Ecology and Evolution. 2017;7:10467–10481.  | 10467www.ecolevol.org

Received:24September2017  |  Accepted:8October2017DOI:10.1002/ece3.3576

O R I G I N A L R E S E A R C H

Predictions of response to temperature are contingent on model choice and data quality

Etienne Low-Décarie1  | Tobias G. Boatman1  | Noah Bennett1 | Will Passfield1 |  Antonio Gavalás-Olea1,2 | Philipp Siegel1  | Richard J. Geider1

ThisisanopenaccessarticleunderthetermsoftheCreativeCommonsAttributionLicense,whichpermitsuse,distributionandreproductioninanymedium,providedtheoriginalworkisproperlycited.©2017TheAuthors.Ecology and EvolutionpublishedbyJohnWiley&SonsLtd.

1SchoolofBiologicalSciences,UniversityofEssex,Colchester,UK2InstitutodeInvestigacionesMarinas(IIM-CSIC),Vigo,Spain

CorrespondenceEtienneLow-Décarie,SchoolofBiologicalSciences,UniversityofEssex,Colchester,UK.Email: elowde@essex.ac.uk

Funding informationNaturalEnvironmentResearchCouncil,Grant/AwardNumber:NE/J500379/1,NE/L501906/1andNE/P002374/1;MinisteriodeEconomíayCompetitividad,Grant/AwardNumber:BES-2013-065752

AbstractThe equations used to account for the temperature dependence of biological pro-cesses,includinggrowthandmetabolicrates,arethefoundationsofourpredictionsofhowglobalbiogeochemistryandbiogeographychangeinresponsetoglobalclimatechange.Wereviewandtesttheuseof12equationsusedtomodelthetemperaturedependence of biological processes across the full range of their temperature re-sponse, including supra- and suboptimal temperatures. We focus on fitting theseequationstothermalresponsecurvesforphytoplanktongrowthbutalsotestedtheequationsonavarietyoftraitsacrossawidediversityoforganisms.Wefoundthatmanyofthesurveyedequationshavecomparableabilitiestofitdataandequallyhighrequirements fordataquality (numberof test temperatures and rangeof responsecaptured)butleadtodifferentestimatesofcardinaltemperaturesandofthebiologicalratesatthesetemperatures.Whentheserateestimatesareusedforbiogeographicpredictions,differencesbetween theestimatesofeven thebest-fittingmodels canexceedtheglobalbiologicalchangepredictedforadecadeofglobalwarming.Asaresult,studiesofthebiologicalresponsetoglobalchangesintemperaturemustmakecarefulconsiderationofmodelselectionandofthequalityofthedatausedforpara-metrizing these models.

K E Y W O R D S

biogeography,biotrait,globalchange,niche,thermodynamic,warming

1  | INTRODUCTION

Temperature is one of the most important environmental drivers of physiologyandthushasimportantimplicationsforthebiogeographyof allorganismsandhow theywill respond toglobalenvironmentalchange.Predictingthebiologicalresponsetochangesintemperatureisthusakeyendeavorinbiology,andthousandsofstudieshavemea-sured the response of biological processes to temperature. Data on the temperature response of over 200 traits covering a wide taxonomic breadth(>300speciesacrossalldomainsoflife)havebeencompiled

(Dell,Pawar,&Savage,2011;Gillooly,2001;Parent&Tardieu,2012).Even fora single trait anda singlegroupoforganisms, forexamplephytoplanktongrowthrate,over200studieshavebeen inventoried(Thomas,Kremer,Klausmeier,&Litchman,2012;Thomas,Kremer,&Litchman,2016).Thesedatasetshavebeenusedtoestablishfunda-mentalmetabolic scaling rules (Delletal.,2011;Gillooly,2001)andbiogeographictheories(Seto&Fragkias,2007).Inaddition,tempera-tureresponsecurves,whetherderivedfrominsitumeasurementsofabundance along natural temperature gradients or from in vitro mea-surementsfromlaboratoryexperiments,areusedextensivelyforthe

10468  |     LOW- DÉCARIE Et AL.

TABLE  1 Nonexhaustivelistofequationsthathavebeenemployedtodescribetherelationshipbetweengrowthormetabolicratesandtemperature across the full response range

Formula EquationsNumber of parameters References

4 4 (Li&Dickie,1987)citing(Hinshelwood,1947)

5 4 (Li&Dickie,1987)citing(Johnson,Eyring,&Williams,1942)

6 6 (Heitzeretal.,1991)

7 4 (Montagnesetal.,2008)citing(Schoolfield,Sharpe,&Magnuson,1981)

8 3 (Li&Dickie,1987)citing(Stoermer&Ladewski,1976)

9 4 (Montagnesetal.,2008)

10 4 (Thomasetal.,2012)citing(Norberg,2004)

11 3 (Montagnesetal.,2008)

12 3 (Montagnesetal.,2008)citing(Flinn,1991)

13 4 (Ratkowskyetal.,1983)

14 5 (Kamykowski,1986)

15 5 (Boatmanetal.,2017)

R=Universalgas(Boltzmann)constant.

Rate=a ⋅exp

(−b

R ⋅T

)−c ⋅exp

(−d

R ⋅T

)

Rate=

a ⋅T ⋅exp(

−b

R⋅T

)

1+exp(

−c

R

)⋅exp

(−d

R⋅T

)

Rate=

a ⋅(

T

298.15

)⋅exp

(b

R⋅

(1

298.15−

1

T

))

1+exp[c

R⋅

(1

d−

1

T

)]+exp

[e

R

(1

f−

1

T

)]

Rate=

a ⋅(

T

293.15

)⋅exp

(b

R⋅

(1

293.15−

1

T

))

1+exp[c

R⋅

(1

d−

1

T

)]

Rate=a ⋅exp

⎡⎢⎢⎣−0.5 ⋅

��T−Tref

�b

�2⎤⎥⎥⎦

Rate=a ⋅exp

�−0.5 ⋅

�abs⌈T−Tref⌉

b

�c�

Rate=a ⋅exp�c ⋅T

� ⎡⎢⎢⎣1−

�T−Tref

b

�2⎤⎥⎥⎦

Rate=a+b ⋅T+c ⋅T2

Rate=1

1+(a+b ⋅T+c ⋅T2

)

Rate=[a ⋅

(T−Tmin

)]2⋅

[1−exp

(b ⋅

(T−Tmax

))]2

Rate=a ⋅{1−exp

[−b ⋅

(T−Tmin

)]}⋅

{1−exp

[−c ⋅

(Tmax−T

)]}

Rate= Rmax ⋅

{sin

[π ⋅

(T−Tmin

Tmax−Tmin

)a]}b

     |  10469LOW- DÉCARIE Et AL.

prediction of the effects of climate change on the biogeography oforganisms [e.g., (Beaugrand,Goberville,Luczak,&Kirby,2014)], therisksofextinctions(e.g.,Sinervoetal.,2010),andglobalbiogeochem-icalcycling[e.g., (Cox,Betts,Jones,Spall,&Totterdell,2000)].Theseessentialpredictionsdependonourabilitytoaccuratelyandpreciselymodel temperature response and parameterize these equations for a largevarietyoftraitsandadiversityofspecies.

