Mitochondrial Mutations, Cellular Instability and Ageing:
Modelling the Population Dynamics of Mitochondria
A. Kowald and T.B.L. Kirkwood
Laboratory of Mathematical Biology
National Institute for Medical Research
The Ridgeway, Mill Hill, London NW7 1AA
England
Running title: Modelling Mitochondrial Mutations
Keywords: Mitochondrial mutations, ageing, DNA damage
August 1992
Abstract
All eukaryotic cells rely on mitochondrial respiration as their major source of metabolic
energy (ATP). However, the mitochondria are also the main cellular source of oxygen
radicals and the mutation rate of mtDNA is much higher than for chromosomal DNA.
Damage to mtDNA is of great importance because it will often impair cellular energy
production. However, damaged mitochondria can still replicate because the enzymes
for mitochondrial replication are encoded entirely in the cell nucleus. For these reasons,
it has been suggested that accumulation of defective mitochondria may be an important
contributor to loss of cellular homeostasis underlying the ageing process.
We describe a mathematical model which treats the dynamics of a population of
mitochondria subject to radical-induced DNA mutations. The model conrms the
existence of an upper threshold level for mutations beyond which the mitochondrial
population collapses. This threshold depends strongly on the division rate of the mitochondria. The model also reproduces and explains (i) the decrease in mitochondrial
population with age, (ii) the increase in the fraction of damaged mitochondria in old
cells, (iii) the increase in radical production per mitochondrion, and (iv) the decrease
in ATP production per mitochondrion.
All eukaryotic cells contain mitochondria, organelles which are responsible for the
energy (ATP) production of the cell. Mitochondria contain their own genetic material
(mtDNA) and an average of 4.6 mtDNA copies per mitochondrion has been determined
for human cell lines (Satoh and Kuroiwa, 1991). Most of the protein-coding capacity is
used to specify proteins of the oxidative phosphorylation pathway. All other mitochodrial proteins are coded for by the nucleus and have to be imported into the organelle.
Concomitantly with the production of energy the mitochondria are also the main
source of activated oxygen species (O2 : , OH :, 1 O2 , H2 O2 ). Together with the special
composition of the mitochondrial genome this has inspired several theories suggesting
that defective mitochondria contribute to, and may be the cause of, ageing (Miquel
et al., 1980; Harman, 1983; Richter, 1988; Linnane et al., 1989). Recently this idea has
gained support from studies using the technique of polymerase chain reaction (PCR).
Linnane et al. (1990) and Yen et al. (1991) reported an age-related 5 kb deletion in
the mitochondrial genomes of several human patients. Similarly Hattori et al. (1991)
found a 7.4 kb mtDNA deletion in human heart tissue. While in young patients no
deletions could be found, mtDNA deletions were detected in all of the older patients.
The underlying idea of the defective mitochondria theory is that damage to the
mtDNA will mainly aect functions that are important for the energy production.
However, because the genes which are important for mitochondrial reproduction are
located in the nucleus, the aected mitochondrion can still propagate and there is a
risk that the cellular population of healthy mitochondria will eventually be overtaken
by nonfunctional ones.
It has been shown that mitochondrial mutations also occur in higher eukaryotes
and are responsible for diseases like Leber's hereditary optic neuropathy (Nikoskelainen et al., 1987; Wallace et al., 1988) or chronic progressive external opthalmoplegia
(CPEO). While the basis of Leber's hereditary optic neuropathy is a point mutation,
CPEO is caused by a large deletion of the mtDNA (Holt et al., 1988).
There is also evidence that the frequency of mitochondrial mutations is correlated
with age. Piko et al. (1988) analyzed denatured and reannealed mtDNA duplexes using
electron microscopy and found a vefold increase in the number of deletion mutations
in old mice.
To understand better the population dynamics of mitochondria which are prone to
radical-induced mtDNA damage we developed a mathematical model which is described
in the next section.
