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Mitochondrial mutations and ageing: Can mitochondrial deletion mutants accumulate via a size based replication advantage?
Journal of Theoretical Biology 340 (2014) 111–118
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Mitochondrial mutations and ageing: Can mitochondrial deletion mutants accumulate via a size based replication advantage? Axel Kowald a,n, Marcus Dawson a,b, Thomas B.L. Kirkwood a a Centre for Integrated Systems Biology of Ageing and Nutrition, Institute for Ageing and Health, Newcastle University, Newcastle upon Tyne NE4 5PL, United Kingdom b University of Manchester, Manchester M13 9PT, United Kingdom
H I G H L I G H T S
We modelled the accumulation of mitochondrial deletion mutants during ageing. Can the reduced size provide a selection advantage via a shorter replication time? We developed a delay differential equation model and a stochastic simulation. The simulations show that the idea only works for very long lived species.
art ic l e i nf o
a b s t r a c t
Article history: Received 21 February 2013 Received in revised form 22 August 2013 Accepted 9 September 2013 Available online 17 September 2013
The mitochondrial theory of ageing is one of the main contenders to explain the biochemical basis of the ageing process. An important line of support comes from the observation that mtDNA deletions accumulate over the life course in post-mitotic cells of many species. A single mutant expands clonally and finally replaces the wild-type population of a whole cell. One proposal to explain the driving force behind this accumulation states that the reduced size leads to a shorter replication time, which provides a selection advantage. However, this idea has been questioned on the grounds that the mitochondrial half-life is much longer than the replication time, so that the latter cannot be a rate limiting step. To clarify this question, we modelled this process mathematically and performed extensive deterministic and stochastic computer simulations to study the effects of replication time, mitochondrial half-life and deletion size. Our study shows that the shorter size does in principle provide a selection advantage, which can lead to an accumulation of the deletion mutant. However, this selection advantage diminishes the shorter is the replication time of wt mtDNA in relation to its half-life. Using generally accepted literature values, the resulting time frame for the accumulation of mutant mtDNAs is only compatible with the ageing process in very long lived species like humans, but could not reasonably explain ageing in short lived species like mice and rats. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Mitochondrial mutation Mathematical model Ageing
1. Introduction Ageing is an intrinsic deterioration of the homoeostatic capabilities of an organism, leading to a constantly increasing risk of death. Although evolutionary considerations suggest that the ageing process is in principle multifactorial (Kirkwood, 1996), a few main mechanisms have been proposed. Among these the mitochondrial theory of ageing is one of the most popular (Harman, 1972, 1983; Miquel et al., 1980; Richter, 1988; Linnane et al., 1989). The theory suggests that the accumulation of defective mitochondria is a major
n
Corresponding author. Tel.: þ 49 179 3427522. E-mail addresses: [email protected] (A. Kowald), Marcus.Dawson@student. manchester.ac.uk (M. Dawson), [email protected] (T.B.L. Kirkwood). 0022-5193/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2013.09.009
contributor to the ageing process. Reactive oxygen species (ROS) generated during respiration have the potential to damage all kinds of biologically relevant macromolecules such as lipids, proteins and mitochondrial DNA (mtDNA). Damage to mtDNA is quite different from damage to other macromolecules, since mitochondrial DNA represents the ultimate blueprint from which everything else follows. Point mutations and deletions could impair mitochondrial ATP production with negative consequences for all aspects of cellular homoeostasis. And indeed, many studies have shown that mitochondrial deletion mutants accumulate with age in various mammalian species such as rats, monkeys and humans (Brierley et al., 1998; Khrapko et al., 1999; Cao et al., 2001; Gokey et al., 2004; Herbst et al., 2007). These single cell studies have shown that the mitochondrial population of a cell is overtaken by a single deletion mutant type via clonal expansion.
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The mechanism behind the accumulation of defective mitochondria is obviously an important aspect of the theory, but is currently unknown. The “vicious cycle” hypothesis suggests that, because defective mitochondria generate more radicals than intact ones, the cellular radical level progressively rises, leading in turn to an increased production of mitochondrial mutants (Bandy and Davison, 1990; Arnheim and Cortopassi, 1992). In this scenario the individual mutant mitochondria do not need to have a selection advantage; they accumulate because they are constantly generated. However, such a mechanism also implies that one should see a plethora of different mtDNA mutants in a single cell, quite the contrary of what has been observed. It has also been proposed that pure random drift might be sufficient to explain clonal expansion (Chinnery and Samuels, 1999; Elson et al., 2001). Although these authors show that such a process might work for long lived species like humans, we have recently performed computer simulations, which demonstrate that such a process is unsuitable for short lived animals like rodents (Kowald and Kirkwood, 2013), which show a similar pattern of accumulation as in humans but on a twenty times accelerated timescale (Cao et al., 2001). An idea that does identify a selection advantage of deletion mutants is called survival of the slowest (SOS) (Grey, 1997). It notes that the fate of a mitochondrion depends not only on its growth rate, but also on its rate of degradation and proposes that defective mitochondria are degraded less frequently than wild-type organelles. However, this hypothesis has problems with mitochondrial dynamics, since fission and fusion break the required link between genotype and phenotype (Kowald and Kirkwood, 2011). Furthermore, it has been shown that dysfunctional mitochondria are preferentially degraded (Twig et al., 2008; Kim and Lemasters, 2011), instead of being spared. Finally, it has been proposed that the selection advantage simply stems from the reduced genome size of the deletion mutant (Wallace, 1992; Lee et al., 1998). Deletions encompassing more than half of the mitochondrial genome have been observed (Cao et al., 2001), which could result in a 50% reduced replication time. However, this idea has been criticised because early studies have shown that the time required for the replication of the mtDNA is only 1–2 h (Berk and Clayton, 1974; Clayton, 1982), while the half-life of mtDNA is in the order of 1–3 weeks (Gross et al., 1969; Huemer et al., 1971; Korr et al., 1998). Therefore it has been argued that it is difficult to see how mtDNA replication could be a rate limiting step for mitochondrial growth (Grey, 1999; Elson et al., 2001). To put these verbal arguments on more solid grounds we developed in this study a detailed mathematical model that
describes the fate of a population of wild-type and mutant mtDNA molecules that undergo replication and degradation. Using delay differential equations as well as stochastic simulations we studied the effects of replication time, mitochondrial half-life and deletion size on the outcome of the competition between mutant and wildtype. In recent years it became clear that mitochondria undergo a constant and rapid process of fusion and fission (Duvezin-Caubet et al., 2006; Twig et al., 2008), the evolutionary and functional significance of which we have considered elsewhere (Kowald and Kirkwood, 2011). As a consequence all mtDNAs, wild-type and mutant, effectively exist in a single common mitochondrial compartment. The competing entities are therefore not complete organelles, but individual mtDNA molecules. Our simulations reflect this situation.
2. Model development Although a reduced size has often been proposed as giving a selection advantage to deleted mtDNAs, the exact molecular mechanism has never been spelled out explicitly. Fig. 1 shows the mechanism that we propose and which we used as the basis to develop the mathematical model. The idea is that mtDNAs can be in a “free”, non-replicating, state or in a “busy”, replicating, state. We assume that only mtDNAs from the “free” pool can respond to a replication signal, begin replication, and thus enter the “busy” pool. The replicating molecule remains in the busy state for a certain time, Δt, after which replication is finished and two molecules are returned into the free pool. Wild-type and mutant mtDNAs in the free pool have equal probabilities to respond to a replication signal, but deletion mutants will spend a shorter time in the busy pool and return earlier into the free pool than wildtype molecules. This should result in an overall selection advantage, leading to the accumulation of deleted mtDNAs. In our simulation study all types of mtDNAs (wild type, mutant, busy and free) are degraded at the same rate, leading to an identical half-life for wild type and mutant. Replication and degradation together also ensure that the population of mtDNA molecules reaches a stable steady state. The system contains four variables: wtF: number of wild type mtDNAs in the free pool. wtB: number of wild type mtDNAs in the busy pool. mtF: number of mutant mtDNAs in the free pool. mtB: number of mutant mtDNAs in the busy pool.
Fig. 1. Model overview. The system consists of a pool of free mtDNAs that can respond to a replication signal and from which a mtDNA molecule can enter the busy state in which replication takes place. After a certain replication time, Δt, two mtDNA molecules return into the free pool. All mtDNAs are degraded at the same rate, resulting in the same half-life for wild type and mutant forms.
