Bulletn of Mathematcal Bology (21 DOI 1.17/s11538-1-9517-4 ORIGINAL ARTICLE Product-Form Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz Department of Mathematcs, Unversty of Wsconsn, Madson, WI 5376, USA Receved: 14 Aprl 29 / Accepted: 1 February 21 Socety for Mathematcal Bology 21 Abstract We consder stochastcally modeled chemcal reacton systems wth massacton knetcs and prove that a product-form statonary dstrbuton exsts for each closed, rreducble subset of the state space f an analogous determnstcally modeled system wth mass-acton knetcs admts a complex balanced equlbrum. Fenberg s defcency zero theorem then mples that such a dstrbuton exsts so long as the correspondng chemcal network s weakly reversble and has a defcency of zero. The man parameter of the statonary dstrbuton for the stochastcally modeled system s a complex balanced equlbrum value for the correspondng determnstcally modeled system. We also generalze our man result to some non-mass-acton knetcs. Keywords Product-form statonary dstrbutons Defcency zero 1. Introducton There are two commonly used models for chemcal reacton systems: dscrete stochastc models n whch the state of the system s a vector gvng the number of each molecular speces, and contnuous determnstc models n whch the state of the system s a vector gvng the concentraton of each molecular speces. Dscrete stochastc models are typcally used when the number of molecules of each chemcal speces s low and the randomness nherent n the makng and breakng of chemcal bonds s mportant. Conversely, determnstc models are used when there are large numbers of molecules for each speces and the behavor of the concentraton of each speces s well approxmated by a coupled set of ordnary dfferental equatons. Typcally, the goal n the study of dscrete stochastc systems s to ether understand the evoluton of the dstrbuton of the state of the system or to fnd the long term statonary dstrbuton of the system, whch s the stochastc analog of an equlbrum pont. The Kolmogorov forward equaton (chemcal master equaton n the chemstry lterature descrbes the evoluton of the dstrbuton and so work has been done n tryng to analyze or solve the forward equaton for certan classes of systems (Gadgl et Correspondng author. E-mal address: anderson@math.wsc.edu (Davd F. Anderson.
Anderson et al. al., 25. However, t s typcally an extremely dffcult task to solve or even numercally compute the soluton to the forward equaton for all but the smplest of systems. Therefore, smulaton methods have been developed that wll generate sample paths so as to approxmate the dstrbuton of the state va Monte Carlo methods. These smulaton methods nclude algorthms that generate statstcally exact (Anderson, 27; Gllespe, 1976, 1977; Gbson and Bruck, 2 and approxmate (Anderson, 28b; Anderson et al., 21; Gllespe, 21; Cao et al., 26 sample paths. On the other hand, the contnuous determnstc models, and n partcular mass-acton systems wth complex balancng states, have been analyzed extensvely n the mathematcal chemstry lterature, startng wth the works of Horn, Jackson, and Fenberg (Horn, 1972, 1973; Horn and Jackson, 1972; Fenberg, 1972, and contnung wth Fenberg s defcency theory n Fenberg (1979, 1987, 1989, 1995. Such models have a wde range of applcatons n the physcal scences, and now they are begnnng to play an mportant role n systems bology (Cracun et al., 26; Gunawardena, 23; Sontag, 21. Recent mathematcal analyss of contnuous determnstc models has focused on ther potental to admt multple equlbra (Cracun and Fenberg, 25, 26 and on dynamcal propertes such as persstence and global stablty (Sontag, 21; Angel et al., 27; Anderson, 28a; Anderson and Cracun, 21; Anderson and Shu, 21. One of the major theorems pertanng to determnstc models of chemcal systems s the defcency zero theorem of Fenberg (1979, 1987. The defcency zero theorem states that f the network of a system satsfes certan easly checked propertes, then wthn each compatblty class (nvarant manfold n whch a soluton s confned there s precsely one equlbrum wth strctly postve components, and that equlbrum s locally asymptotcally stable (Fenberg, 1979, 1987. The surprsng aspect of the defcency zero theorem s that the assumptons of the theorem are completely related to the network of the system whereas the conclusons of the theorem are related to the dynamcal propertes of the system. We wll show n ths paper that f the condtons of the defcency zero theorem hold on the network of a stochastcally modeled chemcal system wth qute general knetcs, then there exsts a product-form statonary dstrbuton for each closed, rreducble subset of the state space. In fact, we wll show a stronger result: that a product-form statonary dstrbuton exsts so long as there exsts a complex balanced equlbrum for the assocated determnstcally modeled system. However, the equlbrum values guaranteed to exst by the defcency zero theorem are complex balanced and so the condtons of that theorem are suffcent to guarantee the exstence of the product-form dstrbuton. Fnally, the man parameter of the statonary dstrbuton wll be shown to be a complex balanced equlbrum value of the determnstcally modeled system. Product-form statonary dstrbutons play a central role n the theory of queueng networks where the product-form property holds for a large, naturally occurrng class of models called Jackson networks (see, for example, Kelly, 1979, Chap. 3, and Chen and Yao, 21, Chap. 2 and a much larger class of quas-reversble networks (Kelly, 1979, Chap. 3, Chen and Yao, 21, Chap. 4, Serfozo, 1999, Chap. 8. Kelly (1979, Secton 8.5, recognzes the possble exstence of product-form statonary dstrbutons for a subclass of chemcal reacton models and gves a condton for that exstence. That condton s essentally the complex balance condton descrbed below, and our man result asserts that for any mass-acton chemcal reacton model the condtons of the defcency zero theorem ensure that ths condton holds.
