Non-Markovian%20dynamics%20of%20small%20genetic%20circuits

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Non-Markovian dynamics of small genetic circuits

• Lev Tsimring
• Institute for Nonlinear Science
• University of California, San Diego

Le Houches, 9-20 April, 2007
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Outline

• Deterministic and stochastic descriptions of
  genetic circuits with very different time scales
• Non-Markovian effects in gene regulation
• transcriptional delay-induced stochastic
  oscillations

3
Gene regulatory networks

• Proteins affect rates of production of other
  proteins (or themselves)
• This leads to formation of networks of
  interacting genes/proteins
• Different reaction channels operate at vastly
  different time scales and number densities
• Sub-networks are non-Markovian, even if the whole
  system is
• Compound reactions are non-Markovian

B
B
A
A
C
D
E
D
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Transients in gene regulation

• Genetic circuits are never at a fixed point
• Cell cycle volume growth division
• External signaling
• Intrinsic noise
• Extrinsic noise
• Circadian rhythms ultradian rhythms

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Interesting design (modeling) issues arise
naturally

• Separation of timescales multiple time-scale
  analysis
• Nonlinearity due to multimerization,
  cooperativity and feedback bifurcation analysis
  
• Time delays
• Spatial compartments and cell signaling -
  spatial models
• Cell-to-cell variations are large

In order to build gene circuits to perform
cellular tasks, we need to understand the
origins of the variability
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External signaling ?-Phage Life Cycle
M.Ptashne, 2002
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Engineered Toggle Switch
Construction/experiments
Gardner, Cantor Collins, Nature 403339 (2001)
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Circadian clock in Neurospora crassa
P.Ruoff
WC-1
WCC
FRQ
WC-2
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Ultradian clock at yeast
Klevecz et al, 2004
Synchronized culture
5,329 expressed genes
Respiratory phase
Average peak-to-trough ratio 2
Reductive phase
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The Repressilator
Model
Construction/experiments
Elowitz and Leibler, Nature 403335 (2001)
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Auto-repressor A cartoon
DNA
promoter
gene
RNAP
RNAP
Binding/unbinding rate lt1 sec Transcription
rate 103 basepairs/min Translation rate 102
aminoacids/min mRNA degradation rate
3min Transport in/out nucleus 10 min Protein
degradation rate 30min ..hours
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Oscillations in gene regulation
DNA
promoter
gene
RNAP
RNAP
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Single gene autoregulation
Fast
Slow
Binding/unbinding rate (k-1,k-2) 1
sec-1 Transcription rate (kt) 1 min-1..0.01
min-1 Protein degradation (kx) 0.01 min-1
?
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Single gene autoregulation
Fast
Slow
?
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Quasi-steady-state approximation (naïve approach)
Correct?
Fixed points yes, Dynamics no x is a fast
variable also!
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Separation of scales(Correct projection)
Kepler Elston, 2001 Bundschuh et al,
2003 Bennett et al, 2007, in press
slow variable

• Prefactor is important if x2/x1,
• i.e. lots of dimers
• Prefactor makes transients slower

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Genetic toggle switch
Gardner, Cantor, Collins, Nature 2000
GFP
Protein A

On
Gene A
on
Gene
B
off
Reporter
Protein B

Off
Gene
A
Gene
on
off
B
Reporter
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Multiple-scale analysis
Fast reactions
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Multiple-scale analysis (contd)
Local equilibrium for fast reactions
Nullspace of the adjoint linear operator
2 eigenvectors
From orthogonality conditions
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Prefactor
w/o prefactor
with prefactor
full model
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Stochastic gene expression Master Equation
approach
Two reactions
production
degradation
Probability of having x molecules of X at time t,
Dynamics of
Continuum limit (xgtgt1)
Fokker-Planck equation
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Stochastic gene expression Langevin equation
approach
Two reactions
production
degradation
Deterministic equation
Each reaction is a noisy Poisson process,
meanvariance
Separately
Langevin equations
Since reactions are uncorrelated, variances add
From Langevin equation to FPE (van Kampen,
Stochastic Processes
in Chemistry
and Physics,1992)
or from FPE to Langevin!
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Autoregulation stochastic description
n total of monomers u of unbound dimers
b - of bound dimers
Master equation for
Projection using
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Back to ODE
In the continuum limit (large n) Fokker-Planck
equation
(no prefactor)
Corresponding Langevin equation
with
Fast reaction noise is filtered out
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Multiscale stochastic simulations(turbo-charged
Gillespie algorithm)

