Supporting Information for The Stochastic Quasi-Steady-State Assumption: Reducing the Model but Not the Noise

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1 TWCCC Texas Wisconsin California Control Consortium Technical report number Supporting Information for The Stochastic Quasi-Steady-State Assumption: Reducing the Model but Not the Noise Rishi Srivastava Dept. Chem. & Biol. Eng. Univ. of Wisconsin. (USA) Ethan Mastny Dept. Chem. & Biol. Eng. Univ. of Wisconsin. (USA) Eric L. Haseltine Div. Chemi. & Chem. Eng Cali. Inst. Tech., Pasadena, CA March 31, 2011 James B. Rawlings Dept. Chem. & Biol. Eng. Univ. of Wisconsin. (USA) We report in the statement of some results and the proofs omitted from the paper The Stochastic Quasi-Steady-State Assumption: Reducing the Model but Not the Noise [3]. All numberings are referred to the paper [3]. S1 Example 1: Derivation for Pap operon regulation Dividing both sides of equations (1) (4) of the paper by r 1 + r 3 and defining ɛ = 1/(r 1 + r 3 ) gives ɛ dp 1 ɛ dp 2 ɛ dp 3 ɛ dp 4 = P 1 + ɛr 2 P 2 + ɛr 4 P 3 (1) = ɛ(r 2 + r 5 )P 2 + K 1 P 1 + ɛr 6 P 4 (2) = ɛ(r 4 + r 7 )P 3 + K 3 P 1 + ɛr 8 P 4 (3) = ɛ(r 6 + r 8 )P 4 + ɛr 5 P 2 + ɛr 7 P 3 (4) (5) in which K 1 = r 1 /(r 1 + r 3 ) and K 3 = r 3 /(r 1 + r 3 ) are O(1) terms. rsrivastava2@wisc.edu. Author to whom all correspondence should be addressed. Eric L. Haseltine eamastny@wisc.edu rawlings@engr.wisc.edu 1

2 TWCCC Technical Report S1.1 Fast time scale solution In the fast time scale we rescale time t to τ = t/ɛ. Using τ we rewrite (1) (4) on the fast time scale as d ˆP 1 = ˆP 1 + ɛr 2 ˆP2 + ɛr 4 ˆP3 (6) d ˆP 2 = ɛ(r 2 + r 5 ) ˆP 2 + K 1 ˆP1 + ɛr 6 ˆP4 (7) d ˆP 3 = ɛ(r 4 + r 7 ) ˆP 3 + K 3 ˆP1 + ɛr 8 ˆP4 (8) d ˆP 4 = ɛ(r 6 + r 8 ) ˆP 4 + ɛr 5 ˆP2 + ɛr 7 ˆP3 (9) in which we denote probability on the fast time scale by ˆP i. We expand ˆP i in a power series Substituting (11) into (6) and retaining terms up to order ɛ 1 give (10) ˆP i = Ŵi0 + ɛŵi1 + ɛ 2 Ŵ i2 + O(ɛ 3 ) (11) dŵ10 + ɛ dŵ11 + = (Ŵ10 + ɛŵ11 + ) + ɛr 2 (Ŵ20 + ɛŵ21 + )+ ɛr 4 (Ŵ30 + ɛŵ31 + ) (12) Comparing O(ɛ 0 ) terms in (12) gives dŵ10 with initial condition Ŵ10(τ = 0) = 1. The solution of (13) is = Ŵ10 (13) Ŵ 10 = e τ (14) which is the equation (5) of the paper. Rewriting equation (7) using equation (11) gives dŵ20 + ɛ dŵ20 + = ɛ(r 2 + r 5 )(Ŵ20 + ɛŵ21 + ) + K 1 (Ŵ10 + ɛŵ11 + )+ ɛr 6 (Ŵ40 + ɛŵ41 + ) (15) Comparing O(ɛ 0 ) terms from (15) gives dŵ20 = K 1 Ŵ 10 (16) with initial condition Ŵ20(τ = 0) = 0. We substitute Ŵ10 from (14) into (16) and solve the differential equation to obtain Ŵ 20 = K 1 (1 e τ ) (17) which is the equation (6) of the paper. We obtain the expansions of the probabilities of states g 3 and g 4, i.e., Ŵ 30 and Ŵ40, analogously.

