Title: Evolution of phenotype-environment associations by genetic responses to selection and phenotypic plasticity in a temporally autocorrelated environment Running Title: Modeling Phenotype-Environment Associations

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Authors: Matt J. Michel1,2, Luis-Miguel Chevin3, and Jason H. Knouft1 Authors’ affiliations: 1 – Department of Biology, Saint Louis University, St. Louis, Missouri, USA, 63103; 2 – Department of Biology and Marine Biology, University of North Carolina, Wilmington, NC 28401; 3 – CEFE-UMR 5175 1919 route de Mende, F-34293 Montpellier, CEDEX 5, France Authors’ emails: [email protected], [email protected], [email protected] Keywords: quantitative genetics model, environmental change, environmental predictability, environmental sensitivity of selection, computer simulation, linear reaction norms Word Count: 5653 Table Count: 0 Figure Count: 5 Data Archival: Not Applicable

This article has been accepted for publication and undergone full peer review but has not been through the copyediting,typesetting, pagination and proofreading process, which may lead to differences between this version and the Version of Record. Please cite this article as doi: 10.1111/evo.12371.

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Abstract Covariation between population-mean phenotypes and environmental variables,

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sometimes termed a „phenotype-environment association‟ (PEA), can result from phenotypic plasticity, genetic responses to natural selection, or both. PEAs can potentially provide information on the evolutionary dynamics of a particular set of populations, but this requires a full theoretical characterization of PEAs and their evolution. Here, we derive formulas for the expected PEA in a temporally fluctuating environment for a quantitative trait with a linear reaction norm. We compare several biologically relevant scenarios, including constant versus evolving plasticity, and the situation where an environment affects both development and selection but at different time periods. We find that PEAs are determined not only by biological factors (e.g., magnitude of plasticity, genetic variation), but also environmental factors, such as the association between the environments of development and of selection, and in some cases the level of temporal autocorrelation. We also describe how a PEA can be used to estimate the relationship between an optimum phenotype and an environmental variable (i.e., the environmental sensitivity of selection), an important parameter for determining the extinction risk of populations experiencing environmental change. We illustrate this ability using published data on the predator-induced morphological responses of tadpoles to predation risk.

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Introduction The observation that population-mean phenotypes covary with certain environmental variables is a core concept of evolutionary ecology and biogeography (Mayr 1942; Clausen et

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al. 1940, 1948; Cain and Sheppard 1954; Endler 1977; Cody and Mooney 1978; Endler 1982). Such relationships, often termed phenotype-environment associations (hereafter PEAs; Langerhans et al. 2007), have historically been identified for populations spanning large geographic areas, for instance, the increase in body mass at higher latitudes (Bergmann‟s rule, Bergmann 1847), or the change in morphology of plant ecotypes across elevational gradients (Clausen et al. 1940, 1948). Recently, PEAs have been extended to the temporal covariation between phenotypes and environment within a population, for example, the increase in breeding time in response to increasing spring temperatures in birds (Husby et al. 2011), or the decrease in body mass with recent climate change (Teplitsky et al. 2008). Correctly measuring and understanding relationships between environments and phenotypes is crucial, because it allows the quantification of the extent phenotypic change is predictable from environmental change. Importantly, this predictability of evolution is a pre-requisite for any practical applications of evolutionary biology. PEAs emerge primarily from two effects of the environment on phenotypes. First,

through natural selection, environments favor individual phenotypes with the highest fitness (Levins 1968; Endler 1986), allowing their genotypes to contribute a greater proportion to subsequent generations. Second, environments can directly affect the development or expression of individual phenotypes through phenotypic plasticity (Pigliucci 2001; WestEberhard 2003; DeWitt and Scheiner 2004), which averaged over the population will also change the mean phenotype. The interplay of these two types of phenotypic responses to the environment (plasticity and genetic evolution) has become a matter of intense empirical research (e.g., Reale et al. 2003; Charmantier et al. 2008; Phillimore et al. 2010; reviewed in

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Gienapp et al. 2007; Hendry et al. 2008), and recent theory has focused on how these processes interact with population growth and extinction risk under anthropogenic climate change (Chevin and Lande 2010; Chevin et al. 2010, 2013; Reed et al. 2010). Empirical work

