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Age-Specific Properties of Spontaneous Mutations Affecting Mortality in Drosophila melanogaster
Scott D. Pletchera, David Houleb, and James W. Curtsingeraa Department of Ecology, Evolution and Behavior, University of Minnesota, St. Paul, Minnesota 55108,
b Department of Zoology, University of Toronto, Toronto, Ontario, Canada M5S 3G5
Corresponding author: Scott D. Pletcher, Department of Ecology, Evolution and Behavior, University of Minnesota, 1987 Upper Buford Circle, St. Paul, MN 55108, plet0005{at}tc.umn.edu (E-mail).
Communicating editor: A. G. CLARK
| ABSTRACT |
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An analysis of the effects of spontaneous mutations affecting age-specific mortality was conducted using 29 lines of Drosophila melanogaster that had accumulated spontaneous mutations for 19 generations. Divergence among the lines was used to estimate the mutational variance for weekly mortality rates and the covariance between weekly mortality rates at different ages. Significant mutational variance was observed in both males and females early in life (up to ~30 days of age). Mutational variance was not significantly different from zero for mortality rates at older ages. Mutational correlations between ages separated by 1 or 2 wk were generally positive, but they declined monotonically with increasing separation such that mutational effects on early-age mortality were uncorrelated with effects at later ages. Analyses of individual lines revealed several instances of mutation-induced changes in mortality over a limited range of ages. Significant age-specific effects of mutations were identified in early and middle ages, but surprisingly, mortality rates at older ages were essentially unaffected by the accumulation procedure. Our results provide strong evidence for the existence of a class of polygenic mutations that affect mortality rates on an age-specific basis. The patterns of mutational effects measured here relate directly to recently published estimates of standing genetic variance for mortality in Drosophila, and they support mutation accumulation as a viable mechanism for the evolution of senescence.
As the source of all genetic variance, mutation provides the basis for both variation and response to selection. Knowledge about the properties of spontaneous mutation is crucial to understanding the maintenance of genetic variance within populations and the genetic divergence between them (![]()
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The number of empirical studies investigating the properties of spontaneous mutations is growing (![]()
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Genetic models that combine age-structure and natural selection to predict age-related changes in life history characters are highly dependent on assumptions about age-specific mutational effects (![]()
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Genetic variance for mortality and reproduction has been shown to vary as a function of age in Drosophila (![]()
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Age-specific patterns of mutational effects also have a profound effect on predictions about the evolutionary dynamics of mortality. Although it is clear that senescence, defined as an increase in age-specific mortality rates with age, can be explained as a consequence of the age-related decline in the intensity of natural selection, it cannot evolve without the age specificity of genetic effects (![]()
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Recent life history investigations using large sample sizes find that mortality rates in very old individuals level off and may even decline (![]()
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Here, we present results from a large demographic studywith observations of 109,860 Drosophila melanogasterdesigned to estimate the age-specific properties of spontaneous mutations on mortality rates. Data have been obtained from 29 inbred lines derived from one isogenic population that accumulated spontaneous mutations for 19 generations, and from three control lines representing the base population before mutation accumulation. We present estimates of mutational variances and covariances for mortality rates at various ages and investigate the age-specific effects of spontaneous mutations that produced large changes in mortality. The large number of animals permits accurate estimates of mortality rates and genetic variance components throughout life, even after a large proportion of the flies have died (![]()
| MATERIALS AND METHODS |
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Stocks:
Mutation accumulation lines were provided by DAVID HOULE, who maintained 100 independent lines of D. melanogaster derived from a single isogenic stock. The isogenic population was derived from a laboratory stock, the Ives population, which was started with 400 isofemale lines in 1975 (![]()
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The inbred isogenic stock was used as a base population for founding 100 independent lines. Each generation, three vials per line were set up: two with a single male and female in each, and a third with four flies of each sex. Progeny from the first of the single-pair vials were used to found the next generation whenever possible. If the first vial did not contain enough flies, progeny from the second and then third vial were used. The overall failure rate for A vials (the target vial for collection) was 8.1% (91/1125 subline generations); 1.5% (17/1125) of the time, this required going to the C or four-pair vial because the second single-pair vial (the B vial) also failed. Logistic regression of the failure rates on generation, with B vials scored as 1, C as 2, and A vials as 0, showed a significant positive effect of generation on failure rate (analysis in Proc Logistic of SAS;
2 = 17.66, 1 d.f., P < 0.0001). This may reflect mutation accumulation in the sublines but possibly environmental factors. There was no significant overall effect of subline on failure rate (analysis in Proc Catmod in SAS; P > 0.9). Such failures allow natural selection to influence the allele frequencies of nonneutral mutations, leading to biased estimates of mutational properties. The few times this occurred and the apparently random distribution among accumulation lines suggest this effect is small. No lines were lost during the 19 generations of accumulation. To control for further mutation during experiments and to control for effects of common environment, at generation 19 of mutation accumulation, samples of flies from each line were used to generate two replicates per line. These replicates were maintained separately and with large population sizes.
