Genotype-Phenotype Maps Maximizing Evolvability: Modularity Revisited

The mechanisms translating genetic to phenotypic variation determine the distribution of heritable phenotypic variance available to selection. Pleiotropy is an aspect of this structure that limits independent variation of characters. Modularization of pleiotropy has been suggested to promote evolvability by restricting genetic covariance among unrelated characters and reducing constraints due to correlated response. However, modularity may also reduce total genetic variation of characters. We study the properties of genotype-phenotype maps that maximize average conditional evolvability, measured as the amount of unconstrained genetic variation in random directions of phenotypic space. In general, maximal evolvability occurs by maximizing genetic variance and minimizing genetic covariance. This does not necessarily require modularity, only patterns of pleiotropy that cancel on average. The detailed structure of the most evolvable genotype-phenotype maps depends on the distribution of molecular variance. When molecular variance is determined by mutation-selection equilibrium either highly pleiotropic or highly modular genotype-phenotype maps can be optimal, depending on the mutation rate and the relative strengths of stabilizing selection on the characters.     

Short‐term insurance versus long‐term bet‐hedging strategies as adaptations to variable environments

Understanding how organisms adapt to environmental variation is a key challenge of biology. Central to this are bet‐hedging strategies that maximize geometric mean fitness across generations, either by being conservative or diversifying phenotypes. Theoretical models have identified environmental variation across generations with multiplicative fitness effects as driving the evolution of bet‐hedging. However, behavioral ecology has revealed adaptive responses to additive fitness effects of environmental variation within lifetimes, either through insurance or risk‐sensitive strategies. Here, we explore whether the effects of adaptive insurance interact with the evolution of bet‐hedging by varying the position and skew of both arithmetic and geometric mean fitness functions. We find that insurance causes the optimal phenotype to shift from the peak to down the less steeply decreasing side of the fitness function, and that conservative bet‐hedging produces an additional shift on top of this, which decreases as adaptive phenotypic variation from diversifying bet‐hedging increases. When diversifying bet‐hedging is not an option, environmental canalization to reduce phenotypic variation is almost always favored, except where the tails of the fitness function are steeply convex and produce a novel risk‐sensitive increase in phenotypic variance akin to diversifying bet‐hedging. Importantly, using skewed fitness functions, we provide the first model that explicitly addresses how conservative and diversifying bet‐hedging strategies might coexist.     

Evolutionary divergence in directions of high phenotypic variance in the ostracode genus poseidonamicus

Trait variation and covariation are understood to influence the response of populations to natural selection on generational time scales, but their role, if any, in shaping long-term macroevolutionary divergence is still unclear. The present study uses the rich fossil record of the ostracode genus Poseidonamicus to reconstruct in great detail the evolutionary history of a set of landmark-based morphometric characters. This reconstruction included two kinds of evolutionary inferences: ancestor-descendant transitions among populations repeatedly sampled at the same location and divergence between lineages measured as independent contrasts on a phylogeny. This reconstructed history was then used to test if evolutionary changes were concentrated in directions (traits or combinations of traits) with high phenotypic variance. Two different statistics of association between evolution and variation tested the null hypothesis that evolutionary changes occur in random directions with respect to trait variability. The first of these measured the similarity between the directions of evolutionary change and the axis of maximum variance, and the second measured the degree to which evolutionary changes were concentrated in directions of high phenotypic variation. Randomization tests indicated that both kinds of evolutionary inferences (ancestor-descendant and phylogenetic contrasts) occurred preferentially in directions of high phenotypic variance (and close to the axis of maximal variation), suggesting that within-population variation can structure long-term divergence. This effect decayed after a few million years, but at least for one metric, never disappeared completely. These results are consistent with Schluter's genetic constraints model in which evolutionary trajectories on adaptive landscapes are deflected by variation within and covariation among traits.     

