Extending the Tools of Chemical Reaction Engineering to the Molecular Scale

Transcription

1 Extending the Tools of Chemical Reaction Engineering to the Molecular Scale Multiple-time-scale order reduction for stochastic kinetics James B. Rawlings Department of Chemical and Biological Engineering March 31, 29 Model Reduction in Reacting Flows University of Notre Dame Rawlings Molecular reaction engineering 1 / 35

2 Outline 1 Introduction to stochastic kinetics 2 Model reduction fast reactions and reactive intermediates 3 Catalyst example with fast diffusion 4 Virus example with fast fluctuation 5 Further reading Rawlings Molecular reaction engineering 2 / 35

3 Introduction to stochastic kinetics Stochastic kinetics Small species populations Species numbers are integers, reactions cause integer jumps Large fluctuations in species numbers and reaction rates Biological networks and catalyst particles Model reduction Develop reduced models from stochastic chemical reactions. These models must meet the following requirements: Simpler than the full model (fewer reactions, fewer parameters, or faster simulation times) Converge to the full model as a specified parameter goes to zero Rawlings Molecular reaction engineering 3 / 35

4 Stochastic simulation method kinetic Monte Carlo n A r 2 r 1 B A k 1 k2 k 1 = 2 k 2 = 1 n A = 6 n B = time (sec) Random number r 1 r 1 +r 2 = KMC Algorithm 1 Choose which reaction 2 Choose time step 3 Repeat 1 r 2 r 1 +r 2 = Which reaction: Time step: Sample from an exponential distribution where the distribution mean is the sum of reaction rates. Rawlings Molecular reaction engineering 4 / 35

5 KMC simulations and probability ni Multiple KMC simulations, A k1 B n B k2 Count Histogram at t=2sec, for 5 simulations n A time (sec) n A KMC simulations are samples of a probability distribution that evolves in time. We can write the evolution equation for the probability density (master equation). Rawlings Molecular reaction engineering 5 / 35

6 Chemical master equation dp(x) dt dp dt = = AP N rxn r j (x ν j )P(x ν j ) r j (x)p(x) }{{}}{{} j=1 rate into state x rate out of state x Master equation example P(n A).15 B A k 1 k2 n A = 1, n B = k 1 = 2, k 2 = 1 11 possible states 11 Coupled ODEs Time (sec) Rawlings Molecular reaction engineering 6 / 35 n A

7 Master equation Important points Chemical master equation dp(x) dt dp dt = = AP N rxn r j (x ν j )P(x ν j ) r j (x)p(x) }{{}}{{} j=1 rate into state x rate out of state x Often the dimensionality of the master equation makes direct solution infeasible The master equation shows what probability distribution is sampled in a KMC simulation A reduced master equation can lead to a new/faster simulation schemes Rawlings Molecular reaction engineering 7 / 35

8 Kinetics of multiple time scales A k 1 k 1 B k 2 C 1 Deterministic - One Time Scale k1 = 2 k 1 =.5 k2 =.5 1 Deterministic - Two Time Scales k1 = 1 k 1 = 1 k2 =.5 1 Deterministic - Two Time Scales k1 = 2 k 1 = 2 k2 = na nc 6 na nb nc 6 na nc nb time (sec) KMC - One Time Scale k1 = 2 k 1 =.5 k2 = time (sec) KMC - Two Time Scales k1 = 1 k 1 = 1 k2 =.5 1 nb time (sec) KMC - Two Time Scales k1 = 2 k 1 = 2 k2 = na nb nc 8 6 na nc 8 6 na nc time (sec) One time scale 2 nb time (sec) Reaction equilibrium 2 nb time (sec) Reactive intermediate Rawlings Molecular reaction engineering 8 / 35

