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Population dynamics of mitochondria II. Turnover and ageing of rat liver mitochondria
J. theor. Biol. (1976) 58, 131-142
Population Dynamics of Mitochondria II. Turnover and Ageing of Rat Liver Mitochondria NOBUO FUKUDA The Division of Clinical Research, National Institute of Radiological Sciences, Chiba 280, Japan AND NAGASUMI YAGO~ The Division of Physiology and Pathology, National Institute of Radiological Sciences, Chiba 280, Japan (Received 10 February 1975, and in revisedform
21 May 1975)
Membrane-bound multi-protein complexes in mitochondria are provisionally classified into four categories based on possible mechanism of their assembly and degradation. These mechanisms may be investigated by the use of pulse-labeled radioactive markers which are not re-utilizable. Age dependent assembly is defined as that mechanism by which one or more of the pulse-labeled subunits are assembled into a complex, only while this complex is assembled. If the labeled sub-units can be taken up by the complex randomly during its life-span, then the mechanism is called age-independent assembly. Age-dependent degradation was defined as that mechanism by which the labeled subunits are decomposed, only when the complex is being degraded as an entity. If the labeled subunits are decomposed randomly, the mechanism is called age-independent degradation. Four categories are made by combining each of the assembly and degradation mechanisms. A differential equation was obtained to describe the fate of labeled sub-units that follow the age-dependent assembly and age-dependent degradation. Also derived was an equation for the age-independent assembly and age-dependent degradation. The other two categories which involve the age-independent degradation after age-dependent or age-independent assembly are described by single exponential kinetics. Practical application of the equations is illustrated with the use of experimental data on mitochondrial turnover found in the literature which suggests that the pulse-labeled proteins in rat liver mitochondria may follow the agedependent assembly and degradation. The present attempts to introduce the concept of ageing into multi-protein complexes in mitochondria are the extensions of the steady state theory of mutation by Eyring & Stover (1970). t Present address: St Marianna University School of Medicine, Kawasaki 214, Japan 131
132
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In a previous paper (Fukuda & Yago, 1974), the decay process of rat adrenocortical mitochondria after hypophysectomy was described by the following differential equation,
dN(t) -=dt
b
___ 1 fea-bt N(t),
where a and b are the constants which are specific to the mitochondrial population that has a density of N(t) at time t. Equation (1) was obtained from a model which constituted a new, plausible biochemical mechanism of ACTH action on the maintenance of mitochondrial population in the rat adrenal cortex. There have been quite a few biochemical studies on mitochondrial turnover using radioactive tracers since Fletcher & Sanadi (196 l), and single exponential decay kinetics have been traditionally applied to the analysis of the experimental data. It has been believed that those components which follow the single exponential decay kinetics do not age themselves, even when they are associated with membranes (Siekevitz, 1972). However, attention must be paid to the fact that there have been no kinetic models available which involve any concept of ageing in organized multi-protein complexes. Therefore, if a proper theoretical method is made available, then some kind of ageing might still be detected in the membranous components, especially in protein complexes, because of their structural complexities. In an effort to explore possible models which involve a concept of ageing in multi-protein complexes, it was found that equation (1) can describe the mechanism by which labeled subunits are assembled into a multi-protein complex only when it is being assembled and the labeled subunits are decomposed only when the complex is being degraded as an entity. The concept of ageing is defined as a decline in the hypothetical equilibrium established between stable and labile states of a newly assembled complex. The decline was approximated by first order reaction kinetics and its rate constant was named the ageing constant. Equation (1) was also found to be useful in describing the other mechanisms of turnover of multi-protein complexes after proper mathematical manipulations. By applying the present hypothesis it was found that some components in hepatic mitochondria in the rat may follow the age-dependent assembly and degradation mechanism. The present trials are one of the possible approaches to the mechanism of ageing in intracellular organelles. It should be stressed here that the approach used is an extension of the steady state theory of mutation by Eyring & Stover (1970).
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2. Definition of Terms and General Assumptions It is postulated that the fate of multi-protein complexes be traced by measuring the change in specific activity of non-reutilizable, radioactive labels in one or more subunits after the single administration of label. Then, the mechanism by which a radioactive precursor is finally incorporated into a multi-protein complex can be correlated with the age of that particular complex. Age is defined as the time elapsed since it was assembled. It may either be dependent or independent of the age of the complex. It is necessary to assume that at least one of the sub-units be labeled and assembled in a short or negligible time as compared to the average life-span of the complex itself. The average life-span of cells should be sufficiently longer than that of the complex so that the cellular physiological conditions would stay rather constant. Regarding these assumptions, some approximate values for rat liver mitochondria may be found in the literature; effective labeling time after the single administration of radioactive amino acids is about 10 min (Bergeron & Droz, 1969); average life-span of mitochondrial components in rat liver, less than 10 days (e.g. see Gross, Getz & Rabinowitz, 1969); and average life-span of rat-liver parenchymal cells, 400-450 days (MacDonald, 1961). With the above basic assumptions in mind, the age-dependent assembly of a multi-protein complex is defined as that mechanism by which the precursor can be incorporated, only while the complex is being assembled. The ageindependent assembly is defined as that mechanism by which the labeled subunits can be assembled randomly during the life-span of the complex. With regard to the fate of labeled sub-units after assembly, there would be two mechanisms. The radio-active label would disappear, in one case, agedependently and, in another, age-independently. These mechanisms are thus named age-dependent degradation and age-independent degradation, respectively. Having classified the mechanisms by which a labeled sub-unit is assembled and degraded, fate of the radioactive label in a multi-protein complex should now be described by one of the following categories; (1) age-dependent assembly followed by age-dependent degradation; (2) age-dependent assembly followed by age-independent degradation; (3) age-independent assembly followed by age-dependent degradation; (4) age-independent assembly followed by age-independent degradation. Both the categories (2) and (4) that involve the age-independent degradation can be described by random loss kinetics. For the time being, however, there is no convenient way to distinguish between them. To the contrary, categories that involve the age-dependent degradation have been fully explored in the present study.
