Modelling the effect of temperature on phytoplankton growth across the global ocean★

8th Vienna International Conference on Mathematical Modelling 8th Vienna18International Conference on Mathematical Modelling February - 20, 2015. Vienna University of Technology, Vienna, 8th Vienna Vienna International International Conference on Mathematical Mathematical Modelling 8th Conference on Modelling February 18 - 20, 2015. Vienna University of Technology, Vienna, Austria Available online at www.sciencedirect.com February 18 20, 2015. Vienna University of Technology, February 18 - 20, 2015. Vienna University of Technology, Vienna, Vienna, Austria Austria Austria

ScienceDirect

IFAC-PapersOnLine 48-1 (2015) 228–233

Modelling the effect of temperature on Modelling of temperature on Modelling the the effect effect ofacross temperature on phytoplankton growth the global phytoplankton growth across the global phytoplankton growth ocean across the global ocean ocean

Ghjuvan Micaelu GRIMAUD ∗∗ , Val´ erie LE GUENNEC ∗,∗∗ ∗,∗∗ Ghjuvan Micaelu GRIMAUD , Val´ e rie LE GUENNEC ∗∗,∗∗∗ ∗ ∗,∗∗ ∗ ∗ , Val´ ∗,∗∗ Sakina-Doroth´ ee AYATA ∗∗,∗∗∗ , Francis MAIRET, Ghjuvan Micaelu GRIMAUD e rie LE GUENNEC Ghjuvan Micaelu GRIMAUD , Val´ e rie LE GUENNEC ∗ Sakina-Doroth´ e ,, Francis MAIRET, ∗∗,∗∗∗ ∗∗,∗∗∗ ∗∗,∗∗∗ Antoine SCIANDRA , Olivier BERNARD Sakina-Doroth´ ee e AYATA AYATA Francis MAIRET,∗∗ ∗∗ Sakina-Doroth´ e e AYATA , Francis MAIRET, ∗∗,∗∗∗ Antoine SCIANDRA ∗∗,∗∗∗ , Olivier BERNARD ∗ Antoine Antoine SCIANDRA SCIANDRA ∗∗,∗∗∗ ,, Olivier Olivier BERNARD BERNARD ∗ ∗ BIOCORE-INRIA,BP93,06902 Sophia-Antipolis Cedex, France ∗ Sophia-Antipolis Cedex, France ∗∗ ∗ ∗ BIOCORE-INRIA,BP93,06902 UPMC Univ Paris 06, UMR 7093, LOV, Observatoire BIOCORE-INRIA,BP93,06902 Sophia-Antipolis Cedex, BIOCORE-INRIA,BP93,06902 Sophia-Antipolis Cedex, France France ∗∗ UPMC Univ Paris 06, UMR 7093, LOV, Observatoire ∗∗ ∗∗ UPMC oc´eanologique, F-06234, Villefranche/mer, France Univ Paris Paris 06, UMR UMR 7093, LOV, LOV, Observatoire Observatoire UPMC Univ 06, 7093, e anologique, F-06234, Villefranche/mer, France ∗∗∗ oc´ CNRS, UMR 7093, LOV,Villefranche/mer, Observatoire oc´eanologique, F-06234, France oc´eeanologique, anologique, F-06234, Villefranche/mer, France ∗∗∗ oc´ UMR 7093, LOV, Observatoire oc´ eeanologique, ∗∗∗ ∗∗∗ CNRS, F-06234,Villefranche/mer, France CNRS, UMR 7093, LOV, Observatoire oc´ anologique, CNRS, UMR 7093, LOV, Observatoire oc´ e anologique, F-06234,Villefranche/mer, France F-06234,Villefranche/mer, F-06234,Villefranche/mer, France France Abstract: : Phytoplankton are ectotherms and are thus directly influenced by temperature. Abstract: :: Phytoplankton are ectotherms and arewhich thus directly influenced temperature. They experience temporal variation in temperature a selectionby pressure. Using Abstract: are and thus directly influenced by temperature. Abstract: : Phytoplankton Phytoplankton are ectotherms ectotherms and are arewhich thus results directlyin influenced by temperature. They experience temporal variation in temperature results in a selection pressure. Using the Adaptive Dynamics theory and an optimizationwhich method, we study phytoplankton thermal They experience temporal variation in temperature temperature which results in aa selection selection pressure. Using They experience temporal variation in results in pressure. Using the Adaptive Dynamics theory and an optimization method, we study phytoplankton thermal adaptation (more particulary the evolution of the optimal growth temperature) to temperature the Adaptive Dynamics theory and an optimization method, we study phytoplankton thermal the Adaptive Dynamics theory and an optimization method, we study phytoplankton thermal adaptation (more particulary the evolution of the optimal growth temperature) to temperature fluctuations. We use this method the scale global ocean and compare two existing models. adaptation (more particulary the evolution of the optimal growth temperature) to adaptation (more particulary the at evolution of of the optimal growth temperature) to temperature temperature fluctuations. We use this method at the scale of global ocean and compare two existing models. We validate our approach by comparing model predictions with experimental data sets from fluctuations. We use this method at the scale of global ocean and compare two existing models. fluctuations. We use this method at the scale of global ocean with and compare two existing models. We validate our approach by comparing model predictions experimental data sets from 57 we showby temperature driveswith evolution and thatdata the sets optimum Wespecies. validateFinally, our approach approach bythat comparing modelactually predictions with experimental data sets from We validate our comparing model predictions experimental from 57 species. Finally, we show that temperature actually drives evolution and that the optimum temperature for phytoplankton growth is strongly linked to thermal amplitude variations. 