Celine Hauzy

Florence D Hulot

Audrey Gins

Michel Loreau

INTRA-AND INTERSPECIFIC DENSITY-DEPENDENT DISPERSAL IN AN AQUATIC PREY-PREDATOR SYSTEM

Dispersal intensity is a key process for the persistence of prey-predator metacommunities. Consequently, knowledge of the ecological mechanisms of dispersal is fundamental to understanding the dynamics of these communities. Dispersal is often considered to occur at a constant per capita rate; however, some experiments demonstrated that dispersal may be a function of local species density.

Here we use aquatic experimental microcosms under controlled conditions to explore intra-and interspecific density-dependent dispersal in two protists, a prey Tetrahymena pyriformis and its predator Dileptus sp.

We observed intraspecific density-dependent dispersal for the prey and interspecific density-dependent dispersal for both the prey and the predator. Decreased prey density lead to an increase in predator dispersal, while prey dispersal increased with predator density.

Additional experiments suggest that the prey is able to detect its predator through chemical cues and to modify its dispersal behaviour accordingly.

Density-dependent dispersal suggests that regional processes depend on local community dynamics. We discuss the potential consequences of density-dependent dispersal on metacommunity dynamics and stability.

Knowledge of dispersal mechanisms is crucial to understanding the dynamics of spatially structured populations and metacommunities [START_REF] Leibold | The metacommunity concept : a framework for multi-scale community ecology[END_REF]. Such knowledge may also be useful for explaining the response of communities to fragmentation and climate change. Metacommunity dynamics can be influenced by local processes such as intra-and interspecific interactions (Lotka 1925;[START_REF] Rosenzweig | Graphical representation and stability conditions of predator-prey interactions[END_REF][START_REF] Volterra | Variations and fluctuations in the numbers of individuals in animal species living together[END_REF] and regional processes such as dispersal, that link the dynamics of several local communities [START_REF] Cadotte | Metacommunity influences on community richness at multiple spatial scales: A microcosm experiment[END_REF]. Dispersal is the movement of individuals from one patch (emigration) to another (immigration). Intermediate intensities of dispersal can increase the persistence of prey-predator metacommunities [START_REF] Crowley | Dispersal and the stability of predator-prey interactions[END_REF]Holyoak & Lawler 1996a, b;[START_REF] Huffaker | Experimental studies on predation : dispersion factors and predatorprey oscillations[END_REF]Nachman 1987a;[START_REF] Reeve | Environmental variability, migration, and persistence in host-parasitoid systems[END_REF][START_REF] Zeigler | Persistence and patchiness of predator-prey systems induced by discrete event population exchange mechanisms[END_REF]. Dispersal rate is often considered a constant trait of species, but it may be condition-dependent. In particular, it may depend on the density of species in the local community. Density-dependent dispersal implies a direct interaction between local (population dynamics) and regional (dispersal) processes, which could influence metacommunity dynamics and stability.

Many studies have explored dispersal in the context of a single species. They have shown that dispersal often depends upon a species' own local density [START_REF] Diffendorfer | Testing models of source-sink dynamics and balanced dispersal[END_REF].

We call this effect intraspecific density-dependent dispersal. Dispersal may either increase (positive density-dependent dispersal) or decrease (negative density-dependent dispersal) as population density increases. Positive and negative intraspecific density-dependent dispersal has been observed in mites [START_REF] Bernstein | Prey and predator emigration responses in the acarine system Tetranychus urticae-Phytoseiulus persimilis[END_REF], insects [START_REF] Fonseca | Density-dependent dispersal of black fly neonates is by flow[END_REF] and vertebrates (French & Travis 2001;[START_REF] Galliard | Mother-offspring interactions affect natal dispersal in a lizard[END_REF][START_REF] Matthysen | Density-dependent dispersal in birds and mammals[END_REF]; see for review [START_REF] Matthysen | Density-dependent dispersal in birds and mammals[END_REF], but not in protists (Holyoak & Lawler 1996a, b). In a mite preypredator system, [START_REF] Bernstein | Prey and predator emigration responses in the acarine system Tetranychus urticae-Phytoseiulus persimilis[END_REF] found positive intraspecific density-dependent dispersal

INTRODUCTION

in the prey, but not in the predator. Conversely, French & Travis (2001) observed densityindependent prey dispersal but density-dependent parasitoid dispersal in a beetle-wasp system.

A few studies have experimentally explored how dispersal of one species is affected by the density of another species. We refer to this type of dispersal as interspecific densitydependent dispersal. The presence of a predator or parasitoid has enhanced prey dispersal in some insect communities [START_REF] Holler | Enemy-induced dispersal in a parasitic wasp[END_REF][START_REF] Kratz | Effects of Stoneflies on Local Prey Populations: Mechanisms of Impact Across Prey Density[END_REF][START_REF] Wiskerke | Larval parasitoid uses aggregation pheromone of adult hosts in foraging behavior -a solution to the reliability-detectability problem[END_REF]. By contrast, in aquatic ciliates, dispersal of the prey (Colpidium striatum) was not affected by the presence of the predator (Didinium nasutum) (Holyoak, personal communication;Holyoak & Lawler 1996a, b). However, these studies considered predator presence or absence and not predator density. [START_REF] Bernstein | Prey and predator emigration responses in the acarine system Tetranychus urticae-Phytoseiulus persimilis[END_REF] showed with terrestrial mites that prey emigration had a positive relationship with predator density and that predator emigration had a negative relationship with prey density. Similarly, [START_REF] Kratz | Effects of Stoneflies on Local Prey Populations: Mechanisms of Impact Across Prey Density[END_REF] found that a decrease in prey density enhanced predator emigration in aquatic insect larvae. French & Travis (2001) observed a decrease in parasitoid swap dispersal as prey dispersal increased but no interspecific density-dependent dispersal for the prey. Thus, overall, dispersal seems to be a function of local densities in several experimental models. However, only two studies [START_REF] Bernstein | Prey and predator emigration responses in the acarine system Tetranychus urticae-Phytoseiulus persimilis[END_REF]French & Travis 2001) have considered the full set of intra-and interspecific effects of density on dispersal in prey-predator systems, in spite of their great interest in the perspective of metacommunity theory.

Interspecific density-dependent dispersal in prey may be considered as a predatorinduced defence [START_REF] Lima | Behaviorial decisions made under the risk of predation : a review and prospectus[END_REF]. Other predator-induced responses include morphological changes in vertebrates [START_REF] Kishida | Flexible architecture of inducible morphological plasticity[END_REF] and invertebrates [START_REF] Kuhlmann | The ecology and evolution of inducible defenses[END_REF][START_REF] Tollrian | Inducible defences in cladocera: constraints, costs, and multipredator environments[END_REF]. Predator-induced dispersal suggests that the prey is able to assess the presence of its predator. Several experiments in aquatic systems showed that prey may detect their predator because of organic compounds they release in the medium, for instance Daphnia [START_REF] Lampert | Chemical induction of colony formation in a green alga (Scenedesmus acutus) by grazers (Daphnia)[END_REF][START_REF] Stibor | Predator-induced phenotypic variation in the pattern of growth and reproduction in Daphnia hyalina (Crustacea; Cladocera)[END_REF] and ciliates [START_REF] Karpenko | Feeding Behavior of Unicellular Animals. II. The Role of Prey Mobility in the Feeding Behavior of Protozoa[END_REF][START_REF] Kuhlmann | The ecology and evolution of inducible defenses[END_REF]. By contrast, perception in ciliates may require encounter between individuals: two mechanisms have been reported in ciliates: (1) detection of their predators by direct membrane contact [START_REF] Kuhlmann | Escape response of Euplotes octocarinatus to turbellarian predators[END_REF][START_REF] Kusch | Behavioural and morphological changes in ciliates induced by the predator Amoeba proteus[END_REF]), and (2) detection of local hydrodynamic disturbances created by the motion of cilia [START_REF] Karpenko | Feeding Behavior of Unicellular Animals. II. The Role of Prey Mobility in the Feeding Behavior of Protozoa[END_REF].

Consequently, interspecific density-dependent dispersal in ciliates may occur through waterborn chemical cues or may require direct contact.

Here we explore intra-and interspecific density-dependent dispersal in freshwater protists. These organisms are often patchily distributed in ponds and lakes at the scale of millimetres or centimetres [START_REF] Arlt | Vertical and horizontal microdistribution of the meifauna in the Greifswalder Bodden[END_REF][START_REF] Smirnov | Spatial distribution of gymnamoebae (Rhizopoda, Lobosea) in brackish-water sediments at the scale of centimeters and millimeters[END_REF][START_REF] Taylor | Microspatial heterogeneity in the distribution of ciliates in a small pond[END_REF][START_REF] Wiackowski | Small-scale distribution of psammophilic ciliates[END_REF]). We use a prey-predator couple, in aquatic experimental microcosms under controlled conditions and investigate the effects of population density on dispersal, and address three questions. First, does a species' own density affect its dispersal (intraspecific density-dependent dispersal)? We test this hypothesis for the prey and the predator separately. Second, does prey density affect predator dispersal, and does predator density affect prey dispersal (interspecific density-dependent dispersal)? If prey dispersal is positively related to predator density, our third question investigates the effects of predator organic compounds on prey dispersal. In addition, we explore these effects at low and high initial prey density to assess the interaction between prey and predator densities on prey dispersal.

Study Organisms

Tetrahymena pyriformis Ehrenberg, a bacterivorous protist, and its protist predator Dileptus sp. were obtained from Carolina Biological Supply (Burlington, NC, USA). Prey and predator were cultured in 50 mL microcosms containing medium inoculated with a mixed bacterial suspension. The medium was prepared by sterilizing mineral water with 0•75 g.L -1 of Protozoan Pellet (Carolina Biological Supply). Cultures were maintained at 18•0 ± 0•5 °C under controlled light (14:10 h light:dark cycle). One day after bacterial inoculation, each culture was inoculated with 1 mL of T. pyriformis to give about 240 cells.mL -1 . Three days later, T. pyriformis cultures reached a stationary phase; they were then used to feed Dileptus sp. The same culturing method was used in all experiments.

Under our standard culture conditions, the minimal generation times of T. pyriformis and Dileptus sp. were 8,18 h and c. 24 h, respectively (Hauzy C. & Hulot F.D., unpublished data).

Experimental design

To measure dispersal, we used microcosms made of two 100 mL bottles (55 mm internal diameter) connected by a 10 cm tube (5 mm internal diameter). We defined dispersal as migration from a bottle initially containing organisms (donor patch) to a bottle free of organisms (recipient patch). We conducted six independent experiments according to the following design. The tube of each microcosm was initially clamped and donor patches were assigned randomly.

MATERIALS AND METHODS

Initial densities in all experiments were adjusted by serial dilution in 1-day-old bacterial culture after counting the 3-day-old T. pyriformis and the 1-day-old Dileptus sp. cultures.

