a combination of mutation accumu-
lation and pleiotropy (Charlesworth, 2001),
a state-based approach (Mangel & Bonsall,
2004), and individual heterogeneity within
the population (Pletcher & Curtsinger,
1998).
4. Human Menopause
The rapid reproductive senescence asso-
ciated with menopause in human
females occurs well in advance of gen-
eral somatic senescence and poses an
interesting evolutionary problem. The
reproductive life span of human females
is limited, as in almost all other mam-
mals, by a finite pool of oocytes estab-
lished in the developing fetus.
Menopause is manifest when this pool is
near to exhaustion, and mechanistic
models have focused on the follicular
dynamics (Faddy et al., 1992) and prob-
lems of fetal loss that increase in fre-
quency as the oocyte pool is depleted
(O’Connor et al., 1998). Other modeling
has focused on determining whether
at some age mothers may increase their
fitness by diverting investment from
continued reproduction to existing
offspring and grand-offspring (Hawkes
et al., 1998; Lee, 2003; Peccei, 1995;
Rogers, 1993; Shanley & Kirkwood,
2001). To date the results have not been
conclusive, but given the importance of
intergenerational assistance—for exam-
ple, as seen in the two-fold improvement
in mortality for infants with a living
grandmother (Sear et al., 2002) combined
with the particularly high risk of mortal-
ity in childbirth for human females—an
evolutionary explanation remains a clear
possibility and the development and
testing of further models appears likely.
V. Models, Data Collection, and
Experimental Design
Models are developed based on the
collective understanding of the scientific
community regarding the underlying
mechanisms driving the processes of
interest. The related activity of designing
experiments to provide data to falsify or
refine the models is essentially just a for-
malization of “the scientific method.”
However, there are a number of issues
that arise in the context of complex
dynamic models that make this proce-
dure far from straightforward in practice.
Models are typically concerned with
underlying mechanisms that are difficult
or impossible to measure directly through
conventional experimental procedures.
Consequently, they often contain param-
eters (such as various kinds of rate
constants) whose values are not known.
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342 T. B. L. Kirkwood et al.
If a model cannot make predictions
regarding quantities that can (at least in
principle) be measured experimentally,
then it is not falsifiable, and in an impor-
tant sense is not “scientific.” There is
therefore a requirement to develop mod-
els that are predictive, as this then
affords an opportunity to compare the
model predictions with experimentally
determined reality. As well as providing
the opportunity for falsification, this pre-
dictive behavior also potentially allows
“calibration” of model parameters by
finding combinations of parameters that
reduce the discrepancy between the
model predictions and reality.
These kinds of “inverse” problems
have long been recognized in the physical
and engineering sciences, and there is a
large literature concerned with attempts
to solve them. The problem can be under-
stood generally as follows: a complex
system has a range of inputs, and based
on these, produces a range of outputs.
The inverse problem is to find a set of
inputs to the system that closely matches
a given set of outputs (desired target or
experimental observation). In the context
of an attempt to match an experimentally
observed history of a physical system,
attempts to solve the inverse problem are
often referred to as history matching or
calibration, of which more is presented
in the next section. Such calibration tech-
niques are generally applied post hoc,
after the experimental data have been
collected. However, in the context of
biological modeling, there is often an
opportunity to go back to the lab to col-
lect appropriate data for model validation
and calibration.
A question then naturally arises as to
exactly what data should be collected,
and how much. In the context of com-
plex biological models, such experimen-
tal design questions are difficult to
tackle within a formal statistical frame-
work, but it is relatively straightforward
to justify some guiding principles. First,
it is necessary to collect data that are (or
can be) predicted by the model. This may
require model refinement but is neces-
sary; otherwise, there is nothing to link
the model and the data that is collected.
Second, one must collect data correspon-
ding to model predictions that are sensi-
tive to underlying model assumptions
(structural and otherwise). This is neces-
sary on the grounds of falsifiability.
