Longtime behavior of stochastically modeled biochemical reaction networks

Transcription

1 Longtime behavior of stochastically modeled biochemical reaction networks David F. Anderson Department of Mathematics University of Wisconsin - Madison ASU Math Biology Seminar February 17th, 217

2 Overview Biochemical interaction systems are used to model intracellular processes: 1. Models of gene and protein regulatory systems, 2. neuronal networks, 3. models of virus infection 4. And can be used for population models: SIR. Typically these systems are mathematically modeled in one of two ways: Deterministic models ODEs with deterministic mass-action kinetics. Stochastic models continuous-time Markov chains.

3 Overview Type of mathematical question I am interested in: Can we prove general theorems relating dynamical properties of the mathematical model to the structure of the interaction network. Key point of research: We want theorems that are applicable to whole classes of reaction networks; we are not studying a single model. Outline: After introducing the models, I will focus on two results one is about stationary distributions, one is about possible extinctions/medium time behavior. This work is part of chemical reaction network theory (CRNT).

4 Reaction networks Focus is on reaction networks. A + B 2B B A

5 Reaction networks 1 1 Guy Shinar and Martin Feinberg, Structural Sources of Robustness in Biochemical Reaction Networks, Science, 21

6 Ligand-Receptor model for T-cell - Omer Dushek (Oxford) is L + R kon k off C k off k p C 2 ϕkp C 1. which will be related to the simple models A k 1 B or α A k 2 β

7 Reaction networks Each network is composed of three sets, {S, C, R}: 1. S, species constituent components 2. C, complexes linear combinations of the species 3. R, reactions ordered pairs of complexes. Example: A + B 2B B A S = {A, B}. C = {A + B, 2B, B, A}. R = {A + B 2B, B A}.

8 Reaction networks: {S, C, R} 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}. ζ 1 = 1 2, ζ 2 = 1 2, ζ 3 = 1 1 1,...

9 Dynamics deterministic with mass-action kinetics Example: A + B κ 1 2B B κ 2 A (R1) (R2) Let x(t) R 2 give concentrations of molecules of A and B: [ 1 ẋ = r 1 (x) 1 ] [ 1 + r 2 (x) 1 ]. Deterministic mass-action kinetics says: so r 1 (x) = κ 1 x A x B, and r 2 (x) = κ 2 x B. ẋ A = κ 1 x A x B + κ 2 x B ẋ B = κ 1 x A x B κ 2 x B. Nonlinear if any reaction requires two or more molecules.

10 Dynamics deterministic Example:

11 Dynamics: discrete stochastic Example: α β A (R1/R2) X(t) = X() + N 1 (t) N 2 (t). where N 1 and N 2 are counting processes satisfying P(N 1 (t + h) N 1 (t) = 1) = αh + o(h) P(N 2 (t + h) N 2 (t) = 1) = βx A (t)h + o(h) P(more than one jump in [t, t + h)) = o(h). So this is a discrete space, continuous time Markov chain. Can show (Tom Kurtz) N 1 (t) = Y 1 (αt) N 2 (t) = Y 2 (β t ) X A (s)ds where Y 1, Y 2 are independent unit-rate Poisson processes. t ) X(t) = X() + Y 1 (αt) Y 2 (β X A (s)ds.

12 Dynamics: discrete stochastic Example: A + B κ 1 2B B κ 2 A (R1) (R2) For Markov models can take [ 1 X(t) = X() + N 1 (t) 1 N 1 (t) = Y 1 ( κ 1 t N 2 (t) = Y 2 ( κ 2 t ] [ 1 + N 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. ].

13 Model of viral infection: Srivistava, You, Summers, and Yin, J. Theor. Biol., 218 (22) R 1 ) T 1 T + G, R 2 ) G.25 T, R 3 ) T 1 T + S, R 4 ) T.25, R 5 ) S 2, R 6 ) G + S V, X G (t) = X G () + Y 1 ( t t X S (t) = X S () + Y 3 (1 t X T (t) = X T () + Y 2 (.25 ( t X V (t) = X V () + Y ) ( X T (s)ds Y 2.25 ) ( X T (s)ds Y 5 2 t t ) ( X G (s)ds Y 4.25 ) X G (s)x S (s)ds ) t ) X G (s)ds Y 6 ( X G (s)x S (s)ds ) ) t X S (s)ds Y 6 ( X G (s)x S (s)ds t ) X T (s)ds structure Counts virus genome 1 1 template Time (Days)

14 Dynamics: discrete stochastic For general system, we have The rate (or intensity or propensity) of kth reaction is λ k : Z d R. As before: X(t) = X() + k N k (t)ζ k, with X(t) = X() + k Y k ( t Y k are independent, unit-rate Poisson processes. ) λ k (X(s))ds ζ k,

15 Mass-action kinetics The standard intensity function chosen is stochastic mass-action kinetics: λ k (x) = κ k i x i! (x i y ki )!. Example: If S 1 anything, then λ k (x) = κ k x 1. Example: If S 1 + S 2 anything, then λ k (x) = κ k x 1 x 2. Example: If 2S 2 anything, then λ k (x) = κ k x 2 (x 2 1) κ 2 x 2 2 if x 2 1. Note: rates are nonlinear if two more molecules are required.

