Maximum likelihood estimation of biophysical parameters of synaptic receptors from macroscopic currents

Bayesian inference of kinetic schemes for ion channels by Kalman filtering

Inferring adequate kinetic schemes for ion channel gating from ensemble currents is a daunting task due to limited information in the data. We address this problem by using a parallelized Bayesian filter to specify hidden Markov models for current and fluorescence data. We demonstrate the flexibility of this algorithm by including different noise distributions. Our generalized Kalman filter outperforms both a classical Kalman filter and a rate equation approach when applied to patch-clamp data exhibiting realistic open-channel noise. The derived generalization also enables inclusion of orthogonal fluorescence data, making unidentifiable parameters identifiable and increasing the accuracy of the parameter estimates by an order of magnitude. By using Bayesian highest credibility volumes, we found that our approach, in contrast to the rate equation approach, yields a realistic uncertainty quantification. Furthermore, the Bayesian filter delivers negligibly biased estimates for a wider range of data quality. For some data sets, it identifies more parameters than the rate equation approach. These results also demonstrate the power of assessing the validity of algorithms by Bayesian credibility volumes in general. Finally, we show that our Bayesian filter is more robust against errors induced by either analog filtering before analog-to-digital conversion or by limited time resolution of fluorescence data than a rate equation approach.

Parameter Optimization for Ion Channel Models: Integrating New Data with Known Channel Properties

Ion channels play a central role in membrane physiology, but to fully understand how they operate, one must have accurate kinetic mechanisms. Estimating kinetics is not trivial when the mechanism is complex, and a large number of parameters must be extracted from data. Furthermore, the information contained in the data is often limited, and the model may not be fully determined. The solution is to reduce the number of parameters and to estimate them in such a way that they not only describe well the new data but also agree with the existing knowledge. In a previous study, we presented a comprehensive formalism for estimating kinetic parameters subject to a variety of explicit and implicit constraints that define quantitative relationships between parameters and describe specific mechanism properties. Here, we introduce the reader to the QuB software, which implements this constraining formalism. QuB features a powerful visual interface and a high-level scripting language that can be used to formulate kinetic models and constraints of arbitrary complexity, and to efficiently estimate the parameters from a variety of experimental data.

Estimating kinetic mechanisms with prior knowledge II: Behavioral constraints and numerical tests

In their preceding paper, Salari et al. describe a formalism that allows existing knowledge to be enforced into kinetic models. Here, Navarro et al. present a penalty-based optimization mechanism to incorporate arbitrary parameter relationships and constraints that quantify the behavior of the model.

Kinetic mechanisms predict how ion channels and other proteins function at the molecular and cellular levels. Ideally, a kinetic model should explain new data but also be consistent with existing knowledge. In this two-part study, we present a mathematical and computational formalism that can be used to enforce prior knowledge into kinetic models using constraints. Here, we focus on constraints that quantify the behavior of the model under certain conditions, and on constraints that enforce arbitrary parameter relationships. The penalty-based optimization mechanism described here can be used to enforce virtually any model property or behavior, including those that cannot be easily expressed through mathematical relationships. Examples include maximum open probability, use-dependent availability, and nonlinear parameter relationships. We use a simple kinetic mechanism to test multiple sets of constraints that implement linear parameter relationships and arbitrary model properties and behaviors, and we provide numerical examples. This work complements and extends the companion article, where we show how to enforce explicit linear parameter relationships. By incorporating more knowledge into the parameter estimation procedure, it is possible to obtain more realistic and robust models with greater predictive power.

Estimating kinetic mechanisms with prior knowledge I: Linear parameter constraints

New mathematical tools are needed to incorporate existing knowledge into kinetic models of ion channels and other proteins. Salari et al. describe an algebraic transformation that can enforce linearly interdependent parameters into kinetic models in order to test new hypotheses.

To understand how ion channels and other proteins function at the molecular and cellular levels, one must decrypt their kinetic mechanisms. Sophisticated algorithms have been developed that can be used to extract kinetic parameters from a variety of experimental data types. However, formulating models that not only explain new data, but are also consistent with existing knowledge, remains a challenge. Here, we present a two-part study describing a mathematical and computational formalism that can be used to enforce prior knowledge into the model using constraints. In this first part, we focus on constraints that enforce explicit linear relationships involving rate constants or other model parameters. We develop a simple, linear algebra–based transformation that can be applied to enforce many types of model properties and assumptions, such as microscopic reversibility, allosteric gating, and equality and inequality parameter relationships. This transformation converts the set of linearly interdependent model parameters into a reduced set of independent parameters, which can be passed to an automated search engine for model optimization. In the companion article, we introduce a complementary method that can be used to enforce arbitrary parameter relationships and any constraints that quantify the behavior of the model under certain conditions. The procedures described in this study can, in principle, be coupled to any of the existing methods for solving molecular kinetics for ion channels or other proteins. These concepts can be used not only to enforce existing knowledge but also to formulate and test new hypotheses.