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Mitochondrial DNA damage and efficiency of ATP biosynthesis: mathematical model
J. theor. Biol. (1995) 172, 161–168
Mitochondrial DNA Damage and Efficiency of ATP Biosynthesis: Mathematical Model N B† R M‡ † Research Center for Biotechnological Systems ‘‘SONAR’’, Ukrainian Academy of Sciences and ‡ Kiev State University, Ukraine (Received on 19 January 1994, Accepted in revised form on 14 June 1994)
The role of mitochondrial DNA (mtDNA) damage in ageing processes and in malignant transformation of a cell is discussed. A mathematical model of the mtDNA population in a cell and in tissue is constructed. The model describes the effects of mtDNA damages accumulated during ageing and some features of malignant transformation and regeneration.
of respiration-deficient cells in human hearts can limit the lifespan of an individual (Kadenbach & Mu¨ller-Ho¨cker, 1990). Alteration in mitochondrial function is an integral component of the organic process (Warburg, 1967). As tumor growth rate increases, the amount of cellular ATP generated by glycolytic reactions increases (Pedersen, 1978). The number of mitochondria decreases in tumor cells (Pedersen et al., 1970). There are also genetical changes in malignant cells mitochondria: the number of deletions and duplications increase (Ebner et al., 1991). A number of point mutations and deletions in mtDNA are recognized as causes of human diseases, such as Leber’s Hereditary Optic Neuropathy (LHON), Myoclonic Epilepsy and Ragged-Red Fiber Disease (MERRF) (Wallace, 1992). It is noted that the percentage of mutated mtDNA in cells can vary greatly, but old individuals contain low levels of a specific mtDNA deletion, i.e. random mutations in the population of mtDNA molecules of each cell occur throughout the lifespan. The content of mtDNA alters in the regenerating organs. Hence, after partial hepatectomy, the content of mtDNA in rat liver cells increases initially and then decreases to the preoperational level within 1 week. This effect is more significant in young rats (Asano et al., 1992).
1. Introduction , In higher eucaryotes, cell mitochondria are the most autonomous and most ‘‘rigidly’’ genetically coded organelles. They play a principal role in the cellular energetic system. About 90% the of the body’s oxygen is consumed by mitochondria in the process of oxidative phoshorylation (Chance et al., 1979). This process of ATP production is provided by five molecular complexes, known as the mitochondrial respiratory chain (see Fig. 1). Mitochondria possess their own genome. The human mitochondrial genome is a circular DNA molecule of 16569 bp. In the last decade the role of mtDNA damages has become a subject of intensive studies. It was shown that the mutations in mtDNA are accumulated during ageing process (Miquel et al., 1980; Murray & Baleavage, 1982; Arnheim & Cortopassi, 1992; Linnane et al., 1992; Miquel, 1992). This process is connected with changes in activity of respiratory chain complexes (Trounce et al., 1989), but the correlation of ageing and the level of respiratory chain protein is not significant (Byrne et al., 1991). The loss of cytochrome c oxidase activity in cardiomyocytes as a function of age has a mosaic pattern (Mu¨ller-Ho¨cker, 1989). The accumulation 1.1.
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Most cells normally contain thousands of mtDNA molecules (three to ten molecules in each mitochondrion). The mitochondria occupy approximately 20% of the cell volume (for example, in hepatocytes). Thus, their number and the number of mtDNA molecules are limited. The mtDNA replications and mitochondrial propagation are independent from the cell cycle. The half-life of mtDNA is between 6.7 days in the heart and 31 days in the brain (Gross et al., 1969). MtDNA is not covered by histones and lacks non-coding fragments. Therefore, its spontaneous mutability is much higher than that of nuclear DNA and the mutations are more harmful. About 1–2% of consumed oxygen is converted in mitochondria to the superoxide radical (Chance et al., 1979). This and other reactive oxygen species are considered as the main factors in mtDNA damage (Richter, 1992). Apparently, mtDNA do not have the reparation system that nuclear DNA has. Mitochondrial DNA recombinants cannot be demonstrated in interspecific somatic cell hybrids or within human populations (Wallace, 1992). So any defect in the mtDNA molecules is irreversible. It will be inherited by all its descendants. The spreading of the faulty mtDNA molecules is limited only by the ‘‘life struggle’’ with other molecules. We can consider mtDNA molecules in the cell as a population with different genotypes. The rates of propagation are different for normal (wild) molecules and for faulty ones. The molecules with deletions can be 30% shorter than normal and this sometimes gives them advantages in replication (Arnheim & Cortopassi, 1992). Also, any defects in complex I of the respiratory chain must lead to
the increase of electron transport through alternative pathways (e.g. succinate dehydrogenase). Thus the superoxide radical production also increases (Gutman et al., 1970; Storey, 1980). Therefore, normal mitochondria will perish faster. Damage of the complex I has been shown as a cause of malignant transformations and of radiation effects by Sidorik & Beregovskaya (1983) and Beregovskaya & Savich (1988). 1.2.
Our goal is to construct a mathematical model of the accumulation of mtDNA mutations in both a single cell and in tissue. Using this model, we can describe many features of ageing, malignant transformation and regeneration. The above allow us to formulate the following basic postulates of the mtDNA population mathematical model: (i) The total amount of mtDNA molecules in the cell is limited by a definite value, which we shall call the cell capacity. (Note that cell capacity in normal conditions is more than the number of mtDNA in the cell. The difference of these qualities is the free capacity of the cell, reserved for stressful conditions.) (ii) In the cell there are molecules that represent different genotypes. There is a competition for the free capacity of the cell between the mtDNA molecules. (iii) Excluding the competition, mtDNA molecules mutate, propagate themselves and perish independently from the other ones.
Fe–S1ab
F. 1. Scheme of respiratory chain (after B. Chance, 1977). I–IV, the complexes of respiratory chain; I, NADH ubiquinone oxidoreductase; II, succinate ubiquinone oxidoreductase; III, ubiquinol cytochrome c oxidoreductase; IV, cytochrome c oxidoreductase. The bracketed portions of the diagram list the total of components found in Complexes I, II, III and IV. Abbreviations used: NADH, nicotinamide adenine dinucleotide in reduced form; FMN, flavin mononucleotide; FAD, flavin adenine dinucleotide; Q, coenzyme Q or ubiquinone; Fe–S, iron–sulfur center; b, c1 , c, a and a3 are cytochromes.
