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Temperature- and size-scaling of phytoplankton population growth
rates: Reconciling the Eppley curve and the metabolic theory of
ecology
Colin T. Kremer,
1,2,a
* Mridul K. Thomas ,
3
Elena Litchman
1,4
1
W. K. Kellogg Biological Station, Michigan State University, Hickory Corners, Michigan
2
Department of Plant Biology, Michigan State University, East Lansing, Michigan
3
Department of Aquatic Ecology, Eawag: Swiss Federal Institute of Aquatic Science and Technology, D€
ubendorf, Switzerland
4
Department of Integrative Biology, Michigan State University, East Lansing, Michigan
Abstract
Quantifying how environmental factors control the growth of phytoplankton communities is essential for
building a mechanistic understanding of global biogeochemical cycles and aquatic food web dynamics. The
strong effects of temperature on population growth rate have inspired two frameworks—the Eppley curve
and the metabolic theory of ecology—that produce different quantitative relationships and employ distinct
statistical approaches. Reconciling these relationships is necessary to ensure the accuracy of ecosystem mod-
els. In this paper, we develop ways to compare these frameworks, overcoming their methodological differ-
ences. Then, analyzing an extensive dataset (>4200 growth rate measurements), we find that increases in
population growth rate with temperature are consistent with metabolic theory, and weaker than previous
estimates of the Eppley curve. A 108C temperature increase will increase growth rates by a factor of 1.53, rath-
er than 1.88 as in previous studies of the Eppley curve. Size and functional group membership are also criti-
cal. Population growth rates decrease with size, but much less strongly that metabolic theory predicts. The
growth rates of different functional groups scale similarly with temperature, but some groups grow faster
than others, independent of temperature. Our results reconcile the analytical methods of the Eppley curve
and metabolic theory, demonstrate that metabolic theory’s temperature-scaling predictions are more accu-
rate, and provide new insights into the factors controlling phytoplankton growth. To avoid over-estimating
the effects of temperature on primary productivity, the parameterization of ecosystem models should be
revised.
Phytoplankton communities regulate global biogeochemi-
cal cycles and support most aquatic food webs. It is critically
important to understand the factors that control their
growth, influencing current and future primary productivity.
Temperature is one such factor, strongly affecting the growth
of phytoplankton populations, mediated by its effects on the
rate of biochemical reactions. Temperature effects are the
primary focus of this paper; however, population growth
rates (hereafter, simply “growth rate”) also depend on cell
size: phytoplankton species with larger cells typically grow
more slowly than those with smaller cells. Relationships
between size, temperature, and growth rate have been docu-
mented for many years, in both the lab and the field, but
the precise representation of these relationships has been
disputed. The dominant bodies of thought, or frameworks,
addressing the temperature-dependence of population
growth are the highly influential Eppley curve (Eppley 1972)
and the metabolic theory of ecology, which we abbreviate as
“MTE” (Gillooly et al. 2001; Brown et al. 2004). The two
frameworks have historically provided quantitatively differ-
ent estimates of the strength of temperature-dependence
(Table 1). Both approaches also differ in their equations, sta-
tistical methods, and foundations: the Eppley curve is based
on characterizations of an empirical relationship, while MTE
provides theoretical predictions emerging from first princi-
ples. These differences have inhibited rigorous comparisons
of the two frameworks. In this paper, we reconcile the
*Correspondence: colin.kremer@yale.edu
a
Present address: Department of Ecology and Evolutionary Biology, Yale
University, New Haven, Connecticut
Additional Supporting Information may be found in the online version
of this article.
1658
LIMNOLOGY
and
OCEANOGRAPHY Limnol. Oceanogr. 62, 2017, 1658–1670
V
C2017 Association for the Sciences of Limnology and Oceanography
doi: 10.1002/lno.10523
contrasting Eppley curve and MTE relationships, making use
of an extensive compilation of growth rate data. This repre-
sents a critical step in developing a mechanistic, quantita-
tively accurate, and predictive understanding of
phytoplankton communities and marine ecosystems at large.
An exponential relationship between maximum growth
rate of phytoplankton and temperature was reported first by
Eppley (1972). Since this seminal paper, the relationship
Eppley estimated has become a fundamental component of
our understanding of how primary productivity is con-
strained by temperature. For example, the “Eppley curve” is
used in algorithms estimating primary productivity from sat-
ellite observations (Morel 1991; Antoine et al. 1996; and
modified in Behrenfeld and Falkowski 1997), and imple-
mented in a wide range of marine ecosystem models (select
examples include Tett et al. 1985; Doney et al. 1996; Palmer
and Totterdell 2001; Taucher and Oschlies 2011; Thomas
et al. 2012; Stock et al. 2014). Together, the data derived
from satellite observations and the predictions made by eco-
system models are essential to studying aquatic ecology,
including detecting and predicting the effects of climate
change on primary productivity. Thus, the Eppley curve’s
accuracy is vital, including both its parameters and the ideas
it embodies. Mathematically, the Eppley curve can be writ-
ten as:
l5aexp bTðÞ (1)
In this equation, lis the maximum growth rate of any phy-
toplankton species, which varies with temperature, T. Param-
eter acontrols the height of the exponential function at 08C
while bdetermines how strongly lrises with temperature, a
value often referred to as the Eppley exponent. Eppley’s ini-
tial estimates of these parameters (a50.59 and b50.0633)
have been updated using more data and rigorous quantile
regression methods (a50.81 and b50.0613), the values we
use throughout the rest of this paper (Table 1; Bissinger et al.
2008; but see also Brush et al. 2002). Ultimately, the Eppley
curve is a phenomenological description of the temperature-
dependence of growth, whose parameters are estimated from
data rather than predicted from first principles.