Control populations were constructed through the use of cryopreservation. Concurrent to the initiation of the 100 mutation accumulation lines, a large number of embryos were sampled from the base population and cryopreserved at Cornell University (see ![]()
To insure that cross contamination did not occur between the mutation accumulation lines or between the mutation accumulation lines and the controls, analyses of transposable element positions were carried out on the control populations as well as 20 of the accumulation lines10 with the highest fitness and 10 with the lowest (D. HOULE, personal communication). Two lines were identified as potentially contaminated and are not included in the analysis.
The base population and all mutation accumulation lines bear the R and weak P cytotype (![]()
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Culture conditions in Minnesota:
At generation 19 of mutation accumulation, a sample of each control population and of 41 randomly chosen accumulation lines (each consisting of two replicates) was sent from the HOULE lab at the University of Toronto to the University of Minnesota. Of the 41 accumulation lines, a random sample of 31 lines was chosen for mortality analysis. Flies from each control population and from each of the two replicates of each accumulation line were transferred into half-pint milk bottles containing standard agar-yeast-molasses-cornmeal medium. Bottles were kept for three generations in a constant temperature (24°) and constant light incubator at ~68% relative humidity. During this time, each experimental and control population was expanded into six half-pint milk bottles to generate sufficient numbers of flies for the mortality measurements.
Mortality measurement:
For mortality measurements, flies were kept in 3.8-liter plastic "population cages" designed specifically for estimating mortality rates (![]()
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For each of the two replicates for the 31 mutation accumulation lines, ~1600 flies emerging within a 30-hr period were collected without anesthesia. Flies were weighed to estimate the ~800 individuals that were placed into each of two cages. Nearly equal numbers of males and females were placed in each cage although there was a slight but consistent female bias. For each of the three control populations, six cages were established. In total, 142 cagesaveraging 775 fliesand 109,860 flies were involved in the experiment. The large numbers of genetically identical flies in each cage allows accurate measures of mortality at each age and makes possible the estimation of mortality rates late in life after a large proportion of the flies have died. With reference to a more standard genetic analysis, each cage is treated as a single observation with an associated mortality rate at each age (![]()
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Mortality estimation:
Each day, all cages were examined for dead flies, which were removed, sexed, and recorded until the last death occurred. Summing over the duration of the experiment provides the number of individuals in the initial cohort, N0, as well as the number alive at the start of each day, Nx. The probability of surviving from age x to age x + 1 given the individual is alive at the start of age x is
x = Nx + 1/Nx. The age-specific rate of mortality is estimated as:
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(1) |
(![]()
x expresses the instantaneous rate of mortality or "hazard." It is not a probability measure; therefore, it is not bounded above (![]()
Age-specific variance components analysis:
All age-specific variance component analyses were carried out on the natural logarithm of mortality. The transformation has two important effects: (1) within an age class, log mortality rates are normally distributed, and (2) the logarithmic transformation normalizes the variance within age classes (![]()
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Relevant components of (co)variance were estimated using maximum likelihood procedures. For each week, the data were analyzed using a modified version of the module nf3.p of QUERCUS, a software package produced by R. and F. SHAW (![]()
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(2) |
i (i = 1, 2, ..., 29) is the line random effect, ßj(i) (j = 1, 2) is the random effect for the two genetic replicates nested within each line, and
k(ij) (k = 1, 2) is the residual error corresponding to the mortality rates in the ijkth cage. The design allows the total variation in mortality rates at any given age to be partitioned into three sources of variation. Parallel analyses were carried out for components of covariation. Thus, the phenotypic variation in log mortality at age x, Vp,(x), is the sum of variation between lines, Vl,(x), between replicates within a line, Vr,(x), and between cages within a replicate, Ve,(x). The same representation stands for covariance components between ages x and y by replacing Vi,(x) with Covi,(x,y). Identical procedures were used to estimate the covariance in mortality rates between males and females of the same age.
