A shout-out to IMA Year on Probability and Statistics in Complex Systems. Ball + P opovic + Rempala + Kurtz

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1 A shout-out to IMA Year on Probability and Statistics in Complex Systems. Ball + P opovic + Rempala + Kurtz IMA F RG F RG F RG + Ball F RG + Craciun + Y in F RG F RG+!!!!!!!!!!!W illiams(unfunded collaborator)!!!!!!!!!!!! SuperF RG SuperF RG + postdocs + grad students SuperDuperF RG Homotopy methods for counting reaction network equilibria Craciun, Helton, and Williams (28) First Prev Next Go To Go Back Full Screen Close Quit 1

2 First Prev Next Go To Go Back Full Screen Close Quit 2 At the interface of stochastic networks and systems biology Enzyme reactions Heavy traffic approximations Product form stationary distributions Fluid approximations References Abstract

3 First Prev Next Go To Go Back Full Screen Close Quit 3 Enzyme reactions Mather, Cookson, Hasty, Tsimring, and Williams (21) D i S i + E λ i Si + D i η S i E µ E + B i X Di = 1, X S = m i=1 X S i, X SE = m i=1 X S i E, X B = m i=1 X B i, X E = M X SE. m Af(s, c, b) = λ i (f(s + e i, c, b) f(s, c, b)) i=1 + + m ηs i (M c )(f(s e i, c i + e i, b) f(s, c, b)) i=1 m µc i (f(s, c i e i, b + e i ) f(s, c, b)) i=1 Conditional distribution of X(t) = (X S (t), X SE (t), X B (t)) given X = ( X S (t), X SE (t), X B (t) ) should be multinomial.

4 First Prev Next Go To Go Back Full Screen Close Quit 4 A martingale fact If X is a solution of the martingale problem for A, that is f(x(t)) f(x()) Af(X(s))ds is a {F t }-martingale for all f D(A), and π t (C) = P {X(t) C G t } (e.g. G t = σ( X(s), s t)) then (µf = fdµ) π t f π f π s Afds is a {G t }-martingale. Under mild conditions, the converse holds. Kurtz and Ocone (1988); Kurtz (1998); Kurtz and Nappo (211) Rogers and Pitman (1981)

5 First Prev Next Go To Go Back Full Screen Close Quit 5 Reduced model Setting λ = m i=1 λ i, p i = λ 1 λ i, s = m i=1 s i, c = m i c i, b = m i=1 b i, define Let α( s, c, b, ds, dc, db) ( ) s = p s 1 1 s 1,..., s ps m m m ( αf( s, c, b ) = c c 1,..., c m ) p c 1 1 pc m m ( b b 1,..., b m f(s, c, b)α( s, c, b, ds, dc, db) ) p b 1 1 pb m m and observe that αaf = Cαf Cg(x, y.z) = λ(g(x + 1, y, z) g(x, y, z) +ηx(m y)(g(x 1, y + 1, z) g(x, y, z)) +µy(g(x, y 1, z + 1) g(x, y, z))

6 First Prev Next Go To Go Back Full Screen Close Quit 6 Check the calculation for m m m f(s, c, b) = exp{ α i s i β i c i γ i b i } i=1 i=1 i=1

7 First Prev Next Go To Go Back Full Screen Close Quit 7 Relationship of processes Let X be a solution of the martingale problem for C and set π t (ds, dc, db) = α( X(t), ds, dc, db) A solution of the martingale problem for C, that is, the Markov chain with generator C, with initial distribution ν(dx, dy, dz) is the projection of the solution of the martingale problem for A with initial distribution α(x, y, z, ds, dc, db)ν(dx, dy, dz).

8 First Prev Next Go To Go Back Full Screen Close Quit 8 Stationary distributions In particular, ifx we ignore the B i, and set ( ) s α( s, c, ds, dc) = p s 1 1 s 1,..., s ps m m m ( c c 1,..., c m the stationary distribution for (X S1,..., X Sm, X C1,..., X Cm ) is π(ds, dc) = α(x, y,, ds, dc, )π (dx, dy) ) p c 1 1 pc m m where π is the stationary distribution for (X S, X C ), and for i j Cov(X Si, X Sj ) = (V ar( X S ) E[ X S )p i p j.