(iv) The rates of mutation, propagation and perishing are different for the molecules of different types, but they can be considered as stable for the molecules of the same type (if the environmental conditions are the same). 2. Description of the Model 2.1.
t -
mtDNA contains the structural genes for 13 proteins of the respiratory chain (seven proteins of complex I, a single subunit protein of complex III, three subunits of complex IV and two subunits of ATP-synthetase). The remaining mtDNA codes the organellar rRNAs and tRNAs, specific to mitochondrial protein synthesis (McConkey, 1993). The absence of complex I decreases the ATP synthesis efficiency, but allows a roundabout way through complex II coded by nuclear DNA (Fig. 1). Without normal complexes III and IY, the respiratory chain is unable to synthesize ATP, so we shall consider three types of mtDNA molecules (Beregovskaya et al., 1992): X—normal mtDNA molecules that code all mitochondrial enzymes, used in the respiratory chain (except nuclear coded enzymes); Y—partly-faulty molecules that do not code enzymes of respiratory complex I, but provide other complexes; Z—completely faulty molecules that can live and replicate themselves, but do not produce active resiratory enzymes. By X, Y and Z we shall denote also the number of corresponding mtDNA in the cell. The capacity of the cell will be denoted by N. Clearly X e 0, Y e 0 and Z e 0. From our first postulate, X + Y + Z E N. The changes of X, Y and Z are connected with the mutation, replication and perishing of individual mtDNA molecules (see Fig. 2). Since the repair
F. 2. Diagram of X, Y and Z genotypes interchanging in the mitochondrial population in the cell. a, b and g: rates of replication; x, l and m: death rates for X, Y and Z types of mtDNA molecules, respectively; a, b and c: corresponding rates of conversion.
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system is absent, mutations can only transform X-type mtDNA to Y or Z-type, and Y-type to Z-type. The reverse changes are impossible. From our third postulate, mutations of each molecule are independent from other molecules. Hence the rate of change of X, connected with transformation into Y, is negative and proportional to X—namely it is equal to −aX, where a is the intensity of the transformation. Analogously the rate of transformation of X into Z is −cX and the rate of perishing is xX. The replicational growth rate is proportional to X and to the free capacity of the cell N − X − Y − Z. (Each molecule needs a ‘‘free place’’ in which it is able to replicate itself.) Therefore the rate of replication is aX (N − X − Y − Z ). The total rate of change of X is the sum of the rates of mutation, replication and perishing. Similarly the rates of change of Y and Z can be computed. Denoting total rates of change of X, Y and Z by X , Y , Z , respectively, we have X = −aX − cX + aX (N − X − Y − Z ) − xX Y = aX − bY + bY (N − X − Y − Z ) − lY Z = cX + bY + gZ (N − X − Y − Z ) − mZ. (1) This system of differential equations describes the life of an mtDNA population in the non-dividing cell. By no means should it be considered as an absolutely correct model. Another model for a smilar problem was proposed by Kowald & Kirkwood (1993). It takes into account some additional parameters of the process, such as the levels of ATP and reactive oxygen in the cell. System (1) can be used as a first approximation to the ‘‘true’’ model, allowing a qualitative mathematical analysis of the situation. The analysis of these equations shows that there can be from two to four stationary points, some of which can be stable. Stable stationary points are the sets of X, Y and Z values to which a cell can tend during its life. In the three-dimensional co-ordinate system (XYZ ) the development of the mtDNA population can be displayed by a curve (a trajectory of the solution of the system of equations), which tends to a stable point. In other words, the stable points attract cells. Clearly, solutions of system (1) cannot leave some bounded (according to our first postulate) area of the space, which we call the permissible field. Our model is completely correct in this sense. One can obtain the stable points of system (1), equating the right-hand side of the equations with zero. In the general case there exist four stable points. We denote them as S0 , S1 , S2 and S3 (Fig. 3). The values of parameters in Fig. 3 and subsequent figures are irrelevant. They
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. . plex I (Metzler, 1977, ch. 10, section E.1, figs 10 and 11). We assume that the same is true for the other control mechanisms (for these reasons they will be briefly referred to as ‘‘hormonal control’’). Therefore, the mitochondria that lack complex I are outside hormonal control. The power of such uncontrollable ATP synthesis in the cell is Pnc = py Y + pz Z. Cell division requires energy. Therefore only cells with sufficient ATP production power can divide. We shall consider two types of division: controlled and noncontrolled (malignant). Controlled division takes place when the hormonal control allows it. The other condition of controlled division is Pmax q Pdiv , where Pdiv is the minimal power needed for division. Our guess is that in organs that can regenerate (e.g. the liver), the hormonal division control is carried out by the complex I respiratory chain activity modulation. As a consequence we have the condition of malignant division Pnc q Pdiv .
F. 3. Phase portraits of the mathematical model for a non-dividing cell (a = b = c = a = b = 1, Nx = 5). (a) Four stable points S0 = 0; S1 , S2 , S3 (g = 2, Ny = 3, Nz = 1) are in the permissible field. (b) The points S0 and S3 are in the permissible field only (g = 1, Ny = 5, Nz = 5).
are chosen for the sake of visual clarity only. (For details see Appendix.) 2.2. t The capacity of the mitochondrial system of ATP synthesis depends on the number of the mtDNA molecules of each type. Denote by px , py and pz the power of ATP synthesis that one mtDNA molecule of the corresponding type can provide. Although completely faulty mitochondria do not produce ATP in the respiratory chain, pz takes into account glycolytic ATP production. Thus pz q 0, but pz Q py 1 2/3px . Then the total power of mitochondrial ATP synthesis in the cell is
Pmax = px X + py Y + pz Z. Mitochondria ATP production is controlled by different cellular control mechanisms, hormones (steroid and other), growth factors and so on. It is known that the target of steroid hormonal control (progesterone and other steroids with rotenon similar structure) in respiratory chain processes is, in general, com-
This statement may be formulated as a postulate of our model: (v) Cells can divide free from the organism’s hormonal control when they have faulty mitochondria sufficient to support division processes by ATP. Connection of complex I defects with uncontrollable mitosis of malignant tumor cells can play a role in the specific activity of tumor necrosis factor (TNF). It has been shown by Schulze-Osthoff et al. (1992) that respiratory chain inhibitors strongly modulate the cytotoxicity of TNF. This modulation is different depending on which respiratory chain complex is inhibited—the first or the following. Unfortunately, the biochemical basis of TNF cytotoxic action against tumor cells is still largely unknown. To simplify our model in what follows we shall suppose that a cell always divides when the stated division conditions hold. A possible mechanism which initiates uncontrollable mitosis is connected with ‘‘mitotic proteins’’ and glycolytic enzymes noted by Warburg (1967). We shall not consider this mechanism in detail. After the division the descendants receive the mitochondria of their ancestor. We assume initially that mtDNA of each type divide into two equal parts. Then both descendants obtain X/2 normal, Y/2
165 2 . 3.
