. Osmotic stress technique [Parsegian, Method. Enzymol., 1986] = casein micelle from "native phosphocaseinate powder", NPC

  final quality of products (safety, functional, sensorial and nutritional) To contribute to create new products / concepts (competitivness of the industry) L h -1 m -2  w = 100 Pa Ceramic membrane 0.1µm, UTP system, VRR=2, T= 50°C Le Berre et Daufin, 1996; Gésan-Guiziou et al., validate these models at the local and global scales To Control, Improve and Predict the process performances and the product properties

•

  the kinetics of accumulation at the membrane surface ? -What are the structure and organisation of the matter in these concentrated layers spatio-temporal evolutions of the concentration and structure of micelles in the accumulated layers at the membrane surface -Promising methodology => to better understand and model filtration Langmuirto determine a priori & precisely the parameters of spray drying for food products  necessity of expensive and time consuming experiments -One major difficulty: drying is non isenthalpe ( drying of « pure » water in ideal conditions) the product is « bounded » to constituants  the removal of water requires extra energy  Decrease in drying performance Questions -How to evaluate the energy required to compensate the association of water with the product ? -How can we predict the spray drying parameters ? and energy balances established over the entire spray dryer  prediction of parameters  1-5 % DE Software SD 2 P : Spray Drying Parameters Simulation & Heat treatment (pasteurization to UHT-sterilization) • Homogenization, cream separation, melting… • Membrane separation modules :  Microfiltration  Ultrafiltration  Nanofiltration and Reverse osmosis • Two cheese production lines (soft-and hard-type cheeses) • Spray drying facilities : pilot (1-7 kg of water per hour) and semi-industrial (BIONOV) units Platform : wide and flexible range of processing technologies Dairy Technology Platform

J. Food Eng. 2009 Drying Technol. 2011

Availability of well equipped and characterised pilots

 Dairy Technology Platform


domain: sdv shs

text:
François Casabianca

Panel III -Raw material production -Production strategies based upon the respect of environment

Following the example of Iberian pigs reared in Dehesa system, several areas of Mediterranean Europe have spent efforts in recovering traditional local pig breeds, characterized by slow growing and high adiposity. The strategy of local actors is focusing on the pigs' way of life for ensuring high added value for dry cured hams. Sometimes, the traditional production system was still in use with sylvopastoral logics (chestnuts and acorns consumption). But in most cases, breeders had to re-invent production systems in order to valorize such animals, emphasizing the free range in forests and pastureland areas and the role of grass income. We present some situations from France and Italy where Geographical Indication is applied or obtained. The main traits of the production systems and the role of the natural resources in such specifications are underlined. Stocking rates but also types and quantities of additional feedstuffs are often required as criteria for managing raw material qualities but not really impact of pigs on their environment. Seasonality of production with a compulsory use of forests during a finishing period seems to be a main issue for a sustainable and environment-friendly way. We discuss the potential unbalanced use of natural resources, and the importance of control plan for preventing environment wastes. Outdoor production fashion induced by modern consumer preferences should take into account the potential negative consequences on the natural resources.

Introduction

The example of Iberian pigs reared in Dehesa system and dedicated to dry cured ham production -the famous "pata negra"of high level of quality, has given some new credibility to the local pig breeds still existing. These traditional local pig breeds are characterized by slow growing and high adiposity, considered as defects for the conventional lean pig industry. So, numerous traditional breeds already disappeared and the remaining breeds' census has shown severe decreasing. Due to the Iberian example, several areas of Mediterranean Europe have spent efforts in recovering their local traditional pig breed oriented toward the production of dry cured hams [START_REF] Pugliese | Quality of meat and meat products produced from southern European pig breeds[END_REF]. We decided to focus our analysis on the French and Italian situations where a production of dry cured ham is connected to the mandatory use of local pig breeds in Geographical Indication.

Rationale

For most of these local traditional pig breeds, the strategy of local actors is focusing on the pigs' way of life for ensuring high added value for dry cured hams. A large part of the arguments and communication are showing the pigs in their natural environment. As targeted consumer cluster, urban people ready to pay more for typical products has integrated that such outdoor production seems better for the pigs' welfare, in opposition to the industrial pig production where animals are reared in indoor units without any contact with natural resource.

Sometimes, the traditional production system was still in use with sylvo-pastoral logics (chestnuts and acorns consumption) and the outdoor production system is more or less a modern adaptation of the traditional system. But in most cases, breeders had to re-invent production systems in order to valorize such animals, emphasizing the free range in forests and pastureland areas and/or the role of grass income.

To what extent the outdoor pig production of local traditional breeds dedicated to typical dry cured hams is using the natural resources in a sustainable way?

Material and methods

We present some situations from France and Italy where Geographical Indication is applied or obtained. In France, 3 cases with 2 of them are still applying, "Noir de Bigorre" ham (NB) based on the Gascon pig breed and "Kintoa" ham (KT) based on the Basque pig breed; and one already obtained at national level and to be registered very soon at European level ("Corsican" ham (CO) names "prisuttu" based on the Nustrale pig breed. In Italy, 2 cases with one still applying, "Nero dei Nebrodi" ham (NN) named "prippuni" based on the "Nero Siciliano" pig breed; and one already registered at European level, "Cinta Senese" pork meat (CS) based on the Cinta Senese pig breed and used for producing dry cured hams.

The main traits of the production systems and the role of the natural resources in such specifications are object of analysis. The code of practices is interesting to consider as a way to connect the animals with their natural environment and, at the same time, as a way to assess the impact of herds on the areas as well as the impact back on the animals for raw material quality and for health and welfare.

Main findings

A linkage between raw material qualities and the use of natural resources

In the usual presentation of the traditional local breeds, genetics seem to be determinant for the typical characteristics of the hams. But, if so, terroir does not exist anymore. The very first justification of protecting the denomination is the genotypeenvironment interaction. Intra-muscular fat is often presented as an evidence of compensatory growth expression of such interaction. Several values are proposed for the various hams: 6% for CO and 2.5% for NB and CS. For KT, high level of IMF is mentioned but without any targeted value. No indication is mentioned for NN.

According to the work of [START_REF] Petron | Effect of duration of the Montanera diet on the hydrocarbon fraction of intramuscular lipids from Iberian dry-cured ham; characterization by gas chromatography[END_REF], neophytadiene content is presented as evidence for grazing activity. This molecule is mentioned for NB, with an obligation for the 6 last months of finishing period to let the pigs on pasturelands. But no targeted value is given for the moment. And it seems to disappear all along the ageing of the ham.

Oleic acid is given as an evidence for the consumption of acorns and chestnuts with a minimum value of 45% for the CO. For NB, a minimum of 51% of monounsaturated acids is required with a minimum of 47% of oleic acid.

Seasonality of production with a compulsory use of forests during a finishing period seems to be a main issue for a sustainable and environment-friendly way, only for the CO. The finishing period (autumn) is determining the slaughtering period (end of November till end of March).

As another issue of the analysis of the 5 specifications, the types and quantities of additional feedstuffs are often required as criteria for managing raw material qualities.

CO has requirements for the feedstuffs content. Maize is forbidden for the last 90 days before slaughter. For NB, feedstuff must contain at least 70% of cereals and less than 3kg are distributed by animal by day. In several cases, GMO are forbidden for environmental but also commercial reasons.

For CS, we notice a limited feedstuff distribution: the permitted daily food ration is allowed for pigs over four months of age, and must not exceed 3 % of the live weight of the animal so as to guarantee the animals a normal dietary input. In the case of CO, after one month of finishing pasture with only chestnuts and/or acorns, the producers are allowed to distribute barley with a maximum of 4 kg/day/pig. But, in this case, they lose the mention "only with chestnuts and acorns" supposed to give more added value to the dry cured ham.

Few elements for the management of natural resources

But the code of practices is not really including the impact of pigs on their environment. In general, GI specifications are not conceived as a way to manage such impact and public authorities are refusing to include into the code of practices any element of livestock management not directly responsible of the product final quality.

As an indirect influence, we could consider that natural resources are under control by the stocking rates. For NB, 25 pigs by hectare of grassland. For KT, 40 pigs by hectare as a maximum. For CS, the animals intended for slaughter must be reared in the wild/semi-wild once they are four months old. The maximum limit per hectare is 1 500 kg live weight. The pigs must be allowed to forage daily in areas of land that are either fenced off or not, and that may include shelters for the night and/or for inclement weather. No precision is given for NN. And for CO, apparently more extensive, only 5 pigs by hectare are allowed during growing period and for the finishing period 8 pigs by hectare is a maximum. Such rule seems completely natural for the breeders according to their experience.

In the specification, the criteria for identify the pasturelands have to be analyzed. For CS, the "defined geographical area is distinguished by its mixed woodland, rich in types of oak tree, perfect for acorn production, and for its small amount of arable land". For NN, free range is required after 5 month but more classical outdoor system is allowed. For KT, this obligation starts at 7 month old. And the pasturelands are of 2 types: i) pasture with grass available for the pigs (spontaneous or cultivated grass) and ii) forests with chestnut-trees and/or oak-trees.

Finally, we must mention the level of activities for the animals obliged to explore the lands and to make some km per day. Such activity is evidenced by the intense color of the muscles [START_REF] Lopez-Bote | Effect of exercise on skeletal muscle proteolytic enzyme activity and meat quality characteristics in Iberian pigs[END_REF]. It should be objectified by the heminic iron, but no targeted value is given in the specifications.