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indicates that many of the rapid phenotypic changes observed in natural populations include components of both plasticity and genetic responses, while theory indicates that these two processes produce very different predictions for population persistence in a changing environment (Chevin et al. 2013). The complete set of measurements required to jointly estimate the relevant parameters, such as genetic variances, phenotypic plasticity, and environmental changes in selection, may be difficult to collect for some species (Chevin et al. 2013), especially those with long generation times or large body sizes (but see Gienapp et al. 2013 for a notable exception). It would therefore be useful to be able to estimate some of these parameters indirectly by using other variables that are easier to obtain, for instance, the correlation between population-mean phenotypes and environmental variables. Here, we show how PEAs can be used for such a purpose. One important parameter that PEAs have the potential to estimate is the

environmental sensitivity of selection (often denoted as B in theoretical models), which is the amount of change in the optimum phenotype with a change in the environment (Chevin et al. 2010, 2013). A recent theoretical study (Chevin et al. 2010) has identified the environmental sensitivity of selection as a crucial parameter determining the ability of populations to persist under environmental change: in the absence of costs of plasticity (DeWitt et al. 1998), population persistence is most likely if the magnitude of plasticity equals B. Because this parameter is difficult to estimate in natural populations (Chevin et al. 2013), and apart from a few examples (Gienapp et al. 2013), has only been estimated thus far using laboratory studies that may be impractical for some species (Chevin et al. 2010), this potential ability of PEAs to estimate B is especially critical.

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Understanding the conditions in which the slope of a PEA will estimate B and other parameters requires a theoretical framework for the evolution of PEAs. Currently, theory has made very few specific predictions about PEAs under genetic evolution and phenotypic

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plasticity. Phillimore et al. (2010) used PEAs across time and space to estimate what proportion of the covariance between spawning date and temperature was caused by plasticity versus genetic change in the common frog, assuming that all temporal change was plastic. However their approach was purely statistical rather than based on an actual dynamical model, so it did not allow understanding the mechanism through which the PEA is established. Adaptation to a moving optimum phenotype has been thoroughly investigated theoretically. In the absence of plasticity, the extent to which mean phenotypes track their optimum depends on (i) the strength of stabilizing selection, (ii) their adaptive potential (genetic variances and covariances), and (iii) the pattern of environmental change, such as its speed and predictability (Charlesworth 1993; Lande and Shannon 1996; Bürger and Krall 2004; Chevin 2013; Lynch & Lande 1993; Gomulkiewicz & Houle 2009). Phenotypic plasticity should alter these predictions in two ways: although it causes a direct covariance between the trait and the environment, it also influences the response to selection by the nonplastic component of the trait (Gavrilets and Scheiner 1993; Chevin and Lande 2010; Lande 2007). Furthermore, the environment that induces the plastic response may differ from the one exerting selection. The correlation between these two environments is critical to the benefits of plasticity: in general, higher correlations select for higher plasticity (steeper reaction norms; de Jong 1999, 2005; Lande 2009; Scheiner and Holt 2012). However, how this correlation affects the PEA has not been investigated formally in a temporally fluctuating environment. In this study, we analyze a model of evolution of a plastic quantitative trait in a randomly fluctuating environment with a specific focus on the covariation between the mean

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phenotype of a population and the environment (i.e., the PEA). First, we derive a set of formulas that relate the PEA to characteristics of the population (genetic variance, strength of selection, plasticity) and of the environment (temporal autocorrelation, variance,

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predictability) using parameters that can be measured empirically. Then, we apply these formulas to more specific scenarios and determine the conditions for which the slope of a PEA can be used to estimate B, the environmental sensitivity of selection. We also use numerical simulations to test the robustness of analytical predictions and explore ranges of parameters for which these predictions should not hold. Finally, we apply these equations to the well-studied system of tadpole phenotypic responses to predation risk in order to estimate the environmental sensitivity of selection on tadpole tailfin depth along a predation risk gradient. The Model We focus on a measured environmental variable  (i.e., sum of temperatures, precipitation, shading, etc.), whose covariance with a quantitative (polygenic) trait z defines the PEA (see Table S1 in Supporting Material for a description of all parameters). We assume that  fluctuates in time with a stationary distribution with autocorrelation   . The variable  is correlated to another fluctuating environmental parameter  S that affects phenotypic selection for our focal trait by modifying the position of an optimum trait value  . We assume a linear relationship between  and  S with slope B (i.e., the environmental sensitivity of selection). Additionally, is correlated to an environmental parameter  D , which induces a plastic phenotypic response by trait z. That  D may differ from  S is a classic property of phenotypic plasticity, reflecting the fact that the environmental cue causing the development or expression of a plastic trait often provides only indirect and partly reliable information about selective pressures on this trait, notably (but not only) because selection generally