F. 4. Dividing cell (X–Y plot, a = 1, b = c = 0, a = 1, b = 3, g = 0, Nx = 5, Ny = 2.5, Nz = 0).
partly faulty and Z/2 completely faulty mtDNA molecules. In the (XYZ )-co-ordinate space this is displayed as a jump of the cell from one trajectory to another. The cell then begins its new life on the new trajectory. According to our fifth postulate, such a jump takes place when at some moment the Pnc of our cell reaches the Pdiv value (see Fig. 4). Applying this condition to our equations we alter possible ways of cell evolution. In this case some stable stationary points (for example, S2 or S3 ) can generate stable closed cycles. In these cycles, the cell, after its jump (division), hits the same trajectory on which it was previously. The stable cycle attracts cells as well as a stable point. The whole co-ordinate space can be divided into domains of attraction of stable points and cycles. The cell in the domain of attraction of the stable cycle can be considered as the malignant one. It divides without control. The cells of stable-point domains of attraction are divided into two groups. If, in the stable point, Pmax q Pdiv then cell division is possible by means of hormonal control. (The point S1 can play such a role.) This cell also can perform its specific function. We call such cells ‘‘normal’’. If Pmax Q Pdiv then the cell cannot divide. We refer to the stable point that satisfies this condition e.g. S3 as a ‘‘trap’’. Cells in a trap are absolutely stable. They cannot divide and are unable to leave this area of space. We call these cells ‘‘weakened’’. Note that ‘‘normality’’, ‘‘weakness’’ and ‘‘malignancy’’ are stable qualities of the cell. Small perturbations do not change them. Some external action is needed to move a cell from one domain of attraction to the other.
Cell division should be considered as a stochastic process. The distribution of mtDNA between descendants is not absolutely equal. One of them can obtain more mutant molecules and the other will be more normal. Thus, the normal cell can produce a malignant or a weakened descendant. The malignant cells can also bear a normal descendant but the probability of this is small. Weakened cells cannot divide, so normal tissue will accumulate weakened cells during ageing. These cells cannot respond to the demands of an organism as well as normal cells. They occupy a place in the tissue and consume all the nutrients that the organism provides. This is an explanation of the growth of mozaism in the tissues of older individuals. The total number of mutated mtDNA also increases in the tissue, but some normal cells continue their life and perform their functions. We consider the decrease of their proportion in the tissue as a general cause of senescence effects. The weakened cells can arise not only because of division but also under the influence of unfavorable factors (e.g. ionizing radiation, chemical carcinogens, intensification of intracellular oxidation and superoxide radicals production) and also as a result of spontaneous mutations. The possibility of the apparition of weakened cells under unfavorable conditions was considered by Toussaint et al. (1991) from a thermodynamic perspective. While ageing is connected with the steady accumulation of cells in traps, the tumor can be caused by one cell that falls into the stable cycle domain of attraction. If this cell does not die it will circulate in the loop with division at each cycle. The growth of tumor is unlimited in this model. We use computer simulation to investigate this model. The example of such a simulation is represented in Fig. 5. The level of uncontrolled ATP production and the number of cells in tissue are plotted as functions of lifetime. One can see that malignant division, initiated at 8.5 years, is connected with steady growth of glycolytic energetics. At 9.7 years the tissue consists only of malignant cells. 3. Comment and Discussion The crucial point of mitochondrial genetic ageing theories is a question of heredity. Why is old age not inherited? Why do old women bear young children? Two possible answers are found in the literature. The first is given by Miquel (1992), who posits that mtDNA mutations are accumulated in post-mitotic 3.1.
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. . weakened cells, which do not divide. But normal cells (in which there is no accumulation) can also live without divisions. Cell division accelerates the accumulation of mtDNA not in each cell but in the entire tissue. 3.2.
Our model can explain two features of malignant transformation: the latent period and the different effects of momentary and prolonged carcinogen influence. The latent period is the time between the exposure to the carcinogen and the appearance of the tumor. In our model each sufficiently strong influence causes a shift of the cell in the co-ordinate space. If the shift occurs from the domain of normal attraction to the domain of stable cycle attraction then a cell becomes malignant. There can be a long time between the shift and the first cell division. The momentary influence of a large carcinogen dose often makes the effect completely different to the prolonged influence of small doses. To explain this fact, one may note that in our model the mechanisms of such influences are quite different. A large dose causes the shift of the cell into new domain of attraction. In contrast, the long time influence alters the coefficients in eqn (1). Such a change can lead to different dynamics of the mtDNA population development.
F. 5. Simulation model of tissue development. The tissue develops from one cell containing normal mtDNA. Results are represented for the case when a stable stationary cycle exists. (a) Part of the glycolytic (uncontrolled) ATP production in total tissue ATP production. (b) Number of cells in tissue.