57 species. Finally, we show that temperature actually drives evolution and that the optimum 57 species. Finally, we show that temperature actually drives evolution and that the optimum temperature for growth is strongly linked to thermal amplitude variations. temperature for phytoplankton phytoplankton growth is linked to amplitude variations. temperature phytoplankton growth is strongly strongly linkedHosting to thermal thermal amplitude © 2015, IFAC for (International Federation of Automatic Control) by Elsevier Ltd. Allvariations. rights reserved. Keywords: phytoplankton, temperature, evolution, Adaptive Dynamics, microalgae, modelling Keywords: phytoplankton, temperature, evolution, Adaptive Dynamics, microalgae, modelling Keywords: Keywords: phytoplankton, phytoplankton, temperature, temperature, evolution, evolution, Adaptive Adaptive Dynamics, Dynamics, microalgae, microalgae, modelling modelling 1. INTRODUCTION model in an evolutionary perspective. Nevertheless, we 1. INTRODUCTION INTRODUCTION model evolutionary perspective. Nevertheless, we propose original approach based on function optimiza1. model in inanan an evolutionary perspective. Nevertheless, we 1. INTRODUCTION model in an evolutionary perspective. Nevertheless, we propose an original approach based on function optimization which allows to predict the evolutionary outcomes propose an original approach based on function optimizaPhytoplankton gathers planktonic autotrophic organisms, propose an original approach based on function optimization which allows predict evolutionary outcomes Phytoplankton gathers planktonic autotrophic organisms, the global oceanto Wethe validate our approach on tion which allows to predict the evolutionary outcomes mostly unicellulars, andplanktonic forms theautotrophic base of marine food at Phytoplankton gathers organisms, tion which allows toscale. predict the evolutionary outcomes Phytoplankton gathers planktonic autotrophic organisms, at the global ocean scale. We validate our approach on mostly unicellulars, and forms the base of marine food a data set of 194 observations (extracted from Thomas at the global ocean scale. We validate our approach on web. Phytoplankton is the input point of inorganic carbon mostly unicellulars, and forms the base of marine food the global ocean scale. We validate ourfrom approach on mostly unicellulars, is and forms the base of marine food at aa data set of 194 observations (extracted Thomas web. Phytoplankton the input point of inorganic carbon et al. (2012)) of the temperature response for different data set of 194 observations (extracted from Thomas in the trophic net, thanks to photosynthesis. It plays web. Phytoplankton is the input point of inorganic carbon a data set of 194 observations (extracted from Thomas web. Phytoplankton is the input point of inorganic carbon et al. (2012)) of the temperature response for different in the the trophic trophic net, thanks to photosynthesis. photosynthesis. It global plays species for which isolation sites areresponse known. We al. of temperature for different therefore a key net, role thanks in biogeochemical cycles atIt in to plays et al. (2012)) (2012)) of the the temperature response for compare different in the trophic thanks to photosynthesis. plays et species for which isolation sites are known. We compare therefore a key key net, role in biogeochemical biogeochemical cycles atIt global global the different existing models, and we address the questions species for which isolation sites are known. We compare scale (Falkowski et al., 1998). Its activity depends on therefore a role in cycles at species for which isolation sites are known. We compare therefore a key role in biogeochemical cycles at global different existing we address the scale (Falkowski (Falkowski et al., al., 1998). Its activity activity dependsand on the of how phytoplankton adaptsand to in temperature varithe different existing models, models, and wesitu address the questions questions many factors, primarily light, nutrient availability scale et Its depends on different existing models, and we address the questions scale (Falkowski et al., 1998). 1998). Its activity dependsand on the of how phytoplankton adapts to in situ temperature varimany factors, primarily light, nutrient availability ations, and investigate the implications at global scale. We of how phytoplankton adapts to in situ temperature varitemperature (Falkowski and Raven, 2007). In the context many factors, primarily light, nutrient availability and of how phytoplankton adapts to in situ temperature varimany factors, primarily light, nutrient availability and ations, and investigate the implications at global scale. We temperature (Falkowski and Raven, 2007). In the context firstly present a model of the direct temperature effect on ations, and investigate the implications at global scale. We of global warming, predicting how the ocean temperature temperature (Falkowski and 2007). In the and investigate the implications at global scale. We temperature (Falkowski and Raven, Raven, 2007). Intemperature the context context ations, firstly present a model of the direct temperature effect on of global warming, predicting how the ocean phytoplankton growth, we then use it in an evolutionary firstly present a model of the direct temperature effect increase will affect marine phytoplankton is a challenging of global warming, predicting how the ocean temperature firstly present agrowth, model ofwethe direct temperature effect on on of global will warming, predicting how the ocean temperature phytoplankton then use it in an evolutionary increase affect marine phytoplankton is a challenging perspective andgrowth, finally we to experiphytoplankton growth, then use use the it in inresults an evolutionary evolutionary issue. increase will affect marine phytoplankton is aa challenging phytoplankton we compare then it an increase will affect marine phytoplankton is challenging perspective and finally we compare the results to experiissue. data.and perspective issue. perspective and finally finally we we compare compare the the results results to to experiexperiissue. models exist to take into account the direct tem- mental mental data. Several mental data. mental data. Several models exist to take into account the direct temperature effect exist on phytoplankton growth the (Eppley, Several models to direct tem2. MODELLING THE DIRECT EFFECT OF Several models exist to take take into into account account the direct 1972; temperatureet effect on phytoplankton phytoplankton growth (Eppley, 1972; 2. DIRECT OF Geider al., 1998; Norberg, 2004; growth Bernard(Eppley, and R´emond, perature