# Stochastic simulation algorithms

## Presentation on theme: "Stochastic simulation algorithms"— Presentation transcript:

Stochastic simulation algorithms
ESE680: Systems Biology

Relevant talks/seminars this week!
Prof. Mustafa Khammash (UCSB) “Noise in Gene Regulatory Networks: Biological Role and Mathematical Analysis ” Friday 23 Mar, 12-1pm, Berger Auditorium Dr. Daniel Gillespie (Dan Gillespie Consultant) “Stochastic Chemical Kinetics” Friday 23 Mar, 2-3pm, Berger Auditorium

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

Erlang distribution

Erlang  Gaussian

Stochastic simulation with Gaussian rv

Stochastic simulation with Gaussian rv
Ito stochastic integral

Chemical Langevin equation
White noise driving the original system

Stochastic fluctuations triggered persistence in bacteria
ESE680: Systems Biology

Bacterial persistence
Discovered as soon as antibiotics were used (Bigger, 1944) A fraction of an isogenic population survives antibiotic treatment significantly better than the rest If cultured, the surviving fraction gives rise to a population identical to the original one Bimodal kill curves Persisters are a very small fraction of the initial population ( ) (from Balaban et al, Science, 2003)

Persisters are an alternative phenotype Similar to dormancy or stasis Since they do not grow, they are less vulnerable Presence of multiple phenotypes has an evolutionary advantage in survival in varying environments Transitions between phenotypes are of stochastic nature – Random events, triggered by noise What is the underlying molecular mechanism?

Persistence as a phenotypic switch
Recent work due to Balaban et al showed that there are two types of persisters: Type I – generated by an external triggering event such as passage through stationary phase Type II – generated spontaneously from cells exhibiting ‘normal’ phenotype

Stringent response and growth control
Triggered by adverse conditions, e.g. starvation Transcription control (p)ppGpp: Lack of nutrients Stalled ribosomes ppGpp synthesis Reprogramming of transcription Translation shutdown Proteases (p)ppGpp involved Activation of toxin-antitoxin modules Toxin reversibly disables ribosomes ppGpp Lon Toxins RAC TRANSCRIPTION TRANSLATION GROWTH NUTRIENT AVAILABILITY

Tox Ant mRNA tmRNA Ribosome Ribosome Ribosome Toxin Antitoxin

Toxin-antitoxin modules
Toxin and antitoxin are part of an operon Overexpression of toxin leads to ‘stasis’ Toxin cleaves mRNA at the stop codon Cleaved mRNA disables translating ribosomes Ribosomes can be ‘rescued’ by tmRNA One example: RelB and RelE (Gerdes 2003)

Toxin-antitoxin modules
TA module provides an emergency brake Normally all toxin is bound to antitoxin Antitoxin binds toxin at a ratio > 1 Antitoxin has a shorter half-life Shutdown can be triggered by fluctuations: Toxin excess  reduced translation  more excess toxin .. translation shutdown Recovery from shutdown facilitated by tmRNA which reverses

Reaction kinetics Variables: T = Toxin concentration
A = Antitoxin concentration R = ribosome activity Transcription:

Reaction kinetics Translation:

Reaction kinetics Ribosome dynamics:

Deterministic simulation result
Toxin Antitoxin Ribosome activity

Stochastic simulation result
Toxin Antitoxin Ribosome activity