Background
Parameter Identifiability in Nonlinear Biophysical Models
Modeling Single Molecule Time Series Using Nonparametric Bayesian Inference
Jensen et al., 2012, Mechanism of Voltage Gating in Potassium Channels, Science , 336, 6078.
Physiological relevance- we only need to account for some of this complexity
Imagery stolen from Jim Crutchfield lecture notes
Stefan et al, 2009. Computing phenomenologic Adair-Klotz constants from microscopic MWC parameters. BMC Systems Biology. 3:68.
Large regions of this parameter space can fit any data extremely well
Practical Non-identifiability
Parameters cannot be inferred accurately even with noiseless data
Analytical methods exist, but can only be used in special cases. Worse, such methods can be misleading, as in the case of practical non-identifiability.
We might calculate the Error (likelihood) over the whole parameter space, but this is infeasible for many parameters.
We need an efficent way to identify the regions of parameter space that lead to good agreement with the data.
The posterior distribution quantifies which regions of the parameter space provide a good explanation of the data.
Bayes' rule specifies how to calculate posterior probability, and Markov chain Monte Carlo provides an efficient method to estimate high-dimensional posterior distributions.
Dynamical Systems
Dynamical Systems
Dynamical Systems
Dynamical Systems (Non-Identifiable)
Dynamical Systems (Non-Identifiable)
Modeling now becomes an iterative process, where non-identifiability forces innovative experimentation
Rely on a flexible class of infinite dimensional probability distributions - the Dirichlet process
We can extract structure from data, instead of assuming models beforehand
Structure might refer to hidden closed and open states in single channel recordings, or to conformational states in FRET traces, or to bleaching events in photobleaching traces
Provides an infinite dimensional probability distribution
But has useful clustering properties when modeling finite data
We now model the transition matrix as the bi-infinite Hierarchical Dirichlet Process
Our model now specifies transitions to and from an infinite number of hidden states
Using this infinite model allows us to discover the number of hidden states in a time series
In each of these cases, we can detect the presence of distinct hidden states.
We think there's only two conductance states (open and closed), but an unknown number of hidden states
Again, we now model the transition matrix as the bi-infinite Hierarchical Dirichlet Process
Our model now specifies transitions to and from an infinite number of hidden states, each of which appears as either open or closed
Using this infinite model allows us to discover the number of hidden states in single channel recording
With simulated data, we can indeed discover hidden states based only on their different dynamics
Application to BK channel data
Nonparametric Bayes approach uncovers the complexity of channel gating