Averaging fast subsystems

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1 First Prev Next Go To Go Back Full Screen Close Quit 1 Martingale problems General approaches to averaging Convergence of random measures Convergence of occupation measures Well-mixed reactions Michaelis-Menten equation Another enzyme reaction model Network reversibility Zero deficiency theorem Zero deficiency and product form References Abstract Averaging fast subsystems

2 First Prev Next Go To Go Back Full Screen Close Quit 2 Intensities for continuous-time Markov chains Assume X is a continuous time Markov chain in Z d. Then P {X(t + t) X(t) = l X(t) = k} β l (k) t, and hence E[f(X(t + t)) f(x(t)) F X t ] l β l (X(t))(f(X(t) + l) f(x(t)) t Af(X(t)) t Then Af(k) = l β l (k)(f(k + l) f(k)) is the generator for the chain

3 First Prev Next Go To Go Back Full Screen Close Quit 3 Martingale problems is made precise by the requirement that f(x(t)) f(x()) Af(X(s))ds be a {Ft X }-martingale for f in an appropriate domain D(A). X is called a solution of the martingale problem for A.

4 First Prev Next Go To Go Back Full Screen Close Quit 4 Martingale problem E state space (a complete, separable metric space) A generator (a linear operator with domain and range in B(E) µ P(E) X is a solution of the martingale problem for (A, µ) if and only if µ = P X() 1 and there exists a filtration {F t } such that f(x(t)) is an {F t }-martingale for each f D(A) Af(X(s))ds

5 First Prev Next Go To Go Back Full Screen Close Quit 5 Examples Standard Brownian motion (E = R d ) Af = 1 2 f, D(A) = C2 c (R d ) Poisson process (E = {, 1, 2...}, D(A) = B(E)) Af(k) = λ(f(k + 1) f(k)) Pure jump process (E arbitrary) Af(x) = λ(x) (f(y) f(x))µ(x, dy) E Diffusion (E = R d ) Af(x) = 1 2 a ij (x) f(x) + 2 x i x j i i,j b i (x) x i f(x), D(A) = C 2 c (R d )

6 First Prev Next Go To Go Back Full Screen Close Quit 6 Uniqueness and the Markov property Theorem 1 If any two solutions of the martingale problem for A satisfying P X 1 () 1 = P X 2 () 1 also satisfy P X 1 (t) 1 = P X 2 (t) 1 for all t, then the f.d.d. of a solution X are uniquely determined by P X() 1 If X is a solution of the MGP for A and X a (t) = X(a + t), then X a is a solution of the MGP for A. Theorem 2 If the conclusion of the above theorem holds, then any solution of the martingale problem for A is a Markov process.

7 First Prev Next Go To Go Back Full Screen Close Quit 7 General approaches to averaging Models with two time scales: (X, V ), V is fast Occupation measure: Γ V (C [, t]) = 1 C(V (s))ds Replace integrals involving V by integrals against Γ V f(x(s), V (s))ds = f(x(s), v)γ V (dv ds) How do we identify η s? E V [,t] E V f(x(s), v)η s (dv)ds

8 First Prev Next Go To Go Back Full Screen Close Quit 8 Generator approach Suppose B r f(x, v) = rcf(x, v) + Df(x, v) where C operates on f as a function of v alone. f(x r (t), V r (t)) r Cf(X r (s), v)γ V r (dv ds) E V [,t] Df(X r (s), v)γ V r (dv ds) E V [,t] Assuming (X r, Γ V r ) (X, Γ V ), dividing by r, we should Cf(X(s), v)γ V (dv ds) = Cf(X(s), v)η s (dv)ds = E V [,t] E V [,t] Suppose that for each x, the solution µ x P(E V ) of E V Cf(x, v)µ x (dv) =, f D, is unique. Then η s (dv) = µ X(s) (dv) (1)

9 First Prev Next Go To Go Back Full Screen Close Quit 9 Prohorov metric The Prohorov metric on M f (S), the space of finite measures on a complete, separable metric space S, is ρ(µ, ν) = inf{ɛ > : µ(b) ν(b ɛ ) + ɛ, ν(b) µ(b ɛ ) + ɛ, B B(S)}, (2) where B ɛ = {x S : inf y B d(x, y) < ɛ}. Lemma 3 (M f (S), ρ) is a complete, separable metric space. Lemma 4 Convergence in the Prohorov metric is equivalent to weak convergence, that is, ρ(µ n, µ) if and only if fdµ n fdµ, f C(S).

