C. elegans reproduction is exquisitely sensitive to temperature changes
Compared to the well-understood heat shock response, less is known about how organisms respond to chronic, moderate temperature stress. It is well established that the average number of eggs laid by C. elegans
hermaphrodites is dependent on temperature 
. We asked whether reproduction is more temperature sensitive than other vital processes and how individual
worms respond to temperature stress. We examined viability, movement, and reproductive output over a range of temperatures (, Table S1
). We developed an experimental protocol in which nematodes were reared at the commonly used cultivation temperature of 20°C, and then, just prior to the onset of reproduction, individually
shifted to various elevated temperatures. This treatment—chronically exposing worms to temperatures between 20°C and 30°C—is qualitatively
different from the standard acute heat shock experiments, which involve brief exposure to nearly fatal temperatures (33°C) 
. Whereas the average number of eggs laid at 28°C was substantially reduced compared to temperatures at which worms are routinely raised (see below), at 30°C reproduction ceased completely (). In contrast, neither viability nor motility was comparably affected ().
Experiments performed to determine the dynamics of C. elegans reproductive behavior.
Reproduction is sensitive to chronic temperature changes.
We documented the reproductive performance of 3,418 individual worms, which laid a total of 144,092 embryos (, Figure S1
, Text S1
). Importantly, we collected dynamic, time-resolved egg-laying curves, not simply overall brood sizes. The temperatures used in our studies (20–30°C) are likely to be physiologically relevant because C. elegans
have been isolated from tropical and equatorial locales 
where temperatures routinely exceed 30°C. Furthermore, nematodes appear to dwell in compost and rotting vegetable matter 
, where temperatures can be even higher than in the ambient environment 
. Brood size of animals cultivated at 20 and 25°C were normally distributed (, S2
, Text S1
). While the means of the brood size distributions varied with temperature, they had indistinguishable coefficients of variation (p
0.58±0.01, permutation test). These results suggest that while the mean output of the reproductive system is temperature-dependent, increasing temperature does not lead to an appreciable increase in the individual-to-individual variability (Figure S4
Chronic temperature stress exposes heterogeneous physiological response of the reproductive system in C. elegans.
At 28°C, however, we observed a qualitatively different behavior—there were more individuals laying low numbers of eggs than would be expected from a normally distributed population (). This was accompanied by a coefficient of variation (Figure S4
) that was significantly higher at 28°C than at 25°C (p
, permutation test). Furthermore, these data could not be captured by a single normal distribution (p<10−4
, Kolmogorov-Smirnov test), but could be well described by a mixture of two distributions (). The relative proportion of animals laying a lower than expected number of eggs increased at higher temperatures (), as evidenced by the increase in the coefficient of variation (Figure S4
). These results suggest that whereas across a range of lower temperatures reproductive systems of all worms are robust, at higher temperatures, only a fraction of individuals continue to act in a robust manner, revealing an inherent heterogeneity in physiological response
Simple macro-level model closely reproduces experimental results
We developed a macro-level model of the C. elegans
reproductive system. Our model is both simple (it includes a small set of essential features and parameters) and falsifiable (designed to be experimentally testable). The reproductive system () can be abstracted as a pipeline for the serial maturation and subsequent fertilization of oocytes. We conceptualized it as a series of interconnected compartments—the gonad (which is encapsulated by the gonadal sheath), spermatheca, and uterus—through which gametes flow (). This process can be likened to a chemical reaction because transitions between compartments can be modeled as the conversion of precursors to products. We made two simple but plausible assumptions (a list of major model assumptions is given in ). First, all gametes in the model are conserved and can be explicitly accounted for 
. Second, all transitions between states obey mass-action kinetics. The latter is a typical assumption for dynamic systems, used in analysis of chemical reaction kinetics 
. It states that a process proceeds at a rate that is proportional to the availability of each of its inputs.
Modeling the dynamics of C. elegans reproduction.
Major assumptions of the model.
Although oocyte development and maturation involves a number of discrete steps and processes 
, for simplicity, we subsume them into a single state. This mathematical abstraction simplifies the subsequent calculations and reflects the difference between a fine-grained molecular model and a macro-level approach. We represent the number of oocytes, that are generated de novo, as O
. Experimental data suggest that the total number of germ cells in adults 
and the rate of oocyte production 
are constant. Therefore our model treats the rate at which oocytes are generated as a constant, subject to saturation that prevents O
from increasing beyond an upper limit established by gonad size 
. Together, these assumptions define the rate
of oocyte creation (),
is a rate constant describing the generation of O
, and ks
is a rate constant pertaining to the carrying capacity of the gonad.
