TY - JOUR
AU - Polly, P, David
AB - Synopsis Functional tradeoffs are often viewed as constraints on phenotypic evolution, but they can also facilitate evolution across the suboptimal valleys separating performance peaks. I explore this process by reviewing a previously published model of how disruptive selection from competing functional demands defines an intermediate performance optimum for morphological systems that cannot simultaneously be optimized for all of the functional roles they must play. Because of the inherent tradeoffs in such a system, its optimal morphology in any particular environmental context will usually be intermediate between the performance peaks of the competing functions. The proportional contribution of each functional demand can be estimated by maximum likelihood from empirically observed morphologies, including complex ones measured with multivariate geometric morphometrics, using this model. The resulting tradeoff weight can be mapped onto a phylogenetic tree to study how the performance optimum has shifted across a functional landscape circumscribed by the function-specific performance peaks. This model of tradeoff evolution is sharply different from one in which a multipeak Ornstein–Uhlenbeck (OU) model is applied to a set of morphologies and a phylogenetic tree to estimate how many separate performance optima exist. The multi-peak OU approach assumes that each branch is pushed toward one of two or more performance peaks that exist simultaneously and are separated by valleys of poor performance, whereas the model discussed here assumes that each branch tracks a single optimal performance peak that wanders through morphospace as the balance of functional demands shifts. That the movements of this net performance peak emerge from changing frequencies of selection events from opposing functional demands are illustrated using a series of computational simulations. These simulations show how functional tradeoffs can carry evolution across putative performance valleys: even though intermediate morphologies may not perform optimally for any one function, they may represent the optimal solution in any environment in which an organism experiences competing functional demands. Introduction Functional tradeoffs have often been viewed as constraints on the evolution of phenotypes. If a trait fulfills two or more functions each with a different performance optimum, selection cannot simultaneously maximize its performance for all of those functions (Wainwright 2007; Walker 2007; Bergmann and McElroy 2014). Tradeoff constraints have been demonstrated mathematically and empirically. Charnov and Stephens (1988), for example, derived mathematical life-history models for the joint optimization of a phenotypic trait in the context of several environmentally specific performance tradeoffs and showed how the optimal value would change from one environment to another. Muñoz (2019) in an empirical phylogenetic study of fish jaw apparatuses found that systems with stronger tradeoffs had less morphological disparity and slower rates of evolution than systems with weaker tradeoffs. Even though functional tradeoffs impose constraints on phenotypic evolution, they may also facilitate evolutionary transitions across putative “performance valleys.” For example, tradeoffs in limb lever mechanics prevent a mammal from being simultaneously optimized for sprinting, which requires high mechanical advantage, and digging, which requires high efficiency (Polly 2007). Each of these locomotor specializations has a different function-specific performance that cannot both be maximized because each requires a different ratio between the lengths of proximal and distal limb segments. But both activities may be needed, in which case selection may favor an intermediate morphology that provides some proficiency at both while maximizing neither. For example, the burrowing swift fox (Vulpes velox) of the Great Plains pursues rabbits and hares across open areas (Egoscue 1979) and digs deep dens to escape daytime heat (Cutter 1958), activities that it jointly accomplishes with an intermediate limb ratio (Polly 2010). With competing functional demands, the morphological optimum for functional performance will be context dependent just as a winning poker hand depends on what cards the other players hold (Wade 2016). In this article, I review a previously published likelihood method for estimating the proportional contribution of two or more functions to the evolution of complex phenotypes (Polly et al. 2016). This method is based on the concept of a net performance optimum defined by the tradeoff between competing functional demands. As the balance between the competing functions changes, the net performance peak moves fluidly in morphospace along paths that lie between the performance peaks of each individual function. The method assumes that a performance surface can be estimated for each function, the weighted sum of which describes the net performance surface (see below). The weighting that maximizes the likelihood of an observed phenotype (its height on the net performance surface) represents the proportion of selection events from each competing function that would have been required to produce that phenotype. I show with a computational simulation that the net performance peak, which resembles a peak on an adaptive landscape, is not defined by selection for an intermediate phenotype but is an entirely emergent phenomenon whose location and contours arise from the relative frequencies of independent selective events in opposite directions, each derived from one of the respective function-specific performance surfaces. The scenario used in the simulation is overly simplistic, but it can be conceptually extended to a variety of more realistic scenarios. Finally, I contrast this model of a net performance peak whose position shifts in response to the changing balance of functional demands with a more commonly applied model for evolutionary tradeoffs in which each function is assumed to have a different selective peak, the locations, and widths of which can be estimated by fitting a multipeak Ornstein–Uhlenbeck (OU) model (Hansen 1997). I show that the multipeak OU model of tradeoffs is quite different, both mathematically and conceptually, from the shifting net performance peak model and that the former may not adequately represent a shifting net performance peak that crosses performance valleys between function-specific optima. A performance landscape for functional tradeoffs Polly et al. (2016) proposed a likelihood method for estimating the tradeoff between two or more performance factors that best explain the evolution of an empirically observed phenotype. The method built on the work of Stayton (2009, 2011), who applied finite element analysis (FEA) to a geometric morphometric morphospace to estimate