Discussion

Diversity of situations and need for assessment

We discuss the potential unbalanced use of natural resources. A distinction has to be made between traditional production systems and re-invented systems. Field studies can show degradation of the forests and pastureland by the herds, in particular during the wet period. So, we must emphasize the importance of control plan for preventing environment wastes. For example, Pugliese (Com. Pers.) as member of the certification body for CS, observed several problems in assessing the stocking rate as animals' weight is varying a lot and very quickly.

Global or local environmental issues?

Life-cycle analysis can be proposed as a relevant method for comparing several production systems. One situation (KT) has been part of a study for environmental impact. Following the methodological proposal from [START_REF] Edwards | Product quality attributes associated with outdoor pig production[END_REF], the study has compared conventional intensive systems with traditional ones (KT and 2 other systems) and organic ones within a European project (QPorkChain). [START_REF] Dourmad | Évaluation par analyse de cycle de vie de la durabilité environnementale de systèmes contrastés de production porcine en Europe[END_REF] reported the results of such comparison. By estimating the equivalent of Green House Gas emission by animal, the traditional systems show a very bad performance: They are producing a lot of GHG (+54% in average) when expressed by kg of pork meat. But, expressed as equivalent of GHG by hectare, traditional systems using a great surface of pasturelands and forests show an excellent performance. So, the way to calculate the impact (by animal vs by hectare) is influencing a lot the results of our traditional systems. And we could say with [START_REF] Dourmad | Évaluation par analyse de cycle de vie de la durabilité environnementale de systèmes contrastés de production porcine en Europe[END_REF], that such traditional systems are more sustainable for the local or territorial impact than for the global impact.

Conclusion

We need to improve our methodologies for assessing the environmental impact of pig production systems when based upon natural resources [START_REF] Edwards | Perception and reality of product quality from outdoor production systems in Northern and Southern Europe[END_REF]. The future of this association local breed and outdoor system is not only commercial (as dry cured hams are sold at very good prices) but also environmental.

Methods like life-cycle analysis should be developed in the next future in order to help managing the impact of pig production on natural resources. This issue is of real interest as outdoor pig production is becoming a real fashion induced by modern consumer preferences. But, such new value scale commonly accepted in our societies should take into account the potential negative consequences on the natural resources.


domain: math info

text:
During 2010 influenza season the previous circulating A/H3N2 subtypes came back

Context and motivation (2)

Three models have been proposed to explain the distinctive phylodynamic pattern observed in human A/H3N2 viruses: viral strains are characterized by a short-lived strain-transcending immunity (Ferguson et al., Nature 2003) HA evolves in a punctuated manner among antigenic types that are linked by a network of neutrally evolving sites (Koelle et al., Science 2006) the virus continually reuses a limited number of antigenic combinations (Recker et al., PNAS 2007) Hypothesis explaining influenza phylodynamics supported by complex simulations -→ difficult to assess the consequences of modelling assumptions -→ difficult to understand the conditions supporting co-circulation and replacement dynamics following the occurence of a new pandemic strain

Outline

My lab: research topics

Influenza charateristics and motivation for this work

Simple models for multi-strain dynamics Influenza A gradual and epochal evolution: insights from simple models

Framework and conditions of coexistence and replacement in the case of a pandemic Notation and construction: HBRS model (Andreasen et al., JMB 1997) Ω = {1, 2, 3, ..., n} the set of all possible strains R J the set of individuals currently uninfected and previously infected by the strains in the set J ⊂ P(Ω) -→ 2 n classes R∅ the set of individuals who had not encountered any infections I i J the set of individuals currently infected by strain i and previously recovered from infectious with strains in J -→ n2 n-1 classes J\j = J\{j} β i transmission rate of strain i σ i J reduction in suceptibility when infected with strain i when immune history

J (σ i ∅ = 1, σ i J ≤ σ i L for L ⊆ J) ν i recovery rate from infection with strain i ṘJ = X j∈J ν i I j J\j - X i / ∈J σ i J R J β i X J⊆Ω\i I i J İ J i = σ i J R J β i X J⊆Ω\i I i J -ν i I i
Notation and construction: SBRS model (Gog and Swinton, MB 2002)

R J the set of individuals currently uninfected and immune to the strains in the set J ⊂ P(Ω) -→ 2 n classes R∅ the set of individuals immune to none strain I i the set of individuals currently infected by strain i -→ n classes C (K , J, i) the proportion of hosts that recover to a state J havong started in K and been infected by strain i Polarized immunity and reduced-infectivity (Gog and Grenfell, PNAS 2002)

(i / ∈ K , i ∈ J, K ⊂ J, P J C (K , J, i) = 1) ṘJ = X i,K C (K , J, i)β i I i R K - X i / ∈J β i I i R J İ i = β i I i X J:i / ∈J R J -ν i I i
Ṙ∅ = -β1R∅I 1 -β2R∅I 2 Ṙ1 = (1 -σ)β1R∅I 1 -β1R1I 1 + (1 -σ)β1R1I 1 -β2R1I 2 Ṙ2 = (1 -σ)β2R∅I 2 -β2R2I 2 + (1 -σ)β2R2I 2 -β1R2I 1 Ṙ 12 = σβ1R∅I 1 + σβ2R∅I 2 + σβ1R1I 1 + σβ2R2I 2 + β2R1I 2 + β1R2I 1 İ 1 = β1R∅I 1 + β1R2I 1 -νI 1 İ 2 = β2R∅I 2 + β2R1I 2 -νI 2
Reformulating the model (new variables S i ) -→ order reduction:

S 1 = R ∅ + R 2 (hosts susceptible to strain 1) S 2 = R ∅ + R 1 (hosts susceptible to strain 2)
2K variables instead of 2 K However, cross-immune boosting appears to be problematic

Polarized immunity and reduced-infectivity (Gog and Grenfell, PNAS 2002)

Ṙ∅ = -β1R∅I 1 -β2R∅I 2 Ṙ1 = (1 -σ)β1R∅I 1 -β1R1I 1 + (1 -σ)β1R1I 1 -β2R1I 2 Ṙ2 = (1 -σ)β2R∅I 2 -β2R2I 2 + (1 -σ)β2R2I 2 -β1R2I 1 Ṙ 12 = σβ1R∅I 1 + σβ2R∅I 2 + σβ1R1I 1 + σβ2R2I 2 + β2R1I 2 + β1R2I 1 İ 1 = β1R∅I 1 + β1R2I 1 -νI 1 İ 2 = β2R∅I 2 + β2R1I 2 -νI 2
Reformulating the model (new variables S i ) -→ order reduction:

S 1 = R ∅ + R 2 (hosts susceptible to strain 1) S 2 = R ∅ + R 1 (hosts susceptible to strain 2)
2K variables instead of 2 K However, cross-immune boosting appears to be problematic

Outline

My lab: research topics

Influenza charateristics and motivation for this work

Simple models for multi-strain dynamics Influenza A gradual and epochal evolution: insights from simple models

Framework and conditions of coexistence and replacement in the case of a pandemic

Framework and conditions of coexistence and replacement in the case of a pandemic Model (1): main modelling assumptions, key processes of influenza dynamics

One subtype Model (1): main modelling assumptions, key processes of influenza dynamics

R∅ E 1 1 E 1 2 I 1 1 I 1 2 Q 1 R1 inclusion of
One subtype Model (1): main modelling assumptions, key processes of influenza dynamics

R∅ E 1 1 E 1 2 I 1 1 I 1 2 Q 1 R1 antigenic drift
Invasion of the second subtype

R∅ E 1 1 E 1 2 I 1 1 I 1 2 Q 1 R1 antigenic drift

Pandemics ( 1 )Figure

 1 Figure courtesy of S. Ballesteros

Outline

  My lab: research topics Influenza charateristics and motivation for this work Simple models for multi-strain dynamics Influenza A gradual and epochal evolution: insights from simple models Framework and conditions of coexistence and replacement in the case of a pandemic Influenza A gradual and epochal evolution: insights from simple models Ballesteros S, Vergu E, Cazelles B. PLoS One 2009, 4(10):e7426 Comparison of the consequences of SB and HB formulations with RS and RI assumptions on the transient dynamics of antigenic clusters Study of the invasion of a new antigenic cluster within a population where a resident antigenic cluster is at the endemic equilibrium Illustration on a two-strain model ), SBRI (middle), HB (right) models Infection with strain 1 (red arrows) and with strain 2 (blue arrows). im. escape) Cluster replacement only for the SBRI model Vergu et al. (MIA UR341, INRA) im. escape) Cluster replacement only for the SBRI model Invasion condition for the mutant cluster Easily obtained after algebraic manipulation from İ 2 > 0 SBRI model: 1 + σ(β/ν -1)/((1 -σ)(β/ν -1) + 1) all other models: 1 + σ(β/ν -1) β: transmission rate (assumed equal for both strains); ν: recovery rate; σ ∈ [0, 1]: immune escape rate

Vergu,

  Ballesteros et al., work in progress Another formulation than the problematic SBRI model -→ HB model New assumptions are necessary Aim: exploring the possible outcomes after the introduction of the new virus To provide a general framework capturing the dynamics of co-circulating subtypes To characterize replacement and coexistence conditions in terms of basic reproductive number, R 0 illustrates the ability of a pathogen to invade a population antigenic drift rate ( notation = g = 1/D ) describes the continuous process of genetic and antigenic change among flu strains accounts for the virus ability to escape immune system seasonal forcing underlying biological mechanisms

  inclusion of a latent state E not realistic assumption of exponential distribution for latency and infectious durations -→ Erlang distribution temporary period of full cross protection (Q)(Ferguson et al.,