Likelihood ratio tests were used to test hypotheses concerning between-line variances and covariances. To test the hypotheses Vl,(x) = 0, we calculated log likelihoods for Equation 2 with the between-line component of variance either free or constrained to zero. Twice the difference in the log likelihoods is asymptotically distributed as
2 with 1 d.f. when parameters are tested one at a time. To test the hypothesis Vl,(x) = Vl,(y) (x
y), we used the module pcrf1.p in QUERCUS. This module was designed to compare variance components from two separate, genetically independent populations. In the present case, mortality rates at each age are clearly not independent, so significance levels from this analysis should be viewed with caution (![]()
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Estimates of mutational variance were obtained from the between-line components of variance, assuming the mutations were neutral, additive, and of small effect. In this case, for each age x,
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(3) |
m,(x)2 is the mutational variance at age x, t is the number of generations of divergence, and Ne is the effective population size in the accumulation lines through the experiment (
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(4) |
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m,(x)2 and Covl,(x,y) for Vl,(x). Mutational correlations between mortality at ages x and y (rm,(x,y)) are calculated as Covm,(x,y)/(
m,(x)
m,(y)).
Variance components for mean longevity:
To allow a comparison of results between the data presented here and other mutation accumulation experiments, variance components were estimated for average longevity. Relevant variance components were estimated using the varcomp procedure in S-Plus (MathSoft, Cambridge, MA). The data consist of the age at death for 95,721 individual flies (control lines are not included in this analysis). In this case, there is an additional level of variance representing variation in age at death among flies within the same cage. Mutational variance is again calculated using Equation 4. The distribution of these data is very nearly normal, and because of the large sample size, hypothesis tests concerning the values of the estimates are based on the asymptotic standard errors provided by the inverse of the information matrix (![]()
Variance components analysis for parameters of mortality models:
A number of mathematical formulae have been proposed for describing the relationship between age and mortality rates (![]()
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(5) |
describes mortality rate at birth, and ß is the rate of exponential increase in mortality with age. Recent experimental work involving large cohorts has documented a significant deceleration of mortality rates in advanced ages (
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(6) |
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and ß. The parameter s determines the extent to which mortality rates decelerate late in life. Higher values of s indicate greater decleration. Note that when s = 0, Equation 6 reduces to Equation 5.
Genetic and phenotypic components of variance for the parameters of these two models were estimated using maximum likelihood methods (S. D. PLETCHER, unpublished results). For each population cage, the parameters of the appropriate mortality model were estimated by maximizing the likelihood over individual deaths. Because likelihood estimates are asymptotically normally distributed, components of variance for each parameter were estimated using QUERCUS, and hypothesis testing was carried out in the same manner as described for measures of age-specific mortality. Estimates of mutational variance for the parameters were calculated using Equation 4.
Estimation of age-specific mutational effects:
The effects of spontaneous mutations on age-specific mortality were examined by comparing mortality rates in 3-day intervals between the control lines and each mutation accumulation line. Mortality rates for each line were determined by calculating the arithmetic average, on a log scale, of the mortality rates in each replicate population cage (four cages for each accumulation line and 16 cages for the control lines). There are two complications in testing differences in mortality rates at any particular age. First, because each individual contributes to observed levels of mortality throughout its lifetime, mortality rates at different ages are not independent. Therefore, treating each 3-day interval as a separate character is not justified. Second, within any age interval, there are 29 comparisons (one for each accumulation line) to the control, and with a type I error rate of 0.05, we would expect one line to exhibit "significantly" different mortality by chance alone.
To alleviate these problems, a bootstrapping procedure (![]()
Mutation accumulation lines with confidence intervals that did not overlap those of the control in at least two age intervals were considered as having mutations affecting mortality. Although it controls for the dependence in mortality rates between ages and for the large number of comparisons at each age, the nonparametric determination of the confidence intervals makes this procedure quite conservative in detecting mortality differences.
To identify small mutational effects on mortality throughout life, log mortality rates, in 3-day intervals, were calculated for each of the four cages within each mutation accumulation line and for the 16 control line cages. Log mortality was then regressed on age using time as a nonlinear covariate. Thus, for each mutation accumulation line and for each sex, we used the following model:
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(7) |
j(i) (j = 1, ..., 4) is the residual error corresponding to the 3-day mortality rates in the ijth cage, and s(t) is the nonlinear covariate estimated using a lowess smooth (| RESULTS |
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Divergence of mortality rates:
Daily estimates of age-specific mortality rates, averaged over replicate cages, are presented for the 29 mutation accumulation lines and three control lines in Figure 1. Although mutational effects in individual lines are difficult to discern, some trends are clear. For both males and females, there is greater variation in mortality rates early in life than at older ages. In addition, female mortality curves in the accumulation lines are nearly all above the control lines early in life, suggesting that, on average, spontaneous mutations tend to increase mortality. This is not so in males because there are approximately the same number of accumulation lines with higher mortality as there are with lower mortality. Lastly, mortality rates for all lines in the experiment show a marked deceleration late in life, as evidenced by the departure from linearity in the mortality curves. Interestingly, the point of deceleration is nearly coincident with the reduction in variation; mortality curves converge at older ages.