THE EVOLUTION OF CANALIZATION AND THE BREAKING OF VON BAER'S LAWS: MODELING THE EVOLUTION OF DEVELOPMENT WITH EPISTASIS

Evolution can change the developmental processes underlying a character without changing the average expression of the character itself. This sort of change must occur in both the evolution of canalization, in which a character becomes increasingly buffered against genetic or developmental variation, and in the phenomenon of closely related species that show similar adult phenotypes but different underlying developmental patterns. To study such phenomena, I develop a model that follows evolution on a surface representing adult phenotype as a function of underlying developmental characters. A contour on such a “phenotype landscape” is a set of states of developmental characters that produce the same adult phenotype. Epistasis induces curvature of this surface, and degree of canalization is represented by the slope along a contour. I first discuss the geometric properties of phenotype landscapes, relating epistasis to canalization. I then impose a fitness function on the phenotype and model evolution of developmental characters as a function of the fitness function and the local geometry of the surface. This model shows how canalization evolves as a population approaches an optimum phenotype. It further shows that under some circumstances, “decanalization” can occur, in which the expression of adult phenotype becomes increasingly sensitive to developmental variation. This process can cause very similar populations to diverge from one another developmentally even when their adult phenotypes experience identical selection regimes.     

Thermoregulation, Flight, and the Evolution of Wing Pattern in Pierid Butterflies: The Topography of Adaptive Landscapes

SYNOPSIS. This paper describes a case study of adaptation, constraint,and evolutionary innovation in pierid butterflies. I developa framework for discussing these issues that focuses on thequestions: What is the form of the adaptive landscape relatingfitness to phenotypic characters? How do such landscapes differfor evolutionarily related groups? I examine the evolution ofwing pigment patterns and thermoregulatory behavior for butterfliesin two subfamilies in the family Pieridae, with three principalresults. First, I show that thermoregulation can be an importantcomponent of fitness in pierids, and that wing color and thermoregulatorybehavior are important phenotypic characters determining thermoregulatoryperformance and the adaptive landscape. Second, I show how limitson possible variation in wing color and behavior constrain evolutionwithin one subfamily of pierids, and how these constraints areset by the physical and biochemical mechanisms of adaptation.Third, I show how evolutionary innovation may have resultedfrom the addition of a new, behavioral dimension to the landscape,and how this addition has altered the functional interrelationsamong various elements of the wing color pattern. I suggestthat comparative analyses of the form and determinants of theadaptive landscape may be useful in identifying evolutionaryinnovations, and complement theoretical analyses of evolutionarydynamics on such fitness surfaces.     

Resolving the paradox of stasis: models with stabilizing selection explain evolutionary divergence on all timescales

We tested the ability of six quantitative genetic models to explain the evolution of phenotypic means using an extensive database compiled by Gingerich. Our approach differs from past efforts in that we use explicit models of evolutionary process, with parameters estimated from contemporary populations, to analyze a large sample of divergence data on many different timescales. We show that one quantitative genetic model yields a good fit to data on phenotypic divergence across timescales ranging from a few generations to 10 million generations. The key feature of this model is a fitness optimum that moves within fixed limits. Conversely, a model of neutral evolution, models with a stationary optimum that undergoes Brownian or white noise motion, a model with a moving optimum, and a peak shift model all fail to account for the data on most or all timescales. We discuss our results within the framework of Simpson's concept of adaptive landscapes and zones. Our analysis suggests that the underlying process causing phenotypic stasis is adaptation to an optimum that moves within an adaptive zone with stable boundaries. We discuss the implication of our results for comparative studies and phylogeny inference based on phenotypic characters.     

