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STOCHASTIC CHEMICAL KINETICS
Dan Gillespie
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′+ → .
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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.
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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.
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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
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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
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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
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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
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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.