Currently, there is no consensus on the “best” equation to em-ploy formodeling the thermal responseofabundanceand/ormeta-bolic rates, and it is likely that different processes require differentequations.Here,wereviewtheequationsavailableformodelingthethermal response and test them on highly resolved measurementsfor seven phytoplankton species and published data covering a di-versity of physiological traits across a large taxonomic breadth.Weused subsampling from the highly resolved phytoplankton growthmeasurementstoassesstheeffectofdataqualityontheerrorintheestimate of temperature response parameters and rates. The results of thisanalysiswereused toestablishnominaldataquality require-ments and to include robustness in the choice of equations. The effect ofmodelchoiceanddataqualityisthencomparedtotheamountofchangepredictedinthebiogeographyofaphytoplanktoninresponseto global warming.

1.1 | Review of temperature response equations

The features of the temperature response that is of paramount impor-tance include the cardinal temperatures that define the temperature range (Tmin,Tmax), the optimum temperature atwhich the responseismaximal (Topt),andthesensitivityoftheresponsetotemperaturechange around Topt or as the temperature of the environment ap-proaches Tmin or Tmax. Inadditiontothreeequationsofresponsetosuboptimal temperatures (Tmin to Topt, Equations 1–3, SupportingInformation),at least12differentequationshavebeenproposedtoaccount for the temperaturedependenceofgrowth rate,metabolicrates,orabundanceacrossthefull rangefromTmin to Tmax (Table1,Equations4–15).Differentequationsmayleadtodifferentpredicted

responsestoglobalwarmingorimplythatdifferentmechanismsun-derlie the temperature response. Furthermore, different traits (e.g.,growthandspeedofmovement)havedifferentactivationrates,cur-vature, and skew (Dell etal., 2011), although these differences de-pendbothonmodelchoiceandondataquality(Pawar,Dell,Savage,&Knies,2016).Ithasalsobeensuggestedthatactivationratesdifferbetweentaxa,butthatthesedifferencesarealsopartlydependentontheequationused(Chen&Laws,2016).

A number of studies have tested the quality of a few of theseequationsforaspecificprocess(e.g.,growthrateorphotosynthesis)andspecies(Angilletta,2006;Li&Dickie,1987;Montagnes,Morgan,Bissinger,Atkinson,&Weisse,2008).Inthesestudies,modelselectionwasbasedonameasureofequationfit tothedata (e.g., likelihood)withapenaltyforthenumberofparameters(e.g.,byuseoftheAkaikeinformationcriterion–AIC-).Inadditiontolikelihood-basedselection,oneneedstoconsidertheaccuracyoftheestimatesofkeyparameterssuchasthecardinal temperatures (e.g., theoptimum,minimum,andmaximum temperatures Topt,Tmin,Tmax) and the robustnessof theseestimates to changes in data quality. For example, equations withfewparametersthatassumeasymmetricresponsearoundTopt would underestimate the ToptofanegativelyskewedresponsebutmaystillhavethelowestAIC(beselectedasthe“best”equation)fordatasetswith few measurements.

Boththetemperaturerangeand/orthetemperatureresolutionofexperimentalorobservationalstudiesmaybeconstrainedbylo-gistical considerationsand/orexperimental goals (Figure1).Theseconstraintsondataquantityandqualitycanaffectmodelselectionand the associated mechanistic biological interpretations of fitted parameterssuchastheactivationenergy,whichprovidesanindexof the increase in performance with increasing temperature when temperature issuboptimal (Knies&Kingsolver,2010;Pawaretal.,2016).

Eventheminimalrequirementtoavoidoverfitting,thatthenum-ber of temperatures measured must exceed the number of parame-tersinanequation,isoftennotmet.Thereisariskthatfundamentalpostulates, such as the existence of a strong relationship between

F I G U R E 1 Characteristicsofexistingdatasetsforthedeterminationofthermalresponsecurves.(a)Numberoftemperaturesinthemostcomprehensivemeta-analysisdatabasecurrentlycompiled,excludingstudieswithtwoorfewertemperaturesandthreestudieswithmorethan75temperatures(Delletal.,2013).Medianandmeannumberoftemperaturesis3and5.7,respectively.71%oftemperatureresponsesonlycoverthesupra-orsuboptimalpartofthetemperaturerangeand84%donothavemorethan7temperaturesandthuscannotbeusedtoparameterizeallequationsinTable1.(b)Numberoftemperaturesineachstudyofthegrowthresponseofphytoplanktontotemperature(Thomasetal.,2012).Themediannumberoftemperaturesis6and69%oftemperatureresponsesdonothavemorethan7temperaturesandcannotbeusedtoparameterizeallequationsinTable1.Alargeproportionofstudiesdonotcoversupra-andsuboptimaltemperatureranges

0

100

200

300

0 20 40 60

Number of temperatures

Num

ber

of th

erm

al r

espo

nses

0

10

20

30

40

5 10 15

Number of temperatures

Num

ber

of s

tudi

es

(a) (b)

10470  |     LOW- DÉCARIE Et AL.

microbialbiogeographyandthermalniche,andpredictionsofthere-sponsetoglobalchangemaybebiasedbyfittingequationstodataofinsufficientquality.Thisisbecauseestimatesofthenumericalvaluesof equation parameters are expected to depend on both the tempera-tureresolutionofthedataandthelocation(relativetoTopt)andextentofthetemperaturerange(relativetoTmin and Tmax)overwhichdataarecollected.However,theeffectofdataqualityontheinferencesthatcan be made when modeling temperature response across the range from Tmin to Tmaxhasnotbeentestedpreviously.

Althoughsuboptimaltemperatureresponsesareusuallyexplainedbythermodynamicactivationandhavebeenextensivelystudied,sev-eralputativemechanismsareproposedforthesupra-optimaldeclineinbiologicalactivityand these remain tobeextensively tested.Thedecline can be attributed to the denaturation of one or more rate lim-itingenzymes(Corkrey,Olley,Ratkowsky,McMeekin,&Ross,2012).However, enzymedenaturation usually occurs atmuch higher tem-peraturesthantheoptimal temperatureformostphysiological ratesmeasured.Thedecline in rateat supra-optimal temperatures for in-dividual enzymes (Hobbs etal., 2013) or bulk processes (Schipper,Hobbs, Rutledge, & Arcus, 2014) may be explained by changes inheatcapacityofthesystemdrivenbyproteindynamics(thenumberof available modes associated with covalent bonds). Ecological ex-planations have also been suggested for the supra-optimal decline,astemperaturealtersabioticandbioticconditions.Forexample,gassolubilitydecreaseswith temperature. Increasing temperaturecouldthus lead to increasingCO2 limitation for photosynthetic processesinaquaticphotoautotrophsorincreasingoxygenlimitationforrespi-rationacrossallaquaticorganisms (Pörtner,2010;Pörtner&Knust,2007).Thislimitationcouldpotentiallyextendtoterrestrialorganismsintermsofchangesinpartialpressurewithtemperature,butfindingsareinconclusive(Klok,Sinclair,&Chown,2004).

