Ageing and survival after different doses of heat shock: the results of analysis of data from stress experiments with the nematode worm Caenorhabditis elegans

Mechanisms of Ageing and Development

122 (2001) 1477–1495

Ageing and survival after different doses ofheat shock: the results of analysis of data from

stress experiments with the nematode wormCaenorhabditis elegans

Anatoli I. Yashin b,*, James R. Cypser c,Thomas E. Johnson c, Anatoli I. Michalski d,

Sergei I. Boyko a, Vasili N. Novoseltsev d

a Max Planck Institute for Demographic Research, Doberaner Strasse 114, 18057, Rostock, Germanyb Center for Demographic Studies, Duke Uni�ersity, Box 90408 Durham, NC 27708-0408, USA

c Institute for Beha�ioral Genetics Campus Box 447, Boulder, CO 80309-0447, USAd Institute of Control Sciences, Russian Academy of Sciences, Profsoyuznaya 81, Moscow, Russia

Received 10 October 2000; received in revised form 8 January 2001; accepted 14 January 2001

Abstract

Stress experiments performed on a population of sterilised nematode worms (Caenorhabdi-tis elegans) show a clear hormesis effect after short exposure and clear debilitation effectsafter long exposure to heat shock. An intermediate duration of exposure results in a mixtureof these two effects. In this latter case the survival curves for populations in the stress andcontrol groups intersect. In this paper we develop an adaptation model of stress and applyit to the analysis of survival data from three such stress experiments. We show that the modelcan be used to explain empirical age-patterns of mortality and survival observed in theseexperiments. We discuss possible biological mechanisms involved in stress response anddirections for further research. © 2001 Elsevier Science Ireland Ltd. All rights reserved.

Keywords: Heat shock; Stress experiments; Nematode worm; Caenorhabditis elegans

www.elsevier.com/locate/mechagedev

* Corresponding author. Present address: Max Planck Institute for Demographic Research, Dober-aner Strasse 114, 18057, Rostock, Germany. Tel.: +49-381-2081106; fax: +49-381-2081169.

E-mail address: [email protected] (A.I. Yashin).

0047-6374/01/$ - see front matter © 2001 Elsevier Science Ireland Ltd. All rights reserved.

PII: S0047 -6374 (01 )00273 -1

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1. Introduction

Several recent experimental studies have evaluated the effects of stress on ageingand life span in laboratory organisms. These studies show that the survival of suchorganisms can be improved by exposure to small doses of stress. This property isreferred to as hormesis (Calabrese et al., 1992; Calabrese and Baldwin, 1998; Foran,1998; Johnson and Bruunsgaard, 1998). Hormesis has been observed in yeast(Jazwinski et al., 1998; Jazwinski, 1999), Drosophila (Maynard Smith, 1958;Khazaeli et al., 1997), nematodes worms (Caenorhabditis elegans) (Lithgow et al.,1994, 1995), and mice (Lorentz et al., 1955; Yakovlev et al., 1993). Murakami andJohnson (1996), Johnson (1997) discussed the role of genes in ageing and stressresistance. Selye (1974)and Regelson and Kalimi (1997) analysed effects of inducedstress resistance in humans. Sapolsky (1993a)and Sapolsky (1993b) analysed associ-ation between stress and neuroendocrine changes during ageing. Seeman et al.(1997a)and Seeman et al. (1997b) analysed the health consequences of adaptation tostress. McEwen (1998a) discussed the protective and damaging effects of stressmediators. Munck et al. (1984), Munck and Leung (1997)and McEwen (1998b)studied the role of glucocorticoids in stress and ageing. Sterling and Eyer (1981)andSterling and Eyer (1988) analysed biological basis of stress-related mortality.Paperiello (1998) discussed the role of hormesis in risk assessment and riskmanagement procedures. Calabrese et al. (1999) indicated the importance ofhormesis in establishing standards for environmental protection.

Despite numerous confirmations of the presence of hormesis in different experi-mental studies its mechanism is still not well understood. Several attempts todevelop a mathematical model of hormesis were performed. Yakovlev et al. (1993)formulated a model of radiation hormesis and applied it to survival data on mice.In this model the effect of hormesis was explained by the fact that radiation can notonly activate hidden defence mechanisms but also kill damaged cells.

