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## Non-linear relationships between Topt and μopt

In chapter 5, and on the contrary to Bissinger et al. [2008], we used non-linear quantile regression to describe the non-linear relationship between Topt and μopt for the whole data set and for each sub-group (defined in chapter 5). Non-linear quantile regression is based on the least absolute deviations regression (LAD) which uses the median rather than the mean, and is therefore less sensitive to extreme outlying values than ordinary least squares (OLS) regression. The quantile q of interest (here q is the 99th quantile) is estimated using an optimization function that minimizes the sum of weighted absolute deviations [Koenker and d’Orey, 1987] (see also the review of Cade and Noon [2003] for biological applications of quantile regression). A linear quantile regression would give, for example: Q(βq) = XN i:yi≥x′ iβ q|yi − x′ iβq| + XN i:yi<x′ iβ (1 − q)|yi − x′ iβq| (2.11).

**The Arrhenius model from Van’t-Hoff to Eyring:**

Since the xixe century, chemists are aware of the prodigious effect of temperature change on chemical reaction rates. As early as 1850, Ludwig Ferdinand Wilhelmy published an article dealing with ‘The law by which the action of acids on cane sugar occurs’ and its temperature dependence [Wilhelmy, 1850]. At that time, Maxwell’s law of distribution of molecular velocities already established that the proportion of molecules having energies greater than the average at ordinary temperatures were very small, but increased with temperature. In 1867, Leopold Pfaundler von Hadermur applied Maxwell’s law to chemical equilibrium [Pfaundler, 1867]. Because at chemical equilibrium, reverse and forwards reactions occur at the same rate, he assumed, using Maxwell’s law, that only a fraction of molecules having energy greater than a critical parameter E could undergo chemical-changes. Moreover, this fraction would exponentially change with temperature. In 1884, the famous physical chemist Jacobus van’t Hoff contemplated a thermodynamical version of Pfaundler thought [van’t Hoff, 1889]. Consider the following chemical equilibrium between two species B and C: B k1 ⇋ k2 C (3.2).

**The protein thermal stability challenge**

Protein thermal stability plays a key role in the UO thermal growth curve [Johnson and Lewin, 1946, Pena et al., 2010, Rosenberg et al., 1971, Zeldovich et al., 2007] (see section 5.1 and 6). In line with the master reaction model, some publications went further in the comprehension of protein thermal stability and its consequences on UO growth.

The modified master reaction model: The master reaction model assumes that G, the Gibbs free energy difference between the native and denatured protein, is temperature independent. Based on Murphy et al. [1990] work, Ross [1993] and then Ratkowsky et al. [2005] remarked that G should vary with T in eq.3.19 following eq. 3.20. Moreover, [Murphy et al., 1990] showed that globular proteins (including enzymes) share common thermodynamic properties. For any protein, the denaturation enthalpy change H and the denaturation entropy change S, normalized to the number of amino-acids residues of this protein, both converge to a fixed value H∗ and S∗ at T∗H and T∗ S respectively [Privalov, 1979]. The reason for such a temperature convergence is still unclear. Nonetheless, it has been shown that at T∗H and T∗ S, the hydrophobic contribution to H and S approaches zero [Robertson and Murphy, 1997]. At that stage, we have to introduce a novel thermodynamics parameter, the heat capacity Cp.

According to [Murphy et al., 1990], H and S can be expressed as a function of the heat capacity change:

H = H∗ + Cp(T − T∗ H ) (3.29).

S = S∗ + Cpln(T/T ∗ S) (3.30).

**The special case of unicellular photosynthetic organisms**

Unicellular photosynthetic organisms perform oxygenic photosynthesis. This metabolic particularity is ensured by special structures and enzymes, such as the rubisco enzyme involved in CO2 fixation, and the photosystems protein complexes which harvest photons.

The thermal sensitivity of unicellular photosynthetic organisms is thus expected to be distinct and will be discussed in this section.

### The Eppley point of view for phytoplankton

In 1972, Richard Eppley published a review dealing with the effect of temperature on phytoplankton growth in the sea [Eppley, 1972]. Comparing different thermal growth curves for a variety of phytoplankton species in non-limiting conditions (nearly 200 data points), Eppley determined that the maximal growth rate μopt for each species is constrained by a virtual envelope along the optimal temperature trait Topt, the so-called ‘Eppley curve’(see Fig.3.6). Eppley stated that for any phytoplankton species growing under 40◦C, ‘hotter is faster’.

In 2004, Jon Norberg used Eppley’s hypothesis to develop a temperature-growth model, the ‘Eppley-Norberg’ model [Norberg, 2004]: μ(T) = » 1 − T − z w 2 # aebT (3.41).

#### The link between photosynthesis and temperature in the models

Photosynthesis involves two distinct parts with different sensitivities to temperature (the dark phase and the light phase, see section 10.1.2). The light phase is virtually not affected by temperature, leading to a possible desequilibrium between the photons harvested and their conversion into chemical energy whenever the dark phase is affected by temperature (see for example the review by Ras et al. [2013]). The model supported by data related to the light phase of photosynthesis (O2 production of Pulse Amplitude-Modulated (PAM) fluorescence) therefore includes a moderated temperature effect [B´echet et al., 2013].

Photosynthesis and temperature uncoupled: These models assume that temperature and light are independent factors. For example, the model developed by Bernard and R´emond [2012] assumes that the growth rate is expressed as: μ(T, I) = f(I).φ(T) (3.43).

