SEARCH

SEARCH BY CITATION

Keywords:

  • Caenorhabditis elegans ;
  • fitness;
  • isogenic;
  • lifetime fecundity;
  • reproduction;
  • variance

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Conflict of Interest
  9. References
  10. Appendix: Worm and food strains
  11. Appendix
  12. Appendix: WinBugs code

Organisms live in heterogeneous environments, so strategies that maximze fitness in such environments will evolve. Variation in traits is important because it is the raw material on which natural selection acts during evolution. Phenotypic variation is usually thought to be due to genetic variation and/or environmentally induced effects. Therefore, genetically identical individuals in a constant environment should have invariant traits. Clearly, genetically identical individuals do differ phenotypically, usually thought to be due to stochastic processes. It is now becoming clear, especially from studies of unicellular species, that phenotypic variance among genetically identical individuals in a constant environment can be genetically controlled and that therefore, in principle, this can be subject to selection. However, there has been little investigation of these phenomena in multicellular species. Here, we have studied the mean lifetime fecundity (thus a trait likely to be relevant to reproductive success), and variance in lifetime fecundity, in recently-wild isolates of the model nematode Caenorhabditis elegans. We found that these genotypes differed in their variance in lifetime fecundity: some had high variance in fecundity, others very low variance. We find that this variance in lifetime fecundity was negatively related to the mean lifetime fecundity of the lines, and that the variance of the lines was positively correlated between environments. We suggest that the variance in lifetime fecundity may be a bet-hedging strategy used by this species.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Conflict of Interest
  9. References
  10. Appendix: Worm and food strains
  11. Appendix
  12. Appendix: WinBugs code

Evolutionary success is achieved by maximizing fitness in spatially and temporally variable environments. Key to this is phenotypic variation, which can be due to genetic differences among individuals and/or due to environmentally induced effects, i.e. phenotypic plasticity. However, in a constant environment genetically identical individuals are assumed to be phenotypically constant. Phenotypic differences among such individuals are usually attributed to stochastic events that would generally result in equal degrees of phenotypic variance among different genotypes. Importantly, implicit in the assumption of stochasticity underlying phenotypic variance among genetically identical individuals is that the phenotypic variance is not genetically controlled, not subject to selection, and thus not adaptive.

Contrary to this assumption, phenotypic variance in isogenic populations in constant environments is observed (Viney and Reece 2013). Beyond stochastically generated differences, microenvironmental differences among individuals, such that each individual is in effect in its own microenvironment, could also cause such phenotypic variance. However, the relative roles of stochasticity and microenvironmental differences in causing phenotypic variance are not known.

One adaptive explanation of the phenomenon of phenotypic variance in isogenic populations is the evolution of bet-hedging or risk-spreading strategies, which makes the link between environmental variability, fitness variance and mean fitness (Gillespie 1974). Bet-hedging strategies are those that minimize fitness variation across generations in the face of environmental variation. Thus, when comparing phenotypically invariant and variant genotypes, an invariant genotype would have a certain fitness in one environment to which it was adapted, but a potentially steep fitness decline outside of that environment. In contrast, a phenotypically variant genotype could have a comparatively greater fitness in heterogeneous environments because its fitness would not decline so quickly as the environment changed. Therefore, phenotypic variance increases the fit phenotypic space of a genotype. The repeated observation of microorganismal phenotypic variance within a clone has been interpreted as maximizing fitness in variable environments (Dubnau and Losick 2006). Other, related approaches to understanding phenotypic variance center around the concept of canlization of phenotypic traits, defined as the robustness of phenotypes to perturbation (Flatt 2005). Despite much working demonstrating different degrees of canalization of different phenotypes, the significance of different degrees of phenotypic variation among genotypes has largely not been considered (Gibson and Dworkin 2004).

Theoretical studies have shown how selection can change phenotypic variance (e.g. Gillespie 1974; Hill and Zhang 2004) and mechanistically phenotypic variance can be genetically controlled, changing the phenotypic robustness of a population (Hermisson and Wagner 2004). Much of the empirical work describing the molecular and genetic mechanism of phenotypic variance of genetically identical individuals in constant environments has focused on unicellular systems (Davidson and Surette 2008; Viney and Reece 2013). In many of these systems there is variability in gene expression (Maamar et al. 2007), often affecting whole gene networks, which if manifest as phenotypic variance may allow selection to act on trait variance (Viney and Reece 2013). The effect of mutations on trait variance (as well as on trait values) has been observed in multicellular organisms too (Dunn and Fraser 1958). For example, wild-type mice have 18–19 whiskers, but the Tabby mutation results in fewer whiskers (mean 12) and a greater variance in whisker number (range 8–16) (Dunn and Fraser 1958). There are also empirical data showing the genetic control of phenotypic variance among genetically identical individuals in a constant environment. For example, Caenorhabditis elegans mutations that affect the developmental cell lineage (which in wild-type is essentially invariant among individuals) are incompletely penetrant, i.e. genetically identical individuals vary phenotypically (Horvitz and Sulston 1980; Braendle and Félix 2008; Braendle et al. 2010). In this system, there is a molecular understanding of how incomplete penetrance of C. elegans phenotypes occurs in certain traits. The specification of gut cells is controlled by a small transcription network and variation in gene expression near a threshold alters the phenotype of individuals, thereby altering penetrance (Raj et al. 2010). Trait variance is of increasing interest in animal breeding settings where low trait variance can be desirable as a means to maximize production gains and agricultural efficiency. The study of trait variance has extended to studies in humans, for example looking at the genetic control of variability in body mass index (Yang et al. 2012).

