Stochastic Modeling of Chemical Reactionstwccc.che.wisc.edu/new/haseltine.pdf · 2001-12-17 · Stochastic Modeling of Chemical Reactions Eric Haseltine Department of Chemical Engineering

Stochastic Modeling of Chemical Reactions

Eric HaseltineDepartment of Chemical Engineering

University of Wisconsin-Madison

TWMCC

27 September 20012TWMCC ? Texas-Wisconsin Modeling and Control Consortium 11

Outline

• Why stochastic chemical kinetics?

• Stochastic simulation

• Handling reactions with different time scales

• Examples

1. Simple motivating example2. Hepatitis B infection3. Particle engineering

• Conclusions

• So what can you do with these tools?

TWMCC—27 September 2001 2

Why stochastic chemical kinetics?

• Stochastic kinetic models treatreactions as molecular events

• Consider the well-mixed reaction:

A+ Bk1zk−1CAoBo

Co

=10

500

molecules

• Scale probabilities by reaction rates– r1 = k1AB– r2 = k−1C– rtot = r1 + r2

• We randomly select:

1. When the next reaction occurs

τ = −log(p1)rtot

2. Which reaction occurs

10

p2

r1rtot

r2rtot


Stochastic Simulation

One simulation Average of many simulations

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Exte

nt

Time

DeterministicStochastic

0

0.02

0.04 0

0.5

10

0.2

0.4

0.6

0.8

1

ExtentTime

Pro

babili

ty


Connection to Deterministic Kinetics (Ω = System Size)

Ω Ω × 10

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Ext

ent

Time


0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Ext

ent

Time


Ω × 100

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Ext

ent

Time


As Ω increases:Stochastic → Deterministic

Computing burden increases!


What can be done when reactions occur over drasticallydifferent time scales?

0.1

1

10

100

1000

10000

0 50 100 150 200

Nu

mb

er

of

Mo

lecu

les

Time (Days)

cccDNArcDNA

Envelope

Make a stochastic approximation...

1. Partition the reactions into twosubsets:

• a fast reaction subset• a slow reaction subset

2. Fast reactions

• ODE approximation• Eliminates fluctuations

3. Slow reactions

• Time-varying reaction rates• Exact solution or approximate

solution?


Solution Techniques

to

rtot

τt+τ Time

• Exact solution

∫ t+τt

rtot(t′)dt′ + log(p1) = 0

May be computationally expensiveto solve!

to

τapp

rtot

τt+τ Time

• Approximate solution

– Artificially introduce a probabilityof no reaction, a0dt

– Let τ ≈ − log(p1)rtot

– As a0 →∞, this simulationmethod becomes exact


A Simple Example

Reaction System

2Ak1-→ B, ε1

A+ C k2-→ D, ε2

k1 = k2 = 1E-7A0

B0

C0

D0

=

1E+60

100

• Approximate ε1 deterministically

• Reconstruct the mean and standarddeviation with 10,000 simulations

• Stochastic simulation CPU time:∼ 8 sec per simulation

• Hybrid simulation CPU time:∼ 0.6 sec per simulation


Exact Stochastic-Deterministic Results

0

200000

400000

600000

800000

1e+06

0 10 20 30 40 50 60 70 80 90 100

Num

ber

of M

olec

ules

Time

Mean for A and B

Stochastic AExact A

Stochastic BExact B

0.01

0.1

1

10

100

1000

0 10 20 30 40 50 60 70 80 90 100

Num

ber

of M

olec

ules

Time

Standard Deviation for A and B

Stochastic AExact A

Stochastic BExact B

0123456789

1011

0 10 20 30 40 50 60 70 80 90 100

Num

ber

of M

olec

ules

Time

C

Stochastic MeanStochastic +/-1 SD

Exact MeanExact +/-1 SD

0

2

4

6

8

10

0 20 40 60 80 100

Num

ber

of M

olec

ules

Time

D




Approximate Stochastic-Deterministic Results, a0 = 4E-2

0

200000

400000

600000

800000

1e+06

0 10 20 30 40 50 60 70 80 90 100

Num

ber

of M

olec

ules

Time

Mean for A and B

Stochastic AExact A

Stochastic BExact B

0.01

0.1

1

10

100

1000

0 10 20 30 40 50 60 70 80 90 100

Num

ber

of M

olec

ules

Standard Deviation for A and B

Stochastic AExact A

Stochastic BExact B

0

2

4

6

8

10

0 20 40 60 80 100

Num

ber

of M

olec

ules

C



0

2

4

6

8

10

0 20 40 60 80 100

Num

ber

of M

olec

ules

D




Squared Error Trends

Error in Mean Error in Standard Deviation

0.1

1

10

0.01 0.1 1 10

Squ

ared

Err

or

a0

Approximate CExact C

0.01

0.1

1

10

0.01 0.1 1 10S

quar

ed E

rror

a0

Approximate CExact C


Hepatitis B Infection

• Consider the infection of a cell by a virus

• System model:

nucleotidescccDNA-→ rcDNA (1)

nucleotides + rcDNA -→ cccDNA (2)

nucleotides + amino acidscccDNA-→ envelope (3)

cccDNA -→ degraded (4)

envelope -→ secreted (5)

rcDNA + envelope -→ virus (6)