Michalski et al. (2001) extended the Strehler and Mildvan (1960) model ofmortality to study the hormesis effect observed in stress experiments with thenematode worm C. elegans. In this model the ageing of organisms is characterisedby a linearly declining capacity to produce energy in response to the stresses of life.The death of an organism occurs when the amount of energy required to restore thefunctioning of the organism after stress exceeds the capacity of the organism. It wasshown that, in the case of hormesis, an organism increases its capacity to producean adequate response to further energy demands associated with the expectedstresses of life. This effect manifests itself as a decrease in the rate of decline of theadaptive capacity (vitality) index or as a weakening of the deleterious effects ofstress, which is probably due to the activation of reserved defence or protectivemechanisms.

Yashin et al. (2001) suggested a model of stress response in heterogeneouspopulations. This model is capable of reproducing all observed effects of stress,including hormesis, debilitation, and the mixture of hormesis and debilitationeffects (incomplete hormesis). This model was applied to the analysis of survivaldata from stress experiments with the nematode worm C. elegans. It was shown

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that an exposure to stress at the beginning of a worm’s adult life changes the initialdiscrete distribution of heterogeneity, i.e. the proportions of worms in the frail,normal, and robust sub-cohorts. These changes depend on the duration of exposureto heat shock. In the case of hormesis, these changes can be interpreted as anadaptation of the organism due to the activation of available reserves.

Although the hormesis effects were related to the adaptation properties of anorganism in many studies, the respective mechanisms of adaptation were notdescribed explicitly. In this paper, we formulate a model of stress that provides uswith such a description. We base our model on a simplified version of the dynamicmodel of mortality and ageing (Yashin and Manton, 1997). This model was usedsuccessfully in the analysis of human data from longitudinal studies of ageing inwhich information about physiological and biological age-related changes in indi-vidual organisms have to be combined with survival data. Such a model allows usto explain the effects on survival observed in stress studies of nematodes by changesin adaptive capacity influenced by stress. In analogy to human ageing, we assumethe presence of a hidden physiological process within the worm, which influences itssurvival chances in accordance with the quadratic hazard model (Yashin andManton, 1997). The model can easily be extended to the case when the data on agetrajectories of such indices are available.

2. Materials and methods

Data. Worms TJ1060 (spe-9; fer-15) were raised on solid medium (NGM platesprespotted with E. coli ) at 25.5°C for 3 days, at which temperature they developinto sterile but otherwise phenotypically wild-type adults. At 3 days of age, wormswere divided into 11 groups and exposed to heat shock at 35°C for periods of0, 1, 2, 4, 6, 8, 10, 12, 16, and 24 h (synchronous start, asynchronous stops). Imme-diately following the heat shock, the worms were permitted to recover for 24 h at20°C. They were then transferred to liquid survival medium and maintained at 20°Con prespotted NGM for the remainder of the experiment. Beginning at the 5th dayof life, the number of worms that had died was counted daily for all groups. Nosurvivors were observed after 16 and 24 h of heat shock. Two other experimentswere performed with the same strain of worms, at two different times, to replicatethe results of the first experiment. In experiment c2, sterilised worms were dividedinto nine groups and exposed to heat shock at 35°C for periods of0, 0.5, 1, 2, 3, 4, 6, 8 and 10 h. In experiment c3 worms were divided into 10groups and exposed to heat shock at 35°C for periods of 1, 2, 3, 4, 5, 6, 7, and 8 h.

Adaptation model: We assume that the chances of death of a worm at age xdepend on the value of some physiological index Yx, so the mortality rate �(x, Yx)conditional on this index is

�(x, Yx)=Yx2ebx. (1)

We assume that Yx is a solution of the following differential equation


Y� x=a0−a1Yx, Y0, (2)

where a0�0 and a1�1 are parameters and Y0 is an initial condition. This equationdescribes the behaviour of a linear system with a negative feedback. In Yashin andManton (1997) stochastic version of (Eq. (2)) has been used. Adding stochasticityinto the model makes it more appropriate for the description of individualdifferences in physiological dynamics. This is especially important when individualobservations of physiological processes are available at certain points in time (as inlongitudinal or follow-up studies, for example). For the purposes of this paper thedeterministic version of the model is sufficient since it captures the adaptiveproperties of the process using ordinary differential equation with negative feed-back. The strength of a feedback mechanism is characterised by the value offeedback coefficient a1.