Model Description
The following list summarizes the features that are taken into account by the model.
Mitochondria are polyploid
Mitochondria produce ATP
Mitochondria produce oxygen radicals
Mitochondria suer DNA damage depending on the radical level
Mitochondria produce more radicals and less ATP after DNA damage
Mitochondria multiply by division
The growth rate is controlled by the cellular ATP concentration
There is a random distribution of mitochondrial genomes during division
There is a turn-over of mitochondria
It is assumed that the mitochondrial genome consists of ve copies of mtDNA. This
value has been chosen because it corresponds most closely to the degree of polyploidy
in higher animals (Satoh and Kuroiwa, 1991) and because preliminary simulations have
shown that this parameter is of only minor importance for the results of the model. The
mtDNA as a whole is regarded as the target for damage, which occurs with a certain
probability and gives rise to six dierent classes of mitochondria termed M0 (none of
the mtDNA's damaged) and M1 : : : M5 (1 to 5 copies damaged, respectively). Figure 1
shows the reactions which are described by the model.
Damage Level and Transcription Rate
It is assumed that the synthesis of active proteins decreases as a consequence of damage
to individual mtDNA molecules. It is also assumed that mitochondria counteract this
decrease by a feedback loop that enhances the transcription rate. If D is the fraction
of damaged DNA molecules (ranging from 0 to 1) and the transcription rate increases
linearly with D then the increase in transcription (SF) is calculated by:
SF = D (SFmax 1) + 1
(1)
SFmax is the maximum factor by which the transcription rate can be increased
(for D = 1). As a consequence of the altered transcription rate the amount of ATP
synthesized by a given mitochondrion is a fraction (1 D) SF of the ATP synthesized
by a healthy mitochondrion (M0 ). Figure 2 shows the resulting, non-linear, relationship
between energy production and damage level.
Radical Production and Mutation Rate
Most injuries to the mitochondrial genome are likely to result in an increased generation of reactive oxygen species. The highest rates are 10-30 fold greater than that
released during normal respiration (Bandy and Davison, 1990). The radical production
is correlated with the degree of damage (D) in our model in the following way.
Undamaged mitochondria (D = 0) produce radicals at a rate of Rad0 per second. For mitochondria which have all mtDNA's damaged (D = 1) the excess radical generation is proportional to the increased transcription (SF) and a constant RE
(Rad0 SF RE ). The biochemical basis of the constant RE is the observed increase in
the radical generation following damage to components of the respiratory chain (Bandy
and Davison, 1990). For values of D between zero and one it is assumed that the radical production depends in a linear way on D. Under these assumptions it holds that
the radical production is given by Equation 2. If SF is substituted by Equation 1, the
non-linear relationship results which is shown in Figure 2.
Rad = D Rad0 (SF RE 1) + Rad0
(2)
It is furthermore assumed that an increase in radical production is linearly proportional to the probability (p) that one copy of mtDNA becomes damaged per day
with the probability p = k3 for a radical level Rad0 of a non-damaged mitochondrion.
This is an approximation, because for very high radical levels p has to approach unity.
However, for the purpose of this model the approximation is valid, because the radical
level under physiological conditions is always so low that the probability of a mutation
is several orders of magnitude below unity. Although oxygen radicals have a high diusion rate (because of their small size) the major species, the superoxide radical, cannot
easily cross the mitochondrial membrane because of its negative charge. This results
in dierent radical levels (Rad0 : : : Rad5 ) being associated with the dierent classes of
mitochondria (M0 : : : M5 ). For convenience Rad0 : : : Rad5 are expressed in multiples of
Rad0 .
While p is the probability that one copy of mtDNA gets damaged per day this is
not the same as the probability that a given mitochondrion traverses from one damage
class to the next. Because the mitochondrial genome is polyploid, the probability of a
mitochondrion of class i entering class i+1 is:
pi = (5 i) p (1 p)4
i
(3)
Synthesis and Growth of Mitochondria
The model assumes that the synthesis of new mitochondria is controlled by the cellular
ATP level. A negative feedback is implemented so that an elevated ATP concentration
suppresses the formation of new organelles.