I2

  inclusion of a latent state E not realistic assumption of exponential distribution for latency and infectious durations -→ Erlang distribution temporary period of full cross protection (Q) (Ferguson et al., pandemic subtype → co-circulation of three subtypes interacting only via a temporary period of full cross-immunity (Q); no co-infections Model (1): main modelling assumptions, key processes of influenza dynamics Co-circulation of three subtypes Additional important assumption: seasonality in transmission → the transmission parameter has a sinusoidal formulation Three age classes within each city realistic contact matrix M and demographic data (N is the population size and pa i is the proportion of population in age class a i ) an Euler-multinomial approximation (Breto et al., Ann App Stat 2009) Number of state variables: A * C * (4 * n * 2 n-1 + 2 * 2 n ) = 3 * 52 * 64 = 9984 where A is the number of age classes, C the number of cities in the network, n the number of strains Preliminary study important to account for co-circulation processes the impact of R0, D and seasonality on invasion and replacement strongly depends on the number of subtypes in competition Vergu et al. (MIA UR341, INRA) Influenza dynamics ANR MANEGE 35 / 43 Analysis (2): R 0 fixed, D variable Distinct regions in parameter space corresponding to several outcomes replacement obtained for plausible ranges of parameter values: R0 = 1.5, D = 3y, low seasonality large area of replacement for R0 = 5 and high seasonality (counterbalancing mechanisms) extinction after replacement obtained for large ranges of parameters: (?) need of additional immunity based mechanisms Analysis (3): trajectories of global incidence dynamics for realistic parameter values -→ replacement Plausible patterns in terms of peaks succession and amplitude replacement of the seasonal subtypes by the new pandemic subtype two pandemic waves in Southern hemisphere, one large wave in Northern zones followed by a refractory period Analysis (4): other trajectories of global incidence dynamics Various patterns in terms of peaks succession and amplitude and with respect to the geographic zone for high R0 and low seasonality: depletion of susceptibles after the first epidemic wave leading to quasi-extinction the high seasonality: (i) stops the tansmission and prevents susceptibles pool exhaustion, (ii) has an impact on the initial conditions of the new sub-type emergence Results (6): R 0 and D fixed, Age susceptibility variable Majority of extinctions without replacement but also successful replacement replacement could occur for different values of susceptibility of elderly given that adults are completely susceptible

Vergu et al. (MIA UR341, INRA) Influenza dynamics ANR MANEGE

Metapopulation of 52 major cities in the world coupling of local dynamics through transportation flows (daily air passengers) → weighted network of mean degree of connectivity=8 (838 connexions / 2652) Model ( 5): simulation of the continuous time Markov chain via the limit of coupled discrete-time multinomial processes with random rates Breto et al., Ann App Stat 2009 X (t) = (X1(t), ..., Xc (t)) is the vector-valued process denoting the (integer or real-valued) counts in each of c compartments.

Multinomial(X i (nδ), p i1 , ..., p i,i-1 , p i,i+1 , ..., p ic , 1 -

Analysis of simulations

Exploration of large ranges of values for the basic reproductive number, R0 the antigenic drift rate, 1/D the susceptibility σ of specific age-classes to the new subtype in terms of most likely event (on average) after the introduction of a new subtype in a two-subtype system at equilibrium (i.e. for each parameter combination, the proportion of trajectories leading to a specific outcome: coexistence, replacement, etc) Analysis ( 5


Romain Yvinec
email: yvinec@math.univ-lyon1.fr

Adiabatic reduction of models of stochastic gene expression with bursting

Keywords: AMS 2000 subject classifications: Primary 92C45, 60Fxx; secondary 92C40, 60J25, 60J75 adiabatic reduction, piecewise deterministic Markov process, stochastic bursting gene expression, quasi-steady state assumption, scaling limit

This paper considers adiabatic reduction in both discrete and continuous models of stochastic gene expression. In gene expression model, the concept of bursting is a production of several molecules simultaneously and is generally represented as a jump terms of random size. In a general two-dimensional birth and death discrete model, we prove that under specific assumptions and scaling (that are characteristics of the mRNAprotein system) an adiabatic reduction leads to a one-dimensional discretestate space model with bursting production. The burst term appears then through the reduction of the first variable. In a two-dimensional continuous model, we also prove that an adiabatic reduction can be performed in a stochastic slow/fast system. In this gene expression model, the production of mRNA (the fast variable) is assumed to be bursty and the production of protein (the slow variable) is linear as a function of mRNA. When the dynamics of mRNA is assumed to be faster than the protein dynamics (due to a mRNA degradation rate larger than for the protein) we prove that, with the appropriate scaling, the bursting phenomena can be transmitted to the slow variable. We show that the reduced equation is either a stochastic differential equation with a jump Markov process or a deterministic ordinary differential equation depending on the scaling that is appropriate.

These results are significant because adiabatic reduction techniques seem to have not been applied to a stochastic differential system containing a jump Markov process. Last but not least, for our particular system, the adiabatic reduction allows us to understand what are the necessary conditions for the bursting production-like of protein to occur.

Introduction

The adiabatic reduction techniques give results that allow to reduce the dimension of a system and justify the use of an effective set of reduced equations in lieu of dealing with a full, higher dimensional model, if different time scales occur in the system. Adiabatic reduction results for deterministic systems of ordinary differential equations have been available since the very precise results of [START_REF] Tikhonov | Systems of differential equations containing small parameters in the derivatives[END_REF] and [START_REF] Fenichel | Geometric singular perturbation theory for ordinary differential equations[END_REF]. The simplest results, in the hyperbolic case, give an effective construction of an uniformly asymptotically stable slow manifold (and hence a reduced equation) and prove the existence of an invariant manifold near the slow manifold, with (theoretically) any order of approximation of this invariant manifold. Such precise and geometric results have been generalized to random systems of stochastic differential equation with Gaussian white noise [START_REF] Berglund | Noise-Induced Phenomena in Slow-Fast Dynamical Systems,A Sample-Paths Approach[END_REF], see also [START_REF] Gardiner | Handbook of stochastic methods, for physics, chemistry and the natural sciences[END_REF] for previous work on the Fokker-Planck equation). However, to the best of our knowledge, analogous results for stochastic differential equations with jumps have not been obtained.

The present paper gives a theoretical justification of an adiabatic reduction of a particular piecewise deterministic Markov process [START_REF] Davis | Piecewise-Deterministic Markov Processes: A General Class of Non-Diffusion Stochastic Models[END_REF]. The results we obtain do not give a bound on the error of the reduced system, but they do allow us to justify the use of a reduced system in the case of a piecewise deterministic Markov process. In fact, we prove limit theorems using martingale strategy, in a similar manner than in recent papers such as [START_REF] Crudu | Convergence of stochastic gene networks to hybrid piecewise deterministic processes[END_REF], Kang and Kurtz and Riedler, Thieullen and Wainrib (2012), where general convergence results for discrete models of stochastic reaction networks are given. In particular, these papers give alternative scaling of the traditional ordinary differential equation and the diffusion approximation depending on the different scaling chosen (see [START_REF] Ball | Asymptotic Analysis of Multiscale Approximations to Reaction Networks[END_REF] for some examples in a reaction network model). After the scaling, the limiting models can be deterministic (ordinary differential equation), stochastic (jump Markov process), or hybrid (piecewise deterministic process). For illustrative and motivating examples given by a simulation algorithm, see [START_REF] Haseltine | Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics[END_REF]; [START_REF] Rao | Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm[END_REF]; [START_REF] Goutsias | Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems[END_REF]. However, we emphasize that we do not consider here a continuous approximation of a discrete model. Rather, we perform adiabatic reduction on both discrete state-space and continuous state-space models. Time-scale reduction have been considered in Kang and Kurtz, but not on the kind we perform here.

Our particular model is meant to describe stochastic gene expression with explicit bursting [START_REF] Friedman | Linking Stochastic Dynamics to Population Distribution: An Analytical Framework of Gene Expression[END_REF]. In discrete state-space bursting models, the variables evolve under the action of a discrete birth and death process, interrupted by discrete positive jumps of random sizes. In continuous state-space bursting models, the variables evolve under the action of a continuous deterministic dynamical system, interrupted by positive jumps of random sizes. In both cases, the positive jumps model the burst production of several molecules instantaneously. In that sense, the convergence theorems we obtain in this paper can be seen as an example in which there is a reaction with size between 0 and ∞. We hope that the results here are generalizable to give insight into adiabatic reduction methods in more general stochastic hybrid systems [START_REF] Hespanha | Modelling and analysis of stochastic hybrid systems[END_REF][START_REF] Bujorianu | General stochastic hybrid systems: modelling and optimal control[END_REF]. We note also that more geometrical approaches have been proposed to reduce the dimension of such systems in [START_REF] Bujorianu | Symmetry reduction for stochastic hybrid systems[END_REF].