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operates later (Moran 1992; Gavrilets and Scheiner 1993; de Jong 1999; Tufto 2000; Lande 2009; Reed et al. 2010). Furthermore, the distinction between   D , and  S is necessary in studies of natural populations where multiple (putatively unmeasured) environmental

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parameters may change jointly and phenotypic plasticity is often investigated through correlative approaches, in contrast to controlled experiments where the environments are manipulated intentionally. The relationships between environmental parameters are characterized by several

correlation coefficients and regression slopes. The first is the autocorrelation   of the measured environment  over one generation, which together with the variance of fluctuations  2 is the main descriptor of measured environmental variation. The second is the correlation  DS between the environmental cue affecting development of the plastic trait and the environment of selection, which determines to what extent plastic responses allow the mean phenotype to track the optimum. Low  DS implies that the cue is an unreliable indicator of the selective pressure, for instance because it occurs too long before selection operates on the trait. Finally, the regression slope of the environment affecting selection (respectively, of the environment affecting development) against the measured environment is denoted as  S (respectively,  D ). The variances and covariance of the environments of selection and development are related to the variance of the measured environment by the formulas

 S2   S2 2   eS2 2  D2   D2  2   eD

cov( S ,  D )   S  D 2  cov(eS , eD )

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where e S (respectively, e D ) is the residual term from the regression of the environment of selection (respectively, development) against that of the measured environment, with mean of

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2 0 and variance of  eS2 (respectively,  eD ).

We assume a linear reaction norm, such that the phenotype of an individual

responding to the environmental cue  D is z  a  b D  e

with e a residual component of phenotypic variation with mean 0. The reaction norm parameters a and b are properties of the genotype. The elevation a (or intercept) is the breeding value measured in a reference environment, and the slope b measures how the breeding value changes with the environment, which quantifies phenotypic plasticity. Assuming all individuals in a local population experience the same environment in a given generation, the mean phenotype in the population is z  a  b D , with a and b the mean elevation and slope, respectively. From these descriptions of environmental and population parameters, we can formally

define the slope of a PEA as:  PEA  cov( z,  )  2 . The numerator (i.e., the covariance between the mean trait and the measured environment) is

cov( z ,  )  cov(a,  )  b cov( D ,  )  cov(a,  )  be 2

(1)

with be  b D . If plasticity is constant, the first term in eq. (1) is the covariance caused by genetic responses to selection by the mean reaction norm elevation, a , while the second term is the covariance caused by phenotypic plasticity. In eq. (1), the effective plasticity be accounts for the fact that the measured environment may differ from the one that actually affects trait expression. This distinction is important for natural populations, where multiple environmental variables may change jointly, such that estimation of plasticity generally

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involves finding the environmental variable yielding the highest be by maximizing  D (see van de Pol and Cockburn 2011 for estimating the best time window affecting plasticity of phenological traits).

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We also introduce the effective environmental sensitivity of selection, Be  B S , which depends both on the association between the optimum  and the environment of selection  S (with slope B), and on the association between environment of selection  S and the measured environment  (with slope  S ). The former is large if selection on the focal trait changes substantially with the environment, while the latter is large if the environmental variable under study is the relevant one for phenotypic selection. In practice, Be and be are the only parameters that can be measured empirically in the wild where environments are not controlled, even though B and b are biologically more meaningful and drive the evolutionary dynamics. In what follows, we will mostly investigate the case of constant plasticity, b  b , for

which all the genetic response to selection is due to changes in the mean reaction norm elevation a (as in Chevin and Lande. 2010); we briefly address evolving plasticity below (see “Specific scenarios: Evolving plasticity”). The covariance between the non-plastic component of the trait a and the environment  depends on (i) how much selection on a changes with  , and (ii) the cumulative responses of a to selection in all past generations. For a polygenic trait with normally distributed genetic values, the response to selection in each generation is the product of the additive genetic variance G a and the selection gradient

   ln W /  z , with W the mean fitness (Lande 1976). If selection towards an optimum trait value  is modeled by making individual fitness a Gaussian function of phenotype, then mean fitness is also a Gaussian function of the mean phenotype, and the selection gradient is

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   ( z   ) (Lande 1976), where  is the strength of stabilizing selection. With constant phenotypic plasticity, genetic responses to selection by the reaction norm elevation a in each generation are determined by the phenotypic deviations from the optimum, after accounting

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for the mean plastic responses of phenotypes to their environment of development. The selection gradient thus becomes

