stochastic chemical kineticsendy/scraps/dg/seminar.gillespie.28... · 2004. 5. 26. · 1 stochastic...

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STOCHASTIC CHEMICAL KINETICS

Dan Gillespie

GillespieDT@mailaps.org

Current Support: Caltech (DARPA / AFOSR) The Molecular Sciences Institute (Sandia / DOE)

Past Support: ONR

Collaborators: Linda Petzold, Yang Cao (UCSB) John Doyle (Caltech) Muruhan Rathinam (UMBC) Larry Lok (MSI)

A Chemically Reacting System

• Molecules of N chemical species 1, , NS S .

- In a volume Ω , at temperature T .

- Different conformations or excitation levels are considered different species if they behave differently.

• M “elemental” reaction channels 1, , MR R .

- each jR describes a single instantaneous physical event, which

changes the population of at least one species. Thus, jR is either

iS∅ → ,

or something elseiS → ,

or something elsei iS S′+ → .

2

Question: How does this system evolve in time?

The traditional answer, for spatially homogeneous systems:

“According to the reaction rate equation (RRE).”

• A set of coupled, first-order ODEs.

• Derived using ad hoc, phenomenological reasoning.

• Implies the system evolves continuously and deterministically.

• Empirically accurate for large systems.

• Often not adequate for small systems.

* * *

The question deserves a more carefully considered answer.

Molecular Dynamics (MD)

• The most exact way of describing the system’s evolution.

• Tracks the position and velocity of every molecule in the system.

• Simulates every collision, non-reactive as well as reactive.

• Shows changes in species populations and their spatial distributions.

• But . . . it’s unfeasibly slow for nearly all realistic systems.

3

A great simplification occurs if successive reactive collisions tend to be separated in time by very many non-reactive collisions.

• The overall effect of the non-reactive collisions is to randomize the positions of the molecules (and also maintain thermal equilibrium).

• The non-reactive collisions merely serve to keep the system well-stir red or spatially homogeneous for the reactive collisions.

• Can describe the state of the system by ( )1( ) ( ), , ( )Nt X t X tX ,

( )iX t the number of iS molecules at time t .

But this well-stirred simplification, which . . .

• ignores the non-reactive collisions, • truncates the definition of the system’s state,

. . . comes at a price:

( )tX must be viewed as a stochastic process.

In fact, the system was never deterministic to begin with! Even if molecules moved according to classical mechanics . . . - monomolecular reactions always involve QM. - bimolecular reactions require collisions, whose extreme sensitivity to initial conditions renders them essentially random. - bimolecular reactions usually involve QM too.

But stochastic processes can be handled.

4

For well-stir red systems, each jR is completely characterized by:

• a propensity function ( )ja x : Given the system in state x ,

( )ja dtx the probability that one jR event will occur in the next dt .

- The existence and form of ( )ja x follow from kinetic theory.

- ( )ja x is roughly equal to, but is not derived from, the RRE “rate”.

• a state change vector ( )1 , ,j j N jvν≡ : i jν the change in the

number of iS molecules caused by one jR event.

- jR induces j→ +x x . i jν ≡ the “stoichiometric matrix.”

E.g.,

1

21 2 12

c

cS S S+

1 1 1 2 1

1 12 2 2

( ) , ( 1, 1,0, ,0)

( 1)( ) , ( 1, 1,0, ,0)

2

a c x x

x xa c

= = + − −= = − +

x

x

5

Two exact, r igorously der ivable consequences . . .

1. The chemical master equation (CME):

0 00 0 0 0

1

( , | , )( ) ( , | , ) ( ) ( , | , )

M

j j j jj

P t ta P t t a P t t

t =

∂ = − − − ∂

x xx x x x x x .

• Gives 0 0 0 0( , | , ) Prob ( ) , given ( )P t t t t= =x x X x X x for 0t t≥ .

• The CME follows from the probability statement

( )

( )

0 0 0 01

0 01

( , | , ) ( , | , ) 1 ( )

( , | , ) ( ) .

M

jj

M

j j jj

P t dt t P t t a dt

P t t a dt

=

=

+ = × −

+ − × −

x x x x x

x x x

• But it’s practically always intractable (analytically and numerically).

• With ( ) 0 0( ) ( ) ( , | , )f t f P t tx

X x x x , can show from CME that

( )1

( )( )

M

j jj

d ta t

dt ==

XX .

• I f there were no fluctuations, we would have

( ) ( ) ( )( ) ( ) ( )f t f t f t= =X X X ,

and the above would reduce to the reaction-rate equation (RRE):

( )1

( )( )

M

j jj

d ta t

dt ==

XX .

