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A mathematical analysis of the mortality kinetics of Drosophila melanogaster exposed to gamma radiation
Mechanisms of Ageing and Development, 4 (1975) 59-69 © Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
59
A M A T H E M A T I C A L A N A L Y S I S OF T H E M O R T A L I T Y K I N E T I C S OF D R O S O P H I L A M E L A N O G A S T E R EXPOSED TO G A M M A R A D I A T I O N
C. B. DOLKAS, H. ATLAN*, G. DOLKAS and J. M1QUEL NASA, Ames Research Center, Moffett Field, Calif. 94035 (U.S.A.) and University of Calilornia, Berkeley, Calif. 94720 (U.S.A.) (Received March 15, 1974; in revised form December 5, 1974)
SUMMARY A mathematical model with nonlinear time-varying characteristics has been developed which describes the relationship between the kinetics of natural aging and radiation-induced delayed mortality. Based on this model, it appears that there is an immediate effect of radiation which is continuously, but nonlinearly, increasing in severity. Two phases appear in this variation, corresponding to the two phases (plateau and dying phase) of the mortality curves for control populations. Accordingly, S / E (survival time post-irradiation/further expectation of life) can best be interpreted as an increasing function during the plateau phase of normal mortality curves, which levels off during the ensuing dying phase.
INTRODUCTION Previous work has shown that g a m m a irradiation of Drosophila melanogaster imagoes 1 to 20 days old with a single dose of 50 kR results in death at a constant postexposure time 1. However, various investigators conducting similar studies of lethality kinetics (i.e., overall death patterns) have developed conflicting interpretations from their respective experimental data. Baxter and Blair 2,3 concluded from their data that a given single dose of radiation produces a constant life shortening (i.e., kills the flies at a constant latent period) irrespective of age at irradiation. L a m b 4 presented data indicating that life shortening is proportional to further expectation of life, i.e., decreases with age at irradiation in such a way that S/E ratio (postexposure survival time/further expectation of life for nonirradiated flies sham-treated at the same time) is constant. Blair and Baxter 5 found that "after-survivals" of adult Drosophila melanogaster to X-ray exposure at various ages are constant fractions of the remaining life expectancies at those ages. The conflict between advocates of an "accelerated aging" and that of a "precocious aging" theory as a mechanism for radiation-induced life shortening has been revived by L a m b and Maynard Smith 6, * Present address: Service de Biophysique, Facult6 de M6decine Broussais, 45 rue des Sts-P6res, Paris-6, France.
60 who proposed a mathematical model based on a theory supported by Lamb's data 4. They rejected a previous model by Baxter and Blair2, 3 because it was based on the "precocious aging" theory which they considered to be contradictory to experimental data. However, the Lamb and Maynard Smith 6 conclusion is similarly questionable since their observed slope of regression line was 0.482 ~ 0.029 as compared (apparently to their satisfaction) to the calculated slope of 0.38. In contrast, the data of Atlan et al. 1 and Miquel et al. 7 support the view that a specific radiation syndrome is responsible for life shortening and is different from natural aging, whether accelerated or precocious. It seems that the latters' data can accommodate the statement by Lamb and Maynard Smith 6 that the "precocious aging" theory is in conflict with experimental evidence. The only known data supporting the theory of precocious aging are those of Baxter and Blair 2,a but these data are suspect because of the large standard error in estimating survival times (as was acknowledged by the authors). In this respect, Lamb's data 4 are less objectionable, but they leave open the question of what happens before natural mortality becomes nonnegligible. Regardless of whether normal aging and radiation injury are completely independent processes, if each is able to produce death at a reproducibly fixed time, the effects on the population mortality are necessarily compounded. Hence, it is important to test the effects of radiation b e f o r e natural mortality becomes measurable in the population. This is possible only if the population exhibits a natural mortality curve of a quasi-rectangular kind, or at least one showing a long zero mortality plateau. The difficulty of obtaining accuracy in delayed death kinetics without acute effects can then be eliminated. This was done by Atlan et aL 1 in whose experiments no acute death occurred after a single dose of 50 kR when ages at irradiation were within the range of zero natural mortality. This was not the case in Lamb's experiments; therefore, the discrepancy with the results of Atlan et al. 1 is not surprising. The latter were able to show clearly a complete independence of postexposure survival time from age at irradiation only when the radiation-induced delayed mortality was separated completely from natural mortality by advantageous use of a long zero mortality plateau (ages at irradiation from 1 to 20 days). This is the experimental basis for the central assumption in the model, namely, that the mechanisms of natural mortality can be separated from those of radiation-induced mortality. In instances where flies of older age were irradiated, the results of Atlan et al. 1 were somewhat ambiguous. This was due to the inherent increase in the standard error spread as the age at irradiation increased; their data could fit a constant S / E ratio curve, similar to that by Lamb 4. Conversely, in Lamb's Fig. 9 ~, one can see clearly that the first point lies well below the horizontal line, which could indicate an increasing variation of S / E during the early phase of the life span, similar to the one observed by Atlan et al. 1. Moreover, histological studies by Miquel et al. s of irradiated and non-irradiated control flies, during their respective periods of death, show significant differences in the pathology of radiation-induced delayed mortality (29-33 days post-irradiation) and of natural aging. The glycogen deposits of muscle and fat body were more abundant in the irradiated than in the normal senescent flies sacrified at 30 and 84 days after eclosion, respectively. However, age at time of irradiation clearly influences survival time when the flies are irradiated
61 at 30 to 90 days after eclosion. It appears on the basis of mortality kinetics that synergism between radiation-induced mortality and normal aging mortality may occur when mortality rates due to natural senescence have reached a measurable value. The present article deals with a model that describes, in mathematical terms, the relationship of the kinetics in mortality due to natural aging and irradiation. The mathematical model is useful as a tool in clarifying the analysis and interpretation of the data, especially of those relating to flies of older age, where mortality occurs as a result of both natural aging and radiation injury. As for interpreting the survival time of irradiated young flies, it can be said with confidence that there is no interaction between natural aging and radiation and that mortality is due to irradiation only, provided death occurs during the period of a long zero mortality plateau of the natural survival curve. DEVELOPMENT OF THE MODEL
Let M, the age-specific mortality rate, be equivalent to the probability, P, that a fly alive at some time, t, will die in some small time interval around t. The probability of death due to radiation is Pr, and the probability of a fly surviving radiation effects, St, in that time interval is 1 - - Pr. The probability, S, of a fly surviving independent effects of aging and radiation in that interval is S = &S~ where Sr = probability of survival due to radiation, and Sn = probability of.survival due to natural aging. (1)
S = (1 - - P r ) (1 - - e n ) The probability of mortality due to either cause is l--S=
1 - - ( 1 - - Pr) ( 1 - - Pn) = P r
+Pn--PrPn
(2)
OF
M -- Mn + Mr--MnMr
(3)
Two " b o u n d a r y " conditions pertinent to eqn. (3) exist. One applies to flies that are not exposed to radiation, and the other to flies that are irradiated on the day of eclosion. In the case of the nonirradiated flies, Mr equals zero, and M -- Mn, the mortality due to natural aging
(4)
In the case of flies that are irradiated on the day of eclosion, the effect of natural aging is nil, and M = Mr
(5)
that is, the population falls to zero before the effects of natural aging become signific-
62 .~ I0 CALCULATED SUM OF INTEGRAL OF i n ( t ) + i r ( t , x ) OR Eq. 17
NO _ [n{~t't - 2 )
\
I*
(30)* (40)* (50)*(60)~70)*
-I
/ /d
~> i0-i
.I
Mr
/ RADIATION~4
Y-
/~i5/ /
~,NORTALITY)l q . "
_z
20
10
30
/ /
/ /
."
