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Mitochondrial mutations, cellular instability and ageing: modelling the population dynamics of mitochondria
Mutation Research, 295 (1993) 93-103
93
© 1993 Elsevier SciencePublishers B.V. All rights reserved 0921-8734/93/$06.00
MUTAGI 09095
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, London NW7 1AA, UK
(Received 14 August1992) (Revisionreceived30 October 1992) (Accepted 19 November1992) Keywords: Mitochondrialmutations;Ageing;DNA damage
Summary 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 homoeostasis 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 confirms 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 oxida-
Correspondence: Dr. A. Kowald,Laboratoryof Mathematical Biology,National Institutefor MedicalResearch, The Ridgeway, Mill Hill, LondonNW7 1AA,UK.
tive phosphorylation pathway. All other mitochondrial 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 ( O 2 , O H , 102, H202). 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
94 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 affect functions that are important for the energy production. However, because the genes which are important for mitochondrial reproduction are located in the nucleus, the affected mitochondrion can still propagate and there is a risk that the cellular population of healthy mitochondria will eventually be overtaken by non-functional 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 ophthalmoplegia (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) analysed denatured and reannealed mtDNA duplexes using electron microscopy and found a fivefold 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 summarises the features that are taken into account by the model. • Mitochondria are polyploid • Mitochondria produce ATP • Mitochondria produce oxygen radicals • Mitochondria suffer 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 five 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 different classes of mitochondria termed M 0 (none of the mtDNAs damaged and M1... M 5 (1-5 copies damaged, respectively). Fig. 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: S F = D ' ( S F m a x - 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). Fig. 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
95
r
Ml
ATP 02
t
"-
RAD~2 RAD ~
M 3
r
M4
r
M5
~
r
Fig. 1. Reactions described by the 'Defective Mitochondria' (DM) model. M 0 represents undamaged mitoehondria while M l - M 5 represent mitochondria with one to five copies of the genome being damaged. All mitoehondria 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.
1.2
12
1.0 ~ . ~ , , , ~
.10
, o,,
0.8
8
i
~ 0.4o oO''° 0.2 .'" .'" 0.0: 0.0
-°
~
I
0.4
. . . .
"4 i "2
~
.., . . . . . . ~ 0.2
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 R a d o per second. For mitochondria which have all mtDNAs damaged (D = 1) the excess radical generation is proportional to the increased transcription ( S F ) and a constant R E ( R a d o • S F . R E ) . 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 0 and 1 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 S F is substituted by Equation 1, the non-linear relationship results which is shown in Fig. 2.
I
0.6 D
. . . .
I
0.8
. . . .
I
1.0
"' 0 1.2
Fig. 2. ATP production ( ) and radical generation ( . . . . . . ) depending on the amount of damage (D). Both variables have been normalised by the corresponding production rates of damage-free mitochondria. For the calculation SFma x = 2 and RE = 5 have been used.
R a d = D . Rado( S F . R E - 1) + R a d o
(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 = k 3 for a radical level R a d o 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 diffusion 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 different radical levels ( R a d o . . . R a d 5) being associated with the different classes of mitochondria (M0... Ms). For convenience R a d o . . . R a d 5 are expressed in multiples of R a d o. 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 poly-
96 ploid, 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 (zl~) (Schleyer et al., 1982) across the inner mitochondrial membrane are 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 defined as the Growth Difference Factor between M 0 and M 5 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 first order decay term is used (k 2.M) to ensure a continuous degradation and thereby turn-over 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 mtDNAs remains constant the offspring of a M 1 mitochondrion (one mtDNA damaged) can be a M 0 mitochondrion (no mtDNA damaged). In general it holds that the probability for a M i mitochondrion to produce a Mj offspring is
(10-2i).
(2i)
(10) where ( ~ ) is the standard combinatorial term for
the number of ways of choosing b objects from a set of a. Model equations Using the information of the last sections it is now possible to develop the necessary differential equations (Fig. 3). The model consists of six formulae for mitochondria belonging to the different 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 different classes of existing organelles and terms which describe the migration of mitochondria from one class to the next. For convenience aCb has been used to de-
scribe the combinatorial term (b)" Equation 11 consists of three terms describing the generation and consumption of energy in the form of ATP. The first 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 influence 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 A T P d if there is a surplus of ATP and declining gradually if there is a low cellular energy charge. Auxiliary equations Fig. 3 shows the seven differential equations which constitute the core of the model. However, extra information can be obtained by additional non-differential equations. The following equations are derived from Equation 11 and describe the cellular energy consumption and the energy expended for synthesising new mitochondria, respectively. Hence they are rates (ATP/s) 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
97 dMo
dMl
kt
_
"(Mo4
(.4TP~ k~ 1 + \ ATP~/
dt _
dl
[
kl
1 + (a--T'~) .4TP k~ x
_
5
8C~.2Co t°Cs
5
81~'4 " ~C 1
( G D F - 1) + 5
toc5
6C~'4C'o
MI+ 2(GDF-I)+5
5
. 6C4. 4C1
• Mi + 2(GDF - 1) + 5
)
t o c - - - - - ~ • M_-, - ( 5 . k 3 ( l - k 3 ) 4 + k 2 ) , ~ l o
- - .I°C5 M~-~
5
(5)
)
4(-'4 " 6C'1
3(GDF-I)+5
~oc-~~- .Ma
/
+5 • k3( 1
dM._ dt
5 (GDF-I)+5
-
k3)4 Mo
kl l _ {are~ ~
-
(4. Radt
•
.- kA-r~7,/
(6)
k3( 1 - Radx - k3) 3 + k~)M,
( 5 \(GDF-1)+5
sC3. 'C.. 5 I°C-'----~'MI+ 2 ( G D F - I ) + 5 "
~C3" 4C._ - -l°C's . M 2 +
5
4Ca. ~C.~
3(GDF_I)+5
~ 1o6,
) .Ma
(7)
+ 4 - Radl • k3(1 - Radl • ka)aMt - (3- Rad2 . k3(1 - Rad2 • k3) 2 + k2)M2
dMz - -dt
=
kl { 5 T p ~) k'" ~ , 2 ( G D F - - 1 ) + 5 1 + {~aaa1%,
6C~. ., 4Cz 5 l°Cs "M~'+3(GDF-I)+5
5
46'2 • sC3
- -l°Cs . M a +
4(GDF_I)+5.
