Extending the Tools of Chemical Reaction Engineering to ...jbr · Extending the Tools of Chemical Reaction Engineering to the Molecular Scale Multiple-time-scale order reduction for

Extending the Tools of Chemical Reaction Engineeringto the Molecular Scale

Multiple-time-scale order reduction for stochastic kinetics

James B. Rawlings

Department of Chemical and Biological Engineering

March 31, 2009Model Reduction in Reacting Flows

University of Notre Dame

Rawlings Molecular reaction engineering 1 / 35

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


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 modelsmust meet the following requirements:

Simpler than the full model (fewer reactions, fewer parameters, orfaster simulation times)

Converge to the full model as a specified parameter goes to zero


Stochastic simulation method — kinetic Monte Carlo

Ak1−−k2

Bk1 = 2 k2 = 1nA0 = 6 nB0 = 3

0

1

2

3

4

5

6

7

8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8time (sec)

r2

r1nA

KMC Algorithm

1 Choose which reaction

2 Choose time step

3 Repeat

Which reaction:

Random number0 1r2

r1+r2= 3

12+3r1

r1+r2= 12

12+3

Time step: Sample from an exponential distribution where thedistribution mean is the sum of reaction rates.


KMC simulations and probability

0

20

40

60

80

100

0 0.5 1 1.5 2 2.5 3

n i

time (sec)

Multiple KMC simulations, Ak1−−k2

B

nA

nB

0

100

200

300

400

500

20 30 40 50 60

Count

nA

Histogram at t=2sec, for 5000 simulations

KMC simulations are samples of a probability distribution that evolvesin time.

We can write the evolution equation for the probability density(master equation).


Chemical master equation

dP(x)

dt=

Nrxn∑j=1

rj (x − νj )P(x − νj )︸︷︷︸rate into state x

− rj (x)P(x)︸︷︷︸rate out of state x

dP

dt= AP

Master equation example

Ak1−−k2

B

nA0 = 100, nB0 = 0

k1 = 2, k2 = 1

101 possible states

101 Coupled ODEs

P(nA)

0

25

50

75

100

nA

0 1 2 3 4Time (sec)

0

0.05

0.1

0.15


Master equation — Important points

Chemical master equation

dP(x)

dt=

Nrxn∑j=1

rj (x − νj )P(x − νj )︸︷︷︸rate into state x

− rj (x)P(x)︸︷︷︸rate out of state x

dP

dt= AP

Often the dimensionality of the master equation makes direct solutioninfeasible

The master equation shows what probability distribution is sampled ina KMC simulation

A reduced master equation can lead to a new/faster simulationschemes


Kinetics of multiple time scales

Ak1−−

k−1

Bk2−→ C

0

20

40

60

80

100

0 1 2 3 4 5time (sec)

Deterministic - One Time Scale

nAnB

nC

k1 = 2 k−1 = 0.5 k2 = 0.5

0

20

40

60

80

100

0 1 2 3 4 5time (sec)

KMC - One Time Scale

nA

nB nC

k1 = 2 k−1 = 0.5 k2 = 0.5

One time scale

0

20

40

60

80

100

0 1 2 3 4 5time (sec)

Deterministic - Two Time Scales

nA

nB

nC

k1 = 10 k−1 = 10 k2 = 0.5

0

20

40

60

80

100

0 1 2 3 4 5time (sec)

KMC - Two Time Scales

nA

nB

nC

k1 = 10 k−1 = 10 k2 = 0.5

Reaction equilibrium

0

20

40

60

80

100

0 1 2 3 4 5time (sec)

Deterministic - Two Time Scales

nA

nB

nC

k1 = 2 k−1 = 20 k2 = 20

0

20

40

60

80

100

0 1 2 3 4 5time (sec)

KMC - Two Time Scales

nA

nB

nC

k1 = 2 k−1 = 20 k2 = 20

Reactive intermediateRawlings Molecular reaction engineering 8 / 35

Deterministic model reductions

x non-QSSA species, y QSSA species

dx

dt= f (x , y) ε

dy

dt= g(x , y)

Classical QSSA

dx

dt= f (x , y)

0 = g(x , y)

DAE reduced model

Singular Perturbation QSSA

x = X0 + εX1 + ε2X2 +O(ε3)

y = Y0 + εY1 + ε2Y2 +O(ε3)

Collect like powers of ε

Equations for dX0dt is the

reduced model

Separate models for fast andslow time scale


SPA on the master equation

Our objective

Apply singular perturbation analysis to develop a reduced master equation.