10 First Prev Next Go To Go Back Full Screen Close Quit 1 Convergence of random measures Lemma 5 Let {Γ n } be a sequence of M f (S)-valued random variables. Then Γ n is relatively compact if and only if {Γ n (S)} is relatively compact as a family of R-valued random variables and for each ɛ >, there exists a compact K S such that sup n P {Γ n (K c ) > ɛ} < ɛ. Corollary 6 Let {Γ n } be a sequence of M f (S)-valued random variables. Suppose that sup n E[Γ n (S)] < and that for each ɛ >, there exists a compact K S such that Then {Γ n } is relatively compact. lim sup E[Γ n (K c )] ɛ. n

11 First Prev Next Go To Go Back Full Screen Close Quit 11 Space-time measures Let L(S) be the space of measures on [, ) S such that µ([, t] S) < for each t >, and let L m (S) L(S) be the subspace on which µ([, t] S) = t. For µ L(S), let µ t denote the restriction of µ to [, t] S. Let ρ t denote the Prohorov metric on M([, t] S), and define ˆρ on L(S) by ˆρ(µ, ν) = e t 1 ρ t (µ t, ν t )dt, that is, {µ n } converges in ˆρ if and only if {µ t n} converges weakly for almost every t. In particular, if ˆρ(µ n, µ), then ρ t (µ t n, µ t ) if and only if µ n ([, t] S) µ([, t] S).

12 First Prev Next Go To Go Back Full Screen Close Quit 12 Relative compactness in L m (S) Lemma 7 A sequence of (L m (S), ˆρ)-valued random variables {Γ n } is relatively compact if and only if for each ɛ > and each t >, there exists a compact K S such that inf n E[Γ n ([, t] K)] (1 ɛ)t. Lemma 8 If V r takes values in a locally compact space E V, ψ 1 and {v E V : ψ(v) c} is compact for each c > 1, and sup E[ r ψ(v r (s))ds] = sup r E[ψ(V r (s))]ds <, then the family of occupation measures {Γ r } is relatively compact in L m (E V ).

13 First Prev Next Go To Go Back Full Screen Close Quit 13 Disintegration of measues Lemma 9 Let Γ be an (L(S), ˆρ)-valued random variable adapted to a complete filtration {F t } in the sense that for each t and H B(S), Γ([, t] H)) is F t -measurable. Let λ(g) = Γ(G S). Then there exists an {F t }-optional, P(S)-valued process γ such that h(s, y)γ(ds dy) = h(s, y)γ s (dy)λ(ds). (3) [,t] S for all h B([, ) S) with probability one. If λ([, t]) is continuous, then γ can be taken to be {F t }-predictable. S

14 First Prev Next Go To Go Back Full Screen Close Quit 14 Convergence of integrals Lemma 1 Let {(x n, µ n )} D E [, ) L(S), and (x n, µ n ) (x, µ). Let h C(E S). Define u n (t) = h(x n (s), y)µ n (ds dy), u(t) = h(x(s), y)µ(ds dy) [,t] S z n (t) = µ n ([, t] S), and z(t) = µ([, t] S). [,t] S a) If x is continuous on [, t] and lim n z n (t) = z(t), then lim n u n (t) = u(t). b) If (x n, z n, µ n ) (x, z, µ) in D E R [, ) L(S), then (x n, z n, u n, µ n ) (x, z, u, µ) in D E R R [, ) L(S). In particular, lim n u n (t) = u(t) at all points of continuity of z.