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3. Age-dependent Assembly followed by Age-dependent Degradation A differential equation to describe the fate of multi-protein complexes in this category is derived as follows. Figure 1 illustrates this model in a schematic way.
cl A
-D
++ 0
1. Hypothetical model for the ageingin the multi-protein complex in mitochondria. S stands for the stable state of a complex, L the labile state, A the active complex, and D the damaged one which is to be eliminated from the population. V1and VZare the rate FIG.
constants.
Multi-protein complexes are assumed to take reversibly two biochemically different states, i.e. stable and labile states as follows: Stable state 2
Labile state,
(2)
where V1 and V, are the rate constants. Then, the ratio between rate constants is expressed by
v, F=e”y
(3)
where a is the constant. Equation (3) describes the equilibrium of equation (2) at the time when the multi-protein complex was just assembled or at time zero (t = 0). If the labile fraction is denoted byf,(t), then the equilibrium at time zero may be expressed by equation (4),
Cl--fimh
=f@>v29
(4)
hence,
f,(O)= (1+2)-l.
(5)
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Assumption 2 The equilibrium established for a multi-protein complex at t = 0 is assumed to be shifted gradually to the labile side with lapse of time, i.e. age of the complex. Thus, it is assumed that this shift be described by a first order reaction kinetics with the rate constant b as follows, Yz -=e Vl
a-bt
and from equations (5) and (6), the size of labile fraction at time t is expressed by equation (7), fr(t) = (1,
z)-’
= (l+e”-b’)-l.
(7)
Assumption 3 Finally, it is assumed that the rate of degradation of the labeled complex is proportional to the size of labile fraction and the population density of pulse labeled complexes,
where N(t) is the population density of labeled complexes at time t and a the proportionality constant. As will be discussed later, it is important to define the force of mortality (Lotka, 1956) in a proper way. Since the size of labile fraction obviously takes part in the force of mortality showing its age-dependent character, it was considered proper to define the constant b as its age-independent component which may be called the ageing constant. Now, the proportionality constant CIcan be decomposed as follows, a = fib,
(9) where fl is the new proportionality constant and b stands for the decay constant of initially established equilibrium now called the ageing constant. Thus, equation (8) is re-written as follows, Wt) dt
= -/?bf,(t)N(t).
Looking at this equation, it would be conceivable that the constant B should represent the sensitivity of the surviving complexes to the force of mortality which is now described as.the product of b and fr(t). In the present
136
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paper, the constant /3 is tentatively normalized is reduced to the following simple form,
and substituting
equation (7) forf(t),
to 1 *O. Then, equation
(10)
equation (12) is obtained,
b dNtt) ___ = pN0, dt
(12)
which is the same as equation (1). The analytical solution of equation (12) and the normalized surviving fraction ,.5’(t), e.g. the normalized specific radioactivity at time t after labeling, can be found in our previous paper (Fukuda & Yago, 1974) as follows, (13) and
respectively, where c is the integration constant. At t = 0, S(O) becomes 1, thus satisfying the definition of survival in general. The half-life (T+) and average life-span ((z)) of the complexes may be calculated by the following equations (Fukuda & Yago, 1974) : T, =t
r
a+ ln(1+2e-“)
I
and
[a+ In (l+e-31.
4. Age-independent Assembly followed by Age-dependent Degradation Any population of complexes in this category may be considered randomized in a perfect manner with respect to the age of individual multi-protein complexes. Therefore, if an age-independent assembly occurs, then the agedistribution of the pulse-labeled complexes should be the same as that of the complexes themselves. The number of the labeled complexes, however, should approach zero with time. Hence, it is possible to determine the probability for a labeled complex to exist at time t after it was pulse-labeled at time zero.
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This probability, as follows,
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denoted here as P(t), has been obtained by Bergner (1962) if&‘)
dt’
P(t) = “--1 S(t) dt”
(17)
and from equation (14), P(t) =
In (l+e-a+br)
a-btf
a+ In (lfe-“)
’
(W
Thus, in the case of age-independent assembly followed by age-dependent degradation, the normalized time-course of the decay curve of the radioactivity in the complex is now described by equation (18).