Ak1−−

k−1

Bk2−→ C

dP(a, b, c)

dt= k1(a + 1)P(a + 1, b − 1, c) + k−1(b + 1)P(a− 1, b + 1, c)

+ k2(b + 1)P(a, b + 1, c − 1)− (k1a + k−1b + k2b)P(a, b, c)

P(a, b, c) = W0(a, b, c) + εW1(a, b, c) + · · ·

ε0 terms:

W0(a, b, c) = 0 if b > 0

In this limit b is always zero


SPA on the master equation

ε1 terms: Reduced master equation

dW0(a, 0, c)

dt= k(a + 1)W0(a + 1, 0, c − 1)− kaW0(a, 0, c)

Reduced mechanism

A −→ C r = k1k2k−1+k2

a

Stochastic same as deterministic SPA mechanism

Same mechanisms due to linearity

First-order correction, 〈b〉〈b〉 = f (W0(a, 0, c)) +O(ε2)〈b〉 = k1

k−1+k2〈a〉


Comparison of mechanisms

A −− 2B r1 = k1a r−1 = k−1

2 b(b − 1)

B −→ C r2 = k2b

Stoch SPA

A −→ 2C

r =

(k1k2

k−12

+k2

)a

Det SPA

A −→ 2C

r = k1a

Det QSSA

A −→ 2C

r = k2

−k2 +√

k22 + 8k1k−1a

4k−1

0

5

10

15

20

25

30

35

40

45

0 0.5 1 1.5 2Time (sec)

〈c〉

nA0 = 25, nB0 = 0, nC0 = 0

k1 = 1

k−1 = 1000

k2 = 1000

Full ModelStoch SPADet QSSA

Det SPA


Catalyst Example

Ak1−→ B

k2−→ C

B + Dk3−→ B + E

k2, k3 k1

0

2

4

6

8

10

12

14

16

0 0.05 0.1 0.15 0.2 0.25 0.3

Num

ber

ofmolecules

(n)

Time (sec)

b

a

e

Stoch SPA mechanism

A −→ C

D + A −→ E + C

2D + A −→ 2E + C· · ·

nD + A −→ nE + C

r0 =k1a

1 + K3dK3 =

k3

k2

r1 = r0K3d

1 + K3(d − 1)

r2 = r1K3(d − 1)

1 + K3(d − 2)

rn = rn−1K3(d + 1 − n)

1 + K3(d − n)

Deterministic SPA mechanism

A −→ C

D + A −→ E + A

r0 = k1a

r1 =k3k1

k2ad


Conclusions — Stochastic quasi-steady-stateapproximation

QSSA species are removed from stochastic models with SPA

Stochastic QSSA mechanisms different than deterministic QSSAmechanisms

Application of stochastic QSSA:I Reduces the number of kinetic parametersI Speeds up KMC simulations (fewer events)


Conclusions — Stochastic quasi-steady-stateapproximation

sQSPA

sQSPA

dQCdQSPA

<1

1103

>>103

1 103 >>103

nonQSSA species (# of molecules)

QS

SA

spe

cies

(# o

f mol

ecul

es)


Catalytic surface reaction modeling

Assumptions for this talk

Two dimensional surface witha lattice for adsorption,diffusion, reaction, anddesorption.

Square lattice, Z=4

All sites have identicalproperties

Constant temperature

Adsorbed CO moleculesexhibit nearest neighborrepulsions

CO + 12 O2 −→ CO2

CO-black, O-gray, Empty-white.