Several equations have been proposed to model the full functional response of biological rates to temperature from the minimum to max-imumtemperaturesthatwillsupportgrowth(Table1,nonexhaustiveandnewmodelsemerging,DeLongetal.,2017).Smalldifferencesinthe shape of the response curve can have major implications for pre-dictingperformanceinthefield[reviewedin(Dowd,King,&Denny,2015)]andforinterpretationofthemechanism(s)drivingtheactiva-tionanddeactivationprocess.Fourofthe12equationsinTable1arebasedonthermodynamicsofchemicalreactions(Equations4,5,6,7,reviewofequations forenzyme-catalyzed reaction rates in (DeLongetal.,2017)]andinvolvevariouscombinationsofexponentialdepen-dencies on temperature. Two other equations that include exponential functionsmakenoclaimtoamechanisticunderpinningandarepurelyempirical (Equations11,12).Equations8and9aremodificationsofa Gaussian function, while Equations 13 and 14 are second-orderpolynomial, and all four are again strictly empirical. Finally, the lastequation inTable1 (Equation15) isalsoempiricalbutuses thesinefunction.Someofthesimplerequations(threeparameters)aresym-metricaroundtheoptimaltemperature,butmostequationspresentedcancapturethecommonlyobservednegativeskewfoundintempera-tureresponsecurves(steeperinactivationattemperaturesaboveTopt than activation at temperature below Topt).

Thefirstattemptstoquantifythefunctionalresponseofrate(μ)totemperature(T),theμ-Tcurve,werebasedonanalogiesbetweenmicrobial growth rates and chemical reaction kinetics. Recent studies suggest that all biological growth rates can be modeled as if growth is controlled by the activation and denaturation of a single limitingenzyme(Corkreyetal.,2012).Thesimplestofthese(Equation4)as-sumes that the observed rate is the difference between two opposing processes,bothofwhichfollowtheArrheniusequation;inthisequa-tion,thecoefficientswithintheexponential functionsareactivationenergies.Whenappliedtoachemicalreaction,theparameter“a”isarateconstantwithunitsofinversetimeperdegreeKelvin(e.g.,s/°K),b = ΔH‡(enthalpyofactivation;unitsofkilocalories/mole),c = ΔH(en-thalpyofreaction;unitsofkilocalories/mole),d = ΔS(entropyofreac-tion;unitsofkilocalories/moleper°K).Anearlierequation(Equation5)describesthesituationwhereactiveandthermallydenaturedformsofanenzymeexistinareversiblethermodynamicequilibrium.Themostcomplicatedoftheseequationsisthe“masterequation”(Equation6)ofHeitzer,Kohler,Reichert, andHamer (1991),whichassumes thattheactive formof the rate-limitingmasterenzyme is inequilibriumwith two inactive states that result from high-temperature or low-temperature denaturation. When low-temperature denaturationis excluded, this master equation simplifies to Equation 7. In bothEquations 6 and 7, “a” is the rate at the reference temperature of298.15°K(=25°C).

Despitecleardeviationsfromthispattern,includingskew,mod-eling the temperature dependence of biological rate as a Gaussian distribution(Equation8)hasbeenattractivetoecologistsinpartbe-cause of its simple parameterization (Angert, Sheth,&Paul, 2011;Dowdetal.,2015).TheGaussianequationmaybespecificallysuitedto modeling aggregated responses that are the sum of individual re-sponses.Forexample,althoughitmaynotbeanadequateequationforthetemperatureresponseforasinglespecies,itmaybethecor-rectequationfortheresponseofacommunitythatconsistsofmanyspecies with different values of Topt. Equation 8 describes a normal distribution,wheretheparameter“a”istherateattheoptimaltem-perature (Topt)which is found at themidpoint of the temperaturerangeandtheparameter“b”isthestandarddeviation(alsoinunitsof temperature).Montagnesetal. (2008)modified thisequation toobtain a modified Gaussian function that allows for the asymme-try around the optimum temperature often seen in theμ-T curve(Equation9).

Thomas etal. (2012) referencing (Norberg, 2004)multiplied thequadratic by an exponential function to obtain Equation 10. In thisequation,there isareferencetemperature(Tref) thatdeterminesthelocation of the maximum of the quadratic portion of the function. This isageneralizationof thefunctionproposedbyNorberg (2004)inwhichthevaluesof“a”and“c”werebasedontheEppleyfunction(a=0.59/d;c=0.0633/°C).

All of the equations considered to this pointwere either basedon theoretical considerations related to chemical reaction kinetics (Equations4–7)oralloweddirectestimationofecologicallyrelevantparameters such as Toptorthethermalnichewidth(Equations8–10).Twootherequationsdonothaveatheoreticalbasisnordotheyallow

     |  10471LOW- DÉCARIE Et AL.

ecologicallyrelevanttemperaturestobeestimateddirectly.Thesearebasedonasecond-orderpolynomial (Equations11,12) (Montagnesetal.,2008).

None of the equations examined to this point include the lower anduppertemperaturelimitsforbiologicalrates(Tmin,Tmax)asfittedparameters.However,Tmin and Tmax, alongwith the temperature atwhichthebiologicalrateismaximum(Topt)arethecardinaltempera-tures that are often of most interest to ecologists. Some of these equationsmaybereformulatedtoincludesomeofthecardinaltem-peratures, for example Equation 10 to includeTmin and Tmax (Bakeretal.,2016).Forequationslackingspecificcardinaltemperatures,thecardinaltemperaturescanbeestimatedfromthefittedequation(seeMethodssection).

Finally,weturntothreeequationswhereTmin and Tmax are among the parameters found directly in the equation (fitted parameters),rather than needing to be calculated from the equation. These are theempiricalequationsofRatkowsky,Lowry,McMeekin,Stokes,andChandler(1983)(Equation13)andKamykowski(1985)(Equation14),andanempiricalequationthat isamodifiedsinefunctionBoatman,Lawson,andGeider(2017)(Equation15).Themodifiedsinefunctionalsoreturnsthemaximumrate(Rmax)attheoptimumtemperatureasadirectlyfittedparameter,andTopt can be calculated from the other fitted parameters. This equation also includes parameters that charac-terizetheskewness(a)andkurtosis(b).

This is not a comprehensive account of all available equations to equation temperature response. Some equations have been pro-posed for the purpose of simulation and are difficult to fit to data (e.g.,Follows,Dutkiewicz,Grant,&Chisholm,2007).Otherequationsareminorvariationsofequationswehaveincluded[e.g.,(Beaugrandetal.,2014)containsanequationthatiscomparabletoEquation8].

2  | MATERIAL AND METHODS

2.1 | Measurement of phytoplankton growth rate

Wemeasuredthetemperaturedependenceofgrowthrateforseventaxonomically distinct phytoplankton.Growth ratesweremeasuredatahigh-temperatureresolution (in0.4–0.5°C increments)withex-tensive thermal coverage on either side of the temperature optima (18–39 individual temperatures per species;with at least two tem-peratureswithpositivegrowthoneithersideoftheoptima).Thedif-ferentspeciesprovidedifferentexpectedtemperatureoptima,skew,andspreadonwhichtotesttheequations(specificratesreportedinFig.S1).