Product-Form Statonary Dstrbutons for Defcency Zero Chemcal The outlne of the paper s as follows. In Secton 2, we formally ntroduce chemcal reacton networks. In Secton 3, we develop both the stochastc and determnstc models of chemcal reacton systems. Also, n Secton 3, we state the defcency zero theorem for determnstc systems and present two theorems that are used n ts proof and that wll be of use to us. In Secton 4, we present the frst of our man results: that every closed, rreducble subset of the state space of a stochastcally modeled system wth mass-acton knetcs has a product-form statonary dstrbuton f the chemcal network s weakly reversble and has a defcency of zero. In Secton 5, we present some examples of the use of ths result. In Secton 6, we extend our man result to systems wth more general knetcs. 2. Chemcal reacton networks Consder a system wth m chemcal speces, {S 1,...,S m }, undergong a fnte seres of chemcal reactons. For the kth reacton, denote by ν k,ν k Zm the vectors representng the number of molecules of each speces consumed and created n one nstance of that reacton, respectvely. We note that f ν k = then the kth reacton represents an nput to the system, and f ν k = then t represents an output. Usng a slght abuse of notaton, we assocate each such ν k (and ν k wth a lnear combnaton of the speces n whch the coeffcent of S s ν k,theth element of ν k. For example, f ν k =[1, 2, 3] T for a system consstng of three speces, we assocate wth ν k the lnear combnaton S 1 + 2S 2 + 3S 3. For ν k =, we smply assocate ν k wth. Under ths assocaton, each ν k (and ν k s termed a complex of the system. We denote any reacton by the notaton ν k ν k,where ν k s the source, or reactant, complex and ν k s the product complex. We note that each complex may appear as both a source complex and a product complex n the system. The set of all complexes wll be denoted by {ν k }:= k ({ν k} {ν k }. Defnton 2.1. Let S ={S }, C ={ν k }, and R ={ν k ν k } denote the sets of speces, complexes, and reactons, respectvely. The trple {S, C, R} s called a chemcal reacton network. The structure of chemcal reacton networks plays a central role n both the study of stochastcally and determnstcally modeled systems. As alluded to n the ntroducton, t wll be condtons on the network of a system that guarantee certan dynamcal propertes for both models. Therefore, the remander of ths secton conssts of defntons related to chemcal networks that wll be used throughout the paper. Defnton 2.2. A chemcal reacton network, {S, C, R}, s called weakly reversble f for any reacton ν k ν k, there s a sequence of drected reactons begnnng wth ν k as a source complex and endng wth ν k as a product complex. That s, there exst complexes ν 1,...,ν r such that ν k ν 1,ν 1 ν 2,...,ν r ν k R. A network s called reversble f ν k ν k R whenever ν k ν k R. Remark. The defnton of a reversble network gven n Defnton 2.2 s dstnct from the noton of a reversble stochastc process. However, n Secton 4.2, we pont out a connecton between the two concepts for systems that are detaled balanced.
Anderson et al. To each reacton network, {S, C, R}, there s a unque, drected graph constructed n the followng manner. The nodes of the graph are the complexes, C. A drected edge s then placed from complex ν k to complex ν k f and only f ν k ν k R. Each connected component of the resultng graph s termed a lnkage class of the graph. We denote the number of lnkage classes by l. It s easy to see that a chemcal reacton network s weakly reversble f and only f each of the lnkage classes of ts graph s strongly connected. Defnton 2.3. S = span {νk ν k R}{ν k ν k} s the stochometrc subspace of the network. For c R m, we say c + S and (c + S R m > are the stochometrc compatblty classes and postve stochometrc compatblty classes of the network, respectvely. Denote dm(s = s. It s smple to show that for both stochastc and determnstc models, the state of the system remans wthn a sngle stochometrc compatblty class for all tme, assumng that one starts n that class. Ths fact s mportant because t changes the types of questons that are reasonable to ask about a gven system. For example, unless there s only one stochometrc compatblty class, and so S = R m, the correct queston s not whether there s a unque fxed pont for a gven determnstc system. Instead, the correct queston s whether wthn each stochometrc compatblty class there s a unque fxed pont. Analogously, for stochastcally modeled systems t s typcally of nterest to compute statonary dstrbutons for each closed, rreducble subset of the state space (each contaned wthn a stochometrc compatblty class wth the precse subset beng determned by ntal condtons. The fnal defnton of ths secton s that of the defcency of a network (Fenberg, 1979. It s not a dffcult exercse to show that the defcency of a network s always greater than or equal to zero. Defnton 2.4. The defcency of a chemcal reacton network, {S, C, R},s δ = C l s,where C s the number of complexes, l s the number of lnkage classes of the network graph, and s s the dmenson of the stochometrc subspace of the network. Whle the defcency s, by defnton, only a property of the network, we wll see n Sectons 3.2, 4, and 6 that a defcency of zero has mplcatons for the long-tme dynamcs of both determnstc and stochastc models of chemcal reacton systems. 3. Dynamcal models The noton of a chemcal reacton network s the same for both stochastc and determnstc systems and the choce of whether to model the evoluton of the state of the system stochastcally or determnstcally s made based upon the detals of the specfc chemcal or bologcal problem at hand. Typcally, f the number of molecules s low, a stochastc model s used, and f the number of molecules s hgh, a determnstc model s used. For cases between the two extremes, a dffuson approxmaton can be used or, for cases n whch the system contans multple scales, peces of the reacton network can be modeled stochastcally, whle others can be modeled determnstcally (or, more accurately, absolutely contnuously wth respect to tme. See, for example, Ball et al. (26 and Secton 5.1.
Product-Form Statonary Dstrbutons for Defcency Zero Chemcal 3.1. Stochastc models The smplest stochastc model for a chemcal network {S, C, R} treats the system as a contnuous tme Markov chan whose state X Z m s a vector gvng the number of molecules of each speces present wth each reacton modeled as a possble transton for the state. We assume a fnte number of reactons. The model for the kth reacton, ν k ν k, s determned by the vector of nputs, ν k, specfyng the number of molecules of each chemcal speces that are consumed n the reacton, the vector of outputs, ν k, specfyng the number of molecules of each speces that are created n the reacton, and a functon of the state, λ k (X, that gves the rate at whch the reacton occurs. Specfcally, f the kth reacton occurs at tme t, the new state becomes X(t = X(t + ν k ν k. Let R k (t denote the number of tmes that the kth reacton occurs by tme t. Then the state of the system at tme t can be wrtten as X(t = X( + k R k (t(ν k ν k, (1 where we have summed over the reactons. The process R k s a countng process wth ntensty λ k (X(t (called the propensty n the chemstry lterature and can be wrtten as ( ( R k (t = Y k λ k X(s ds, (2 where the Y k are ndependent, unt-rate Posson processes (Kurtz, 1977/1978, Ether and Kurtz, 1986, Chap. 11. Note that (1 and(2 gve a system of stochastc equatons that unquely determnes X up to sup{t : k R k(t < }. The generator for the Markov chan s the operator, A, defnedby Af (x = k λ k (x ( f(x+ ν k ν k f(x, (3 where f s any functon defned on the state space. A commonly chosen form for the ntensty functons λ k s that of stochastc massacton, whch says that for x Z m the rate of the kth reacton should be gven by ( m ( x λ k (x = κ k ν lk! l=1 ν k = κ k m l=1 x l! (x l ν lk! 1 {x l ν lk }, (4 for some constant κ k, where we adopt the conventon that!=1. Note that the rate (4 s proportonal to the number of dstnct subsets of the molecules present that can form the nputs for the reacton. Intutvely, ths assumpton reflects the dea that the system s well strred n the sense that all molecules are equally lkely to be at any locaton at any tme. For concreteness, we wll assume that the ntensty functons satsfy (4 throughout most of the paper. In Secton 6, we wll generalze our results to systems wth more general knetcs.
