Thermodynamics of aging in Drosophila melanogaster

Mechanisms of Ageing and Development, 5 (1976) 371-387 © ElsevierSequoia S.A., Lausanne - Printed in the Netherlands

T H E R M O D Y N A M I C S O F A G I N G IN D R O S O P H I L A

371

MELANOGASTER

HENRI ATLAN*, JAIME MIQUEL, LELAND C. HELMLE and CONSTANTIN B. DOLKAS Ames Research Center, NASA, Moffett Field, Calif. 94035 (U.S.A.) (Received May 10, 1976)

SUMMARY The data on mortality kinetics and decline in functions reported in the preceding article are used to calculate temperature coefficients for the aging process(es) in Drosophila. Different values are found, according to the model chosen to account for the mortality kinetics. The respective implications of three equally suitable models are discussed. Thus, organization parameters on two different levels can be identified: rates of changes assumed to occur at the elementary molecular level, and redundancy factors at a more integrated level. Their temperature coefficients are compared with those of protein denaturations and lipid peroxidation pigment accumulation. It is suggested that elementary molecular processes responsible for aging can indeed 'oe protein denaturations, whereas the known lipid peroxidation pigment accumulation is more likely to be a secondary effect, resulting from a failure of the overall cellular organization at a more integrated, supramolecular, level.

1. MORTALITYKINETICSAND ORGANIZATION The measurements of "vitality" based on negative geotaxis and mating frequency in adult Drosophila melanogaster (see preceding article, Figs. 7 and 8) have revealed the same classical pattern of an apparently linear decrease with age, known to take place for most parameters of physiological performance [1 ]. Although this decrease is unlikely to be truly linear since it is not a single elementary zero order kinetics phenomenon, its apparent linearity followed by a rapid change towards a low, constant value can be explained as the result of compensating positive and negative effects, in agreement with previously expressed concepts [2]. In addition, the notion of an optimum temperature is supported by the initial period of improvement in performance (adaptation and "learning"?) which was found only at 21 °C, the temperature at which the flies were *Present address: Service de Biophysique, Facult6 de M6decine Broussais-H6teloDieu,45, rue des Saints P~res, Paris 6~, France.

372 raised for many generations and developed in their pre-imaginal phases. We can see from the data obtained at 18 °C (preceding article, Figs. 7 and 8) that although the flies lived longer, their initial performance was decreased, although not so much as for flies kept at 27°C. A linear decrease of the vitality after its maximum - in the form of V = Vo- qt is responsible for the mortality curves with increasing rate of mortality according to the widely accepted Strehler-Mildvan theory [1], which postulates that the individuals are killed by random stresses which they are not able to overcome; the probability of being killed (Le. the mortality rate R) is an exponential function of the decreasing vitality

R = C exp (-V(~ta))= Ro exp(at)

(1)

with the following relations between the constants

and ~ = q/a. In other words, the mortality rate is determined by the varying level of the vitality, not by its (constant) rate of decrease. In a previous article we have suggested [2] that the emphasis in this theory should be shifted from the energetic to the informational aspect of "vitality". By studying the effects of temperature on both the slopes of decrease in physiological performances and the rate of increase of the mortality rate, it will be possible to show that the formal basic assumption of the Strehler-Mildvan theory is not contradicted by our present data. In our previous work, the only kind of mortality kinetics analyzed was that of the Gompertz law (1825) [1,2] according to which the mortality rate R = dN/N dt, where N is the number of survivors at age t, is an increasing exponential function of time according to eqn. (1). Recently, Rosenberg et al. [3], looking for one single temperature dependent coefficient of aging, have questioned the validity and the convenience of the Gompertz law. They have proposed a power law as an alternative, in which the mortality rate is an increasing power function of time, i.e. .dN R = = A tn

N dt

(2)

with A and n constants. They found the power law both more convenient and better fitting the data in their Drosophila mortality experiments, and in human mortality curves a s well. On the other hand, Johnson and Pavelec [4] studied the temperature dependent rate of aging in hamster cells in culture, by means of the decrease of their colony forming ability in time. They were able to fit their data with an equation transposed from the multihit theory of radiation injury, i.e.