Metabolic theory (MTE) offers a contrasting, mechanistic
description of the same empirical pattern, derived from
studying the factors constraining individual metabolic rates
(Gillooly et al. 2001; Brown et al. 2004). It focuses specifically
on the biochemical reactions essential for life, which depend
on temperature and the supply of substrate(s). Assuming
substrates are not limiting, reaction rates will depend only
on temperature, and can be described by the Arrhenius-van’t
Hoff equation: k/e2E=RT . This states that the rate constant
of a reaction, k, is proportional to an exponential function
of: temperature, T, the Boltzmann constant (R58.617 3
10
25
eV K
21
), and the activation energy of the reaction, E.
Conceptually, Edescribes the energy required to initiate a
specific chemical reaction and determines the sensitivity of k
to changes in temperature. MTE further assumes that the
rate of the slowest chemical reaction (lowest E) that is essen-
tial for life will determine just how quickly individuals and
populations can grow, such that l/k(Savage et al. 2004a).
As such, Eis analogous to the Eppley exponent b, in that
they both control the sensitivity of maximum growth rate to
temperature. Despite this, it is impossible to directly com-
pare these coefficients because the Eppley curve is a function
Table 1. Comparing the predictions of the Eppley and MTE approaches to understanding the effects of temperature on growth.
Eppley MTE
Temperature dependence l5aexp bTðÞ l5l0exp 2E=kT1273ðÞðÞðÞ
Predicted coefficients *a50.81, b50.0631 †E50.32 eV
Implications Q
10
51.88 Q
10
51.62
Data All growth rates measured across
temperatures and species
Maximum growth rates of individual species
and corresponding temperatures
Statistical method ‡Quantile regression Multiple linear regression, mixed models, or SMA regression
Full regression equation ln l99
½5ln a1bT1c2M1c3Gln l½5ln l01E21
kT1273ðÞ
1alnM1c3G
Covariates T5temperature (8C), M5mass, G5functional group identity
Parameters l
99
599
th
quantile growth rate
b5Eppley coefficient or exponent
a5maximum growth rate at 08C
c
2
5mass effect
c
3
5functional group effect
l5maximum growth rate
E5activation energy (eV)
l
0
5normalization constant
a5mass scaling coefficient
c
3
5functional group effect
k5Boltzman factor, 8.6173 310
25
eV/K
* Bissinger et al. (2008), Eppley (1966).
†
Savage et al. (2004a), Allen et al. (2005).
‡
Koenker (2005), Cade and Noon (2003).
Kremer et al.Temperature-scaling of phytoplankton growth
1659
of T, while the Arrhenius-van’t Hoff equation depends on
21/T. Fortunately, as we will see, an approximation enables
us to compare these competing frameworks (Gillooly et al.
2002; Supporting Information: Converting between MTE and
Eppley).
MTE makes explicit predictions for the value of E,
depending on which biochemical process is rate-limiting for
growth. In heterotrophs, this is respiration (ATP synthesis),
which has an activation energy of 0.65 eV (Allen et al. 2005;
L
opez-Urrutia et al. 2006). Photoautotrophs, such as phyto-
plankton, are limited instead by photosynthesis, with the
lower activation energy of 0.32 eV (Allen et al. 2005; L
opez-
Urrutia et al. 2006). This provides an explicit prediction of
the temperature-dependence of the maximum growth rate of
phytoplankton, given saturating light and nutrients. Rela-
tionships between maximum growth rate and temperature
have been widely reported (see meta-analysis of Angilletta
et al. 2010), although many of the studies examined report
variation from the theoretical predictions of Savage et al.
(2004a). There is evidence in plankton that the population
growth rate of herbivorous and bacterivorous protists and
copepods scales more strongly with temperature than photo-
trophic protists (Rose and Caron 2007), supporting the gen-
eral idea that heterotrophs are more sensitive to temperature
than autotrophs.
Unlike the Eppley curve, MTE also accounts for the effects
of the size of organisms on their metabolic rate and ability
to grow. Just as the effect of temperature on individual meta-
bolic rate can be scaled up to provide insights on population
growth rate, the effects of an individual’s size on its metabol-
ic rate can be propagated similarly (Savage et al. 2004a). Spe-
cifically, the maximum growth rate, l
max
, of a population at
a fixed temperature is related to the average mass of individ-
uals in the population, M, following the relationship l
max
/
M
a
(Savage et al. 2004a). In this expression, agoverns the
strength of the mass scaling relationship. It has a theoretical-
ly predicted value of 21/4, based on properties related to the
optimization of resource uptake and distribution networks
(West et al. 1997, 1999a,b; Banavar et al. 1999; Brown et al.
2004; Savage et al. 2004). This implies that populations con-
sisting of larger individuals will grow more slowly than those
comprised of smaller individuals (or cells, in the case of phy-
toplankton). Overall, this predicted relationship has been
upheld for organisms ranging from vertebrates to unicellular
eukaryotes (Savage et al. 2004a).
However, investigations of size-scaling relationships in
phytoplankton tend to reveal lower values of athan MTE
predicts. Exact estimates vary depending on details including
the taxonomic composition of communities, the metric of
size examined (volume vs. mass in carbon), and the location
(lab or field) and experimental methods of studies (L
opez-
Urrutia et al. 2006; Mara~
n
on 2008; Chen and Liu 2010,
2011; Sal & L
opez-Urrutia 2011; Sal et al. 2015). As phyto-
plankton sizes span more than seven orders of magnitude
(biovolume in lm
3
, Finkel et al. 2009), even a weak relation-
ship would result in substantial predicted differences in
growth rate between large and small species. Aside from
identifying the value of a, several studies suggest that the
relationship between size and growth rate may actually be
unimodal, rather than linear (Bec et al. 2008; Chen and Liu
2010; Mara~
n
on et al. 2013; Mara~
n
on 2015). These patterns
may be driven by differences in evolutionary history among
species (Sal et al. 2015), although other evidence suggests
unimodal relationships occur within taxonomic groups
(Raven 1994). While the precise nature of the effects of cell
size on phytoplankton growth rate remains a topic of debate,
the overall existence of a relationship is not in question. As
a result, it is important to jointly consider the effects of size
and temperature when studying the factors limiting popula-
tion growth.