'C2. SCa 1o6,~
) .M4
(8)
+ 3 - Rad~ • kz(1 - Rad~ • ka) ~-M~, - (2- Rada . ka(1 - Rada • kz) + k2)Ma
dM4 - -dt
=
5
kl ( 5 6 C 1 . 4C 4 5 4C1- 6C4 . M 3 + ( a r / , ~ k~ " ~2(GDF--- 1 ) + 5 " 1oC-~-. M 2 + 3 ( G D F - 1 ) + 5 " - -1°C5 1 + ~,a-'T'P:,}
4(GDF_I)+5
"~Cl. *C4 ~ loc,
) .M4
(9)
+ 2 - Rad3 • ka(l - Rada • ka )Ma - ( Rad4 • kz + k2 )M4
dM5 dt
kl =
f
5
4Co- 6C's
l + ~[a_Yl~¢~ ATP ) ks " 3 ( G D F - - 1 ) + 5 "
5
'ZCo - SCs
5
l°C-----~'Ma÷ 4 ( G D F - I ) + 5 "
- - . t°6'5 M4"+
5(GDF- 1)+5
+(Rad4 • k3 + k2)M4 - k2Mn
dt
s
.
/=0
"~
°C0. l°C.~ t0C~
) M5 (10)
s
5
.= i . ( G D F - I ) + 5
kl
1 + (\ aATP r e , ] ~ ks k4 ~[i
i ,
SF~
=
g ( b F , .... - 1 ) + 1
(12)
Rad:
=
i F i'RE-I)+I 7.)(S
(13)
Fig. 3.
Differential equations of the 'Defective Mitochondria' model. M 0 . . . M 5 represent subclasses of mitochondria containing an increasing amount of damage. R a d i (i = 1 - 5 ) is given in multiples of Rad o.
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 flux of energy in different cellular compartments. ATP ATPceu = A T P a A T P + k 6
s ATPmito = E
i=O
(14)
dRad dt
5 i" ( G D F - 1) + 5
kl ATP I k5
l+
Arpc]
k,M,
The next equation describes the total cellular radical generation per unit time. d R a d / d t is given in multiples of Rad o 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 (Rado... Rad 5) it is important to know d R a d / d t to relate it to experimental results.
5 M o + }-'~Radi.M i
(16)
i=l
(15)
The final three equations calculate the fraction of damaged mitochondria with respect to intact mitochondria (DM), the amount of radicals generated per mitochondrion in multiples of Rad o
98 (RM), and the amount of ATP produced per m i t o c h o n d r i o n in m u l t i p l e s o f A T P o ( A M ) .
2000-
SCALING:
18001600-
. . . .
"~a~-
- - .--~
?,
1400-
M1 + M 2 + M 3 + M 4 + M 5 5 Y~i=oMi
DM=
Mo
=
ATPeel RAD
ffi 0.5 = 1.5
1
1200-
(17)
1000-
i
800600-
M o + Y'.Si=lRadiM i RM = 5
400-
(18)
Ei=0M/
200. 0
o.0~s 0.0oi oi0~ls 0A02 01~s K8
Mo+ E5 AM =
SFi
rs--E
~=0M,
(19)
Results T h e d y n a m i c b e h a v i o u r o f t h e m o d e l is illust r a t e d with s i m u l a t i o n s t h a t r e v e a l its m a i n c h a r acteristics.
Standard simulation W i t h a given set o f initial c o n c e n t r a t i o n s a n d using t h e d e f a u l t p a r a m e t e r v a l u e s as d e s c r i b e d in t h e A p p e n d i x t h e system c o n v e r g e s quickly to a stable, s t e a d y s t a t e c o n d i t i o n . T h i s is shown in Fig. 4 a n d will b e r e f e r r e d to as t h e s t a n d a r d simulation. Because different molecular species have very d i f f e r e n t s t e a d y s t a t e values, s e p a r a t e scaling f a c t o r s have b e e n u s e d to d i s p l a y all vari-
2000-
SCALING:
1800^a'r~
Mo
16001400-
~
1200 1000-
Mo
800600-
400200 01
MI
s -
M2
=
M3 M4 M5
-
ATP ATPcel ATP~t
-
0.5 1
RAD
=
1.5
-
Al'l~t M I M2 M~ ~ .... i .... 200
IdB ~ .... 400
A'I~ i .... 800
i .... 800
1000
Time (days)
Fig. 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: M 0 = 1000, M1-M s = 0, and ATP = 10. Note that all values referring to ATP or radical concentrations or consumptions are given for convenience in units of 10 9.