N/No = 1 - (1 - exp(-kt)) p

(3)

373 where N(t) is the number of cells having retained their colony forming ability (survivors) at time t, and No the initial number of living cells, k is the probability dN/N dt of one cell being "hit" once per unit time, so that the cell probability of having been hit once at time t is (N O - N)/N = 1 - exp(-kt). If p "hits" are necessary to inactivate a cell, the inactivated cell fraction at time t becomes (1 - e x p ( - k t ) ) p, which gives the form of eqn. (3) for the survivor fraction. The goal of the studies of Rosenberg et al. and of Johnson and Pavelec was to measure a temperature dependent parameter of aging, preferably with a rate (i.e. t -~) dimension, which would allow for an estimation of the activation enthalpy and entropy of the main chemical reaction(s) underlying the aging process(es), along the lines of the reaction rate theory [5]. Previous studies (Strehler [6], Sacher [7], Hollingsworth [8]) used the mean longevity as the desired parameter, transformed into a rate by simply taking its reciprocal. With no further assumptions, by comparing the Arrhenius plots of metabolic rates (i.e.

t b°

nn

~pe

10-1 "7

< re

~ 10 - 2 ,.1 < I--

o

o/

II

o

re

10-3

/~o o / Q®

,' ;;iii O 18°C e

1 0 Q

10-4 ~

_ lO

_ sO

,

AGE,days

loo

15o

Fig. 1. Mortality data from the preceding article (Table 1 and Fig. 3). The semi log plot of the mortality rate shows how the Gompertz law fits the data.

374

oxygen consumption) and aging rate (i.e. the reciprocal of mean longevity), Sacher [7] was able to demonstrate the generality of the concept of an optimum temperature for insects (poikilotherms). This was based on an estimate of what he called the organizational entropy, which is a measure of the efficiency of the energy utilization to maintain life at a given temperature (see further). We have used our data on groups of 378 to 959 flies, each exposed to 18 °, 21 °, 27 ° or 30 °C (preceding article) for Arrhenius plots of aging parameters calculated according to the four different methods proposed so far, namely - the mean longevity - parameters from Gompertz functions (eqn. (1)) fitted with the data - parameters from a power law (eqn. (2)) fitted with the same data - parameters from a random multihit function (eqn. (3)) fitted similarly. Usually the adequacy of the fit and the values of the parameters are estimated graphically. As can be seen in Figs. 1-3, it was possible to fit our data reasonably well with these three functions. Since we wanted to compare the adequacy of the fit of the various proposed equations, we used a non-linear least square method [9, 10] which, for each function, gave us the most probable values of the parameters and their standard 10 POWER LAW No

=

Ioe A 1 A

@e 8 •

o

* I.+lHo~t

I

A

30~C Z?'C

rn

21=C lsoc

•

t:J 0

[]

0

•

Q O

z°J z 10_1 .E=

&

0

z~O Q•

0

•

g

n

O

zx

[]

~

I-:1

•

°St •

•

r~•

el

10=2 o 121 OA

10-3

I

10 AGE, days

I

I

I

50 100 150

Fig. 2. Mortality data from the preceding article (Table 1 and Fig. 3). The log-log plot of the survivorship reciprocal shows how the power law fits the data [3 ].

375

10-1

10-2

M U L T I H I T EQUATION ~" = 1 - (1-e'kt) P NO O

30~C 27'C 2~°C

• O

0

ls°c

I

I

I

50

100

150

AGE, days Fig. 3. Mortality data from the preceding article (Table 1 and Fig. 3). The semi log plot of the survivorship shows how the multglit equation fits the data [4].

errors for confidence interval limits of 0.05. In addition, the estimated standard deviation of the residuals allowed us to appreciate the relative adequacy of the fit of our data with these three functions. The values of the most satisfactorily fitted parameters, their S. E. and the S. D. of the residuals appear in Table I. Since the most general feature which is destroyed in the aging process is the organization which makes the system exist and function, we can assume that age dependent mortality kinetics represent in some way the dynamic organization of the flies. It has been demonstrated on purely formal grounds [11, 12] that attempts to quantitate biological organization cannot lead to less than three parameters. Therefore, it is not surprising that laws with two parameters do not provide the best possible fit. A recent analysis of mortality curves in mice [13] has also shown that three parameters are a minimum for a good fit of the mortality data. However, before a deductive theory of mortality kinetics with three mea_ningful parameters of organization is found to fit the data much better than these two parameter models, something can be learned from the comparative analysis of these models. It can be seen in Table I that the Gompertz function fits the data best for all the four temperatures. The power law fits better than the multihit function at 21 °C and 27 °(2, and not as well at 18 °C and 30 °12.However, the differences in the standard deviations of the residuals are not dramatic enough to make us disqualify the power law and the

376 TABLE I MORTALITY DATA OF D. MELANOGASTER AT 4 DIFFERENT TEMPERATURES, FROM THE PRECEDING ARTICLE (TABLE 1 AND FIG. 3). THREE DIFFERENT FUNCTIONS PROPOSED IN THE LITERATURE SO FAR HAVE BEEN FOUND TO FIT THE DATA (SEE TEXT). ESTIMATES OF THE BEST FITTED PARAMETERS OF THE THREE FUNCTIONS HAVE BEEN CALCULATED BY A NON-LINEAR LEAST SQUARE METHOD [9, 10], TOGETHER WITH THEIR STANDARD ERRORS (S. E.) AT 0.05 CONFIDENCE INTERVALS AND THE STANDARD DEVIATIONS OF THE RESIDUALS (S. D.0)

T°C

Ro

18 21 27 30

0.364 0.359 0.287 0.166

S.E. X X X X

10-7 10 --4 10 --4 10-6

S.E.