Anderson et al. A probablty dstrbuton {π(x} s a statonary dstrbuton for the chan f π(xaf(x = x for a suffcently large class of functons f or, takng f(y= 1 x (y and usng Eq. (3, f π(x ν k + ν kλ k (x ν k + ν k = π(x λ k (x (5 k k for all x n the state space. If the network s weakly reversble, then the state space of the Markov chan s a unon of closed, rreducble communcatng classes. (Ths fact follows because f the Markov chan can proceed from state x to state y va a sequence of reactons, weak reversblty of the network mples those reactons can be undone n reverse sequental order by another sequence of reactons. Also, each closed, rreducble communcatng class s ether fnte or countable. Therefore, f a statonary dstrbuton wth support on a sngle communcatng class exsts t s unque and lm P ( X(t = x X( = y = π(x, t for all x,y n that communcatng class. Thus, the statonary dstrbuton gves the longterm behavor of the system. Solvng Eq. (5 s n general a formdable task. However, n Secton 4 we wll do so f the network s weakly reversble, has a defcency of zero, and f the rate functons λ k (x satsfy mass-acton knetcs, (4. We wll also show that the statonary dstrbuton s of product form. More specfcally, we wll show that for each communcatng class there exsts a c R m > and a normalzng constant M> such that c x π(x = M π (x := M x! =1 =1 satsfes Eq. (5. The c n the defnton of π wll be shown to be the th component of an equlbrum value of the analogous determnstc system descrbed n the next secton. In Secton 6, we wll solve (5 for more general knetcs. 3.2. Determnstc models and the defcency zero theorem Under an approprate scalng lmt (see Secton 4.1 the contnuous tme Markov chan (1, (2, (4 becomes x(t = x( + k ( ( f k x(s ds (ν k ν k := x( + f ( x(s ds, (6 where the last equalty s a defnton and f k (x = κ k x ν 1k 1 x ν 2k 2 x ν mk m, (7 whereweusetheconventon = 1. We say that the determnstc system (6 hasmassacton knetcs f the rate functons f k have the form (7. The proof of the followng
Product-Form Statonary Dstrbutons for Defcency Zero Chemcal theorem by Fenberg can be found n Fenberg (1979 or Fenberg (1995. We note that the full statement of the defcency zero theorem actually says more than what s gven below and the nterested reader s encouraged to see the orgnal work. Theorem 3.1 (The Defcency Zero Theorem. Consder a weakly reversble, defcency zero chemcal reacton network {S, C, R} wth dynamcs gven by (6 (7. Then for any choce of rate constants {κ k }, wthn each postve stochometrc compatblty class there s precsely one equlbrum value, and that equlbrum value s locally asymptotcally stable relatve to ts compatblty class. The dynamcs of the system (6 (7 take place n R m. However, to prove the defcency zero theorem, t turns out to be more approprate to work n complex space, denoted R C, whch we wll descrbe now. For any U C, let ω U : C {, 1} denote the ndcator functon ω U (ν k = 1 {νk U}. Complex space s defned to be the vector space wth bass {ω νk ν k C}, where we have denoted ω {νk } by ω νk. If u s a vector wth nonnegatve nteger components and w s a vector wth nonnegatve real components, then let u!= u! and w u = wu, where we nterpret = 1 and!=1. Let Ψ : R m R C and A κ : R C R C be defned by Ψ(x= ν k C x ν k ω νk, A κ (y = ν k ν k R κ k y νk (ω ν k ω νk, where the subscrpt κ of A κ denotes the choce of rate constants for the system. Let Y : R C R m be the lnear map whose acton on the bass elements {ω νk } s defned by Y(ω νk = ν k. Then Eqs. (6 (7 can be wrtten as the coupled set of ordnary dfferental equatons ẋ(t = f ( x(t = Y ( A κ ( Ψ ( x(t. Therefore, n order to show that a value c s an equlbrum of the system, t s suffcent to show that A κ (Ψ (c =, whch s an explct system of equatons for c. In partcular, A k (Ψ (c = f and only f for each z C {k:ν k =z} κ k c νk = {k:ν k =z} κ k c ν k, (8 where the sum on the left s over reactons for whch z s the product complex and the sum on the rght s over reactons for whch z s the source complex. The followng has been shown n Horn and Jackson (1972 and Fenberg (1979 (see also Gunawardena, 23. Theorem 3.2. Let {S, C, R} be a chemcal reacton network wth dynamcs gven by (6 (7 for some choce of rate constants, {κ k }. Suppose there exsts a c R m > for whch A κ (Ψ (c =, then the followng hold:
Anderson et al. 1. The network s weakly reversble. 2. Every equlbrum pont wth strctly postve components, x R m > wth f(x=, satsfes A κ (Ψ (x =. 3. If Z ={x R m > f(x= }, then ln Z := {y Rm x Z and y = ln(x } s a coset of S, the perpendcular complement of S. That s, there s a k R m such that ln Z ={w R m w = k + u for some u S }. 4. There s one, and only one, equlbrum pont n each postve stochometrc compatblty class. 5. Each equlbrum pont of a postve stochometrc compatblty class s locally asymptotcally stable relatve to ts stochometrc compatblty class. Thus, after a choce of rate constants has been made, the conclusons of the defcency zero theorem pertanng to the exstence and asymptotc stablty of equlbra (ponts 4 and 5 of Theorem 3.2 hold so long as there exsts at least one c R m > such that A κ (Ψ (c =. The condton that the system has a defcency of zero only plays a role n showng that there does exst such a c R m >. A proof of the followng can be found n Fenberg (1979, 1987, 1995. Theorem 3.3. Let {S, C, R} be a chemcal reacton network wth dynamcs gven by (6 (7 for some choce of rate constants, {κ k }. If the network has a defcency of zero, then there exsts a c R m > such that A κ(ψ (c = f and only f the network s weakly reversble. A chemcal reacton network wth determnstc mass-acton knetcs (and a choce of rate constants that admts a c for whch A κ (Ψ (c = s called complex balanced n the lterature. The second concluson of Theorem 3.2 demonstrates why ths notaton s approprate. The equvalent representaton gven by (8 shows the orgn of ths termnology. The surprsng aspect of the defcency zero theorem s that t gves smple and checkable suffcent condtons on the network structure alone that guarantee that a system s complex balanced for any choce of rate constants. We wll see n the followng sectons that the man results of ths paper have the same property: product-form statonary dstrbutons exst for all stochastc systems that are complex balanced when vewed as determnstc systems, and δ = s a suffcent condton to guarantee ths for weakly reversble networks. 4. Man result for mass-acton systems The collecton of statonary dstrbutons for a countable state space Markov chan s convex. The extremal dstrbutons correspond to the closed, rreducble subsets of the state space; that s, every statonary dstrbuton can be wrtten as π = Γ α Γ π Γ, (9 where α Γ, Γ α Γ = 1, and the sums are over the closed, rreducble subsets Γ of the state space. Here π Γ s the unque statonary dstrbuton satsfyng π Γ (Γ = 1. We now state and prove our man result for systems wth mass-acton knetcs.
