The breeder's equation as a framework

The breeder's equation (see Glossary) is a fundamental tool used in quantitative genetics for understanding phenotypic evolution in response to selection, and has been used by evolutionary biologists for over 50 years. Quantitative traits have phenotypes that are continuously distributed in natural populations, and include morphological, physiological, behavioural and molecular phenotypes. Like other quantitative traits, metabolic rates are likely to be genetically complex and sensitive to environmental conditions. Quantitative genetic variation underlies phenotypic evolution – measuring the genetic basis of variation in quantitative traits is therefore essential to understanding variation in phenotypes, such as metabolic rates. The univariate breeder's equation predicts the amount of change in a single trait from one generation to the next in response to selection. The response of a quantitative trait to selection, R, is described by the breeder's equation: (1)where h² is narrow-sense heritability [the ratio of additive genetic variance (see Glossary) to total phenotypic variance; see ‘Genetic variation in metabolic rates’ section below], and S is the selection differential (the change in population mean after selection) (see Glossary). The breeder's equation serves as a simple, but powerful, tool for understanding variation in metabolic rate and other physiological traits.

Univariate selection on metabolic rate

Selection is the phenotypic covariance between a trait and fitness, where fitness of an individual is determined by the contribution of offspring to the next generation (Falconer and Mackay, 1996). If fitness covaries with a trait, then that trait is said to be under selection. This relative difference in fitness among phenotypes (selection) forms one half of the breeder's equation and provides a standardized estimate of the strength and direction in which evolution is expected to occur, if the trait has adequate genetic variation. The slope of the relationship between relative fitness and a particular character (i.e. selection coefficient), weighted by the phenotype distribution, represents standardized estimates of selection.

Two general forms of univariate selection can occur: linear and quadratic selection (Box 1). Linear selection (see Glossary) occurs when fitness (w) consistently increases or decreases with the value of a trait (z), and is fitted by a linear function: (2)

where α is the intercept of the fitness function, and β is the coefficient giving the direction (positive or negative) and magnitude of selection. If a trait exhibits sufficient genetic variation (i.e. if it is heritable) and not constrained by other traits that are also correlated with fitness (see ‘Metabolic rate is more than a single trait’ section below), persistent directional selection (see Glossary) should result in a shift in the mean trait of a population (Kingsolver and Pfennig, 2007). Quadratic selection is characterized by a non-linear fitness function that can also be positive (disruptive) or negative (stabilizing), and is described by the quadratic fitness function: (3)where γ is the degree of curvature in the fitness function. Selection is stabilizing when β is 0 and γ is negative, such that intermediate values of a trait possess highest fitness while extreme trait values have lowest fitness. Selection is disruptive when β is 0 and γ is positive. Where disruptive selection (see Glossary) is maintained across generations, population variance will increase as selection favours trait values on the tail ends of the trait distribution. Under constant stabilizing selection (see Glossary), there is a single optimal value for a phenotype, hence variance in population traits would be expected to decrease over generations. Note that quadratic selection can occur when β≠0; this is termed either concave (stabilizing) or convex (disruptive) selection. By providing comparable estimates of selection on metabolic rate, selection analyses have the potential to leverage comparative data (i.e. comparing values of β and γ) that vary across spatial and temporal scales, study systems and phenotypic characters (Kingsolver et al., 2001). Indeed, the idea that a single value of a trait is consistently beneficial under all circumstances seems unlikely, and the same is true of metabolic rate. Spatial and temporal variation in selection therefore seems likely to be a mechanism by which variance in metabolic rate is maintained. To some extent, selection analyses have already been implemented in the general mechanisms that have been proposed to explain variation in metabolic rates, i.e. covariance between metabolic rate and some measure of performance (fitness). The increased-intake and compensation hypotheses point towards positive and negative directional selection on (basal or standard) metabolic rate, respectively, for example, while the ‘context dependent’ hypothesis (Burton et al., 2011) points toward selection gradients (see Glossary) that vary in space and time. The approach we advocate is therefore not incompatible with proximate mechanistic approaches; rather selection analyses provide the formalism and standardized measures required to make comparable estimates for the relationship between metabolic rate and fitness.

