Structural identifiability of equilibrium ligand-binding parameters

Classical structural identifiability methodology applied to low-dimensional dynamic systems in receptor theory

Mathematical modelling has become a key tool in pharmacological analysis, towards understanding dynamics of cell signalling and quantifying ligand-receptor interactions. Ordinary differential equation (ODE) models in receptor theory may be used to parameterise such interactions using timecourse data, but attention needs to be paid to the theoretical identifiability of the parameters of interest. Identifiability analysis is an often overlooked step in many bio-modelling works. In this paper we introduce structural identifiability analysis (SIA) to the field of receptor theory by applying three classical SIA methods (transfer function, Taylor Series and similarity transformation) to ligand-receptor binding models of biological importance (single ligand and Motulsky-Mahan competition binding at monomers, and a recently presented model of a single ligand binding at receptor dimers). New results are obtained which indicate the identifiable parameters for a single timecourse for Motulsky-Mahan binding and dimerised receptor binding. Importantly, we further consider combinations of experiments which may be performed to overcome issues of non-identifiability, to ensure the practical applicability of the work. The three SIA methods are demonstrated through a tutorial-style approach, using detailed calculations, which show the methods to be tractable for the low-dimensional ODE models.

Identifiability of equilibrium constants for receptors with two to five binding sites

Ligand-gated ion channels (LGICs) are regularly oligomers containing between two and five binding sites for ligands. Neither in homomeric nor heteromeric LGICs the activation process evoked by the ligand binding is fully understood. Here, we show on theoretical grounds that for LGICs with two to five binding sites, the cooperativity upon channel activation can be determined in considerable detail. The main requirements for our strategy are a defined number of binding sites in a channel, which can be achieved by concatenation, a systematic mutation of all binding sites and a global fit of all concentration-activation relationships (CARs) with corresponding intimately coupled Markovian state models. We take advantage of translating these state models to cubes with dimensions 2, 3, 4, and 5. We show that the maximum possible number of CARs for these LGICs specify all 7, 13, 23, and 41 independent model parameters, respectively, which directly provide all equilibrium constants within the respective schemes. Moreover, a fit that uses stochastically varied scaled unitary start vectors enables the determination of all parameters, without any bias imposed by specific start vectors. A comparison of the outcome of the analyses for the models with 2 to 5 binding sites showed that the identifiability of the parameters is best for a case with 5 binding sites and 41 parameters. Our strategy can be used to analyze experimental data of other LGICs and may be applicable to voltage-gated ion channels and metabotropic receptors.

Benzodiazepine Modulation of GABAA Receptors: A Mechanistic Perspective

Benzodiazepines (BZDs) are a class of widely prescribed psychotropic drugs that target GABA_A receptors (GABA_ARs) to tune inhibitory synaptic signaling throughout the central nervous system. Despite knowing their molecular target for over 40 years, we still do not fully understand the mechanism of modulation at the level of the channel protein. Nonetheless, functional studies, together with recent cryo-EM structures of GABA_A(α1)₂(βX)₂(γ2)₁ receptors in complex with BZDs, provide a wealth of information to aid in addressing this gap in knowledge. Here, mechanistic interpretations of functional and structural evidence for the action of BZDs at GABA_A(α1)₂(βX)₂(γ2)₁ receptors are reviewed. The goal is not to describe each of the many studies that are relevant to this discussion nor to dissect in detail all the effects of individual mutations or perturbations but rather to highlight general mechanistic principles in the context of recent structural information.

A strategy for determining the equilibrium constants for heteromeric ion channels in a complex model

Ligand-gated ion channels are oligomers containing several binding sites for the ligands. However, the signal transmission from the ligand binding site to the pore has not yet been fully elucidated for any of these channels. In heteromeric channels, the situation is even more complex than in homomeric channels. Using published data for concatamers of heteromeric cyclic nucleotide-gated channels, we show that, on theoretical grounds, multiple functional parameters of the individual subunits can be determined with high precision. The main components of our strategy are (1) the generation of a defined subunit composition by concatenating multiple subunits, (2) the construction of 16 concatameric channels, which differ in systematically permutated binding sites, (3) the determination of respectively differing concentration-activation relationships, and (4) a complex global fit analysis with corresponding intimately coupled Markovian state models. The amount of constraints in this approach is exceedingly high. Furthermore, we propose a stochastic fit analysis with a scaled unitary start vector of identical elements to avoid any bias arising from a specific start vector. Our approach enabled us to determine 23 free parameters, including 4 equilibrium constants for the closed-open isomerizations, 4 disabling factors for the mutations of the different subunits, and 15 virtual equilibrium-association constants in the context of a 4-D hypercube. From the virtual equilibrium-association constants, we could determine 32 equilibrium-association constants of the subunits at different degrees of ligand binding. Our strategy can be generalized and is therefore adaptable to other ion channels.