A final consideration is the extreme diversity of the phy-
toplankton, consisting of groups such as cyanobacteria and
diatoms whose evolutionary histories diverged long ago.
Many recent studies have found that the relatedness and
functional group identity of species is an important predictor
of their physiology, ranging from light and nutrient uptake
traits (Litchman et al. 2007; Edwards et al. 2012, 2015), to
thermal tolerance traits (Thomas et al. 2016). Similar taxo-
nomic effects have been found in the metabolic theory liter-
ature across wide swaths of life. For example, groups from
unicellular organisms to fish and amphibians have identical
slopes relating their mass-corrected metabolic rate to temper-
ature, but their normalization constants (intercepts) are
quite different (Gillooly et al. 2001; Brown et al. 2004). At
best, ignoring these effects may obscure relationships across
species, while in some cases the inferred shape of relation-
ships may change entirely (e.g., Bickel et al. 1975; Clark
et al. 2011; Sal et al. 2015). Understanding how and why the
traits of different functional groups differ is also of practical
use, given their unique effects on global biogeochemical
cycles (Litchman et al. 2015). Relevant to our study, func-
tional group identity has already been shown to explain vari-
ation in maximum growth rate of marine and freshwater
phytoplankton, even after accounting for differences in cell
volume (Litchman et al. 2007; Edwards et al. 2012). Conse-
quently, testing the importance of functional group identity
is a critical component of our efforts to reconcile metabolic
theory and the Eppley curve.
In the current paper, we show that the temperature-
dependence of growth rate is consistent with MTE, and
weaker than suggested by earlier estimates of the Eppley
curve, after correcting for the effects of cell size and func-
tional group identity. This conclusion is based on a recently
expanded data set (with >4200 observations), which
includes observations used in earlier analyses (Eppley 1972;
Bissinger et al. 2008) as well as additional values from the
primary literature. It also required developing a way to com-
pare these two competing frameworks, given their divergent
Kremer et al.Temperature-scaling of phytoplankton growth
1660
statistical approaches. Our results suggest that the
temperature-dependence utilized in many areas of aquatic
ecology (including satellite observation algorithms and eco-
system models) need to be adjusted to provide accurate
results. In particular, models of marine primary productivity
that rely on the Eppley curve (e.g., Doney et al. 1996; Palmer
and Totterdell 2001; Taucher and Oschlies 2011; Toseland
et al. 2013; Stock et al. 2014), as previously parameterized
(Eppley 1972; Bissinger et al. 2008) are likely to over-
estimate productivity increases driven by rising ocean
temperatures.
Methods
Data sources
Thermal tolerance curves
We previously compiled data on estimates of population
growth rates at varying temperatures from primary literature
sources (Thomas et al. 2012, 2016). Subjected to rigorous
quality control, these data consist of laboratory measure-
ments that were made under reasonable light, nutrient, and
salinity conditions (for details, see Thomas et al. 2012,
2016). In addition to those criteria, for the current analysis
we also excluded any growth rates reported to be less than
0.1 (d
21
), removing growth rate values near or below zero,
which are very difficult to estimate reliably. We also exclud-
ed data from phytoplankton belonging to functional groups
for which we had few measurements, leaving only diatoms,
dinoflagellates, green algae, and cyanobacteria. Our data set
includes marine, estuarine, and freshwater strains. Marine
and freshwater species share many similarities in their ther-
mal traits, although freshwater species have higher optimum
temperatures (temperatures at which they achieve their max-
imum growth rate) and potentially wider thermal niches
(Thomas et al. 2016). To focus on a biologically relevant
range of temperatures, we also excluded a small number of
growth rate estimates from temperatures >408C. Finally, we
excluded observations of taxa for which we were unable to
find size estimates (next section). The remaining data serve
as the foundation of subsequent analyses (4208 measure-
ments in total; Supporting Information Table S.1) and are
provided in the Supplementary Information.
Cell size data
Because papers supplying data on temperature-dependent
growth rates rarely include corresponding estimates of cell
size, we gathered estimates from a range of alternative sour-
ces, similar to the approach of Sal et al. (2015). The most
common metric of size in phytoplankton is cell volume, typ-
ically estimated by calculating the mean volume of cells in a
culture based on their three-dimensional shape and linear
dimensions (Hillebrand et al. 1999). Previous papers have
assembled cell volumes for freshwater and marine species
(Litchman et al. 2009; Edwards et al. 2011). With these
resources, and a new compilation of size data for >1200
freshwater species (Kremer et al. 2014), we obtained cell vol-
ume estimates for 92% of the taxa represented in our
growth rate data set. Within functional groups, coverage of
size estimates in our original growth data set ranged from a
low of 81% (dinoflagellates) to a high of 98% (diatoms). Bio-
volume data and metadata are provided in the Supporting
Information. Cell volume estimates (in lm
3
) were converted
into estimates of dry weight (in lg) using the relationship
Mass 50.47*(Volume)
0.99
310
26
, after Reynolds (2006, p.
25). This is a standard relationship, although the exact con-
nection between cell size and dry weight can vary signifi-
cantly between cells and species, depending on the size of
internal vacuoles and storage (Reynolds 2006), and other
conversions have been proposed (e.g., Strathmann 1967;
Menden-Deuer and Lessard 2000). In the analyses we pre-
sent, we have focused on ln(mass) as our measure of phyto-
plankton body size, using the Reynolds relationship.
However, analyses based on ln(cell volume) and estimates of
mass following Strathmann (1967) yielded similar results.