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During the experiment, two of the 142 population cages exhibited high (>20 times the average) mortality beginning at eclosion. In large experiments like this, we often see a small number of anomalous cages. This is likely to be caused by local problems with the cage environment during set up. Such cages are treated as outliers and removed from the analysis. Also, on day 3 of the experiment, a technical problem with the fly medium caused abnormally high male mortality in 17 of the 142 cages (12%). High mortality was random with respect to genotype and position in the incubator, and cages experiencing this mortality increase did not show any lasting effects. Female mortality was unaffected.
Age-specific variance components:
The REML procedures require that the data be normally distributed, and after log transformation, weekly mortality rates are normally distributed (P > 0.10, Shapiro-Wilks test). In one case (females week 6) P = 0.01, but this is not significant after correcting for multiple hypothesis tests. The log transformation forced us to consider cages with zero deaths during any period as undefined, and omitting zero mortality cages early in life can introduce a significant bias in the estimation of genetic variance for mortality (![]()
The pattern of age-specific genetic variance in log mortality rates created by 19 generations of mutation accumulation is consistent with the pattern seen in Figure 1. Females show significant genetic variance in mortality over the first four weeks, after which there is a sudden decline in variance (Figure 2). Males show significant genetic variance in mortality from week 2 through week 4, after which genetic variance is not significantly different from zero (Figure 2). These ages remain significant after a Bonferroni correction for multiple tests (P < 0.007). Although we were unable to detect significant genetic variance for males in week 1, this is most likely caused by the large amount of random mortality that occurred on day 3 of the experiment.
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We used the pcrf1.p module of QUERCUS to compare variance in log mortality rates between two separate ages. All comparisons between genetic variance in mortality rates early in life (<5 wk) and late in life were corrected for 15 multiple comparisons using the Bonferroni criterion. For an
= 0.05, this requires a P < 0.003 to indicate significance. In females, the genetic variance for log mortality is significantly higher in the first 4 wk of life than it is in weeks 5 or 7. The pattern is less clear for comparisons involving week 6 mortality. The algorithm did not converge for comparisons between week 1 with week 6 and week 2 with week 6, and comparisons between week 3 and week 4 with week 6 were not significant after correction for multiple comparisons (P = 0.02 and P = 0.05, respectively). In males, genetic variance in mortality rates during weeks 2 and 3 is significantly greater than it is for weeks 57 (P < 0.003), and variance in week 3 is significantly greater than variance in week 4. There is some evidence for greater genetic variance in week 4 mortality over weeks 6 or 7, but these comparisons are not significant after correction for multiple comparisons (P = 0.04 and P = 0.009, respectively). There is weak evidence for an initial increase in genetic variance from week 1 to week 3 (P = 0.09).
Variance component analyses were also carried out on weekly control line mortality. Total phenotypic variation in mortality rates was partitioned into variance between the three control lines and error variance. In both males and females, there were no instances of significant between line variance in weekly mortality rates (P > 0.20; data not presented). In most cases, this estimate was essentially zero, i.e., <10-8. Error variance was qualitatively very similar to that estimated in the analysis of the mutation accumulation lines (data not presented). This provides evidence that the cryopreservation process did not induce random mutations that affect mortality rates, but it does not rule out the unlikely possibility that freezing results in very specific and consistent changes in the genome (also see ![]()
Age specificity of mutational effects:
The genetic cor- relation in weekly mortality rates generated by 19 generations of mutation accumulation is presented in Table 1. One striking feature of the correlation structure is the preponderance of positive correlations. For both males and females, mortality rates are highly correlated between ages separated by 1 or 2 wk. Although in both sexes there is strong evidence for a number of correlations being greater than zero, none remained significant after a Bonferroni correction for 21 multiple comparisons (see Table 1). Point estimates of the genetic correlation decline essentially monotonically, and they are not significantly different from zero (
= 0.05) between ages separated by
4 wk. Furthermore, maximum likelihood methods designed to test for a decline in genetic correlation over time in an age-dependent character provide strong evidence in favor of a significant decline in both sexes (PLETCHER and GEYER, manuscript in review). Although in females there is a small number of negative genetic correlations involving mortality in weeks 6 and 7, these are not significantly different from zero (P > 0.10). Assuming a normal distribution of random error, we expect half of the estimates not different from zero to give negative point estimates by chance. This expectation is borne out because five of nine estimates are negative in sign. The genetic variance from mutation in male mortality during week 5 was estimated to be zero; therefore, we are unable to estimate correlations involving this trait.
