Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Switching Regulatory Models of Cellular Stress
Reaction
Guido Sanguinetti
Joint work with M. Opper, A. Ruttor and C. Archambeau
Computer Science/ ChELSI - The University of Sheffield
StoMP, Jul 2009
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Outline of the talk
1
Stochastic models and inference
2
Basic problem
3
Single Input Motif
Model
Results
4
Conclusions and future work
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Where does stochasticity come from?
My personal view: three types of stochasticity
Intrinsic stochasticity: a deterministic description of the
system is not appropriate regardless of the amount of
information available (e.g. quantum mechanics, perhaps single
cell protein production)
Induced stochasticity: the system is truly deterministic but we
have no information about some parts of the system, hence
for all practical purposes it should be modelled as a stochastic
system
Noise: the system is deterministic but observations are noisy,
hence stochasticity
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Bayesian inference
A stochastic model defines a probability distribution over its
variables x
We can therefore use observations x̂ to update the model
using Bayes’ theorem
p (x|x̂) =
p (x̂|x) p (x)
p(x̂)
p(x) is our prior model
p (x̂|x) is the likelihood, connecting observations and model
The updated belief or posterior is the prior re-weighted by the
likelihood
The bottleneck
is computing the marginal
P
p (x̂) = x p (x̂|x) p (x)
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Basic problem
We wish to predict dynamics of transcription factors (TFs)
during adaptation to stress, based on time-course mRNA
profiles
Example: E. coli transition between aerobic and anaerobic
states (see previous talk)
Not just interested in steady states
Due to difficulties in measuring TF activity, treat it as an
inference problem with TFs as latent variables (functions)
Not the first one to think of it: see Liao et al, Sabatti and
James, Khanin et al, Barenco et al, Rogers et al, Lawrence et
al...
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Model
Results
Single Input Motif
In general, most genes have complex promoter structures with
several TFs interacting
The single input motif (SIM) is a specific network motif where
several genes are controlled by a single TF
The TF input to the SIM are generally Master regulators, TFs
who control hundreds of genes and generally are associated
with large shifts in cellular behaviour
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Model
Results
Basic problem
Consider an ODE model of SIM dynamics
dxi (t)
= g (f (t) , θi ) + bi − λi xi (t)
dt
Given time course observations of the expression levels of the
target genes xi , infer the profile of the transcription factor f
and the model parameters θi , bi and λi
Problem originally considered by Barenco et al. (linear
dependence on the TF), and then Lawrence et al., Khanin et
al., Rogers et al.,...
We wish to hardwire into the model the fast dynamics of
stress response
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Model
Results
Swicthing latent process
We assume TF activity jumps quickly from zero to saturation
level
dxi (t)
= Ai µ (t) + bi − λi xi (t)
(1)
dt
where µ(t) ∈ {0, 1}
The driving process µ(t) is modelled as a two-states Markov
jump process, also known as a telegraph process. NB: This is
not a logical approximation to Michaelis-Menten.
Given transition rates f0,1 (t) for the process, the probability
p1 (t) of µ(t) = 1 at a given time is given by the following
Master equation
dp1 (t)
= −(f1 + f0 )p1 (t) + f1 (t) .
dt
Guido Sanguinetti
(2)
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Model
Results
Why?
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Activation
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Left: Michaelis-Menten activation as function of f . Right:
Michaelis-Menten activation as function of t, with f starting
to change exponentially at t = 200.
Biologically meaningful? More identifiable?
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Model
Results
Inference
Exact inference is theoretically possible for this model
It relies on a forward-backward procedure, involving solving
iteratively PDEs numerically
An alternative is to use a variational formulation and find an
optimal approximate solution
We compute the Kullback-Leibler (KL) divergence between
the posterior process and an approximating telegraph
(Markov) process q (µ|g± )
KL [qkppost ] = ln Z + KL [qkpprior ] −
N
X
Eq [ln p (x̂j |x (tj ))] .
j=1
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Model
Results
Toy example
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Figure: Left: variational inference A = 2.3 ± 0.5 × 10−3 ,
b = 1.0 ± 0.2 × 10−3 , λ = 4 ± 0.3 × 10−3 . Right: exact inference
A = 3.2 ± 1.1 × 10−3 , b = 0.08 ± 0.6 × 10−3 , λ = 3.1 ± 1.3 × 10−3 .
True values A = 3.7 × 10−3 , b = 0.8 × 10−3 , λ = 5 × 10−3 .
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Model
Results
FNR regulation
As a real example on which to test our approach, we
considered transcriptomic measurements of the reaction of
E.coli to sudden oxygen starvation
When oxygen is removed, Fe-S clusters are generated which
dimerise and activate the master regulator FNR
FNR activation is thought to be the main channel used by the
bacterium to switch between aerobic and nitrate metabolism
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Model
Results
FNR regulation
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ompW
yjiD
hypB
moaA
aspA
Figure: Results on E.coli data: (a) posterior mean FNR profile; (b) half
lives of targets (in minutes) with uncertainty, inferred (triangles on the
right) versus experimentally measured. No measurement of the half life
of yjiD or moaA is available.
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Conclusions
We have proposed a novel TF inference framework which
arguably could describe better some biological conditions
It is of interest in its own right as an example of hybrid
discrete-continuous (and stochastic/deterministic) model
It can identify both the time profile of TF activity and the
model parameters, including non-trivial interaction terms
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction
Stochastic models and inference
Basic problem
Single Input Motif
Conclusions and future work
Future work
What type of data do we need for a large-scale application?
Hierarchical models of transcriptional regulation (e.g. FFL)
Try to model dynamics of the signal too (e.g., oxygen
metabolism)
Consider SDEs driven by telegraph noise, look at single cell
data
Guido Sanguinetti
Switching Regulatory Models of Cellular Stress Reaction