A POPULATION GENETIC THEORY OF CANALIZATION

Canalization is the suppression of phenotypic variation. Depending on the causes of phenotypic variation, one speaks either of genetic or environmental canalization. Genetic canalization describes insensitivity of a character to mutations, and the insensitivity to environmental factors is called environmental canalization. Genetic canalization is of interest because it influences the availability of heritable phenotypic variation to natural selection, and is thus potentially important in determining the pattern of phenotypic evolution. In this paper a number of population genetic models are considered of a quantitative character under stabilizing selection. The main purpose of this study is to define the population genetic conditions and constraints for the evolution of canalization. Environmental canalization is modeled as genotype specific environmental variance. It is shown that stabilizing selection favors genes that decrease environmental variance of quantitative characters. However, the theoretical limit of zero environmental variance has never been observed. Of the many ways to explain this fact, two are addressed by our model. It is shown that a “canalization limit” is reached if canalizing effects of mutations are correlated with direct effects on the same character. This canalization limit is predicted to be independent of the strength of stabilizing selection, which is inconsistent with recent experimental data (Sterns et al. 1995). The second model assumes that the canalizing genes have deleterious pleiotropic effects. If these deleterious effects are of the same magnitude as all the other mutations affecting fitness very strong stabilizing selection is required to allow the evolution of environmental canalization. Genetic canalization is modeled as an influence on the average effect of mutations at a locus of other genes. It is found that the selection for genetic canalization critically depends on the amount of genetic variation present in the population. The more genetic variation, the stronger the selection for canalizing effects. All factors that increase genetic variation favor the evolution of genetic canalization (large population size, high mutation rate, large number of genes). If genetic variation is maintained by mutation-selection balance, strong stabilizing selection can inhibit the evolution of genetic canalization. Strong stabilizing selection eliminates genetic variation to a level where selection for canalization does not work anymore. It is predicted that the most important characters (in terms of fitness) are not necessarily the most canalized ones, if they are under very strong stabilizing selection (k > 0.2V_e). The rate of decrease of mutational variance V_m is found to be less than 10% of the initial V_m. From this result it is concluded that characters with typical mutational variances of about 10^–3 V_e are in a metastable state where further evolution of genetic canalization is too slow to be of importance at a microevolutionary time scale. The implications for the explanation of macroevolutionary patterns are discussed.     

Phenotypic noise and the cost of complexity

Experimental studies demonstrate the existence of phenotypic diversity despite constant genotype and environment. Theoretical models based on a single phenotypic character predict that during an adaptation event, phenotypic noise should be positively selected far from the fitness optimum because it increases the fitness of the genotype, and then be selected against when the population reaches the optimum. It is suggested that because of this fitness gain, phenotypic noise should promote adaptive evolution. However, it is unclear how the selective advantage of phenotypic noise is linked to the rate of evolution, and whether any advantage would hold for more realistic, multidimensional phenotypes. Indeed, complex organisms suffer a cost of complexity, where beneficial mutations become rarer as the number of phenotypic characters increases. Using a quantitative genetics approach, we first show that for a one-dimensional phenotype, phenotypic noise promotes adaptive evolution on plateaus of positive fitness, independently from the direct selective advantage on fitness. Second, we show that for multidimensional phenotypes, phenotypic noise evolves to a low-dimensional configuration, with elevated noise in the direction of the fitness optimum. Such a dimensionality reduction of the phenotypic noise promotes adaptive evolution and numerical simulations show that it reduces the cost of complexity.     

The effect of phenotypic plasticity on evolution in multipeaked fitness landscapes

When facing the challenge of developing an individual that best fits its environment, nature demonstrates an interesting combination of two fundamentally different adaptive mechanisms: genetic evolution and phenotypic plasticity. Following numerous computational models, it has become the accepted wisdom that lifetime acclimation (e.g. via learning) smooths the fitness landscape and consequently accelerates evolution. However, analytical studies, focusing on the effect of phenotypic plasticity on evolution in simple unimodal landscapes, have often found that learning hinders the evolutionary process rather than accelerating it. Here, we provide a general framework for studying the effect of plasticity on evolution in multipeaked landscapes and introduce a rigorous mathematical analysis of these dynamics. We show that the convergence rate of the evolutionary process in a given arbitrary one-dimensional fitness landscape is dominated by the largest descent (drawdown) in the landscape and provide numerical evidence to support an analogous dominance also in multidimensional landscapes. We consider several schemes of phenotypic plasticity and examine their effect on the landscape drawdown, identifying the conditions under which phenotypic plasticity is advantageous. The lack of such a drawdown in unimodal landscapes vs. its dominance in multipeaked landscapes accounts for the seemingly contradictory findings of previous studies.     

Constraints on the evolution of adaptive phenotypic plasticity in plants

The high potential fitness benefit of phenotypic plasticity tempts us to expect phenotypic plasticity as a frequent adaptation to environmental heterogeneity. Examples of proven adaptive plasticity in plants, however, are scarce and most plastic responses actually may be 'passive' rather than adaptive. This suggests that frequently requirements for the evolution of adaptive plasticity are not met or that such evolution is impeded by constraints. Here we outline requirements and potential constraints for the evolution of adaptive phenotypic plasticity, identify open questions, and propose new research approaches. Important open questions concern the genetic background of plasticity, genetic variation in plasticity, selection for plasticity in natural habitats, and the nature and occurrence of costs and limits of plasticity. Especially promising tools to address these questions are selection gradient analysis, meta-analysis of studies on genotype-by-environment interactions, QTL analysis, cDNA-microarray scanning and quantitative PCR to quantify gene expression, and two-dimensional gel electrophoresis to quantify protein expression. Studying plasticity along the pathway from gene expression to the phenotype and its relationship with fitness will help us to better understand why adaptive plasticity is not more universal, and to more realistically predict the evolution of plastic responses to environmental change.     