• Assume:

1. nucleotides and amino acids areavailable at constant concentrations

2. cccDNA catalyzes reactions (1) and (3)


Hepatitis B Infection

• Reduced state

· x =

cccDNArcDNA

envelope

=ABC

· xo =

100

· After ∼ 200 days:

xss =

20200

10000

• System reaction channels H:

H(x) =

k1Ak2Bk3Ak4Ak5Ck6BC

,

k1

k2

k3

k4

k5

k6

=

10.02510000.25

1.99857.5E-6

day−1

• When A>0 and C>100, reactions(3) and (5) dominate. Partition thesystem accordingly.


cccDNA Comparisons

ApproximateStochastic Results Stochastic-Deterministic Results

Average of 1000 Simulations Average of 1000 Simulations


cccDNA Comparisons

Prob(cccDNA = 0,t) Prob(cccDNA,t = 199 days)

0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Time (Days)

Pro

babili

ty

StochasticApproximate

0 10 20 30 400

0.05

0.1

0.15

0.2

0.25

cccDNA Molecules

Pro

babili

ty

StochasticApproximate


rcDNA Comparisons

Comparison of theComparison of the Mean Standard Deviation

0 50 100 150 2000

50

100

150

200

250

rcD

NA

Mole

cule

s

Time (Days)

StochasticApproximationODE

0 50 100 150 2000

20

40

60

80

100

120

rcD

NA

Mole

cule

s

Time (Days)

StochasticApproximation

An ODE model gets the mean wrong!


Envelope Comparisons

Comparison of theComparison of the Mean Standard Deviation

0 50 100 150 2000

2000

4000

6000

8000

10000

12000

14000

Time (Days)

Envelo

pe M

ole

cule

s

StochasticApproximationODE

0 50 100 150 2000

1000

2000

3000

4000

5000

6000

Envelo

pe M

ole

cule

s

Time (Days)

StochasticApproximation


Why the deterministic solution is wrong...

Typical Infection Aborted Infection

0.1

1

10

100

1000

10000

0 50 100 150 200

Num

ber

of M

ole

cule

s

Time (Days)

cccDNArcDNA

Envelope

0.1

1

10

100

1000

0 1 2 3 4 5

Num

ber

of M

ole

cule

s

Time (Days)

cccDNArcDNA

Envelope

The deterministic model cannot express both cases.


Particle Engineering

• Size-independent nucleation and growth

– Empirical deterministic formulation– Stochastic formulation– Examples

• Size-independentnucleation, growth and agglomeration


more

more

more

more

more

more

more


Empirical Deterministic Formulation

Population Balance

∂f(L, t)∂t

= −G∂f(L, t)∂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

dCdt

= −3ρckvhG∫∞

0fL2dL ρVCp

dTdt

= −3∆HcρckvVG∫∞

0fL2dL−UA(T − Tj(t))

Nucleation and growth in the bulk

B = kb(C − Csat(T)Csat(T)

)b= kbSb G = kg

(C − Csat(T)Csat(T)

)g= kgSg


Stochastic Formulation

Reaction Mechanism for Nucleation and Growth

2M kn-→ Nl1 ∆Hnrxn

Nln +Mkg-→ Nln+1 ∆Hg

rxn

Nln is the number of crystals of size ln per volume.

Mass Balance Energy Balance

Mtot = Msat +M dTdt =

UAρCpV (Tj − T)

(between reaction events)

Solubility Relationship

log10Msat = a log10 T +bT+ c


Isothermal, Size-Independent Nucleation and Growth

Stochastic Solution Deterministic SolutionAverage of 100 Simulations Via Orthogonal Collocation

Discrete particle sizes Continuous particle sizesInteger-valued particle accounting Real-valued particle accounting


Nonisothermal, Size-Independent Nucleation and Growth

Stochastic Solution Deterministic SolutionAverage of 500 Simulations Via Orthogonal Collocation


Isothermal, Size-Independent Nucleation, Growth, andAgglomeration

Stochastic Solution Deterministic Solution

Add one reaction: Nlp +Nlqka-→ Nlp+q Via Adaptive Mesh Methods?

Average of 500 Simulations Large time investment!


Conclusions

Stochastic models are important when:

1. 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:

1. New modeling approximation for handling reactions occurring overdifferent time scales

2. Application of modeling techniques


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:1. State estimation for fluctuating systems2. Chemical reaction models

– Other approaches for computing conditional densities of nonlineardynamical systems


Documents

Stochastic Modeling of Chemical Reactionstwccc.che.wisc.edu/new/haseltine.pdf · 2001-12-17 · Stochastic Modeling of Chemical Reactions Eric Haseltine Department of Chemical Engineering