Third, it is necessary to gather measure-
ments that help to answer questions of
key scientific interest. For example, if a
key model parameter of interest is the
degradation rate of a particular protein,
then measurements should be taken on
data that are sensitive to the choice of
rate rather than data that are relatively
robust to this choice. This will ensure
that the data collected provide useful
information. Fourth, enough data should
be gathered to ensure that an adequate
assessment can be made of inter- and
intra-experimental variation; otherwise,
there is no way to be sure that the data
are representative and that the model is
not being calibrated to fit an atypical
data set. If the model being calibrated is
stochastic, more data are probably
required, as it is likely to be necessary to
reliably determine a good approximation
to the full probability distribution of key
observables. These basic principles are
fairly self-evident, but effectively opera-
tionalizing them for a complex dynamic
simulation model is not necessarily
easy. However, there are many excellent
texts on experimental design that can
provide further guidance (see for example
Clarke & Kempson, 1997; Cochran &
Cox, 1992; and Mead, 1988).
VI. Parameter Inference
A. The Calibration Problem
Many approaches have been applied to
the calibration problem within the
traditional engineering context. First, a
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CHAPTER 12 / Computer Modeling in the Study of Aging 343
(deterministic) computer simulator
(model) of the physical (or biological) sys-
tem under study is built, so that for a
given set of inputs, a corresponding set of
outputs may be computed. Next, a sim-
ple measure of “distance” for how far the
output is from the desired match is
defined. The input space is then searched
for a set that minimizes the distance
measure. This optimization problem can
be approached using a variety of tech-
niques for multivariate function mini-
mization—for example, steepest descent,
Newton methods, conjugate gradients,
simulated annealing, genetic program-
ming, and so on (Koza et al., 2001). Such
minimization techniques can work well
if the computer simulator is very fast,
but for a simulator of a large, complex
system, which may take hours or days
for a single run, such naïve approaches
generally fail.
Simple search methods are very waste-
ful of information. Typically, a very small
number of runs are used to decide on a
new “best guess” for the target input
parameters, and then all existing infor-
mation is discarded as the search contin-
ues from this new input set. In contrast,
statistical approaches to the calibration
problem attempt to use all available runs
from the simulator in order to infer a
model for the relationship between the
inputs to and outputs from the simulator.
In this context, it is not necessarily opti-
mal to always evaluate the simulator at
the current “best guess” at the optimal
input set, but instead to evaluate at an
input set that gives the most information
regarding the relationship between the
inputs and outputs in the vicinity of the
predicted optimal set. Thus, such statisti-
cal approaches to calibration need to
combine both non-parametric statistical
inference techniques and experimental
design algorithms in order to effectively
solve the problem (Sacks et al., 1989).
A range of different approaches can be
taken to carrying out statistical inference
for model parameters given data, and
these correspond to different schools of
statistical thought. Classical frequentist
approaches seek to construct estimators
that are a function of the data and have
desirable properties (such as consistency
and lack of bias) under repeated sam-
pling. However, even leaving aside the
serious philosophical objections many
people have to the repeated sampling
framework at the heart of frequentist
inference, there are many practical diffi-
culties associated with applying such
techniques in the context of complex
dynamic models. Consequently, few stat-
isticians would consider a frequentist
framework in this scenario.
Approaches based on the likelihood
function of the data provide a more
powerful and natural way of addressing
the simulation model parameter inference
problem. These can be divided into two
main camps. The first is the maximum
likelihood school, which attempts to be
“objective” by using only the likelihood
function of the data and seeks combina-
tions of parameters, which makes the
data as likely as possible conditional on
those parameters. The likelihood (or log-
likelihood) function is used as a way of
“scoring” the goodness of fit, which can
then be optimized. Although this sounds
straightforward in principle, the likeli-
hood function is typically not analytically
tractable for complex models, and this
introduces a variety of complications.
