Stochastic Phenomena
in
Biological Systems
Kyung H. Kim
4/25/2008
1
Stochastic Phenomena
• Stochastic Focusing
• Stochastic Switching
• Single Events
• Multiplicative Noise Effect
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2
Stochastic Gene Regulatory Network
• Stochastic Focusing
• Stochastic Switching
• Single Events
• Multiplicative Noise Effect
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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)
where we used
dx
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1 df ( x)
,
f ( x) dx
d ln f ( x)
1
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 ln S2
d S1
S2
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
X
v(S)
v( S )
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k2 S
KM
S
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Stochastic Focusing
• We can change mean S by changing r0.
• The change of mean rate
Deterministic
Case
increases with stochastic
noise.
• Sensitivity increases.
 “Stochastic Focusing”.
r0
S
v(S)
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k1S
Stochastic
Case
Deterministic
Case
X
Stochastic
Case
7
Stochastic De-Focusing
• The change of v decreases with stochastic
Deterministic
noise.
Case
• Sensitivity decreases.
 “Stochastic De-Focusing”.
Deterministic
Case
Stochastic
Case
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Stochastic Focusing-Defocusing
Compensation
• Stochastic focusing can occur in one region of
the curve and stochastic defocusing in another
region.
Stochastic
Focusing
Stochastic
De-Focusing
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Stochastic Focusing
Two step cascade reactions [Paulsson, et al. PNAS 97, 7148 (2000)]
Sensitivit y
d v3
S1
d ln v3
d S1 v3
d ln S1
.
Stochastic Case
Deterministic Case
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Stochastic Focusing
Consider these effects in a pathway such as the one below.
r0
k1S
S
X
v(S)
v( S )
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k2 S
KM
S
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Stochastic Focusing
Two step cascade reactions [Paulsson, et al. PNAS 97, 7148 (2000)]
Sensitivit y
Mean (v3 )
p4 Mean ( S 2 ).
Mean ( S 2 )
Mean (v3 )
.
p4
d v3
S1
d ln v3
d S1 v3
d ln S1
.
ln[ Mean ( S 2 )] ln[ Mean (v3 )] ln[ p4 ].
d ln[ Mean(S2 )] d ln[ Mean(v3 )] d ln[ p4 ].
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Stochastic Focusing
Two step cascade reactions [Paulsson, et al. PNAS 97, 7148 (2000)]
Sensitivit y
d S2
S1
d ln S2
d S1
S2
d ln S1
.
Stochastic Case
Deterministic Case
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Stochastic Gene Regulatory Network
• Stochastic Focusing
• Stochastic Switching
• Single Events
• Multiplicative Noise Effect
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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
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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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Stochastic Gene Regulatory Network
• Stochastic Focusing
• Stochastic Switching
• Single Events
• 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.
vX
X*
vX *
dX *
dt
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X
E X
,
X Km
E X*
,
*
X
Km
vX
X*
vX *
X
.
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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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