Finally, we also acknowledge that cell size varies significantly
due to intra-specific variation and environmental effects. As
a result, the literature-based size estimates we have obtained
may differ from the true size of cells during the actual
growth assays. This likely introduces additional uncertainty
to our analyses, but is unavoidable.
Eppley re-analysis
Prior analyses
The Eppley curve attempts to capture an upper envelope
bounding maximum growth rates as an exponential function
of temperature. Bissinger et al. (2008) used regression techni-
ques to rigorously estimate this envelope. Whereas standard
linear regression models the relationship between covari-
ate(s) and the mean of a response variable, quantile regres-
sion models variation in specific quantiles of the response
variable. Models of extreme quantiles can be used to gener-
ate envelope functions (Cade and Noon 2003; Koenker
2005). Quantile regression is a useful approach, but it has
some important limitations. In particular, acceptable fitting
of extreme quantiles requires a great deal of data (Cade and
Noon 2003; Koenker 2005). To avoid sample size issues, in
our analyses we did not consider models with interactions
between our covariates (size, temperature, functional group),
for which we also lacked a priori hypotheses.
Current analysis
We used quantile regression to examine linear relation-
ships between the 99
th
quantile of ln(l) and covariates
including temperature, ln(M), and functional group (Table
1). Linear models relating ln(l) and temperature are equiva-
lent to assuming that growth rate, l, is exponentially related
to temperature (Bissinger et al. 2008). We fit and evaluated
all quantile regression models using the [R] package quantreg
(version 5.19; Koenker 2015). To estimate the parameters of
the Eppley curve, we used the entire growth rate data set
Kremer et al.Temperature-scaling of phytoplankton growth
1661
described in 2.1 above. As with previous data used to address
this question, our data set includes many examples where
there are multiple growth rate observations per strain (rang-
ing from 2 to 97), including sub-optimal growth rates esti-
mated at temperatures above or below a strain’s optimum
temperature. Previous studies ignore this issue, but these
measurements create a lack of independence among observa-
tions. Such situations are typically handled using a mixed
effects model; however, there are no accessible methods
combining mixed effects models and quantile regression. As
a compromise, we used weighted quantile regression to
ensure that each species contributes equally to the analysis
independent of how many times they were measured. We
calculated weights, w
i
, for the observations associated with a
particular strain ias w
i
51/(# of observations for strain i). In
strains with many observations, each individual observation
will consequently contributes less to the overall analysis.
Ultimately, this enables us to avoid biasing quantile regres-
sions towards intensively studied species or strains of phyto-
plankton. Code for these, and subsequent analyses, is
provided in the Supporting Information. We also repeated
our analyses after removing a few very high growth rate esti-
mates, to check the sensitivity of the quantile regression fits
to potential outliers (results not shown). The results did not
change significantly, perhaps due to the higher level of
uncertainty already inherent in regressions of extreme
quantiles.
Metabolic theory analysis
Prior analyses
Tests of metabolic theory differ in several significant ways
from the Eppley-style analyses described in 2.2, including
both the data and statistical methods they employ. Under-
standing and resolving these differences is essential to our
work reconciling MTE with the Eppley curve, so we discuss
the three major differences here. First, while Eppley analyses
use all available growth rate data, MTE analyses ideally focus
on just the highest observed growth rate per strain or spe-
cies. While this dramatically reduces the amount of data
used, MTE analyses do not depend on extreme quantile
regressions, so they are inherently less sensitive to small
sample sizes. Second, while the Eppley curve models maxi-
mum growth rate, l
max
, as a function of exp(T), MTE uses a
Boltzmann term, which depends on exp(1/T). More specifi-
cally, combining the temperature- and size- (or mass-)
dependence of growth yields (after Savage et al. 2004a):
lmax 5l0Mae2E=kT (2)
where l
0
is a normalization constant or intercept term, and
other variables are as defined previously. We reconcile these
differences using an approximation discussed later, in
“Connecting Eppley and MTE analyses” section. Third,
although the effects of mass and temperature can be
combined in a single equation as in (Eq. 2), tests of MTE typ-
ically examine each effect separately. For example, a mass-
corrected growth rate will be regressed against temperature
(assuming the theoretical value of a521/4 is true), while a
temperature-corrected growth rate will be regressed against
mass (assuming E50.32 eV; or 0.65 eV for heterotrophs).
These bivariate analyses are necessary in MTE analyses that
rely on “standardized major axis regression” or SMA (Warton
et al. 2006; Edwards et al. 2015), a commonly used MTE
technique that does not easily accommodate multiple covari-
ates. While there are good reasons for conducting MTE tests
this way (Warton et al. 2006), the approach has drawbacks:
(1) estimating size and temperature effects separately, rather
than conducting a multiple regression, can lead to parameter
estimates with incorrect confidence intervals, and (2) SMA is
incompatible with mixed effects models, yet data sets are
often structured hierarchically, with repeated measurements
of one or more species (Edwards et al. 2015).
Current analysis
To test metabolic theory’s predictions, in light of these
methodological issues and our over-arching goal of comparing
MTE and the Eppley curve, we have adopted different
approaches. First, we reduced our data so that it included only
measurements of the highest observed growth rate for each
strain, yielding n5425 values across 194 unique species
(some species were represented by multiple strains). Second,
rather than using SMA, we use linear mixed effects models.
These have been used occasionally in MTE type analyses (e.g.,
O’Connor et al. 2007; Yvon-Durocher et al. 2012). With this
approach, we can simultaneously investigate the effects of
multiple covariates, while using a species-level random effect
to account for the fact that 37% of the species in our data
set are represented by multiple, separately measured strains.
We examined models containing main effects of size, temper-
ature, and functional group, as well as models where function-
al group interacted with size and/or temperature.