performance surfaces for pond turtle shells (Fig. 1). Every point in the morphospace corresponds to a unique shape. Stayton used three-dimensional landmarks of the carapace and plastron to deform a finite-element mesh to match the shape at regularly spaced intervals across the morphospace. From the meshes, he was able to estimate performance for swimming and predator defense based on the cross-sectional area of the mesh (a proxy for drag and thus hydrodynamic efficiency) and an FEA-derived index of the shell’s resistance to breakage during predator attack. Aquatic locomotion, which is important for foraging and predator escape in the water, is more efficient in species with streamlined shell shapes (e.g., Davenport et al. 1984), whereas high-domed shells are better at resisting terrestrial predators like coyotes and domestic dogs that break the shell to get at the animal inside it (Minckley 1966; Esque et al. 2010). After estimating performance at regular intervals across the morphospace, Stayton fit quadratic surfaces to create a performance surface for each function. The low, flattened carapaces that minimize cross sectional areas were in a different region of morphospace than the high domed shells that maximize resistance to crushing (Fig. 1A–C). Stayton showed that if fitness requires efficient swimming, we would expect selection to drive evolution toward the performance peak that minimizes cross-sectional area, but if fitness involves predator defense, selection should push evolution in the opposite direction toward the performance peak that maximizes strength against crushing. Taxa with shells lying between the function-specific performance peaks experience a classic functional tradeoff that pushes them in opposite directions if fitness involves both predator defense and efficient swimming. Fig. 1 Open in new tabDownload slide An overview of the net performance peak model. Turtle shell morphologies across a multivariate morphospace (A) are measured for function-specific performance resisting predator-induced stresses (B), and hydrodynamic efficiency (C). These are the weighted contributions of the two function-specific performance surfaces produce a tradeoff such that net performance is optimized somewhere between their two peaks (D). The combination of the two function-specific performances that best explains the morphology of a real turtle can be found by estimating the relative weights that maximize its net performance (E) using a likelihood function (F). Fig. 1 Open in new tabDownload slide An overview of the net performance peak model. Turtle shell morphologies across a multivariate morphospace (A) are measured for function-specific performance resisting predator-induced stresses (B), and hydrodynamic efficiency (C). These are the weighted contributions of the two function-specific performance surfaces produce a tradeoff such that net performance is optimized somewhere between their two peaks (D). The combination of the two function-specific performances that best explains the morphology of a real turtle can be found by estimating the relative weights that maximize its net performance (E) using a likelihood function (F). If one or both performances contribute to fitness, then the position of a real turtle shell in the morphospace should be related to the relative contribution of each function to the evolution of that taxon’s morphology. If selection exerted by only one function during its history, then, all other things being equal, the shell morphology should lie close to that function-specific performance peak. But if both functions were equally important, selection should have produced a shell that lies midway between the respective function-specific performance peaks on an intermediate net performance peak whose position is defined by the functional tradeoff. The weighting of the two functions that maximize the likelihood of evolving a shell (or group of shells) with a particular morphology can thus be estimated using this principle and some quantitative genetic theory (Polly et al. 2016). Let’s start with Lande’s (1976) model of multivariate phenotypic evolution, which describes evolution of the population mean of multivariate trait z from one generation to the next as Δz-=Gβ+GN,(1) where G is the additive genetic covariance matrix, β is a vector of selection gradients, and N is effective population size (Lande 1976; Arnold et al. 2001). The first term of Equation (1) describes the response of the trait to selection, the second term is random genetic drift. Drift only makes a small contribution when selection is strong but dominates when selection is weak (such as at a fitness peak). In subsequent equations, we will ignore drift as an error term since it adds to the variance of Δz- but does not change the expected direction or magnitude of the selection component. The selection gradient β is the slope of the selection surface at the trait mean, which in Lande’s model is the slope of the natural log of the reproductive fitness landscape W- ⁠, β= ∂ln⁡W-∂z-.(2) The fitness function W- can theoretically have any form, including a rugged one, but for purposes of discussion we will consider it to have a multivariate normal distribution consistent with standard OU models of trait evolution: W-∝exp⁡(-12z-- θTω+G-1(z-- θ)),(3) where θ is a vector of the optimal values of multivariate trait z, ω is a covariance matrix describing the size and orientation of the surface, and G is the genetic covariance matrix of z in the evolving population (Lande 1980; Hansen 1997; Butler and King 2004). Arnold (1983) pointed out that Lande’s fitness landscape can be decomposed into functional performance and reproductive fitness components, each of which is environmentally specific. Walker (2007) usefully formalized Arnold’s idea by decomposing β into an F matrix of the slopes of traits regressed onto performance (thus representing a functional performance surface such as the ones described above) and a vector w- of the partial regression coefficients of reproductive fitness onto performance in a particular environment (thus describing a fitness surface): β=Fw-.