**Table of contents :**

**1 Introduction **

1.1 An ode to phytoplankton

1.2 Phytoplankton and temperature

1.2.1 The direct effect of temperature

1.2.2 Phytoplankton in a changing climate

1.2.3 A decisive parameter in biotechnological applications

1.3 From acclimation to adaptation

1.4 Objectives of the thesis

**2 Material and methods **

2.1 Culture device and selection procedure

2.1.1 The selectiostats

2.1.2 Cultivation mode for selection experiments

2.1.3 The TIP device

2.2 Data compilation, parameters identification and models calibration

2.2.1 Thermal growth curves compilation

2.2.2 Species biovolume

2.2.3 CTMI parameters determination

2.2.4 Data sets selection

2.2.5 Hinshelwood model calibration

2.2.6 Data analysis

2.2.6.1 Linear relationships between the cardinal temperatures .

2.2.6.2 Non-linear relationships between Topt and μopt

2.2.6.3 Statistical tools for models comparison

**3 Modelling the temperature effect on unicellular organisms from heterotrophic bacteria to autotrophic eukaryotes: a review **

3.1 Introduction

3.2 Modelling the specific growth rate of unicellular organisms as a function of temperature

3.2.1 Methodological clarification

3.2.2 The Arrhenius model from Van’t-Hoff to Eyring:

3.2.3 Empirical approach

3.2.4 Mechanistic approach

3.2.5 The protein thermal stability challenge

3.3 The special case of unicellular photosynthetic organisms

3.3.1 The Eppley point of view for phytoplankton

3.3.2 The link between photosynthesis and temperature in the models .

3.4 The dynamical effect of temperature on unicellular organisms

3.4.1 The metabolic response to temperature acclimation

3.4.2 The thermally-induced death

3.5 Conclusion

**4 Correlation between the cardinal temperatures: insight into thermal adaptation **

4.1 Introduction

4.2 Relation between the cardinal temperatures

4.2.1 Linear relationships

4.2.2 Differences among the phylogenetic groups

4.2.3 A closer look at microalgae

4.3 Thermal adaptation and the thermal niche width

4.4 Conclusion

**5 Revisiting the Eppley hypothesis **

5.1 Introduction

5.2 Hotter is not always faster

5.2.1 Describing the relationship between Topt and μopt using quantile regression

5.2.2 Group specificities

5.2.3 Scaling law in the thermal growth curve

5.3 The phytoplankton paradigm

5.3.1 The revisited Eppley curve for phytoplankton

5.3.2 A case study: the cyanobacteria Synechococcus sp.

5.4 Conclusion

**6 Towards understanding the thermodynamical fundament of the thermal growth curve: a modelling approach **

6.1 Introduction

6.2 The Hinshelwood model as a theoretical framework

6.2.1 Metabolism represented as a set of n autocatalytic reactions

6.2.2 Data normalization and calibration of the Hinshelwood model

6.3 Accounting for the enthalpy-entropy compensation

6.3.1 Theoretical approach

6.3.2 Calibration

6.4 Accounting for the activity-stability trade-off

6.4.1 Theoretical approach

6.4.2 Calibration

6.5 The two parameters Hinshelwood model

6.5.1 Reducing the parameter number of the Hinshelwood model down to 2

6.5.2 Comparison between the reduced Hinshelwood and the reduced CTMI models

6.5.3 Correlation between the cardinal temperatures

6.6 Conclusion

**7 Modelling thermal adaptation in microalgae: an adaptive dynamics point of view **

7.1 Introduction

7.2 Simple dynamical model describing the temperature effect on microalgae in chemostat

7.2.1 The Monod model in chemostat

7.2.2 The specific case of the Droop model in chemostat

7.3 Evolutionary Model

7.3.1 General case

7.3.2 Modelling the evolution of the optimal temperature trait

7.3.3 Structural link between adaptive traits

7.4 Fluctuating temperature

7.4.1 Ecological timescale

7.4.2 Evolutionary timescale

7.4.2.1 Case 1: Topt is varying only

7.4.2.2 Case 2: the thermal niche width is kept constant

7.4.3 Evolutionary Branching conditions

7.5 Conclusion

**8 Selecting thermal tolerant strains of the Haptophyceae Tisochrysis lutea **

8.1 Introduction

8.2 Selection experiment in controlled systems

8.2.1 Summary of the experiment

8.2.2 Main results

8.3 Modelling selection during the experiment

8.3.1 Re-identification of the final thermal tolerance parameters

8.3.1.1 Identification at the population scale

8.3.1.2 Competition between two thermal phenotypes

8.4 Modelling thermal adaptation during the experiment

8.4.1 Determining the invasion fitness

8.4.1.1 Population dynamics, invasion fitness and selection gradient in a turbidostat growth model

8.4.1.2 Population dynamics, invasion fitness and selection gradient in a fed-batch growth model

8.4.2 Evolutionary dynamics

8.4.3 Evolutionary equilibrium

8.5 Conclusion

**9 Modelling the effect of temperature on phytoplankton growth across the global ocean **

9.1 Introduction

9.2 Evolutionary model for thermal adaptation

9.2.1 Slow-fast dynamical system

9.2.2 Evolutionary model using Adaptive Dynamics theory

9.3 Global ocean scale simulations

9.3.1 Evolutionary model with realistic temperature signal

9.3.2 Global scale simulations

9.3.3 Comparison with experimental data

9.3.4 The warming scenario

9.4 Conclusion

**10 Conclusion & Perspectives **

10.1 The physiological impacts of temperature on phytoplankton

10.1.1 From empirical models to thermodynamical insights

10.1.2 The submerged iceberg of unknown: future works

10.2 Capturing the evolutionary trajectories

10.2.1 Selection experiments and evolutionary modelling

10.2.2 Evolution in the ocean

10.3 Conclusion

**Annexes **

Determining Tmin, Topt, Tmax and muopt in the Hinshelwood model

Autocatalytic view of the Hinshelwood model

**Bibliography **