15 First Prev Next Go To Go Back Full Screen Close Quit 15 c) The continuity assumption on h can be replaced by the assumption that h is continuous a.e. ν t for each t, where ν t M(E S) is the measure determined by ν t (A B) = µ{(s, y) : x(s) A, s t, y B}. d) In both (a) and (b), the boundedness assumption on h can be replaced by the assumption that there exists a nonnegative convex function ψ on [, ) satisfying lim r ψ(r)/r = such that sup n ψ( h(x n (s), y) )µ n (ds dy) < (4) for each t >. [,t] S

16 Well-mixed reactions Consider A + B κ C. The generator for the Markov chain model is Af(m, n) = κmn(f(m 1, n 1) f(m, n)) Spatial model U i V j state (location and configuration) of ith molecule of A state of jth molecule of B Bf(u, v) = m n rc A u i f(u, v) + rc B v j f(u, v) i=1 j=1 + ρ(u i, v j )(f(θ i u, θ j v) f(u, v)) i,j where rc A is a generator modeling the evolution of a molecule of A and rc B models the evolution of a molecule of B. First Prev Next Go To Go Back Full Screen Close Quit 16

17 First Prev Next Go To Go Back Full Screen Close Quit 17 Independent evolution of molecules If there was no reaction rcf(u, v) = m rc A u i f(u, v) + i=1 n rc B v j f(u, v) would model the independent evolution of m molecules of A and n molecules of B. j=1

18 First Prev Next Go To Go Back Full Screen Close Quit 18 Averaging: Markov chain model Assume that the state spaces E A, E B for molecules of A and B are compact and let E = m,n E m A En B. Let Γ r be the occupation measure so f(u r (t), V r (t)) Γ r (C [, t]) = E [,t] 1 C (U r (s), V r (s))ds, (rcf(u, v) + Df(u, v))γ r (du dv ds) is a martingale. Then {(Γ r, XA r, Xr B )} is relatively compact, and assuming all functions are continuous, any limit point (Γ, X A, X B ) of Γ r as r satisfies Cf(u, v)γ(du, dv, ds) =. E [,t] cf. Kurtz [4]

19 First Prev Next Go To Go Back Full Screen Close Quit 19 Averaged generator If f depends only on the numbers of molecules the martingale becomes f(x A (t), X B (t)) E [,t] i,j ρ(u i, v j )(f(x A (s) 1, X B (s) 1) f(x A (s), X B (s)))γ(du, dv, ds). If C A and C B have unique stationary distributions µ A, µ B, then for f(u, v) = m i=1 g(u i) n j=1 h(u j), f(u, v)γ(du, dv, t) = g, µ A XA(s) h, µ B XB(s) ds and setting κ = ρ(u, v )µ A (du )µ B (dv ), f(x A (t), X B (t)) is a martingale. κx A (s)x B (s)(f(x A (s) 1, X B (s) 1) f(x A (s), X B (s)))ds

20 First Prev Next Go To Go Back Full Screen Close Quit 2 Averaging: Michaelis-Menten kinetics Consider the reaction system A + E AE B + E modeled as a continuous time Markov chain satisfying Z N A (t) = Z N A () N 1 Y 1 (N X N E (t) = X N E () Y 1 (N X N B (t) = Y 3 (N +Y 3 (N κ 3 X N AE(s)ds κ 1 Z N A (s)x N E (s)ds + N 1 Y 2 (N κ 1 Z N A (s)x N E (s)ds + Y 2 (N κ 3 X N AE(s)ds Note that M = XAE N (t) + XN E (t) is constant. κ 2 X N AE(s)ds) κ 2 X N AE(s)ds)

21 First Prev Next Go To Go Back Full Screen Close Quit 21 Quasi-steady state Then f(x N E (t)) f(x N E ()) Nκ 1 Z N A (s)x N E (s)(f(x N E (s) 1) f(x N E (s)))ds N(κ 2 + κ 3 )(M X N E (s))(f(x N E (s) + 1) f(x N E (s)))ds At least along a subsequence Z N A = N 1 X N A Z A, and by (1), M η s (k)(κ 1 Z A (s)k(f(k 1) f(k)+(κ 2 +κ 3 )(M k)(f(k+1) f(k)) = k= so η s is binomial(m, p s ), where p s = κ 2 + κ 3 κ 2 + κ 3 + κ 1 Z A (s).

