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Genetic parameters of royal jelly production and behavioural traits of honey bees

Florence Phocas, C. Le Bihan, B. Basso

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Florence Phocas, C. Le Bihan, B. Basso. Genetic parameters of royal jelly production and behavioural traits of honey bees. 68. The French association of royal jelly producers implemented a breeding scheme at a national scale to improve a honey bee population for traits of economic interest for royal jelly production. This contribution aims to estimate genetic parameters for those traits because selective breeding requires knowledge of heritabilities and genetic correlations between the relevant traits. Data from 817 colonies were collected from 2011 to 2016 in 31 apiaries.

Queens of those colonies were produced by 88 inseminated dam queens. Quantity of royal jelly of the first two harvests in the production season were collected for the 817 colonies and the average performance was the production trait studied (PROD). Three behavioural traits were recorded for 497 to 582 of those colonies at the beginning of the harvest season: gentleness (GENT), swarming (SWAR) and sanitary score (SANI). They were assessed by subjective scoring on a 4-mark scale with the mark 4 being the most favourable one. Genetic parameters were estimated with a multiple trait animal model considering the performance of the colony as a trait of the queen. The bee genetic (male haploïdy) and reproductive (polyandry) specificities were accounted for in the derivation of the relationship matrix. Heritabilities of PROD, GENT, SWAR and SANI were close to 30%, 40%, 40% and 10% respectively. Standard errors of heritability estimates ranged between 0.03 (SANI) to 0.13 (SWAR) while those of the estimates of genetic correlations were higher and ranged between 0.14 and 0.55 due to the small size of the dataset. Even if most of the genetic correlations were not significantly different from zero, a clear tendency toward unfavourable genetic correlations between SANI and all other traits was shown. The two correlations that were significantly different from zero were estimated to -0.71±0.14 between SANI and GENT and to -0.48±0.17 between SANI and PROD. These preliminary results have to be confirmed by a future analysis on a larger dataset. In conclusion, genetic improvement for royal jelly production is possible by selective breeding but attention should be paid not to deteriorate the hygienic behavior of the bees.

Genetic parameters of royal jelly production and behavioural traits of honey bees.

  Genetic parameters of royal jelly production and behavioural traits of honey bees. In: Book of Abstracts of the EAPP annual meeting, Tallinn 2017. INRA, GABI, 78350 Jouy-en-Josas, France GPGR, Agrapole, 23 rue Jean Baldassini, 69 364 Lyon cedex 7, France ITSAP, Domaine Saint Paul, Site Agroparc, 84914 Avignon cedex 9, France

	Presented at 68. Annual Meeting of the European Associationfor Animal Production (EAAP),
	Tallinn, EST (2017-08-28 -2017-08-31).

Annual Meeting of the European Associationfor Animal Production (EAAP), Aug 2017, Tallinn, Estonia. hal-02736436 Comment citer ce document : Phocas, F., Le Bihan, C., Basso, B. (2017). F. Phocas*, C. Le Bihan, B. Basso. *


Damien P Prévéraud

Charles Desmarchelier

David Chapron

Ali Makky

Véronique Rosilio

Patrick Borel

palmitate (vitamin A) display different solubilization capacities in mixed micelle solutions: effect of interactions with mixed micelle components and of cholecalciferol self-associating properties

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membrane proteins (2) enzymatic hydrolysis  Digestion process in the upper gastrointestinal tract (1) :

 Micellization is a key step for bioavailability  but it is also due to the fact that, conversely to RP, D3 can self-associate in structures that are readily soluble in an aqueous phase.

 Whether these D3 self-assemblies are available for absorption by the intestinal cell needs further studies.

Scope

The results obtained show that D3 displays a higher solubility in mixed micelles than RP. This difference is mainly due to the different abilities of the two vitamins to solubilize into mixed micelles based on their interactions with micelle components, but it is also due to the fact that, conversely to RP, D3 can self-associate in structures that are readily soluble in an aqueous phase. Whether these D3 self-assemblies are available for absorption by the intestinal cell needs further studies.

Conclusion

-Vitamin A (as retinyl palmitate, RP) and vitamin D (as cholecalciferol, D3) : the 2 main fat-soluble vitamins found in foods of animal origin. The fate of these vitamins in the human upper gastrointestinal tract during digestion is assumed to follow that of dietary lipids (1) ; this includes emulsification, solubilization in mixed micelles, diffusion across the unstirred water layer and uptake by the enterocyte (2) .

-Solubilization in mixed micelles (micellarization) : a key step for their bioavailability.

-Mixed micelles are mainly made of a mixture of bile lipids (bile salts, cholesterol, phospholipids, lysophospholipid, and their digestion products) and dietary lipids (phospholipids, lysophospholipids, cholesterol, fatty acids and monoglycerides).

-Mixed micelle lipid composition has been shown to significantly affect vitamin solubility (3)(4) . Moreover, the possibility for D3 to form self-aggregates in water (5) has led to question the need of mixed micelles for its solubilization in the aqueous environment of the intestinal tract lumen (6)(7) .

 This study was designed to compare in vitro the relative solubility of D3 and RP in the aqueous phase rich in mixed micelles that exists in the upper intestinal lumen during digestion, and to study by surface tension and surface pressure measurements the interactions between these vitamins and the mixed micelle components. This research may have important implications for understanding the determinants of D3 and RP bioaccessibility.

Methods Results

1 Borel, P., Caillaud, D., and Cano, N. J. ( 2013



  Vitamin A  retinyl palmitate (RP) 1 Borel, P., Caillaud, D. and Cano, N. J. (2013) Crit. Rev. Food Sci. Nutr. 55.; 2 Reboul, E. and Borel, P. (2011) Prog. Lipid Res. 50.

InteractionsFibers

  compare the relative solubility of D3 and RP in the aqueous phase rich in mixed micelles (ii) study the interactions between these vitamins and the mixed micelle components 3 Mixed micelle lipid composition affects the solubility of fat-soluble vitamins: -Substitution of lysophospholipids by phospholipids = lymphatic absorption of vitamin E  in rat (Koo and Noh, 2001 J Nutr) -Change in FA profile (chain length and saturation degree) = absorption of D3  in perfused rat (Hollander, 1978 Gut) -Addition of oleic and linoleic acids = absorption of D3  (Hollander, 1978 Gut) Methods www.fsv-paris2017.com 16th Fat Soluble Vitamins Congress -March 21-23, 2017 -Paris Incorporation efficiency of RP and D3 in mixed micelles Vitamin analysis by HPLC Desmarchelier et al. (2013) MNFR LDP:NaTC 1.1:5 www.fsv-paris2017.com 16th Fat Soluble Vitamins Congress -March 21-23, 2017 -Paris interactions between RP and D3, and the mixed micelle components JEOL 2200FS + CCD camera Copper grid (TedPella) D3 solubilization was linear (R²=0.98, regression slope=0.71) and higher than that of RP, which reached a plateau with a maximum concentration around 125 µM www.fsv-paris2017.com 16th Fat Soluble Vitamins Congress -March 21-23, 2017 LDP showed better surface parameters (CMC and surface tension) than NaTC  LDP aimed to improved NaTC surface tension and CMC by inserting well in NaTC domain  D3 and RP solubilization depend on their capacity to penetrate the LDP and NaTC domains "working" concentration (6.1 mM) www.fsv-paris2017.com 16th Fat Soluble Vitamins Congress -March 21-23, 2017 -Paris 10 Interfacial behaviour of the 2 vitamins and lipids  Based on K max , the lipid monolayers can be classified into poorly organized (K < 100 mN/m, for lyso-PC, monoolein, and oleic acid), liquid condensed (100 < K < 250 mN/m, for POPC and the LDP mixture) and highly rigid monolayers (K > 250 mN/m, for cholesterol)  D3 and LDP exhibit similar interfacial behavior, whereas RP show very different behavior compared to the micelle lipids RP D3 LDP A c : collapse molecular area, π c : collapse surface pressure. K max : maximum compressibility modulus. -paris2017.com 16th Fat Soluble Vitamins Congress -March 21-23, 2017 -Paris 11 Interaction of the 2 vitamins with NaTC and LDP  NaTC micelle barely affect by D3 or RP. D3 and RP insert into NaTC domain (same effect on its organization) D3 conferred a better rigidity of LDP monolayers than RP:  Micelles are more stable  Better capacity to solubilize at low concentration For LDP/D3, surface pressure was controlled by LDP  For LDP/RP, surface pressure was controlled by RP www.fsv-paris2017.com 16th Fat Soluble Vitamins Congress -March 21-23, 2017 -Paris 12 Interaction of the 2 vitamins with mixed micelle  Low impact of vitamin concentration in mixed micelles Is that also true for higher concentration ? www.fsv-paris2017.com 16th Fat Soluble Vitamins Congress -March 21-23, 2017 -Paris 13 cholecalciferol self-associating properties  D3 can self-associate to form micelles at concentrations ≥ 0.1 mM  D3 forms various organized systems at higher concentrations (1 mM): Since (i) D3 an interfacial behavior similar to that of the lipid mixture, and since (ii) it could be solubilized at very high concentrations in an aqueous phase rich in mixed micelles, then its self-assembling properties were more specifically investigated: Conclusion www.fsv-paris2017.com 16th Fat Soluble Vitamins Congress -March 21-23, 2017 -Paris 15 Summary of Findings -Conclusion  D3 displays a higher solubility in mixed micelles than RP… WHY ?  Solubility of vitamins in mixed micelles is function of their interfacial behavior and affinity with micelle components:  Even if [LDP] < [NaTC] in micelle, solubility of RP and D3 is dependent of their capacity to insert into both NaTC and LDP domains:  D3 has similar behavior than LDP, whereas RP is very different  Both RP and D3 can insert into NaTC  the different abilities of the two vitamins to solubilize into mixed micelles is based on their interactions with micelle components,

  XVI th Fat Soluble Vitamins Congress Paris (France), March 21-23, 2017 Cholecalciferol (vitamin D3) and Retinyl Palmitate (vitamin A) display different solubilization capacities in mixed micelle solutions: effect of interactions with mixed micelle components and of cholecalciferol self-associating properties Damien P. Prévéraud 1* , Charles Desmarchelier 2 , David Chapron 3 , Ali Makky 3 , Véronique Rosilio 3 , Patrick Borel 2 1 Adisseo France SAS, Centre of Expertise and Research in Nutrition, Commentry, France; 2 UMR 1260 INRA/1062 INSERM/Aix-Marseille University, "Nutrition, Obesity and Risk of Thrombosis", Marseille, France; 3 University Paris-Sud, CNRS UMR 8612, Institut Galien Paris Sud, Châtenay-Malabry, France; *damien.preveraud@adisseo.com

Figure 1 .Figure 3 .

