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Experimental Gerontology 37 (2001) 57±66 www.elsevier.com/locate/expgero Hormesis and debilitation effects in stress experiments using the nematode worm Caenorhabditis elegans: the model of balance between cell damage and HSP levels Alexander Butov a, Thomas Johnson b, James Cypser b, Igor Sannikov a, Maxim Volkov a, Mary Sehl c, Anatoli Yashin d,* a Ulyanovsk State University, Ulyanovsk, Russia Institute for Behavioural Genetics, University of Colorado at Boulder, CO, USA c Brown University, USA d Max Planck Institute for Demographic Research, 114 Doberaner Str., 18057 Rostock, Germany b Received 9 April 2001; received in revised form 25 July 2001; accepted 20 August 2001 Abstract In this article, we discuss mechanisms responsible for the effects of heat treatment on increasing subsequent survival in the nematode worm Caenorhabditis elegans. We assume that the balance between damage associated with exposure to thermal stress and the level of heat shock proteins produced plays a key role in forming the age-pattern of mortality and survival in stress experiments. We propose a stochastic model of stress, which describes the accumulation of damage in the cells of the worm as the worm ages. The model replicates the age trajectories of experimental survival curves in three experiments in which worms were heat-treated for 0, 1, 2, 4, 6, or 8 h. We also discuss analytical results and directions of further research. The proposed method of stochastic modelling of survival data provides a new approach that can be used to model, analyse and extrapolate experimental results. q 2001 Elsevier Science Inc. All rights reserved. Keywords: C. elegans; Hormesis; Heat shock proteins; Longevity 1. Introduction The relationship between thermotolerance and longevity in Caenorhabditis elegans is well known (Lithgow et al., 1995; Walker et al., 1998; Michalski et al., 2000). It has also been demonstrated that heat shock proteins are induced in response to a variety of stressors in C. elegans, including heat (Jones and Candido, 1999), oxidative stress (Yanase et al., * Corresponding author. Tel.: 149-381-208-11-06; fax: 149381-208-11-69. E-mail address: yashin@demogr.mpg.de (A. Yashin). 1999; Link et al., 1999), electromagnetic ®elds (Junkersdorf et al., 2000), and immunological stress (Nowell et al., 1999). Heat shock proteins function during cell stress as molecular chaperones, interacting with diverse protein substrates to assist in repairing damaged proteins, by refolding, or in degrading them, thereby restoring protein homeostasis and promoting cell survival (Jolly and Morimoto, 2000; Frydman and Harti, 1994). Hyperthermia shifts cells into a state of oxidative stress (Finkel and Holbrook, 2000), and the synthesis of stress proteins is modulated by antioxidant status (Peng et al., 2000). The role of oxidative damage and 0531-5565/01/$ - see front matter q 2001 Elsevier Science Inc. All rights reserved. PII: S 0531-556 5(01)00161-9 58 A. Butov et al. / Experimental Gerontology 37 (2001) 57±66 antioxidant defence in the aging process has been intensely studied (Harman, 1957; Sohal et al., 1995; Cutler 1991; Orr and Sohal, 1994; Sohal and Weindruch, 1996). Increased resistance to oxidative stress has been shown to be associated with increased longevity in C. elegans, perhaps mediated by higher levels of the antioxidant enzymes superoxide dismutase and catalase (Larsen, 1993; Van¯eteren, 1993). In contrast, low doses of stress have been shown to slow aging and prolong life span for a variety of species and stresses: hypergravity in Drosophila melanogaster (Le Bourg and Minois, 1999); radiation in C. elegans (Johnson and Hartman, 1988); and heat stress in Drosophila (Khazaeli et al., 1997; Kurapti, 2000), yeast (Shama et al., 1998), and C. elegans (Michalski et al., 2000), to name a few. The mechanism underlying this increased longevity is thought to be associated with the protective chaperone function of heat shock proteins induced during the exposure to stress. These proteins are thought to provide a longlived protective function, persisting after the exposure to stress and functioning to repair damage in addition to those caused by the stressor itself (Kurapti et al., 2000). In this paper, we propose a mathematical model describing a stressor such as heat shock in the nematode worm C. elegans. We assume that each unit of stress load produces equivalent intracellular damage in the worm. The same stress also stimulates the production of heat-shock proteins, which tend to reduce cellular damage. Imbalance between the damage production and its removal, results in damage accumulation shortening the life span of the worm, as is observed in populations of worms exposed to different durations of heat shock. We assume the presence of some (random) initial