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It also contains the following five parameters:
2.1. Model based on delay differential equations (DDE)
mutS: size of the mutant given as fraction of the wild type size. Default ¼ 0.5. halfL: half-life of mtDNAs. Default ¼ 10 days. replT: the replication time of a wild type mtDNAs. Default ¼2 h. replS: parameter controlling the replication probability as function of the total number of mtDNAs. If there are only very few mtDNAs, the replication probability is equal to replS. carryC: parameter controlling the replication probability as function of the total number of mtDNAs. If the number of mtDNAs is equal to carryC, the replication probability drops to 1/2*replS.
A standard approach to model biochemical reactions is the use of differential equations. Each equation describes the time course of one variable of the system. Thus, our model consists of four differential equations for wtF, wtB, mtF and mtB. For replication, mtDNAs enter into the busy state from which they reappear after a certain fixed delay, which is given by the replication time, replT. To describe this process mathematically it is necessary to use delay differential equations (DDE), which refer not only to the current value of variables, but also to values that the variables had some time ago. Each of the four equations consists of three terms describing the transfer of mtDNAs to or from the busy pool, the transfer of mtDNAs to or from the free pool and the degradation process. The first term is the most complex one and deserves some explanation. In the first equation for instance, the amount of wild type mtDNAs in the free state, wtF, is increased by wtF that started replicating replT hours ago and are now returning from the busy state. Hence the reference to “treplT” in the first term. However, since we assume that also mtDNAs in the busy state are subject to degradation, the quantity of returning mtDNAs has to be diminished by an appropriate amount, given by the exponential expression. And finally, the factor 2 stems from the fact that each of the replicating molecules has produced a copy of itself. Please note that the DDE model does not contain de novo mutations. It describes the competition between existing mutant and wild-type molecules. For dealing with mutations we developed the stochastic model described in the next section. The system of delay differential equations has been implemented and numerically solved using the software Mathematica 8 by Wolfram Research.
Based on the above description, the following list provides a summary of the reactions included in the model. Parameters and variables that influence the reaction are shown on top of the reaction arrows. The next sections describe in detail how these quantities enter into the kinetics of the different models, but in general we assumed that the replication probability declines with the total amount of existing mtDNAs. The idea is that some form of negative feedback exists that limits the production of new mtDNAs if already plenty exist. The mathematical expression that we used to model this behaviour is of the form ðreplS=ððΣmtDNAsÞ=carry CÞ þ 1ÞÞ. This construct has the property that the replication probability is equal to replS if only very few mtDNAs are present and drops to replS/2 when the number of molecules has reached carryC. wt F wt B mt F mt B
carryC; replS; wtF; wtB; mtF; mtB
⟹
wt B
carryC; replS; wtF; wtB; mtF; mtB; half L; replT
⟹
carryC; replS; wtF; wtB; mtF; mtB
⟹
2 wt F 2.2. Model based on stochastic simulations
mt B
carryC; replS; wtF; wtB; mtF; mtB; half L; replT; mutS
⟹
half L
wt F ⟹removal half L
wt B ⟹removal half L
mt F ⟹removal half L
mt B ⟹removal
2mt F
Models based on differential equations are a versatile and successful approach to study biological systems. However, this technique assumes that the modelled variables are continuous quantities and that the system behaviour is completely deterministic. The larger the number of molecules, the more realistic are these assumptions, but if the molecule number is very small, random fluctuations and discreteness effects can become important. For our investigation of the competition between wild type and mutant mtDNA molecules this means that differential equations are appropriate if both types of mtDNA are present in large numbers, but that stochastic simulations are appropriate if the simulation starts with a single mutant mtDNA. Although the
dwtF 2replS wtFðt replTÞ ¼ e ðln 2 replT=half LÞ dt ðwtFðt replTÞ þ wtBðt replTÞ þ mtFðt replTÞ þ mtBðt replTÞÞ=carryC þ 1 replS wtFðtÞ ln 2 wtFðtÞ ðwtFðtÞ þ wtBðtÞ þ mtFðtÞ þ mtBðtÞÞ=carryC þ 1 half L dwtB replS wtFðt replTÞ ¼ e ðln 2 replT=half LÞ dt ðwtFðt replTÞ þ wtBðt replTÞ þmtFðt replTÞ þ mtBðt replTÞÞ=carryC þ 1 replS wtFðtÞ ln 2 wtBðtÞ þ ðwtFðtÞ þ wtBðtÞ þ mtFðtÞ þ mtBðtÞÞ=carryC þ 1 half L dmtF 2replS mtFðt replT mutSÞ ¼ e ðln 2 replT mutS=half LÞ dt ðwtFðt replT mutSÞ þ wtBðt replT mutSÞ þ mtFðt replT mutSÞ þ mtBðt replT mutSÞÞ=carryC þ 1 replS mtFðtÞ ln 2 mtFðtÞ ðwtFðtÞ þ wtBðtÞ þ mtFðtÞ þ mtBðtÞÞ=carryC þ 1 half L dmtB replS mtFðt replT mutSÞ ¼ e ðln 2 replT mutS=half LÞ dt ðwtFðt replT mutSÞ þwtBðt replT mutSÞ þ mtFðt replT mutSÞ þ mtBðt replT mutSÞÞ=carryC þ1 replS mtFðtÞ ln 2 mtBðtÞ þ ðwtFðtÞ þ wtBðtÞ þ mtFðtÞ þ mtBðtÞÞ=carryC þ 1 half L
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mutant might have a selection advantage, random effects could lead to its degradation before it can spread. We therefore developed a computer programme that explicitly models the fate of a population of 1000 mtDNA molecules by maintaining a list of molecules in the free state and a list of molecules in the busy state. While the DDE model describes the competition between existing mtDNA molecules, we designed the simulation programme to also handle de novo mutations. We assumed that mitochondrial deletions happen via slippage replication at perfect and imperfect direct repeats (Shoffner et al., 1989; Guo et al., 2010) and therefore each replication event leads with a certain probability, Pmut, to a deletion and thus a new, unique, mtDNA mutant. All probabilistic decisions (mutation, degradation, replication) are realised by generating a uniformly distributed random number between 0 and 1 and testing if the number is smaller than the respective probability value. A stochastic simulation starts with a pure population of wild-type mtDNAs and calculates a single trajectory of the system until the end of the specified simulation time. The programme loops through time in intervals of 1 h and performs within each time step the following tasks:
replication time and mutant size. Stochastic simulations are then used to study the consequences of de novo mutations and the system behaviour at the tissue level. The primary outcome of the DDE simulation is the “extinction time”, the time it takes until the mutant has replaced the wild type. To make these times comparable between simulations it is important that the steady state level of the total mtDNA population remains the same, since it can be expected that extinction times also depend on population size. Of course, the steady state level also depends on the parameters we want to study and thus the free parameter carryC, which controls the replication probability, is adjusted appropriately so that the desired population size is maintained. The relationship between the mtDNA steady state level, Mss, and the model parameters can be obtained by setting the left hand side of the DDE system to zero and then solving it for the steady state values of mtDNAs in the free and busy state. The sum of these values is Mss and given by the following expression:
Check if the replication time of any of the mtDNAs in the busy
For our simulations we chose Mss to be equal to 1000 and adjusted carryC correspondingly. Fig. 2 shows a typical simulation result of the system. Using generally accepted literature values for replication time and half-life, the simulation was started with 999 molecules of wild-type mtDNA and a single mutant mtDNA (the standard starting conditions for all DDE simulations). Under these conditions the wild-type went extinct after ca. 95 years. The exact extinction time was defined as the time it takes until the amount of wild-type drops below 1. For the simulation in Fig. 2 the mutant size was set to 50% of the wild type, replication time was 2 h and mtDNA half-life was assumed to be 10 days. The diagram on the right side provides information about the fraction of replicating mutant (fracReplMt) and wild type mtDNAs (fracReplWt) in the total mtDNA population as well as the fraction of mutant molecules in the free pool (fracMtFree). The system is started with all mtDNAs in the non-replicating state but very quickly the wildtype molecules reach a semi equilibrium with ca. 0.58% of the mtDNAs in the replicating state. After this initial phase the fraction of replicating wild type, fracReplWt, drops continuously since wild type is constantly displaced from the population. As a consequence the fraction of mutant in the free pool, fracMtFree, increased from an initial 0.1% to finally 100%. During the displacement process also the total fraction of replicating mtDNAs declines from 0.58% to 0.29%. The reason is that the mutant mtDNAs only have half the size of the wild-type and therefore also only spend half of the time in the replicating state.
list is over. If yes, move that molecule into the free list. Check if a deletion mutation happened during the replication by testing if a random number is below the mutation probability, Pmut. If yes, add a new unique mutant into the free list, if not add a copy of the replicating mtDNA into the free list. Check if any of the mtDNAs in the free or busy list should be degraded by testing for each molecule if a random number is below the degradation probability (ln 2/halfL). If yes, remove those molecules from the list. Check if any of the mtDNAs in the free list starts replicating by testing if a random number is below the replication probability. If yes, move those molecules into the busy list and record the time the replication started.