50 / 6 0 // l/
--" /
/?/ /
IO-2
40
/
70/
/ ,~/ / off/
{NATURAL'~
/ ~"/
/ AGING)
Mn
2,:. .
G F-io-3
~1; /,.,?'/ / / / ~1; J / I l." / A/, , /o Y/ ,,/ / ~// / . ~ . f / ,'/ / *NOMBE,S,0,CATE
o
//
/
10-4 I / 0
1
//
/
/ "~// / / /
~11 20
I
, /
/,' / L/'' 40
AGEATTINEOF
;
Rt,D,ATION(SOkR)
I 60
j
,
80
I00
120
TIME, ), days --c--o--o- EXPERIMENTAL DATA - - - - - - INTEGRAL OF EXPERIMENTAL DATA -----INTEGRAL OFM r SHIFTED | -X days - CALCULATED VALUES, SEE LEGEND FOR EXPLANATION
Fig. 1. Semilog plot showing separate effects of Mn and Mr and their summation. Solid lines for ages enclosed in ( ) near top of the figure denote summation of integrals, Mn(t) + Mr(t, Z) at integral of mortality when t = T1/2for 5 0 ~ survival, or No/Nt = 2. Note, In(No/N) = 0.693.
ant. To obtain the relations for Mn and Mr we c o m p u t e mortality for nonirradiated controls and for flies irradiated at one day of age, using the following equation : M(t) --
1
AN
Nt
At
(Note that--
AN At
< 0, M >
0)
(6)
where Nt = number of flies at time t, and t = time after eclosion, days. The results were plotted on semilog paper (Fig. 1) fo: normal aging and delayed postirradiation mortality. The control mortality rates Mn(t) follow a straight line except for the very last portion, in agreement with the k n o w n departure from G o m p e r t z i a n kinetics at the very end of the mortality curve. The mortality rates for irradiated Drosophila melanogaster Mr(t) also follow a straight line with an appreciably greater slope. Hence, from the plot we obtain: Mn = Mn0 exp (ant) Mn = 6.62 × 10 -7 exp (0.124t)
(7a) (7b)
63 where: Mn0 = initial mortality for natural aging, and an = slope of mortality rate for natural aging. For flies irradiated on the first day after eclosion:
(8a)
Mr = Mro exp (art) Mr -- 1.645 X 10-2 exp (0.304 t)
(8b)
where : initial mortality rate to radiation-induced life-shortening, i.e., immediately after radiation, and Ctr = slope of mortality rate to radiation-induced life-shortening. Of course, in Gompertzian kinetics we are tacitly assuming that a n and at, the slopes of the mortality rates, are constant throughout the life span. Although this may not be necessarily true, a reliable calculation of an or at, particularly in the terminal phase, is made inherently difficult because of the low number of surviving flies. However, the argument is somewhat academic because, as will be shown later, the expression for mortality kinetics, M, will be integrated to obtain the number of survivors. In that process, the parameters a n and a r are part of a coefficient that is derived from experiment. Tile problem now reduces to that of predicting the effects of radiation and aging at intermediate ages. This is described below. From A N / A t ~ - - N t M ( t ) , and taking the limit of this as At -+ 0 and transposing, we have: Mr 0 --
dN Nt
(9)
M(t)dt
To describe the effects in terms of the ratios of initial to surviving numbers of flies, eqn. (9) is integrated: .t
,.Nt dN
(lO) j No N
j to
where Nt ~ number of flies at time t, N 0 = number of flies at to, t time after eclosion, days, and t o ~ day of eclosion, or initial day of computation. Substituting eqns. (3), (7a) and (Sa) into (10), gives:
rio =
Mno exp(ant) + Mr~ exp(art)
-
-
64
MnoMro exp(ant) exp(art) ] dt
For t o :
(12)
0:
No -Art
ln-
Mno
Mro exp(ant) + - ar
an
Mno
Mro
(MnoMro)
an
ar
(an -~ ar)
exp(art)
exp(ant)exp(art) +
MnoMro (an -~ ar)
(13)
where it is understood that the flies are irradiated upon eclosion. The effect of radiation, imposed at an arbitrary age, Z, in days, is described by a shift in g in the exponent of the radiation term Mr in eqn. (8a). Therefore, Mr = Mr0 exp[ar (t - - Z)]
(14)
where Mr = 0, when t < Z, i.e., effects occur only in irradiated flies. The computation of M then, for different values of Z, provides the experimental survival times. For this purpose, the following approximations seem justified: (a) Because of experimental errors and the fact that radiation procedures were administered late in the afternoon, we assimilate the above expression of Mr, eqn. (14), for Z = 1 day to the one for Z = 0.5 and therefore add 0.5 day on subsequent days of radiation. (b) Because of the shapes of the survival curves, the mean longevities, which were determined experimentally, were always very close to times for 50 percent survival, T1/2. Therefore, we consider these mean longevities as experimental values for T1/2. From eqns. (11) and (14), noting that t : T1/2 when N o / N t = 2, we have: Mno 0.693 -- - an
+
Mro
exp(anT1/2) + - ar
Mno
Mro
Mn.Mro
an
ar
(an ~- ar)
[ Mn°Mr° ] (an + ar) exp(arZ)
exp[ar(T1/2 - - Z)] - -
exp(anTa/2)exp[ar(T1/2 - - Z)] +
(15)
for Tale >_ 7.: Mn0 an Mr0 ar
~ = = =
6.62 × 10-v, 0.124, 1.645 X 10 -~, and 0.304.
