Stochastic Processes
in
Gene Regulatory Circuits
Kyung Hyuk Kim
Advanced Synthetic Biology
BIOEN 424/524
Feb. 25 2011
Stochastic Gene Circuits
• Today – Experimental evidence of stochastic
gene expressions
• Monday – Modeling stochastic circuits:
Gillespie stochastic simulation algorithm
Deterministic vs. Stochastic
• “Deterministic”?
• “Stochastic”?
– Future events are determined by probability
E.g.,
Weather forecast
Stock price change
Brownian motion

Chaotic behavior
Quantum mechanical effects
Lack of information
Overview
• Experiments showing stochasticity in gene
circuits
– How do transcription factors find their specific
promoter regions?
 Stochastic diffusion.
– How do phage-λ infected cells decide lysis or
lysogeny?
 Stochastic gene expression.
– Intrinsic and extrinsic noise in gene expressions
How do transcription factors find their
target promoter regions?
• 1-D diffusion along DNA segments.
• 3-D diffusion through cytoplasm.
Elf, Li, and Xie, Science 316 1191 (2007), E-coli
• Life time in 1-D diffusion ~ 5 msec.
D1 = 0.046 ¹m2/sec, 85 bp per 1-D diffusion.
• Time duration in 3-D diffusion before any binding ~ 0.5 msec.
D3 = 3 ¹m2/sec.
• Search time ~ 3 min. (One TF  One Promoter)
How do transcription factors find their
target promoter regions?
E.coli
Elf, Li, and Xie,
Science 316 1191 (2007)
w/ LacI::venus integrated
plasmids
1 sec exposure
Specific binding
lacIOZ - Non-specific binding
¸ Phage
λ phage is a virus that can infect E. coli.
It consists of a head, containing
double-stranded DNA, and a tail that allows
it to attach to the surface of E. coli.
http://www.steve.gb.com/images/scie
nce/lambdaphage.jpg
www.asm.org/division/m/foto/LamAttack.html
¸ Phage
What determines
its initial decision
on whether to go
lytic or lysogenic?
Stochastic
processes of gene
expressions.
http://www.waksmanf
oundation.org/labs/roc
hester/Image3.gif
¸ Phage
Circuit
Arkin, Ross, and
McAdams,
Gnetics 149 1633.
N=
Antiterminator
Lysogeny
Lysis
¸ Phage
Circuit
CII degrades
very fast:
lifetime ~2min.
Arkin, Ross, and
McAdams,
Gnetics 149 1633.
Lysogeny
Lysis
Positive
feedback
¸ Phage
Circuit
Arkin, Ross, and
McAdams,
Gnetics 149 1633.
CIII protects CII
from degrading.
Lysogeny
Lysis
Positive
feedback
¸ Phage
Circuit
Lysogeny
Arkin, Ross, and
McAdams,
Gnetics 149 1633.
Lysogeny
Lysis
Time evolution of CII and CIII
• Simultaneous high concentrations of CII and CIII induces cI
expression.  Lysogeny.
• Gillespie stochastic simulation algorithm used.
Arkin, Ross, and McAdams, Genetics 149 1633.
Stochastic Gene Expression in a
Single Cell
Repressor – YFP
(Green) Fusion
RFP
(Red)
http://www.elowitz.caltech.edu/publications.html
Different Kinds of Noise in Cellular
Circuits
• Two gene expressing Yellow and Cyan fluorescent proteins are
controlled by the identical lac promoters.
• Two sources of potential noise in the system:
1.
2.
Extrinsic – External noise from outside
E.g., cell replication, ribosome number fluctuations, etc.
Intrinsic – Internal noise generated by the circuit.
Low IPTG.
High IPTG.
Low levels of
High levels of
expressions.
expressions.
Large noise.
Less noise.
CFP = Green, YFP = Red
Gillespie Algorithm
Simulating a stochastic model is quite different from an
ODE model. In a stochastic model we take account of individual
reactions as they convert one molecule into another. Solving a
stochastic model is a two stage process.
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.
The most well known implementation of this approach is
the Gillespie method (Gillespie, 1977).
Simulating a Simple System
Consider the following simple system:
Simulating a Simple System
1. Set t = 0, initialize concentrations (molecule numbers)
A = 50 molecules; B = 0; k1 = 0.1; k2 = 0.2;
2. Compute all reaction rates and their total rate, rtot
Simulating a Simple System
3. Generate two random numbers, p1 and p2:
p1=urnd()  when the next reaction to fire
p2=urnd()  which reaction to fire
4. Compute the time of next reaction:
τ is the time taken for the next reaction to occur:
Simulating a Simple System
5. Compute the relative probability rates:
Simulating a Simple System
Determine which reaction will occur.
6. Compute which reaction will fire:
7. Update the current time:
120
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8. Go back to step 2
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