• The computational analog of the projection
  procedure stochastic partial equilibrium (Cao,
  Petzold, Gillespie, 2005)
• Identify slow and fast variables
• Fast reactions at quasi-equilibrium
• distribution for fixed is
  assumed known
• Compute propensities for slow reactions

Easy for zero- and first-order reactions,
more tricky for higher
order reactions
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Regulatory delay in genetic circuits
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Single gene autoregulation transcriptional delay
Fast
Slow
cf. Santillán Mackey, 2001
Delayed
After projection
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Genetic oscillations Hopf bifurcation
Fixed point
Complex eigenvalues
Instability
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Transcriptional delay a non-Markovian process
Stochastic simulations (modified Gillespie
algorithm)
update
update

• Markovian reactions dimerization, degradation,
  binding
• exponential next reaction time distribution

• which reaction to choose?

Non-Markovian channels transcription,
translation Gaussian time
distribution
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Scheme of numerical simulation
delay
time steps
Modified Direct Gillespie algorithm (Gillespie,
1977)

• Input values for initial state , set t0
• Compute propensities
• Generate random numbers
• Compute time step until next reaction
• Check if there has been a delayed reaction
  scheduled in
• a) if yes, then last steps 2,3,4 are
  ignored, time advances to ,
• update in accordance
  with delayed reaction
• b) if not, go to the step 6
• Find the channel of the next reaction
• Update time and
  

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Stochastic simulations
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Analytical results
Deterministic model
Reactions
(no dimerization)
No Hopf bifurcation!
probability to have n monomers at time t given
the state s at time t-?
Stochastic model (Master equations)
Approximation
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Boolean model
Two-state gene
at time t depends on the state at t-T
Transition probability
if
if
positive feedback
negative feedback
For double-well quartic potential
-1
1
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Master equations
the probability of having value s(t) ?1 at time
t
s ?1 to 1
within (t,tdt)
probability of transition from
s 1 to ?1
Delayed master equation
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Autocorrelation function
Linear equation!
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Autocorrelation function
T1000, p10.1 p20.3
Stochastic oscillations!
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Power spectrum two-state model
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Analytical results
Deterministic model
Reactions
(no dimerization)
No Hopf bifurcation!
probability to have n monomers at time t given
the state s at time t-?
Stochastic model (Master equations)
Approximation
39
Analytical results
Correlation function
Result
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Effect of stochasticity and delay on regulation
Standard deviation/mean
Time delay increases noise level
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Conclusions

• Fast binding-unbinding processes can be
  eliminated both in deterministic and stochastic
  modeling, however an accurate averaging procedure
  has to be used leads to prefactors affecting
  transient times and noise distributions
• Multimerization increases time scales of genetic
  regulation
• Deterministic and stochastic description of
  regulatory delays developed, delays of
  transcription/translation of auto-repressor may
  lead to increased fluctuations levels and
  oscillations even when deterministic model shows
  no Hopf bifurcation
• Modified Gillespie algorithm is developed for
  simulating time-delayed reactions

L.S. Tsimring and A. Pikovsky, Phys. Rev. Lett.,
87, 2506021 (2001). D.A. Bratsun, D. Volfson,
L.S. Tsimring, and J. Hasty, PNAS, 102,
14593-12598 (2005). M. Bennett, D. Volfson, L.
Tsimring, and J. Hasty, Biophys. J., 2007, in
press.