damage in the cell. When the stress is small, the amount of damage produced by it is also small. In our model, the production of heat shock proteins is not strictly proportional to the amount of damage. After a small stress, heat shock proteins (HSPs) are over-produced; these HSPs not only remove damage produced by the stress, but also remove part of the initial preexisting damage. This improves function of the entire organism resulting in increased life span. This scenario corresponds to the hormesis effect observed in populations of worms exposed to two or less hours of heat shock (Michalski et al., 2000; Yashin et al., 2001). Long exposure to heat produces substantial damage, which cannot be entirely repaired, even by the higher levels of HSPs produced during this shock. This amount, however, is not enough to completely compensate for the damage produced by the stress, so the functioning of the cells deteriorates, and life span decreases. This is observed in the populations exposed to six or more hours of heat shock. Intermediate duration of heat results in a delay in restoration of function by HSPs. As a result, the survival probability is lower earlier in life and higher later in life when the damage associated with the stress as well as part of the initial damage are eliminated. This is the scenario in a population exposed to 4 h of heat stress. Exciting methods involving the use of DNA-chip microarrays are underway to study kinetics of HSP mRNA synthesis, which can be used in conjunction with these models to examine the dynamics of HSP gene expression and the mechanisms underlying the increase in longevity seen in organisms exposed to low levels of thermal stress. 2. Materials and methods 2.1. Experimental data Worms TJ1060 (spe-9; fer-15) were raised on NGM plates, prespotted with E. coli, at 25.58C for 3 days at which time they had developed into sterile, but otherwise phenotypically wild-type adults. Then, populations were divided into 11 groups and exposed to heat shocks at 358C for periods of 0, 1, 2, 4, 6, 8, 10, 12, 16, or 24 h (synchronous start, asynchronous stops). Immediately following the longest heat shock, the animals were permitted to recover for an additional 24 h at 208C. They were then transferred to liquid survival medium and maintained at 208C for the remainder of the experiment. Beginning with the ®fth day of life, the number of live and dead worms was counted daily for all groups. No survivors were observed after 16 and 24 h of heat shock. Two other experiments were performed with the same strains of worms, at two different times, to replicate the results of experiment 1. In experiment 2, sterilized worms were divided into nine groups and exposed to heat shocks at 358C for periods of 0, 0.5, 1, 2, 3, 4, 6, 8 or 10 h. In experiment 3, worms were divided into ten A. Butov et al. / Experimental Gerontology 37 (2001) 57±66 groups and exposed to heat shock at 358C for periods of 1, 2, 3, 4, 5, 6, 7, or 8 h. Here, we consider only those results in which worms were shocked for 0, 1, 2, 4, 6 or 8 h. 2.2. Modeling assumptions We introduce a stochastic model describing the response to heat shock. We assume that the response to stress can be described by two characteristic components: processes A1 and A2 . The A1 component represents the accumulated damage resulting from a given thermal shock. Such damage may result from increased oxidative stress following exposure to thermal shock (Finkel and Holbrook, 2000). We assume that the changes in A1 are determined by t the rate of damage D1 associated with the level and t duration of thermal shock, the activity of repair mechanisms associated with the production of HSPs ht and the random component which magnitude is proportional to the accumulated level of damage A1 . The effect of HSPs is proportional to t the level of damage A1 . This allows us to describe t component A1 using the following stochastic differential equation dA1 D1 dt 2 A1 H 1 ht dt 1 s 1 A1 dWt1 ; A1 . 0: t t t t 0 1 Here, H 1 and s 1 are parameters of the model, and Wt1 is a standard Wiener process. The A2 component represents the innate accumulation of oxidative damage, and other forms of damage resulting from various processes required for life. We suppose that the changes in A2 are determined by the rate of t damage D2 associated with base damage level (other than that induced by thermal stress), the activity of repair mechanisms associated with the production of HSPs ht and the random component, the magnitude of which is proportional to the accumulated level of damage A2 . The effect of HSPs is proportional to the t level of damage A2 . This allows us to describe compot nent A2 using the following stochastic differential equation dA2 D2 dt 2 A2 H 2 ht dt 1 s 2 A2 dWt2 ; A2 . 