Statistical evaluation of ion-channel gating models based on distributions of log-likelihood ratios

Multiple conductance states of lipid pores during Voltage-Clamp electroporation

We used voltage-clamp electroporation to obtain single-channel recordings of lipid pores and analyzed the idealized dwell-time sequences using Maximum-Likelihood Fitting. We observed traces with multiple current levels and determined whether they were a result of the presence of multiple pores or a single pore with multiple conductance states. We found that, within the same recording, the bilayer can have a single pore with multiple conductance states or multiple independent pores. Using high sampling rates (100 kHz) we were able to observe pores with 40 μs lifetimes, and in experiments using high-voltage pulses we observed the existence of long-lived fluctuations minutes after the removal of the electric field. These results come closer to reconciling the nanosecond lifetime pores in molecular dynamics simulations and the long-lived permeability of cells after electroporation.

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.

Computing rates of Markov models of voltage-gated ion channels by inverting partial differential equations governing the probability density functions of the conducting and non-conducting states

Markov models are ubiquitously used to represent the function of single ion channels. However, solving the inverse problem to construct a Markov model of single channel dynamics from bilayer or patch-clamp recordings remains challenging, particularly for channels involving complex gating processes. Methods for solving the inverse problem are generally based on data from voltage clamp measurements. Here, we describe an alternative approach to this problem based on measurements of voltage traces. The voltage traces define probability density functions of the functional states of an ion channel. These probability density functions can also be computed by solving a deterministic system of partial differential equations. The inversion is based on tuning the rates of the Markov models used in the deterministic system of partial differential equations such that the solution mimics the properties of the probability density function gathered from (pseudo) experimental data as well as possible. The optimization is done by defining a cost function to measure the difference between the deterministic solution and the solution based on experimental data. By evoking the properties of this function, it is possible to infer whether the rates of the Markov model are identifiable by our method. We present applications to Markov model well-known from the literature.

Analyzing single-molecule time series via nonparametric Bayesian inference

The ability to measure the properties of proteins at the single-molecule level offers an unparalleled glimpse into biological systems at the molecular scale. The interpretation of single-molecule time series has often been rooted in statistical mechanics and the theory of Markov processes. While existing analysis methods have been useful, they are not without significant limitations including problems of model selection and parameter nonidentifiability. To address these challenges, we introduce the use of nonparametric Bayesian inference for the analysis of single-molecule time series. These methods provide a flexible way to extract structure from data instead of assuming models beforehand. We demonstrate these methods with applications to several diverse settings in single-molecule biophysics. This approach provides a well-constrained and rigorously grounded method for determining the number of biophysical states underlying single-molecule data.

Single-channel analysis of glutamate receptors

This is a companion to UNIT 11.16: Single-Channel Recording of Glutamate Receptors. Described here are techniques for analyzing single-channel currents recorded from glutamate receptors to characterize their properties. In addition, issues that need to be taken into account when analyzing glutamate receptor single-channel recording results are discussed.

Putative chanzyme activity of TRPM2 cation channel is unrelated to pore gating

Transient receptor potential melastatin 2 (TRPM2) is a Ca(2+)-permeable cation channel expressed in immune cells of phagocytic lineage, pancreatic β cells, and brain neurons and is activated under oxidative stress. TRPM2 activity is required for immune cell activation and insulin secretion and is responsible for postischemic neuronal cell death. TRPM2 is opened by binding of ADP ribose (ADPR) to its C-terminal cytosolic nudix-type motif 9 (NUDT9)-homology (NUDT9-H) domain, which, when expressed in isolation, cleaves ADPR into AMP and ribose-5-phosphate. A suggested coupling of this enzymatic activity to channel gating implied a potentially irreversible gating cycle, which is a unique feature of a small group of channel enzymes known to date. The significance of such a coupling lies in the conceptually distinct pharmacologic strategies for modulating the open probability of channels obeying equilibrium versus nonequilibrium gating mechanisms. Here we examine the potential coupling of TRPM2 enzymatic activity to pore gating. Mutation of several residues proposed to enhance or eliminate NUDT9-H catalytic activity all failed to affect channel gating kinetics. An ADPR analog, α-β-methylene-ADPR (AMPCPR), was shown to be entirely resistant to hydrolysis by NUDT9, but nevertheless supported TRPM2 channel gating, albeit with reduced apparent affinity. The rate of channel deactivation was not slowed but, rather, accelerated in AMPCPR. These findings, as well as detailed analyses of steady-state gating kinetics of single channels recorded in the presence of a range of concentrations of ADPR or AMPCPR, identify TRPM2 as a simple ligand-gated channel that obeys an equilibrium gating mechanism uncoupled from its enzymatic activity.

Catalyst-like modulation of transition states for CFTR channel opening and closing: new stimulation strategy exploits nonequilibrium gating

Two gating transition states determine open probability of CFTR (the chloride channel mutated in cystic fibrosis), defining strategic targets for therapeutic intervention.