Fig. 5. An increase in the mutation r a t e ( k 3) 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 different maximum growth rates (k 1). The simulation with k~ = 0.055 (continuous lines) leads to a collapse at k 3 = 10 - 3 . When k I is increased to 0.06 (dashed lines) the point of breakdown is close to 2.10 - 3 .
a b l e s in o n e d i a g r a m . T h e o r d i n a t e v a l u e shown r e f e r s to a scaling f a c t o r o f 1. W i t h M 0 = 890 a n d M 1 = 8.7, m i t o c h o n d r i a with o n e d a m a g e d g e n o m e r e p r e s e n t just 1% o f t h e p o p u l a t i o n a n d t h e o t h e r subclasses exist only at levels close to zero. T h e total r a d i c a l g e n e r a tion r e a c h e s a c o n s t a n t v a l u e o f ca. 908 which is in t h e e x p e c t e d r a n g e since e r r o r - f r e e o r g a n e l l e s constitute the majority of the mitochondrial popu l a t i o n a n d t h e r a d i c a l g e n e r a t i o n is given in units o f R a d o. F u r t h e r m o r e t h e d e f a u l t p a r a m e t e r s a r e chosen so t h a t t h e A T P s t e a d y state level is very low ( A T P = 5.7) c o m p a r e d to t h e r a t e o f e n e r g y cons u m p t i o n by c e l l u l a r (hTecell = 3200) a n d m i t o c h o n d r i a l p r o c e s s e s (ATPmi t = 180). This reflects t h e fact t h a t t h e daily A T P e x p e n d i t u r e is m u c h g r e a t e r t h a n c e l l u l a r s t e a d y s t a t e levels o f A T P . N o t e t h a t t h e v a l u e s for t h e A T P a n d r a d i c a l s t e a d y state levels a n d c o n s u m p t i o n a r e given in units o f 109 .
Mutation rate and steady state concentrations T h e m a i n s t a t e m e n t o f t h e o r i e s which invoke d e f e c t i v e m i t o c h o n d r i a to explain t h e a g e i n g p r o cess is t h a t a high m u t a t i o n r a t e l e a d s to a g r a d ual loss o f intact m i t o c h o n d r i a a n d e v e n t u a l l y to a c o l l a p s e o f vital c e l l u l a r functions. Fig. 5 shows a s i m u l a t i o n t h a t investigates the i n f l u e n c e o f d i f f e r e n t m u t a t i o n r a t e s on t h e s t e a d y state concentrations.
99 The model confirms that with increasing mutation rate the population n u m b e r of mitochondria declines and eventually drops to zero when the mutation rate reaches a certain threshold. This threshold is modulated by the p a r a m e t e r kl which represents the maximum growth rate (per day) of the mitochondrial population. An increase of k 1 from 0.055 to 0.060 renders the system more robust and shifts the maximum endurable mutation rate from ca. 0 . 9 - 1 0 -3 to 1.9.10 -3 (per day).
Mutation rate and loss o f homoeostasis T r u e steady state levels can exist only if cellular homoeostasis can be maintained indefinitely. In the real world this condition is not fulfilled, 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. Fig. 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 k 3 = 10 -4 (which corresponds to a stable point in p a r a m e t e r space) have been calculated and then used as initial values for the computation with k 3 = 2 . 2 . 1 0 -3. Mutation rate and model lifespan From simulations like the one shown in Fig. 6, information about the relationship between mito-
soooo
ii
J
i o.os o.og2oog4
~
10~0
0.ool°° MutmlonRateIK3)
GrowthRate(K1)
Fig. 7. The model lifespan depends on the mutation rate (k 3) and the maximum mitochondrial growth rate (kl). The diagram summarises 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.
chondrial mutation rate, the maximum mitochondrial growth rate and the model lifespan can be obtained. This information is summarized in Fig. 7. T h e lifespan was defined as the time from the start of the simulation until the m o m e n t 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 kl. 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
x./
1~0] 1400
SCAL~G: Mo
- 1
Time(l:~ys)
Fig. 6. Behaviour of selected variables during a gradual breakdown of the system. The mutation rate (k 3) has been shifted from k3= 10-4 to k3= 2.2"10 -3, which puts too high a mutational load on the ceil. 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.
O f fundamental importance for ageing theories based on defective organelles is the mutation rate (Linnane et al., 1989; Miquel, 1991). In Fig. 5 the steady state values for damage-free mitochondria, the radical production and the cellular A T P 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 k 3 caused the complete breakdown ( M 0 = 0) of the system. F u r t h e r m o r e the ability of the system to tolerate mutations increases with increasing k 1. In fact it is the net growth rate ( k l - k2) of intact mitochondria which is the important factor. The point of the collapse is reached when the mutation rate
100 exceeds the net growth rate. As a good approximation this condition is fulfilled when 5 "k3(1 - k3) 4 > k 1 - k 2
(20)
If k 3 is increased further, the population of intact mitochondria decreases exponentially until M 0 = 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 Von Borstel, 1992) from which an estimate of ca. 10-2-10 -4 per day can be obtained. Depending on the growth rate k 1 the model simulations show a breakdown at 1 • 10 - 4 2 - 1 0 -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 k 3 and the maximum growth rate k 1 (Fig. 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 density of a cell population is controlled by two opposite factors: the proliferation rate of the cells, and the gene damage accumulation rate. Our finding is also in agreement with a 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 k 1 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 k 1 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 Fig. 6. As explained above, this exponential decrease is caused by too high a mutation rate. Concomitantly with the decline of M 0, 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 specific 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 flies (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 artificially 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
101 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.