(4) Note that Walker’s w- differs from Lande’s W- in that the former is a vector of fitness slopes for a particular mean phenotype, not a multivariate function like Lande’s, and because w- excludes the performance component of fitness which Lande’s does not. These equations can be used to derive a new equation for Δz- that is decomposed into separate selection gradients for functional performance and for fitness. For the turtle shell example, F would be a matrix consisting of two rows associated with the traits (Principal Component (PC) 1 and PC2, which are independent linear combinations of the shell landmark coordinates) and two columns associated with the functional performances (cross-sectional area and shell stress) and w- would be a vector of fitness gradients for the two performances in a particular environment. F is a therefore the matrix of slopes from the two performance surfaces illustrated in Fig. 1B and C. Because the multiplication of matrix F and vector w- is the sum of their products, Equation (1) can be rewritten as Δz-=G(βF1βW1+βF2βW2)+ε(5) G is a constant, therefore the change in the trait is proportional to the sum of performance-based selection gradients for the two functions. The two selection parameters for functional performance (⁠ βF) in Equation (5) are the gradients of the performance surfaces F. The two selection parameters for reproductive fitness (⁠ βW) can be thought of as the proportional weight w that each performance variable has on Δz- where w1 = 1 − w2 and w1 ranges from 0 to 1. This implies a selection surface ln⁡W- that is the sum of the two performance surfaces weighted by their relative contribution to fitness: lnW-= w1F1+w2F2.(6) The slope of this surface describes the gradient produced by the net contributions of selection from the two functional performance surfaces. The net performance peak necessarily lies somewhere between the function-specific performance peaks. Figure 1D shows the combined peak when w1 = w2 = 0.5. Note that with two function-specific performance peaks, the net performance peak lies on a line connecting the two known as the Pareto front (Shoval et al. 2012). When there are more than two function-specific peaks the Pareto front becomes a convex hull polygon connecting them. The weighting factor w is a measure of the tradeoff between the function-specific performance surfaces. It can be estimated using likelihood by finding the values of w that maximize the height of a real shell on the net performance peak, in other words, the values that maximize its fitness with respect to the two performance factors (Polly et al. 2016). Because intermediate net performance peaks are lower than either of the function-specific peaks, its height must be standardized as part of the maximization procedure. If w2 = 1 – w1, the weighting that maximizes the likelihood of the observed mean phenotype(s) can be found by maximizing the following equation: ln⁡w1F1z-+(1-w1)F2[z-]Max[w1F1+(1-w1)F2].(7) For the shell of the painted turtle, Chrysemys picta, the values that maximize its net performance are w1 = 0.46 and w2 = 0.54 (Fig. 1E and F). Note that the weights reported here are different than the ones reported by Polly et al. (2016) because in that article w1 was held constant at 1.0 and w2 was allowed to vary from zero to infinity as it would if the coefficient directly represented selection intensity instead of the proportion of selection events contributed by each function-specific performance. Once estimated for each taxon in the analysis, the weights w effectively describe the positions of their individual net performance peaks relative to the positions of the function-specific peaks. This approach was used to reconstruct the history of functional tradeoffs in the evolution of shell shape in emydid turtles (Polly et al. 2016). That study revealed that aquatic species were more influenced by hydrodynamic optimization than terrestrial ones, but the difference was not statistically significant. It also showed that emydids occupied a constrained area of morphospace nearer the Pareto front than expected by chance, indicating that the tradeoff does indeed constrain shell morphology, but that hydrodynamic efficiency and shell strength were not collectively sufficient to explain all of the variations in shape even when drift and phylogenetic factors were taken into account. Stayton (2019) later extended the analysis to show that performance for self-righting accounted for additional variance in shell shape. The net performance peak model can be used to describe the evolutionary history of a functional tradeoff by estimating w for each tip taxa and mapping it onto a phylogenetic tree (Polly et al. 2016). This is done in Fig. 2 using a least squares method assuming Brownian motion (Martins and Hansen 1997; Blomberg et al. 2003), which is equivalent to the maximum-likelihood estimation under the same model (Schluter et al. 1997; Revell and Collar 2009). In the turtles, the phylogenetic mapping revealed that hydrodynamic performance (small cross-sectional area) became independently more important in two or three aquatic clades (Fig. 2). In general, the phylogenetic mapping of w shows how the importance of the function-specific performances has changed over the evolutionary history of the clade, and thus how the position of the net performance peak has shifted. Fig. 2 Open in new tabDownload slide The evolution of the tradeoff between the two function-specific performance peaks can be studied phylogenetically by estimating the optimal tradeoff weight (w1, which equals 1 − w2) and mapping it onto a phylogenetic tree. Fig. 2 Open in new tabDownload slide The evolution of the tradeoff between the two function-specific performance peaks can be studied phylogenetically by estimating the optimal tradeoff weight (w1, which equals 1 − w2) and mapping it onto a phylogenetic tree. Net performance peaks are different from Simpson’s original adaptive peak model This model of a shifting net performance peak that emerges solely from the competing demands of two different function-specific performance peaks is fundamentally different from the earliest notions of what an “adaptive peak” represents. While most current mathematical models of adaptive landscapes are derived from Lande’s work (Lande 1976, 1979; Arnold et al. 2001), the concept of the adaptive peak has its roots in Simpson’s vision of the role of selection in macroevolution (Simpson 1944). Drawing on Wright’s metaphorical portrayal of a topographically complex fitness landscape for genotypes (Wright 1932), Simpson argued that selection for phenotypes was directly related to their performance in a particular environment. Simpson linked the evolution of the high-crowned grazing molars and distally elongated limbs of modern horses from the low-crowned browsing teeth and five-toed feet of their Eocene ancestors to the spread of grasslands in the Miocene where long-lasting dentitions and efficient long-distance locomotion conveyed a performance advantage and thus increased fitness. The horse clade split along environmental lines into woodland and grassland clades, each with phenotypes that performed well in their respective habitats. Adaptive peaks for Simpson thus were associated with environmentally circumscribed adaptive zones, each of which supported lineages with phenotypes that performed well in that environment. Evolution was comparatively slow within the bounds of one of Simpson’s adaptive zones but occasionally proceeded in rapid “quantum” bursts across the fitness valleys between them, dynamics that are described by today’s multiple