Journal of Theoretical Biology 2
de
ra, R
Th
S-Ge
Tyne,
of N
form
s a sy
cerevisiae, yield a limited number of daughter cells
1990). The ‘‘lifespan’’ of a yeast cell is expressed as the
number of generations, or daughter cells produced,
(e.g. temperature), whereas chronological lifespan (time
daughter cells (Egilmez and Jazwinski, 1989). In
subsequent generations, daughters born to such prema-
ARTICLE IN PRESSturely aged cells re-acquire a normal lifespan, indicating
that the senescence factor can be cleared through further
divisions.
Sinclair and Guarente (1997) showed that a strong
candidate for the senescence factor in S. cerevisiae is the
*Corresponding author. Tel.: +44-0-191-256-3467; fax: +44-0-191-
256-3445.
E-mail address: c.gillespie@ncl.ac.uk (C.S. Gillespie).
0022-5193/$ - see front matter r 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtbbefore senescence (Mortimer and Johnston, 1959). This
phenomenon has been increasingly studied as a model of
cellular ageing (Jazwinski, 2001). Although individual
cells senesce and die, the continual generation of new
daughter cells ensures that a yeast colony can be
maintained indefinitely. During ageing, various mor-
phological and physiological changes take place in yeast
cells, such as an increase in cell size, an increase in bud
scar number (Bartholomew and Mittwer, 1953), an
increase in generation time (Egilmez and Jazwinski,
1989) and a decrease in metabolic activity (Jazwinski,
between formation and eventual senescence) may alter
considerably.
The average lifespan of wild-type S. cerevisiae is
around 25 divisions, with the maximum being around
40. When a yeast cell divides, the mother’s replicative
age is increased by one but the age of the daughter is
usually reset to zero. However, it has been observed that
daughter cells produced by very old mother cells may
exhibit reduced lifespans as though they are prematurely
old at birth. This has given rise to the suggestion that
there is a ‘‘senescence factor’’ which may ‘‘leak’’ intoage ‘‘reset’’ to zero. Accumulation of extrachromosomal ribosomal DNA circles (ERCs) appears to be an important contributor to
ageing in yeast, and we describe a mathematical model that we developed to examine this process. We show that an age-related
accumulation of ERCs readily explains the observed features of yeast ageing but that in order to match the experimental survival
curves quantitatively, it is necessary that the probability of ERC formation increases with the age of the cell. This implies that some
other mechanism(s), in addition to ERC accumulation, must underlie yeast ageing. We also demonstrate that the model can be used
to gain insight into how an extra copy of the Sir2 gene might extend lifespan and we show how the model makes novel, testable
predictions about patterns of age-specific mortality in yeast populations.
r 2004 Elsevier Ltd. All rights reserved.
Keywords: Mathematical model; Saccharomyces cerevisiae; Ageing; ERC; Sir2
1. Introduction
Individual cells of the budding yeast, Saccharomyces
before a newly formed cell loses the capacity to form
new buds. This replicative lifespan remains more or less
constant under a change in environmental conditionsonly a limited number of divisions before they undergo senescence, whereas newly formed daughters usually have their replicativeA mathematical mo
Colin S. Gillespiea,*, Carole J. Procto
Darren J. Wilkinsonb,
aHenry Wellcome Laboratory for Biogerontology Research, SCM
Newcastle upon
bSchool of Mathematics & Statistics, University
Received 4 December 2003; received in revised
Abstract
Budding yeast, Saccharomyces cerevisiae, is commonly used ai.2004.03.01529 (2004) 189–196
l of ageing in yeast
ichard J. Boysb, Daryl P. Shanleya,
omas B.L. Kirkwooda
rontology, University of Newcastle, Newcastle General Hospital,
NE4 6BE, UK
ewcastle, Newcastle upon Tyne, NE1 7RU, UK
27 February 2004; accepted 15 March 2004
stem to study cellular ageing. Yeast mother cells are capable of

of one of the repeat sequences of the rDNA, followed by
ARTICLE IN PRESS
heorehomologous recombination. Each ERC contains an
autonomously replicating sequence (ARS) and so is able
to replicate independently during S phase (Larionov
et al., 1984). This allows the copy number of ERCs to
build up faster than the chromosomal DNA, leading
to an intracellular over-accumulation. To prevent this
