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Adiabatic reduction of models of stochastic gene expression with bursting 3 Romain Yvinec∗ 1 Université de Lyon CNRSUMR 5208 Université Lyon1 0 Institut Camille Jordan 43 blvd. du 11 novembre 1918 2 F-69622 Villeurbanne Cedex France n e-mail: [email protected] a Abstract: Thispaper considers adiabatic reduction inboth discrete and J continuousmodelsofstochasticgeneexpression.Ingeneexpressionmodel, 7 theconceptofburstingisaproductionofseveralmoleculessimultaneously and is generally represented as a jump terms of random size. In a gen- ] eraltwo-dimensional birthanddeath discrete model,weprovethat under R specific assumptions and scaling (that are characteristics of the mRNA- P proteinsystem)anadiabaticreductionleadstoaone-dimensionaldiscrete- . state spacemodelwithburstingproduction. Theburstterm appearsthen h through the reduction of the first variable. In a two-dimensional continu- at ousmodel,wealsoprovethatanadiabaticreductioncanbeperformedina m stochasticslow/fastsystem.Inthisgeneexpressionmodel,theproduction ofmRNA(thefastvariable)isassumedtobeburstyandtheproductionof [ protein(theslowvariable)islinearasafunctionofmRNA.Whenthedy- namicsofmRNAisassumedtobefasterthantheproteindynamics(dueto 1 amRNAdegradationratelargerthanfortheprotein)weprovethat,with v theappropriatescaling,theburstingphenomenacanbetransmittedtothe 3 slowvariable.Weshowthatthereducedequationiseitherastochasticdif- 9 ferentialequationwithajumpMarkovprocessoradeterministicordinary 2 differentialequationdependingonthescalingthatisappropriate. 1 Theseresultsaresignificantbecauseadiabaticreductiontechniquesseem . to have not been applied to a stochastic differential system containing a 1 jump Markov process. Last but not least, for our particular system, the 0 adiabaticreductionallowsustounderstandwhatarethenecessarycondi- 3 tionsfortheburstingproduction-likeofproteintooccur. 1 : AMS 2000 subject classifications: Primary 92C45, 60Fxx; secondary v 92C40,60J25,60J75. i X Keywordsandphrases:adiabaticreduction,piecewisedeterministicMarkov process,stochasticburstinggeneexpression,quasi-steadystateassumption, r scalinglimit. a Introduction The adiabaticreductiontechniques giveresultsthat allowto reducethe 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 ordinarydifferentialequationshavebeen availablesince the veryprecise results ∗correspondingauthor 1 R. Yvinec1/adiabatic reduction for PDMP 2 ofTikhonov(1952)andFenichel (1979).The simplestresults,in the hyperbolic case, give an effective construction of an uniformly asymptotically stable slow manifold(andhenceareducedequation)andprovetheexistenceofaninvariant manifold near the slow manifold, with (theoretically) any order of approxima- tion of this invariant manifold. Such precise and geometric results have been generalizedtorandomsystemsofstochasticdifferentialequationwithGaussian white noise (Berglund and Gentz (2006), see also Gardiner (1985) for previous work on the Fokker-Planck equation). However, to the best of our knowledge, analogousresultsfor stochasticdifferentialequationswith jumps havenotbeen obtained. Thepresentpapergivesatheoreticaljustificationofanadiabaticreductionof a particular piecewise deterministic Markov process (Davis, 1984). 