9 Deterministic model reductions x non-qssa species, y QSSA species dx dt = f (x, y) ɛdy dt = g(x, y) Classical QSSA dx dt = f (x, y) = g(x, y) DAE reduced model Singular Perturbation QSSA x = X + ɛx 1 + ɛ 2 X 2 + O(ɛ 3 ) y = Y + ɛy 1 + ɛ 2 Y 2 + O(ɛ 3 ) Collect like powers of ɛ Equations for dx dt is the reduced model Separate models for fast and slow time scale Rawlings Molecular reaction engineering 9 / 35

10 SPA on the master equation Our objective Apply singular perturbation analysis to develop a reduced master equation. dp(a, b, c) dt A k 1 k 1 B k 2 C = k 1 (a + 1)P(a + 1, b 1, c) + k 1 (b + 1)P(a 1, b + 1, c) + k 2 (b + 1)P(a, b + 1, c 1) (k 1 a + k 1 b + k 2 b)p(a, b, c) P(a, b, c) = W (a, b, c) + ɛw 1 (a, b, c) + ɛ terms: W (a, b, c) = if b > In this limit b is always zero Rawlings Molecular reaction engineering 1 / 35

11 SPA on the master equation ɛ 1 terms: Reduced master equation dw (a,, c) dt Reduced mechanism = k(a + 1)W (a + 1,, c 1) kaw (a,, c) A C r = k 1k 2 k 1 +k 2 a Stochastic same as deterministic SPA mechanism Same mechanisms due to linearity First-order correction, b b = f (W (a,, c)) + O(ɛ 2 ) b = k 1 k 1 +k 2 a Rawlings Molecular reaction engineering 11 / 35

12 Comparison of mechanisms A 2B r 1 = k 1 a r 1 = k 1 b(b 1) B C r 2 = k 2 b 2 Stoch SPA A 2C ( r = k 1 k 2 k 1 2 +k c ) a na = 25, nb =, nc = k1 = 1 k 1 = 1 k2 = 1 Det SPA A 2C r = k 1 a Det QSSA A 2C k 2 + k k 1k 1 a r = k 2 4k 1 Full Model Stoch SPA Det QSSA Det SPA Time (sec) Rawlings Molecular reaction engineering 12 / 35

13 Catalyst Example Number of molecules (n) A k 1 B k 2 C B + D k 3 B + E k 2, k 3 k 1 a 2 b Time (sec) e Stoch SPA mechanism A D + A C E + C 2D + A 2E + C nd + A ne + C k 1 a r = 1 + K 3 d K 3 = k 3 k 2 K 3 d r 1 = r 1 + K 3 (d 1) r 2 = r 1 K 3 (d 1) 1 + K 3 (d 2) r n = r n 1 K 3 (d + 1 n) 1 + K 3 (d n) Deterministic SPA mechanism A C D + A E + A r = k 1a r 1 = k3k1 ad k 2 Rawlings Molecular reaction engineering 13 / 35

14 Conclusions Stochastic quasi-steady-state approximation QSSA species are removed from stochastic models with SPA Stochastic QSSA mechanisms different than deterministic QSSA mechanisms Application of stochastic QSSA: Reduces the number of kinetic parameters Speeds up KMC simulations (fewer events) Rawlings Molecular reaction engineering 14 / 35

15 Conclusions Stochastic quasi-steady-state approximation QSSA species (# of molecules) >> <1 sqspa sqspa dqc dqspa >>1 3 non QSSA species (# of molecules) Rawlings Molecular reaction engineering 15 / 35

16 Catalytic surface reaction modeling Assumptions for this talk Two dimensional surface with a lattice for adsorption, diffusion, reaction, and desorption. Square lattice, Z=4 All sites have identical properties Constant temperature Adsorbed CO molecules exhibit nearest neighbor repulsions CO O 2 CO 2 CO-black, O-gray, Empty-white. Rawlings Molecular reaction engineering 16 / 35

17 Model mechanism and time scales Adsorption CO(g)+ i α CO i O 2(g)+ i + j β O i + O j Desorption CO i γ CO(g) + i O i + O j ρ O 2(g) + i k Reaction CO i + O r j CO2(g)+ i d Diffusion CO i + 1 j i +CO j d O i + 2 j i + O j + j + j 1/sec α = 1.6 β =.8 γ =.8 ρ =.1 k r = 1 d d Rawlings Molecular reaction engineering 17 / 35