effect on 1972; TEMPERATURE ON THE PHYTOPLANKTON GROWTH perature effect on phytoplankton growth (Eppley, 1972; 2. MODELLING MODELLING THE DIRECT EFFECT EFFECT OF 2. MODELLING THE DIRECT EFFECT OF Geider et al., 1998; Norberg, 2004; Bernard and R´ e mond, TEMPERATURE ON PHYTOPLANKTON GROWTH 2012). The models from Eppley (1972) and Norberg Geider et al., 1998; Norberg, 2004; Bernard and R´ ee(2004) mond, Geider et al., 1998; Norberg, 2004; Bernard and R´ mond, TEMPERATURE ON PHYTOPLANKTON GROWTH TEMPERATURE ON PHYTOPLANKTON GROWTH 2012). The models from Eppley (1972) and Norberg (2004) are mostly used in from global scale (1972) studiesand (Dutkiewicz et al., The direct temperature effect on phytoplankton growth 2012). The Eppley Norberg 2012). The models models from Eppley (1972) and Norberg (2004) (2004) are mostly mostly used in global scale studies (Dutkiewicz et the al., The direct temperature effect on phytoplankton growth 2009; Thomas et al., 2012). However, they assume that are used in global scale studies (Dutkiewicz et al., are mostly used in global scale studies (Dutkiewicz et the al., is andirect asymmetric curve effect with three key parameters, the The temperature on growth The direct temperature effect on phytoplankton phytoplankton growth 2009; Thomas et al., 2012). However, they assume that is an asymmetric with three key parameters, the maximum growth rate of aHowever, given species (µopt ) increases 2009; Thomas et 2012). they that 2009; Thomas et al., al., 2012). However, they assume assume that the the minimal, optimal, curve and maximal temperatures for growth is an asymmetric curve with three key parameters, the is an asymmetric curve with three key parameters, the ) increases maximum growth rate of a given species (µ opt ) increases minimal, optimal, and maximal temperatures for growth exponentially with rate the optimal growth temperature (T ). maximum growth of aa given species (µ opt opt ) increases maximum growth rate of given species (µ (T , T , T respectively) called cardinal temperaminimal, optimal, and maximal temperatures for growth opt min opt max and maximal temperatures for growth minimal, optimal, exponentially with the optimal growth temperature (T ). opt ). (T T ,, T called exponentially with the optimal temperature (T min ,,(Fig. opt 1). max opt ). exponentially optimal growth growth (Topt tures The respectively) maximal growth ratecardinal (µopt ) istemperadefined T respectively) called cardinal temperamin We focus herewith on the temperature as an temperature evolution driver in (T (T Topt , T Tmax respectively) called cardinal temperamin ,(Fig. opt 1). max tures The maximal growth rate (µ )) is defined opt we We focus here on temperature as an evolution driver in as the growth rate at T . As a first step, use the tures (Fig. 1). The maximal growth rate (µ is defined opt opt phytoplankton at global scale. Evolution of phytoplankton We focus here on temperature as an evolution driver in (Fig. 1). The maximal growth ratestep, (µopt we ) is use defined We focus here at onglobal temperature as an evolution driver in tures as the growth rate at T . As a first the opt phytoplankton scale. Evolution of phytoplankton model developped by Bernard and R´ e mond (2012) based as the the growth growth rate rate at at T Topt . As a first step, we use the facing realistic temperature conditions has already been phytoplankton at global scale. Evolution of phytoplankton as . As a first step, we use the opt phytoplankton at global scale.conditions Evolution has of phytoplankton model developped by Bernard and R´ eethe mond (2012) based facing realistic temperature already been on Rosso et al. (1993) to represent thermal growth model developped by Bernard and R´ mond (2012) based modeled by Thomas et al. (2012) using Norberg (2004) facing realistic temperature conditions has already been model developped by Bernard and R´ e mond (2012) based facing realistic temperature conditions has already(2004) been on Rosso et al. (1993) to represent the thermal growth modeled by Thomas et (2012) using Norberg (Fig.et on Rosso model and Grimaud et al. (2014) using Bernard and curve modeled byby Thomas et al. al. (2012) using Norberg (2004) on Rosso et1):al. al. (1993) (1993) to to represent represent the the thermal thermal growth growth modeled by Thomas et al. (2012) using Norberg (2004) curve model and by Grimaud et al. (2014) using Bernard and curve (Fig. (Fig. 1): 1): R´ e mond (2012) model. In line with Thomas et al. (2012) model and by Grimaud et al. (2014) using Bernard and curve (Fig. 1): model and by Grimaud et al. (2014) using Bernard and  R´ eemond (2012) In line with et al. (2012) and Grimaud et model. al. (2014), we use theThomas Adaptive R´ (2012) model. In with Thomas et al. 0 if T ≤ Tmin   R´ emond mond (2012) model. In line line with Thomas et Dynamics al. (2012) (2012)  and Grimaud et al. (2014), we use the Adaptive Dynamics  0 if if T T ≤ ≤ Tmin  theory to study a given temperature-dependent growth and Grimaud et al. (2014), we use the Adaptive Dynamics 0 min ifmin Tmin < T < Tmax and Grimaud et al. (2014), we use the Adaptive Dynamics µ µB (T ) =  (1) 0 opt if φ(T T ≤) T T theory to study a given temperature-dependent growth  µ φ(T if T T < T µ (T ) = (1) theory to to study study a a given given temperature-dependent temperature-dependent growth growth opt φ(T ) min < max B (T ) = theory µ ) if T < T < T µ (1) opt min max 0 if T ≥ T  B  µ φ(T ) if T < T < T max µ (T ) = (1) opt min max Corresponding authors [email protected], valerie.leB  0 if T ≥ T  Corresponding authors [email protected], valerie.le max 0 if if T T ≥ ≥T Tmax  Corresponding authors [email protected], valerie.lewith [email protected] 0  max Corresponding authors [email protected], valerie.lewith [email protected] with [email protected] with [email protected]

