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Computational Science and Engineering
Stiffness
Yang Cao
Department of Computer Science
http://courses.cs.vt.edu/~cs6404
Computational Science and Engineering
Summary
• What is stiffness
• Stiffness in deterministic regime
– Test Problem
– Explicit Method and Implicit Method
• Stiffness in Stochastic regime
– Test Problem
– Explicit tau-leaping and Implicit tau-leaping
– Damping Effect
– Trapezoidal tau-leaping and Down-Shifting Method
Computational Science and Engineering
Stiffness
Stiffness presents in most multiscale systems, particularly, when the system has the following feature.
• Exhibit slow and fast time scales.
• The fast scales are stable.
• Fast reactions almost cancel each other while slow reactions determine the trend.
• In both the deterministic and stochastic regimes. stiffness is a major reason that makes a simulation unnecessarily slow.
Computational Science and Engineering
Stiffness in Deterministic Simulation
32
112
211
1
4
3
2
1
SS
SSS
SSS
S
c
c
c
c
→
+→
→+
∅→
=
−−=
+−−=
243
2423
2
122
23
2
12111
2
2
xcx
xcxcxx
xcxcxcx
c
&
&
&
1.0
1000
10
1
4
3
2
1
=
=
=
=
c
c
c
c
Solved by ODE45 and Ploted at every 100 steps (total 50353 steps)
Solved by ODE15s Plot at every step (32 steps in total)
0)0(
798)0(
400)0(
3
2
1
=
=
=
x
x
x
Computational Science and Engineering
DimerDecay Reactions
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
A(J
) /
A0
(J
=1 G
rn;
J=
2 B
lu;
J=
3 R
ed
; J
=4 P
ur)
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 - 2000 steps (reactions) per dot - 527,928 steps (reaction events) total
- X(3) ends at 17125
- T ends at 43.86
– Plot of reaction propensities against time using SSA.
– The fast red and blue reactions almost cancel each other.
Stiff example : Decaying-dimerizing reaction set
Computational Science and Engineering
Deterministic Case
For any ODE (ordinary differential equation)
by local linearization, it can be reduced to
Suppose has eigenvalues , there exists
nonsingular matrix such that
Let . We see a process to convert any ODE to a simple linear test problem
We assume the original ODE is stable, thus
),(' txfx =
xx A' =
nλλ ,,1 LAT
},,diag{ATT n1
-1 λλ L=
xy-1T=
xx λ='
( ) 0Re ≤λ
Computational Science and Engineering
Absolute Stability
Think about the case when explicit Euler method applied to the test
problem.
We note that the original system should have the following property
The absolute stability requires that the numerical solution for test problem also satisfy this property. Thus
This leads to
When is large, the stepsize will have to be very small.
( ) nnnn xhxhfxx λ+=+=+ 1)(1'
nn xx ≤+1
11 ≤+ hλ
( )λRe
2
−≤h
)Re( λ
Computational Science and Engineering
Euler Method Applied to Test Problem
xx 100' −=For a test problem
Stepsize = 0.0001, 0.01, 0.018, 0.02 Stepsize = 0.021
Computational Science and Engineering
More General Test Problem
For any nonlinear function , one can always construct the test problem
where . Or even
See numerical results (Forward Euler Method) for
)(tg
)('))((' tgtgxx +−= λ
0)Re( <<λ ))((' tgxx −= λ
Stepsize = 0.001, 0.01, 0.018, 0.02 Stepsize = 0.021
1)0( )),sin((100' =−−= xtxx
Computational Science and Engineering
Stiffness
When the absolute stability requirement is far more strict than the accuracy requirement, it is considered stiff. For example, consider the following system:
−=
−=
22
11
100'
1.0'
xx
xx
This system has two time scale.
The fast variable converges
to its stable state very quickly. The
dynamics of the whole system is
then represented by the slow
variable . To accurately
calculate , the stepsize can be
as large as 0.5 (5% error control).
But the absolute stability for the
whole system requires a stepsizethat is smaller than 0.02.
2x
1x1x
Computational Science and Engineering
Implicit Method
nn xh
xλ−
=+1
11
( ) 0Re ≤λ
Backward (Implicit) Euler method is given by
When applied to the test problem, it leads to
Again, the absolute stability requires
(*)
But if , (*) is satisfied for all
Since the absolute stability does not add extra requirement on the stepsize for implicit method, implicit method is a natural choice for stiff problems.