Laboratory studies of unicellular species show that among genetically identically individuals there is phenotypic variation that can be genetically controlled and that can be adaptive (Viney and Reece 2013). For multicellular species, though, there has been much less investigation of these phenomena. Also, empirical data supporting the idea that phenotypic variance amongst genetically identical individuals in a constant environment may contribute to reproductive success is limited. Here, we quantified the phenotypic variance among genetically identical individuals in the model nematode system C. elegans. This system allows accurate quantitation of many aspects of individuals' phenotypes, crucially including lifetime fecundity. We have therefore examined the lifetime fecundity of C. elegans and, more particularly, how this varies among individuals. Specifically, we asked for an inbred line of C. elegans in one environment how much phenotypic variance in lifetime fecundity there was among individuals. We compared 20 recently wild isolates of C. elegans in this way and found that the among individual variation in lifetime fecundity of each line varied significantly among the C. elegans isolates. This means that the C. elegans genotype – a genetic effect – affects phenotypic variance. We then repeated this but using different food environments, which tests how the environment itself affects this phenotypic variance. These different environments changed the phenotypic variance of the lines. Also, within each food environment we found that there was a negative relationship between the lifetime fecundity and the among individual variance in lifetime fecundity. Furthermore, we found that these variances in lifetime fecundity correlated positively across the two food environments. We suggest that these results may show that these C. elegans isolates display a gradient of risk-spreading strategies.

Materials and Methods

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Conflict of Interest
  9. References
  10. Appendix: Worm and food strains
  11. Appendix
  12. Appendix: WinBugs code

Worms and food

We used 20 recently wild (see Appendix 1) C. elegans isolates and the standard N2 wild-type. For each isolate, one isogenic line was made by single-worm inbreeding for at least 10 generations (inbreeding coefficient > 0.9) and cryopreserved; hereafter these isogenic lines are referred to as lines. Each experiment used a new cryopreserved stock of the isogenic line. We used two food sources, Escherichia coli OP50 and Bacillus pumilus B215 (see Appendix 1).

Assays

Lifetime fecundity was the total number of viable progeny produced by individual hermaphrodites grown with excess food at 19°C. To measure this, arrested first stage larvae were generated by hypochlorite treatment from which synchronized L1s hatched, and after 24 hours 10 individuals of each line were individually introduced to food (day 0), with five individuals for each food source. Worms grew into adult hermaphrodites, which were transferred every other day to NGM fresh plates (Hope 1999) for their reproductive life. Egg-containing plates (from which adult hermaphrodites had been removed) were maintained for 48 hours, when the number of viable progeny were counted. We conducted the observations in three experimental blocks, and all lines and both food sources were included in each block; thus, 3 blocks × 21 lines × 5 worm replicates = 315 individual observations per food source (see Appendix 2: Tables A2 and A3). We found no differences among the blocks (see below) and so the block data for each food source was pooled.

Statistical models and analyses

Our objectives were to analyse how the (i) mean lifetime fecundity (LF) and (ii) the variance in LF varied among the C. elegans lines and between the two food sources. We used a Bayesian modelling approach, using a hierarchical linear model, to compare different distributions of the mean LF and the variance in LF. From this we then concluded how the LF and variance in LF varied among the lines and between the two food sources.

Our initial Model 1 was:

  • display math(Model 1)

where LFi is the LF of the ith individual worm of all of the observations that we made, N denotes a normal distribution, μ is the mean LF across the i observations (= 315), and σ2 is the variance across LF observations. μ has an N(0, 1e10) prior distribution and the precisions τ = 1/σ2 has G (0.001, 0.001) prior (so the gamma distribution has extremely large standard deviations). The relevant computer code for these models is available in Appendix 3.

The worms were grown in two different food environments so we applied Model 1 to the LF data from these two food environments separately. This showed that the mean LF of the lines differed significantly between the two food environments, shown by the non overlap of the credible intervals (CI) of the mean LF; mean LF on B. pumilus 98.76, CI 91.31–106.3; mean LF on E. coli 244.9, CI 236.3–253.6. We therefore analysed the LF from the two food environments separately.

We next extended Model 1 to make Model 2:

  • display math(Model 2)

which models the data such that each C. elegans line, j (= 1…21), has its own mean LF, denoted by μj, but all lines have the same variance in LF, denoted by σ2, referred to as the residual variance. This model has a μj with an N (θ, σ2among) prior distribution, where the hyperparameter θ has a N (0, 1e10) hyperprior, and the precisions τamong = 1/σ2among and τ = 1/σ2 both have G (0.001, 0.001) priors. θ represents the mean LF among lines, and σ2among the among line variance which quantifies how the mean LF varies among the lines. The relative contributions (interclass correlation coefficient) that each component contributes to the total variance are, for example, σ2among/(σ2among + σ2); these are presented expressed as percentages. If Model 2 was a better model than Model 1 then we concluded that the C. elegans lines have different LF. We describe later in this section how we compared models.