adaptive peak OU models of evolution (e.g., Hansen 1997). For Simpson, there was a straightforward, one-to-one correspondence between environment, performance, phenotype, fitness, and adaptive peak. In the tradeoff model presented above, the net fitness peak, which is analogous to Simpson’s adaptive peak, does not necessarily correspond to any function-specific performance peak. Even when fitness is fully conditional on performance, as is assumed in the derivations above, conflicting functional demands can create a situation in which the fittest phenotype is one that objectively underperforms in all of its functions. The intermediate net performance peak in Fig. 1E is truly a tradeoff between irreconcilable functional needs, not the fleeting results of a dash across as fitness valley between adaptive zones. The net performance peak may be evolutionarily stable even though there is no selection toward it, only away from it toward the function-specific performance peaks. Their tug of war causes the phenotype to stall somewhere between them at a position that maximizes fitness. The net performance peak thus lies objectively in a performance valley, making it quite different from Simpson’s concept in which an intermediate peak would necessarily be associated with an intermediate function. Net performance peaks emerge from unrelated selective events That intermediate net performance peaks are not associated with intermediate functions but emerge from the selective pulls of irreconcilable functions can be illustrated by a simple stochastic simulation in which the phenotypes of local populations are subject to selection toward one performance optimum or another depending on the conditions of the local habitat. In these simulations, the mean morphology of the species as a whole converges on an intermediate value for which there is no direct selection. Figure 3 illustrates the main components of the simulations, which are fully described in Supplementary Data S1. Performance surfaces were modeled to emulate the real turtle example shown above using Equation (3). A hypothetical geographic landscape was created with watery cells (blue) where fitness depends on hydrodynamic efficiency surrounded by grassland (yellow) or forest (green) cells where fitness depends on shell strength to resist predatory bites. At each step, local populations were subjected to selection based on the habitat they occupy, they undergo genetic drift, they disperse and interbreed with adjacent cells, and they can become extirpated if their phenotype is too different from the local performance optimum. Fitness in any given cell depends on one and only one function-specific performance peak. In water cells fitness is entirely dependent on hydrodynamic efficiency, in grassland or forest cell, fitness is entirely dependent on shell strength. Nowhere on the grid was their selection for both performances on the same local population and nowhere was selection toward an intermediate phenotype. Nevertheless, the phenotypes local populations could end up anywhere because of the interaction between dispersal, the strength of local selection, and the likelihood of extirpation. Fig. 3 Open in new tabDownload slide Simulation showing how the frequency of selection from each of the two performance surfaces relates to the tradeoff weightings estimated by likelihood. A virtual geographic landscape of 2271 hexagonal cells was created in which 210 (9.2%) were water (blue) and the remaining 2061 (90.8%) were terrestrial (forest = green, grassland = yellow). In watery cells, selection was determined by performance surface 1 (hydrodynamic efficiency), which favors a flat carapace, and in terrestrial cells selection was chosen by performance surface 2 (shell strength), which favors a domed carapace. The morphospace and performance surfaces are analogous to those in Fig. 1. Simulation code and output are in Supplementary Data S1. Fig. 3 Open in new tabDownload slide Simulation showing how the frequency of selection from each of the two performance surfaces relates to the tradeoff weightings estimated by likelihood. A virtual geographic landscape of 2271 hexagonal cells was created in which 210 (9.2%) were water (blue) and the remaining 2061 (90.8%) were terrestrial (forest = green, grassland = yellow). In watery cells, selection was determined by performance surface 1 (hydrodynamic efficiency), which favors a flat carapace, and in terrestrial cells selection was chosen by performance surface 2 (shell strength), which favors a domed carapace. The morphospace and performance surfaces are analogous to those in Fig. 1. Simulation code and output are in Supplementary Data S1. Despite the spatial complexity, the evolving morphology reaches a stable equilibrium that is almost but not quite what is expected from the frequency of selection. After an initial phase in which the species expands across the landscape, the average phenotype (the mean of all the local populations) asymptotically approaches the equilibrium expected from the proportion of selective events from each of the two function-specific performance surfaces, which is itself determined the number of landscape cells where selection favors hydrodynamic efficiency (9.2%) versus those selecting for shell strength (90.8%; Fig. 4). Even though only one performance surface contributes to selection for any given local population, the phenotypic mean of the species reaches an equilibrium positing along the Pareto front that is ∼91% from the hydrodynamic peak and 9% from the shell strength peak. The same equilibrium is approached regardless of whether local performance selection was weak or strong (Fig. 4). Fig. 4 Open in new tabDownload slide Results of simulations in a static environment. Three simulations were performed: weak selection (A–C), medium selection (D–F), and strong selection (G–I). The graphs in the right column show the distribution of phenotypes as a bivariate histogram in the shell morphospace. The center column shows the location of the mean phenotype at the end of each simulation. The right column shows how the mean phenotype changed over the course of each simulation (curved orange line) with respect to the proportion of watery cells on the landscape (straight blue line). Fig. 4 Open in new tabDownload slide Results of simulations in a static environment. Three simulations were performed: weak selection (A–C), medium selection (D–F), and strong selection (G–I). The graphs in the right column show the distribution of phenotypes as a bivariate histogram in the shell morphospace. The center column shows the location of the mean phenotype at the end of each simulation. The right column shows how the mean phenotype changed over the course of each simulation (curved orange line) with respect to the proportion of watery cells on the landscape (straight blue line). To see why the