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\begin{document}

\title{Divergence in rates of phenotypic plasticity among ectotherms}



\author[1]{Sigurd Einum}%
\author[1]{Tim Burton}%
\affil[1]{Norwegian University of Science and Technology}%


\vspace{-1em}



  
  \date{May 24, 2024}


\begingroup
\let\center\flushleft
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\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
An individual's fitness cost associated with environmental change likely
depends on the rate of adaptive phenotypic plasticity, and yet our
understanding of plasticity rates in an ecological and evolutionary
context remains limited. We provide the first quantitative synthesis of
existing plasticity rate data, focusing on acclimation of temperature
tolerance in ectothermic animals, where we demonstrate applicability of
a recently proposed analytical approach. The analyses reveal
considerable variation in plasticity rates of this trait among species,
with half-times (how long it takes for the initial deviation from the
acclimated phenotype to be reduced by 50\% when individuals are shifted
to a new environment) ranging from 3.7 to 770.2 h. Furthermore, rates
differ among higher taxa, being higher for amphibians and reptiles than
for crustaceans and fishes, and with insects being intermediate. We
argue that a more comprehensive understanding of phenotypic plasticity
will be attained through increased focus on the rate parameter.%
\end{abstract}%



\sloppy


\textbf{Divergence in rates of phenotypic plasticity among ectotherms}

\par\null

Sigurd Einum\textsuperscript{1*}and Tim Burton\textsuperscript{1,2}

\textsuperscript{1}Centre for Biodiversity Dynamics, Department of
Biology, Norwegian University of Science and Technology, Realfagbygget,
NO-7491 Trondheim, Norway

\textsuperscript{2}Norwegian Institute for Nature Research, PO Box 5685,
Torgarden, NO-7485, Trondheim, Norway.

*corresponding author: sigurd.einum@ntnu.no

Type of article: Letter

Number of words in abstract: 150

Number of words in main text: 4969

Number of references: 29

Number of figures: 3

Number of tables: 2

Statement of authorship: both authors conceived the study and identified
data sources, SE extracted the data, conducted the analyses, and wrote
the first draft of the manuscript, both authors contributed to
revisions.

Data accessibility statement: The data will be published in Dryad upon
acceptance

Running title: Variation in phenotypic plasticity rates

Keywords: Phenotypic plasticity, acclimation rate, comparative,
temperature, time course, body size, selection, fitness, exponential
decay, critical



\textbf{Abstract}

An individual's fitness cost associated with environmental change likely
depends on the rate of adaptive phenotypic plasticity, and yet our
understanding of plasticity rates in an ecological and evolutionary
context remains limited. We provide the first quantitative synthesis of
existing plasticity rate data, focusing on acclimation of temperature
tolerance in ectothermic animals, where we demonstrate applicability of
a recently proposed analytical approach. The analyses reveal
considerable variation in plasticity rates of this trait among species,
with half-times (how long it takes for the initial deviation from the
acclimated phenotype to be reduced by 50\% when individuals are shifted
to a new environment) ranging from 3.7 to 770.2 h. Furthermore, rates
differ among higher taxa, being higher for amphibians and reptiles than
for crustaceans and fishes, and with insects being intermediate. We
argue that a more comprehensive understanding of phenotypic plasticity
will be attained through increased focus on the rate parameter.



\textbf{Introduction}

Phenotypic plasticity, or the expression of different phenotypes across
environments by a single genotype, is an important process by which
organisms can minimize environmental impacts on fitness (Gabriel, 2005;
Gabriel et al. 2005; Padilla \& Adolph, 1996; Siljestam \& \selectlanguage{ngerman}Östman,
2017). Such plasticity can be described by two parameters. First,
the\emph{capacity for plasticity} determines the amount by which an
individual can actively change its phenotype following a shift in the
environment. The term ``actively change'' is used here to separate
between passive and active phenotypic plasticity. An example of passive
plasticity is the increase in respiration that typically occurs in
response to an increase in temperature, which organisms may counteract
over time by active plasticity responses (Kielland et al., 2017). Thus,
it is active plasticity that is of interest when studying the responses
that organisms have evolved to minimize the effect of environmental
change on fitness. This parameter can be measured as the change in the
slope of the relationship between trait value and environment (i.e. the
slope of the reaction norm) as plasticity proceeds, from acute exposure
until the full plastic response has been achieved (see Einum et al.,
2019 for arguments why it is this change in the slope, and not
slope\emph{per se} , that describes \emph{capacity for plasticity} ).