Product-Form Statonary Dstrbutons for Defcency Zero Chemcal Theorem 4.1. Let {S, C, R} be a chemcal reacton network and let {κ k } be a choce of rate constants. Suppose that, modeled determnstcally, the system s complex balanced wth complex balanced equlbrum c R m >. Then the stochastcally modeled system wth ntenstes (4 has a statonary dstrbuton consstng of the product of Posson dstrbutons, π(x = =1 c x x! e c, x Z m. (1 If Z m s rreducble, then (1 s the unque statonary dstrbuton, whereas f Zm s not rreducble then the π Γ of Eq. (9 are gven by the product-form statonary dstrbutons π Γ (x = M Γ m =1 c x x!, x Γ, and π Γ (x = otherwse, where M Γ s a postve normalzng constant. Proof: Let π satsfy (1 wherec R m > satsfes A κ(ψ (c =. We wll show that π s statonary by verfyng that Eq. (5 holds for all x Z m. Pluggng π and (4 nto Eq. (5 and smplfyng yelds κ k c ν k ν k 1 m (x ν k! k l=1 1 {xl ν lk } = k κ k 1 (x ν k! 1 {xl ν lk }. (11 l=1 Equaton (11 wll be satsfed f for each complex z C, κ k c νk z 1 1 {xl z (x z! l } = 1 κ k 1 {xl z (x z! l }, (12 {k:ν k =z} l=1 {k:ν k =z} l=1 where the sum on the left s over reactons for whch z s the product complex and the sum on the rght s over reactons for whch z s the source complex. The complex z s fxed n the above equaton, and so (12 s equvalent to (8, whch s equvalent to A κ (Ψ (c =. To complete the proof, one need only observe that the normalzed restrcton of π to any closed, rreducble subset Γ must also be a statonary dstrbuton. The followng theorem gves smple and checkable condtons that guarantee the exstence of a product-form statonary dstrbuton of the form (1. Theorem 4.2. Let {S, C, R} be a chemcal reacton network that has a defcency of zero and s weakly reversble. Then for any choce of rate constants {κ k } the stochastcally modeled system wth ntenstes (4 has a statonary dstrbuton consstng of the product of Posson dstrbutons, π(x = =1 c x x! e c, x Z m,
Anderson et al. where c s an equlbrum value for the determnstc system (6 (7, whch s guaranteed to exst and be complex balanced by Theorems 3.1 3.3. If Z m s rreducble, then π s the unque statonary dstrbuton, whereas f Z m s not rreducble then the π Γ of Eq.(9 are gven by the product-form statonary dstrbutons π Γ (x = M Γ m =1 c x x!, x Γ, and π Γ (x = otherwse, where M Γ s a postve normalzng constant. Proof: Ths s a drect result of Theorems 3.3 and 4.1. We remark that Theorems 4.1 and 4.2 gve suffcent condtons under whch Z m beng rreducble guarantees that when n dstrbutonal equlbrum the speces numbers: (a are ndependent and (b have Posson dstrbutons. We return to ths pont n Examples 5.2 and 5.3. 4.1. The classcal scalng Defnng ν k = ν k and lettng V be a scalng parameter usually taken to be the volume of the system tmes Avogadro s number, t s reasonable to scale the rate constants of the stochastc model wth the volume lke κ k = ˆκ k V ν k 1, (13 for some ˆκ k >. Ths follows by consderng the probablty of a partcular set of ν k molecules fndng each other n a volume proportonal to V n a tme nterval [t,t + t. In ths case, the ntensty functons become λ V k (x = ˆκ ( k ν V ν k! k 1 ( x ν k 1 x! = V ˆκ k V ν k (x ν k!. (14 Snce V s the volume tmes Avogadro s number and x gves the number of molecules of each speces present, c = V 1 x gves the concentratons n moles per unt volume. Wth ths scalng and a large volume lmt, λ V k (x V ˆκ k c ν k = V ˆκ k c ν k V ˆλ k (c. (15 Snce the law of large numbers for the Posson process mples V 1 Y k (V u u, (2 and (15, together wth the assumpton that X( = VC( for some C( R m >,mply C(t = V 1 X(t C( + k ˆκ k C(s ν k ds (ν k ν k,
Product-Form Statonary Dstrbutons for Defcency Zero Chemcal whch n the large volume lmt gves the classcal determnstc law of mass acton detaled n Secton 3.2. For a precse formulaton of the above scalng argument, termed the classcal scalng ; see Kurtz (1972, 1977/1978, 1981. Because the above scalng s the natural relatonshp between the stochastc and determnstc models of chemcal reacton networks, we expect to be able to generalze Theorem 4.1 to ths settng. Theorem 4.3. Let {S, C, R} be a chemcal reacton network. Suppose that, modeled determnstcally wth rate constants {ˆκ k }, the system s complex balanced wth complex balanced equlbrum c R m >. For some V>, let {κ k} be related to {ˆκ k } va (13. Then the stochastcally modeled system wth ntenstes (4 and rate constants {κ k } has a statonary dstrbuton consstng of the product of Posson dstrbutons, π(x = (V c x e Vc, x Z m x!. (16 =1 If Z m s rreducble, then (16 s the unque statonary dstrbuton, whereas f Zm s not rreducble then the π Γ of Eq. (9 are gven by the product-form statonary dstrbutons π Γ (x = M Γ m =1 (V c x, x Γ, x! and π Γ (x = otherwse, where M Γ s a postve normalzng constant. Proof: The proof s smlar to before, and now conssts of makng sure the V s cancel n an approprate manner. The detals are omtted. We see that Theorem 4.1 follows from Theorem 4.3 by takng V = 1. Theorem 4.2 generalzes n the obvous way. 4.2. Reversblty and detal balance An equlbrum value, c R m >, for a reversble, n the sense of Defnton 2.2, chemcal reacton network wth determnstc mass-acton knetcs s called detaled balanced f for each par of reversble reactons, ν k ν k,wehave κ k c ν k = κ k cν k, (17 where κ k,κ k are the rate constants for the reactons ν k ν k,ν k ν k, respectvely. Fenberg (1989, p. 182 shows that f one postve equlbrum s detaled balanced then they all are; a result smlar to the second concluson of Theorem 3.2 for complex balanced systems. A reversble chemcal reacton system wth determnstc mass acton knetcs s therefore called detaled balanced f t admts one detaled balanced equlbrum. It s mmedate that any system that s detaled balanced s also complex balanced. The fact that a product-form statonary dstrbuton of the form (1 exsts for the stochastc systems whose determnstc analogs are detaled balanced s well known. See, for example, Whttle (1986. Theorems 4.1 and 4.2 can therefore be vewed as an extenson of that