The surprising difficulty of "simple" equilibrium binding measurements on ligand-gated ion channels

Godellas and Grosman revisit equilibrium binding assays to shed new light on pentameric ligand-gated ion channels.

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.

Continuous Injection Isothermal Titration Calorimetry for In Situ Evaluation of Thermodynamic Binding Properties of Ligand-Receptor Binding Models

We utilize a continuous injection approach (CIA) rather than the traditional incremental injection approach (IIA) to deliver ligand (or receptor) to the calorimeter cell to evaluate thermodynamic binding parameters for three common ligand-receptor binding models-single independent, competitive, and two independent binding sites-using isothermal titration calorimetry (ITC). A general mathematical expression for the binding isotherm for any binding stoichiometry under continuous delivery of ligand (or receptor) resulting in an analytical solution for the thermodynamic binding parameters is presented. The advantages of CIA include reduction in experimental time, estimation of thermodynamic binding parameter values, and automation of the experiment since thermodynamic parameters are estimated in situ. We demonstrate the inherent advantages of CIA over IIA for the three binding models. For the single independent site model, we utilized the binding of Ba²⁺ ions to ethylenediaminetetraacetic acid (EDTA), while competitive binding was captured by titration of Ca²⁺ ions into a buffered solution of Ba²⁺ and EDTA. We experimentally simulated a two independent binding site system by injecting Ca²⁺ into a solution of EDTA and 1,3-diaminopropane-N,N,N',N'-tetraacetic acid (DPTA). The results demonstrate estimation of thermodynamic parameters with greater confidence and simultaneous reduction in the experimental time of 83% and titrating reagent of 50%, as compared to IIA.

Non-Additive Effects of Binding Site Mutations in Calmodulin

Despite decades of research on ion-sensing proteins, gaps persist in the understanding of ion binding affinity and selectivity even in well-studied proteins such as calmodulin. Site-directed mutagenesis is a powerful and popular tool for addressing outstanding questions about biological ion binding and is employed to selectively deactivate binding sites and insert chromophores at advantageous positions within ion binding structures. However, even apparently nonperturbative mutations can distort the binding dynamics they are employed to measure. We use Fourier transform infrared (FTIR) and ultrafast two-dimensional infrared (2D IR) spectroscopy of the carboxylate asymmetric stretching mode in calmodulin as a mutation- and label-independent probe of the conformational perturbations induced in calmodulin's binding sites by two classes of mutation, tryptophan insertion and carboxylate side-chain deletion, commonly used to study ion binding in proteins. Our results show that these mutations not only affect ion binding but also induce changes in calmodulin's conformational landscape along coordinates not probed by vibrational spectroscopy, remaining invisible without additional perturbation of binding site structure. Comparison of FTIR line shapes with 2D IR diagonal slices provides a clear example of how nonlinear spectroscopy produces well-resolved line shapes, refining otherwise featureless spectral envelopes into more informative vibrational spectra of proteins.

Models to determine the kinetic mechanisms of ion-coupled transporters

Although high-resolution structures are now available for many ion-coupled, or secondary, transporters, the mechanisms by which coupling is achieved remain to be determined. Lolkema and Slotboom derive new mathematical models that can be used to analyze transport data and determine kinetic mechanisms.

With high-resolution structures available for many ion-coupled (secondary active) transporters, a major challenge for the field is to determine how coupling is accomplished. Knowledge of the kinetic mechanism of the transport reaction, which defines the binding order of substrate and co-ions, together with the sequence with which all relevant states are visited by the transporter, will help to reveal this coupling mechanism. Here, we derived general mathematical models that can be used to analyze data from steady-state transport measurements and show how kinetic mechanisms can be derived. The models describe how the apparent maximal rate of substrate transport depends on the co-ion concentration, and vice versa, in different mechanisms. Similarly, they describe how the apparent affinity for the transported substrate is affected by the co-ion concentration and vice versa. Analyses of maximal rates and affinities permit deduction of the number of co-ions that bind before, together with, and after the substrate. Hill analysis is less informative, but in some mechanisms, it can reveal the total number of co-ions transported with the substrate. However, prior knowledge of the number of co-ions from other experimental approaches is preferred when deriving kinetic mechanisms, because the models are generally overparameterized. The models we present have wide applicability for the study of ion-coupled transporters.

An Information-Theoretical Approach for Calcium Signaling Specificity

Calcium is a key signaling agent in animals and plants. Its involvement in the regulation of a wide range of processes has led to the question of how calcium signals can activate stimulus-specific responses. We introduce a computational framework for studying intracellular calcium signaling using elements of information theory. We use mutual information to quantify the differential activation of proteins in response to different calcium signals to provide an operational definition of specificity. Using optimization procedures this framework allows us to explore the biochemical determinants of calcium decoding. We explore simple toy models and general binding kinetics approaches to demonstrate the utility and limitations of the proposed framework. Unravelling signaling specificity is key for understanding information processing within cells and for the future design of synthetic nanodevices for molecular communications.