The second parameter of active phenotypic plasticity is the \emph{rate
of plasticity} , which represents how quickly the change in phenotype
(and hence the change in the reaction norm slope) occurs following a
change in the environment (Fig. 1). Whereas the \emph{capacity for
plasticity} has received considerable theoretical (e.g. Lande, 2014) and
empirical interest (e.g. Seebacher et al., 2015; Pottier et al. 2022)
from ecologists and evolutionary biologists, empirical support for
certain predictions regarding the evolution of this plasticity parameter
remain equivocal. For example, while it has been proposed that organisms
inhabiting more variable environments should evolve
greater\emph{capacity for plasticity} , this is rarely supported by
empirical data (Gunderson \& Stillman, 2015; Kelly et al., 2012; MacLean
et al., 2019; Pereira et al., 2017; Phillips et al., 2016; Sgro et al.,
2010; van Heerwaarden et al., 2016; van Heerwaarden et al., 2014).
Recently, Burton et al. (2022) suggested that this discrepancy between
theoretical expectations and empirical data gives reason for pause, and
that greater considerations of the second parameter, the \emph{rate of
plasticity} , which addresses the timescale over which plastic
phenotypic change occurs, might aid in bringing this field of research
forward.

If the plasticity of a trait is an adaptive response, the fitness cost
that an organism incurs following a change in its environment should be
minimized once the phenotype becomes fully adjusted to the new
environment. Hence, the rate at which the phenotype approaches this
state should determine how long the individual expresses a sub-optimal
phenotype, and in part, determine the magnitude of the fitness cost
associated with that change in the environment. Given that organisms are
unlikely to be able to predict changes in all of the relevant
environmental variables they are exposed to, it seems plausible that
individuals may actually spend a considerable proportion of their time
having a phenotype that is not fully adjusted to their current
environment. This mismatch between environment and phenotype, and
associated cumulative fitness costs, will be exacerbated if the
adjustment of the phenotype is slow relative to the timescale of
environmental change. Furthermore, as pointed out by Burton et al.
(2022), the \emph{rate of plasticity} might even influence how
the\emph{capacity for plasticity} evolves, because the evolution of
capacity depends on the predictability of the environment. Organisms
that can rapidly implement their phenotypic response to a new
environment can postpone the onset of this process closer to the time of
selection in that environment than organisms that do so at a slower
rate. Thus, in a temporally autocorrelated environment, a
faster\emph{rate of plasticity} might effectively increase
predictability in the environment, which in turn should favour the
evolution of greater\emph{capacity for plasticity} (Lande, 2014).

Presently, a quantitative synthesis of data on the \emph{rate of
plasticity} is lacking, and consideration of how this parameter of
phenotypic plasticity might be expected to evolve in response to
environmental change is absent from current theoretical models (Lande,
2014; Siljestam \& Östman, 2017). Although a substantial number of
empirical studies document how phenotypes change over time when
introduced into new environments, these studies remain largely
descriptive, fail to address evolutionary hypotheses, and very rarely
(four out of 166 studies surveyed by Burton et al., 2022) attempt to
provide any formal statistical quantification of the time course of
plasticity. Thus, advancing our understanding of the evolution of
phenotypic plasticity might arguably benefit from a shift in focus
from\emph{capacity for plasticity} to \emph{rates of plasticity} . To
stimulate such a shift, we provide the first comparative analysis of
published data describing \emph{rates of plasticity} . In doing so, we
follow recent suggestions (Burton et al., 2022) regarding the estimation
of plasticity rates in a (i) standardized way, which is (ii) consistent
with theory and (iii) directly comparable across taxa and traits.

We draw upon published data from studies of acclimation to temperature
among ectotherms. Temperature is an environmental variable that affects
all organisms, varies substantially in space and time, and which has
particularly pervasive effects on biochemical, physiological and
ecological processes in this group of animals (Daufresne et al., 2009).
We focus our synthesis on traits describing temperature tolerance. We
first determine the shape of how temperature tolerance changes over time
(exponential vs. linear decay) in response to a shift in ambient
temperature, as this is the first step required when calculating
the\emph{rate of plasticity} . After calculating rates of plasticity for
each published dataset, we then investigate relationships
between\emph{rates of plasticity} and taxonomic class, body size, and
acclimation temperature. By providing clear evidence that \emph{rates of
plasticity} have diverged among ectotherm classes we show how this rate
can, and does, evolve, and that increased empirical and theoretical
focus on the rate parameter is likely to provide a way forward in
understanding evolution of phenotypic plasticity.