18 Singular perturbation on the master equation Surface reaction master equation dp(n, x) dt x - microscopic configuration n - number of each species X rxn = k j a j (n ν j, x ν x,j )P(n ν j, x ν x,j ) k j a j (n, x)p(n, x) j=1 X diff + d j a j (n, x ν x,j )P(n, x ν x,j ) d j a j (n, x)p(n, x) j=1 Singular perturbation P(n, x) = W (n, x) + ɛw 1 (n, x) + ɛ 2 W 2 (n, x) + ɛ = 1/d ɛ terms: Diffusion equilibration equations for W (x n) Rawlings Molecular reaction engineering 18 / 35

19 Slow time-scale evolution equation ɛ 1 terms : Reduced master equation dw (n) dt N rxn = k i s i (n ν i ) W (n ν i ) k i s i (n) W (n) i=1 What have we gained? Removed micro-states from the master equation Lattice Size Species Micro-states Coverage states N s = N s = N s = Tractable number of states, master equation can be solved Rawlings Molecular reaction engineering 19 / 35

20 Slow time-scale evolution equation dw (n) dt Reaction propensities N rxn = k i s i (n ν i ) W (n ν i ) k i s i (n) W (n) i=1 s i (x) number of reaction i on configuration x: n CO =45 black, n O =8 gray s CO O =26 s CO O =22 s CO O =26 s CO O =23 s CO O = 24.9 s i (n) = x s i(x)w (x n) Calculate with diffusion only KMC Rawlings Molecular reaction engineering 2 / 35

21 Reduced master equation solution (5x5 lattice) Steady-state probability distribution (5x5).25.4 Probability of n CO, 5x5 lattice Probability n CO Time (sec) n CO Rawlings Molecular reaction engineering 21 / 35

22 Verification of perturbation method P(n, x) = W (n) + ɛw 1 (n) + O(ɛ 2 ) x ɛ = 1/d As the diffusion rate increases P(n) approaches W (n) Steady-state probability distribution (5 5) Probability Reduced ME d=1 d=1 d=1 d=1 d=.1 d=.1 d= n CO Rawlings Molecular reaction engineering 22 / 35

23 Conclusions Surface reactions in the infinite diffusion limit SPA can be used to eliminate spatial configuration states in a reduced master equation. The reduced master equation has sufficiently few states to be simulated on small lattices. Reduced master equations of surface reactions can be used to motivate reduced KMC and reduced ODE models. Rawlings Molecular reaction engineering 23 / 35

24 Model for Vesicular Stomatitis Virus (VSV) infection!!! " # $%& # "$%& "$%& "$%& I is encapsidation of viral genome II is replication of encapsidated genome III is transcription of genome to messenger RNA " # $%&! '()* '()* '()* '()* '()* '()* Rawlings Molecular reaction engineering 24 / 35

25 Onset of fast fluctuations in the N protein Molecule level I III N mrna ( )RNA 1258 II N protein (+)RNA genomes (+)RNA 1258 ( )RNA genomes Time (hours) Features of simulation Presence of fast fluctuating and rapidly rising species Fast fluctuations slow the full KMC simulation Motivates the formulation of a simpler example to understand this phenomenon Rawlings Molecular reaction engineering 25 / 35

26 Fast fluctuation and rapid rise in VSV biology k ( )RNA + L 1 1 ( )RNA + L2 k ( )RNA + L 2 2 2( )RNA + L1 2( )RNA k 3 ( )RNA The viral genome is amplified by first two reactions The free viral proteins and messages are not amplified Values of parameters k 1, k 2 and k 3 may cause fast fluctuation in polymerases along with rapid amplification of viral genome Rawlings Molecular reaction engineering 26 / 35