(Usually written in terms of the concentration ( ) ( )t t ΩZ X .)

But as yet, we have no justification for ignoring fluctuations.

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2. The stochastic simulation algorithm (SSA):

• A procedure for constructing sample paths or realizations of ( )tX .

• Approach: Generate the time to the next reaction and the index of that reaction.

• Theoretical justification: With ( , | , )p j tτ x defined by

( , | , )p j t dτ τx prob, given ( )t =X x , that the next reaction will

occur in [ , )t t dτ τ τ+ + + , and will be an jR ,

can prove that

( )0 01

( , | , ) ( ) exp ( ) , where ( ) ( )M

j jj

p j t a a a aτ τ ′′=

= − x x x x x .

This implies that the time τ to the next reaction event is an exponential random variable with mean 01 ( )a x , and the index j of

that reaction is an integer random variable with prob 0( ) ( )ja ax x .

The “ Direct” Version of the SSA

1. With the system in state x at time t, evaluate 01

( ) ( )M

jj

a a ′′=x x .

2. Draw two unit-interval uniform random numbers 1r and 2r , and

compute τ and j according to

• 0 1

1 1ln

( )a rτ

=

x,

• j = the smallest integer satisfying 2 01

( ) ( )j

jj

a r a′′=

> x x .

3. Replace t t τ← + and j← +x x .

4. Record ( , )tx . Return to Step 1, or else end the simulation.

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A Simple Example: 11 0cS → .

1 1 1 1 1( ) , 1a x c x ν= = − . Take 01 1(0)X x= .

RRE: 11 1

( )( )

dX tc X t

dt= − . Solution is 10

1 1( ) e c tX t x −= .

CME: 0

0 01 11 1 1 1 1 1 1

( , | ,0)( 1) ( 1, | ,0) ( , | ,0)

P x t xc x P x t x x P x t x

t

∂ = + + − ∂

.

Solution: ( )01 1

1 1 1

00 01

1 1 1 101 1 1

!( , | ,0) e 1 e ( 0,1, , )

!( )!

x xc x t c txP x t x x x

x x x

−− −= − =−

which implies 101 1( ) e c tX t x −= , ( )1 10

1 1sdev ( ) e 1 ec t c tX t x − −= − .

SSA: Given 1 1( )X t x= , generate 1 1

1 1ln

c x rτ =

, then update:

1 1, 1t t x xτ← + ← − .

0 1 2 3 4 5 60

20

40

60

80

100

S1 → 0

c1 = 1, X1(0) = 100

8

0 1 2 3 4 5 60

20

40

60

80

100

S1 → 0

c1 = 1, X1(0) = 100

0 1 2 3 4 5 60

20

40

60

80

100

S1 → 0

c1 = 1, X1(0) = 100

9

0 1 2 3 4 5 60

20

40

60

80

100

S1 → 0

c1 = 1, X1(0) = 100

The SSA . . .

• Is exact.

• Is equivalent to (but is not derived from) the CME.

• Does not entail approximating “dt” by “ t∆ ” .

• Is procedurally simple, even when the CME is intractable. • Has been redesigned to be faster and more efficient for very large

systems (though more complicated to code) by Gibson and Bruck. • Remains too slow for most practical problems: Simulating every

reaction event, one at a time, is just too much work if any reactant is present in very large numbers.

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We would be willing to sacr ifice a little exactness . . . . . . if that would buy us a faster simulation.

One way of doing this: Tau-Leaping

• Approximately advances the process by a pre-selected time τ , which may encompass more than one reaction event.

• The size of τ is limited by the Leap Condition: The changes induced in the propensity functions during τ must be “ small” .

• If τ can also be taken large enough to encompass many reaction events, tau-leaping will be faster than the SSA.

• Real speed-ups of 100× obtained for some simple model systems.

• But must use with care. Not as automatic and foolproof as the SSA.

Basics of Tau-Leaping

• Some math: The Poisson random variable ( , )a τ the number of events that will occur in time τ , given that the probability of an event occurring in any dt is adt .

• So, with ( )t =X x , if τ is such that ( )ja x ≈ constant in [ , ]t t τ+ ,

then the number of jR reactions that will occur in [ , ]t t τ+ is

approximately ( )( ),ja τx ; hence,

( )1

( ) ( ),M

j j jj

t aτ τ=

+ +X x x .

• Often feasible because . . . - Reliable procedures exist for generating samples of ( , )a τ .

- A way has been found to estimate in advance the largest τ for a specified degree of adherence to the Leap Condition.