 13 Figure 1. Solubilization of D 3 () and RP () in mixed micelle solution

Figure 4 .

 4 Figure 4. Interaction of D3 (A) and RP (B) with LDP


Bertrand Cloez

Renaud Dessalles

Alexandre Genadot

Florent Malrieu

Aline Marguet

AND Romain Yvinec

ALEXANDRE GENADOT ‡ , FLORENT MALRIEU § Aline Marguet ⊥

Probalistic and piecewise deterministic models in biology

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Introduction

The piecewise deterministic Markov processes (denoted PDMPs) were first introduced in the literature by Davis [START_REF] Davis | Piecewise-deterministic markov processes: a general class of nondiffusion stochastic models[END_REF][START_REF] Davis | Markov models and optimization[END_REF]. The key point of his work was to endow the PDMP with rather general tools, similar as such that already exist for diffusion processes. Indeed, PDMPs form a family of non-diffusive càdlàg Markov processes, involving a deterministic motion punctuated by random jumps. The motion of the PDMP {X(t)} t≥0 depends on three local characteristics, namely the jump rate λ, the deterministic flow ϕ and the transition measure Q according to which the location of the process at the jump time is chosen. The process starts from x and follows the flow ϕ(x, t) until the first jump time T 1 which occurs either spontaneously in a Poisson-like fashion with rate λ(ϕ(x, t)) or when the flow ϕ(x, t) hits the boundary of the state-space. In both cases, the location of the process at the jump time T 1 , denoted by Z 1 = X(T 1 ), is selected by the transition measure Q(ϕ(x, T 1 ), •) and the motion restarts from this new point as before. This fully describes a piecewise continuous trajectory for {X(t)} t≥0 with jump times {T k } and post jump locations {Z k }, and which evolves according to the flow ϕ between two jumps.

Since the seminal work of Davis, PDMPs have been heavily studied from the theoretical perspective, we may refer for instance the readers to [START_REF] Costa | Stability and ergodicity of piecewise deterministic Markov processes[END_REF][START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF][START_REF] Azaïs | Piecewise deterministic Markov process -recent results[END_REF][START_REF] Malrieu | Some simple but challenging Markov processes[END_REF][START_REF] Benaïm | Quantitative ergodicity for some switched dynamical systems[END_REF][START_REF] Benaïm | Qualitative properties of certain piecewise deterministic Markov processes[END_REF] among many others.

From the applied point of view, and more precisely in biological applications, these processes are sometimes referred as hybrid, and are especially appealing for their ability to capture both continuous (deterministic) dynamics and discrete (probabilistic or deterministic) events. First applications date back at least to the studies of the cell cycle model [START_REF] Lasota | The statistical dynamics of recurrent biological events[END_REF][START_REF] Lasota | Cell division and the stability of cellular populations[END_REF], and more recent works, to name just a few, in neurobiology [START_REF] Pakdaman | Fluid limit theorems for stochastic hybrid systems with application to neuron models[END_REF][START_REF] Ditlevsen | Multi-class oscillating systems of interacting neurons[END_REF][START_REF] Ditlevsen | Multi-class oscillating systems of interacting neurons[END_REF], cell population and branching models [START_REF] Bansaye | Limit theorems for Markov processes indexed by continuous time Galton-Watson trees[END_REF][START_REF] Cloez | Limit theorems for some branching measure-valued processes[END_REF][START_REF] Marguet | Uniform sampling in a structured branching population[END_REF][START_REF] Doumic | Statistical estimation of a growth-fragmentation model observed on a genealogical tree[END_REF][START_REF] Lambert | The contour of splitting trees is a Lévy process[END_REF][START_REF] Champagnat | Moments of the frequency spectrum of a splitting tree with neutral poissonian mutations[END_REF], gene expression models [START_REF] Yvinec | Adiabatic reduction of a model of stochastic gene expression with jump Markov process[END_REF][START_REF] Mackey | Dynamic behavior of stochastic gene expression models in the presence of bursting[END_REF][START_REF] Dessalles | A stochastic analysis of autoregulation of gene expression[END_REF], food contaminants [START_REF] Bouguet | Quantitative speeds of convergence for exposure to food contaminants[END_REF][START_REF] Bertail | A storage model with random release rate for modeling exposure to food contaminants[END_REF] or multiscale chemical reaction network models [START_REF] Crudu | Convergence of stochastic gene networks to hybrid piecewise deterministic processes[END_REF][START_REF] Ball | Asymptotic analysis of multiscale approximations to reaction networks[END_REF][START_REF] Kurtz | Separation of time-scales and model reduction for stochastic reaction networks[END_REF][START_REF] Hepp | Adaptive hybrid simulations for multiscale stochastic reaction networks[END_REF]. See also [START_REF] Kouretas | Stochastic Hybrid Systems, chapter Stochastic hybrid modeling of biochemical processes[END_REF][START_REF] Bressloff | Stochastic processes in cell biology[END_REF][START_REF] Rudnicki | chapter Piecewise deterministic Markov processes in biological models[END_REF] for selected reviews on hybrid models in biology.

The paper is organized as follows. In Section 1, we present long time behavior results for switched flows in population dynamics. In Section 2, we present many-to-one formulas and description of the trait of an individual uniformly sampled in a general branching model applied to structured population dynamic. In Section 3, we 2964), Université Franc ¸ois-Rabelais, Parc de Grandmont, 37200 Tours, France.

⊥ CMAP, École Polytechnique, Université Paris-Saclay, Palaiseau, France.

Physiologie de la Reproduction et des Comportements, Institut National de la Recherche Agronomique (INRA) UMR85, CNRS-Université Franc ¸ois-Rabelais UMR7247, IFCE, Nouzilly, 37380 France present limit theorems and time scale separation in integrate-and-fire models used in neuroscience. Finally, in Section 4, we derive asymptotic moments formulas in a stochastic model of gene expression.

Long time behavior of some PDMP for population dynamics

In this section, we present recent results on long time behavior and exponential convergence of some PDMP used in population dynamics, in particular, for modeling population growth in a varying environment.

We consider processes (X t , I t ) t≥0 , where the first component X t has continuous paths in a continuous space E, namely a subset of R d , for some integer d, and the second component I t is a pure jump process on a finite state space F or in N. The continuous variable represents the number of individuals (or more precisely the population density or the relative abundance) of a population and the discrete variable models the population environment. In a fixed environment I t = i ∈ F , we assume that (X t ) t≥0 evolves as the solution of an ordinary differential equation, that is

∀t ≥ 0 , ∂ t X t = F (i) (X t ), (1.1)
where F (i) is some smooth function. For instance, the oldest and maybe the most famous model to represent the evolution of a population is the choice E = R + and

F (i) : x → a i x, a i ∈ R. (1.2)
When there is no variability in the environment, solutions are given by exponential functions and there is either explosion or extinction according to the sign of a i . We develop in Subsection 1.1 this toy model when the environment varies. Almost all properties (almost-sure convergence, moments...) can be derived and it gives a first hint to understand more complicated models such as multi-dimensional linear models or non-linear models. These generalizations are described in Subsection 1.2, and 1.3, respectively. For non-linear population models, maybe one of the most famous models is given by the Lotka-Volterra equations (also known as the predator-prey equations), with state-space E = R 2 + and

F (i) (x, y) = α i x(1 -a i x -b i y) β i x(1 -c i x -d i y) , a i , b i , c i , d i , α i , β i ∈ R + .
To end the introduction of this section, let us mention that all of these models are particular instance of a class of switching Markov model, see for instance [START_REF] Benaïm | Qualitative properties of certain piecewise deterministic Markov processes[END_REF][START_REF] Yin | Hybrid switching diffusions, volume 63 of Stochastic Modelling and Applied Probability[END_REF] for general references. Some other examples of application are developed in [START_REF] Malrieu | Some simple but challenging Markov processes[END_REF][START_REF] Fontbona | Long time behavior of telegraph processes under convex potentials[END_REF][START_REF] Monmarché | Piecewise deterministic simulated annealing[END_REF].

1.1. Malthus model. Let us consider, in this subsection, that (I t ) t≥0 is an irreducible continuous time Markov chain on a finite state space F . We denote by ν its unique invariant probability measure. The process (X t ) t≥0 will be the solution of

∀t ≥ 0, ∂ t X t = a It X t .
This means that F (i) is given by (1.2). Thus, we have

∀t ≥ 0, X t = X 0 e t 0 a Is ds .
The almost-sure behavior of this process is easy to understand. Indeed, by the ergodic theorem, we have that

lim t→+∞ 1 t t 0 a Is ds = i∈F a i ν({i}). = ν(a).
Then, we have the following dichotomy: either ν(a) > 0 and (X t ) t≥0 tends almost-surely to infinity at an exponential rate, or ν(a) < 0 and (X t ) t≥0 tends to 0. The second statement can be understood as the extinction of the population, even if, in contrast with classical birth and death processes, (X t ) t≥0 never hits 0; that is the extinction time (or the hitting time of 0) is almost-surely infinite.