16 Other ways to understand model Could just say that for x Z d, x + ζ 1, x + ζ 2, x. x + ζ r, with rate λ 1 (x) with rate λ 2 (x) with rate λ r (x) where ζ k Z d. a continuous time Markov chain with infinitesimal generator Af (x) = k λ k (x)(f (x + ζ k ) f (x)). Markov model with forward (chemical master) equation p t (x) = k p t(x ζ k )λ k (x ζ k ) k p t(x)λ k (x) Can simulate with Gillespie s algorithm or the next reaction method.

17 Story #1: stationary distributions Stationary distributions play a central role in understanding (all) stochastic models. They: 1. Characterize the long-time behavior of the model. 2. Are used heavily in stochastic averaging (and, therefore, transient dynamics). They are a fixed point of the chemical master equation: d dt pµ(x, t) = k p µ(x ζ k, t)λ k (x ζ k ) k p µ(x, t)λ k (x), Thus, they solve λ k (x ζ k )π(x ζ k ) = k k λ k (x)π(x). Problems: 1. there is one such equation for each state, x, in the state space. 2. the functions λ k are often nonlinear.

18 Story #1: stationary distributions Wide open question: when does a stationary distribution exist for this class of models? Maybe: if the ode model has a globally stable fixed point, then stochastic model has stationary distribution? False. Ode: ȧ = k 1 k 2 ab k 1 A k 1 B A + B k 2. Stable equilibrium for each initial condition. Stochastic: X a(t) X b (t) is a random walk. ḃ = k 1 k 2 ab

19 Deficiency Zero Theorem - stochastic Theorem (Anderson, Craciun, Kurtz, 21) Let {S, C, R} be a chemical reaction network with rate constants κ k. Suppose the deterministic, mass-action model has complex-balanced equilibrium c. Then, for any irreducible communicating equivalence class, Γ Z d, the stochastic system has a product form stationary distribution π(x) = 1 Z V where Z V is a normalizing constant. d i=1 e c c x i i i, x Γ, (1) x i! Horn, Jackson, Feinberg: ODE has a complex balanced equilibrium if 1. the network is weakly reversible, and 2. has a deficiency of zero. David F. Anderson, Gheorghe Craciun, and Thomas G. Kurtz, Product-form stationary distributions for deficiency zero chemical reaction networks, Bulletin of Mathematical Biology, Vol. 72, No. 8, , 21.

20 Connectivity Definition A chemical reaction network, {S, C, R}, is called weakly reversible if each connected component of the network 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

21 Connectivity The following is not weakly reversible: C 2 C 3 C 5 C 6 C 7 C 1 C 4 C 8 Neither is the following: A + B α 2B (Linkage Class 1) B β A (Linkage Class 2)

22 Deficiency. deficiency of {S, C, R} = δ = n l s, where 1. n = # of complexes. 2. l = # of connected components of the network. 3. s = dimension of span of reaction vectors. So it is easy to check. Example A + B 2B B A (R1) (R2) n = 4, l = 2, s = 1 = δ = 1.

23 Deficiency Example n = 5 l = 2 s = 3 = δ = =.

24 Deficiency Of course, you are thinking I have no idea what deficiency is that was pointless. In reality, ẋ = f (x) = YA κψ(x), where 1. Y and A κ are matrices, 2. and deficiency = δ = dim(ker(y ) imagea κ).

25 Deficiency Zero Theorem - stochastic Theorem (Anderson, Craciun, Kurtz, 21) Let {S, C, R} be a chemical reaction network with rate constants κ k. Suppose the deterministic, mass-action model has complex-balanced equilibrium c. Then, for any irreducible communicating equivalence class, Γ Z d, the stochastic system has a product form stationary distribution π(x) = 1 Z V where Z V is a normalizing constant. d i=1 e c c x i i i, x Γ, (2) x i! Horn, Jackson, Feinberg: ODE has a complex balanced equilibrium if 1. the network is weakly reversible, and 2. has a deficiency of zero. Proof: just check. Greatly generalizes linear results. David F. Anderson, Gheorghe Craciun, and Thomas G. Kurtz, Product-form stationary distributions for deficiency zero chemical reaction networks, Bulletin of Mathematical Biology, Vol. 72, No. 8, , 21.

26 Example: Enzyme kinetics Consider the possible model of enzyme kinetics given by E + S ES E + P, E S Easy to see that deficiency is and that Γ = Z 4. Thus, in distributional equilibrium, the species numbers are independent and have Poisson distributions.

27 Enzyme kinetics Consider the slightly different enzyme kinetics given by E + S ES E + P, E We see S + ES + P = N Tot. In distributional equilibrium: E has Poisson distribution, S, ES, P have a multinomial distribution, and E is independent from S, ES, and P.