Counts were done under a binocular microscope in 10 µL drops for T. pyriformis, and 100 µL drops for Dileptus sp. Several drops were examined until a minimum number of 400 individuals was counted. The donor patch received 50 mL of the experimental treatment culture. The recipient patch received 50 mL of standardized 1-day-old bacterial culture. The experiments were initiated by releasing the clamp off the tube. Organisms dispersed freely during a time that was shorter than the generation time of the species studied. Treatments were replicated five times, except experiment 5, which was replicated four times.

At the end of the experiment, the content of each bottle was fixed with formaldehyde at a final concentration of 0•2%. Because the recipient patches did not contain high population densities, they were concentrated by centrifugation (5 min, 2000 r.p.m., 425 g).

Organisms were counted under a binocular microscope in 10 µL drops for T. pyriformis, and 100 µL drops for Dileptus sp. Several drops were examined in accordance with the following two procedures: (1) in experiments 1-4 and 6 (see below) up to 100 or 400 individuals were counted, respectively, and (2) in experiment 5, individuals were counted in 800 µL. Dispersal was measured by the dispersal rate per capita and per generation, and was calculated as the ratio of the density of the focal species in the recipient patch at the end of the experiment to its initial density in the donor patch. Initial, not final, density in the donor patch was used to avoid the potentially confounding factor of prey depletion in experiments testing prey dispersal in the presence of the predator (see experiments 4, 5 and 6 below).

In experiment 1 we tested the effect of T. pyriformis density on its own dispersal in the absence of Dileptus sp. Density treatments corresponded to cultures with 12 700 cells.mL -1 , 1270 cells.mL -1 and 43.1 cells mL -1 . The dispersal time was 4 h.

In experiment 2 we tested the effect of Dileptus sp. density on its own dispersal.

Treatments correspond to three levels of density: 61.3 cells mL -1 , 30.6 cells.mL -1 and 15.3 cells.mL -1 . T. pyriformis density was adjusted to 3.3 cells.mL -1 in all treatments. The dispersal time was 18 h.

Interspecific density-dependent dispersal

Experiment 3 tested the effect of T. pyriformis density on Dileptus sp. dispersal. A Dileptus sp. culture was mixed 50:50 with a T. pyriformis culture of varying density. We obtained three treatments with the same initial Dileptus sp. density (20.8 cells.mL -1 ) but different initial T. pyriformis densities: 5400 cells.mL -1 , 540 cells.mL -1 and 54.0 cells.mL -1 .

The dispersal time was 18 h. Experiment 4 tested the effect of Dileptus sp. density on T. pyriformis dispersal.

Cultures with different Dileptus sp. densities were mixed 50:50 with a T. pyriformis culture.

T. pyriformis initial density was 1120 cells.mL-1 in all treatments, and Dileptus sp. densities were 37.5 cells.mL -1 , 18.8 cells.mL -1 and 9.4 cells.mL -1 . The dispersal time was 5 h.

Mechanism of detection

In order to test whether T. pyriformis is able to detect Dileptus sp. via a chemical signal, we compared prey dispersal rate in the presence of the predator (treatment «with»), in a filtered medium of predator culture (treatment «filtered») and in the absence of predator (treatment «without»). This hypothesis was tested independently for two initial T. pyriformis densities (experiment 5: 550 cells.mL -1 ; experiment 6: 6600 cells.mL -1 ). In the treatment «with», we added the Dileptus sp. culture to the T. pyriformis culture (initial density of Dileptus sp. in experiment 5: 63.5 cells mL -1 ; in experiment 6: 22.1 cells mL -1 ). In the treatment «filtered», we replaced the Dileptus sp. culture of the treatment «with» with the same Dileptus sp. filtered with a 1.2 µm Whatman GF/C filter permeable to chemical compounds and bacteria. In the treatment «without», the T. pyriformis culture was diluted with a 1-day-old bacterial culture. Each treatment was replicated five and four times in experiments 5 and 6, respectively. The dispersal time in both experiments was 8 h.

Statistical Analysis

Data were analysed with linear (LM) or linear mixed effects models in R vs. 2•2•0.

For experiments 1-4, data were considered as continuous variables whereas data of experiments 5 and 6 were considered categorical. When homoscedasticity of variances (Bartlett's test) was satisfied (experiments 2, 3, 5 and 6), we used the LM procedure. When variances were heteroscedastic (experiments 1 and 4), we used the Generalized Least Squares procedure of the linear mixed effects model, which accounts for heteroscedasticity.

The Generalized Least Squares procedure gave the same qualitative results as the LM procedure. Tukey's post hoc tests were used to determine the differences between treatments and groups of treatments.

In experiment 1, no T. pyriformis individuals could be detected in the recipient patch for three of five replicates of the low density treatment. T. pyriformis density had a strong significant effect on its own dispersal rate (Fig. 1a; t = 4.17, d.f. = 13, P = 0.001). The treatment with the highest density (12 700 cells.mL -1 ), which corresponded to the beginning of the stationary phase, was significantly different (P < 0.001) from the lower density treatments (1270 and 43.1 cells.mL -1 ).

Experiment 2 (Fig. 1d) showed no significant effect of Dileptus sp. density on its per capita dispersal rate ( F = 2•45, d.f. = 14, P = 0•141).

Interspecific density-dependent dispersal

In experiment 3 (Fig. 1c), T. pyriformis density had a strong significant effect on Dileptus sp. dispersal rate (F = 7.07, d.f. = 14, P = 0.019). The average Dileptus sp. dispersal rate was significantly higher at the lowest prey density (54.0 cells mL -1 ) than at higher prey densities (5400.0 cells.mL -1 and 540.0 cells.mL -1 ) (P < 0.0001).

In experiment 4 (Fig. 1b), the initial T. pyriformis density (1120 cells.mL -1 ) was chosen such that it does not affect its own dispersal rate (see Results of experiment 1). Dileptus sp. density had a strong significant effect on the dispersal rate of its prey (F = 22.28, d.f. = 14, P < 0.001) and the dispersal rate of T. pyriformis was significantly higher at the two highest Dileptus sp. densities (37.5 cells.mL -1 and 18.8 cells.mL -1 ) than at the lowest density (9.8 cells mL -1 ) (P < 0.0001).

RESULTS

Mechanism of detection

Experiments 5 and 6 were conducted at a predator density that induces prey dispersal (see Results of experiment 4). When the density of T. pyriformis was low (experiment 5), the differences among treatments on T. pyriformis dispersal rate were significant (Fig. 2a; F = 165.4, d.f. = 12, P < 0.001). Tukey's post hoc test indicated that prey dispersal rate in the treatments «filtered» and «with» were significantly higher than in the treatment «without» (P < 0.001). Prey dispersal rate was also significantly higher in the treatment «filtered» than in the treatment «with» (P < 0.005). When initial T. pyriformis density was high (experiment 6), the effects of the treatments «without», «filtered» and «with» on T. pyriformis dispersal rate were marginally significant (Fig. 2b; F = 3.623, d.f. = 9, P = 0.070). Tukey's post hoc test shows that the prey dispersal rate in the treatment «filtered» was marginally higher than in the treatment «without» (P = 0•060). The results of our study suggest that in aquatic prey-predator systems, the dispersal of a species can be a plastic trait that depends on population densities. We observed intraspecific density dependence in dispersal for the prey T. pyriformis. By contrast, there was no significant intraspecific density dependence in dispersal for the predator Dileptus sp.

Interspecific density-dependent dispersal was observed for both the prey and the predator. A decrease in T. pyriformis density led to a significant increase in Dileptus sp. dispersal rate, while T. pyriformis dispersal was higher when Dileptus sp. density was higher.

The two previous studies [START_REF] Bernstein | Prey and predator emigration responses in the acarine system Tetranychus urticae-Phytoseiulus persimilis[END_REF]French & Travis 2001) that have exhaustively explored density-dependent dispersal in a prey-predator system revealed two different patterns (Fig. 3). French & Travis (2001) observed that predator dispersal depended on its own density and on prey density, but prey dispersal was densityindependent. By contrast, [START_REF] Bernstein | Prey and predator emigration responses in the acarine system Tetranychus urticae-Phytoseiulus persimilis[END_REF] showed interspecific density-dependent dispersal for both the prey and the predator, and intraspecific density-dependent dispersal for the prey only. Our results follow the same pattern as [START_REF] Bernstein | Prey and predator emigration responses in the acarine system Tetranychus urticae-Phytoseiulus persimilis[END_REF]. Thus, only two patterns of density-dependent dispersal in prey-predator systems have received experimental support. An increase in the prey dispersal rate when predator density increases, suggests that the prey is able to detect its predator and avoid it. Studies on ciliates' perception have shown that two different detection mechanisms are possible:

recognition through chemical cues released in the medium [START_REF] Karpenko | Feeding Behavior of Unicellular Animals. II. The Role of Prey Mobility in the Feeding Behavior of Protozoa[END_REF][START_REF] Kuhlmann | The ecology and evolution of inducible defenses[END_REF][START_REF] Seravin | Feeding Behavior of Unicellular Animals. I. The Main Role of Chemoreception in the Food Choise of Carnivorous Protozoa[END_REF] and recognition that requires direct contact [START_REF] Karpenko | Feeding Behavior of Unicellular Animals. II. The Role of Prey Mobility in the Feeding Behavior of Protozoa[END_REF][START_REF] Kuhlmann | Escape response of Euplotes octocarinatus to turbellarian predators[END_REF][START_REF] Kusch | Behavioural and morphological changes in ciliates induced by the predator Amoeba proteus[END_REF]). Our results suggest that the prey is able to detect its predator through chemical cues. At a low initial prey density, prey dispersal was significantly higher when prey was in the presence of predators or in the presence of a filtered medium of predator cultures than in the control. At a high initial prey density, prey dispersal was marginally higher when prey was in the presence of a predatorfiltered culture than in the control or in the presence of the predator. The difference in prey dispersal between the predator-filtered culture and the predator culture may be a result of prey depletion by the predator in the latter treatment.

Two hypotheses may explain the discrepancy between the experiments at low and high densities. First, at a low initial prey density (550 cells.mL -1 ), there is no effect of prey density on its own dispersal (see experiment 1, Fig. 1a). The dispersal observed in the presence or simulated presence of the predator is only due to the predator. By contrast, at a high initial prey density (6600 cells.mL -1 ), prey density may have an effect on its own dispersal. Therefore, in the absence of the predator, prey dispersal is high and the effect of a predator (whether real or simulated) on dispersal is reduced in comparison with the prey's intraspecific density effect. This result suggests an upper bound on prey dispersal. Second, the discrepancy between the two experiments might be a consequence of different predator densities in experiments 5 (63.5 cells.mL -1 ) and 6 (22.1 cells.mL -1 ). However, these two densities are both in the range of predator densities that induce prey dispersal (see experiment 4, Fig. 1b). Therefore the latter hypothesis is not supported by our data. [START_REF] Travis | Density-dependent dispersal in host-parasitoid assemblages[END_REF], and in (b) [START_REF] Bernstein | Prey and predator emigration responses in the acarine system Tetranychus urticae-Phytoseiulus persimilis[END_REF] and present experiments.