27 Fast fluctuation and rapid rise Idealized problem A + G k 1 C + G r1 = 1 Ω k 1ag C + G 2G k 2 2G + A r2 = 1 Ω k 2cg k 3 G r3 = 1 Ω k g(g 1) 3 2 Species Initial number Rate constant ( m3 /mol s) A 3 k 1 = C k 2 = G 1 k 3 = Rawlings Molecular reaction engineering 27 / 35

28 The full SSA on the system No. of C molecules t (sec) No. of G molecules t (sec) time-step (sec) t (sec) Rawlings Molecular reaction engineering 28 / 35

29 The hybrid SSA - Ω technique At large population of G we want to switch to a continuous description for it: g = Ωφ G + Ω 1/2 ξ φ G is the deterministic evolution term and ξ is the continuous noise in the evolution of G We can obtain approximation for the evolution of system using hybrid SSA - Ω technique Approximation of pdf of C W (c) = (1 + q) N ( N c ) q (N c) Deterministic evolution of G dφ G dt = γ 1 c φ G k 3 2 φ2 G N q = k 2 k 1 Initial number of polymerases Ratio of rate constants Rawlings Molecular reaction engineering 29 / 35

30 Comparison of full SSA with hybrid SSA - Ω Full SSA Hybrid SSA - Ω 3 3 No. of C molecules 2 1 No. of C molecules 2 1 No. of G molecules t (sec) t (sec) No. of G molecules t (sec) t (sec) Rawlings Molecular reaction engineering 3 / 35

31 Comparison of full SSA with hybrid SSA - Ω Probability densities of C from SSA and from hybrid SSA - Ω Probability Hybrid SSA Ω SSA No. of C particles Rawlings Molecular reaction engineering 31 / 35

32 Conclusions QSSA and fast fluctuations Hybrid SSA Ω expansion matches closely the full SSA Computation speed increases by factor of 45 Application to kinetic virus infection models Rawlings Molecular reaction engineering 32 / 35

33 Acknowledgments Dr. Ethan A. Mastny, BP Alaska Dr. Eric L. Haseltine, Vertex Pharmaceuticals Rishi Srivastava, UW NSF #CNS Rawlings Molecular reaction engineering 33 / 35

34 Further reading Stochastic reaction equilibrium Haseltine, E. L. and J. B. Rawlings. Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys., 117(15): , October 22. Haseltine, E. L. and J. B. Rawlings. On the origins of approximations for stochastic chemical kinetics. J. Chem. Phys., 123:164115, October 25. Cao, Y., D. T. Gillespie, and L. R. Petzold. The slow-scale stochastic simulation algorithm. J. Chem. Phys., 122(1):14116, January 25. Samant, A. and D. G. Vlachos. Overcoming stiffness in stochastic simulation stemming from partial equilibrium: A multiscale Monte Carlo algorithm. J. Chem. Phys., 123:144114, 25. Salis, H. and Y. Kaznessis. Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions. J. Chem. Phys., 122(5):5413, February 25. Mastny, E. A., E. L. Haseltine, and J. B. Rawlings. Stochastic simulation of catalytic surface reactions in the fast diffusion limit. J. Chem. Phys., 125(19):194715, November 26. Rawlings Molecular reaction engineering 34 / 35

35 Further reading Stochastic QSSA Rao, C. V. and A. P. Arkin. Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm. J. Chem. Phys., 118(11): , March 23. van Kampen, N. G. Stochastic Processes in Physics and Chemistry. Elsevier Science Publishers, Amsterdam, The Netherlands, second edition, Mastny, E. A., E. L. Haseltine, and J. B. Rawlings. Two classes of quasi-steady-state model reductions for stochastic kinetics. J. Chem. Phys., 127(9):9416, September 27. Hensel, S., J. B. Rawlings, and J. Yin. Stochastic kinetic modeling of vesicular stomatitis virus intracellular growth. Submitted for publication in Bulletin of Math. Bio., May 28. Rawlings Molecular reaction engineering 35 / 35