9 First Prev Next Go To Go Back Full Screen Close Quit 9 Heavy traffic limit Consider D λ S + D S + E η SE µ E + B. The Markov chain model can be obtained as the solution of X S (t) = X S () + Y 1 (λt) Y 2 ( X SE (t) = X SE () + Y 2 ( X E (t) = X E () Y 2 ( where M = X E + X SE ηx S (s)x E (s)ds) ηx S (s)x E (s)ds) Y 3 (µ ηx S (s)x E (s)ds) + Y 3 (µ Assume that λ varies with N so that N(µM λ) α. X SE (s)ds) (M X E (s))ds),

10 First Prev Next Go To Go Back Full Screen Close Quit 1 Rescaled equations Speed up time by a factor of N, and scale X S by N. X N S (t) = X N S () + 1 N Y 1 (Nλt) 1 N Y 2 (N 3/2 XE N (t) = XE N () Y 2 (N 3/2 ηxs N (s)xe N (s)ds) +Y 3 (µn N ηxs N (s)xe N (s)ds µmt (M X N E (s))ds) ηx N S (s)x N E (s)ds)

11 First Prev Next Go To Go Back Full Screen Close Quit 11 Limiting equations X N S (t) X N S () + 1 N Y 1 (Nλt) 1 N Y 3 (µn = XS N () + 1 Ỹ 1 (Nλt) 1 Ỹ 3 (µn N N + N(λt = X N S () + 1 N Ỹ 1 (Nλt) 1 N Ỹ 3 (µn + N(λ µm)t + N (M X N E (s))ds) (M X N E (s))ds) µ(m X N E (s))ds) (M X N E (s))ds) µx N E (s)ds

12 First Prev Next Go To Go Back Full Screen Close Quit 12 Limiting equation X S (t) = X S () + 2µMW (t) αt + Λ + (t) + Λ (t) η µ X S(s)dΛ + (s) = µmt where Λ + and Λ are nondecreasing, Λ + increases only when X S > and Λ increases only when X S =. Equivalently, X S (t) = X S () + 2µMW (t) αt + If η µ, X S does not hit zero. µ 2 M ηx S (s) ds + Λ (t).

13 First Prev Next Go To Go Back Full Screen Close Quit 13 Chemical network models 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. νki A i ν kia Deterministic model (law of mass action) Ċ = k κ k C ν k (ν k ν k ) c ν = i cν i i Stochastic model X(t) = X() + k ( ) Xi (s) Y k (κ k ds)(ν k ν k ) i ν ki

14 First Prev Next Go To Go Back Full Screen Close Quit 14 Weak reversibility Definition 1 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.

15 First Prev Next Go To Go Back Full Screen Close Quit 15 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.

16 First Prev Next Go To Go Back Full Screen Close Quit 16 Stoichiometric subspace Definition 2 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. Note that X(t) X() 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.

17 First Prev Next Go To Go Back Full Screen Close Quit 17 Deficiency of a network Definition 3 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 4 (Feinberg (1987)) 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

18 First Prev Next Go To Go Back Full Screen Close Quit 18 Deficiency zero theorem Theorem 5 (The Deficiency Zero Theorem, Feinberg (1987)) 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. (1) k:ν k =η k:ν k =η c ν k = m i=1 c ν ki i

19 First Prev Next Go To Go Back Full Screen Close Quit 19 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, (2) Theorem 6 ( Kelly (1979),Anderson, Craciun, and Kurtz (21)) 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 π(x) = M cx, x! x Γ, (3) where M is a normalizing constant.

20 First Prev Next Go To Go Back Full Screen Close Quit 2 Michaelis-Menten (cf. Darden (1982); Kang, Kurtz, and Popovic (213)) X N E (t) = X E () Y 1 (N S + E SE E + B +Y 3 (Nκ 3 XSE(s)ds) N Z N S (t) = Z N S () N 1 Y 1 (N κ 1 ZS N (s)xe N (s)ds) + Y 2 (Nκ 2 XSE(s)ds) N κ 1 Z N S (s)x N E (s)ds) +N 1 Y 2 (Nκ 2 XSE(s)ds) N m XE N(t) + XN SE (t) does not depend on t.

21 First Prev Next Go To Go Back Full Screen Close Quit 21 Averaging M N,1 (t) = Z N S (t) Z N S () and (κ 2 X N SE(s) κ 1 Z N S (s)x N E (s))ds κ 1 ZS N (s)xe N (s)ds κ 2 (m XE N (s))ds κ 3 (m XE N (s))ds) = Side calculation: [M N,1 ] t = N 2 Y 1 (N (κ 1 Z N S (s) + κ 2 + κ 3 )X N E (s)ds (κ 2 + κ 3 )mt X N E (s)ds m(κ 2 + κ 3 ) κ 1 Z N S (s) + κ 2 + κ 3 ds κ 1 ZS N (s)xe N (s)ds) + N 2 Y 2 (Nκ 2 XSE(s)ds) N

22 First Prev Next Go To Go Back Full Screen Close Quit 22 Deterministic limit so Z N S Z N S (t) = Z N S () N 1 Y 1 (N Z N S () κ 1 Z N S (s)x N E (s)ds) +N 1 Y 2 (Nκ 2 XSE(s)ds) N ZS N () κ 3 (m = Z N S () Z S satisfying Z S (t) = Z S () κ 1 ZS N (s)xe N (s)ds) + κ 2 XSE(s)ds) N m(κ 2 + κ 3 ) κ 1 Z N S (s) + κ 2 + κ 3 )ds mκ 1 κ 3 Z N S (s) κ 1 Z N S (s) + κ 2 + κ 3 ds mκ 1 κ 3 Z S (s) κ 1 Z S (s) + κ 2 + κ 3 ds

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

24 First Prev Next Go To Go Back Full Screen Close Quit 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 SE ] + 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 SE ] + 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

25 First Prev Next Go To Go Back Full Screen Close Quit 25 Fast subnetwork S + E SE B + E E F + G G But we can treat S as constant in abundance, so E SE B + E E F + G G But B has no further effect on the network, so E SE E E F + G G This network is weakly reversible. There are four species and possible changes in species numbers are 1 1 ± 1, ± 1, ± 1 1 so s = 3. Sine C = 5 and l = 2, δ =.

26 First Prev Next Go To Go Back Full Screen Close Quit 26 References David F. Anderson, Gheorghe Craciun, and Thomas G. Kurtz. Product-form stationary distributions for deficiency zero chemical reaction networks. Bull. Math. Biol., 72(8): , 21. Gheorghe Craciun, J. William Helton, and Ruth J. Williams. Homotopy methods for counting reaction network equilibria. Math. Biosci., 216(2):14 149, 28. ISSN doi: 1.116/j.mbs URL http: //dx.doi.org/1.116/j.mbs Thomas A. Darden. Enzyme kinetics: stochastic vs. deterministic models. In Instabilities, bifurcations, and fluctuations in chemical systems (Austin, Tex., 198), pages Univ. Texas Press, Austin, TX, 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): , Hye-Won Kang, Thomas G. Kurtz, and Lea Popovic. Central limit theorems and diffusion approximations for multiscale Markov chain models. Ann. Appl. Probab., 213. to appear. Frank P. Kelly. Reversibility and stochastic networks. John Wiley & Sons Ltd., Chichester, ISBN Wiley Series in Probability and Mathematical Statistics. Thomas G. Kurtz. Martingale problems for conditional distributions of Markov processes. Electron. J. Probab., 3:no. 9, 29 pp. (electronic), ISSN Thomas G. Kurtz and Giovanna Nappo. The filtered martingale problem. In Dan Crisan and Boris Rozovskii, editors, Handbook on Nonlinear Filtering, chapter 5, pages Oxford University Press, 211. Thomas G. Kurtz and Daniel L. Ocone. Unique characterization of conditional distributions in nonlinear filtering. Ann. Probab., 16(1):8 17, ISSN

27 First Prev Next Go To Go Back Full Screen Close Quit 27 William H. Mather, Natalie A. Cookson, Jeff Hasty, Lev S. Tsimring, and Ruth J. Williams. Correlation resonance generated by coupled enzymatic processing. Biophysical J., 99: , 21. L. C. G. Rogers and J. W. Pitman. Markov functions. Ann. Probab., 9(4): , ISSN URL 2-G&origin=MSN.

28 First Prev Next Go To Go Back Full Screen Close Quit 28 Abstract At the interface of stochastic networks and systems biology Beginning with a model of a system of enzyme reactions used by Ruth and her collaborators, connections between results on queueing network models and chemical reaction network models used in systems biology will be explored, including product form stationary distributions, Burke s theorem, fluid limit approximations, heavy traffic limits, and, perhaps, large deviations.