We note, from Fig. 2 or Table 2 of Ref. 1, that the bounds of T1/2 are equal to 31 days and 95.3 days for flies irradiated on age day one, and for nonirradiated flies, respectively. The constants of integration are negligible compared to 0.693.
65 The above equation must be solved for T1/2 for fixed values of g. This would represent the mean lifetime of the population that has been irradiated at Z days of age. Although eqn. (15) may be solved analytically, its solution is made more convenient by the use of the semilog plot (Fig. 1). In this plot, the intersection of the horizontal line corresponding to ln(No/Nt ) = 0.693 with the summation of the terms on the righthand side of eqn. (15) should yield T1/z for any g within the life-span of the Drosophila. In the region of the T1/e, eqn. (15) can be closely approximated by the summation of the terms Mn and Mr it seems. This would allow one to use the graph to add these terms. Since the right-hand side of eqn. (15) is the integral of the total mortality rate, (see eqns. 9-15) the experimental data, Mn and Mr, of Fig. 1 must be divided by an and ar, respectively to obtain the heavier lines "integral of Mn and Mr" as shown. In Fig. 1 the arrows denote these summations and are labeled with the age (in parenthesis) at time of irradiation. Referring to Fig. 3 of Ref. 1 for S/E (i.e. survival time over further expectation of life for control groups sham-treated at the same time as the experimental flies) one can calculate the experimental T1/2 for various ages at irradiation, since S = T1/2 - - Z and E = 95.3 - - g. When comparing the experimentalvalues thus found with the calculated values (or intersections mentioned above), one notes that there are differences particularly at large values of Z. The differences of T1/2 found experimentally, compared with values found by summing the terms of equation (15,) can be due either to the aging term or to the radiation term. If we choose the first hypothesis, it would mean that the natural aging term must be changed due to irradiation, which would mean a genuine radiationinduced accelerated aging. However, this change would appear to be negligible for irradiations at young ages (where discrepancies are neglibible, if not zero) and to increase continuously with increasing age at irradiation. This makes little sense, since this would imply that there is a radiation-induced accelerated aging only at old ages, and something else at young ages. The second hypothesis, therefore, seems more plausible; it is the radiation term that must be changed because of age at irradiation. This change is very small or negligible for young ages at irradiation and becomes increasingly larger for older ages at irradiation, when post-irradiation death occurs during the natural dying period. We can express these thoughts in the function Mr0 which would increase with age, •, rather than remain constant with age. The meaning of such a characteristic would be that the probability of the immediate death of a fly after irradiation is due to radiation and the probability of this increases with the age of the fly at the time of irradiation. We should note that the single dose (50 kR) did not produce acute death during the plateau phase of the mortality curve. As was shown clearly by Baxter and BlairZ, 3, the radio-sensitivity for acute death increases with age. Therefore, it is possible that the contribution of an acute death syndrome, which may be nonexistent for a 50 kR dose in young flies, becomes noticeable for the same dose in older flies. This is a possible meaning, in biological terms, of Mr0 as an increasing function of age at irradiation. The specific nature of this characteristic, Mro, is developed by the following procedure. The use of eqn. (15) tacitly assumes the independence of aging and radiation. However, acknowledging the discrepancies of T1/z at the larger values of Z, we
66 10-2
/
I
/
// IIll
10-5
o
d >_ I0 -4
.J 0: 0
! __.- .........
IO'5
ii / J
/
/// top exp(aop X) //
rod exp (aodX)
(X=AGE AT IRRADIATION, 5OkR) 10-6
0
.~ ~o ;o .o ~o ~o ;o ;o ;o
,oo,,o
AGE AT IRRADIATION, X, doys
Fig. 2. Semilog plot showing variation of Mr0 with age at irradiation. Note that the referenced experimental data, Atlan et al. l, exhibits a plateau region until approximately 50 days. The above plot, obtained from the referenced data, shows a continuously variable increase from the start in the
mortality rate immediately after radiation. The increase in sensitivity is substantial even before the end of the plateau.
ask what would the nature of Mr0 be if the actual experimental values of the Ta/2's are used? Reversing the previous procedure, we equate the values at the intersections with the horizontal line of l n ( -N° Nt
-- 2)
in Fig. 1, and the experimental values of Tz/2. N o t e that a shift of any plot in the t (or 2;) direction can be brought to the identical position by a change instead in the intercept at t = 0, i.e., the "initial conditions". When we assume these experimental values, and compute back to find what the value of Mr0 must be to make this agreement, we obtain the plot of Fig. 2. These values are plotted, for convenience, on a semi-log plot. This figure shows that the characteristic, Mr0, is a function of age at irradiation. We note that this initial condition increases nonlinearly with age and the radiation-induced life shortening is enhanced accordingly. In order to describe Mr0 in
67 160 140 12o
,,-3
Q
zQ IO0 80
~ so 0
d 4o 20 I0
20
30
40 50 60 "?0 80 90 TIME AFTER ECLOSION, t, doys
I00
I10
120
Fig. 3. Comparison of calculated survival curves (solid lines) with experimental results (symbols). Calculations were made using the integral eqn. (19), or more conveniently from Figs. 1 and 2. © - flies irradiated (50 kR) at 0.5 day of age (at A)*. [] - - flies irradiated (50 kR) at 27.5 days of age (at B)*. ~ - - f l i e s irradiated (50kR) at 50.5 days of age (at C)*. /~ - - controls - - naturally aging flies. * All irradiation during zero death plateau.
