Stochasticity in molecular systems biology

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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 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

Biochemical reactions Yildirim & Mackey, Biophys J, 2003.

Two stables states mRNA concentration 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 l), hence not that many molecules. -16 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. Interarrival times are independent random variables, with exponential distribution. Memoryless property. event event event time

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 AB time reaction reaction reaction time

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

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

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

The stochastic model concentration discrete, stochastic # molecules

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