In order to gain reliable estimates of selection, studies need to measure actual fitness (see Glossary). So far, selection studies on metabolic rate have relied almost exclusively on the use of fitness proxies, such as survival, growth or reproductive traits such as clutch size, rather than the ultimate measure of fitness: lifetime reproductive output (Box 2). This view is illustrated by the compilations of Biro and Stamps (2010), Burton et al. (2011) and White and Kearney (2013). The tables summarizing the known phenotypic correlations between metabolic rate and fitness proxies in these papers do not provide any examples of a correlation between metabolic rate and actual fitness.

Using fitness proxies can create misleading or incomplete interpretations of the strength and direction of selection if these proxies trade-off with actual fitness. For example, Pettersen et al. (2016) show that metabolic rates through ontogeny covary with actual fitness (lifetime reproductive output) as well as several fitness proxies, but the direction and magnitude of the covariance differs among measurements of metabolic rate. Fitness was maximized when individuals had low metabolic rates early in ontogeny (MR_E) but high metabolic rates later (MR_L) (or vice versa). Although we found evidence for correlational selection (see Glossary) alone based on true fitness, estimates based on fitness proxies incorrectly implied that directional selection was operating. For example, individuals with higher MR_E reproduced sooner, but individuals with lower MR_L were longer lived, and growth rate was maximized when MR_E was high and MR_L was low. In this case, using any of the commonly used proxies for fitness (growth rate, longevity, age at the onset of reproduction) would lead to wildly different, and incorrect, conclusions about the expected evolutionary trajectory of metabolic rate.

Genetic variation in metabolic rates

As the breeder's equation elegantly illustrates, selection on a trait will not generate evolution of that trait unless the trait is heritable. The capacity for metabolic rates to evolve thus depends not only on covariation between metabolic rate and fitness, but also on the other half of the breeder's equation – the genetic basis of variation in metabolic rate. The total phenotypic variance of a trait (V_P) is the sum of the variances attributable to genetic (V_G) and environmental (V_E) influences (including maternal effects), and the variance associated with the interaction between genetic and environmental influences (V_GE). V_G can be further subdivided into three components: additive (V_A), dominance (V_D) and interaction (V_I) variance, where collectively V_D and V_I are known as non-additive genetic variance and are not easily disentangled using standard quantitative genetics designs. V_A quantifies deviations from the mean phenotype attributable to the additive contribution of particular alleles to the phenotype; V_D quantifies interactions between alleles (dominance) and V_I quantifies interactions between alleles (epistasis). Heritability in the broad sense (H²) is calculated as V_G/V_P, whereas heritability in the narrow sense (h²) – the metric of interest for the breeder's equation – quantifies the contribution of additive genetic variance to total phenotypic variance and is calculated as V_A/V_P.

The heritability of a trait can be estimated in multiple ways (Box 3), but a common feature of all approaches is that they require the measurement of usually hundreds or thousands of individuals of known pedigree. The requirement to measure so many individuals means that estimates of h² for metabolic rate are historically rare, but are becoming much more common: we are aware of only two estimates published prior to 2000 (Lacy and Lynch, 1979; Lynch and Sulzbach, 1984), and most (43) of the remaining 64 estimates we were able to locate have been published since 2010. The available estimates range from 0 to 0.72; h² is significantly higher for endotherms than for ectotherms, and h² is significantly higher for active metabolic levels than for resting metabolic levels, defined here as the rate of oxygen consumption of an inactive, non-reproductive, post-absorptive animal (Box 4). These heritability estimates suggest that metabolic rate is, in many cases and especially for endotherms and for active metabolic rates, likely to be free to evolve under selection. In support of this suggestion, artificial selection experiments have yielded responses to selection on basal metabolic rate (Ksiazek et al., 2004) and maximum metabolic rate in laboratory mice (Gebczynski and Konarzewski, 2009; Wone et al., 2015), and maximum metabolic rate in bank voles Clethrionomys glareolus (Sadowska et al., 2015).

