Stochastic Modeling of Chemical Reactions

Transcription

1 Stochastic Modeling of Chemical Reactions Eric Haseltine Department of Chemical Engineering University of Wisconsin-Madison TWMCC 27 September Texas-Wisconsin Modeling and Control Consortium 22TWMCC

2 Outline Why stochastic chemical kinetics? Stochastic simulation Handling reactions with different time scales Examples. Simple motivating example 2. Hepatitis B infection 3. Particle engineering Conclusions So what can you do with these tools? TWMCC 27 September 2 2

3 Why stochastic chemical kinetics? Stochastic kinetic models treat reactions as molecular events Consider the well-mixed reaction: A o B o C o = A + B k k C 5 molecules We randomly select:. When the next reaction occurs τ = log(p ) r tot 2. Which reaction occurs p 2 Scale probabilities by reaction rates r = k AB r 2 = k C r tot = r + r 2 r r tot r 2 r tot TWMCC 27 September 2 3

4 Stochastic Simulation.8 One simulation Deterministic Stochastic Average of many simulations.8 Extent.6.4 Probability Time.2 Time.4.5 Extent TWMCC 27 September 2 4

5 Connection to Deterministic Kinetics (Ω = System Size) Ω Ω Deterministic Stochastic Deterministic Stochastic.8.8 Extent.6.4 Extent Time Time Ω Extent Deterministic Stochastic As Ω increases: Stochastic Deterministic Computing burden increases! Time TWMCC 27 September 2 5

6 What can be done when reactions occur over drastically different time scales? Number of Molecules cccdna rcdna Envelope. Partition the reactions into two subsets: a fast reaction subset a slow reaction subset 2. Fast reactions ODE approximation Eliminates fluctuations Time (Days) Make a stochastic approximation Slow reactions Time-varying reaction rates Exact solution or approximate solution? TWMCC 27 September 2 6

7 Solution Techniques r tot r tot t o τ t+τ Time τ app τ t o t+τ Time Exact solution t+τ r tot (t )dt + log(p ) = t May be computationally expensive to solve! Approximate solution Artificially introduce a probability of no reaction, a dt Let τ log(p ) rtot As a, this simulation method becomes exact TWMCC 27 September 2 7

8 A Simple Example Reaction System Approximate ε deterministically 2A k B, ε A + C k 2 D, ε 2 Reconstruct the mean and standard deviation with, simulations k = k 2 = E-7 A E+6 B C = D Stochastic simulation CPU time: 8 sec per simulation Hybrid simulation CPU time:.6 sec per simulation TWMCC 27 September 2 8

9 Number of Molecules e Exact Stochastic-Deterministic Results Mean for A and B Time Stochastic A Exact A Stochastic B Exact B Number of Molecules Standard Deviation for A and B Stochastic A. Exact A Stochastic B Exact B Time Number of Molecules C Stochastic Mean Stochastic +/- SD Exact Mean Exact +/- SD Time Number of Molecules D Stochastic Mean 2 Stochastic +/- SD Exact Mean Exact +/- SD Time TWMCC 27 September 2 9

10 Number of Molecules Approximate Stochastic-Deterministic Results, a = 4E-2 e Mean for A and B Time Stochastic A Exact A Stochastic B Exact B Number of Molecules Standard Deviation for A and B Stochastic A. Exact A Stochastic B Exact B Number of Molecules C Stochastic Mean Stochastic +/- SD Exact Mean Exact +/- SD Number of Molecules D Stochastic Mean 2 Stochastic +/- SD Exact Mean Exact +/- SD TWMCC 27 September 2

11 Squared Error Trends Error in Mean Error in Standard Deviation Approximate C Exact C Approximate C Exact C Squared Error Squared Error a a TWMCC 27 September 2

12 Hepatitis B Infection Consider the infection of a cell by a virus System model: nucleotides cccdna rcdna () nucleotides + rcdna cccdna (2) nucleotides + amino acids cccdna envelope (3) cccdna degraded (4) envelope secreted (5) rcdna + envelope virus (6) Assume:. nucleotides and amino acids are available at constant concentrations 2. cccdna catalyzes reactions () and (3) TWMCC 27 September 2 2