The species assayed include a coccolithophorid, Emiliania hux-leyi (CCMP 370); a cyanobacterium Trichodesmium erythraeum IMS101; and two diatoms, Thalassiosira pseudonana (CCMP 1335);Phaeodactylum tricornutum(CCMP2561);twochlorophytesDunaliella tertiolecta (CCAP1320) andPycnococcus provasolii (CCMP1203); and aprymnesiophyte, Isochrysis galbana (Ply 546). Specific details of the media and light for each species are provided in the data file. The num-ber of replicates at each temperature is in parenthesis next to each genus below.

Growth rates for Trichodesmium[publishedpreviouslyin(Boatmanetal., 2017)], Emiliania, Thalassiosira, and Phaeodactylum were mea-suredusingthemethoddescribedby(Boatmanetal.,2017).Briefly,culturesweregrownat lowvolumes (5ml) in12mlglass test tubesinathermalgradientblock(temperatureiscontrolledatbothendsofan aluminum block using circulating water baths and a linear tempera-ture gradient forms across theblock).As a proxy for biomass, dailymeasurementsoffluorescence(Fo)weremadeondark-adaptedcells(20min) using a FRRfII Fastact Fluorometer (Chelsea TechnologiesGroupLtd,UK).Cultureswerekeptatthelowersectionoftheexpo-nential growth phase andoptically thin to avoid nutrient limitation,self-shadingandtominimizeCO2 drift.

ForDunaliella (rep=2),Pycnococcus (2) and Isochrysis (2) cultureswere grown in 24-well microtiter plates sealedwith air permeablemembranes.Similar tocultures thatweregrown inglass test tubes,theseplateswerealsogrownonathermalgradientblock(describedabove).Thesurfaceofthegradientwascoveredwith1cmofwatertoenhance thermal conductance between the block and the well plates. Growthofthecultureswasassessedbyadailymeasurementofopti-caldensityat660nmusingamultiparameterplatereader(FLUOstarOmega).

Growth was monitored during early exponential growth phase,andtheexponentialgrowthrate(μ)wascalculatedfromtheslopeofthenaturallogoffluorescenceorthenaturallogofopticaldensityasa function of time.

2.2 | Published data

Inordertoprovidearobusttestofthethermalresponsebetweentaxaandallow for a comparisonof fit between traits,we supplementedour measured data (described above) with existing published data.Weusedthebiotraitsdatabase(Dell,Pawar,&Savage,2013),ada-tabase of temperature response in phytoplankton growth (Thomasetal., 2012), and additional data from the literature (sources citedindatafile).Datasetswithpositiveratesforat leastsevendifferenttemperatures with at least two temperatures being above and two being below the optimal temperature were selected from the data-bases. Datasets were not selected based on our proposed data qual-ity requirements (see sectionon “Dataquality requirements” in theresultssectionbelow)astoofewdatasetsmetthesemorestringentrequirements.

2.3 | Equation fitting

WeimplementedthefittingofallequationsinanRpackageavailableonComprehensiveRArchiveNetwork(CRANtemperatureresponse).The equations were fit to data using a modified Levenberg–Marquardtalgorithm(Elzhov,Mullen,Spiess,Bolker,&Mullen,2015;More,1978).Thisalgorithmallowsrobustfittingofnonlinearequa-tions,evenwhenreliablestartingparameterscannotbeestablished.Whenequationparametervaluesrepresentfeaturesofthedataset,the starting valueswere estimated from the dataset (e.g., thea in Equations8–10wassetasthemaximumrateinthedataset,Tref,Topt,

10472  |     LOW- DÉCARIE Et AL.

Tmin,Tmaxweresettothemean,themedian,theminimum,andthemaximumtemperatureofthedataset,respectively).Whenthiswasnotpossible,startingvaluesfortheparameterswerethefittedpa-rametersfromthesourcepublicationsfortheequation,oraparam-etersetthatensuredadownwardparabola-likeshape.Inequationsrequiring inputs in°K,valueswereconverted in theequation from°C.Theequationswerefittopositivenonzerodataaveragedacrossreplicates at each temperature. This is essential for equations with eitherasymptoticorexponentialrelationshipsofratewithtempera-tureattheextremes,becausezerovaluesreportedfromaboveTmax or below Tmin have high leverage on the equation fit and lead to poor predictionswithinthebiokineticrange.Forappropriateequationfits,theonlynullratesthatshouldbeincludedareTmax and Tmin,whichcannotbedeterminedbeforefitting.Asaresult,nozerovalueswerekept.However,measurementsextendingtothelimitsofthegrowthrange,thatis,includingzerovalues,wouldbenecessaryforthemostaccurate parametrization of some equations.

Fromequationfits,cardinaltemperatureswereextracted(Sinclairetal.,2016).Theseincluded:

Topt: the temperature at which the maximum rate is predicted to be achieved,whichwasdeterminedusingnumericoptimization.

T50min and T50max:thelowestandhighesttemperaturesatwhich50%of the maximum rate is predicted to be achieved. This was calcu-latedastherootsofthefunctionwhen50%ofthepredictedmaxi-mumratewasremoved(RpackagerootSolve).

Tmin and Tmax(CTmin and CTmax):temperatureswithinwhichapositiverate is predicted. This was calculated as the roots of the function. Some equations are asymptotic and therefore would not pre-dictzeroornegativerates, inwhichcaseTmin and Tmax cannot be determined.

Activation and deactivation rateswere calculated from themeanofvalueof thederivativeacrosssub- (Tmin to Topt) andsupra- (Topt to Tmax) optimal temperatures, respectively. Skewwas calculated as the

differencebetweenactivationanddeactivation(i.e.,anegativeskewin-dicatesthatdeactivationissteeperthanactivation).

Equationswere rankedoneachdatasetusingBayesian informa-tioncriterion(BIC).Thedifferencebetweenequationsinmodelqualityacrossdatasetswas testedusingaKruskal–Wallis ranksumtestonBIC-basedranksfollowedbytheassociatedposthocpairwisecom-parison(Giraudoux,2017;Siegel&Castellan,1988).Thesameconclu-sionsarisewhenothermeasuresofmodelqualitywereused;valuesforAkaikeinformationcriterion(AIC)andtheAICcorrectedforfinitesamplesizes(AICc)areavailableinsupplementalmaterial(Fig.S2).

Reported deviations in cardinal temperatures were calculated as thedifferencefromtheweightedmeanacrossallequations(weightedbyAkaikeweights). Reported deviations in growthwere calculatedabsolute deviation from the weighted mean across all equations (weightedbyAkaikeweights).

Differences between the different equations in their prediction of cardinal temperatures were assessed using analysis of variance(ANOVA)andaTukey-HSD.AnANOVAandaTukey-HSDwerealsoused to compare equations for the temperature range required to staywithinthedesignatedthresholdsfordeviationfromthefittothefulldata(0.5°CforToptand5%forgrowthrate).Differencesbetweenequations for sample size required to stay within these thresholdswereassessedusingageneralized linearequation(GLM)witha log-linkforthePoissondistributionofcountdataandTukeycontrasts.

Toassesssimilaritybetweenequationpredictionsacrossthetem-perature range, the Euclidian distancewas calculated based on theratepredictedbytheequationateachexperimentaltemperatureandclustering was done using Ward’sminimumvariancemethod(Fig.S3).

