Chemical reaction network theory for stochastic and deterministic models of biochemical reaction systems

Transcription

1 Chemical reaction network theory for stochastic and deterministic models of biochemical reaction systems University of Wisconsin at Madison MBI Workshop for Young Researchers in Mathematical Biology August 26th, 214

2 I expect this will be new to you tutorial. This is an accessible theory. Martin Feinberg, Lectures on Chemical Reaction Networks, (Delivered at the Mathematics Research Center, U. of Wisconsin, 1979) (These helped me get through graduate school!) Jeremy Gunawardena, Chemical reaction network theory for in-silico biologists, and Thomas G. Kurtz, Continuous time Markov chain models for chemical reaction networks, chapter in Design and Analysis of Biomolecular Circuits: Engineering Approaches to Systems and Synthetic Biology, H. Koeppl et al. (eds.), Springer, 211. (But also available on my personal webpage!)

3 Big picture Biochemical/population networks can range from simple to very complex. Example 1: A. Example 2: A+B C. Example 3: A + B 2B B A Example 4: Gene transcription & translation: G κ 1 G + M M κ 2 M + P M κ 3 P κ 4 G + P κ 5 κ 5 B transcription translation degradation degradation Binding/unbinding of Gene Cartoon representation: 1 1

4 Big picture Example 5: EnvZ/OmpR signaling system 2 2 Guy Shinar and Martin Feinberg, Structural Sources of Robustness in Biochemical Reaction Networks, Science, 21

5 Big picture Big picture Hanahan and Weinberg, The Hallmarks of Cancer, Cell, 2.!

6 Big picture Metabolic Pathways Roche Applied Science!

7 Big picture For complex models, simulation is often used to explore the possible dynamics (for both deterministic and stochastic models). One problem with this: key system parameters are oftentimes unknown, or known only up to an order of magnitude. Want an alternative approach: discover what pieces of the network architecture determine overall system behavior. This research is part of chemical reaction network theory, which is part of systems biology.

8 Will attempt to do four things (probably won t have time for all my abstract was ambitious): 1 ( ) Describe network structure and introduce both the deterministic and stochastic models. 2 Show how they are related via a scaling limit (law of large numbers). 3 Provide network conditions that guarantee both an especially stable deterministic model and an especially stable stochastic model. When will deterministic and stochastic models give similar behavior? 4 Provide network conditions that guarantee deterministic model has component which is Absolutely robust, stochastic model has extinction event. When will deterministic and stochastic models give different behavior? Isolated examples abound (Keizer s paradox, models in ecology, etc). This characterizes broad class.

9 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Definition A chemical reaction network is given by a triple of (finite) sets (S, C, R): Species, S := {S 1,..., S d }: constituent molecules or species undergoing a series of chemical reactions. Complexes, C: linear combinations of the species representing those used, and produced, in each reaction. A set of reactions, R := {y k y k}, with y k, y k C.

10 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example A S = {A}. C = {, A}. R = { A, A }.

11 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example A + B 2B B A S = {A, B}. C = {A + B, 2B, B, A}. R = {A + B 2B, B A}.

12 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example Species: S = {A, B, C, D, E}. Complexes: C = {A, 2B, A + C, D, B + E}. Reactions: R = {A 2B, 2B A, A+C D, D A+C, D B+E, B+E A+C}.

13 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Dynamics may be modeled deterministically or stochastically. Deterministic: Keep track of reactant concentrations: c i R Reactions occur continuously and simultaneously Modeled with system of ODEs Stochastic: Keep track of reactant abundances: X i {, 1, 2,...} Reactions occur discretely and at separate times Modeled as a continuous time Markov chain (CTMC) Gillespie algorithm, chemical master equation.

14 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example B 1 3 2B yields the ordinary differential equation (ODE) c B(t) = 1 3 c B(t) 1. You can solve this: c B (t) = x e 1 3 t.

15 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example B 2B yields the ordinary differential equation (ODE) c B(t) = 1 3 c B(t) 1 5 c B(t) = 2 15 c B(t) You can solve this: c B (t) = x e 2 15 t.

16 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example A + B α 2B B β A, yields the ODE or [ ca c B ] = αc A c B [ 1 1 ] + βc B [ 1 1 ], c A(t) = αc A c B + βc B c B(t) = αc A c B βc B, which is nonlinear and we can not immediately solve.