process of ERC accumulation from driving the popula-
tion to extinction it appears that when the cell divides,
there is a strong bias for any ERCs that may be present
within the mother cell to be retained there, and
not transmitted to the daughter. The asymmetrical
segregation of ERCs appears to be mediated by proteins
including SIR2 that tether circular DNAs to the
mother cell (Ansari and Gartenberg, 1997). However,
in old mothers, where the numbers of intracellular
ERCs are greatest, it seems that the capacity of this
mechanism to retain ERCs within the mother cell
becomes saturated and a few ERCs are passed on to
daughter cells. Since it may take some time for an
ERC to be formed by de novo excision in a daughter cell
that has not inherited any from its mother, any daughter
cell that inherits an ERC from its mother is already
advanced along the pathway to senescence. However,
in the early divisions, while its ERC burden is still
small, the mechanism of asymmetrical segregation
permits ERC-inheriting daughter cells to produce
ERC-free daughters themselves, thus explaining
how the prematurely aged daughters of old mothers
are nevertheless able to produce normal, young
offspring.
Sinclair and Guarente (1997) reported that when an
ERC is released into a young mother cell, the cell divides
an average of 15 times before senescing. They estimated
that the number of ERCs formed during these 15
generations is between 500 and 1000. It is thought that
in the end this large number of ERCs renders the mother
cell incapable of replicating and/or transcribing its
genomic DNA by competing against the chromosomal
DNA for interactions with the replication and tran-
scription machinery.
In this paper, we develop a mathematical model of
ERC accumulation in S. cerevisiae to explore whether or
not this mechanism is sufficient to account for the
ageing process in this species.
2. Model description
In order for ERCs to accumulate and eventuallyaccumulation of extrachromosomal ribosomal DNA
circles (ERCs) in old cells. Yeast rDNA is located on
chromosome XII and consists of 100–200 copies of a
9.1 kb repeat sequence. ERCs are formed by the excision
C.S. Gillespie et al. / Journal of T190stall the cellular replication and/or transcriptionalcomponents, three processes are necessary (see Fig. 1):
1. The cell acquires an ERC either through excision
from the chromosome or by inheritance from its
mother.
2. ERCs replicate during successive generations of
individual mother cells.
3. ERCs are distributed between mother and daughter
cells at division.
The data indicate that process 2 is exponential and that
process 3 is strongly asymmetrical. By inheriting ERCs
from their mother, daughters can be prematurely aged
(see Fig. 1). Formation of a new ERC in a cell that does
not already contain one (process 1) appears to occur
randomly at a relatively low frequency. Our modelling
of these three processes is now described. We develop
the model using a discrete time framework where one
time unit corresponds to one cell division. Events like
ERC generation and replication are treated as discrete
events that occur with specified probabilities in each
generation.
2.1. ERC generation
ERCs are initially formed by the excision of one of the
repeat sequences of rDNA. The rate at which this occurs
is not known. Therefore, we investigate three possible
scenarios of ERC formation using a constant, a linear,
or a quadratic probability structure. These three cases
can be expressed as
Pfor ¼ minðaixi; 1Þ for i ¼ 0; 1 or 2;
where Pfor is the probability that a new ERC is formed
in an ERC-free mother cell in the next generation, x is
the number of generations the mother cell has already
completed, and ai is a constant. If i was allowed to be
any real, positive number, a better fit of the data may
have been achieved. However, by constraining i to 0, 1,
or 2, a clearer understanding of ERC generation is
obtained. The case i ¼ 0 corresponds to the situation
where the probability of ERC formation is independent
of the age of the mother cell, whereas if i ¼ 1 or 2 this
probability increases with age with a linear or quadratic
rate, respectively.
2.2. ERC replication
Once an ERC is present within the cell, it replicates to
between 500 and 1000 ERCs in 15 generations. We
assume that existing ERCs replicate with a constant
probability Prep per ERC per generation.