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 deterministicMarkovprocess.Infact,weprovelimittheoremsusingmartingale strategy,in a similar manner thanin recentpapers suchas Cruduet al. (2012), KangandKurtzandRiedler,ThieullenandWainrib(2012),wheregeneralcon- vergence results for discrete models of stochastic reaction networks are given. In particular, these papers give alternative scaling of the traditional ordinary differentialequationandthediffusionapproximationdependingonthedifferent scaling chosen (see Ball et al. (2006) 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 deterministicprocess).Forillustrativeandmotivatingexamplesgivenbyasim- ulation algorithm, see Haseltine and Rawlings (2002); Rao and Arkin (2003); Goutsias(2005).However,weemphasizethatwedonotconsiderhereacontinu- ous approximationof a discrete model. Rather, we performadiabatic reduction on both discrete state-space and continuous state-space models. Time-scale re- ductionhavebeenconsideredinKangandKurtz,butnotonthekindweperform here. Our particular model is meant to describe stochastic gene expression with explicit bursting (Friedman, Cai and Xie, 2006). 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 generalstochastic hybrid systems (Hespanha, 2006; Bujorianu and Lygeros, 2004). We note also that more geometrical approaches have been proposed to reduce the dimension of such systems in Bujorianu and Katoen (2008). Biologically,theburstingofmRNAorproteinmoleculesisdefinedasthepro- duction of several molecules within a very short time, indistinguishable within R. Yvinec1/adiabatic reduction for PDMP 3 thetimescaleofthemeasurement.Inthebiologicalcontextofmodelsofstochas- tic gene expression, explicit models of bursting mRNA and/or protein production have been analyzed recently, either using a discrete (Shahrezaei and Swain,2008;Lei,2009)oracontinuousformalism(Friedman,CaiandXie,2006; Mackey,Tyran-Kamin´skaandYvinec,2011)asmoreandmoreexperimentalev- idencefromsingle-moleculevisualizationtechniqueshasrevealedtheubiquitous natureofthisphenomenon(Ozbudaketal.,2002;Goldingetal.,2005;Rajetal., 2006; Elf, Li and Xie, 2007; Xie et al., 2008; Raj and van Oudenaarden, 2009; Suter et al., 2011). 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 stationarydistributioninShahrezaeiandSwain(2008).However,itisclearthat two different models may havethe same stationarydistribution but verydiffer- ent behavior (continuous or discontinuous trajectories, monostable or bistable, etc; for an example in that context, see Mackey, Tyran-Kamin´ska and Yvinec (2011)).Hence, our results are of importance to rigorouslyprove 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 (Schwanhäusser et al. (2011)). 