Coordination to lanthanide ions distorts binding site conformation in calmodulin

The Ca²⁺-sensing protein calmodulin (CaM) is a popular model of biological ion binding since it is both experimentally tractable and essential to survival in all eukaryotic cells. CaM modulates hundreds of target proteins and is sensitive to complex patterns of Ca²⁺ exposure, indicating that it functions as a sophisticated dynamic transducer rather than a simple on/off switch. Many details of this transduction function are not well understood. Fourier transform infrared (FTIR) spectroscopy, ultrafast 2D infrared (2D IR) spectroscopy, and electronic structure calculations were used to probe interactions between bound metal ions (Ca²⁺ and several trivalent lanthanide ions) and the carboxylate groups in CaM's EF-hand ion-coordinating sites. Since Tb³⁺ is commonly used as a luminescent Ca²⁺ analog in studies of protein-ion binding, it is important to characterize distinctions between the coordination of Ca²⁺ and the lanthanides in CaM. Although functional assays indicate that Tb³⁺ fully activates many Ca²⁺-dependent proteins, our FTIR spectra indicate that Tb³⁺, La³⁺, and Lu³⁺ disrupt the bidentate coordination geometry characteristic of the CaM binding sites' strongly conserved position 12 glutamate residue. The 2D IR spectra indicate that, relative to the Ca²⁺-bound form, lanthanide-bound CaM exhibits greater conformational flexibility and larger structural fluctuations within its binding sites. Time-dependent 2D IR lineshapes indicate that binding sites in Ca²⁺-CaM occupy well-defined configurations, whereas binding sites in lanthanide-bound-CaM are more disordered. Overall, the results show that binding to lanthanide ions significantly alters the conformation and dynamics of CaM's binding sites.

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.

Identifiability, reducibility, and adaptability in allosteric macromolecules

Bohner and Venkataraman propose a link between the sensitivity of allosteric macromolecules to their underlying biophysical parameters, the interrelationships between these parameters, and macromolecular adaptability. They argue that “emergent” combinations of parameters yield mechanistic insight that individual parameters cannot.

The ability of macromolecules to transduce stimulus information at one site into conformational changes at a distant site, termed “allostery,” is vital for cellular signaling. Here, we propose a link between the sensitivity of allosteric macromolecules to their underlying biophysical parameters, the interrelationships between these parameters, and macromolecular adaptability. We demonstrate that the parameters of a canonical model of the mSlo large-conductance Ca²⁺-activated K⁺ (BK) ion channel are non-identifiable with respect to the equilibrium open probability-voltage relationship, a common functional assay. We construct a reduced model with emergent parameters that are identifiable and expressed as combinations of the original mechanistic parameters. These emergent parameters indicate which coordinated changes in mechanistic parameters can leave assay output unchanged. We predict that these coordinated changes are used by allosteric macromolecules to adapt, and we demonstrate how this prediction can be tested experimentally. We show that these predicted parameter compensations are used in the first reported allosteric phenomena: the Bohr effect, by which hemoglobin adapts to varying pH.

The structure of binding curves and practical identifiability of equilibrium ligand-binding parameters

In their preceding paper, Middendorf and Aldrich describe a method to determine the accuracy of binding parameters estimated from models of agonist binding. Here, they present an approach to determine whether binding parameters can be accurately estimated from experimental, or noisy, data.

A critical but often overlooked question in the study of ligands binding to proteins is whether the parameters obtained from analyzing binding data are practically identifiable (PI), i.e., whether the estimates obtained from fitting models to noisy data are accurate and unique. Here we report a general approach to assess and understand binding parameter identifiability, which provides a toolkit to assist experimentalists in the design of binding studies and in the analysis of binding data. The partial fraction (PF) expansion technique is used to decompose binding curves for proteins with n ligand-binding sites exactly and uniquely into n components, each of which has the form of a one-site binding curve. The association constants of the PF component curves, being the roots of an n-th order polynomial, may be real or complex. We demonstrate a fundamental connection between binding parameter identifiability and the nature of these one-site association constants: all binding parameters are identifiable if the constants are all real and distinct; otherwise, at least some of the parameters are not identifiable. The theory is used to construct identifiability maps from which the practical identifiability of binding parameters for any two-, three-, or four-site binding curve can be assessed. Instructions for extending the method to generate identifiability maps for proteins with more than four binding sites are also given. Further analysis of the identifiability maps leads to the simple rule that the maximum number of structurally identifiable binding parameters (shown in the previous paper to be equal to n) will also be PI only if the binding curve line shape contains n resolved components.