It is intuitively clear that the fate of a cell depends in a crucial way on the reproduction rates of healthy and damaged mitochondria, respectively. Energy (Harmey et al.,
1977) and a membrane potential () (Schleyer et al., 1982) across the inner mitochondrial membrane is required for the transport of nuclear-coded mitochondrial proteins
into the organelles. Since these prerequisites are impaired in damaged mitochondria
it is assumed that they have a replication disadvantage which is proportional to their
degree of damage. GDF is dened as the Growth Dierence Factor between M0 and
M5 mitochondria. Mitochondria containing a level of damage D grow D (GDF 1)+1
times slower than error-free organelles (a linear relationship between D and GDF is
assumed). Furthermore a rst order decay term is used (k2 M ) to ensure a continuous
degradation and thereby turn-over of the mitochondrial population.
Finally a last characteristic of mitochondrial biochemistry has to be taken into
account. During mitochondrial biogenesis there is a binomial segregation of mtDNA
copies to the daughter mitochondria. This means that although the copy number of
mtDNA's remains constant the ospring of a M1 mitochondrion (one mtDNA damaged) can be a M0 mitochondrion (no mtDNA damaged). In general it holds that the
probability for a Mi mitochondrion to produce a Mj ospring is
0
B@ 10
Pi!j =
5
1 0 1
CA B@ 2i CA
j
j
0 1
B@ 10 CA
2i
5
(4)
0 1
a
where B
@ CA is the standard combinatorial term for the number of ways of choosing
b
b objects from a set of a.
Model Equations
Using the information of the last sections it is now possible to develop the necessary
dierential equations (Fig. 3). The model consists of six formulae for mitochondria
belonging to the dierent damage classes and one equation for the cellular ATP level.
The equations for the mitochondria consist basically of terms which describe the synthesis of new organelles by the dierent classes of existing organelles and terms which
describe the migration of mitochondria from one class
0 to1the next. For convenience a Cb
a
has been used to describe the combinatorial term B
@ CA.
b
Equation 11 consists of three terms describing the generation and consumption of
energy in the form of ATP. The rst two terms are straightforward, describing the
ATP generation by each subpopulation of mitochondria and the energy consumed for
the synthesis of new mitochondria. The third term is less obvious. It deals with the
fact that mitochondria supply the rest of the cell with energy. That means there is an
additional energy consumption by cellular mechanisms that lies outside the scope of
this particular model. However, this energy consumption has to be considered because
the ATP level has a direct inuence on the behaviour of the mitochondrial population.
It is assumed that the cellular energy consumption depends on the cellular ATP level,
reaching a maximum of ATPd if there is a surplus of ATP and declining gradually if
there is a low cellular energy charge.
Auxiliary Equations
Figure 3 shows the seven dierential equations which constitute the core of the model.
However, extra information can be obtained by additional non-dierential equations.
The following equations are derived from Equation 11 and describe the cellular
energy consumption and the energy expended for synthesizing new mitochondria, respectively. Hence they are rates (ATP/sec) and not steady state levels. The rationale
is that the ATP steady state value is the result of energy generation and energy consumption and therefore gives no insight into the cellular energy requirements, which is
an important issue when addressing evolutionary theories of ageing (Kirkwood, 1981).
Although both rates together determine the total energy usage the two equations are
kept separate to monitor the ux of energy in dierent cellular compartments.
ATP
ATP + k6
5
X
5
k1
=
k4 Mi
ATP k5
i =0 i (GDF 1 ) + 5 1 + ATP
c
ATPcell = ATPd
(14)
ATPmito
(15)
The next equation describes the total cellular radical generation per unit time.
dRad=dt is given in multiples of Rad0 which makes it easier to compare the radical
production with the radical output of an intact mitochondrion. Although the mutation
rate does not depend on the total radical generation but on the generation rates of the
individual subclasses (Rad0 : : : Rad5 ) it is important to know dRad=dt to relate it to
experimental results.
