Stochastic Processes
in
Gene Regulatory Circuits (III)
Kyung Hyuk Kim
Systems and Synthetic Biology, 498A
Overview
• Stochastic phenomena
• Chemical Langevin equations
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2
Stochastic Phenomena
• Stochastic Focusing
• Stochastic Switching
• Single Events (¸ Phage)
• Multiplicative Noise Effects
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3
Stochastic Focusing
• Stochastic Focusing:
Sensitivity increase due to stochastic effects.
• Sensitivity:
% Change of Response Signal
Sensitivit y 
% Change of Source Signal
The sensitivity can be used to estimate how a system
responds due to changes in the environment.
dX Y d ln X
Sensitivit y 

dY X d ln Y .
d ln f ( x)
1 df ( x)

,
where we used
dx
f ( x) dx
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1
d ln f ( x) 
df ( x).
f ( x)
4
Stochastic Focusing
• Stochastic Focusing:
Sensitivity increase due to stochastic effects.
[Paulsson, et al. PNAS 97, 7148-7153 (2000)]
Two step cascade reactions
Source Signal = S1
Response Signal = S2
Sensitivit y 
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d S2
S1
d S1
S2

d ln S 2
d ln S1
x  Mean(x)
5
Stochastic Focusing
Fluctuations in the concentration of S leads to fluctuations in the
reaction rate v(S).
How does the mean rate of reaction change with the noise?
E.g.,
S
v(S)
X
k2 S
v( S ) 
KM  S
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Stochastic Focusing
• We can change mean S.
• The change of mean rate
increases with stochastic
noise.
• Sensitivity increases.
 “Stochastic Focusing”.
S
v(S)
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Deterministic
Case
Stochastic
Case
Deterministic
Case
X
Stochastic
Case
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Stochastic Focusing-Defocusing
Compensation
v0
S
v(S)
kS
X
Stochastic focusing
can occur in one
region of the curve
and stochastic
defocusing in another
region.
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Stochastic Focusing
Consider these effects in a pathway such as the one below.
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Stochastic Focusing
Two step cascade reactions [Paulsson, et al. PNAS 97, 7148 (2000)]
d lnhS3 i
d lnhv3 i
Sensitivity =
=
d lnhS1 i
d lnhS1 i
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Stochastic Focusing
Two step cascade reactions [Paulsson, et al. PNAS 97, 7148 (2000)]
d lnhS3 i
d lnhv3 i
Sensitivity =
=
d lnhS1 i
d lnhS1 i
Stochastic Case
Deterministic Case
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Stochastic Phenomena
• Stochastic Focusing
• Stochastic Switching
• Single Events (¸ Phage)
• Multiplicative Noise Effects
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Stochastic Switching
• Stochastic Switching:
Bistability in a stochastic framework means double
peaks in the probability distribution function of
concentrations. Jumping from one peak to another is
possible with a finite probability.
Thermal
Noise
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On
13
Stochastic Switching
Stochastic Switching
On
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14
Stochastic Gene Regulatory Network
• Stochastic Focusing
• Stochastic Switching
• Single Events: Bifurcation in Phage ¸–infected
E. coli. [Arkin, et al. Genetics 149 1633 (1998)]
• Multiplicative Noise Effect
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Multiplicative Noise Effect
• Multiplicative Noise: Noise strength depends on the state of the
system.
• Noise-induced Bistability.
• Enzyme futile cycle reaction.
[Samoilov, et al. PNAS 102 2310 (2005)]
E+ is allowed to fluctuate.
External
Noise
Deterministic
Dynamics
vX  X *
E X

,
X  Km
vX *  X
E X *
 *
,
X  Km
dX *
 vX  X *  vX *  X .
dt
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Multiplicative Noise Effect
• Var[E+ ] / Mean(E+)2p
• p=1/2 for Poisson
distribution.
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Multiplicative Noise Effect
• Detailed version of the enzyme futile cycle
system……
p=0.96
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Chemical Langevin Equation (CLE)
• A Stochastic differential equation
– Continuous space (number of occurrences of each reaction)
• Internal noise is modeled by Gaussian white noise.
• E.g.,
X +Y !Z
p
dz
= kxy + kxy»(t)
dt
p
dx
dy
=
= kxy + kxy»(t)
dt
dt
»: Gaussian white noise
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h»(t)»(t0)i = ±(t ¡ t0 ).
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Euler-Maruyama Algorithm for Solving CLEs
• Euler method for chemical Langevin equations
 Fast, but causing overshoot in stiff systems
[Salis, Sotiropoulos and Kaznessis, BMC Bioinformatics, 7:93 (2006)]
• Numerical integration accuracy = O(p¿)
• E.g.,
X +Y !Z
p
dz
= kxy + kxy»(t)
dt
Normal Gaussian random number with
a mean of zero and a variance of ¿
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Euler-Maruyama Method
A
1.
k1 a
k2 b
B
Set t = 0, initialize concentrations
a = 0.5 ; b = 0; k1 = 0.1; k2 = 0.2;
2.
Set ¿ = 1.2 .
3.
Compute reaction rates: k1 a , k2 b.
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Euler-Maruyama Method
3. Generate two normal Gaussian random numbers (»1¿,
»2 ¿) with their means of zero and their variances of ¿.
* Box-Muller transform sampling method
Generate two random numbers in (0,1] = p1, p2.
Compute
4. Compute the concentrations
at the next time step
a(t+¿) and b(t+¿) by using
the formula in the previous slide.
5. Update the current time t = t + ¿.
6. Go back to step 3.
A 2-dimension Gaussian
distribution will be projected
onto x-axis.
»2 / 2 is distributed
exponentially.
Euler-Maruyama Method
xi (t + ¿) = xi (t) +
M
X
j=1
Nij vj (x(t))¿ +
M
X
j=1
q
Nij vj (x(t)) dWj
Nij: Stoichiometic matrix
dWj:
A normal Gaussian random number with a
mean of zero and a variance of ¿
Chemical Langevin Equation - Assumptions
X +Y !Z
p
dz
= kxy + kxy»(t)
dt
Mean
Rate
Standard
deviation
of rate
Mean = Variance
Number of events in a time step ¿ is
assumed to be distributed as a
normal Gaussian distribution
with its mean equal to its variance.
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X and Y decrease after reaction
firing.
Thus, number of events in time
step ¿ is not a Poisson
distribution.
But, we can assume it is similar
to a Poisson distribution, actually
further assume a Gaussian
distribution.

X and Y should not change
significantly.
Sufficient condition:
The number of X and Y
molecules need to be large
enough.
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Chemical Langevin Equation
Time step ( ¿ ) needs to be
[Gillespie, Journal of Chemical Physics 113 297 (2000)]
– Small enough that number changes due to
reactions during ¿ is so small that all reaction rates
are not changed appreciably.  Poisson
distribution
– Large enough that the number of occurrence of
each reactions is >> 1.  Gaussian distribution
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Better Algorithm - Milstein Method
• Increased accuracy but reduced speed.
• Numerical integration accuracy = O(¿)
•
E.g.,
X +Y !Z
p
dz
= kxy + kxy»(t)
dt
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