Cystic fibrosis transmembrane conductance regulator (CFTR) is the chloride ion channel mutated in cystic fibrosis (CF) patients. It is an ATP-binding cassette protein, and its resulting cyclic nonequilibrium gating mechanism sets it apart from most other ion channels. The most common CF mutation (ΔF508) impairs folding of CFTR but also channel gating, reducing open probability (P_o). This gating defect must be addressed to effectively treat CF. Combining single-channel and macroscopic current measurements in inside-out patches, we show here that the two effects of 5-nitro-2-(3-phenylpropylamino)benzoate (NPPB) on CFTR, pore block and gating stimulation, are independent, suggesting action at distinct sites. Furthermore, detailed kinetic analysis revealed that NPPB potently increases P_o, also of ΔF508 CFTR, by affecting the stability of gating transition states. This finding is unexpected, because for most ion channels, which gate at equilibrium, altering transition-state stabilities has no effect on P_o; rather, agonists usually stimulate by stabilizing open states. Our results highlight how for CFTR, because of its unique cyclic mechanism, gating transition states determine P_o and offer strategic targets for potentiator compounds to achieve maximal efficacy.

Conformational changes in the catalytically inactive nucleotide-binding site of CFTR

A central step in the gating of the cystic fibrosis transmembrane conductance regulator (CFTR) chloride channel is the association of its two cytosolic nucleotide-binding domains (NBDs) into a head-to-tail dimer, with two nucleotides bound at the interface. Channel opening and closing, respectively, are coupled to formation and disruption of this tight NBD dimer. CFTR is an asymmetric adenosine triphosphate (ATP)-binding cassette protein in which the two interfacial-binding sites (composite sites 1 and 2) are functionally different. During gating, the canonical, catalytically active nucleotide-binding site (site 2) cycles between dimerized prehydrolytic (state O₁), dimerized post-hydrolytic (state O₂), and dissociated (state C) forms in a preferential C→O₁→O₂→C sequence. In contrast, the catalytically inactive nucleotide-binding site (site 1) is believed to remain associated, ATP-bound, for several gating cycles. Here, we have examined the possibility of conformational changes in site 1 during gating, by studying gating effects of perturbations in site 1.

Previous work showed that channel closure is slowed, both under hydrolytic and nonhydrolytic conditions, by occupancy of site 1 by N⁶-(2-phenylethyl)-ATP (P-ATP) as well as by the site-1 mutation H1348A (NBD2 signature sequence). Here, we found that P-ATP prolongs wild-type (WT) CFTR burst durations by selectively slowing (>2×) transition O₁→O₂ and decreases the nonhydrolytic closing rate (transition O₁→C) of CFTR mutants K1250A (∼4×) and E1371S (∼3×). Mutation H1348A also slowed (∼3×) the O₁→O₂ transition in the WT background and decreased the nonhydrolytic closing rate of both K1250A (∼3×) and E1371S (∼3×) background mutants. Neither P-ATP nor the H1348A mutation affected the 1:1 stoichiometry between ATP occlusion and channel burst events characteristic to WT CFTR gating in ATP. The marked effect that different structural perturbations at site 1 have on both steps O₁→C and O₁→O₂ suggests that the overall conformational changes that CFTR undergoes upon opening and coincident with hydrolysis at the active site 2 include significant structural rearrangement at site 1.

Efficient maximum likelihood estimation of kinetic rate constants from macroscopic currents

A new method is described that accurately estimates kinetic constants, conductance and number of ion channels from macroscopic currents. The method uses both the time course and the strength of correlations between different time points of macroscopic currents and utilizes the property of semiseparability of covariance matrix for computationally efficient estimation of current likelihood and its gradient. The number of calculation steps scales linearly with the number of channel states as opposed to the cubic dependence in a previously described method. Together with the likelihood gradient evaluation, which is almost independent of the number of model parameters, the new approach allows evaluation of kinetic models with very complex topologies. We demonstrate applicability of the method to analysis of synaptic currents by estimating accurately rate constants of a 7-state model used to simulate GABAergic macroscopic currents.

Strict coupling between CFTR's catalytic cycle and gating of its Cl- ion pore revealed by distributions of open channel burst durations

CFTR, the ABC protein defective in cystic fibrosis, functions as an anion channel. Once phosphorylated by protein kinase A, a CFTR channel is opened and closed by events at its two cytosolic nucleotide binding domains (NBDs). Formation of a head-to-tail NBD1/NBD2 heterodimer, by ATP binding in two interfacial composite sites between conserved Walker A and B motifs of one NBD and the ABC-specific signature sequence of the other, has been proposed to trigger channel opening. ATP hydrolysis at the only catalytically competent interfacial site is suggested to then destabilize the NBD dimer and prompt channel closure. But this gating mechanism, and how tightly CFTR channel opening and closing are coupled to its catalytic cycle, remains controversial. Here we determine the distributions of open burst durations of individual CFTR channels, and use maximum likelihood to evaluate fits to equilibrium and nonequilibrium mechanisms and estimate the rate constants that govern channel closure. We examine partially and fully phosphorylated wild-type CFTR channels, and two mutant CFTR channels, each bearing a deleterious mutation in one or other composite ATP binding site. We show that the wild-type CFTR channel gating cycle is essentially irreversible and tightly coupled to the ATPase cycle, and that this coupling is completely destroyed by the NBD2 Walker B mutation D1370N but only partially disrupted by the NBD1 Walker A mutation K464A.