Specifically, models were fit in [R] using the function lmer
in the lme4 package (version 1.1-10; Bates et al. 2014). Mod-
el comparison was used to select the best model based on
AICc, which corrects for small sample sizes and converges on
AIC for large sample sizes (Burnham and Anderson 2002).
Estimates of p-values were obtained using a parametric boot-
strapping approach, PBmodcomp, in the pbkrtest package
(version 0.4-4; Halekoh and Højsgaard 2014). This method is
resilient to violations of the asymptotic assumptions inher-
ent in likelihood ratio and Wald tests (Halekoh and
Højsgaard 2014), although all three methods yielded similar
results. To assess the overall goodness of fit of models, we
calculated estimates of conditional and marginal R
2
using
the function r.squaredGLMM in the R package MuMIn (ver-
sion 1.13.4; Nakagawa and Schielzeth 2013). Finally, Tukey
post-hoc comparisons were conducted using glht, also in the
multcomp package (version 1.4-1; Hothorn et al. 2008), to
Kremer et al.Temperature-scaling of phytoplankton growth
1662
test for differences between functional groups as determined
by our mixed effects model.
Connecting Eppley and MTE analyses
There are two remaining differences between the Eppley
and MTE analyses that make it more difficult to compare their
results. First, as we have mentioned above, these analyses use
different temperature scales (Tvs. 1/T), so their temperature
coefficients (band E) are not directly comparable. This issue
can be resolved using a Taylor series approximation to express
the MTE relationship as a function of temperature, T(see Lai-
dler 1984 and Supporting Information: Converting between
MTE and Eppley). This leads to a simple conversion between
the Eppley exponent band MTE’s activation energy E:
bE
kT02or EbkT02(3)
where T
0
is 273. Mathematically, (Eq. 3) allows us to compare
numerical estimates of the temperature-scaling of growth
obtained from Eppley and MTE analyses. For example, we can
take estimates of b, its standard error, and its 95% confidence
interval (all obtained by quantile regression), and convert them
into activation energies. Similarly, an estimated activation ener-
gy coefficient from an MTE regression can be converted into an
Eppley exponent. To test whether these values differ significant-
ly from their theorized or previously estimated values (Table 1),
we can use a simple two-tailed test of differences in slope (Zar
1999). Second, we note that the conversion in Eq. 3 simply
allows us to quantitatively compare Eppley and MTE coeffi-
cients. Differences in these coefficients may be affected by the
different statistical methods (quantile regression vs. linear
regression) and data (all growth rates vs. maximum growth
rates of individual strains) employed by each approach (Eppley
and MTE, respectively). This is important to recall when inter-
preting results presented in the following sections.
Results
Eppley results
We found significant effects of temperature, cell mass,
and functional group identity on the maximum growth rates
of phytoplankton (Table 2, Supporting Information Table
S.2), using methods from “Current analysis” section. The
model including these three main effects performed better
than a set of simpler models (Supporting Information Table
S.3), based on AICc comparison. As expected, growth rates
increased with temperature (Fig. 1A; b50.047, p<0.0001,
95% CI 5{0.034, 0.060}), but decreased with cell size (Fig.
1B; slope 520.084, p<0.0001, 95% CI 5{20.125, 20.043}).
Functional groups displayed significantly different baseline
capacities for growth: across temperatures and cell masses,
the cyanobacteria and dinoflagellates exhibited much lower
maximum growth rates than diatoms and green algae (Fig. 1;
Supporting Information Table S.2). In fact, cyanobacteria
and dinoflagellates never display maximum growth rates as
high as in previous estimates of the Eppley curve (Fig. 2),
while diatoms and green algae do so only at low tempera-
tures. These differences among groups cannot be attributed
to differences in size between groups (Supporting Informa-
tion Fig. S.1), as cell mass was included in the same model.
We were unable to test for an interaction between tempera-
ture and functional group (i.e., distinct Eppley exponents for
each group), due to the data-hungry nature of extreme quan-
tile regressions. However, this question is addressed in the
MTE analyses below.
MTE results
Performing an MTE analysis (see “Prior analyses” section
above), we again found significant main effects of tempera-
ture, cell mass, and functional group identity on maximum
growth rates (Fig. 3; Table 2, Supporting Information Table
S.4). In particular, temperature increased growth rates, with
an activation energy of E50.30 eV (p<0.001, 95%
CI 50.233 to 0.368; Supporting Information Table S.4),
which is quite similar to the theoretically predicted value of
0.32 eV (Table 1). While mass had a negative effect on
growth rate (a520.054; p<0.005, 95% CI 520.089 to
20.018), the strength of this effect was much weaker than
predicted (a520.25; Savage et al. 2004a,b). As with the Epp-
ley analysis, functional groups differed from each other in
their capacity for growth (i.e., growth rate at 08C; p<0.001),
except for cyanobacteria and dinoflagellates (Supporting
Table 2. Results of the Eppley curve and MTE analyses. Additional details on model fits are presented as Supporting Information
Tables S.2 and S.4. Variables match those defined in Table 1.
Eppley curve results MTE results
Coefficient Variable Estimate Coefficient Variable Estimate
Temperature b0.04738 1/(kT) E20.300
ln(Mass) c
2
20.08408 ln(Mass) a20.054
Intercept (Cyanobacteria) a21.52173 Intercept (Cyanobacteria) l
0
10.50
Diatoms c
3
0.69629 Diatoms c
3
0.992
Dinoflagellates 0.05037 Dinoflagellates 0.135
Greens 0.81941 Greens 0.530
Kremer et al.Temperature-scaling of phytoplankton growth
1663
Information Table S.5). We also fit two additional models
testing for an interaction effect between functional group
identity and temperature, or size and temperature, respec-
tively. However, AICc model comparison showed that nei-
ther of these more complex models were improvements
(Supporting Information Table S.6). This implies that phyto-
plankton share a common activation energy across function-
al groups and cell sizes. This is consistent with the MTE
literature, which has repeatedly identified similar scaling
exponents across major branches of life, and does not pre-
dict interactive effects between size and temperature (e.g.,
Brown et al. 2004; Savage et al. 2014a).