Biologically, the bursting of mRNA or protein molecules is defined as the production of several molecules within a very short time, indistinguishable within the time scale of the measurement. In the biological context of models of stochastic gene expression, explicit models of bursting mRNA and/or protein production have been analyzed recently, either using a discrete [START_REF] Shahrezaei | Analytical distributions for stochastic gene expression[END_REF][START_REF] Lei | Stochastic in single gene expression with both intrinsic noise and fluctuation in kinetic parameters[END_REF] or a continuous formalism [START_REF] Friedman | Linking Stochastic Dynamics to Population Distribution: An Analytical Framework of Gene Expression[END_REF][START_REF] Mackey | Molecular distributions in gene regulatory dynamics[END_REF] as more and more experimental evidence from single-molecule visualization techniques has revealed the ubiquitous nature of this phenomenon [START_REF] Ozbudak | Regulation of noise in the expression of a single gene[END_REF][START_REF] Golding | Real-Time Kinetics of Gene Activity in Individual Bacteria[END_REF][START_REF] Raj | Stochastic mRNA Synthesis in Mammalian Cells[END_REF][START_REF] Elf | Probing Transcription Factor Dynamics at the Single-Molecule Level in a Living Cell[END_REF][START_REF] Xie | Single-Molecule Approach to Molecular Biology in Living Bacterial Cells[END_REF][START_REF] Raj | Single-Molecule Approaches to Stochastic Gene Expression[END_REF][START_REF] Suter | Mammalian Genes Are Transcribed with Widely Different Bursting Kinetics[END_REF]. Traditional models of gene expression are composed of at least two variables (mRNA and protein, and sometimes the DNA state). The use of a reduced one-dimensional model (that has the advantage that it can be solved analytically) has been justified so far by an argument concerning the stationary distribution in [START_REF] Shahrezaei | Analytical distributions for stochastic gene expression[END_REF]. However, it is clear that two different models may have the same stationary distribution but very different behavior (continuous or discontinuous trajectories, monostable or bistable, etc; for an example in that context, see [START_REF] Mackey | Molecular distributions in gene regulatory dynamics[END_REF]). Hence, our results are of importance to rigorously prove the validity of using a reduced model. Our results are based on the standard assumption that the mRNA molecules have a shorter lifetime than the protein molecules, that is widely observed in both prokaryotes and eukaryotes [START_REF] Schwanhäusser | Global quantification of mammalian gene expression control[END_REF]). Depending on the assumed scaling of other kinetic parameters within the mRNA degradation rates, different limiting models are obtained.

The paper is organized as follows. In the first section, we prove a reduction results for a discrete state-space model, that is a two-dimensional birth and death process. Assumptions on the birth and death rates are in agreement with a standard model of gene expression for the mRNA-protein system. That is both variables remain positive and birth of the second variable can occur only if the first variable is positive. Using an appropriate scaling of birth and death rates, we prove that this model converges to a general one-dimensional discrete bursting model.

In the second section, we prove a reduction for a continuous state-space model, that is a two-dimensional piecewise deterministic model of gene expression with a jump production term for the first variable. Using appropriate scaling on parameters, we prove that this model converge either to a deterministic ordinary differential equation or to a one-dimensional continuous bursting model.

A bursting model from a two-dimensional discrete model

The fact that bursting models arise as a reduction procedure of a higher dimensional model was already observed in [START_REF] Shahrezaei | Analytical distributions for stochastic gene expression[END_REF] and [START_REF] Crudu | Convergence of stochastic gene networks to hybrid piecewise deterministic processes[END_REF]. In [START_REF] Shahrezaei | Analytical distributions for stochastic gene expression[END_REF], the authors show that, within an appropriate scaling, the stationary distribution of a 2-dimensional discrete model converge to the stationary distribution of a 1-dimensional bursting model. The authors used analytic methods through the transport equation on the generating function. Their result seems to be restricted to first-order kinetics. The first variable is a fast variable that induces infrequent kicks to the second one. In [START_REF] Crudu | Convergence of stochastic gene networks to hybrid piecewise deterministic processes[END_REF], the authors show that, within an appropriate scaling, a fairly general discrete state space model with a binary variable converge to a bursting model with continuous state space. The authors obtained a convergence in law of the solution through Martingale techniques. The binary variable is a fast variable that, when switching in an "ON" state, induces kicks to the other variable.

We present below analogous result of [START_REF] Crudu | Convergence of stochastic gene networks to hybrid piecewise deterministic processes[END_REF] when the fast variable is similar to the one of [START_REF] Shahrezaei | Analytical distributions for stochastic gene expression[END_REF]. Our limiting model is still a discrete state space model. These results are more precise than the one of [START_REF] Shahrezaei | Analytical distributions for stochastic gene expression[END_REF], and more general (some kinetics rates can be nonlinear). We use martingales techniques, with a proof that is similar to [START_REF] Crudu | Convergence of stochastic gene networks to hybrid piecewise deterministic processes[END_REF] and also inspired by results from Kang and Kurtz. We present below the model, then state our result in the subsection 1.1, and divide the proof in the three next subsections 1.2-1.4.

We consider the following two-dimensional stochastic kinetic chemical reaction model

∅ λ1(X1,X2) -------→ X 1 , Production of X 1 at rate λ 1 (X 1 , X 2 ) X 1 γ1(X1,X2) -------→ ∅, Destruction of X 1 at rate γ 1 (X 1 , X 2 ) ∅ λ2(X1,X2) -------→ X 2 , Production of X 2 at rate λ 2 (X 1 , X 2 ) X 2 γ2(X1,X2) -------→ ∅, Destruction of X 2 at rate γ 2 (X 1 , X 2 ) (1)
with γ 1 (0, X 2 ) = γ 2 (X 1 , 0) = 0 to ensure positivity. This model can be represented by a continuous time Markov chain in N 2 , and is then a general birth and death process in N 2 . It can be described by the following set of stochastic differential equations

X 1 (t) = X 1 (0) + Y 1 t 0 λ 1 (X 1 (s), X 2 (s))ds -Y 2 t 0 γ 1 (X 1 (s), X 2 (s))ds , X 2 (t) = X 2 (0) + Y 3 t 0 λ 2 (X 1 (s), X 2 (s))ds -Y 4 t 0 γ 2 (X 1 (s), X 2 (s))ds ,
where Y i , for i = 1...4 are independent standard poisson processes. The generator of this process is given by

Bf (X 1 , X 2 ) =λ 1 (X 1 , X 2 ) f (X 1 + 1, X 2 ) -f (X 1 , X 2 ) + γ 1 (X 1 , X 2 ) f (X 1 -1, X 2 ) -f (X 1 , X 2 ) + λ 2 (X 1 , X 2 ) f (X 1 , X 2 + 1) -f (X 1 , X 2 ) + γ 2 (X 1 , X 2 ) f (X 1 , X 2 -1) -f (X 1 , X 2 ) , (2)
for every bounded function f on N 2 .

Exemple 1. We have in mind the standard mRNA-Protein system given by the following choice:

γ i (X 1 , X 2 ) = g i X i with g i > 0 for i = 1, 2, λ 1 (X 1 , X 2 ) = λ 1 (X 2 ) and λ 2 (X 1 , X 2 ) = k 2 X 1 with k 2 > 0.
Note however that even in the context of models of gene expression, different models have been proposed, that includes nonlinear feedback of mRNA and/or nonlinear degradation terms [START_REF] Bose | Binary Response in MicroRNA-mediated Regulation of Gene Expression[END_REF].

Statement of the result

We suppose the following scaling holds

γ N 1 (X 1 , X 2 ) = N γ 1 (X 1 , X 2 ) λ N 2 (X 1 , X 2 ) = N λ 2 (X 1 , X 2 )
where N → ∞ that is degradation of X 1 and production of X 2 occurs at a faster time scale than the two other reactions. Then X 1 is degraded very fast, and induces also as a very fast production of X 2 . The rescaled model is given by

X N 1 (t) = X N 1 (0) + Y 1 t 0 λ 1 (X N 1 (s), X N 2 (s))ds -Y 2 t 0 N γ 1 (X N 1 (s), X N 2 (s))ds , X N 2 (t) = X N 2 (0) + Y 3 t 0 N λ 2 (X N 1 (s), X N 2 (s))ds -Y 4 t 0 γ 2 (X N 1 (s), X N 2 (s))ds , (3)
and the generator of this process is given by

B N f (X 1 , X 2 ) =λ 1 (X 1 , X 2 ) f (X 1 + 1, X 2 ) -f (X 1 , X 2 ) + N γ 1 (X 1 , X 2 ) f (X 1 -1, X 2 ) -f (X 1 , X 2 ) + N λ 2 (X 1 , X 2 ) f (X 1 , X 2 + 1) -f (X 1 , X 2 ) + γ 2 (X 1 , X 2 ) f (X 1 , X 2 -1) -f (X 1 , X 2 ) . (4)
We can prove the following reduction holds: Theorem 1. We assume that 1. The degradation function on X 2 satisfies γ 2 (X 1 , 0) ≡ 0.