Implications for prey-predator metacommunities

In a seminal paper, [START_REF] Huffaker | Experimental studies on predation : dispersion factors and predatorprey oscillations[END_REF] showed that prey-predator interactions persist longer in a large fragmented landscape than in a small fragmented landscape or isolated patches. His experiment stimulated theoretical studies that have explicitly addressed the role of spatial heterogeneity in the persistence of prey-predator interactions that are prone to extinction when isolated de Roos, 1991 #29;Hassell, 1991 #30;Sabelis, 1988 #31;Sabelis, 1991 #32}. Several experimental studies showed that individuals' migration between local communities allows regional persistence because of the asynchrony of local dynamics (Holyoak & Lawler 1996a;[START_REF] Janssen | Metapopulation dynamics of a persisting predator-prey system in the laboratory: Time series analysis[END_REF][START_REF] Taylor | Metapopulations, Dispersal, and Predator-Prey Dynamics: An Overview[END_REF][START_REF] Van De Klashorst | A demonstration of asynchronous local cycles in an acarine predator-prey system[END_REF].

Theoretical studies focused on the essential role of dispersal intensity in prey-predator metacommunities [START_REF] Crowley | Dispersal and the stability of predator-prey interactions[END_REF]Nachman 1987a, b;[START_REF] Reeve | Environmental variability, migration, and persistence in host-parasitoid systems[END_REF][START_REF] Zeigler | Persistence and patchiness of predator-prey systems induced by discrete event population exchange mechanisms[END_REF]. These models (reviewed in (Holyoak & Lawler 1996a, b) predict that an intermediate dispersal level of prey and predator enables metacommunity persistence. A low dispersal rate reduces the probability of recolonization of locally extinct patches and cannot prevent local extinctions, whereas a high dispersal rate tends to synchronize local dynamics [START_REF] Brown | Turnover Rates in Insular Biogeography : Effect of Immigration on Extinction[END_REF][START_REF] Levins | Extinction. In Some mathematical questions in biology[END_REF][START_REF] Yodzis | The Indeterminacy of Ecological Interactions as Perceived through Perturbation Experiments[END_REF]. Experiments have confirmed that moderate dispersal extends the persistence of prey-predator systems (Holyoak & Lawler 1996a, b). However, in these theoretical studies the dispersal ability of species from one patch to another is regarded as an unconditional process described by a single parameter.

Our results add to the body of experiments [START_REF] Bernstein | Prey and predator emigration responses in the acarine system Tetranychus urticae-Phytoseiulus persimilis[END_REF]French & Travis 2001; for review see Matthysen 2005) that show that dispersal is density-dependent, and hence that regional processes depend upon local population dynamics. This strong interaction between local and regional processes is likely to affect the dynamics and stability of communities and metacommunities.

Recent models that incorporate density-dependent dispersal behaviour show different community-level effects of dispersal (reviewed in [START_REF] Bowler | Causes consequences of animal dispersal strategies: relating individual behavior to spatial dynamics[END_REF]. Most of these models explored the effects of intraspecific density-dependent dispersal on the stability of single-species metapopulations. Models that incorporate positive densitydependent dispersal behaviour, as we showed here with T. pyriformis, have found a stabilizing effect of dispersal on population dynamics, whereas models that have simpler dispersal rules do not observe stabilizing effects [START_REF] Janosi | On the Evolution of Density Dependent Dispersal in a Spatially Structured Population Model[END_REF][START_REF] Ruxton | Fitness-dependent dispersal in metapopulations and its consequences for persistence and synchrony[END_REF]; but see [START_REF] Ruxton | Density-dependent migration and stability in a system of linked populations[END_REF]. Other models have shown that the form of the relationship between dispersal and density is important for predicting its consequences for stability [START_REF] Amarasekare | Interactions between local dynamics and dispersal: Insights from single species models[END_REF][START_REF] Ruxton | Density-dependent migration and stability in a system of linked populations[END_REF][START_REF] Ylikarjula | Effects of Patch Number and Dispersal Patterns on Population Dynamics and Synchrony[END_REF].

The effects of interspecific density-dependent dispersal on the stability of preypredator metacommunities are still unclear. French & Travis (2001) parameterized a model and found no differences in species persistence and community dynamics between a fixed mean dispersal and interspecific density-dependent dispersal for the predator (parasitoid).

By contrast, taking into account intra-and interspecific density-dependent dispersal improves the ability of prey-predator metacommunity models to predict metacommunity dynamics in experiments [START_REF] Bernstein | A Simulation Model for an Acarine Predator-Prey System (Phytoseiulus persimilis-tetranychus urticae)[END_REF][START_REF] Ellner | Habitat structure and population persistence in an experimental community[END_REF]Nachman 1987a, b). Thus, density-dependent dispersal may be fundamental for our understanding of prey-predator metacommunity dynamics. At present, several questions remain unanswered. Is there an interaction between the effects of intra-and interspecific density-dependent dispersal on prey-predator metacommunities? Do different density-dependent dispersal patterns (Fig. 3)

have different effects at the metacommunity level? What are the implications of the interaction between local and regional processes for conservation and biological control?

Our microcosm experiments demonstrate that the dispersal of prey and predator protists can depend on both intra-and interspecific density. Our results may be fundamental and general because they were obtained with relatively simple organisms (unicellular eukaryotes). We further show that prey can detect predator presence through organic compounds that the predator releases in the medium. Therefore chemical signals among organisms may play an important role in species dispersal, and density-dependent dispersal may be a pivotal process in metacommunity dynamics. Understanding and testing the effects of density-dependent dispersal on metacommunity dynamics, is a challenge for future studies.

Figure 1 .

 1 Figure 1. Effects of (a) Tetrahymena pyriformis density and (b) Dileptus sp. density on T. pyriformis dispersal rate, and effects of (c) T. pyriformis density and (d) Dileptus sp. density on Dileptus sp. dispersal (mean ± 1 SE). Letters indicate significant differences in dispersal rate among density treatments.

Figure 2 .

 2 Figure 2. Tetrahymena pyriformis detects Dileptus sp. presence through chemical cues (mean ± 1 SE). (a) Low initial density of T. pyriformis; (b) high initial density of T. pyriformis. Letters indicate significant differences in dispersal rate among treatments.

Figure 3 .

 3 Figure 3. Density-dependent dispersal patterns in prey-predator systems. Arrows indicate positive (+) or negative (-) significant effect of density on dispersal observed in (a) French & Travis (2001), and in (b)[START_REF] Bernstein | Prey and predator emigration responses in the acarine system Tetranychus urticae-Phytoseiulus persimilis[END_REF] and present experiments.

We thank M. Huet for her help in maintaining the protist cultures and T. Tully for advice on the statistical analysis. We thank G. Lacroix, S. Leroux and anonymous reviewers for theirs remarks that permit progress of the manuscript. C.H. thanks B. Boublil for his constant support.

ACKNOWLEDGEMENTS REFERENCES


Sylvain Landron

Marie-Bernadette Lepetit

The crucial importance of the t 2g -e g hybridization in transition metal oxides

We studied the influence of the trigonal distortion of the regular octahedron along the (111) direction, found in the CoO2 layers. Under such a distortion the t2g orbitals split into one a1g and two degenerated e ′ g orbitals. We focused on the relative order of these orbitals. Using quantum chemical calculations of embedded clusters at different levels of theory, we analyzed the influence of the different effects not taken into account in the crystalline field theory ; that is metal-ligand hybridization, long-range crystalline field, screening effects and orbital relaxation. We found that none of them are responsible for the relative order of the t2g orbitals. In fact, the trigonal distortion allows a mixing of the t2g and eg orbitals of the metallic atom. This hybridization is at the origin of the a1g-e ′ g relative order and of the incorrect prediction of the crystalline field theory.

I. INTRODUCTION

Since the discovery of super-conductivity in the hydrated Na 0.35 CoO 2 -1.3H 2 O 1 compound and of the very large thermopower in the Na 0.7±δ CoO 2 2 members of the same family, the interest of the community in systems built from CoO 2 layers has exploded. The first step in the understanding of the electronic properties of transition metal oxides, such as the CoO 2 -based compounds, is the analysis of the crystalline field splitting of the d orbitals of the transition metal atom. Indeed, depending on this splitting, the spin state of the atom, the nature of the Fermi level orbitals, and thus the Fermi level properties will differ.

The CoO 2 layers are built from edge-sharing CoO 6 octahedra (see figure 1). In these layers, the first coordina- tion shell of the metal atom differs from the regular octahedron by a trigonal distortion along the three-fold (111) axis (see figure 6). In all known materials (whether cobalt oxides or other metal oxides such as LiVO 2 , NaTiO 2 , NaCrO 2 , etc. . . ), this distortion is in fact a compression. The local symmetry group of the metal atom is lowered from O h to D 3d . The T 2g irreducible representation of the O h group is thus split into one E g and one A 1g representations. The relative energies of the resulting e ′ g and a 1g orbitals (see figure 6) has been a subject of controversy in the recent literature, as far as the low spin Co 4+ ion is concerned. At this point let us point out the crucial importance of the knowledge of this energetic order for the understanding of the low energy properties of the CoO 2 layers. Indeed, the possible existence of an orbital order, as well as the minimal model pertinent for the description of these systems depend on this order.

Authors such as Maekawa 3 , following the crystalline field theory, support that the a 1g orbital is of lower energy than the two degenerated e g ones, leading to an orbital degeneracy for the Co 4+ ion. On the contrary, ab initio calculations, both using periodic density functional methods 4 and local quantum chemical methods for strongly correlated systems 5 yield an a 1g orbital of higher energy than the e ′ g ones, and a non degenerated Fermi level of the Co 4+ ion. Angle Resolved Photoemis- sion Spectroscopy (ARPES) experiments were performed on several CoO 2 compounds 6 . This technique probes the Fermi surface and clearly shows that the Fermi surface of the CoO 2 layers is issued from the a 1g orbitals, and not at all from the e ′ g orbitals (orbitals of E g symmetry, issued from the former t 2g orbitals), supporting the ab-initio results.

In the present work, we will try to understand the reasons why the crystalline field model is unable to find the good energetic order of t 2g orbitals in such trigonal distortions. Several hypotheses can be made to explain the orbital order : the delocalization of the metal 3d orbitals toward the ligands, the fact that the electrostatic potential of the whole crystal differs from the one assumed in the crystalline field model, the correlation effects within the 3d shell, the screening effects, etc. All these hypotheses will be specifically tested on the Co 4+ (3d 5 ) ion that is subject in this work to a more thorough study than other metal fillings. Nevertheless, other metal fillings (3d 1 to 3d 3 , that can be found in vanadium, titanium chromium, . . . oxides) will also be studied. We will see the crucial importance of the band filling on the t 2g orbitals order. In this work we will focus only on the O h to D 3d trigonal distortion, subject of the controversy.