After accounting for genetic contributions to phenotypic variance, there remains a significant proportion of unexplained variation in metabolic rate that needs to be considered. Variation in metabolic rate may also be a consequence of environmental effects, which can affect metabolic rate either directly (e.g. temperature effects on metabolic rate in ectotherms; Angilletta et al., 2002), or indirectly (e.g. nutritional state on standard metabolic rate; Auer et al., 2015). Parental effects are also known to influence physiological traits (e.g. Bacigalupe et al., 2007; Sadowska et al., 2013). For example, brown trout may alter the routine metabolic rates of their offspring in order to control timing of emergence and therefore dispersal in larvae (Régnier et al., 2010). Addressing the relative importance of heritable versus non-heritable components of variation in metabolic rate will provide a more complete picture of how we expect variation in metabolic rate to evolve.

Multivariate breeder's equation

The univariate breeder's equation is a useful heuristic tool for understanding how microevolutionary processes (see Glossary) work. Increasingly, however, it seems that a more complex approach to predicting microevolution is necessary. The univariate breeder's equation necessarily treats each trait in isolation but it has long been recognized that no trait is an island (Dobzhansky, 1956). Traits covary with each other genetically such that evolution in one trait will necessarily cause evolution in another, and selection often acts on multiple traits simultaneously such that the fitness returns of one trait value depend on the value of other traits. The multivariate breeder's equation reflects this complexity and connectedness of traits in terms of both genetics and selection.

Consider the response to selection of a trait, which we will call trait 1 (z₁). As described by the univariate breeder's equation, the evolution of that trait will of course depend on the selection on that trait (β₁) and the genetic variation in that trait (which we will denote as G_1,1). However, let us suppose that another trait (trait 2) covaries genetically with trait 1; such a covariance would be denoted as G_1,2. Let us also suppose that trait 2 is under selection (β₂). The response of trait 1 (Δz₁) will therefore be the sum of the evolution due to direct selection on trait 1 and the indirect selection (see Glossary) on trait 1 via the genetic covariance (see Glossary) with trait 2 and selection on trait 2, or formally: (4)Furthermore, the covariance between traits 1 and 2 will also be affected by the correlational selection on these traits, formally represented as γ_1,2 (see below). Eqn 4 can be extended to as many traits that genetically covary and experience selection: (5)where the column vector of changes in phenotypic trait values for n traits, Δz={Δz₁, Δz₂, … Δz_n}^T, is a function of a column vector of selection gradients β={β₁, β₂, …}^T and a matrix of genetic variances and covariances [the G matrix (see Glossary)]. Although more complicated, a quick consideration of a realistic but simple example reveals why the multivariate equation provides a more complete understanding of the microevolutionary forces acting on metabolic rate. Suppose for example that trait 1 is metabolic rate and trait 2 is running speed in a hypothetical lizard species. Further assume that metabolic rate is subjected to strong negative directional selection (i.e. β₁ is negative) and that the heritability of metabolic rate is high, because the trait has significant additive genetic variance (G_1,1>0). The univariate breeder's equation and the first component of the multivariate breeder's equation (β₁×G_1,1) would therefore predict that metabolic rate would decrease from one generation to the next. However, further suppose that metabolic rate covaries positively with running speed (G_1,2 is positive) and there is strong positive directional selection for faster running speeds (β₂ is positive). The second component of the multivariate breeder's equation (β₂×G_1,2) would therefore be highly positive and might ‘cancel out’ the selection for lower metabolic rate in the first term. Thus by considering more traits, we move from a misleading prediction of an evolutionary response to a more accurate one. Unfortunately, there is no magic number of traits that should be considered; instead we are left with the rather unsatisfying statement that more traits are likely to be more informative than fewer traits. A multivariate view of evolution is particularly important for considerations of metabolic rate specifically for at least two reasons: first, because metabolic rate is likely to be more than just a single trait, and second, because metabolic rate is almost certainly under multivariate selection.