simple mathematical terms, we approximated it for low and high values of X, each respectively by the exponential expressions shown in Fig. 7. When these functions are summed they closely approximate the plot for Mr0 even for intermediate values of g. It is a time-varying coefficient for the radiation effect, as opposed to the assumed constant value used previously. The term at the lower values of Z, r0v, is the initial mortality rate for the plateau range due to radiation, whereas the second term, r0o, has a much greater slope and is pertinent to the dying phase, again due to radiation. Therefore, the coefficient of the Mr term becomes: Mro : rop exp(aopZ) + rOd exp(a0d) Z
(16a)
where: Mro = = Ct0d = r0p =
see eqn. (16a) and Fig. 2. slope, in Mro, or rate of exponential, for the plateau phase, slope, in Mr0, or rate of exponential, for the dying phase, intercept, or initial mortality rate due to irradiation corresponding to a process taking place during the plateau, and rOd = intercept, or initial mortality rate for the period during dying. Equation (15a), with numerical values from Fig. 2 then becomes Mro (Z) :
1.505 × 10-5 exp(5.06 × 10-3Z)+ 1.197 × 10-s exp(0.156Z) (16b)
Substituting eqn. (7a), (14), and (16a) into eqn. (3), we have M = Mn0 exp(ant) + Mr0(:~) exp[ar(t - - Z)] - - Mno exp(ant)Mro(Z) × exp [ a r ( t - - Z)]
(17)
68 The first term on the right-hand side of eqn. (17) describes the effect of natural aging on total mortality rate. The second term describes the effect on irradiation, at age Z, on mortality independent of natural aging. The coefficient of this radiation term describes the time-varying "initial condition" due to radiation-induced life-shortening effect (i.e. radiation effect on mortality rate soon after irradiation). In the factor exp [ar(t - - Z)] of eqn. (17) above, the parameter t - - Z has the effect of delaying the increase in mortality due to radiation, and the overall effect is that, the later in life the flies are irradiated, the closer their characteristics are to the natural aging curve (since, for t < Z, the whole term is equal to zero, i.e. Mr0 = 0). From eqns. (10) ,(16b), (17) and the experimental data of Fig. 3, the final equation, after integration, becomes:
j
No In
-
-
--
Nt
~t
to ~¢Mno exp(ant) + Mro(g) exp[ar(t - - Z)]
- - MnoMro(g) exp(ant) exp[ar(t - - Z)] i dt
(18)
or:
In
No
--
Nt
Mn0
exp(ant) +
O~n
Mr0(Z)
exp[ar(t - - Z)]
Otr
MnoMro
(Z) exptant + a r ( t - Z)I + [Mn.M,.(~'~][ ... _2_o_.m [ exp(arZ) (an @ ar) [ an @ ar J (19)
where: Mno an Mro(Z) ar
= = = =
6.62 × 10-7, 0.12, 1.506 × 10-5 exp(6.06 × 1 0 - z Z ) ÷ 1.197 × 10-8 exp(0.156Z), and 0.304.
RESULTS A N D DISCUSSION
A comparison of the calculated effects of a single exposure to 50 kR gamma radiation on mortality with experimental findings on Drosophila melanogaster is shown in Fig. 3. Of course, the plotted values of Fig. 2 can be substituted, instead of eqn. (16b), into eqn. (19) and be solved for Art, the number of surviving Drosophila. Figure 3 shows that a good relationship between calculated and experimental values was achieved, and that prediction of the lifeshortening effect of irradiation at various ages can be obtained using the mathematical model developed. Moreover, it appears that there is a continuously variable initial effect responsible for a latent radiationinduced mortality (Fig. 2). It is broadly divided into two phases : one, a slow, essentially constant, low-level effect during the plateau phase; second, a much higher rate dominating in the mortality phase. The two "components" are a function of age at irradiation and are, therefore, time-varying. The result implies an enhanced immediate
69 effect o f i r r a d i a t i o n with age, which could be related to the r e p o r t e d increase in r a d i o sensitivity for acute d e a t h with age (Baxter a n d BlairS), b u t as f o u n d by M i q u e l et al. 7 the life-shortening effects o f g a m m a r a d i a t i o n seem to be the result o f a r a d i a t i o n s y n d r o m e u n r e l a t e d to aging. A n i m p o r t a n t f u n c t i o n o f this " m o d e l " is its usefulness in o b t a i n i n g the best i n t e r p r e t a t i o n for the d a t a on S / E r a t i o for old ages. As n o t e d in a previous article by A t l a n et al. 1 a n d in the b e g i n n i n g o f this article, the statistics are n o t as g o o d for g r o u p s o f old flies as for y o u n g ones, a n d the d a t a can fit e i t h e r an increasing linear function, as a c o n t i n u a t i o n o f the one given b y accurate d a t a on y o u n g ages, o r a h o r i z o n t a l f u n c t i o n similar to t h a t f o u n d by L a m b 4. I f the first version o f the m o d e l in which Mr0(Z) was a s s u m e d to be c o n s t a n t was satisfactory, then the first m a t c h i n g (or fit) w o u l d have been acceptable, because it w o u l d have m e a n t t h a t the same kinetics h o l d for all phases o f the life span. However, this was n o t the case, a n d this r e q u i r e d the a s s u m p t i o n o f an increasing function for Mr0(Z), which results in survival times, S, s h o r t e r t h a n in the first version when r a d i a t i o n d e a t h occurs d u r i n g the dying phase o f the n o n i r r a d i a t e d p o p u l a t i o n . Since survival times for old ages are s h o r t e r t h a n expected when a c o n t i n u a t i o n o f an increasing linear function for S / E is assumed, a n d since E r e m a i n s identical for either schemes, it means t h a t the increasing " l i n e a r funct i o n " S / E m u s t level off for old ages. This is c o m p a t i b l e with a h o r i z o n t a l line (or at least a linear function o f smaller slopes) for these ages as was f o u n d by L a m b 4.
REFERENCES 1 H. Atlan, J. Miquel and R. Binnard, Differences between radiation-induced life shortening and natural aging in Drosophila melanogaster, J. Gerontol., 24 (1969) 1~-. 2 R. C. Baxter and H. A. Blair, Kinetics of aging as revealed by X-ray-dose-lethality relations in Drosophila, Radiat. Res., 30 (1967) 48-70. 3 R. C. Baxter and H. A. Blair, Age of death in Drosophila following sublethal exposure to gamma radiation, Radiat. Res., 31 (1967) 287-303. 4 M. J. Lamb, The relationship between age at irradiation and life-shortening in adult Drosophila. In P. J. Lindop and G. A. Sacher (eds.), Radiation and aging, Taylor and Francis, London, 1966, pp. 163-174. 5 H. A. Blair and R. C. Baxter, Radiation life shortening in Drosophila as a function of age and prior exposure, Radiat. Res., 43 (1970) 439~151. 6 M. J. Lamb and J. Maynard Smith, Radiation induced life-shortening in Drosophila, Radiat. Res., 40 (1969) 45(P464. 7 J. Miquel, K. G. Bensch, D. E. Philpott and H. Atlan, Natural aging and radiation-induced life shortening in Drosophila melanogaster, Mech. Age. Dev., 1 (1972) 71-97. 8 J. Miquel, H. Atlan and R. Binnard, Comparison of the histopathological effects of radiation and normal aging on Drosophila melanogaster, 20th Ann. Meeting of the Gerontological Soc., The Gerontologist, 7 (3, II) (1967).
