Modelling ion channels with a view towards identifiability

TL;DR

This paper analyzes the properties and limitations of aggregated Markov models for ion channels, focusing on their non-identifiability issues and implications for mechanistic modeling.

Contribution

It provides a detailed analysis of the non-identifiability of aggregated Markov models with multiple states and discusses alternative modeling approaches based on additional data sources.

Findings

Models with more states than 2 n_O n_C are non-identifiable.

Two derivations of the classical non-identifiability result are presented.

Non-identifiability impacts the interpretability of mechanistic ion channel models.

Abstract

Aggregated Markov models provide a flexible framework for stochastic dynamics that develops on multiple timescales. For example, Markov models for ion channels often consist of multiple open and closed state to account for "slow" and "fast" openings and closings of the channel. The approach is a popular tool in the construction of mechanistic models of ion channels - instead of viewing model states as generators of sojourn times of a certain characteristic length, each individual model state is interpreted as a representation of a distinct biophysical state. We will review the properties of aggregated Markov models and discuss the implications for mechanistic modelling. First, we show how the aggregated Markov models with a given number of states can be calculated using P\'olya enumeration However, models with $n_O$ open and $n_C$ closed states that exceed the maximum number $2 n_O n_C$ of parameters are non-identifiable. We will present two derivations for this classical result and investigate non-identifiability further via a detailed analysis of the non-identifiable fully connected three-state model. Finally, we will discuss the implications of non-identifiability for mechanistic modelling of ion channels. We will argue that instead of designing models based on assumed transitions between distinct biophysical states which are modulated by ligand binding, it is preferable to build models based on additional sources of data that give more direct insight into the dynamics of conformational changes.