17 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example XD k 1 X k 3[T ] XT k 5 X p k 2 [D] k 4 X p + Y k 6 X py k 8 X + Y p k 7 ċ X = k 9 k XD + Y p XDY p 11 XD + Y k 1 k 1 c XD (k 2 [D] + k 3 [T ])c X + k 4 c XT + k 8 c XpY ċ XD = k 1 c XD + k 2 [D]c X k 9 c XD c Yp + (k 1 + k 11 )c XDYp ċ XT = k 3 [T ]c X (k 4 + k 5 )c XT ċ Xp = k 5 c XT k 6 c Xp c Y + k 7 c XpY ċ Y = k 6 c Xp c Y + k 7 c XpY + k 11 c XDYp ċ XpY = k 6 c Xp c Y (k 7 + k 8 )c XpY ċ Yp = k 8 c XpY k 9 c XD c Yp + k 1 c XDYp ċ XDYp = k 9 c XD c Yp (k 1 + k 11 )c XDYp,

18 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic For general system, we have S = {S 1,..., S d }, with This is cumbersome. Set, to get, R : c (t) = k d y ki S i i=1 κ k ( d i=1 c y k c (t) = k What about stochastic dynamics d i=1 c y ki i d i=1 ) c y ki i, y kis i (y k y k ), κ k c(t) y k (y k y k ).

19 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example α A β X(t) = X() + R 1 (t) R 2 (t). (R1/R2) R 1 ( ) is a counting process with intensity/propensity α: P(R 1 (t + t) R 1 (t) = 1) = α t + o( t) P(R 1 (t + t) R 1 (t) 2) = o( t). R 2 ( ) is a counting process with intensity/propensity βx A (t): P(R 2 (t + t) R 2 (t) = 1) = βx A (t) t + o( t) P(R 2 (t + t) R 2 (t) 2) = o( t). Can represent R 1 and R 2 via t R 1 (t) = Y 1 (αt), R 2 (t) = Y 2 (β ) X A (s)ds

20 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Will view a Poisson process, Y ( ), through the lens of an underlying point process. (a) Let {e i } be i.i.d. exponential random variables with parameter one. (b) Now, put points down on a line with spacing equal to the e i : x x x x x x x x e 1 e2 e3 t Let Y 1 (t) denote the number of points hit by time t. In the figure above, Y 1 (t) = λ =

21 The Poisson process Big picture Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Let Y 1 be a unit rate Poisson process. Define Y λ (t) Y 1 (λt), Then Y λ is a Poisson process with parameter λ. x x x x x x x x e 1 e2 e3 t Intuition: The Poisson process with rate λ is simply the number of points hit (of the unit-rate point process) when we run along the time frame at rate λ λ =

22 The Poisson process Big picture Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic There is no reason λ needs to be constant in time, in which case ( t ) Y λ (t) Y λ(s)ds is a non-homogeneous Poisson process with propensity/intensity λ(t). Thus P{Y λ (t + t) Y λ (t) > } = 1 exp { t+ t } λ(s)ds λ(t) t. t Point: We have changed time to convert a unit-rate Poisson process to one which has rate or intensity or propensity λ(t).

23 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example α A β X(t) = X() + R 1 (t) R 2 (t). (R1/R2) t ) X(t) = X() + Y 1 (αt) Y 2 (β X A (s)ds. where Y 1, Y 2 are independent unit-rate Poisson processes. ODE is t c A (t) = c A () + αt βc A (s) ds

24 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example ([ X(t) = X() + R 1 (t) 2 A + B α 2B B β A ] [ 1 1 ]) ([ 1 + R 2 (t) ] [ 1 ]) (R1) (R2) [ 1 = X() + R 1 (t) 1 ] [ 1 + R 2 (t) 1 ]. R 1 ( ) is a counting process with intensity/propensity αx A (t)x B (t): P(R 1 (t + t) R 1 (t) = 1) = αx A (t)x B (t) t + o( t) P(R 1 (t + t) R 1 (t) 2) = o( t). R 2 ( ) is a counting process with intensity/propensity βx B (t): P(R 2 (t + t) R 2 (t) = 1) = βx B (t) t + o( t) P(R 2 (t + t) R 2 (t) 2) = o( t).