Inverse effects on gating and modulation caused by a mutation in the M2-M3 Linker of the GABA(A) receptor gamma subunit

M2-M3 linkers are receptor subunit domains known to be critical for the normal function of cysteine-loop ligand-gated ion channels. Previous studies of alpha and beta subunits of type "A" GABA receptors suggest that these linkers couple extracellular elements involved in GABA binding to the transmembrane segments that control the opening of the ion channel. To study the importance of the gamma subunit M2-M3 linker, we examined the macroscopic and single-channel effects of an engineered gamma2(L287A) mutation on GABA activation and propofol modulation. In the macroscopic analysis, we found that the gamma2(L287A) mutation decreased GABA potency but increased the ability of propofol to enhance both GABA potency and efficacy compared with wild-type receptors. Indeed, although propofol had significant effects on GABA potency in wild-type receptors, we found that propofol produced no corresponding increase in GABA efficacy. At the single-channel level, mutant receptors showed a loss in the longest of three open-time components compared with wild-type receptors under GABA activation. Furthermore, propofol reduced the duration of one closed-time component, increased the duration of two open-time components, and generated a third open component with a longer lifetime in mutant compared with wild-type receptors. Taken together, we conclude that although the gamma subunit is not required for the binding of GABA or propofol, the M2-M3 linker of this subunit plays a critical role in channel gating by GABA and allosteric modulation by propofol. Our results also suggest that in wild-type receptors, propofol exerts its enhancing effects by mechanisms extrinsic to channel gating.

Real-time kinetic modeling of voltage-gated ion channels using dynamic clamp

We propose what to our knowledge is a new technique for modeling the kinetics of voltage-gated ion channels in a functional context, in neurons or other excitable cells. The principle is to pharmacologically block the studied channel type, and to functionally replace it with dynamic clamp, on the basis of a computational model. Then, the parameters of the model are modified in real time (manually or automatically), with the objective of matching the dynamical behavior of the cell (e.g., action potential shape and spiking frequency), but also the transient and steady-state properties of the model (e.g., those derived from voltage-clamp recordings). Through this approach, one may find a model and parameter values that explain both the observed cellular dynamics and the biophysical properties of the channel. We extensively tested the method, focusing on Na(v) models. Complex Markov models (10-12 states or more) could be accurately integrated in real time at >50 kHz using the transition probability matrix, but not the explicit Euler method. The practicality of the technique was tested with experiments in raphe pacemaker neurons. Through automated real-time fitting, a Hodgkin-Huxley model could be found that reproduced well the action potential shape and the spiking frequency. Adding a virtual axonal compartment with a high density of Na(v) channels further improved the action potential shape. The computational procedure was implemented in the free QuB software, running under Microsoft Windows and featuring a friendly graphical user interface.

The N-terminal transmembrane domain (TMD0) and a cytosolic linker (L0) of sulphonylurea receptor define the unique intrinsic gating of KATP channels

ATP-sensitive potassium (K(ATP)) channels comprise four pore-forming Kir6 and four regulatory sulphonylurea receptor (SUR) subunits. SUR, an ATP-binding cassette protein, associates with Kir6 through its N-terminal transmembrane domain (TMD0). TMD0 connects to the core domain of SUR through a cytosolic linker (L0). The intrinsic gating of Kir6.2 is greatly altered by SUR. It has been hypothesized that these changes are conferred by TMD0. Exploiting the fact that the pancreatic (SUR1/Kir6.2) and the cardiac (SUR2A/Kir6.2) K(ATP) channels show different gating behaviours, we have tested this hypothesis by comparing the intrinsic gating of Kir6.2 with the last 26 residues deleted (Kir6.2Delta26) co-expressed with SUR1, S1-TMD0, SUR2A and S2-TMD0 at -40 and -100 mV (S is an abbreviation for SUR; TMD0/Kir6.2Delta26, but not TMD0/Kir6.2, can exit the endoplastic reticulum and reach the cell membrane). Single-channel kinetic analyses revealed that the mean burst and interburst durations are shorter for TMD0/Kir6.2Delta26 than for the corresponding SUR channels. No differences were found between the two TMD0 channels. We further demonstrated that in isolation even TMD0-L0 (SUR truncated after L0) cannot confer the wild-type intrinsic gating to Kir6.2Delta26 and that swapping L0 (SUR truncated after L0)between SUR1 and SUR2A only partially exchanges their different intrinsic gating. Therefore, in addition to TMD0, L0 and the core domain also participate in determining the intrinsic gating of Kir6.2. However, TMD0 and L0 are responsible for the different gating patterns of full-length SUR1 and SUR2A channels. A kinetic model with one open and four closed states is presented to explain our results in a mechanistic context.