   [a  ] ,

(2)

where   B S  b D is the optimum for the mean reaction norm elevation a (with the assumption that over the long run in a stationary environment, the average mean breeding value equals the average optimum). The temporal variance of  , denoted as  2 , quantifies the effective magnitude of

fluctuations as „perceived‟ by the non-plastic component of the trait, after accounting for plastic responses, and the temporal autocorrelation of  over a generation,  , quantifies the predictability of selection on this non-plastic component. Assuming constant genetic variance G a , these two parameters can be used to predict the covariance between the mean genetic component of the trait a and the environment in the long run, using methods that were used previously to derive the expected lag load in a fluctuating environment in the absence of plasticity (Charlesworth 1993; Lande and Shannon 1996; Chevin 2013). Under a continuous-time approximation (similar to that in Lande and Shannon 1996; Chevin 2013), and assuming the environment is a first order autoregressive process (such that the autocorrelation is positive and decreases exponentially with time) and that there are moderately strong responses to selection, we show in app. A that the long-term expectation of the covariance between a and  is

 Ga2 cov(a, )  .  Ga  ln 

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From the definition of  , we also have cov(a, )  ( Be  be ) cov(a,  ) , so

 Ga2 . cov(a,  )  ( Be  be )( Ga  ln  )

(3)

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Equations (1) and (3) show how the covariance between the mean trait value and the environment depends on the plasticity and additive genetic variance of a quantitative trait, as well as on the effective magnitude and autocorrelation of fluctuations in the optimum phenotype for this trait. However these effective parameters of fluctuating selection themselves depend on plasticity, since b is included in the definition of  . Expanding terms in the variance of  yields, after some rearrangement,

 2   2 [1    (   2  DS )]

(4)

where      D  S , with   b / B the relative plasticity (as in Chevin et al. 2010), and

 2  B 2 S2 the raw variance of fluctuations in the optimum for z (not accounting for

plasticity). In the absence of plasticity (   0 ), eq. (4) simplifies to 2   2 , that is, fluctuations in the optimum directly influence the response to selection. Phenotypic plasticity affects the response to selection by changing the effective magnitude of fluctuations, which is multiplied by the factor in square brackets. When  D and  S are the same environmental variable but experienced at different time points, then  S   D and eq. (4) reduces to

 2 [1   (  2 DS )] . Furthermore, with perfect correlation between the cue and the

environment of selection (  DS  1 ), we recover the result by Lande (2007, Appendix) that partially adaptive plasticity (i.e., B and b of same sign) reduces the variance of fluctuations by a factor 1    . However with arbitrary  DS , whether fluctuating selection is effectively 2

weaker or stronger with plasticity depends on whether B and b are of the same sign, and on the correlation  DS between the cue and the selective environment (which may depend on the

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temporal autocorrelation of the environment). For instance, plasticity increases the variance of fluctuations whenever   2  DS (assuming DS  0 ), that is, if relative plasticity is larger than twice the reliability of the cue (measured as its correlation with the environment of

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selection), because it causes overshoots of the optimum (see also Reed et al. 2010; Chevin et al. 2013). The effective magnitude of fluctuating selection is at a minimum when    DS , that is, when plasticity scaled to the environmental sensitivity of selection equals the reliability of the cue. The last required element is the temporal autocorrelation of  . The autocovariance

of  over a generation is cov( , ' )  B 2 cov( S ,  S ' )  b 2 cov( D ,  D ' )  Bb(cov( D ,  S ' )  cov( S ,  D ' )) ,

where primes denote values in the next generation. Using the relationships between environmental variables this can be rewritten as





cov , '   2  S   2  D     DS '   SD'  ,

where  S and  D are the autocorrelations of the selective and developmental environments over the generation time, and  DS ' is the correlation between the environmental cue in the current generation and the environment of selection in the next (and reciprocally for  SD ' ). The autocorrelation of  is obtained by dividing by  2 (eq. 4b), yielding

 

 S   2  D     DS '   SD'  . 1       2  DS 

(5)

Equation (5) shows that the effective autocorrelation of fluctuating selection (in terms of its effect on the response to selection by the non-plastic component of the trait), depends in a complex way on the parameters of environmental variation and on plasticity. In the absence of plasticity (   0 ),    S , such that the autocorrelation of selection equals that of the selective environment. Somewhat counter-intuitively, the effective autocorrelation of

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fluctuations decreases with increasing correlation between the cue and the environment of selection across generations (  DS ' and  SD ' ).