The convergence of moments of (X t ) t≥0 is more tricky than the almost-sure convergence. Indeed, one cannot use a dominated convergence theorem or a Jensen inequality (which is in the wrong way

∀q > 0, lim t→+∞ E[X q t ] = +∞.
If ν(a) < 0 then there exists p > 0 such that

∀q ≤ p, lim t→+∞ E[X q t ] = 0.
The proof which follows comes from [START_REF] Bardet | Long time behavior of diffusions with Markov switching[END_REF]; see also [START_REF] De Saporta | Tail of a linear diffusion with Markov switching[END_REF].

Proof. If we denote by A the generator of the process (I t ) t≥0 and µ 0 the vector corresponding to its initial law, then the Feynman-Kac formula states that, for every p > 0,

E[X p t ] = E[X p 0 ]E e t 0 pa Is ds = E[X p 0 ]µ 0 e t(A+pD) 1,
where it is the classical exponential of matrices, D is the diagonal matrix such that D i,i = a i and 1 is the vector made of ones. Now, classical results on matrices (including Perron-Frobenius Theorem) give the existence of two constants c p , C p > 0 such that

c p e λpt ≤ µ 0 e t(A+pD) 1 ≤ C p e λpt ,
where λ p is the (unique) eigenvalue of largest real part of the matrix (A + pD). It then remains to understand the sign of λ p to understand the moment behaviour. By definition of λ p , there exists v p such that

v p (A + pD) = λ p v p .
Moreover, for p = 0, we have λ 0 = 0 and v 0 = ν. Again classical results on matrices ensure that p → λ p and p → v p are smooth and then differentiating in p, we obtain

v p D + ∂ p v p (A + pD) = v p ∂ p λ p + λ p ∂ p v p .
In particular, for p = 0 and multiplying by 1 by the right, we have ν(a) = (∂ p λ p ) |p=0 . As a consequence, if ν(a) > 0, then p → λ p is increasing in a neighborhood of 0 and for all p > 0 small enough we have λ p > 0.

Thus, E[X p t ] tends to infinity for p small enough, and then for all p > 0 by the Jensen inequality. In contrary, if ν(a) < 0 then there exists p > 0 such that lim t→+∞ E[X p t ] = 0 and also for all q < p, again by Jensen inequality.

Comment citer ce document : Cloez, B., Dessalles, R., Genadot, A., Malrieu, F., Marguet Understanding the long time behavior of moments is primordial when we are interested in the convergence to some invariant law more general than δ 0 . Indeed, moments provide Lyapunov-type functionals ensuring that the process returns frequently in a compact-set.

1.2. Others linear models. From a modeling point of view, the results of the previous section are easily understandable. Indeed, it is enough that the environment has in mean a positive effect to ensure an exponential growth, while if the environment is, in mean, not favorable, then the population goes to extinct.

However, note that this interpretation only holds because the growth parameter is well-chosen. Indeed, let us first consider a discrete time analogous of the Malthus model: the sequence (Y n ) n≥0 satisfying

∀n ≥ 0, Y n+1 = Θ n Y n ,
where (Θ n ) n is a sequence of i.i.d. non-negative random variables. Here Θ n represents the mean number of children per individuals at generation n (all individuals have the same number of children which depends on the environment). If (Θ n ) n≥0 is a constant sequence then Y n = Θ n 1 Y 0 and a dichotomy occurs with respect to a critical value Θ 1 = 1. When (Θ n ) n≥0 is a sequence of i.i.d random variable, then there is also a dichotomy but the critical value (to be smaller or larger than 1) is no longer E[Θ 1 ] but exp (E[ln(Θ 1 )]). Indeed, we have

∀n ≥ 0, Y n = Y 0 n-1 k=0 Θ k = X 0 e n-1 k=0 ln(Θ k ) .
This is exactly the same phenomenon for Galton-Watson chain in random environment, see for instance [START_REF] Bansaye | On the survival probability for a class of subcritical branching processes in random environment[END_REF] and [42, Section 2.9.2].

There is a similar (but more complex, maybe) problem in upper dimension. Let us consider that E = R 2 , F = {0, 1}, and (I t ) t≥0 jumps from 0 to 1 and from 1 to 0 with the same rate λ > 0, and

F (0) : v → -1 4 -1/4 -1 • v, F (1) : v → -1 -1/4 4 -1 • v. (1.3)
Then the solutions of the equation ∂ t x y = F (1) x y are given by ∀t ≥ 0, x(t)=e -t (cos(t)x(0) -sin(t)y(0)/4) y(t)=e -t (4 sin(t)x(0) + cos(t)y(0)).

(1.4)

It is easy to see that there exists some constants C > 0 such that for all t ≥ 0, (x(t), y(t)) ≤ Ce -t (x(0), y(0)) .

This property is also satisfied by the solutions of ∂ t x y = F (0) x y . However, we have the following theorem.

Theorem 1.2 ([8]

). Let (X t ) t≥0 be the solution of Equation (1.1) with flows defined in (1.3). There exists

β c > 0 such that if λ < β c then lim t→+∞ X(t) = 0,
and if λ > β c then lim t→+∞ X(t) = +∞. Moreover, the speed of convergence is exponential.

For this type of results for more general vector fields, see for instance [START_REF] Benaïm | On the stability of planar randomly switched systems[END_REF][START_REF] Malrieu | Some simple but challenging Markov processes[END_REF][START_REF] Lawley | Sensitivity to switching rates in stochastically switched ODEs[END_REF]. Let us briefly explain this phenomenon. One can see that, whatever the initial conditions of the solutions (x, y) of (1.4), the map t → (x(t), y(t)) is not decreasing; see for instance Figure [START_REF] Bakhtin | Invariant densities for dynamical systems with random switching[END_REF].

Then, heuristically, we see that if I jumps sufficiently fast (before the decay of the distance) then the distance may grow. However, if I is too slow, then (X t ) t≥0 essentially follows one of the two flows and goes rapidly to 0.

Switching linear models can have even more unpredictable behavior than the previous one. Indeed, in [START_REF] Lawley | Sensitivity to switching rates in stochastically switched ODEs[END_REF], the author exhibits two different linear functions F (0) , F (1) on R 2 and two critical values β 2 > β 1 such that the switched process (X t ) t≥0 tends to infinity for at least one λ ∈ (β 1 , β 2 ) and to 0 if λ ∈ (0, β 1 ) ∪ (β 2 , +∞).

Comment citer ce document : Cloez, B., Dessalles, R., Genadot, A., Malrieu, F., Marguet

1.3.

A general theorem for convergence. In this subsection, we expose a condition similar to Subsection 1.1 for ensuring convergence to an equilibrium (not necessary δ 0 ). As pointed out in the previous section, we have to measure precisely the growth of each underlying flow. So we assume that for all i ∈ F , there exists

ρ(i) ∈ R such that x -y, F (i) (x) -F (i) (y) ≤ -ρ(i) x -y 2 .
(1.5) The parameter ρ(i) measures the contraction of a solution u = F (i) (u) with respect to the norm • . Indeed, for two solutions, u 1 and u 2 , we have

∀t ≥ 0, u 1 (t) -u 2 (t) 2 ≤ e -ρ(i)t u 1 (0) -u 2 (0) 2 .
Note that the next result does not rely on a specific norm, but it has to be the same for all flows. A sufficient condition to prove (1.5) is given by the next lemma.

Lemma 1.3. Let G : R d → R d be a C 1 map and let J x G be its Jacobian matrix at the point x ∈ R d . If for all u, x ∈ R d , u, J x G • u ≤ -ρ u 2 , then, for all x, y ∈ R d , x -y, G(x) -G(y) ≤ -ρ x -y 2 .
(1.6) In particular, if G = ∇V , for some C 2 function V , then (1.6) holds under the classical condition that V is strongly convex with constant ρ; namely for all x ∈ R 2 , Hess x V ≥ ρI, in the sense of quadratic form.

Proof. It is classic to derive the previous bounds using the mean value theorem for the following map

ϕ : t → x -y, G(x) -G(ty + (1 -t)x) x -y 2 .
Comment citer ce document : Cloez, B., Dessalles, R., Genadot, A., Malrieu, F., Marguet We can now state the main result of this subsection.

Theorem 1.4 ([24]). Assume that (I t ) t≥0 is an irreducible Markov process on a finite state space F with invariant measure ν, (X t ) t≥0 satisfies (1.1), and that Eq.

(1.5) holds. If i∈F ρ(i)ν(i) > 0,
then (X t , I t ) converges in law to a unique invariant law.

The proof of this theorem is given in [START_REF] Cloez | Exponential ergodicity for Markov processes with random switching[END_REF][START_REF] Benaïm | Quantitative ergodicity for some switched dynamical systems[END_REF]. The proof of [START_REF] Cloez | Exponential ergodicity for Markov processes with random switching[END_REF] is based on a generalization (developed in [START_REF] Hairer | Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations[END_REF]) of the classical Harris Theorem [START_REF] Hairer | Yet another look at Harris' ergodic theorem for Markov chains[END_REF], which does not hold in general for PDMP (see for instance the introduction of [START_REF] Benaïm | Quantitative ergodicity for some switched dynamical systems[END_REF]). One of the main point of the proof is the construction of a Lyapunov function V , in the sense that there exist,

C, γ, K > 0 such that ∀t ≥ 0, E[V (X t , I t )] ≤ Ce -γt V (X 0 , I 0 ) + K.
(1.7) The construction of V and the control of its moments are based on Lemma 1.1.