Eppley vs. MTE comparison
Finally, we turn to comparing our newly estimated effects
of temperature on maximum growth rate (after accounting
for both mass and functional group effects) with the values
provided by MTE and previous estimates of the Eppley curve.
We find that the temperature-dependence of population
growth rate is significantly weaker than implied by previous
estimates of the Eppley curve, but not statistically different
than MTE predicts (Fig. 4; Supporting Information Table
S.7). This result applies whether we use the parameters
estimated from the quantile regression approach, or MTE
methods (Supporting Information Table S.7). Interestingly,
when we fit a simple, un-weighted quantile regression of
maximum growth rate against temperature alone (reproduc-
ing Bissinger et al. 2008’s re-analysis of the Eppley curve), we
obtain a higher estimate of b(0.055; p<0.00001, 95%
CI 5{0.049, 0.061}; Supporting Information Table S.8). This
corresponds to a temperature-scaling effect that is weaker
than prior estimates of the Eppley curve, stronger than MTE
predictions, and significantly distinct from both (Supporting
Information Table S.7).
Discussion and conclusion
Characterizing the effects of temperature on organisms is
central to understanding fundamental patterns in aquatic
communities and ecosystems. This task is impeded by the
existence of two popular frameworks (the Eppley curve and
the metabolic theory of ecology) that describe the
temperature-dependence of growth rate differently. Figuring
out how to directly compare these paradigms is essential for
advancing the study of phytoplankton communities and for
refining contemporary models of primary productivity and
0246
Cyanobacteria
Growth rate (d-1)
Diatoms
-15 -10 -5 0
0246
Dinoflagellates
Mass, ln(μg)
Growth rate (d-1)
-15 -10 -5 0
Greens
Mass, ln(μg)
0246
Cyanobacteria
Growth rate (d-1)
Diatoms
0 10203040
0246
Dinoflagellates
Growth rate (d-1)
0 10203040
Greens
99th quantile
regression
Bissinger
et al. 2008
A B
Fig. 1. (A) The Eppley curves of phytoplankton functional groups vary substantially (solid lines). Plotted regressions are from a weighted, 99
th
quan-
tile regression including main effects of temperature, functional group, and mass. For visualization purposes, each curve was drawn using the mean
mass of the corresponding functional group. Individual observations contributing to the analysis are indicated by partially transparent points. Notably,
temperature-scaling in our new estimate of the Eppley curve is weaker than previously estimated by Bissinger et al. (2008) (dotted line). For reference,
we also plot the results of applying Bissinger et al.’s approach, ignoring size and functional group, to our data set (dashed line). (B) Growth rates
simultaneously decline with increasing cell mass; these plots show the size effect produced by the same 99
th
quantile regression as displayed in panel
A. For visualization, each curve was drawn using the mean temperature of the corresponding functional group within our data set.
Kremer et al.Temperature-scaling of phytoplankton growth
1664
global ecosystems. We have demonstrated how these two
frameworks, with divergent histories and separate literatures,
can be related (“Connecting Eppley and MTE analyses” sec-
tion and “Eppley vs. MTE comparison section” above). From
this foundation, our analyses show that the increase of pop-
ulation growth rate with temperature is weaker than previ-
ous characterizations of the Eppley curve, yet consistent
with MTE predictions (Supporting Information Table S.7),
suggesting that maximum growth rate is indeed limited by
the temperature sensitivity of photosynthesis. For a 108C
increase in temperature, previous estimates of the Eppley
curve suggested that growth rates would increase by a factor
of 1.88; in contrast, our new results suggest that this increase
is smaller, around a factor of 1.53 (given Supporting Infor-
mation Table S.3). We also found evidence that differences
in size explain less variation in growth rate than MTE predic-
tions suggest. Under the assumed MTE value of 20.25 for
the size-scaling relationship, a decrease in cell mass of 3
orders in magnitude would increase growth rates by a factor
of 3.5; in contrast, if the weaker relationship we obtained
is true, growth rates would only increase by a factor of 1.5
(given Supporting Information Table S.3).
Critically, our results also show that the slope of the
temperature-scaling relationship is consistent across func-
tional groups (Supporting Information Table S.6), although
groups differ in their intercepts (maximum growth rate at
T508C). The invariant nature of this relationship across
groups supports a central tenet of metabolic theory—that
the temperature dependence of growth rate under replete
conditions emerges from the shared biochemistry and ther-
modynamic constraints of all photoautotrophs. It also
implies that increases in ocean temperatures will not change
the growth rate hierarchy of functional groups, as their
growth rate vs. temperature relationships will never cross.
Despite the our new estimate of a weaker effect of tempera-
ture on growth rate, we found that diatoms and green algae
have intercepts that are higher than the previously estimated
intercept of the Eppley curve (Table 1 vs. Supporting Infor-
mation Table S.2). This implies that models using older
parameterizations of the Eppley curve to describe the growth
of these functional groups will underestimate their growth
rates at low temperatures, while over estimating their growth
0 10203040
-6 -5 -4 -3 -2 -1 0
Change in growth rate
Diatoms
Green algae
Cyanobacteria
Dinoflagellates
Fig. 2. Difference in predicted maximum population growth rate (d
21
)
relative to the Eppley curve as estimated by Bissinger et al. (2008).
Although our results indicate that growth rate increases more slowly with
temperature than in previous versions of the Eppley curve, functional
groups have different intercepts. Consequently, at low temperatures, fast
growing green algae and diatoms grow faster than previously predicted.