22 First Prev Next Go To Go Back Full Screen Close Quit 22 Substrate dynamics f(z N A (t)) f(z N A ()) Nκ 1 Z N A (s)x N E (s)(f(z N A (s) N 1 ) f(z N A (s)))ds Nκ 2 (M X N E (s))(f(z N A (s) + N 1 ) f(z N A (s)))ds Noting that M k= kη s(k) = Mp s, so the averaged generator becoms f(z A (t)) f(z A ()) is a martingale (actually ), so Z A (t) = Z A () + = Z A () + (κ 2 M(1 p s ) κ 1 Mp s Z A (s))f (Z A (s))dx (κ 2 M(1 p s ) κ 1 Mp s Z A (s))ds Mκ 1 κ 3 Z A (s) κ 2 + κ 3 + κ 1 Z A (s) ds

23 First Prev Next Go To Go Back Full Screen Close Quit 23 Another enzyme reaction model A + E AE B + E E F + G G Z N A (t) = Z N A () N 1 Y 1 (N X N E (t) = X N E () Y 1 (N +Y 3 (N X N F (t) = X N F () + Y 5 (N κ 1 Z N A (s)x N E (s)ds + N 1 Y 2 (N κ 1 Z N A (s)x N E (s)ds + Y 2 (N κ 3 X N AE(s)ds + Y 4 (N X N G (t) = X N G () + Y 6 (Nκ 6 t) + Y 5 (N κ 5 X N E (s)ds) Y 4 (N κ 2 X N AE(s)ds) κ 4 X N F (s)x N G (s))ds Y 5 (N κ 4 X N F (s)x N G (s))ds) Y 7 (N κ 7 X G (s)ds) κ 5 X N E (s)ds) Y 4 (N κ 2 X N AE(s)ds) κ 5 X N E (s)ds) κ 4 X N F (s)x N G (s))ds)

24 Stationary expectations for fast process Need the stationary expectations for the fast subsystem (κ 1 z + κ 5 )E[X E ] + (κ 2 + κ 3 )E[X AE ] + κ 4 E[X F X G ] = κ 5 E[X E ] κ 4 E[X F X G ] = κ 6 + κ 5 E[X E ] κ 4 E[X F X G ] κ 7 E[X G ] = E[X E ] + E[X AE ] + E[X F ] = M Claim: and hence E[X F X G ] = E[X F ]E[X G ] E[X E ] = κ 4 κ 6 M κ 5 κ 7 + κ 4 κ 6 + κ 1κ 4 κ 6 z. κ 2 +κ 3 First Prev Next Go To Go Back Full Screen Close Quit 24

25 First Prev Next Go To Go Back Full Screen Close Quit 25 Network reversibility conditions S = {A i : i = 1,..., m} chemical species C = {ν k, ν k R = {ν k ν k : k = 1,..., n} complexes : k = 1,..., n} reactions determine a chemical reaction network. Definition 11 A chemical reaction network, {S, C, R}, is called weakly reversible if for any reaction ν k ν k, there is a sequence of directed reactions beginning with ν k as a source complex and ending with ν k as a product complex. That is, there exist complexes ν 1,..., ν r such that ν k ν 1, ν 1 ν 2,..., ν r ν k R. A network is called reversible if ν k ν k R whenever ν k ν k R.

26 First Prev Next Go To Go Back Full Screen Close Quit 26 Linkage classes Let G be the directed graph with nodes given by the complexes C and directed edges given by the reactions R = {ν k ν k }, and let G 1,..., G l denote the connected components of G. {G j } are the linkage classes of the reaction network. Intuition for probabilists: If the network is weakly reversible, then, thinking of the complexes as states of a Markov chain, the linkage classes are the irreducible communicating equivalence classes of classical Markov chain theory. BUT, these equivalence classes do not correspond to the communicating equivalence classes of the Markov chain model of the reaction network.