Like the system of delay differential equations, this stochastic simulation has also been implemented in Mathematica 8 of Wolfram Research.
3. Results The aim of the deterministic simulations is to show how the outcome of the competition between existing mutant and wild type mtDNAs is influenced by the three parameters, half-life,
M ss ¼
carryCðhalf LðreplSð1 21 replT=half L ÞÞ þ ln 2Þ ln 2
Fig. 2. Numerical solutions of the set of delay differential equations, showing the time course of wild type (wt), mutant (mt) and total mtDNA (left). The right panel shows additional information such as the fraction of replicating wild type molecules (fracReplWt), the fraction of replicating mutant molecules (fracReplMt), the total fraction of replicating mtDNAs (totalFracRepl) and the fraction of mutant molecules in the free pool (fracMtFree).
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3.1. Influence of mtDNA half-life To test the effect of mtDNA half-life on extinction times, we performed a series of simulations with a replication time of 2 hours and half-lives ranging from 2 to 10 days. Fig. 3 shows a summary of the results. Increasing the half-life leads to an exponential increase in the time it takes for the mutant to displace the wild-type. This is accompanied by an exponential decline in the total fraction of replicating mtDNAs. This decline is a logical consequence of increasing the half-life while the replication time remains constant. More time is now available for the mtDNA population to be doubled and consequently the fraction of mtDNAs that have to be in the replicating state to achieve this, is smaller. But shorter deletion mutants have a selection advantage against wild type molecules only in the replicating state, since they can return from this pool after a shorter time. Thus, increasing the ratio of half-life to replication time, decreases the selection advantage and increases the time for the mutant to win. With a half-life of 2 days the wild type mtDNA went extinct after approx. 3.9 years, while the extinction time increased to 95 years for a half-life of 10 days.
Fig. 4. Effects of varying the replication time of mtDNAs (replT) on extinction time of the wild-type and the total fraction of replicating molecules. Increasing replT leads to a linear increase in the fraction of replicating molecules, which in turn leads to a drastic decrease of the time it takes to displace the wild-type. The simulation was started with 999 wild-type and a single mutant mtDNA, which had 50% of the wild-type size. A half-life of 10 days was used.
3.2. Influence of mtDNA replication time The replication time, replT, is another important parameter in our model. To study its effect we performed another series of simulations with replication times increasing from 2 to 10 h, while keeping the mtDNA half-life at 10 days. Fig. 4 shows that increasing the replication time, and thus decreasing the ratio of half-life to replication time, exponentially shortens the extinction time. It also increases the total fraction of replicating mtDNAs linearly to exactly the same values that were obtained during the variation of half-life (Fig. 3). But despite the same values for totalFracRepl, the extinction times are much longer. In the last section we saw that a ratio replT/halfL¼2/(2n24)¼ 1/24 results in a totalFracRepl of 0.0145 and an extinction time of 3.9 years. The results in Fig. 4 show that the same ratio replT/halfL¼10/(10n24)¼1/24 also results in the same totalFracRepl of 0.0145, but in an extinction time of 19.33 years. The reason is that the selection advantage per mtDNA generation time is the same in both simulations, but because the generation time (half-life) in the second case is 5 times longer, the replacement process also takes 5 times longer.
Fig. 5. Effects of varying the size of the mtDNA mutant (mutS) on extinction time and the total fraction of replicating molecules. Increasing mutS reduces the selection advantage of the deletion mutant and leads to a strong rise of the extinction time. The fraction of replicating mtDNAs rises linearly, since a larger mutant needs more time for replication. The simulation was started with 999 wild-type and a single mutant mtDNA. A half-life of 10 days and a replication time of wt of 2 h was used.
3.3. Influence of the size of the mtDNA mutant
Fig. 3. Effects of varying the half-life of mtDNAs (halfL) on extinction time of the wild-type and the total fraction of replicating molecules. Increasing halfL fivefold from 2 to 10 days, leads to an almost 25 fold increase in extinction time from 3.9 to 95 years. The simulation was started with 999 wild-type and a single mutant mtDNA, which had 50% of the wild-type size. A replication time of 2 h was used.
The mutant size, mutS, is the last major parameter of the model. For the simulations so far, we used a value of 0.5, which corresponds to a deletion of 50% of the wild type genome. This provides the largest selection advantage, but also comes close to the largest deletion that is possible without deleting one of the two origins of replication of mammalian mtDNA. It is therefore interesting also to study the behaviour of larger mutants. For this purpose we varied the mutant size from 0.5 to 0.7 with a replication time of 2 h and a half-life of 10 days (Fig. 5). The larger the mutant, i.e. the smaller the deletion, the longer it takes for the mutant to replace the wild-type. A mutant of 50% of the wild type size requires ca. 95 years but a mutant with 70% of the wild type size takes almost 160 years! Extinction times approach infinity for mutS ¼1, since in that case there is no selection advantage for the mutant. Furthermore, the larger the mutant the higher is the total fraction of replicating mtDNA. The reason is that the replication
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time (time spent in the busy state) increases proportionally to the mutant size and hence the total fraction of mtDNAs in the busy state increases linearly. 3.4. Stochastic tissue simulations Next we used our stochastic model to study the effects of mutant accumulation at the tissue level in animals of various lifespans. A cytochrome c oxidase (COX) deficient phenotype has been observed in post mitotic cells of old animals and is generally regarded as consequence of a non-functional oxidative phosphorylation (Cao et al., 2001). As in other studies (Elson et al., 2001; Kowald and Kirkwood, 2013), we regard a cell as COX-deficient if it contains at least 60% mutant mtDNAs at the end of the simulation time. Clonally expanded deletion mutants have been found in short and long lived animals (from rat to man) and we therefore performed tissue simulations for 3, 10, 40 and 80 years with a halflife of 10 days, replication time of 2 h and a mutant size of 0.5. For this purpose we performed 1000 repetitions for each lifespan and counted the fraction of simulations that resulted in a COX deficient phenotype. Then mutation rates were adjusted in such a way, that at the end of the simulation 10 70.5% of the cells were COX deficient. Fig. 6A shows that there is a roughly 500 fold difference in mutation rate that is required to achieve the same percentage of COX deficient cells in animals with a lifespan of 3 years compared
to 80 years. Especially intriguing is the low degree of heteroplasmy that has been observed experimentally, i.e. it is in most cases a single type of mutant that has displaced the wild-type mtDNA. In the stochastic model we assume that each new deletion results in a new, unique, mutation type and Fig. 6B presents the number of different mtDNAs per COX deficient cell as a measure for the degree of heteroplasmy. As can be seen, heteroplasmy dramatically increases for short lifespans, leading on average to almost 30 different types of mtDNA per cell for a lifespan of 3 years. These results can hardly be reconciled with experimental findings. In Fig. 3 it was shown that the extinction time is drastically shortened if the mtDNA half-life is reduced, presumably because this increases the ratio of replication time to half-life. What influence has this for stochastic simulations at the tissue level? We studied this by performing a number of simulations for a lifespan of 3 years with different half-lives, but the standard values for replication time (2 h) and mutant size (0.5). Fig. 7A shows that the mutation rate required to achieve 10% COX negative cells after 3 years, strongly decreases with a decreasing half-life. Similarly, also the degree of heteroplasmy decreases to only 1.16 different mtDNAs for a half-life of 2 days (Fig. 7B). These simulations confirm the DDE calculations shown in Fig. 3 and emphasise the relevance of the ratio of replication time to half-life for the strength of the selection advantage that a shorter genome can confer in the replication process.
Fig. 6. Stochastic simulation of a population of 1000 mtDNAs for 3, 10, 40 and 80 years. The left diagram shows the mutation rate required to result in 10 7 0.5% of COX deficient cells (defined as harbouring 4 60% mutant mtDNAs) after the indicated time with replT¼ 2 h, halfL ¼10 d and mutSize¼ 0.5. Each point is calculated from 1000 simulation runs. The right diagram shows the degree of mtDNA heteroplasmy in COX deficient cells, measured by the number of different types of mtDNA mutants per cell.