3 TWCCC Technical Report S1.2 Slow time scale solution In (1) we substitute expansion of P i, 1 i 4 P i = W i0 + ɛw i1 + ɛ 2 W i2 + O(ɛ 3 ) (18) giving ɛ ( dw 10 + ɛ dw 11 + ) = ( W 10 + ɛw 11 + ) + ɛr 2 (W 20 + ɛw 21 + )+ ɛr 4 (W 30 + ɛw 31 ) (19) Collecting O(ɛ 0 ) terms from (19) gives W 10 = 0 (20) which is equation (9) of the paper. Next we collect O(ɛ 1 ) terms from (19) to obtain W 11 = r 2 W 20 + r 4 W 30 (21) A better approximation for the probability of state g 1, P 1, is obtained by approximating P 1 up to O(ɛ). P 1 = W 10 + ɛw 11 (22) Using equations (20) and (21) in equation (22) gives which is equation (13) of the paper. Next we substitute (18) into (2) to obtain P 1 = ɛ(r 2 W 20 + r 4 W 30 ) (23) ɛ ( dw 20 + ɛ dw 21 + ) = ɛ(r 2 + r 5 )(W 20 + ɛw 21 + ) + K 1 (W 10 + ɛw 11 + )+ ɛr 6 (W 40 + ɛw 41 + ) (24) Comparison of O(ɛ 0 ) terms from (24) leads to equation (20). Thus comparison of O(ɛ 0 ) terms from (24) does not lead to any new information. Comparing O(ɛ 1 ) terms from (24) we obtain: dw 20 = (r 2 + r 5 )W 20 + r 6 W 40 + K 1 W 11 (25) Substituting W 11 from (21) into (25) gives dw 20 = [ r 1 + r 5 ]W 20 + r 2 W 30 + r 6 W 40 (26) which is the same as equation (10) of the paper. Evolution equations for W 30 and W 40 can be obtained similarly.

4 TWCCC Technical Report S2 Example 2: Derivations for robustness to noise in a biochemical oscillator. S2.1 sqspa reduction We consider reaction network (15) of the paper These reactions include the transcription events to form M A, translation to form A, and degradation of M A. These reactions do not lead to change in the populations of species D A and D A, hence we write P (D A, D A, M A, A) as P (M A, A).The master equation for this system is dp (M A, A) = α A D A P (M A 1, A) + α AD AP (M A 1, A) + δ MA (M A + 1)P (M A + 1, A)+ β A M A P (M A, A 1) [ α A D A + α AD A + (δ MA + β A )M A ] P (MA, A) (27) When the condition β A, δ MA α A, α A holds, the mrna species M A is a QSSA species. We further define 1 ɛ = δ MA = ɛδ MA βa = ɛβ A (β A + δ MA ) Multiplying (27) with ɛ gives us ɛ dp (M A, A) = ɛ(α A D A + α AD A)P (M A 1, A) + δ MA (M A + 1)P (M A + 1, A)+ β A M A P (M A, A 1) [ ɛ(α A D A + α AD A) + ( δ MA + β A )M A ] P (MA, A) (28) We expand P (M A, A) in a power series P (M A, A) = W 0 (M A, A) + ɛw 1 (M A, A) + ɛ 2 W 2 (M A, A) + (29) Then by grouping like powers of ɛ in (28), we obtain ɛ 0 : 0 = δ MA (M A + 1) W 0 (M A +1, A)+ β A M A W 0 (M A, A 1) ( δma + β ) A MA W 0 (M A, A) (30) Expressing this result for M A = 0, 1,... yields W 0 (M A, A) = 0 for M A > 0 for all A (31) Similarly, equating terms of O(ɛ 1 ) yields ɛ 1 dw 0 (0, A), M A = 0 : = MA W 1 (1, A) ( α A D A + α AD A) W0 (0, A) (32) ɛ 1, M A = 1 : W 1 (1, A) = β A W 1 (1, A 1) + (α A D A + α AD A)W 0 (0, A) (33) Solving the last equation for W 1 (1, A) gives A W 1 (1, A) = (α A D A + α AD A)W 0 (0, A) + β j A (α AD A + α AD A)W 0 (0, A j) Substituting this result into (32) gives j=1 dw 0 (0, A) = δ A MA β j A (α AD A + α AD A)W 0 (0, A j) β A (α A D A + α AD A)W 0 (0, A) (34) j=1 The catalytic reactions in the paper, reaction sets (16) and (17), follow from this equation.