Hermaphrodites of the standard laboratory strain (Bristol or N2) of C. elegans
produce approximately 300 sperm during development before the germline irreversibly transitions to oogenesis 
. Because animals produce oocytes continuously until their cache of sperm is depleted, the number of sperm determines the overall fecundity 
. A dedicated mechanism communicates the presence of sperm to the developing oocytes. Sperm release major sperm protein (MSP) into the proximal gonad 
, where it induces meiotic maturation of the proximal oocyte 
. Concomitantly, MSP promotes sheath cell contraction, leading to ovulation 
. As the oocyte is pulled into the spermatheca, fertilization takes place 
. After the spermatheca, the embryo passes to the uterus where it completes the first several cell divisions before being laid 
. The dynamics of egg-laying are known to be bursty, but the time intervals between these bursts are typically on the order of minutes 
, much shorter than the time intervals at which we counted eggs. Therefore we need not consider these dynamics in our model.
The reproductive rate, while approximately constant early in adulthood, decreases as the animals age 
. This decline in reproductive function likely has multiple causes. In the first several days of reproductive maturity it likely reflects the decreasing number of sperm and the coupling of ovulation to sperm number 
, because mating during this period can produce substantially more progeny 
. About 5 days after the onset on reproduction, oocyte quality becomes compromised 
, and mating of week-old hermaphrodites does not increase their brood size 
. At lower temperatures (e.g., 20°C), within 4–5 days of reproductive maturity nearly all of a hermaphrodite's sperm have been used to fertilize eggs 
. However, it is reasonable to expect that chronic exposure to higher temperatures will result in gamete death. While developing oocytes are likely damaged by chronic temperature stress, they can be continuously generated, therefore their destruction is difficult to decouple from a decrease in their production rate. We thus captured this process by allowing net oocyte production rate in the model to vary with temperature. These assumptions, and their related mass action kinetics, yield expressions for the rate of ovulation
and the rate of sperm death
is the number of active sperm,
is a rate constant of ovulation, and
is a rate constant of sperm death.
rapidly achieves a steady state 
, we simplified the model specified in Equations 1 and 2 using a quasi-steady-state approximation 
. We found that this reformulation results in a model that captures the system dynamics equally well (see next section and Text S1
). We explicitly solved the steady-state mass balance equation to obtain
(see Text S1
). This allowed us to express the dynamics of the system using a smaller subset of parameters. In the interest of parsimony, we used the parameter kmax
to summarize the intrinsic maximum rate of oogenesis,
depends weakly on Sa
, and can be treated as a constant (see Text S1
Together, these assumptions can be combined into a system of mass balance equations describing the dynamics of C. elegans
In our experiments, we observed substantial variability in both the overall fecundity and the dynamics of egg-laying among individuals. We hypothesized that this variability arises from differences in the intrinsic capacity (kmax
) for oogenesis and the number of sperm produced by each animal, both of which we surmised are normally distributed (, S2
, Text S1
). The rate of sperm production is approximately constant over time 
, and high sperm count is associated with delayed onset of oogenesis 
. To capture this, when simulating our model, the number of sperm of each individual and the timing of the onset of embryo production were determined by the same variable drawn from a normal distribution.
Recalling the heterogeneity of brood sizes at higher temperatures (), we reasoned that the fraction (δ
) of animals that exhibit a non-robust reproductive output varies with temperature, and treated δ
as a free parameter. Although the mean-field behavior of our model can be analytically solved (Text S1
), we solved it numerically. We used maximum likelihood estimation 
to determine the kinetic parameters for our model. Interestingly, our estimates of kmax
were substantially different for the two classes.
We used time-resolved, densely sampled egg-laying curves collected at 20, 25, and 29°C (, ) to train our model for both the robust and non-robust classes of animals. Noting the narrow range of relevant temperatures, we hypothesized exponential dependence of the model parameters on temperature. Because δ is only nonzero at 28°C and above, we used curves collected at 20, 28, and 29°C to estimate its value more robustly. The estimated coefficients of these exponential functions () result in model predictions that closely recapitulate the empirical data ().
Fitting the model to experimental data.