Note that another proof of the previous theorem is given in [START_REF] Benaïm | Quantitative ergodicity for some switched dynamical systems[END_REF]. Nevertheless, the proof of [START_REF] Cloez | Exponential ergodicity for Markov processes with random switching[END_REF] takes the advantage to be generalizable in many contexts (dependent environment, non-deterministic underlying dynamics, asymptotic assumptions...). See [START_REF] Cloez | Exponential ergodicity for Markov processes with random switching[END_REF] for details.

Closely related articles [START_REF] Benaïm | Lotka Volterra in fluctuating environment or "how switching between beneficial environments can make survival harder[END_REF][START_REF] Malrieu | Lotka-Volterra with randomly fluctuating environments: a full description[END_REF][START_REF] Malrieu | On the persistence regime for Lotka-Volterra in randomly fluctuating environments[END_REF] deal with a fluctuating Lotka-Volterra model. They show that random switching between two environments both favorable to the same specie can lead to the extinction of this specie or coexistence of both species.

As Theorem 1.4, their proofs are based on the construction of a Lyapunov function. Nevertheless instead of controlling the exit from a compact set of R to infinity, this Lyapunov function is used to control the exit from a compact set included in (0, ∞) 2 to the axes {(0, y)|y ≥ 0} or {(x, 0, )|x ≥ 0}. A complete description of this model is given in [START_REF] Malrieu | Lotka-Volterra with randomly fluctuating environments: a full description[END_REF]. Note however that their counter-intuitive result is similar to the one of Theorem 1.2. It is based on the fact that, when I jumps sufficiently fast (namely λ being large in Theorem 1.2), then X is close to follow the following deterministic dynamics:

∀t ≥ 0, ∂ t X t = i∈F ν(i)F (i) (X).
Finally, we end this section mentioning that such hybrid framework can also be applied to models where the population is represented by a discrete variable, and/or the environment fluctuates continuously. One can cite for instance [START_REF] Costa | A piecewise deterministic model for a prey-predator community[END_REF], where the author studies a model of interaction between a population of insects and a population of trees. Another example is the chemostat model (a type of bioreactor), introduced in [START_REF] Crump | Some stochastic features of bacterial constant growth apparatus[END_REF] and studied in [START_REF] Collet | Stochastic models for a chemostat and long-time behavior[END_REF][START_REF] Campillo | Stochastic modeling of the chemostat[END_REF][START_REF] Campillo | Weak convergence of a mass-structured individual-based model[END_REF][START_REF] Cloez | Gaussian approximations for chemostat models in finite and infinite dimensions[END_REF]. Lyapunov functions are again a key theoretical tool to study long time behavior for such models. However, we point out that extinction events may occur in discrete population models, and that the interplay between the discrete and continuous components makes the problem more difficult than standard absorbing time studies in purely discrete population models. Nevertheless, it was proved in [START_REF] Cloez | Gaussian approximations for chemostat models in finite and infinite dimensions[END_REF]Theorem 3.4] that extinction event is almost-surely finite and has furthermore exponential tails. This generalizes part of a result of [START_REF] Collet | Stochastic models for a chemostat and long-time behavior[END_REF].

Uniform sampling in a branching structured population

In this section, which is a short version of [START_REF] Marguet | Uniform sampling in a structured branching population[END_REF], we give a characterization of the trait of a uniformly sampled individual among the population at time t in a structured branching population.

2.1. Introduction. We consider a branching Markov process where each individual u is characterized by a trait (X u t , t ≥ 0) which dynamic follows an X -valued Markov process, where X := X × R + and X ⊂ (R + ) d for some d ≥ 1. For any x ∈ X , the last coordinate corresponds to a time coordinate and we denote by x ∈ X the vector of the first d coordinates. We assume that this trait influences the lifecycle of each individual (in terms of lifetime, number of descendants and inheritance of the trait). Then, an interesting problem consists in the characterization of the trait of a "typical" individual in the population. This is the question we address here.

Comment citer ce document : Cloez, B., Dessalles, R., Genadot, A., Malrieu, F., Marguet Let us describe the process in more details. We consider a strongly continuous contraction semi-group with associated infinitesimal generator G :

D(G) ⊂ C b (X ) → C b (X )
, where C b (X ) denotes the space of continuous bounded functions from X to R. Then, each individual u has a trait (X u t , t ≥ 0) which evolves as a Markov process defined as the unique X -valued càdlàg solution of the martingale problem associated with (G, D(G)). An individual with trait x dies at an instantaneous rate B(x), where B is a continuous function from X to R + . It is replaced by A u (x) children, where A u (x) is a N-valued random variable with distribution (p k (x) , k ≥ 0). For convenience, we assume that p 1 (x) ≡ 0 for all x ∈ X . The trait of the descendants at birth depends on the trait of the mother at death. For all k ∈ N, let P (k) (x, •) be the probability measure on X k corresponding to the trait distribution at birth of the k descendants of an individual with trait x. We denote by P (k) j (x, •) the jth marginal distribution of P (k) for all k ∈ N and j ≤ k.

Finally, we denote by M P (X ) the set of point measures on X . Following Fournier and Méléard [START_REF] Fournier | A microscopic probabilistic description of a locally regulated population and macroscopic approximations[END_REF], we work in D (R + , M P (X )), the state of càdlàg measure-valued Markov processes. For any Z ∈ D (R + , M P (X )), we write:

Z t = u∈Vt δ X u t , t ≥ 0,
the measure-valued process describing the dynamic of the population where V t denotes the set of all individuals in the population at time t. We will denote by N t the cardinality of V t and by:

m(x, s, t) = E N t Z s = δ x , its expected value, for 0 ≤ s ≤ t.
An example of such a process is the size-structured population where each individual grows exponentially fast. For more details we refer the reader to [START_REF] Doumic | Statistical estimation of a growth-fragmentation model observed on a genealogical tree[END_REF] or [START_REF] Bertoin | Compensated fragmentation processes and limits of dilated fragmentations[END_REF]. Then, at rate B(X u t ), the individual u splits at time t in two daughter cells of size X u t /2. Figure 3 is a realization of such a process. The diameter of each circle corresponds to the size of the individual at division and the length of the branch represents the lifetime of each individual. In particular, we notice the link between the lifetime and the size of each individual: the bigger the cell is, the shorter its lifetime is.

In Section 2.2, we give the chosen assumptions on the model to ensure that our process is well-defined. Then, in Section 2.3, we describe the process corresponding to the trait of a "typical" individual in the population. Finally, in Section 2.4, we explain why this process corresponds to the trait of a uniformly sampled individual in a large population approximation.

Comment citer ce document : Cloez, B., Dessalles, R., Genadot, A., Malrieu, F., Marguet

Definition and existence of the structured branching process.

To define rigorously the structured branching process on R + , we need to ensure that almost surely the number of individuals in the population does not blow up in finite time.

As explained in the previous description of the process, the lifetime of each individual depends on the dynamic of the trait through the division rate B. One way of dealing with this dependency is to assume that the division rate is upper bounded by some constant B. Then, in the case of a binary division, the expected number of individuals in the population at time t is upper bounded by the expected value of a Yule process with birth rate B at time t which is equal to exp(Bt) and which is bounded on compact sets. In the case of an unbounded division rate, the same reasoning applies if we can control the excursions of the dynamic of the trait. Thus, we consider two sets of hypotheses: the first one controls what happens regarding divisions (in term of rate of division and of mass creation) and the second one controls the dynamic of the trait between divisions.

Assumption 1. We consider the following assumptions:

(1) There exist b 1 , b 2 ≥ 0 and γ ≥ 0 such that for all x ∈ X ,

B(x) ≤ b 1 |x| γ + b 2 .
(2) There exists x ∈ X such that for all x ∈ X , k ∈ N:

X k k i=1 y i P (k) (x, dy 1 . . . dy k ) ≤ x ∨ x, componentwise.
(3) There exists m ≥ 0 such that for all x ∈ X ,

m(x) := k kp k (x) ≤ m.
The first point controls the division rate. The second point means that we consider a fragmentation process with a possibility of mass creation at division when the mass is small enough. In particular, clones are allowed in the case of bounded traits and bounded number of descendants and any finite type branching structured process can be considered.

As explained before, we make a second assumption to control the behavior of traits between divisions.

Assumption 2. There exist c 1 , c 2 ≥ 0 such that for all x ∈ X :

Gh γ (x) ≤ c 1 h γ (x) + c 2 ,
where γ is defined in Assumption 1 and for

x ∈ (R + ) d , h γ (x) = |x| γ = d i=1 x i γ .
This assumption ensures in particular that the trait does not blow up in finite time in the case of a deterministic dynamic for the trait. Assumptions 1(1) and 2 are linked via the parameter γ which controls the balance between the growth of the population and the dynamic of the trait. In particular, if Assumption 1 is satisfied for γ = 0, the division rate is bounded and Assumption 2 is always satisfied.

Under Assumptions 1 and 2, the previously described measure-valued process is well-defined as the strongly unique solution of a stochastic differential equation.

The proof of existence relies on a recursive construction of a solution via the sequence of jumps time of the population. Then, we prove that this sequence is unique conditionally to the Poisson point measure determining the jumps in the population and to the stochastic flows corresponding to the dynamic of the trait. Finally, using Assumptions 1 and 2, we prove that the number of individuals in the population does not blow up in finite time. This is detailed in [START_REF] Marguet | Uniform sampling in a structured branching population[END_REF] (Theorem 2.3.).

2.3.