The second camp is the Bayesian
school, which corrects the conditioning
from data on parameters to parameters on
data, and thereby seeks parameters that
are likely given the data. This is done at
the expense of introducing prior distribu-
tions into the problem but has a range of
benefits as a result. These include the
fact that the resulting framework is fully
probabilistic, and that probabilistic infor-
mation regarding all parameters can be
obtained from the posterior distribution.
The use of priors is also valuable, as they
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344 T. B. L. Kirkwood et al.
help regularize the problem and allow
the modeler to incorporate information
regarding realistic parameter ranges
into the inference algorithm. A further
benefit of the Bayesian framework is that
because it is probabilistic, powerful com-
putational algorithms may be naturally
applied to problems where the likeli-
hood is analytically intractable. Markov
chain Monte Carlo (MCMC) algorithms
(Gamerman, 1997) use stochastic simula-
tion techniques to obtained realizations
from the (complex) posterior distribution,
which are then used to draw inferences
about model parameters. For more infor-
mation about Bayesian inference, see
Bernardo and Smith (2000), O’Hagan and
Forster (2004), and references therein.
B. Statistical Approaches to Simulation
Model Calibration
Although non-Bayesian approaches to
the calibration problem are possible, the
complexity and dimensionality of the
problem, together with the need to incor-
porate available expert prior information
regarding, inter alia, plausible ranges for
rate constants and information on data
quality, mean that a Bayesian approach is
particularly attractive. Typically, (in the
context of deterministic processes), a
model is specified in the following form:
(4)
Here z
(z1, z2, . . . , zn) represents the
available experimental data, obtained
from n different experimental conditions
x1, x2, . . . xn; (xi) is the real behavior of
the biological system under experimental
condition xi; i is the measurement error
associated with the ith experiment;  is a
bias associated with the computer simula-
tor of the biochemical system; (xi, ) is
the result of running the computer simu-
lator under experimental condition xi with
the “perfect” set of calibration parameters
; and (xi) represents model inadequacy
(xi)
(xi,) (xi)zi
(xi) i ,
that is independent of the calibration
issue. In addition to the experimental data,
there will be data y
(y1, y2, . . . , yN)
obtained from N runs of the computer
simulator (where N will typically be larger
than n, even in the case of an expensive
simulator), where
(5)
is the result of the jth computer experi-
ment, xj* is the experimental condition
associated with the jth computer experi-
ment and tj is the set of calibration
parameters associated with the jth com-
puter experiment (Kennedy & O’Hagan,
2001).
Note that within this framework, the
computer simulator of the biological
model is represented by a (deterministic)
function (.,.), which can be evaluated at
any combination of experimental condi-
tions and calibration parameters by run-
ning the simulator with the specified
input. If the simulator were very fast, so
that evaluating (.,.) were cheap, then
standard Bayesian inference techniques
could be used in order to make direct
inferences for  using (1), generating data
of the form (2) as and when required.
However, due to the expense of evaluat-
ing (.,.) for large complex models, (.,.) is
often regarded as an unknown function,
modeled using a Gaussian process. Thus
inference may proceed for  using only
the N computer simulator runs available.
Bayesian inference is typically carried
out using a mixture of analytic direct
matrix computations related to Gaussian
processes together with computationally
intensive techniques, using Markov
chain Monte Carlo (MCMC) methods.
Note that in the context of biological
modeling, choice of the experimental
conditions for the n “wet lab” experiments
will often (though not always) be predeter-
mined. However, the choice of conditions
for the N computer simulator runs will be
at least partly under the calibrator’s control
yj
(x*j, tj)
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CHAPTER 12 / Computer Modeling in the Study of Aging 345
and will be of key importance to the over-
all effectiveness of the procedure. This is a
nontrivial (sequential) experimental design
problem, but limited literature already
exists that will provide guidance in this
area (Craig et al., 1996; Currin et al., 1991;
Kennedy & O’Hagan, 2001; Sacks et al.,
1989).