The next section will present the method used in this work, section three and four will reports the calculations and analyze them, finally the last section will be devoted to the conclusion.

II. COMPUTATIONAL METHOD AND DETAILS

The energy of the atomic 3d orbitals is an essentially local value, as supposed in the crystalline field model. However its analysis exhibits some non local contributions. Indeed, orbitals energies can be seen as resulting from the following terms:

• the electrostatic potential due to the first coordination shell -in the present case, the six oxygen atoms of the octahedron, further referred as nearest neighbor oxygens (NNO) -,

• the electrostatic potential due to the rest of the crystal,

• the kinetic energy that includes the hybridization of the metal orbitals with nearest neighbor ligands,

• the Coulomb and exchange contributions within the 3d shell,

• the radial relaxation of the 3d orbitals,

• and finally the virtual excitations from the other orbitals that are responsible for the screening effects.

All these contributions, excepts for the electrostatic potential due to the rest of the crystal (nucleus attractions and Coulomb interactions), are essentially local contributions 7 and known to decrease very rapidly with the distance to the metal atom. In fact, they are mostly restricted to the first coordination shell of the cobalt. On the contrary, the Madelung potential retains the resulting non local contributions from the nucleus attraction and the Coulomb electron-electron repulsion. It is known to be very slowly convergent with the distance. We thus made calculations at different levels, including first all the above effects, and then excluding them one at the time, in order to end up with the sole effects included in the crystalline field model. The calculations will thus be done on CoO 6 or Co fragments. Different embedding and different levels of calculation will be used. The Co -O distance will be fixed to the value of the super-conducing compound, i.e. R Co-O = 1.855 Å. The angle θ between the Co -O direction and the z axis (see figure 6) will be varied from 0 to 90 • . The calculations will be done at the Complete Active Space Self Consistent Field + Difference Dedicated Configurations Interaction 8,9 (CASSCF+DDCI, see subsection II A) level for the most involved case, using the core pseudopotential and basis set of Barandiaran et al. 10 . The fragment used will include all the first coordination oxygens in addition to the cobalt atom. The embedding will be designed so that to properly represent the full Madelung potential of the super-conducting material, and the exclusion effects of the rest of the crystal on the computed fragment electrons (see reference 5 for further details). For the simplest case a minimal basis set derived from the preceeding one will be used and only the cobalt atom will be included in the computed fragment. The effect of the crystalline field will be described by -2 point charges located at the positions of the first coordination shell oxygens. The calculations will be done at the CASSCF level only. Between these two extreme cases, several intermediate ones will be considered, in order to check the previously enumerate points.

The electrostatic potential due to the cobalt first oxygen neighbors (NNO), as well as the unscreened Coulomb and exchange contributions within the 3d shell, are included in all calculations. The electrostatic potential is treated either through the inclusion of the NNO in the computed fragment or through -2 point charges. The Coulomb and exchange contributions are treated through the CASSCF calculation. The electrostatic contribution of the rest of the crystal is included only in the most involved calculations, using an appropriated embedding of point charges and Total Ions pseudo-Potential 11 . The hybridization of the metal 3d orbitals is treated by including explicitely the NNO in the considered fragment (CoO 6 ). The radial relaxation of the 3d orbitals is treated when extended basis set are used. When a minimal basis set is used, the radial part of the orbitals is frozen as in the high spin state of the isolated Co 4+ ion. Finally, the screening effects are treated only when the calculation is performed at the CASSCF+DDCI level.

A. The CASSCF and DDCI methods

Let us now described shortly the CASSCF and DDCI ab initio methods. These methods are configurations interaction (CI) methods, that is exact diagonalization methods within a selected set of Slater's determinants. These methods were specifically designed to treat strongly correlated systems, for which there is no qualitative single-determinant description. The CASSCF method treats exactly all correlation effects and exchange effects within a selected set of orbitals (here the 3d shell of the cobalt atom). The DDCI method treats in addition the excitations responsible for the screening effects on the exchange, repulsion, hopping, etc. integrals. These methods are based on the partitioning of the fragment orbitals into three sets the occupied orbitals that are always doublyoccupied in all determinants of the Complete Active Space or CAS (here the cobalt inner electrons and the NNO ones), the active orbitals that can have all possible occupations and spins in the CAS (here the cobalt 3d orbitals),

the virtual orbitals that are always empty in the CAS.

The CASCI method is the exact diagonalization within the above defined Complete Active Space. The CASSCF method optimizes in addition the fragment orbitals in order to minimize the CASCI wave function energy. This is a mean-field method for the occupied orbitals but all the correlation effects within the active orbitals are taken into account. Finally the DDCI method uses a diagonalization space that includes the CAS, all single-and double-excitations on all determinants of the CAS, except the ones that excite to occupied orbitals into two virtual orbitals. Indeed, such excitations can be shown not to contribute -at the second order of perturbation -to the energy differences between states that differ essentially by their CAS wave function. Therefore, they have little importance for the present work. The DDCI method thus accurately treats both the correlation within the CAS and the screening effects.

Compared to the very popular density functional methods, the CAS+DDCI method presents the advantage of treating exactly the correlation effects within the 3d shell. This is an important point for strongly correlated materials such as the present ones. Indeed, even if the DFT methods should be exact provided the knowledge of the correct exchange-correlation functional, the present functionals work very well for weakly correlated systems, but encounter more difficulties with strong correlation effects. For instance the LDA approximation finds most of the sodium cobaltites compounds ferromagnetic 4 in contradiction with experimental results. LDA+U functionals try to correct these problems by using an ad hoc on-site repulsion, U, within the strongly correlated shells. This correction yields better results, however it treats the effect of the repulsion within a mean field approximation, still lacking a proper treatment of the strong correlation. The drawbacks of the CAS+DDCI method compared to the DFT methods are its cost in term of CPU time and necessity to work on formally finite and relatively small systems. In the present case however, this drawback appear to be an advantage since it decouples the local quantities under consideration from the dispersion problem.

III. RESULTS AND ANALYSIS

Let us first attract the attention of the reader on what is supposed to be the energy difference between the e ′ g and a 1g orbitals of the Co 4+ ion in an effective model. In fact, the pertinent parameters for an effective model should be such that one can reproduce by their means the exact energies or, in the present case, the ab-initio calculation of the different Co 4+ atomic states. It results, that within a Hubbard type model, the pertinent effective orbital energies should obey the following set of equations

E (|a 1g ) = 4ε(e ′ g ) + ε(a 1g ) + 2U + 8U ′ -4J H E |e ′ g = 3ε(e ′ g ) + 2ε(a 1g ) + 2U + 8U ′ -4J H ∆E = E |e ′ g -E (|a 1g ) = ε(a 1g ) -ε(e ′ g )
where the schematic picture of the |e ′ g and |a 1g states is given in figure 3, ε(e ′ g ) and ε(a 1g ) are the effective orbital energies of the e ′ g and a 1g atomic orbitals, U is the effective electron-electron repulsion of two electrons in the same cobalt 3d orbital, U ′ the effective repulsion of two electrons in different cobalt 3d orbitals and J H the atomic Hund's exchange effective integrals within the cobalt 3d shell. g is doubly-degenerated, the hole being located either on the e ′ g1 or on the e ′ g2 orbitals.

A. The reference calculation

The reference calculation includes all effects detailed in the preceding section. For the super-conducting com-pound the effective t 2g splitting was reported in reference 5 to be

∆E = ε(a 1g ) -ε(e ′ g ) = 315 meV
This point corresponds to θ ≃ 61.5 • (that is a value of θ larger than the one of the regular octahedron θ 0 ≃ 54.74 • ) where the crystalline field theory predicts a reverse order between the t 2g orbitals.

B. Screening effects

The effect of the screening on the t 2g orbital splitting can be evaluated by doing a simple CASCI calculation using the same fragment, embedding, basis set and orbitals as the preceding calculation. Without the screening effects, one finds a t 2g splitting of

∆E = ε(a 1g ) -ε(e ′ g ) = 428 meV
Obviously the screening effects cannot be taken as responsible for the qualitative energetic order between the a 1g and e ′ g orbitals.

C. Cobalt 3d -oxygen hybridization

The effect of the hybridization of the cobalt 3d orbitals with the neighboring oxygen ligands can be evaluated by taking out the oxygen atoms from the quantum cluster, and treating them as simple -2 point charges at the atomic locations. The other parameters of the calculation are kept as in the preceding case. The new orbitals are optimized at the average-CASSCF level between the two |e ′ g and the |a 1g states. It results in a t 2g splitting of ∆E = ε(a 1g )ε(e ′ g ) = 40 meV for the super-conducting compound. Again the hybridization of the cobalt 3d orbitals with the neighboring oxygens cannot be taken as responsible for the inversion of the splitting between the a 1g and e ′ g orbitals.

D. Long-range electrostatic potential

The effect of the long-range electrostatic potential can be evaluated by restricting the embedding to the NNO point charges only, that is to the electrostatic potential considered in the crystalline field method. One finds a t 2g splitting of

∆E = ε(a 1g ) -ε(e ′ g ) = 124 meV
Once again the results is positive and thus the long-range electrostatic potential is not the cause of the crystalline field inversion of the t 2g splitting.

E. Orbital radial relaxation

At this point only few effects on top of the crystalline field theory are still treated in the calculation. One of them is the radial polarization effect of the 3d orbitals, that allows their adaptation to the different occupations in the specific |a 1g and |e ′ g states. This polarization is due to the use of an extended basis set. We thus reduce the basis set to a minimal basis set (only one orbital degree of freedom per (n, l) occupied or partially occupied atomic shell). The minimal basis set was obtained by the contraction of the extended one ; the radial part of the orbitals being frozen as the one of the the isolated Co 4+ high spin state. This choice was done in order to keep a basis set as close as possible to the extended one, and because only for the isolated atom all 3d orbitals are equivalent, and thus have the same radial part. One obtains in this minimal basis set a t 2g splitting of

∆E = ε(a 1g ) -ε(e ′ g ) = 41 meV
At this point we computed the effective orbital energies in the sole crystalline field conditions, however the result is still reverse than what is usually admitted within this approximation. Indeed, the Co 4+ ion was computed in the sole electrostatic field of the NNO, treated as -2 point charges, the calculation is done within a minimal basis set, and at the average-CASSCF level.

F. Further analysis

In order to understand this puzzling result, we plotted the whole curve ∆E(θ) (see figure 4) at this level of calculation and analyzed separately all energetic terms involved in this effective orbital energy difference.

One sees on figure 4 that the ∆E(θ) curve is not monotonic, as expected from the crystalline field theory. Indeed, while for θ = 0 the relative order between the a 1g and e ′ g orbitals is in agreement with the crystalline field predictions, for θ = 90 • the order is reversed. One should also notice that, in addition to the θ 0 value of the regular octahedron, there is another value of θ for which the three t 2g orbitals are degenerated. In the physically realistic region of the trigonal distortion (around the regular octahedron θ 0 value) the relative order between the a 1g and e ′ g orbitals is reversed compared to the crystalline field predictions.

