stochastic simulation algorithms ese680: systems biology
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
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Chemical reactions are random events
A
B
A + B AB A + B AB
A
B
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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.
A + B AB
k
Rate of Poisson process
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Stochastic reaction kinetics
reaction
time
time
A
AB
reaction reaction
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Multiple reactions
Multiple reactions are seen as concurrent Poisson processes.
Gillespie simulation algorithm: determine which reaction happens first.
A + B ABk1
k2
Rate 1 Rate 2
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Multiple reactions
reaction 1
time
time
A
AB
reaction 2 reaction 1
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– leaping scheme
r1
time
time
AB
r2 r1 r1r1
r2 r2
A
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Erlang distribution
0 2 4 6 8 10 12 14 16 18 200
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Erla
ng d
istr
ibut
ion
n
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Erlang Gaussian
0 5 10 15 20 25 30 35 40 45 500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Normal distribution Erlang distribution
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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
Stochastic fluctuations triggered persistence in
bacteria
ESE680: Systems Biology
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)
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?
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
ribosomesppGpp
Lon Toxins
TRANSLATIONTRANSCRIPTION
RAC
GROWTH
NUTRIENTAVAILABILITY
Tox Ant
RibosomeRibosome
Ribosome
mRNA
ToxinAntitoxin
tmRNA
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 activityTranscription:
Reaction kinetics
Translation:
Reaction kinetics
Ribosome dynamics:
Deterministic simulation result
Toxin Antitoxin Ribosome activity
Stochastic simulation result
Toxin Antitoxin Ribosome activity