2.3. ERC segregation
A critical aspect of the yeast life cycle is the
tical Biology 229 (2004) 189–196asymmetrical segregation of ERCs between the mother

ARTICLE IN PRESS
heoreC.S. Gillespie et al. / Journal of Tand the daughter. It is observed that in a steady-state
population the mother typically retains all the ERCs in
78% of divisions. Moreover, the cases where ERCs are
passed on to the daughter cell occur towards the end of
the mother cell’s lifespan (Sinclair and Guarente, 1997).
rDNA

Young
No ERCs
Excision
ERC
Asymmetrical
segregation
Old moth
a prematu
daughter
cell: of an produces
old
Fig. 1. Schematic representation of accumulation of ERCs during yeast agein
a cell carrying no ERCs. After some generations an ERC is formed by random
causing the mother cell to senesce. When new daughters are formed by buddin
of ERCs between mother and daughter cells means that daughters usually in
from an old mother cell containing many ERCs, one or more ERCs may be in
daughter to be reduced, since they effectively mean that the daughter is alreatical Biology 229 (2004) 189–196 191Although we know the frequency (22%) with which one
or more ERCs are passed on to the daughter, we do not
know the numbers of ERCs that are transmitted.
Two schemes of mother–daughter ERC segregation
are considered. Let N be the number of ERCs in the
er
rely

Asymmetrical
segregation
a healthy
Old cell:
ERC
causes

produces
daughter
overcrowding
senescence
g (adapted from Sinclair and Guarente, 1997). The process begins with
excision of rDNA, and this replicates within the mother cell eventually
g from young and middle-aged mothers, the asymmetrical segregation
herit no ERCs. However, when a daughter cell is formed by budding
herited by the daughter. These cause the lifespan or an ERC-inheriting
dy aged (in terms of ERC accumulation) at its formation.

3. Results
3.1. Canonical model
The results of the model which best fits the experi-
mental data (Sinclair et al., 1998) is shown in Fig. 2a.
We will use this as the canonical case for illustrating the
effects of varying the different parameters and assump-
tions of the model. To fit the canonical model, we use
the following procedure. Since it is assumed ERCs have
to be present in large numbers for a cell to senesce, once
ARTICLE IN PRESS
0
20
40
60
80
100
0 50
Generations
Su
rv
iv
al
(%
)
Exp. Data
Simulation
0
20
40
60
80
100
0
Generations
Su
rv
iv
al
(%
)
60
80
100
vi
va
l (%
)
10 20 30 40
5010 20 30 40
(a)
(b)
heoretical Biology 229 (2004) 189–196mother cell and R the number of ERCs retained by the
mother after cell division.
Scheme 1: Suppose that each ERC has a constant
probability of being retained (independently of other
ERCs) then R has a binomial BinðNmax; y1Þ distribution.
Scheme 2: Suppose that the first Nmax ERCs are
retained (independently) by the mother cell with
probability y2, and any additional ERCs above Nmax
are equally distributed between the mother and daughter
cells, then
R ¼
R1 if NpNmax;
R1 þ R2 otherwise;


where R1 and R2 are independent binomial BinðNmax; y2Þ
and BinðN  Nmax; 0:5Þ random variables.
2.4. Cell senescence
It is believed that ERCs eventually reach levels of
around 1000 ERCs, thereafter stalling the cellular
replication machinery. Therefore, we assume that once
a cell reaches 1000 ERCs the cell ceases to divide.
2.5. Simulation of a cell
To simulate a single yeast cell we combine ERC
formation, replication and segregation into the follow-
ing algorithm:
1. Commence the simulation with a healthy mother cell,
i.e. 0 ERCs;
2. A new ERC is generated through excision with
probability Pfor;
3. Existing ERCs are replicated with probability Prep;
4. If the number of ERCs in a cell is greater than 1000,
then the cell senesces and the simulation ends;
otherwise proceed to 5;
5. Cell divides, with R ERCs being retained using
scheme i, where i ¼ 1; 2;
6. Return to step 2.
To calculate a survival curve it is necessary to follow a
population of cells. This is achieved by commencing
with a single cell, then following the course of all of its
daughters and grand-daughters. This allows for the
possibility that a yeast cell inherits ERCs from its
mother, which may induce premature senescence, and
thus simulates the spectrum of states that might exist in
a culture.