28 Ligand-Receptor model for T-cell - Omer Dushek (Oxford) is L + R kon k off C k off k p C 2 ϕkp C 1. Also used in spatial models: works by (i) Reed, Popovic and McKinley, and (ii) Per Lötstedt.

29 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) ẋ A (t) = αx A (t)x B (t) + βx B (t) ẋ B (t) = αx A (t)x B (t) βx B (t) M = def x A () + x B (), Solving for equilibria: x A = β α, x B = M β α, Network has absolute concentration robustness in species A.

30 We also made similar measur osmolarity, in which the culture me Once again we found that ompc om insensitive to the expression level o below WT, and that the ratio decre for EnvZ levels above WT (Fig normalized the fluorescence and E ing low osmolarity WT values. W EnvZ at high osmolarity is larger th by a factor of 2.1, which is similar in ref. 38. Fig. 4b also includes the lo asenseofscaleforthechangei between low and high osmolarity transcription at fixed osmolarity. Absolute concentration robustness OmpC OmpF Transcription Is Insensiti OmpR. To test whether the EnvZ respect to OmpR, we constructed carries a copy of ompr that lacks it and is under control of the lac p Methods). When we transformed 2 with peb15 we were able to vary O orders of magnitude ( 1-fold be 2 Batchelor and Goulian, Fig. 4. (a) Robustness CFP YFP fluorescence and the ratios cycle versus ofenvz phosphorylation levels for culturesand of dephosphorylation levels) by adjusting the in concentra MDG135 penvz at low osmolarity (minimal medium) with varying amounts of medium. The levels of EnvZ did n a two-component regulatory IPTG. Fluorescence system, and EnvZ PNAS, levels22 were normalized by WT values, which were this range except at the very highe

31 Theorems: deterministic and stochastic Theorem (M. Feinberg and G. 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.

32 Differing in one species 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

33 Terminal and non-terminal complexes 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.

34 Theorems: deterministic and stochastic Theorem (M. Feinberg and G. 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. What about stochastically modeled systems? Theorem (A., Enciso, Johnston 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 a last time a nonterminal reaction fires. 2 David F. Anderson, Germán Enciso, and Matthew Johnston, Stochastic analysis of biochemical reaction networks with absolute concentration robustness, J. Royal Society Interface, Vol. 11, , February 12, 214.

35 Example Reaction network A + B 2B B A has state space # B Molecules # A Molecuels

36 Example

37 Idea of proof 1. Suppose not true: then there is a positive recurrent state for which a nonterminal complex has positive intensity (rate). 2. This provides a series of stopping times, τ N. 3. Use stochastic equation: = X (τ N ) X () = k = [ ( 1 τn κ k τn κ k k (X (s)) y k ds Y k κ k ( τn ) Y k κ i (X (s)) y k ds ζ k ) (X (s)) y k ds 1 t N ] τn (X (s)) y k ds ζ k 4. In limit N, find something in kernel of YA κ that can not be there.

38 Extinction can be rare event. What s real behavior? A + B α 2B B β A X A () + X B () = M, # B Molecules # A Molecuels MATLAB: β = 1, α = 1, M big Can show that quasi-stationary distribution for A converges to Poisson πm(x) Q e (β/α) (β/α) x, as M. x!

39 Quasi-stationary distribution: EnvZ-OmpR signaling system 3 3 Guy Shinar and Martin Feinberg, Structural Sources of Robustness in Biochemical Reaction Networks, Science, 21

40 Quasi-stationary distribution 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

41 Behavior on compact time intervals Let N be a scaling parameter (total conserved value), and consider N 1. Let q be the value of the ACR equilibrium value and let J Poisson(q). (3) Theorem (Anderson, Cappelletti, Kurtz, ) Suppose t > and that some technical assumptions hold on the reaction network and let J be as in (3). Then, for k, ( ) P XACR,i(t) N N = k P(J i = k) = e q q k i i k!. 4 David F. Anderson, Daniele Cappelletti, and Thomas G. Kurtz, Finite time distributions of stochastically modeled chemical systems with absolute concentration robustness, submitted, 217

42 That is the story. Thanks! Main References: 1. David F. Anderson, Gheorghe Craciun, Thomas G. Kurtz, Product-form stationary distributions for deficiency zero chemical reaction networks, Bulletin of Mathematical Biology, Vol. 72, No. 8, , Guy Shinar and Martin Feinberg, Structural Sources of Robustness in Biochemical Reaction Networks, Science, David F. Anderson, Germán Enciso, and Matthew Johnston, Stochastic analysis of biochemical reaction networks with absolute concentration robustness, J. Royal Society Interface, Vol. 11, , February 12, David F. Anderson, Daniele Cappelletti, and Thomas G. Kurtz, Finite time distributions of stochastically modeled chemical systems with absolute concentration robustness, submitted, David F. Anderson and Thomas G. Kurtz, Stochastic analysis of biochemical systems, Springer, 215.