Stochastic Processes
in
Gene Regulatory Circuits (II)
Kyung Hyuk Kim
Systems and Synthetic Biology, 498A
Overview
• Modeling of stochastic processes.
– Assumptions
– Reaction rate equations: Deterministic vs.
stochastic
• When to implement stochastic models
• Gillespie Algorithm
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The Origin of Stochasticity
Stochasticity:
• Thermal motion: Intracellular macro-molecules collide with each
other and with water molecules.
• At a time scale of 10-6 sec, velocity information of typical proteins is
lost due to the random collisions.
• Interactions between water molecules and macromolecules are
often modeled with white noise and friction force.
Homogeneity:
• At a time scale of 1 msec, typical small proteins floating in the
cytoplasm of E. coli will diffuse throughout the cell. Position
information of the proteins is therefore quickly lost due to rapid
diffusion.
• Non-specific binding of TF’s to DNA strands ~ 5msec (lifetime).
• At a time scale of minutes, we assume that TF’s are uniformly
distributed in cytoplasm.
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Model of
the Homogeneous System
•
•
•
•
•
•
•
The state of a Homogeneous System is described by numbers of molecules
of each species.
Reactions occur randomly with a probability given by rate constants and
the number of combinatory cases.
No force fields
No velocities
No spatial variation
Markov process : all the
previous information is lost and
current information determines
the next future events.
V
Nred , Nblue
Rate Constants
Very idealized systems!
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Stochastic Reaction Systems
When the numbers of molecules (n) are
small:
• Concentrations fluctuate in a discrete
manner.
• Concentrations fluctuate significantly.
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Discrete Noise for Small n
– The number of LacI tetrameric repressor protein in E.coli
~ 10 molecules.
– If one LacI repressor binds to a promoter region, the
number of free LacI repressors = 9.
– 10% change in its concentration and number! The
change becomes discrete.
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Significant Noise for Small n
• Consider proteins that are translated from
mRNA and degrade.
mRNA
r0
Protein
r1 =k1 n
r0 is assumed to be constant.
r1 is proportional to the number of protein
molecules, n.
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Poisson Distributions in Steady States
mRNA
r0
Protein
r1 =k1 n
The stationary statistics of n follows a Poisson distribution.
The variance of n [Var(n)] = The mean of n [Mean(n)].
Standard deviation of n [Std(n)] = [Var(n)]1/2 = [Mean(n)]1/2.
As the Mean(n) gets larger, the distribution becomes
narrower in the relative terms.
Mean=100
Mean=10
0.12
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
1.4
0.1
1.2
0.08
Mean=10
0.06
Mean=100
0.04
1
0.8
0.6
0.4
0.02
0.2
0
12
24
36
48
60
72
84
96
108
120
132
144
0
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0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.15
0.3
0.45
0.6
0.75
0.9
1.05
1.2
1.35
1.5
1.65
1.8
1.95
0.14
Reaction Rate Equations
•
•
•
When the number of reactant X is 4,
v=k2 4*3.
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Gillespie Algorithm
Exact simulation algorithm for chemical master equations.
At each time point we must answer the following two questions:
1. Determine when the next reaction will occur.
2. Determine which reaction will occur.
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Gillespie Algorithm
1. Determine when the next reaction will occur.
Compute the time of next reaction:
mRNA
r0
Protein
r1 =k1 n
What is the probability distribution of ¿?
y
The probability that the next reaction will occur at [¿ , ¿ + dt]
= exp[-rtot ¿ ] rtot dt. (rtot : normalization factor)
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¿
11
Gillespie Algorithm
2. Determine which reaction will occur.
The probability to choose a reaction ri among all possible reactions
Therefore,
The probability that next reaction r0 will occur during
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Alternative Simulation Approach
mRNA
r0
Protein (n)
r1 =k1 n
The number of proteins are updated at each time step tm = m dt.
Probability to jump from n to n+1 = r0 dt
n-1 n n+1
Probability to jump from n to n-1 = k1 n dt
n-1 n n+1
Probability to stay = [1- (k1 n + r0 ) dt ]
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n-1 n n+1
1.
2.
3.
4.
5.
6.
7.
8.
At t=0, start from n=10.
Generate a random number
“rand” between (0, 1].
If rand < r0 dt, n=11.
Else if rand < (r0 + k1 n) dt, n=9.
Else if n=10.
This new n is the number of
protein at t1=dt.
Repeat step 3-6 and t2=2dt.
…..
This can be less efficient than the
Gillespie algorithm. It is possible
that reactions do not occur.
13
Alternative Simulation Approach
k1 n
r0
n-1 n n+1
• Initially, the number of proteins = n.
• What is the probability that no reaction occurs up to time ¿ =
m dt?
The probability that next reaction r0 will occur during
[¿ , ¿ + dt] is
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Two different simulation approaches
• Gillespie
– Prob[ No reaction occurs until ¿ and any single
reaction occurs between ¿ and ¿ + dt] *
Prob[ a specific reaction (i-th) should have occurred ]
• Alternative approach
– Prob[ No reaction occurs until ¿] *
Prob[ a specific reaction (i-th) occurs between ¿ and ¿
+ dt ]
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Example 2
rg
X
Y (nY)
r0=k0nY
Z (nZ)
r1 =k1 nz
• State: “n”  “(nY , nZ)”
nZ + 1
r0dt
nZ + 1
rgdt
nZ
nZ - 1
r1dt
nY - 1
nY nY + 1
nZ
nZ - 1
1- (rg+r0+r1)dt
nY - 1
nY nY + 1
• The probability that next reaction r0 will occur during [¿ , ¿ +
dt] is
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Gillespie Algorithm
Exact simulation algorithm for chemical master equations.
At each time point we must answer the following two questions:
1. Determine when the next reaction will occur.
2. Determine which reaction will occur.
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Stochastic Phenomena
• Stochastic Focusing
• Stochastic Switching
• Single Events (¸ Phage)
• Multiplicative Noise Effects
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Stochastic Focusing
• Stochastic Focusing:
Sensitivity increase due to stochastic effects.
[Paulsson, et al. PNAS 97, 7148-7153 (2000)]
• Two step cascade reactions:
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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 lnX
Sensitivit y 

dY X d lnY .
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)
20
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)
21
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
23
Stochastic Focusing-Defocusing
Compensation
• Stochastic
focusing can
occur in one
region of the
curve and
stochastic
defocusing in
another region.
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Stochastic Focusing
Two step cascade reactions [Paulsson, et al. PNAS 97, 7148 (2000)]
Sensitivit y 
d v3
S1
d S1
v3

d ln 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.
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Stochastic Focusing
Two step cascade reactions [Paulsson, et al. PNAS 97, 7148 (2000)]
Sensitivit y 
d v3
S1
d S1
v3

d ln v3
d ln S1
.
Mean(v3 )  p4 Mean( S 2 ).
Mean( S 2 ) 
Mean(v3 )
.
p4
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 S1
S2

d ln S 2
d ln S1
.
Stochastic Case
Deterministic Case
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