Copyright 2015,IFAC IFAC (International Federation of Automatic Control) 228Hosting by Elsevier Ltd. All rights reserved. 2405-8963 ©© 2015, Copyright © 2015, IFAC 228 Copyright © 2015, IFAC 228 Peer review under responsibility of International Federation of Automatic Copyright © 2015, IFAC 228Control. 10.1016/j.ifacol.2015.05.059

MATHMOD 2015 Ghjuvan Micaelu GRIMAUD et al. / IFAC-PapersOnLine 48-1 (2015) 228–233 February 18 - 20, 2015. Vienna, Austria

λ(T ) β(T ) λ(T ) = (T − Tmax )(T − Tmin )2 β(T ) = (Topt − Tmin )[(Topt − Tmin )(T − Topt ) −(Topt − Tmax )(Topt + Tmin − 2T )]

A

φ(T ) =

with

229

(2)

(3) Topt > (Tmin + Tmax )/2 ◦ and where T is the temperature in C. This model does not make any assumption of structural link between the optimal growth temperature Topt and the optimal growth rate µopt (an increase of Topt is not followed by an increase of µopt ). Moreover, the model has only four parameters with biological meanings, which facilitates their estimation with experimental data. All along this study, we compare Bernard and R´emond (2012) model with Eppley-Norberg temperature model (Eppley, 1972; Norberg, 2004) defined as follow:  2   T −z bT 1− (4) µE (T ) = ae w/2 where a and b are parameters without biological meanings (a = 0.81, b = 0.0631), and w is the thermal niche width corresponding to Tmax − Tmin in Bernard and R´emond model. Instead of considering the Topt parameter, this model uses a parameter called z. It corresponds to the maximum of the quadratic expression of eq. (4) and it is determined by the ”Eppley curve” T → aebT (Eppley, 1972) (Fig. 1 B). However, Topt can be expressed as a function of z:  bz − 1 + (w/2)2 b2 + 1 (5) Topt = b Using parameter z, the model assumes that there is a structural link between Topt and µopt : the higher Topt , the higher µopt . Using data sets for 26 species, Grimaud et al. (2014) showed that there exists a linear link between cardinal temperatures: Tmax = mTopt + p (6) Tmin = rTopt − n and with m = 0.93, p = 9.83 and r = 0.97, n = 21.85. We thus replaced Tmax and Tmin in eq.(1) by their function of Topt in eq.(6). Given the parameter values, w is almost constant, in accordance with Thomas et al. (2012) approach (w = 30). We thus obtain two different temperature models depending only on one parameter Topt , µB (T, Topt ) and µE (T, Topt ). 3. EVOLUTIONARY MODEL FOR THERMAL ADAPTATION