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The genetic correlation in log mortality rates between males and females of the same age were large and positive for early ages (Table 2). Correcting for multiple tests, significant positive correlations were detected in weeks 3 and 4 (P
0.01). Correlations in week 1 and week 2 were marginally significant (P < 0.10). Male mutational variance was estimated to be zero in week 5, precluding the calculation of between sex genetic correlations involving this trait.
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For each sex, each of the 29 mutation accumulation lines was examined for significant differences in mortality rates (3-day intervals) from the control lines. In females, 12 lines were identified as exhibiting significantly different mortality over at least two age intervals. Another two lines showed significantly different mortality in one age class. For males, six lines were identified with significantly different mortality in at least two age classes, and one additional line was found with different mortality at a single age. Six of these seven lines were also observed to have significant age-specific effects in females. A sample of mortality estimates relative to the control lines for females from eight lines and males from eight lines is presented in Figure 3. Most of the mutations affect mortality rates early in life, and a few show significant effects on middle ages. No lines were identified as showing mortality effects of mutation late in life, i.e., >37 days.
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Evidence for mutations that have effects on mortality throughout life were identified in both sexes (Table 3). Although there were no lines that remained significant after a strict Bonferroni correction for the 29 hypothesis tests in each sex, we would expect only 1.45 lines to show significance at the P = 0.05 level by chance alone. In females, eight accumulation lines showed mortality rates consistently higher than controls, and one showed lower mortality throughout life (at the P = 0.05 level). Of these eight lines, five were also identified as having significant effects at specific ages. The evidence is less convincing in males: Two lines were considered to have higher mortality than the controls throughout life, while one line showed a consistent decrease in mortality. The two lines with higher mortality also exhibited significant effects of mutation in at least two age intervals (Table 3).
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Mean mortality curves for the 29 mutation accumulation lines and the three control lines are presented in Figure 4. The prevalence of mutations increasing early-age mortality in females is reflected in the significantly greater average mortality for the accumulation lines. The mutational bias toward effects that increase mortality is not seen at older ages. In males, the nearly identical average mortality rates in the accumulation and control lines is evidence for a lack of mutational bias toward mutations that increase mortality at any age.
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Mortality models:
We find significant mutational variance for the baseline mortality parameter and the rate of increase parameter for the Gompertz model (Equation 5) for females only (Table 4). Males show no significant effects of mutations on these general mortality patterns. Although the Gompertz model is commonly used to analyze mortality data, it is clear that our data do not follow the log-linear Gompertz trajectory (Figure 1). The logistic frailty model (Equation 6), which predicts a deceleration of age-specific mortality late in life, produced a better fit (likelihood ratio test, P < 0.01) for males in 99/116 cages (85%) and for females in 92/116 cages (79%). Under this model, females have significant mutational variance in baseline mortality as well as in the rate of mortality increase with age (Table 4). There is marginal evidence for genetic variance in the s parameter of the logistic model (P = 0.08), suggesting that mutations are affecting the rate of deceleration of mortality at older ages. Males show no significant effects of mutations on any parameters of the logistic model.
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Mean longevity:
The amount of genetic variance for longevity created each generation by mutation is 0.30 for males and 0.46 for females (Table 5). The ratio Vm/Ve for female mean longevity is twice that of males (7.3 x 10-3 and 3.5 x 10-3, respectively), but both are of the order of magnitude commonly reported for quantitative traits (![]()
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| DISCUSSION |
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Significant genetic variance for weekly mortality rates caused by recent spontaneous mutations was observed in both males and females early in life (up to approximately day 30) but not at older ages. Mutational correlations are highly positive between mortality rates separated by a week or two, and they decline monotonically as the ages in question become further separated in time. Correlations are not significantly different from zero between ages separated by
4 wk. An examination of individual mutation accumulation lines provides strong evidence for a class of mutations with age-specific effects on mortality. Surprisingly, late-life mortality was essentially unaffected by the mutation accumulation procedure.