Metabolic rate is more than a single trait

What is metabolic rate? Measures of metabolic rate integrate the rates at which organisms expend energy to do metabolic work, and so incorporate energy expenditure for a wide range of processes including the maintenance of homeostasis, growth and reproduction, movement and digestion. Metabolic rate is measured as the rate of heat production by direct calorimetry, or – more often – is estimated from rates of oxygen consumption or carbon dioxide production measured by indirect calorimetry (Lighton, 2008). Metabolic rate can be measured for animals that are free-living in the field; for animals at rest; for animals experiencing elevated metabolic rates due to exercise, digestion, lactation, thermogenesis or osmoregulation; or for animals exhibiting depressed metabolism due to hibernation or torpor, hypoxia or anoxia, desiccation or aestivation (Suarez, 2012). The major contributors to whole-organism metabolic rate will change as animals transition through these metabolic states, raising the important question of the extent to which they are constrained to always evolve together (‘metabolic rate’ is a single trait), or free to evolve independently (‘metabolic rate’ is many traits). In mammals, most metabolic activity during basal metabolism is associated with the internal organs including liver, kidney, gastrointestinal tract, heart and brain. Whereas during exercise-induced maximal metabolism, most (ca. 90%) metabolic activity is associated with work done by the locomotor muscles and the work done to deliver substrates and oxygen to these (reviewed by White and Kearney, 2013). From a mechanistic perspective, it therefore seems reasonable to conclude that these metabolic states represent different traits. From a quantitative genetics perspective, however, what matters is the extent to which two putative traits covary genetically. Published mass-independent additive genetic correlations between basal and running-induced maximal metabolic rate range from 0.21 to 0.72 (Dohm et al., 2001; Wone et al., 2009, 2015). Thus these traits – basal and maximal metabolic rate – are at least somewhat free to evolve independently, as has been demonstrated in selection experiments (Sadowska et al., 2015; Wone et al., 2015). What is less clear, however, is the extent to which measurements of a single metabolic state, but taken at different times, represent the same trait.

Two measurements of the same phenotype can be considered a single trait genetically only if they covary perfectly. Resting metabolic rate (as defined earlier) is perhaps the most widely measured physiological phenotype. Resting metabolic rate is repeatable (Nespolo and Franco, 2007; White and Kearney, 2013; Auer et al., 2016a) and heritable (see ‘Genetic variation in metabolic rates’ section above, and Box 4), but not perfectly so. It varies during ontogeny due to changes in size and growth (e.g. Moran and Wells, 2007; Rosenfeld et al., 2015), seasonally (e.g. Smit and McKechnie, 2010), geographically (e.g. Broggi et al., 2007), with food deprivation (e.g. Schimpf et al., 2012), due to changes mitochondrial coupling (Salin et al., 2015), and in response to a range of other biotic and abiotic variables (reviewed by Konarzewski and Książek, 2013; White and Kearney, 2013). Furthermore, not only does metabolic rate vary over time in the same individuals, but individuals can vary in the flexibility of their metabolic rate – in other words, the reaction norm of metabolic rate varies among individuals (Auer et al., 2015, 2016b). Thus an organism has no single metabolic rate, even for a single well-defined metabolic state (e.g. resting metabolic rate), and metabolic rate is therefore likely to be more than one single trait. Even if differences in metabolic rate throughout the life history were trivial, we know from a previous study that selection perceives metabolic rates (and their combinations) differently (Pettersen et al., 2016). In Pettersen et al. (2016), metabolic rate was only measured at two time points in the life history – both during early stages of development, which is unlikely to capture a complete picture of selection. We therefore suggest that the field should work towards gaining multiple measures of metabolic rate if we are to gain an accurate representation of net selection on metabolic rates. We acknowledge the considerable logistical challenges associated with doing so, but we nonetheless advocate treating metabolic rate at different times as separate traits as a useful heuristic for future studies.

