Bayesian analysis of isothermal titration calorimetry for binding thermodynamics

Maximizing the Accuracy of Equilibrium Dissociation Constants for Affinity Complexes: From Theory to Practical Recommendations

The equilibrium dissociation constant (Kd) is a major characteristic of affinity complexes and one of the most frequently determined physicochemical parameters. Despite its significance, the values of Kd obtained for the same complex under similar conditions often exhibit considerable discrepancies and sometimes vary by orders of magnitude. These inconsistencies highlight the susceptibility of Kd determination to large systematic errors, even when random errors are small. It is imperative to both minimize and quantitatively assess the systematic errors inherent in Kd determination. Traditionally, Kd values are determined through nonlinear regression of binding isotherms. This analysis utilizes three variables: concentrations of two reactants and a fraction R of unbound limiting reactant. The systematic errors in Kd arise directly from systematic errors in these variables. Therefore, to maximize the accuracy of Kd, this study thoroughly analyzes the sources of systematic errors within the three variables, including (i) non-additive signals to calculate R, (ii) mis-calibrated experimental instruments, (iii) inaccurate calibration parameters, (iv) insufficient incubation time, (v) unsaturated binding isotherm, (vi) impurities in the reactants, and (vii) solute adsorption onto surfaces. Through this analysis, we illustrate how each source contributes to inaccuracies in the determination of Kd and propose strategies to minimize these contributions. Additionally, we introduce a method for quantitatively assessing the confidence intervals of systematic errors in concentrations, a crucial step toward quantitatively evaluating the accuracy of Kd. While presenting original findings, this paper also reiterates the fundamentals of Kd determination, hence guiding researchers across all proficiency levels. By shedding light on the sources of systematic errors and offering strategies for their mitigation, our work will help researchers enhance the accuracy of Kd determination, thereby making binding studies more reliable and the conclusions drawn from such studies more robust.

Bayesian Mechanistic Inference, Statistical Mechanics, and a New Era for Monte Carlo

On the one hand, much of computational chemistry is concerned with "bottom-up" calculations which elucidate observable behavior starting from exact or approximated physical laws, a paradigm exemplified by typical quantum mechanical calculations and molecular dynamics simulations. On the other hand, "top down" computations aiming to formulate mathematical models consistent with observed data, e.g., parametrizing force fields, binding or kinetic models, have been of interest for decades but recently have grown in sophistication with the use of Bayesian inference (BI). Standard BI provides an estimation of parameter values, uncertainties, and correlations among parameters. Used for "model selection," BI can also distinguish between model structures such as the presence or absence of individual states and transitions. Fortunately for physical scientists, BI can be formulated within a statistical mechanics framework, and indeed, BI has led to a resurgence of interest in Monte Carlo (MC) algorithms, many of which have been directly adapted from or inspired by physical strategies. Certain MC algorithms─notably procedures using an "infinite temperature" reference state─can be successful in a 5-20 parameter BI context which would be unworkable in molecular spaces of 103 coordinates and more. This Review provides a pedagogical introduction to BI and reviews key aspects of BI through a physical lens, setting the computations in terms of energy landscapes and free energy calculations and describing promising sampling algorithms. Statistical mechanics and basic probability theory also provide a reference for understanding intrinsic limitations of Bayesian inference with regard to model selection and the choice of priors.

From Average Transient Transporter Currents to Microscopic Mechanism─A Bayesian Analysis

Electrophysiology studies of secondary active transporters have revealed quantitative mechanistic insights over many decades of research. However, the emergence of new experimental and analytical approaches calls for investigation of the capabilities and limitations of the newer methods. We examine the ability of solid-supported membrane electrophysiology (SSME) to characterize discrete-state kinetic models with >10 rate constants. We use a Bayesian framework applied to synthetic data for three tasks: to quantify and check (i) the precision of parameter estimates under different assumptions, (ii) the ability of computation to guide the selection of experimental conditions, and (iii) the ability of our approach to distinguish among mechanisms based on SSME data. When the general mechanism, i.e., event order, is known in advance, we show that a subset of kinetic parameters can be "practically identified" within ∼1 order of magnitude, based on SSME current traces that visually appear to exhibit simple exponential behavior. This remains true even when accounting for systematic measurement bias and realistic uncertainties in experimental inputs (concentrations) are incorporated into the analysis. When experimental conditions are optimized or different experiments are combined, the number of practically identifiable parameters can be increased substantially. Some parameters remain intrinsically difficult to estimate through SSME data alone, suggesting that additional experiments are required to fully characterize parameters. We also demonstrate the ability to perform model selection and determine the order of events when that is not known in advance, comparing Bayesian and maximum-likelihood approaches. Finally, our studies elucidate good practices for the increasingly popular but subtly challenging Bayesian calculations for structural and systems biology.