5 TWCCC Technical Report S2.2 Deterministic quasi steady state classical (dqc) reduction Again we consider the reaction network (15) of the paper. The master equation describing the species M A can be written as: dm A = α A D A + α AD A δ MA M A β A M A (35) Under the condition β A, δ MA α A, α A, δ A, M A is a QSSA species. To apply dqc we set left hand side of equation (35) to 0 to obtain following expression for the population of M A. M A = α AD A + α A D A δ MA + β A (36) The rate of formation of A is equal to the rate of last reaction of the network da =β AM A = β A (α A D A + α AD A) (37) Since M A is already removed from the network. The only species that changes with time is species A. Equations (36) and (37) give rise to the dqc reduced reactions (18) and (19) of the paper. S3 Example 3: Derivation for Fast fluctuation We apply Ω expansion as outlined by van Kampen [4] on the master equation of reactions (20) (22) of the paper. The master equation for these reactions is dp (a, c, g) = k 1 Ω (a + 1)gP (a + 1, c 1, g) + k 2 (c + 1)(g 1)P (a 1, c + 1, g 1) Ω + k ( 3 k1 (g + 1)gP (a, c, g + 1) Ω Ω ag + k 2 Ω cg + k ) 3 g(g 1) P (a, c, g) (38) Ω There are two phases of time evolution of the species population. In the first phase, species G increases and species A and C start fluctuating. After species G has risen above a certain threshold value, we switch the simulation methodology to the SSA-Ω approach. We write g as a continuous random variable: g = Ωφ G + Ω 1/2 ξ (39) where φ G is the deterministic mean concentration of species G in the regime of SSA-Ω and Ω 1/2 ξ is the noise in G. For the transformed variable g, the probability P (a, c, g) corresponds to Π(a, c, ξ). In (38) we can write dp (a, c, g) Π(a, c, ξ) = Under this change of variables, (38) becomes Π(a, c, ξ) Ω 1/2 dφ g Π(a, c, ξ) = k 1 Ω (a+1)(ωφ G + Ω 1/2 ξ)π(a+1, c 1, ξ) Ω 1/2 dφ g Π(a, c, ξ) + k 2 Ω (c+1)[1 Ω 1/2 2 2 ](Ωφ G+Ω 1/2 ξ)π(a 1, c+1, ξ) + k 3 Ω [1 Ω 1/2 2 2 ](Ωφ G+Ω 1/2 ξ)(ωφ G +Ω 1/2 ξ 1)Π(a, c, ξ) ( k1 Ω a(ωφ G+Ω 1/2 ξ) + k 2 Ω c(ωφ G+Ω 1/2 ξ)+ k ) 3 Ω (Ωφ G+Ω 1/2 ξ)(ωφ G +Ω 1/2 ξ 1) Π(a, c, ξ) (40)