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 deathprocess.Assumptions onthe birthanddeathratesarein agreementwith 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 convergesto a generalone-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 expres- sionwithajumpproductiontermforthefirstvariable.Usingappropriatescaling on parameters, we prove that this model converge either to a deterministic ordinary differentialequationorto a one-dimensionalcontinuous bursting model. 1. A bursting model from a two-dimensional discrete model Thefactthatburstingmodelsariseasareductionprocedureofahigherdimen- sional model was already observed in Shahrezaei and Swain (2008) and Crudu et al. (2012). In Shahrezaei and Swain (2008), the authors show that, within an appropriate scaling, the stationary distribution of a 2-dimensional discrete modelconvergetothestationarydistributionofa1-dimensionalburstingmodel. The authors used analytic methods throughthe transportequationon the gen- erating function. Their result seems to be restricted to first-order kinetics. The R. Yvinec1/adiabatic reduction for PDMP 4 first variable is a fast variable that induces infrequent kicks to the second one. In Crudu et al. (2012), 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 conver- genceinlawofthesolutionthroughMartingaletechniques.Thebinaryvariable is a fast variable that, when switching in an ”ON” state, induces kicks to the other variable. We present below analogousresult of Cruduet al. (2012)when the fast vari- ableis similarto the oneofShahrezaeiandSwain(2008).Ourlimiting modelis stilladiscretestatespacemodel.Theseresultsaremoreprecisethantheoneof ShahrezaeiandSwain(2008),andmoregeneral(somekineticsratescanbenon- linear). We use martingales techniques, with a proof that is similar to Crudu et al. (2012)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 , Production of X at rate λ (X ,X ) 1 1 1 1 2 ∅ −−−−−−−→ X γ1(X1,X2) , Destruction of X at rate γ (X ,X ) 1 −−−−−−−→ ∅ 1 1 1 2 (1) λ2(X1,X2) X , Production of X at rate λ (X ,X ) 2 2 2 1 2 ∅ −−−−−−−→ X γ2(X1,X2) , Destruction of X at rate γ (X ,X ) 2 2 2 1 2 −−−−−−−→ ∅ with γ (0,X ) = γ (X ,0) = 0 to ensure positivity. This model can be rep- 1 2 2 1 resented by a continuous time Markov chain in N2, and is then a general birth and death process in N2. It can be described by the following set of stochastic differential equations t t X (t)=X (0)+Y λ (X (s),X (s))ds Y γ (X (s),X (s))ds , 1 1 1 1 1 2 2 1 1 2 − (cid:16)Z0 (cid:17) (cid:16)Z0 (cid:17) t t X (t)=X (0)+Y λ (X (s),X (s))ds Y γ (X (s),X (s))ds , 2 2 3 2 1 2 4 2 1 2 − (cid:16)Z0 (cid:17) (cid:16)Z0 (cid:17) where Y , for i=1...4 are independent standardpoisson processes.The genera- i tor of this process is given by Bf(X ,X )=λ (X ,X ) f(X +1,X ) f(X ,X ) 1 2 1 1 2 1 2 1 2 − h i +γ (X ,X ) f(X 1,X ) f(X ,X ) 1 1 2 1 2 1 2 − − (2) h i +λ (X ,X ) f(X ,X +1) f(X ,X ) 2 1 2 1 2 1 2 − h i +γ (X ,X ) f(X ,X 1) f(X ,X ) , 2 1 2 1 2 1 2 − − h i for every bounded function f on N2. R. Yvinec1/adiabatic reduction for PDMP 5 Exemple 1. We have in mind the standardmRNA-Protein system givenby the following choice: γ (X ,X ) = g X with g > 0 for i = 1,2, λ (X ,X ) = i 1 2 i i i 1 1 2 λ (X ) and λ (X ,X ) = k X with k > 0. Note however that even in the 1 2 2 1 2 2 1 2 contextofmodelsofgeneexpression,differentmodelshavebeenproposed,that includesnonlinearfeedbackofmRNAand/ornonlineardegradationtermsBose and Ghosh (2012). 