59
A M A T H E M A T I C A L A N A L Y S I S OF T H E M O R T A L I T Y K I N E T I C S OF D R O S O P H I L A M E L A N O G A S T E R EXPOSED TO G A M M A R A D I A T I O N
C. B. DOLKAS, H. ATLAN*, G. DOLKAS and J. M1QUEL NASA, Ames Research Center, Moffett Field, Calif. 94035 (U.S.A.) and University of Calilornia, Berkeley, Calif. 94720 (U.S.A.) (Received March 15, 1974; in revised form December 5, 1974)
SUMMARY A mathematical model with nonlinear time-varying characteristics has been developed which describes the relationship between the kinetics of natural aging and radiation-induced delayed mortality. Based on this model, it appears that there is an immediate effect of radiation which is continuously, but nonlinearly, increasing in severity. Two phases appear in this variation, corresponding to the two phases (plateau and dying phase) of the mortality curves for control populations. Accordingly, S / E (survival time post-irradiation/further expectation of life) can best be interpreted as an increasing function during the plateau phase of normal mortality curves, which levels off during the ensuing dying phase.
INTRODUCTION Previous work has shown that g a m m a irradiation of Drosophila melanogaster imagoes 1 to 20 days old with a single dose of 50 kR results in death at a constant postexposure time 1. However, various investigators conducting similar studies of lethality kinetics (i.e., overall death patterns) have developed conflicting interpretations from their respective experimental data. Baxter and Blair 2,3 concluded from their data that a given single dose of radiation produces a constant life shortening (i.e., kills the flies at a constant latent period) irrespective of age at irradiation. L a m b 4 presented data indicating that life shortening is proportional to further expectation of life, i.e., decreases with age at irradiation in such a way that S/E ratio (postexposure survival time/further expectation of life for nonirradiated flies sham-treated at the same time) is constant. Blair and Baxter 5 found that "after-survivals" of adult Drosophila melanogaster to X-ray exposure at various ages are constant fractions of the remaining life expectancies at those ages. The conflict between advocates of an "accelerated aging" and that of a "precocious aging" theory as a mechanism for radiation-induced life shortening has been revived by L a m b and Maynard Smith 6, * Present address: Service de Biophysique, Facult6 de M6decine Broussais, 45 rue des Sts-P6res, Paris-6, France.
60 who proposed a mathematical model based on a theory supported by Lamb's data 4. They rejected a previous model by Baxter and Blair2, 3 because it was based on the "precocious aging" theory which they considered to be contradictory to experimental data. However, the Lamb and Maynard Smith 6 conclusion is similarly questionable since their observed slope of regression line was 0.482 ~ 0.029 as compared (apparently to their satisfaction) to the calculated slope of 0.38. In contrast, the data of Atlan et al. 1 and Miquel et al. 7 support the view that a specific radiation syndrome is responsible for life shortening and is different from natural aging, whether accelerated or precocious. It seems that the latters' data can accommodate the statement by Lamb and Maynard Smith 6 that the "precocious aging" theory is in conflict with experimental evidence. The only known data supporting the theory of precocious aging are those of Baxter and Blair 2,a but these data are suspect because of the large standard error in estimating survival times (as was acknowledged by the authors). In this respect, Lamb's data 4 are less objectionable, but they leave open the question of what happens before natural mortality becomes nonnegligible. Regardless of whether normal aging and radiation injury are completely independent processes, if each is able to produce death at a reproducibly fixed time, the effects on the population mortality are necessarily compounded. Hence, it is important to test the effects of radiation b e f o r e natural mortality becomes measurable in the population. This is possible only if the population exhibits a natural mortality curve of a quasi-rectangular kind, or at least one showing a long zero mortality plateau. The difficulty of obtaining accuracy in delayed death kinetics without acute effects can then be eliminated. This was done by Atlan et aL 1 in whose experiments no acute death occurred after a single dose of 50 kR when ages at irradiation were within the range of zero natural mortality. This was not the case in Lamb's experiments; therefore, the discrepancy with the results of Atlan et al. 1 is not surprising. The latter were able to show clearly a complete independence of postexposure survival time from age at irradiation only when the radiation-induced delayed mortality was separated completely from natural mortality by advantageous use of a long zero mortality plateau (ages at irradiation from 1 to 20 days). This is the experimental basis for the central assumption in the model, namely, that the mechanisms of natural mortality can be separated from those of radiation-induced mortality. In instances where flies of older age were irradiated, the results of Atlan et al. 1 were somewhat ambiguous. This was due to the inherent increase in the standard error spread as the age at irradiation increased; their data could fit a constant S / E ratio curve, similar to that by Lamb 4. Conversely, in Lamb's Fig. 9 ~, one can see clearly that the first point lies well below the horizontal line, which could indicate an increasing variation of S / E during the early phase of the life span, similar to the one observed by Atlan et al. 1. Moreover, histological studies by Miquel et al. s of irradiated and non-irradiated control flies, during their respective periods of death, show significant differences in the pathology of radiation-induced delayed mortality (29-33 days post-irradiation) and of natural aging. The glycogen deposits of muscle and fat body were more abundant in the irradiated than in the normal senescent flies sacrified at 30 and 84 days after eclosion, respectively. However, age at time of irradiation clearly influences survival time when the flies are irradiated
61 at 30 to 90 days after eclosion. It appears on the basis of mortality kinetics that synergism between radiation-induced mortality and normal aging mortality may occur when mortality rates due to natural senescence have reached a measurable value. The present article deals with a model that describes, in mathematical terms, the relationship of the kinetics in mortality due to natural aging and irradiation. The mathematical model is useful as a tool in clarifying the analysis and interpretation of the data, especially of those relating to flies of older age, where mortality occurs as a result of both natural aging and radiation injury. As for interpreting the survival time of irradiated young flies, it can be said with confidence that there is no interaction between natural aging and radiation and that mortality is due to irradiation only, provided death occurs during the period of a long zero mortality plateau of the natural survival curve. DEVELOPMENT OF THE MODEL
Let M, the age-specific mortality rate, be equivalent to the probability, P, that a fly alive at some time, t, will die in some small time interval around t. The probability of death due to radiation is Pr, and the probability of a fly surviving radiation effects, St, in that time interval is 1 - - Pr. The probability, S, of a fly surviving independent effects of aging and radiation in that interval is S = &S~ where Sr = probability of survival due to radiation, and Sn = probability of.survival due to natural aging. (1)
S = (1 - - P r ) (1 - - e n ) The probability of mortality due to either cause is l--S=
1 - - ( 1 - - Pr) ( 1 - - Pn) = P r
+Pn--PrPn
(2)
OF
M -- Mn + Mr--MnMr
(3)
Two " b o u n d a r y " conditions pertinent to eqn. (3) exist. One applies to flies that are not exposed to radiation, and the other to flies that are irradiated on the day of eclosion. In the case of the nonirradiated flies, Mr equals zero, and M -- Mn, the mortality due to natural aging
(4)
In the case of flies that are irradiated on the day of eclosion, the effect of natural aging is nil, and M = Mr
(5)
that is, the population falls to zero before the effects of natural aging become signific-
62 .~ I0 CALCULATED SUM OF INTEGRAL OF i n ( t ) + i r ( t , x ) OR Eq. 17
NO _ [n{~t't - 2 )
\
I*
(30)* (40)* (50)*(60)~70)*
-I
/ /d
~> i0-i
.I
Mr
/ RADIATION~4
Y-
/~i5/ /
~,NORTALITY)l q . "
_z
20
10
30
/ /
/ /
."