25 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example ([ X(t) = X() + R 1 (t) 2 A + B α 2B B β A ] [ 1 1 ]) ([ 1 + R 2 (t) ] [ 1 ]) (R1) (R2) Can take [ 1 = X() + R 1 (t) 1 t R 1 (t) = Y 1 (α t R 2 (t) = Y 2 (β ] [ 1 + R 2 (t) 1 ]. ) X A (s)x B (s)ds ) X B (s)ds where Y 1, Y 2 are independent unit-rate Poisson processes: t ) [ ] ( 1 t X(t) = X() + Y 1 (α X A (s)x B (s)ds + Y 1 2 β ) [ 1 X B (s)ds 1 ]

26 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example A + B α 2B (R1) Stochastic equations t X(t) = X() + Y 1 (α Deterministic equations t x(t) = x() + α B β A ) [ 1 X A (s)x B (s)ds 1 [ 1 x A (s)x B (s)ds 1 ] + Y 2 ( β ] + β t t ) [ 1 X B (s)ds 1 [ 1 x B (s)ds 1 ]. (R2) ].

27 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example A + B α 2B B β A (R1) (R2) Deterministic B * * * * * * * * * 1 Stochastic * * * * * 2 * * * * * * * * * * A 3 4

28 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic For general system, we have S = {S 1,..., S d }, with R : d y ki S i i=1 d y ki S i i=1 The intensity/propensity of kth reaction is λ k : Z d R. As before: X(t) = X() + k R k (t)(y k y k ), with X(t) = X() + ( t ) Y k λ k (X(s))ds (y k y k ), k Y k are independent, unit-rate Poisson processes.

29 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Could just say that for n Z d, n + y 1 y 1, n + y 2 y 2, n. n + y K y K, with rate λ 1 (n) with rate λ 2 (n) with rate λ K (n) where y k y k Z d. I.e. a continuous time Markov chain with infinitesimal generator Af (n) = k λ k (n)(f (n + y k y k ) f (n)). Kolmogorov forward equations (chemical master equation) p t (n) = k λ k (n y k + y k )p t(n y k + y k ) p t(n) k λ k (x), n Z d

30 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example Consider a linear growth model (bacterial colony): Deterministic model: B 1 3 2B with solution x (t) = 1 x(t) x() = 1, (1) 3 x(t) = 1e 1 3 t.

31 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic Example Consider a linear growth model (bacterial colony): Stochastic model: Stochastic equation: Forward (master) equation B 1/3 2B X(t) = X + Y ( t ) 1 X(s) ds. 3 p t (n) = 1 (n 1) pt(n 1) 1 n pt(n), n {1, 2,... } 3 3 with p t(), which means: p t (1) = 1 3 pt(1) p t (2) = 1 3 pt(1) pt(2) p t (3) = 1 3 2pt(2) pt(3).

32 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic 7 6 Colony size Time Stochastic realizations/experiments appear to follow the deterministic system in a noisy way.

33 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic p n () p n (1) n n.7.6 p n (2) p n (3) n n

34 Chemical reaction networks Associated dynamical systems deterministic Associated dynamical systems stochastic 1 Introduce both the deterministic and stochastic models. 2 Show how they are related via a scaling limit. 3 Provide network conditions that guarantee both an especially stable deterministic model and an especially stable stochastic model. When will deterministic and stochastic models give similar behavior? 4 Provide network conditions that guarantee deterministic model has component which is Absolutely robust, stochastic model has extinction event. When will deterministic and stochastic models give different behavior? Isolated examples abound (Keizer s paradox, models in ecology, etc). This characterizes broad class.

35 Assuming: Then, Big picture V is a scaling parameter (volume times Avogadro s number), X i = O(V ), and X V (t) = def X(t)/V, λ k (X(t)) V ( κ k X V (t) y k ), X V (t) 1 V X + k ( 1 t ) V Y k V κ k X V (s) y k ds (y k y k ) LLN for Y k says 1 V Y k(vu) u ( lim sup V 1 Y k (Vu) u =, V u T a.s. ) so as V, X V converges (on compact time interval) to solution of c(t) = c() + k t κ k c(s) y k ds (y k y k ),

36 LLN: Example Big picture Stochastic models: A + B 2/V 2B B 1 A (R1) (R2) with X() = [3V, V ] so that [A V, B V ] = X/V satisfies A V () = 3, B V () = 1. ODE model of A + B 2 2B B 1 A, with x() = [3, 1].

37 V= V=1 A B AB V=1 A B V=1 A B

38 Story 1: deficiency zero Story 2: Absolute Concentration Robustness 1 Introduce both the deterministic and stochastic models. 2 Show how they are related via a scaling limit. 3 Provide network conditions that guarantee both an especially stable deterministic model and an especially stable stochastic model. When will deterministic and stochastic models give similar behavior? 4 Provide network conditions that guarantee deterministic model has component which is Absolutely robust, stochastic model has extinction event. When will deterministic and stochastic models give different behavior? Isolated examples abound (Keizer s paradox, models in ecology, etc). This characterizes broad class.