5. Algorithm A practical according to as described values of the
algorithm for curve-fitting of S’(t) and P(t) has been written the self-consistent iterative method of Iinear regression analysis previously (Fukuda & Yago, 1974) starting from appropriate parameters a and b.
6. Category (1) as an Extension of Eyring-Stover Theory In one of a series of papers entitled “Dynamics of Life”, Eyring & Stover (1970), have derived the steady state theory of mutations. It has been presented as a general theory to explain the survival, rate of death, ageing, and effects of drugs as well as of ionizing radiations. Based on the theory of absolute reaction rate, these authors have derived the following equation for the surviving fraction S(t) of a population, S(‘(t)= 1 - fd “i El.--
=
1 1 Sea-” 1 1 +e-(n-b’)’
where a and b are the constants and fd is the damaged fraction which is essentially equivalent to the labile fraction defined in the present study. Equation (19) was then shown to fit experimental data very well.
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However, it has the following mathematical
N.
YAGO
difficulty at time zero,
and this does not satisfy the definition of the survival at t = 0, i.e. S(0) should be 3. It is clear that this difficulty has arisen from the fact that the force of mortality was not properly defined in the construction of the Eyring-Stover theory. Then, in view of the previous success in describing the decay process of adrenocortical mitochondria (Fukuda & Yago, 1974), it became clear that the above difficulty can be avoided by simply defining fa as an age-dependent component of the force of mortality which is exerted on the surviving members of the population. In the present study, this was expressed in assumption 3. By assuming the involvement of another component in the force of mortality, i.e. age-independent constant b, equation (14) was obtained which satisfied the definition of survival at time zero. As shown already, it described the age-dependent assembly and degradation. Moreover, utilizing equation (14), another equation which describes the age-independent assembIy and age-dependent degradation was obtained. Thus, the present hypothesis has modified and extended the Eyring-Stover theory. Assumption 1, i.e. an equilibrium be established for a multi-protein complex between stable and labile state, is another extension of one of the assumptions in the steady state theory of mutation. Although the molecular basis for this assumption remains to be elucidated, there have been some intriguing hypotheses. Citing the experiments by Schimke (1970) and his own group, Siekevitz (1973) has suggested two possibilities with regard to the protein turnover, namely: (1) equilibrium states of protein exist between stable and unstable configurations, and (2) a component of membrane may be unstable when it is loosely attached and stable when firmly attached. If these suggestions be extended to a system with higher complexity, then assumption 1 may be subscribed to. 7. Population Dynamics of Rat Liver Mitochondria In the present study, age-dependency of assembly and/or degradation has been considered to be present in multi-protein complexes in mitochondria. Of special interest would be enzyme complexes like cytochrome c oxidase (Mason & Schatz, 1973), oligomycin-sensitive ATPase (Tzagoloff, Rubin & Sierra, 1973) and succinate-cytochrome c reductase (Yu, Yu & Ring, 1974): they are all composed of several protein sub-units and associated with the inner membrane as building stones with particular sidedness (Schneider,
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1974). Other membrane components, namely lipid molecules, have been excluded from candidates for such age-dependency because of their fluidity (Singer & Nicolson, 1972), exchange reaction between mitochondria and endoplasmic reticulum (McMurray & Dawson, 1969), and re-utilization (Gross, Getz & Rabinowitz, 1969). One of the best ways to test the validity of the present hypothesis would be to incorporate data on those multi-protein complexes mentioned above into the equations obtained in the present study. Unfortunately there has been no such an experiment that will allow the immediate use of the present equations. Nevertheless, the present authors expected to detect even traces of age-dependency, if any, in mitochondria by using presently available data. Aschenbrenner, Druyan, Albin, & Rabinowitz (1970) have studied the half-lives of the pulse-labeled total protein and haemoprotein in the liver mitochondria of rat by analysing the experimental data according to single exponential decay kinetics. Their data on the decay of the specific radioactivity were thus incorporated into the equations derived in the present study. Figure 2 illustrates the results of the curve-fitting for total protein labeled with [guanido- “C]arginine which is not re-utilizable (Swick & Handa, 1956). The degree of fitness of curves with the experimental data was checked by calculating the sum of squares of the differences between the theoretical and experimental values. The curve computed by the category (l), (age-dependent assembly and age-dependent degradation) had a smaller value of the sum of the squares than did those by the category (3), (ageindependent assembly and age-dependent degradation) and single exponential decay kinetics. Data for total protein labeled with 3H,-leucine also fit the curve by the category (1) best of all with a little longer life-span than [guanido-14C]arginine-labeled total protein. However, leucine has been shown to be re-utilizable (Poole, 1971) and the results of computation was not included here. In addition, the experimental data on cytochrome c which was labeled with b-amino-[3,5-3Hz]laevulate were also shown to fit best the curve by category (1). This precursor has also been reported to be non-reutilizable (Druyan, DeBernard & Rabinowitz, 1969). With regard to heme a labeled with Ci-amino-[3,5-3H,]laevulate, the fluctuation between the experimental data was unfortunately too large for the present algorithm to perform the computation. Table 1 compares the results of calculation of half-life and average life-span between the present computation and the single exponential decay model. Aschenbrenner et al. (1970) have already drawn a conclusion that the inner membrane or a significant fraction thereof is synthesized and degraded as a
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IO
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i 20 2 2
4
6
ii!