We then extended Model 2 to make Model 3:

  • display math(Model 3)

which models the data such that each C. elegans line, j, also has its own variance in LF, denoted by σ2within line j. This model had with an N (θ, σ2) prior distribution, a N (0, 1e10) hyperprior and the precisions are τwithin line j = 1/σ2within line j and τ = 1/σ2, both have G (0.001, 0.001) priors. If Model 3 was a better model than Model 2 then we concluded that the C. elegans lines have different variances in LF.

We used the same approach to investigate whether there were differences in LF among the different experimental blocks. We did this by extending Model 3, making Model 4:

  • display math(Model 4)

which accounts for the three experimental blocks by including the term αk, where k is the number of experimental blocks (= 3), with a N (0, 1) prior. When we used this model it was unable to converge on a solution for this coefficient (i.e. an identifiability issue, Spiegelhalter et al. 2002). This suggests either that the block effect was negligible or that the dataset was too small to allow for an analysis of block effects. We therefore reduced the number of terms in the model to test for an effect of experimental block by extending Model 2 to include the experimental block parameter, αk, making Model 5:

  • display math(Model 5)

If Model 5 was a better model than Model 2 we concluded that there are significant effects of the three experimental blocks.

We compared models by calculating the differences in deviance information criterion (ΔDIC) between two models. To choose the best model, a ΔDIC of less than 2 was taken to indicate substantial support for the simpler model, a ΔDIC of between 4 and 7 was taken to indicate considerably less support for the simpler model, and a ΔDIC of greater than 10 was taken to indicate essentially no support for the simpler model (Spiegelhalter et al. 2002).

We constructed the models using WinBUGS, which is a software package that uses Markov chain Monte Carlo (MCMC) methods to fit Bayesian statistical models (Lunn et al. 2000); this code is available in Appendix 3; Table A1 presents the DIC values of models 1 – 3. For the results, the reported parameters are after 10,000 iterations, then followed by a further 10,000 updates for the parameters. This produced a Monte Carlo error of less than 5% of the posterior standard deviation. Throughout the variance is described as standard deviations unless otherwise stated; estimates are presented as the mean with its respective 2.5 and 97.5% CI.

Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Conflict of Interest
  9. References
  10. Appendix: Worm and food strains
  11. Appendix
  12. Appendix: WinBugs code

We firstly consider whether there was any effect of the experimental block. There was no effect of experimental block on the mean LF of the lines, shown by the ΔDIC, on the B. pumilus food source (ΔDIC Model 3 – Model 4 = 0.11; ΔDIC Model 2 – Model 5 = −0.1) or on the E. coli food source (ΔDIC Model 3 – Model 4 = 0.74; ΔDIC Model 2 – Model 5 = 0.53), and so data from all three blocks were analysed together.

The mean LF varied between the two food sources with those worms eating B. pumilius having approximately two-fifths of the progeny of worms eating E. coli (see Appendix 2 Table A1). The results from Model 1 showed that the mean LF of the lines differed significantly between the two food environments, shown by the nonoverlap of the credible intervals (CI) of the mean LF; mean LF on B. pumilus 98.76, CI 91.31–106.3; mean LF on E. coli 244.9, CI 236.3–253.6.

Considering the LF of worms when fed B. pumilius, the mean LF of the isogenic lines ranged from 35–167 [Fig. 1A; Appendix 2 Table A1(A)]. The lines differed in their LF in this food environment (difference in mean LF among lines, ΔDICModel 1 – Model 2 = 86.7, Table 1, Fig. 1A). Analysis of variance components showed that the among line variance was greater than the residual variance (108.4 [79.8–149.6 CI] and 57.5 [52.9–62.3 CI], σamong and σ, respectively, Model 2), suggesting that c. 65% of the observed variance in LF was among lines, and the remainder the residual variance.

Table 1. DIC results of the models for B. pumilus B215 and E. coli OP50. The results show Dbar, Dhat and pD. Dbar is −2 times the sample average of the log-likelihoods; Dhat is −2 times the log-likelihood evaluated at the posterior mean of the parameters; pD, calculated as Dbar − Dhat, is the effective number of parameters in the model. The most parsimonious model according to ΔDIC is shown in bold.
 DbarDhatpDDIC
B. pumilus
Model
Model 13551.983549.92.03553.9
Model 23445.43423.721.73467.2
Model 3 3373.2 3332.3 40.9 3414.1
E. coli
Model
Model 13643.23641.21.93645.2
Model 23531.53509.522.03553.5
Model 3 3447.1 3406.3 40.9 3488.0
image

Figure 1. (A) Mean lifetime fecundity (± 2SE) of C. elegans lines fed B. pumilus B215. 1 = N2, 2 = CB4853, 3 = JU1400, 4 = JU1401, 5 = JU1409, 6 = JU1410, 7 = JU1411, 8 = JU1416, 9 = JU1442, 10 = JU1494, 11 = JU262, 12 = JU319, 13 = JU345, 14 = JU362, 15 = JU393, 16 = JU400, 17 = MY1, 18 = MY16, 19 = MY2, 20 = PX174 and 21 = PX179. Open circles represent the individual observations, colored by block (black, red and green, for block one, two and three respectively). (B) Relationship between a line's mean lifetime fecundity and its coefficient of variation (CV). A line's mean and CV of lifetime fecundity are estimates across blocks' pooled data.