Appendix Parameter definitions and default values k I = 0.06 d-1
k 2 = 0.05 d - 1
k 3 = 10 -4 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 synthesised. Using this value the whole mitochondrial population could doubl~ within 12 days. Fraction of all mitochondria which is degraded per day. With this value the first 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 m t D N A by H u e m e r et al. (1971). Probability (per day) for a copy of m t D N A to suffer damage inflicted by radicals. Although it is known that the mutation rate of mitochondrial D N A in man~mals is approximately 10 times that of nuclear D N A (Brown et
k 4 = 4.109
k5=3
al., 1979), the rate of somatic mutations of m t D N A in man is unknown. T h e r e f o r e the yeast petite mutation rate ( 1 0 - x - 1 0 -3) (Ferguson and Von Borstel, 1992) has been used as a point of orientation which is equivalent to a mutation rate of approximately 10 -4 d - l . Molecules of ATP needed to synthesise one mitochondrion. It has been assumed that the main costs consist of synthesising the matrix proteins and the membrane proteins. To calculate the costs the following assumptions have been made. A mitochondrion is a rod-like structure 1 /xm in diameter and 2 /zm long. 18% of its weight is made up to matrix proteins. The average weight of a protein is 40,000 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 A T P are required to synthesise a protein. Under these assumptions, which are based on data from AIberts et al. (1983) and Stryer (1988), an estimated cost of approximately 4 . 1 0 9 molecules of A T P can be calculated. Constant controlling the strength of negative feed-
102
k 6 = 10 9
GDF--- 1.1
RE = 5
SFma x = 2
A T P c = 10" 109
A T P d -- 3.8 • 1012
back for the synthesis of new mitochondria. If the ATP level dropped t o k 6 then the cellular A T P consumption is also decreased to 50% of its maximum A T P d. This value has to be seen in relation to A T P c. Growth difference factor between healthy ( M 0) and completely damaged ( M 5) mitochondria. 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). Maximum by which transcription can be increased. Cellular ATP level which the cell tries to maintain. The A T P steady state level varies from species to species and tissue to tissue. The ATP concentration in rat heart can vary from 1000 to 5600 /zM (Albe et al., 1990) which is equivalent to 2.4 • 109 to 13.5" 109 ATP molecules per cell, if a cell volume of 4000 ~ m 3 is assumed. Amount of ATP consumed per day per cell. This value is calculated under the assumption that the specific metabolic rate for humans is 0.25 ml of 0 2 per g / h (Adelman et
R a d o = 86.4.106 s -
Radl-5
al., 1988), that six molecules of A T P are generated per molecule of oxygen, and that the cell volume is 4000/zm 3. 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 R a d o is equal to 86.4" 10 6 S - 1. Amount of radicals produced by the different classes of mitochondria per day given in multiples of R a d o.
Acknowledgements A. Kowald thanks the Carl Duisberg Stiftung, the British Council and the Max Buchner Forschungsstiftung for financial support.
References Adelman, R., R.L. Saul and B.N. Ames (1988) Oxidative damage to DNA: Relation to species metabolic rate and life span, Proc. Natl. Acad. Sci. USA, 85, 2706-2708. Albe, K.R., M.H. Butler and B.E. Wright (1990) Cellular concentrations of enzymes and their substrates, J. Theor. Biol., 143, 163-195. Alberts, B., D. Bray, J. Lewis, M. Raft, K. Roberts and J.D. Watson (1983) Molecular Biology of the Cell, Garland, New York, NY. Bandy, B., and A.J. Davison (1990) Mitochondrial mutations may increase oxidative stress: Implications for carcinogenesis and ageing?, Free Radical Biol. Med., 8, 523-539. Bodenteich, A., L.G. Mitchell and C.R. Merril (1991) A lifetime of retinal light exposure does not appear to increase mitochondrial mutations, Gene, 108, 305-310. Brown, W.M., M. George Jr, and A.C. Wilson (1979) Rapid evolution of animal mitochondrial DNA, Proc. Natl. Acad. Sci. USA, 76, 1967-1971. Byrne, E., X. Dennett and I. Trounce (1991) Oxidative energy failure in post-mitotic cells: A major factor in senescence, Rev. Neurol. (Paris), 147, 6-7. Cardellach, F., J. Galofre, R. Cusso and A. Urbano-Marquez (1989) Decline in skeletal muscle mitochondrial respiratory chain function with ageing, Lancet, ii, 44-45.
103 Farmer, K.J., and R.S. Sohal (1989) Relationship between superoxide anion radical generation and aging in the housefly, Musca domestica, Free Radical Biol. Med., 7, 23-29. Ferguson, L.R., and R.C. Von Borstel (1992) Induction of the cytoplasmic petite mutation by chemical and physical agents in Saccharomyces cerevisiae, Mutation Res., 265, 103-148. Fleming, J.E. (1986) Role of mitochondria in Drosophila aging, in: K. Collatz and R. Sohal (Eds.), Insect Aging, Strategies and Mechanisms, Springer, Berlin, pp. 130-141. Harman, D. (1983) Free radical theory of aging: consequences of mitochondrial aging, Age, 6, 86-94. Harmey, M.A., G. Hallermayer, H. Korb and W. Neupert (1977) Transport of cytoplasmically synthesized proteins into the mitochondria in a cell free system from Neurspora crassa, Eur. J. Biochem., 81,533-544. Hattori, K., M. Tanaka, S. Sugiyama, T. Obayashi, T. Ito, T. Satake, Y. Hanaki, J. Asai, M. Nagano and T. Ozawa (1991) Age-dependent increase in deleted mitochondrial DNA in the human heart: Possible contributory factor to preshycardia, Am. Heart J., 121, 1735-1742. Holt, I.J., J.M. Cooper and J.A. Morgan-Hughes (1988) Deletions of muscle mitochondrial DNA, Lancet, i, 1462. Huemer, R.P., K.D. Lee, A.E. Reeves and C. Bickert (1971) Mitochondrial studies in senescent mice. II. Specific activity, buoyant density, and turnover of mitochondrial DNA, Gerontology, 6, 327-334. Joenje, H., J.J.P. Giile, A.B. Oostra and P. van der Valk (1985) Some characteristics of hyperoxia-adapted HeLa cells, Lab. Invest., 52, 420-428. Kirkwood, T.B.L. (1981) Repair and its evolution: Survival versus reproduction, in: C. Townsend and P. Calow (Eds.), Physiological Ecology: An Evolutionary Approach to Resource Use, Blackwell Scientific, Oxford, pp. 165-189. Linnane, A.W., S. Marzuki, T. Ozawa and M. Tanaka (1989) Mitochondrial DNA mutations as an important contributor to ageing and degenerative diseases, Lancet, i, 642-645. Linnane, A.W., A. Baumer, R.J. Maxwell, H. Preston, C. Zhang and S. Marzuki (1990) Mitochondrial gene mutation: The ageing process and degenerative diseases, Biochem. Int., 22, 1067-1076. Manzelmann, M.S., and H.J. Harmon (1987) Lack of age-dependent changes in rat heart mitochondria, Mech. Ageing Dev., 39, 281-288. Massie, H.R., M.B. Baird and M.M. McMahon (1975) Loss of mitochondrial DNA with aging in Drosophila melanogaster, Gerontologia, 21, 231-238. Menzies, R.A., and P.H. Gold (1971) The turnover of mitochondria in a variety of tissues of young adult and aged rats, J. Biol. Chem., 246, 2425-2429. Miquel, J. (1991) An integrated theory of aging as the result of mitochondrial-DNA mutation in differential cells, Arch. Gerontol. Geriatr., 12, 99-117.