The trait of sampled individuals at a fixed time : Many-to-One formula. In order to characterize the trait of a uniformly sampled individual, the spinal approach ( [START_REF] Chauvin | KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees[END_REF], [START_REF] Lyons | Conceptual proofs of L log L criteria for mean behavior of branching processes[END_REF]) consists in following a "typical" individual in the population whose behavior summarizes the behavior of the entire population. In this section, we give the dynamic of the trait of a typical individual: it is a Markov process called the auxiliary process. From now on, we assume that for all x ∈ X , t ≥ 0 and s ≤ t, m(x, s, t) = 0. With a slight abuse of notation, we denote by X u s the trait of the unique ancestor living at time s of u. Let f be a non-negative measurable function and x ∈ X . The idea is to notice that the following operator:

P (t) r,s f (x) := E u∈Vt f (X u s ) Z r = δ x m(x, r, t) , ( 2.1)
is a conservative (non-homogeneous) semi-group. Then, the auxiliary process is defined as the time-inhomogeneous Markov process with associated family of semi-groups

P (t)
r,s , r ≤ s ≤ t . It describes the dynamic of the trait of a "typical" individual in the population in the sense that it satisfies a so-called "Many-to-One" formula. This formula is true under Assumptions 1, 2 and under the following technical assumptions: Assumption 3. There exists a function C such that for all j ≤ k, j, k ∈ N and 0 ≤ s ≤ t, we have:

sup x∈X sup s∈[0,t] X m(y, s, t) m(x, s, t) P (k) j (x, dy) ≤ C(t), ∀t ≥ 0.
This assumption tells us that we control uniformly in x the benefit or the penalty of a division.

Assumption 4. For all t ≥ 0 and x ∈ X we have:

-s → m(x, s, t) is differentiable on [0, t] and its derivative is continuous on [0, t], -for all f ∈ D(A) := {f ∈ D(G) s.t. m(•, s, t)f ∈ D(G) ∀t ≥ 0, ∀s ≤ t}, s → G(m(•, s, t)f )(x) is contin- uous,
We can now state the Many-to-One formula.

Theorem 2.1. Under Assumptions 1, 2, 3 and 4, for all t ≥ 0, for all x 0 ∈ X and for all non-negative measurable functions f : X → R + , we have:

E δx 0 u∈Vt f (X u s ) = m(x 0 , 0, t)E x0 f Y (t) s , (2.2)
where

Y (t)
s , s ≤ t is an inhomogeneous-Markov process with infinitesimal generator

A (t)
s , D(A) given for f ∈ D(A) and x ∈ X by:

A (t) s f (x) = G (t) s f (x) + B (t) s (x) X (f (y) -f (x)) P (t) s (x, dy) ,
where: .

G (t) s f (x) = G (m(•, s, t)f ) (x) -f (x) G (m(•, s, t)) (x) m(x, s, t) , B (t) s (x) = B(x) X m(y,
Comments on the proof. To prove the Many-to-One formula (2.2), we first show that the family of semi-groups

P (t)
r,s , r ≤ s ≤ t is uniquely defined as the unique solution of an integro-differential equation (see Lemma 3.2 in [START_REF] Marguet | Uniform sampling in a structured branching population[END_REF]). In particular, we need Assumption 3 to prove the uniqueness. Then, the infinitesimal generator Remark 2.2. We can also prove a pathwise version of (2.2) using a monotone class argument (Theorem 3.1. in [START_REF] Marguet | Uniform sampling in a structured branching population[END_REF]).

A (t) s s≤t , D (A)
Unlike previous works on this subject ( [START_REF] Georgii | Supercritical multitype branching processes: the ancestral types of typical individuals[END_REF], [START_REF] Hardy | A spine approach to branching diffusions with applications to Lp-convergence of martingales[END_REF], [START_REF] Cloez | Limit theorems for some branching measure-valued processes[END_REF]), the existence of our auxiliary process does not rely on the existence of spectral elements for the mean operator of the branching process. In particular, we can apply this result to models with a varying environment. The dynamic of this auxiliary process heavily depends on the comparison between m(x, s, t) and m(y, s, t), for x, y ∈ X . It emphasizes several bias due to growth of the population. First, the auxiliary process jumps more than the original process, if jumping is beneficial in terms of number of descendants. This phenomenon of time-acceleration also appears for examples in [START_REF] Chauvin | KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees[END_REF], [START_REF] Lyons | Conceptual proofs of L log L criteria for mean behavior of branching processes[END_REF] or [START_REF] Hardy | A spine approach to branching diffusions with applications to Lp-convergence of martingales[END_REF]. Moreover, the reproduction law favors the creation of a large number of descendant as in [START_REF] Bansaye | Limit theorems for Markov processes indexed by continuous time Galton-Watson trees[END_REF] and the non-neutrality favors individuals with an "efficient" trait at birth in terms of number of descendants. Finally, a new bias appears on the dynamic of the trait because of the combination of the random evolution of the trait and non-neutrality. Indeed, if the dynamic of the trait is deterministic, we have

G (t) s f (x) = Gf (x).
2.4. Ancestral lineage of a uniform sampling at a fixed time in a large population. The Many-to-One formula (2.2) gives the law of the trait of a typical individual in the population. But so far, we did not prove that it corresponds to the trait of a uniformly sampled individual. In particular, we have to take into account that the number of individuals in the population is stochastic and depends on the dynamic of the trait. Using the law of large numbers, we can approximate the number of individuals in a population starting for n individuals, divided by n, by the mean number of individual in the population. That is why we now look at the ancestral lineage of a uniform sampling in a large population.

It only makes sense to speak of a uniformly sampled individual at time t if the population does not become extinct before time t. For all t ≥ 0, let Ω t = {N t > 0} denote the event of survival of the population. Let ν ∈ M P (X ) be such that:

P ν (Ω t ) > 0.
We set

ν n := n i=1 δ Xi ,
where X i are i.i.d. random variables with distribution ν. For t ≥ 0, we denote by U (t) the random variable with uniform distribution on V t conditionally on Ω t and by X U (t) s , s ≤ t the process describing the trait of a sampling along its ancestral lineage. If X is a stochastic process, we denote by X ν the process with initial distribution ν ∈ M P (X ). Theorem 2.3. Under Assumptions 1,2, 3 and 4, for any t ≥ 0, the following convergence in law in D ([0, t], X ) holds:

X U (t),νn [0,t] -→ n→+∞ Y (t),πt [0,t] , where π t (dx) = m(x, 0, t)ν(dx) X m(x, 0, t)ν(dx) .
Remark 2.4. If we start with n individuals with the same trait x, we obtain:

E F X U (t),νn [0,t] -→ n→+∞ E x F Y (t) [0,t]
.

Therefore, the auxiliary process describes exactly the dynamic of the trait of a uniformly sampled individual in the large population limit, if all the starting individuals have the same trait. If the initial individuals have different traits at the beginning, the large population approximation of a uniformly sampled individual is a linear combination of the auxiliary process.

Comment citer ce document : Cloez, B., Dessalles, R., Genadot, A., Malrieu, F., Marguet 3. Averaging for some integrate-and-fire models 3.1. The model. We examine the following slow-fast hybrid version of integrate-and-fire models [START_REF] Burkitt | Averaging for some simple constrained markov processes[END_REF], used in mathematical neuroscience. In such a setting, X represents the membrane potential of a neural cell which is increasing until it reaches some threshold c, corresponding to the time where a nerve impulse is triggered, and then the potential is reset to some lower value. More precisely, the process (X(t), t ∈ [0, T ]), with T some time horizon, obeys the following dynamic:

(1) Initial state: At time T * 0 = 0, the process starts at X(T * 0 ) = ξ 0 , a random variable with support included in (m, c) where {c} is considered as a boundary and m < c is some real.

T * 1 = inf t > 0 ; ξ 0 + t T * 0 α(Y (s))F (X(s))ds = c ,
where α is a positive measurable function such that α(Y) is bounded from above and F is a positive continuous function. (3) Piecewise deterministic motion: For t ∈ [T * 0 , T * 1 ), we set

X(t) = ξ 0 + t T * 0 α(Y (s))F (X(s))ds.
The dynamic of X is thus continuous here, and given by the differential equation:

dX dt (t) = α(Y (t))F (X(t)), X(0) = ξ 0 .
(4) Jumping measure: Then, at time T * - 1 , the process is constrained to stay inside (m, c) by jumping according to the Y -dependent measure µ Y T * -

1
whose support is included in (m, c):

∀A Borel subset of (m, c), P(X(T * 1 ) ∈ A) = µ Y T * - 1 (A).
(5) And so on: Go back to step 1 in replacing T * 0 by T * 1 and ξ 0 by ξ 1 = X(T * 1 ). This is not hard to see that the process (X, Y ) is a piecewise deterministic Markov process. Moreover, due to the presence of a boundary c, this is also a constrained Markov process. Our aim is, in this quite simple situation, to describe the behavior of the process X at the boundary c, when the dynamic of the underlying process Y is infinitely accelerated.

Acceleration and averaged process.

From now on, we assume that the underlying process of celerities, that is the continuous time Markov chain Y , has a fast dynamic, by introducing a (small) timescale parameter ε such that ∀t ≥ 0, Y ε (t) = Y (t/ε) . In the same time, to insure a limiting behavior, we assume that Y is positive recurrent with intensity matrix Q = (q zy ) z,y∈Y and invariant probability measure π, considered as a row vector. For convenience, let us also define by α(V ) the diagonal matrix such that diag(α(V )) = {α(y) ; y ∈ Y}. As ε goes to zero, the process Y ε converges towards the stationary state associated to Y in the sense that, by the ergodic theorem, ∀t ≥ 0, ∀y ∈ Y,

lim ε→0 P(Y ε (t) = y) = π(y).
Therefore, as ε goes to zero, the process X ε , defined as X by replacing Y by Y ε , should have its dynamic averaged with respect to the measure π. The behavior of the limiting process away from the boundary is indeed not hard to describe, see for example [START_REF] Kurtz | Applied Stochastic Analysis, chapter Averaging for martingale problems and stochastic approximation[END_REF]

X(t) c ξ

0 ξ 1 ξ 2 Y (t) t 1 1 2 T * 1 T * 2 Figure 4.
A trajectory of X, in a piecewise linear case, dX(t)/dt = Y (t), with Y switching between 1/2 and 1.