6 TWCCC Technical Report For the rate constants of the system, we have k 1, k 2 k 3. We define the following parameters K 1 = k 1 Ω = Ωk 1 2 (41) Following Mastny et al. [2], we express Π as a power series in Ω 1, Π(a, c, ξ) = W 0 (a, c, ξ) + Ω 1 W 1 (a, c, ξ) + Ω 2 W 2 (a, c, ξ) + and substitute into (40) to obtain the following expanded master equation ( Ω 1 W0 (a, c, ξ) +Ω 1 W ) 1(a, c, ξ) + Ω 1/2 dφ ( G W0 (a, c, ξ) +Ω 1 W ) 1(a, c, ξ) + = K 1 Ω 1 (a+1)(ωφ G + Ω 1/2 ξ) ( W 0 (a+1, c 1, ξ)+ω 1 W 1 (a + 1, c 1, ξ)+ ) + (Ω) 1 (c+1)[1 Ω 1/2 2 2 ](Ωφ G+Ω 1/2 ξ) ( W 0 (a 1, c+1, ξ)+ω 1 W 1 (a 1, c+1, ξ)+ ) + k 3 Ω 2 [1 Ω 1/2 2 2 ](Ωφ G+Ω 1/2 ξ)(ωφ G +Ω 1/2 ξ 1) ( W 0 (a, c, ξ) + Ω 1 W 1 (a, c, ξ) + ) ( ) (W0 K 1 Ω 1 a(ωφ G +Ω 1/2 ξ) + 1 Ω 1 c(ωφ G +Ω 1/2 ξ) (a, c, ξ) + Ω 1 W 1 (a, c, ξ) + ) k 3 Ω 2 (Ωφ G +Ω 1/2 ξ)(ωφ G +Ω 1/2 ξ 1) ( W 0 (a, c, ξ) + Ω 1 W 1 (a, c, ξ) + ) (42) Ω 0 terms. Collecting Ω 0 terms from (42) gives 0 = K 1 (a + 1)φ G W 0 (a + 1, c 1, ξ) ( K 1 aφ G + 1 cφ G ) W0 (a, c, ξ) + 1 (c + 1)φ G W 0 (a 1, c + 1, ξ) (43) Since none of the terms multiplying joint densities in (43) contain ξ, we can integrate (43) to obtain an equation for marginal density as a function of a and c only. Also we note that because of the stoichiometry of the problem a + c = N 0 and we can replace c with N 0 a. Making these substitutions in (43), we obtain 0 = K 1 (a + 1)W 0a (a + 1) K 1 aw 0a (a) 1 N 0 W 0a (a) + 1 aw 0a (a) + 1 N 0 W 0a (a 1) 1 (a 1)W 0a (a 1) (44) An analytical expression for marginal density W 0a (a) can be obtained by taking the z-transform of the equation (44). That produces equation (23) of the paper. The distribution in (23) is a binomial distribution W 0a Bin( q 1+q, N 0) in which q = 1/(K 1 ). Since a + c = N 0 equation (24) is an immediate consequence of (23). From equation (43), it is also clear that W 0 (a, ξ) = W 0a (a)w 0ξ (ξ). Ω 1/2 terms. Collecting Ω 1/2 terms from equation (42) gives dφ G W 0 (a, c, ξ) =K 1 (a+1)ξw 0 (a+1, c 1, ξ) K 1 aξw 0 (a, c, ξ)+ 1 (c+1)ξw 0 (a 1, c+1, ξ) 1 cξw 0 (a, c, ξ) + k 3 ( φ 2 G W 0(a, c, ξ) ) (45)

7 TWCCC Technical Report We recall that argument c in function W 0 is redundant because a + c = N 0 at all times. Summing equation (45) over all the values of a and c along with using the independence of a and ξ gives us dφ G = 1 c φ G + k 3 φ 2 G (46) which is (25) in the paper. Equations (23) and (46) together give an approximation of the populations of A, C and G when G has risen over a threshold value. Ω 1 terms. Collecting Ω 1 terms from equation (42) gives W 0 (a, c, ξ) =K 1 (a + 1)φ G W 1 (a + 1, c 1, ξ) K 1 aφ G W 1 (a 1, c + 1, ξ) + c + 1 (ξw 0(a 1, c + 1, ξ)) + c (φ GW 0 (a 1, c + 1, ξ) cφ G W 1 (a, c, ξ)) + 2k 3 φ G (ξw 0(a, c, ξ)) + k φ2 G 2 W 0(a, c, ξ) Summing equation (47) for all values of a and c gives 2 (47) W 0ξ (ξ) = c (ξw 0ξ(ξ))+ 1 2 c φ G 2 2 W 0ξ(ξ)+2k 3 φ G (ξw 0ξ(ξ))+ k φ2 G Rearranging equation (48) gives W 0ξ (ξ) = [( ) ] c 2k 3φ G ξw 0ξ (ξ) [( ) ] c φg k 3 φ 2 G W 0ξ (ξ) 2 W 0ξ(ξ) (48) Equation (49) is a linear Fokker-Planck equation and is the same as equation (26) of the paper. Its equivalent stochastic differential equation is given by ( ) c dξ = 2k c φ G 3φ G ξ + + k 3 φ 2 G dw (50) in which dw is a normally distributed random variable with zero mean and variance [1]. Equation (50) here is the same as equation (27) of the paper. References [1] Peter E. Kloeden and Eckhard Platen. Numerical solution of stochastic differential equations. Springer Verlag, Berlin, [2] Ethan A. Mastny, Eric L. Haseltine, and James B. Rawlings. Two classes of quasi-steady-state model reductions for stochastic kinetics. J. Chem. Phys., 127(9):094106, September [3] Rishi Srivastava, Eric L. Haseltine, Ethan A. Mastny, and James B. Rawlings. The stochastic quasi-steady-state assumption: Reducing the model but not the noise. J. Chem. Phys., Accepted for publication. [4] N. G. van Kampen. Stochastic Processes in Physics and Chemistry. Elsevier Science Publishers, Amsterdam, The Netherlands, second edition, (49)