The above algorithm can be coded using most
standard computing languages. In our simulations we
used the language ‘C’ under a Linux Debian environ-
ment. The time required to simulate a survival curve for
population of yeast cells, up to the grand-daughter
stage, is about a few seconds using a standard home
computer. The source code is available upon request
C.S. Gillespie et al. / Journal of T192from the authors.0
20
40
0
Generations
Su
r
5010 20 30 40
(c)
Fig. 2. (a) Survival curves of a simulated and experimental yeast
population. The experimental data are from Sinclair et al. (1998) while
the simulated data are obtained from a single run using the canonical
model (see text). (b) Survival curves for 4 simulations of the canonical
model including all cells up to and including the grand-daughters. (c)
Survival curves for 4 simulations of the canonical model including
cells up to the daughter generation only.

all cases ði ¼ 0; 1; 2Þ lifespan decreases monotonically
with increasing ai; as would be expected, providing a
unique value of Pfor for any chosen lifespan.
When we choose parameter values ai for each of the
cases i ¼ 0; 1; 2 so that each has the same mean lifespan
as the experimental data, we find that the simulated
survival curves have different shapes (Fig. 3). The
closest fit to the experimental data is given by the case
i ¼ 2; with a2 ¼ 0:0009 (quadratically increasing prob-
ability of excision with increasing cell generation
number), while the case i ¼ 0; with a0 ¼ 0:074 (constant
probability of excision with increasing cell generation
number) gives the poorest fit predicting that some cells
should have lifespans almost three times greater than the
experimentally observed maximum of 40 generations.
This observation is very important because it means that
ARTICLE IN PRESS
heoretical Biology 229 (2004) 189–196 193an ERC is present in the cell, ERC formation
contributes little in the build up of ERCs compared to
ERC replication. Therefore, we can choose a value of
Prep that allows the cell to reach approximately 1000
ERCs after 15 generations. The segregation scheme is
chosen to allow the mother cell to retain all ERCs in
78% of cell divisions. Since few ERCs are passed on to
the daughter (compared to those already present in the
mother), the choice of scheme has little effect on the
lifespan of the mother cell. The remaining parameter
Pfor is then calculated to allow the average lifespan of
the cell to be 25 generations. Following this method
yields a canonical model in which the probability of
ERC formation is Pfor ¼ minð0:0009x2; 1Þ; the prob-
ability of an ERC replicating is Prep ¼ 0:60; and the
segregation probabilities follow Scheme 2 with Nmax ¼
700 and y2 ¼ 0:998: The canonical model predicts an
average replicative lifespan of 25 generations and that
mother cells inherit all ERCs (i.e. do not transmit to
their daughters) in 78% of cell divisions. Note that in
this model, once a cell acquires an ERC (by excision or
inheritance) the average remaining lifespan is 15
generations, so the reduced lifespan of daughters
produced from very old mothers is also explained.
In simulating the model, we initiated our cell
population with a single mother cell containing no
ERCs and we recorded the lifespans of all daughters and
grand-daughters of the founder cell. Using this informa-
tion we constructed a survival curve for these lifespans.
Since the model is stochastic in nature there is some run-
to-run variability in the resulting curve. However, a
large number of simulation runs indicates that this
variability is small when recording up to the grand-
daughter level; see Fig. 2b for the survival curves for
four such runs. We also plotted the survival curves for
just the daughter generation (Fig. 2c). It may be seen
that the shapes of the survival curves in the Figs. 2b and
c are similar, indicating that the distribution within the
population quickly reached equilibrium. However, there
was a greater run-to-run variance in Fig. 2c when only
daughters were considered. Hence, all subsequent
simulations included the grand-daughters.
Having established that the canonical model provided
a good fit to the experimental data, we examined the
effects of varying the assumptions of this model in order
to discover which features of the model were the most
critical.