• 1 F is integrable over (m, c), • α(Y) is bounded from above, • E(sup ε∈(0,1] p * ε (T )
) is finite, where p * ε (T ) is the counting measure at the boundary for the process X ε . Proposition 3.1. Assume that ξ 0 is deterministic. Then, for any η > 0, the process X ε converges in law towards a process X on D 0, c-ξ0 max α(Y) -η , R defined as:

X(t) = ξ 0 + y∈Y α(y)π(y) t 0 F ( X (
f ( X(t)) -f (ξ 0 ) - y∈Y α(y)π(y) t 0 f ( X(s))F ( X(s))ds - t 0 c m [f (u) -f ( X(s -))]μ(du)p * (ds) (3.1)
for t ∈ [0, T ], defined a martingale, with p * the counting measure at the boundary for X. The averaging measure at the boundary μ is defined by

μ(du) = y∈Y µ y (du)π * (y)
where, for y ∈ Y, π * is given by

π * (y) = π(y)α(y) y ∈Y π(y )α(y )
.

We can read in (3.1) the behavior of the averaged process X. Away from the boundary, the dynamic of X is the one of the averaged flow, as indeed described in Proposition 3. measure is not given by the averaged version of the jumping measure, that is

μ(du) = y∈Y µ y (du)π(y),
but the values of the celerity at the boundary are taken into account through the presence of α. There is a balance between the celerity intensities and the invariant probabilities. For example, consider a piecewise linear motion whose speed switches at rate 1/ε between 1 and 2. When, ε goes to zero, the process will spend as much time increasing with celerities 1 and 2, but when increasing with celerity 2, the process obviously increases twice more than with celerity one in a same time window, and then approaches the boundary twice more rapidly, and therefore will have in fact twice more chance to hit the boundary with this celerity. Note that because of the separation of variables in the form of the flow, the value of the process X at the boundary does not appear in the expression of π * , as it could be the case in more general situations. Theorem such as 3.2 can be derived using the Prohorov program : prove the tightness of the family {X ε , ε ∈ (0, 1)} and then identify the limit. Tightness for constrained Markov processes, as the process described in the present section, has an interesting and rich literature, see [START_REF] Kurtz | Recent Advances in Stochastic Calculus, chapter Martingale problems for constrained Markov processes[END_REF] and references therein. We refer to [START_REF] Genadot | Averaging for some simple constrained markov processes[END_REF] for more details about the proof of Theorem 3.2.

Stochastic models of protein production with cell division and gene replication

The production of proteins is the main mechanism of the prokaryotic cells (like bacteria). These functional molecules represent about 3.6 × 10 6 molecules of 2000 different types and compose about half of the dry mass of the cell [START_REF] Bremer | Modulation of chemical composition and other parameters of the cell at different exponential growth rates[END_REF] and this pool needs to be duplicated in the cell cycle time. In total, it is estimated that 67% of the energy of the cell is dedicated to the protein production [START_REF] Russell | Energetics of bacterial growth: balance of anabolic and catabolic reactions[END_REF]. The protein production is a two steps process by which the genetic information is first transcribed into an intermediate molecule, the messenger-RNA (mRNA), and then translated into a protein. Experimental studies have shown that this production is subject to a large variability [START_REF] Elowitz | Stochastic gene expression in a single cell[END_REF][START_REF] Ozbudak | Regulation of noise in the expression of a single gene[END_REF][START_REF] Taniguchi | Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. This heterogeneity can either be explained by the stochastic nature of the production mechanism (both transcription and translation are due to random encounter between molecules) or the effect of cell scale phenomenon like the DNA replication, cell division, or fluctuation in the resources.

In particular, the extensive experimental study of [START_REF] Taniguchi | Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] [START_REF] Taniguchi | Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] measured the variability of protein concentration for more than 1000 different proteins in E. coli. For each of them, the authors have measured the average protein concentration, the average mRNA concentration and their lifetime. They interpret different sources of proteins, either from the protein production mechanism itself (i.e. the transcription of mRNAs and the translation of proteins) or from external factors, like the gene replication, division or the sharing of common resources in the production.

One can interpret the experimental results in the light of classical stochastic models of protein production [START_REF] Berg | A model for the statistical fluctuations of protein numbers in a microbial population[END_REF][START_REF] Rigney | Stochastic model of linear, continuous protein synthesis in bacterial populations[END_REF][START_REF] Paulsson | Models of stochastic gene expression[END_REF] that predict the variability of protein production. But these classical models do not consider important phenomenon that may have an impact on the production: they usually do not consider the replication of the gene at some point in the cell cycle and also do not take into account the random assignment of each protein in either of the two daughter cells at division. Moreover, they only consider the number of mRNAs and proteins without any explicit notion of volume, and therefore the notion of concentration is unclear in this case. Since the experimental study only considers concentrations, it would be difficult to have quantitative comparisons between the variances predicted by these models and the ones experimentally measured.

We therefore propose here a more realistic model of protein production that takes into account all the basic features that can be expected for the production of a type of protein inside a cell cycle: the transcription, the translation, the gene replication and the division. We also explicitly represent the number of mRNAs and proteins in the growing volume of the bacteria, so that we can consider the concentration of each of these entities in the cell. We will then estimate the theoretical contribution of these features to the protein variance through mathematical analysis. We will also make a biological interpretation of these results with respect to experimental measures.

Comment citer ce document : Cloez, B., Dessalles, R., Genadot, A., Malrieu, F., Marguet

Division Replication

Time in cell cycle

Gene copy

0 τ R τ D (a) M P ∅ Periodic divisions every τ D λ 1 • (1 + 1 s>τ R ) σ 1 M λ 2 M B(M, 1/2) B(P, 1/2)
mRNAs Proteins 4.1. Model and Results. Our model presented here is adapted from a simple gene constitutive model that does not consider any gene regulation (it is a particular case of the model presented in [START_REF] Paulsson | Models of stochastic gene expression[END_REF]). Like this classical model, our model is gene centered (it represents the production of only one protein), and has two stages (transcription and translation) where both the number of mRNAs, M , and proteins, P , are explicitly represented. Each event (creation or degradation of an mRNA, creation of a protein, etc.) is supposed to occur at random times. The time intervals are considered as exponentially distributed. In addition to this classical framework, we introduce a notion of cell cycle represented by cell growth, division and gene replication. Periodically two events occurs: the replication at a deterministic time τ R in the cell cycle that doubles the mRNA production rate and the division after a time τ D starting a new cell cycle. We can then consider the evolution of one type of mRNA and protein through a cell lineage (see Figure 5a). Overall, the model represents five aspects that intervene in protein production (see Figure 5b): mRNAs: Messenger-RNAs are transcribed at rate λ 1 before the replication and at rate 2λ 1 after gene replication. When transcription happens, the number M of mRNAs increases by 1. As in the previous model, each mRNA has a lifetime of rate σ 1 (so the global mRNA degradation rate is σ 1 M ). Proteins: Each mRNA can be translated into a protein at rate λ 2 (so the global protein production rate is λ 2 M ). The number of proteins P is increased by 1. As the protein lifetime is usually of the order of several cell cycles, we consider that the proteins do not degrade. Gene replication: After a deterministic time τ R after each division (with τ R < τ D ), the gene replication occurs. The gene responsible for the mRNA transcription is replicated, hence doubling the mRNA transcription rate until next division. Division: Divisions occur periodically after deterministic time interval τ D . After each division, we only follow one of the two daughter cells (see Figure 5a) so that each mRNA and each protein have a probability 1/2 to be in this next considered cell. The result is a binomial sampling: for instance, with M mRNAs just before division, the number of mRNAs after division follows a binomial distribution B (n, p) with parameters n = M and p = 1/2. Moreover, as there is only one copy of the gene in the newborn cell, the mRNA transcription rate is anew set to λ 1 until the next gene replication. Volume growth: The volume V (s) considers the entire volume of the cell at any moment s of the cell cycle and it increases as the cell grows. One considers the volume growth of the cell as deterministic.

In real life experiments, a bacteria volume globally grows exponentially (see [START_REF] Wang | Robust growth of Escherichia coli[END_REF]) and approximately doubles its volume at the time of division τ D . As a consequence, in our model, for a time s in the cell cycle, then the volume grows as V (s) = V 0 2 s/τ D with V 0 the typical size of a cell at birth. From this, if M s and P s denote respectively the number of mRNAs and proteins at time s, the concentrations are M s V (s) and P s V (s) .

Our main results are the analytical expressions for the distribution of the number of mRNAs and the first moments of the number of proteins at any time s of the cell cycle, when the system is at equilibrium (ie after many cell divisions).

x s = λ 1 σ 1 1 - e -(s+τ D -τ R )σ1 2 -e -τ D σ1 + 1 s≥τ R 1 -e -(s-τ R )σ1 .
The proof of this proposition uses the framework of Marked Point Poisson Processes to determine the mRNA distribution at any time of the cell cycle knowing the random variable M 0 , the number of mRNA at birth. Then, the distribution of M 0 can be determined as the equilibrium stipulates that the distribution of M is the same at time 0 and just after the next division. The proof will be given in [START_REF] Dessalles | Stochastic models for gene expression with cell division, gene replication and protein production interactions[END_REF] (in preparation). Proposition 4.2. At any time s ∈ [0, τ R [ before replication, the mean and the variance of P s are given as a function of the moments of (P 0 , M 0 ) by:

E [P s ] = E [P 0 ] + λ 2 λ 1 σ 1 s + x 0 - λ 1 σ 1 1 -e -σ1s σ 1 , Var [P s ] = Var [P 0 ] + 2λ 2 1 -e -σ1s σ 1 Cov [P 0 , M 0 ] λ 2 1 -e -σ1s σ 1 2 x 0 + x 0 λ 2 σ 1 1 -e -σ1s + λ 2 σ 1 1 -e -σ1s e -σ1s + 2sσ 1 + + λ 1 λ 2 σ 2 1 sσ 1 -1 + e -σ1s + 2 λ 2 σ 1 σ 1 s 1 + e -σ1s -2 1 -e -σ1s .
with x 0 defined in Proposition 4.1.