0: t t t 0 2 Here, H 2 and s 2 are parameters of the model, and Wt2 is a standard Wiener process. We assume that the A2 type of damage, although not induced by the heat shock, can be repaired by the same processes that 59 are induced by the heat stress. Note that the model can be created and almost properly adjusted without the A2 component. However, the presence of the stable phenomenon of `heavy tails' in the survival curves after 2 or 4 h heating (and respective crossing of the survival curves) cannot be modelled only by a single component, A1 . We explain these phenomena by `side effects' of HSP expression induced by stress which diminishes the level of A2 . To model the action of stress, we assume that each cell can be in one of the two states: stressed wexp ; or unstressed w1 . The switch of the state of the cell wt from 0 unstressed to stressed state (from w1 to wexp ) occurs 0 immediately following the start of heat exposure. Process wt is de®ned as i V b w1 ; when t Ó 3; 3 1 Texp ` 0 wt 3 i; b exp X w ; when t [ 3; 3 1 Texp where parameter Texp is the duration of heating (for example, heating for 2 h approximately corresponds to the part of one day, equal to Texp 0.08 < 2 h/ 24 h), and it changes in accordance with every period of heat-treatment in the experiment. Shortly after heat shock, antioxidant defences are expressed, as long as the temperature is below some critical value. Accordingly, we suppose that the accumulation of damage from a thermal shock occurs after some delay (the amount of time it takes the antioxidant defence system to become saturated). The intensity of damage D1 t from thermal shock for the A1 component is de®ned by 4 dD1 k wt 2 D1 dt; t t with initial value D1 w1 . Coef®cient k in Eq. (4) 0 0 characterizes the average lag time before the organism begins to react to the thermal shock by producing reactive oxygen species. It is worthy of note that this lag time is here also equal to the amount of time the organism takes to stop producing reactive oxygen species once the thermal shock is removed. For the second component (the accumulation of damage and toxins), the intensity of damages D2 w2 is held constant (i.e. equal to some base damage 0 level), which results from the assumption that environmental conditions (other than thermal stresses) leading to the accumulation of toxins are kept 60 A. Butov et al. / Experimental Gerontology 37 (2001) 57±66 Fig. 1. Survival curves for C. elegans hermaphrodites calculated from experiment 1 data. H0, control group (no heating); H1, groups for heated 1 h; H2, for 2 h; H3, for 3 h; H4, for 4 h; H6, for 6 h; H8, for 8 h. constant over time in stress experiments with heat shocks. We hypothesize that the additional effects of HSPs (i.e. repair of the A2 damage) are responsible for the life span extension seen after heat stress. We draw the assumption about minimum and maximum levels for HSPs in the organism from the evidence that under usual conditions HSPs genes of various types (molecular mass) show low levels of expression that ensure respectively low levels of concentration (in the model hMIN 0.07) of these proteins in the total amount of all protein production (in different species including the nematode worm C. elegans up to 0.3±2.0% of the total amount), and during heat stress the output increases considerably and can achieve 50% (in the model hMAX 10.0) (Link et al., 1999; Langer and Neupert, 1994; McKay et al., 1994; Morimoto et al., 1994). The defence mechanisms initiated by heat shock are often associated with the production of additional HSPs, above a baseline level hMIN , and approach the level hMAX during exposure to thermal stress. We make further assumptions about the activity of HSPs in the absence of thermal shock. After thermal stress is completed, the residual activity of the induced HSPs reduces damage produced by innate metabolism as represented by A2 . The process ht characterizes the amount of HSPs in a cell and is described by À Á dht b1 hMAX 2 ht I wt . D dt À Á 1 b2 hMIN 2 ht I wt # D dt; 5 with the initial value h0 hMIN (where I(´) is the indicator function: I true 1; I false 0). Parameter b1 is the time while the production of HSPs during heat stress reaches the level hMAX . Parameter b2 is the time while the production of HSPs decreases to the level hMIN after heating. Parameter b1 10.0 is taken greater than b2 2.0 in accordance with the assumption that the production of HSPs during thermal stress requires 5±15 min, while the sequestration of HSPs in the cytoplasm of cells takes approximately 12 h after termination of the heat in¯uence (Hahn and Li, 1990; Parsell and Lindquist, 1994). We choose such values of parameters b1 and b2 because they are exponentially dependent. Adjustment of other model parameters is carried out on the empirical and the modelled survival functions. When duration of heat shock is small, an increase in ht reduces the amount