Age-specific properties of mutations
Our results provide strong evidence for the existence of a class of polygenic mutations that differ with respect to their effects on mortality at various ages. The consistent decline in genetic correlation between increasingly separated ages suggests that mutations with temporary effects are not uncommon and may even form the dominant component of the spectrum of mutations that affect mortality. Mutations affecting mortality throughout life were also identified, but their contribution to measured levels of mutational variance was small. All mutation accumulation lines exhibiting significant effects of mutation showed changes in mortality over a range of adjacent ages. We did not find evidence for the occurrence of "antagonistic" mutations with beneficial effects on mortality early in life and detrimental effects in later ages (or vice versa).
The pattern of age-specific mutational variance suggests the following: (1) loci that affect mortality at early ages are more likely to experience mutations, (2) a larger proportion of the genome influences mortality at earlier ages, and/or (3) the average effects of spontaneous mutations are larger in loci that influence mortality at early ages. It is interesting to note that our estimates of Vm/Ve for mean longevity are the same order of magnitude as those previously reported for longevity (![]()
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Many of the lines identified as showing age-specific effects of mutations have large effects in both sexes (Figure 3). This observation is consistent with the high positive mutational correlations between sexes (Table 2). In all cases, mutation-induced increases (decreases) in female mortality was mirrored by increases (decreases) in male mortality (Figure 3). The age range of effects seen in each line is roughly equivalent in both males and females, although the magnitude of the mortality changes generated by mutation is larger in females than males. This result is consistent with the larger estimates of mutational variance seen in females (Figure 2) and with previous reports of sex-specific genetic variance for mortality rates in laboratory populations of Drosophila (![]()
There is no evidence for antagonistically pleiotropic effects of mutations across sexes for mortality rates at the same age. At any particular age, there were no accumulation lines with a significant increase in mortality in one sex and decrease in the other, and all genetic correlations across sexes were positive (Table 2). It is possible that, for example, high male mortality early in life might generate relatively high female mortality at older ages because of the deleterious effects of reproductive activity on females (![]()
The deceleration and convergence of mortality rates at old ages in the mutation accumulation and the control lines is probably not caused by a decline in density as the flies age. It has been argued (![]()
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Mutations affecting genes with age-dependent patterns of expression provide a possible mechanism for generating the range of mutational effects in our data. Using enhancer trap-marked genes that express ß-galactosidase when transcriptionally active, ![]()
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The mutation accumulation lines used in this experiment were initiated from the Ives stock (![]()
In addition to mutational effects on mortality per se, we can envision four other factors that have the potential to influence our results. First, there may be a reproduction effect. Because we measured mortality on flies kept in mixed-sex cages, we cannot distinguish between mutations that affect mortality directly and those that influence mortality through costs of reproduction (![]()
Second, cryopreservation may have introduced unmeasured genetic changes. Our estimates of mutational effects rely on the cryopreserved control lines to accurately represent the genetic aspects of age-specific mortality in the base population before mutation accumulation was started. At the present time, there is no detailed information on the genetic consequences of freezing Drosophila embryos, but the lack of a significant "between-line" variance in control mortality is evidence against cryopreservation, causing random genetic changes (see also ![]()
Third, nongenetic and developmentally acquired variation may have age-specific effects. Environmentally induced variation in overall physiological quality can lead to age-specific changes in observed variance components and to departures from log-linear (Gompertz) mortality dynamics (![]()
Fourth, the reduced sample size at older ages may result in a loss of statistical power to detect significant effects of mutations. Although deaths in each cohort (population cage) result in a progressive reduction in sample size, the power to detect age-specific genetic variation in mortality rates depends onin addition to the number of flies alive at a particular agethe observed mortality rate at each age and the number of mortality observations at each age along with their genetic relationships. To investigate the effect of declining sample size and increasing mortality rate on our ability to detect significant genetic variance, we conducted a large number of computer simulations in which genetic variance was held constant for all ages and our ability to detect it was evaluated (details of the simulation algorithm are given in the APPENDIX). Three different combinations of between-line (genetic), between-replicate, and error variance were used in the simulations. A high-variance combination used values similar to those measured at early ages (Vl = 0.4, Vr = 0.2, and Ve = 0.3), a low-variance set of parameters was similar to values observed at late ages (Vl = 0.2, Vr = 0.1, and Ve = 0.2), and the third set consisted of a combination of the two (Vl = 0.2, Vr = 0.2, and Ve = 0.3). Simulated data sets were generated using the observed average female mortality rate at each age and the average number of females alive at each age (see APPENDIX). For each age and combination of variance parameters, 500 data sets were generated. Variance component estimation was then carried out on each set using the varcomp procedure in S-Plus (MathSoft). Asymptotic standard errors on the estimates allowed us to determine the fraction of simulated data sets that would have resulted in the detection of significant genetic variance. In addition, the average of the 500 estimates was obtained to determine if we would expect an age-related bias in the observed levels of genetic variance.
For all three combinations of variance parameters, the power to detect genetic variance in log mortality is greatest at intermediate ages, and it is essentially equivalent among ages 1 and 2 wk and age 6 wk (Figure 5). This is because the average mortality rate increases as the cohort ages, and as a result, smaller numbers of flies are required to obtain accurate estimates of mortality. Such is the case until week 7, when the average number of flies in each cage becomes too small (<20), and there are few cages with intermediate levels of mortality (many cages become extinct and must be treated as missing data). Genetic variance in log mortality tends to be underestimated at older ages (Figure 5). As a proportion of the actual parameter value, the average estimated genetic variance underestimates the actual variance by 24, 10, and 12% in the high, low, and combination simulations, respectively. A major source of bias in these cases is the generation of large mortality rates late in life (after week 4) causing many cages to become extinct; this is reflected in the larger underestimates in the high-variance simulations. In our actual data, there were no extinctions of cages before week 6, suggesting that this source of bias is minimal. In summary, the power to detect any fixed level of genetic variance does not decline dramatically in our experiment until week 7, and the age-related bias in the estimates of genetic variance, though considerable, are not sufficient to generate the magnitude of age-specific changes seen in our data. Therefore, it is not likely that the observed decline in genetic variance after week 4 is an artifact of declining sample size in each cohort.
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Mortality models: Mathematical models are useful for summarizing general differences in age-specific mortality between experimental cohorts. For example, under the Gompertz model, mortality variation can be divided into differences in baseline mortality and differences in the rate of aging. The logistic frailty model adds a third term describing the degree of mortality deceleration, and, therefore, differences in this aspect of age-specific mortality can be examined as well. The large mutational variance in mortality for female mortality rates early in life (Figure 2) likely accounts for the significant genetic variation detected in the baseline mortality parameter of the Gompertz and logistic models (Table 4). Since mortality rates converge at older ages, this variation would be translated into variation in the rate parameter of these models. Furthermore, despite strong evidence for age-specific effects of mutations in males (Figure 2), we fail to detect significant mutational variance for patterns of mortality under either of the models (Table 4). It is not clear how transient, age-specific changes in mortality rates would affect the parameters of these models. It is likely that age-specific "bumps" in the mortality curves would result in somewhat unstable estimates of model parameters. New mathematical mortality models that describe how mortality rates should change with age and that incorporate the possibility of age-specific changes in mortality are needed to further investigate different classes of mutational effects.
Mutation and life histories:
The primary motivation behind this study was to determine how spontaneous mutations affect mortality. Is there a class of mutations that affect mortality in only a subset of ages? What is the pleiotropic structure of mutations in regard to mortality rates at various ages? If mutations do have age-specific properties, are mutations with effects at one age as likely to occur as mutations with effects at another? These questions relate directly to current thought about the evolution of life history characters in general and the evolution of senescence in particular. Investigations into the genetics of age-specific characters are revealing inconsistencies between theoretical predictions and experimental data (![]()
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Relationship between mutation and levels of standing genetic variance:
The high mutational variance early in life and low mutational variance at older ages provide insight into the factors that influence estimates of variance in equilibrium populations. Observed levels of genetic variance and covariance in outbred laboratory populations of Drosophila result from an interaction between selection and mutation (![]()
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Additional information concerning the nature of standing levels of genetic variance for mortality can be obtained by examining a response to selection. A number of laboratories have selected for increased longevity in Drosophila and have observed a significant response within 1520 generations (![]()
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The evolution of senescence:
Currently, there are two predominant evolutionary models of senescencemutation accumulation (MA) and antagonistic pleiotropy (AP) (![]()
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The MA hypothesis requires that there be a substantial input into a population of deleterious mutations that affect mortality and that there is a class of such mutations whose effects are restricted to a narrow range of ages (![]()
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The AP theory predicts the existence of mutations with opposite effects on fitness at two separate ages. Our failure to observe mutations of this kind cannot be used to reject this hypothesis for two reasons. First, because mutations that increase early fitness at the expense of late can be extremely rare and still contribute to senescence (![]()
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Results from this study and others examining the genetics of age-specific life-history characters (![]()
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Future prospects in the evolutionary study of aging:
Ideally, we are interested in estimating the rates of mutation at loci with age-specific effects and the age-specific changes in mortality resulting from mutation at a single locus. Traditional inferences of minimum mutation rate and maximum gene effect using models of equal gene effect (![]()
Despite statistical difficulties, recent work investigating the genetic properties of age-specific characters has produced a number of exciting and unexpected results (![]()
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| ACKNOWLEDGMENTS |
|---|
We thank A. KHAZAELI, D. PROMISLOW, R. SHAW, M. SIMMONS, M. TATAR, and two anonymous reviewers for comments on the manuscript. Thanks especially to K. BROZ, C. GENDRON, G. KANTOR, and K. KELLY for their technical assistance and endless patience in counting dead flies. This work was supported by National Institutes of Health grants AG-0871 and AG-11722 to J.C.