\textbf{Methods}

\emph{Collection of data}

We identified published studies documenting the rate of change in
whole-body measurements of the thermal tolerance of ectotherms in
response to manipulation of ambient temperature (Supplementary
Information). For most of the papers, data were presented in figure
format only, usually as the mean tolerance of groups of individuals that
had experienced the `new' temperature for differing lengths of time
prior to being measured for the chosen tolerance trait. We digitized
such data using the WebPlotDigitizer (https://apps.automeris.io/wpd/).
In addition to thermal tolerance data, we extracted information for the
following variables: taxonomic class, species name, mean body mass of
individuals used in the experiment, life stage (juvenile vs. adult),
acclimation temperature, and type of thermal tolerance measure. For the
latter variable, a range of different terms that describe behavioural
responses to either high or low acute temperature exposure are employed
in different studies, and are typically taxon-specific (e.g. knock-down,
loss of equilibrium, paralysis and spasms, loss of righting response).
All such endpoints were categorized as a behavioural response.
Experiments using some measure of mortality (time until or temperature
at which signs of death such as cessation of breath or heart beating
occurs) were categorized as a mortality response. For experiments where
body mass information was absent we used the following procedure to
obtain such data. First, we searched for information on minimum and
maximum adult body size (length or mass) for the species in question. If
only data on body length were available, we searched for allometries
between length and mass for that species, or for species within the same
genus, and used these to convert body lengths to mass. We then
calculated the mean of the minimum and maximum body mass values. If no
data on minimum or maximum size was found, we searched for studies that
presented body sizes of adult individuals during field surveys and used
the mean of these. The following classes of ectotherms were represented
in the data set: amphibians, reptiles, insects, thecostraca,
turbellaria, osteichtyes and malacostraca. For simplicity the latter two
will be referred to as fishes and crustaceans, respectively

\emph{Calculating rates of plasticity}

Rates of plasticity were calculated according to the approach outlined
by Burton et al. (2022). Thus, for each observation of temperature
tolerance at times \emph{t} we first calculated the proportion of the
full plastic response that remained to be
achieved,\emph{D\textsubscript{t}} , as (\emph{z\textsubscript{t}}
-\selectlanguage{english}\emph{z\textsubscript{[?]}} )/(\emph{z\textsubscript{0}}
--\selectlanguage{english}\emph{z\textsubscript{[?]}} ), where\emph{z\textsubscript{0}} is the
first measurement of the phenotype (typically measured prior to the
onset of acclimation), \emph{z\textsubscript{t}} is the phenotype after
acclimating for a time period \emph{t} (for the current study
\emph{t}was standardized to unit h), and \selectlanguage{english}\emph{z\textsubscript{[?]}} is
the fully adjusted phenotype. In theory, \selectlanguage{english}\emph{z\textsubscript{[?]}} is
the final measurement of thermal tolerance. However, given measurement
noise this may not be the case. Thus, we defined
\selectlanguage{english}\emph{z\textsubscript{[?]}} as the maximum (for acclimation to higher
temperature) or minimum (for acclimation to lower temperature) value for
a given measurement of tolerance observed. For each experiment, we
considered two potential responses in plasticity of thermal tolerance.
First,\emph{D\textsubscript{t}} may have a value of 1 at \emph{t} = 0
(first measurement) and decline linearly with time with a
rate\selectlanguage{greek}\emph{λ\textsubscript{L}}\selectlanguage{english} given by \selectlanguage{greek}\emph{λ\textsubscript{L}}\selectlanguage{english} =
(1-\emph{D\textsubscript{t}} )/\emph{t} until it reaches 0, after which
it will be constant. Alternatively, \emph{D\textsubscript{t}} may have a
value of 1 at \emph{t} = 0 and decline as an exponential decay function
towards 0 with rate \selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} , such
that\(D_{t}=e^{{-\lambda}_{E}t}\).

\emph{Statistical analyses - Comparison of linear vs. exponential decay
functions}

For each experiment, we fitted both linear and exponential decay models
to the observed data. For the linear decline we fitted a piecewise
regression with an intercept of 1 (at \emph{t} = 0), which estimates the
breakpoint \emph{b} (i.e. the time at which
\emph{D\textsubscript{t}}reaches 0), from which the rate
\selectlanguage{greek}\emph{λ\textsubscript{L}}\selectlanguage{english} can be calculated as 1/\emph{b} . Thus, both
models estimate a single parameter, and their relative fits for a given
experiment can be compared directly using their residual standard
errors. Models were fitted using the function \emph{nls\_multstart} from
the \emph{nls.multstart} package v.1.2.0 (Padfield \& Matheson, 2020) in
R v.4.1.2 (R Core Team, 2021).