3.2. Effects of varying assumptions about ERC formation
Since ERCs are excised at a low rate from the
chromosome, the time of formation of the first one has a
relatively large effect on the average lifespan of a cell.
Therefore, the values of ai in Pfor can be chosen by
determining in each instance which value allows the
C.S. Gillespie et al. / Journal of Taverage lifespan of a yeast cell to be 25 generations. Inthe model strongly indicates that ERC formation is not
governed by a constant probability but by one that
increases with the age of the mother. This suggests that
some intrinsic ageing process, other than ERC accumu-
lation alone, must contribute to the ageing of yeast cells.
3.3. Effects of varying assumptions about ERC
replication
The probability Prep determines the lifespan of a yeast
cell from the point at which an ERC is introduced,
either by de novo excision or inheritance. It has been
shown that the average lifespan of a cell after an ERC
has been introduced is 15 generations. Fig. 4 demon-
strates that, as expected, altering Prep in the canonical
model has a straightforward scaling effect on the later
sections of the survival curve, this being the age range
that is most affected by the accumulation of ERCs. This
probability has little other effect on the shape of the
survival curve and the best-fitting parameter value can
therefore be determined easily from the data.
0
20
40
60
80
100
Generations
Su
rv
iv
al
(%
)
Exp. Data
Constant
Linear
Quadratic
0 10 20 30 40 50 60
Fig. 3. Comparison of experimental and simulated survival curves
when a constant, linear or quadratic form for the dependence of the
probability of ERC formation on cell generation number was
assumed.

ERCs, when the cell contains 1000 ERCs. This allows
old mothers to produce old daughters.
3.5. Effects of varying assumptions about cell senescence
To model cellular senescence, we simply assumed that
cells which had more than 1000 ERCs automatically
stalled the replication machinery, and that senescence
did not occur before this happened. Although it is
unlikely that there is such a precise threshold for all
cells, we found (results not shown) that other, more
complicated forms of this assumption did not signifi-
cantly alter the results. For instance, the probability that
a cell ceases dividing at a given number N of ERCs
could be modelled via a term
Prðcell senescenceÞ ¼ Psen ¼
N  500
1000
ARTICLE IN PRESS
heoretical Biology 229 (2004) 189–1963.4. Effects of varying assumptions about ERC
segregation
The two schemes for mother–daughter ERC segrega-
tion described above were examined subject to the
requirement that mother cells should inherit all ERCs in
an average of 78% of cases, as observed experimentally.
Scheme 1 assumes that the probability of an ERC
being retained by the mother cell is independent of both
the total number of ERCs present and the number of
generations a cell has completed. For the mother cell to
retain all ERCs in 78% of cases, a value of y1 ¼ 0:998 is
required. If y1 is slightly reduced to 0.995, then ERCs
would be passed on to 32% of daughter cells. Reducing
the value of y1 further to 0.99, results in ERCs being
passed on to almost 50% of daughter cells. A critical
aspect of the yeast ageing cycle is that old mothers may
0
20
40
60
80
100
Generations
Su
rv
iv
al
(%
)
Exp. Data
Prep = 0.5
Prep = 0.6
Prep = 0.7

0 10 20 30 40 50
Fig. 4. Comparison between the experimental survival curve to the
simulated survival curves where the ERC replication parameter, Prep,
has been altered. By increasing the rate of replication, the mean
lifespan of a cell is reduced.
C.S. Gillespie et al. / Journal of T194produce prematurely old daughters (see Fig. 1). For a
daughter cell to senesce within, say, 5 generations,
approximately 100 ERCs must be inherited. However
when y1 ¼ 0:988 the probability of passing on over 20
ERCs from a total of 1000 ERCs is 5 1014; if y1 were
0.995, passing on over 20 ERCs would still only occur
with probability 7 107. Therefore the probability of
ERC retention cannot be constant throughout the cell’s
life.
