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Stochasticity in molecular
systems biology
ESE680 – 003 Systems Biology
Spring Semester 2007
University of Pennsylvania
Lactose regulation system of E. coli
lac Z
lac Y
lac A
lac I
mRNA
mRNA
B
repressor
M
b galactosidase
P
permease
external
TMG
Internal TMG
T
E
by: Vijay Kumar
Biochemical reactions
Yildirim & Mackey, Biophys J, 2003.
mRNA concentration
Two stables states
High state
Low state
External TMG concentration
Two modes of equilibria, low concentration (uninduced)
and high concentration (induced).
Experimental observation
Nature, vol. 427, pp. 737-740, 2004.
Bacteria can spontaneously switch between the two states!!!
Biochemical reactions
DETERMINISTIC!!
Good approximation when the number of molecules is LARGE.
Cells are small (10 -16 l), hence not that many molecules.
Yildirim & Mackey, Biophys J, 2003.
Stochasticity in genetic expression
Examples of possible stochastic influences on phenotype
J. M. Raser et al., Science 309, 2010 -2013 (2005)
Phenotype vs genotype

Phenotype: physical manifestation of the
individuals.
 Genotype: genetic information in the genome.
 Identical genotype can lead to different
phenotype.

Genotype + environment + noise  phenotype.

Compare: identical PCs with the same
software running different programs.
Chemical reactions are random events
B
B
A
A
A+B
AB
A+B
AB
Poisson process

Poisson process is used to model the occurrences of
random events.
event
event
event
time

Interarrival times are independent random variables,
with exponential distribution.

Memoryless property.
Stochastic reaction kinetics

Quantities are measured as #molecules
instead of concentration.
 Reaction rates are seen as rates of Poisson
processes.
k
A + B  AB
Rate of Poisson process
Stochastic reaction kinetics
A
AB
time
reaction
reaction
reaction
time
Multiple reactions

Multiple reactions are seen as concurrent
Poisson processes.
k1
A + B  AB
k2
Rate 1

Rate 2
Gillespie simulation algorithm: determine
which reaction happens first.
Multiple reactions
A
AB
time
reaction 1
reaction 2
reaction 1
time
t – leaping scheme
A
AB
r1
r2
D
r2
r1
D
time
r2
r1
r1
D
D
time
The stochastic model
concentration
discrete, stochastic
# molecules
The stochastic model
35
30
# mRNA molecules
mRNA concentration
25
20
15
10
5
0
0
500
1000
1500
Time
(min)TMG concentration
External
Increase E