equilibrium value departs from the expected morphology, let us have a closer look at the events that led up to the equilibrium phase. The simulations started by randomly placing a founder population in one of the terrestrial cells with a phenotype set to the shell strength optimum (3,3) (Fig. 4). Over the first half of each simulation, the populations replicate and disperse until they have filled the landscape (see PhenotypeMap animations in Supplementary Data S2). When the species first encounters watery cells, some local populations start to be selected toward the hydrodynamic optimum (0,0) and the average morphology begins moving away from its starting point on the shell strength peak and eventually reaches an equilibrium after the point when all the cells are contributing selection events. Figure 4 compares the outcomes of this process when local selection is weak, medium, and strong. The left column shows the distribution of local phenotypes in the principal components morphospace at the end of each simulation, which is aligned along the Pareto front with most at or near the shell strength performance peak, as would be expected because most cells select toward that peak (animated versions of the graphs are in Supplementary Data S2). Only in the strong selection, simulation is local phenotypes found at the hydrodynamic peak. A stream of intermediate phenotypes lies scattered between the two performance peaks, most clustering near one of them. The middle column of Fig. 4 shows where the average phenotype for the entire species lies at the end of the simulation (orange dot) and the values it took on earlier in the simulation (gray line; animated versions of these graphs are in Supplementary Data S2). The proportional position of the morphological mean along the Pareto front was 7.3% for weak selection, 7.4% for medium selection, and 8.1% for strong selection, approaching 9.2% but not quite realizing that value. The right column of graphs in Fig. 4 shows the mean phenotype at each step of the model using the same proportional distance. The mean always starts at 0 (the shell strength optimum) and rises toward 1 (the hydrodynamic mean) as drift and selection. When selection is weak, the shift is slow and smooth as the species encounters more hydrodynamic cells (Fig. 4C), but when selection is stronger the curve makes rapid jumps toward the hydrodynamic peak when the expansion front hits each of the two lakes on the landscape and the frequency of hydrodynamic selection events increases (Fig. 4F and I). In all three simulations, the mean phenotype equilibrates close to the 9.2% distance expected from the number of watery cells, but slightly less. There are two reasons why the morphological equilibrium value is not reached, both of which represent important caveats for interpreting estimated values of w. First, drift causes the phenotypes of local populations to vary around their optimum values (Lande 1976). But the effect may be small if selection is non-negligible. The magnitude of drift in these simulations was on the order of 0.003% of the length of the Pareto front, some 35–60 times smaller than the discrepancy of the trait means from the expected value. The magnitude of drift in real data can be approximated from Equation (1) using the phenotypic variance, which can be measured in almost any sample, and choosing a range of plausible heritabilities and effective population sizes to give maximum and minimum estimates of the drift rate (e.g., Cheverud 1988; Wójcik et al. 2006). Most of the departure from the equilibrium point in these simulations can be explained by the interaction of gene flow, the strength of local selection, and the geographic configuration of the landscape. Local populations living in water cells are selected toward the hydrodynamic peak, but their phenotypes are simultaneously pulled toward the shell strength peak by gene flow from populations inhabiting the surrounding terrestrial cells. Conversely, gene flow from aquatic populations pulls nearby terrestrial populations toward the hydrodynamic morphology (see animated phenotype maps in Supplementary Data S2). The deviation due to gene flow is, therefore, greatest near the interfaces between water and land where the local populations along this ecotone end up with intermediate phenotypes. The degree of mixing arises from the balance of the strength of selection and the rate of gene flow, as well as the proportion of ecotone cells that are strongly affected (c.f., Levins 1964; Holt 1987; Hanski and Mononen 2011). The bias caused by gene flow is toward the shell strength peak in these simulations because most terrestrial cells are far from water and thus virtually unaffected, but all of the water cells lie within six steps of a terrestrial cell thus preventing them from ever completely reaching the hydrodynamic optimum (Fig. 3). The net effect is that selection for hydrodynamic efficiency is differentially dampened relative to selection for shell strength, an effect that is stronger with weaker selection. The bias in these simulations arises because of the geographical configuration of the landscape, but the same kind of bias will occur in real world if one of the function-specific performance peaks exerts selection events that are either much more frequent or more intense than the other. Thus we can expect w to be biased toward its nearest function-specific performance peak unless it lies fairly equidistant between them. This bias is thus generalizable to real world data, but its magnitude will be difficult to estimate except in rare cases where selection is well studied in many local populations. Change in frequency of selection carries the net performance peak across the performance valley That the relative frequency of selection events from the function-specific performance peaks determines the location of the net performance peak is even more clearly demonstrated when the proportion changes systematically over time. Changes in the proportion cause the net performance peak to shift its position along the Pareto front, potentially crossing the performance valley between the function-specific peaks. To demonstrate this, the previous simulations were repeated in a dynamically changing environment that started with selection solely in one direction and ended entirely in the other direction. At the start, the landscape had only terrestrial cells each inhabited by a local population optimized for shell strength. As the simulation progressed, cells were randomly flooded such that by halfway through the simulation the entire landscape was water (see animated habitat maps in Supplementary Data S3). Everything else was the same as in the previous simulations (see Supplementary Data S1 for a more detailed description). As the proportion of watery cells increased, and thus the frequency of hydrodynamic selection, the mean phenotype