2.4 | Data quality sensitivity analysis

Toensurethatthehigh-resolutiondatasetswereofsufficientqual-ity to distinguish between equations,we conducted a simulationbasedonequation fits toeachdataset.Normallydistributed ran-dom noise was added to the predicted growth rate value from each equation at each temperature. The noise was centered on 0 and its standard deviation was the square root of the mean residuals squared arising from the fit of the equation. Each equation was thenfittothesimulateddatasetsgeneratedbyeachequationandrankedbasedonBIC.Eachsimulationwasreplicatedfivetimes.

TomeasuresensitivityoftheestimateforTopt and the estimate of growth rate at each temperature to the temperature resolution of a dataset,adecreasingproportionofthemeasuredtemperatureswereremovedbasedon:(1)randomsamplingacrossthetemperaturerangetoestablishthenumberoftemperaturesrequiredand(2)limitingthetemperatures included in the analysis to thosewhere theobservedgrowth rates were above a predetermined proportion of the maximum growthrate,thuscapturingaproportionofthetemperaturerange.Arangeof100%isexpectedtoextendfromTmin to Tmax,whilearangeof50%includestemperaturesallowingatleast50%ofthemaximumgrowthratetobeachieved(fromCT50min to CT50min).Topt is expected toalwaysbewithinthetemperaturerangeofthedatasampledusingaproportion of the maximum growth rate.

F IGURE  2 EquationrankingbasedonBICforeachdataset.Equationsareorderedbymedianrank(bestequationsatleftwithlowerrank).Pointisthemedianrankanderrorbarsare95%confidence interval across datasets

2

4

6

8

10

12

15 06 10 14 07 13 09 08 05 11 12 04

Model

Ran

k (b

ased

on

BIC

)

Genus

Dunaliella

Emiliania

Isochrysis

Phaeodactylum

Pycnococcus

Thalassiosira

Trichodesmium

     |  10473LOW- DÉCARIE Et AL.

DataqualityrequirementsforprecisionandaccuracyofTopt and the estimateofgrowthrateateachtemperaturewereassessedbyfittingtheequationstosubsamplesofthephytoplanktongrowthdatasetsandcom-paring these values to values obtained from fits to the complete data. Error was measured as the absolute deviation compared to values obtained from the fits to the complete dataset of cardinal temperature measure-ments(Topt)andthemeandeviationinpredictedrateatalltemperatures.The temperature response of each individual species was treated as a

replicateinthisanalysis,andconfidenceintervalswerecalculatedacrossthesereplicates.Anerrorof0.5°CinToptoranaverageerrorof5%ofthemaximumgrowthrateswassetastheminimumqualitythresholds.Thecritical number of temperatures was defined as the maximal number of temperatures at which the threshold was exceeded plus 1. The critical range was the maximum range at which the threshold was exceeded or met.Insomecases,thiswasthelowestvaluefornumberoftemperaturesor range at which equations could be fit to the subsampled data.

F I G U R E 3  (a)Equationfittoanexampledatasetofphytoplanktongrowthrateasafunctionoftemperature(Phaeodactylumtricornutum).Thepointsarethemeasuredgrowthrate(samevaluesacrosspanels),andthelinesaretheequationpredictedgrowthrates.(b)Equationresidualsasfunctionoftemperature.(c)Valueofthefirstderivative(gradient)ateachmeasuredtemperature.Numberswithinthefigureindicatetheequationnumber.Equationsaregroupedasafunctionoftheirnumberofparameters(3–6).Equationswithfourparametersarefurtherdividedbetweenempiricalandmechanisticequationstominimizeclutterwithintheplots.Linesforindividualequationsarelabeledwithcolorand the equation number. Similar patterns can be observed for other species

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2.5 | Predicting changes in biogeography with global warming

Giventhecentralityoftheseequations(Table1)tothepredictionofthebioticresponsetoglobalwarming,touraimwastoassesswhetherdifferences among the equations used to account for the tempera-ture dependence of growth rate can affect predictions of the effect ofglobalwarmingonthebiogeographyofphytoplankton.Todothis,wemakethesimplifyingassumptionthatthegeographicalrangeofaspecies depends on the response of its growth rate to temperature. Seasurfacetemperature(SST)datawereusedtomodelthedistribu-tion of a species based on the response of its growth to temperature. Each equation was parameterized using the experimental data for the species,andtheparameterizedequationwasappliedtopredictionofgrowth from SST.

ContemporarySSTforthemonthofAugustfortheyears2006to2016was obtained fromMODIS data accessed using theGiovanni

onlinedatasystem,developedandmaintainedbytheNASAGESDISC(Acker& Leptoukh, 2007). Predicted SST forAugust 2100was ob-tainedfromNCDC-NOMADS.ThispredictedSSTwasbasedonIPCCSRESA1B emission scenario for CO2 emissions and modeled using theGeophysicalFluidDynamicsLaboratory(GFDL)CoupledClimateModel (CCM2.1) (Delworthetal.,2006).Values fromthemonthofAugust are used as an example, and similar observationswould bemadeifanothermonthoftheyearwasselectedorifcalculationswerebasedonmeanannualtemperature,althoughthelatterwouldnotac-countforseasonality.

We recognize that any inferences based on such an analy-sis are subject to the caveats that (1) phytoplankton abundancemay not correlatewith growth rate, (2) biogeography is affectedby many other factors that may change in concertwith or inde-pendentofglobalwarming,and(3)giventheirrapidgrowthrates,phytoplanktoncanbeexpectedtoevolveinresponsetosustainedwarming.

F I G U R E 4 EquationrankbasedonBICacross(a)traittype[datacompiledin(Delletal.,2013)]andfor(b)growthrateacrossalgalclassesorphyla[datacompilationof(Thomasetal.,2012)]foreachequation.Onlytraitsorclasses/phylawithmorethantwotaxonomicunitsareincludedinthefigure.Pointsindicatethemedianandtheerrorbarsindicatethe95%confidenceintervalcalculatedacrossexperiments(asingletaxonomicunitcanbeinmultipleexperiments).Equationorderisbasedonmedianequationrankforthephytoplanktongrowthdataset(asinFigure2).Numbersinparenthesesindicatethenumberoftaxonomicunits(uptospecieswhenidentified)withineachtraitorclass.Notallequations converged on a solution for all individual published datasets

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3  | RESULTS

3.1 | Differences between equations

Allequationscouldbefittoeachphytoplanktongrowthdataset,butno singleequationconsistentlyprovided thebest fit (i.e., couldnotaccountforthemajorityofvariance)acrossallphytoplanktongrowthdatasets(Figure2).Mostequationscouldnotbedistinguishedacrossdatasetsbasedonrank,althoughEquations6,14,and15hadbetterranksthan4,andEquation15alsosignificantlyoutranked12(p<.05,Figure2).

Simulationsindicatethatthequalityofthephytoplanktongrowthdatasets is sufficient to for the selectionof abestmodel.All equa-tionshadbetterrankingsonthesimulateddatathattheyhadgener-ated thanondata generatedby anyother equation (Kruskal–Wallisp < 10−3,Fig.S4).