Moreover, we can show a similar result for s ∈ [τ R , τ D [, a time after gene replication. Then, at equilibrium, we have that the distribution of the couple (M, P ) after division is the same as at the beginning of the cell cycle. This allows us to determine the explicit values for E [P 0 ], Var [P 0 ] and Cov [P 0 , M 0 ]. These complementary results and the proof of Proposition 4.2 will be available in [START_REF] Dessalles | Stochastic models for gene expression with cell division, gene replication and protein production interactions[END_REF] (in preparation). 4.2. Discussion. In order to have realistic values for the parameters λ 1 , σ 1 , λ 2 and τ R , we fix them to correspond to the average production of each protein measured in [START_REF] Taniguchi | Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] [START_REF] Taniguchi | Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. It gives groups of parameters that represent a large spectrum of different genes. Then, using Propositions 4.1 and 4.2, as well as the additional results, we can compute the concentration for every gene at any time of the cell cycle.

An example of the evolution of one protein concentration is shown in Figure 6. It appears that the mean concentration at a given time s of the cell cycle E [P s /V (s)] is not constant during the cell cycle. The curve of E [P s /V (s)] fluctuates around 2% of the global average protein production (denoted by µ p in the graph). Experimental measures of average protein expression during the cell cycle show similar results: for instance, the expression of genes at different positions on the chromosome have been measured in [START_REF] Walker | Generation and filtering of gene expression noise by the bacterial cell cycle[END_REF]; it shows a similar profile during the cell cycle and depicts a fluctuation also around 2% of the global average (see Figure 1 Overall, for all the genes considered, the effect of the cell cycle (the fluctuation of E [P s /V (s)] around µ p ) is small compared to the standard deviation at each moment of the cell cycle Var [P s /V (s)] (the coloured areas in Figure 6). We have determined the range of parameters where, on the contrary, the fluctuations due to the cell cycle would be higher than the inherent variability of the system. It appears that such regime is possible the variance and µ p the mean) experimentally measured in [START_REF] Taniguchi | Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF]. It clearly appears two regimes: the first one (for µ p < 10) shows a CV inversely proportional to the mean; the second regime (for µ p > 10) shows a CV independent of the average production. (B) Same graph obtained with the proteins of our model. Even if the CV is roughly inversely proportional to the average production, there is no lower bound limit for highly expressed proteins.

only for highly active gene (high λ 1 ) and for mRNAs not very active (low λ 2 ) and which exist for short periods of time (high σ 1 ). Biologically, such regime seems unrealistic because the cost of mRNA production would be very high. In Figure 7, we compare the experimental results obtained in the article of [START_REF] Taniguchi | Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells[END_REF] [68] and our analytical expressions obtained in our model, for the global coefficient of variation (CV) of protein concentration as a function of the average concentration. In the experimental measures, it clearly appears two regimes for the CV of protein concentration: for low expressed proteins (mean protein concentration < 10), the CV roughly scales inversely with the average concentration; for genes with higher protein production (mean protein concentration > 10), the CV becomes independent of the average protein production level, the plateau is around 10 -1 . In our model, the global tendency of the CV approximately inversely scales the average protein concentration, and there is no lower bound limit, as in the experiments.

We have therefore shown through this model that both DNA replication and division can not explain the second regime observed in Figure 7a. Yet the authors of the experimental study propose another possible origin for the noise observed in highly produced proteins; they propose that it is due to fluctuations in the availability of RNA-polymerases and ribosomes. These two macro-molecules are shared among the productions of all the different proteins because they respectively participate to the transcription of mRNAs and the translation of proteins. Since they are present in limited quantities, fluctuations in their availability could indeed have repercussions on the protein noise. We will present in [START_REF] Dessalles | Stochastic models for gene expression with cell division, gene replication and protein production interactions[END_REF] an extension of the model presented here that takes into account this sharing of ribosomes and RNA-polymerases in the production of all proteins of the bacteria, and the impact it has on the protein noise.

Comment citer ce document : Cloez, B., Dessalles, R., Genadot, A., Malrieu, F., Marguet

Figure 1 .Lemma 1 . 1 (

 111 Figure 1. Trajectory of (X t ) t≥0 in the Malthus model.

Figure 2 .

 2 Figure 2. Graph of t → (x(t), y(t)) = e -t 15 sin 2 (t) + 1 for (1.4) when (x 0 , y 0 ) = (1, 0).

Figure 3 .

 3 Figure 3. Descending genealogy from an individual with size 1 until time T = 50 of a sizestructured population. Each cell grows exponentially at rate 0.01 and divide at rate B(x) = x.

  Comment citer ce document : Cloez, B., Dessalles, R., Genadot, A., Malrieu, F., Marguet, A., Yvinec, R. (2017). Probalistic and piecewise deterministic models in biology . In: Journées MAS 2016. Phénomènes complexes et hétérogènes (p. 1-19). Presented at Journées MAS 2016. Phénomènes complexes et hétérogènes, Grenoble, FRA (2016-08-29 -2016-08-31).

( 2 )

 2 First jumping time: Let Y be a continuous time Markov chain valued in a countable space Y. This chain starts at Y (0) = ζ, a Y-valued random variable. The first hitting time of the boundary occurs at time T * 1 defined as

Theorem 3 . 2 .

 32 s))ds. Proposition 3.1 gives the behavior of the limiting process away from the boundary: celerities are averaged against the measure π. But what happens at boundary? This is what we want to characterize. The following result holds. The process X ε converges in law in D([0, T ], R) towards a process X such that for any sufficiently regular function f : (m, c) → R the process

1 .

 1 At the boundary, the averaged jump Comment citer ce document : Cloez, B., Dessalles, R., Genadot, A., Malrieu, F., Marguet, A., Yvinec, R. (2017). Probalistic and piecewise deterministic models in biology . In: Journées MAS 2016. Phénomènes complexes et hétérogènes (p. 1-19). Presented at Journées MAS 2016. Phénomènes complexes et hétérogènes, Grenoble, FRA (2016-08-29 -2016-08-31).

Figure 5 .

 5 Figure 5. Model with gene replication and division. (A) The concept of cell cycle in the model. (B) Model of production for one protein.

Proposition 4 . 1 .

 41 At equilibrium, the mRNA number M s at time s in the cell cycle follows a Poisson distribution of parameter

  .

  d and Figure S6.b of the article).

  /V(s)] ± std[P s /V(s)]

Figure 6 .Figure 7 .

 67 Figure 6. Profile example of the gene Adk, the central line represents the mean protein concentration over the cell cycle, and the coloured areas represent the standard-deviation in the models.

  Comment citer ce document : Cloez, B., Dessalles, R., Genadot, A., Malrieu, F., Marguet, A., Yvinec, R. (2017). Probalistic and piecewise deterministic models in biology . In: Journées MAS 2016. Phénomènes complexes et hétérogènes (p. 1-19). Presented at Journées MAS 2016. Phénomènes complexes et hétérogènes, Grenoble, FRA (2016-08-29 -2016-08-31).

  , A., Yvinec, R. (2017). Probalistic and piecewise deterministic models in biology . In: Journées MAS 2016. Phénomènes complexes et hétérogènes (p. 1-19). Presented at Journées MAS 2016. Phénomènes complexes et hétérogènes, Grenoble, FRA (2016-08-29 -2016-08-31).

  , A., Yvinec, R. (2017). Probalistic and piecewise deterministic models in biology . In: Journées MAS 2016. Phénomènes complexes et hétérogènes (p. 1-19). Presented at Journées MAS 2016. Phénomènes complexes et hétérogènes, Grenoble, FRA (2016-08-29 -2016-08-31).

  , A., Yvinec, R. (2017). Probalistic and piecewise deterministic models in biology . In: Journées MAS 2016. Phénomènes complexes et hétérogènes (p. 1-19). Presented at Journées MAS 2016. Phénomènes complexes et hétérogènes, Grenoble, FRA (2016-08-29 -2016-08-31). Comment citer ce document : Cloez, B., Dessalles, R., Genadot, A., Malrieu, F., Marguet, A., Yvinec, R. (2017). Probalistic and piecewise deterministic models in biology . In: Journées MAS 2016. Phénomènes complexes et hétérogènes (p. 1-19). Presented at Journées MAS 2016. Phénomènes complexes et hétérogènes, Grenoble, FRA (2016-08-29 -2016-08-31).

Acknowledgements Bertrand Cloez would like to thank Romain Yvinec for his enthusiastic and stimulating organization of the "Journées MAS 2016" and Florent Malrieu for many interesting discussions on the subject of his section. The research of Bertrand Cloez was partially supported by the Agence Nationale de la Recherche PIECE 12-JS01-0006-01 and by the Chaire "Modélisation Mathématique et Biodiversité " . The research of Aline Marguet was partially supported by the Chaire "Modélisation Mathématique et Biodiversité". Renaud Dessalles would like to thank Philippe Robert and Vincent Fromion for their support.