5
dRad = M + X
Radi Mi
0
dt
i=1
(16)
The nal three equations calculate the fraction of damaged mitochondria with respect to intact mitochondria (DM ), the amount of radicals generated per mitochondrion in multiples of Rad0 (RM ), and the amount of ATP produced per mitochondrion
in multiples of ATP0 (AM ).
+ M3 + M4 + M5
DM = M1 + M2 P
5
i=0 Mi
(17)
P5 Rad M
M
0+
RM =
P5i=1M i i
i=0 i
P
M + 5 SF M
AM = 0 P5i=1 TSF i
i=0 Mi
i
i
(18)
(19)
Results
The dynamic behaviour of the model is illustrated with simulations that reveal its main
characteristics.
Standard Simulation
With a given set of initial concentrations and using the default parameter values as
described in the appendix the system converges quickly to a stable, steady state condition. This is shown in Figure 4 and will be referred to as the standard simulation.
Because dierent molecular species have very dierent steady state values, separate
scaling factors have been used to display all variables in one diagram. The ordinate
value shown refers to a scaling factor of one.
With M0 = 890 and M1 = 8:7, mitochondria with one damaged genome represent
just 1% of the population and the other subclasses exist only at levels close to zero. The
total radical generation reaches a constant value of ca. 908 which is in the expected
range since error-free organelles constitute the majority of the mitochondrial population
and the radical generation is given in units of Rad0 .
Furthermore the default parameters are chosen so that the ATP steady state level
is very low (ATP=5.7) compared to the rate of energy consumption by cellular (ATPcel
= 3200) and mitochondrial processes (ATPmit = 180). This reects the fact that the
daily ATP expenditure is much greater than cellular steady state levels of ATP. Note
that the values for the ATP and radical steady state levels and consumption are given
in units of 109 .
Mutation Rate and Steady State Concentrations
The main statement of theories which invoke defective mitochondria to explain the
ageing process is that a high mutation rate leads to a gradual loss of intact mitochondria
and eventually to a collapse of vital cellular functions. Figure 5 shows a simulation that
investigates the inuence of dierent mutation rates on the steady state concentrations.
The model conrms that with increasing mutation rate the population number of
mitochondria declines and eventually drops to zero when the mutation rate reaches a
certain threshold. This threshold is modulated by the parameter k1 which represents
the maximum growth rate (per day) of the mitochondrial population. An increase of k1
from 0.055 to 0.060 renders the system more robust and shifts the maximum endurable
mutation rate from ca. 0:9 10 3 to 1:9 10 3 (per day).
Mutation Rate and Loss of Homeostasis
True steady state levels can exist only if cellular homeostasis can be maintained indenitely. In the real world this condition is not fullled, but instead animals age and
eventually die. Most experimental studies have therefore concentrated on the change of
mitochondrial properties such as their number, the radical production or the fraction
of damaged mitochondria.
Figure 6 shows how the corresponding variables of the model develop over time
during a breakdown of the system. For this simulation the steady state values for k3 =
10
4
(which corresponds to a stable point in parameter space) have been calculated
and then used as initial values for the computation with k3 = 2:2 10 3 .
Mutation Rate and Model Lifespan
From simulations like the one shown in Figure 6, information about the relationship
between mitochondrial mutation rate, the maximum mitochondrial growth rate and
the model lifespan can be obtained. This information is summarized in Figure 7. The
lifespan was dened as the time from the start of the simulation until the moment when
the level of intact mitochondria fell below 10% of the initial level.