Fitness landscapes emerging from pharmacodynamic functions in the evolution of multidrug resistance

Adaptive evolution often involves beneficial mutations at more than one locus. In this case, the trajectory and rate of adaptation is determined by the underlying fitness landscape, that is, the fitness values and mutational connectivity of all genotypes under consideration. Drug resistance, especially resistance to multiple drugs simultaneously, is also often conferred by mutations at several loci so that the concept of fitness landscapes becomes important. However, fitness landscapes underlying drug resistance are not static but dependent on drug concentrations, which means they are influenced by the pharmacodynamics of the drugs administered. Here, I present a mathematical framework for fitness landscapes of multidrug resistance based on Hill functions describing how drug concentrations affect fitness. I demonstrate that these ‘pharmacodynamic fitness landscapes’ are characterized by pervasive epistasis that arises through (i) fitness costs of resistance (even when these costs are additive), (ii) nonspecificity of resistance mutations to drugs, in particular cross‐resistance, and (iii) drug interactions (both synergistic and antagonistic). In the latter case, reciprocal drug suppression may even lead to reciprocal sign epistasis, so that the doubly resistant genotype occupies a local fitness peak that may be difficult to access by evolution. Simulations exploring the evolutionary dynamics on some pharmacodynamic fitness landscapes with both constant and changing drug concentrations confirm the crucial role of epistasis in determining the rate of multidrug resistance evolution.     

The spatial scale of local adaptation in a stochastic environment

The distribution of phenotypes in space will be a compromise between adaptive plasticity and local adaptation increasing the fit of phenotypes to local conditions and gene flow reducing that fit. Theoretical models on the evolution of quantitative characters on spatially explicit landscapes have only considered scenarios where optimum trait values change as deterministic functions of space. Here, these models are extended to include stochastic spatially autocorrelated aspects to the environment, and consequently the optimal phenotype. Under these conditions, the regression of phenotype on the environmental variable becomes steeper as the spatial scale on which populations are sampled becomes larger. Under certain deterministic models – such as linear clines – the regression is constant. The way in which the regression changes with spatial scale is informative about the degree of phenotypic plasticity, the relative scale of effective gene flow and the environmental dependency of selection. Connections to temporal models are discussed.     

Constraints on evolution and postcopulatory sexual selection: trade-offs among ejaculate characteristics

Ejaculates function as an integrated unit to ensure male fertility and paternity, can have a complex structure, and can experience multiple episodes of selection. Current studies on the evolution of ejaculates typically focus on phenotypic variation in sperm number, size, or related traits such as testes size as adaptations to postcopulatory male-male competition. However, the evolution of the integrated nature of ejaculate structure and function depends on genetic variation in and covariation between the component parts. Here we report a quantitative genetic study of the components of the ejaculate of the cockroach Nauphoeta cinerea, including those we know to experience postcopulatory sexual selection, in the context of functional integration of ejaculate characters. We use the patterns of genetic variation and covariation to infer how the integration of the functions of the ejaculate constrain and shape its evolution. Ejaculate components were highly variable, showed significant additive genetic variance, and moderate to high evolvability. The level of genetic variation in these characters, despite strong directional or truncating selection, may reflect the integration of multiple episodes of selection that occur in N. cinerea. There were few significant phenotypic correlations, but all the genetic correlations among ejaculate characters were significantly different from zero. The patterns of genetic variation and covariation suggest that there are important trade-offs among individual traits of the ejaculate and that evolution of ejaculate characteristics will not proceed unconstrained. Fully describing the genetic relationships among traits that perform as an integrated unit helps us understand how functional relationships constrain or facilitate the evolution of the complex structure that is the ejaculate.     