14
I6
18
t (days) FIG. 2. Fate of [guanido-14C] arginine in total protein of rat liver mitochondria. Theoretical N(t)-curve computed by category (1) (----) and curve by single exponential decay kinetics () are shown with the experimental values (@) which were taken from the study by Aschenbrenner et al. (1970). Values of sum of square of the difference between experimental and theoretical values were 102.37 for category (l), 243.09 for single exponential kinetics and 969.74 for category (3) (curve for this not shown here). Values for constants a and b (ageing constant) were 05929 and 0.1835 for category (1).
TABLE
1
Half--lives and average life span for two dFerent radioactive labels in rat liver mitochondria. Experimental dat
Values by category (1)
(5)
5.0 7.22
7.3 8.75
Cytochrome c 3
5-S 7.94
7.65 9.05
Total protein
T*
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unit, because the half-lives for [guanido-r4C]arginine and &amino-[l,S3H,]laevulate were both calculated to be several days by the single exponential kinetics. All of the present values lead to the same qualitative conclusion, although each individual value for half-life and average life-span is different from the corresponding values obtained by Aschenbrenner et al. (1970).
8. Concluding Remarks Bednarek & Gear (1974) have recently reported that shortly after in viva pulse-labeling of rat liver mitochondria the highest radioactivity of proteins, lipids and nucleic acids was found in a particular population of mitochondria which had lower densities than did the main population. This would be one of the first, biochemical reports that directly indicated the possible presence of the age-dependent assembly of mammalian mitochondria in terms of the present study. It is thus hopefully expected that the age-dependency may be detected even at the level of whole organelles. Also, it is expected that the present categories of assembly and degradation of multi-protein complexes would be applicable to other membranous proteins like receptors for hormones, drugs and lectins, and that the physiological meaning of ageing constant could be further studied at the molecular level. The authors express their sincere gratitude to Professor Murray Radinowitz, Department of Medicine, The University of Chicago, Chicago, and Dr J. S. Killip, Editorial Secretariat, The Biochemical Journal, London, for their kind permission to use the data of Professor Rabinowitz and his colleagues for the purpose of the present computations. The authors are also grateful to Professor Rabinowitz for his valuable discussions. The authors are grateful to Drs Brian Poole and Donald G. Lindmark, Department of Biochemical Cytology, The Rockefeller University, New York, and to Drs Donald L. Schneider and Thomas L. Mason, Department of Biochemistry, University of Massachusetts, Amherst, Mass., for their valuable discussions and help in preparing the manuscript. The authors also thank Mr Kenjiro Fukuhisa, Data Processing Laboratory, National Institute of Radiological Sciences, Chiba, for his help in the present computations. REFERENCES ASCHENBRENNER, V., DRUYAN, R., ALBIN, R., & RABINOWITZ, R. (1970). Biochem. J. 119, 157. BEDNAREK, J. M. & GEAR, A. R. L. (1974). Fed Proc. Fedn. Am. Sots. cap. Biol. 33, 1270. BERGERON, M. & DROZ, B. (1969). J. Ultrastruct. Res. 26, 17. BERGNER, P. E. (1962). J. them-. Biol. 2, 279. DRWYAN, R., DEBERNARD, B. & RABINOWITZ, M. (1969). J. Biol. Chem. 244, 5874. EYRING, H. & STOVER, B. J. (1970). Proc. nntn. Acad. Sci. U.S.A. 66, 441.
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FLETCHER, M. J. & SANADI, D. R. (1961).Biochem. biophys. Acta 51, 356. FUKUDA.N. & YAW. N. (1974).J. theor. Biol. 46. 21. GROSS, k. J., GETZ, 6. S. ‘& R~INOWITZ, M. (1929).J. bioI. Chem. 244, 1552. LOTKA, A. J. (1956).In Elements of Mathematical Biology, p. 102,NewYork: Dover. MACDONALD, R. A. (1961). Arch. int. Med. 107, 335. MASON, T. L. & SCHATZ, G. (1973). J. biol. Chem. 248, 1355. MCMURRAY, W. C. & DAWSON, R. M. C. (1969).Biochem. J. 112, 91. POOLE, B. (1971). J. biol. Chem. 246, 6587. SCHIMKE, R. T. (1970).In Mammalian Protein Metabolism, Vol. 4, p. 177(H. N. Mum-o,ed.)
New York: AcademicPress.
SCHNEIDER,D. L. (1974). In Progress in Surface and Membrane Science, Vol. 8, p. 209.
New York: AcademicPress.
SIEKEVITZ, P. (1972). J. theor. BioI. 37, 321. SINGER, S. J. & NICOIBON, G. L., (1992). Science, N. Y. 175, 720. SWICK. R. & HANDA. D. T. (1956). J. biol. Chem.. 218. 557. TZAGO~OFF, A., RUB&, M. S. ‘& SI&A, M. F. (197j).Biochim. biophys. Acta. 301,71. Yu, C. A., Yu, L. & KING, T. E. (1974). J. biol. Chem. 249,4905.