cells only. From this hypothesis, it follows that dividing cells save all normal mitochondria good. This concept is too restrictive to be general in our view. Malignant cells divide very quickly but they have faulty mitochondria. The regeneration of the liver is connected with divisions of hepatocytes but it accelerates ageing (and supposedly the accumulation of mutations of the mtDNA molecules). The second hypothesis (Linnane et al., 1992) states that the accumulation of defects has the same rate in all cells but the heredity is improved during oocyte generation. In our model such mechanisms are not needed. The mutations are accumulated in the
3.3. It was mentioned above that the content of mtDNA in rat liver cells increases after partial hepatectomy (Asano, 1992). This fact seems paradoxical. In the regenerated organ, cell division is very fast and one would assume that normal mtDNA replication would fall behind. But in our model this effect can be explained by taking into account the different rates of replication of different types of molecules. If the fast replicating molecules have a high perishing rate, they will initially overtake other cell molecules after division. But after that they will perish. The lower number of low perishing rate mtDNA molecules will be defeated in the life struggle. The decrease of this effect in the tissues of old individuals is caused by the presence of weakened cells that do not divide. We do not suppose here that fast-replicating mtDNA molecules are always faulty. Existence of two normal mtDNA types in the cell with different replication and perishing rates will cause the same effect. It could be a regulatory mechanism which provides an increase of ATP production for the regeneration process or in other stressful situations. However,
accelerated ageing and the increase of malignant transformation probability after partial hepatectomy indicate the accumulation of only faulty mtDNA in liver. Computation by eqns (1) with appropriately chosen parameters illustate these effects. The next step should be to find real parameters of some tissue cells, substitute them into the equations to obtain quantitative results. We are grateful to Prof. W. Wertelecki and both referees for helpful comments. The work was supported in part by Ukrainian State Foundation for Fundamental Researches and by ISF. REFERENCES A, N. & C, G. (1992). Deleterious mitochondrial DNA mutations accumulate in aging human tissues. Mutat. Res. 275, 157–167. A, K., N, M., A, A., S, T. & T, H. (1992). Quantitation of changes in mitochondrial DNA during ageing and regeneration of rat liver using non-radioactive DNA probes. Mech. Ageing Dev. 62, 85–98. B, N. N., M, R. E., S, A. D. & Y, T. N. (1992). The role of mitochondrial DNA violations in realization of remote effects of different genotoxic actions. Rep. Ukrain. Acad. Sci. 12, 129–135. (In Russian.) B, N. N. & S, A. V. (1988). Possibility of iron–sulfur proteins coding in mammalian mitochondrial genome. Biopolim. i kletka 4(5), 238–245. (In Russian.) B, E., T, I. & D, X. (1991). Mitochondrial theory of senescence: respiratory chain protein studies in human skeletal muscle. Mech. Ageing Dev. 60, 295–302. C, B., S, H. & B, A. (1979). Hydroperoxide metabolism in mammalian organs. Physiol. Rev. 59, 527–605. E, D., R, G., P, I. & H, O. (1991). Functional and molecular analysis of mitochondria in thyroid oncocytoma. Virchows Arch. B. Cell Path. 60, 139–144. G, N. J., G, G. S. & R, M. (1969). Apparent turnover of mitochondrial deoxyribonucleic acid and mitochondrial phospholipids in the tissues of the rat. J. biol. Chem. 244, 1552–1562. G, M., S, T. P. & B, H., et al. (1970). Reaction sites of rotenone, piericidin A, and amytal in relation to the nonheme iron components of NADH dehydrogenase. Proc. natn. Acad. Sci. U.S.A. 65(3), 763–770. K, B. & M¨-H¨, J. (1990). Mutations of mitochondrial DNA and human health. Naturwissenshaften 27, 221–225. K, A. & K, T. B. L. (1993). Mitochondrial mutations, cellular instability and ageing: modelling the population dynamics of mitochondria. Mutat. Res. 295, 93–103. L, A. W., Z, C., B, A. & N, P. (1992). Mitochondrial DNA mutation and the ageing process: bioenergy and pharmacological intervention. Mutat. Res. 275, 195–208. MC, E. H. (1993). Human Genetics. The Molecular Revolution. Boston and London: Jones and Bartlett. M, D. E. (1977). Biochemistry. The Chemical Reactions of Living Cells. New York: Academic Press. M, J., E, A. C., F, J. & J, J. E. (1980). Mitochondrial role in cell aging. Expl Gerontol. 15, 575–591. M, J. (1992). An update on the mitochondrial-DNA mutation hypothesis of cell aging. Mutat. Res. 275, 209–216. M¨-H¨, J. (1989). Cytochrome c oxidase deficient cardiomyocyte in the human heart—an age related phenomenon. Am. J. Pathol. 134, 1167–1173.
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M, M. A. & B, W. X. (1982). Changes in mitochondrial DNA during ageing. Mech. Ageing Dev. 20, 233–242. P, P. L. (1978). Tumor mitochondria and the bioenergetics of cancer cells. Prog. expl Tumor Res. 22, 190–274. P, P. L., G, J. W., C, T. L. & M, H. P. (1970). A comparison of some ultrastructural and biochemical properties of mitochondria from Morris hepatomas 9618A, 7800 and 3924A. Cancer Res. 30, 2620–2626. R, C. (1992). Reactive oxygen and DNA damage in mitochondria. Mutat. Res. 275, 249–255. S-O, K., B, A. C., V, B. & B, R. (1992). Cytotoxic activity of tumor necrosis factor is mediated by early damage of mitochondrial functions. J. biol. Chem. 267, 5317–5323. S, E. P. & B, N. N. (1983). Iron–sulfur proteins and free radicals of the electron transfer chains of mitochondria in chemical and hormonal carcinogenesis. Ukrain. Biochem. J. 55(5), 544–547. (In Russian.) S, B. T. (1980). Inhibitors of energy-coupling site I of the mitochondrial respiratory chain. Pharm. Ther. 10(2), 399–406. T, O., R, M. & R, J. (1991). Ageing as multi-step process characterized by lowering of entropy production leading the cell to a sequence of defined stages. Mech. Aging Dev. 61, 45–64. T, I., B, E. & M, S. (1989). Mitochondrial respiratory chain function: possible factor in ageing. Lancet 1, 637–639. W, D. C. (1992). Diseases of the mitochondrial DNA. A. Rev. Biochem. 61, 1175–1212. W, O. (1967). Oxygen, the creator of differentiation. In: Aspects of Yeast Metabolism (Mills, A. K. & Krebs, H., eds) pp. 327–337. Oxford: Blackwell Scientific Publications.
APPENDIX Stationary Points of Non-dividing Cell Model Consider the stable points of system (1) in the main text. It may be represented in more convenient form: X = −aX − cX + aX (Nx − X − Y − Z ) Y = aX − bY + bY (Ny − X − Y − Z ) Z = cX + bY + gZ (Nz − X − Y − Z ),
(A.1)
whereNx = N − x/a,Ny = N − l/bandNz = N − m/g, are the effective capacities of the corresponding type of mtDNA molecules in the cell. These capacities represent the maximum quantity of mtDNA molecules of corresponding type that may be accumulated in the cell, taking into account its constant decay. Equating the right-hand sides of eqns (A.1) with zero we obtain, in general, four stable points S0 , S1 , S2 and S3 . The point S0 = (0, 0, 0) corresponds to the absence of any type of mtDNA molecules. At the point S3 = (0, 0, Nz ) there are only completely-faulty mtDNA and the energy ensurance is realized by glycolytic means. Points S0 and S3 are always in the permissible field.
. .
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The point S2 = (0, Y2 , Z2 ), where
0
g Nz − N y + Y2 = −
b b
10
Ny −
0
b − g Nz − Ny +
0 0
b b Ny − b Z2 =
b b
1
b b − g N z − Ny + b
1
b b
1
1 ,
belongs to the permissible field when Nz − Ny + b/ b Q 0.
The point S1 = (X1 , Y1 , Z1 ), co-ordinates of which satisfy the following system of linear equations X1 + Y1 + Z1 = Nx −
0
aX1 + b Ny − Nx +
0
a+c a
1
a+c b − Y1 = 0 a b
cX1 + bY1 + g Nz − Nx +
1
a+c Z1 = 0, a
belongs to the permissible field if Ny − Nx + (a+c)/ a − b/b Q 0 and Nz − Nx + (a + c)/a Q 0.