Liberating genetic variance through sex

Genetic variation in fitness is the fundamental prerequisite for adaptive evolutionary change. If there is no variation in survival and reproduction or if this variation has no genetic basis, then the composition of a population will not evolve over time. Consequently, the factors influencing genetic variation in fitness have received close attention from evolutionary biologists. One key factor is the mode of reproduction. Indeed, it has long been thought that sex enhances fitness variation and that this explains the ubiquity of sexual reproduction among eukaryotes. Nevertheless, theoretical studies have demonstrated that sex need not always increase genetic variation in fitness. In particular, if fitness interactions among beneficial alleles (epistasis) are positive, sex can reduce genetic variance in fitness. Empirical data have been sorely needed to settle the issue of whether sex does enhance fitness variation. A recent flurry of studies[1-4] has demonstrated that sex and recombination do dramatically increase genetic variation in fitness and consequently the rate of adaptive evolution. Interpreted in light of evolutionary theory, these studies rule out positive in these experiments epistasis as a major source of genetic associations. Further studies are needed, however, to tease apart other possible sources.     

Developmental plasticity and the origin of novel forms: unveiling cryptic genetic variation via "use and disuse"

Natural selection eliminates phenotypic variation from populations, generation after generation-an observation that haunted Darwin. So, how does new phenotypic variation arise, and is it always random with respect to fitness? Repeated behavioral responses to a novel environment-particularly those that are learned-are typically advantageous. If those behaviors yield more extreme or novel morphological variants via developmental plasticity, then previously cryptic genetic variation may be exposed to natural selection. Significantly, because the mean phenotypic effect of "use and disuse" is also typically favorable, previously cryptic genetic variation can be transformed into phenotypic variation that is both visible to selection and biased in an adaptive direction. Therefore, use-induced developmental plasticity in a very real sense "creates" new phenotypic variation that is nonrandom with respect to fitness, in contrast to the random phenotypic effects of mutation, recombination, and "direct effects" of environment (stress, nutrition). I offer here (a) a brief review of the immense literature on the effects of "use and disuse" on morphology, (b) a simple yet general model illustrating how cryptic genetic variation may be exposed to selection by developmentally plastic responses that alter trait performance in response to "use and disuse," and (c) a more detailed model of a positive feedback loop between learning (handed behavior) and morphological plasticity (use-induced morphological asymmetry) that may rapidly generate novel phenotypic variation and facilitate the evolution of conspicuous morphological asymmetries. Evidence from several sources suggests that handed behaviors played an important role both in the origin of novel forms (asymmetries) and in their subsequent evolution.     

Evolution and maintenance of the environmental component of the phenotypic variance: benefit of plastic traits under changing environments

How environmental variances in quantitative traits are influenced by variable environments is an important problem in evolutionary biology. In this study, the evolution and maintenance of phenotypic variance in a plastic trait under stabilizing selection are investigated. The mapping from genotypic value to phenotypic value of the quantitative trait is approximated by a linear reaction norm, with genotypic effects on its phenotypic mean and sensitivity to environment. The environmental deviation is assumed to be decomposed into environmental quality, which interacts with genotypic value, and residual developmental noise, which is independent of genotype. Environmental quality and the optimal phenotype of stabilizing selection are allowed to randomly fluctuate in both space and time, and individuals migrate equally before development and reproduction among different niches. Analyses show that phenotypic plasticity is adaptive within variable environments if correlations have become established between the optimal phenotype and environmental quality in space and/or time. The evolved plasticity increases with variances in optimal phenotypes and correlations between optimal phenotype and environmental quality; this further induces increases in mean fitness and the environmental variance in the trait. Under certain circumstances, however, the environmental variance may decrease with increase in variation in environmental quality.     

The long-term evolution of multilocus traits under frequency-dependent disruptive selection