Miquel, J., and J. Fleming (1986) Theoretical and experimental support for an 'Oxygen Radical-Mitochondrial Injury' hypothesis of cell aging, J. Modern Aging Res., 8, 51-74. Miquel, J., A.C. Economos, J. Fleming and J.E. Johnson (1980) Mitochondrial role in cell ageing, Exp. Gerontol., 15, 575-591. Munkres, K.D. (1985) Aging of fungi, Rev. Biol. Res. Aging, 2, 29-43. Nikoskelainen, E.K., M.L. Savontaus, O.P. Wanne, M.J. Katila and K.U. Nummelin (1987) Leber's hereditary optic neuroretinopathy, a maternally inherited disease. A genealogic study in four pedigrees, Arch. Ophthalmol., 105, 665. Piko, L , A.J. Hougham and K.J. Bulpitt (1988) Studies of sequence heterogeneity of mitochondrial DNA from rat and mouse tissues: evidence for an increased frequency of deletions/additions with aging, Mech. Ageing Dev., 43, 279-293. Richter, C. (1988) Do mitochondrial DNA fragments promote cancer and aging?, FEBS Lett., 241, 1-5. Satoh, M., and T. Kuroiwa (1991) Organization of multiple nucleoids and DNA molecules in mitochondria of a human cell, Exp. Cell Res., 196, 137-140. Sawada, M., and J.C. Carlson (1987) Changes in superoxide radical and lipid peroxide formation in the brain, heart and liver during the lifetime of the rat, Mech. Ageing Dev., 41, 125-137. Schleyer, M., B. Schmidt and W. Neupert (1982) Requirement of a membrane potential for the posttranslational transfer of proteins into mitochondria, Eur. J. Biochem., 125, 109116. Sohal, R.S., and B.H. Sohal (1991) Hydrogen peroxide release by mitochondria increases during aging, Mech. Ageing Dev., 57, 187-202. Stryer, L. (1988) Biochemistry, 3rd edn., Freeman, New York, NY. Tauchi, H., and T. Sato (1968) Age changes in size and number of mitochondria of human hepatic cells, J. Geront., 23, 454-461. Wallace, D.C., G. Singh, M.T. Lott, J.A. Hodge, T.G. Schurr, A.M.S. Lezza, L.J. Elsas II and E.K. Nikoskelainen (1988) Mitochondrial DNA mutation associated with Leber's hereditary optic neuropathy, Science, 242, 1427. Yen, T.C., Y.S. Chen, K.L. King, S.H. Yeh and Y.H. Weih (1990) Liver mitochondrial respiratory functions decline with age, Biochem. Biophys. Res. Commun., 165, 9941003. Yen, T.-C., J.-H. Su, K.-L. King and Y.-H. Wei (1991) Ageing-associated 5 kb deletion in human liver mitochondrial DNA, Biochem. Biophys. Res. Commun., 178, 124131. Zheng, T. (1991) A mathematical model of proliferation and aging of cells in culture, J. Theor. Biol., 149, 287-315.
93
© 1993 Elsevier SciencePublishers B.V. All rights reserved 0921-8734/93/$06.00
MUTAGI 09095
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, London NW7 1AA, UK
(Received 14 August1992) (Revisionreceived30 October 1992) (Accepted 19 November1992) Keywords: Mitochondrialmutations;Ageing;DNA damage
Summary 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 homoeostasis 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 confirms 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 oxida-
Correspondence: Dr. A. Kowald,Laboratoryof Mathematical Biology,National Institutefor MedicalResearch, The Ridgeway, Mill Hill, LondonNW7 1AA,UK.
tive phosphorylation pathway. All other mitochondrial 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 ( O 2 , O H , 102, H202). 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
94 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 affect functions that are important for the energy production. However, because the genes which are important for mitochondrial reproduction are located in the nucleus, the affected mitochondrion can still propagate and there is a risk that the cellular population of healthy mitochondria will eventually be overtaken by non-functional 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 ophthalmoplegia (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) analysed denatured and reannealed mtDNA duplexes using electron microscopy and found a fivefold 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 summarises the features that are taken into account by the model. • Mitochondria are polyploid • Mitochondria produce ATP • Mitochondria produce oxygen radicals • Mitochondria suffer 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 five 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 different classes of mitochondria termed M 0 (none of the mtDNAs damaged and M1... M 5 (1-5 copies damaged, respectively). Fig. 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: S F = D ' ( S F m a x - 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). Fig. 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
95
r
Ml
ATP 02
t
"-
RAD~2 RAD ~
M 3
r
M4
r
M5
~
r
Fig. 1. Reactions described by the 'Defective Mitochondria' (DM) model. M 0 represents undamaged mitoehondria while M l - M 5 represent mitochondria with one to five copies of the genome being damaged. All mitoehondria 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.