B ley Epp

ve

cur

Fig. 1. Thermal growth rate curves using model from Bernard and R´emond (2012) (A) and from Eppley (1972) (B). Both models are calibrated on experimental data from Synechococcus strain WH8501 extracted from Thomas et al. (2012), in black circles. However, it is not possible to adjust the maximal growth rate in Eppley (1972) model due to its structural dependence to the ”Eppley curve” (dashed line). S is the nutrient concentration in the chemostat, Sin is the inflow substrate concentration, X is the algal biomass concentration, µ(T (t)) is either µB (T (t)) or µE (T (t)), fT (t) is a periodic function (reflecting seasonality), D is the dilution rate with D < µ(T (t))ρ(Sin ) ∀t, and ρ(S) is the substrate uptake defined as: S (8) K +S where K is a half-saturation coefficient. We consider that growth, in term of carbon fixation, is fast compared to temperature fluctuations, i.e.  is a small positive parameter. It is possible to analyze system (M  ) using the Singular Perturbation Theory (Tikhonov, 1952). The fast dynamics, where T (t) = T (0), corresponds to the classical chemostat model:  S˙ = fS (S, X, T (t)) (9) X˙ = fX (S, X, T (t)) ρ(S) =

System (9) has a unique positive globally asymptotically 2 stable equilibrium (S ∗ (T (t)), X ∗ (T (t))) ∈ R+ (see e.g. Grimaud et al. (2014)), where: S ∗ (T (t)) = ∗

KD µ(T (t)) − D

(10)

∗

X (T (t)) = (Sin − S (T (t)))

3.1 Slow-fast dynamical system

In line with Grimaud et al. (2014), we include eq.(1) The slow dynamics is given by: and eq.(4) in a simple chemostat model of phytoplankton  KD  growth with varying temperature:  S ∗ (T (t)) = µ(T (t)) − D 0 M : X ∗ (T (t)) = (S − S ∗ (T (t))) (11)  in    S˙ = fS (S, X, T (t)) = D(Sin − S) − µ(T (t))ρ(S)X ˙ T = fT (t), T (0) = T0 M  : X˙ = fX (S, X, T (t)) = [µ(T (t))ρ(S) − D]X (7)  ˙ 0 The reduced system (M ) admits a unique solution T ∗ (t): T = fT (t) =  cos(ωt) 229

MATHMOD 2015 230 Ghjuvan Micaelu GRIMAUD et al. / IFAC-PapersOnLine 48-1 (2015) 228–233 February 18 - 20, 2015. Vienna, Austria

 sin(ωt) (12) ω Tikhonov’s theorem (Tikhonov, 1952) allows us to conclude: Proposition 1. For sufficiently small values of  > 0, system (M  ) admits a unique positive solution (X  (t); S  (t); T  (t)) on [0; τ ], where 0 < τ < +∞. Moreover: T ∗ (t) = T0 +

lim→0 S  (t) = S ∗ (t) lim→0 X  (t) = X ∗ (t)

(13)

From a biological point of view, prop.1 assumes that phytoplankton populations are always at equilibrium because growth is faster than long-term temperature variations. Assuming small annual temperature fluctuation of amplitude δ, we obtain  = δω << 1. The long-term (i. e. annual) dynamics of the algal biomass can thus be approximated by: KD µ(T (t)) − D X ∗ (T (t)) = (Sin − S ∗ (T (t))) T (t) = T0 + δ sin(ωt)

S ∗ (T (t)) =

(14)