1.1. Statement of the result We suppose the following scaling holds γN(X ,X )=Nγ (X ,X ) 1 1 2 1 1 2 λN(X ,X )=Nλ (X ,X ) 2 1 2 2 1 2 where N that is degradation of X and production of X occurs at a 1 2 → ∞ faster time scale than the two other reactions. Then X is degraded very fast, 1 and induces also as a very fast production of X . The rescaled model is given 2 by t t XN(t)=XN(0)+Y λ (XN(s),XN(s))ds Y Nγ (XN(s),XN(s))ds , 1 1 1 1 1 2 − 2 1 1 2 (cid:16)Z0 (cid:17) (cid:16)Z0 (cid:17) t t XN(t)=XN(0)+Y Nλ (XN(s),XN(s))ds Y γ (XN(s),XN(s))ds , 2 2 3 2 1 2 − 4 2 1 2 (cid:16)Z0 (cid:17) (cid:16)Z0 (3) (cid:17) and the generator of this process is given by B f(X ,X )=λ (X ,X ) f(X +1,X ) f(X ,X ) N 1 2 1 1 2 1 2 1 2 − h i +Nγ (X ,X ) f(X 1,X ) f(X ,X ) 1 1 2 1 2 1 2 − − (4) h i +Nλ (X ,X ) f(X ,X +1) f(X ,X ) 2 1 2 1 2 1 2 − h i +γ (X ,X ) f(X ,X 1) f(X ,X ) . 2 1 2 1 2 1 2 − − h i We can prove the following reduction holds: Theorem 1. We assume that 1. The degradation function on X satisfies γ (X ,0) 0. 2 2 1 ≡ 2. The degradation function on X satisfies γ (0,X ) 0, and 1 1 2 ≡ inf γ (X ,X )=γ >0. 1 1 2 X1≥1,X2≥0 3. The production rate of X satisfies λ (0,X )=0. 2 2 2 4. The production rate function λ and λ are linearly bounded by X +X . 1 2 1 2 5. Either λ or λ is bounded. 1 2 R. Yvinec1/adiabatic reduction for PDMP 6 Let (XN,XN) the stochastic process whose generator is B (defined in eq. (4)). 1 2 N Assumethattheinitialvector(XN(0),XN(0))convergesindistributionto(0,X(0)), 1 2 as N . Then, for all T >0, (XN(t),XN(t)) converges in L1(0,T) (and in Lp→, 1∞ p < ) to (0,X(t)) w1here X2(t) ist≥t0he stochastic process whose ≤ ∞ generator is given by B ϕ(X)=λ (0,X) ∞P (γ (1,.)ϕ(.))(X)dt ϕ(X) 1 t 1 ∞ (cid:16)Z0 − (cid:17) +γ (0,X) ϕ(X 1) ϕ(X) , (5) 2 − − h i where Ptg(X)=E g(Y(t,X)e−R0tγ1(1,Y(s,X))ds , and Y(t,X) is the stochastic p(cid:2)rocess starting at X at t = 0(cid:3) whose generator is given by Ag(Y)=λ (1,Y) g(Y +1) g(Y) . 2 − Remark 2. The firstthree hypotheses o(cid:0)ftheorem1 areth(cid:1)e maincharacteristics ofthemRNA-proteinsystem(seeexample1).Basically,theyimposethatquan- tities 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 variablecannotincreasewhenthefirstvariableis0.Hencethesethreehypothe- ses will guaranteethat (with our particular scaling) the first variable converges to 0, and will lead to an intermittent production of the second variable. The lasttwohypotheses aremore technical,andguaranteethatthe Markovchainis not explosive, and hence well defined for all t 0, and that the limiting model ≥ is well defined too. Remark 3. Theaboveexpressioneq.(5)isageneratorofaburstingmodelfora “generalburstingsizedistribution“.Forinstance,forlinearfunctionγ (X ,X )= 1 1 2 g X , and λ (X ,X )=k X , we have 1 1 2 1 2 2 1 P (γ (.)ϕ(.))(p) = g P (ϕ)(p), t 1 1 t = g1E ϕ(Yty)e−g1t , = g1e−hg1t ϕ(z)Pi Yty =z , z y X≥ (cid:8) (cid:9) = g1e−g1t ϕ(z)(k2t)z−ye−k2t. (z y)! z y − X≥ It follows by integration integration by parts that ∞ g1 k2 z P (γ (.)