0.269 0.071 0.075 0.120

× X × X

10- 7 10--4 10 --4 10-6

= = = =

74% 20% 26% 72%

S.D.¢

0.1102 0.0846 0.2044 0.7740

0.00595 0.00255 0.00699 0.03920

n

S.E.

9.73 8.034 8.473 9.716

0.105 0.188 0.049 0.122

k

S. E.

0.1084 0.0852 0.2166 0.7216

0.00231 0.0078 0.0245 0.0332

= 5% = 3% = 3.4% = 5%

0.035 0.022 0.016 0.020

Atn+ll Power law N/No = exp \ - ~ / T °C

A

18 21 27 30

1.345 1.77 3.03 1.325

S.E. X X X X

10-22 10 -17 10 - i s 10 -13

Multihit function N/N o T°C

P

18 21 27 30

1.045 0.992 4.987 6.305

0.90 0.98 1.65 1.76 = 1 -

(1

-

× 10-22 X 10-17 X 10 - i s X 10 -~3 exp(-kt))

= 66.4% = 55.2% = 54.4% = 133%

104 103 103 105

0.335 0.649 5.013 3.980

X × × X

106 103 103 lO s

= 1% = 2.3% = 0.6% = 1.25%

0.047 0.043 0.026 0.054

p

S. E. X x x ×

S.D.~

= 32% = 65.4% = 100.5% = 63.1%

S. D.q~ = 2% -- 9% = 11.3% = 4.6%

0.042 0.065 0.051 0.025

m u l t i h i t f u n c t i o n . a s possible valuable m o d e l s for the m o r t a l i t y kinetics. T h e r e f o r e a m o r e r e f i n e d analysis is n e c e s s a r y , to c o m p a r e the a c c u r a c y r e a c h e d o n the various p a r a m e t e r s , t o g e t h e r w i t h their t e m p e r a t u r e sensitivity.

2. TEMPERATURE DEPENDENCE IN DIFFERENT MODELS The l o g a r i t h m o f the t e m p e r a t u r e d e p e n d e n t p a r a m e t e r ( s ) was p l o t t e d versus the reciprocal o f the a b s o l u t e t e m p e r a t u r e ( A r r h e n i u s p l o t ) and the slope o f the p l o t w i t h i n a given t e m p e r a t u r e range allowed us to calculate the activation energy w i t h i n this range a c c o r d i n g t o the r e a c t i o n rate t h e o r y [ 5 ] .

377

~ "~E

a = 45 136 cal/mole °K

r-] MEAN LO1-NGEVITY "day-1

O

'T I- lC I

-

-

O

100 Z

> Lu

"o -4

Z

o

Z

Z <

Lu

1 3.300

I

I

I

3.333

3.401

3.436

lO

1

,'~'-

10-3

OK-1

Fig. 4. Arrhenius plot of the mean rate of aging (reciprocal of the mean longevity) and mean oxygen

consumption. Data from the preceding article (Tables 1 and 5, Figs. 3 and 6). The activation energies E a are calculated according to the reaction rate theory [5] by multiplying the slope of the linear plot b y the gas constant R.

The results are the following: Figure 4 shows the Arrhenius plot of the reciprocal of the mean longevity. The points from 18 °C to 27 °C are on a straight line the slope of which gives an activation energy of 22,315 cal/mole °K. The plot departs from this straight line for the higher temperature (30 °C), which indicates a different mechanism taking place with a much higher activation energy. Within the 18 ° to 27 °C range this figure is consistent with the data of Hollingsworth [8], Strehler [6] and Sacher [7]. Much higher activation energies have been found at higher temperatures and are often correlated (Hollingsworth) with activation energies for protein denaturation, of the order of magnitude of 200,000 cal/mole OK. Comparing our data with Hollingsworth [8], we can see that our 30 °C point is likely to lie already in the higher activation energy temperature range with a transition between 27 °C and 30 °C. The same plot for the rate of oxygen consumption at 18 ° 21 ° and 27 °C (Fig. 4) shows the tendency towards a different direction in the curvature as compared with that of the mean longevity rate. This phenomenon was found by Sacher [7] from data in the literature on a wide temperature range (4.7 ° to 34.7 °C) and led him to the measurement