3.2 Evolutionary model using Adaptive Dynamics theory We study system (M  ) in an evolutionary perspective using the Adaptive Dynamics theory (Dieckmann and Law, 1996). To do so we allow one parameter to evolve, called the adaptive trait, here Topt . One mutant Xmut appears in the resident population at equilibrium with a mut different value of Topt , Topt :  S = S ∗ (T (t), Topt )    X = X ∗ (T (t), Topt )  mut  ˙ )Xmut (15) Mmut : Xmut = fXmut (T (t), Topt , Topt   = [µ (T (t))ρ(S) − D]X mut mut   T˙ = fT (t) Assuming that the mutant is initially rare, we compute the mutant growth rate in the resident populamut ). Depending on the sign of tion, fXmut (T (t), Topt , Topt mut fXmut (T (t), Topt , Topt ), the mutant can invade and replace the resident or not. Prop.1 insures that resident population is actually at equilibrium during mutant invasion. Here, because T is a periodically time varying variable of period τ , we use the time average mutant growth rate (Ripa and Dieckmann, 2013): mut ) >= < fXmut (Topt , Topt

1 τ

τ 0

mut ) dt fXmut (T (t), Topt , Topt

(16)

mut We then compute the selection gradient g(Topt , Topt ) which gives the selection direction (e.g. growing or decreasing values of Topt are selected through evolution). The selection gradient is defined as the partial derivative of the mut time average mutant growth rate with respect to Topt mut evaluated in Topt = Topt :  mut ) >  ∂ < fXmut (Topt , Topt mut (17) g(Topt , Topt ) =  mut  mut ∂Topt Topt =Topt

At the evolutionary equilibrium, the selection gradient is equal to zero: 230

 mut ∂ < fXmut (Topt , Topt ) >   mut  ∂Topt

=0

(18)

mut =T ∗ Topt opt =Topt

The evolutionary outcome of the model is thus given by the selection gradient. It is possible to simplify the way to ∗ find Topt : ∗ is given Proposition 2. The evolutionary equilibrium Topt by: ∗ Topt = arg max < ln(µ(T(t), Topt )) > (19) Topt

Proof. According to Grimaud et al. (2014):  mut ) >  ∂ < fXmut (Topt , Topt D = .  mut  mut ∂Topt τ Topt =Topt



τ

0

(20)



µ (T (t), Topt ) dt µ(T (t), Topt )

Moreover:  1 τ µ (T (t), Topt ∂ < ln(µ(T(t), Topt )) > dt (21) = ∂Topt τ 0 µ(T (t), Topt ) ∗ If Topt = Topt (evolutionary equilibrium), eq.(20)=0. Thus: ∂ < ln(µ(T(t), T∗opt )) > =0 ∂Topt which is equivalent to say that: ∗ Topt = arg max < ln(µ(T(t), Topt )) > Topt

4. GLOBAL OCEAN SCALE SIMULATIONS 4.1 Evolutionary model with realistic temperature signal We will now study phytoplankton thermal evolution at global ocean scale. Let us consider an ubiquitous phytoplankton species which has evolved locally at each sea surface location (i, j) in response to environmental pressure. In a first assumption, each point of latitude/longitude (i, j) can be viewed as a chemostat with growth equations given by eq. (7). Assuming that the sea surface temperature is a proxy of the temperature experienced by the phytoplankton cells, we use a realistic temperature signal T (t, i, j) from in situ observations. The sea surface temperature data for the global ocean have been downloaded from the European short term meteorological forecasting website (http://apps.ecmwf.int). The data cover the years 2010 to 2012 and the spatial resolution is 1◦ in latitude and longitude with a temporal resolution of 3 hours. We calculate for each time step (3 hours) at a given location, the value of the function φ(T (t, i, j), Topt ), depending on the perceived in situ temperature T (t, i, j) and the optimum growth temperature Topt . We then calculate the average of the integrated function over 3τ = 3 years (2010, 2011, 2012):  3τ 1 · ψ(Topt ) = ln(µ(T(t, i, j), Topt ))dt (22) 3τ 0 Using prop.2, we search the maximum of eq.(22) and the corresponding Topt to find the evolutionary optimum temperature Topt *.

MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Ghjuvan Micaelu GRIMAUD et al. / IFAC-PapersOnLine 48-1 (2015) 228–233

A

Bernard and Rémond model

231

shape, it is more suitable to have higher Topt when temperature fluctuates. This assumption is in good aggreement with Kimura et al. (2013) observations for Archae; these organisms live near their maximum temperature, with a Topt much higher than the environmental T¯.

-

∗ ���� −�

Simulations for the Eppley-Norberg model (Fig. 2 B) show similar results, but the evolutionary temperature is always higher than the average temperature (for about 6◦ C). 4.3 Comparison with experimental data

B

C

Eppley-Norberg model

Temperature range

Using model (1), we determine the cardinal temperatures (Tˆmin , Tˆopt , Tˆmax ) for 194 phytoplankton strains thanks to growth rate versus temperature data sets from Thomas et al. (2012). Because µopt is a function of light and nutrient availability, we normalize the data sets. For each strain, an algorithm developped by Bernard and R´emond (2012) based on the Quasi-Newton Broyden-Fletcher-GoldfarbShanno minimizing error method is used to determine the cardinal temperatures. The calibration is coupled to a Jackkniffe statistical test evaluating the confidence interval of the parameters as in Bernard and R´emond (2012). We then only consider strains associated with data sets providing a confidence interval smaller than 5◦ C for the estimated Tˆopt , i. e. 57 strains.