The maximal and current accuracy of rigorous protein-ligand binding free energy calculations

Computational techniques can speed up the identification of hits and accelerate the development of candidate molecules for drug discovery. Among techniques for predicting relative binding affinities, the most consistently accurate is free energy perturbation (FEP), a class of rigorous physics-based methods. However, uncertainty remains about how accurate FEP is and can ever be. Here, we present what we believe to be the largest publicly available dataset of proteins and congeneric series of small molecules, and assess the accuracy of the leading FEP workflow. To ascertain the limit of achievable accuracy, we also survey the reproducibility of experimental relative affinity measurements. We find a wide variability in experimental accuracy and a correspondence between binding and functional assays. When careful preparation of protein and ligand structures is undertaken, FEP can achieve accuracy comparable to experimental reproducibility. Throughout, we highlight reliable protocols that can help maximize the accuracy of FEP in prospective studies.

Quantifying cooperative multisite binding in the hub protein LC8 through Bayesian inference

Multistep protein-protein interactions underlie most biological processes, but their characterization through methods such as isothermal titration calorimetry (ITC) is largely confined to simple models that provide little information on the intermediate, individual steps. In this study, we primarily examine the essential hub protein LC8, a small dimer that binds disordered regions of 100+ client proteins in two symmetrical grooves at the dimer interface. Mechanistic details of LC8 binding have remained elusive, hampered in part by ITC data analyses employing simple models that treat bivalent binding as a single event with a single binding affinity. We build on existing Bayesian ITC approaches to quantify thermodynamic parameters for multi-site binding interactions impacted by significant uncertainty in protein concentration. Using a two-site binding model, we identify positive cooperativity with high confidence for LC8 binding to multiple client peptides. In contrast, application of an identical model to the two-site binding between the coiled-coil NudE dimer and the intermediate chain of dynein reveals little evidence of cooperativity. We propose that cooperativity in the LC8 system drives the formation of saturated induced-dimer structures, the functional units of most LC8 complexes. In addition to these system-specific findings, our work advances general ITC analysis in two ways. First, we describe a previously unrecognized mathematical ambiguity in concentrations in standard binding models and clarify how it impacts the precision with which binding parameters are determinable in cases of high uncertainty in analyte concentrations. Second, building on observations in the LC8 system, we develop a system-agnostic heat map of practical parameter identifiability calculated from synthetic data which demonstrates that the ability to determine microscopic binding parameters is strongly dependent on both the parameters themselves and experimental conditions. The work serves as a foundation for determination of multi-step binding interactions, and we outline best practices for Bayesian analysis of ITC experiments.

Bayesian regression and model selection for isothermal titration calorimetry with enantiomeric mixtures

Bayesian regression is performed to infer parameters of thermodynamic binding models from isothermal titration calorimetry measurements in which the titrant is an enantiomeric mixture. For some measurements the posterior density is multimodal, indicating that additional data with a different protocol are required to uniquely determine the parameters. Models of increasing complexity—two-component binding, racemic mixture, and enantiomeric mixture—are compared using model selection criteria. To precisely estimate one of these criteria, the Bayes factor, a variation of bridge sampling is developed.

A discussion on the affecting factors of the fitting procedures' reliability in isothermal titration calorimetry analysis

The standard isothermal titration calorimetry (ITC) curve, characterized as a typical sigmoid is strictly confined by the so-called c value, which is a ratio of titrand concentration to K_D. The proper c value with a range from 5 to 500 is commonly recommended as a standard protocol in routine detection process for acquiring the reliable fitting results in 1:1 binding mode. However, if the c value is less than "1" due to the weak binding or low concentration of analyte, fitting precision gets unstable and susceptible to the data noise. Herein, we first got a deep discussion into the reliability of the fitting procedure for 1:1 binding mode by data simulation, then quantized the effect of several affecting factors on the precision of parameters estimation through mathematical analysis. Finally, we proposed the value of 2~4 times K_D for final ligand concentration is optimal for the ITC titration in low c system (c < 1). All the theoretical derivations were further verified by a practical experiment of Magnesium-EDTA binding test.