ϕ(.))(y)dt = ϕ(z+y) , t 1 g +k k +g Z0 1 2 Xz≥0 (cid:16) 2 1(cid:17) which gives then an additive geometric burst size distribution of parameter p= k2 , as expected Shahrezaei and Swain (2008). k2+g1 R. Yvinec1/adiabatic reduction for PDMP 7 We divide the proof in three steps: moment estimates, tightness and identification of the limit. 1.2. Moment estimates Because production rates are linearly bounded, it is straightforward that with f(X ,X ) = X +X in eq. (4), there is a constant C (that depends on N 1 2 1 2 N and other parameters) such that B f(X ,X ) C (X +X ). N 1 2 N 1 2 ≤ Then E XN(t)+XN(t) is bounded on any time interval [0,T] and 1 2 (cid:2) (cid:3) t f(XN(t),XN(t)) f(XN(0),XN(0)) B f(XN(s),XN(s))ds 1 2 − 1 2 − N 1 2 Z0 is a L1-martingale. 1.3. Tightness Clearly, from the stochastic differential equation on XN, we must have 1 XN(t) 0. We can show in fact that the Lebesgue measure of the set 1 → t T :XN(t)0 convergesto 0. Indeed, taking f(X ,X )=X in eq. (4), we { ≤ 1 } 1 2 1 have t XN(t) XN(0) (λ (XN(s),XN(s)) Nγ (XN(s),XN(s)))ds (6) 1 − 1 − 1 1 2 − 1 1 2 Z0 is a martingale. Thanks to the lower bound assumption on γ , we have 1 t t γE 1 ds E γ (XN(s),XN(s))ds. Z0 {X1N(s)≥1} ≤ Z0 1 1 2 (cid:2) (cid:3) Then, by the martingale property, we deduce from (6) t t γNE 1 ds E XN(0) + E λ (XN(s),XN(s)) ds. (7) Z0 {X1N(s)≥1} ≤ 1 Z0 1 1 2 (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) Now for XN we obtain from eq. (3), 2 t XN(t) XN(0)+Y N1 λ (XN(s),XN(s))ds . 2 ≤ 2 3(cid:16)Z0 {X1N(s)≥1} 2 1 2 (cid:17) Let us now distinguish between the two cases. Suppose first that λ is bounded (say by K). Then 2 • t E XN(t) E XN(0) +KNE 1 ds . 2 ≤ 2 Z0 {X1N(s)≥1} (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) R. Yvinec1/adiabatic reduction for PDMP 8 As λ is linearly bounded (say by K) by XN +XN, the upper bound 1 1 2 eq. (7) becomes t t γNE 1 ds E XN(0) +K E XN(s) +E XN(s) ds. (cid:2)Z0 {X1N(s)≥1} (cid:3)≤ (cid:2) 1 (cid:3) Z0 (cid:16) (cid:2) 1 (cid:3) (cid:2) 2 (cid:3)(cid:17) Finally, with eq. (6), it is clear that t E XN(t) E XN(0) +K E XN(s) +E XN(s) ds. 1 ≤ 1 1 2 (cid:2) (cid:3) (cid:2) (cid:3) Z0 (cid:16) (cid:2) (cid:3) (cid:2) (cid:3)(cid:17) Hence, with the three last inequalities, we can conclude by the Grönwall lemma that E XN(t) is bounded on [0,T], uniformly in N. Then 2 (cid:2) (cid:3) T NE 1 ds XN(s) 1 Z0 { 1 ≥ } (cid:2) (cid:3) is bounded and XN 0 in L1([0,T],N). By the law of large number, 1 → 1Y (N) is almost surely convergent, and hence almost surely bounded. N 3 We deduce then there exists a random variable C such that t XN(t) XN(0)+NC 1 ds, 2 ≤ 2 Z0 {X1N(s)≥1} almost everywhere. By Grönwall lemma and Markov inequality P sup XN(t) M 0 2 ≥ → t [0,T] ∈ (cid:8) (cid:9) as M , uniformly in N. →∞ Now suppose λ is bounded (say K). By the martingale eq. (6) (and the 1 • same lower bound hypothesis on γ , it is clear that 1 T NE 1 ds XN(s) 1 Z0 { 1 ≥ } (cid:2) (cid:3) is bounded and XN 0 in L1([0,T],N). Now, let us denote UN(t) = 1 → 1XN(t),VN = 1XN(t)andWN =N1 (whichisthenbounded iNn L11([0,T[)). FroNm e2q. (3), and from the{Xli1Nn(eta)≥r1b}ound on λ (say by K) 2 1 t VN(t) VN(0)+ Y NKWN(UN(s)+VN(s))ds . 