We assume that (Eq. (2)) characterises the homeostatic system involved in theresponse of an organism to stress, where the latter disturbs an initial value, Y0. So,Y0 is interpreted as damage produced by an incidence of stress. Note that thestationary solution of this equation is

Y�=a0

a1

.

When Y0=Y�, Yx=Y0 (Eq. (2)) describes the Gompertz mortality rate. WhenY0�Y�, Yx�Y�, x�0 the mortality rate differs from the Gompertz curve. Weassume that an exposure to stress at the beginning of life increases Y0�Y�. Whenthe duration of heat shock increases, Y0 increases as well. The trajectory of Yx

shows how the homeostatic system of an organism returns to its steady state.

3. Results of statistical analysis

First, we performed an empirical analysis of the data and tested the hypothesisabout the presence of hormesis and debilitation effects in the series of stress-exper-iments. Then we fit the adaptation model to the data. Then we tested thehypotheses about the presence of hormesis and debilitation effects using theadaptation model.

Empirical analysis of data: In the empirical analysis of experimental stress datafor nematode worms, three distinct patterns of survival curve emerged. The firstcorresponds to the hormesis effect. Here the survival curve is higher than that in thecontrol group for all ages. Heat shocks with durations of 1 and 2 h produce thispattern of survival. The second pattern characterises a population exposed to 4 h ofheat shock. The survival function intersects that of the control group. It is lowerthan in the control group earlier in life and higher later in life. We call this pattern‘incomplete hormesis’. The third pattern characterises a debilitation effect. Here thepopulation is exposed to 6 or more hours of heat shock. The survival function inthis population is lower than that of the control group for all ages. Fig. 1 (left-handpanel) shows three-dimensional plots of survival functions in the three experiments.

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Fig. 1. Three-dimensional plots of smoothed empirical survival functions considered as function of age and the duration of heat shock for experiments c1, 2and 3 (left panel). The trajectories of Yt in the adaptation model calculated from survival data on populations of nematode worms C. elegans exposed toheat shock of different duration in experiments c1, 2, and 3 (right panel).


Fig

.1.

(Con

tinu

ed)


Fig

.1.

(Con

tinu

ed)


One can see that the effect of survival improvement in the case of hormesis (after1 and 2 h of heat shock) becomes gradually replaced by the incomplete hormesis (afterabout 3–4 h of heat shock) and then by debilitation (after 6 h of heat shock). Thethree-dimensional plots allow us to predict the results of stress experiments withintermediate durations of heat shocks. Fig. 2 (left-hand panels) shows empiricalestimates of survival functions in three experiments together with estimates of survivalfunctions obtained using an adaptation model (right-hand panels).

All patterns observed in experiments can be explained using an adaptation model,where low and moderate levels of stress activate a powerful adaptation mechanismthat increases the feedback coefficient and decreases the time of recovery. Thestationary level of the physiological process after stress also decreases. The higherdoses of stress produce substantial damage. They increase recovery time significantly,and they may increase the stationary level of physiological process as well. Table 1summarises the results of our empirical analysis.

One can see from this table that the experimental data on survival in theexperimental groups subjected to 1 and 2 h of heating in experiment c1 do notcontradict the hypothesis about the presence of an hormesis effect. The data onsurvival in the groups subjected to 6 or more hours of heating in all three experimentsdo not contradict the hypothesis about the presence of debilitation effects.

We estimated parameters of the model for the data from each stress experimentusing the maximum likelihood method. The empirical conditional probabilities ofdeath on the jth day of age, given survival to age j−1 days, were estimated usingrelationships

mj=dj

nj−1

. (3)

Here dj is the number of deaths observed during the jth day of life, and nj−1 isthe number of worms alive at the end of the previous day.