Download figure to PowerPoint

We then further explored the differences among lines by describing the variation within each line; specifically, we wanted to test whether including an extra parameter to describe the within line variance in LF for each line improved the fit of the model. The analysis showed that Model 3, which included the within line variance, was the preferred model (ΔDICModel 2 – Model 3 = 53.1, Table 1), consistent with the within line variance in LF varying significantly among the C. elegans lines. The within line variance in LF ranged between 525.8 and 11310.0 [σwithin line line 3 and 7, respectively, Fig. 1A; Appendix 2 Table A1(A) and Fig. A1]. Line 7 showed a variance 21 times higher than that of line 3. Therefore, we conclude that in the Bpumilus food source there were significant differences in the mean LF among lines, and in the within line variance in LF. We also found that the lines' coefficient of variation (CV) and mean lifetime fecundity were correlated (Spearman coefficient = −0.59, < 0.01) (Fig. 1B). The lines did not differ in their schedule of reproduction, all commenced reproduction on day 2 and had completed it by day 8.

The E. coli food environment was significantly better than the B. pumilus environment, with the lines producing an average of 151 more offspring (Fig. 2A). In this better environment the mean LF [range 162–299, Fig. 2; Appendix 2 Table A1(B)] was different among the lines (ΔDICModel 1 – Model 2 = 91.7, Table 1). The analysis of variance components showed that the among line variance was greater than the residual variance (256.4 [189.8–251.2 CI] and 65.8 [60.7–71.5 CI], σamong and σ, respectively, Model 2), suggesting that c. 80% of the observed variance in LF was among lines, and the remainder the residual variance. Also, Model 3, that included the within line variance, was the preferred model (ΔDICModel 2 – Model 3 = 65.5, Table 1), again consistent with the within line variance in LF varying among the C. elegans lines. For example, line 6 showed the lowest variance (σwithin line = 473.6 [218–1019 95% CI]; Appendix 2 Table A1(B) and Fig. A2), whereas line 18 showed a c. 28 times greater variance (σwithin line = 13180.0 [6076–27480 CI]; Appendix 2 Table A1(B) and Fig. A2). Therefore, similarly to the B. pumilus food environment, we conclude that in the Ecoli food environment there were significant differences in the mean LF among lines and in the within line variance in LF.

image

Figure 2. (A) Mean lifetime fecundity (± 2SE) and (B) the relationship between mean and CV of lifetime fecundity of C. elegans lines fed E. coli OP50; isogenic lines as Figure 1.

Download figure to PowerPoint

In the E. coli environment there was a significant negative relationship between the mean and CV LF of the lines (= −0.88, < 0.001) (Fig. 2B). Reproduction occurred earlier on the E. coli food source (E. coli and B. pumilus maximum daily fecundity on day 2 and 4 respectively).

The within line variance in LF was greater in the poorer quality B. pumilus environment (shown by the CV values in each environment) showing that within line variance in LF is phenotypically plastic (Appendix 2 Fig. A3). There was a relationship in the CVs of the lines in the two food environments (= 0.62, = 0.002) (Fig. 3), but there was no such relationship between the LF of the lines (= 0.29, = 0.20).

image

Figure 3. The relationship between the CV of lifetime fecundity of the C. elegans lines fed B. pumilus and E. coli.

Download figure to PowerPoint

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Conflict of Interest
  9. References
  10. Appendix: Worm and food strains
  11. Appendix
  12. Appendix: WinBugs code

Our results show that variance in lifetime fecundity within an isogenic line differs among C. elegans isogenic lines, thus suggesting there is a genetic effect on phenotypic variance. Specifically, we demonstrate that in the same environment, genetically identical individuals show different levels of among individual phenotypic variation depending on the genotype. There are genotypes with low phenotypic variation and others with high phenotypic variation. We therefore conclude that this phenotypic variance is a property of a genotype.

These results show that the assumption of the equality of phenotypic variance among genotypes is, in this case, false. Furthermore, that we observed that this phenotypic variance differed among recently wild genotypes suggests that this variance may be due to the genotypes' different evolutionary histories. Consistent with this notion is the fact that this phenotypic variance was plastic between the two food environments (Viney and Diaz 2012). Specifically, we find that the absolute degree of phenotypic variance of each of the lines changes as the food environment changes. However, each isogenic line's among-individual variation is similar in both of the two food environments as shown by the positive correlation between CVs. This suggests that although the environment affects the absolute level of variation (i.e. higher in the poorer quality environment) isogenic lines show a similar level of variation across environments.

