1. Stochasticity in molecular systems biology
ESE680 – 003 Systems Biology
Spring Semester 2007
University of Pennsylvania

2. Lactose regulation system of E. coli
lac Z
lac Y
lac A
lac I
mRNA
mRNA B M
b galactosidase
repressor
The lac operon is under “negative” control (lac repressor, LacI) and “positive” control (CAP or CAMP receptor protein - CRP)
The ‘original’ lac network P
permease E
external
TMG
Internal TMG T
by: Vijay Kumar

3. Biochemical reactions
Yildirim & Mackey, Biophys J, 2003.

4. Two stables states
mRNA concentration
High state
Low state
External TMG concentration
Two modes of equilibria, low concentration (uninduced) and high concentration (induced).

5. Experimental observation
Nature, vol. 427, pp , 2004.
Bacteria can spontaneously switch between the two states!!!

6. Biochemical reactions
DETERMINISTIC!!
Good approximation when the number of molecules is LARGE.
Cells are small ( l), hence not that many molecules.
-16
Yildirim & Mackey, Biophys J, 2003.

7. Stochasticity in genetic expression
Examples of possible stochastic influences on phenotype
J. M. Raser et al., Science 309, (2005)

8. 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.

9. Chemical reactions are random eventsB B A A
A + B AB
A + B AB

10. Poisson process
Poisson process is used to model the occurrences of random events.
Interarrival times are independent random variables, with exponential distribution.
Memoryless property.
event
event
event
time

11. 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

12. Stochastic reaction kineticsAB
time
reaction
reaction
reaction
time

13. Multiple reactions A + B  AB
Multiple reactions are seen as concurrent Poisson processes.
Gillespie simulation algorithm: determine which reaction happens first. k 1
A + B  AB k 2
Rate 1
Rate 2

14. Multiple reactions A AB
time
reaction 1
reaction 2
reaction 1
time

15. t – leaping scheme A AB
time r2 r1 r2 r1 r1 r2 r1 D D D D
time

16. The stochastic model
concentration
discrete, stochastic
# molecules

17. The stochastic model mRNA concentration # mRNA molecules Time (min)
Increase E
External TMG concentration
mRNA concentration