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The Tau-Leap Simulation Algorithm

• 1. In state x at time t , choose τ so that the expected change in every propensity function in [ , ]t t τ+ is 0( )aε≤ x . (This can be done.)

• 2. Generate the number of firings jk of channel jR in [ , ]t t τ+ as

( )( ), ( 1, , )j jk a j Mτ= =x .

• 3. Leap: Replace t t τ← + and 1

M

j jj

k=

← +x x .

• 4. Record ( , )tx . Then return to Step 1, or else end the simulation.

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Time

mo

no

mer

X1,

un

stab

le d

imer

X2,

sta

ble

dim

er X

3

X2

X3

X1

Decaying-Dimerizing Reaction Set S1--> 0 c1 = 1 S1 + S1 --> S2 c2 = 0.002 S2 --> S1 + S1 c3 = 0.5 S2 --> S3 c4 = 0.04 - Exact SSA Run - 500 steps (reactions) per plotted dot. - Intially: X1=100,000; X2=X3=0. - Last reaction at T=48.07. - Final X(3)=17,106. - 524,112 steps (reactions) total.

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0

5000

10000

15000

20000

25000

30000

35000

40000

45000

50000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Time

mo

no

mer

X1,

un

stab

le d

imer

X2,

sta

ble

dim

er X

3

X2

X3

X1

Decaying-Dimerizing Reaction Set S1--> 0 c1 = 1 S1 + S1 --> S2 c2 = 0.002 S2 --> S1 + S1 c3 = 0.5 S2 --> S3 c4 = 0.04 - Explicit Tau Leaping Run (Epsilon=0.03) - 1 leap per plotted dot. - Intially: X1=100,000; X2=X3=0. - Last reaction at T=44.52. - Final X(3)=17,033. - 592 leaps total.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 5 10 15 20 25 30

Time

AJ

/ A0

A1/A0

A2/A0

A3/A0

A4/A0

Decaying-Dimerizing Reaction Set

R1: S1 ---> 0 A1 = C1*X1 R2: S1 + S1 ---> S2 A2 = C2*X1*(X1-1)/2 R3: S2 ---> S1 + S1 A3 = C3*X2 R4: S2 ---> S3 A4 = C4*X2

Plots of AJ/A0, where A0=A1+A2+A3+A4. Exact SSA run. 500 reactions per plotted point.

A2/A0

A3/A0

13

1800 2100 2400 2700 3000

X1(t=10)

0.000

0.002

0.004

0.006

0.008

rela

tive

freq

uenc

y

3 x 10,000 Simulation Runs

t=0: X1=4150, X2=39565, X3=3445

Gaussian-windowed histograms ofX1 at t=10

Exact SSA (32 hours).

Tau-leaping, =0.02, (11 minutes).

Tau-leaping, =0.03, (6.5 minutes).

13200 13550 13900 14250 14600

X2(t=10)

0.000

0.001

0.002

0.003

0.004

0.005

rela

tive

freq

uenc

y

3 x 10,000 Simulation Runs

t=0: X1=4150, X2=39565, X3=3445

Gaussian-windowed histograms ofX2 at t=10

Exact SSA

Tau-leaping ( =0.02)

Tau-leaping ( =0.03)

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13100 13375 13650 13925 14200

X3(t=10)

0.000

0.002

0.004

0.006

rela

tive

freq

uenc

y

3 x 10,000 Simulation Runs

t=0: X1=4150, X2=39565, X3=3445

Gaussian-windowed histograms ofX3 at t=10

Exact SSA

Tau-leaping ( =0.02)

Tau-leaping ( =0.03)

We can push Tau-Leaping fur ther . . .

• Some more math: ( , )a τ has mean and variance aτ . And when

1aτ , can approximate ( , ) ( , ) (0,1)a a a a aτ τ τ τ τ≡ + .

• So, with ( )t =X x , suppose we can choose τ to satisfy the Leap

Condition, and also the conditions ( ) 1,ja jτ ∀x . Then

( )1

( ) ( ),M

j j jj

t aτ τ=

+ +X x x

can be further approximated to

( ) ( )1 1

( ) (0,1)M M

j j j j jj j

t a aτ τ τ= =

+ + + X x x x .

• Valid iff τ is chosen small enough to satisfy the Leap Condition, yet large enough that every jR fires many more times than once in τ .

• It’s not always possible to find such a τ ! But it usually is if all the reactant populations are “sufficiently large”.