For most studies, \emph{z\textsubscript{0}} was measured in individuals
prior to transfer to the new temperature. It was often less clear
whether experimenters had been able to measure a `true' value
of\emph{z\textsubscript{[?]}} , i.e. thermal tolerance after full
acclimation to the new temperature had been obtained. This is a key
point when measuring rates of plasticity because estimates
of\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} will be biased if full acclimation to the
new environment has not been achieved (Fig. S2). However, an advantage
of the exponential decay function is that achievement of full
acclimation can be assessed by calculating the slope of the estimated
function (i.e.\({{-\lambda}_{E}e}^{\lambda_{E}t}\)) at the final acclimation time point
\emph{t\textsubscript{n}} (this slope has an asymptotic value of 0).
Thus, this value was calculated for each experiment and included as a
covariate in our analysis (see below) to control for any bias introduced
by variation in maximum acclimation time among studies.

\emph{Statistical analyses -- Variation in rates of phenotypic
plasticity}

Comparisons of exponential and linear decay models showed that the
former was superior to the latter in most cases (see Results). Thus, the
following statistical analyses use estimates of\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english}
as the dependent variable.

Inspection of the distribution of standard errors (SE) for estimates
of\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} were used to exclude experiments that had
poor model fit for the exponential decline models (Fig. S3). This was
based on judgement, trading-off maintaining sample size while excluding
potentially biased estimates. Using a threshold SE value of 0.01 enabled
us to retain 78\% of the experiments (n = 240 out of 308). The
statistical analyses described below use estimates obtained below this
threshold. Increasing the threshold SE value to 0.02 increased the
inclusion rate to 88\% (n = 271) without resulting in qualitative
changes in the results (Table S1). The bias in
estimated\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} for experiments with SE
\textgreater{} 0.02 and within the range 0.01-0.02 can be observed by
comparing\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} in experiments within different SE
bins (Fig. S4).

Variation in \selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} was analysed using
meta-analytical models, fitted using the function \emph{rma.mv} in the
package \emph{metafor} v.3.8-1 (Viechtbauer, 2010), while accounting for
the sampling variances (i.e. the squared standard error of
the\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} estimates). The full model contained the
fixed effects of taxonomic class, body mass, acclimation temperature,
type of thermal tolerance measure, and slope of the estimated
exponential decay function at \emph{t\textsubscript{n}} . Life stage was
given in less than one third of the identified experiments, and was
therefore not included in the model. Random effects included effects of
species, study, and experiment. The latter was included because up to
several experiments were included per species and study, and
meta-analytical models do not contain an error term. In addition, we
included the possibility for a phylogenetic signal in
\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} beyond that imposed by taxonomic class. This
was done by building a tree using the package\emph{rotl} v.3.0.12
(Michonneau et al., 2016) and the Open Tree of Life (Open Tree et al.)
which was made ultrametric using the function\emph{compute.brlen} in
package \emph{ape} v.5.6-2 (Paradis \& Schliep, 2019). This tree was
used to compute the species relatedness variance-covariance matrix which
was included as a second species-level random effect. Thus, this model
accounts for heterogeneity in estimated\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} due to
differences between species unrelated to phylogeny, and a random species
effect that accounts for the influence of phylogenetic relatedness
within taxonomic classes. All statistical analyses were conducted in R
v.4.1.2 (R Core Team, 2021)

The evidence for the full model relative to simplified ones was
evaluated by comparing AICc values. We first evaluated the relative
support for a model that included the phylogenetic random effect against
one that did not contain this term (fitted using REML). After choosing
the appropriate random effect structure we proceeded by comparing the
relative support for all alternative models (fitted using ML) with
simpler fixed structures using the function \emph{dredge} from the
package \emph{MuMIn} v.1.47.1 (Barton, 2022). Since we found no evidence
for a body mass effect (see Results), and several experiments lacked
this information (Table S2), we repeated the comparison of models with
different fixed effect structures while excluding the body mass variable
to allow for the potential inclusion of alternative fixed effects.
Finally, we present the estimated model parameters from the model
receiving the strongest support (fitted using REML). For all analyses,
inspection of residual plots suggested that assumptions of their
normality and homogeneity were satisfied.

\textbf{Results}

\emph{Comparison of linear vs. exponential decay functions}

A total of 290 experiments could be fitted with both piecewise
regression and exponential decay models (Fig. S5). The remaining 18
could not be fitted with piecewise regression, which most frequently
occurred because the minimum value of \emph{D} (i.e. complete
acclimation) was reached at the second observation, in which case
multiple breakpoints can yield an identical fit. Of these 290
experiments, comparisons of their residual standard errors revealed that
the data in 62\% (n=180) of cases was best explained by the exponential
decay model. Furthermore, superior fit by piecewise regressions (N = 110
experiments) was observed primarily in experiments where thermal
tolerance was measured at relatively few time points following exposure
of the study organisms to the new temperature (Fig. S6). Thus, for more
comprehensive experiments (i.e. greater number of measurement points in
time), the shape of the plasticity rate response is better described by
an exponential decay function. Thus, in all further analyses we use the
estimates of \selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} (Table S2).