378

of the ratio of the two rates as a function of temperature. This ratio is used to define what the author calls an organizational entropy Sorg, by Sorg = R In #/p + constant, where /~ is the metabolic rate (i.e. the oxygen consumption), p the mean longevity rate (Le. the reciprocal of the mean longevity), and R the gas constant. Sorg was found [7] to have a maximum value for the optimum temperature range, corresponding roughly to the region where the two curves lie closest to one another. In our data, limited to a much narrower range of temperature not too far from that of development, this phenomenon is only indicated here by the tendency for the two curves to come closer, around 21 °C, but it will appear more clearly in more sophisticated analysis where the temperature dependence of the rate of aging is studied on other parameters aerived from the mortality kinetics. Figure 5 shows the Arrhenius plot of the a coefficient in the Gompertz function (eqn. (1)), of the k coefficient in the multihit function (eqn. (2)) and of the n coefficient in the power law (eqn. 3)). The plots of a and k are superimposed, whereas n shows very small variations, as in Rosenberg et al. [3], which prompted these

\

~ ~ (GOMPERTZ LAW) - K (MULTIHIT EQUATION) ~ n X 10-1 (POWER LAW)

~

~ _

°K 24,280 cal/mole ° K

Ea = 81,060 cal/mole

~.~E~ 10-1

~

)-

¢{ t~

LXMATING GATIVE GEOTAXIS (b) a

10

10-2

gt. ttl

RATE OF FLUORESCENCE ACCUMULATION

_1

> Ill

3,928 cal/mole ° K

li

3.300

/ 3.337 3.333

/

/

I

3.401

3.388

10_3 OK_1

I

3.436 1

Z

1

o Id.

T

Fig. 5. (a) Arrhenius plot of the parameters a (Gompertz law), k (multihit equation) aaad n (power law). (b) The rates of decrease in negative geotaxis and in mating (from rough estimates of the slopes in Figs. 7 and 8 of the preceding article) are plotted versus l/T: the theory implies that these Arrhenins plots should be parallel to one another and to the similar plot (curve a) of the a coefficient in the Gompertz function (see text and refs. 1 and 2). This appears to be the case in the 21-27 °C range. (c) The Arrhenius plot of the rate of fluorescent pigment accumulation from Sheldahl and Tappel's data [14] (see text and preceding article), is parallel to (a) and (b) within the same temperature range.

379 authors to consider it as a constant. However, the standard errors calculated on these n values show that their temperature dependence is real, although it is small. We shall come back to this point later when we discuss the features of the three different laws with their respective advantages. The activation energy given from the plot of c~ and k from 21°to 27 °C is 24,980 cal/mole °K, i.e. almost the same as calculated from the mean longevity rate. A similar change in slope towards a higher value appears between 27 ° and 30 °C, being here more manifest. The most interesting additional feature is the change from 21 ° to 18 °C, which is such that 21 °C appears as the temperature at which both a and k (and n as well) are minimum. As will be seen further, it is tempting to interpret this minimum along the lines of what we know about this temperature being the optimum as far as functional performance (negative geotaxis, mating) is concerned, probably as a result of a previous adaptation during preimaginal development.

Decrease in vitality and Gompertz law The Arrhenius plot of a gives us the opportunity of a rough check of the theoretical basis of the Gompertz law which we briefly recalled above. Assuming a linear decrease in time of the vitality shown by the physiological performances (preceding article, Figs. 7 and 8), with a given slope q, the a coefficient of the rate function R(t) (eqn. (1)) must be proportional to q (see above). Therefore, since q and a are temperature dependent, their Arrhenius plots should be parallel. The accuracy of the estimates of q cannot be very high - as can be seen from the shapes of the curves on negative geotaxis and mating (preceding article, Figs. 7 and 8) - as usual in these kinds of physiological "vitality" measurements [ 1]. However, Arrhenius plots of crude estimates of the rate of decrease in negative geotaxis and in mating rate with time are in fairly good agreement with the theory (see Fig, 5)within the "normal" aging temperature range, i.e. 21-27 °C. Moreover, in a similar analysis of the data of Sheldahl and Tappel [14] on pigment accumulation (see preceding article), we find (Fig. 5 curve c) a parallel line for the Arrhenius plot of the rate of fluorescent pigment accumulation at 22.2 °C and 26.7 °C with a similar activation energy of 23,928 cal/mole °K. In the light of this evidence it may be pertinent to correlate the temperature effects on aging with the kinetics of ceroidlipofuscin accumulation, at least in the range of 21 ° to 27 °C. Other parameters However, the Arrhenius plot of the second parameter in the three different functions, respectively Ro, A and p, shows a different pattern (Fig. 6). Whereas the variations of log p and (-log Ro) - from the multihit and Gompertz law respectively versus lIT show an upward curvature with a minimum at 21 °C, log A (from the power law) decreases monotonously from 30 °C, with a large change in slope at 21 °C. The activation energy calculated from this latter plot varies between 30 °C and 21 °C from 222,820 to 152,260 cal/mole °K. But if we draw one single straight line along the three points in this range - which is justified by taking into account the estimated standard errors on the computed values ofA - we end up with an approximate averaged activation energy of 175,680 cal/mole °K in the range of 21-30 °(2. This number agrees fairly well

380 A l o g A (POWER LAW) E] log n (POWER LAW) (EXPANDED SCALE) O l o g 1/Ro (GOMPERTZ LAW) • log p (MULTIHIT EQUATION) 222,820 cal/mole ° K

A -12

log

-13' 1

/

-

-14

.98

-

.97

-

.96

-

-15

-

-16

==

_o

.92 - -17 .91 ,90 - -18

7

-19

5 a.