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Specific scenarios Equations (1 – 5), while providing a general description of the problem, are not easy to interpret owing to their large number of parameters. In order to gain a deeper understanding of the results, it is worth investigating simplified cases. In this section, we apply the model to specific scenarios that may be common for natural populations or more applicable to experimental studies. Because we were interested in determining the ability of PEAs to estimate the environmental sensitivity of selection, B, we present equations in terms of the regression slope of the population-mean phenotype against the measured environment,  PEA . Predictions from these formulas were compared to computer simulations, which allowed assessing the robustness of the approximations used for their derivation, as well as exploring parameter ranges beyond those where formulas are not expected to apply (see app. B, available online, for information regarding the structure and parameters of the simulations).

No Plasticity – In the absence of plasticity, a simplified formula for the slope of the PEA can be found by combining eq. (1) with eqs. (3 – 5), dividing by  2 , and setting be to zero, yielding

 PEA 

B S Ga , 2 R Ga  ln(  S )

where R 2 is the coefficient of determination of the environment of selection (i.e., the fraction of its variance explained by the measured environment, or  S2 2  S2 ). The PEA slope increases as the temporal autocorrelation in the measured environment, the strength of stabilizing selection, and additive genetic variance increase (Fig. 1; effects of increasing Ga

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not shown). In addition, as the environment of selection more closely matches the measured environment (i.e., a high  S ), the PEA slope also increases and more closely approximates B. This implies that a low empirical PEA may simply mean that the measured environmental

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variable correlates weakly with those that actually affect selection. For instance, Gienapp et al. (2013) recently found unreliable predictions using the model of Chevin et al. (2010) because the environment used for projections did not capture changes in the optimum accurately. This can be attributed to a reduced  S rather than an actual change in B: the optimum still changed, but in response to another environmental variable. Interestingly, a somewhat strong PEA can form in the absence of plasticity. However,

this requires both strong selection (large  ) and a highly predictable environment (large  S ). The simulations closely followed these analytical results, except at high selection strength, high  S , and low levels of temporal autocorrelation, where the analytical solutions were greater than the simulation results (Fig. 1). This difference likely occurs because the analytical formulas were developed under the assumption of weak stabilizing selection (small

 ), and positive autocorrelation (    0 ).

Plasticity and one environmental variable – We next consider the situation where individuals are allowed to exhibit a plastic response to an environment, that (i) also directly affects natural selection, such that  DS  1 , and (ii) is measured directly as , such that  D   S  1 . This scenario could describe experiments where an environment known to cause both phenotypic plasticity and selection is manipulated (reviewed in Scheiner 2002). These assumptions further entail that  DS '   SD '     S   D , leading to     . Combining eqs. (1 – 5) with these conditions yields

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 PEA  b  B  b 

Ga Ga  ln(   )

.

(6)

This shows that with a perfectly reliable cue measured accurately, the contribution of constant plasticity to the PEA equals the slope of the (linear) reaction norm. The contribution

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of genetic evolution is proportional to the difference B  b between the reaction norm slope and the slope of the optimum, and increases with higher genetic variance G a , strength of stabilizing selection  , and temporal autocorrelation in the environment   (Fig. 2). Stronger plasticity (larger b) results in a stronger PEA, and reduces the influence of selection on the PEA. As in the above scenario, the simulation results tracked the analytical results, except at high selection strength (larger  ) and low values of temporal autocorrelation (lower   ).

Delay between development and selection – Often, the same environment influences both development and selection, but at different moments in the life cycle. For instance, the environment may affect the expression of the plastic trait at a critical time in life, some fraction of a generation  before it influences phenotypic selection, as modeled by Lande (2009). An example could be spring temperature acting as a cue for birds, which tune their phenology such that their eggs later hatch during a peak of caterpillar food abundance (Charmantier et al. 2008). For an autoregressive process, this means that the environmental correlation between development and selection is simply  DS    , and the cross-generation correlations are  DS '   1 and  SD '   1 . Furthermore, because the environments of selection and development are the same, we can assume that  D   S and thus,     . Lastly, if the selection environment is the one being measured, then  S  1 and

 D   DS    . Replacing in (4) and (5) yields, after some rearrangement,

2   2 1     2 

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(7)

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        .    1   1     2      Note that both the denominator and numerator of the fraction in  are necessarily positive

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for all 0     1 , which indicates that in this case, partially adaptive plasticity always decreases the autocorrelation of fluctuating selection on the non-plastic component of the trait. A possible reason is that plastic responses capture part of the fluctuation pattern at the time scale of a th of a generation, therefore eliminating this component of the autocorrelation for the non-plastic component of the trait. The expected PEA in this scenario is obtained by inserting (7) into (3) and noting that

Be  be  B  b if the environment is measured upon selection. We plot this expected PEA

slope under different values of plasticity and τ in Fig 3. Again, increases in plasticity and temporal autocorrelation of the measured environment resulted in larger PEA slopes that were closer to the set value of B. Smaller values of τ (i.e., shorter timeframe between development and selection) resulted in greater PEA slope values. This result is due to the greater correlation between developmental and selective environments when τ is small. Generally, the simulation results corresponded well with analytical results at various levels of plasticity and fractional generation time (Fig. 3); however, at high levels of temporal autocorrelation the simulations produced PEA slopes that were greater than those predicted by the analytical formulas.