Let us now decompose ∆E(θ) into

• its two-electron part within the 3d shell -∆E 2 (θ) -

• and the rest referred as 3d single-electron part -∆E 1 (θ). ∆E 1 includes the kinetic energy, the electron-nucleus and electron-charge interaction, and the interaction of the 3d electrons with the inner shells electrons. FIG. 4: Orbital splitting between the a1g and e ′ g orbitals when only the nearest neighbor ligands electrostatic field is included. The dotted red curve corresponds to the singleelectron part of the orbital energy difference : ∆E1, that is the kinetic energy (equation ( 1)), the electron-charge interaction (equation ( 2)) and the interaction with the core electrons (equation ( 3)) . The dashed green curve corresponds to the two-electron part of the orbital energy difference : ∆E2, that is the repulsion and exchange terms within the 3d shell (equation ( 4)). The solid vertical line points out the regular octahedron θ value and the dashed vertical line the θ value for the super-conducting compound.

One thus has ∆E

= ∆E 1 + ∆E 2 = ε(a 1g ) -ε(e ′ g1 ) = ε(a 1g ) -ε(e ′ g2 )
with

∆E1 = a1g - ∇ 2 2 a1g -e ′ g - ∇ 2 2 e ′ g (1)
+ a1g

N -ZN RN a1g -e ′ g N -ZN RN e ′ g (2)
+ χ : occ 2 a1g χ 1 r12 a1g χ -a1g χ 1 r12 χ a1g - χ : occ 2 e ′ g χ 1 r12 e ′ g χ -e ′ g χ 1 r12 χ e ′ g (3)
and

∆E2 = a1g a1g 1 r12 a1g a1g -e ′ g e ′ g 1 r12 e ′ g e ′ g +2 a1g e ′ g 1 r12 a1g e ′ g -a1g e ′ g 1 r12 e ′ g a1g (4)
- where the equations are given in atomic units. Z N refers to the nucleus charge of the cobalt atom and the -2 point charges located at the NNO positions. R N is the associated electron-charge distance. The sum on χ runs over all the orbitals of the cobalt inner-shells.

Let us now examine the dependence on θ of each of the terms of ∆E 1 and ∆E 2 .

Kinetic energy : the radial part of each of the 3d orbitals being identical due the the minimal basis set restriction, the kinetic part is identical for all 3d orbitals and thus its contribution to ∆E 1 (terms labeled 1 of ∆E 1 ) vanishes.

Nuclear interaction : obviously this contribution to ∆E 1 (terms labeled 2 of ∆E 1 ) strongly depends on θ through the position of the -2 charges.

Interaction with the inner-shells electrons : this term (terms labeled 3 of ∆E 1 ) depends only on the shape of the t 2g and inner-shells orbitals. However, the minimal basis set does not leave any degree of freedom for the relaxation of the inner-shells orbital whose shapes are thus independent of θ.

Similarly, the 3d radial part of the 3d orbitals is totally frozen.

∆E 2 : finally, the dependence of ∆E 2 can only go through the shape of the a 1g and e ′ g orbitals whose radial part is totally frozen due to the use of a minimal basis set.

If one accepts that the a 1g and e ′ g orbitals are issued from the t 2g orbitals of the regular octahedron, their angular form is totally given by the symmetry (see eq. 5, 6) and both ∆E 2 and the third contribution of ∆E 1 should be independent of θ.

e g    e • g1 = 1 √ 3 d xy + √ 2 √ 3 d xz e • g2 = 1 √ 3 d x 2 -y 2 + √ 2 √ 3 d yz (5) t 2g          a • 1g = d z 2 e •′ g1 = √ 2 √ 3 d xy -1 √ 3 d xz e •′ g2 = √ 2 √ 3 d x 2 -y 2 -1 √ 3 d yz (6)
where the x, y and z coordinates are respectively associated with the a, b and c crystallographic axes. Figure 4 displays both ∆E 1 (dotted red curve) and ∆E 2 (dashed green curve) contributions to ∆E. One sees immediately that ∆E 2 is not at all independent of θ but rather monotonically increasing with θ. It results that the above hypotheses of the t 2g exclusive origin for the e ′ g orbitals is not valid. Indeed, out of the θ = θ 0 point, the only orbital perfectly defined by the symmetry is the a 1g orbital. The e ′ g and e g orbitals belong to the same irreducible representation (E g ) and can thus mix despite the large t 2g -e g energy difference. If we name this mixing angle α, it comes Figure 5 displays α as a function of θ. One sees that the t 2g -e g hybridization angle α is non null -except for the regular octahedron -and a monotonic, increasing function of θ. Even if very small (±0.6 • ), this t 2g -e g hybridization has an important energetic effect, since it lowers the the e ′ g orbital energy while increasing the e g one. α is very small but it modulates large energetic factors in ∆E 2 : on-site Coulomb repulsions of two electrons in the 3d orbitals. The result is a monotonic increasing variation of ∆E 2 as a function of θ. The variation of the ∆E 1 term is dominated by its nuclear interaction part and exhibits a monotonic decreasing variation as a function of θ, as expected from the crystalline field theory. The nuclear interaction and t 2g -e g hybridization have thus opposite effects on the a 1g -e ′ g splitting. The failure of the crystalline field theory thus comes from not considering the t 2g -e g hybridization.

In the calculations presented in figures 4 and 5, the screening effects on the on-site Coulomb repulsions and exchange integrals were not taken into account. Thus, the absolute value of ∆E 2 as a function of the hybridization α, is very large and α is very small. When the screening effects are properly taken into account, the absolute value of ∆E 2 as a function of α is reduced by a factor about 6, and the t 2g -e g hybridization is much larger than the values presented in figure 5. Indeed, in the superconducting compound, for a realistic calculation including all effects, one finds α ≃ 13 • (θ = 61.5 • ).

At this point we would like to compare the a 1g -e ′ g splitting found in the present calculations and the one found using DFT methods. Indeed, our splitting (315 meV for the superconducting compound) is larger than the DFT evaluations (always smaller < 150 meV). This point can be easily understood using the single-electron and two-electron part analysis presented above. Indeed, while the single-electron part is perfectly treated in DFT calculations, the two-electron part is treated within the exchange-correlation kernel. However these kernels are well known to fail to properly reproduce the strong correlation effects present in the transition metal opened 3d shells. One thus expect that while the singleelectron part of the atomic orbital energies is well treated, the two-electron part is underestimated, resulting in an under-evaluation of the a 1g -e ′ g splitting, as can be clearly seen from figure 4.

IV. OTHER CASES

We considered up to now a Co 4+ ion, that is five electrons in the 3d shell, and a fixed metal-ligand distance, R M-O . Let us now examine the effect of the distance RM -O and the band filling on the a 1g -e ′ g splitting. The calculations presented in this section follow the same procedure as in sections III E, III F. For different fillings a typical example in the transition metal oxides family was used to define the type of metallic atom and metal oxygen distances. Minimal basis set issued from full contraction of the basis set given in reference 10 will be used. sees immediately that despite the large variation of the metal-ligand distance, the relative order of the a 1g and e ′ g orbitals remains identical. The main effect of RM -O is thus to renormalize the amplitude of the splitting, low-ering the splitting for larger distances and increasing it for smaller ones.

A. The effect of the Co-O distance

B. 3d 1

The simplest filling case corresponds to only one electron in the 3d shell. This is, for instance, the case of the NaTiO 2 compound. The calculations were done using the average Ti-O distance found in NaTiO 2 12 : R Ti-O = 2.0749 Å.

In this case, ∆E 2 = 0 and ∆E(θ) = ∆E 1 (θ) behaves as pictured in figure 4. The a 1g orbital is of lower energy than the e ′ g for θ > θ 0 and of higher energy for θ < θ 0 . This result is in perfect agreement with the crystalline field theory.

C. 3d 2

A simple example of the 3d 2 filling in transition metal oxides is the LiVO 2 compound. Indeed, the vanadium atom is in the V 3+ ionization state. We thus used a metal oxygen distance of R V-O = 1.9787 Å13 . Figure 7 displays the a 1g -e ′ g splitting as well as its decomposition into the single-electron and two-electron parts. As in the FIG. 7: Orbital splitting between the a1g and e ′ g orbitals for a 3d 2 transition metal. Only the nearest neighbor ligands electrostatic field is included in the calculation. The dotted red curve corresponds to the single-electron part of the orbital energy difference : ∆E1, that is the kinetic energy (equation (1)), the electron-charge interaction (equation ( 2)) and the interaction with the core electrons (equation ( 3)) . The dashed green curve corresponds to the two-electron part of the orbital energy difference : ∆E2, that is the repulsion and exchange terms within the 3d shell (equation ( 4)). 3d 5 case (figure 4), the single-electron and two-electron parts behave in a monotonic way as a function of θ, and in an opposite manner. In the present case, however, the two-electron part always dominates over the one-electron part and the a 1g -e ′ g orbital splitting is always reversed compared to the crystalline field predictions. As for the 3d 5 system, there is a slight e ′ g -e g hybridization that is responsible for the t 2g orbitals order.

D. 3d 3

Examples of 3d 3 transition metal oxides are found easily in the chromium compounds. Let us take for instance the NaCrO 2 system 14 . The metal oxygen distance is thus : R Cr-O ≃ 1.901 Å. Figure 8 displays the a 1g -e ′ g orbital splitting as well as its decomposition into singleand two-electron parts. As usual the single-electron part

0 20 40 60 80 θ ο ∆E = ε(a 1g ) - ε(e g ') 2-elec. part 1-elec. part ε(a 1g ) -ε(e g ')
t 2g orbital splitting 3d 3 system FIG. 8: Orbital splitting between the a1g and e ′ g orbitals for a 3d 3 transition metal. Only the nearest neighbor ligands electrostatic field is included in the calculation. The dotted red curve corresponds to the single-electron part of the orbital energy difference : ∆E1, that is the kinetic energy (equation (1)), the electron-charge interaction (equation ( 2)) and the interaction with the core electrons (equation ( 3)) . The dashed green curve corresponds to the two-electron part of the orbital energy difference : ∆E2, that is the repulsion and exchange terms within the 3d shell (equation ( 4)).

and the two-electron part are monotonic as a function of θ but with slopes of opposite signs. This case is quite similar to the 3d 5 case since none of the single-and twoelectron parts dominates the t 2g orbital splitting over the whole range. Indeed, for small values of θ, the crystalline field effect dominates and the a 1g orbital is above the e ′ g ones while, for large values of θ, the two-electron part dominates and the a 1g orbital is again above the e ′ g ones. In a small intermediate region the order is reversed. In the realistic range of θ (θ ≃ θ 0 ) there is a strong competition between the two effects (quasi-degeneracy of the a 1g and e ′ g orbitals) and no simple theoretical prediction can be made. The crystalline field theory is not predictive but the present calculations cannot be considered as predictive either, since all the neglected effects may reverse the a 1g -e ′ g order.