1.2
12
1.0 ~ . ~ , , , ~
.10
, o,,
0.8
8
i
~ 0.4o oO''° 0.2 .'" .'" 0.0: 0.0
-°
~
I
0.4
. . . .
"4 i "2
~
.., . . . . . . ~ 0.2
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 R a d o per second. For mitochondria which have all mtDNAs damaged (D = 1) the excess radical generation is proportional to the increased transcription ( S F ) and a constant R E ( R a d o • S F . R E ) . 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 0 and 1 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 S F is substituted by Equation 1, the non-linear relationship results which is shown in Fig. 2.
I
0.6 D
. . . .
I
0.8
. . . .
I
1.0
"' 0 1.2
Fig. 2. ATP production ( ) and radical generation ( . . . . . . ) depending on the amount of damage (D). Both variables have been normalised by the corresponding production rates of damage-free mitochondria. For the calculation SFma x = 2 and RE = 5 have been used.
R a d = D . Rado( S F . R E - 1) + R a d o
(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 = k 3 for a radical level R a d o 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 diffusion 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 different radical levels ( R a d o . . . R a d 5) being associated with the different classes of mitochondria (M0... Ms). For convenience R a d o . . . R a d 5 are expressed in multiples of R a d o. 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 poly-
96 ploid, 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 (zl~) (Schleyer et al., 1982) across the inner mitochondrial membrane are 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 defined as the Growth Difference Factor between M 0 and M 5 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 first order decay term is used (k 2.M) to ensure a continuous degradation and thereby turn-over 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 mtDNAs remains constant the offspring of a M 1 mitochondrion (one mtDNA damaged) can be a M 0 mitochondrion (no mtDNA damaged). In general it holds that the probability for a M i mitochondrion to produce a Mj offspring is
(10-2i).
(2i)
(10) where ( ~ ) is the standard combinatorial term for
the number of ways of choosing b objects from a set of a. Model equations Using the information of the last sections it is now possible to develop the necessary differential equations (Fig. 3). The model consists of six formulae for mitochondria belonging to the different 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 different classes of existing organelles and terms which describe the migration of mitochondria from one class to the next. For convenience aCb has been used to de-
scribe the combinatorial term (b)" Equation 11 consists of three terms describing the generation and consumption of energy in the form of ATP. The first 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 influence 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 A T P d if there is a surplus of ATP and declining gradually if there is a low cellular energy charge. Auxiliary equations Fig. 3 shows the seven differential equations which constitute the core of the model. However, extra information can be obtained by additional non-differential equations. The following equations are derived from Equation 11 and describe the cellular energy consumption and the energy expended for synthesising new mitochondria, respectively. Hence they are rates (ATP/s) 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
97 dMo
dMl
kt
_
"(Mo4
(.4TP~ k~ 1 + \ ATP~/
dt _
dl
[
kl
1 + (a--T'~) .4TP k~ x
_
5
8C~.2Co t°Cs
5
81~'4 " ~C 1
( G D F - 1) + 5
toc5
6C~'4C'o
MI+ 2(GDF-I)+5
5
. 6C4. 4C1
• Mi + 2(GDF - 1) + 5
)
t o c - - - - - ~ • M_-, - ( 5 . k 3 ( l - k 3 ) 4 + k 2 ) , ~ l o
- - .I°C5 M~-~
5
(5)
)
4(-'4 " 6C'1
3(GDF-I)+5
~oc-~~- .Ma
/
+5 • k3( 1
dM._ dt
5 (GDF-I)+5
-
k3)4 Mo
kl l _ {are~ ~
-
(4. Radt
•
.- kA-r~7,/
(6)
k3( 1 - Radx - k3) 3 + k~)M,
( 5 \(GDF-1)+5
sC3. 'C.. 5 I°C-'----~'MI+ 2 ( G D F - I ) + 5 "
~C3" 4C._ - -l°C's . M 2 +
5
4Ca. ~C.~
3(GDF_I)+5
~ 1o6,
) .Ma
(7)
+ 4 - Radl • k3(1 - Radl • ka)aMt - (3- Rad2 . k3(1 - Rad2 • k3) 2 + k2)M2
dMz - -dt
=
kl { 5 T p ~) k'" ~ , 2 ( G D F - - 1 ) + 5 1 + {~aaa1%,
6C~. ., 4Cz 5 l°Cs "M~'+3(GDF-I)+5
5
46'2 • sC3
- -l°Cs . M a +
4(GDF_I)+5.
'C2. SCa 1o6,~
) .M4
(8)
+ 3 - Rad~ • kz(1 - Rad~ • ka) ~-M~, - (2- Rada . ka(1 - Rada • kz) + k2)Ma
dM4 - -dt
=
5
kl ( 5 6 C 1 . 4C 4 5 4C1- 6C4 . M 3 + ( a r / , ~ k~ " ~2(GDF--- 1 ) + 5 " 1oC-~-. M 2 + 3 ( G D F - 1 ) + 5 " - -1°C5 1 + ~,a-'T'P:,}
4(GDF_I)+5
"~Cl. *C4 ~ loc,
) .M4
(9)
+ 2 - Rad3 • ka(l - Rada • ka )Ma - ( Rad4 • kz + k2 )M4
dM5 dt
kl =
f
5
4Co- 6C's
l + ~[a_Yl~¢~ ATP ) ks " 3 ( G D F - - 1 ) + 5 "
5
'ZCo - SCs
5
l°C-----~'Ma÷ 4 ( G D F - I ) + 5 "
- - . t°6'5 M4"+
5(GDF- 1)+5
+(Rad4 • k3 + k2)M4 - k2Mn
dt
s
.
/=0
"~
°C0. l°C.~ t0C~
) M5 (10)
s
5
.= i . ( G D F - I ) + 5
kl
1 + (\ aATP r e , ] ~ ks k4 ~[i
i ,
SF~
=
g ( b F , .... - 1 ) + 1
(12)
Rad:
=
i F i'RE-I)+I 7.)(S
(13)
Fig. 3.