Evolving Plasticity – When there is genetic variance in reaction slopes, it is more difficult to predict the relative contributions of plasticity and genetic adaptation to PEAs. Indeed in that case, the response to selection can occur through both components of the reaction norm (i.e., elevation and slope); that is, plasticity itself can evolve (reviewed in Scheiner 1993). In an

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environment that fluctuates with a stationary distribution and if  S and  D have equal variances, the mean plasticity will evolve in the long run to b eq   DS B , such that plasticity equals the reliability of the cue (or predictability of the environment of selection) multiplied

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by the environmental sensitivity of selection (Gavrilets and Scheiner 1993; Lande 2009). However, in natural populations, the different environments affecting development and selection are likely to have unequal variances. In this more realistic scenario, plasticity will evolve to  S|D B , the regression slope of the environment of selection against the environment of development multiplied by the environmental sensitivity of selection. An approximation can thus be made by using b eq   S|D B instead of b in the formulas

for constant plasticity to account for the long-term effect of evolving plasticity on PEA. In essence, this consists in assuming that even though plasticity can evolve, after it has reached its expected equilibrium it will respond less to selection than the mean reaction norm elevation in each generation. This approximation will be all the more accurate as the magnitude of fluctuations in the environment of development multiplied by the genetic variance in reaction slopes is smaller than the genetic variance in reaction norm elevation, such that most of the response to selection in any generation is caused by the evolution of the non-plastic component of the trait (Lande 2009). For the scenario involving multiple environmental variables (  ,  S , and  D ), the regression slope  S|D will equal  DS  S  D . For the scenario involving only one environment that influences development and selection at different time periods in the individual‟s life-history (i.e., the „delay between development and selection‟ scenario),  S|D   DS    . It is also worth noting that the equilibrium plasticity beq is also the plasticity that minimizes the effective magnitude of fluctuating selection on the non-plastic component of the trait in eq. (4), when the variances of the developmental and selective environment are equal.

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In the simulations, plasticity was allowed to evolve by setting Gb > 0, where Gb is the additive genetic variance in the reaction norm slope. The number of generations was also increased to 10,000. For both types of environmental scenarios (multiple environments and

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delay between development and selection), plasticity quickly evolved to  S|D B (about 500 generations), and the results closely matched predictions from the analytical formulas (Fig. 4). For the „multiple environments‟ scenario, beq is independent of the degree of temporal autocorrelation and the PEA slopes do not differ much as the environment becomes more autocorrelated (Fig. 4A). However, the PEA slope increases with the correlation between the environments of development and selection,  DS , partly because plasticity evolves to greater values as  DS increases. In contrast, for the „delay between development and selection‟ scenario, beq is a function of the level of autocorrelation and the PEA slopes increase with increasing   (Fig. 4B). As in the constant plasticity scenario (Fig. 3), PEA slopes were closer to B as the timeframe between development and selection decreased. Using the PEA Slope to Estimate B: An Example The environmental sensitivity of selection B can be found indirectly from a PEA if empirical estimates are available for all other parameters. As an example of this procedure, we use data on the well-studied relationship between tadpole tailfin depth and predation risk for the wood frog (Lithobates sylvaticus). Environmental cues indicating the presence of predators induce L. sylvaticus tadpoles to develop deeper tailfins (Relyea 2001, 2003). In addition, stabilizing selection likely acts on tailfin depth in general, as tadpoles with shallow and very deep tailfins are attacked more by predators than tadpoles with a typical predator-induced tailfin (Van Buskirk et al. 2003). Using data collected from published sources (Relyea 2005; Michel 2011, 2012), we obtained the following parameter estimates for plasticity and the PEA: be = 0.011 and  z | =

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0.0164 (Michel 2011, 2012); and for phenotypic variance: G a = 0.00067 and  z2 = 0.001 (Relyea 2005). Note that the PEA from Michel (2011) was obtained across populations at a given time, to which our model can be applied if these populations exchange very few