\emph{Variation in rates of phenotypic plasticity}

For one experiment the estimate of \selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} was
negative, and six experiments lacked information on acclimation
temperature (Table S2). These experiments were excluded from all
subsequent analyses. Additionally, the classes thecostraca and
turbellaria were represented by only one and two species, respectively
(Table S2), and were therefore also excluded from subsequent analyses.
Finally, for the first set of analyses where we allowed for the
inclusion of a body mass effect, an additional 10 experiments that
lacked such data (Table S2) were excluded. Thus, the dataset used for
these analyses that had an SE of \selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} of less than
0.01 consisted of 219 experiments from a total of 84 species (12
insects, 6 crustaceans, 39 amphibians, 9 reptiles and 18 fishes).
Species-specific mean (SD) estimated \selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} of this
data set was 0.0345 (0.0305) h\textsuperscript{-1}, and ranged from
0.0009 to 0.1892 h\textsuperscript{-1}. When comparing the fit of models
with and without a random phylogenetic effect, the evidence for such an
effect was weak (AICc = -899.15 and -901.05 for models with and without,
respectively). Thus, we proceeded with comparison of models with
different fixed structures while including the random effects of
species, study, and experiment. Variation in
\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english}was best explained by a model that included
taxonomic class, acclimation temperature, the slope of the estimated
exponential decay function at\emph{t\textsubscript{n}} , and the type of
thermal tolerance measure, whereas evidence for an effect of body mass
was weak (Table 1). We therefore repeated the comparison of models with
different fixed effects while excluding body mass from the full model.
This allowed us to include experiments that lacked body size
information, increasing the total number of experiments to 229,
representing 91 species (13 insects, 7 crustaceans, 44 amphibians, 9
reptiles and 18 fishes). Species-specific mean (SD) estimated
\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} of this data set was 0.0347 (0.0302)
h\textsuperscript{-1}, with the same range as in the previous data set.
Strong evidence was found for effects of taxonomic class, acclimation
temperature, and the slope of the estimated exponential decay function
at \emph{t\textsubscript{n}} , whereas removing the effect of type of
thermal tolerance measure from the model caused a slight decrease in
AICc (Table 1).

Inspection of coefficients from the best ranked model showed that
plasticity rates were highest among amphibians and reptiles, lowest
among fishes and crustaceans, and intermediate in insects (Table 2).
This pattern persisted when examining distributions of
observed\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} among these classes without correcting
for the covariates fitted in the best ranked model (Fig. 2). We also
observed that plasticity rates increased with acclimation temperature
(Table 2, Fig. 3). Finally, plasticity rates were observed to be higher
when the slope of the estimated exponential decay function
at\emph{t\textsubscript{n}} was shallower (Table 2). In other words,
experiments where complete acclimation was more likely to have been
obtained were associated with higher estimated plasticity rates. This
pattern was mainly driven by the amphibian data, which had a large
number of experiments with relatively steep slopes
at\emph{t\textsubscript{n}} (Fig. S7). This is in the opposite direction
to what might be predicted if a relationship between this slope and
estimated acclimation rate is a statistical artefact (Fig. S2). Rather,
the causality of this relationship is likely in the opposite direction,
i.e. experiments on species that had a low acclimation rate were
stopped\emph{before} complete acclimation to the new temperature had
occurred. Nevertheless, to evaluate if this anomaly could have
influenced our results, we repeated the model comparison summarized in
Table 1 but including only experiments where the slope of the estimated
exponential decay function at \emph{t\textsubscript{n}} was larger than
-0.002 and -0.001, respectively. In both cases, strong evidence was
found for effects of taxonomic class, acclimation temperature, and the
slope of the estimated exponential decay function at
\emph{t\textsubscript{n}}(Table S3, S4). Parameter estimates for the
different taxonomic classes showed only minor quantitative changes in
comparison to the main analysis (Table S5, S6 vs. Table 2).

\textbf{Discussion}

Here, we provide the first quantitative synthesis of plasticity rates
(\selectlanguage{greek}\emph{λ} \selectlanguage{english}) across taxa and apply a novel method to obtain standardized
and thus comparable measures of this plasticity parameter. For our focal
trait, acclimation of temperature tolerance to temperature change, the
shape of the plasticity responses was well described by an exponential
decay function (i.e. with the rate \selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} ). In other
words, the absolute rate of change in temperature tolerance when an
individual is shifted to a new temperature is proportional to the
deviance from the phenotype when completely acclimated to that
temperature. We thus validate an assumption that has previously been
made in theory describing the evolution of phenotypic plasticity (Lande
2014). In contrast, superior fit by piecewise linear regression was
primarily observed in experiments with poor temporal resolution, thus
demonstrating the importance of measuring phenotypes at multiple time
points in the new environment to determine the optimal model to use when
estimating the shape of the plasticity response.