~ .

-20

652 000 ole °K

~

-21

3 "T

-22

2 t

3.300 3.333 (30°C) (27°C)

i

I

3.401 (21°C)

3.436 (18°C)

10-3 oK-1

1_ T

Fig. 6. Arrhenius plot of the parameters A and n (power law), R o (Gompertz law), p (multihit equation). An averaged activation energy of A in the 21-30 °C range, computed by drawing a straight line across the standard errors of the three points, gives a value o f E a = 175,680 cal/mole °K.

with that found by Rosenberg et al. [3] (198,600 cal/mole °K) who investigated a temperature range from 33 °C to 25 °C. As these authors and others [4, 8] pointed out, this number is of the same order of magnitude as activation energies found for protein denaturation in vitro, and for thermal killing of microorganisms. Comparing the temperature dependence of A (Fig. 6) with that of a and k (Fig. 5) all three of which will be identified further as rate parameters - we can notice that 21 °C appears as a peculiar point in all cases. But whereas log A versus l I T keeps decreasing between 21 °C and 18 °C with a large increase in slope, ct and k increase from 21 °C to 18 0(2. This phenomenon will be interpreted as an additional indication on the intricate character of a and k, representing integrated phenomena rather than elementary reactions : their temperature "dependence seems to be linked to that of Ro and p respectively, and can hardly be reduced to that of simple elementary physicochemical reactions slowed down at lower temperatures. On the contrary, the temperature dependence of A will be interpreted as more directly determined by elementary reactions, with much less influence of the response to temperature of supramolecular, integrated levels of organization. But the change in slope from 21 °C to 18 °C for the Arrhenius plot of A indicated a phenomenon different from - and much more temperature sensitive than - that in the range of 21-27 °C. Phase transitions with increase in the viscosity of the lipidic and lipoprotidic constituents of membranes would be good candidates for such physicochemical primary effects of abnormal, low temperatures. -

381 On the other hand, as mentioned above, the n parameter in the power law shows small variations of no more than 915% around a mean value of 8.99, which is also consistent with the n values found by Rosenberg et al. [3] with a maximum deviation of 6.6% in the range of 25-33 °C*. These authors considered the low sensitivity of n to temperature, together with the very large variations observed on A, as the main advantage of the power law over the Gompertz law where the two parameters are temperature dependent. However, each of the three laws proposed so far to account for the mortality kinetics shows a reasonable fit with the experimental data (Table I). On the other hand, the temperature dependence of the parameters, mentioned above, appears quite different according to which law we choose to fit the data. Therefore some discussion on the theoretical meaning of these laws seems appropriate if we want to be able to interpret these results.

3. DISCUSSION: TWO LEVELS OF ORGANIZATION: RATE PARAMETERS (or, k, A) AND REDUNDANCYFACTORS(Ro, p, n) Let us come back to the fact that the n parameter actually varies slightly but significantly within the temperature range which we studied. In fact, an Arrhenius plot of n with an expanded scale represented in Fig. 6 shows that its variation parallels that of (-log Ro) from the Gompertz law, and to some extent that of log p from the multihit function. Therefore, it seems that whatever the equation chosen to fit the data, we are dealing with two different kinds of parameters, which give us two different kinds of information on the mortality kinetics: Ro, p and n on the one hand, a, k and A on the other. The latter three parameters represent rates of variation in their respective equations: in the multihit analysis (eqn. (3)), k is the rate of the single hit denaturation process which has to be repeated p times in order to kill the organism. In the Gompertz analysis, a is the rate constant of the exponential increase of the mortality rate. It is a rate of a rate, as A in the power law is a rate of increase of the mortality rate to the n power. (As Rosenberg et al. [3] put it, A has the dimension of a reciprocal time to the (n + 1) power; therefore, A 1/n in the power law would give us a parameter more closely related to the rate constants in the two other laws.) We propose to consider Ro, p and n on the one lland, and the rate constants (a, k and A t/n) on the other, as parameters of organization at two different levels. In effect, the thermal killing effect, as well as the "normal" aging process - the temperature of which is always high in relation to the absolute zero - can be thought of as the result of a progressive disorganization of the living system leading to death [2]. But the organization of an organism, such as any heterogeneous functioning system, must be

*The mean value for n quoted in Rosenberg et al. [3] is 4.5 ± 6.6% which, after the necessary correction by a In 10 = 2.3 factor, is consistent with our own data in the range of 18-30 °C giving ff = 8 . 9 9 ± 9 . 5 % .