Mitochondrial DNA Damage and Efficiency of ATP Biosynthesis: Mathematical Model N B† R M‡ † Research Center for Biotechnological Systems ‘‘SONAR’’, Ukrainian Academy of Sciences and ‡ Kiev State University, Ukraine (Received on 19 January 1994, Accepted in revised form on 14 June 1994)
The role of mitochondrial DNA (mtDNA) damage in ageing processes and in malignant transformation of a cell is discussed. A mathematical model of the mtDNA population in a cell and in tissue is constructed. The model describes the effects of mtDNA damages accumulated during ageing and some features of malignant transformation and regeneration.
of respiration-deficient cells in human hearts can limit the lifespan of an individual (Kadenbach & Mu¨ller-Ho¨cker, 1990). Alteration in mitochondrial function is an integral component of the organic process (Warburg, 1967). As tumor growth rate increases, the amount of cellular ATP generated by glycolytic reactions increases (Pedersen, 1978). The number of mitochondria decreases in tumor cells (Pedersen et al., 1970). There are also genetical changes in malignant cells mitochondria: the number of deletions and duplications increase (Ebner et al., 1991). A number of point mutations and deletions in mtDNA are recognized as causes of human diseases, such as Leber’s Hereditary Optic Neuropathy (LHON), Myoclonic Epilepsy and Ragged-Red Fiber Disease (MERRF) (Wallace, 1992). It is noted that the percentage of mutated mtDNA in cells can vary greatly, but old individuals contain low levels of a specific mtDNA deletion, i.e. random mutations in the population of mtDNA molecules of each cell occur throughout the lifespan. The content of mtDNA alters in the regenerating organs. Hence, after partial hepatectomy, the content of mtDNA in rat liver cells increases initially and then decreases to the preoperational level within 1 week. This effect is more significant in young rats (Asano et al., 1992).
1. Introduction , In higher eucaryotes, cell mitochondria are the most autonomous and most ‘‘rigidly’’ genetically coded organelles. They play a principal role in the cellular energetic system. About 90% the of the body’s oxygen is consumed by mitochondria in the process of oxidative phoshorylation (Chance et al., 1979). This process of ATP production is provided by five molecular complexes, known as the mitochondrial respiratory chain (see Fig. 1). Mitochondria possess their own genome. The human mitochondrial genome is a circular DNA molecule of 16569 bp. In the last decade the role of mtDNA damages has become a subject of intensive studies. It was shown that the mutations in mtDNA are accumulated during ageing process (Miquel et al., 1980; Murray & Baleavage, 1982; Arnheim & Cortopassi, 1992; Linnane et al., 1992; Miquel, 1992). This process is connected with changes in activity of respiratory chain complexes (Trounce et al., 1989), but the correlation of ageing and the level of respiratory chain protein is not significant (Byrne et al., 1991). The loss of cytochrome c oxidase activity in cardiomyocytes as a function of age has a mosaic pattern (Mu¨ller-Ho¨cker, 1989). The accumulation 1.1.
0022–5193/95/020161 + 08 $08.00/0
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7 1995 Academic Press Limited
. .
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Most cells normally contain thousands of mtDNA molecules (three to ten molecules in each mitochondrion). The mitochondria occupy approximately 20% of the cell volume (for example, in hepatocytes). Thus, their number and the number of mtDNA molecules are limited. The mtDNA replications and mitochondrial propagation are independent from the cell cycle. The half-life of mtDNA is between 6.7 days in the heart and 31 days in the brain (Gross et al., 1969). MtDNA is not covered by histones and lacks non-coding fragments. Therefore, its spontaneous mutability is much higher than that of nuclear DNA and the mutations are more harmful. About 1–2% of consumed oxygen is converted in mitochondria to the superoxide radical (Chance et al., 1979). This and other reactive oxygen species are considered as the main factors in mtDNA damage (Richter, 1992). Apparently, mtDNA do not have the reparation system that nuclear DNA has. Mitochondrial DNA recombinants cannot be demonstrated in interspecific somatic cell hybrids or within human populations (Wallace, 1992). So any defect in the mtDNA molecules is irreversible. It will be inherited by all its descendants. The spreading of the faulty mtDNA molecules is limited only by the ‘‘life struggle’’ with other molecules. We can consider mtDNA molecules in the cell as a population with different genotypes. The rates of propagation are different for normal (wild) molecules and for faulty ones. The molecules with deletions can be 30% shorter than normal and this sometimes gives them advantages in replication (Arnheim & Cortopassi, 1992). Also, any defects in complex I of the respiratory chain must lead to
the increase of electron transport through alternative pathways (e.g. succinate dehydrogenase). Thus the superoxide radical production also increases (Gutman et al., 1970; Storey, 1980). Therefore, normal mitochondria will perish faster. Damage of the complex I has been shown as a cause of malignant transformations and of radiation effects by Sidorik & Beregovskaya (1983) and Beregovskaya & Savich (1988). 1.2.
Our goal is to construct a mathematical model of the accumulation of mtDNA mutations in both a single cell and in tissue. Using this model, we can describe many features of ageing, malignant transformation and regeneration. The above allow us to formulate the following basic postulates of the mtDNA population mathematical model: (i) The total amount of mtDNA molecules in the cell is limited by a definite value, which we shall call the cell capacity. (Note that cell capacity in normal conditions is more than the number of mtDNA in the cell. The difference of these qualities is the free capacity of the cell, reserved for stressful conditions.) (ii) In the cell there are molecules that represent different genotypes. There is a competition for the free capacity of the cell between the mtDNA molecules. (iii) Excluding the competition, mtDNA molecules mutate, propagate themselves and perish independently from the other ones.
Fe–S1ab
F. 1. Scheme of respiratory chain (after B. Chance, 1977). I–IV, the complexes of respiratory chain; I, NADH ubiquinone oxidoreductase; II, succinate ubiquinone oxidoreductase; III, ubiquinol cytochrome c oxidoreductase; IV, cytochrome c oxidoreductase. The bracketed portions of the diagram list the total of components found in Complexes I, II, III and IV. Abbreviations used: NADH, nicotinamide adenine dinucleotide in reduced form; FMN, flavin mononucleotide; FAD, flavin adenine dinucleotide; Q, coenzyme Q or ubiquinone; Fe–S, iron–sulfur center; b, c1 , c, a and a3 are cytochromes.