39 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Example α β A I know the ODE system has a unique fixed point (α/β), and it is stable! c A(t) = α βc A (t). I know the stochastic model has a stationary distribution that charges all states {, 1, 2,... } and it is Poisson with parameter α/β! M/M/ queue. Known for 1 years. Not so impressive or amazing.

40 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Example

41 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Consider the possible enzyme kinetics given by E + S ES E + P, E S In distributional equilibrium the specie numbers are independent and have Poisson distributions.

42 Enzyme kinetics Big picture Story 1: deficiency zero Story 2: Absolute Concentration Robustness Consider the slightly different enzyme kinetics given by E + S ES E + P, E We see S + ES + P = N. In distributional equilibrium E has Poisson distribution, S, ES, P have a multinomial distribution, and E is independent from S, ES, and P.

43 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Theorem (Deterministic - Horn, Jackson, Feinberg, 197 s) Suppose we have a biochemical reaction system whose network satisfies the following two conditions, Deficiency of zero, weakly reversible. Then, the associated deterministic model satisfies: for any choice of rate constants κ k, within each stoichiometric compatibility class there is precisely one equilibrium value c, and that equilibrium value is locally asymptotically stable. (globally?) Actually have stronger result: for each η C, κ k c y k = k:y k =η c is said to be a complex balanced equilibrium. k:y k =η κ k c yk. (2)

44 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Theorem (A., Craciun, Kurtz, 21) Suppose we have a biochemical reaction system whose network satisfies the following two conditions, Deficiency of zero, weakly reversible. Then, the associated stochastic model satisfies: There is a stationary distribution which is the product of Poisson distributions: d c x i i π(x) = M, x Γ, (3) x i! where M is a normalizing constant. i=1

45 Story 1: deficiency zero Story 2: Absolute Concentration Robustness We introduce some elements from Chemical Reaction Network Theory Definition The connected components of the reaction network are called the linkage classes. Example Has two linkage classes. A + B α 2B (Linkage Class 1) B β A (Linkage Class 2)

46 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Example Has two linkage classes.

47 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Definition A chemical reaction network, {S, C, R}, is called weakly reversible if each linkage class is strongly connected. A network is called reversible if y k y k R whenever y k y k R. Weakly Reversible C 1 Reversible C 1 C 3 C 2 C 3 C 2

48 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Example Is weakly reversible.

49 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Definition (Stoichiometry) S = span {yk y k R}{y k y k } is the stoichiometric subspace of the network. Denote dim(s) = s. Translations, c + S, with c R d are stoichiometric compatibility classes. Example: Reaction network A + B 2B B A # B Molecules # A Molecuels

50 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Definition The deficiency of a chemical reaction network, {S, C, R}, is δ = n l s, where n is the number of complexes, l is the number of linkage classes of the network graph, and s is the dimension of the stoichiometric subspace. Example A + B 2B B A (R1) (R2) n = 4, l = 2, s = 1 = δ = 1. But, A + B C B A (R1) (R2) n = 4, l = 2, s = 2 = δ =.

51 Deficiency Big picture Story 1: deficiency zero Story 2: Absolute Concentration Robustness Example: n = 5 l = 2 s = 3 = δ = =.

52 Story 1: deficiency zero Story 2: Absolute Concentration Robustness deficiency of {S, C, R} = δ = n l s, Now you are probably thinking: Fiiiiine, but that was utterly useless to me. I have no idea what it means! Attempt 2: a measure of nonlinearity We define f (c) = def k κ k c y k (y k y k ), and we can find other functions, Y, A κ, and Ψ for which f (c) = Y A κ Ψ(c). Key point: Y and A κ are matrices!

53 Story 1: deficiency zero Story 2: Absolute Concentration Robustness The hunt for linearity: f = Y A κ Ψ Example has ODE ċ(t) = [ A + B κ 1 κ 2 ] 2B κ3 2A κ 4 κ 1 κ 2 κ 1 (κ 2 + κ 3 ) κ 4 κ 3 κ 4 c A c B c 2 B c 2 A

54 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Deficiency: attempt 2 The deficiency satisfies f (x) = Y A κ Ψ(x). δ dim(ker Y imagea κ). You are probably thinking: Oh my, that did not help at all... in fact, I think it made things significantly worse. My response: think about fixed points to ODE model: f ( x) = Y A κ Ψ( x) = with x R d >. This can happen in one of two ways: (i) A κ(ψ( x)) ker Y or (ii) Ψ( x) ker A κ. The second is a very nice condition: Complexed Balanced Equilibrium

55 Story 1: deficiency zero Story 2: Absolute Concentration Robustness 1 Introduce both the deterministic and stochastic models. 2 Show how they are related via a scaling limit. 3 Provide network conditions that guarantee both an especially stable deterministic model and an especially stable stochastic model. When will deterministic and stochastic models give similar behavior? 4 Provide network conditions that guarantee deterministic model has component which is Absolutely robust, stochastic model has extinction event. When will deterministic and stochastic models give different behavior? Isolated examples abound (Keizer s paradox, models in ecology, etc). This characterizes broad class.