As can be seen the lifespan depends strongly on the interplay of the mitochondrial
mutation rate, k3 , and the maximum mitochondrial growth rate k1 . To survive for a
given amount of time a low growth rate requires a low mutation rate while a cell with
a higher growth rate can tolerate a higher mutation rate.
Discussion
Of fundamental importance for ageing theories based on defective organelles is the
mutation rate (Linnane et al., 1989; Miquel, 1991). In Figure 5 the steady state values
for damage-free mitochondria, the radical production and the cellular ATP consumption
were shown for an increasing mutational load (k3 ). The simulations showed that the
variables change only moderately until a point was reached where a slight increase of
k3 caused the complete breakdown (M0 = 0) of the system. Furthermore the ability
of the system to tolerate mutations increases with increasing k1 . In fact it is the net
growth rate (k1 k2 ) of intact mitochondria which is the important factor. The point
of the collapse is reached when the mutation rate exceeds the net growth rate. As a
good approximation this condition is fullled when
5 k3 (1 k3 )4 > k1 k2
(20)
If k3 is increased further, the population of intact mitochondria decreases exponentially until M0 = 0. Unfortunately the rate of somatic mutations of mtDNA in
humans is unknown (Linnane et al., 1989). However, yeast petite mutants occur with a
frequency of 10 1 {10 3 per generation (Ferguson and Vonborstel, 1992) from which an
estimate of ca. 10 2 {10 4 per day can be obtained. Depending on the growth rate k1
the model simulations show a breakdown at 1 10 4 {2 10 4 which is in good agreement
with the petite data.
As a consequence of the above mentioned condition the stability behaviour of the
system depends strongly on both parameters, the mutation rate k3 and the maximum
growth rate k1 (Figure 7). This dependence is not surprising. The model employed to
describe the dynamics of the mitochondrial population is in some respects similar to
mathematical models which have been developed to account for the limited proliferative capacity of mammalian cells in culture. For instance Zheng (1991) came to the
conclusion that the destiny of a cell population is controlled by two opposite factors:
the proliferation rate of the cells, and the gene damage accumulation rate. Our nding
is also in agreement with an hypothesis by Miquel and Fleming (1986) who argued
that there is no mitochondrial ageing in rapidly dividing cells, because mitochondria in
those cells replicate faster than mitochondria in non-dividing cells.
It would appear from the simulations that a simple way to cope with mitochondrial
mutations would be to increase k1 to whatever level is required. This solution might,
however, have its limitations. Mitochondria are large, complex objects and most of the
mitochondrial proteins and membrane lipids have to be synthesized by the cytoplasmic
machinery. The maximum level to which k1 can be increased is therefore a question of
how many resources the cell invests into the apparatus for generating new mitochondria.
There may very well be an upper limit to the growth rate of mitochondria.
A continual loss of mitochondria with age is a result found for several species such as
humans (Tauchi and Sato, 1968) and Drosophila Massie et al. (1975); Fleming (1986).
This phenomenon is also observed in the simulation shown in Figure 6. As explained
above, this exponential decrease is caused by too high a mutation rate.
Concomitantly with the decline of M0 , a rise in the fraction of damaged mitochondria (MD) occurs. The basic mechanism of the proposed model, genomic instability
of mitochondrial DNA (point mutations, deletions, duplications, rearrangements), has
been observed in one or the other way in many species, ranging from fungi (Munkres,
1985) to mice (Piko et al., 1988) and humans (Linnane et al., 1990; Hattori et al., 1991;
Yen et al., 1991). The qualitative results of the model are therefore not restricted to
specic organisms, but are valid for a broad range of species.
As a consequence of the continual rise in the fraction of defective organelles, the
radical production per mitochondrion (RM) rises and the ATP generation per organelle
(AM) decreases with time (Fig. 6). The same behaviour was observed for the radical
generation in ies (Farmer and Sohal, 1989; Sohal and Sohal, 1991) and rats (Sawada
and Carlson, 1987) and a decline of respiratory activity has been found in human muscle
(Cardellach et al., 1989; Byrne et al., 1991) and liver (Yen et al., 1990).