We have also found that among the C. elegans iosgenic lines the variance in lifetime fecundity is negatively related to mean lifetime fecundity. Why might this pattern exist? We suggest that this may be evidence of different bet hedging strategies. Bet-hedging strategies maximize geometric mean fitness, but at a cost to arithmetic mean fitness. A detailed analysis of bet-hedging strategies has shown that this can be achieved (i) by reducing among individual variance in fitness across generations and (ii) by reducing the correlation in fitness among individuals within a generation (because this effectively decreases the variance in fitness across generations) with these approaches being considered conservative and diversified bet-hedging strategies respectively (Starrfelt and Kokko 2012). Conservative and diversified bet-hedging strategies are extremes along a continuum and both strategies can occur together because they are not mutually exclusive (Starrfelt and Kokko 2012).

To consider our observations with the perspective of bet-hedging strategies, let us assume that the variance in lifetime fecundity is a measure of the variance in the quality of the worms. With this assumption, the C. elegans lines then have different ranges of offspring quality. Considering the negative relationship that we observed between the mean and CV of lifetime fecundity, then among the 21 lines there are a gradient of phenotypes where the extremes are (i) lines of high fecundity, low among individual variance-in-quality through to (ii) lines of low fecundity, high among-individual variance-in-quality.

With this assumption, the low variance lines may be pursuing a conservative bet-hedging strategy, perhaps adapted to a stable, benign environment, such that the variance in fitness across generations is small. In contrast, the high variance lines may be using a diversified bet-hedging strategy, perhaps adapted to a spatially variable environment (Starrfelt and Kokko 2012) where within a generation there is variance among individuals in their fitness, but across generations the fitness variance is minimized, thereby maximizing geometric mean fitness.

The existence of high variance, low fecundity, lines might be puzzling because it would be seem to better for these high variance lines to have a high fecundity too. One reason for this may be that there is some limit or trade-off among individual progeny, such that if one individual offspring is high quality (observed here as having a high lifetime fecundity) other individuals have to be of low quality. This could be achieved in C. elegans, for example, by adult hermaphrodites differently allocating resources among offspring, which results in differences among individuals in their quality, which we observe as differences in lifetime fecundity.

There are caveats to interpreting the phenotypes of these lines as bet-hedging strategies. Thus, it is possible that in other environments the lines' lifetime fecundity and variance in lifetime fecundity will differ in other ways, and present other patterns (or no pattern at all), which might then argue against such a bet-hedging strategy existing in this species. More needs to be known about the natural environment of C. elegans, which is a subject of current work (Félix and Duveau 2012).

Our results can also be interpreted from the perspective of trait canalization (Flatt 2005). We have found that the lines differ in their variability in lifetime fecundity, which is consistent with some lines being strongly canalized for this trait whereas others are not (Baer 2008; Braendle et al. 2010). This view is then tacit about the adaptive value, if any, of this.

Other studies have shown how among individual, within genotype phenotypic variance can occur (e.g. Horvitz and Sulston 1980; Braendle and Félix 2008). A common understanding of the control of a trait is that it is tightly genetically controlled, and that relaxation of that control (for example by mutation) degrades the trait, seen as developmental errors or incomplete penetrance. Our results show that phenotypic variance in lifetime fecundity is a property of a genotype.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Conflict of Interest
  9. References
  10. Appendix: Worm and food strains
  11. Appendix
  12. Appendix: WinBugs code

We thank Marie-Anne Félix, the Caenorhabditis Genetics Center, Robbie Rae and Ralf Sommer for materials; Louise Hughes and Laura Weldon for technical support; Charlie Baer, Andrew Beckerman, Andy Fenton, Andy Higginson, T.J. McKinley, Tom Little, Sarah Reece, Olivier Restif and Bristol colleagues for discussions. MEV would also like to thank the Wissenschaftskolleg zu Berlin for a fellowship that enhanced this work. This work was funded by NERC UK.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Conflict of Interest
  9. References
  10. Appendix: Worm and food strains
  11. Appendix
  12. Appendix: WinBugs code

Appendix: Worm and food strains

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Conflict of Interest
  9. References
  10. Appendix: Worm and food strains
  11. Appendix
  12. Appendix: WinBugs code

C. elegans recently wild isolates were obtained from two sources: JU1400, JU1401, JU1409, JU1410, JU1411, JU1416, JU1442, JU1494 from Marie-Anne Félix (Paris); CB4853, MY1, MY2, MY16, JU262, JU319, JU345, JU362, JU393, JU400, PX174, PX179 and the standard N2 wild-type was obtained from the Caenorhabditis Genetics Centre (CGC). These isolates were collected in France in 2001 (JU262), 2002 (JU319, JU345, JU362, JU393, JU400) or 2008 (JU1494); Spain in 2008 (JU1400, JU1401, JU1409, JU1410, JU1411, JU1416, JU1442); northwest Germany in 2002 (MY1, MY2, MY16); Oregon, USA in 2004 (PX174, PX179) and California, USA in 1974 (CB4853). This information is available at www.justbio.com/worms/index.php and www.cgc.cbs.umn.edu. The single-worm inbreeding was done on an E. coli OP50 food source. After inbreeding worms were cryopreserved (Hope 1999) and maintained at −80°C.