382 considered at least at two different levels: that of its components (Le. elementary biochemical reactions, specific macromolecular structures), and that of the functional relationship between the components (i.e. how the enzymatic reactions are coupled together in a functioning network) specific for this particular organism and different from many other possible networks which could be made from the same components [15]. Temperature, as a random molecular agitation, is going to act at these two levels of organization. At the first level, there is no reason to believe that the elementary physicochemical reactions should not be temperature dependent in a simple, monotonous way, according to the reaction rate theory. At the second level, the temperature dependence of the functional properties of the network is an overall response of the integrated network to an agent working both as a source of non-directed energy (from a thermodynamics point of view) and of random noise in the communication channels (from an information theory point of view). We have pointed out elsewhere [2, 11, 12] that the organization of any system can be defined from a formal information theory point of view by the kinetics of its responses to accumulated noise producing factors in time. The fact that these factors may also have a positive effect, in addition to their detrimental, disorganizing effect, was taken into account, as obviously in the case of temperature which is needed for any biological system to function. In fact, the very existence of an optimum (relatively high) temperature for the functional organization of even poikilotherms, outlines the fact that the temperature dependence of the overall functional performances of the organism is not the result of a straightforward simple application of the rate constant theory. (The known existence of optimum temperatures for enzymatic reactions can be viewed as the result of the selective pressure on the overall organization of the organisms as they have evolved towards their present functioning state. The enzymatic control of the biochemical reactions would be, from a cybernetic point of view, the necessary link between the overall organization of the system and the functioning of its parts [ 15 ] .) Coming back to our three equations for mortality kinetics, we suggest that the rate constants in each be considered as determined mainly by the physico-chemical processes at the level of the elementary parts (Le. reaction rates, denaturation rates, etc.), whereas the three other parameters (p, Ro-and n) would be determined by functional organization at the more integrated, supramolecular, level. More precisely, p, Ro and n would somehow measure the functional redundancy of the system, which was shown in earlier publications [11, 12] to be one of the most important features in the definition of organization, determining the reliability and the lifespan of a system. This will help us to understand the differences between the different models of organization on which the three equations are based. In the multihit equation (eqn. (3)), the number p (the extrapolation number in similar radiation biology equations) is a straightforward measure of redundancy since it represents the number of hits necessary to produce inactivation. This can be visualized as a system of p different parallel pathways used to perform the measured function, so that the break of one of them is not enough to inactivate the system. On the other hand, the n parameter in the power law is also a straightforward measure of redundancy since it also

383 represents a number of "blows" (in the metaphor of Rosenberg et al. [3] ) necessary to destroy a link in a chain. The different form of the equation is due to the fact that this number is a minimum value, characteristic of one link in a chain made of several consecutive links, each needing a different number of blows to be destroyed. Without entering into the details of the statistics used in both cases, it can easily be seen that an important difference between this and the previous case lies in that we are dealing with a structure in series, where the destruction of one link is enough to break the chain. As to the Gompertz equation, although the meaning of Ro as a redundancy parameter is not so obvious, it can be seen via the relationship between redundancy and reliability [11, 16] in the following way. Ro is an initial condition in the integration of the Gompertz equation, i.e. the initial value of the mortality rate at time t = 0. This initial value, which is actually very low - and which is responsible for the initial shoulder in the sigmoid form of the mortality 'curves - is an overall failure rate characteristic of the organization of the system in its response to a given environment. Therefore its reciprocal is a characteristic time which measures a reliability parameter of this organization known to express its functional redundancy [11, 12]. Thus, each particular model implies two parameters, a rate constant for the processes at the elementary part level, and a redundancy factor for the overall network. But the particular form of each of the different equations implies a particular relationship between these two levels so that, for example, the temperature effects on the one are likely to be felt by the other. Moreover, along this line, we can understand how these effects on the two levels of organization are more or less clearly separatec' according to the particular form of the equation. The power law minimizes the variations on n while it amplifies those on A (partly owing to the fact that A is a rate to the n power). This suggests that the effects of temperature on the elementary part (molecular) level of the organization appear more .conspicuously in the temperature dependence of A, whereas the effects at the more integrated level are minimized. Thus the power law best separates the sensitivities of the two level parameters, one being almost constant, the other very sensitive. Since the two other laws do not achieve such a clear cut separation, their respective rate constants are likely to show a composite temperature dependence, mixture of temperature effects on the two levels of organization.