Fig. 7. Stochastic simulation of a population of 1000 mtDNAs for 3 years (replT¼ 2 h, mutSize¼ 0.5) and different values for the half-life. The left diagram shows the mutation rate required to result in 10 7 0.5% of COX negative cells (defined as harbouring 4 60% mutant mtDNAs) and the right diagram shows the degree of mtDNA heteroplasmy in COX negative cells, measured by the number of different types of mtDNA mutants.
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4. Discussion In this work we studied the question of whether a shorter replication time of mitochondrial deletion mutants can provide them with a selection advantage, so that they can accumulate and replace the wild type. This idea had been put forward to explain the experimentally observed clonal expansion of deletion mutants (Wallace, 1992; Lee et al., 1998). We proposed a specific molecular mechanism of how a shorter replication time can result in a selection advantage by assuming that currently replicating molecules are not available to respond to further replication signals, since they are in a busy state until they finish their current round of replication. We used models based on delay differential equations as well as stochastic simulations to investigate this question by varying (i) replication time of mtDNA, (ii) half-life of mtDNA, and (iii) the size of the mutant and by calculating the degree of heteroplasmy observable at the tissue level. The simulations confirm that the half-life of mtDNA has a major influence on the extinction time, i.e. the time required for the mutant to replace the wild type population (Fig. 3). Using parameter values that are compatible with accepted literature values (replication time 2 h, half-life 10 days), it takes approximately 95 years until the wild type is replaced by a deletion mutant that has only 50% of the wild type size. Our simulations also show why the selection advantage dwindles away with increasing half-life. After a period of time equivalent to the half-life, 50% of the population of mtDNA molecules has been degraded and consequently the population has to double in this time to maintain a steady state. If half-life increases, there is more time for the population to double and consequently at any given time there will be a smaller fraction of mtDNAs in the replicating state. This decline of the total fraction of replicating molecules is also shown in Fig. 3. However, the smaller size of the mutant is only an advantage in this busy state (via a faster exit time) and consequently the selection advantage decreases with increasing half-life. How reliable and comprehensive are the experimental data regarding mtDNA half-life and replication time? Unfortunately the experimental basis is not very satisfactory. To our knowledge mtDNA half-life has only been measured in mice or rats, but never in humans. Furthermore, studies are both sparse and old (Gross et al., 1969; Huemer et al., 1971; Korr et al., 1998), or gave results completely incompatible with other studies (Collins et al., 2003). The situation regarding measurements of the mtDNA replication time is even worse. Clayton (1982) and Davis and Clayton (1996) are often cited as reference for a 1–2 h timespan for the mitochondrial replication time, but those articles seem to refer back to a single article by Berk and Clayton (1974). Given this uncertainty we regarded it appropriate also to perform simulations covering longer replication times. As expected, increasing replication times also increased the fraction of replicating mtDNAs and consequently reduced extinction times (Fig. 4). But even with a replication time of 10 h (and 10 days half-life), a newly emerging single mutant still requires 19 years before the wild type has been replaced. These time frames are evidently incompatible for explaining the ageing process in short lived animals. This becomes even clearer if the simulations are extended to mtDNAs with smaller deletions (Fig. 5). Using standard parameters a mutant that is 30% shorter than the wild-type takes almost 150 years to take over the mitochondrial population. If the parameters used for halflife and replication time are reliable, these numbers clearly rule out that deletion mutants accumulate based on a reduced genome size within a time frame that is compatible with mammalian lifespans. We also performed stochastic simulations to take random fluctuations into account and model the appearance of de novo mutations for species with lifespans ranging from 3 to 80 years.
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The simulations show that short lifespans require high mutation rates to generate a level of COX negative cells that is compatible with experimental findings (Fig. 6A). Even more important, the calculated degree of heteroplasmy also rises strongly for short lifespans (Fig. 6B). On average almost 30 different types of mtDNAs are present in COX negative cells after a 3 year simulation. Also these results disagree strongly with experimental observations. The results of the deterministic model have already indicated that the ratio between replication time and mtDNA half-life is critical for the selection advantage of the mutant. To further corroborate this finding we also performed stochastic simulations for a lifespan of 3 years with a variation of the half-life from 2–10 days (Fig. 7). A reduction of the half-life strongly reduced the necessary mutations rate as well as the degree of heteroplasmy. These effects are caused by the increasing selection advantage that accompanies the parameter change. In summary, our modelling investigation shows that despite all the uncertainty regarding experimental measurements of mtDNA half-life and replication time, it is highly unlikely that mitochondrial deletion mutants gain a selection advantage from a faster replication time based on the reduced size. Our simulations show that the large differences between replication time and half-life would lead to such extremely long extinction times (4 90 years), that such a process cannot serve as general mechanism to explain the clonal expansion of mitochondrial deletion mutants in long and short lived animal species.
Acknowledgements This work was funded by a Glenn Foundation award to T.K. The research was also supported by the National Institute for Health Research (NIHR) Newcastle Biomedical Research Centre based at Newcastle upon Tyne Hospitals NHS Foundation Trust and Newcastle University. The views expressed are those of the authors and not necessarily those of the NHS, the NIHR or the Department of Health. References Arnheim, N., Cortopassi, G., 1992. Deleterious mitochondrial DNA mutations accumulate in aging human tissues. Mutation Research 275, 157–167. Bandy, B., Davison, A.J., 1990. Mitochondrial mutations may increase oxidative stress: implications for carcinogenesis and aging? Free Radical Biology and Medicine 8, 523–539. Berk, A.J., Clayton, D.A., 1974. Mechanism of mitochondrial DNA replication in mouse L-cells: asynchronous replication of strands, segregation of circular daughter molecules, aspects of topology and turnover of an initiation sequence. Journal of Molecular Biology 86, 801–824. Brierley, E.J., Johnson, M.A., Lightowlers, R.N., James, O.F., Turnbull, D.M., 1998. Role of mitochondrial DNA mutations in human aging: implications for the central nervous system and muscle. Annals of Neurology 43, 217–223. Cao, Z., Wanagat, J., McKiernan, S.H., Aiken, J.M., 2001. Mitochondrial DNA deletion mutations are concomitant with ragged red regions of individual, aged muscle fibers: analysis by laser-capture microdissection. Nucleic Acids Research 29, 4502–4508. Chinnery, P.F., Samuels, D.C., 1999. Relaxed replication of mtDNA: A model with implications for the expression of disease. American Journal of Human Genetics 64, 1158–1165. Clayton, D.A., 1982. Replication of animal mitochondrial DNA. Cell 28, 693–705. Collins, M.L., Eng, S., Hoh, R., Hellerstein, M.K., 2003. Measurement of mitochondrial DNA synthesis in vivo using a stable isotope–mass spectrometric technique. Journal of Applied Physiology 94, 2203–2211. Davis, A.F., Clayton, D.A., 1996. In situ localization of mitochondrial DNA replication in intact mammalian cells. Journal of Cell Biology 135, 883–893. de Grey, A.D.N.J., 1997. A proposed refinement of the mitochondrial free radical theory of aging. BioEssays 19, 161–166. de Grey, A.D.N.J., 1999. The Mitochondrial Free Radical Theory of Aging, Austin. R.G. Landes Company. Duvezin-Caubet, S., Jagasia, R., Wagener, J., Hofmann, S., Trifunovic, A., Hansson, A., Chomyn, A., Bauer, M.F., Attardi, G., Larsson, N.G., Neupert, W., Reichert, A.S., 2006. Proteolytic processing of OPA1 links mitochondrial dysfunction to alterations in mitochondrial morphology. Journal of Biological Chemistry 281, 37972–37979.