Variation in estimated \selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} among species was
considerable. To put these values into perspective they can be
translated into half-times, or how long it takes for the initial
deviation from the fully acclimated phenotype to be reduced by 50\%
after being shifted to a new environment, which is given as
ln(2)/\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} . The mean
observed\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} of 0.0347 h\textsuperscript{-1}
corresponds to a half-time of 20.0 h, whereas the minimum and maximum
species-specific\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} , estimated to 0.0009 and
0.1892 h\textsuperscript{-1}, correspond to half-times of 770.2 and 3.7
h, respectively (based on data where SE
\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english}\textless{} 0.01). Using model estimates (Table
2), half-times at 20 \selectlanguage{ngerman}°C when the final slope of the decay function is
zero are 12.0, 16.9, 27.7, 36.5 and 38.5 h for amphibians, reptiles,
insects, crustaceans and fishes, respectively. Thus, our analyses
demonstrate considerable systematic variation in rates of phenotypic
plasticity among these taxa. This begs the question of why such
variation has evolved. One might speculate that due to the higher heat
capacity of water compared to air, fishes (all species) and crustaceans
(5 out of 7 species in our data) which inhabit aquatic environments
experience higher temperature stability than taxa inhabiting terrestrial
environments. Furthermore, aquatic habitats are often thermally
stratified (lentic habitats such as lakes and oceans) or provide
cold-water plumes (rivers), which allows for behavioural
thermoregulation under periods of stressful temperatures (Kurylyk et
al., 2015; Freitas et al., 2016; Harrison et al., 2016). This may reduce
the strength of selection on rapid plasticity compared to in the
terrestrial environment occupied by reptiles and insects (all species in
our data). Although the amphibians also inhabit aquatic environments
(particularly during juvenile life stages), their utilization of thermal
refugia in deep or fast flowing water is likely limited. Unfortunately,
habitat use is confounded with phylogeny in this data set, preventing
direct analysis of the effect of habitat on rate of plasticity in
thermal tolerance. Thus, although we do demonstrate evolutionary
divergence in plasticity rates among these taxa, it remains an open
question as to whether this pattern results from evolutionary adaptation
to environmental conditions. However, this question could be addressed
in future work that targets populations or species that experience known
and contrasting patterns of environmental variability.

We observed a positive relationship between acclimation temperature and
rate of plasticity in temperature tolerance. This pattern may be
explained by the general relationship that exists between developmental
rate and body temperature in ectotherms, which is driven by the positive
effect of temperature on biochemical reactions and metabolic rate (Brown
et al., 2004). It may also explain the observation that within a
species, acclimation to high temperature is achieved faster than
acclimation to low temperature (Burton et al., 2020). A relationship
between metabolic rate and the rate of plasticity was previously
hypothesized and addressed by Rohr et al. (2018), but in a less direct
manner. Specifically, Rohr et al. (2018) argued that the effect of
metabolic rate on rates of thermal plasticity in ectotherms should be
evident as a negative relationship between body size and plasticity
rate, because smaller organisms tend to have a higher mass-specific
metabolic rate than larger ones. They did not however, calculate rates
of plasticity from experiments that were explicitly designed to do so.
Rather, they used data from experiments that measure the phenotype at
only two time points (\emph{z\textsubscript{0}}
and\selectlanguage{english}\emph{z\textsubscript{[?]}} in our terminology), and from this inferred
how the bias in acclimation capacity caused by insufficient acclimation
time was influenced by body size. Based on their results it was
concluded that rates of plasticity appeared to be higher for smaller
organisms. Using a more direct approach we failed to find support for a
general relationship between body size and rate of thermal plasticity.
Yet, our observation that the rate of plasticity in temperature
tolerance is positively related to acclimation temperature suggests a
role for metabolic rate in causing some of the variation in plasticity
rate across experiments.