Scheme 2 assumes that only a limited number of
ERCs can be strongly retained by the mother cell, with
any ERCs above this amount being equally distributed
between cells. Since we wish scheme 2 to retain all ERCs
in the same number of cases as in scheme 1, y2 is set
equal to y1 ¼ 0:998: A value of Nmax ¼ 700 then allows
old mother cells to pass on significant amounts of ERCs
to their daughters but still make it unlikely that the
mother cell could ‘‘off-load’’ all her ERCs into her
offspring, thereby rejuvenating herself. Using the second
scheme, a mother cell will pass on approximately 150for N between 500 and 1500. For No500 Psen is zero
and for N > 1500 Psen is one. While this would increase
the variability in the predicted lifespan of a single cell, it
had negligible effect on the distribution of lifespans
within the population and hence on the predicted
survival curve for a population, indicating that the
other stochastic factors within the model were of much
greater significance.
3.6. Examining the possible role of Sir2 gene action
It was observed experimentally that when an addi-
tional copy of the Sir2 ð
%
silent
%
information
%
regulatorÞ
gene was added to a healthy mother yeast cell, a 30–40%
increase in lifespan was produced (Kaeberlein et al.,
1999). Whilst Sir2 has been shown to repress recombi-
nation in rDNA (Gottlieb and Esposito, 1989), it is
unclear whether Sir2 influences lifespan by affecting
ERC segregation, replication or formation.
0
20
40
60
80
100
Generations
Su
rv
iv
al
(%
)
Wild-type
aSir2+
bSir2+
Simulation 1
Simulation 2
0 10 20 30 40 50
Fig. 5. Effects on experimental and simulated survival curves of an
additional copy of the Sir2 gene. aSir2+ and bSir2+ refers to the
data in Fig. 7 of Kaeberlein et al. (1999). For reference, the wild-type
survival curve is also shown. Simulation 1 shows the effect of reducing
Prep to 0.4 within the canonical model. Simulation 2 shows the effect
of reducing Pfor to min(0.00045x
2,1).

ARTICLE IN PRESS
heoreAddition of an extra copy of Sir2 changes the shape
of the survival curve (see Fig. 5). Since we showed in 3.3
that reducing the rate of ERC replication simply scales
the survival curve, but does not alter its shape, Sir2 gene
action is unlikely to affect ERC replication. It is also
unlikely that Sir2 acts through changing ERC segrega-
tion (by improving the efficiency of screening against
mother–daughter transmission), since such an effect
would alter the lifespans of relatively few individuals
within the population. However, if Sir2 reduces the rate
of ERC formation, this both increases the lifespan and
changes the shape of the survival curve. It can be seen
from Fig. 5 that halving the value of Pfor provides a
good match to the experimental effects of adding an
additional copy of Sir2.
3.7. Effect of initial ERC number on profiles of age-
0
0.2
0.4
0.6
0.8
1
Generations
M
or
ta
lit
y(%
)
Nerc (0) = 0
Nerc (0) = 1
Nerc (0) = 10
Nerc (0) = 10
0 10 20 30 40 50
Fig. 6. Age-specific mortality (ASM) of yeast cells which begin with 0,
1, 10 or 100 ERCs, obtained by simulating 500,000,000 individual
cells. As expected, when a cell has 10 or 100 ERCs introduced into the
first generation, the ASM is increased. However, when a single ERC is
introduced, there is a small probability that it may pass on the repeat,
thereby rejuvenating itself.
C.S. Gillespie et al. / Journal of Tspecific mortality
Sinclair and Guarente (1997) examined the effects of
adding additional ERCs into healthy yeast mother cells,
effectively ageing them prematurely. For technical
reasons, they could not specify the exact number of
added ERCs but we could use our model to examine this
question. Additionally the age-specific mortality (ASM),
that is, the proportion of individuals of generation x
who die by generation x þ 1; can be obtained. To
calculate the ASM across an extended age range
requires a large number of simulation runs, since the
fraction of survivors at the oldest ages becomes very
small. Fig. 6 shows the ASM curves obtained by
simulating 500,000,000 cells, each of which commenced
with 0, 1, 10 or 100 ERCs.