Foragivendataset, the12equations (Table1)didnotconvergeonthesameoptimaltemperatureormaximumgrowthrate(Figure3).Predictedoptimaltemperatureswereonaverage−1.18°C[rangefrom−2.28to−0.18°C]fromtheweightedmean(Akaikeweights)predictedoptimaltemperatureacrossequations(S4),andthemeanabsolutede-viation in growth rate at each temperaturewas 0.018 day−1 [range 0.015day−1 to0.022day−1]whencompared to theweightedmeanacross equations. Equations 4 and 6 consistently predicted higheroptimal temperatures compared to other equations. Equations with a highnumberofparameters(5–6)ledtosimilarpredictions,butequa-tionsbasedonsimilarmechanisms,similarfunctionalforms,orsimilarrankintermsofBICdidnotleadtomoresimilarpredictions(Fig.S3).

Equations differed in their skew (deviation from median skewacrossequations,F11,81=2.87,p<0.01),withtheaverageskewbeing−0.017[−0.030,−0.005]acrossallequationsanddatasets.Asacon-sequence,T50min and T50maxwerehighlyvariablebetweenequationsand datasets. The median distance between equations for each data-setwas1.0°CforCT50minand2.9°CforCT50max.However,forsomeofthespeciesinourdataset,someequations(Equations6,7,12,and14)producedestimatesgreaterthan10°Cfromtheweightedmeanvalueacross equations for these cardinal temperatures.

There was no individual equation that outperformed all other equationsconsistentlyacrossorwithintraits,norwithinanalgalclass(forgrowthrate)wheretherewastaxonomicreplication(Figure4).Allequations represent the best equation for at least one of the responses (for a trait of a given taxa), except forEquation4which performedpoorlyingeneral.

3.2 | Data quality requirements

Forallequations(Table1),therewasanapproximatelylinearincreasein the error of cardinal temperatures estimates with a decrease in temperature resolution (i.e., numberof experimental temperatures).Similarly,theerrorincreasedlinearlywithadecreaseinthemeasuredrangeofgrowthrates (differencebetweentheminimumrate inthesubsample and maximum rate). Only the most extreme equationsdiffered significantly in termsof theirdataquality requirements for

numberoftemperatures.Onaverageacrossallequations,aminimumof16[rangeof15–17]temperaturepointsarerequiredtomaintainthe predicted Toptwithin0.5°Cofthevaluepredictedonthefulldata-setincludingalltemperaturesmeasured(Equation6differedfrom5,8,10,and14,p<.05,Figure5a).Aminimumof8[7–9]temperaturepoints was required to maintain predictions of growth rate to within 5%ofthevaluepredictedfromthefulldataset(Equation4differedfrom Equation 8 p<.05).Fortherangeinratesmeasured,56%[50%–60%]ofthefullrange(0tomaximumrate)isrequiredtomaintainthepredicted Toptwithin0.5°Cofthevaluepredictedonthefulldatasetand29% [24%–34%]maintain predictionsof growth rate towithin5%ofthevaluepredictedfromthefulldataset(Figure5b).BasedonBIC,someofthe“best”fittingequationsrequiredataofthehighestresolutionandrangeinordertomaintainthequalityoftheirfit(e.g.,Equation 6 had the highest number of temperatures for the accurate prediction of Topt),whilesomeoftheweakestfittingequationsarethemostrobustto loss indataquality (e.g.,Equation5),althoughthesedifferencesareonlymarginal.

3.3 | Implications for predictions of the response to global warming

Wefoundthatdifferencesamongequationsinthepredictedgrowthratesforourstudiedphytoplanktonspeciestranslateintolargedif-ferences in expected contemporary biogeography evenwhen thetwobestfittingequationsarecompared.Equations6and15havesimilarqualityscores (Figure2)and leadtosimilarpredicationsofrate (Fig.S3);however,Equation6predictsaglobalmeangrowthof1.9%lessthanEquation15(Figure6a–c).Thisdifferenceiscom-parable to the change predicted from a decade of global warming.

F IGURE  5  (a)Numberoftemperaturepointsand(b)therangein growth rate required to maintain the predicted Toptwithin0.5°Cofthevaluepredictedonthefulldataset(bluetriangles),andtomaintainratepredictionsonaveragewithin5%ofthevaluepredictedfromthefulldataset(redcircles)foreachequation.Equationsareorderedbasedonmedianrankinthefulldatasets(matchingFigure2)

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Basedonaprojectionof futureSST, all equations lead topredic-tions of large-scale changes in biogeography for the studied spe-cies;however,themagnitudeofchangediffersbetweenequations.Forexample,theglobalmeandeclineingrowthforP. tricornutum of 20.75%overthemodeledperiodor2.3%perdecadewithEquation6and25.5%overtheperiodor2.9%perdecadeforEquation15,Figure6g–h).

4  | DISCUSSION

4.1 | Scale of difference in predicted biogeography and response to global warming

Thedifferencebetweenequations forpredicted rates (meandiffer-enceof0.018day−1 across the temperature range for growth of our phytoplankton)andcardinaltemperature(meandifferenceinTopt of 0.44°Cforgrowthofourphytoplankton)maybeperceivedassmallbutareecologicallysignificant.Whenscaledtopredictionsofchangesinglobalprocesses, suchasbiogeography,differencesbetween thebest models can be larger than changes predicted over decades (Figure6). The importance of data quality and modeling approachis recognized across disciplines which attempt to predict responses

toglobalchange.Differences indatasetsandmethodologycanleadto opposing predictions of the change in biogeographywith globalwarming (Brownetal.,2016).Changes in thescaleof theobserveddifference between equations can alter predictions of species extinc-tionorchangesintheepidemiologyofmajordiseases(Mordecaietal.,2013).

The difference between equations in global average predicted ratesandourthresholdfordataqualityarebothontheorderofre-sponsestoglobalclimatechange,includingobservedchangesinter-restrialprimaryproductionof3.3%perdecadefrom1982to1999(Nemani,2003)andpredictedincreasesinabundances(andassoci-atedchangeindistribution)of2.9%perdecadeforProchlorococcus and 1.4% per decade for Synechococcus (Flombaum etal., 2013).Theyarealsoofsimilarscaletothedifferencebetweenequationsproposedtoaccountforthecolimitationofphytoplanktongrowthbytemperatureandnutrients(Thomasetal.,2017).Differencesbe-tweenequations are smaller than theestimateddecline inphyto-planktonbiomassgloballyof10%perdecadeoverthelastcentury(Boyce, Lewis,&Worm, 2010).However, these trends have beendisputed (McQuatters-Gollop etal., 2011), and growth rate andstandingphytoplanktonbiomassarenotexpectedtobecorrelated(Behrenfeld,2014).

F IGURE  6 BiogeographicdistributionPhaeodactylumtricornutumbasedontwobest-fittingequations(Equation6a,d,gandEquation15b,e,h)appliedtoseasurfacetemperatureforAugust,inpresentday(averagefrom2006to2016,a–b)andmodeledforthefuture(averagesfor2095–2105,d–e);theaveragechangeingrowthperdecade(additive,notcompounded)fortheperiod2006–2016to2095–2105(g–h);andthedifferencebetweenthepredictionsfromthesetwoequationsforgrowth(c–f)andforchange(i).BluecontourlinesforgrowthareforCTmin/max and T50min/max.ThisisnotintendedtobeanaccuraterepresentationofthebiogeographyofP.tricornutum.Rather,itisprovidedtoillustratethescaleofdifferencesbetweenequations,andwenotethatsimilardifferencesbetweenequationsariseindependentofspecies

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     |  10477LOW- DÉCARIE Et AL.