We wished to measure the lines' lifetime fecundity on two different food sources, E. coli OP50 and B. pumilus B215. The E. coli OP50 (the standard laboratory C. elegans food source) was obtained from CGC; the Bacillus pumilus B215 isolate which was obtained from Robbie Rae and Ralf Sommer (Tübingen) (Rae et al. 2010). We chose the B. pumilus B215 because pilot studies showed it to be not overtly pathological to the C. elegans lines and not to induce behaviors which would interfere with measurement of worms' lifetime fecundity. The E. coli OP50 and B. pumilus B215 were cultured as described previously (Hope 1999; Rae et al. 2010).

We observed minimal sterile hermaphrodites throughout out the experiment, specifically, on E. coli OP50 JU1409, JU1411, JU319, MY16 each had one; on B. pumilus B215 N2, JU1442, JU319, JU393, JU400 each had one, and JU1401, JU1411, JU1494, JU262, JU362, MY16, MY2, and PX174 between two and five. These data were included in the analyses. We use the following line codes to report the statistics: 1 = N2, 2 = CB4853, 3 = JU1400, 4 = JU1401, 5 = JU1409, 6 = JU1410, 7 = JU1411, 8 = JU1416, 9 = JU1442, 10 = JU1494, 11 = JU262, 12 = JU319, 13 = JU345, 14 = JU362, 15 = JU393, 16 = JU400, 17 = MY1, 18 = MY16, 19 = MY2, 20 = PX174, and 21 = PX179.

Appendix

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Conflict of Interest
  9. References
  10. Appendix: Worm and food strains
  11. Appendix
  12. Appendix: WinBugs code

Model results: Table A1.

Table A1. Results of preferred Model 3 (see main text, Table 1). It includes the mean, SD and the 2.5 and 97.5% credible intervals (CI) of the posterior distribution for θ (theta), μ (mu), and σwithin line j (sigma.with.line) of each line for (A) B. pumilus B215 and (B) E. coli OP50. These results are of 30000 sampled values at 10001 post burn-in.
NodeMeanSD2.5% CI97.5% CI
(A) B. pumilus
Theta97.649.2079.71116.00
mu[1]126.5018.9588.37162.90
mu[2]86.5010.2266.49107.00
mu[3]52.485.8641.0364.34
mu[4]111.9018.3875.10148.10
mu[5]83.3914.8654.16112.90
mu[6]139.808.55122.20156.20
mu[7]124.6022.5679.17168.40
mu[8]107.8015.8375.98138.60
mu[9]48.577.9633.2364.78
mu[10]167.3019.26126.30202.30
mu[11]35.168.0919.7851.80
mu[12]105.0017.3270.98139.40
mu[13]126.1013.2399.02151.90
mu[14]65.0213.1439.5392.04
mu[15]80.999.4262.4099.70
mu[16]120.8013.7892.75147.60
mu[17]116.1010.5194.86136.80
mu[18]59.8815.3730.4391.21
mu[19]68.1321.1127.12110.70
mu[20]91.5912.5367.13116.80
mu[21]132.509.40113.40150.90
sigma.residual38.647.3526.9555.48
sigma.with.line[1]6926.003091.003212.0014760.00
sigma.with.line[2]1714.00748.70801.903603.00
sigma.with.line[3]525.80234.90242.901113.00
sigma.with.line[4]6719.002957.003113.0014240.00
sigma.with.line[5]3986.001766.001848.008491.00
sigma.with.line[6]1126.00503.30519.302427.00
sigma.with.line[7]11310.004958.005224.0023820.00
sigma.with.line[8]4590.002026.002131.009615.00
sigma.with.line[9]963.30427.40441.202039.00
sigma.with.line[10]5765.002759.002528.0012760.00
sigma.with.line[11]984.90446.10449.702121.00
sigma.with.line[12]5870.002547.002731.0012460.00
sigma.with.line[13]2942.001307.001337.006272.00
sigma.with.line[14]2932.001305.001341.006194.00
sigma.with.line[15]1435.00637.00665.403023.00
sigma.with.line[16]3253.001429.001505.006919.00
sigma.with.line[17]1816.00796.60839.503817.00
sigma.with.line[18]4120.001820.001882.008700.00
sigma.with.line[19]9274.004025.004354.0019330.00
sigma.with.line[20]2728.001205.001251.005788.00
sigma.with.line[21]1405.00626.50647.302980.00
(B) E. coli
Theta248.5010.50227.30268.90
mu[1]284.8010.41263.80305.00
mu[2]229.2013.43203.00256.20
mu[3]276.408.19259.90292.50
mu[4]236.3020.35196.20276.90
mu[5]175.0020.33136.90217.80
mu[6]292.605.61281.20303.50
mu[7]215.6020.44176.10256.90
mu[8]299.3010.62277.50319.70
mu[9]272.209.69252.60291.20
mu[10]275.8014.45246.40303.70
mu[11]233.5014.97204.20263.60
mu[12]247.4021.46205.30290.00
mu[13]232.7016.58200.00266.10
mu[14]162.6025.75116.50218.20
mu[15]260.5010.77238.80281.50
mu[16]267.2021.51223.70309.10
mu[17]298.2012.51272.40322.30
mu[18]222.7024.52174.80271.60
mu[19]211.3021.02170.50254.00
mu[20]295.507.34280.80310.00
mu[21]228.2018.98191.30266.50
sigma.residual43.778.6729.5363.69
sigma.with.line[1]1701.00764.30786.803625.00
sigma.with.line[2]3035.001325.001419.006382.00
sigma.with.line[3]1055.00468.40488.402243.00
sigma.with.line[4]8152.003576.003768.0017210.00
sigma.with.line[5]6476.003057.002878.0014270.00
sigma.with.line[6]473.60211.30218.701019.00
sigma.with.line[7]7855.003446.003620.0016670.00
sigma.with.line[8]1742.00789.10798.603703.00
sigma.with.line[9]1464.00644.80672.603086.00
sigma.with.line[10]3529.001564.001622.007536.00
sigma.with.line[11]3856.001702.001778.008202.00
sigma.with.line[12]9578.004147.004461.0020310.00
sigma.with.line[13]4882.002148.002235.0010360.00
sigma.with.line[14]9887.004963.004191.0022680.00
sigma.with.line[15]1883.00833.00872.403973.00
sigma.with.line[16]9301.004038.004330.0019590.00
sigma.with.line[17]2513.001115.001152.005317.00
sigma.with.line[18]13180.005713.006076.0027480.00
sigma.with.line[19]8282.003619.003874.0017360.00
sigma.with.line[20]840.70376.40383.501794.00
sigma.with.line[21]6765.002974.003131.0014230.00