What implications has this model for future research ? It shows that many changes
which are observed in old mitochondria are consistent with one underlying mechanism:
Radical induced damage to the mitochondrial DNA. In long lived animals and germline cells the mutation rate would be kept low by either a lower radical generation or
a more elaborate antioxidant system. Consequently the model predicts that articially
increasing the level of antioxidant enzymes within the mitochondrion (e.g. in transgenic
animals) should lengthen the lifespan. It also predicts that a considerable fraction of the
population consists of impaired mitochondria. Although several studies support such a
prediction other investigations failed to detect age dependent changes in mitochondria
(Manzelmann and Harmon, 1987; Bodenteich et al., 1991). Clearly more experimental
and theoretical work is necessary to elucidate the role of defective mitochondria in the
ageing process.
Acknowledgements
A. Kowald thanks the Carl Duisberg Stiftung, the British Council and the Max Buchner
Forschungsstiftung for nancial support.
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Appendix
Parameter Denitions and Default Values
k1 =
0.06 d
1
The maximum growth rate of the mitochondrial population is assumed to be 6% per day. This takes account of the fact that mitochondria are large objects which require a considerable amount
of time and resources to be synthesized. Using this value the
whole mitochondrial population could double within 12 days.
k2 =
0.05 d
1
Fraction of all mitochondria which is degraded per day. With
this value the rst order decay function results in a half-life of
ca. 14 days. This value was chosen in agreement with studies
of Menzies and Gold (1971) who determined the half-life of mitochondria in a variety of rat tissues to be between 10 and 30
days. Similar values (10{15 days) have been obtained for mouse
mtDNA by Huemer et al. (1971).
k3 =
10 4 d
1
Probability (per day) for a copy of mtDNA to suer damage
inicted by radicals. Although it is known that the mutation
rate of mitochondrial DNA in mammals is approximately ten
times that of nuclear DNA (Brown et al., 1979), the rate of
somatic mutations of mtDNA in man is unknown. Therefore
the yeast petite mutation rate (10 1 {10 3 ) (Ferguson and Vonborstel, 1992) has been used as a point of orientation which is
equivalent to a mutation rate of approximately 10 4 d 1 .
4 109
k4 =
Molecules of ATP needed to synthesize one mitochondrion. It
has been assumed that the main costs consist of synthesizing
the matrix proteins and the membrane proteins. To calculate
the costs the following assumptions have been made. A mitochondrion is a rod like structure 1m in diameter and 2m
long. 18% of its weight is made up of matrix proteins. The
average weight of a protein is 40000 daltons. The mitochondrial membrane consists of 75% protein which is equivalent to
ca. 25 lipid molecules per protein. The average diameter of a
lipid is 1.5 nm and 5.5 nm for a membrane protein. Finally
800 molecules of ATP are required to synthesize a protein.
Under these assumptions, which are based on data from Alberts et al. (1983) and Stryer (1988), an estimated cost of
approximately 4 109 molecules of ATP can be calculated.
k5
=
3
Constant controlling the strength of negative feedback for the
synthesis of new mitochondria.
k6
=
109
It the ATP level dropped to k6 then the cellular ATP consumption is also decreased to 50% of its maximum ATPd .
This value has to be seen in relation to ATPc .
GDF =
1.1
Growth dierence factor between healthy (M0 ) and completely damaged (M5 ) mitochondria.
RE
=
5
Radical enhancement factor. Completely damaged mitochondria produce RE times more radicals than intact mitochondria. This is a conservative value since it has been
estimated that mitochondrial mutations can raise the radical production 10{30 fold (Bandy and Davison, 1990).
SFmax =
2
ATPc =
10 109
Maximum by which transcription can be increased.