Minimd in redundancy and optimum temperature The intricate temperature dependence of the multihit and Gompertz laws could explain why the inverse temperature dependence observed consistently from 21 °C to 18 °C in the three redundancy parameters (Fig. 6) is also seen in the variations of the rate parameters k and a of these two laws (Fig. 5). Otherwise this increase in rate parameters from 21 °C to 18 °C would be difficult to understand since it is accompanied by an increase in longevity. However, the fact that the redundancy factors are minimum (and not maximum) in the "normal" aging range, below 27 °C down to the optimum 21 't3, also seems paradoxical and needs to be explained. To understand that the optimum does not correspond to a maximum value for a parameter of organization, we must remember that the parameters obtained by mortality kinetics experiments measure the

384

organization needed to sustain survival only. Discussing what organization is, we were led to propose a definition based on the kinetics of the change in information content of a system under the effects of noise producing factors [2, 11, 12]. As can be seen in the present article, comparisons of mortality curves with physiological and biochemical data at various temperatures could lead us to actual measurements of organization parameters previously defined by us only in formal way. But we must remember that the organization is used not only to survive as individuals but also to perform functions. In our experiments, 21 °C appears to be optimum not for plain survival - which increases monotonously with decreasing temperature - but for function, as seen from the data on negative geotaxis and mating. In other words, at the optimum temperature set up by the previous phases of development, what is maximum is not simpie survival but function. This is exemplified by the initial period of improvement in functions which is more obvious at 21 °C, as mentioned above. It is also exemplified by the temperature dependence of a and k, and of the redundancy factors: the rate parameters a and k, owing to the relationship of their temperature dependence with that of the redundancy factors, are found to be higher at 18 °C although the flies live longer. This phenomenon is an indication of the intricate character of these two parameters which express additional features of organization beside elementary reaction rates. At 18 °C, the flies have to adapt themselves to this new temperature after their eclosion, through some change in organization. In other words, flies at 18 °C are submitted to a detrimental environment from the point of view of their adaptation needs and this appears as an increase in their "rate" parameter comparable with that produced by detrimental higher temperatures, although the rate of their elementary biochemical processes is decreased, which makes them live longer! Thus a departure of the temperature from the optimum necessitates additional responses of the organism for survival adaptation*. Therefore, what is needed to sustain survival on!y, is minimum at the optimum temperature. We have shown elsewhere [11, 12, 17] that adaptation (and learning in the broadest sense) implies a use in redundancy. What is measured in our mortality kinetics experiments is a redundancy needed for survival only. This is why it is minimum at the temperature to which the preimaginal stages of organization have previously been adapted: at this temperature, there is no need for additional survival adaptation and most of the overall redundancy can be available for the observed improvement of functions.

Distinctive features of the power law If our interpretation is correct, it would be easier to reach the molecular level of the organization, responsible for the responses of the organism to aging and temperature, by the power law analysis than by the two other proposed equations; contrary to t~ and k, A seems to be almost independent of a redundancy factor. At the same time, however,

*This seems to be even m o r e the case at a lower temperature w h e n the elementary reaction rates are slowed down, leading to an overall lengthening of life: contrary to ,~ and k, the redundancy factors are not increased from 21 ° to 27 °C; owing to a " n o r m a l " increase in the reaction rates there would be no need for additional survival adaptation since the lifespan is shortened anyway.

385 this analysis makes the features of the supramolecular level of organization - namely its redundancy - almost disappear. This distinctive feature of the power law, as compared with the two others, can be understood as a consequence of the series model statistics on which it is based, where more weight is given to the extreme (i.e. the weaker link) whereas the effects on the other links are minimized [3]. Interfly variability and optimum temperature Put in another way, the difference between the power law and the two others can be understood from the point of view of how they deal with the variability among the Drosophila population. The variability between flies in a highly inbred population is likely to be found in their redundancy or reliability parameter, at the level of their overall organization, rather than at the simple biochemical level. Now, the optimum environment for a population is that in which most of the flies can function in spite of the variability between them. Therefore the differences in function and longevity due to built in variability will be more evident in the optimum environment where the stresses from outside are, by definition, minimum. Any departure from this optimum will increase the stress so that the variability between the flies will be wiped out by the intensity of the stress. This shows up in the shape of the mortality curves (preceding article Fig. 3) by a more pronounced steepness of the slope in the dying phase, for any departure from the optimum temperature of 21 °C. This effect on the interfly variability obviously disappears when one considers only the mean longevities, as in Fig. 4, where no optimum temperature is found. We suggest that, for the same reason, in the power law analysis no optimum is found in the variation of A, owing to the minimization of the variations of n; the optimum is indeed found in the variations of n (Figs. 5 and 6), but since these variations are minimized, the temperature function of A (i.e. the rate constant) is left almost completely independent of that of n (the redundancy factor). Therefore, assuming that the effects on the variability appear only at the integrated level, Le. on the redundancy, it can be understood that the optimum does not show up for the elementary rate constant parameter given by the power law. On the contrary, the information given by the temperature dependences of the rate constants in the multihit and the Gompertz law is misleading if one is interested in the molecular elementary mechanisms, since their temperature function seems to be dependent also on the redundancy factor. Therefore, in our present data the order of magnitude of the activation energies, computed from the rate constants in these two laws, is as much of an overall integrated process as that of an actual molecular or biochemical mechanism which would be the "primary" agent of aging. Along these lines, it is not surprising that, within the "normal" aging temperature range of 21-27 °C, we find identical slopes for the Arrhenius plot of these parameters and for those which obviously measure overall integrated phenomena such as physiological performances (oegative geotaxis and mating) and mean longevity. Therefore, if our interpretation is correct, the fact that the fluorescent lipopigment accumulation also seems to follow a similar temperature dependence, as mentioned above (Fig. 5), shouM be a reason to consider this phenomenon not as a primary elementary process in aging b u t rather as a result o f the progressive failure o f the overall organization. On the con-