Differential equations of the 'Defective Mitochondria' model. M 0 . . . M 5 represent subclasses of mitochondria containing an increasing amount of damage. R a d i (i = 1 - 5 ) is given in multiples of Rad o.
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 flux of energy in different cellular compartments. ATP ATPceu = A T P a A T P + k 6
s ATPmito = E
i=O
(14)
dRad dt
5 i" ( G D F - 1) + 5
kl ATP I k5
l+
Arpc]
k,M,
The next equation describes the total cellular radical generation per unit time. d R a d / d t is given in multiples of Rad o 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 (Rado... Rad 5) it is important to know d R a d / d t to relate it to experimental results.
5 M o + }-'~Radi.M i
(16)
i=l
(15)
The final three equations calculate the fraction of damaged mitochondria with respect to intact mitochondria (DM), the amount of radicals generated per mitochondrion in multiples of Rad o
98 (RM), and the amount of ATP produced per m i t o c h o n d r i o n in m u l t i p l e s o f A T P o ( A M ) .
2000-
SCALING:
18001600-
. . . .
"~a~-
- - .--~
?,
1400-
M1 + M 2 + M 3 + M 4 + M 5 5 Y~i=oMi
DM=
Mo
=
ATPeel RAD
ffi 0.5 = 1.5
1
1200-
(17)
1000-
i
800600-
M o + Y'.Si=lRadiM i RM = 5
400-
(18)
Ei=0M/
200. 0
o.0~s 0.0oi oi0~ls 0A02 01~s K8
Mo+ E5 AM =
SFi
rs--E
~=0M,
(19)
Results T h e d y n a m i c b e h a v i o u r o f t h e m o d e l is illust r a t e d with s i m u l a t i o n s t h a t r e v e a l its m a i n c h a r acteristics.
Standard simulation W i t h a given set o f initial c o n c e n t r a t i o n s a n d using t h e d e f a u l t p a r a m e t e r v a l u e s as d e s c r i b e d in t h e A p p e n d i x t h e system c o n v e r g e s quickly to a stable, s t e a d y s t a t e c o n d i t i o n . T h i s is shown in Fig. 4 a n d will b e r e f e r r e d to as t h e s t a n d a r d simulation. Because different molecular species have very d i f f e r e n t s t e a d y s t a t e values, s e p a r a t e scaling f a c t o r s have b e e n u s e d to d i s p l a y all vari-
2000-
SCALING:
1800^a'r~
Mo
16001400-
~
1200 1000-
Mo
800600-
400200 01
MI
s -
M2
=
M3 M4 M5
-
ATP ATPcel ATP~t
-
0.5 1
RAD
=
1.5
-
Al'l~t M I M2 M~ ~ .... i .... 200
IdB ~ .... 400
A'I~ i .... 800
i .... 800
1000
Time (days)
Fig. 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: M 0 = 1000, M1-M s = 0, and ATP = 10. Note that all values referring to ATP or radical concentrations or consumptions are given for convenience in units of 10 9.
Fig. 5. An increase in the mutation r a t e ( k 3) 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 different maximum growth rates (k 1). The simulation with k~ = 0.055 (continuous lines) leads to a collapse at k 3 = 10 - 3 . When k I is increased to 0.06 (dashed lines) the point of breakdown is close to 2.10 - 3 .
a b l e s in o n e d i a g r a m . T h e o r d i n a t e v a l u e shown r e f e r s to a scaling f a c t o r o f 1. W i t h M 0 = 890 a n d M 1 = 8.7, m i t o c h o n d r i a with o n e d a m a g e d g e n o m e r e p r e s e n t just 1% o f t h e p o p u l a t i o n a n d t h e o t h e r subclasses exist only at levels close to zero. T h e total r a d i c a l g e n e r a tion r e a c h e s a c o n s t a n t v a l u e o f ca. 908 which is in t h e e x p e c t e d r a n g e since e r r o r - f r e e o r g a n e l l e s constitute the majority of the mitochondrial popu l a t i o n a n d t h e r a d i c a l g e n e r a t i o n is given in units o f R a d o. F u r t h e r m o r e t h e d e f a u l t p a r a m e t e r s a r e chosen so t h a t t h e A T P s t e a d y state level is very low ( A T P = 5.7) c o m p a r e d to t h e r a t e o f e n e r g y cons u m p t i o n by c e l l u l a r (hTecell = 3200) a n d m i t o c h o n d r i a l p r o c e s s e s (ATPmi t = 180). This reflects t h e fact t h a t t h e daily A T P e x p e n d i t u r e is m u c h g r e a t e r t h a n c e l l u l a r s t e a d y s t a t e levels o f A T P . N o t e t h a t t h e v a l u e s for t h e A T P a n d r a d i c a l s t e a d y state levels a n d c o n s u m p t i o n a r e given in units o f 109 .
Mutation rate and steady state concentrations T h e m a i n s t a t e m e n t o f t h e o r i e s which invoke d e f e c t i v e m i t o c h o n d r i a to explain t h e a g e i n g p r o cess is t h a t a high m u t a t i o n r a t e l e a d s to a g r a d ual loss o f intact m i t o c h o n d r i a a n d e v e n t u a l l y to a c o l l a p s e o f vital c e l l u l a r functions. Fig. 5 shows a s i m u l a t i o n t h a t investigates the i n f l u e n c e o f d i f f e r e n t m u t a t i o n r a t e s on t h e s t e a d y state concentrations.
99 The model confirms that with increasing mutation rate the population n u m b e r of mitochondria declines and eventually drops to zero when the mutation rate reaches a certain threshold. This threshold is modulated by the p a r a m e t e r kl which represents the maximum growth rate (per day) of the mitochondrial population. An increase of k 1 from 0.055 to 0.060 renders the system more robust and shifts the maximum endurable mutation rate from ca. 0 . 9 - 1 0 -3 to 1.9.10 -3 (per day).
