The structure of binding curves and practical 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.

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.

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.

UL25 capsid binding facilitates mechanical maturation of the Herpesvirus capsid and allows retention of pressurized DNA

The maturation process that occurs in most viruses is evolutionarily driven as it resolves several conflicting virion assembly requirements. During herpesvirus assembly in a host cell nucleus, micron-long double-stranded herpes DNA is packaged into a nanometer-sized procapsid. This leads to strong confinement of the viral genome with resulting tens of atmospheres of intra-capsid DNA pressure. Yet, the procapsid is unstable due to weak, reversible interactions between its protein subunits, which ensures free energy minimization and reduces assembly errors. In this work we show that herpesviruses resolve these contradictory capsid requirements through a mechanical capsid maturation process facilitated by multi-functional auxiliary protein UL25. Through mechanical interrogation of herpes simplex virus type 1 (HSV-1) capsid with atomic force microscopy nano-indentation, we show that UL25 binding at capsid vertices post-assembly provides the critical capsid reinforcement required for stable DNA encapsidation; the absence of UL25 binding leads to capsid rupture. Furthermore, we demonstrate that gradual capsid reinforcement is a feasible maturation mechanism facilitated by progressive UL25 capsid binding, which is likely correlated with DNA packaging progression. This work provides insight into elegantly programmed viral assembly machinery where targeting of capsid assembly mechanics presents a new antiviral strategy that is resilient to development of drug resistance. Importance: Most viruses undergo a maturation process from a weakly assembled particle to a stable virion. Herpesvirus capsid undergoes mechanical maturation to withstand tens of atmospheres of DNA pressure. We demonstrate that this mechanical capsid maturation is mainly facilitated through binding of auxiliary protein UL25 in HSV-1 capsid vertices. We show that UL25 binding provides the critical capsid reinforcement required for stable DNA encapsidation. Our data also suggests that gradual capsid reinforcement by progressive UL25 binding is a feasible capsid maturation mechanism, correlated with DNA packaging progression.

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.

Deciphering cell-cell interactions and communication from gene expression

Cell-cell interactions orchestrate organismal development, homeostasis and single-cell functions. When cells do not properly interact or improperly decode molecular messages, disease ensues. Thus, the identification and quantification of intercellular signalling pathways has become a common analysis performed across diverse disciplines. The expansion of protein-protein interaction databases and recent advances in RNA sequencing technologies have enabled routine analyses of intercellular signalling from gene expression measurements of bulk and single-cell data sets. In particular, ligand-receptor pairs can be used to infer intercellular communication from the coordinated expression of their cognate genes. In this Review, we highlight discoveries enabled by analyses of cell-cell interactions from transcriptomic data and review the methods and tools used in this context.

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.

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.

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.

Structural identifiability of equilibrium ligand-binding parameters

Precise mathematical descriptions of ligand–protein interactions are hindered by the inability to experimentally measure affinity and cooperativity, although these parameters can be estimated from agonist binding models. Middendorf and Aldrich present a method to determine the accuracy of parameters estimated in this way.

Understanding the interactions of proteins with their ligands requires knowledge of molecular properties, such as binding site affinities and the effects that binding at one site exerts on binding at other sites (cooperativity). These properties cannot be measured directly and are usually estimated by fitting binding data with models that contain these quantities as parameters. In this study, we present a general method for answering the critical question of whether these parameters are identifiable (i.e., whether their estimates are accurate and unique). In cases in which parameter estimates are not unique, our analysis provides insight into the fundamental causes of nonidentifiability. This approach can thus serve as a guide for the proper design and analysis of protein–ligand binding experiments. We show that the equilibrium total binding relation can be reduced to a conserved mathematical form for all models composed solely of bimolecular association reactions and to a related, conserved form for all models composed of arbitrary combinations of binding and conformational equilibria. This canonical mathematical structure implies a universal parameterization of the binding relation that is consistent with virtually any physically reasonable binding model, for proteins with any number of binding sites. Matrix algebraic methods are used to prove that these universal parameter sets are structurally identifiable (SI; i.e., identifiable under conditions of noiseless data). A general approach for assessing and understanding the factors governing practical identifiability (i.e., the identifiability under conditions of real, noisy data) of these SI parameter sets is presented in the companion paper by Middendorf and Aldrich (2017. J. Gen. Physiol.https://doi.org/10.1085/jgp.201611703).