Cellular ATP level which the cell tries to maintain. The
ATP steady state level varies from species to species and
tissue to tissue. The ATP concentration in rat heart can
vary from 1000 5600M (Albe et al., 1990) which is equivalent to 2:4 109 : : : 13:5 109 ATP molecules per cell, if a
cell volume of 4000m3 is assumed.
ATPd =
3:8 1012
Amount of ATP consumed per day per cell. This value is
calculated under the assumption that the specic metabolic rate for humans is 0.25 ml of O2 per g/hr (Adelman
et al., 1988), that six molecules of ATP are generated per
molecule of oxygen, and that the cell volume is 4000m3 .
Rad0 = 86:4 106 s
1
Amount of radicals produced per day by one intact mitochondrion. Assuming that 106 radicals are produced per
second per cell (Joenje et al., 1985) and that a cell has an
average of 1000 mitochondria Rad0 is equal to 86:4 106 s 1 .
Rad1 5 :
Amount of radicals produced by the dierent classes of
mitochondria per day given in multiples of Rad0 .
Figure 1
M
0
M
1
M
2
RAD.
RAD.
O2
ATP
RAD.
RAD
M
3
M
4
M
5
RAD.
RAD.
.
1.2
12
1.0
10
0.8
8
0.6
6
0.4
4
0.2
2
0.0
0.0
0.2
0.4
0.6
D
0.8
1.0
0
1.2
Radical Generation
ATP Synthesis
Figure 2
Figure 3
=
=
=
=
=
=
=
=
=
dM0
dt
dM1
dt
dM2
dt
dM3
dt
dM4
dt
dM5
dt
dATP
dt
SFi
Radi
5
(5
1) + 5
4
C103 C C2 M2 + 3(GDF 5
5
6
6C 4C
4
1
M2 + 3(GDF 5
10 C
1) + 5
5
6C 4C
5
0
M
2
10 C
1) + 5
5
ATPc
ATPc
5
max
i=0
i (SF RE
i
5
5
i (SF
+(
1) + 1
1) + 1
i=0
Rad4 k3 + k2 )M4 k2 M5
5
5
X
X
5
ATP0 (1 5i )SFi Mi
i (GDF
ATP
ATPc
1) + 5
ATPc
5
1) + 5
8
C100 C C5 M4 + 5(GDF 5
5
2
ATP
1
kATP
k k4Mi ATPd ATP
+ k6
1+
Rad3 k3 (1 Rad3 k3 )M3 (Rad4 k3 + k2 )M4
4
6
k1
k5 3(GDF 5 1) + 5 C100 C C5 M3 + 4(GDF 5
1+
+2
+3
Rad2 k3 (1 Rad2 k3 )2 M2 (2 Rad3 k3 (1 Rad3 k3 ) + k2 )M3
6
4
4
6
1
kATP
k5 2(GDF 5 1) + 5 C101 C5C4 M2 + 3(GDF 5 1) + 5 C101 C5C4 M3 + 4(GDF 5
1+
+4
Rad1 k3 (1 Rad1 k3 )3 M1 (3 Rad2 k3 (1 Rad2 k3 )2 + k2 )M2
6
4
4
6
k1
ATP k5 2(GDF 5 1) + 5 C102 C5C3 M2 + 3(GDF 5 1) + 5 C102 C5C3 M3 + 4(GDF 5
1+
ATP
ATPc
k3 (1 k3 )4 M0 (4 Rad1 k3 (1 Rad1 k3 )3 + k2 )M1
8
2
k1
k5 (GDF 5 1) + 5 C103 C C2 M1 + 2(GDF 5
1+
+5
ATPc
k1
8
2
ATP k5 (GDF 5 1) + 5 C104 C5C1 M1 + 2(GDF 5
1+
ATPc
8
2
ATP k5 M0 + (GDF 5 1) + 5 C105 C5C0 M1 + 2(GDF 5
1+
k1
C103 C C2 M3
5
6
1) + 5
10
C010C C5 M5
5
0
8
C1 C4 M
4
10 C
1) + 5
5
2
2C 8C
2
3
M4
10 C
1) + 5
5
1) + 5
4
4C 6C
4
1
M
3
10 C
1) + 5
5
k3 (1 k3 )4 + k2 )M0
(13)
(12)
(11)
(10)
(9)
(8)
(7)
(6)
(5)
Figure 4
2000
SCALING:
1800
ATPcel
1600
Mo
M1
M2
M3
M4
M5
ATP
ATPcel
ATPmit
RAD
RAD
1400
1200
1000
Mo
800
600
400
ATPmit
200
M1 M2 M3 M4 M5
0
0
200
400
600
Time (days)
ATP
800
1000
=
=
=
=
=
=
=
=
=
=
1
1
1
1
1
1
5
0.5
1
1.5
Figure 5
2000
SCALING:
1800
1600
Mo
ATPcel
RAD
ATPcel
ATPcel
1400
RAD
RAD
1200