More complicated models do not offer an improved description of the system
To obtain Equation 3, we surmised that the dynamics of oocyte development are steady-state 
, and the number of developed oocytes O
is constant (also see Text S1
). To ensure that this approximation does not lead to an overly simplistic model that fails to capture aspects of reproductive dynamics, we evaluated predictions for two distinct model formulations. The first assumed that O
reaches a quasi-steady-state according to Equation 3. This simplified model is fully described in Equation 4. The second was more complicated, explicitly accounting for oocyte generation and development (Equations 1 and 2a) and allowing O
to vary. Only subtle quantitative differences existed in the predictions of these two models, justifying the use of the parsimonious version ().
More complicated models do not offer an improved description of the system.
To ensure that the parsimonious model (Equation 4) does not omit other details that could improve the description of the system, we constructed an alternative model with an additional component that plausibly exists in the reproductive system: oocyte death. In a model that explicitly included discrete states for dead oocytes (Od
) (), the rate of oocyte accumulation becomes,
is the rate of oocyte death and
is the rate constant of oocyte death. Reformulating Equation 5, we obtain,
. Because this expression is mathematically equivalent to Equation 4a, it is difficult to differentiate between this model that accounts for oocyte death from the more parsimonious model formulated above (Equation 4).
Testing predictions of the model
Our modeling framework provides the basis for predicting the behavior of animals treated under different conditions and having different genetic backgrounds. As a first test, we generated predictions of the dynamics of reproductive output following chronic temperature shifts conducted under the same experimental protocol that was used to train the model, but at three different temperatures. At 23, 28, and 30°C, we observed a close correspondence between predicted values and experimental results (). Predictions were obtained using parameters estimated from the training data (); the only additional information that was specified was the temperature to which the animals were exposed. Importantly, in addition to the quantitative matches obtained for the population means, we also observed a correspondence between predicted and experimentally measured animal-to-animal variances of brood sizes.
Predicting the dynamics of C. elegans reproduction.
As a second test, we probed the reproductive dynamics of two mutants, tra-3
, that produce different numbers of offspring than the wild-type N2 strain (Table S2
). In our experimental paradigm, at 20°C these two mutants produced 437±40 and 238±115 progeny, respectively. At least two lines of evidence suggest that availability of sperm is the limiting factor in C. elegans
reproduction. First, self-fertile hermaphrodites continue laying unfertilized eggs once their cache of sperm becomes exhausted 
. Second, hermaphrodites that are mated to males generate up to four times the number of progeny as their unmated counterparts because male ejaculate provides many more sperm than the number produced by a hermaphrodite 
. Relevantly, the cdc-48.1
(tm544) mutant animals lay approximately as many eggs as the wild type, but a substantial fraction of these oocytes are not fertilized 
. We therefore reasoned that the number of progeny of individual animals accurately reflected the number of sperm they produced. Using these inferred sperm counts and the model parameters estimated from the training data (), we predicted the dynamics of the reproductive output of the two mutants. At 20 and 25°C, predictions for the cdc-48.1
mutants matched the experimental results, as did predictions for the tra-3
animals at 20°C (). At 25°C, however, the tra-3
mutants laid fewer embryos than predicted by our model ().
Predicting behavior of C. elegans reproductive mutants.
We investigated the plausible causes of this discrepancy. At 20°C the embryos of both the wild-type N2 and tra-3 animals were arranged in an orderly fashion within the uterus (). At 25°C () the embryos in wild-type animals were more numerous than at 20°C, but this effect was far more pronounced in the tra-3 mutants, which had retained embryos that were older than the age at which they are typically laid (). The number of embryos retained by individuals correlated with the sperm count, such that retention in the tra-3 animals was substantially higher than in the wild-type (). We interpreted this as an indication that our model over-predicted the number of eggs laid because it did not consider the accumulation of eggs in the uterus and its possible consequences. The total number of eggs laid and retained in the uterus of the tra-3 animals at 25°C was indistinguishable from that in the wild-type N2 animals under the same conditions. In contrast, at 20°C tra-3 mutants produced nearly 50% more offspring (437 vs. 302) reflecting a greater number of sperm. Together, these results suggest that a higher aggregate egg production rate at 25°C results in higher egg retention which causes a mechanical impediment to the passage of eggs and therefore disrupts reproduction.
The accumulation of embryos inside the uterus led to a “bagging” phenotype 
and eventual hatching within the parent (, Table S3
). Significantly, the bagging phenotype of the tra-3
mutants was completely suppressed by an egl-19
(ad695) mutation that causes constitutive egg-laying 
. This suggests that the mechanical elements of the egg-laying apparatus were compromised by chronic heat stress, serving as a physical impediment to achieving the maximum rate of egg-laying and, therefore, the highest brood size given the number of available sperm.