Implementing Bayesian Methods for Ion Channel Modelling
Michael Epstein, Prof. Lucia Sivilotti, Dr Ben Calderhead and Prof. Mark Girolami
CoMPLEX,UCL
Department of Statistical Science, UCL
Department of Neuroscience, Physiology & Pharmacology, UCL
What are Ligand Gated Ion Channels?
Modelling Single Ion Channels as Stochastic Processes
Example Model Fitting employing MCMC
This gating process is statistically modelled as an aggregated Markov
process [1]:
• The underlying mechanism is a finite state space, continuous time
Markov process, S(t), t ≥ 0, parameterised by:
• a generator matrix, Q
• initial state distribution p(0)
• The observed signal is an aggregated process, where sets of states
typically can have equal conductance levels. Consider an example
scheme below for the acetylcholine receptor:
An example model fitting is shown below for the simple del Castillo &
Katz [3] mechanism, employing an wide uniform prior for the rate
constants.
A Metropolis-Hastings [4] sampler
was employed to generate posterior
samples from two simulated data
sets of 1 and 5 seconds in length.
Parameter estimates, correlations
and mixing quality for the synthetic
data sets are shown below:
AR*
α1
β1
AR k+1xA
R
k−1
k+2xA
Posterior parameter distributions
A2R*
Open States
k+1
AR*
k−1
α
β
0.035
0.030
density
0.025
Ion channels are transmembrane proteins which are crucial for cell
excitability. Ligand-gated channels open and close by undergoing
conformational changes in the protein shape in response to the binding
of a chemical, such as a neurotransmitter.
0.020
0.015
k−2
0.010
0.005
0.000
α2
50
α1
β2
100
150
200
50
100
150
200
Parameter Value
β1
900
1000
1100
1200
800
850
900
950
1000
1050
1100
1150
sample
1 second
5 seconds
Posterior parameter sample mixing
k−2
AR
k+1
k−1
200
R
k−1
Parameter Value
Understanding dynamical changes in ion channel structure is key to
understanding channel function. Structural data gives static pictures of
some of the channel’s most stable states but cannot account for
dynamical changes as the channel opens and closes in the membrane.
• Modelling the gating process therefore helps suggest and
subsequently validate conformational changes which occur during
channel gating
• From this, we can explain how different agonists or mutations affect
the speed or the nature of conformational changes
• This will help predict the impact of novel agonists - leading to a
process of rational drug design
Closed States
A2R
2k+1xA
α
β
1150
200
1100
1200
1050
150
150
1000
1100
950
100
1000
100
900
850
900
50
800
50
1000
2000
3000
4000
1000
2000
3000
4000
1000
2000
3000
4000
1000
2000
3000
4000
Sample
sample
1 second
5 seconds
Fitting Ion-Channel Models using a Bayesian Approach
Posterior pairwise k+1 correlations
k−1
alpha
beta
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1200
Fitting a hypothesised mechanism (Colquhoun et al. [2]) is currently
achieved by maximising the likelihood of the recorded data (y) for a
given model (H) and set of rate constant parameters (θ). However, ML
model parameterisations have limitations:
Parameter Value
Why is Modelling Ion Channel Structure Important?
k+2xA
1000
800
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k+1
• ML estimation produces only point estimates of parameters
●
sample
1 second
5 seconds
●
• The fitting has difficulties with multi-modal likelihood surfaces
Gathering Data from Ion Channels
• It is difficult to compare and choose between competing models
However, adopting a Bayesian framework provides a natural way of
taking model and parameter uncertainty into account in a
mathematically rigorous way.
Consider Bayes’ rule:
P(y | θ, H)P(θ | H)
R
P(θ | y, H) =
P(y | θ0 , H)P(θ0 | H) dθ0
Typically, the marginal likelihood is analytically intractable so the
application of Bayesian methods typically requires the use of sampling
techniques, such as Markov chain Monte Carlo methods, to draw
samples from posterior distributions. These samples can be used for
parameter inference and estimating marginal likelihoods.
Time series data is gathered from patch-clamp recordings of cells.
Glycine receptors are good models for the nicotinic channel superfamily
as they express well in recombinant cell lines, there is no agonist
blocking of the membrane pore and the signal-to-noise ratio is large.
Additional Challenges and Future Investigations
Significant challenges remain in sampling more complicated ion channel
models using experimental data. In particular we need to
• accurately estimate marginal likelihoods for model comparison
• improve sampling ( reduce autocorrelations, optimise step-sizes)
• model the noise process and "missed events" accurately
Future work will use sophisticated MCMC methods (e.g. RMHMC, SMC,
ABC, pMCMC) for investigating higher dimensional models.
Bibliography
[1]
D. Colquhoun and A. G. Hawkes. “On the stochastic properties of bursts of single ion channel openings and of clusters of bursts”. In: Philosophical
Transactions of the Royal Society of London. B, Biological Sciences 300.1098 (1982), pp. 1–59.
[2]
D. Colquhoun, A. G. Hawkes, and K. Srodzinski. “Joint distributions of apparent open and shut times of single-ion channels and maximum likelihood
fitting of mechanisms”. In: Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
354.1718 (1996), pp. 2555–2590.
[3]
J. Del Castillo and B. Katz. “Interaction at end-plate receptors between different choline derivatives”. In: Proceedings of the Royal Society of London.
Series B-Biological Sciences 146.924 (1957), pp. 369–381.
[4]
W. K. Hastings. “Monte Carlo sampling methods using Markov chains and their applications”. In: Biometrika 57.1 (1970), pp. 97–109.