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migrants and their environments are uncorrelated. The strength of stabilizing selection,  , is unknown for this system, so we set  = 500 for a plausible fitness function (i.e., a standardized selection gradient of ~  0.5 , Lande and Arnold 1983). Because little data are available regarding the environmental parameters (i.e,   ,  D ,  S ,  DS ), we calculated B under various values of these parameters. Environmental sensitivity of selection ranged from 0.03 to 0.10 (the data are log-transformed, so this means that for every increase in one predator/tadpole, the logarithm of the tadpole tailfin in centimeters should increase from 0.03 to 0.10), and relative plasticity ranged from 0.20 to 0.70 (Fig. 5). The environmental sensitivity of selection is greater (and, thus, relative plasticity is lower) when the temporal autocorrelation in predation risk is close to zero, and when the correlation between the environments of development and selection is larger. Discussion To relate empirically-measured parameters to biological ones, we introduce the

concepts of effective plasticity, be , and effective environmental sensitivity of selection, Be . These parameters scale b and B by the relationship between the measured environment and the environments of development (  D ) or of selection (  S ), respectively. An important implication of this scaling is that reduced  S can cause spurious detection of reduced environmental sensitivity of selection (as in Gienapp et al. 2013), and similarly for  D and plasticity. We also found that the evolution of PEAs was dependent on other environmental parameters that have been shown previously to affect the evolution of plasticity: the level of temporal variation and predictability (de Jong 1995, 2005; Scheiner and Holt 2012). We

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found that, in general, PEAs were stronger when environmental predictability was large. Unfortunately, values of these parameters are not commonly reported among the literature, despite their prominence in both theoretical and empirical studies (e.g., Reed et al. 2010).

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This lack of coverage is especially surprising given the large amounts of environmental data that cover extensive spatial and temporal extents. A deeper understanding of how easilymeasured environmental variables relate to those environmental cues that affect phenotypic development or selection is clearly needed. Our model indicates that the presence of plasticity can affect a PEA in two contrasting

ways: (i) plasticity directly adds to the covariation between a phenotype and its environment (eq. 1), and (ii) phenotypic plasticity can influence the genetic response to selection of the non-plastic component by changing its distance from the optimum phenotype (eq. 2). However, the results from the model and the simulations suggest that this former effect of plasticity predominates over the latter effect, and that plasticity increases the slope of a PEA towards the environmental sensitivity of selection (compare Fig. 1, with Figs. 2-4). Thus, it is not surprising that most examples of PEAs from natural populations include phenotypes that are known to be highly plastic, such as morphology (e.g., Clausen et al. 1948; Langerhans et al. 2007; Michel 2011) or color patterning (e.g., Endler 1982). PEAs have been documented for populations across a species‟ geographic range (a

spatial PEA; e.g., Clausen et al. 1940, 1948; Langerhans et al. 2007; Michel 2011), as well as for a population inhabiting temporal environmental variation (a temporal PEA; e.g., Husby et al. 2011). Our formulas were derived to describe the evolution of temporal PEAs. However in our simulations, the PEA was obtained by regressing each population‟s mean phenotype against its measured environment in a given generation, rather than a single population across generations, relying on the property of ergodicity whereby the long-term distribution of a stationary process is similar to the distribution of this process across replicates. The similarity

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between the simulation results and the analytical formulas suggest that the formulas could be applicable for spatial PEAs, under some assumptions. More precisely, they could be a good approximation in cases where (i) the sampled populations experience environments that are

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essentially independent in space, and with the same stationary distribution (i.e., negligible spatial autocorrelation, no geographic gradient); and (ii) there is little influence of gene flow on the local adaptation and plasticity in each population (i.e., migration is negligible relative to the average strength of directional selection). In contrast, if the environment covaries with space, such as over a latitudinal gradient, then the evolution of spatial PEAs would be more likely driven by local adaptation than by the pattern of temporal variation of environmental variables (Blanquart et al. 2012). Besides, strong gene flow among populations should affect our results by changing both the genetic variance and the dynamics of the mean phenotype in each deme. This sets a geographic scale of application of the formulas: populations need to be far enough or encounter sufficient dispersal barriers in order to experience limited gene flow, but still exist in a relatively restricted geographic range so that they do not experience largescale gradients. Future models that incorporate the spatial aspects of PEAs could also assess how population-level processes such as dispersal and local adaptation and environmental processes such as spatial autocorrelation would affect the formation of PEAs. The environmental sensitivity of selection, B, has recently been identified as an

important parameter in determining the extinction risk of a population in response to future environmental change (Chevin et al. 2010), but has rarely been estimated in natural populations (but see Gienapp et al. 2013, Vedder at al. 2013). We demonstrate that unless the environmental variables have near-perfect temporal autocorrelation (≈ 1), the slope of a PEA measured in natural populations will almost always underestimate the environmental sensitivity of selection. However, if estimates of other environmental parameters are available, B can be obtained from our models, and we demonstrate this ability by estimating