Anderson et al. result. However, more can be sad n the case when the determnstc system s detaled balanced, and whch we nclude here for completeness (no orgnalty s beng clamed. As mentoned n the remark followng Defnton 2.2, the term reversble has a meanng n the context of stochastc processes that dffers from that of Defnton 2.2. Before defnng ths, we need the concept of a transton rate. For any contnuous tme Markov chan wth state space Γ,thetranston rate from x Γ to y Γ (wth x y s a nonnegatve number α(x,y satsfyng P ( X(t + t = y X(t = x = α(x,y t + o( t. Thus, n the context of ths paper, f y = x + ν k ν k for some k, thenα(x,y = λ k (x, and zero otherwse. Defnton 4.4. A contnuous tme Markov chan X(t wth transton rates α(x,y s reversble wth respect to the dstrbuton π f for all x,y n the state space Γ π(xα(x,y = π(yα(y,x. (18 It s smple to see (by summng both sdes of (18 wth respect to y over Γ, that π must be a statonary dstrbuton for the process. A statonary dstrbuton satsfyng (18 s even called detaled balanced n the probablty lterature. The followng s proved n Whttle (1986, Chap. 7. Theorem 4.5. Let {S, C, R} be a reversble 2 chemcal reacton network wth rate constants {κ k }. Then the determnstcally modeled system wth mass-acton knetcs has a detaled balanced equlbrum f and only f the stochastcally modeled system wth ntenstes (4 s reversble wth respect to ts statonary dstrbuton. 3 Succnctly, ths theorem says that reversblty and detaled balanced n the determnstc settng s equvalent to reversble (and hence, detaled balanced n the stochastc settng. 4.3. Non-unqueness of c For stochastcally modeled chemcal reacton systems any rreducble subset of the state space, Γ, s contaned wthn (y + S Z m for some y Rm. Therefore, each Γ s assocated wth a stochometrc compatblty class. For weakly reversble systems wth a defcency of zero, Theorems 3.2 and 3.3 guarantee that each such stochometrc compatblty class has an assocated equlbrum value for whch A κ (Ψ (c =. However, nether Theorem 4.1 nor Theorem 4.2 makes the requrement that the equlbrum value used n the product-form statonary measure π Γ ( be contaned wthn the stochometrc compatblty class assocated wth Γ. Therefore, we see that one such c can be used to construct a product-form statonary dstrbuton for every closed, rreducble subset. Conversely, for a gven rreducble subset Γ any postve equlbrum value of the system 2 In the sense of Defnton 2.2. 3 In the sense of Defnton 4.4.
Product-Form Statonary Dstrbutons for Defcency Zero Chemcal (6 (7 can be used to construct π Γ (. Ths fact seems to be contrary to the unqueness of the statonary dstrbuton; however, t can be understood through the thrd concluson of Theorem 3.2 as follows. Let Γ be a closed, rreducble subset of the state space wth assocated postve stochometrc compatblty class (y + S Z m,andletc 1,c 2 R m > be such that A κ (Ψ (c 1 = A κ (Ψ (c 2 =. For {1, 2} and x Γ,letπ (x = M c x/x!, wherem 1 and M 2 are normalzng constants. Then for each x Γ π 1 (x π 2 (x = M 1c1 x x! x! M 2 c x 2 = M 1 c1 x. M 2 c x 2 For any vector u,wedefne(ln(u = ln(u. Then for x Γ y + S c1 x c2 x = e x (ln c 1 ln c 2 = e y (ln c 1 ln c 2 = cy 1 c y, (19 2 where the second equalty follows from the thrd concluson of Theorem 3.2. Therefore, π 1 (x π 2 (x = M 1 c y 1 M 2 c y 2. (2 Fnally, ( /( 1 = M 1 c1 x /x! M 2 c2 x /x! x Γ x Γ = M ( y 1 c /( 1 M 2 c y c2 x /x! c2 x /x! 2 x Γ x Γ = π 1(x π 2 (x, where the second equalty follows from Eq. (19 and the thrd equalty follows from Eq. (2. We therefore see that the statonary measure s ndependent of the choce of c, as expected. 5. Examples Our frst example ponts out that the exstence of a product-form statonary dstrbuton for the closed, rreducble subsets of the state space does not necessarly mply ndependence of the speces numbers. Example 5.1 (Nonndependence of speces numbers. Consder the smple reversble system S 1 k 1 k 2 S 2,
Anderson et al. where k 1 and k 2 are nonzero rate constants. We suppose that X 1 ( + X 2 ( = N, andso X 1 (t + X 2 (t = N for all t. Ths system has two complexes, one lnkage class, and the dmenson of the stochometrc compatblty class s one. Therefore, t has a defcency of zero. Snce t s also weakly reversble, our results hold. An equlbrum to the system that satsfes the complex balance equaton s ( k2 c =, k 1 + k 2 k 1 k 1 + k 2, and the product-form statonary dstrbuton for the system s π(x = M cx 1 1 c x 2 2 x 1! x 2!, where M> s a normalzng constant. Usng that X 1 (t + X 2 (t = N for all t yelds π 1 (x 1 = M cx 1 1 c N x 1 2 x 1! (N x 1! = M x 1!(N x 1! cx 1 1 (1 c 1 N x 1, for x 1 N. After settng M = N!, we see that X 1 s bnomally dstrbuted. Smlarly, ( N π 2 (x 2 = c x 2 2 (1 c 2 N x 2, x 2 for x 2 N. Therefore, we trvally have that P(X 1 = N= c N 1 and P(X 2 = N= c N 2, but P(X 1 = N,X 2 = N= c N 1 cn 2,andsoX 1 and X 2 are not ndependent. Remark. The concluson of the prevous example, that ndependence does not follow from the exstence of a product-form statonary dstrbuton, extends trvally to any network wth a conservaton relaton among the speces. Example 5.2 (Frst order reacton networks. The results presented below for frst order reacton networks are known n both the queueng theory and mathematcal chemstry lterature. See, for example, Kelly (1979 and Gadgl et al. (25. We present them here to pont out how they follow drectly from Theorem 4.2. We begn by defnng v = v for any vector v R m. We say a reacton network s a frst order reacton network f ν k {, 1} for each complex ν k C. Therefore, a network s frst order f each entry of the ν k are zeros or ones, and at most one entry can be a one. It s smple to show that frst order reacton networks necessarly have a defcency of zero. Therefore, the results of ths paper are applcable to all frst order reacton networks that are weakly reversble. Consder such a reacton network wth only one lnkage class (for f there s more than one lnkage class we may consder the dfferent lnkage classes as dstnct networks. We say that the network s open f there s at least one reacton, ν k ν k,forwhchν k =. Hence, by weak reversblty, there must also be a reacton for whch ν k =. If no such reacton exsts, we say the network s closed. If the network s open we see that S = R m, Γ = Z m s rreducble, and so by Theorem 4.2 the unque
