Higher calcification costs at lower temperatures do not break the temperature-size rule in an intertidal gastropod with determinate growth

Abstract

The vast majority of ectothermic organisms grow larger when developing at cooler environmental temperatures, a pattern frequently referred to as the temperature-size rule (TSR). Assuming that this reaction norm has adaptive significance, life history theory predicts that converse patterns may evolve if favored by natural selection, namely if the costs associated with complying to the TSR outweigh the benefits. Calcifying ectotherms may comprise such an exception not following the TSR, because calcification is expected to be more costly at lower temperatures thus increasing associated costs. To test this hypothesis, we reared wild-caught juveniles of the intertidal gastropod Monetaria annulus and compared their shell sizes at the end of the juvenile stage between two rearing temperatures. Contrary to our prediction, M. annulus does follow the TSR, suggesting that increased calcification costs at lower temperatures are not high enough to break the TSR. Such plastic responses should be considered when interpreting geographical patterns such as latitudinal size clines, which may be caused at least partly by phenotypic plasticity.

Appendices

Appendix 1: Mathematical proof for the working hypothesis

Assuming that the instantaneous mortality rate, μ, is independent of age or size, the lifetime reproductive success (R ₀) of a determinate grower is simply given by

$$R_{0} = M_{x} \times L_{x} \times \int_{0}^{\infty } {{\text{e}}^{ - \mu t} {\text{d}}t} $$

(1)

where M _x is the age-independent fecundity rate of an individual that matures at age x, L _x is the probability that the individual survives until maturity (i.e., L _x  = e^−μx), and t is the age after maturity (i.e., t = −x at birth). Taking the natural logarithm of R ₀ leads to

$$\ln R_{0} = \ln M_{x} + \ln L_{x} - \ln \mu , $$

(2)

where the first term on the right-hand side can be regarded as the adult size advantage (ASA), if fecundity increases with increasing size. On the other hand, the second term is the delayed maturation disadvantage (DMD), because a longer growth period decreases the survivorship until maturity. Let v(a,x) be the size at maturity of an individual that matures at age x with growth rates, a, where growth rates (a) is passively and uniquely determined by temperature, θ. If $x^{*}$ and $v^{*}$ designate the age and size at maturity that maximize the fitness R ₀, then the dependency of the optimal size at maturity ($v^{*}$) on temperature (θ) is

$$\frac{{{\text{d}}v^{*} }}{{{\text{d}}\theta }} = \frac{{\partial v^{*} }}{\partial a}\frac{\partial a}{\partial \theta } + \frac{{\partial v^{*} }}{\partial \mu }\frac{\partial \mu }{\partial \theta } , $$

(3)

where ∂μ/∂θ > 0 is assumed to reflect adaptive significance of the TSR.

To illustrate the above results, we assign specific functions to the growth and fecundity functions: dv/dx = av ^k and M _x  = v(x)^k, respectively. k is a coefficient for metabolism that often takes a value of two-thirds. Approximating the initial size v(0) to zero, maturation size is explicitly given as v(x) = (a(1 − k)x)^1/(1−k). From Eq. (2), ASA = k ln v(x) and DMD = − μx = − μv(x)^1−k/a(1 − k) in this case (see Fig. 1). Taking the derivative of ln R ₀ with respect to x, equation it to zero, and solving it for x gives the optimal age at maturity, $x^{*}$ = k/(1 − k)μ, and thus $v^{*}$ = v($x^{*}$) = (ak/μ)^1/(1−k). The necessary condition that the TSR is selected against is

$$\frac{{\partial v^{*} }}{\partial a}\frac{\partial a}{\partial \theta } > - \frac{{\partial v^{*} }}{\partial \mu }\frac{\partial \mu }{\partial \theta } , $$

(4)

which is obtained by setting the right-hand side of Eq. (3) to be positive and then rearranging it. When dv/dx = av ^k and M _x  = v(x)^k, we have

$$\frac{1}{a}\frac{\partial a}{\partial \theta } > \frac{1}{\mu }\frac{\partial \mu }{\partial \theta } , $$

(5)

or equivalently,

$$\frac{\partial \ln a}{\partial \theta } > \frac{\partial \ln \mu }{\partial \theta } . $$

(6)

This inequality suggests that the TSR is more likely to be broken when (1) growth rates (a) are small, (2) growth rates (a) decrease more steeply with temperature (θ), or (3) both. The same conclusion is reached when assuming the linear growth function [i.e., v(x) = ax], for example.