11
1≤
− hλ
)( 11' ++ += nnn xhfxx
0>h
Computational Science and Engineering
Implicit Euler Method Applied to Test Problem
))sin((100' txx −−=
For test problems
Stepsize = 0.0001, 0.01, 0.02, 0.04, 0.1 Stepsize = 0.001, 0.02, 0.05, 0.1
xx 100' −=
Computational Science and Engineering
Another Point of View: Explicit vs Implicit Method
Explicit Euler Method
Backward Euler Method
)(1' nnn xhfxx +=+
)( 11' ++ += nnn xhfxx
nx )(1' nnn xhfxx +=+ 1+nx
nx 1+nx
feedback
)( 11' ++ += nnn xhfxx
Computational Science and Engineering
Newton Iteration
To solve any nonlinear equation
Newton iteration gives
Implicit Euler method leads to
To solve it, Newton iteration yields
where
0)( =xF
[ ] K,1,0 ),()('1
1 =−=−
+ kxFxFxx kkkk
0)()( =+−= xhfxxxF n
( ))()]([ 1
1 knknkk xhfxxxhJIxx +−−−= −+
0x1x2x
)(xF
Newton Iteration
)()( nn xx
FxJ
∂
∂=
Computational Science and Engineering
Multiscale Challenges for Heat Shock Response
σ DNAKσ
RNAP
RNAP
σ
DnaK FtsH
DNAK
Computational Science and Engineering
Formulate the Multiscale Difficulty in Simple Models
• The multiscale behavior can be modeled in the following simple model:
or a simpler model
• Features– Fast and slow reactions
– Fast reactions usually “less important” than slow ones
– SSA will spend most of its time simulating the fast reactions
541
321
SSS
SSS
→+
←→+
31
21
SS
SS
→
←→
Computational Science and Engineering
Explicit tau leaping applied to stiff problem
( )τν )a(P n1 xxX nn +=+
Explicit Tau-leaping
Test Problem (Reversible Isomerization) not a decaying process
Sample trajectories – stepsizes .005, .009, .02
1001
1
1
=
∅ →
c
Sc
Yes No
10021
21
==
←→
cc
SS
Computational Science and Engineering
Stability Analysis
For reversible reactions
The conservation law is satisfied
Thus the state can be represented by
Tau-leaping method yields
Take the mean value at both sides
Let The stability requires:
which leads to
21 SS ←→
( ) ( ).P)(cP 121 ττ nntnn xcxxxX −−+=+
txXX =+ 21
. , 21 XxXXX t −==
( )[ ] tnn xcXccX ττ 2211 1 ++−=+
.21 cc +=λ
11 ≤− λτ
λτ
2≤
Computational Science and Engineering
Implicit Tau-Leaping
• Based on the (explicit) tau method
• Only the mean part is implicit
• Reduces to the Backward Euler scheme in the SDE and ODE regimes
• Agrees with SSA in the small step size limit
• Better suited for stiff problems
( )[ ] )()()( 11 ττντν nnnnn xaxaPXaxX −++= ++
The update formula
Computational Science and Engineering
Implicit tau applied to stiff problem
Reversible Isomerization Reaction
Sample trajectories, stepsizes .005, .05
10021
21
==
←→
cc
SS
Computational Science and Engineering
Damping Effect
Although the implicit tau-leaping does not have the stability problem, it does have a side effect called “damping effect”. It will damp the variance when the stepsize is relatively large.
Stepsize = 0.01 Stepsize = 0.05
Stepsize = 0.1 Stepsize = 1
Computational Science and Engineering
Trapezoidal Formula and Down-Shifting Method
There are two ways to deal with the damping effect.
1. Trapezoidal Formula
2. Down-Shifting Method
( )[ ] ( ) . )()(2
111 τντν nnnnn xaPxaXaxX +−+= ++
Computational Science and Engineering
Time for a break?!!!
Computational Science and Engineering
Stiffness II
Yang Cao
Department of Computer Science
http://courses.cs.vt.edu/~cs6404
Computational Science and Engineering
Multiscale Challenges for Heat Shock Response
σ DNAKσ
RNAP
RNAP
σ
DnaK FtsH
DNAK
What if is at a very low copy number?