-

∗ ���� −�

max(T(t)) -min(T(t))

Fig. 2. World map of the difference between the optimal temperature for growth and the mean temperature ∗ ∆ = Topt − T¯ for Bernard and R´emond model (A) and Eppley-Norberg model (B) (red correspond to areas where ∆ ≥ 6◦ C for (A) and ∆ ≥ 12◦ C for (B)), and temperature range max(T (t))−min(T (t)) for the three years 2010, 2011, 2012 (C). 4.2 Global scale simulations Global scale simulations for Bernard and R´emond model (Fig. 2 A) show that for any range of temperature experienced by phytoplankton, the evolutionary temperature ∗ Topt at a given place (i, j) is always higher or equal to the average temperature T¯(i, j). In the tropical zone, where ∗ (i, j)  the average temperature is high (near 26◦ C), Topt ¯ T (i, j). In temperate and coastal zones, where the average ∗ (i, j) > T¯(i, j). temperature is between 10 and 20◦ C, Topt This corresponds to areas where the temperature range max(T(t)) − min(T(t)) is higher than 10◦ C (Fig. 2 C). We suppose that due to the thermal growth curve asymetrical 231

Because the geographical coordinates of the isolation of the 194 stains are known, it is possible to compare Tˆopt to in situ T¯(i, j) (Fig. 3 A). Results support the fact that Topt is much higher than T¯ (here the maximum difference is 10◦ C) in temperate areas and almost the same as T¯ in tropical and polar areas. We simulate the two models at the isolation coordinates of the 57 selected strains (Fig. 3 B, green points). Bernard and R´emond model presents the same non-linear trend previously stated (Fig. 3 B, blue points) whereas Eppley-Norberg model mostly captures a ∗ and T¯ (Fig. 3 B, more flattened relationship between Topt red points). For Bernard and R´emond model, ∆ (defined as ∗ ∆ = Topt − T¯) is awlays higher or equal to zero due to the assumption that µopt does not depend on Topt . Because it is observed that ∆ is nearly equal to zero in tropical zones, this assumption seems justified. Quite the opposite, ∆ is always higher than 6◦ C for Eppley-Norberg model. The linear shape obtained with Eppley-Norbeg model plus ∆ which does not allow to match the points situated in tropical zones are due to the ”Eppley curve” hypothesis, which should probably be reparametrised. ∗ to observed Tˆopt (Fig. 4). We then compare predicted Topt For Bernard and R´emond model, mean error calculated ∗ − Tˆopt |/Tˆopt is 21.7% whereas for Eppley-Norberg as |Topt model, mean error is 25.5%. Error is thus quite similar for the two models, but the Eppley-Norberg model overesti∗ mates Topt because of its constraint linked to the ”Eppley curve”. Model predictions are accurate despite some sharp simplifications. Thus, direct temperature effect must drive evolution at global scale, contrary to what is claimed by Mara˜ n´on et al. (2014).

There are some unavoidable biases in our approach. The first one is due to the age of the cultures which have been used to provide the data. In general, the measurements were not performed right after in situ isolation. It results

MATHMOD 2015 232 Ghjuvan Micaelu GRIMAUD et al. / IFAC-PapersOnLine 48-1 (2015) 228–233 February 18 - 20, 2015. Vienna, Austria

A

A 40

40

35

35

∗ (°C) Predicted ����

Observed ���� (°C)

35

30

30

30

25

^

25

20

15

10 5 0

25

20

15

20

10

15

5

10

0

−5

0 50

5 5

10 10

15 15

20 20

25 25

30 30

B

5

B

∗ Predicted ���� (°C)

35

5

10

15

20

25

30

35

35 35

30

3030

∗ (°C) Predicted ����

25

2525

20

15

20 20

10

1515

5 0

0

5

10

15

20

25

10 10

30

Average temperature � (°C)

55 00

Fig. 3. (A) Observed Tˆopt for 194 phytoplankton strains as a function of T¯. The 57 selected strains are indicated ∗ as a function of in green points. (B) Predicted Topt ¯ T for Bernard and R´emond model (blue points) and Eppley-Norbeg model (red points). The y = x curve is indicated in black. The green points correspond to ∗ for the 57 selected strains (light the predicted Topt green for Bernard and R´emond model, dark green for Eppley-Norberg model). that the strains may have evolved, due to the temperature where the strains are stored in the culture collection. Second, we only consider the effect of temperature. Effects of light and nutrients as a master of µopt are not taken into account. Finally, we use sea surface temperature. Phytoplankton can migrate and are advected along the water column, and experience temperatures different from the surface. ∗ Despite these biases, we estimated accurately the Topt , e.g. for the cyanobacteria group, which lives in areas where the average temperature is high. It is thus possible that the sea surface temperature, where light is available, is actually a good proxy to predict thermal adaptation.

