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Stochastic simulation algorithms

Join us for ESE680: Systems Biology this week featuring two informative seminars. Prof. Mustafa Khammash from UCSB will discuss "Noise in Gene Regulatory Networks: Biological Role and Mathematical Analysis" on Friday, March 23, from 12-1 PM in Berger Auditorium. Later, Dr. Daniel Gillespie will present on "Stochastic Chemical Kinetics," highlighting the randomness in chemical reactions, interarrival times, and their representation through Poisson processes. Both talks will explore the vital role of stochastic processes in biological systems.

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Stochastic simulation algorithms

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  1. Stochastic simulation algorithms ESE680: Systems Biology

  2. 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

  3. A + B AB A + B AB Chemical reactions are random events B B A A

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

  5. 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

  6. Stochastic reaction kinetics A AB time reaction reaction reaction time

  7. k k 1 2 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

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

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

  10. Erlang distribution

  11. Erlang  Gaussian

  12. Stochastic simulation with Gaussian rv

  13. Stochastic simulation with Gaussian rv Ito stochastic integral

  14. Chemical Langevin equation White noise driving the original system

  15. Stochastic fluctuations triggered persistence in bacteria ESE680: Systems Biology

  16. Bacterial persistence • 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 (10-5-10-6) • Discovered as soon as antibiotics were used (Bigger, 1944) • A fraction of an isogenic population survives antibiotic treatment significantly better than the rest (from Balaban et al, Science, 2003)

  17. Persistence as an evolutionary advantage • 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?

  18. 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

  19. Stringent response and growth control • Triggered by adverse conditions, e.g. starvation • Transcriptioncontrol (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

  20. tmRNA mRNA Toxin Antitoxin Tox Ant Ribosome Ribosome Ribosome

  21. 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)

  22. 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

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

  24. Reaction kinetics • Translation:

  25. Reaction kinetics • Ribosome dynamics:

  26. Deterministic simulation result Toxin Antitoxin Ribosome activity

  27. Stochastic simulation result Toxin Antitoxin Ribosome activity

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