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Elson, J.L., Samuels, D.C., Turnbull, D.M., Chinnery, P.F., 2001. Random intracellular drift explains the clonal expansion of mitochondrial DNA mutations with age. American Journal of Human Genetics 68, 802–806. Gokey, N.G., Cao, Z., Pak, J.W., Lee, D., McKiernan, S.H., McKenzie, D., Weindruch, R., Aiken, J.M., 2004. Molecular analyses of mtDNA deletion mutations in microdissected skeletal muscle fibers from aged rhesus monkeys. Aging Cell 3, 319–326. Gross, N.J., Getz, G.S., Rabinowitz, M., 1969. Apparent turnover of mitochondrial deoxyribonucleic acid and mitochondrial phospholipids in the tissues of the rat. Journal of Biological Chemistry 244, 1552–1562. Guo, X., Popadin, K.Y., Markuzon, N., Orlov, Y.L., Kraytsberg, Y., Krishnan, K.J., Zsurka, G., Turnbull, D.M., Kunz, W.S., Khrapko, K., 2010. Repeats, longevity and the sources of mtDNA deletions: evidence from “deletional spectra”. Trends in Genetics 26, 340–343. Harman, D., 1972. The biologic clock: the mitochondria? Journal of the American Geriatrics Society 20, 145–147. Harman, D., 1983. Free radical theory of aging: consequences of mitochondrial aging. Age 6, 86–94. Herbst, A., Pak, J.W., McKenzie, D., Bua, E., Bassiouni, M., Aiken, J.M., 2007. Accumulation of mitochondrial DNA deletion mutations in aged muscle fibers: evidence for a causal role in muscle fiber loss. Journals of Gerontology Series A: Biological Sciences and Medical Sciences 62, 235–245. Huemer, R.P., Lee, K.D., Reeves, A.E., Bickert, C., 1971. Mitochondrial studies in senescent mice – II. Specific activity, bouyant density, and turnover of mitochondrial DNA. Experimental Gerontology 6, 327–334. Khrapko, K., Bodyak, N., Thilly, W.G., van Orsouw, N.J., Zhang, X., Coller, H.A., Perls, T.T., Upton, M., Vijg, J., Wei, J.Y., 1999. Cell by cell scanning of whole mitochondrial genomes in aged human heart reveals a significant fraction of myocytes with clonally expanded deletions. Nucleic Acids Research 27, 2434–2441. Kirkwood, T.B.L., 1996. Human senescence. BioEssays 18, 1009–1016. Kim, I., Lemasters, J.J., 2011. Mitophagy selectively degrades individual damaged mitochondria after photoirradiation. Antioxidants and Redox Signaling 14, 1919–1928.
Korr, H., Kurz, C., Seidler, T.O., Sommer, D., Schmitz, C., 1998. Mitochondrial DNA synthesis studied autoradiographically in various cell types in vivo. Brazilian Journal of Medical and Biological Research 31, 289–298. Kowald, A., Kirkwood, T.B., 2011. Evolution of the mitochondrial fusion-fission cycle and its role in aging. Proceedings of the National Academy of Sciences USA 108, 10237–10242. Kowald, A., Kirkwood, T.B.L., 2013. Mitochondrial mutations and aging: random drift is insufficient to explain the accumulation of mitochondrial deletion mutants in short-lived animals. Aging Cell. Lee, C.M., Lopez, M.E., Weindruch, R., Aiken, J.M., 1998. Association of age-related mitochondrial abnormalities with skeletal muscle fiber atrophy. Free Radical Biology and Medicine 25, 964–972. Linnane, A.W., Marzuki, S., Ozawa, T., Tanaka, M., 1989. Mitochondrial DNA mutations as an important contributor to ageing and degenerative diseases. Lancet 1, 642–645. Miquel, J., Economos, A.C., Fleming, J., Johnson Jr., J.E., 1980. Mitochondrial role in cell aging. Experimental Gerontology 15, 575–591. Richter, C., 1988. Do mitochondrial DNA fragments promote cancer and aging? FEBS Letters 241, 1–5. Shoffner, J.M., Lott, M.T., Voljavec, A.S., Soueidan, S.A., Costigan, D.A., Wallace, D.C., 1989. Spontaneous Kearns-Sayre/chronic external ophthalmoplegia plus syndrome associated with a mitochondrial DNA deletion: a slip-replication model and metabolic therapy. Proceedings of the National Academy of Sciences USA 86, 7952–7956. Twig, G., Elorza, A., Molina, A.J., Mohamed, H., Wikstrom, J.D., Walzer, G., Stiles, L., Haigh, S.E., Katz, S., Las, G., Alroy, J., Wu, M., Py, B.F., Yuan, J., Deeney, J.T., Corkey, B.E., Shirihai, O.S., 2008. Fission and selective fusion govern mitochondrial segregation and elimination by autophagy. EMBO Journal 27, 433–446. Wallace, D.C., 1992. Mitochondrial genetics: a paradigm for aging and degenerative diseases? Science 256, 628–632.
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Mitochondrial mutations and ageing: Can mitochondrial deletion mutants accumulate via a size based replication advantage? Axel Kowald a,n, Marcus Dawson a,b, Thomas B.L. Kirkwood a a Centre for Integrated Systems Biology of Ageing and Nutrition, Institute for Ageing and Health, Newcastle University, Newcastle upon Tyne NE4 5PL, United Kingdom b University of Manchester, Manchester M13 9PT, United Kingdom
H I G H L I G H T S
We modelled the accumulation of mitochondrial deletion mutants during ageing. Can the reduced size provide a selection advantage via a shorter replication time? We developed a delay differential equation model and a stochastic simulation. The simulations show that the idea only works for very long lived species.
art ic l e i nf o
a b s t r a c t
Article history: Received 21 February 2013 Received in revised form 22 August 2013 Accepted 9 September 2013 Available online 17 September 2013
The mitochondrial theory of ageing is one of the main contenders to explain the biochemical basis of the ageing process. An important line of support comes from the observation that mtDNA deletions accumulate over the life course in post-mitotic cells of many species. A single mutant expands clonally and finally replaces the wild-type population of a whole cell. One proposal to explain the driving force behind this accumulation states that the reduced size leads to a shorter replication time, which provides a selection advantage. However, this idea has been questioned on the grounds that the mitochondrial half-life is much longer than the replication time, so that the latter cannot be a rate limiting step. To clarify this question, we modelled this process mathematically and performed extensive deterministic and stochastic computer simulations to study the effects of replication time, mitochondrial half-life and deletion size. Our study shows that the shorter size does in principle provide a selection advantage, which can lead to an accumulation of the deletion mutant. However, this selection advantage diminishes the shorter is the replication time of wt mtDNA in relation to its half-life. Using generally accepted literature values, the resulting time frame for the accumulation of mutant mtDNAs is only compatible with the ageing process in very long lived species like humans, but could not reasonably explain ageing in short lived species like mice and rats. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Mitochondrial mutation Mathematical model Ageing
1. Introduction Ageing is an intrinsic deterioration of the homoeostatic capabilities of an organism, leading to a constantly increasing risk of death. Although evolutionary considerations suggest that the ageing process is in principle multifactorial (Kirkwood, 1996), a few main mechanisms have been proposed. Among these the mitochondrial theory of ageing is one of the most popular (Harman, 1972, 1983; Miquel et al., 1980; Richter, 1988; Linnane et al., 1989). The theory suggests that the accumulation of defective mitochondria is a major
n
Corresponding author. Tel.: þ 49 179 3427522. E-mail addresses: [email protected] (A. Kowald), Marcus.Dawson@student. manchester.ac.uk (M. Dawson), [email protected] (T.B.L. Kirkwood). 0022-5193/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2013.09.009
contributor to the ageing process. Reactive oxygen species (ROS) generated during respiration have the potential to damage all kinds of biologically relevant macromolecules such as lipids, proteins and mitochondrial DNA (mtDNA). Damage to mtDNA is quite different from damage to other macromolecules, since mitochondrial DNA represents the ultimate blueprint from which everything else follows. Point mutations and deletions could impair mitochondrial ATP production with negative consequences for all aspects of cellular homoeostasis. And indeed, many studies have shown that mitochondrial deletion mutants accumulate with age in various mammalian species such as rats, monkeys and humans (Brierley et al., 1998; Khrapko et al., 1999; Cao et al., 2001; Gokey et al., 2004; Herbst et al., 2007). These single cell studies have shown that the mitochondrial population of a cell is overtaken by a single deletion mutant type via clonal expansion.
