Stochastic simulation algorithms

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Stochastic simulation algorithms

• ESE680 Systems Biology

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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

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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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Erlang distribution

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Erlang ? Gaussian

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Stochastic simulation with Gaussian rv
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Stochastic simulation with Gaussian rv
Ito stochastic integral
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Chemical Langevin equation
White noise driving the original system
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Stochastic fluctuations triggered persistence in
bacteria

• ESE680 Systems Biology

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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 (10-5-10-6)

(from Balaban et al, Science, 2003)
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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?

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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

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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
TRANSLATION
TRANSCRIPTION
GROWTH
NUTRIENT AVAILABILITY
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Tox
Ant
Ribosome
Ribosome
Ribosome
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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)

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Toxin-antitoxin modules

• TA module provides an emergency brake
• Normally all toxin is bound to antitoxin
• Antitoxin binds toxin at a ratio gt 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

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Reaction kinetics

• Variables
• T Toxin concentration
• A Antitoxin concentration
• R ribosome activity
• Transcription

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Reaction kinetics

• Translation

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Reaction kinetics

• Ribosome dynamics

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Deterministic simulation result
Toxin
Antitoxin
Ribosome activity
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Stochastic simulation result
Toxin
Antitoxin
Ribosome activity