13 Hepatitis B Infection Reduced state cccdna x = rcdna = envelope x o = After 2 days: 2 x ss = 2 A B C System reaction channels H: k A k 2 B k 3 A H(x) =, k 4 A k 5 C k 6 BC k k 2 k 3 k 4 k 5 k 6.25 = day E-6 When A> and C >, reactions (3) and (5) dominate. Partition the system accordingly. TWMCC 27 September 2 3

14 cccdna Comparisons Stochastic Results Average of Simulations Approximate Stochastic-Deterministic Results Average of Simulations TWMCC 27 September 2 4

15 cccdna Comparisons Prob(cccDNA =,t) Prob(cccDNA,t = 99 days).8.7 Stochastic Approximate.25 Stochastic Approximate.6 Probability Probability Time (Days) cccdna Molecules TWMCC 27 September 2 5

16 rcdna Comparisons Comparison of the Mean Comparison of the Standard Deviation 25 2 Stochastic Approximation ODE 2 Stochastic Approximation rcdna Molecules 5 5 rcdna Molecules Time (Days) An ODE model gets the mean wrong! Time (Days) TWMCC 27 September 2 6

17 Envelope Comparisons Comparison of the Mean Comparison of the Standard Deviation 4 2 Stochastic Approximation ODE 6 5 Stochastic Approximation Envelope Molecules Envelope Molecules Time (Days) Time (Days) TWMCC 27 September 2 7

18 Why the deterministic solution is wrong... Typical Infection Aborted Infection Number of Molecules cccdna rcdna Envelope Number of Molecules cccdna rcdna Envelope Time (Days) Time (Days) The deterministic model cannot express both cases. TWMCC 27 September 2 8

19 Particle Engineering Size-independent nucleation and growth Empirical deterministic formulation Stochastic formulation Examples Size-independent nucleation, growth and agglomeration TWMCC 27 September 2 9

20 more more more more more more more TWMCC 27 September 2 2

21 Empirical Deterministic Formulation Population Balance f (L, t) t f (L, t) = G L in which f (L, t) is the number of crystals of size L and G is the crystal growth rate. Mass and Energy Balances dĉ dt = 3ρ ck v hg f L 2 dl dt ρv C p dt = 3 H cρ c k v V G f L 2 dl UA(T T j (t)) Nucleation and growth in the bulk B = k b (Ĉ Ĉsat (T ) Ĉ sat (T ) ) b (Ĉ ) g = k b S b Ĉsat (T ) G = k g = k g S g Ĉ sat (T ) TWMCC 27 September 2 2

22 Stochastic Formulation Reaction Mechanism for Nucleation and Growth 2M kn N l N ln + M k g N ln+ H n rxn H g rxn N ln is the number of crystals of size l n per volume. Mass Balance M tot = M sat + M Energy Balance dt dt = UA ρc p V (T j T ) (between reaction events) Solubility Relationship log M sat = a log T + b T + c TWMCC 27 September 2 22

23 Isothermal, Size-Independent Nucleation and Growth Stochastic Solution Average of Simulations Deterministic Solution Via Orthogonal Collocation Discrete particle sizes Integer-valued particle accounting Continuous particle sizes Real-valued particle accounting TWMCC 27 September 2 23

24 Nonisothermal, Size-Independent Nucleation and Growth Stochastic Solution Average of 5 Simulations Deterministic Solution Via Orthogonal Collocation TWMCC 27 September 2 24

25 Isothermal, Size-Independent Nucleation, Growth, and Agglomeration Stochastic Solution Add one reaction: N lp + N lq N lp+q Average of 5 Simulations k a Deterministic Solution Via Adaptive Mesh Methods? Large time investment! TWMCC 27 September 2 25

26 Conclusions Stochastic models are important when:. Fluctuations in the numbers of particles are important 2. In regions where multiple steady states are possible 3. Solution or formulation of the deterministic problem is difficult Our advances to the current technology:. New modeling approximation for handling reactions occurring over different time scales 2. Application of modeling techniques TWMCC 27 September 2 26

27 So what can you do with these tools? Wide array of possible applications Biotechnology Particle engineering Bridging the gap from the microscopic to the macroscopic Engineering at interfaces Modeling site interactions on catalysts Nanomaterials Further insight into the estimation problem Better understanding of:. State estimation for fluctuating systems 2. Chemical reaction models Other approaches for computing conditional densities of nonlinear dynamical systems TWMCC 27 September 2 27