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The mechanism behind the accumulation of defective mitochondria is obviously an important aspect of the theory, but is currently unknown. The “vicious cycle” hypothesis suggests that, because defective mitochondria generate more radicals than intact ones, the cellular radical level progressively rises, leading in turn to an increased production of mitochondrial mutants (Bandy and Davison, 1990; Arnheim and Cortopassi, 1992). In this scenario the individual mutant mitochondria do not need to have a selection advantage; they accumulate because they are constantly generated. However, such a mechanism also implies that one should see a plethora of different mtDNA mutants in a single cell, quite the contrary of what has been observed. It has also been proposed that pure random drift might be sufficient to explain clonal expansion (Chinnery and Samuels, 1999; Elson et al., 2001). Although these authors show that such a process might work for long lived species like humans, we have recently performed computer simulations, which demonstrate that such a process is unsuitable for short lived animals like rodents (Kowald and Kirkwood, 2013), which show a similar pattern of accumulation as in humans but on a twenty times accelerated timescale (Cao et al., 2001). An idea that does identify a selection advantage of deletion mutants is called survival of the slowest (SOS) (Grey, 1997). It notes that the fate of a mitochondrion depends not only on its growth rate, but also on its rate of degradation and proposes that defective mitochondria are degraded less frequently than wild-type organelles. However, this hypothesis has problems with mitochondrial dynamics, since fission and fusion break the required link between genotype and phenotype (Kowald and Kirkwood, 2011). Furthermore, it has been shown that dysfunctional mitochondria are preferentially degraded (Twig et al., 2008; Kim and Lemasters, 2011), instead of being spared. Finally, it has been proposed that the selection advantage simply stems from the reduced genome size of the deletion mutant (Wallace, 1992; Lee et al., 1998). Deletions encompassing more than half of the mitochondrial genome have been observed (Cao et al., 2001), which could result in a 50% reduced replication time. However, this idea has been criticised because early studies have shown that the time required for the replication of the mtDNA is only 1–2 h (Berk and Clayton, 1974; Clayton, 1982), while the half-life of mtDNA is in the order of 1–3 weeks (Gross et al., 1969; Huemer et al., 1971; Korr et al., 1998). Therefore it has been argued that it is difficult to see how mtDNA replication could be a rate limiting step for mitochondrial growth (Grey, 1999; Elson et al., 2001). To put these verbal arguments on more solid grounds we developed in this study a detailed mathematical model that
describes the fate of a population of wild-type and mutant mtDNA molecules that undergo replication and degradation. Using delay differential equations as well as stochastic simulations we studied the effects of replication time, mitochondrial half-life and deletion size on the outcome of the competition between mutant and wildtype. In recent years it became clear that mitochondria undergo a constant and rapid process of fusion and fission (Duvezin-Caubet et al., 2006; Twig et al., 2008), the evolutionary and functional significance of which we have considered elsewhere (Kowald and Kirkwood, 2011). As a consequence all mtDNAs, wild-type and mutant, effectively exist in a single common mitochondrial compartment. The competing entities are therefore not complete organelles, but individual mtDNA molecules. Our simulations reflect this situation.
2. Model development Although a reduced size has often been proposed as giving a selection advantage to deleted mtDNAs, the exact molecular mechanism has never been spelled out explicitly. Fig. 1 shows the mechanism that we propose and which we used as the basis to develop the mathematical model. The idea is that mtDNAs can be in a “free”, non-replicating, state or in a “busy”, replicating, state. We assume that only mtDNAs from the “free” pool can respond to a replication signal, begin replication, and thus enter the “busy” pool. The replicating molecule remains in the busy state for a certain time, Δt, after which replication is finished and two molecules are returned into the free pool. Wild-type and mutant mtDNAs in the free pool have equal probabilities to respond to a replication signal, but deletion mutants will spend a shorter time in the busy pool and return earlier into the free pool than wildtype molecules. This should result in an overall selection advantage, leading to the accumulation of deleted mtDNAs. In our simulation study all types of mtDNAs (wild type, mutant, busy and free) are degraded at the same rate, leading to an identical half-life for wild type and mutant. Replication and degradation together also ensure that the population of mtDNA molecules reaches a stable steady state. The system contains four variables: wtF: number of wild type mtDNAs in the free pool. wtB: number of wild type mtDNAs in the busy pool. mtF: number of mutant mtDNAs in the free pool. mtB: number of mutant mtDNAs in the busy pool.
Fig. 1. Model overview. The system consists of a pool of free mtDNAs that can respond to a replication signal and from which a mtDNA molecule can enter the busy state in which replication takes place. After a certain replication time, Δt, two mtDNA molecules return into the free pool. All mtDNAs are degraded at the same rate, resulting in the same half-life for wild type and mutant forms.
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It also contains the following five parameters:
2.1. Model based on delay differential equations (DDE)
mutS: size of the mutant given as fraction of the wild type size. Default ¼ 0.5. halfL: half-life of mtDNAs. Default ¼ 10 days. replT: the replication time of a wild type mtDNAs. Default ¼2 h. replS: parameter controlling the replication probability as function of the total number of mtDNAs. If there are only very few mtDNAs, the replication probability is equal to replS. carryC: parameter controlling the replication probability as function of the total number of mtDNAs. If the number of mtDNAs is equal to carryC, the replication probability drops to 1/2*replS.
A standard approach to model biochemical reactions is the use of differential equations. Each equation describes the time course of one variable of the system. Thus, our model consists of four differential equations for wtF, wtB, mtF and mtB. For replication, mtDNAs enter into the busy state from which they reappear after a certain fixed delay, which is given by the replication time, replT. To describe this process mathematically it is necessary to use delay differential equations (DDE), which refer not only to the current value of variables, but also to values that the variables had some time ago. Each of the four equations consists of three terms describing the transfer of mtDNAs to or from the busy pool, the transfer of mtDNAs to or from the free pool and the degradation process. The first term is the most complex one and deserves some explanation. In the first equation for instance, the amount of wild type mtDNAs in the free state, wtF, is increased by wtF that started replicating replT hours ago and are now returning from the busy state. Hence the reference to “treplT” in the first term. However, since we assume that also mtDNAs in the busy state are subject to degradation, the quantity of returning mtDNAs has to be diminished by an appropriate amount, given by the exponential expression. And finally, the factor 2 stems from the fact that each of the replicating molecules has produced a copy of itself. Please note that the DDE model does not contain de novo mutations. It describes the competition between existing mutant and wild-type molecules. For dealing with mutations we developed the stochastic model described in the next section. The system of delay differential equations has been implemented and numerically solved using the software Mathematica 8 by Wolfram Research.
Based on the above description, the following list provides a summary of the reactions included in the model. Parameters and variables that influence the reaction are shown on top of the reaction arrows. The next sections describe in detail how these quantities enter into the kinetics of the different models, but in general we assumed that the replication probability declines with the total amount of existing mtDNAs. The idea is that some form of negative feedback exists that limits the production of new mtDNAs if already plenty exist. The mathematical expression that we used to model this behaviour is of the form ðreplS=ððΣmtDNAsÞ=carry CÞ þ 1ÞÞ. This construct has the property that the replication probability is equal to replS if only very few mtDNAs are present and drops to replS/2 when the number of molecules has reached carryC. wt F wt B mt F mt B
carryC; replS; wtF; wtB; mtF; mtB
⟹
wt B
carryC; replS; wtF; wtB; mtF; mtB; half L; replT
⟹
carryC; replS; wtF; wtB; mtF; mtB
⟹
2 wt F 2.2. Model based on stochastic simulations
mt B
carryC; replS; wtF; wtB; mtF; mtB; half L; replT; mutS
⟹
half L
wt F ⟹removal half L
wt B ⟹removal half L
mt F ⟹removal half L
mt B ⟹removal
2mt F
Models based on differential equations are a versatile and successful approach to study biological systems. However, this technique assumes that the modelled variables are continuous quantities and that the system behaviour is completely deterministic. The larger the number of molecules, the more realistic are these assumptions, but if the molecule number is very small, random fluctuations and discreteness effects can become important. For our investigation of the competition between wild type and mutant mtDNA molecules this means that differential equations are appropriate if both types of mtDNA are present in large numbers, but that stochastic simulations are appropriate if the simulation starts with a single mutant mtDNA. Although the