The degradation function on

X 1 satisfies γ 1 (0, X 2 ) ≡ 0, and inf X1≥1,X2≥0 γ 1 (X 1 , X 2 ) = γ > 0.
3. The production rate of X 2 satisfies λ 2 (0, X 2 ) = 0. 4. The production rate function λ 1 and λ 2 are linearly bounded by

X 1 + X 2 . 5. Either λ 1 or λ 2 is bounded. Let (X N 1 , X N 2 )
the stochastic process whose generator is B N (defined in eq. ( 4)). Assume that the initial vector

(X N 1 (0), X N 2 (0)) converges in distribution to (0, X(0)), as N → ∞. Then, for all T > 0, (X N 1 (t), X N 2 (t)) t≥0 converges in L 1 (0, T ) (and in L p , 1 ≤ p < ∞) to (0, X(t))
where X(t) is the stochastic process whose generator is given by

B ∞ ϕ(X) = λ 1 (0, X) ∞ 0 P t (γ 1 (1, .)ϕ(.))(X)dt -ϕ(X) + γ 2 (0, X) ϕ(X -1) -ϕ(X) , (5) where P t g(X) = E g(Y (t, X)e -t 0 γ1(1,Y (s,X)
)ds , and Y (t, X) is the stochastic process starting at X at t = 0 whose generator is given by

Ag(Y ) = λ 2 (1, Y ) g(Y + 1) -g(Y ) .
Remark 2. The first three hypotheses of theorem 1 are the main characteristics of the mRNA-protein system (see example 1). Basically, they impose that quantities remains non-negative, that the first variable has always the possibility to decrease to 0 (no matter the value of the second variable), and that the second variable cannot increase when the first variable is 0. Hence these three hypotheses will guarantee that (with our particular scaling) the first variable converges to 0, and will lead to an intermittent production of the second variable. The last two hypotheses are more technical, and guarantee that the Markov chain is not explosive, and hence well defined for all t ≥ 0, and that the limiting model is well defined too. Remark 3. The above expression eq. ( 5) is a generator of a bursting model for a "general bursting size distribution". For instance, for linear function γ 1 (X 1 , X 2 ) = g 1 X 1 , and λ 2 (X 1 , X 2 ) = k 2 X 1 , we have

P t (γ 1 (.)ϕ(.))(p) = g 1 P t (ϕ)(p), = g 1 E ϕ(Y y t )e -g1t , = g 1 e -g1t z≥y ϕ(z)P Y y t = z , = g 1 e -g1t z≥y ϕ(z) (k 2 t) z-y e -k2t (z -y)! .
It follows by integration integration by parts that

∞ 0 P t (γ 1 (.)ϕ(.))(y)dt = g 1 g 1 + k 2 z≥0 ϕ(z + y) k 2 k 2 + g 1 z ,
which gives then an additive geometric burst size distribution of parameter p = k2 k2+g1 , as expected [START_REF] Shahrezaei | Analytical distributions for stochastic gene expression[END_REF].

We divide the proof in three steps: moment estimates, tightness and identification of the limit.

Moment estimates

Because production rates are linearly bounded, it is straightforward that with f (X 1 , X 2 ) = X 1 + X 2 in eq. ( 4), there is a constant C N (that depends on N and other parameters) such that

B N f (X 1 , X 2 ) ≤ C N (X 1 + X 2 ). Then E X N 1 (t) + X N 2 (t) is bounded on any time interval [0, T ] and f (X N 1 (t), X N 2 (t)) -f (X N 1 (0), X N 2 (0)) - t 0 B N f (X N 1 (s), X N 2 (s))ds is a L 1 -martingale.

Tightness

Clearly, from the stochastic differential equation on X N 1 , we must have X N 1 (t) → 0. We can show in fact that the Lebesgue measure of the set {t ≤ T : X N 1 (t) 0} converges to 0. Indeed, taking f (X 1 , X 2 ) = X 1 in eq. ( 4), we have

X N 1 (t) -X N 1 (0) - t 0 (λ 1 (X N 1 (s), X N 2 (s)) -N γ 1 (X N 1 (s), X N 2 (s)))ds (6)
is a martingale. Thanks to the lower bound assumption on γ 1 , we have

γE t 0 1 {X N 1 (s)≥1} ds ≤ E t 0 γ 1 (X N 1 (s), X N 2 (s))ds.
Then, by the martingale property, we deduce from ( 6)

γN E t 0 1 {X N 1 (s)≥1} ds ≤ E X N 1 (0) + t 0 E λ 1 (X N 1 (s), X N 2 (s)) ds. (7)
Now for X N 2 we obtain from eq. ( 3),

X N 2 (t) ≤ X N 2 (0) + Y 3 t 0 N 1 {X N 1 (s)≥1} λ 2 (X N 1 (s), X N 2 (s))ds .
Let us now distinguish between the two cases.

• Suppose first that λ 2 is bounded (say by K). Then

E X N 2 (t) ≤ E X N 2 (0) + KN E t 0 1 {X N 1 (s)≥1} ds .
As λ 1 is linearly bounded (say by K) by X N 1 + X N 2 , the upper bound eq. ( 7) becomes

γN E t 0 1 {X N 1 (s)≥1} ds ≤ E X N 1 (0) +K t 0 E X N 1 (s) +E X N 2 (s) ds.
Finally, with eq. ( 6), it is clear that

E X N 1 (t) ≤ E X N 1 (0) + K t 0 E X N 1 (s) + E X N 2 (s) ds.
Hence, with the three last inequalities, we can conclude by the Grönwall lemma that E X N 2 (t) is bounded on [0, T ], uniformly in N .

Then

N E T 0 1 {X N 1 (s)≥1} ds is bounded and X N 1 → 0 in L 1 ([0, T ], N)
. By the law of large number,

1 N Y 3 (N )
is almost surely convergent, and hence almost surely bounded. We deduce then there exists a random variable C such that

X N 2 (t) ≤ X N 2 (0) + N C t 0 1 {X N 1 (s)≥1} ds,
almost everywhere. By Grönwall lemma and Markov inequality

P sup t∈[0,T ] X N 2 (t) ≥ M → 0 as M → ∞, uniformly in N . • Now suppose λ 1 is bounded (say K).
By the martingale eq. ( 6) (and the same lower bound hypothesis on γ 1 , it is clear that 3), and from the linear bound on λ 2 (say by K)

N E T 0 1 {X N 1 (s)≥1} ds is bounded and X N 1 → 0 in L 1 ([0, T ], N). Now, let us denote U N (t) = 1 N X N 1 (t), V N = 1 N X N 2 (t) and W N = N 1 {X N 1 (t)≥1} (which is then bounded in L 1 ([0, T [)). From eq. (
V N (t) ≤ V N (0) + 1 N Y 3 t 0 N KW N (U N (s) + V N (s))ds .
Then, still by the law of the large number there exists a random variable C such that

V N (t) ≤ V N (0) + C t 0 W N (U N (s) + V N (s))ds,
and hence

X N 2 (t) ≤ X N 2 (0) + C t 0 W N (X N 1 (s) + X N 2 (s))ds. By Grönwall lemma, sup [0,T ] X N 2 (t) ≤ (X N 1 (0) + X N 2 (0)) exp C t 0 W N (s)ds ,
which is then bounded, uniformly in N .

For any subdivision of [0, T ],

0 = t 0 < t 1 < • • • < t n = T , n-1 i=0 | X N 2 (t i+1 ) -X N 2 (t i ) | ≤ n-1 i=0 Y 3 ti+1 ti N 1 {X N 1 (s)≥1} λ 2 (X N 1 (s), X N 2 (s))ds ≤ Y 3 T 0 N 1 {X N 1 (s)≥1} λ 2 (X N 1 (s), X N 2 (s))ds
so by a similar argument as above, we also get the tightness of the BV norm [START_REF] Giusti | Minimal surfaces and functions of bounded variations[END_REF]).

P X N 2 [0,T ] ≥ K → 0 as K → 0, independently in N . Then X N 2 is tight in L p ([0, T ]), for any 1 ≤ p < ∞ (

Identification of the limit

We choose an adherence value (0, X 2 (t)) of the sequence (X N 1 (t), X N 2 (t)) in L 1 ([0, T ]) × L p ([0, T ]). Then a subsequence (again denoted by) (X N 1 (t), X N 2 (t)) converge to (0, X 2 (t)), almost surely and for almost t ∈ [0, T ]. We are looking for test-functions such that

f (X N 1 (t), X N 2 (t)) -f (X N 1 (0), X N 2 (0) - t 0 B N f (0, X N 2 (s))1 X N 1 (s)=0 ds - t 0 B N f (X N 1 (s), X N 2 (s))1 X N 1 (s)≥1 ds is a martingale and B N f (X N 1 (s), X N 2 (s)) is bounded independently of N when X 1 ≥ 1.
The following choice is inspired by [START_REF] Crudu | Convergence of stochastic gene networks to hybrid piecewise deterministic processes[END_REF]. We introduce the stochastic process Y x,y t , starting at y and whose generator is

A x g(y) = λ 2 (x, y) g(y + 1) -g(y) ,
for any x ≥ 1. and we introduce the semigroup P x t defined on bounded function, for any x ≥ 1, by

P x t g(y) = E g(Y x,y t )e -t 0 γ1(x,Y x,y s )ds . ( 8
)
Then the semigroup P x t satisfies the equation dP x t g(y) dt = A x P x t g(y)γ 1 (x, y)P x t g(y).