In contrast, at high temperatures, all groups show substantially lower
growth rates. [Color figure can be viewed at wileyonlinelibrary.com]
AB
Fig. 3. Partial regression plots from the metabolic theory analysis showing the effects of: (A) temperature and (B) cell mass on growth rate, separated
by a significant effect of functional group.
Kremer et al.Temperature-scaling of phytoplankton growth
1665
rates at high temperatures (Fig. 2). Collectively, our results
highlight the importance of developing a conceptually and
quantitatively accurate framework describing interactions
between cellular physiology, environmental factors (like
temperature), and key ecological parameters (such as species’
population growth rate).
Our efforts to resolve the differences between the Eppley
and MTE frameworks depended critically on accounting for
the effects of cell size and functional group. Although the
possibility of these effects was discussed in previous work, it
was not directly addressed (Eppley 1972; Bissinger et al.
2008). When we performed a 99
th
quantile regression relat-
ing population growth rate to temperature, but ignoring
both of these factors, we obtained an estimate of the temper-
ature effect that was significantly larger than the MTE pre-
diction (“Eppley vs. MTE comparison” section; Supporting
Information Table S.7). We believe that this discrepancy like-
ly arises from what is essentially a model specification error.
Basically, there is heterogeneity in the observed growth rates
that a temperature-only model cannot capture, as it is attrib-
utable to functional group identity and cell size. Otherwise
unexplained, this heterogeneity ends up biasing estimates of
the temperature-scaling coefficient in quantile regressions,
especially at extreme quantiles (e.g., the 99
th
quantile), yield-
ing higher estimates of the temperature-scaling coefficient.
We present additional results and discussion of this statisti-
cal issue (see Supporting Information: Evaluation of quantile
regression stability). This effect may explain why previous
analyses of the Eppley curve (e.g., Bissinger et al. 2008)
found temperature coefficients that are larger than predicted
by MTE analyses.
Functional group identity and cell size are also interesting
in their own right, not just because they are important for
obtaining rigorous estimates of temperature effects. We turn
now to discussing cell size and mass. Although Eppley noted
that cell size could have significant effects on growth rate
(1972), his analysis and the resulting Eppley curve do not
account for size. In contrast, metabolic theory makes explic-
it, quantitative predictions for the effect of mass on individ-
ual metabolism and, subsequently, population growth rate.
While we found that growth rate decreases with cell size,
consistent with general knowledge, the rate of decline was
much weaker than MTE predicts (20.054 rather than 20.25).
This result is consistent with previous publications on phyto-
plankton (review of Chisholm 1992; more recently, Mara~
n
on
2008; Chen and Liu 2011; Sal et al. 2015). Note that in these
and other studies, it is important to distinguish between
examples of the temperature sensitivity of production, respi-
ration, and growth rate.
There are several reasons why we might observe size scal-
ing relationships in phytoplankton that deviate from MTE
predictions: (1) Cell size varies significantly within and
among strains and species, often in response to environmen-
tal conditions such as temperature (e.g., Finkel et al. 2009).
We used mean cell size estimates taken from the literature,
rather than values taken during growth rate assays, as do
other studies (e.g., Edwards et al. 2011, 2012; Sal et al. 2015).
This introduces noise to our analysis, which would reduce
the strength of observed relationships. (2) The strength of
size-scaling relationships can be sensitive to the metric of
size used. For example, based on in situ data, L
opez-Urrutia
et al. (2006) found that rates of both net production and res-
piration scaled nearly isometrically (i.e., with slope 51) with
the total mass of carbon in cells, deviating from MTE predic-
tions. However, using cell biovolume instead of mass yielded
a relationship more consistent with the 3
=
4prediction, which
0.04
0.05
0.06
Eppley
(
g
roup & mass)
MTE Eppley
(classic)
Eppley exponent
0.25
0.30
0.35
0.40
Eppley
(
g
roup & mass)
MTE Eppley
(classic)
Activation energy (eV)
AB
Fig. 4. We can illustrate the temperature-scaling coefficients obtained through Eppley or MTE analyses as either (A) an Eppley exponent or (B)an
activation energy, due to the conversion derived in the Supporting Information. This allows us to compare our newly estimate coefficients (black dots,
with 95% confidence interval error bars) with predictions based on previous Eppley analyses (red dotted line; Bissinger et al. 2008) and theoretical
MTE predictions (blue dashed line). Our Eppley analysis (accounting for functional group and mass) and the MTE analysis yield coefficients that are
significantly lower than previous Eppley estimates, but not significantly different from the MTE prediction (Supporting Information Table S.7). Howev-
er, an Eppley analysis ignoring functional group and mass (“Epply classic”) yields a larger temperature-scaling coefficient that differs significantly from
the MTE and Eppley predictions (Supporting Information Table S.7). [Color figure can be viewed at wileyonlinelibrary.com]
Kremer et al.Temperature-scaling of phytoplankton growth
1666
they attributed to the allometric relationship between the
volume and carbon content of cells. These conversions are
problematic: multiple relationships exist for converting
between cell volume and mass in carbon (e.g., Strathmann
1967; Menden-Deuer and Lessard 2000; Reynolds 2006), and
the choice of conversion can affect estimates of scaling rela-
tionships (Mara~
n
on 2008). However, in our analysis, we
found very little difference in the size-scaling coefficient
whether we used ln(biovolume) or ln(mass), calculating mass
using two different conversions (Supporting Information
Table S.9; Strathman 1967; Reynolds 2006). In the end, due
to variation in the physiology of different species and func-
tional groups (e.g., vacuoles affect cell size, but hold little
carbon), a universally applicable conversion based on allom-
etry may not exist; (3) Finally, there are important aspects of
the biology of phytoplankton not incorporated into MTE’s
predictions that could lead to a weaker size-scaling relation-
ship (Chisholm 1992; Mara~
n
on 2015). Larger cells are gener-
ally assumed to have lower surface area to volume ratios and
to experience increased self-shading. As a result, growth rate is
thought to decrease with size, as larger cells struggle to use
light efficiently and to obtain sufficient nutrients and distrib-
ute them internally through diffusion. However, changes in
cell shape with increasing size can alleviate these constraints
and weaken the size-scaling relationship (Mara~
n
on 2008).