Differentiable biology: using deep learning for biophysics-based and data-driven modeling of molecular mechanisms

Deep learning using neural networks relies on a class of machine-learnable models constructed using 'differentiable programs'. These programs can combine mathematical equations specific to a particular domain of natural science with general-purpose, machine-learnable components trained on experimental data. Such programs are having a growing impact on molecular and cellular biology. In this Perspective, we describe an emerging 'differentiable biology' in which phenomena ranging from the small and specific (for example, one experimental assay) to the broad and complex (for example, protein folding) can be modeled effectively and efficiently, often by exploiting knowledge about basic natural phenomena to overcome the limitations of sparse, incomplete and noisy data. By distilling differentiable biology into a small set of conceptual primitives and illustrative vignettes, we show how it can help to address long-standing challenges in integrating multimodal data from diverse experiments across biological scales. This promises to benefit fields as diverse as biophysics and functional genomics.

waveRAPID-A Robust Assay for High-Throughput Kinetic Screens with the Creoptix WAVEsystem

Surface-based biophysical methods for measuring binding kinetics of molecular interactions, such as surface plasmon resonance (SPR) or grating-coupled interferometry (GCI), are now well established and widely used in drug discovery. Increasing throughput is an often-cited need in the drug discovery process and this has been achieved with new instrument generations where multiple interactions are measured in parallel, shortening the total measurement times and enabling new application areas within the field. Here, we present the development of a novel technology called waveRAPID for a further-up to 10-fold-increase in throughput, consisting of an injection method using a single sample. Instead of sequentially injecting increasing analyte concentrations for constant durations, the analyte is injected at a single concentration in short pulses of increasing durations. A major advantage of the new method is its ability to determine kinetics from a single well of a microtiter plate, making it uniquely suitable for kinetic screening. We present the fundamentals of this approach using a small-molecule model system for experimental validation and comparing kinetic parameters to traditional methods. By varying experimental conditions, we furthermore assess the robustness of this new technique. Finally, we discuss its potential for improving hit quality and shortening cycle times in the areas of fragment screening, low-molecular-weight compound screening, and hit-to-lead optimization.

Estimation of non-constant variance in isothermal titration calorimetry using an ITC measurement model

Isothermal titration calorimetry (ITC) is the gold standard for accurate measurement of thermodynamic parameters in solution reactions. In the data processing of ITC, the non-constant variance of the heat requires special consideration. The variance function approach has been successfully applied in previous studies, but is found to fail under certain conditions in this work. Here, an explicit ITC measurement model consisting of main thermal effects and error components has been proposed to quantitatively evaluate and predict the non-constant variance of the heat data under various conditions. Monte Carlo simulation shows that the ITC measurement model provides higher accuracy and flexibility than variance function in high c-value reactions or with additional error components, for example, originated from the fluctuation of the concentrations or other properties of the solutions. The experimental design of basic error evaluation is optimized accordingly and verified by both Monte Carlo simulation and experiments. An easy-to-run Python source code is provided to illustrate the establishment of the ITC measurement model and the estimation of heat variances. The accurate and reliable non-constant variance of heat is helpful to the application of weighted least squares regression, the proper evaluation or selection of the reaction model.

CALX-CBD1 Ca2+-Binding Cooperativity Studied by NMR Spectroscopy and ITC with Bayesian Statistics

The Na⁺/Ca²⁺ exchanger of Drosophila melanogaster, CALX, is the main Ca²⁺-extrusion mechanism in olfactory sensory neurons and photoreceptor cells. Na⁺/Ca²⁺ exchangers have two Ca²⁺ sensor domains, CBD1 and CBD2. In contrast to the mammalian homologs, CALX is inhibited by Ca²⁺ binding to CALX-CBD1, whereas CALX-CBD2 does not bind Ca²⁺ at physiological concentrations. CALX-CBD1 consists of a β-sandwich and displays four Ca²⁺-binding sites at the tip of the domain. In this study, we used NMR spectroscopy and isothermal titration calorimetry (ITC) to investigate the cooperativity of Ca²⁺ binding to CALX-CBD1. We observed that this domain binds Ca²⁺ in the slow exchange regime at the NMR chemical shift timescale. Ca²⁺ binding restricts the dynamics in the Ca²⁺-binding region. Experiments of ¹⁵N chemical exchange saturation transfer and ¹⁵N R₂ dispersion allowed the determination of Ca²⁺ dissociation rates (∼30 s^-1). NMR titration curves of residues in the Ca²⁺-binding region were sigmoidal because of the contribution of chemical exchange to transverse magnetization relaxation rates, R₂. Hence, a novel, to our knowledge, approach to analyze NMR titration curves was proposed. Ca²⁺-binding cooperativity was examined assuming two different stoichiometric binding models and using a Bayesian approach for data analysis. Fittings of NMR and ITC binding curves to the Hill model yielded n_Hill ∼2.9, near maximal cooperativity (n_Hill = 4). By assuming a stepwise model to interpret the ITC data, we found that the probability of binding from 2 up to 4 Ca²⁺ is approximately three orders of magnitude higher than that of binding a single Ca²⁺. Hence, four Ca²⁺ ions bind almost simultaneously to CALX-CBD1. Cooperative Ca²⁺ binding is key to enable this exchanger to efficiently respond to changes in the intracellular Ca²⁺ concentration in sensory neuronal cells.