of damage described by the components A1 t and A2 . In our model, we consider the dynamics of the t processes A1 and A2 from birth until death. The time of death of the modelled worm occurs in the case when either component A1 or A2 exceeds the threshold A. Butov et al. / Experimental Gerontology 37 (2001) 57±66 61 Fig. 2. Calculated survival curves for C. elegans hermaphrodites from simulated data. H0, control group (no heating); H1, groups for heated 1 h; H2, for 2 h; H3, for 3 h; H4, for 4 h; H6, for 6 h; H8, for 8 h. AMAX Ðthe maximum quantity of the accumulated damages for each component. So, we simulated the set of worm life histories and appropriate life spans. These simulated life spans are then used for construction of the modelled survival functions. As a result, the life span of worms exposed to such heat shock increases. This is observed in the groups of worms heated for less than 2 h. After four or more hours of heating, the level of HSPs still increases, however, the level of damage D1 also increases, in accordance with t Eq. (4). The dynamics of ht and D1 are such that the t amount of HSPs accumulated during the exposure to heat shock is not enough to reduce component A1 . 3. Results 3.1. Survival curves for experiment 1 In the model, the number of animals in each cohort Fig. 3. Survival curves for C. elegans hermaphrodites calculated from experiment 2 data. H0±H8 correspond to groups heated for 0±8 h, respectively. 62 A. Butov et al. / Experimental Gerontology 37 (2001) 57±66 Fig. 4. Survival curves for C. elegans hermaphrodites calculated from simulated data for experiment 2. H0±H8 correspond to groups heated for 0±8 h, respectively. was chosen to correspond to the population size in each cohort in the actual experiment in which the duration of thermal stress was varied. Experimental survival curves of the worms from the ®rst experiment with various duration of thermal stress are shown in Fig. 1. From the experimental data, one can see that when worms are exposed to heat for 1 or 2 h, life expectancy is increased (survival curves are shifted to the right of the control group). After longer exposure to heat, the death rate at early ages sharply increases (survival curves are shifted to the left of the control group). Thus, the effect of hormesis is observed only after 1 or 2 h of thermal stress. However, it should be noted that after longer duration of heat exposure, there are still some individuals for which the thermal stress has caused an increase in life expectancy. These effects are also re¯ected in the results of the model. Average survival curves, resulting from applying our model to the ®rst set of experiments, are shown Fig. 5. Survival curves for C. elegans hermaphrodites calculated from experiment 3 data. H0±H8 correspond to groups heated for 0±8 h, respectively. A. Butov et al. / Experimental Gerontology 37 (2001) 57±66 63 Fig. 6. Survival curves for C. elegans hermaphrodites calculated from simulated data for experiment 3. H0±H8 correspond to groups heated for 0±8 h, respectively. in Fig. 2. In this ®gure, one can see that modelled survival curves behave as well as the experimental survival curves corresponding to each duration of thermal stress. Note the presence of `heavy tails' of the survival curves, resulting from repair of the A 2 component. 3.2. Survival curves for experiment 2 Experimental survival curves for worms in the second experiment with 1, 2, 4, 6 and 8 h of thermal stress and the control group (on which there was no thermal in¯uence) are shown in Fig. 3. In this experiment, there are some prominent features that were also observed in the ®rst experiment, namely, the effect of hormesis and the clear increase of life expectancy (when compared with control) in those groups receiving short thermal exposures (about 1±4 h). Modelled survival curves for experiment two are shown in Fig. 4. As in experiment 1, the modelled curves for experiment 2 behave in a way similar to experimental data. 3.3. Survival curves for experiment 3 Experimental survival curves of worms in the third experiment and modeling curves after various thermal in¯uences are shown in Figs. 5 and 6. For all experiments both the empirical and the modelled survival curves show the increase of the average and maximum life spans for worms heated for 1 and 2 h. 3.4. The effect of hormesis Using a stochastic model of hormesis, we have been able to develop a mathematical model that corresponds to the results of experimental observations from heat-treatments of worms for various periods of time. The effect of hormesis can be observed when using only one component, A1 (damage due to thermal stress). The main part of empirical survival curves obtained from simulated data when only component A1 is considered looks similar to that produced from experimental data: at short heating until 