Given the general patterns in rate of plasticity observed here, further
efforts in studying this plasticity parameter may be fruitful and
provide a better foundation for understanding how plasticity evolves in
response to environmental variation. From an empirical perspective,
including a temporal-dimension in experiments that study plasticity may
be included without large costs. In this respect, we make two
recommendations. First, a proper choice of model (linear vs. exponential
decay) for estimating lambda requires multiple measurements of the
phenotype as it responds to the new environment. Our analyses indicate
that five or more measurements may be required to adequately establish
the shape of the plasticity response (Fig. S6). Superficially, this
requirement might appear to substantially increase the workload of such
studies in comparison to studies that only estimate the capacity for
plastic phenotypic change in a trait. However, once the model that best
describes the shape of the plastic response to the new environment is
established, a single measurement \emph{z\textsubscript{t}} after
time\emph{t} (which must be prior to achievement of full acclimation) in
addition to those typically measured (\emph{z\textsubscript{0}} in
non-acclimated individuals and \selectlanguage{english}\emph{z\textsubscript{[?]}} after the full
acclimation response has been obtained) is sufficient to accurately
estimate \emph{D\textsubscript{t}} , which in turn can be used to
calculate the rate of plasticity (\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} =
ln(\emph{D\textsubscript{t}} )/\emph{t} for exponential decay
or\selectlanguage{greek}\emph{λ\textsubscript{L}}\selectlanguage{english} = (1-\emph{D\textsubscript{t}} )/\emph{t}
for linear decay). Thus, the workload in such experiments can be greatly
reduced by performing a pilot experiment with sufficient temporal
resolution (in terms of measurement time points) that provides a precise
description of the shape of plasticity response to the new environment
before performing more replicates at a lower temporal resolution to
obtain the desired estimates of \selectlanguage{greek}\emph{λ} . \selectlanguage{english}It should be noted
that\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} and \selectlanguage{greek}\emph{λ\textsubscript{L}}\selectlanguage{english} are not
directly comparable, because the initial approach towards the fully
adjusted phenotype is more rapid under exponential decay. Thus, the
relative support for these two types of plasticity responses should be
reported. As a second recommendation, experimenters should strive to
ensure that complete acclimation to the new environment is achieved
prior to measuring \emph{z\textsubscript{[?]}} . Our analyses show that
failing to do so can, and does, lead to bias in estimation
of\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} (Fig. S2, S7). Ideally this is achieved by
rearing individuals in all the alternative environments for the whole
duration of the experiment (i.e. both prior to and after some of the
individuals are transferred into new environments). As pointed out by
Burton et al. (2022), this has rarely been done in studies of rates of
plasticity. Rather, the majority of studies first acclimate the animals
to a single initial environment before shifting them to a new
environment and then performing repeated measures of phenotype in this
new environment for what typically appears to be a pre-determined (and
potentially insufficient) duration.

Natural next steps in research on evolution of plasticity would be to
test for links between environmental variation and the evolution of
rates of plasticity, and to provide theoretical models that address the
co-evolution of plasticity rates and capacity (see Introduction).
Although this is beyond the scope of the current paper, our work
provides both methodology and novel insights that should stimulate
future work along these lines. We also re-emphasize a point made
previously (Burton et al., 2022) - that selection on the rate of
plasticity might be stronger than selection on the capacity for
plasticity. Evolutionary theory posits a central role for phenotypic
plasticity in mitigating the fitness impact of environmental variation,
but that possessing the potential for such a response is associated with
a fitness cost in stable environments (Lande, 2009). Fitness costs of
plasticity can be categorized into costs of maintenance and costs of
production. Costs of maintenance represent the investment of resources
into maintaining the machinery required for detecting and responding to
a change in the environment and will be paid at a constant rate
independent of environmental conditions (Auld et al., 2010). In
contrast, production costs are only paid when the plastic response is
triggered and are compensated by the fitness benefits associated with
changing the phenotype. If one assumes that the capacity for plasticity
can be increased by operating the `machinery' required to change a trait
for a longer duration, this will increase production costs but not
maintenance costs. Populations living in less variable environments may
therefore pay a small price for maintaining their capacity for
plasticity (as shown by Van Buskirk \& Steiner, 2009), and adaptation of
this parameter of plasticity to levels of environmental fluctuations may
therefore be relatively modest in magnitude. In contrast, increasing the
rate of change in the same trait would require increasing the size or
output of that `machinery', with corresponding increases in maintenance
costs. Populations living in less variable environments should therefore
experience strong selection against maintaining rapid plasticity due to
higher maintenance costs, and adaptative evolution across populations
may then be expected to be more pronounced for the rate of plasticity.
This line of reasoning is also consistent with theoretical results
showing that maintenance costs shape the evolution of plasticity to a
greater extent than production costs (Sultan \& Spencer, 2002). Given
these considerations, and the results presented in the current study, it
seems prudent to address the hypothesis that adaptation to environmental
variation may be more pronounced in terms of rates of plasticity rather
than capacity of plasticity. By providing clear evidence that rates of
plasticity have diverged among ectotherm classes we show that it is a
trait that evolves, and that increased empirical and theoretical focus
on the rate parameter is likely to provide a way forward for a more
comprehensive understanding of phenotypic plasticity.