4.2 | Constraints on cardinal temperatures

TherelativelysmalldifferencebetweenequationsintheestimatesofToptforagivendatasetmayinpartbeattributedtothefactthatthefitting of these equations has a bias to solutions that return values for the optimal temperature that fall within the temperature range measured or even at themean temperature. In published datasets(Thomasetal.,2012),theestimatesofoptimaltemperaturewerecor-relatedwith themean temperature ofmeasurements (R2 =0.8, Fig.S6).Thismayreflectabiasof theunderlyingequationto forceTopt to approach the mean of the temperatures at which measurements weremade.Alternatively,experimentalistsmayusepriorknowledgeof temperatures where their species can grow to select experimental temperatures that are centered around Topt.Finally,negativegrowthratesmaynotbereported.However,whenfittingallequationstoran-domlygenerateddata,strongcorrelationsbetweenthemidpoint(ormean)oftherangeinmeasurementtemperaturesandcalculatedTopt remainformostequations(Fig.S7).ThefactthatequationscanbiasTopt toward mean values of the dataset can have important implica-tions for the studies attempting to find a mechanistic explanation for differences in optimal temperatures [e.g., (Sal, Alonso-Saez, Bueno,Garcıa,&Lopez-Urrutia,2015)].

The constraints on Topt estimates from equation fits pose a major challenge for the estimation of confidence around estimates of Topt. Bootstrappingmethods (modelingonsamplesarising fromrandomsampling from the original complete dataset with replacement re-sulting inequation fitsonevensmaller subsetsofdata)commonlyused to estimatewill greatly underestimate parametervariance. Incontrast,MonteCarlosimulationscangreatlyoverestimatethesizeof the confidence interval around fits because the parameters do notfollowanestablishedmultivariatedistributionthatcaneasilybesimulatedfromthevariance/covariancematricesandthusimpossiblylarge or small rates can be predicted from simulations that ignore this issue.

The other cardinal temperatures (CTmin, CTmax, T50min, T50max)are less constrained by the temperatures at which measurementswere obtained (S7). To ensure accurate estimates of the extremecardinaltemperatures(CTmin,CTmax),extremely lowgrowthrates(μ/μmax<0.05) must be includedwithin the data. This is because thelowerandupperthermaltolerancelimits(i.e.,CTminandTmax)arelessconstrainedbythemeanexperimentaltemperaturesthanTopt and are moredependentonthe“shape”implicitintheequations(e.g.,sinevs.Gaussian).These limitationsmaycombinetoyielda largedifferencebetween equations in the estimation of these cardinal temperatures. This may partially explain why correlations between maximal (andminimaltemperatures)andambienttemperatureorlatitudeareoftenabsent or weaker than those found for Topt inmeta-analyses basedonreportedcardinaltemperatures(Araújoetal.,2013;Sundayetal.,2014),althoughcorrelationwith latitudeofequalstrengthhasbeenfound for Topt,Tmin, and Tmax when the same equation is applied across alldatasets(Thomasetal.,2016).

The larger differences between equations at the upper and lower temperature regionsof thecurves (Tmin,T50min,T50max,Tmax)

are particularly problematic for the prediction of the response oforganismstoglobalchange. Inadditionto implicationsofshifts inrangelimits,thesevalueswill influencehowanorganismcancopewithfluctuatingtemperatures.Increasedtemperaturevariation,andthusthecapacitytodealwiththesemoreextremetemperatures,isexpected to pose a greater threat to species survival than warming (Vasseur etal., 2014).Thermal variability can also alter the shapeandthescaleofthethermalresponseoforganisms(Paaijmansetal.,2013).Inavariableenvironment,basedonJensen’sinequality,theoptimal mean temperature is expected to be lower than in a constant environment [reviewed inDowdetal., 2015)] leading toobserva-tions of optimal temperature higher than the mean temperature of the environment in more variable temperate habitats compared to lessvariabletropicalhabitats(Amarasekare&Johnson,2017).Thetemperature response is also dependent on prior exposure to the measurementtemperature,allowingforacclimation,andthedura-tionoftheexposure(Schulte,Healy,&Fangue,2011).Asaresult,temperature fluctuations and acclimation need to be accounted for bothinstrategiesformeasurementandpotentiallyinthedesignofequations.

4.3 | Implications for evolution under global change

In addition to the difference in estimates of cardinal temperatures,the shape of the temperature response curve will influence manypredictedresponses(Dowdetal.,2015), includingtheprobabilityofanevolutionaryresponsetoglobalwarming.Iftheabsolutevalueofthe firstderivativeof the curve (Figure3) ishigh (i.e., a steep tem-peratureresponse,high-temperaturesensitivity,ahighQ10),asmallchange in temperature would be expected to lead to a large change in biologicalprocess,whichinturnwouldbeexpectedtotranslateintoa largechangeinselection.Theevolutionaryoutcomeofthisselec-tion pressure will depend on numerous factors including the standing geneticdiversityofthepopulation,thepopulationsize,thetempera-turehistoryofthepopulation(Bell,2013;Bell&Collins,2008),andthedirectionofthechange(Low-Décarieetal.,2014).Forexample,Equation15willpredictasteepertemperatureresponseatextremetemperaturesthanEquation9andthusleadtoapredictionofgreaterthermalsensitivityandahigherselectionpressure.Influctuatingen-vironments,evolutionshouldleadtoareductionintemperaturesen-sitivity(i.e.,anincreaseinplasticityandaflatteningoftheresponsecurve;(Clarke&Fraser,2004).

Acrossourdatasets,activationhasalowerslopethaninactiva-tion(negativeskew).Theskewisfoundtocorrelatepositivelywithoptimal temperature (Pawar etal., 2016), consistent with a fixedupper limit to biological activity.This leads to the expectation ofhigher selection at the upper limits of thermal tolerance. This is compatible with the observation that the upper limits of heat tol-erance in terrestrial ectotherms are highly conserved across tax-onomic groups,whereas there is large variation in cold tolerance(Araújo etal., 2013) and that upper limits of heat tolerance cor-relatewithlatitude,whereaslowertemperaturetolerancemaynot(Sundayetal.,2014).

10478  |     LOW- DÉCARIE Et AL.

4.4 | There is no “Best” equation

Despite the importance of these differences between equations,thebestequationfortheresponsetotemperatureofphytoplanktongrowthrateorotherbiologicaltraitscannotbereliablyestablishedon a single criterion. Notwithstanding penalties in the metric of equationquality,manyequationswithhighernumbersofparame-tershadlowerBIC,butmorecomplexequationswerelessrobusttolossindataquality.Equation15fortheresponseofphytoplanktongrowth to temperature performed well in terms of fit and robust-ness to data resolution but not robustness to limitations in the range of relative growth rates captured within the experiment. The fact thatwe could not identify the “best” equationmay be related toimportant biological phenomena, such as fundamental differencesin the shape of the biological response among taxa or among the biologicalprocessesofinterest,orissueswiththedata,fitting,andmodel selection.