2 h, the life expectancy is increased; after longer heating, survival curves are displaced to the left. However, the effect of increase in survival at advanced ages (i.e. the `heavy tail' effect) is only observed when a second component, A2 (associated with other damages) is added. Thus, by performing simulation experiments using our model, we can examine the underlying mechanism of the effects of hormesis on survival curves. The results of such modelling can in turn be used to suggest new experiments to uncover fundamental properties of the action of hormesis in aging. In particular, the model forecasts that in the case of heating in the oldest ages (for example with the 10th day of life) the appearance of 64 A. Butov et al. / Experimental Gerontology 37 (2001) 57±66 the more `heavy tails' in the survival curves along with an increase of mortality in the early ages for the intermediate levels of heating (here, for 4 h). 3.5. Parameters of the model The choice of numerical values of parameters used in the stochastic model is based on consideration of physiological mechanisms appropriate for the nematode worm C. elegans and a result of model adjustment to the data. The following parameters are ®xed for all experiments: s 2 0:02; b1 10:0; b2 2:0; wexp 10:5; AMAX 3:0; hMAX 10:0; hMIN 0:07; k 1:2; D 0:2; Texp 0:0 for not heated;Texp 0:4 for heated 1 h;Texp 0:08 for heated 2 h;Texp 0:17 for heated 4 h;Texp 0:25 for heated 6 h;Texp 0:33 for heated 8 h: For the ®rst experiment the following parameters are used: H 1 0:09; H2 0:095; s 1 0:08; w1 0:07; w2 0 0 0:11; a1 0:8; a2 0:9; b1 0:7; b2 0:2: For the second experiment: 1 2 1 H 0:135; H 0:125; s 1 2 dynamic processes of accumulated damage and the threshold model of mortality. Two sources of damage are considered. One is associated with the action of thermal shock per se. Another deals with `natural' accumulations of damage produced by other factors. The thermal shock induces the production of heat shock proteins in the cells. This induction helps reduce the level of accumulated damage from both sources by activation repair mechanisms. As a result, both the effect of longevity hormesis and the effect of increase in survival for the oldest old become explained. In this model, we used a simplifying assumption that cells in different tissues of the worm organism have the same sensitivity to heat. This assumption is partly justi®ed by the fact that experimental worms were sterilized. During the sterilization procedure, cells with mitotic activity possessing the highest sensitivity to stress were destroyed. However, the effect of different sensitivity of other tissues to stress on longevity is an interesting question which could be the subject of future studies. It is clear that our model cannot describe the development of the nematode worm C. elegans and its entire biology. It is focused on explanation of longevity hormesis arising in heating experiments. We show that simple assumptions about the role of heat shock proteins in the accumulation of damages in the organism of the nematode worm can be used for explanation of survival patterns observed in stress experiments. Acknowledgements 0:05; w1 0 1 0:073; w2 0 2 0:067; a 1:3; a 1:6; b 0:6; b 0:25: For the third experiment: H 1 0:11; H2 0:12; s 1 0:05; w1 0:07; w2 0 0 0:085; a1 1:2; a2 1:4; b1 0:64; b2 0:2: 4. Discussion In this paper, we explain the effect of longevity hormesis in the nematode worm C. elegans using The authors wish to thank Professor James W. Vaupel for the opportunity to use the facilities of the Max Planck Institute for Demographic Research in Rostock, Germany, during work on this paper. We also thank anonymous reviewers for valuable comments. Appendix The experimental analysis was carried out using methods of stochastic simulation modelling. The À Á À Á À Á 1 processesÀ AÁ A1 t$0 , A2 A2 t$0 , D1 D1 t$0 t t t and h ht t$0 are time dependent and are de®ned by the stochastic differential equations. In the stochastic differentials for the processes A1 t A. Butov et al. / Experimental Gerontology 37 (2001) 57±66 A2 t A1 0 A2 0 and the initial values and are random variables determining the initial level of the damage, as shown below: A1 a1 1 b1 e1 1 0 A2 a2 1 b2 e2 1 0 where e1 ; e2 are independent Gaussian random variables with mean zero and variance one. Variables a1 ; b1 ; a2 ; b2 are parameters of the model (the notation a1 refers to the positive part of the value a (i.e. a1 a, if a . 0, and a1 0, if a # 0). In the stochastic differential equations (1) and (2), W 1 Wt1 t$0 and W 2 Wt2 t$0 are standard independent Wiener processes and s 1 ; s 2 ; H 1 ; H 2 are parameters. Finally, we determine the time of death, t , as follows: t minft1 ; t2 g; È É where ti min t : Ait $ AMAX ; i 1; 2: References Cutler, R.G., 1991. 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