When the cells started with 10 or 100 ERCs, the ASM
increases rapidly. Conversely, when no ERCs are
present initially, the mortality rate remains low before
increasing more rapidly at about generation 18. How-
ered as the necessary components for the transcriptional
repression of the silent mating type loci, HML and
HMR (Ivy et al., 1986; Rine and Herskowitz, 1987).
Loss of silencing at these loci points results in
coexpression of a and a mating type genes and sterility
in haploid strains. Additionally, it is believed that the
SIR proteins are involved in non-homologous end
joining, which is used to repair breaks in DNA by
ligation of the free ends (for review, see Critchlow and
Jackson, 1998). When an additional copy of the Sir2p
was introduced into yeast cells, a 40% increase in
lifespan was observed (Kaeberlein et al., 1999). How-
ever, it is unclear whether the increase in lifespan wasever, we noticed a curious effect when the cell starts with
a single ERC. The ASM initially increased, but then
decreased, before finally increasing again. The explana-
tion for this surprising behaviour is that although a cell
initially commences with a single ERC, it is able
sometimes to pass this ERC onto its daughter, thereby
gaining ERC-free status and significantly extending its
lifespan. The probability of this event is approximately
one cell in a thousand, so to attempt to test this model
prediction experimentally would be time consuming but
nevertheless interesting.
4. Discussion
We have shown in our model how yeast cell
senescence can be partly explained by accumulation of
ERCs. However, the model also showed that in order to
match the observed survival curves, the rate of ERC
formation needed to increase as the number of cell
divisions increased. This strongly suggests that ERC
formation is insufficient alone to account for yeast
ageing and that another underlying process must act to
cause deterioration in the mechanisms that inhibit ERC
formation.
It has been noted by Sinclair and Guarente (1997)
that the ERC segregation properties of yeast cells
cannot be a passive process, since the nuclear volume
is partitioned approximately equally to mother and
daughter cells (Robinow and Marak, 1966; Gordon,
1977). Through modelling the retention of ERCs
between mother and daughter cells, we found that for
a mother cell to retain all ERCs in 78% of cell divisions,
the probability of retaining a single ERC must be high.
Furthermore, this mechanism of retention must break
down for old mother cells to yield old daughter cells.
This could be the ERCs themselves which overload the
segregation machinery, thereby allowing ERCs to be
passed on.
In budding yeast, Saccharomyces cerevisiae, the Sir
family of genes has been identified as possessing several
cellular functions. For instance, they were first discov-
tical Biology 229 (2004) 189–196 195due to an effect on ERC segregation, replication or

formation properties. By slowing down the rate of ERC
formation in a healthy yeast cell, we were able to obtain
survival curves similar to those observed experimentally,
and we were also able to show that this match could not
be reproduced by altering the ERC replication rate or
segregation properties.
It would be naive to assume that ERC accumulation
can wholly explain ageing in yeast. Indeed, other ageing
mechanisms apparently independent of ERC accumula-
experimental system for studying ageing, we believe
ERCs are needed to fit the data for ageing yeast
genome stability, and heterochromatin. Microbiol. Mol. Biol.
Rev. 67, 376–399.
Critchlow, S.E., Jackson, S.P., 1998. DNA-end-joining: from yeast to
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Egilmez, N.K., Jazwinski, S.M., 1989. Evidence for the involvement of
a cytoplasmic factor in the aging of the yeast Saccharomyces
cerevisiae. J. Bacteriol. 171, 37–42.
Falcon, A.A., Aris, J.P., 2003. Plasmid accumulation reduces life span
in Saccharomyces cerevisiae. J. Biol. Chem. 278, 41607–41617.
Gershon, H., Gershon, D., 2000. The budding yeast, Saccharomyces
cerevisiae, as a model for ageing research: a critical review. Mech.
ARTICLE IN PRESS
C.S. Gillespie et al. / Journal of Theoretical Biology 229 (2004) 189–196196supports the idea of ‘network’ models of the kind we are
developing in the Biology of Ageing e-Science Integra-
tion and Simulation (BASIS) system (Kirkwood et al.,
2003).
Acknowledgements
We thank the BBSRC, MRC and DTI for financial
support and Leonard Guarente for helpful comments.
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