dwtF 2replS wtFðt replTÞ ¼ e ðln 2 replT=half LÞ dt ðwtFðt replTÞ þ wtBðt replTÞ þ mtFðt replTÞ þ mtBðt replTÞÞ=carryC þ 1 replS wtFðtÞ ln 2 wtFðtÞ ðwtFðtÞ þ wtBðtÞ þ mtFðtÞ þ mtBðtÞÞ=carryC þ 1 half L dwtB replS wtFðt replTÞ ¼ e ðln 2 replT=half LÞ dt ðwtFðt replTÞ þ wtBðt replTÞ þmtFðt replTÞ þ mtBðt replTÞÞ=carryC þ 1 replS wtFðtÞ ln 2 wtBðtÞ þ ðwtFðtÞ þ wtBðtÞ þ mtFðtÞ þ mtBðtÞÞ=carryC þ 1 half L dmtF 2replS mtFðt replT mutSÞ ¼ e ðln 2 replT mutS=half LÞ dt ðwtFðt replT mutSÞ þ wtBðt replT mutSÞ þ mtFðt replT mutSÞ þ mtBðt replT mutSÞÞ=carryC þ 1 replS mtFðtÞ ln 2 mtFðtÞ ðwtFðtÞ þ wtBðtÞ þ mtFðtÞ þ mtBðtÞÞ=carryC þ 1 half L dmtB replS mtFðt replT mutSÞ ¼ e ðln 2 replT mutS=half LÞ dt ðwtFðt replT mutSÞ þwtBðt replT mutSÞ þ mtFðt replT mutSÞ þ mtBðt replT mutSÞÞ=carryC þ1 replS mtFðtÞ ln 2 mtBðtÞ þ ðwtFðtÞ þ wtBðtÞ þ mtFðtÞ þ mtBðtÞÞ=carryC þ 1 half L
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mutant might have a selection advantage, random effects could lead to its degradation before it can spread. We therefore developed a computer programme that explicitly models the fate of a population of 1000 mtDNA molecules by maintaining a list of molecules in the free state and a list of molecules in the busy state. While the DDE model describes the competition between existing mtDNA molecules, we designed the simulation programme to also handle de novo mutations. We assumed that mitochondrial deletions happen via slippage replication at perfect and imperfect direct repeats (Shoffner et al., 1989; Guo et al., 2010) and therefore each replication event leads with a certain probability, Pmut, to a deletion and thus a new, unique, mtDNA mutant. All probabilistic decisions (mutation, degradation, replication) are realised by generating a uniformly distributed random number between 0 and 1 and testing if the number is smaller than the respective probability value. A stochastic simulation starts with a pure population of wild-type mtDNAs and calculates a single trajectory of the system until the end of the specified simulation time. The programme loops through time in intervals of 1 h and performs within each time step the following tasks:
replication time and mutant size. Stochastic simulations are then used to study the consequences of de novo mutations and the system behaviour at the tissue level. The primary outcome of the DDE simulation is the “extinction time”, the time it takes until the mutant has replaced the wild type. To make these times comparable between simulations it is important that the steady state level of the total mtDNA population remains the same, since it can be expected that extinction times also depend on population size. Of course, the steady state level also depends on the parameters we want to study and thus the free parameter carryC, which controls the replication probability, is adjusted appropriately so that the desired population size is maintained. The relationship between the mtDNA steady state level, Mss, and the model parameters can be obtained by setting the left hand side of the DDE system to zero and then solving it for the steady state values of mtDNAs in the free and busy state. The sum of these values is Mss and given by the following expression:
Check if the replication time of any of the mtDNAs in the busy
For our simulations we chose Mss to be equal to 1000 and adjusted carryC correspondingly. Fig. 2 shows a typical simulation result of the system. Using generally accepted literature values for replication time and half-life, the simulation was started with 999 molecules of wild-type mtDNA and a single mutant mtDNA (the standard starting conditions for all DDE simulations). Under these conditions the wild-type went extinct after ca. 95 years. The exact extinction time was defined as the time it takes until the amount of wild-type drops below 1. For the simulation in Fig. 2 the mutant size was set to 50% of the wild type, replication time was 2 h and mtDNA half-life was assumed to be 10 days. The diagram on the right side provides information about the fraction of replicating mutant (fracReplMt) and wild type mtDNAs (fracReplWt) in the total mtDNA population as well as the fraction of mutant molecules in the free pool (fracMtFree). The system is started with all mtDNAs in the non-replicating state but very quickly the wildtype molecules reach a semi equilibrium with ca. 0.58% of the mtDNAs in the replicating state. After this initial phase the fraction of replicating wild type, fracReplWt, drops continuously since wild type is constantly displaced from the population. As a consequence the fraction of mutant in the free pool, fracMtFree, increased from an initial 0.1% to finally 100%. During the displacement process also the total fraction of replicating mtDNAs declines from 0.58% to 0.29%. The reason is that the mutant mtDNAs only have half the size of the wild-type and therefore also only spend half of the time in the replicating state.
list is over. If yes, move that molecule into the free list. Check if a deletion mutation happened during the replication by testing if a random number is below the mutation probability, Pmut. If yes, add a new unique mutant into the free list, if not add a copy of the replicating mtDNA into the free list. Check if any of the mtDNAs in the free or busy list should be degraded by testing for each molecule if a random number is below the degradation probability (ln 2/halfL). If yes, remove those molecules from the list. Check if any of the mtDNAs in the free list starts replicating by testing if a random number is below the replication probability. If yes, move those molecules into the busy list and record the time the replication started.
Like the system of delay differential equations, this stochastic simulation has also been implemented in Mathematica 8 of Wolfram Research.
3. Results The aim of the deterministic simulations is to show how the outcome of the competition between existing mutant and wild type mtDNAs is influenced by the three parameters, half-life,
M ss ¼
carryCðhalf LðreplSð1 21 replT=half L ÞÞ þ ln 2Þ ln 2
Fig. 2. Numerical solutions of the set of delay differential equations, showing the time course of wild type (wt), mutant (mt) and total mtDNA (left). The right panel shows additional information such as the fraction of replicating wild type molecules (fracReplWt), the fraction of replicating mutant molecules (fracReplMt), the total fraction of replicating mtDNAs (totalFracRepl) and the fraction of mutant molecules in the free pool (fracMtFree).
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3.1. Influence of mtDNA half-life To test the effect of mtDNA half-life on extinction times, we performed a series of simulations with a replication time of 2 hours and half-lives ranging from 2 to 10 days. Fig. 3 shows a summary of the results. Increasing the half-life leads to an exponential increase in the time it takes for the mutant to displace the wild-type. This is accompanied by an exponential decline in the total fraction of replicating mtDNAs. This decline is a logical consequence of increasing the half-life while the replication time remains constant. More time is now available for the mtDNA population to be doubled and consequently the fraction of mtDNAs that have to be in the replicating state to achieve this, is smaller. But shorter deletion mutants have a selection advantage against wild type molecules only in the replicating state, since they can return from this pool after a shorter time. Thus, increasing the ratio of half-life to replication time, decreases the selection advantage and increases the time for the mutant to win. With a half-life of 2 days the wild type mtDNA went extinct after approx. 3.9 years, while the extinction time increased to 95 years for a half-life of 10 days.
Fig. 4. Effects of varying the replication time of mtDNAs (replT) on extinction time of the wild-type and the total fraction of replicating molecules. Increasing replT leads to a linear increase in the fraction of replicating molecules, which in turn leads to a drastic decrease of the time it takes to displace the wild-type. The simulation was started with 999 wild-type and a single mutant mtDNA, which had 50% of the wild-type size. A half-life of 10 days was used.
3.2. Influence of mtDNA replication time The replication time, replT, is another important parameter in our model. To study its effect we performed another series of simulations with replication times increasing from 2 to 10 h, while keeping the mtDNA half-life at 10 days. Fig. 4 shows that increasing the replication time, and thus decreasing the ratio of half-life to replication time, exponentially shortens the extinction time. It also increases the total fraction of replicating mtDNAs linearly to exactly the same values that were obtained during the variation of half-life (Fig. 3). But despite the same values for totalFracRepl, the extinction times are much longer. In the last section we saw that a ratio replT/halfL¼2/(2n24)¼ 1/24 results in a totalFracRepl of 0.0145 and an extinction time of 3.9 years. The results in Fig. 4 show that the same ratio replT/halfL¼10/(10n24)¼1/24 also results in the same totalFracRepl of 0.0145, but in an extinction time of 19.33 years. The reason is that the selection advantage per mtDNA generation time is the same in both simulations, but because the generation time (half-life) in the second case is 5 times longer, the replacement process also takes 5 times longer.
Fig. 5. Effects of varying the size of the mtDNA mutant (mutS) on extinction time and the total fraction of replicating molecules. Increasing mutS reduces the selection advantage of the deletion mutant and leads to a strong rise of the extinction time. The fraction of replicating mtDNAs rises linearly, since a larger mutant needs more time for replication. The simulation was started with 999 wild-type and a single mutant mtDNA. A half-life of 10 days and a replication time of wt of 2 h was used.
3.3. Influence of the size of the mtDNA mutant
Fig. 3. Effects of varying the half-life of mtDNAs (halfL) on extinction time of the wild-type and the total fraction of replicating molecules. Increasing halfL fivefold from 2 to 10 days, leads to an almost 25 fold increase in extinction time from 3.9 to 95 years. The simulation was started with 999 wild-type and a single mutant mtDNA, which had 50% of the wild-type size. A replication time of 2 h was used.