σ
Computational Science and Engineering
Michaelis Menton Equation
][k[E][S] 1-1 ESk =
][][k[E][S] 21-1 ESkESk +=
PSE→
ESE + S k1
k-1
k2
P + E
Partial Equilibrium Assumption
Steady State Assumption
Overall reaction rate
][
][][2
SK
SEkv
M
T
+= where
Henrici 1903Michaelis-Menton 1913
Briggs-Haldane 1925
Computational Science and Engineering
Michaelis Menton Equation
][k[E][S] M ES=
PSE→
][
][][2
SK
SEkv
M
T
+=
We obtain the overall reaction rate equation
where
][k[ES])[S]-([E] MT ES=
Let . The above equation becomes][][][ EESE T +=
Thus
][
][][][
SK
SEES
M
T
+=
][2
][ ESkdt
Pd =
Since the production rate satisfies:
Computational Science and Engineering
Michaelis Menton Equation
.][][ SE <<
Second Assumption: .[S][S] Thus .][][ T≈<< SE
Otherwise, species S will stay in the form of ES most time. All we know is
M-M equation matches with the data very well when substrate is in excess of enzyme.
T[S]
When substrate is in excess When enzyme is in excess
Computational Science and Engineering
Related Work
� SSA
� Direct Method and First Reaction Method, Gillespie, 1976, 1977
� Next Reaction Method, Gibson and Bruck, 2000
� Optimized Direct Method, Cao et al. 2004
� Poisson Tau-Leaping Methods
� Developed by Gillespie, 2001.
� Implicit tau, Rathinam, Petzold, Cao, Gillespie, 2003
� Trapezoidal tau, Cao et. al. 2004
� Binomial tau-leaping, Tian and Burrage 2004
� Modified adaptive Poisson tau-leaping, Cao et al. 2005, 2006
� Hybrid Methods
� Rao and Arkin, 2003
� Haseltine and Rawlings, 2002
� Slow-scale SSA, Cao, Gillespie and Petzold. 2004, 2005
Computational Science and Engineering
Formulate the Multiscale Difficulty in Simple Models
• The multiscale behavior can be modeled in the following simple model:
or a simpler model
• Features– Fast and slow reactions
– Fast reactions usually “less important” than slow ones
– SSA will spend most of its time simulating the fast reactions
541
321
SSS
SSS
→+
←→+
31
21
SS
SS
→
←→
Computational Science and Engineering
Partition the System
• Partition the reactions:
– Fast reactions:
– Slow reactions: – Criterion:
• Partition the species:
– Fast species:
– Slow species:
– Criterion: A species is fast if its population gets changed by at least one fast reaction; otherwise, this species is slow.
( )sfRRR ,=
{ }f
M
ff
fRRR ,,1 K=
{ }s
M
ss
sRRR ,,1 K=
xjjaxas
j
fj most""for and ', ,)( ' ∀<<
( ) ,, sfSSS =
{ }f
N
ff
fSSS ,,1 K=
{ }s
N
ss
sSSS ,,1 K=
( ))(),()( tXtXtXsf=
Computational Science and Engineering
Virtual Fast Process
• = the fast species populations driven by only the fast reaction channels
• is with all the slow reactions “turned off”
• is a physically real but non-Markovianprocess. It does not satisfy an ordinary master equation.
• is a physically fictitious but Markovianprocess. It does satisfy an ordinary master equation: the ME for the fast species, driven by only the fast reactions, with all slow species populations held constant.
)(ˆ tXf
)(ˆ tXf
)(ˆ tXf )(tX
f
)(tXf
)(ˆ tXf
Computational Science and Engineering
(Stochastic) Partial Equilibrium Approximation
� In deterministic simulation of chemical systems, the partial equilibrium approximation assumes that the fast reactions are always in equilibrium and the fast state variables are at quasi-steady state.
� In stochastic simulation, the state variables keep changing. The stochastic partial equilibrium approximation is based on the assumption that the distributions of the fast species remain unchanged by the fast reactions.
R1,...,RM '
S1,...,SN '
SN '+1,...,SN
Computational Science and Engineering
Stochastic Partial Equilibrium
• In the virtual system, the states change with time. The behavior is quite different from the situation in the deterministic model.
• But the distribution of the states at different time points are similar to each other. Moreover, the temporal distribution is similar to the ensemble distribution.
321 SSS ←→+
Computational Science and Engineering
Stochastic Partial Equilibrium Assumption (SPEA)
• Assumption 1: must be stable; I.e.,
• Assumption 2: in a time that is very small compared to the time between slow reaction events.
• Reasoning of the two assumptions: – The two assumptions should be satisfied whenever stiffness is
present. Then the fast reactions will be “less important” than the slow ones, so skipping over them will make sense.