progressively shifted from the shell strength peak to the hydrodynamic peak (Fig. 5 shows this transition using the same format as the graphs in the right column of Fig. 4). The proportion of watery cells is shown as a blue curve and the mean phenotype, measured as the proportional distance along the Pareto front, is shown in orange. In all three simulations, the phenotype shifted from its start at the shell strength peak to and endpoint at (or near) the hydrodynamic peak, traversing all intermediate morphologies along its way. With weak selection, the mean phenotype lagged behind the flooding of the landscape (Fig. 5A), but nevertheless converged on the hydrodynamic optimum by the end of the simulation. While it did not quite reach the peak, it clearly would have if the simulation had lasted longer. With strong selection, the mean phenotype tracks the shift in environment almost perfectly, albeit with a lag at the start. The lags of the morphology behind the change in selection frequency are due to the same effects of gene flow that affected the equilibrium point in the static environment simulations. When selection is weak, it takes considerable time to shift all the local populations to their new optimum point. When selection is strong, the shift is almost simultaneous, but there is a small lag at the beginning because most cells are selecting for shell strength and gene flow buffers the populations that are being selected for hydrodynamic efficiency. Fig. 5 Open in new tabDownload slide Results of simulations in a dynamic environment. Three simulations were performed: weak selection (A), medium selection (B), and strong selection (C). Each simulation started with all terrestrial cells which were randomly flooded until the entire landscape was covered in water (dashed blue curves). The phenotype of every local population was optimized for shell strength at the start of the simulation, and selection with increasing frequency for hydrodynamic efficiency caused the mean phenotype to track the spread of water cover (solid orange curves). Fig. 5 Open in new tabDownload slide Results of simulations in a dynamic environment. Three simulations were performed: weak selection (A), medium selection (B), and strong selection (C). Each simulation started with all terrestrial cells which were randomly flooded until the entire landscape was covered in water (dashed blue curves). The phenotype of every local population was optimized for shell strength at the start of the simulation, and selection with increasing frequency for hydrodynamic efficiency caused the mean phenotype to track the spread of water cover (solid orange curves). What exactly is the tradeoff weight w? These simulations confirm that competing selection for two function-specific performances can produce an intermediate net performance peak in a tradeoff situation. The proportion of selection from each of the two functions determines where the morphological system ends up. The two weights, w1 and w2, can thus be thought of either as the balance of selection from the two function-specific performance surfaces or as the position of the net performance peak relative to the function-specific peaks. In the simulations, the tradeoff balance was controlled by the proportion of habitat where each performance held sway. When turtles were in a watery habitat, selection was entirely toward hydrodynamic efficiency, but when they were in a terrestrial habitat, selection was entirely toward greater shell strength. Local populations were carried toward one adaptive peak or the other, but the mean of the larger panmictic species converged on a predictable morphological position between the peaks that were proportional to the number of selection events from each performance surface. As stated before, no selective process in the simulations was optimized for an intermediate morphology; instead, the morphological outcome emerges from the interaction of the tradeoff and the balance between two sets of conflicting selection events. In these simulations, w1 and w2 are estimates not only of the position of the phenotype, but also the proportion of watery and terrestrial habitat in the species’ range and thus the proportion of selection events for the two functions. But interpreting w1 and w2 in real data is not quite so straightforward. The tradeoff weights should be generalized as an unknown balance between the frequency of selection events and their proportional intensity. In the simulations, the frequency of selection events from the two function-specific performance peaks differed, but their intensities were equal. The same outcomes would have arisen if the frequencies had been equal and the intensities varied by the same proportion because the frequency and intensity of selection events have an interchangeable relationship (Whitlock 1996; Van Dyken and Wade 2010). A 10-to-1 difference in the frequency of opposing selection events of the same intensity will have the same expected outcome as a 10-to-1 difference in their intensity. In real datasets, selection is likely to vary both in intensity and frequency. Intense low-frequency events for one function may make as much of a contribution to the evolution of the morphology as frequent but less-intense events from the other (e.g., Grant et al. 2017). Consequently, the tradeoff weights should be viewed as a generic, dimensionless measure of the relative contribution of the two function-specific performance peaks in terms of frequency and intensity during the evolutionary history of the taxon. What model of evolution best represents the consequences of functional tradeoffs? The tradeoff model discussed here describes a single, mobile net performance peak that emerges from the proportion of conflicting function-specific selection events. It is fundamentally different from a tradeoff model in which one function or the other is maximized. The distinction is of practical importance for choosing a phylogenetic comparative model for inferring whether a functional tradeoff has influenced the pattern of morphological evolution. Some authors have chosen multi-peak OU models to determine whether phenotypes in a clade have a multimodal distribution around a small number of distinct adaptive peaks (e.g. Anderson et al. 2014; Muir 2015; Rebelo and Measey 2019). These methods assess whether the distribution of empirical phenotypes is best explained by multiple fitness peaks, as might be expected if the tradeoff results in one of the competing functions being optimized, or whether it can be better explained by a single peak or a pure Brownian motion process. Like the shifting-peak model reviewed here, these multi-peak models are ultimately derived from theoretical work on the evolutionary expectations of shifting adaptive landscapes (Hansen 1997; Butler and King 2004; Hansen et al. 2008) but as implemented in the