Now for any bounded function g, define recursively

f (0, y) = g(y), f (x, y) = ∞ 0 P x t (γ 1 (x, .)f (x -1, .))(y)dt.
Such a test function is well defined by the assumption on γ 1 . We then verify that

B N f (0, y) = λ 1 (0, y) ∞ 0 P 1 t (γ 1 (1, .)g(.))(y)dt -g(y) + γ 2 (0, y) g(y -1) -g(y) , B N f (x, y) = λ 1 (x, y) f (x + 1, y) -f (x, y) + γ 2 (x, y) f (x, y -1) -f (x, y) .
Indeed, for any x ≥ 1,

A x f (x, y) -γ 1 (x, y)f (x, y) = ∞ 0 A x P x t (γ 1 (x, .)f (x -1, .))(y) -γ 1 (x, y)P x t (γ 1 (x, .)f (x -1, .))(y)dt, = ∞ 0 d dt P x t (γ 1 (x, .)f (x -1, .))(y)dt, = lim t→∞ P x t (γ 1 (x, .)f (x, .))(y) -γ 1 (x, y)f (x -1, y), = -γ 1 (x, y)f (x -1, y). Then λ 2 (x, y) f (x, y + 1) -f (x, y) + γ 1 (x, y) f (x -1, y) -f (x, y) = 0.
Hence B N f (x, y) is independent of N, and, taking the limit N → ∞ in

f (X N 1 (t), X N 2 (t)) -f (X N 1 (0), X N 2 (0)) - t 0 B N f (X N 1 (s), X N 2 (s))ds, we deduce g(X 2 (t)) -g(X 2 (0)) - t 0 B ∞ g(X 2 )
is a martingale where

B ∞ g(y) = λ 1 (0, y) ∞ 0 P 1 t (γ 1 (1, .)g(.))(y)dt -g(y) + γ 2 (0, y) g(y -1) -g(y)
.

Uniqueness Due to assumption on k 1 and k 2 , the limiting generator defines a pure-jump Markov process in N which is not explosive. Uniqueness of the martingale then follows classically.

Continuous-state bursting model

The model we consider now is a continuous state-space model that explicitly assume the production of several molecules instantaneously, through a jump Markov process, in agreement with experimental observations [START_REF] Golding | Real-Time Kinetics of Gene Activity in Individual Bacteria[END_REF]; [START_REF] Raj | Stochastic mRNA Synthesis in Mammalian Cells[END_REF]). In line with experimental observations, it is standard to assume a Markovian hypothesis (an exponential waiting time between production jumps) and that the jump sizes are exponentially distributed (geometrically in the discrete case) as well [START_REF] Suter | Mammalian Genes Are Transcribed with Widely Different Bursting Kinetics[END_REF]). The intensity of the jumps can be a linearly bounded function, to allow for self-regulation.

For simplicity, we will only consider the standard model of gene expression, that is with linear degradation rates and the production rate of the second variable is linear with respect to the first variable (as in example 1). Note that more general rates as in the previous section could be considered as well. Here, we ask the question of what is the correct scaling so that the bursting production term is transmitted from the first variable to the second one, when the first variable is eliminated through an adiabatic limit. The propagation of bursting property in a gene network is an important question in molecular biology [START_REF] Kaern | Stochasticity in gene expression: From theories to phenotypes[END_REF].

This section is structured as follows. We first present the model in the rest of this paragraph, then state the results in subsection 2.1, and divide the proofs in the remaining three subsection 2.2,2.3,2.4.

Let x 1 and x 2 denote the concentrations of mRNA and protein respectively. A simple model of single gene expression with bursting transcription is given by

dx 1 dt = -g 1 x 1 + N (h, λ 1 (x 2 )), ( 9
)
dx 2 dt = -g 2 x 2 + k 2 x 1 . (10)
Here g 1 and g 2 are the degradation rates for the mRNA and protein respectively, k 2 is the mRNA translation rate, and N (h, λ 1 (x 2 )) describes the transcription that is assumed to be a compound Poisson white noise occurring at a rate λ 1 (x 2 ) with a non-negative jump size ∆x 1 distributed with density h. The equations ( 9)-( 10) are a short hand notation for

x 1 (t) = x 1 (0) - t 0 g 1 x 1 (s -)ds (11) + t 0 ∞ 0 ∞ 0 1 {r≤λ1(x2(s -))} zN (ds, dz, dr), x 2 (t) = x 2 (0) - t 0 g 2 x 2 (s -)ds + t 0 k 2 x 1 (s -)ds. ( 12
)
where X s-= lim t→s -X(t), and N (ds, dz, dr) is a Poisson random measure on (0, ∞) × [0, ∞) 2 with intensity dsh(z)dzdr, where s denotes the times of the jumps, r is the state-dependency in an acceptance/rejection fashion, and z the jump size. Note that (x 1 (t)) is a stochastic process with almost surely finite variation on any bounded interval (0, T ), so that the last integral is well defined as a Stieltjes-integral.

Hypothesis 2. The following discussion is valid for general rate functions λ 1 and density functions h(•) that satisfy

• λ 1 ∈ C 1 , λ 1 is globally lipschitz and linearly bounded with

0 ≤ λ 1 (x) ≤ c + Kx. • h ∈ C 0 and ∞ 0 xh(x)dx < ∞.
For such a general density function h, we denote the average burst size by

b = ∞ 0 xh(x)dx. ( 13
)
Remark 4. Hill functions are often used to model gene self-regulation, so that λ 1 is given by

λ 1 (x 2 ) = 1 + x α 2 L + Dx α 2
where L, D are positive parameters and α is a positive integer (see Mackey, Tyran-Kamińska and Yvinec (2011) for more details). An exponential distribution of the bursting transcription is often used in modeling gene expression, in accordance with experimental findings [START_REF] Xie | Single-Molecule Approach to Molecular Biology in Living Bacterial Cells[END_REF]), so that the density function h is given by

h(x) = 1 b e -x/b
, with b the average burst size.

If λ 1 (x 2 ) ≡ k 1 is independent of the state x 2 , the average transcription rate is bk 1 , and the asymptotic average mRNA and protein concentrations are

x eq 1 := E x 1 (t → ∞) = bk 1 g 1 . x eq 2 := E x 2 (t → ∞) = k 2 g 2 x eq 1 = bk 1 k 2 g 1 g 2 .

Statement of the results

Although Equations ( 9)-( 10) are simple, they are not analytically solvable.

Hence, for pratical use to interpret experimental data, and to avoid numerical simulations, one looks for a reduced, analytically solvable, one-dimensional equation. In the following discussion, we consider the situation when mRNA degradation is a fast process, i.e. g 1 is "large enough", but the average equilibrium protein concentration x eq 2 remains unchanged. In most organisms and for most genes, mRNA has a smaller lifetime than protein [START_REF] Schwanhäusser | Global quantification of mammalian gene expression control[END_REF]. In what follows, we denote by g n 1 , k n 2 sequences of parameters, λ n 1 sequence of functions and h n sequence of density function that will replace g1, k 2 , λ 1 , h in ( 11)-( 12) and satisfy hypothesis 2. We then denote (x n 1 , x n 2 ) its associated solution. We will always assume one of the following three scaling relations:

(S1) Frequent production rate of mRNA, namely g n 1 = ng 1 , λ n 1 = nλ 1 , and

k n 2 = k 2 h n = h are independent of n; (S2) Large burst of mRNA, namely g n 1 = ng 1 , h n (z) = 1 n h( z n ) and λ n 1 = λ 1 ,k n 2 = k 2 remain unchanged; (S3) Large production rate of protein, namely g n 1 = ng 1 , k n 2 = nk 2 , and λ n 1 = λ 1 h n = h are independent of n;
These three different scaling are then associated with different behaviors of the biological systems given by (x 1 , x 2 ). As different genes may have different kinetics, each one of the possible scaling are of importance [START_REF] Suter | Mammalian Genes Are Transcribed with Widely Different Bursting Kinetics[END_REF][START_REF] Schwanhäusser | Global quantification of mammalian gene expression control[END_REF].

In this section we determine an effective reduced equation for equation ( 10) for each of the three scaling conditions (S1)-( S3). In particular, we show that under assumption (S1), equation ( 10) can be approximated by the deterministic ordinary differential equation

dx 2 dt = -g 2 x 2 + λ 2 (x 2 ) ( 14
)
where

λ 2 (x 2 ) = bk 2 λ 1 (x 2 )/g 1 .
We further show that under the scaling relations (S2) or (S3), equation ( 10) can be reduced to the stochastic differential equation

dx 2 dt = -g 2 x 2 + N ( h, λ 1 (x 2 )). ( 15
)
where h is a suitable density function in the jump size ∆x 2 (to be detailed below).

We first explain, using some heuristic arguments, the differences between the three scaling relations and the associated results. When n → ∞, g n 1 → ∞ and applying a standard quasi-equilibrium assumption we have

dx n 1 dt ≈ 0, which yields x n 1 (t) ≈ 1 g n 1 N (h n (.), λ n 1 (x n 2 )) = N (g n 1 h n (g n 1 •), λ n 1 (x n 2 )),
and therefore the second equation ( 10) becomes

dx n 2 dt ≈ -g 2 x n 2 + k n 2 g n 1 N (h n (.), λ n 1 (x n 2 )), ≈ -γ 2 x n 2 + N g n 1 k n 2 h n ( g n 1 k n 2 ), λ n 1 (x n 2 ) .
Hence in (15), h(x 2 ) = (k 2 /g 1 ) -1 h((k 2 /g 1 ) -1 x 2 ) under the scaling (S2) and (S3). Furthermore, we note that the scaling (S2) also implies nh n (n•) = h(•), while in (S1), nh n (n•) = nh(n•) so that the jumps become more frequent and smaller.