Product-Form Statonary Dstrbutons for Defcency Zero Chemcal statonary dstrbuton s π(x = =1 c x x! e c, x Z m, where c R m > s the complexed balanced equlbrum of the assocated (lnear determnstc system. Therefore, when n dstrbutonal equlbrum, the speces numbers are ndependent and have Posson dstrbutons. Note that nether the ndependence nor the Posson dstrbuton resulted from the fact that the system under consderaton was a frst order system. Instead both facts followed from Γ beng all of Z m. In the case of a closed, weakly reversble, sngle lnkage class, frst order reacton network, t s easy to see that there s a unque conservaton relaton X 1 (t+ +X m (t = N, for some N. Thus, n dstrbutonal equlbrum X(t has a multnomal dstrbuton. That s for any x Z m satsfyng x 1 + x 2 + +x m = N ( π(x = N x 1,x 2,...,x m c x = N! x 1! x m! cx 1 1 cxm m, (21 where c R m > s the equlbrum of the assocated determnstc system normalzed so that c = 1. As n the case of the open network, we note that the form of the equlbrum dstrbuton does not follow from the fact that the network only has frst order reactons. Instead, (21 follows from the structure of the closed, rreducble communcatng classes. Example 5.3 (Enzyme knetcs I. Consder the possble model of enzyme knetcs gven by E + S ES E + P, E S, (22 where E represents an enzyme, S represents a substrate, ES represents an enzymesubstrate complex, P represents a product, and some choce of rate constants has been made. We note that both E and S are beng allowed to enter and leave the system. The network (22 s reversble and has sx complexes and two lnkage classes. The dmenson of the stochometrc subspace s readly checked to be four, and so the network has a defcency of zero. Theorem 4.2 apples and so the stochastcally modeled system has a product-form statonary dstrbuton of the form (1. Orderng the speces as X 1 = E, X 2 = S, X 3 = ES, andx 4 = P, the reacton vectors for ths system nclude 1 1, 1, 1 1, 1 1 1. We therefore see that Γ = Z 4 s the unque closed, rreducble communcatng class of the stochastcally modeled system and Theorem 4.2 tells us that n dstrbutonal equlbrum the speces numbers are ndependent and have Posson dstrbutons wth parameters c, whch are the complex balanced equlbrum values of the analogous determnstcally modeled system.
Anderson et al. Example 5.4 (Enzyme knetcs II. Consder the possble model for enzyme knetcs gven by E + S k 1 k 1 ES k2 E + P, k 2 k3 k 3 E, (23 where the speces E, S, ES, andp are as n Example 5.3. We are now allowng only the enzyme E to enter and leave the system. The network s reversble, there are fve complexes, two lnkage classes, and the dmenson of the stochometrc compatblty class s three. Therefore, Theorem 4.2 mples that the stochastcally modeled system has a product-form statonary dstrbuton of the form (1. The only conserved quantty of the system s S + ES + P,andsoX 2 (t + X 3 (t + X 4 (t = N for some N>and all t. Therefore, after solvng for the normalzng constant, we have that for any x Z 4 satsfyng x 2 + x 3 + x 4 = N π(x = e c 1 cx 1 1 N! x 1! x 2!x 3!x 4! cx 2 2 cx 3 3 cx 4 4 = e c1 cx 1 1 ( N x 1! x 2,x 3,x 4 c x 2 2 cx 3 3 cx 4 4, where c = (k 3 /k 3,c 2,c 3,c 4 has been chosen so that c 2 + c 3 + c 4 = 1. Thus, when the stochastcally modeled system s n dstrbutonal equlbrum we have that: (a E has a Posson dstrbuton wth parameter k 3 /k 3,(bS, ES, and P are multnomnally dstrbuted, and (c E s ndependent from S, ES, and P. 5.1. The multscale nature of reacton networks Wthn a cell, some chemcal speces may be present n much greater abundance than others. In addton, the rate constants κ k may vary over several orders of magntude. Consequently, the scalng lmt that gves the classcal determnstc law of mass acton detaled n Secton 4.1 may not be approprate, and a dfferent approach to dervng a scalng lmt approxmaton for the basc Markov chan model must be consdered. As a consequence of the multple scales n a network model, t may be possble to separate the network nto subnetworks of speces and reactons, each domnated by a tme scale of a specfc magntude. Wthn each subnetwork, the graph structure and stochometrc propertes may determne propertes of the asymptotc solutons of the subnetwork. Example 5.5. Consder the reacton network S + E 1 C P + E 1, E 1 A + E 2, E 2, where E 2 and E 2 represent producton and degradaton of E 2, respectvely, S s a substrate beng converted to a product P, E 1 and E 2 are enzymes, and A s a substrate that reacts wth E 2 allostercally to transform t nto an actve form. We suppose that ( the enzymes E 1, E 2, and the substrate A are n relatvely low abundances, ( the substrate S has a large abundance of O(V, and ( the reacton rates are also of the order O(V. We change notaton slghtly and denote the number of molecules of speces A at tme t as XA V (t, and smlarly for the other speces. Further, we
Product-Form Statonary Dstrbutons for Defcency Zero Chemcal denote XS V (t/v = ZV S (t. Combned wth the conservaton relaton XV E 1 + XC V + XV A = M Z >, the scaled equatons for the stochastc model are ZS V (t = ZV S ( V 1 Y 1 (V κ 1 ZS V (sxv E 1 (s ds + V 1 Y 2 (V X V E 1 (t = X V E 1 ( Y 1 (V + Y 3 (V + Y 5 (V X V A (t = XV A ( + Y 4 κ 2 X V C (s ds, ( κ 1 ZS V (sxv E 1 (s ds + Y 2 V κ 3 X V C (s ds Y 4 (V κ 5 XA V (sxv E 2 (s ds, ( V X V E 2 (t = X V E 2 ( + Y 6 (V κ 6 t+ Y 4 (V Y 5 (V ( κ 4 XE V 1 (s ds Y 5 V κ 5 X V A (sxv E 2 (s ds κ 2 XC V (s ds κ 4 XE V 1 (s ds κ 4 XE V 1 (s ds Y 7 ( V κ 5 XA V (sxv E 2 (s ds, κ 7 XE V 2 (s ds, where the Y are unt-rate Posson processes. The frst equaton satsfes ZS V (t = ZV S ( V 1 Y 1 (V κ 1 ZS V (s xμ V s (dx ds + V 1 Y 2 (V κ 2 xηs, V (dx ds where μ V s (A = I {XE V (s A} and ηv s (A = I {X V 1 C (s A} are the respectve occupaton measures. Usng methods from stochastc averagng (see, for example, Ball et al., 26; Kurtz, 1992, as V the fast system s averaged out : Z S (t = Z S ( κ 1 Z S (s xμ s (dx ds + κ 2 xη s (dx ds, (24 where μ s and η s are the statonary dstrbutons of X E1 and X C, respectvely, of the fast subsystem wth Z S (s held constant (assumng a statonary dstrbuton exsts. Ths reduced network (.e., the fast subsystem s κ 5 κ 1 Z S (s A + E 2 E 1 C, κ6 E 2. (25 κ 4 κ 2 +κ 3 κ 7
Anderson et al. Settng z = Z S (s, we have the followng equlbrum relatons for the moments of the above network: κ 4 E[X E1 ] κ 5 E[X A X E2 ]=, (κ 1 z + κ 4 E[X E1 ]+(κ 2 + κ 3 E[X C ]+κ 5 E[X A X E2 ]=, κ 6 + κ 4 E[X E1 ] κ 5 E[X A X E2 ] κ 7 E[X E2 ]=, E[X E1 ]+E[X C ]+E[X A ]=M. (26 E[X E1 ] and E[X C ], whch are both functons of z and needed n Eq. (24, cannot be explctly solved for va the above equatons wthout extra tools as (26 s a system of four equatons wth fve unknowns. Ths stuaton arses frequently as t stems from the nonlnearty of the system. However, the network (25 conssts of fve complexes, two connected components, and the dmenson of ts stochometrc subspace s three. Therefore, ts defcency s zero. As t s clearly weakly reversble, Theorem 4.1 apples and, due to the product form of the dstrbuton and the unboundedness of the support of X E2, t s easy to argue that X E2 s ndependent of X A, X E1,andX C when n equlbrum. Thus, E[X A X E2 ]=E[X A ]E[X E2 ] and the frst moments can be solved for as functons of Z S (s. After solvng and nsertng these moments, (24 becomes κ 1 κ 3 κ 5 κ 6 MZ S (s Z S (t = Z S ( (κ 5 κ 6 + κ 7 κ 4 (κ 2 + κ 3 + κ 1 κ 5 κ 6 Z S (s ds, whch s Mchaels Menten knetcs. 6. More general knetcs In ths secton, we extend our results to systems wth more general knetcs than stochastc mass acton. The generalzatons we make are more or less standard for the types of results presented n ths paper (see, for example, Kelly, 1979, Secton 8.5, Whttle, 1986, Chap. 9. What s surprsng, however, s that the condtons of the defcency zero theorem of Fenberg (whch are condtons on mass-acton determnstc systems are also suffcent to guarantee the exstence of statonary dstrbutons of stochastcally modeled systems even when the ntensty functons are not gven by (4. It s nterestng to note that the generalzatons made here for the stochastc defcency zero Theorem 4.2 are smlar to those made n Sontag (21, whch generalzed Fenberg s defcency zero Theorem 3.1 n the determnstc settng. Suppose that the ntensty functons of a stochastcally modeled system are gven by λ k (x = κ k m ν k 1 =1 j= θ (x j= κ k m =1 θ (x θ (x 1θ ( x (ν k 1, (27 where the κ k are postve constants, θ : Z R, θ (x = fx, and we use the conventon that 1 j= a j = 1forany{a j }. Note that the fnal condton allows us to drop the ndcator functons of (4. As ponted out n Kelly (1979, the functon θ should be thought of as the rate of assocaton of the th speces. We gve a few nterestng choces
Product-Form Statonary Dstrbutons for Defcency Zero Chemcal for θ.ifθ (x = x for x, then (27 s stochastc mass-acton knetcs. However, f for x, θ (x = v x, (28 k + x for some postve constants k and v, then the system has a type of stochastc Mchaels Menten knetcs (Keener and Sneyd, 1998, Chap. 1. Fnally, f ν k {, 1} and θ (x = mn{n,x } for x, then the dynamcal system models an M/M/n queueng network n whch the th speces (and n ths case complex represents the queue length of the th queue, whch has n servers who work on a frst come, frst serve bass. The man restrcton mposed by (27 s that for any reacton for whch the th speces appears n the source complex, the rate of that reacton must depend upon X va θ (X only. Therefore, f, say, the th speces s governed by the knetcs (28, then the constants k and v must be the same for each ntensty whch depends upon X (although the v may be ncorporated nto the rate constants κ k, and so the real restrcton s on the constant k. However, systems wth ntenstes gven by (27 are qute general n that dfferent knetcs can be ncorporated nto the same model through the functons θ. For example, f n a certan system speces S 1 s modeled to be governed by Mchaels Menten knetcs (28 and speces S 2 s modeled to be governed by mass-acton knetcs, then the reacton S 1 + S 2 ν k would have ntensty v 1 x 1 λ k (x = κ k x 2, k 1 + x 1 for some constant κ k. In followng we use the conventon that j=1 a j = 1 for any choce of {a j }. Theorem 6.1. Let {S, C, R} be a stochastcally modeled chemcal reacton network wth ntensty functons (27. Suppose that the assocated mass-acton determnstc system wth rate constants {κ k } has a complex balanced equlbrum c R m >. Then the stochastcally modeled system admts the statonary dstrbuton π(x = M =1 c x x j=1 θ (j, x Zm, (29 where M> s a normalzng constant, provded that (29 s summable. If Z m s rreducble, then (29 s the unque statonary dstrbuton, whereas f Z m s not rreducble then the π Γ of equaton (9 are gven by the product-form statonary dstrbutons π Γ (x = M Γ m =1 c x x j=1 θ, x Γ, (3 (j and π Γ (x = otherwse, where M Γ > s a normalzng constant, provded that (3 s summable. Proof: The proof conssts of pluggng (29 and(27 nto equaton (5 and verfyng that c beng a complex balanced equlbrum s suffcent. The detals are smlar to before and so are omtted.
Anderson et al. Remark. We smply remark that just as Theorem 4.2 followed drectly from Theorem 4.1, the results of Theorem 6.1 hold, ndependent of the choce of rate constants κ k, so long as the assocated network s weakly reversble and has a defcency of zero. Example 6.2. Consder a network, {S, C, R}, that s weakly reversble and has a defcency of zero. Suppose we have modeled the dynamcs stochastcally wth ntensty functons gven by (27 wth each θ gven va (28 for some choce of v > andk a nonnegatve nteger. That s, we consder a system endowed wth stochastc Mchaels Menten knetcs. Then x j=1 x θ (j = j=1 v j k + j = vx / ( k + x. x Thus, our canddate for a statonary dstrbuton s π(x = M =1 c x x j=1 θ (j = M ( k + x =1 x ( c v x. (31 Notng that ( k + x = O(x k, x x, we see that π(x gvenby(31 s summable f c <v for each speces S whose possble abundances are unbounded. In ths case, (31 s ndeed a statonary dstrbuton for the system. We note that the condton c <v for each speces S s both necessary and suffcent to guarantee summablty f Z m s rreducble, as n such a stuaton the speces numbers are ndependent. Example 6.3. Levne and Hwa (27 computed and analyzed the statonary dstrbutons of dfferent stochastcally modeled chemcal reacton systems wth Mchaels Menten knetcs (28. The models they consdered ncluded among others: drected pathways ( S 1 S 2 S L, reversble pathways ( S 1 S 2 S L, pathways wth dluton of ntermedates (S, and cyclc pathways (S L S 1. Each of the models consdered n Levne and Hwa (27 s bologcally motvated and has a frst order reacton network ( ν k {, 1}, see Example 5.2, whch guarantees that they have a defcency of zero. Further, the networks of the models consdered are weakly reversble; therefore, the results of the current paper, and n partcular Theorem 6.1 and the remark that follows, apply so long as the restrctons dscussed n the paragraph precedng Theorem 6.1 are met. Whle these restrcton are not always met (for example, dluton s typcally modeled wth a lnear ntensty functon and there s no reason for the k of a forward and a backward reacton for a speces S n a reversble pathway to be the same, they found that the statonary dstrbutons for these models are ether of product form (when the restrctons are met or near product form (when the restrctons are not met. Further, because Z m s rreducble n each of these models, the product form of the dstrbuton mples that the speces numbers are ndependent. It s then postulated that the ndependence of the speces numbers could play an mportant, benefcal, bologcal role
Product-Form Statonary Dstrbutons for Defcency Zero Chemcal (see Levne and Hwa, 27, for detals. Smlar to the conclusons we drew n Example 5.2, Theorem 6.1 and the remark that follows pont out how the models analyzed n Levne and Hwa (27 are actually specal cases of a qute general famly of systems that have both the product form and ndependence propertes, and that these propertes may be more wdespread, and taken advantage of by lvng organsms, than prevously thought. We return to the result of Example 6.2 pertanng to the summablty of (31andshow that ths can be generalzed n the followng manner. Theorem 6.4. Suppose that for some closed, rreducble Γ Z m, π Γ : Γ R satsfes π Γ (x = M =1 c x x j=1 θ (j, for some c R m > and M>, where θ : Z R for each. Then π Γ (x s summable f for each for whch sup{x x Γ }= we have that θ (j>c + ɛ for some ɛ> and j suffcently large. Proof: The condtons of the theorem mmedately mply that there are postve constants C and ρ for whch π Γ (x < Ce ρ x,forallx Γ, whch mples that π Γ (x s summable. It s temptng to beleve that the condtons of Theorem 6.4 are n fact necessary, as n the case when Z m s rreducble. The followng smple example shows ths not to be the case. Example 6.5. Consder the reacton system wth network S 1 + S 2, where the rate of the reacton S 1 + S 2 s λ 1 (x = 1, and the rate of the reacton S 1 + S 2 s λ 2 (x = 1 θ 1 (x 1 θ 2 (x 2,where θ 1 (x 1 = 3x 1 1 + x 1, θ 2 (x 2 = (1/2x 2 1 + x 2. Assume further that X 1 ( = X 2 (. For the more physcally mnded readers, we note that ths model could descrbe a reacton system for whch there s a chemcal complex C = S 1 S 2 that sporadcally breaks nto ts chemcal consttuents, whch may then reform. The complex C may be present n such hgh numbers relatve to free S 1 and S 2 that we choose to model t as fxed, whch leads to the above reacton network. We note that n ths case, the reacton rates {κ k } for the correspondng mass-acton determnstc system are both equal to one, and so an equlbrum value guaranteed to exst for the determnstcally modeled system by the defcency zero theorem s c = (1, 1. Ths system does not satsfy the assumptons of Theorem 6.4 because both X 1 and X 2
Anderson et al. are unbounded and lm j θ 2 (j = 1/2 < 1 = c 2. However, for any x Γ ={x Z 2 : x 1 = x 2 }, ( ( 1 + x1 1 x1 ( ( 1 + x2 1 x2 ( 1 + 2 ( x1 2 x1 π Γ (x = =, x 1 3 x 2 (1/2 x 1 3 whch s summable over Γ. For the most general knetcs handled n ths paper, we let the ntensty functons of a stochastcally modeled system be gven by θ(x λ k (x = κ k 1 {xl ν θ(x ν k lk }, (32 l=1 where the κ k are postve constants, and θ : Z m R >.Notethatf θ(x= x θ (j, =1 j=1 for some functons θ,then(32 s equvalent to (27, and so the followng theorem mples Theorem 6.1. It s proof s smlar to the prevous theorems and so s omtted. Theorem 6.6. Let {S, C, R} be a stochastcally modeled chemcal reacton network wth ntensty functons (32. Suppose that the assocated mass-acton determnstc system wth rate constants {κ k } has a complex balanced equlbrum c R m >. Then the stochastcally modeled system admts the statonary dstrbuton π(x = M 1 θ(x =1 c x, x Z m, (33 where M> s a normalzng constant, provded that (33 s summable. If Z m s rreducble, then (33 s the unque statonary dstrbuton, whereas f Z m s not rreducble then the π Γ of equaton (9 are gven by the product-form statonary dstrbutons π Γ (x = M Γ 1 θ(x =1 c x, x Γ, (34 and π Γ (x = otherwse, where M Γ > s a normalzng constant, provded that (34 s summable. Remark. Smlar to the remark followng Theorem 6.1, we pont out that the results of Theorem 6.6 hold, ndependent of the choce of rate constants κ k, so long as the assocated network s weakly reversble and has a defcency of zero. Acknowledgements We gratefully acknowledge the fnancal support of the Natonal Scence Foundaton through grant NSF-DMS-553687.