,Investigation of the relationship between longevity and environmental factors such as the study of reactions of organisms to stress, is a key to understanding the nature of ageing, senescence and death processes. Studies of many factors on different animals are in progress now. The investigated factors range from ionising radiation to environment temperature and dietary restriction. A diverse set of species are used in these experiments including yeast, worms, flies and mammals. It is not surprising that large deviations from the "normal" conditions, such as high radiation, extreme temperature and so on, lead to the distruction of living organisms. At the same time, however, it was found that small deviations and low levels of detrimental chemicals can lead to prolongation of life and slowing senescence. Ordy et al. (1967) found that mice, exposed daily to electric shock, demonstrate increased longevity. Holloszy and Smith (1986) obtained the same result for rats, exposed daily to four hours of cool water immersion. The positive effect of dietary restriction on longevity and external stress resistance is reported by Heydari et al. (1993) and Duffy et al. (1995). Consideration of this effect in terms of hormesis - beneficial effect of detrimental factors, is given by Masoro (1998).
Experiments in invertebrate species are attractive because of the short life span of the animals and ability to study large population in controlled experiments. Additionally, low doses of radiation have been reported to slow aging and prolong life span in some insects ( Sohal and Allen, 1984 ) and in Caenorhabditis elegans ( Johnson and Hartman, 1988 ), hypergravity postpones aging in Drosophila melanogaster ( Le Bourg and Minois, 1999 ), thermal stress lead to increase in life expectancy in C.elegans ( Lithgow et al., 1995 ) and in D.melanogaster ( Tatar et al., 1997 ).
In this study, we propose a mathematical model for post-heat stress survival in C.elegans . The model is based on empirically observed mortality and describes survival in phenotypically heterogeneous population, composed of three homogeneous groups. Survival in each homogenous group is determined by an "aging protection system" with different efficiency, which is reflected in different values for parameters of the classical mortality model, based on the Gompertz equation. The sub-groups parameters depend on the duration of heat stress. The model, identified on empirical data, fits well both the hormetic and debilitation effects, observed in the experiment.
C.elegans worms (strain TJ1060 (spe-9; fer-15)) were raised for three days at 25.5oC degrees and therefore sterile. At three days of age the worms were divided into ten groups and exposed to heat (35oC) 0, 1, 2, 4, 6, 8, 10, 12, 16 or 24 hours. Worms were permitted to recover for 24 hours at temperature 20oC degrees and then transferred to liquid survival medium at 20oC for the reminder of the experiment. Starting from day 5, the number of alive and dead worms were counted daily for all groups. During 1 and 2 hours heating no worms died. No worms survived 16 or 24 hours of heat.
Statistical Analysis
The life expectancy for worms, surviving heat, was estimated as the
arithmetic mean for the experimental life spans given in days
,
.
The standard error for LE estimate is given by the formula
| Heating duration (hours) | 0 | 1 | 2 | 4 | 6 | 8 | 10 | 12 |
| Number of deaths during heat stress | 0 | 0 | 0 | 1 | 1 | 14 | 63 | 121 |
| Total deaths after | 137 | 100 | 152 | 133 | 164 | 152 | 198 | 178 |
| LE (days) | 21.1 | 22.7 | 22.1 | 19.1 | 11.3 | 8.7 | 6.3 | 5.3 |
| SE (days) | 0.4 | 0.6 | 0.4 | 0.6 | 0.4 | 0.2 | 0.1 | 0.04 |
| p-value for LE | --- | 0.012 | 0.055 | 0.999 | 1.0 | 1.0 | 1.0 | 1.0 |
Table 1. Statistical estimates for different duration of heating. The last row presents p-value for hypothesis that there is no hormesis - extension of life expectancy by heat stress.
The classical mathematical model for the probability to survive till age x is Gompertz model (
Gompertz, 1997
), defined by equations
The Gompertz model has been justified on theoretical grounds in terms of vitality and energy demand, needed for life support, ( Strehler and Mildvan, 1960 ), which reflect the level of environmental challenge and the efficiency of life protection systems. Modifications of this model are widely used in mathematical demography ( Wilson, 1994; Mueller et al., 1995 ) ; some modifications are related to non-homogeneous populations ( Vaupel et al., 1979, 1987 ) which permits the description of experimental results of slowed ageing ( Brooks et al., 1994 ) as an effect of altered heterogeneity. The concept of heterogeneity also has been employed in exploring the results of other stress experiments ( Khazaeli et al., 1995 ; Curtzinger and Khazaeli, 1997 ).
In the present investigation, the observed survival curves were approximated by composition of three Gompretz survival curves
denotes cumulative hazard of death in the experiment after heating
for h hours at three days of age
. Model (
1
) is constructed
assuming that the investigated population of worms
consists of three sub-
groups: "frail", "normal" and "robust", depending on mortality parameters
ajh and bjh.