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how the optimum tailfin depth of L. sylvaticus tadpoles should change with variation in predation risk. We were also able to estimate relative plasticity, or α, although it is difficult to determine how our measure of α for the tadpole tailfin system fits in with other studies as

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few, if any, studies have estimated this parameter. As for predictions from any model, it should be noted that application of our formulas to estimate B in natural system will depend on the accuracy of estimates of input parameters, such as environmental (auto)correlation, strength of stabilizing selection, additive genetic variance, and reaction norm slopes. Our main aim in this paper was to introduce a framework with which to model the

formation of phenotype-environment associations. It is our expectation that this model will lead to a greater understanding of how population-mean phenotypes can both spatially and temporally covary with environmental gradients. In addition, by disentangling the mechanisms for the relationship between a population‟s mean phenotype and the environment, we expect that analysis of PEAs will be useful in determining if and how populations will be able to adapt to environmental change.

Acknowledgments: This work was supported by funding from the United States National Science Foundation to JHK (DEB-0844644) and from the ContempEvol grant from the Agence National de la Recherche to LMC.

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Figure 1. Analytical (lines) and simulation (points; mean ± 1 STD) results for the phenotypeenvironment slope for the “No Plasticity” scenario. Dotted line indicates set value of the environmental sensitivity of selection (B = 1). Black and gray indicate weak (κS = 0.2) and

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strong (κS = 0.8) relationships between environment of selection and measured environment, respectively. Smooth and dashed lines and filled and open points indicate weak (γ = 0.1) and strong (γ = 0.6) stabilizing selection, respectively. Other parameter values: n = 50, simulation iterations = 50, g = 1000,  2 = 4.0, and Ga = 0.5.

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Figure 2. Analytical (lines) and simulation (points; mean ± 1 STD) results for the phenotypeenvironment slope in the “One Environmental Variable” scenario. Dotted line indicates set value of the environmental sensitivity of selection (B = 1). Grayscale (from black to gray)

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indicate increasing levels of plasticity. Smooth and dashed lines and filled and open points indicate weak (γ=0.1) and strong (γ=0.6) stabilizing selection, respectively. Other parameter values: n = 50, simulation iterations = 50, g = 1000,  2 = 4.0, and Ga = 0.5.

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Figure 3. Analytical (lines) and simulation (points; mean ± 1 STD) results for the phenotypeenvironment slope for the “Delay between development and selection” scenario with τ = 1/10(A) and τ = 1/5 (B). Dotted line indicates set value of the environmental sensitivity of

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selection (B = 1). Grayscale (from black to gray) indicate increasing levels of plasticity. Other parameter values: n = 50, simulation iterations = 50, g = 2000, Ga = 0.5,  2 = 4.0, and γ =

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0.1.

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Figure 4. Analytical (lines) and simulation (points, mean ± 1 STD) results for the phenotypeenvironment slope for the “Evolving Plasticity” scenario under two types of environmental conditions: A) Base model in which there are multiple environmental variables and B)

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“Delay between development and selection” model. Dotted line indicates set value of the environmental sensitivity of selection (B = 1). Grayscale (from black to gray) indicates increasing levels of the correlation between the environments of development and selection (panel A) or decreasing fraction of the generation in which the environment influences development before imposing selection (panel B). Initial plasticity (b) was set to 0.6 in every scenario, but evolved to the value of κS|D in each scenario. Other parameter values: n = 50, simulation iterations = 25, g = 10,000,  2 = 4.0, γ = 0.1, Ga = 0.5, Gb = 0.2, and Gab = 0.

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Figure 5. Determination of the environmental sensitivity of selection (black; left y-axis) and relative plasticity (gray; right y-axis) for the tadpole tailfin/predation risk example across various degrees of temporal autocorrelation and correlation levels between environments of

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selection and development, ρDS. Dotted line indicates the actual, observed phenotypeenvironment slope used to calculate the environmental sensitivity of selection. Other parameter values: Ga = 0.000672, Gb = 0, be = 0.011, γ = 500, κD = κS = 0.5.

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Evolution of phenotype-environment associations by genetic responses to selection and phenotypic plasticity in a temporally autocorrelated environment.

Covariation between population-mean phenotypes and environmental variables, sometimes termed a "phenotype-environment association" (PEA), can result f...
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