Population Dynamics of Mitochondria II. Turnover and Ageing of Rat Liver Mitochondria NOBUO FUKUDA The Division of Clinical Research, National Institute of Radiological Sciences, Chiba 280, Japan AND NAGASUMI YAGO~ The Division of Physiology and Pathology, National Institute of Radiological Sciences, Chiba 280, Japan (Received 10 February 1975, and in revisedform
21 May 1975)
Membrane-bound multi-protein complexes in mitochondria are provisionally classified into four categories based on possible mechanism of their assembly and degradation. These mechanisms may be investigated by the use of pulse-labeled radioactive markers which are not re-utilizable. Age dependent assembly is defined as that mechanism by which one or more of the pulse-labeled subunits are assembled into a complex, only while this complex is assembled. If the labeled sub-units can be taken up by the complex randomly during its life-span, then the mechanism is called age-independent assembly. Age-dependent degradation was defined as that mechanism by which the labeled subunits are decomposed, only when the complex is being degraded as an entity. If the labeled subunits are decomposed randomly, the mechanism is called age-independent degradation. Four categories are made by combining each of the assembly and degradation mechanisms. A differential equation was obtained to describe the fate of labeled sub-units that follow the age-dependent assembly and age-dependent degradation. Also derived was an equation for the age-independent assembly and age-dependent degradation. The other two categories which involve the age-independent degradation after age-dependent or age-independent assembly are described by single exponential kinetics. Practical application of the equations is illustrated with the use of experimental data on mitochondrial turnover found in the literature which suggests that the pulse-labeled proteins in rat liver mitochondria may follow the agedependent assembly and degradation. The present attempts to introduce the concept of ageing into multi-protein complexes in mitochondria are the extensions of the steady state theory of mutation by Eyring & Stover (1970). t Present address: St Marianna University School of Medicine, Kawasaki 214, Japan 131
132
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FUKUDA
AND
N.
YAGO
In a previous paper (Fukuda & Yago, 1974), the decay process of rat adrenocortical mitochondria after hypophysectomy was described by the following differential equation,
dN(t) -=dt
b
___ 1 fea-bt N(t),
where a and b are the constants which are specific to the mitochondrial population that has a density of N(t) at time t. Equation (1) was obtained from a model which constituted a new, plausible biochemical mechanism of ACTH action on the maintenance of mitochondrial population in the rat adrenal cortex. There have been quite a few biochemical studies on mitochondrial turnover using radioactive tracers since Fletcher & Sanadi (196 l), and single exponential decay kinetics have been traditionally applied to the analysis of the experimental data. It has been believed that those components which follow the single exponential decay kinetics do not age themselves, even when they are associated with membranes (Siekevitz, 1972). However, attention must be paid to the fact that there have been no kinetic models available which involve any concept of ageing in organized multi-protein complexes. Therefore, if a proper theoretical method is made available, then some kind of ageing might still be detected in the membranous components, especially in protein complexes, because of their structural complexities. In an effort to explore possible models which involve a concept of ageing in multi-protein complexes, it was found that equation (1) can describe the mechanism by which labeled subunits are assembled into a multi-protein complex only when it is being assembled and the labeled subunits are decomposed only when the complex is being degraded as an entity. The concept of ageing is defined as a decline in the hypothetical equilibrium established between stable and labile states of a newly assembled complex. The decline was approximated by first order reaction kinetics and its rate constant was named the ageing constant. Equation (1) was also found to be useful in describing the other mechanisms of turnover of multi-protein complexes after proper mathematical manipulations. By applying the present hypothesis it was found that some components in hepatic mitochondria in the rat may follow the age-dependent assembly and degradation mechanism. The present trials are one of the possible approaches to the mechanism of ageing in intracellular organelles. It should be stressed here that the approach used is an extension of the steady state theory of mutation by Eyring & Stover (1970).
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2. Definition of Terms and General Assumptions It is postulated that the fate of multi-protein complexes be traced by measuring the change in specific activity of non-reutilizable, radioactive labels in one or more subunits after the single administration of label. Then, the mechanism by which a radioactive precursor is finally incorporated into a multi-protein complex can be correlated with the age of that particular complex. Age is defined as the time elapsed since it was assembled. It may either be dependent or independent of the age of the complex. It is necessary to assume that at least one of the sub-units be labeled and assembled in a short or negligible time as compared to the average life-span of the complex itself. The average life-span of cells should be sufficiently longer than that of the complex so that the cellular physiological conditions would stay rather constant. Regarding these assumptions, some approximate values for rat liver mitochondria may be found in the literature; effective labeling time after the single administration of radioactive amino acids is about 10 min (Bergeron & Droz, 1969); average life-span of mitochondrial components in rat liver, less than 10 days (e.g. see Gross, Getz & Rabinowitz, 1969); and average life-span of rat-liver parenchymal cells, 400-450 days (MacDonald, 1961). With the above basic assumptions in mind, the age-dependent assembly of a multi-protein complex is defined as that mechanism by which the precursor can be incorporated, only while the complex is being assembled. The ageindependent assembly is defined as that mechanism by which the labeled subunits can be assembled randomly during the life-span of the complex. With regard to the fate of labeled sub-units after assembly, there would be two mechanisms. The radio-active label would disappear, in one case, agedependently and, in another, age-independently. These mechanisms are thus named age-dependent degradation and age-independent degradation, respectively. Having classified the mechanisms by which a labeled sub-unit is assembled and degraded, fate of the radioactive label in a multi-protein complex should now be described by one of the following categories; (1) age-dependent assembly followed by age-dependent degradation; (2) age-dependent assembly followed by age-independent degradation; (3) age-independent assembly followed by age-dependent degradation; (4) age-independent assembly followed by age-independent degradation. Both the categories (2) and (4) that involve the age-independent degradation can be described by random loss kinetics. For the time being, however, there is no convenient way to distinguish between them. To the contrary, categories that involve the age-dependent degradation have been fully explored in the present study.