(iv) The rates of mutation, propagation and perishing are different for the molecules of different types, but they can be considered as stable for the molecules of the same type (if the environmental conditions are the same). 2. Description of the Model 2.1.
t -
mtDNA contains the structural genes for 13 proteins of the respiratory chain (seven proteins of complex I, a single subunit protein of complex III, three subunits of complex IV and two subunits of ATP-synthetase). The remaining mtDNA codes the organellar rRNAs and tRNAs, specific to mitochondrial protein synthesis (McConkey, 1993). The absence of complex I decreases the ATP synthesis efficiency, but allows a roundabout way through complex II coded by nuclear DNA (Fig. 1). Without normal complexes III and IY, the respiratory chain is unable to synthesize ATP, so we shall consider three types of mtDNA molecules (Beregovskaya et al., 1992): X—normal mtDNA molecules that code all mitochondrial enzymes, used in the respiratory chain (except nuclear coded enzymes); Y—partly-faulty molecules that do not code enzymes of respiratory complex I, but provide other complexes; Z—completely faulty molecules that can live and replicate themselves, but do not produce active resiratory enzymes. By X, Y and Z we shall denote also the number of corresponding mtDNA in the cell. The capacity of the cell will be denoted by N. Clearly X e 0, Y e 0 and Z e 0. From our first postulate, X + Y + Z E N. The changes of X, Y and Z are connected with the mutation, replication and perishing of individual mtDNA molecules (see Fig. 2). Since the repair
F. 2. Diagram of X, Y and Z genotypes interchanging in the mitochondrial population in the cell. a, b and g: rates of replication; x, l and m: death rates for X, Y and Z types of mtDNA molecules, respectively; a, b and c: corresponding rates of conversion.
163
system is absent, mutations can only transform X-type mtDNA to Y or Z-type, and Y-type to Z-type. The reverse changes are impossible. From our third postulate, mutations of each molecule are independent from other molecules. Hence the rate of change of X, connected with transformation into Y, is negative and proportional to X—namely it is equal to −aX, where a is the intensity of the transformation. Analogously the rate of transformation of X into Z is −cX and the rate of perishing is xX. The replicational growth rate is proportional to X and to the free capacity of the cell N − X − Y − Z. (Each molecule needs a ‘‘free place’’ in which it is able to replicate itself.) Therefore the rate of replication is aX (N − X − Y − Z ). The total rate of change of X is the sum of the rates of mutation, replication and perishing. Similarly the rates of change of Y and Z can be computed. Denoting total rates of change of X, Y and Z by X , Y , Z , respectively, we have X = −aX − cX + aX (N − X − Y − Z ) − xX Y = aX − bY + bY (N − X − Y − Z ) − lY Z = cX + bY + gZ (N − X − Y − Z ) − mZ. (1) This system of differential equations describes the life of an mtDNA population in the non-dividing cell. By no means should it be considered as an absolutely correct model. Another model for a smilar problem was proposed by Kowald & Kirkwood (1993). It takes into account some additional parameters of the process, such as the levels of ATP and reactive oxygen in the cell. System (1) can be used as a first approximation to the ‘‘true’’ model, allowing a qualitative mathematical analysis of the situation. The analysis of these equations shows that there can be from two to four stationary points, some of which can be stable. Stable stationary points are the sets of X, Y and Z values to which a cell can tend during its life. In the three-dimensional co-ordinate system (XYZ ) the development of the mtDNA population can be displayed by a curve (a trajectory of the solution of the system of equations), which tends to a stable point. In other words, the stable points attract cells. Clearly, solutions of system (1) cannot leave some bounded (according to our first postulate) area of the space, which we call the permissible field. Our model is completely correct in this sense. One can obtain the stable points of system (1), equating the right-hand side of the equations with zero. In the general case there exist four stable points. We denote them as S0 , S1 , S2 and S3 (Fig. 3). The values of parameters in Fig. 3 and subsequent figures are irrelevant. They
164
. . plex I (Metzler, 1977, ch. 10, section E.1, figs 10 and 11). We assume that the same is true for the other control mechanisms (for these reasons they will be briefly referred to as ‘‘hormonal control’’). Therefore, the mitochondria that lack complex I are outside hormonal control. The power of such uncontrollable ATP synthesis in the cell is Pnc = py Y + pz Z. Cell division requires energy. Therefore only cells with sufficient ATP production power can divide. We shall consider two types of division: controlled and noncontrolled (malignant). Controlled division takes place when the hormonal control allows it. The other condition of controlled division is Pmax q Pdiv , where Pdiv is the minimal power needed for division. Our guess is that in organs that can regenerate (e.g. the liver), the hormonal division control is carried out by the complex I respiratory chain activity modulation. As a consequence we have the condition of malignant division Pnc q Pdiv .
F. 3. Phase portraits of the mathematical model for a non-dividing cell (a = b = c = a = b = 1, Nx = 5). (a) Four stable points S0 = 0; S1 , S2 , S3 (g = 2, Ny = 3, Nz = 1) are in the permissible field. (b) The points S0 and S3 are in the permissible field only (g = 1, Ny = 5, Nz = 5).
are chosen for the sake of visual clarity only. (For details see Appendix.) 2.2. t The capacity of the mitochondrial system of ATP synthesis depends on the number of the mtDNA molecules of each type. Denote by px , py and pz the power of ATP synthesis that one mtDNA molecule of the corresponding type can provide. Although completely faulty mitochondria do not produce ATP in the respiratory chain, pz takes into account glycolytic ATP production. Thus pz q 0, but pz Q py 1 2/3px . Then the total power of mitochondrial ATP synthesis in the cell is
Pmax = px X + py Y + pz Z. Mitochondria ATP production is controlled by different cellular control mechanisms, hormones (steroid and other), growth factors and so on. It is known that the target of steroid hormonal control (progesterone and other steroids with rotenon similar structure) in respiratory chain processes is, in general, com-
This statement may be formulated as a postulate of our model: (v) Cells can divide free from the organism’s hormonal control when they have faulty mitochondria sufficient to support division processes by ATP. Connection of complex I defects with uncontrollable mitosis of malignant tumor cells can play a role in the specific activity of tumor necrosis factor (TNF). It has been shown by Schulze-Osthoff et al. (1992) that respiratory chain inhibitors strongly modulate the cytotoxicity of TNF. This modulation is different depending on which respiratory chain complex is inhibited—the first or the following. Unfortunately, the biochemical basis of TNF cytotoxic action against tumor cells is still largely unknown. To simplify our model in what follows we shall suppose that a cell always divides when the stated division conditions hold. A possible mechanism which initiates uncontrollable mitosis is connected with ‘‘mitotic proteins’’ and glycolytic enzymes noted by Warburg (1967). We shall not consider this mechanism in detail. After the division the descendants receive the mitochondria of their ancestor. We assume initially that mtDNA of each type divide into two equal parts. Then both descendants obtain X/2 normal, Y/2
165 2 . 3.