386 trary, a mere protein denaturation process, suggested by the p o w e r law analysis, would be more likely to represent a truly rate-determining phenomenon at the molecular level. CONCLUSION Whereas plain statistical analysis would not tell us unambiguously which of the models best fits the data, the study o f the temperature dependence o f the parameters given by these models helps us to visualize their respective advantages and disadvantages: the power law seems to be valuable in giving us information on the elementary, biochemical reaction, level o f organization, measured by a rate parameter, whereas the multihit and Gompertz laws provide us with a more intricate analysis in which greater emphasis is put on a more integrated, supramolecular, level o f organization, measured by a redundancy factor. This type of analysis can help us to distinguish the biochemical changes correlated with aging which are likely to be primary molecular events (here, protein denaturations) from those (in our case accumulation o f lipoperoxide pigments) which are likely to be secondary effects, resulting from the overall failure of the organization at a more integrated level.

ACKNOWLEDGEMENTS The work o f H. A. was supported by INSERM (Paris) and the Fundations Claude Pompidou and Philippe.

REFERENCES 1 B.L. Strehler, Time Cells and Aging, Academic Press, New York, 1962. 2 H. Atlan, Strehler's theory of mortality and the second principle of thermodynamics, J. Gerontol., 23 (1968) 196-200. 3 B. Rosenberg, G. Kemeny, L. G. Smith, I. D. Skurnick and M. J. Bandurski, The kinetics and thermodynamics of death in multicellular organisms, Mech. Ageing Develop., 2 (1973)275-293. 4 H. A. Johnson and M. Pavelec, Thermal injury due to normal body temperature, Am. J. PathoL, 66 (1972) 557-564. 5 S. Glasstone, K. J. Laidler and H. Eyring, The Theory o f Rate Processes, McGraw-Hill, New York, 1941~ 6 B. L. Strehler, Further studies on the thermally induced aging of Drosophila melanogaster, J. Gerontol., 17 (1962) 347-352. 7 G. A. Sacher, The complementary of entropy terms for the temperature dependence of development and aging. Ann iV. Y. Acad. ScL, 138 (1967) 680-712. 8 M. J. HoUingsworth, Temperature and length of life in Drosophila, Exptl. Gerontoi., 4 (1969) 49-55. 9 P. R. Berington, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York, 1969. 10 D. W. Morquardt, Least square estimation of non linear parameters, Shore Distribution No. 3094, E. L Dupont de Nemours, Eng. Dept., Wilmington, Delaware.

387 11 H. Atlan, L 'organisation biologique et la th~orie de l'in[ormation, Hermann, Paris, 1972. 12 H. Atlan, A formal definition of organization,J. Theoret. Biol., 45 (1974) 295-304. 13 I. Kunstyr and H. G. W. Leuenberger, Gerontological data of C 57 BL[6 J mice. L Sex differences in survival curves, J. Gerontol., 30 (1975) 157-162. 14 J. A. Sheldahl and A. L. Tappel, Fluorescent products from aging Drosophila melanogaster: An indicator of free radical lipid peroxydation damage, Exptl. Gerontol., 9 (1974) 33-41. 15 H. Atlan, Source and transmission of information in biological networks, in I. Miller (ed.), Stability and Origin o f Biological Information, Ist A. Katzir-Katchalsky Memorial Symp., Rehovot, 1973, Wiley-Interscience, New York, 1975, pp. 95-118. 16 S. Winograd and J. D. Cowan, Reliable Computation in the Presence o f Noise, M.I.T. Press, Cambridge, Mass., 1963. 17 H. Atlan, Le principe d'ordre ~ partir de bruit, l'apprentissage non dixig6 et le r6ve, in E. Morin and S. M. Piatelli-Palmerini (eds.), L 'Unit~ de I'Homme, Seuil, Paris, 1974, pp. 469-475.