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3. Age-dependent Assembly followed by Age-dependent Degradation A differential equation to describe the fate of multi-protein complexes in this category is derived as follows. Figure 1 illustrates this model in a schematic way.
cl A
-D
++ 0
1. Hypothetical model for the ageingin the multi-protein complex in mitochondria. S stands for the stable state of a complex, L the labile state, A the active complex, and D the damaged one which is to be eliminated from the population. V1and VZare the rate FIG.
constants.
Multi-protein complexes are assumed to take reversibly two biochemically different states, i.e. stable and labile states as follows: Stable state 2
Labile state,
(2)
where V1 and V, are the rate constants. Then, the ratio between rate constants is expressed by
v, F=e”y
(3)
where a is the constant. Equation (3) describes the equilibrium of equation (2) at the time when the multi-protein complex was just assembled or at time zero (t = 0). If the labile fraction is denoted byf,(t), then the equilibrium at time zero may be expressed by equation (4),
Cl--fimh
=f@>v29
(4)
hence,
f,(O)= (1+2)-l.
(5)
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Assumption 2 The equilibrium established for a multi-protein complex at t = 0 is assumed to be shifted gradually to the labile side with lapse of time, i.e. age of the complex. Thus, it is assumed that this shift be described by a first order reaction kinetics with the rate constant b as follows, Yz -=e Vl
a-bt
and from equations (5) and (6), the size of labile fraction at time t is expressed by equation (7), fr(t) = (1,
z)-’
= (l+e”-b’)-l.
(7)
Assumption 3 Finally, it is assumed that the rate of degradation of the labeled complex is proportional to the size of labile fraction and the population density of pulse labeled complexes,
where N(t) is the population density of labeled complexes at time t and a the proportionality constant. As will be discussed later, it is important to define the force of mortality (Lotka, 1956) in a proper way. Since the size of labile fraction obviously takes part in the force of mortality showing its age-dependent character, it was considered proper to define the constant b as its age-independent component which may be called the ageing constant. Now, the proportionality constant CIcan be decomposed as follows, a = fib,
(9) where fl is the new proportionality constant and b stands for the decay constant of initially established equilibrium now called the ageing constant. Thus, equation (8) is re-written as follows, Wt) dt
= -/?bf,(t)N(t).
Looking at this equation, it would be conceivable that the constant B should represent the sensitivity of the surviving complexes to the force of mortality which is now described as.the product of b and fr(t). In the present
136
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paper, the constant /3 is tentatively normalized is reduced to the following simple form,
and substituting
equation (7) forf(t),
to 1 *O. Then, equation
(10)
equation (12) is obtained,
b dNtt) ___ = pN0, dt
(12)
which is the same as equation (1). The analytical solution of equation (12) and the normalized surviving fraction ,.5’(t), e.g. the normalized specific radioactivity at time t after labeling, can be found in our previous paper (Fukuda & Yago, 1974) as follows, (13) and
respectively, where c is the integration constant. At t = 0, S(O) becomes 1, thus satisfying the definition of survival in general. The half-life (T+) and average life-span ((z)) of the complexes may be calculated by the following equations (Fukuda & Yago, 1974) : T, =t
r
a+ ln(1+2e-“)
I
and
[a+ In (l+e-31.
4. Age-independent Assembly followed by Age-dependent Degradation Any population of complexes in this category may be considered randomized in a perfect manner with respect to the age of individual multi-protein complexes. Therefore, if an age-independent assembly occurs, then the agedistribution of the pulse-labeled complexes should be the same as that of the complexes themselves. The number of the labeled complexes, however, should approach zero with time. Hence, it is possible to determine the probability for a labeled complex to exist at time t after it was pulse-labeled at time zero.
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This probability, as follows,
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denoted here as P(t), has been obtained by Bergner (1962) if&‘)
dt’
P(t) = “--1 S(t) dt”
(17)
and from equation (14), P(t) =
In (l+e-a+br)
a-btf
a+ In (lfe-“)
’
(W
Thus, in the case of age-independent assembly followed by age-dependent degradation, the normalized time-course of the decay curve of the radioactivity in the complex is now described by equation (18).
5. Algorithm A practical according to as described values of the
algorithm for curve-fitting of S’(t) and P(t) has been written the self-consistent iterative method of Iinear regression analysis previously (Fukuda & Yago, 1974) starting from appropriate parameters a and b.