Model mechanism and time scales

Adsorption CO(g)+ ∗iα−→ COi

O2(g)+ ∗i + ∗jβ−→ Oi + Oj

Desorption COiγ−→ CO(g) + ∗i

Oi + Ojρ−→ O2(g) + ∗i + ∗j

Reaction COi + Ojkr−→ CO2(g)+ ∗i + ∗j

Diffusion COi + ∗jd1−→ ∗i +COj

Oi + ∗jd2−→ ∗i + Oj

1/sec

α = 1.6

β = 0.8

γ = 0.8

ρ = 0.001

kr = 1

d1 ≈ 1010

d2 ≈ 108


Singular perturbation on the master equation

Surface reaction master equation

x - microscopic configuration n - number of each species

dP(n, x)

dt=

Xrxn∑j=1

kjaj (n − νj , x − νx,j )P(n − νj , x − νx,j )− kjaj (n, x)P(n, x)

+

Xdiff∑j=1

djaj (n, x − νx,j )P(n, x − νx,j )− djaj (n, x)P(n, x)

Singular perturbation

P(n, x) = W0(n, x) + εW1(n, x) + ε2W2(n, x) + · · ·ε = 1/d

ε0 terms: Diffusion equilibration equations for W0(x |n)


Slow time-scale evolution equation

ε1terms : Reduced master equation

dW0(n)

dt=

Nrxn∑i=1

ki 〈si (n − νi )〉W0(n − νi )− ki 〈si (n)〉W0(n)

What have we gained?

Removed micro-states from the master equation

Lattice Size Species Micro-states Coverage states

Ns = 4 1 16 5

Ns = 25 2 1012 325

Ns = 100 2 1048 5050

Tractable number of states, master equation can be solved


Slow time-scale evolution equation

dW0(n)

dt=

Nrxn∑i=1

ki 〈si (n − νi )〉W0(n − νi )− ki 〈si (n)〉W0(n)

Reaction propensities

si (x) number of reaction i on configuration x : nCO=45 black, nO=8 gray

sCO−O=26 sCO−O=22 sCO−O=26 sCO−O=23〈sCO−O〉 = 24.9

〈si (n)〉 =∑

x si (x)W0(x |n) – Calculate with diffusion only KMC


Reduced master equation solution (5x5 lattice)

Probability of nCO, 5x5 lattice

0 5 10 15 20 25nCO 02

46

810

Time (sec)

0

0.1

0.2

0.3

0.4

0

0.05

0.1

0.15

0.2

0.25

0 5 10 15 20 25P

roba

bilit

y

nCO

Steady-state probability distribution (5x5)


Verification of perturbation method∑x

P(n, x) = W0(n) + εW1(n) +O(ε2)

ε = 1/d

As the diffusion rate increases P(n) approaches W0(n)

0

0.05

0.1

0.15

0.2

0 5 10 15 20 25

Pro

babi

lity

nCO

Steady-state probability distribution (5× 5)

Reduced MEd=1000d=100d=10d=1

d=0.1d=0.01

d=0.0010.1

1

10100

1000


Conclusions — Surface reactions in the infinite diffusionlimit

SPA can be used to eliminate spatial configuration states in a reducedmaster equation.

The reduced master equation has sufficiently few states to besimulated on small lattices.

Reduced master equations of surface reactions can be used tomotivate reduced KMC and reduced ODE models.


Model for Vesicular Stomatitis Virus (VSV) infection

!

!

"#$%&#! "$%& "$%& "$%& "#$%& !

'()* '()* '()* '()* '()* '()*

I is encapsidation of viralgenome

II is replication of encapsidatedgenome

III is transcription of genome tomessenger RNA


Onset of fast fluctuations in the N protein

0 0.5 1 1.5 2 2.5 3

100

102

104

Time (hours)

Mol

ecul

e le

vel

(−)RNA genomes(+)RNA genomes

(−)RNA1258

N protein

(+)RNA1258

N mRNA

II

III

I

Features of simulation

Presence of fastfluctuating and rapidlyrising species

Fast fluctuations slowthe full KMC simulation

Motivates theformulation of a simplerexample to understandthis phenomenon


Fast fluctuation and rapid rise in VSV biology

(−)RNA + L1k1−→ (−)RNA + L2

(−)RNA + L2k2−→ 2(−)RNA + L1

2(−)RNAk3−→ (−)RNA

The viral genome is amplified by first two reactions

The free viral proteins and messages are not amplified

Values of parameters k1, k2 and k3 may cause fast fluctuation inpolymerases along with rapid amplification of viral genome