Frequency-dependent disruptive selection is widely recognized as an important source of genetic variation. Its evolutionary consequences have been extensively studied using phenotypic evolutionary models, based on quantitative genetics, game theory, or adaptive dynamics. However, the genetic assumptions underlying these approaches are highly idealized and, even worse, predict different consequences of frequency-dependent disruptive selection. Population genetic models, by contrast, enable genotypic evolutionary models, but traditionally assume constant fitness values. Only a minority of these models thus addresses frequency-dependent selection, and only a few of these do so in a multilocus context. An inherent limitation of these remaining studies is that they only investigate the short-term maintenance of genetic variation. Consequently, the long-term evolution of multilocus characters under frequency-dependent disruptive selection remains poorly understood. We aim to bridge this gap between phenotypic and genotypic models by studying a multilocus version of Levene's soft-selection model. Individual-based simulations and deterministic approximations based on adaptive dynamics theory provide insights into the underlying evolutionary dynamics. Our analysis uncovers a general pattern of polymorphism formation and collapse, likely to apply to a wide variety of genetic systems: after convergence to a fitness minimum and the subsequent establishment of genetic polymorphism at multiple loci, genetic variation becomes increasingly concentrated on a few loci, until eventually only a single polymorphic locus remains. This evolutionary process combines features observed in quantitative genetics and adaptive dynamics models, and it can be explained as a consequence of changes in the selection regime that are inherent to frequency-dependent disruptive selection. Our findings demonstrate that the potential of frequency-dependent disruptive selection to maintain polygenic variation is considerably smaller than previously expected.     

Rare ecomorphological convergence on a complex adaptive landscape: Body size and diet mediate evolution of jaw shape in squirrels (Sciuridae)

Convergence is widely regarded as compelling evidence for adaptation, often being portrayed as evidence that phenotypic outcomes are predictable from ecology, overriding contingencies of history. However, repeated outcomes may be very rare unless adaptive landscapes are simple, structured by strong ecological and functional constraints. One such constraint may be a limitation on body size because performance often scales with size, allowing species to adapt to challenging functions by modifying only size. When size is constrained, species might adapt by changing shape; convergent shapes may therefore be common when size is limiting and functions are challenging. We examine the roles of size and diet as determinants of jaw shape in Sciuridae. As expected, size and diet have significant interdependent effects on jaw shape and ecomorphological convergence is rare, typically involving demanding diets and limiting sizes. More surprising is morphological without ecological convergence, which is equally common between and within dietary classes. Those cases, like rare ecomorphological convergence, may be consequences of evolving on an adaptive landscape shaped by many‐to‐many relationships between ecology and function, many‐to‐one relationships between form and performance, and one‐to‐many relationships between functionally versatile morphologies and ecology. On complex adaptive landscapes, ecological selection can yield different outcomes.     

Correlated evolution of thermal niches and functional physiology in tropical freshwater fishes

The role of ecology in phenotypic and species diversification is widely documented. Nonetheless, numerous nonadaptive processes can shape realized niches and phenotypic variation in natural populations, complicating inferences about adaptive evolution at macroevolutionary scales. We tested for evolved differences in thermal tolerances and their association with the realized thermal niche (including metrics describing diurnal and seasonal patterns of temperature extremes and variability) across a genus of tropical freshwater fishes reared in a standardized environment. There was limited evolution along the thermal niche axis associated with variation in maximum temperature and in upper thermal limits. In contrast, there was considerable diversification along the first major axis of the thermal niche associated with minimum temperatures and in lower thermal limits. Across our adaptive landscape analyses, 70% of species exhibited evidence of divergence in thermal niches. Most importantly, the first two major axes of thermal niche variation were significantly correlated with variation in lower thermal limits. Our results indicate adaptation to divergent thermal niches and adaptive evolution of related functional traits, and highlight the importance of divergence in lower thermal limits for the evolution of tropical biodiversity.     

Components of uncertainty in clutch-size optimization

Environmental uncertainty can be both a cause and consequence of chance variation in many of the phenotypic factors associated with the control of clutch size in birds. When such uncertainty inflates or otherwise influences the variance associated with expected reproductive success for any genotype, it will also influence the resulting phenotypic optima. Random variation that affects the evolution of clutch size optima explicitly may occur both within (intra-) and across (inter-) generations. Examples of intra-generational uncertainty could include chance variation in: (1) the quality and quantity of offspring, (2) parental quality, and (3) temporal resources like food. Inter-generational uncertainty would include chance variation in demographic and population characters. With respect to clutch (or litter) size, almost all forms of uncertainty tend to favor an optimum (genetic) strategy with a clutch that is smaller than the clutch associated with the apparent or actual maximal fitness of an individual parent. The overall effect of all the components of uncertainty can be evaluated through the integration of all this phenotypic variation: however each step of the integration is a conditional expectation of each component. Therefore, a single factor analysis may indicate a false optimum, and an integrated analysis of all components is necessary to evaluate the importance of their individual and joint effects on the adaptive evolution of clutch size.