Appendix 2: Incorporating NJW into the ANCOVA model

Prior to the statistical analyses on metamorphic size, we defined the “native shell width at metamorphosis” (NJW) as the shell width at metamorphosis of the individuals staying in the wild until metamorphosis (Fig. 4; see also Irie and Morimoto 2008). The NJW should be identical between the two treatments, because the individuals who stayed in the wild until metamorphosis were subject to the common environmental conditions. We incorporated this into our statistical model in order to improve the estimation of regression lines. NJW can be estimated as the intersection point between the regression line of JW on IW and the diagonal line (Fig. 4). As the individuals with the same sex and from the same experiment have the same expected NJW, the regression coefficients, a and b, can be estimated by minimizing the total sum of residual squares, ${\text{RSS}} = \phi_{\text{F}} + \phi_{\text{M}}$. The sum of residual squares for females ($\phi_{\text{F}}$) and males ($\phi_{\text{M}}$) are given as

$$\phi_{\text{F}} = \sum\limits_{i = 1}^{{N_{\text{FL}} }} {\left\{ {{\text{JW}}_{i} - \left( {a_{\text{FL}} {\text{IW}}_{i} + b_{\text{FL}} } \right)} \right\}^{2} } + \sum\limits_{i = 1}^{{N_{\text{FH}} }} {\left\{ {{\text{JW}}_{i} - \left( {a_{\text{FH}} {\text{IW}}_{i} + b_{\text{FH}} } \right)} \right\}^{2} } $$

(7)

and

$$\phi_{\text{M}} = \sum\limits_{i = 1}^{{N_{\text{ML}} }} {\left\{ {{\text{JW}}_{i} - \left( {a_{\text{ML}} {\text{IW}}_{i} + b_{\text{ML}} } \right)} \right\}^{2} } + \sum\limits_{i = 1}^{{N_{\text{MH}} }} {\left\{ {{\text{JW}}_{i} - \left( {a_{\text{MH}} {\text{IW}}_{i} + b_{\text{MH}} } \right)} \right\}^{2} } , $$

(8)

where IW _i and JW _i are initial shell width and juvenile shell width at metamorphosis, respectively, for i-th individual out of N total individuals. Subscripts F and M signify females and males, respectively. Similarly, subscripts L and H denote the lower and the higher temperature treatments conducted in the same year, respectively. Using the least-square estimates of the regression slope ($\hat{a}$) and the y-intercept ($\hat{b}$), the expected NJW is calculated as:

$${\text{E}}\left[ {\text{NJW}} \right] = {{\left( {1 - \hat{a}} \right)} \mathord{\left/ {\vphantom {{\left( {1 - \hat{a}} \right)} {\hat{b}}}} \right. \kern-0pt} {\hat{b}}} . $$

(9)

Since the expected NJW is identical between the two temperature treatments (but not between the sexes) within one experiment, it follows that:

$$\frac{{1 - \hat{a}_{\text{L}} }}{{\hat{b}_{\text{L}} }} = \frac{{1 - \hat{a}_{\text{H}} }}{{\hat{b}_{\text{H}} }} , $$

(10)

which reduces the number of parameters to be estimated. All the least-square estimates were numerically computed using Mathematica (version 7 for Windows, Wolfram Research, Champaign, IL).