Data summary: Table A2 and A3.

Table A2. The mean and standard deviation (SD) lifetime fecundity of the lines for each block on the B. pumilus B215 food source.
LineBlock 1Block 2Block 3
MeanSDMeanSDMeanSD
1. N297.8066.34206.6038.55103.0074.68
2. CB485364.0018.84126.2027.0966.6031.62
3. JU140043.4027.0860.0017.3950.6019.19
4. JU1401119.0068.72112.80106.94116.8065.37
5. JU140960.6052.69120.4064.5361.2046.66
6. JU1410144.0026.33142.2043.94140.2027.07
7. JU1411185.2078.7138.0053.57191.4078.75
8. JU1416110.4074.82107.6054.52112.2073.01
9. JU144237.0022.8665.0038.8837.0013.62
10. JU1494202.6037.03160.00110.48195.4032.92
11. JU26225.6021.2050.4037.6720.2019.85
12. JU319140.8058.1537.6048.72142.6055.67
13. JU345156.4032.0083.6053.22150.4030.11
14. JU36259.8039.2852.2071.5069.0044.02
15. JU39383.6015.3175.8056.8980.0028.79
16. JU400155.6023.6863.2041.78154.6021.50
17. MY1126.8019.73106.4068.43120.0011.42
18. MY1666.4067.7034.4044.5356.6071.41
19. MY28.205.54155.6096.013.207.16
20. PX174109.0023.3558.6072.68104.6022.41
21. PX179113.4023.86171.2029.30120.0016.40
Table A3. The mean and standard deviation (SD) lifetime fecundity of the lines for each block on the E. coli OP50 food source.
LineBlock 1Block 2Block 3
MeanSDMeanSDMeanSD
1. N2325.4032.54278.805.50257.4031.63
2. CB4853218.8020.86206.8077.71255.4033.25
3. JU1400295.0029.78254.6027.28283.0021.64
4. JU1401286.8050.80151.2077.02259.2059.05
5. JU1409212.2020.10153.0099.19107.0033.08
6. JU1410288.2026.44303.605.18288.4022.33
7. JU1411224.6051.93186.4078.71207.40117.48
8. JU1416322.2042.20287.4043.34298.6027.04
9. JU1442284.4017.98276.4024.61260.0055.49
10. JU1494282.4035.18272.2082.10283.6050.15
11. JU262216.4083.66239.8058.02237.0029.39
12. JU319241.80135.55303.2029.58195.4052.51
13. JU345241.6022.22227.2053.68220.60105.57
14. JU362157.6053.7556.0026.95181.80105.99
15. JU393265.2022.06264.6064.77254.0030.27
16. JU400319.8039.42217.2020.66283.40139.95
17. MY1303.6025.58303.6080.33301.4019.63
18. MY16255.6023.73130.6070.57245.00150.27
19. MY2200.2024.79248.4038.65151.40130.09
20. PX174309.8017.57297.2037.67284.0019.46
21. PX179248.2044.91245.8061.81175.60101.90

Supporting figures: Figure A1, A2 and A3.

image

Figure A1. Results of Model 3 (see main text, Table 1) showing the posterior distributions of (A) mean (μj) and (B) standard deviation (σwithin line j) ± CI (Credible Intervals, from Appendix 2 Table A1) of LF of lines fed B. pumilus B215. The C. elegans line number is shown is shown in brackets (Appendix 1). The vertical lines represent the global mean of the posterior means and standard deviations among lines, shown in Appendix 2 Table A1.

Download figure to PowerPoint

image

Figure A2. Results of Model 3 (see main text, Table A1) showing the posterior distributions of each (A) mean (μj) and (B) standard deviation (σ2within line j) ± CI of LF of lines fed E. coli OP50. The C. elegans line number is shown is shown in brackets (Appendix 1). The vertical lines represent the global mean of the posterior means and standard deviations among lines, shown in Appendix 2 Table A1.

Download figure to PowerPoint

image

Figure A3. The CV in LF between the two food environments for each C. elegans line.