We denote (D[0, ∞), S) the cadlad function space of function defined on [0, ∞) at values in R + with the usual Skorohod topology [START_REF] Jacod | Limit theorems for stochastic processes[END_REF]). Similarly (D[0, T ], J) is the cadlag funtion space on [0, T ], with the Jakubowski topology [START_REF] Jakubowski | A non-Skorohod topology on the Skorohod space[END_REF]). Also, L p [0, T ) the space of L p integrable function on [0, T ), with T > 0, which we endowed with total variation norm [START_REF] Giusti | Minimal surfaces and functions of bounded variations[END_REF]), and M (0, ∞) is the space of real measurable function on [0, ∞) with the metric [START_REF] Kurtz | Random time changes and convergence in distribution under the Meyer-Zheng conditions[END_REF])

d(x, y) = ∞ 0 e -t max{1, | x(t) -y(t) |}dt.
Our main results can be stated as follows Theorem 3. Consider the equations ( 9)-( 10) and assume Hypothesis 2. If the scaling (S1) is satisfied, i.e., k n 1 = nk 1 , and if x n 2 (0) → x 0 2 , then when n → ∞, 1. The stochastic process x n 1 (t) does not converge in any functional sense; 2. The stochastic process x n 2 (t) converges in law in (D[0, ∞), S) towards the deterministic solution of the ordinary differential equation

dx 2 dt = -g 2 x 2 + λ 2 (x 2 ), x 2 (0) = x 0 2 , (16)
where

λ 2 (x 2 ) = bk 2 λ 1 (x 2 )/g 1 .
Theorem 4. Consider the equations (9)-( 10) and assume Hypothesis 2. If the scaling (S2) is satisfied, i.e., h n (z) = 1 n h( z n ), and if x n 2 (0) → x 0 2 , then when n → ∞,

The stochastic process

x n 1 (t) n converges in law in L p , 1 ≤ p < ∞ and in (D[0, T ], J) to the (deterministic) fixed value 0; 2. The stochastic process x n 2 (t) converges in law in L p , 1 ≤ p < ∞ and in (D[0, T ], J) to the stochastic process defined by the solution of the stochastic differential equation

dx 2 dt = -g 2 x 2 + N ( h, λ 1 ), x 2 (0) = x 0 2 ≥ 0, ( 17
)
where h(x 2 ) = (k 2 /g 1 ) -1 h((k 2 /g 1 ) -1 x 2 ).
Theorem 5. Consider the equations (9)-( 10) and assume Hypothesis 2. If the scaling (S3) is satisfied, i.e., k n 2 = nk 2 , and if x n 2 (0) → x 0 2 , then when n → ∞, 1. The stochastic process x n 1 (t) converges in law in L p , 1 ≤ p < ∞ and in (D[0, T ], J) to the (deterministic) fixed value 0;

2. The stochastic process x n 2 (t) converges in law in L p , 1 ≤ p < ∞ and in (D[0, T ], J) to the stochastic process determined by the solution of the stochastic differential equation

dx 2 dt = -γ 2 x 2 + N ( h, ϕ), x 2 (0) = x 0 2 ≥ 0, where h(x 2 ) = (k 2 /g 1 ) -1 h((k 2 /g 1 ) -1 x 2 ).
Remark 5. Note that scalings (S2) and (S3) give similar results for the equation governing the protein variable x 2 (t) but very different results for the asymptotic stochastic process related to the mRNA. In particular, in Theorem 4, very large bursts of mRNA are transmitted to the protein, where in Theorem 5, very rarely is mRNA present but when present it is efficiently synthesized into a burst of protein.

In the rest of this paper, we provide proofs of the results mentioned above, using martingale techniques. In a companion paper [START_REF] Yvinec | Adiabatic reduction of jump processes in a model of stochastic gene expression with bursting transcription[END_REF], we use partial differential techniques to prove similar results (see also [START_REF] Haseltine | On the origins of approximations for stochastic chemical kinetics[END_REF]; Zeron and Santillán (2010); [START_REF] Santilln | Irreversible thermodynamics in multiscale stochastic dynamical systems[END_REF]).

The proofs of the three theorems above are divided in three steps. In section 2.2 we first recall generator properties and derive moment estimates associated to ( 9)-(10). In section 2.3 we show the tightness result for all three theorems. We then identify the limit using a martingale approach in section 2.4.

General properties and moment estimates

We first summarize the important background results on the stochastic processes used in the next.

One dimensional equation For the one-dimensional stochastic differential equation ( 15) perturbed by a compound Poisson white noise, of intensity λ(.) and jump size distribution h(.), the extended generator of the stochastic process (x 2 (t)) t≥0 is (Davis, 1984, Theorem 5.5), for any f ∈ D(A ∞ ),

A 1 f (x) = -g 2 x df dx + λ(x) ∞ x h(z -x)f (z)dz -f (x) D(A 1 ) = {f ∈ M(0, ∞) : t → f (xe -γ2t ) is absolutely continuous for t ∈ R + and E Ti≤t |f (x 2 (T i )) -f (x 2 (T - i ))| < ∞ for all t ≥ 0}
where M(0, ∞) denotes a Borel-measurable function of (0, ∞) and the times T i are the instants of the jump of x 2 . It is an extended domain containing all functions that are sufficiently smooth along the deterministic trajectories between the jumps, and with a bounded total variation induced by the jumps.

For any f ∈ D(A 1 ), we have

d dt Ef (x 2 (t)) = EA 1 (f (x 2 (t))).
Two dimensional equation Consideration the two-dimensional stochastic differential equation ( 9)-( 10) perturbed by a compound Poisson white noise, of intensity λ 1 (x 2 ) and jump size distribution h follows along similar lines. Its infinitesimal generator and extended domain are

A 2 g(x 1 , x 2 ) = -g 1 x 1 ∂g ∂x 1 + (k 2 x 1 -g 2 x 2 ) ∂g ∂x 2 + λ 1 (x 2 ) ∞ x1 h(z -x 1 )g(z, x 2 )dz -g(x 1 , x 2 ) , (18)
D(A 2 ) = {g ∈ M((0, ∞) 2 ) : t → g(φ t (x 1 , x 2 )) is absolutely (19) continuous for t ∈ R + and E Ti≤t |g(x 1 (T i ), x 2 (T i )) -g(x 1 (T - i ), x 2 (T - i ))| < ∞ for all t ≥ 0}
where φ t is the deterministic flow given by the deterministic part of equations ( 9)-( 10), namely

dx 1 dt = -g 1 x 1 , dx 2 dt = -g 2 x 2 + k 2 x 1 .
For any f ∈ D(A 2 ), we have

d dt Ef (x 1 (t), x 2 (t)) = EA 2 (f (x 1 (t), x 2 (t))). ( 20
)
Using the stochastic differential equations ( 11)-( 12), we can deduce moment estimates, needed to be able to use unbounded test function (namely f (x 1 , x 2 ) = x 1 and f (x 1 , x 2 ) = x 2 ) in the martingale formulation. By taking the mean into (11)-( 12), neglecting negatives values and using hypothesis 2,

0 ≤ E x 1 (t) ≤ E t 0 bλ 1 (x 2 (s))ds ≤ t 0 b(c + KE x 2 (s) )ds 0 ≤ E x 2 (t) ≤ t 0 k 2 E x 1 (s) ds
where we note b = E h = ∞ 0 zh(z)dz. By Gronwall inequalities, there exist a constant C such that

E sup t∈[0,T ] x 1 (t) ≤ C(E x 1 (0) + e CT ) E sup t∈[0,T ] x 2 (t) ≤ C(E x 2 (0) + e CT ) (21)
Then we claim that f (x 1 , x 2 ) = x 1 is in the domain of the generator A 2 . We only have to verify (see Eq ( 19))

E Ti≤t |x 1 (T i ) -x 1 (T - i )| < ∞ for all t ≥ 0.
By equation ( 11)

E Ti≤t |x 1 (T i ) -x 1 (T - i )| = E t 0 ∞ 0 ∞ 0 1 {r≤λ1(x2(s -))} zN (ds, dz, dr), ≤ bE t 0 c + Kx 2 (s)ds .
which is finite according to the previous estimates.

Tightness

S1

We first show the tightness property for the scaling (S1) corresponding to theorem 3. In such case x n 1 does no converge in any functional sense because it fluctuates very fast, as more and more jumps appears of size that stay of order 1 (given by h). However, E x n 1 (t) remains bounded,

x n 1 n goes to 0, and by eq. ( 12),

| x n 2 (t) |≤| x n 2 (0) | + t 0 k 2 | x n 1 (s) | ds.
For any n, let N n be the compound Poisson process associated to (11), with {T n,i } ∞ i=1 the jump times which occur at a rate nλ 1 (x 2 (s) n ), and {Z n,i } ∞ i=1 the jump sizes that are iid random variables with density h (with the convention T n,0 = 0 and Z n,0 = X 0 ), N n (t) =

Tn,i≤t Z n,i .

Then

x n 1 (t) =

Tn,i≤t

Z n,i e -ng1(t-Tn,i) 1 {t≥Tn,i} .