Mutation rate and loss o f homoeostasis T r u e steady state levels can exist only if cellular homoeostasis can be maintained indefinitely. In the real world this condition is not fulfilled, 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. Fig. 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 k 3 = 10 -4 (which corresponds to a stable point in p a r a m e t e r space) have been calculated and then used as initial values for the computation with k 3 = 2 . 2 . 1 0 -3. Mutation rate and model lifespan From simulations like the one shown in Fig. 6, information about the relationship between mito-
soooo
ii
J
i o.os o.og2oog4
~
10~0
0.ool°° MutmlonRateIK3)
GrowthRate(K1)
Fig. 7. The model lifespan depends on the mutation rate (k 3) and the maximum mitochondrial growth rate (kl). The diagram summarises 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.
chondrial mutation rate, the maximum mitochondrial growth rate and the model lifespan can be obtained. This information is summarized in Fig. 7. T h e lifespan was defined as the time from the start of the simulation until the m o m e n t 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 kl. 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
x./
1~0] 1400
SCAL~G: Mo
- 1
Time(l:~ys)
Fig. 6. Behaviour of selected variables during a gradual breakdown of the system. The mutation rate (k 3) has been shifted from k3= 10-4 to k3= 2.2"10 -3, which puts too high a mutational load on the ceil. 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.
O f fundamental importance for ageing theories based on defective organelles is the mutation rate (Linnane et al., 1989; Miquel, 1991). In Fig. 5 the steady state values for damage-free mitochondria, the radical production and the cellular A T P 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 k 3 caused the complete breakdown ( M 0 = 0) of the system. F u r t h e r m o r e the ability of the system to tolerate mutations increases with increasing k 1. In fact it is the net growth rate ( k l - k2) of intact mitochondria which is the important factor. The point of the collapse is reached when the mutation rate
100 exceeds the net growth rate. As a good approximation this condition is fulfilled when 5 "k3(1 - k3) 4 > k 1 - k 2
(20)
If k 3 is increased further, the population of intact mitochondria decreases exponentially until M 0 = 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 Von Borstel, 1992) from which an estimate of ca. 10-2-10 -4 per day can be obtained. Depending on the growth rate k 1 the model simulations show a breakdown at 1 • 10 - 4 2 - 1 0 -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 k 3 and the maximum growth rate k 1 (Fig. 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 density of a cell population is controlled by two opposite factors: the proliferation rate of the cells, and the gene damage accumulation rate. Our finding is also in agreement with a 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 k 1 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 k 1 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 Fig. 6. As explained above, this exponential decrease is caused by too high a mutation rate. Concomitantly with the decline of M 0, 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 specific 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 flies (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 artificially 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
101 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.
Appendix Parameter definitions and default values k I = 0.06 d-1
k 2 = 0.05 d - 1
k 3 = 10 -4 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 synthesised. Using this value the whole mitochondrial population could doubl~ within 12 days. Fraction of all mitochondria which is degraded per day. With this value the first 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 m t D N A by H u e m e r et al. (1971). Probability (per day) for a copy of m t D N A to suffer damage inflicted by radicals. Although it is known that the mutation rate of mitochondrial D N A in man~mals is approximately 10 times that of nuclear D N A (Brown et
k 4 = 4.109
k5=3
al., 1979), the rate of somatic mutations of m t D N A in man is unknown. T h e r e f o r e the yeast petite mutation rate ( 1 0 - x - 1 0 -3) (Ferguson and Von Borstel, 1992) has been used as a point of orientation which is equivalent to a mutation rate of approximately 10 -4 d - l . Molecules of ATP needed to synthesise one mitochondrion. It has been assumed that the main costs consist of synthesising the matrix proteins and the membrane proteins. To calculate the costs the following assumptions have been made. A mitochondrion is a rod-like structure 1 /xm in diameter and 2 /zm long. 18% of its weight is made up to matrix proteins. The average weight of a protein is 40,000 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 A T P are required to synthesise a protein. Under these assumptions, which are based on data from AIberts et al. (1983) and Stryer (1988), an estimated cost of approximately 4 . 1 0 9 molecules of A T P can be calculated. Constant controlling the strength of negative feed-
102
k 6 = 10 9
GDF--- 1.1
RE = 5
SFma x = 2
A T P c = 10" 109
A T P d -- 3.8 • 1012
back for the synthesis of new mitochondria. If the ATP level dropped t o k 6 then the cellular A T P consumption is also decreased to 50% of its maximum A T P d. This value has to be seen in relation to A T P c. Growth difference factor between healthy ( M 0) and completely damaged ( M 5) mitochondria. 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). Maximum by which transcription can be increased. Cellular ATP level which the cell tries to maintain. The A T P steady state level varies from species to species and tissue to tissue. The ATP concentration in rat heart can vary from 1000 to 5600 /zM (Albe et al., 1990) which is equivalent to 2.4 • 109 to 13.5" 109 ATP molecules per cell, if a cell volume of 4000 ~ m 3 is assumed. Amount of ATP consumed per day per cell. This value is calculated under the assumption that the specific metabolic rate for humans is 0.25 ml of 0 2 per g / h (Adelman et
R a d o = 86.4.106 s -
Radl-5
al., 1988), that six molecules of A T P are generated per molecule of oxygen, and that the cell volume is 4000/zm 3. 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 R a d o is equal to 86.4" 10 6 S - 1. Amount of radicals produced by the different classes of mitochondria per day given in multiples of R a d o.
Acknowledgements A. Kowald thanks the Carl Duisberg Stiftung, the British Council and the Max Buchner Forschungsstiftung for financial support.
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