50 / 6 0 // l/
--" /
/?/ /
IO-2
40
/
70/
/ ,~/ / off/
{NATURAL'~
/ ~"/
/ AGING)
Mn
2,:. .
G F-io-3
~1; /,.,?'/ / / / ~1; J / I l." / A/, , /o Y/ ,,/ / ~// / . ~ . f / ,'/ / *NOMBE,S,0,CATE
o
//
/
10-4 I / 0
1
//
/
/ "~// / / /
~11 20
I
, /
/,' / L/'' 40
AGEATTINEOF
;
Rt,D,ATION(SOkR)
I 60
j
,
80
I00
120
TIME, ), days --c--o--o- EXPERIMENTAL DATA - - - - - - INTEGRAL OF EXPERIMENTAL DATA -----INTEGRAL OFM r SHIFTED | -X days - CALCULATED VALUES, SEE LEGEND FOR EXPLANATION
Fig. 1. Semilog plot showing separate effects of Mn and Mr and their summation. Solid lines for ages enclosed in ( ) near top of the figure denote summation of integrals, Mn(t) + Mr(t, Z) at integral of mortality when t = T1/2for 5 0 ~ survival, or No/Nt = 2. Note, In(No/N) = 0.693.
ant. To obtain the relations for Mn and Mr we c o m p u t e mortality for nonirradiated controls and for flies irradiated at one day of age, using the following equation : M(t) --
1
AN
Nt
At
(Note that--
AN At
< 0, M >
0)
(6)
where Nt = number of flies at time t, and t = time after eclosion, days. The results were plotted on semilog paper (Fig. 1) fo: normal aging and delayed postirradiation mortality. The control mortality rates Mn(t) follow a straight line except for the very last portion, in agreement with the k n o w n departure from G o m p e r t z i a n kinetics at the very end of the mortality curve. The mortality rates for irradiated Drosophila melanogaster Mr(t) also follow a straight line with an appreciably greater slope. Hence, from the plot we obtain: Mn = Mn0 exp (ant) Mn = 6.62 × 10 -7 exp (0.124t)
(7a) (7b)
63 where: Mn0 = initial mortality for natural aging, and an = slope of mortality rate for natural aging. For flies irradiated on the first day after eclosion:
(8a)
Mr = Mro exp (art) Mr -- 1.645 X 10-2 exp (0.304 t)
(8b)
where : initial mortality rate to radiation-induced life-shortening, i.e., immediately after radiation, and Ctr = slope of mortality rate to radiation-induced life-shortening. Of course, in Gompertzian kinetics we are tacitly assuming that a n and at, the slopes of the mortality rates, are constant throughout the life span. Although this may not be necessarily true, a reliable calculation of an or at, particularly in the terminal phase, is made inherently difficult because of the low number of surviving flies. However, the argument is somewhat academic because, as will be shown later, the expression for mortality kinetics, M, will be integrated to obtain the number of survivors. In that process, the parameters a n and a r are part of a coefficient that is derived from experiment. Tile problem now reduces to that of predicting the effects of radiation and aging at intermediate ages. This is described below. From A N / A t ~ - - N t M ( t ) , and taking the limit of this as At -+ 0 and transposing, we have: Mr 0 --
dN Nt
(9)
M(t)dt
To describe the effects in terms of the ratios of initial to surviving numbers of flies, eqn. (9) is integrated: .t
,.Nt dN
(lO) j No N
j to
where Nt ~ number of flies at time t, N 0 = number of flies at to, t time after eclosion, days, and t o ~ day of eclosion, or initial day of computation. Substituting eqns. (3), (7a) and (Sa) into (10), gives:
rio =
Mno exp(ant) + Mr~ exp(art)
-
-
64
MnoMro exp(ant) exp(art) ] dt
For t o :
(12)
0:
No -Art
ln-
Mno
Mro exp(ant) + - ar
an
Mno
Mro
(MnoMro)
an
ar
(an -~ ar)
exp(art)
exp(ant)exp(art) +
MnoMro (an -~ ar)
(13)
where it is understood that the flies are irradiated upon eclosion. The effect of radiation, imposed at an arbitrary age, Z, in days, is described by a shift in g in the exponent of the radiation term Mr in eqn. (8a). Therefore, Mr = Mr0 exp[ar (t - - Z)]
(14)
where Mr = 0, when t < Z, i.e., effects occur only in irradiated flies. The computation of M then, for different values of Z, provides the experimental survival times. For this purpose, the following approximations seem justified: (a) Because of experimental errors and the fact that radiation procedures were administered late in the afternoon, we assimilate the above expression of Mr, eqn. (14), for Z = 1 day to the one for Z = 0.5 and therefore add 0.5 day on subsequent days of radiation. (b) Because of the shapes of the survival curves, the mean longevities, which were determined experimentally, were always very close to times for 50 percent survival, T1/2. Therefore, we consider these mean longevities as experimental values for T1/2. From eqns. (11) and (14), noting that t : T1/2 when N o / N t = 2, we have: Mno 0.693 -- - an
+
Mro
exp(anT1/2) + - ar
Mno
Mro
Mn.Mro
an
ar
(an ~- ar)
[ Mn°Mr° ] (an + ar) exp(arZ)
exp[ar(T1/2 - - Z)] - -
exp(anTa/2)exp[ar(T1/2 - - Z)] +
(15)
for Tale >_ 7.: Mn0 an Mr0 ar
~ = = =
6.62 × 10-v, 0.124, 1.645 X 10 -~, and 0.304.