study of functional tradeoffs, authors have often equated each function with a separate peak whereas the net performance model is applied to a situation where multiple functions are associated with a single peak whose position varies continuously from one taxon to another. Either model can plausibly be applied to the study of functional tradeoffs, so it is worth looking at exactly how they differ. The choice between them depends on how macroevolutionary landscapes and adaptive zones are conceptualized. Figure 6 compares the two models using a simple univariate simulation. An OU model with separate performance peaks consists of what is effectively a single performance function with two local maxima (blue line; Fig. 6A). Adapting Equation (3), which describes a single adaptive peak shaped like a normal distribution, the two peak OU model (⁠ W-2P ⁠) would be the sum of two such functions, each with a separate θ and ω: W-2P∝exp⁡-12z--θ1Tω1+G-1z--θ1+exp⁡-12z--θ2Tω2+G-1z--θ2 (8) Fig. 6 Open in new tabDownload slide A comparison of the evolutionary outcomes of a multi-peak performance model (A) and the net performance peak model described in this article (B). Multi-peak models select lineages to one of the two peaks, but the net performance peak model simultaneously selects lineages toward both performance peaks creating an optimal net performance peak between them, the position of which depends on the relative frequency or intensity of selection for each of the two functions. Fig. 6 Open in new tabDownload slide A comparison of the evolutionary outcomes of a multi-peak performance model (A) and the net performance peak model described in this article (B). Multi-peak models select lineages to one of the two peaks, but the net performance peak model simultaneously selects lineages toward both performance peaks creating an optimal net performance peak between them, the position of which depends on the relative frequency or intensity of selection for each of the two functions. The selection gradient of W-2P would be the derivative of its log (Lande 1976; Butler and King 2004). Presuming the peaks are stationary, the selection gradient will be either positive or negative (or zero) but never both for any given trait value, therefore any lineage will be selected toward one peak or the other but never to an intermediate point (green lines). Under this multi-peak OU model, an evolving lineage never experiences conflicting selection (except arguably at the precise saddle point between the peaks). In contrast, the net performance peak model involves two independent performance functions, which simultaneously act on an evolving population. The trait is selected in two directions at once and ends up at a compromise point determined by the relative frequency or intensity with which the two selection surfaces are applied (Fig. 6B). Using the same notation, the expectation of the net performance peak model (⁠ W-NP ⁠) can be written as a product of the two terms instead of as their sum: W-NP∝exp⁡-12z--θ1Tω1+G-1z--θ1×exp⁡-12z--θ2Tω2+G-1z--θ2 (9) As before, the selection gradient is the derivative of the log of this equation. That this equation describes the outcomes simulated above can be seen by considering that the model posits that two function-specific selection gradients are operating the morphology independently. Their selection surfaces would each be described by the log of just one of these two terms, each of which makes an additive contribution to Δz- (see Equation 5). Because the sum of two logs is equal to the log of their products, the effect of independent selection events results in the distribution described by Equation (9). Note that Hansen’s (1997) multi-peak OU model was originally conceptualized as one in which adaptive peaks moved across a multivariate morphological landscape, and thus shares much in common with the net performance peak model, even though its recent implementations have been applied to situations where several stationary functional performance peaks are expected. The realized evolutionary paths of lineages in the multipeak-peak model follow smooth deterministic paths, unlike the stochastically varying paths produced by the net performance model (Fig. 6). This is because the latter has substantial stochastic component that arises from the randomness of which peak contributes selection events in any given step. The net performance peak would be similarly deterministic if selection for both performances was applied with precisely the same frequency at each step. The variance of the outcomes in the net performance peak model depends on the pattern of stochasticity and on the width of the two function-specific performance peaks. Wide peaks will increase the variance and narrow peaks will lower it. Regardless, the mean expectation of the net performance peak model is predictable from the proportional frequency/intensity of selection events even if its variance is contingent on other factors. Which model should be applied? The choice depends on the functional morphology of the system and the adaptive process that is likely to govern its evolution. If the morphological system varies between discrete states each associated with a different performance optimum, then a multi-peak OU model with more than one peak best describes how the process is expected to play out in a phylogenetic context. However, if the morphological system can vary continuously in response to competing functional demands, then the net performance peak model best describes the scenario. To apply the latter model phylogenetically, one would estimate the tradeoff weight for each tip taxon and map the change in weights onto the tree as described above (Polly et al. 2016). If the process actually operates like the net performance peak model but a researcher chooses to apply a multi-peak OU model, we would expect the best fit to be a single peak centered somewhere between the function-specific performance peaks with a width parameter that encompasses them both because the net performance model will distribute tip morphologies continuously along the Pareto front (or inside the Pareto polygon if there are more than two function-specific peaks). Conclusions Function, performance, and fitness can have a complex relationship in which the evolution of a structure is governed by compromises that depend on its specific environmental and ecological contexts (Fisher 1985; Losos et al 1993; Koehl 1996). This article illustrates how conflicting selection from two opposing function-specific performance peaks can create an intermediate net performance peak. The net peak emerges only because the morphological system has an inherent functional tradeoff that prevents it from simultaneously being optimized for both function-specific performances. When an organism can only have one morphology, but its environment requires that the