Mélanie Deschasaux

Chantal Julia

Laurent Zelek

Emmanuelle Kesse-Guyot

Véronique Gourlet

Lucie Lecuyer

Caroline Méjean

Pauline Ducrot

Sandrine Peneau

Paule Latino-Martel

Prospective association between breast cancer risk and an individual dietary index based on the British Food Standards Agency nutrient profiling system

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Jonathan Lorenzo

Bryan Brancotte

Thomas Lacroix
email: thomas.lacroix@inra.fr

Mohamed Bedri

Jean-Francois Gibrat

Christophe Blanchet

Thomas Lacroix2

Mohammed Behri

Jean-François Gibrat

Multi-Cloud deployment for microbial genomes analysis

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The goal is to deploy the "appliance" in one click over one or more cloud infrastructures. To achieve this, new features to automate deployment of complex application were added to the IFB's cloud portal through the connection to the SlipStream cloud broker. Developed by SixSq, SlipStream is a multi-cloud application management platform. It automates the full application management lifecycle, within Infrastructure as a Service (IaaS) cloud infrastructures. Such complex application deployments can be done over several cloud infrastructures and provide scientists with high-level cloud features such as the dynamic allocation of a dedicated network for the isolation of the virtual machines, with the replication of the user data and with a direct link from the cloud portal to the Insyght web portal.

Conclusion

  adjust resources and target cloud (**) Access by eduGAIN after authentication in the federation identity (no ssh-key needed) or by ssh-key for developer Run analysis


Helene Yken

Sylvain Derick

Hélène Camille Gironde

Taiana Darius

Christophe Furger

Dominique Laurent

Jean-Marie François

Mireille Chinain

Hélène Taiana

Development of new biosensors to detect ciguatoxins

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Future developments : Express the Mammalian VGSC in yeast, per se or as chimera with the yeast ion channel, and follow cellular signaling induced. Measure binding of the different toxins to the receptors by Single Molecule Force Spectroscopy (by AFM or Optical Tweezers). Develop cell-free sensors systems with the toxins receptor (human VGSC or its yeast homolog) integrated in lipid bilayers. Create point mutants of the receptors, to identify the residues involved and then design or search for possible cures.

Hélène Martin-Yken 1 , Sylvain Derick 2 , Camille Gironde 2 , H. Taiana Darius 3 , Christophe Furger 2 , Dominique Laurent 4 , Jean-Marie François 1 and Mireille Chinain 3 .

Development of New Biosensors to detect Ciguatoxins

Design of new biosensors based on yeast cells:

-Why Yeast cells ? well-known, easy to genetically modify, eukaryotic = good conservation of signaling pathways with higher eukaryotes, fast and easy to grow.

-

On-going improvements: Replace colorimetry by Fluorescence, or other value easily measurable in the field Use cell wall and signaling mutants to improve CTXs access and amplify the signal Mutagenesis on the channel protein to improve CTXs binding


Mathilde Le Sciellour

Isabelle Hochu

Olivier Zemb

Juliette Riquet

Hélène Gilbert

Mario Giorgi

Yvon Billon

Jean-Luc Gourdine

David Renaudeau

Effect of heat stress on faecal microbiota composition in swine: preliminary results

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

542

EAAP -69 th Annual Meeting, Dubrovnik 2018

The socio-economic evaluation of vaccines in livestock systems C. Bellet and J. Rushton Institute of Infection and Global Health / University of Liverpool,IC2 Building,146 Brownlow Hill,L3 5RF,Liverpool,United Kingdom;camille.bellet@liverpool.ac.uk Socio-economic evaluations provide a foundation to define resource allocation priorities and disease management strategies, which in turn help to tailor interventions to the specific context of a livestock system. This presentation examines how socio-economic evaluations of vaccines are performed in animal health research in relation to three European cases of respiratory and digestive diseases in cattle, pig and poultry. It explores their limitations and the potential of transdisciplinary approaches to support more robust evaluations of livestock vaccination interventions.

Vaccines not only help to protect against acute diseases but also against chronic diseases that may not hit the headlines, but which are important for livestock businesses and society. Vaccines also benefit unvaccinated animals, their owners and consumers by decreasing disease transmission, which in turn reduces the risk of epizootics and the pressure generated by these epizootics on health care providers and market economy. In addition, vaccines impact on drug use reducing selection pressure for drug resistance, and improving drug efficacy in humans. However, for vaccines to generate these benefits, the associated vaccination strategies need to protect critical proportions of animals on a given time scale. These strategies depend on vaccine quality, availability, accessibility and acceptability to farmers, their animal health advisors, and increasingly consumers. The growing complexity of vaccines, vaccine delivery and their relationship to long and diverse food systems demand interdisciplinary and integrative approaches that address human, animal and ecosystem dimensions. Yet current socio-economic evaluations of livestock vaccines remain mostly limited to one of the many dimensions of a disease problem such as the reduction of disease incidence. The presentation concludes that there is a need to unpack the dynamics of farm practices and the food system to understand how these shape animal health management and vaccine intervention. This requires a transdisciplinary approach with a depth of different forms of expertise and data sources in order to achieve informed evaluations on the trade-offs of vaccine intervention. URZ,INRA,France;mathilde.lesciellour@inra.fr Gut microbiota plays a central role in health and nutrient digestion and would help the host for better coping with environmental perturbations. In tropical conditions or in temperate countries during Summer, elevated ambient temperatures can cause economic losses to the pig industry. During heat stress (HS), the reduction in voluntary feed intake is the main adaptation response for reducing heat production. This lower feed intake has subsequent negative effects on pig performance. The main purpose of this study was to investigate the relationships between HS and gut microbiota composition. A better understanding of the microbiota response to HS could allow the selection for animals well adapted to HS. Genetically related pigs were raised under temperate or tropical farm conditions with mean thermal humidity indexes respectively 23 and 25.5 from 11 to 23 weeks of age. In temperate conditions, pigs were submitted to a 3-week HS challenge at 30 °C. Faecal samples were collected in all pigs at 23 weeks of age in both environments (n=1,200 samples) and at 26 weeks of age in the temperate environment (n=600). Therefore, it was possible to compare microbiota from pigs raised in a temperate environment, a tropical climate, and exposed to HS. Microbiota extracted from pigs under temperate and tropical climate had different compositions whereas pigs exposed to heat challenge or raised in tropical conditions tended to share a common microbiota. HS challenge drastically modified gut microbiota and the groups before and after the challenge could be predicted in a multilevel sparse partial least square discriminant analysis with 30 OTUs and a mean classification error rate of 14%. Our experiment suggests that microbiota can be used as biomarkers of HS exposition. This study is part of the Feed-a-Gene Project funded by the European Union's H2020 Program (grant 633531), and of the PigHeat project funded by the French National Agency of Research (ANR-12-ADAP-0015).

Effect of heat stress on faecal microbiota composition in swine: preliminary results

M. Le Sciellour

Session 56

Theatre 8

Session 56 Theatre 7

  1 , I. Hochu 2 , O. Zemb 2 , J. Riquet 2 , H. Gilbert 2 , M. Giorgi 3 , Y. Billon 4 , J.-L. Gourdine 5 and D. Renaudeau 1 1 PEGASE, INRA, Agrocampus-Ouest, 35042 RENNES, France, 2 GenPhySE, Université de Toulouse, INRA, INPT, INP-ENVT, 31320 Castanet Tolosan, France, 3 PTEA, INRA, 97170 Petit-Bourg, France, 4 GenESI, INRA, 17700 Surgères, France, 5


Efficiency of different BLUP indexes accounting for polyandry in bee genetic evaluation

Bastien Gravellier, Florence Phocas

To cite this version: 2017-08-28 -2017-08-31).

Efficiency of different BLUP indexes accounting for polyandry in bee genetic evaluation

B. GRAVELLIER*, F. PHOCAS INRA, UMR 1313 GABI, 78350

Jouy-en-Josas, France.

The best linear unbiased prediction (BLUP) applied to an animal model is the standard method for genetic evaluation of domestic livestock. It requires modelling the relationship matrix between all the animals registered in the pedigree of candidates for selection. For diploid species the derivation of relationship coefficients are well established. But the honey bee (Apis mellifera) is a haplodiploid species in which several (10 to 20) haploid drones fertilize a diploid queen. In consequence, the father of a female progeny is in most cases unknown (with the exception of a queen insemination with the semen of a single drone). The bee genetic (male haploïdy) and reproductive (polyandry) specificities have therefore to be accounted for in the derivation of the relationship matrix to model relevant paternal gene flows and uncertain paternity of the workers and queen of the colony. The objective of the study was to compare different approaches for deriving the relationship matrix to estimate the bee breeding values by BLUP indexes. The simplest approach is to ignore the paths of paternal gene transmission in a similar way that in diploid species paths of maternal gene transmission have classically been ignored under BLUP -sire model considering a relationship matrix build from triplets "animal -sire -maternal grandsire" in the pedigree file. In a symmetric approach for bee a dam model can be considered with a pedigree file composed of triplets "animal -dam -paternal grand dam". Such a simplified model was compared to different approaches proposed in the literature to model bee relationship matrix under an animal model. In these last strategies, the group of drone-producing queens is considered as a dummy father, the drones being the male gametes. A stochastic simulation was used to produce a bee population of 5000 queens with known phenotypes and true genetic values for a trait of heritability varying from 0.1 to 0.5. The different BLUP indexes were computed assuming various degree of polyandry. Correlations between true and estimated breeding values were derived in order to evaluate the impacts of the parameters describing the paternal origins. The number of drone producing queens appeared to have the highest impact on the estimates.

  Comment citer ce document : Gravellier, B., Phocas, F. (2017). Efficiency of different BLUP indexes accounting for polyandry in bee genetic evaluation. In: Book of Abstracts of the EEAP annual meeting, Tallinn 2017.

Bastien Gravellier, Florence Phocas. Efficiency of different BLUP indexes accounting for polyandry in bee genetic evaluation. 68. Annual Meeting of the European Association for Animal Production (EAAP), Aug 2017, Tallinn, Estonia. hal-02736780 Presented at 68. Annual Meeting of the European Association for Animal Production (EAAP), Tallinn, EST (