Many complex computer codes have
the facility to be run at different levels of
sophistication, and hence accuracy. The
BASIS simulator, for example, may be run
in “exact” mode, where the simulation of
the stochastic process used to model a
given biochemical system is “perfect,”
based on a discrete event simulation strat-
egy similar to the Gillespie algorithm.
Such exact simulation procedures are
desirable but are typically very expensive
to carry out. On the other hand, the sys-
tem may also be run in an approximate
mode, based on a time-discretization of
the process, where both the accuracy
of the procedure and the time taken for a
run depend on the size of time-step
adopted. In this case, it can often be
optimal to combine a large number of fast
(but less accurate) runs with a small num-
ber of slow (but accurate) runs in order to
make most efficient use of computer
time. There is already a sizable literature
in this area; see for example Higdon and
colleagues (2003), Kennedy and O’Hagan
(2001), and references therein.
C. Direct Statistical Parameter Inference
The calibration techniques alluded to
above work well in the deterministic con-
text but are not completely straightfor-
ward to extend to the case of stochastic
simulation models. For a stochastic model
of relatively low dimension, it may be pos-
sible to make a direct attempt to carry out
statistical inference for the parameters of
the system given (for example, time
course) experimental data on the system
dynamics. Here, rather than regarding the
simulator as an “unknown function,” the
stochastic process corresponding to the
simulator is modeled directly, and all
aspects of the process that are not
observed are “filled-in” probabilistically
using appropriate MCMC techniques.
Conditional on complete knowledge of the
stochastic process, inference for any rate
parameters driving the system dynamics is
straightforward. The difficulty of such
methods is in the construction of the
MCMC algorithms to fill in the missing
aspects of the stochastic process. This is
very problem-specific and generally
requires a fairly detailed understanding of
the underlying dynamics, including the
likelihood function, as well as experience
in the use of MCMC algorithms. The use
of these techniques for identification of
biological models is still in its infancy, but
see Boys and colleagues (2004), Gibson &
Renshaw (2001), Golightly & Wilkinson
(2005), and O’Neill (2002) for some suc-
cessful examples.
VII. Conclusions
This chapter has examined the rationale
for the use of computer models in study-
ing the aging process and has reviewed the
range of models that have been developed.
It has also described some of the generic
issues that need to be addressed in terms
of the methodology of modeling. In com-
ing years, it is likely to be essential, if
aging research is to realize its potential,
that modeling studies are greatly extended
and that models are increasingly used to
link together the pieces of the picture that
are revealed by reductionist experimental
techniques. These developments are an
inherent part of the “new” ways of doing
science that are commonly described as
“systems biology.” Whether systems biol-
ogy is really new or not is a matter for
debate, and a spectrum of opinion can be
found. What is unquestionably new is the
mass of detailed information emerging
at accelerating pace from functional
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346 T. B. L. Kirkwood et al.
genomic technologies, the rapid expan-
sion of raw computing power, the devel-
oping connectivity offered by advances in
Internet (soon to be GRID-enabled) Web
services, and the recruitment of increas-
ing numbers of mathematicians, statisti-
cians, and computer scientists into the
life sciences.
Not all areas of biology need to be
taken over by the systems approach, but
few are likely to remain untouched by it.
The biology of aging is one area where it
is hard to envisage the necessary progress
being made without embracing the sys-
tems approach. There are just too many
mechanisms, levels of action, and experi-
mental models for it to be realistic to
anticipate integration without the use of
computer models. Effecting the building
of the cross-disciplinary research pro-
grams to bring this about is going to be a
challenge, but it should also prove to be
intellectually stimulating and fun.
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AU8 Please provide pub city below.
AU9 Please fill in the missing volume or page number.
AU10 Are you able to complete the reference, or is this still in press?
AU11 Can you verify that the reference is now correct?
AU12 Is highlighted word correct?
AU13 Please fill in pub city.
AU14 Please fill in pub city.
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