V. DISCUSSION AND CONCLUSION

In the present work we studied the validity of the crystalline field theory under the application of a trigonal distortion on the regular octahedron. Under such a distortion, the T 2g irreducible representation (irrep) of the O h group spits into A 1g and E g irreps (T 2g -→ A 1g ⊕E g ), while the e g irrep remains untouched (E g -→ E g ). The hybridization between the t 2g and e g orbitals thus become symmetry allowed, even if hindered by energetic factors. This hybridization is not taken into account in the crystalline field theory. It is however of crucial importance for the relative order between the former t 2g orbitals and the reason of the failure of the crystalline field theory to be predictive. Indeed, due to the t 2g -e g orbitals hybridization, the two-electron part of the e ′ g orbital energy becomes dependant of the amplitude of the distortion and of opposite effect to the single-electron part. The relative order of the t 2g orbitals thus depends on the competition between these two effects and as a consequence of the band filling.

In this work we studied the O h to D 3d distortion, however one can expect similar effects to take place for other distortions of the regular octahedron. The condition for these effects to take place is that the T 2g irreducible representation splits into a one-dimensional irrep (A) and the same two-dimensional irrep (E) as the one the e g orbitals are transformed to

T 2g -→ A ⊕ E E g -→ E
Indeed, under such a distortion, t 2g -e g hybridization phenomena are allowed. The distortion should thus transform O h into sub-groups that keep the C 3 (111) symmetry axis : C 3 , C 3v , D 3 , S 6 and D 3d . Examples such deformations are the elongation of the metal-ligand distance of one of the sets of three symmetry related ligands, or the rotation of such a set three ligands around the (111) symmetry axis. For instance, one will expect that t 2g -e g hybridization will also take place in trigonal prismatic coordination.

However, in real systems like the sodium cobaltites, these distortion do not usually appear alone but rather coupled. For instance, in the squeezing of the metal layer between the two oxygen layers observed as a function of the sodium content in Na x CoO 2 , the Co-O bond length and the three-fold trigonal distortion are coupled. Since this composed distortion belongs to the above-cited class, the t 2g -e g hybridization will take place and the relative orbital order between the a 1g and e ′ g orbitals will be qualitatively the same as in figure 4. The bond length modification at equal distortion angle, θ, will only change the quantitative value of the orbital splitting, but not its sign. A bond elongation reduces the splitting a bond compression increases it. One can thus expect in sodium cobaltites that the a 1g -e ′ g orbital energy splitting will decrease with increasing sodium content. The reader should however have in mind that the effects of this splitting reduction will remain relatively small compared to the band width as clearly seen in reference 17 . In fact, one can expect that a large effect will be the modification of the band dispersion due not only to the bond length modification, but also to the t 2g -e g hybridization.

FIG. 1 :

 1 FIG. 1: Schematic representation of the CoO2 layers.

FIG. 2 :arccos 1 √ 3 ≃

 213 FIG. 2: Schematic representation of cobalt 3d splitting. θ represents the angle between the z axis -the 3-fold (111) axis of the CoO6 octahedron -and the Co -O direction. θ0 = arccos 1 √ 3 ≃ 54.74 • is the θ angle for the regular octahedron.

FIG. 3 :

 3 FIG.3: Schematic representation of the Co 4+ states of interest. Let us point out that |e ′ g is doubly-degenerated, the hole being located either on the e ′ g1 or on the e ′ g2 orbitals.

  . part ε(a 1g ) -ε(e g ')

FIG. 5 :

 5 FIG.5: t2g-eg hybridization angle under the trigonal distortion.

Figure 6 FIG. 6 :

 66 Figure 6 displays the a 1g -e ′ g energy splitting as a function of the distortion angle θ and for different distances. The range of variation : from 1.8 Å to 1.95 Å, includes all physically observed distances in CoO 2 layers. One

  . part ε(a 1g ) -ε(e g ')

Acknowledgments

The authors thank Jean-Pierre Doumerc and Michel Pouchard for helpful discussions and Daniel Maynau for providing us with the CASDI suite of programs. These calculations where done using the CNRS IDRIS computational facilities under project n • 1842.


C Hausswirth

C Caillaud

R Lepers

J Brisswalter
email: brisswalter@univ-tln.fr

Influence d'une supplémentation en vitamines sur le rendement de la locomotion après une épreuve d'ultratrail Influence of a vitamin supplementation on locomotion gross efficiency after an ultra-trail race

Keywords: Rendement, Exercice de longue durée, Vitamines, Altération musculaire Gross efficiency, Long duration exercise, Vitamin, Muscle damage

Objectifs. -Le but de ce travail était d'étudier l'importance de la variation du rendement de la locomotion à la suite d'une d'épreuve d'ultratrail. Le second objectif était d'étudier l'effet sur le rendement d'une stratégie de supplémentation avant l'exercice en vitamines selon des doses et une composition correspondantes aux apports nutritionnels conseillés (ANC).

Sujets et méthodes. -Vingt-deux sujets bien entraînés en endurance ont réalisé quatre tests de mesure du rendement avant, 24, 48 et 72 heures après une épreuve de type ultra-« trail » (3000 m en montée suivis de 3000 m en descente) ainsi que quatre tests de mesure de la force maximale volontaire aux mêmes périodes. Ces sujets étaient divisés selon une méthode en double insu en deux groupes expérimentaux (avec ou sans apport nutritionnel en vitamines et micronutriments, Isoxan Endurance ® ).

Résultats.

-Dans les deux groupes on a observé une diminution du rendement de la locomotion 24 et 48 heures après la course (respectivement entre le prétest et 24 heures après : 20,02 ± 0,2 vs 19,4 ± 0,1 %, p < 0,05) et une diminution de la force maximale volontaire immédiatement après l'épreuve. Dans ce cadre, la diminution du rendement, 24 heures après la course est significativement moins importante dans le groupe avec apport nutritionnel.

Conclusion. -Les résultats de cette étude confirment la diminution du rendement à la suite d'un exercice de longue durée observée classiquement dans la littérature. Dans notre étude, l'apport en vitamines et micronutriment est associé à une moindre diminution du rendement et de la force maximale volontaire postexercice suggérant un possible effet de cet apport sur la fonction musculaire. Des travaux ultérieurs devront tester l'effet de ce type d'apport sur une moindre altération de la fonction musculaire notamment à la suite d'exercices excentriques.

Introduction

La dernière décennie a vu l'émergence et le développement chez des pratiquants de différents niveaux d'entraînement d'activités physiques d'endurance de très longue durée (supérieur à cinq heures) dans des profils de terrain et de dénivelés variés « ultra-trails ». Dans ce cadre, comme pour toute activité de longue durée la capacité de l'athlète à dépenser le moins d'énergie pour un même niveau de puissance fournie (rendement) est un facteur de la performance sportive [START_REF] Prampero | Energetics of muscular exercise[END_REF][START_REF] Hausswirth | Le coût énergétique de la course de durée prolongée : étude des paramètres d'influence[END_REF]. La variation du rendement de la locomotion avec la durée de l'exercice a bien été décrite dans la littérature. Pour des exercices d'une durée supérieure à une heure, avec l'apparition de phénomènes de fatigue centrale et périphérique, une diminution du rendement est systématiquement décrite [e.g. , 8]. Plusieurs facteurs impliqués sont cités comme responsables de cette altération tels que la variation de la mobilisation des substrats énergétiques, le stress thermique et la régulation des électrolytes de l'organisme, l'altération de la fonction musculaire, liés à la surcharge de travail notamment de type excentrique ou encore la modification du patron locomoteur.

La dépense énergétique importante (supérieure à 3000 kcal/ j) lors de ce type d'épreuve s'accompagne de la nécessité pour l'athlète d'associer à sa préparation une stratégie d'apport énergétique exogène et de contrôler la composition alimentaire en macro-et micronutriments de ces apports [START_REF] Bigard | Nutrition du sportif[END_REF]. Par ailleurs, dans les épreuves de type « trail » les variations de déclivité et de nature de terrain augmentent la part des contractions excentriques et les risques de microlésions musculaires. Dans ce cadre, il est à présent bien établi dans les épreuves d'endurance que l'augmentation de la consommation d'oxygène et des dommages musculaires se traduisent par l'apparition d'un stress oxydatif néfaste pour l'organisme notamment chez le sujet peu entraîné. Ainsi, des travaux récents se sont intéressés à l'influence de certaines vitamines sur ce stress oxydatif, les résultats semblent suggérer une influence positive de plusieurs vitamines (E et C) sur la capacité antioxydante et un possible effet de cette supplémentation sur l'altération musculaire lors du travail excentrique [START_REF] Maxwell | Changes in plasma antioxydant status during eccentric exercise and the effect of vitamin supplementation[END_REF].

Dans ce cadre, le premier objectif de ce travail est d'observer l'importance de la variation du rendement de la locomotion lors de ce type particulier d'épreuve d'ultratrail. Le second objectif est d'étudier un possible effet bénéfique sur cette variation du rendement d'une stratégie de supplémentation avant l'exercice en vitamines selon des doses et une composition correspondantes aux apports nutritionnels conseillés (ANC) pour la population sportive [START_REF] Martin | Apports nutritionnels conseillés pour la population française[END_REF].

Méthodes

Sujets

Vingt-deux sujets bien entraînés en endurance ont participé à ce travail (âge : 40 ± 1,9 ans, taille : 177 ± 1,3 cm, masse corporelle : 70,4 ± 1 kg. Au cours des deux mois précédant les tests, leur volume d'entraînement hebdomadaire comprenait en moyenne 76 km par semaine. Tous les sujets étaient habitués aux épreuves sur ergocycle en laboratoire. Ils ont rempli un consentement écrit après avoir été informés en détail des procédures de l'expérimentation et cette étude a été agréée par le comité d'éthique pour la protection des individus (Saint-Germain-en-Laye, France)

Protocole expérimental 2.2.1. Test progressif maximal

La première épreuve réalisée par tous les sujets était un test progressif maximal de détermination de la consommation maximale d'oxygène (O 2max ) réalisé sur ergocycle un mois avant l'épreuve d'ultratrail. Après un échauffement de six minutes à 100 W, l'intensité mécanique était augmentée de 30 W par minute, jusqu'à ce que le sujet ne puisse plus maintenir la puissance imposée. Les critères d'atteinte de O 2max étaient les suivants : un plateau de O 2 malgré l'augmentation de la puissance, une fréquence cardiaque (FC) supérieure à 90 % de la FC max théorique et un quotient respiratoire (QR) supérieur à 1,15 [START_REF] Howley | Criteria for maximal oxygen uptake: review and commentary[END_REF]. À partir des valeurs de débit ventilatoire (E), de consommation d'oxygène (O 2 ) et de production de dioxyde de carbone (CO 2 ) le seuil ventilatoire (SV) était déterminé selon la méthode décrite par [START_REF] Wasserman | Anaerobic threshold and respiratory gas exchange during exercise[END_REF] [START_REF] Wasserman | Anaerobic threshold and respiratory gas exchange during exercise[END_REF]. Lors de cette première épreuve les sujets étaient également familiarisés à un test d'évaluation de la force maximale isométrique des membres inférieurs (FMIV). Lors de ce test l'angle de flexion du genou était fixé à 100 degrés. Chaque contraction maximale était maintenue deux à trois secondes.