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\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

Fig. 1. Illustration of the two parameters of phenotypic plasticity.
Upper panel shows the change in phenotype over time following a change
in the environment at Time = 1. The time it takes for the phenotype to
become fully adjusted (in this case at Time = 2) is determined by
the\emph{rate of plasticity} . For this example, the change in phenotype
follows a linear trajectory, but other shapes of response are possible
(addressed in the current study). Bottom panel shows the corresponding
reaction norms for individuals reared in environment 1 acutely exposed
to environment 2 (i.e. at Time = 1, prior to any adjustment by active
plasticity), and for individuals where the active plastic response is
completed (i.e. at Time = 2). The difference in slope of these two
reaction norms represents the \emph{capacity for plasticity} (Einum et
al. 2019).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

\textbf{Fig. 2.} Median observed rates of plasticity
(\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} ) in temperature tolerance for different
classes of ectotherms animals. Data used represent calculated mean
observed values for each species. Boxes represent the
25\textsuperscript{th} and 75\textsuperscript{th} percentiles, and
whiskers represent 1.5 inter-quartile range from the box. Estimates
obtained from a linear mixed effect model that controls for random
effects as well as acclimation temperature and the slope of the
estimated exponential decay function at the final measurement are given
in Table 2.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

\textbf{Fig. 3.} Observed rates of plasticity (\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english}
) in temperature tolerance as a function of acclimation temperature for
different classes of ectothermic animals. Points represent separate
experiments, and regression lines with a common slope are based on the
best fitting model (Table 2), and are given for the median slope of the
estimated exponential decay function at the final acclimation time
point.

\textbf{Table 1.} 1 AICc comparisons of the top candidate models
explaining variation in rates of plasticity (\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} )
in temperature tolerance among different classes of ectothermic animals.
`Class' is taxonomic class, 'Slope' is the slope of the estimated
exponential decay function at the final measurement, `Measure' is the
measurement type of thermal tolerance (behaviour vs. mortality), and
`Mass' is body mass. Species identity, study and observation are
included as random intercepts in all models. \selectlanguage{greek}Δ\selectlanguage{english}AIC\textsubscript{C} =
12.8 and 14.6 for the best model among those not listed for the top and
bottom data set, respectively.\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
& K & AIC\textsubscript{C} & \selectlanguage{greek}Δ\selectlanguage{english}AIC\textsubscript{C} &
w\textsubscript{i}\tabularnewline
\midrule
\endhead
Reduced data set containing body mass data & & & &\tabularnewline
Acclimation temperature + Class + Slope + Measure & 11 & -1044.2 & 0.00
& 0.378\tabularnewline
Acclimation temperature + Class + Slope & 10 & -1044.0 & 0.19 &
0.343\tabularnewline
Acclimation temperature + Class + Slope + Measure + Mass & 12 & -1042.5
& 1.77 & 0.156\tabularnewline
Acclimation temperature + Class + Slope + Mass & 11 & -1042.0 & 2.28 &
0.121\tabularnewline
Full data set including experiments lacking body mass data & & &
&\tabularnewline
Acclimation temperature + Class + Slope & 10 & -1095.0 & 0.00 &
0.541\tabularnewline
Acclimation temperature + Class + Slope + Measure & 11 & -1094.7 & 0.34
& 0.458\tabularnewline
\bottomrule
\end{longtable}

\textbf{Table 2.} Summary of the best fitting model for the full data
set (i.e. including experiments without body mass data, Table 1,
re-fitted using REML) describing variation in rates of plasticity
(\selectlanguage{greek}\emph{λ\textsubscript{E}}\selectlanguage{english} ) in thermal tolerance among different
classes of ectothermic animals. 'Slope at end' is the slope of the
estimated exponential decay function at the final measurement.\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
. & Estimate & 95\% CI\tabularnewline
\midrule
\endhead
Fixed effects & &\tabularnewline
Amphibians & 0.046 & 0.033 -- 0.059\tabularnewline
Reptiles & 0.029 & 0.007 -- 0.050\tabularnewline
Insects & 0.013 & -0.003 -- 0.028\tabularnewline
Fishes & 0.006 & -0.007 -- 0.019\tabularnewline
Crustaceans & 0.007 & -0.012 -- 0.025\tabularnewline
Acclimation temperature & 0.0006 & 0.0003 -- 0.0009\tabularnewline
Slope at end & 9.12 & 6.50 -- 11.75\tabularnewline
Random effects (SD) & &\tabularnewline
Study & 0.0001 &\tabularnewline
Species & 0.0003 &\tabularnewline
Observation & 0.0002 &\tabularnewline
\bottomrule
\end{longtable}

\selectlanguage{english}
\FloatBarrier
\end{document}