Additionally, small cells may experience limited growth rates
because unavoidable investments in non-scalable nitrogenous
compounds such as DNA result in reduced biosynthetic capac-
ity. Depending on their strength and shape, the combined
effects of opposing tradeoffs between being small and large
could produce a size-scaling relationship that is weaker than
the MTE proposed 21/4, or even result in a nonlinear relation-
ship (Mara~
n
on et al. 2013; Mara~
n
on 2015). Additional
research is clearly needed to determine the strength and shape
of size-scaling relationships, efforts that will benefit from con-
trolling for confounding effects of environment and phyloge-
netic relationships among taxa (e.g., Sal et al. 2015).
We turn next to considering functional group effects:
both the Eppley and MTE style analyses indicated that func-
tional group identity explained significant variation in the
intercept (but not the slopes) of the temperature- and size-
scaling relationships (Figs. 1, 3; Supporting Information
Tables S.3, S.6). Variation in intercepts among taxonomic
groups has been frequently documented in MTE studies
(e.g., Brown et al. 2004) and occasionally discussed (Gillooly
et al. 2001). However, to our knowledge, no rigorous mecha-
nisms explaining these differences have been proposed, cre-
ating opportunities for the future development of metabolic
theory. The differences in growth rates we observe among
phytoplankton groups are consistent with previous work
(Edwards 2012), adding to a growing literature on the dis-
tinct traits of functional groups (Litchman and Klausmeier
2008; Finkel et al. 2009; Edwards et al. 2012, 2015; Thomas
et al. 2016). The slow growth rate of dinoflagellates has been
noted previously (Tang 1995), especially relative to diatoms
(Chisholm 1992). The source of these differences may be
linked to fundamental physiological distinctions. For exam-
ple, genome size varies across taxonomic groups: dinoflagel-
lates in particular have enormous genomes, and consistently
lower growth rates (e.g., Litchman et al. 2007; Oliver et al.
2007). Functional groups also differ in their ecological strate-
gies, in ways that may influence their growth rate. Dinofla-
gellates are often mixotrophic, but our data reflect their
ability to grow autotrophically. The ability to both photo-
synthesize and consume other organisms may come at the
expense of rapid growth. In contrast, diatoms can grow very
rapidly, perhaps consistent with selection to take advantage
of pulses of nutrients supplied by seasonal turnover or other
events. In general, because the functional group differences
we document are consistent across dozens of species and a
wide range of temperatures, the underlying mechanisms
seem likely to be tied to evolutionarily conserved features
(Thomas et al. 2016).
Together, developing an accurate knowledge of the
temperature-dependence of phytoplankton growth, as well as the
distinct properties of functional groups, is important for develop-
ing rigorous ecosystem models. These models often attempt to
describethedynamicsofentireecosystemsacrossbroadspatial
and temporal ranges by reducing the complexity of phytoplank-
ton and zooplankton communities into a handful of representa-
tive groups, usually based on size and/or functional group
identity. For these models to accurately forecast ecosystem
dynamics both now and in a future inevitably influenced by cli-
mate change, it is essential that they use a quantitatively accurate
temperature-scaling. Balancing tradeoffs between complexity and
computational cost, they must also employ functional groups
that capture the most important differences among species. Con-
sequently, it is important to know which relationships apply
broadly (ie, growth rates increase with temperature according to
0.32 eV) and which differ by group. For example, our results
suggest that, at the same temperature, communities dominated
by different functional groups will have substantially different
maximum growth rates, and hence, productivities (Fig. 1; growth
rate of green algae >diatoms >cyanobacteria >dinoflagellates).
While growth rates will increase across temperatures, this hierar-
chy will not change. Collectively, these results present an oppor-
tunity for further refining the structure and parameterization of
ecosystem models.
This paper is an attempt to resolve the conflict between
distinct but related bodies of research that describe how phy-
toplankton growth rates scale with temperature, by uniting
divergent methodologies and exploring an extensive empiri-
cal data set. We have shown that metabolic theory provides
a more quantitatively accurate, and conceptually rigorous,
description of the temperature-dependence of growth, sub-
ject to additional effects of cell size and functional group
identity. Understanding the physiological and mechanistic
basis for the differences among functional groups, and
Kremer et al.Temperature-scaling of phytoplankton growth
1667
dissecting these patterns using more nuanced phylogenetic
methods, represent important goals. Other important ave-
nues for further research include challenging the mechanis-
tic basis of metabolic theory (e.g., Clarke 2006), and
attempting to relate the inter-specific scaling patterns
explored in the present paper to the effects of temperature
on the growth of individual species. Ultimately, it is our
hope that the current results will strengthen the theoretical
foundation and parameterization of research, including eco-
system modeling, that depends on understanding how
changing environmental conditions affect phytoplankton
community composition and ecosystem function.
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Acknowledgments
CTK was supported by an NSF GRFP fellowship and NSF PRFB
1402074. Additional support came from NSF grants DEB-0845932 to EL
and OCE-0928819 to EL and C Klausmeier. We appreciate the com-
ments of two reviewers and K Edwards’ assistance with biovolume data.
This is W. K. Kellogg Biological Station contribution number 1919.
Conflict of Interest
None declared.
Submitted 09 August 2016
Revised 24 October 2016
Accepted 04 January 2017
Associate editor: Mikhail Zubkov
Kremer et al.Temperature-scaling of phytoplankton growth
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