Protocole de supplémentation

À la suite du premier test, les sujets ont été divisés en deux groupes d'aptitude aérobie identique et l'apport en vitamines et micronutriments (Isoxan Endurance ® , NHS, Rungis, France) a été randomisé selon une procédure en double insu avec un groupe supplémenté (Iso) et un groupe placebo (Pla). Le traitement a débuté 21 jours avant l'épreuve et a pris fin deux jours après la fin de la course. La composition et les doses en Isoxan Endurance ® étaient conformes aux apports nutritionnels conseillés pour les sportifs.

Tests sous-maximaux d'évaluation du rendement de la locomotion

Le rendement de la locomotion a été évalué lors d'un exercice de pédalage de six minutes sur ergocycle réalisé à une intensité de 100 W (inférieure au seuil ventilatoire pour l'ensemble des sujets) suivi de dix minutes à l'intensité correspondante au seuil ventilatoire. Ces tests ont été réalisés au cours de quatre sessions expérimentales respectivement avant (préexercice), puis 24, 48 et 72 heures après la course (post-24, post-48, post-72). Dix minutes après chaque session, les sujets réalisaient un test de FMVI.

Description de la course

La course a eu lieu à La Plagne sur un parcours totalisant 3000 m de dénivelé positif suivi de 3000 m de dénivelé négatif. Les temps moyens réalisés par les sujets lors de cette course étaient de six heures 34 ± 49 minutes soit une vitesse moyenne de 8,4 km/h. Le rendement mécanique global du cyclisme (en pourcentage) était calculée comme le rapport entre le travail mécanique accompli par minute et l'énergie métabolique dépensée par minute [START_REF] Chavarren | Cycling efficiency and pedalling frequency in road cyclists[END_REF].

Matériel et mesures

Cadence de pédalage

Toutes les épreuves de cyclisme se déroulaient sur un ergocycle à résistance électromagnétique de type SRM (Jülich, Welldorf, Allemagne). Cet ergocycle pouvait s'ajuster précisément à leurs caractéristiques anthropométriques grâce à un réglage horizontal et vertical de la selle et du cintre. Son mode de fonctionnement permettait la production d'une puissance constante indépendamment de la cadence de pédalage naturellement adoptée par les sujets [START_REF] Jones | The dynamic calibration of bicycle power measuring cranks[END_REF][START_REF] Jones | Experimental human muscle damge: morphological changes in relation with other indices of damage[END_REF]. La cadence de pédalage (rév/min) était enregistrée en continu pendant toute la durée des épreuves.

Analyse statistique

Pour chaque variable, la valeur moyenne et l'écart-type étaient calculés. L'effet de la période de mesure et du groupe Les différences avec les valeurs de précourse sont significatives : pour p < 0,05.

de supplémentation sur l'ensemble des variables mesurées était analysé par une analyse de variance (Manova) à deux facteurs.

Pour cette analyse les valeurs étaient exprimées en fonction de la valeur enregistrée au préexercice. Puis les différences entre les conditions expérimentales étaient déterminées par un test posthoc de type Newman-Keuls. Le seuil de signification était fixé à p < 0,05.

Résultats

Rendement de la locomotion

Les valeurs de rendement, de ventilation et de cadence de pédalage sont présentées Tableau 1. Chez tous les sujets, on observe une diminution du rendement de la locomotion et une augmentation de la ventilation 24 et 48 heures après la course. En revanche, aucune différence significative n'est observée entre les valeurs de rendement mesurées préexercice et 72 heures après la course (Fig. 1). Enfin, une diminution significative de la cadence de pédalage est observée 24 heures après la course. Lorsque l'on compare les deux groupes expérimentaux la diminution du rendement (Delta rendement) est significativement moindre dans le groupe supplémenté (Iso) comparé au groupe placebo (Pla) 24 heures après la course (Fig. 2).

Force maximale volontaire

Les valeurs de force maximale volontaires diminuent de façon significative après la course dans les deux groupes (respectivement pour Iso et Pla : -36,5 ± 3 % et -36,9 ± 2 %). Une corrélation significative est observée entre la diminution du rendement et celle de la force maximale isométrique (r = 0,978, p < 0,05). Dans ce cadre les valeurs de FMVI du groupe Iso retournent à des valeurs de repos plus rapidement que dans le groupe Pla.

Discussion

Le premier résultat important de cette étude est l'altération du rendement de la locomotion observée 24 et 48 heures après une épreuve de longue durée de type ultratrail. Ces résultats correspondent à ceux classiquement observés dans la littérature depuis une dizaine d'année. [START_REF] Brisswalter | In: Énergie et performance[END_REF]. Plusieurs facteurs explicatifs sont avancés pour expliquer cette variation : d'une part une modification de l'utilisation des substrats avec une métabolisation accrue de substrats lipidiques, d'autre part l'effet du stress thermique et de la déshydratation associée et enfin une altération des propriétés contractiles notamment dans le cadre d'exercices mettant en jeu une part importante de travail excentrique. Dans notre travail d'une durée moyenne de six heures 34 ± 49 minutes, la moitié de l'épreuve se déroulait en descente, nous aurions pu ainsi émettre l'hypothèse d'une altération plus importante du rendement comparée à des épreuves de durée inférieure et se déroulant en terrain plat. Paradoxalement, nous observons une altération moindre (environ 3 %) que celles observées dans la littérature (de 5-7 %), (pour revue, [START_REF] Hausswirth | Le coût énergétique de la course de durée prolongée : étude des paramètres d'influence[END_REF]). Plusieurs facteurs méthodologiques peuvent expliquer cette différence, en particulier le niveau d'intensité de l'exercice qui correspond ici environ à 40 % de VO 2max , par exemple la place du premier test de mesure du rendement situé 24 heures après la course alors que dans les autres études il est mesuré immédiatement après. Par ailleurs, dans ce travail nous n'observons aucune variation du quotient respiratoire entre le test préexercice et celui postcourse. Dans ce cadre nous pouvons émettre l'hypothèse selon laquelle la diminution du rendement observée ici est principalement liée à un effet résiduel d'altération des propriétés contractiles du muscle qui disparaît dans notre étude 72 heures après l'épreuve. La réalisation d'un exercice physique immédiatement après une course de ce type reste difficile ou impossible à étudier dans des conditions réelles de course, aussi les travaux ultérieurs devront essayer d'analyser les effets de la modification des propriétés contractiles à la suite du travail excentrique sur le rendement immédiatement après l'exercice.

Le second résultat intéressant de ce travail est l'effet significatif et bénéfique d'une supplémentation en vitamines et micronutriments sur l'altération du rendement et sur celle de la force maximale volontaire après l'épreuve. À notre connaissance, il n'a été effectué aucune étude concernant les effets d'un apport en vitamines et micronutriments sur les aspects métaboliques de la locomotion, la majeure partie des travaux ayant étudié les effets de cet apport sur la fonction musculaire [e.g. 13]. À ce jour les résultats restent encore peu clairs. Néanmoins, il est classiquement rapporté dans la littérature une altération de la fibre musculaire lors d'exercice excentriques associée à une perte de force [e.g. 7] .La diminution de la force peut atteindre des valeurs de 50 % et le retour à des valeurs normales perdure plusieurs jours après l'exercice [START_REF] Mackey | Skeletal muscle collagen contents in humans following high force eccentric contractions[END_REF]. Plusieurs facteurs explicatifs semblent impliqués dans cette altération musculaire lors de l'exercice prolongé notamment la production de radicaux libres ou stress oxydatif liés, d'une part à une consommation d'oxygène importante, d'autre part aux microlésions musculaires induites par l'exercice notamment excentrique [pour revue, 1]. Dans ce cadre, il a été proposé une action bénéfique d'un apport en vitamines (notamment C et E) sur ce stress oxydatif. À ce jour, bien que les résultats expérimentaux ayant tenté de valider ces hypothèses restent encore contradictoires, et malgré les conditions de notre étude en situation réelle qui nous limitait à une approche descriptive nous pouvons émettre l'hypothèse selon laquelle chez les sujets du groupe ayant pris un apport en vitamines, la moindre altération de la fonction musculaire a permis également de minimiser la diminution du rendement de la locomotion.

Conclusion

Les résultats de cette étude confirment et précisent les résultats présentés dans la littérature sur la diminution du rendement à la suite d'un exercice de longue durée. Un résultat intéressant de ce travail est la relation significative entre la diminution de la force maximale volontaire observée après l'étude et celle du rendement de la locomotion. Dans le cadre de cette étude descriptive nous observons un effet d'un apport en vitamines et micronutriments sur cette relation. Des travaux ultérieurs portant sur la nature de cet effet, notamment en prenant en compte un possible effet sur le stress oxydatif sont nécessaires pour préciser l'intérêt d'un apport en vitamines sur l'adaptation physiologique dans ce type d'épreuve.

2. 3 . 1 .

 31 Mesure des paramètres ventilatoires et gazeux La fréquence cardiaque était enregistrée en continu pendant la course grâce à un cardiofréquencemètre (POLAR vantage, Finlande). Pendant les épreuves sur ergocycle, la consommation d'oxygène (O 2 ), la fréquence cardiaque (FC) ainsi que les paramètres respiratoires (débit ventilatoire :E, fréquence respiratoire : FR) étaient enregistrés en continu par un système d'analyse télémétrique de type Cosmed K4b 2 (Rome, Italie) validé par Mc Laughlin et al. (2001) [9]. Pour chaque paramètre, une valeur de la moyenne et de l'écart-type étaient calculées entre la troisième et la dixième minute d'exercice.

Fig. 1 .

 1 Fig. 1. Variation du rendement de la locomotion après la course d'ultratrail.

Fig. 2 .

 2 Fig. 2. Comparaison de la variation du rendement (delta rendement) entre les deux groupes expérimentaux (Iso vs Pla).

Remerciements

Cette étude a bénéficié du support des laboratoires NHS (Rungis, France). Nous remercions également les docteurs P. Le Van, J.M. Vallier, E. Joussellin, ainsi que C. Bernard pour leur aide lors de la réalisation de ce projet.