One may argue that our data could be equally well analyzed by applying an ordinary ANCOVA method with JW as the dependent variable, IW as the covariate, and temperature and sex as fixed-effect factors. In this case, the interaction effect between IW and temperature should be significant if sample size is sufficiently large; then the conformance to the TSR is accepted when a higher rearing temperature leads to smaller y-intercept (see Fig. 4). However, regression lines were sometimes unstable (i.e., slopes were too steep to attain a meaningful interpretation) in our data, probably because our sample sizes were relatively low and IW was not scattered broadly enough. This is the main reason why we implemented a model with a mathematical constraint designated by Eq. (10), instead of performing an ordinary ANCOVA. Except for this point, our method is mathematically equivalent with ANCOVA, which should be noticed by looking at using the least-square regression lines and the F-statistic computed by dividing the explained variance between groups by the unexplained variance within the groups.

Appendix 3: ML estimation of the lognormal regression parameters

If Y (>0) is a random variable that follows a lognormal distribution and linearly depends on a positive covariate, x, its probability density function is expressed as:

$$g\left( {y|x;\alpha ,\beta ,\sigma^{2} } \right) = \frac{1}{{y\sqrt {2\pi \sigma^{2} } }}\exp \left[ { - \frac{{\left( {\ln y - \left( {\alpha + \beta x} \right)} \right)^{2} }}{{2\sigma^{2} }}} \right] , $$

(11)

where the parameters are $\alpha \in \Re$, $\beta \in \Re$, and $\sigma^{2} > 0$. In this case, the median of Y is given as ${\text{Med}}\left[ Y \right] = \exp \left[ {\alpha + \beta x} \right]$. The maximum likelihood estimates of the parameters are those that maximize the likelihood function:

$$L\left( {\alpha ,\beta ,\sigma^{2} } \right) = \prod\limits_{i = 1}^{n} {\frac{1}{{y_{i} \sqrt {2\pi \sigma^{2} } }}\exp \left[ { - \frac{{\left( {\ln y_{i} - \left( {\alpha + \beta x_{i} } \right)} \right)^{2} }}{{2\sigma^{2} }}} \right]} , $$

(12)

or its logarithmic form:

$$LL\left( {\alpha ,\beta ,\sigma^{2} } \right) = - \frac{n}{2}\ln \left( {2\pi \sigma^{2} } \right) - \sum\limits_{i = 1}^{n} {\ln y_{i} } - \frac{1}{{2\sigma^{2} }}\sum\limits_{i = 1}^{n} {\left( {\ln y_{i} - \left( {\alpha + \beta x_{i} } \right)} \right)^{2} } , $$

(13)

where y _i is DTM and x _i is IW for the i-th individual out of n individuals. Taking the partial derivative of LL with respect to each parameter and equating it to zero yields:

$$\hat{\alpha } = \frac{1}{n}\left( {\sum\limits_{i = 1}^{n} {\ln y_{i} } - \hat{\beta }\sum\limits_{i = 1}^{n} {x_{i} } } \right) , $$

(14a)

$$\hat{\beta } = {{\left( {\sum\limits_{i = 1}^{n} {x_{i} \ln y_{i} } - \hat{\alpha }\sum\limits_{i = 1}^{n} {x_{i} } } \right)} \mathord{\left/ {\vphantom {{\left( {\sum\limits_{i = 1}^{n} {x_{i} \ln y_{i} } - \hat{\alpha }\sum\limits_{i = 1}^{n} {x_{i} } } \right)} {\sum\limits_{i = 1}^{n} {x_{i}^{2} } }}} \right. \kern-0pt} {\sum\limits_{i = 1}^{n} {x_{i}^{2} } }} , $$

(14b)

and

$$\hat{\sigma }^{2} = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left\{ {\ln y_{i} - \left( {\hat{\alpha } + \hat{\beta }x_{i} } \right)} \right\}^{2} } , $$

(14c)

respectively (the “hat” indicates the maximum likelihood estimate of each parameter). Simultaneous Eqs. (14a) and (14b) provide an explicit function of $\hat{\alpha }$:

$$\hat{\alpha } = \frac{{\sum {x_{i}^{2} } \sum {\ln y_{i} } - \sum {x_{i} } \sum {x_{i} \ln y_{i} } }}{{n\sum {x_{i}^{2} - \left( {\sum {x_{i} } } \right)^{2} } }} , $$

(15)

which also gives the maximum likelihood estimates of $\beta$ and $\sigma^{2}$ from Eqs. (14b) and (14c).