6. Category (1) as an Extension of Eyring-Stover Theory In one of a series of papers entitled “Dynamics of Life”, Eyring & Stover (1970), have derived the steady state theory of mutations. It has been presented as a general theory to explain the survival, rate of death, ageing, and effects of drugs as well as of ionizing radiations. Based on the theory of absolute reaction rate, these authors have derived the following equation for the surviving fraction S(t) of a population, S(‘(t)= 1 - fd “i El.--
=
1 1 Sea-” 1 1 +e-(n-b’)’
where a and b are the constants and fd is the damaged fraction which is essentially equivalent to the labile fraction defined in the present study. Equation (19) was then shown to fit experimental data very well.
138
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However, it has the following mathematical
N.
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difficulty at time zero,
and this does not satisfy the definition of the survival at t = 0, i.e. S(0) should be 3. It is clear that this difficulty has arisen from the fact that the force of mortality was not properly defined in the construction of the Eyring-Stover theory. Then, in view of the previous success in describing the decay process of adrenocortical mitochondria (Fukuda & Yago, 1974), it became clear that the above difficulty can be avoided by simply defining fa as an age-dependent component of the force of mortality which is exerted on the surviving members of the population. In the present study, this was expressed in assumption 3. By assuming the involvement of another component in the force of mortality, i.e. age-independent constant b, equation (14) was obtained which satisfied the definition of survival at time zero. As shown already, it described the age-dependent assembly and degradation. Moreover, utilizing equation (14), another equation which describes the age-independent assembIy and age-dependent degradation was obtained. Thus, the present hypothesis has modified and extended the Eyring-Stover theory. Assumption 1, i.e. an equilibrium be established for a multi-protein complex between stable and labile state, is another extension of one of the assumptions in the steady state theory of mutation. Although the molecular basis for this assumption remains to be elucidated, there have been some intriguing hypotheses. Citing the experiments by Schimke (1970) and his own group, Siekevitz (1973) has suggested two possibilities with regard to the protein turnover, namely: (1) equilibrium states of protein exist between stable and unstable configurations, and (2) a component of membrane may be unstable when it is loosely attached and stable when firmly attached. If these suggestions be extended to a system with higher complexity, then assumption 1 may be subscribed to. 7. Population Dynamics of Rat Liver Mitochondria In the present study, age-dependency of assembly and/or degradation has been considered to be present in multi-protein complexes in mitochondria. Of special interest would be enzyme complexes like cytochrome c oxidase (Mason & Schatz, 1973), oligomycin-sensitive ATPase (Tzagoloff, Rubin & Sierra, 1973) and succinate-cytochrome c reductase (Yu, Yu & Ring, 1974): they are all composed of several protein sub-units and associated with the inner membrane as building stones with particular sidedness (Schneider,
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1974). Other membrane components, namely lipid molecules, have been excluded from candidates for such age-dependency because of their fluidity (Singer & Nicolson, 1972), exchange reaction between mitochondria and endoplasmic reticulum (McMurray & Dawson, 1969), and re-utilization (Gross, Getz & Rabinowitz, 1969). One of the best ways to test the validity of the present hypothesis would be to incorporate data on those multi-protein complexes mentioned above into the equations obtained in the present study. Unfortunately there has been no such an experiment that will allow the immediate use of the present equations. Nevertheless, the present authors expected to detect even traces of age-dependency, if any, in mitochondria by using presently available data. Aschenbrenner, Druyan, Albin, & Rabinowitz (1970) have studied the half-lives of the pulse-labeled total protein and haemoprotein in the liver mitochondria of rat by analysing the experimental data according to single exponential decay kinetics. Their data on the decay of the specific radioactivity were thus incorporated into the equations derived in the present study. Figure 2 illustrates the results of the curve-fitting for total protein labeled with [guanido- “C]arginine which is not re-utilizable (Swick & Handa, 1956). The degree of fitness of curves with the experimental data was checked by calculating the sum of squares of the differences between the theoretical and experimental values. The curve computed by the category (l), (age-dependent assembly and age-dependent degradation) had a smaller value of the sum of the squares than did those by the category (3), (ageindependent assembly and age-dependent degradation) and single exponential decay kinetics. Data for total protein labeled with 3H,-leucine also fit the curve by the category (1) best of all with a little longer life-span than [guanido-14C]arginine-labeled total protein. However, leucine has been shown to be re-utilizable (Poole, 1971) and the results of computation was not included here. In addition, the experimental data on cytochrome c which was labeled with b-amino-[3,5-3Hz]laevulate were also shown to fit best the curve by category (1). This precursor has also been reported to be non-reutilizable (Druyan, DeBernard & Rabinowitz, 1969). With regard to heme a labeled with Ci-amino-[3,5-3H,]laevulate, the fluctuation between the experimental data was unfortunately too large for the present algorithm to perform the computation. Table 1 compares the results of calculation of half-life and average life-span between the present computation and the single exponential decay model. Aschenbrenner et al. (1970) have already drawn a conclusion that the inner membrane or a significant fraction thereof is synthesized and degraded as a
140
N. FUKUDA
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N. YAGO
8
IO
140
i 20 2 2
4
6
ii!