Références


R Lepers

G Y Millet

N A Maffiuletti

C Hausswirth

J Brisswalter

Effect of pedalling rates on physiological response during endurance cycling

Keywords: Cadence, Oxygen uptake, Triathletes, Fatigue

This study was undertaken to examine the effect of different pedalling cadences upon various physiological responses during endurance cycling exercise. Eight well-trained triathletes cycled three times for 30 min each at an intensity corresponding to 80% of their maximal aerobic power output. The first test was performed at a freely chosen cadence (FCC); two others at FCC-20% and FCC +20%, which corresponded approximately to the range of cadences habitually used by road racing cyclists. The mean (SD) FCC, FCC-20% and FCC + 20% were equal to 86 (4), 69 (3) and 103 (5) rpm respectively. Heart rate (HR), oxygen uptake (VO2), minute ventilation (VE) and respiratory exchange ratio (R) were analysed during three periods: between the 4th and 5th, 14th and 15th, and 29th and 30th min. A significant effect of time (P < 0.01) was found at the three cadences for HR, VO 2 . The V E and R were significantly (P < 0.05) greater at FCC + 20% compared to FCC-20% at the 5th and 15th min but not at the 30th min. Nevertheless, no significant effect of cadence was observed in HR and VO 2 . These results suggest that, during high intensity exercise such as that encountered during a time-trial race, well-trained triathletes can easily adapt to the changes in cadence allowed by the classical gear ratios used in practice.

Introduction

During training or racing, experienced cyclists or triathletes usually select a relative high pedalling cadence, close to 80-90 rpm. The reasons behind the choice of such a cadence are still controversial and are certainly multi-factorial. Several assumptions relating to neuromuscular, biomechanical or physiological parameters have previously been proposed.

The concept of a most economical cadence is generally supported by experiments where cadences have been varied from the lowest to the highest rates and a parabolic oxygen uptake (V0 2 )-cadence relationship has been obtained. Nevertheless in reality, extreme cadences such as 50 or 110 rpm are very rarely used by road cyclists or triathletes. Simple observations have shown for example that on a flat road at 40 km•h-1 cadences ranged from 67 rpm with a 53:11 gear ratio (GR) to 103 rpm with a 53:17 GR. During an up-hill climb at 20 km•h-1 , cadences ranged from 70 rpm with a 39:17 GR to 103 rpm with a 39:25 GR. Thus, the range of cadences adopted by cyclists using these common GR may vary from 70 to 100 rpm, which corresponds to approximately 85 rpm ± 20%.

The effect of exercise duration upon cycling cadence has not been well studied. The freely chosen cadence (FCC) has seemed to be relatively stable during high intensity cycling exercise of 30 min duration [START_REF] Brisswalter | Energetically optimal cadence vs. freely-chosen cadence during cycling : effect of exercise duration[END_REF] but the FCC was found to decrease during 2 h of cycling at submaximal intensity [START_REF] Lepers | Evidence of neuromuscular fatigue after prolonged cycling exercise[END_REF]. In a non fatiguing situation, the FCC is known to be higher than the most economical cadence. However, a shift in the energetically optimal rate during exercise towards the FCC has recently been reported by [START_REF] Brisswalter | Energetically optimal cadence vs. freely-chosen cadence during cycling : effect of exercise duration[END_REF]. These observations raise a question concerning the choice of any particular GR by road racing cyclists and thus of a pedalling rate and the physiological consequences of this choice for exercise duration .

Therefore, the purpose of this study was to investigate whether the use of cadences 20% lower or higher than the freely chosen one during high intensity endurance exercise induced different changes in metabolic parameters as fatigue occurred.

Methods

Subjects

Eight well-trained male triathletes volunteered to participate in this study. The physical characteristics of the subjects are given in Table 1. They were informed in detail of the experiment and gave written consent prior to all tests.

Experiment procedures

Each subject completed four tests during a 3 week period. Each session was separated by at least 72 h. AH experiments were conducted using an electromagnetically braked cycle ergometer (Type Excalibur, Lode, Groningen, The Netherlands) of which the seat and handlebars are fully adjustable to the subject's dimensions. The ergometer was also equipped with racing pedals, and toe clips allowing the subjects to wear cycling shoes. The first session was used to determine the maximal oxygen uptake (VO2max) of the subjects. The V0 2max test began with a warm-up at 100 W lasting 6 min, after which the power output was increased by 25 W every 2 min until the subjects were exhausted. The three other sessions were composed of a 10 min warm-up ride followed by a 30 min submaximal test at 80% of the highest power sustained for 2 min (P max ). The first of these three sessions was performed at the FCC which corresponded to the cadence that the subjects spontaneously adopted within the first 5 min. During the last 25 min of this test, subjects were asked to maintain a similar cadence. For the two other tests, subjects rode in a random order at FCC-20% or FCC + 20%. The heart rate (HR) was monitored continuously, and gas exchanges were collected at three periods: between the 4th-5th (period 1), the 14th-15th (period 2), and 29th-30th (period 3) min. The HR, VO 2 , minute ventilation (V E ) and respiratory exchange ratio (R) for these three periods were analysed.

Statistical analysis

A two-way ANOVA (time x cadence) was performed using HR, VO2, VE and R as dependent variables. When a significance of P < 00.05 was obtained using the ANOVA, Tukey post-hoc multiple comparisons were made to determine differences either among pedal rates or among periods.

Results

Mean (SD) FCC were 86 (4) rpm, therefore FCC-20% and FCC + 20% corresponded to 69 (3) and 103 (5) rpm, respectively (Table 1).

A significant time effect (P <0.01) was found at the three cadences in HR, VE (Table 2). The rise in VO 2 between the 5th and the 30th min corresponded to 11.0 (7.4)%, 10.3 (6.9)% and 9.9 (3.7)% at FCC-20%, FCC and FCC + 20%, respectively. Between the 5th and the 30th min VE increased by 35.4 (17.4)%, 28.7 (10.9)% and 21.2 (5.2)% at FCC-20%, FCC and FCC +20%, respectively. No significant differences appeared among the three cadences.

A significant effect of cadence was found in V E and R in the first part of the exercise (Table 2). Posthoc tests showed that V E was significantly greater at FCC + 20% compared to FCC-20% at the 5th and 15th min but not at the 30th min. Similarly, R was significantly greater at FCC + 20% in comparison to FCC-20% and FCC at the 5th and 15th min but not at the 30th min. In VO 2 and HR, no significant effect of cadence was observed.

Discussion

The main finding of this study was the absence of significant differences in physiological parameters among the three different pedalling rates (FCC, FCC-20% and FCC + 20%) despite a significant effect of exercise duration.

Increases in HR, VO 2 and V E at the end of 30 min of cycling at 80% of P max observed in this study were similar to previous observations made in well-trained triathletes [START_REF] Brisswalter | Energetically optimal cadence vs. freely-chosen cadence during cycling : effect of exercise duration[END_REF]. Several hypotheses have been proposed to explain the so-called drift in VO 2 at high power outputs, such as an additional oxygen cost of higher rates of V E , increasing muscle and body temperatures and/or changes in muscle activity patterns and/or in fibre type recruitment (for a review, see [START_REF] Whipp | The slow component of 0 2 uptake kinetics during heavy exercise[END_REF]. [START_REF] Barstow | Influence of muscle fiber type and pedal frequency on oxygen uptake kinetics of heavy exercise[END_REF] examined the physiological responses of subjects to intense exercise (half way between the estimated blond lactate threshold and VO 2max ) lasting 8 min for a range of pedalling frequencies between 45 and 90 rpm. Their results showed that the slow composent of VO2 was significantly affected by fibre type distribution but not by pedalling rates. Similarly, [START_REF] Billat | The role of cadence on the slow VO 2 component in cycling and running exercise[END_REF] have shown that for high intensity exercise (95% VO 2max ), a pedalling rate 10% lower than the freely chosen one induced the same VO 2 slow component.

In the present study, in the range of pedalling rates used habitually by road cyclists (from 70 to 100 rpm), no significant effects of cadence were found upon the rises in VO 2 during 30 min of endurance exercise. Also, our data are quite different to those of [START_REF] Brisswalter | Energetically optimal cadence vs. freely-chosen cadence during cycling : effect of exercise duration[END_REF] who examined cadences between 50 and 110 rpm. However, such a discrepancy could be explained by the relatively small range of cadences used in the present study. The only difference observed between cadences in this study occurred in V E and R in the first part of the exercise. High pedalling rates induced greater V E at the 5th and 15th min of exercise, which were associated with higher R values (> 1.0). These data suggest a higher contribution of anaerobic metabolism to power production in the first 15 min at FCC + 20%. Moreover, they corroborate those of [START_REF] Zoladz | Human muscle power generating capability during cycling at different pedalling rates[END_REF] who showed that beyond 100 rpm there is a decrease in external power that can be delivered at a given V0 2 with an associated earlier onset of metabolic acidosis. Importantly, this could be disadvantageous for maintained high intensity exercise. However, in the present study, such a specificity at the highest pedalling rates did not affect the continuation of the exercise since similar values of V E and R were found at the end of the exercise at all three cadences.

The mean cadence spontaneously adopted by the triathletes in this study [86 (4) rpm] corroborated previous results obtained from trained cyclists or triathletes [START_REF] Brisswalter | Energetically optimal cadence vs. freely-chosen cadence during cycling : effect of exercise duration[END_REF][START_REF] Lepers | Evidence of neuromuscular fatigue after prolonged cycling exercise[END_REF]. Although it has been shown that pedalling rates could affect:

1. The maximal power during a 10 s sprint [START_REF] Zoladz | Human muscle power generating capability during cycling at different pedalling rates[END_REF] 2. The power generating capabilities following high intensity cycling exercise close to 90% VO2max [START_REF] Beelen | Effect of prior exercise at different pedalling frequencies on maximal power in humans[END_REF] the reasons behind the choice of a particular cadence during endurance cycling and the corresponding GR by cyclists remain unclear.

We recently showed that cycling exercise at different pedalling rates induced changes in the neural and contractile properties of the quadriceps muscle but no significant effects of cadence were found when considering a range of FCC ± 20% (Lepers et al., in press). Moreover, in the present study the FCC did not appear to be more energetically optimal than FCC-20% or FCC + 20%, either at the beginning or at the end of the exercise. [START_REF] Brisswalter | Energetically optimal cadence vs. freely-chosen cadence during cycling : effect of exercise duration[END_REF] have recently shown that the theoretical energetically optimal pedalling rate, corresponding to the lowest point of the parabolic V02cadence relationship, shifted progressively over the du-ration of exercise towards a higher pedalling rate (from 70 to 86 rpm) which was closer to the freely chosen one. Therefore, a minimisation of energy cost seems not to be a relevant parameter for the choice of cadence, at least in a non fatigued state. Actually, the choice of cadence adopted by cyclists during endurance exercise seems dependent upon factors other than the metabolic cost. Biomechanical and neuromuscular hypotheses have already been proposed to explain the choice of the pedalling rate during short-term high intensity exercise.

However, such interesting hypotheses need to be explored during prolonged exercise.

In conclusion, the results of the present study showed that, for high intensity endurance exercise corresponding to a time trial race for example, the use of cadences in a range corresponding to the classical GR induced similar physiological effects. These data suggest that well-trained triathletes can easily adapt to the changes in cadence used habitually during racing. Further investigations are necessary to target the mechanisms involved in the choice of pedalling rate during prolonged cycling.