By integration, t 0

x n 1 (s)ds =

Tn,i≤t Z n,i 1 ng 1 (1 -e -g1(t-Tn,i) ) 1 {t≥Tn,i} .
Then,

x n 2 (t) ≤ x n 2 (0) + t 0 k 2 x n 1 (s)ds ≤ x n 2 (0) + k 2 ng 1 Tn,i≤t Z n,i .
Finally we deduce, by definition of the compound Poisson process,

x n 2 (t) ≤ x n 2 (0) + k 2 ng 1 N n (t).
Now, by a time change, there exists a process Y such that

N n (t) = Y t 0 nλ 1 (x n 2 (s))ds ,
where Y is unit rate compound Poisson process of jump size iid (with density h). By the law of large number, 1 n Y (nt) is almost surely convergent, and hence almost surely bounded. We deduce then there exists a random variable C such that

x n 2 (t) ≤ x n 2 (0) + k 2 g 1 C t 0 λ 1 (x n 2 (s))ds.
By Gronwall lemma and Markov inequality

P sup t∈[0,T ] x n 2 (t) ≥ M → 0,
as M → ∞ and uniformly in n. Similarly, for any

t 1 , t 2 ∈ [0, T ], | x n 2 (t 2 ) -x n 2 (t 1 ) |≤ k 2 ng 1 | N n (t 2 ) -N n (t 1 ) | . Again, N n (t 2 ) -N n (t 1 ) = Y t2 t1
nλ 1 (x n 2 (s))ds and, still by the law of large number

| x n 2 (t 2 ) -x n 2 (t 1 ) |≤ k 2 g 1 C t2 t1 λ 1 (x n 2 (s))ds, so that , for any ε > 0 lim θ→0 lim sup n sup S1≤S2≤S1+θ P | x n 2 (S 2 ) -x n 2 (S 1 ) |≥ ε = 0,
where the supremum is over stopping times bounded by T . Then by Aldous' tightness criteria ( (Jacod and Shiryaev, 1987, thm 4.5 p 356)), x n 2 is tight in (D[0, ∞), S).

S3

Now we show the tightness property for the scaling (S3) corresponding to theorem 5, with k n 2 = nk 2 . In such case x n 1 converges to 0 in L 1 , and we get a control over n t 0 x n 1 (s)ds. Indeed using g(x 1 , x 2 ) = x 1 in (18), we get

x n 1 (t) -x n 1 (0) - t 0 (-ng 1 x n 1 (s) + bλ 1 (x 2 (s) n )ds),
is a martingale so that due to Hypothesis 2,

g 1 E n t 0 x n 1 (s)ds ≤ E x n 1 (0) + b(ct + K t 0 E x n 2 (s) ds)
By eq. ( 12),

x n 2 (t) ≤ E x n 2 (0) + k 2 n t 0 x n 1 (s)ds. then sup t∈[0,T ] x n 2 (t) ≤ E x n 2 (0) + k 2 n T 0
x n 1 (s)ds.

Reporting into the estimates for x n 1 yelds

g 1 E n t 0 x n 1 (s)ds ≤ E x n 1 (0) + b(ct + K(E x n 2 (0) + tk 2 n t 0 E x n 1 (s) ds)), ≤ C 1 T + C 2 T E n t 0 x n 1 (s)ds , for two constants C 1 T , C 2
T that depends solely on T . Then E n t 0 x n 1 (s)ds is bounded uniformly in n so that x n 1 converges to 0 in L 1 and

P sup t∈[0,T ] x n 2 (t) ≥ M → 0 as M → ∞ and uniformly in n. Now for any subdivision of [0, T ], 0 = t 0 < t 1 < • • • < t n = T , n-1 i=0 | x n 2 (t i+1 ) -x n 2 (t i ) | ≤ E x n 2 (0) + k 2 n t 0
x n 1 (s)ds, so that we also get the tightness of the BV norm,

P x n 2 [0,T ] ≥ M → 0,
as M → 0, independently in n. Then x n 2 is tight in L p ([0, T ]), for any 1 ≤ p < ∞ [START_REF] Giusti | Minimal surfaces and functions of bounded variations[END_REF]) and also, by a similar criteria, in (D[0, T ], J) [START_REF] Jakubowski | A non-Skorohod topology on the Skorohod space[END_REF]).

S2 Now we show the tightness property for the scaling (S2) corresponding to theorem 4, with h n = 1 n h( 1 n ). Remark that on such case, denoting z n =

x n 1 n , the variables (z n , x n 2 ) satisfies ( 11)-( 12) with the (S3) scaling, so we already now that x n 2 is tight in L p ([0, T ]), for any 1 ≤ p < ∞. For x n 1 , formally, note that each jumps yelds a contribution for x n 1 of b g1 so there's no hope for a convergence to 0 in L 1 . However, we still have

x n 1 (t) = Tn,i≤t
Z n,i e -ng1(t-Tn,i) 1 t≥Tn,i .

where T n,i appears with rate λ 1 (x n 2 (s)), and {Z n,i } ∞ i=1 are iid random variables with density h n . Then Then

M n t = f (x n 1 (t)) -f (x n 1 (0)) -n R+ t 0 C x n 2 (s) f (x 1 )V n 1 (dx 1 × ds)
is a martingale. Dividing by n, for any limiting point (V 1 , x 2 ), we must have, for any f ∈ C b (R + ),

E R+ t 0 C x2(s) f (x 1 )V 1 (dx 1 × ds) = 0.
Because for any x 2 , the generator C x2 is (exponentially) ergodic, V 1 is uniquely determined by the invariant measure associated to C x2 . In particular, for any t > 0

R+ t 0 x 1 V n 1 (dx 1 × ds) → t 0 b g 1 λ 1 (x 2 (s))ds.
Then for f that depends only on x 2 ,

f (x n 2 (t)) -f (x n 2 (0)) - R+ t 0 (k 2 x 1 -g 2 x n 2 (s))f ′ (x n 2 (s))V n 1 (dx 1 × ds) converges to f (x 2 (t)) -f (x 2 (0)) - t 0 ( bk 2 g 1 λ 1 (x 2 (s)) -g 2 x 2 (s))f ′ (x 2 (s))ds
Due to the assumption on λ 1 , there exists a unique solution associated to the (deterministic) equation 14 so x 2 is uniquely determined.

Proof of Theorem 5

We've already seen that x n 1 converges to 0 in L 1 ([0, T ]) and x n 2 is tight in L p ([0, T ]). We then take a subsequence (x n 1 (t), x n 2 (t)) that converges to (0, x 2 (t)), almost surely and for almost t ∈ [0, T ]. Then we consider the fast component of the generator A n 2 , given in this case by

-g 1 x 1 ∂f ∂x 1 + k 2 x 1 ∂f ∂x 2 .
This defines a transport equation. Starting at (x 1 , x 2 ) at time 0, the asymptotic value of the flow associated to the transport equation is (0, y) where

y = x 2 + ∞ 0 k 2 x 1 (s)ds = x 2 + x1 0 k 2 z g 1 z dz = x 2 + k 2 g 1 x 1
We then consider f (x 1 , x 2 ) = g(x 2 + k 2 g 1 x 1 ), that satisfies, for any x 1 , x where h(x 2 ) = (k 2 /g 1 ) -1 h((k 2 /g 1 ) -1 x 2 ). Hence the limiting process x 2 must satisfy the martingale problem associated with the generator

A ∞ g(x) = -g 2 x dg dx + λ 1 (x) ∞ x h(z -x)f (z)dz -f (x) ,
for which uniqueness holds for bounded k 1 (see (Crudu et al., 2012, thm 2.5)).

A truncature argument allows then to conclude for linearly bounded k 1 .

Proof of Theorem 4 As noticed before, (z n , x n 2 ) with z n (t) =

x n 1 (t) n satisfies the scaling (S3) so similar conclusion holds for x n 2 .

  (s) + z)dzg(x 2 (s)) ds,

	Now taking the limit n → ∞ into
	f (x n 1 (t), x n 2 (t)) -f (x n 1 (0), x n 2 (0)) -	0	t	A n 2 f (x n 1 (s), x n 2 (s))ds,
	yelds
	g(x 2 (t)) -g(x 2 (0)) -	0	t	-g 2 x 2 g ′ (x 2 (s))
		+ λ 1 (x 2 (s))	∞	h(z)g(x 2
						0
			2 ,
		-g 1 x 1	∂f ∂x 1	+ k 2 x 1	∂f ∂x 2	= 0.

But for M > 0, by Markov inequality,

for any ε and n sufficiently large. Then, conditionning by the jump times,

for n large. Because t 0 x n 2 (s)ds has been shown to be bounded independently of n, we can drop the conditionning, and t 0 P x n 1 (s) > M is arbitrary small. We show also similarly that Kurtz, 1991, thm 4.1)).

Identification with the martingale problem

The three theorems below can be proved using martingale techniques, with similar spirit. For each scaling, the generator A n 2 can be decomposed into a fast component, or order n, and a slow component, of order 1. In each case, one need to find particular condition to ensure that the fast component vanishes. For the scaling (S1), the fast component acts only in the first variable, so ergodicity of this component will ensure that it vanishes, as in averaging theorems [START_REF] Kurtz | Averaging for martingale problems and stochastic approximation[END_REF]. For the other two scaling, the fast component acts on both variables, and we will have to find the particular relation between both variable that ensures this component vanishes.

Proof of Theorem 3 For any B ∈ B(R + ), t > 0, we define the occupation measure

and we identify V n 1 as a stochastic process with value in the space of finite meaure on R + . Because E x n 1 (t) remains bounded uniformly in n on any [0, T ], it is stochastically bounded and V 1 then satisfies Aldous criteria of tightness. Now take a test function f that depends only on x 1 , so that

h(zx 1 )f (z)dzf (x 1 ) .