Stochasticity in molecular systems biology

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Stochasticity in molecular systems biology

• ESE680 003 Systems Biology
• Spring Semester 2007
• University of Pennsylvania

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Lactose regulation system of E. coli
lac Z
lac Y
lac A
lac I
mRNA
mRNA
M
B
b galactosidase
repressor
P
permease
E
external TMG
Internal TMG
T
by Vijay Kumar
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Biochemical reactions
Yildirim Mackey, Biophys J, 2003.
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Two stables states
mRNA concentration
High state
Low state
External TMG concentration
Two modes of equilibria, low concentration
(uninduced) and high concentration (induced).
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Experimental observation
Nature, vol. 427, pp. 737-740, 2004.
Bacteria can spontaneously switch between the two
states!!!
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Biochemical reactions
Yildirim Mackey, Biophys J, 2003.
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Stochasticity in genetic expression
Examples of possible stochastic influences on
phenotype
J. M. Raser et al., Science 309, 2010 -2013
(2005)
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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.

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Chemical reactions are random events
B
B
A
A
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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
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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
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Stochastic reaction kinetics
A
AB
time
reaction
reaction
reaction
time
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Multiple reactions

• Multiple reactions are seen as concurrent Poisson
  processes.
• Gillespie simulation algorithm determine which
  reaction happens first.

A B ? AB
Rate 1
Rate 2
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Multiple reactions
A
AB
time
reaction 1
reaction 2
reaction 1
time
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t leaping scheme
A
AB
time
r2
r1
r2
r1
r1
r2
r1
D
D
D
D
time
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The stochastic model
concentration
discrete, stochastic
molecules
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The stochastic model