F. 4. Dividing cell (X–Y plot, a = 1, b = c = 0, a = 1, b = 3, g = 0, Nx = 5, Ny = 2.5, Nz = 0).
partly faulty and Z/2 completely faulty mtDNA molecules. In the (XYZ )-co-ordinate space this is displayed as a jump of the cell from one trajectory to another. The cell then begins its new life on the new trajectory. According to our fifth postulate, such a jump takes place when at some moment the Pnc of our cell reaches the Pdiv value (see Fig. 4). Applying this condition to our equations we alter possible ways of cell evolution. In this case some stable stationary points (for example, S2 or S3 ) can generate stable closed cycles. In these cycles, the cell, after its jump (division), hits the same trajectory on which it was previously. The stable cycle attracts cells as well as a stable point. The whole co-ordinate space can be divided into domains of attraction of stable points and cycles. The cell in the domain of attraction of the stable cycle can be considered as the malignant one. It divides without control. The cells of stable-point domains of attraction are divided into two groups. If, in the stable point, Pmax q Pdiv then cell division is possible by means of hormonal control. (The point S1 can play such a role.) This cell also can perform its specific function. We call such cells ‘‘normal’’. If Pmax Q Pdiv then the cell cannot divide. We refer to the stable point that satisfies this condition e.g. S3 as a ‘‘trap’’. Cells in a trap are absolutely stable. They cannot divide and are unable to leave this area of space. We call these cells ‘‘weakened’’. Note that ‘‘normality’’, ‘‘weakness’’ and ‘‘malignancy’’ are stable qualities of the cell. Small perturbations do not change them. Some external action is needed to move a cell from one domain of attraction to the other.
Cell division should be considered as a stochastic process. The distribution of mtDNA between descendants is not absolutely equal. One of them can obtain more mutant molecules and the other will be more normal. Thus, the normal cell can produce a malignant or a weakened descendant. The malignant cells can also bear a normal descendant but the probability of this is small. Weakened cells cannot divide, so normal tissue will accumulate weakened cells during ageing. These cells cannot respond to the demands of an organism as well as normal cells. They occupy a place in the tissue and consume all the nutrients that the organism provides. This is an explanation of the growth of mozaism in the tissues of older individuals. The total number of mutated mtDNA also increases in the tissue, but some normal cells continue their life and perform their functions. We consider the decrease of their proportion in the tissue as a general cause of senescence effects. The weakened cells can arise not only because of division but also under the influence of unfavorable factors (e.g. ionizing radiation, chemical carcinogens, intensification of intracellular oxidation and superoxide radicals production) and also as a result of spontaneous mutations. The possibility of the apparition of weakened cells under unfavorable conditions was considered by Toussaint et al. (1991) from a thermodynamic perspective. While ageing is connected with the steady accumulation of cells in traps, the tumor can be caused by one cell that falls into the stable cycle domain of attraction. If this cell does not die it will circulate in the loop with division at each cycle. The growth of tumor is unlimited in this model. We use computer simulation to investigate this model. The example of such a simulation is represented in Fig. 5. The level of uncontrolled ATP production and the number of cells in tissue are plotted as functions of lifetime. One can see that malignant division, initiated at 8.5 years, is connected with steady growth of glycolytic energetics. At 9.7 years the tissue consists only of malignant cells. 3. Comment and Discussion The crucial point of mitochondrial genetic ageing theories is a question of heredity. Why is old age not inherited? Why do old women bear young children? Two possible answers are found in the literature. The first is given by Miquel (1992), who posits that mtDNA mutations are accumulated in post-mitotic 3.1.
166
. . weakened cells, which do not divide. But normal cells (in which there is no accumulation) can also live without divisions. Cell division accelerates the accumulation of mtDNA not in each cell but in the entire tissue. 3.2.
Our model can explain two features of malignant transformation: the latent period and the different effects of momentary and prolonged carcinogen influence. The latent period is the time between the exposure to the carcinogen and the appearance of the tumor. In our model each sufficiently strong influence causes a shift of the cell in the co-ordinate space. If the shift occurs from the domain of normal attraction to the domain of stable cycle attraction then a cell becomes malignant. There can be a long time between the shift and the first cell division. The momentary influence of a large carcinogen dose often makes the effect completely different to the prolonged influence of small doses. To explain this fact, one may note that in our model the mechanisms of such influences are quite different. A large dose causes the shift of the cell into new domain of attraction. In contrast, the long time influence alters the coefficients in eqn (1). Such a change can lead to different dynamics of the mtDNA population development.
F. 5. Simulation model of tissue development. The tissue develops from one cell containing normal mtDNA. Results are represented for the case when a stable stationary cycle exists. (a) Part of the glycolytic (uncontrolled) ATP production in total tissue ATP production. (b) Number of cells in tissue.