We note, from Fig. 2 or Table 2 of Ref. 1, that the bounds of T1/2 are equal to 31 days and 95.3 days for flies irradiated on age day one, and for nonirradiated flies, respectively. The constants of integration are negligible compared to 0.693.
65 The above equation must be solved for T1/2 for fixed values of g. This would represent the mean lifetime of the population that has been irradiated at Z days of age. Although eqn. (15) may be solved analytically, its solution is made more convenient by the use of the semilog plot (Fig. 1). In this plot, the intersection of the horizontal line corresponding to ln(No/Nt ) = 0.693 with the summation of the terms on the righthand side of eqn. (15) should yield T1/z for any g within the life-span of the Drosophila. In the region of the T1/e, eqn. (15) can be closely approximated by the summation of the terms Mn and Mr it seems. This would allow one to use the graph to add these terms. Since the right-hand side of eqn. (15) is the integral of the total mortality rate, (see eqns. 9-15) the experimental data, Mn and Mr, of Fig. 1 must be divided by an and ar, respectively to obtain the heavier lines "integral of Mn and Mr" as shown. In Fig. 1 the arrows denote these summations and are labeled with the age (in parenthesis) at time of irradiation. Referring to Fig. 3 of Ref. 1 for S/E (i.e. survival time over further expectation of life for control groups sham-treated at the same time as the experimental flies) one can calculate the experimental T1/2 for various ages at irradiation, since S = T1/2 - - Z and E = 95.3 - - g. When comparing the experimentalvalues thus found with the calculated values (or intersections mentioned above), one notes that there are differences particularly at large values of Z. The differences of T1/2 found experimentally, compared with values found by summing the terms of equation (15,) can be due either to the aging term or to the radiation term. If we choose the first hypothesis, it would mean that the natural aging term must be changed due to irradiation, which would mean a genuine radiationinduced accelerated aging. However, this change would appear to be negligible for irradiations at young ages (where discrepancies are neglibible, if not zero) and to increase continuously with increasing age at irradiation. This makes little sense, since this would imply that there is a radiation-induced accelerated aging only at old ages, and something else at young ages. The second hypothesis, therefore, seems more plausible; it is the radiation term that must be changed because of age at irradiation. This change is very small or negligible for young ages at irradiation and becomes increasingly larger for older ages at irradiation, when post-irradiation death occurs during the natural dying period. We can express these thoughts in the function Mr0 which would increase with age, •, rather than remain constant with age. The meaning of such a characteristic would be that the probability of the immediate death of a fly after irradiation is due to radiation and the probability of this increases with the age of the fly at the time of irradiation. We should note that the single dose (50 kR) did not produce acute death during the plateau phase of the mortality curve. As was shown clearly by Baxter and BlairZ, 3, the radio-sensitivity for acute death increases with age. Therefore, it is possible that the contribution of an acute death syndrome, which may be nonexistent for a 50 kR dose in young flies, becomes noticeable for the same dose in older flies. This is a possible meaning, in biological terms, of Mr0 as an increasing function of age at irradiation. The specific nature of this characteristic, Mro, is developed by the following procedure. The use of eqn. (15) tacitly assumes the independence of aging and radiation. However, acknowledging the discrepancies of T1/z at the larger values of Z, we
66 10-2
/
I
/
// IIll
10-5
o
d >_ I0 -4
.J 0: 0
! __.- .........
IO'5
ii / J
/
/// top exp(aop X) //
rod exp (aodX)
(X=AGE AT IRRADIATION, 5OkR) 10-6
0
.~ ~o ;o .o ~o ~o ;o ;o ;o
,oo,,o
AGE AT IRRADIATION, X, doys
Fig. 2. Semilog plot showing variation of Mr0 with age at irradiation. Note that the referenced experimental data, Atlan et al. l, exhibits a plateau region until approximately 50 days. The above plot, obtained from the referenced data, shows a continuously variable increase from the start in the
mortality rate immediately after radiation. The increase in sensitivity is substantial even before the end of the plateau.
ask what would the nature of Mr0 be if the actual experimental values of the Ta/2's are used? Reversing the previous procedure, we equate the values at the intersections with the horizontal line of l n ( -N° Nt
-- 2)
in Fig. 1, and the experimental values of Tz/2. N o t e that a shift of any plot in the t (or 2;) direction can be brought to the identical position by a change instead in the intercept at t = 0, i.e., the "initial conditions". When we assume these experimental values, and compute back to find what the value of Mr0 must be to make this agreement, we obtain the plot of Fig. 2. These values are plotted, for convenience, on a semi-log plot. This figure shows that the characteristic, Mr0, is a function of age at irradiation. We note that this initial condition increases nonlinearly with age and the radiation-induced life shortening is enhanced accordingly. In order to describe Mr0 in
67 160 140 12o
,,-3
Q
zQ IO0 80
~ so 0
d 4o 20 I0
20
30
40 50 60 "?0 80 90 TIME AFTER ECLOSION, t, doys
I00
I10
120
Fig. 3. Comparison of calculated survival curves (solid lines) with experimental results (symbols). Calculations were made using the integral eqn. (19), or more conveniently from Figs. 1 and 2. © - flies irradiated (50 kR) at 0.5 day of age (at A)*. [] - - flies irradiated (50 kR) at 27.5 days of age (at B)*. ~ - - f l i e s irradiated (50kR) at 50.5 days of age (at C)*. /~ - - controls - - naturally aging flies. * All irradiation during zero death plateau.