Fast fluctuation and rapid rise — Idealized problem

A + Gk1−→ C + G r1 =

1

Ωk1ag

C + Gk2−→ 2G + A r2 =

1

Ωk2cg

2Gk3−→ G r3 =

1

Ωk3

g(g − 1)

2

Species Initial number Rate constant (m3/mol·s)

A 3 k1 = 9× 105

C 0 k2 = 5× 105

G 1 k3 = 5× 10−2


The full SSA on the system

0

1

2

3

0 0.4 0.8 1.2 1.6 2 2.4

No.

ofC

mol

ecul

es

t (sec)

100

101

102

103

104

105

106

107

108

0 0.5 1 1.5 2 2.5

No.

ofG

mol

ecul

es

t (sec)

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

tim

e-st

ep(s

ec)

t (sec)


The hybrid SSA - Ω technique

At large population of G we want to switch to a continuous description forit:

g = ΩφG + Ω1/2ξ

φG is the deterministic evolution term and ξ is the continuous noisein the evolution of G

We can obtain approximation for the evolution of system using hybridSSA - Ω technique

Approximation of pdf of C

W0(c) = (1 + q)−N0

(N0

c

)q(N0−c)

Deterministic evolution of G

dφG

dt= γ−1〈c〉φG −

k3

2φ2

G

N0 Initial number of polymerases

q = k2k1

Ratio of rate constants


Comparison of full SSA with hybrid SSA - Ω

Full SSA

0

1

2

3

0 0.4 0.8 1.2 1.6 2 2.4

No.

ofC

mol

ecul

es

t (sec)

100

101

102

103

104

105

106

107

108

0 0.5 1 1.5 2 2.5

No.

ofG

mol

ecul

es

t (sec)

Hybrid SSA - Ω

0

1

2

3

0 0.4 0.8 1.2 1.6 2 2.4

No.

ofC

mol

ecul

es

t (sec)

100

101

102

103

104

105

106

107

108

0.5 1 1.5 2 2.5

No.

ofG

mol

ecul

es

t (sec)


Comparison of full SSA with hybrid SSA - Ω

Probability densities of C from SSA and from hybrid SSA - Ω

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 1 2 3

Pro

babi

lity

No. of C particles

Hybrid SSA ΩSSA


Conclusions — QSSA and fast fluctuations

Hybrid SSA – Ω expansion matches closely the full SSA

Computation speed increases by factor of 450

Application to kinetic virus infection models


Acknowledgments

Dr. Ethan A. Mastny, BP Alaska

Dr. Eric L. Haseltine, Vertex Pharmaceuticals

Rishi Srivastava, UW

NSF #CNS–0540147


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):6959–6969, October 2002.

Haseltine, E. L. and J. B. Rawlings.On the origins of approximations for stochastic chemical kinetics.J. Chem. Phys., 123:164115, October 2005.

Cao, Y., D. T. Gillespie, and L. R. Petzold.The slow-scale stochastic simulation algorithm.J. Chem. Phys., 122(1):014116, January 2005.

Samant, A. and D. G. Vlachos.Overcoming stiffness in stochastic simulation stemming from partial equilibrium: Amultiscale Monte Carlo algorithm.J. Chem. Phys., 123:144114, 2005.

Salis, H. and Y. Kaznessis.Accurate hybrid stochastic simulation of a system of coupled chemical or biochemicalreactions.J. Chem. Phys., 122(5):054103, February 2005.

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 2006.


Further reading — Stochastic QSSA

Rao, C. V. and A. P. Arkin.Stochastic chemical kinetics and the quasi-steady-state assumption: Application to theGillespie algorithm.J. Chem. Phys., 118(11):4999–5010, March 2003.

van Kampen, N. G.Stochastic Processes in Physics and Chemistry.Elsevier Science Publishers, Amsterdam, The Netherlands, second edition, 1992.

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):094106, September 2007.

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 2008.


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Extending the Tools of Chemical Reaction Engineering to ...jbr · Extending the Tools of Chemical Reaction Engineering to the Molecular Scale Multiple-time-scale order reduction for