3 ≤ N (cid:16)Z0 (cid:17) Then, still by the law of the large number there exists a random variable C such that t VN(t) VN(0)+C WN(UN(s)+VN(s))ds, ≤ Z0 R. Yvinec1/adiabatic reduction for PDMP 9 and hence t XN(t) XN(0)+C WN(XN(s)+XN(s))ds. 2 ≤ 2 1 2 Z0 By Grönwall lemma, t supXN(t) (XN(0)+XN(0))exp C WN(s)ds , 2 ≤ 1 2 [0,T] (cid:16) Z0 (cid:17) which is then bounded, uniformly in N. For any subdivision of [0,T], 0=t <t < <t =T, 0 1 n ··· n−1 n−1 ti+1 XN(t ) XN(t ) Y N1 λ (XN(s),XN(s))ds Xi=0 | 2 i+1 − 2 i | ≤ Xi=0 3(cid:16)Zti {X1N(s)≥1} 2 1 2 (cid:17) T Y N1 λ (XN(s),XN(s))ds ≤ 3(cid:16)Z0 {X1N(s)≥1} 2 1 2 (cid:17) so by a similar argument as above, we also get the tightness of the BV norm P XN K 0 k 2 k[0,T] ≥ → as K 0,independently in N(cid:8).Then XN is ti(cid:9)ght in Lp([0,T]), for any 1 p< → 2 ≤ (Giusti (1984)). ∞ 1.4. Identification of the limit We choose an adherence value (0,X (t)) of the sequence (XN(t),XN(t)) in 2 1 2 L1([0,T]) Lp([0,T]).Then a subsequence (again denoted by) (XN(t),XN(t)) × 1 2 converge to (0,X (t)), almost surely and for almost t [0,T]. We are looking 2 ∈ for test-functions such that t f(XN(t),XN(t)) f(XN(0),XN(0) B f(0,XN(s))1 ds 1 2 − 1 2 − N 2 X1N(s)=0 Z0 t B f(XN(s),XN(s))1 ds −Z0 N 1 2 X1N(s)≥1 is a martingale and B f(XN(s),XN(s)) is bounded independently of N when N 1 2 X 1. The following choice is inspired by Crudu et al. (2012). We introduce 1 the≥stochastic process Yx,y, starting at y and whose generator is t Axg(y)=λ (x,y) g(y+1) g(y) , 2 − h i foranyx 1.andweintroducethesemigroupPx definedonboundedfunction, ≥ t for any x 1, by ≥ Ptxg(y)=E g(Ytx,y)e−R0tγ1(x,Ysx,y)ds . (8) h i R. Yvinec1/adiabatic reduction for PDMP 10 Then the semigroup Px satisfies the equation t dPxg(y) t =AxPxg(y) γ (x,y)Pxg(y). dt t − 1 t Now for any bounded function g, define recursively f(0,y)=g(y), f(x,y)= ∞Px(γ (x,.)f(x 1,.))(y)dt. t 1 − Z0 Such a test function is well defined by the assumption on γ . We then verify 1 that B f(0,y)=λ (0,y) ∞P1(γ (1,.)g(.))(y)dt g(y) +γ (0,y) g(y 1) g(y) , N 1 t 1 − 2 − − (cid:16)Z0 (cid:17) h i B f(x,y)=λ (x,y) f(x+1,y) f(x,y) +γ (x,y) f(x,y 1) f(x,y) . N 1 2 − − − h i h i Indeed, for any x 1, ≥ Axf(x,y) γ (x,y)f(x,y) 1 − = ∞AxPx(γ (x,.)f(x 1,.))(y) γ (x,y)Px(γ (x,.)f(x 1,.))(y)dt, t 1 − − 1 t 1 − Z0 = ∞ dPx(γ (x,.)f(x 1,.))(y)dt, dt t 1 − Z0 = lim Px(γ (x,.)f(x,.))(y) γ (x,y)f(x 1,y), t t 1 − 1 − →∞ = γ (x,y)f(x 1,y). 1 − − Then λ (x,y) f(x,y+1) f(x,y) +γ (x,y) f(x 1,y) f(x,y) =0. 2 1 − − − h i h i Hence B f(x,y) is independent of N, and, taking the limit N in N →∞ t f(XN(t),XN(t)) f(XN(0),XN(0)) B f(XN(s),XN(s))ds, 1 2 − 1 2 − N 1 2 Z0 we deduce t g(X (t)) g(X (0)) B g(X ) 2 2 2 − −Z0 ∞ is a martingale where B g(y)=λ (0,y) ∞P1(γ (1,.)g(.))(y)dt g(y) +γ (0,y) g(y 1) g(y) . ∞ 1 (cid:16)Z0 t 1 − (cid:17) 2 h − − i Uniqueness Due to assumption on k and k , the limiting generator defines 1 2 a pure-jump Markov process in N which is not explosive. Uniqueness of the martingale then follows classically.