Manuscript received June 9, 1997; Accepted for publication September 24, 1997.
| APPENDIX |
|---|
This appendix describes the procedure used to calculate the statistical power to detect levels of age-specific genetic variance in mortality rates. The general approach was based on a suggestion provided by D. PROMISLOW.
Each simulated data set was created according to a three-step procedure. First, log mortality rates were generated for each of the 116 cages (29 distinct mutation accumulation lines each with two genetic replicates and each genetic replicate represented by two population cages; see MATERIALS AND METHODS) according to specific levels of between-line (genetic), between-replicate, and error variance. Second, the probability of death for each individual in each population cage was calculated from the simulated mortality rates. Random numbers were drawn for each individual to determine if the individual died in the age interval in question, and the resulting "observed" mortality rates were calculated. Third, variance component analysis was carried out on the "observed" mortality rates to estimate the level of between-line variance and to determine if between-line variance was significant (P < 0.05).
Log mortality rates were simulated using the covariance matrix for all cages in the experiment. Because each of the 29 mutation accumulation lines was composed of two observations from each of two genetic replicates, the phenotypic covariance matrix of log mortality rates at any specific age can be written as follows:
![]() |
(A1) |
represents the Kronecker product,
2l reflects the genetic variance,
2r is the between-replicate variance, and
2e is the random environmental variance (see MATERIALS AND METHODS). The covariance matrix, P, is then decomposed, using the Cholesky factorization, into the lower triangular matrix L such that P = LLT, where T denotes matrix transposition. A sequence, x, of 116 independent, standard normal deviates is then generated, and the vector y is calculated according to y = Lx. The vector y now has covariance matrix Cov[y] = LCov[x]LT = LI116LT = P. For each age, the mean female mortality rate was added to each component of y to obtain the proper simulated mortality rates for that particular age. Values used to simulate data were obtained from the actual female data set. Three distinct levels of variance were used. The first, Vl = 0.4, Vr = 0.2, and Ve = 0.3, was based on the levels of variance estimated for mortality rates early in life. The second, Vl = 0.2, Vr = 0.1, and Ve = 0.2, represented variance estimated from older ages (> week 5), and the third, Vl = 0.2, Vr = 0.2, and Ve = 0.3, was a combination of the two. Mean log mortality rates used for generating mortality values at each age were -3.65, -3.27, -1.55, -0.371, 0.188, 0.385, and 0.673 for weeks 17, respectively.
Once log mortality rates were obtained for each cage (i) and for each age (x), the probability of death for individuals in that cage during that age interval, qxi, was calculated according to the following equation:
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(A2) |
Then, for each individual alive at the start of age x, a uniform random number was drawn. If the number was less than qxi, the individual was considered to have died during the age interval; otherwise, it survived. Thus, for each cage, we were able to calculate the "observed" mortality rate according to Equation 1. These values were then log transformed for the variance component analysis. The number of individuals assumed to be alive at the start of each age was calculated from the actual data as the average number of females alive in each cage at the start of each week. These values are 419, 402, 380, 285, 154, 60, and 16 for weeks 17, respectively.
Variance components were then estimated for each data set using the varcomp procedure in S-Plus (MATHSOFT). This procedure also provides the asymptotic variance of the estimates. To determine statistical power, each data set was considered to show significant genetic variation if the point estimate of between-line variance was larger than twice the standard deviation of the estimate (indicating that it was significantly >0 at the 0.05 level). For each age and combination of variance parameters, the fraction of data sets out of 500 that indicated significance was calculated and is reported as statistical power in Figure 5. In addition, for each set of 500 simulations, the mean estimate of between-line variance was calculated to investigate the potential for age-related bias in parameter estimation (Figure 5).
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