Multivariate selection on metabolic rates

Selection acts on combinations of traits, rather than individual traits in isolation (Lande and Arnold, 1983; Blows and McGuigan, 2015). Multivariate (or non-linear correlational) selection examines how selection affects, and is affected by, correlations between traits (Phillips and Arnold, 1989). Studies measuring selection on metabolic rate have largely focused on relationships between fitness and single traits (although see Artacho et al., 2015); however, univariate analyses provide limited scope for predicting change in phenotypic distribution (Phillips and Arnold, 1989). This is because apparent selection on one trait may be due to selection on another unmeasured, correlated trait – resulting in misleading conclusions about selection on the initial trait. Genetically coupled traits will not evolve independently; selection on one trait is likely to cause evolutionary changes in the other trait. For example, selection on metabolic rate early in ontogeny (MR_E) may yield a correlated response in metabolic rate late in ontogeny (MR_L) if MR_E and MR_L are positively genetically correlated, even if there is no direct selection on MR_L. Metabolic rate is known to show additive genetic correlations with a range of traits including body mass (Rønning et al., 2007; Nilsson et al., 2009; Tieleman et al., 2009; Careau et al., 2011; Schimpf et al., 2013), maximum metabolic rate (Sadowska et al., 2005; Wone et al., 2009, 2015), growth rate (Sadowska et al., 2009), the ability to cope with a poor diet (Sadowska et al., 2009) and exploratory behaviour (Careau et al., 2011). These and other additive genetic correlations may constrain the evolution of metabolic rate, but such constraints would not be identifiable in a univariate framework that considers metabolic rate in isolation. If several traits are measured, however, a multivariate approach can determine relative direct and indirect selection acting on each trait through multiple regression.

Correlational selection favours certain combinations of traits, and is measured using second-order polynomial regression to produce a fitness surface that is a function of linear and squared (quadratic) trait values: (6)where z = {z₁, z₂,… z_n}^T is a column vector of phenotypic values for n traits, β={β₁, β₂, …}^T is the column vector of directional selection gradients, and γ is the matrix of non-linear selection (see Glossary) gradients: (7)where γ_i,i is a stabilizing or disruptive selection gradient for trait i, and γ_i,j is a correlational selection gradient for traits i and j (Stinchcombe et al., 2008). Note that in the univariate case where correlational selection is not considered, Eqn 6 simplifies to: (8)(i.e. Eqn 1). Despite the importance of estimating correlational selection for providing a more complete visualization of the distribution of phenotypes, studies that measure correlational selection on physiological traits are rare.

In a study on a bryozoan, Pettersen et al. (2016) found significant negative correlational selection between metabolic rates across two life stages [early (MR_E) and late (MR_L) in juvenile development], but positive phenotypic covariance between these traits (individuals with high MR_E generally possessed high MR_L and vice versa). In other words, there is a positive covariance between the two metabolic rates but selection ‘wants’ to decrease this covariance. Furthermore, under persistent correlational selection across generations, we might expect the positive covariance among metabolic rates to decrease and become negative over time. However, without an understanding of the degree of genetic covariance among traits (such as metabolic rates across ontogeny), our capacity to make such predictions remains limited.

Conclusions and future directions

Metabolic rate is perhaps the most widely measured physiological trait, and has long been argued to have important implications for life history, ecology and evolution. We argue that more widespread adoption of a microevolutionary quantitative genetics framework is valuable for understanding variation in metabolic rate. In adopting such an approach, we should consider metabolic rate as a multivariate trait and measure actual fitness (lifetime reproductive output) in the field, in order to estimate the genetic covariance between metabolic rates and fitness throughout ontogeny. Such measurements are needed in order to understand the drivers of phenotypic variation in metabolic rate.