– If the two conditions cannot be satisfied, the fast reactions are not less important than the slow ones, and skipping over them isnot a good idea.
)(ˆ tXf
exist.must )|,(ˆ),|,(ˆlim 000 xxPtxtxP fff
t∞=
∞→
)(ˆ)(ˆ ∞→ ffXtX
Computational Science and Engineering
Slow Scale Propensity Theorem
• Theorem: With let be a time increment that is large compared to the time for
, but small compared to the time
between slow reactions. Then the probability that one will occur in is approximately
, where
• is called the slow-scale propensity
function for
• It’s the average of w.r.t. .
• It replaces on the time scale of the slow reactions.
),,()( sfxxtX =
)(ˆ)(ˆ ∞→ ffXtX
s∆
s
jR ),[ stt ∆+
s
fss
j xxa ∆);(
( ) ( ).,,|,ˆ);( ''
'
sf
x
s
j
sfffss
j xxaxxxPxxaf
∑ ∞=
);( fss
j xxas
jR
),( sfs
j xxa )(ˆ ∞fX
),( sfs
j xxa
Computational Science and Engineering
The Slow-Scale SSA
Initialize: Given , set
1. In state at time , evaluate
2. Compute With take
smallest integer satisfying
3. Advance system to the next slow reaction
Then “relax” the fast variables:
4. Record . . Return to 1, or else stop.
),,()( 000
sfxxtX = .,, 000
ffssxxxxtt ←←←
),( sfxx t .,,1for );( s
fss
j Mjxxa K=
.);(10 ∑ =
= sM
j
fss
j
sxxaa ),1,0(, 21 Urr ∈
).;();( 02
1
fss
j
fss
j xxarxxa ≥∑=
µ
,1
ln);(
1
10
=
rxxa fssτ
====µµµµ
=+←
=+←
+←
),,,1(
),,,1(
,
f
fs
i
f
i
f
i
s
ss
i
s
i
s
i
Nixx
Nixx
tt
K
K
µ
µ
ν
ν
τ
).(ˆ of sample ∞← ffxx
),()( sfxxtX =
Computational Science and Engineering
A conservation law is satisfied for the virtual fast process
Examples
31
21
SS
SS
→
←→
txxx =+ 21
3SS tc
t →
The virtual fast process will be:
The whole system can be reduced to a process:
where represents the sum of and
21 SS ←→
tS1S 2S
The overall reaction rate is calculated as
( ) ( )∞=∞
=+
2211
21
xcxc
xxx t
⇒ ( ) tcc
cxx
21
2
1 +=∞
Thus
21
32
cc
cc
tc +=
Computational Science and Engineering
Conservation laws are satisfied for the virtual fast process
Examples (Michaelis- Menton)
413213 SSSSS
c +→←→+
232
131
t
t
xxx
xxx
=+
=+
42 SS tc→
The virtual fast process will be:
The whole system can be reduced to a process:
321 SSS ←→+
The overall reaction rate is calculated as
( ) ( )∞≈∞
=+
32211
131
xcxxc
xxx
t
t
⇒ ( ) 13 221
21
tcxc
xcxx
t
t
+=∞
Then
m
t
kx
xxc
s xa +=2
213)(
We assume . Then 21 xx << 22xx t ≈
1
2
c
ckm =
Let
Computational Science and Engineering
Heat Shock Response Model
Stochastic Model involves 28 species and 61 chemical reactions. This is a moderate-sized model.
CPU time on 1.46 Ghz Pentium IV Linux workstation for one SSA simulation is 90 seconds.
CPU time for 10,000 simulations is more than 10 days.
12 fast reaction channels were chosen for the SPEA. The fast reactions were identified from a single SSA simulation to be the ones that fired most frequently. These 12 reaction channels fire 99% of the total number of times for all reaction channels.
CPU time for the multiscale SSA� Without down-shifting: 3 hours for 10,000 runs� With down-shifting: 4 hours for 10,000 runs
Computational Science and Engineering
Heat Shock Response Model Results
Histogram (10,000 samples) of a slow species mRNA(DNAK) (left) and a fast species DNAK with (right) and without
downshifting method (middle) solved by the original SSA method (purple solid line with ‘o’) and the MSSA method (red
dashed line with ‘+’), for the HSR model.
fast specieswith down shiftingOne fast speciesone slow species
Computational Science and Engineering
Thanks! Questions? Plato is my friend, Aristotle
is my friend, but my best
friend is truth --- Newton