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

3

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

5

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

10

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