1000
800
600
Mo
Mo
400
200
0
0
0.0005
0.001
0.0015
K3
0.002
0.0025
= 1
= 0.5
= 1.5
Figure 6
1600
SCALING:
1400
RM
DM
1200
1000
AM
800
Mo
600
400
200
0
0
200
400
600
Time (Days)
800
1000
Mo
DM
RM
AM
=
=
=
=
1
1500
200
1000
Figure 7
Model Lifespan (Days)
50000
50000
40000
40000
30000
30000
20000
20000
10000
10000
0
0
0
0.001
0.05
0.052
0.054
0.002
0.056
0.058
Growth Rate (K1)
0.06
0.003
Mutation Rate (K3)
Figure Legends
Figure 1:
Reactions described by the \Defective Mitochondria" (DM) model. M0 represents
undamaged mitochondria while M1 to M5 represent mitochondria with one to ve
copies of the genome being damaged. All mitochondria consume oxygen and generate
ATP as well as radicals. Reactions with radicals (damage) cause the advance of a given
mitochondrion from one damage class to the next.
Figure 2:
ATP production (||) and radical generation (- - - -) depending on the amount of
damage (D). Both variables have been normalized by the corresponding production
rates of damage-free mitochondria. For the calculation SFmax = 2 and RE = 5 have
been used.
Figure 3:
Dierential equations of the \Defective Mitochondria" model. M0 : : : M5 represent
subclasses of mitochondria containing an increasing amount of damage. Radi (i = 1 to
5) is given in multiples of Rad0 .
Figure 4:
Standard simulation of the \Defective Mitochondria" model over 1000 days using the
default parameter values described in the appendix. The initial values for this simulation are: M0 = 1000, M1 { M5 = 0, and ATP = 10. Note that all values referring
to ATP or radical concentrations or consumptions are given for convenience in units of
109 .
Figure 5:
An increase in the mutation rate (k3 ) leads to a decline of the steady state concentrations and eventually to a collapse of the system. The diagram shows the results of
two simulations for dierent maximum growth rates (k1 ). The simulation with k1 =
0.055 (continuous lines) leads to a collapse at k3 10 3 . When k1 is increased to 0.06
(dashed lines) the point of breakdown is close to 2 10 3 .
Figure 6:
Behaviour of selected variables during a gradual breakdown of the system. The mutation rate (k3 ) has been shifted from k3 = 10
4
to k3 = 2:2 10 3 , which puts too
high a mutational load on the cell. As described in the text DM stands for the fraction
of damaged mitochondria, RM is the radical production per mitochondrion and AM
represents the ATP generation per mitochondrion.
Figure 7:
The model lifespan depends on the mutation rate (k3 ) and the maximum mitochondrial
growth rate (k1 ). The diagram summarizes the results of 90 simulations. The \cell" is
considered to be dead when the number of intact mitochondria falls below 10% of the
initial level.