Testing the goodness of fit. We tested the goodness of fit using the likelihood ratiotest, where the likelihood of data corresponding to the model parameters wascompared to the likelihood associated with the ‘saturated’ model represented byempirical estimates of survival functions. The large P-values (more than 0.47 and 0.3in the first, and the third experiments, respectively) indicate an appropriate fit of themodel to the data from these experiments. The fit is not so good for the secondexperiment: the P-value is about 0.01. This model allows for testing hypotheses aboutthe presence of hormesis and debilitation effects in the after-stress mortality inexperimental populations using the likelihood ratio test. As in the case of empiricalanalysis this test confirms the presence of hormesis and debilitation effects withP-values �0.001, which again confirms the fact that the model captures the mainproperties of the phenomenon. Fig. 3 shows how the adaptation model fits survivaldata from experiments c3 Table 2.

The graphs associated with experiments c1 and 2 look similar and for this reasonare not shown. One can see that a very simple adaptation model can capture generalpatterns of mortality and survival in stress groups exposed to different duration ofheat shock.


Fig. 2. Empirical survival functions estimated from survival data on populations of nematode worms C.elegans exposed to 0, 1, 4 and 6 h of heat shock in experiments c1 and 3, and to 0, 1, 3 and 6 h of heatshock in experiment c2 (left panel). Survival functions estimated from survival data on populations ofnematode worms C. elegans exposed to 0, 1, 4 and 6 h of heat shock in experiments c1 and 3, and to0, 1, 3 and 6 h of heat shock in experiment c2 using adaptation model (right panel).

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Table 1Survival statistics for the data obtained in stress experiments c1, 2, and 3 with populations of the nematode worm C. elegans with varying durations ofheat shock

0 3 4 5 6 7 8 10 120.5 1Hours 2

14.6 – 6.8 – 4.2LEc1 1.816.6 0.8– 18.2 17.6 –15.8 5.1 3.6 0.918.0720.0LEc2 18.5 20.822.0

21.320.9 18.5 10.9 5.9 3.6 3.0 3.024.5 24.8 24.92LEc3

0.4 0.6 0.4 0.2 0.1 0.04SEc1 0.60.40.7 0.3 0.2 0.081.00.5 0.5SEc2 0.4 0.4

0.70.6 0.8 0.7 0.3 0.1 0.08 0.080.6 0.7 0.7SEc3137 133 164 152 198 178100 152Sizec1

164 200 200 19990Sizec2 139189186174187Sizec3 210174 209 210 242 242197 188 180 193

0.99 – 1 – 10.05 1P-value Ic1 1–0.01––– 0.42 1 – 1 – 1 1 –0.006 0P-value Ic2 0

0.49 1 1 1 1 1P-value Ic3 –– 0 0 0 0.33

0 – 0 – 00.94 0P-value IIc1 0–0.99––– 0.58 0.001 – 0 – 0 0 –0.99 1P-value IIc2 1

1 0.67 0.51 0 0 0 0 0 –P-value IIc3 – 1 1

0.53 – 0 – 00.004 0– 00 –P-value IIIc1 –– 0.008 0.91 – 0 0 0 0 –0.025 0P-value IIIc2 0

0 0.51 0.44 0 0 0 0 0 –P-value IIIc3 – 0.001 0

The rows ‘P-values I’ show P-values calculated by t-test statistics for the hypothesis that heat exposure does not increase life expectancy (H0: no hormesis).The rows with ‘P-values II’ show P-values calculated by t-test statistics for the hypothesis that heat exposure does not decrease life expectancy (H0: nodebilitation). The rows ‘P-values III’show P-values calculated by log-rank test statistics for the hypothesis that heat exposure does not change the survivalcurve.


Fig. 3. Empirical (dots) and estimated using adaptation model (solid lines) survival functions fornematode worms C. elegans obtained in stress experiments c3 with 0, 1, 4 and 6 h of heat shock.


How does the adaptation mechanism work. In the case of hormesis arising from 1h of heat shock, we assume that stress can activate physiological and biologicaladaptation mechanisms, which reverse damage, repair tissue, and furnish anorganism with additional defence mechanisms and preventive capacity. In ourmodel the efficiency of these mechanisms is characterised by the value of thenegative feedback coefficient, a1. The activation of adaptive capacities correspondsto an increase in the absolute value of a1. When the level of stress is small, thedeviation of Y0 from its value under ‘normal conditions’ is also small. However, inthe case of hormesis, an adaptive response to a small amount of stress, i.e. thechange in the feedback coefficient a1, may be substantial. An organism may ‘regard’a low level of stress as a signal that a greater disturbance may also occur soon. Itmobilises energy and other resources to produce an adequate preventive response.The level of mobilisation of adaptive mechanisms may also be reflected in the valueof b in (Eq. (1)). When adaptation mechanisms become activated this value may bereduced. In our case, however, all effects were explained by the changes in thevalues of a1 (see Table 2).

When a1 increases to a1* the stationary value of Yx decreases. Hence, afterrecovery, the mortality rate in the disturbed system becomes lower than in thecontrol group. In the case of hormesis, the change in Y0 is small, but the stationarylevel Y�=a0/a1* is lower than in the control group. The convergence of Yx to thislevel occurs faster than in the group without stress. Thus the resulting mortalityrate is lower and the respective survival function is higher than in the control group(hormesis).

Table 2The values of the parameters of the adaptation model estimated from survival data on populations ofnematode worms C. elegans obtained in experiments c1, 2 and 3

Y0 a0/a1Duration of heat stress (h) ba1a0

0.020 1.000 0.303 0.0200 0.0200.0201 1.450 0.303 0.0140.030

Exp c1 0.0202 1.250 0.303 0.0160.0700.0180.3031.1000.0200.1804

0.0206 0.400 0.303 0.0500.250

0 0.030 0.030 1.000 0.210 0.0300.2101.3000.030 0.0230.0901

2 0.110 0.030Exp c2 1.280 0.210 0.0230.290 0.030 1.100 0.210 0.0273

6 0.340 0.030 0.340 0.210 0.088

0 0.075 0.075 1.000 0.115 0.0751 0.0440.1151.7000.0750.095

0.115 0.075 1.2902 0.115 0.058Exp c30.250 0.0750.1151.0004 0.075

6 0.500 0.880 0.0850.1150.075


When the level of stress increases, the value of Y0, which characterises the levelof damage in the system of an organism, increases as well. The adaptive capacity,however, becomes exhausted, so a1* cannot increase any further. A higher level ofdamage corresponds to higher values of initial stress, Y0. A high degree of deviationin the initial value of Yx from its steady-state level increases the time it takes toapproach this level by the process Yx. So the resulting mortality rate immediatelyafter exposure to stress remains higher than in the control group for a considerableamount of time, and the chances of debilitating effects occurring earlier in lifeincrease. If the value of Y0 is still not too high, an organism may be capable ofrecovering, in which case it will have a higher chance of survival at later ages as incase with 4 h of heating in experiments c1 and 3 and 3 h of heating in case ofexperiment c2.

When the damage is too large, an organism is not able to approach a stationarylevel of Yx fast enough, so mortality remains higher than in the control group fora longer period, and the respective survival function remains lower than in thecontrol group during the entire life course (as is the case with 6 h of heat shock inall three experiments). Higher doses of stress can also damage the adaptivemechanisms. As a result adaptive capacity a1 decreases, the stationary value of Yx

increases. The mortality rate in the stress group exceeds that of the control group,and the respective survival curve is lower than that of the control group.

4. Discussion

Fisher’s Fundamental Theorem of Natural Selection establishes the relationshipbetween changes in average fitness and the additive component of genetic varianceof this fitness. However, in the derivation of this theorem Fisher (1958) did notinclude the influence of environmental changes on the dynamics of average fitness.This generated intense debate about the conditions of validity of this theorem andthe assumptions used in its derivation (Price, 1972; Frank and Slatkin, 1992;Lessard, 1997). Kimura (1958) overcame this limitation by adding new terms to theright side of the equation for average fitness. These terms characterise the contribu-tion of external factors to changes in fitness. Thus the new equation emphasises theimportance of both external factors and genes in contributing to fitness and otherlife history traits. To use the result of Kimura (1958) for testing evolutionaryhypotheses or to describe properties of individual adaptation, the mechanisms ofgenetic and environmental influence on adaptive traits must be specified. Experi-mental studies of stress play an important role in clarifying this mechanism(Hoffmann and Parsons, 1991; Parsons, 1996).

In this paper we used a mathematical model of adaptation of the individualorganism to stress to explain different patterns of survival in the nematode worm C.elegans after exposure to heat shock of varying durations. Three main patterns ofsurvival observed in the experiment were discussed. They include hormesis, incom-plete hormesis and debilitation effects. In the case of hormesis, survival in the stressgroup is better than in the control group. In the case of incomplete hormesis the


initial part of the survival curve in the stress group is lower and the remainder ishigher than that in the control group. In the case of debilitation, survival in thecontrol group is higher than that in the stress group. An important result of thepaper is that all three effects may be explained using the same adaptation model.This does not exclude other explanations based on different hypotheses. Moreoverthe search for such explanations is necessary for development of furtherexperiments.

In this paper we pay special attention to the effect of incomplete hormesis. Thiseffect is intermediate between hormesis and debilitation. It is manifested in theintersection of survival curves for populations in the stress and control groups. Itshows that survival curve does not necessarily change symmetrically when stressload increases. A review of the literature reveals that the intersection of survivalcurves during ageing is not a new phenomenon. It has often observed in the effectsof caloric restriction on longevity (Masoro and Austad, 1996; Masoro, 1998). Suchan effect was obtained in survival experiments with mice exposed to different dosesof chemicals (Anisimov, 2000). The intersection of mortality and survival curves forpopulations of human individuals carrying different genes or genotypes was foundin the analysis of genetic markers data in centenarian studies (Yashin et al., 1999).Despite these findings the mechanism of this phenomenon was not systematicallystudied.

What is going on in a living cell during and after the heat shock? When the heatshock is mild, cells acquire resistance to subsequent stress that would normally belethal. This induced stress resistance (also called acquired stress tolerance) is oftenassociated with the induction of heat shock proteins. In Saccharomices cere�isiae,acquired thermo-tolerance is associated with the induction of HSP104 (Sanchez andLindquist, 1990; Hottiger et al., 1994). Another factor of potential importance forthermo-tolerance in yeast involves threhalose (Hottiger et al., 1994; Tamura et al.,1998). More recent studies suggest the involvement of mechanisms other than heatshock protein production in the regulation of stress tolerance. For example, heatshock proteins HSP16 and HSP70 are produced by Drosophila exposed to heat(Dahlgaard et al., 1998). After mild exposure an increased resistance to heat and toother stresses is observed. However, the times of the highest concentration of theheat shock proteins in the tissues and the maximum thermal resistance (i.e. survivalat 39°C for 85 min) do not coincide. The maximum concentration of HSP70 isobserved 2 h after the exposure to heat (37°C for 55 min). The maximum stressresistance is reached between 8 and 30 h after exposure to stress. The level ofHSP70 (Gabai et al., 1997) in males exceeded that of females during the entireperiod. However, the survival of females exceeded that of males in almost all cases(Dahlgaard et al., 1998). Thus, there must be some additional mechanisms ofinduced stress resistance that involve other biological components contributing tosurvival and longevity. Koehn and Bayne (1989) analysed the genetic and physio-logical aspects of energy production in stress response. Calow (1989) discussed thedifference in responses to stress in biological systems.

High doses (duration and/or magnitude) of heat shock produce a debilitatingeffect. The biological manifestation of debilitation involves the denaturation of


proteins and damage to DNA and other bio-molecules. This disturbs properfunctioning of an organism and damages or destroys its adaptive capacities. As aresult the life span of the organism is reduced. At the population level this effectis manifested as a lower survival curve for the stress group. Debilitation activatesthe production of heat shock proteins and DNA repair enzymes in cells thatsurvive the heat shock. This may partly be responsible for long tails in survivalfunctions observed in populations exposed to debilitative stress.

The intermediate levels of stress (such as 4 h exposure to heat shock inexperiment c1) may result in severe damage to certain systems, which decreasesthe survival chances of an organism during some period of time following heatshock. However, survival may improve later on if the adaptive mechanisms areable to repair the damage, replace destroyed elements and restore the functioningof critical systems. The chances of survival may become higher compared to thecontrol group because of improved protection and enhanced defence mechanismsin those who survived the stress.

An organism’s response to stress is an adaptive reaction. The adaptation in-volves all levels: the whole organism, distinct organs, individual cells, and molec-ular mechanisms (e.g. DNA repair) all protect the functioning of the entireorganism as a reproductive machine. High doses of stress induce mechanisms atthe evolutionary level of adaptation. The recombination frequency and the num-ber of mutations rise with an increase in the severity of stress (Parsons, 1988,1989; Hoffmann and Parsons, 1991, 1993; Parsons, 1995, 1996). At these levels,it is the genetic variability of the progeny that provides the best opportunity forcontinuity of the species. The phenotypic variability may also increase whenstress is applied at certain critical developmental stages (Hoffmann and Parsons,1991).

The dynamic model of adaptation to stress described above shows that thefunctioning of individual biological feedback mechanisms activated by stress maymake substantial contributions to the age patterns of mortality and survival inan exposed population of organisms. Despite the fact that the mortality model(Eqs. (1) and (2)) seems to be too simple to provide an exhaustive explanationof the biological processes developing in an organism during stress, it helps us toelucidate three important conclusions associated with the effects. One is that thehormesis effect is the result of an increased adaptation of an organism activatedby stress-stimuli. The second is that certain combinations of the destructiveeffects of stress and activation of the protective and adaptive mechanisms mayproduce the intersection of the survival curves of the stressed and control popu-lations. The third is that the debilitation effect results from an increase in theamount of damage and a decreased capacity to adapt. This reduction increasesthe vulnerability of an organism to environmental challenges.

The deterministic model used in this paper can easily be extended to astochastic model using the ideas described in Yashin and Manton (1997). Thismodel can also be used in the analysis of follow up data, i.e. data where themeasurements of trajectories of individual biological characteristics are combined


with the measurements of survival times. In particular, measurements of thedynamic changes in metabolic rate and in the activity of antioxidants with age,the accumulation of the toxic by-products of metabolism, and damage associatedwith the activity of free radicals could help us to better understand the regulari-ties of the ageing process. The measurement of the expression of genes responsi-ble for the production of stress proteins might provide insight into the nature ofadaptation mechanisms producing hormetic and debilitative effects.

Several theoretical concepts and models have attempted to relate observedfacts about stress, metabolic activity, and ageing. Sohal and Weindruch (1996)suggested the ‘oxidative stress theory of ageing’, which supplements the rate ofliving theory by emphasising the role of free radicals in the ageing process.Parsons (1995) unified earlier approaches in the stress theory of ageing. None ofthese theories, however, was formulated in terms of mathematical models thatwould be capable of predicting and explaining the results of specific stress exper-iments. Michalski et al. (2000); Novoseltsev et al. (2000, 2001) explain severalcontroversial results of experimental studies of ageing using methods of mathe-matical modelling and statistical analysis. Their studies show that combining themethods and ideas of survival analysis and statistical modelling, with theoreticalconcepts on the biology of ageing, opens a new avenue in the analysis andinterpretation of data obtained in experimental studies of ageing and life span.

Acknowledgements

The work on this paper was partly supported by NIH/NIA grant PO1AG08761-01. The authors thank anonymous reviewers for valuable comments.

Appendix A

The likelihood function. We have used the maximum likelihood approach toobtain the estimates of the model parameters. Let qj

h denote the probability ofdeath on the jth day among worms exposed to heat for h hours who survivedj−1 days after heat exposure. Then the log likelihood function can be expressedin the form of

Log Lik=�h

�j

(mjhlnqj

h+ (njh−mj

h) ln (1−qjh))+c.

Here mjh is the number of death cases observed on the jth day of the experiment

with h hours heat exposure, njh is the number of such worms that survived j−1

days after exposure, and c is a constant term containing no unknown parame-ters. The values qj

h are related to hazard rates by the relationships qjh=1−

(S� h( j+1))/(S� h( j )).


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Ageing and survival after different doses of heat shock: the results of analysis of data from stress experiments with the nematode worm Caenorhabditis elegans