Download figure to PowerPoint

Appendix: WinBugs code

  1. Top of page
  2. Abstract
  3. Introduction
  4. Materials and Methods
  5. Results
  6. Discussion
  7. Acknowledgments
  8. Conflict of Interest
  9. References
  10. Appendix: Worm and food strains
  11. Appendix
  12. Appendix: WinBugs code

Model 1

  • # model's likelihood

  • model{

  • for (i in 1:Nobs){

  • y[i]∼ dnorm(mu, tau)

  • }

  •  

  • #Priors

  • mu ∼ dnorm(theta,tau.residual)

  • tau∼ dgamma(0.001,0.001)

  • sigma <- sqrt(1/tau)

  •  

  • theta ∼ dnorm(0,1e10)

  • tau.residual ∼ dgamma(0.001,0.001)

  • sigma.residual <- sqrt(1/tau.residual)

  • }

Model 2

  • # model's likelihood

  • model{

  • for (i in 1:Nobs){

  • mu[i] <- muS[line[i]]

  • y[i]∼ dnorm(mu[i], tau.among)

  • }

  • #Priors

  • for(j in 1:Nline){

  • muS[j]∼ dnorm(theta, tau.residual)

  • }

  • tau.among ∼ dgamma(0.001,0.001)

  • sigma.among <- sqrt(1/tau.among)

  •  

  • #Priors

  • theta ∼ dnorm(0,1)

  • tau.residual ∼ dgamma(0.1,0.1)

  • sigma.residual <- sqrt(1/tau.residual)

  • }

Model 3

  • # model's likelihood

  • model{

  • for (i in 1:Nobs){

  • mu[i] <- muS[line[i]]

  • y[i]∼ dnorm(mu[i], tau.within.line[line[i]])

  • }

  •  

  • #Priors

  • for(j in 1:Nline){

  • muS[j]∼ dnorm(theta, tau.residual)

  • tau.within.line[j]∼ dgamma(0.001,0.001)

  • sigma.within.line[j]) <- (1/tau.within.line[j])

  • }

  • #Priors

  • theta ∼ dnorm(0,1.0E-10)

  • tau.residual ∼ dgamma(0.001,0.001)

  • sigma.residual <- sqrt(1/tau.residual)

  • }

Model 4

  • #model's likelihood

  • model{

  • for (i in 1:Nobs){

  • mu[i] <- muS[line[i]] + alpha [exp1[i]]

  • y[i]∼ dnorm(mu[i], tau.within.line[line[i]])

  • }

  •  

  • #Priors

  • for(j in 1:Nline){

  • muS[j]∼ dnorm(theta, tau.residual)

  • tau.within.line[j]∼ dgamma(0.001,0.001)

  • sigma.within.line[j] <- sqrt(1/tau.within.line[j])

  • }

  • #Priors

  • theta ∼ dnorm(0,1.0E-10)

  • tau.residual ∼ dgamma(0.001,0.001)

  • sigma.residual <- sqrt(1/tau.residual)

  •  

  • #Priors

  • for(k in 1:Nexp){

  • alpha[k]∼ dnorm(0,1)

  • }

  • }

Model 5

  • #model's likelihood

  • model{

  • for (i in 1:Nobs){

  •  

  • mu[i] <- muS[line[i]]+ alpha[exp1[i]]

  • y[i]∼ dnorm(mu[i], tau.among)

  • }

  •  

  • #Priors

  • for(j in 1:Nline){

  • muS[j]∼ dnorm(theta, tau.residual)

  • }

  • tau.among∼ dgamma(0.001,0.001)

  • sigma.among <- sqrt(1/tauS tau.among)

  •  

  • #Priors

  • theta ∼ dnorm(0,1)

  • tau.residual∼ dgamma(0.1,0.1)

  • sigma.residual <- sqrt(1/tau.residual)

  •  

  • #Priors

  • for(k in 1:Nexp){

  • alpha[k]∼ dnorm(0,1)

  • }

  • }

We wished to run three parallel chains. Below are the sets of initial values we used. For all the models, initial values were the same for both food sources: E. coli and B. subtilis data.

Model 1:

  • list(tau =1, tau.residual =1)

  • list(tau =10, tau.residual =10)

  • list(tau =100, tau.residual =100)

Model 2:

  • list(tau.residual =1, tau.among =1)

  • list(tau.residual =10, tau.among =10)

  • list(tau.residual =100, tau.among =100)

Model 3:

  • list(tau.residual =1, tau.within.line =c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1))

  • list(tau.residual =10, tau.within.line =c(10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10))

  • list(tau.residual =100, tau.within.line =c(100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100))

Model 4:

  • list(tau.residual =1, tau.within.line =c(1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1), alpha=c(0,0,0))

  • list(tau.residual =100, tau.within.line =c(10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10), alpha=c(0,0,0))

  • list(tau.residual =1000, tau.within.line =c(100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100), alpha=c(0,0,0))

Model 5:

  • list(tau.among =1, tau.residual =1, alpha=c(0,0,0))

  • list(tau.among =10, tau.residual =10, alpha=c(0,0,0))

  • list(tau.among =100, tau.residual =100, alpha=c(0,0,0))