5. CONCLUSION We have presented a new method based on Adaptive Dynamics theory to study the outcome of phytoplank232

0

55

10 10

15 15

Observed

20 20

^ ����

25 25

30 30

35 35

(°C)

∗ Fig. 4. Predicted Topt compared to experimental Tˆopt for Eppley-Norberg model (A) and Bernard and R´emond model (B). Phytoplankton phylogenetics groups are indicated in color in (B): green, Dinoflagellates, pink, Diatoms, black, Cyanobacteria, blue, others.

ton adaptation at global ocean scale. We defined a standard ubiquitous phytoplantkon species and we compared two different thermal growth models describing it in the context of evolution. We found, in agreement with our ∗ theory, that the evolutionary optimal temperature Topt is always equal or higher than the average temperature experienced by the phytoplankton T¯. Moreover, the area ∗ with high difference between Topt and T¯ characterizes large temperature fluctuations. When based on Bernard and R´emond, our model successfully fit the data, contrary to ∗ the Eppley-Norberg model which linearly link Topt and T¯, ∗ and overstimates Topt . We inferred that direct temperature effect actually drives evolution at the scale of the global ocean. This work is only a first approach to understand thermal adaptation at global scale. The next step will be to study the evolution of the maximum temperature Tmax in a context of global warming.

MATHMOD 2015 February 18 - 20, 2015. Vienna, Austria Ghjuvan Micaelu GRIMAUD et al. / IFAC-PapersOnLine 48-1 (2015) 228–233

ACKNOWLEDGEMENTS This work was supported by the ANR Facteur 4 project. We are grateful to the anonymous reviewers for their helpful suggestions. REFERENCES Bernard, O. and R´emond, B. (2012). Validation of a simple model accounting for light and temperature effect on microalgal growth. Bioresource Technology, 123, 520– 527. Dieckmann, U. and Law, R. (1996). The dynamical theory of coevolution: A derivation from stochastic ecological processes. Journal of Mathematical Biology, 34, 579– 612. Dutkiewicz, S., Follows, M.J., and Bragg, J.G. (2009). Modeling the coupling of ocean ecology and biogeochemistry. Global Biogeochemical Cycles, 23(4). Eppley, R.W. (1972). Temperature and phytoplankton growth in the sea. Fishery Bulletin, 70, 1063–1085. Falkowski, P., Barber, R., and Smetacek, V. (1998). Biogeochemical controls and feedbacks on ocean primary production. Science, 281, 200–206. Falkowski, P. and Raven, J.A. (2007). Aquatic photosynthesis. Princeton University Press, New Jersey. Geider, R.J., MacIntyre, K.L., and Kana, T.M. (1998). A dynamic regulatory model of phytoplanktonic acclimation to light, nutrients, and temperature. Limnoly and Oceanogry, 43, 679–694. Grimaud, G.M., Mairet, F., and Bernard, O. (2014). Modelling thermal adaptation in microalgae: an adaptive dynamics point of view. 19th IFAC World Congress, Cape Town (South Africa). Kimura, H., Mori, K., Yamanaka, T., and Ishibashi, J. (2013). Growth temperatures of archaeal communities can be estimated from the guanine-plus-cytosine contents of 16S rRNA gene fragments. Environmental microbiology reports, 5, 468–474. Mara˜ no´n, E., Cerme˜ no, P., Huete-Ortega, M., L´opezSandoval, D.C., Mouri˜ no-Carballido, B., and Rodr´ıguezRamos, T. (2014). Resource supply overrides temperature as a controlling factor of marine phytoplankton growth. PloS one, 9(6), e99312. Norberg, J. (2004). Biodiversity and ecosystem functioning: A complex adaptive systems approach. Limnology and Oceanography, 49, 1269–1277. Ripa, J. and Dieckmann, U. (2013). Mutant invasions and adaptive dynamics in variable environments. Evolution, 67, 1279–1290. doi:10.1111/evo.12046. Rosso, L., Lobry, J., and Flandrois, J. (1993). An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model. Journal of Theoretical Biology, 162, 447–463. Thomas, M., Kremer, C., Klausmeier, C., and Litchman, E. (2012). A global pattern of thermal adaptation in marine phytoplankton. Science, 338, 1085–1088. Tikhonov, A.N. (1952). Systems of differential equations containing a small parameter multiplying the derivative. Matematicheskii Sbornik, 31, 575–586.

233

233