27 First Prev Next Go To Go Back Full Screen Close Quit 27 Stoichiometric subspace Definition 12 S = span {νk ν k R} {ν k ν k} is the stoichiometric subspace of the network. For c R m we say c + S and (c + S) R m > are the stoichiometric compatibility classes and positive stoichiometric compatibility classes of the network, respectively. Denote dim(s) = s. If the network is weakly reversible, then the communicating equivalence classes for the Markov chain model are of the form {z + k a k (ν k ν k ) : a = (a 1,..., a n ) Z n } for some z Z m.

28 First Prev Next Go To Go Back Full Screen Close Quit 28 Deficiency of a network Definition 13 The deficiency of a a chemical reaction network, {S, C, R}, is δ = C l s, where C is the number of complexes, l is the number of linkage classes, and s is the dimension of the stoichiometric subspace. Lemma 14 (Feinberg [2]) The deficiency of a network is nonnegative. Proof. Let C i be the complexes in the ith linkage class and let S i be the span of the reaction vectors giving the edges in the ith linkage class. Then dim(s i ) C i 1 and dim(s) i dim(s i ) l C i l = C l. i=1

29 First Prev Next Go To Go Back Full Screen Close Quit 29 Deficiency zero theorem Theorem 15 (The Deficiency Zero Theorem, Feinberg [2]) Let {S, C, R} be a weakly reversible, deficiency zero chemical reaction network with mass action kinetics. Then, for any choice of rate constants κ k, within each positive stoichiometric compatibility class there is precisely one equilibrium value c, k κ kc ν k (ν k ν k) =, and that equilibrium value is locally asymptotically stable relative to its compatibility class. More precisely, for each η C, κ k c ν k = κ k c νk. (5) k:ν k =η k:ν k =η

30 First Prev Next Go To Go Back Full Screen Close Quit 3 Zero deficiency theorem for stochastic models For x Z m, c x = m i=1 cx i i k:ν k =η and x! = m i=1 x i!. If c R m > satisfies κ k c ν k = then the network is complex balanced. k:ν k =η κ k c νk, η C, (6) Theorem 16 (Kelly [3],Anderson, Craciun, and Kurtz [1]) Let {S, C, R} be a chemical reaction network with rate constants κ k. Suppose that the system is complex balanced with equilibrium c R m >. Then, for any irreducible communicating equivalence class, Γ, the stochastic system has a product form stationary measure where M is a normalizing constant. π(x) = M cx, x Γ, (7) x!

31 First Prev Next Go To Go Back Full Screen Close Quit 31 References [1] David F. Anderson, Gheorghe Craciun, and Thomas G. Kurtz. Product-form stationary distributions for deficiency zero chemical reaction networks. [2] Martin Feinberg. Chemical reaction network structure and the stability of complex isothermal reactors i. the deficiency zero and deficiency one theorems. Chem. Engr. Sci., 42(1): , [3] Frank P. Kelly. Reversibility and stochastic networks. John Wiley & Sons Ltd., Chichester, Wiley Series in Probability and Mathematical Statistics. [4] Thomas G. Kurtz. Averaging for martingale problems and stochastic approximation. In Applied stochastic analysis (New Brunswick, NJ, 1991), volume 177 of Lecture Notes in Control and Inform. Sci., pages Springer, Berlin, 1992.

32 First Prev Next Go To Go Back Full Screen Close Quit 32 Abstract Averaging fast subsystems Reducing the complexity of system models by averaging fast subsystems has a long history in applied mathematics in general and for stochastic models in particular. The previous lectures exploited ad hoc, stochastic analytic relationships to derive the desired averages. This lecture will focus on more systematic methods based on the martingale properties of the underlying Markov processes.