56 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Guy Shinar and Martin Feinberg, Structural Sources of Robustness in Biochemical Reaction Networks, Science, 21. A + B α 2B B β A (R1) (R2) ċ A (t) = αc A (t)c B (t) + βc B (t) ċ B (t) = αc A (t)c B (t) βc B (t) M = def c A () + c B (), Solving for equilibria: c A = β/α, c B = M β/α, Network has absolute concentration robustness in species A.

57 Story 1: deficiency zero Story 2: Absolute Concentration Robustness

58 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Theorem (Marty Feinberg and Guy Shinar, Science, 21 deterministic) Consider a deterministic mass-action system that has a deficiency of one. admits a positive steady state and has two non-terminal complexes that differ only in species S, then the system has absolute concentration robustness in S.

59 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Examples: 1 differ in species B. A, A + B 2 differ in species Y p. XT, XT + Y p 3 differ in species G. T, T + G

60 Story 1: deficiency zero Story 2: Absolute Concentration Robustness k 1 XD X XT X p k 2 [ D] k 6 k 3 k4 X p +Y X p Y X+Y p k 7 k 8 k 9 [ T] XD+Y p XDY p XD+Y k 1 k 11 k 5 The orange complexes are called terminal. The blue complexes are called non-terminal.

61 Story 1: deficiency zero Story 2: Absolute Concentration Robustness So what about stochastic models satisfying the same conditions? AIM:

62 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Theorem (Marty Feinberg and Guy Shinar, Science, 21 deterministic) Consider a deterministic mass-action system that has a deficiency of one. admits a positive steady state and has two non-terminal complexes that differ only in species S, then the system has absolute concentration robustness in S. Theorem (A., Enciso, Johnston, Royal Society Interface, 214 stochastic) Consider a reaction network satisfying the following: has a deficiency of one, the deterministic model admits a positive steady state, has two non-terminal complexes that differ only in species S, (new) is conservative, then with probability one there is an extinction event.

63 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Reaction network has state space A + B 2B B A # B Molecules # A Molecuels

64 Story 1: deficiency zero Story 2: Absolute Concentration Robustness

65 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Extinction can be rare event: quasi-stationary distribution: A + B α 2B B β A X A () + X B () = M, # B Molecules # A Molecuels Find π Q M so that for τ absorption time and x transient states, lim Pν(X(t) = x τ > t) = t πq M(x). Satisfies πm(x) Q = P π Q (X(t) = x τ > t). M Can show that quasi-stationary distribution for A converges to Poisson πm(x) Q e (β/α) (β/α) x, as M. x!

66 Story 1: deficiency zero Story 2: Absolute Concentration Robustness 3 3 Guy Shinar and Martin Feinberg, Structural Sources of Robustness in Biochemical Reaction Networks, Science, 21

67 Story 1: deficiency zero Story 2: Absolute Concentration Robustness Quasi stationary probabilities X tot = 1 Y tot = 35 X tot = 1 Y tot = 35 X tot = 1 Y tot = 35 Poisson Molecules of Y p Open question: are all such distributions well approximated by a Poisson?

68 That is the story. Big picture Story 1: deficiency zero Story 2: Absolute Concentration Robustness References: 1 Martin Feinberg, Lectures on Chemical Reaction Networks, (Delivered at the Mathematics Research Center, U. of Wisconsin, 1979) 2 Jeremy Gunawardena, Chemical reaction network theory for in-silico biologists, 3 Guy Shinar and Martin Feinberg, Structural Sources of Robustness in Biochemical Reaction Networks, Science, 21. 4, Germán Enciso, and Matthew Johnston, Stochastic analysis of biochemical reaction networks with absolute concentration robustness, Journal of the Royal Society Interface, Vol. 11, , February 12, , Gheorghe Craciun, Thomas G. Kurtz, Product-form stationary distributions for deficiency zero chemical reaction networks, Bulletin of Mathematical Biology, Vol. 72, No. 8, , 21. And see the chemical reaction network wiki: