2008 Hotelling Lectures

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1 First Prev Next Go To Go Back Full Screen Close Quit 1 28 Hotelling Lectures 1. Stochastic models for chemical reactions 2. Identifying separated time scales in stochastic models of reaction networks 3. Averaging fast subsystems

2 First Prev Next Go To Go Back Full Screen Close Quit 2 Indeed the use of differential equations supplies the statistician with a powerful tool, replacing the purely empirical fitting of arbitrary curves by a reasonable resultant of general considerations with particular data. But this growing statistical use of differential equations must inevitably face the fact that our a priori knowledge can never supply us with a definite relation between a variable and its rate of change, but only with a correlation. In astronomy, physics and chemistry nearly all correlations obtained in good work are very close either to zero or to perfection. Consequently by discarding the first set and identifying the other with complete causation it has been possible and natural to avoid consideration of correlated but not rigidly connected variables. In this way it has come about that, for those fields in which large random fluctuations and merely correlated variables must be dealt with, the existing theory of differential equations is inadequate and needs to be supplemented by a new theory involving probability. Harold Hotelling JASA 1927 [11]

3 First Prev Next Go To Go Back Full Screen Close Quit 3 Stochastic models for chemical reactions Application of the LLN and CLT to Poisson processes Formulating Markov models Reaction networks Classical scaling and the law of mass action Central limit theorem Diffusion approximations Multiple scales Example: Michaelis-Menten equation Example: Model of a viral infection References Abstract Collaboration with David Anderson, Karen Ball, George Craciun, Hye-Won Kang, Lea Popovic, Greg Rempala

4 First Prev Next Go To Go Back Full Screen Close Quit 4 Poisson processes A Poisson process is a model for a series for random observations occurring in time. x x x x x x x x t Let Y (t) denote the number of observations by time t. In the figure above, Y (t) = 6. Note that for t < s, Y (s) Y (t) is the number of observations in the time interval (t, s]. We make the following assumptions about the model. 1) Observations occur one at a time. 2) Numbers of observations in disjoint time intervals are independent random variables, i.e., if t < t 1 < < t m, then Y (t k ) Y (t k 1 ), k = 1,..., m are independent random variables. 3) The distribution of Y (t + a) Y (t) does not depend on t.

5 First Prev Next Go To Go Back Full Screen Close Quit 5 Characterization of a Poisson process Theorem 1 Under assumptions 1), 2), and 3), there is a constant λ > such that, for t < s, Y (s) Y (t) is Poisson distributed with parameter λ(s t), that is, P {Y (s) Y (t) = k} = (λ(s t))k e λ(s t). k! If λ = 1, then Y is a unit (or rate one) Poisson process. If Y is a unit Poisson process and Y λ (t) Y (λt), then Y λ is a Poisson process with parameter λ.

6 First Prev Next Go To Go Back Full Screen Close Quit 6 Application of the LLN and CLT to Poisson processes Theorem 2 If Y is a unit Poisson process, then for each u >, lim sup Y (Ku) u = a.s. K u u K Proof. For fixed u, by the independent increments assumption, the result is just the ordinary law of large numbers. The uniformity follows by monotonicity. The central limit theorem suggests that for large K Y (Ku) Ku K W (u), where W is standard Brownian motion. Y (Ku) K u + 1 K W (u)

7 First Prev Next Go To Go Back Full Screen Close Quit 7 Formulating Markov models Suppose Y λ (t) = Y (λt) and F t represents the information obtained by observing Y λ (s), for s t. P {Y λ (t+ t) Y λ (t) = 1 F t } = P {Y λ (t+ t) Y λ (t) = 1} = 1 e λ t λ t A continuous time Markov chain X taking values in Z d is specified by giving its transition intensities that determine P {X(t + t) X(t) = l F X t } β l (X(t)) t, l Z d.

8 First Prev Next Go To Go Back Full Screen Close Quit 8 Counting process representation If we write X(t) = X() + ln l (t) l where N l (t) is the number of jumps of l at or before time t, then P {N l (t + t) N l (t) = 1 Ft X } β l (X(t)) t, l Z d. N l is a counting process with intensity (propensity in the chemical literature) β l (X(t)) and we can write N l (t) = Y l ( β l (X(s))ds), where the Y l are independent, unit Poisson processes. Consequently, X(t) = X() + ly l ( l β l (X(s))ds).

9 First Prev Next Go To Go Back Full Screen Close Quit 9 Random jump equation Alternatively, setting β(k) = l β l(k), and N(t) = Y ( X(t) = X() + β(x(s))ds) F (X(s ), ξ N(s ) )dn(s) where Y is a unit Poisson process, {ξ i } are iid uniform [, 1], and P {F (k, ξ) = l} = β l(k) β(k).

10 First Prev Next Go To Go Back Full Screen Close Quit 1 Connections to simulation schemes Simulating the random-jump equation gives Gillespie s [7, 8] direct method (the stochastic simulation algorithm SSA). Simulating the time-change equation gives the next jump) method as defined by Gibson and Bruck [6]. reaction (next For = τ (x) < τ 1 (x) <, satisfying τ k (x) = τ k (x τk ), where { x τk x(s) s < τ (s) = k (x) x(τ k (x) ) s τ k (x) (typically, τ k+1 (x) = τ k (x) + g k+1 (x(τ k ))), simulation of ˆX(t) = X() + ( ) ly l β l ( ˆX(τ k ))(τ k+1 t τ k t) l k gives Gillespie s [9] τ-leap method

11 First Prev Next Go To Go Back Full Screen Close Quit 11 Bilingual dictionary Chemistry propensity master equation nonlinear diffusion approximation Langevin approximation Van Kampen approximation quasi steady state/partial equilibrium Probability intensity forward equation diffusion approximation central limit theorem averaging

12 First Prev Next Go To Go Back Full Screen Close Quit 12 Reaction networks Standard notation for chemical reactions A + B k C is interpreted as a molecule of A combines with a molecule of B to give a molecule of C. A + B C means that the reaction can go in either direction, that is, a molecule of C can dissociate into a molecule of A and a molecule of B. We consider a network of reactions involving m chemical species, A 1,..., A m. m m ν ik A i i=1 i=1 ν ika i where the ν ik and ν ik are nonnegative integers.

13 First Prev Next Go To Go Back Full Screen Close Quit 13 Markov chain models X(t) number of molecules of each species in the system at time t. ν k number of molecules of each chemical species consumed in the kth reaction. ν k number of molecules of each species created by the kth reaction. λ k (x) rate at which the kth reaction occurs. (The propensity/intensity.) If the kth reaction occurs at time t, the new state becomes X(t) = X(t ) + ν k ν k. The number of times that the kth reaction occurs by time t is given by the counting process satisfying R k (t) = Y k ( λ k (X(s))ds), where the Y k are independent unit Poisson processes.

14 First Prev Next Go To Go Back Full Screen Close Quit 14 Equations for the system state The state of the system satisfies X(t) = X() + k = X() + k R k (t)(ν k ν k ) Y k ( λ k (X(s))ds)(ν k ν k ) = (ν ν)r(t) ν is the matrix with columns given by the ν k. ν is the matrix with columns given by the ν k. R(t) is the vector with components R k (t).

15 First Prev Next Go To Go Back Full Screen Close Quit 15 Rates for the law of mass action For a binary reaction A 1 + A 2 A 3 or A 1 + A 2 A 3 + A 4 λ k (x) = κ k x 1 x 2 For A 1 A 2 or A 1 A 2 + A 3, λ k (x) = κ k x 1. For 2A 1 A 2, λ k (x) = κ k x 1 (x 1 1). For a binary reaction A 1 +A 2 A 3, the rate should vary inversely with volume, so it would be better to write λ N k (x) = κ k N 1 x 1 x 2 = Nκ k z 1 z 2, where classically, N is taken to be the volume of the system times Avogadro s number and z i = N 1 x i is the concentration in moles per unit volume. Note that unary reaction rates also satisfy λ k (x) = κ k x i = Nκ k z i.

16 First Prev Next Go To Go Back Full Screen Close Quit 16 General form for classical scaling All the rates naturally satisfy λ N k (x) Nκ k i z ν ik i N λ k (z).

17 First Prev Next Go To Go Back Full Screen Close Quit 17 First scaling limit Setting C N (t) = N 1 X(t) C N (t) = C N () + k C N () + k N 1 Y k ( N 1 Y k (N λ N k (X(s))ds)(ν k ν k ) λ k (C N (s))ds)(ν k ν k ) The law of large numbers for the Poisson process implies N 1 Y (Nu) u, C N (t) C N () + κ k Ci N (s) ν ik (ν k ν k )ds, k i which in the large volume limit gives the classical deterministic law of mass action Ċ(t) = κ k C i (t) ν ik (ν k ν k ) F (C(t)). k i

18 First Prev Next Go To Go Back Full Screen Close Quit 18 Central limit theorem/van Kampen approximation V N (t) N(C N (t) C(t)) V N () + N( 1 t N Y k(n λ k (C N (s))ds)(ν k ν k ) k = V N () + 1 t Ỹ k (N λ k (C N (s))ds)(ν k ν k ) k N + N(F (CN (s)) F (C(s)))ds V N () + W k ( λ k (C(s))ds)(ν k ν k ) k + F (C(s)))V N (s)ds F (C(s))ds)

19 First Prev Next Go To Go Back Full Screen Close Quit 19 Gaussian limit V N converges to the solution of V (t) = V () + k W k ( λ k (C(s))ds)(ν k ν k ) + F (C(s)))V (s)ds C N (t) C(t) + 1 N V (t)

20 First Prev Next Go To Go Back Full Screen Close Quit 2 Diffusion approximation where C N (t) = C N () + k N 1 Y k ( C N () + N 1/2 W k ( k + F (C N (s))ds, F (c) = k λ k (X N (s))ds)(ν k ν k ) λ k (c)(ν k ν k ). λ k (C N (s))ds)(ν k ν k ) The diffusion approximation is given by the equation C N (t) = C N ()+ k N 1/2 W k ( λ k ( C N (s))ds)(ν k ν k )+ F ( C N (s))ds.

21 First Prev Next Go To Go Back Full Screen Close Quit 21 Itô formulation The time-change formulation is equivalent to the Itô equation C N (t) = N 1/2 λ k ( C N (s))d W k (s)(ν k ν k ) C N () + k = C N () + k + F ( C N (s))ds N 1/2 σ( C N (s))d W (s) + where σ(c) is the matrix with columns λ k (c)(ν k ν k). F ( C N (s))ds, See Kurtz [12], Ethier and Kurtz [4], Chapter 1, Gardiner [5], Chapter 7, and Van Kampen [14].

22 First Prev Next Go To Go Back Full Screen Close Quit 22 Multiple scales Take N to be of the order of magnitude of the abundance of the most abundant species in the system. For each species i, define the normalized abundances (or simply, the abundances) by Z i (t) = N α i X i (t), where α i 1 should be selected so that Z i = O(1). Note that the abundance may be the species number (α i = ) or the species concentration or something else. The rate constants may also vary over several orders of magnitude κ k = κ kn β k, so for a binary reaction κ kx i x j = N β k+α i +α j κ k z i z j

23 First Prev Next Go To Go Back Full Screen Close Quit 23 A parameterized family of models Let Z N i (t) = Z i () + k N α i Y k ( N β k+ν k α λ k (Z N (s))ds)(ν ik ν ik ). Then the true model is Z = Z N.

24 First Prev Next Go To Go Back Full Screen Close Quit 24 Example: Michaelis-Menten kinetics Consider the reaction system A + E AE B + E modeled as a continuous time Markov chain satisfying X A (t) = X A () Y 1 ( X E (t) = X E () Y 1 ( +Y 3 ( κ 1X A (s)x E (s)ds) + Y 2 ( κ 1X A (s)x E (s)ds) + Y 2 ( κ 3X AE (s)ds) X B (t) = Y 3 ( κ 3X AE (s)ds) κ 2, κ 3 >> κ 1 κ 2X AE (s)ds) κ 2X AE (s)ds)

25 First Prev Next Go To Go Back Full Screen Close Quit 25 Scaling Note that M = X AE (t) + X E (t) is constant. Let N = O(X A ) >> M. Setting β 2 = β 3 = 1, α A = 1, α E = α AE =, κ 1 = κ 1, κ 2 = κ 2N 1, κ 3 = κ 3N 1 V E (t) = M 1 X E (s)ds, Z A (t) = N 1 X A (t) Z A (t) = Z A () N 1 Y 1 (N X E (t) = X E () Y 1 (N +Y 3 (Nκ 3 M(t V E (t))) X B (t) = Y 3 (Nκ 3 M(t V E (t))) κ 1 MZ A (s)dv E (s)) + N 1 Y 2 (Nκ 2 M(t V E (t))) κ 1 MZ A (s)dv E (s)) + Y 2 (Nκ 2 M(t V E (t)))

26 First Prev Next Go To Go Back Full Screen Close Quit 26 Analysis Divide X E (t) = X E () Y 1 (N by N so that and lim N lim N +Y 3 (Nκ 3 M(t V E (t))) ( (κ 2 + κ 3 )M(t V E (t))) κ 1 MZ A (s)dv E (s)) + Y 2 (Nκ 2 M(t V E (t))) ) κ 1 MZ A (s)dv E (s) = ( ) Z A (t) Z A () + κ 1 MZ A (s)dv E (s) κ 2 M(t V E (t)) =

27 First Prev Next Go To Go Back Full Screen Close Quit 27 Derivation of Michaelis-Menten equation Theorem 3 (Darden [2, 3]) Assume that N and ZA N() = X A()/N x A (). Then (ZA N, V E N) converges to (x A(t), v E (t)) satisfying x A (t) = x A () = and hence v E (s) = κ 1 Mx A (s) v E (s)ds + κ 1 x A (s) v E (s)ds + κ 2 +κ 3 κ 2 +κ 3 +κ 1 x A (s) and ẋ A (t) = Mκ 1κ 3 x A (t) κ 2 + κ 3 + κ 1 x A (s). κ 2 M(1 v E (s))ds(1) (κ 2 + κ 3 )(1 v E (s))ds,

28 First Prev Next Go To Go Back Full Screen Close Quit 28 Example: Model of a viral infection Srivastava, You, Summers, and Yin [13], Haseltine and Rawlings [1], Ball, Kurtz, Popovic, and Rampala [1] Three time-varying species, the viral template, the viral genome, and the viral structural protein (indexed, 1, 2, 3 respectively). The model involves six reactions, T + stuff κ 1 T + G G κ 2 T T + stuff κ 3 T + S T κ 4 S κ 5 G + S κ 6 V

29 First Prev Next Go To Go Back Full Screen Close Quit 29 Stochastic system X 1 (t) = X 1 () + Y b ( X 2 (t) = X 2 () + Y a ( X 3 (t) = X 3 () + Y c ( κ 2X 2 (s)ds) Y d ( κ 1X 1 (s)ds) Y b ( Y f ( κ 6X 2 (s)x 3 (s)ds) κ 3X 1 (s)ds) Y e ( Y f ( κ 6X 2 (s)x 3 (s)ds) κ 4X 1 (s)ds) κ 2X 2 (s)ds) κ 5X 3 (s)ds)

30 Figure 1: Simulation (Haseltine and Rawlings 22) First Prev Next Go To Go Back Full Screen Close Quit 3

31 First Prev Next Go To Go Back Full Screen Close Quit 31 Scaling parameters Each X i is scaled according to its abundance in the system. For N = 1, X 1 = O(N ), X 2 = O(N 2/3 ), and X 3 = O(N ) and we take Z 1 = X 1, Z 2 = X 2 N 2/3, and Z 3 = X 3 N 1. Expressing the rate constants in terms of N = 1 κ κ N 2/3 κ 3 1 N κ κ κ N 5/3

32 First Prev Next Go To Go Back Full Screen Close Quit 32 Normalized system With the scaled rate constants, we have Z N 1 (t) = Z N 1 () + Y b ( Z N 2 (t) = Z N 2 () + N 2/3 Y a ( 2.5Z N 2 (s)ds) Y d ( N 2/3 Y f ( Z3 N (t) = Z3 N () + N 1 Y c ( N 1 Y f ( Z N 1 (s)ds) N 2/3 Y b (.25Z N 1 (s)ds).75z N 2 (s)z N 3 (s)ds) NZ N 1 (s)ds) N 1 Y e (.75Z N 2 (s)z N 3 (s)ds), 2.5Z N 2 (s)ds) 2NZ N 3 (s)ds)

33 First Prev Next Go To Go Back Full Screen Close Quit 33 Limiting system With the scaled rate constants, we have Z 1 (t) = Z 1 () + Y b ( Z 2 (t) = Z 2 () Z 3 (t) = Z 3 () + 2.5Z 2 (s)ds) Y d ( Z 1 (s)ds 2Z 3 (s)ds.25z 1 (s)ds)

34 First Prev Next Go To Go Back Full Screen Close Quit 34 Fast time scale Define V N i (t) = Z i (N 2/3 t). V N 1 (t) = V N 1 () + Y b ( V N 2 (t) = V N 2 () + N 2/3 Y a ( V3 N (t) = V3 N () + N 1 Y c ( 2.5N 2/3 V2 N (s)ds) Y d ( N 2/3 V1 N (s)ds) N 2/3 Y b ( N 2/3 Y f (N 2/3 2.5N 2/3 V N 2 (s)ds) N 5/3 V1 N (s)ds) N 1 Y e ( N 1 Y f (.25N 2/3 V N 1 (s)ds).75v2 N (s)v3 N (s)ds).75n 2/3 V N 2 (s)v N 3 (s)ds) 2N 5/3 V N 3 (s)ds)

35 First Prev Next Go To Go Back Full Screen Close Quit 35 Averaging As N, dividing the equations for V1 N and V3 N by N 2/3 shows that V N 1 (s)ds 1 V N 3 (s)ds 5 V N 2 (s)ds V N 2 (s)ds. The assertion for V3 N and the fact that V2 N is asymptotically regular imply V2 N (s)v3 N (s)ds 5 V2 N (s) 2 ds. It follows that V2 N converges to the solution of (2).

36 First Prev Next Go To Go Back Full Screen Close Quit 36 Law of large numbers Theorem 4 For each δ > and t >, where V 2 is the solution of lim P { sup V2 N (s) V 2 (s) δ} =, N s t V 2 (t) = V 2 () + 7.5V 2 (s)ds) 3.75V 2 (s) 2 ds. (2)

37 First Prev Next Go To Go Back Full Screen Close Quit The Whole System The Reduced System when γ= 3 X1:Viral Template X1:Viral Template 2.5 The Number of X The Number of X Time Time The Whole System The Reduced System when γ= 12 X3:Viral Structural Protein 12 X3:Viral Structural Protein The Number of X The Number of X Time Time

38 First Prev Next Go To Go Back Full Screen Close Quit The Whole System The Reduced System when γ= 3 X2:Viral Genome X2:Viral Genome 25 The Number of X The Number of X Time Time

39 First Prev Next Go To Go Back Full Screen Close Quit The Whole System The Reduced System when γ=2/3 25 X2:Viral Genome X2:Viral Genome The Number of X The Number of X Time Time

40 First Prev Next Go To Go Back Full Screen Close Quit The Whole System The Reduced System when γ=2/3 35 X1:Viral Template X1:Viral Template 3 The Number of X The Number of X Time Time 16 The Whole System X3:Viral Structural Protein 16 The Reduced System when γ=2/3 X3:Viral Structural Protein The Number of X The Number of X Time Time

41 First Prev Next Go To Go Back Full Screen Close Quit 41 References [1] Karen Ball, Thomas G. Kurtz, Lea Popovic, and Greg Rempala. Asymptotic analysis of multiscale approximations to reaction networks. Ann. Appl. Probab., 16(4): , 26. [2] Thomas Darden. A pseudo-steady state approximation for stochastic chemical kinetics. Rocky Mountain J. Math., 9(1):51 71, Conference on Deterministic Differential Equations and Stochastic Processes Models for Biological Systems (San Cristobal, N.M., 1977). [3] 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, [4] Stewart N. Ethier and Thomas G. Kurtz. Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, Characterization and convergence. [5] C. W. Gardiner. Handbook of stochastic methods for physics, chemistry and the natural sciences, volume 13 of Springer Series in Synergetics. Springer-Verlag, Berlin, third edition, 24. [6] M. A. Gibson and Bruck J. Efficient exact simulation of chemical systems with many species and many channels. J. Phys. Chem. A, 14(9): , 2. [7] Daniel T. Gillespie. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Computational Phys., 22(4):43 434, [8] Daniel T. Gillespie. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem., 81:234 61, [9] Daniel T. Gillespie. Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics, 115(4): , 21. [1] Eric L. Haseltine and James B. Rawlings. Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys., 117(15): , 22.

42 First Prev Next Go To Go Back Full Screen Close Quit 42 [11] Harold Hotelling. Differential equations subject to error, and population estimates. Journal of the American Statistical Association, 22(159): , [12] Thomas G. Kurtz. Strong approximation theorems for density dependent Markov chains. Stochastic Processes Appl., 6(3):223 24, 1977/78. [13] R. Srivastava, L. You, J. Summers, and J. Yin. Stochastic vs. deterministic modeling of intracellular viral kinetics. J. Theoret. Biol., 218(3):39 321, 22. [14] N. G. van Kampen. Stochastic processes in physics and chemistry. North-Holland Publishing Co., Amsterdam, Lecture Notes in Mathematics, 888.

43 First Prev Next Go To Go Back Full Screen Close Quit 43 Abstract Stochastic models for chemical reactions Attempts to model chemical reactions within biological cells have led to renewed interest in stochastic models for these systems. The classical stochastic models for chemical reaction networks will be reviewed, and multiscale methods for model reduction will be described. The methods will be illustrated with derivation of the Michaelis-Menten model for enzyme reactions and a simple model of viral infection of a cell.