Stirring rate affects thermodynamics and unfolding kinetics in isothermal titration calorimetry

Isothermal titration calorimetry (ITC) directly provides thermodynamic parameters depicting the energetics of intermolecular interactions in solution. During ITC experiments, a titration syringe with a paddle is continuously rotating to promote a homogeneous mixing. Here, we clarified that the shape of the paddles (flat, corkscrew and small-pitched corkscrew) and the stirring rates influence on the thermodynamic parameters of protein-ligand interaction. Stirring with the flat paddle at lower and higher rate both yielded a lower exothermic heat due to different reasons. The complete reaction with no incompetent fractions was achieved only when the stirring was performed at 500 or 750 rpm using the small-pitched corkscrew paddle. The evaluation of the protein solution after 1,500 rpm stirring indicated that proteins in the soluble fraction decreased to 94% of the initial amount, among which 6% was at an unfolded state. In addition, a significant increase of micron aggregates was confirmed. Furthermore, a new approach for the determination of the unfolding kinetics based on the time dependence of the total reaction heat was developed. This study demonstrates that a proper stirring rate and paddle shape are essential for the reliable estimation of thermodynamic parameters in ITC experiments.

C-Terminal Plays as the Possible Nucleation of the Self-Aggregation of the S-Shape Aβ11-42 Tetramer in Solution: Intensive MD Study

Amyloid beta (Aβ) peptides are characterized as the major factors associated with neuron death in Alzheimer's disease, which is listed as the most common form of neurodegeneration. Disordered Aβ peptides are released from proteolysis of the amyloid precursor protein. The Aβ self-assembly process roughly takes place via five steps: disordered forms → oligomers → photofibrils → mature fibrils → plaques. Although Aβ fibrils are often observed in patient brains, oligomers were recently indicated to be major neurotoxic elements. In this work, the neurotoxic compound S-shape Aβ11-42 tetramer (S4Aβ11-42) was investigated over 10 μs of unbiased MD simulations. In particular, the S4Aβ11-42 oligomer adopted a high dynamics structure, resulting in unsuccessful determination of their structures in experiments. The C-terminal was suggested as the possible nucleation of the Aβ42 aggregation. The sequences 27-35 and 39-40 formed rich β-content, whereas other residues mostly adopted coil structures. The mean value of the β-content over the equilibrium interval is ∼42 ± 3%. Furthermore, the dissociation free energy of the S4Aβ11-42 peptide was predicted using a biased sampling method. The obtained free energy is ΔG US = -58.44 kcal/mol which is roughly the same level as the corresponding value of the U-shape Aβ17-42 peptide. We anticipate that the obtained S4Aβ11-42 structures could be used as targets for AD inhibitor screening over the in silico study.

Broad-scale analysis of thermodynamic signatures in medicinal chemistry: are enthalpy-favored binders the better development option?

Thermodynamic profiles of ligand binding, particularly enthalpically favored binding signatures, have been suggested as a criterion to support the decision-making process around which compounds to select for further optimization in drug development. The concept was enthusiastically taken up, but turned out to be too superficial, either because many aspects determining thermodynamic profiles are insufficiently appreciated or because it is difficult to compare such data on a global scale. The impact of water, changes in protonation states, along with buffer dependencies and incompatible measurement conditions that are far from standard conditions hamper such broad-scale comparisons. However, thermodynamic signatures can make us aware of the impact of these aspects and provide important hints for improving our understanding of the binding process and defining criteria for drug optimization.