The mutant size, mutS, is the last major parameter of the model. For the simulations so far, we used a value of 0.5, which corresponds to a deletion of 50% of the wild type genome. This provides the largest selection advantage, but also comes close to the largest deletion that is possible without deleting one of the two origins of replication of mammalian mtDNA. It is therefore interesting also to study the behaviour of larger mutants. For this purpose we varied the mutant size from 0.5 to 0.7 with a replication time of 2 h and a half-life of 10 days (Fig. 5). The larger the mutant, i.e. the smaller the deletion, the longer it takes for the mutant to replace the wild-type. A mutant of 50% of the wild type size requires ca. 95 years but a mutant with 70% of the wild type size takes almost 160 years! Extinction times approach infinity for mutS ¼1, since in that case there is no selection advantage for the mutant. Furthermore, the larger the mutant the higher is the total fraction of replicating mtDNA. The reason is that the replication
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time (time spent in the busy state) increases proportionally to the mutant size and hence the total fraction of mtDNAs in the busy state increases linearly. 3.4. Stochastic tissue simulations Next we used our stochastic model to study the effects of mutant accumulation at the tissue level in animals of various lifespans. A cytochrome c oxidase (COX) deficient phenotype has been observed in post mitotic cells of old animals and is generally regarded as consequence of a non-functional oxidative phosphorylation (Cao et al., 2001). As in other studies (Elson et al., 2001; Kowald and Kirkwood, 2013), we regard a cell as COX-deficient if it contains at least 60% mutant mtDNAs at the end of the simulation time. Clonally expanded deletion mutants have been found in short and long lived animals (from rat to man) and we therefore performed tissue simulations for 3, 10, 40 and 80 years with a halflife of 10 days, replication time of 2 h and a mutant size of 0.5. For this purpose we performed 1000 repetitions for each lifespan and counted the fraction of simulations that resulted in a COX deficient phenotype. Then mutation rates were adjusted in such a way, that at the end of the simulation 10 70.5% of the cells were COX deficient. Fig. 6A shows that there is a roughly 500 fold difference in mutation rate that is required to achieve the same percentage of COX deficient cells in animals with a lifespan of 3 years compared
to 80 years. Especially intriguing is the low degree of heteroplasmy that has been observed experimentally, i.e. it is in most cases a single type of mutant that has displaced the wild-type mtDNA. In the stochastic model we assume that each new deletion results in a new, unique, mutation type and Fig. 6B presents the number of different mtDNAs per COX deficient cell as a measure for the degree of heteroplasmy. As can be seen, heteroplasmy dramatically increases for short lifespans, leading on average to almost 30 different types of mtDNA per cell for a lifespan of 3 years. These results can hardly be reconciled with experimental findings. In Fig. 3 it was shown that the extinction time is drastically shortened if the mtDNA half-life is reduced, presumably because this increases the ratio of replication time to half-life. What influence has this for stochastic simulations at the tissue level? We studied this by performing a number of simulations for a lifespan of 3 years with different half-lives, but the standard values for replication time (2 h) and mutant size (0.5). Fig. 7A shows that the mutation rate required to achieve 10% COX negative cells after 3 years, strongly decreases with a decreasing half-life. Similarly, also the degree of heteroplasmy decreases to only 1.16 different mtDNAs for a half-life of 2 days (Fig. 7B). These simulations confirm the DDE calculations shown in Fig. 3 and emphasise the relevance of the ratio of replication time to half-life for the strength of the selection advantage that a shorter genome can confer in the replication process.
Fig. 6. Stochastic simulation of a population of 1000 mtDNAs for 3, 10, 40 and 80 years. The left diagram shows the mutation rate required to result in 10 7 0.5% of COX deficient cells (defined as harbouring 4 60% mutant mtDNAs) after the indicated time with replT¼ 2 h, halfL ¼10 d and mutSize¼ 0.5. Each point is calculated from 1000 simulation runs. The right diagram shows the degree of mtDNA heteroplasmy in COX deficient cells, measured by the number of different types of mtDNA mutants per cell.
Fig. 7. Stochastic simulation of a population of 1000 mtDNAs for 3 years (replT¼ 2 h, mutSize¼ 0.5) and different values for the half-life. The left diagram shows the mutation rate required to result in 10 7 0.5% of COX negative cells (defined as harbouring 4 60% mutant mtDNAs) and the right diagram shows the degree of mtDNA heteroplasmy in COX negative cells, measured by the number of different types of mtDNA mutants.
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4. Discussion In this work we studied the question of whether a shorter replication time of mitochondrial deletion mutants can provide them with a selection advantage, so that they can accumulate and replace the wild type. This idea had been put forward to explain the experimentally observed clonal expansion of deletion mutants (Wallace, 1992; Lee et al., 1998). We proposed a specific molecular mechanism of how a shorter replication time can result in a selection advantage by assuming that currently replicating molecules are not available to respond to further replication signals, since they are in a busy state until they finish their current round of replication. We used models based on delay differential equations as well as stochastic simulations to investigate this question by varying (i) replication time of mtDNA, (ii) half-life of mtDNA, and (iii) the size of the mutant and by calculating the degree of heteroplasmy observable at the tissue level. The simulations confirm that the half-life of mtDNA has a major influence on the extinction time, i.e. the time required for the mutant to replace the wild type population (Fig. 3). Using parameter values that are compatible with accepted literature values (replication time 2 h, half-life 10 days), it takes approximately 95 years until the wild type is replaced by a deletion mutant that has only 50% of the wild type size. Our simulations also show why the selection advantage dwindles away with increasing half-life. After a period of time equivalent to the half-life, 50% of the population of mtDNA molecules has been degraded and consequently the population has to double in this time to maintain a steady state. If half-life increases, there is more time for the population to double and consequently at any given time there will be a smaller fraction of mtDNAs in the replicating state. This decline of the total fraction of replicating molecules is also shown in Fig. 3. However, the smaller size of the mutant is only an advantage in this busy state (via a faster exit time) and consequently the selection advantage decreases with increasing half-life. How reliable and comprehensive are the experimental data regarding mtDNA half-life and replication time? Unfortunately the experimental basis is not very satisfactory. To our knowledge mtDNA half-life has only been measured in mice or rats, but never in humans. Furthermore, studies are both sparse and old (Gross et al., 1969; Huemer et al., 1971; Korr et al., 1998), or gave results completely incompatible with other studies (Collins et al., 2003). The situation regarding measurements of the mtDNA replication time is even worse. Clayton (1982) and Davis and Clayton (1996) are often cited as reference for a 1–2 h timespan for the mitochondrial replication time, but those articles seem to refer back to a single article by Berk and Clayton (1974). Given this uncertainty we regarded it appropriate also to perform simulations covering longer replication times. As expected, increasing replication times also increased the fraction of replicating mtDNAs and consequently reduced extinction times (Fig. 4). But even with a replication time of 10 h (and 10 days half-life), a newly emerging single mutant still requires 19 years before the wild type has been replaced. These time frames are evidently incompatible for explaining the ageing process in short lived animals. This becomes even clearer if the simulations are extended to mtDNAs with smaller deletions (Fig. 5). Using standard parameters a mutant that is 30% shorter than the wild-type takes almost 150 years to take over the mitochondrial population. If the parameters used for halflife and replication time are reliable, these numbers clearly rule out that deletion mutants accumulate based on a reduced genome size within a time frame that is compatible with mammalian lifespans. We also performed stochastic simulations to take random fluctuations into account and model the appearance of de novo mutations for species with lifespans ranging from 3 to 80 years.
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The simulations show that short lifespans require high mutation rates to generate a level of COX negative cells that is compatible with experimental findings (Fig. 6A). Even more important, the calculated degree of heteroplasmy also rises strongly for short lifespans (Fig. 6B). On average almost 30 different types of mtDNAs are present in COX negative cells after a 3 year simulation. Also these results disagree strongly with experimental observations. The results of the deterministic model have already indicated that the ratio between replication time and mtDNA half-life is critical for the selection advantage of the mutant. To further corroborate this finding we also performed stochastic simulations for a lifespan of 3 years with a variation of the half-life from 2–10 days (Fig. 7). A reduction of the half-life strongly reduced the necessary mutations rate as well as the degree of heteroplasmy. These effects are caused by the increasing selection advantage that accompanies the parameter change. In summary, our modelling investigation shows that despite all the uncertainty regarding experimental measurements of mtDNA half-life and replication time, it is highly unlikely that mitochondrial deletion mutants gain a selection advantage from a faster replication time based on the reduced size. Our simulations show that the large differences between replication time and half-life would lead to such extremely long extinction times (4 90 years), that such a process cannot serve as general mechanism to explain the clonal expansion of mitochondrial deletion mutants in long and short lived animal species.
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