simple mathematical terms, we approximated it for low and high values of X, each respectively by the exponential expressions shown in Fig. 7. When these functions are summed they closely approximate the plot for Mr0 even for intermediate values of g. It is a time-varying coefficient for the radiation effect, as opposed to the assumed constant value used previously. The term at the lower values of Z, r0v, is the initial mortality rate for the plateau range due to radiation, whereas the second term, r0o, has a much greater slope and is pertinent to the dying phase, again due to radiation. Therefore, the coefficient of the Mr term becomes: Mro : rop exp(aopZ) + rOd exp(a0d) Z
(16a)
where: Mro = = Ct0d = r0p =
see eqn. (16a) and Fig. 2. slope, in Mro, or rate of exponential, for the plateau phase, slope, in Mr0, or rate of exponential, for the dying phase, intercept, or initial mortality rate due to irradiation corresponding to a process taking place during the plateau, and rOd = intercept, or initial mortality rate for the period during dying. Equation (15a), with numerical values from Fig. 2 then becomes Mro (Z) :
1.505 × 10-5 exp(5.06 × 10-3Z)+ 1.197 × 10-s exp(0.156Z) (16b)
Substituting eqn. (7a), (14), and (16a) into eqn. (3), we have M = Mn0 exp(ant) + Mr0(:~) exp[ar(t - - Z)] - - Mno exp(ant)Mro(Z) × exp [ a r ( t - - Z)]
(17)
68 The first term on the right-hand side of eqn. (17) describes the effect of natural aging on total mortality rate. The second term describes the effect on irradiation, at age Z, on mortality independent of natural aging. The coefficient of this radiation term describes the time-varying "initial condition" due to radiation-induced life-shortening effect (i.e. radiation effect on mortality rate soon after irradiation). In the factor exp [ar(t - - Z)] of eqn. (17) above, the parameter t - - Z has the effect of delaying the increase in mortality due to radiation, and the overall effect is that, the later in life the flies are irradiated, the closer their characteristics are to the natural aging curve (since, for t < Z, the whole term is equal to zero, i.e. Mr0 = 0). From eqns. (10) ,(16b), (17) and the experimental data of Fig. 3, the final equation, after integration, becomes:
j
No In
-
-
--
Nt
~t
to ~¢Mno exp(ant) + Mro(g) exp[ar(t - - Z)]
- - MnoMro(g) exp(ant) exp[ar(t - - Z)] i dt
(18)
or:
In
No
--
Nt
Mn0
exp(ant) +
O~n
Mr0(Z)
exp[ar(t - - Z)]
Otr
MnoMro
(Z) exptant + a r ( t - Z)I + [Mn.M,.(~'~][ ... _2_o_.m [ exp(arZ) (an @ ar) [ an @ ar J (19)
where: Mno an Mro(Z) ar
= = = =
6.62 × 10-7, 0.12, 1.506 × 10-5 exp(6.06 × 1 0 - z Z ) ÷ 1.197 × 10-8 exp(0.156Z), and 0.304.
RESULTS A N D DISCUSSION
A comparison of the calculated effects of a single exposure to 50 kR gamma radiation on mortality with experimental findings on Drosophila melanogaster is shown in Fig. 3. Of course, the plotted values of Fig. 2 can be substituted, instead of eqn. (16b), into eqn. (19) and be solved for Art, the number of surviving Drosophila. Figure 3 shows that a good relationship between calculated and experimental values was achieved, and that prediction of the lifeshortening effect of irradiation at various ages can be obtained using the mathematical model developed. Moreover, it appears that there is a continuously variable initial effect responsible for a latent radiationinduced mortality (Fig. 2). It is broadly divided into two phases : one, a slow, essentially constant, low-level effect during the plateau phase; second, a much higher rate dominating in the mortality phase. The two "components" are a function of age at irradiation and are, therefore, time-varying. The result implies an enhanced immediate
69 effect o f i r r a d i a t i o n with age, which could be related to the r e p o r t e d increase in r a d i o sensitivity for acute d e a t h with age (Baxter a n d BlairS), b u t as f o u n d by M i q u e l et al. 7 the life-shortening effects o f g a m m a r a d i a t i o n seem to be the result o f a r a d i a t i o n s y n d r o m e u n r e l a t e d to aging. A n i m p o r t a n t f u n c t i o n o f this " m o d e l " is its usefulness in o b t a i n i n g the best i n t e r p r e t a t i o n for the d a t a on S / E r a t i o for old ages. As n o t e d in a previous article by A t l a n et al. 1 a n d in the b e g i n n i n g o f this article, the statistics are n o t as g o o d for g r o u p s o f old flies as for y o u n g ones, a n d the d a t a can fit e i t h e r an increasing linear function, as a c o n t i n u a t i o n o f the one given b y accurate d a t a on y o u n g ages, o r a h o r i z o n t a l f u n c t i o n similar to t h a t f o u n d by L a m b 4. I f the first version o f the m o d e l in which Mr0(Z) was a s s u m e d to be c o n s t a n t was satisfactory, then the first m a t c h i n g (or fit) w o u l d have been acceptable, because it w o u l d have m e a n t t h a t the same kinetics h o l d for all phases o f the life span. However, this was n o t the case, a n d this r e q u i r e d the a s s u m p t i o n o f an increasing function for Mr0(Z), which results in survival times, S, s h o r t e r t h a n in the first version when r a d i a t i o n d e a t h occurs d u r i n g the dying phase o f the n o n i r r a d i a t e d p o p u l a t i o n . Since survival times for old ages are s h o r t e r t h a n expected when a c o n t i n u a t i o n o f an increasing linear function for S / E is assumed, a n d since E r e m a i n s identical for either schemes, it means t h a t the increasing " l i n e a r funct i o n " S / E m u s t level off for old ages. This is c o m p a t i b l e with a h o r i z o n t a l line (or at least a linear function o f smaller slopes) for these ages as was f o u n d by L a m b 4.
REFERENCES 1 H. Atlan, J. Miquel and R. Binnard, Differences between radiation-induced life shortening and natural aging in Drosophila melanogaster, J. Gerontol., 24 (1969) 1~-. 2 R. C. Baxter and H. A. Blair, Kinetics of aging as revealed by X-ray-dose-lethality relations in Drosophila, Radiat. Res., 30 (1967) 48-70. 3 R. C. Baxter and H. A. Blair, Age of death in Drosophila following sublethal exposure to gamma radiation, Radiat. Res., 31 (1967) 287-303. 4 M. J. Lamb, The relationship between age at irradiation and life-shortening in adult Drosophila. In P. J. Lindop and G. A. Sacher (eds.), Radiation and aging, Taylor and Francis, London, 1966, pp. 163-174. 5 H. A. Blair and R. C. Baxter, Radiation life shortening in Drosophila as a function of age and prior exposure, Radiat. Res., 43 (1970) 439~151. 6 M. J. Lamb and J. Maynard Smith, Radiation induced life-shortening in Drosophila, Radiat. Res., 40 (1969) 45(P464. 7 J. Miquel, K. G. Bensch, D. E. Philpott and H. Atlan, Natural aging and radiation-induced life shortening in Drosophila melanogaster, Mech. Age. Dev., 1 (1972) 71-97. 8 J. Miquel, H. Atlan and R. Binnard, Comparison of the histopathological effects of radiation and normal aging on Drosophila melanogaster, 20th Ann. Meeting of the Gerontological Soc., The Gerontologist, 7 (3, II) (1967).