14
I6
18
t (days) FIG. 2. Fate of [guanido-14C] arginine in total protein of rat liver mitochondria. Theoretical N(t)-curve computed by category (1) (----) and curve by single exponential decay kinetics () are shown with the experimental values (@) which were taken from the study by Aschenbrenner et al. (1970). Values of sum of square of the difference between experimental and theoretical values were 102.37 for category (l), 243.09 for single exponential kinetics and 969.74 for category (3) (curve for this not shown here). Values for constants a and b (ageing constant) were 05929 and 0.1835 for category (1).
TABLE
1
Half--lives and average life span for two dFerent radioactive labels in rat liver mitochondria. Experimental dat
Values by category (1)
(5)
5.0 7.22
7.3 8.75
Cytochrome c 3
5-S 7.94
7.65 9.05
Total protein
T*
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unit, because the half-lives for [guanido-r4C]arginine and &amino-[l,S3H,]laevulate were both calculated to be several days by the single exponential kinetics. All of the present values lead to the same qualitative conclusion, although each individual value for half-life and average life-span is different from the corresponding values obtained by Aschenbrenner et al. (1970).
8. Concluding Remarks Bednarek & Gear (1974) have recently reported that shortly after in viva pulse-labeling of rat liver mitochondria the highest radioactivity of proteins, lipids and nucleic acids was found in a particular population of mitochondria which had lower densities than did the main population. This would be one of the first, biochemical reports that directly indicated the possible presence of the age-dependent assembly of mammalian mitochondria in terms of the present study. It is thus hopefully expected that the age-dependency may be detected even at the level of whole organelles. Also, it is expected that the present categories of assembly and degradation of multi-protein complexes would be applicable to other membranous proteins like receptors for hormones, drugs and lectins, and that the physiological meaning of ageing constant could be further studied at the molecular level. The authors express their sincere gratitude to Professor Murray Radinowitz, Department of Medicine, The University of Chicago, Chicago, and Dr J. S. Killip, Editorial Secretariat, The Biochemical Journal, London, for their kind permission to use the data of Professor Rabinowitz and his colleagues for the purpose of the present computations. The authors are also grateful to Professor Rabinowitz for his valuable discussions. The authors are grateful to Drs Brian Poole and Donald G. Lindmark, Department of Biochemical Cytology, The Rockefeller University, New York, and to Drs Donald L. Schneider and Thomas L. Mason, Department of Biochemistry, University of Massachusetts, Amherst, Mass., for their valuable discussions and help in preparing the manuscript. The authors also thank Mr Kenjiro Fukuhisa, Data Processing Laboratory, National Institute of Radiological Sciences, Chiba, for his help in the present computations. REFERENCES ASCHENBRENNER, V., DRUYAN, R., ALBIN, R., & RABINOWITZ, R. (1970). Biochem. J. 119, 157. BEDNAREK, J. M. & GEAR, A. R. L. (1974). Fed Proc. Fedn. Am. Sots. cap. Biol. 33, 1270. BERGERON, M. & DROZ, B. (1969). J. Ultrastruct. Res. 26, 17. BERGNER, P. E. (1962). J. them-. Biol. 2, 279. DRWYAN, R., DEBERNARD, B. & RABINOWITZ, M. (1969). J. Biol. Chem. 244, 5874. EYRING, H. & STOVER, B. J. (1970). Proc. nntn. Acad. Sci. U.S.A. 66, 441.
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FLETCHER, M. J. & SANADI, D. R. (1961).Biochem. biophys. Acta 51, 356. FUKUDA.N. & YAW. N. (1974).J. theor. Biol. 46. 21. GROSS, k. J., GETZ, 6. S. ‘& R~INOWITZ, M. (1929).J. bioI. Chem. 244, 1552. LOTKA, A. J. (1956).In Elements of Mathematical Biology, p. 102,NewYork: Dover. MACDONALD, R. A. (1961). Arch. int. Med. 107, 335. MASON, T. L. & SCHATZ, G. (1973). J. biol. Chem. 248, 1355. MCMURRAY, W. C. & DAWSON, R. M. C. (1969).Biochem. J. 112, 91. POOLE, B. (1971). J. biol. Chem. 246, 6587. SCHIMKE, R. T. (1970).In Mammalian Protein Metabolism, Vol. 4, p. 177(H. N. Mum-o,ed.)
New York: AcademicPress.
SCHNEIDER,D. L. (1974). In Progress in Surface and Membrane Science, Vol. 8, p. 209.
New York: AcademicPress.
SIEKEVITZ, P. (1972). J. theor. BioI. 37, 321. SINGER, S. J. & NICOIBON, G. L., (1992). Science, N. Y. 175, 720. SWICK. R. & HANDA. D. T. (1956). J. biol. Chem.. 218. 557. TZAGO~OFF, A., RUB&, M. S. ‘& SI&A, M. F. (197j).Biochim. biophys. Acta. 301,71. Yu, C. A., Yu, L. & KING, T. E. (1974). J. biol. Chem. 249,4905.