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( ) ( )1 1

( ) (0,1)M M

j j j j jj j

t a aτ τ τ= =

+ + + X x x x

• Is called the Langevin leaping formula. • It’s faster than ordinary tau-leaping. • It directly implies (and is entirely equivalent too) a SDE called the chemical Langevin equation (CLE):

( ) ( )1 1

( )( ) ( ) ( )

M M

j j j j jj j

d ta t a t t

dtΓ

= =+

XX X .

jΓ ’ s are “Gaussian white noises” , ( ) ( ) ( )j j j jt t t tΓ Γ δ δ′ ′′ ′= − .

• Our discrete stochastic process ( )tX has now been approximated as a continuous stochastic process.

• Again, the CLE can’ t always be invoked. But it usually can if all the reactant molecular populations are “ sufficiently large” .

The Thermodynamic Limit

Def: All iX → ∞ , and Ω → ∞ , with iX Ω constant.

• Can prove that, in this limit, all propensity functions grow linearly with the system size.

• So as we approach this limit, in the CLE

( ) ( )1 1

( )( ) ( ) ( )

M M

j j j j jj j

d ta t a t t

dtΓ

= =+

XX X ,

the deterministic term grows like (system size), while the

stochastic term grows like system size .

• This establishes the well know rule-of-thumb: Relative fluctuations scale as the inverse square root of the system size.

• Very near the thermodynamic limit, the CLE approximates to

( )1

( )( )

M

j jj

d ta t

dt =

XX ,

the reaction rate equation (RRE)! ( )tX has now become a continuous deterministic process. And we’ve derived the RRE!

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Summarizing . . .

Star t with Molecular Dynamics • The Gold Standard. • “State” = positions and velocities of all the molecules. • Simulates all collisions, non-reactive as well as reactive. • But is way too slow.

The “ well-stir red” simplification gives the CME/SSA • Assumes the non-reactive collisions occur frequently enough to keep

the system well-stirred (spatially homogeneous) for the reactive collisions.

• Simulates only the reactive collisions. • “State” = the molecular populations of all the species, ( )tX .

• ( )tX is a discrete stochastic process.

The Spectrum of Analytical Approaches for Well-Stirred Systems

,( ) const ( ) 1

constin [ , ],

ij j

i i

Xa a

Xt t j j

ΩτΩτ

→∞ →∞≈→+ ∀ ∀

x x

- → → →CME/SSA Tau Leaping CLE RRE

approximate approximate approximate

continuou

exact

discrete discrete

stochastic stochastic stochastic

s continuous

determinis

tic

Is it simply a matter of picking the “ right tool” from this spectrum?

Not quite.

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Complications from “ Multiscale”

• Some jR occur very much more frequently than others.

• Some iX are very much larger than others.

• Fast & slow & small & large usually all occur coupled – not easy to separate.

• Leads to dynamical stiffness, which usually restricts τ to the smallest time scale in the system.

• One recently proposed fix: Implicit tau-leaping – a stochastic adaptation of the implicit numerical solution method for stiff ODEs.

• Other proposed fixes: stochastic versions of the quasi-steady state and partial equilibrium approximation methods of deterministic chemical kinetics.

Spatially Inhomogeneous Systems

• Spatial homogeneity does not require that all equal-size subvolumes of Ω contain the same number of molecules!

• The CME and SSA require only that the center of a randomly chosen

iS molecule be found with equal probability at any point inside Ω .

• A system consisting of only one molecule can be “well-stirred”.

But the well-stir red assumption can’ t always be made.

In that case, we must do something different; however, the traditional reaction-diffusion equation (RDE) is not always the answer:

• The RDE (like the RRE) is continuous and deterministic. • It assumes that each dΩ contains a spatially homogeneous mixture of

infinitely many molecules. • Not the case in most cellular systems, where spatial inhomogeneity

arises not from slow mixing but rather from compartmentalization caused by highly heterogeneous structures within the cell.

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We can formulate a spatially inhomogeneous CME/SSA: Subdivide Ω into K spatially homogeneous subvolumes kΩ . Mathematically is

the same as in the homogeneous case, except now we have

• KN species i kS , and KM chemical reactions j kR ,

• plus a whole bunch of diffusive transfer reactions diffi kkR ′ .

Challenges:

• Enormous increases in the numbers of species and reactions, so simulations will run orders of magnitude slower.

• No unambiguous definition of spatial homogeneity, hence no clear criterion for making the partitioning kΩ Ω→ .

• Dynamically altering the kΩ will probably be necessary, and will

require bookkeeping that is complicated and time-consuming. • Conflicting ways to compute the diffusive transfer rate constants.