The better performance in terms of likelihood of more complex equations suggests that most responses exhibit taxon-specific pat-terns,suchasskewandconcaveorconvexactivation,thatmusteachbecapturedbyaparameter. Itmaynotbepossibletohaveasinglebest equation. The mechanism of response to temperature of different majortaxonomicgroupsmaydifferandeventheresponseofdifferentdevelopmentalstagesforagiventaxonomicgroupmayexhibitdiffer-encesintheshapeoftheirresponsetotemperature(Mordecaietal.,2013; Paaijmans etal., 2013; Sinclair etal., 2016). Even genotypeswithinaspeciesmaydifferintheshapeoftheirtemperatureresponse(Boydetal.,2013).Forexample,thetemperatureresponsemayfun-damentally differ betweenmajor groups of phytoplankton (Chen &Laws,2016;Lürling,Eshetu,Faassen,Kosten,&Huszar,2013;Thomasetal.,2016).Eachmajortaxonomicgroupwouldrequireanequationthat captures these differences in response.Testing this hypothesiswould require the measurement of the response to temperature of manyminortaxonomicgroups(e.g.,species)withinmajortaxonomicgroupswithequallyhigh-temperatureresolutionandrangecoverageforeachtestedtaxon.Alternatively,anequationmayyetbedevelopedthat outperforms all the equationswe have tested, independent oftaxa,atleastforagiventrait.Thisequationmaybebasedonabetterintegration of interactions between multiple mechanisms for activa-tion(e.g.,accountingfordifferentactivationratesofmultipleenzymes)and inactivation (heat capacity, substrate availability, and ecologicalfactors)orincludeayettobeestablishedmechanisticexplanationforthese processes.

The limitations of current temperature response data for equation selection have been extensively recognized (Knies &Kingsolver,2010;Pawaretal.,2016).Ourresultsshowthatevenfor a single selected equation, very few existing datasets meetdataquality requirements tominimizeerror inpredictionsof car-dinal temperatures and rates across the full biokinetic temperature range.Forrecoveringestimatesfromexistingdatathatarelimitedby resolution and range, a robust equationwith few parameters(e.g.,Equation8)thatmaynotaccuratelyrepresenttheunderlyingprocessandpatterns (suchasskew) ispreferabletobetter fitting

equations for which changes in data range and resolution lead to important changes in estimates (e.g., Equation 6).We did notvary theprecisionof themeasurementof rateorof temperature.A proposed rule of thumb is that the precision of the measure-ment of temperature is at least three times that of the precision ofthemeasurementoftheresponsevariable (Pawaretal.,2016).Another element not tested in our analysis is the location alongthe temperature scale, although measured activation can differbetween organismswith colder orwarmer growth ranges (Pawaretal., 2016), potentially influencingmodel choice, but this couldnotbetestedinourhigh-resolutiondatasetsbecauseoftempera-turerangesforgrowthmostlyoverlapped.Thechallengeofmodelselectionandthelackofqualitydatalimitourabilitytopredict,forexample,changesinthedistributionofspecieswithglobalclimatechange[e.g.,(Gobleretal.,2017)].

Even in simple laboratory experimentswith only a single tro-phic level, the response to temperature of growth rate does notconsistently lead to predictable changes in competitive dynamics(Limberger,Low-Décarie,&Fussmann,2014).Whilethebiogeogra-phyofmarineectothermsmatchesthepredictionsoftheirthermalperformancecurves,thisisnotthecaseforterrestrialectotherms(Sunday,Bates,&Dulvy,2012).Thesedifferencesbetweenthere-sponseofspeciestotemperature,competition,andtheirdistribu-tionmaybeattributedtothecomplexitiesofecologicalinteractionsand the associated need to integrate many concomitant biologi-cal responses with the potential for nonlinear interactions. These differences may also limit the credibility of biogeographic infer-encessuchasthatpresentedinFigure6,whichwouldcompletelychange if, for example, nutrient limitationwas included (Thomasetal., 2017). In models of natural ecosystems, the difference inresponse between trophic levels can cause trophic cascades, ex-acerbating the predicted effect of warming (Chust etal., 2014).However, these differences between single species physiologicalresponsesandecologicalobservationsmayinpartberesolvedbya better measurement and understanding of the individual species responsestotemperature.Ourfindingshighlighttheneedtofocusour measurement and modeling efforts on simple but fundamental aspectsoftheresponseoforganismstotemperature,withtheaimtomakemorerobustpredictionsonthechangesintheecologyoforganisms and associated global biogeochemical processes based on future climate scenarios.

ACKNOWLEDGMENTS

NB was supported by a NERC-funded Undergraduate ResearchPlacement awarded through the EnvEast Doctoral TrainingPartnershiptoELDandRG.PSwassupportedbyastudentshipfromtheUniversityofEssexawardedtoELD.TGBandWPweresupportedbyNERCPhD studentships (NE/J500379/1;NE/L501906/1).AGOwassupportedbyMINECOdoctoralgrant(BES-2013-065752).RG’sworkonthermalacclimationissupportedbyaNERCstandardgrant(NE/P002374/1).We thankKirraleeBaker andMichaelSteinke forcomment on a draft.

     |  10479LOW- DÉCARIE Et AL.

DATA AVAILABILITY STATEMENT

Data and scripts are available on Dryad (https://doi.org/10.5061/dryad.52mc5).

CONFLICT OF INTEREST

None declared.

AUTHOR CONTRIBUTION

Tobias Boatman, Noah Bennett, Will Passfield, Antonio Gavalás-Olea, and Philipp Siegel acquired the experimented data. NoahBennettandEtienneLow-Décarieimplementedthecoding.RichardGeiderandEtienneLow-Décariedesignedtheresearchandwrotethe manuscript. Etienne Low-Décarie, Tobias G. Boatman, NoahBennett,Will Passfield, AntonioGavalásOlea, Philipp Siegel, andRichardJ.Geidereditedandcriticallyreviewedthemanuscript.

ORCID

Etienne Low-Décarie http://orcid.org/0000-0002-0413-567X

Tobias G. Boatman http://orcid.org/0000-0001-7541-3844

Philipp Siegel http://orcid.org/0000-0002-1755-083X

Richard J. Geider http://orcid.org/0000-0003-3276-047X

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Amarasekare,P.,&Johnson,C.(2017).Evolutionofthermalreactionnormsin seasonally varying environments. American Naturalist, 189, E000.https://doi.org/10.1086/690293

Angert,A.L.,Sheth,S.N.,&Paul,J.R. (2011). Incorporatingpopulation-level variation in thermal performance into predictions of geographic range shifts. Integrative and Comparative Biology,51,733–750.https://doi.org/10.1093/icb/icr048

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How to cite this article:Low-DécarieE,BoatmanTG,BennettN,etal.Predictionsofresponsetotemperaturearecontingentonmodelchoiceanddataquality.Ecol Evol. 2017;7:10467–10481. https://doi.org/10.1002/ece3.3576