These three groups describe natural variation in the properties of
life-protection systems. Just-after-
heating proportions of the three groups of animals are
. Proportions of groups and mortality parameters are affected by
the duration of heat stress. These changes are reflected
as hormesis - improvement in survival,
or debilitation-decrement in survival.
From biological point of view, model
(
1
)
describes phenotypic diversity in nematode survival
acquired during heating.
The parameters ajh, bjh and Pjh (j=1, 2, 3) of model ( 1 ) were estimated for different durations of heat stress, h, using a nonlinear least- square regression procedure ( Sharrod, 1998 ) to minimise the square functional
Results of statistical analysis of life expectancy are presented in Table 1 and Figure 1. Table 1 presents the life expectancy estimate (LE) and standard error (SE) for different duration of heat stress. The last row presents p-values, calculated by t-test statistics, for the hypothesis H0: heating does not increase life expectancy (no hormesis). Figure 1 presents the graph for life expectancies after different durations of heat stress together with the corresponding 95% confidence intervals. From the table and figure one can see that an early life stress of 35oC for 1 or 2 hours significantly increases the life expectancy of C.elegans.
Survival and Mortality
Figure 2
shows proportion of survivors till age x after different
heat treatments,
in other words survival
probability - probability to stay alive till specified age.
From the graph one can see that proportion of worms, surviving longer than 22 days in groups
heated for 1 or 2 hours, increases in comparison with non-heated control group.
This demonstrates the positive action of early life heating and explains the increase in life
expectancy, presented in
Table 1
. In the group of worms, heated for 4 hours, the proportion of survivors is less than in the
control group at almost all ages, which is a result of debilitative
effects on the worms during
heating. The longer
the duration of heat stress (6, 8, 10 and 12 hours of heat)
- the more debilitation is exhibited in proportion of
worms surviving the heat and the life expectancy after the end of heat
treatment
. Survival after 0, 1, 2 and 4 hours was modelled
in the present study to describe the result of early life heating when debilitation effects are
not very significant.
Figure 2. Proportion of worms surviving as a
function of age. Survival curves
exclude worms dieing during the heat treatment.
Results of modelling using the three groups
, heterogeneous
survival model (
1
) are presented in Figures 3-5.
Figure 3
presents the chart for the estimated parameters in "frail", "normal" and
"robust" sub-groups, depending on the duration of heating.
Figure 3. Model parameters:
initial mortality (a) and
rate of ageing (b).
Figure 4
presents the charts for the three
proportions in each of the groups after
heat treatment. The Figure shows
redistribution between sub-groups as a result of heating. After heating
for 1 and 2 hours,
the proportion of "normal" subgroup in population increases by diminishing
the proportion of
the "robust" sub-group. When combined with the corresponding changes in
Gompertz survival parameters,
presented in
Figure 3
, this leads to increase in observed life expectancy. Heating for 4 hours
leads to practical
elimination of the "robust" sub-group and increase the proportion of the
"frail" sub-group.
This is the evidence of a negative action of
prolonged heat stress, which leads to decreased life expectancy.
It is important to note, that redistribution between sub-groups
as a result of heating does not result from lose of
weak animals, because during 1 and 2 hours heating no worms died.
For reference see
table 1.
Redistribution between sub-groups
must reflect some underlying change(s) in the life-protection systems in these
worms.
Figure 4. Proportions of sub-groups.
Figure 5
presents probability of death for each day (mjh)
modelled from the parameters derived above,
together
with mortality rate (qjh) and its 95% confidence intervals.
The confidence intervals were calculated to show a "tube" which should cover the real mortality
rate with 95% confidence level at all ages. From the Figure one can see that early life heating
shifts the risk of death to the older ages, which can be interpreted as slowing ageing.
Figure 5. Mortality rates.
Long heating at 35oC early in life is destructive for C.elegans. Table 1 shows decreased life expectancy after heating longer than 4 hours. The inference about negative action of long heating is supported by observed changes in survival curves as well. Figure 1 shows sharp decrease in proportion of surviving worms in experiments with heating longer than 4 hours.
Moderate heating prolongs life of C.elegans. Statistical analysis shows increases in life expectancy after 1 and 2 hours heating, compared to the control non-heated group. The corresponding p-value for t-test statistics under the null hypothesis (no increase in life expectancy) equals 0.012 for 1 hour heat and 0.055 for 2 hours heat. For reference see table 1. This is a clear demonstration of hormesis: heating activates the life protecting systems, which compensate negative influence of 35oC.
Modelling survival in the whole population, composed of "frail", "normal" and "robust" worms, allows the interpretation and estimation of observed changes in survival. From the model, it follows that the positive effect of heating results from the redistribution of animals by groups, which reflects changes in the life protecting systems after heat stress. Moderate heating increases the proportion of worms in the "normal" group which demonstrate slowed ageing, presented by Figure 3.
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