cells only. From this hypothesis, it follows that dividing cells save all normal mitochondria good. This concept is too restrictive to be general in our view. Malignant cells divide very quickly but they have faulty mitochondria. The regeneration of the liver is connected with divisions of hepatocytes but it accelerates ageing (and supposedly the accumulation of mutations of the mtDNA molecules). The second hypothesis (Linnane et al., 1992) states that the accumulation of defects has the same rate in all cells but the heredity is improved during oocyte generation. In our model such mechanisms are not needed. The mutations are accumulated in the
3.3. It was mentioned above that the content of mtDNA in rat liver cells increases after partial hepatectomy (Asano, 1992). This fact seems paradoxical. In the regenerated organ, cell division is very fast and one would assume that normal mtDNA replication would fall behind. But in our model this effect can be explained by taking into account the different rates of replication of different types of molecules. If the fast replicating molecules have a high perishing rate, they will initially overtake other cell molecules after division. But after that they will perish. The lower number of low perishing rate mtDNA molecules will be defeated in the life struggle. The decrease of this effect in the tissues of old individuals is caused by the presence of weakened cells that do not divide. We do not suppose here that fast-replicating mtDNA molecules are always faulty. Existence of two normal mtDNA types in the cell with different replication and perishing rates will cause the same effect. It could be a regulatory mechanism which provides an increase of ATP production for the regeneration process or in other stressful situations. However,
accelerated ageing and the increase of malignant transformation probability after partial hepatectomy indicate the accumulation of only faulty mtDNA in liver. Computation by eqns (1) with appropriately chosen parameters illustate these effects. The next step should be to find real parameters of some tissue cells, substitute them into the equations to obtain quantitative results. We are grateful to Prof. W. Wertelecki and both referees for helpful comments. The work was supported in part by Ukrainian State Foundation for Fundamental Researches and by ISF. REFERENCES A, N. & C, G. (1992). Deleterious mitochondrial DNA mutations accumulate in aging human tissues. Mutat. Res. 275, 157–167. A, K., N, M., A, A., S, T. & T, H. (1992). Quantitation of changes in mitochondrial DNA during ageing and regeneration of rat liver using non-radioactive DNA probes. Mech. Ageing Dev. 62, 85–98. B, N. N., M, R. E., S, A. D. & Y, T. N. (1992). The role of mitochondrial DNA violations in realization of remote effects of different genotoxic actions. Rep. Ukrain. Acad. Sci. 12, 129–135. (In Russian.) B, N. N. & S, A. V. (1988). Possibility of iron–sulfur proteins coding in mammalian mitochondrial genome. Biopolim. i kletka 4(5), 238–245. (In Russian.) B, E., T, I. & D, X. (1991). Mitochondrial theory of senescence: respiratory chain protein studies in human skeletal muscle. Mech. Ageing Dev. 60, 295–302. C, B., S, H. & B, A. (1979). Hydroperoxide metabolism in mammalian organs. Physiol. Rev. 59, 527–605. E, D., R, G., P, I. & H, O. (1991). Functional and molecular analysis of mitochondria in thyroid oncocytoma. Virchows Arch. B. Cell Path. 60, 139–144. G, N. J., G, G. S. & R, M. (1969). Apparent turnover of mitochondrial deoxyribonucleic acid and mitochondrial phospholipids in the tissues of the rat. J. biol. Chem. 244, 1552–1562. G, M., S, T. P. & B, H., et al. (1970). Reaction sites of rotenone, piericidin A, and amytal in relation to the nonheme iron components of NADH dehydrogenase. Proc. natn. Acad. Sci. U.S.A. 65(3), 763–770. K, B. & M¨-H¨, J. (1990). Mutations of mitochondrial DNA and human health. Naturwissenshaften 27, 221–225. K, A. & K, T. B. L. (1993). Mitochondrial mutations, cellular instability and ageing: modelling the population dynamics of mitochondria. Mutat. Res. 295, 93–103. L, A. W., Z, C., B, A. & N, P. (1992). Mitochondrial DNA mutation and the ageing process: bioenergy and pharmacological intervention. Mutat. Res. 275, 195–208. MC, E. H. (1993). Human Genetics. The Molecular Revolution. Boston and London: Jones and Bartlett. M, D. E. (1977). Biochemistry. The Chemical Reactions of Living Cells. New York: Academic Press. M, J., E, A. C., F, J. & J, J. E. (1980). Mitochondrial role in cell aging. Expl Gerontol. 15, 575–591. M, J. (1992). An update on the mitochondrial-DNA mutation hypothesis of cell aging. Mutat. Res. 275, 209–216. M¨-H¨, J. (1989). Cytochrome c oxidase deficient cardiomyocyte in the human heart—an age related phenomenon. Am. J. Pathol. 134, 1167–1173.
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M, M. A. & B, W. X. (1982). Changes in mitochondrial DNA during ageing. Mech. Ageing Dev. 20, 233–242. P, P. L. (1978). Tumor mitochondria and the bioenergetics of cancer cells. Prog. expl Tumor Res. 22, 190–274. P, P. L., G, J. W., C, T. L. & M, H. P. (1970). A comparison of some ultrastructural and biochemical properties of mitochondria from Morris hepatomas 9618A, 7800 and 3924A. Cancer Res. 30, 2620–2626. R, C. (1992). Reactive oxygen and DNA damage in mitochondria. Mutat. Res. 275, 249–255. S-O, K., B, A. C., V, B. & B, R. (1992). Cytotoxic activity of tumor necrosis factor is mediated by early damage of mitochondrial functions. J. biol. Chem. 267, 5317–5323. S, E. P. & B, N. N. (1983). Iron–sulfur proteins and free radicals of the electron transfer chains of mitochondria in chemical and hormonal carcinogenesis. Ukrain. Biochem. J. 55(5), 544–547. (In Russian.) S, B. T. (1980). Inhibitors of energy-coupling site I of the mitochondrial respiratory chain. Pharm. Ther. 10(2), 399–406. T, O., R, M. & R, J. (1991). Ageing as multi-step process characterized by lowering of entropy production leading the cell to a sequence of defined stages. Mech. Aging Dev. 61, 45–64. T, I., B, E. & M, S. (1989). Mitochondrial respiratory chain function: possible factor in ageing. Lancet 1, 637–639. W, D. C. (1992). Diseases of the mitochondrial DNA. A. Rev. Biochem. 61, 1175–1212. W, O. (1967). Oxygen, the creator of differentiation. In: Aspects of Yeast Metabolism (Mills, A. K. & Krebs, H., eds) pp. 327–337. Oxford: Blackwell Scientific Publications.
APPENDIX Stationary Points of Non-dividing Cell Model Consider the stable points of system (1) in the main text. It may be represented in more convenient form: X = −aX − cX + aX (Nx − X − Y − Z ) Y = aX − bY + bY (Ny − X − Y − Z ) Z = cX + bY + gZ (Nz − X − Y − Z ),
(A.1)
whereNx = N − x/a,Ny = N − l/bandNz = N − m/g, are the effective capacities of the corresponding type of mtDNA molecules in the cell. These capacities represent the maximum quantity of mtDNA molecules of corresponding type that may be accumulated in the cell, taking into account its constant decay. Equating the right-hand sides of eqns (A.1) with zero we obtain, in general, four stable points S0 , S1 , S2 and S3 . The point S0 = (0, 0, 0) corresponds to the absence of any type of mtDNA molecules. At the point S3 = (0, 0, Nz ) there are only completely-faulty mtDNA and the energy ensurance is realized by glycolytic means. Points S0 and S3 are always in the permissible field.
. .
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The point S2 = (0, Y2 , Z2 ), where
0
g Nz − N y + Y2 = −
b b
10
Ny −
0
b − g Nz − Ny +
0 0
b b Ny − b Z2 =
b b
1
b b − g N z − Ny + b
1
b b
1
1 ,
belongs to the permissible field when Nz − Ny + b/ b Q 0.
The point S1 = (X1 , Y1 , Z1 ), co-ordinates of which satisfy the following system of linear equations X1 + Y1 + Z1 = Nx −
0
aX1 + b Ny − Nx +
0
a+c a
1
a+c b − Y1 = 0 a b
cX1 + bY1 + g Nz − Nx +
1
a+c Z1 = 0, a
belongs to the permissible field if Ny − Nx + (a+c)/ a − b/b Q 0 and Nz − Nx + (a + c)/a Q 0.