Download State estimation for gene networks with intrinsic and extrinsic noise

State estimation for gene networks with
intrinsic and extrinsic noise:
Models and approaches on a case study
Eugenio Cinquemani
HYCON2-AD3 Workshop on Biological and Medical Systems
Wednesday, 5 June 2013
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Outline
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Introduction : Stochastic models of gene networks, single-cell data, etc
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Case study : arabinose uptake dynamics in E.coli
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Stochastic modelling : CME model and CLE approximation
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State estimation : problem, comparison of CME and CLE-based methods
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Pure intrinsic noise, intrinsic & extrinsic noise
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Analysis in-silico : simulation results on the case study
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With Alfonso Carta (INRIA Sophia-Antipolis), ECC 2013, accepted.
Intrinsic and extrinsic noise
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Elowitz et al., Science 2002 :Yellow and Cyan
reporters under control of same promoter enable
isolation of random transcription/translation
dynamics (intrinsic noise) from other physiological
or environmental fluctuations (extrinsic noise)
Stochastic modelling of biochemical networks
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Different sources of noise well captured by stochastic models
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Markov Chain models : n species react via m probabilistic reactions
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Xi number of molecules of species i, state X=(X1,...,Xn)
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p(Z,t) = Prob{X(t)=Z} evolves according to the Chemical Master Equation
Mixed-effects models : parametric models of cell dynamics where parameters
take different values across individuals according to a known population prior
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Best suited to capture extrinsic noise
Inference methods for both model classes exist (won't discuss here)
Case study: Arabinose uptake in E.coli
(Inspired by Megerle et al,
Biophys. J. 2008 )
A Markov Chain (CME) model
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We model transcription, translation, uptake, whereas lump AraC-ara complex formation
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For x=(x1,x2,x3,x4,x5)=(ara, mUpt, Upt, mGFP, GFP) we have :
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We take this CME model as the reference model ( «true » system )
Random parameters will account for extrinsic noise :
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State estimation
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Time-lapse fluorescence microscopy (gene expression time profiles for individual
cells) + stochastic modelling of gene expression allow for real-time estimation of
unobserved cell states
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For analysis, for use in identification, or for real-time model-based control
Problem formulation : Given gene expression time samples for N cells
, with
compute
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in real time
The problem splits: At every new measurement, update
(Hence we drop
from now on.)
Particle filter approach : The bootstrap filter
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Randomized filtering approach based on N particles (candidate state values)
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Initialization : At time k=0, draw N particles Xi(0) according to p(X,0) ●
Iterates k=0,1,2... : Given the new measurement yk,
(Measurement update) Compute
Form estimate
(Bootstrap resampling) Resample with replacement the N particles according
to probability distribution (w1,... wN)
(Time update) Propagare every new particle from tk to tk+1 by simulation of the
random system dynamics
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Good results for large N (e.g. 1000). But heavy : requires N simulations at all k
Approx : Chemical Langevin Equation
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If most species occur in large numbers with large probability, the CLE is a known viable
approximation of the CME :
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Stochastic differential equation (in concentrations domain) determined by the CME
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How good is the approximation ?
We compared CME and CLE in simulation
for parameter values fixed from literature
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Comparison: CME and CLE simulations
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System is assumed to start from steadystate prior to exhausion of glucose
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no internal arabinose
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abundance of external arabinose
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CLE accumulates bias from small initial molecule numbers
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After mean equalization, CME and CLE distributions differ only in small states
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Promising approximation for state estimation (will feedback compensate for bias?)
CLE-based filtering
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CLE approximation allows one to put in place nonlinear extensions of Kalman filtering
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We consider a Square-Root (continuous-time) version of the Unscented Kalman Filter
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Initialization : At time k=0, we are given a priori state statistics x[0|-1] and P[0|-1]
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Iterates k=0,1,2,... : Given the new measurement yk,
(Measurement update) : Get x[k|k], P[k|k] from x[k|k-1], P[k|k-1] by Kalman update
(Choice of particles) : 2n+1 particles are computed from x[k|k] and factorization of P[k|k]
(Time update) : Propagate particles from tk to tk+1 via the nonlinear system dynamics,
then compute x[k+1|k], P[k+1|k] from these particles
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Square-root version propagates factorization of P directly for numerical stability
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Requires (Euler) integration of system dynamics, but for only 2n+1 particles
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Limited to (conditional) 2nd order moments, but applicable to non-Gaussian distributions
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State estimation on a case study
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Arabinose uptake used as a case study for studying real-time state estimation (with
real-time control applications in mind). We :
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Consider single-cell fluorescence microscopy profiles (tracking)
Assume a nominal Chemical Master Equation (CME) single-cell stochastic gene
expression model (« reasonable » equations and parameter values )
Address the problem of estimating unobserved abundances of mRNAs, proteins,
and internal arabinose from the (noisy) fluorescence readouts
We study the problem in-silico :
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We Gillespie-simulate the nominal model to generate data from many cells
For data from identical individuals, we compare performance of estimation
algorithms based on the nominal CME model and on a more tractable Langevin
approximation
For data from non-identical individuals, we study the performance gain obtained by
including extrinsic noise in the model by a ME approach
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Estimation results (identical individuals)
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From previous result, CME and CLE
differ for species in small copy
numbers by a small bias
We wish to see if a CLE-based state
estimator is able to compensate for
the model mismatch (recall CME is the
« true » system)
In a case where initial state knowledge
is not perfect (arabinose uptake
started before expected) we show that
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the estimator is able to improve
the state knowledge, and
it performs as good as a heavy
estimator (particle filter) based on
the CME
Estimation in presence of extrinsic noise
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To account for random parameters in filtering, we take a state-augmentation approach :
and adapt both the SR-UKF and the PF accordingly.
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In simulation, we consider only some plausible sources of extrinsic noise
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Extrinsic noise on Ku
(Ku depends on unmodelled AraC concentration that may change across
individuals)
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Extrinsic noise on v4
(mGFP synthesis rate proportional to variable plasmid copy number)
Simulation results
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We build the SR-UKF assuming that a ME
extrinsic noise model is known.
We consider two cases :
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Extrinsic noise on Ku (left)
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Extrinsic noise on v4
(mGFP synthesis rate proportional to
variable plasmid copy number)
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We run the augmented version of the SRUKF on data generated from the ME model
In both cases, state estimation explicitly
accounting for extrinsic noise improves :
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the knowledge of the individual
parameters, and
State estimates
with respect to state estimators assuming
identical parameters across individuals
Discussion
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State estimation from single fluorescence output can improve knowledge of the
internal state
Use of stochastic model approximations to simplify estimation problem does not spoil
performance
Modelling extrinsic noise (without knowing the actual individual parameter values!) is
important to improve knowledge of the internal state
Testing & applications to real data on this and other systems
... Thank you !
[email protected]
www.inria.fr