morphology serve two or more competing functions, the fittest morphology may be one that performs suboptimally for both functions. The turtle shells in this example cannot simultaneously be optimized for both maximum strength against crushing and hydrodynamic efficiency. Only when morphological systems become decoupled, can performance and fitness both be optimized for distinct functional roles (Liem 1973; Lauder 1983; Wainwright 2007). The net performance peak model suggests that functional tradeoffs could be a mechanism for evolution to cross performance valleys. More precisely, this model suggests that that fitness peaks can shift from one function-specific performance peak to another and thus cross a performance valley. The decomposition of selection into performance and fitness components (Arnold 1983; Walker 2007) highlights the fact that the two need not be synonymous as they were in Simpson’s original conception of the adaptive landscape (Simpson 1944). The idea that the position and shape of an adaptive peak can emerge from a variety of disruptive selectional demands with different directions and magnitudes is not new (e.g., Kimura 1954; Hansen 1997; Boucher et al. 2018), but more often adaptive peaks are conceptualized as the product of stabilizing selection toward the peak rather than away from it. In the latter concept, fitness valleys between peaks either cannot be crossed or they are crossed when the directions of selection change. In the net performance peak model, the target of selection is never the “adaptive” peak itself but rather in the directions of the function-specific performance peaks. The peak shifts not because these directions change, but because the relative importance of the functions changes. Such a process may arise whenever two competing performance demands simultaneously act on a single morphological system that cannot be optimized for both. This idea is similar to Hansen’s (1997) conceptualization of OU peaks in which “[t]he optimum thus reached is a function of all selective demands on the trait, incorporating numerus functional trade-offs and conflicts” (Hansen 1997, 1343), even though Hansen’s paper is viewed as the intellectual origin of the multi-peak OU model contrasted above. In closing, it is worth noting that I have throughout this article blithely ignored the concept of many-to-one mapping between morphology and function (Wainwright et al. 2005; Wainwright 2007). The existence of many-to-one mappings does not change the conceptual results of this article, but they have a profound effect on the details. What are performance peaks in this article would become performance ridges running along the crests of complex manifolds and the simple line of the Pareto front would become a complex surface connecting those ridges. Nevertheless, the idea that conflicting selection for two performances will produce an intermediate morphology that maximizes net performance would still pertain. Fitness, unlike an absolute index of performance for a particular function, varies with the condition of the environment and interactions with other members of an organism’s ecological community (Wade 2016). If the predatory canids in the simulations above were to be progressively extirpated by a Noachian deluge that inundated the landscape, as in the second set of simulations, performance for shell strength ceases to matter and shell shape evolves steadily toward a hydrodynamic optimum. In this scenario, terrestrial turtles would become increasingly unfit regardless of how strong their shells and would eventually become extinct. Aquatic turtles would become increasingly fit and widespread. Only if the shell system was to evolve a way of decoupling hydrodynamic efficiency from shell strength, could fitness optimize a solution where shells could climb to the top of both performance peaks without producing morphological intermediates. Such functional decoupling would depopulate the performance valley and render it uncrossable after it becomes vacated. Decoupling thus produces an either-or scenario of multiple adaptive peaks, whereas tradeoffs require a compromise in which net performance necessarily rafts morphologies across suboptimal valleys between function-specific performance peaks. What opposing functional demands are to morphological tradeoffs is what Gould and Vrba’s (1982) exaptations are to adaptive radiations. An exaptation is a trait that initially evolves with no particular function, but later provides the performance advantage needed to fuel an adaptive radiation when the fitness context changes. Similarly, a functional demand that is initially so rare that it barely contributes to the net performance peak may be one that later facilitates an evolutionary transition between an old performance optimum and a new one, ferrying the phenotype across the performance valley. From the symposium “Melding Modeling and Morphology: Integrating Approaches to Understand the Evolution of Form and Function” presented at the annual meeting of the Society for Integrative and Comparative Biology, January 3–7, 2020 at Austin, Texas. Acknowledgments Thank you to Lindsay Waldrop and Jonathan Rader for organizing the “Melding Modeling and Morphology” symposium at SICB 2020 and for encouraging me to participate. This article would not have been possible without input from Tristan Stayton, Peter Wainwright, Mike Wade, Jim Parham, Amit Patel, my coauthors from our 2016 paper on combining morphometrics and FEA, and the inspiration of the other presentations in the SICB symposium. Three anonymous reviewers and two editors, Cori Richards-Zawacki and Suzanne Miller, offered suggestions that substantially improved this article. All faults rest with the author. Simulations and other analyses were performed in Mathematica© (Wolfram Research, Inc., 2020) with assistance from the Morphometrics for Mathematica v 12.3 and Phylogenetics for Mathematica v 6.5 packages (Polly 2019a, 2019b). This article’s title refers to Psalm 23:4—“Yea, though I walk through the valley of the shadow of death, I will fear no evil: for thou art with me; thy rod and thy staff they comfort me”—in allusion to the difficulty of crossing the valleys that lie between performance peaks, but also to the global coronavirus pandemic of 2020, during the height of which this article was written. I hope tradeoffs carry everyone safely across that looming valley. Funding This work was supported by Indiana University’s Robert R. 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TI - Functional Tradeoffs Carry Phenotypes Across the Valley of the Shadow of Death
JF - Integrative and Comparative Biology
DO - 10.1093/icb/icaa092
DA - 2020-11-01
UR - https://www.deepdyve.com/lp/oxford-university-press/functional-tradeoffs-carry-phenotypes-across-the-valley-of-the-shadow-vutPVXCU2o
SP - 1268
EP - 1282
VL - 60
IS - 5
DP - DeepDyve
ER -