Measured Equilibrium Constants Research Articles

Moment Analysis Method for Measurement of Reaction Equilibrium and Rate Constants by Using High-Performance Liquid Chromatography.

The moment analysis method was developed for the determination of association equilibrium constant (KA) and association (ka) and dissociation (kd) rate constants of intermolecular interactions between solute and ligand molecules. They are accurately determined by using moment equations from elution peak profiles because they are measured by using high-performance liquid chromatography (HPLC) under preferable conditions that neither immobilization nor chemical modification (i.e., fluorescence labeling) of solute and ligand molecules is required. To demonstrate the effectiveness of the method, it was applied to the inclusion complex formation system between dibenzo-18-crown-6 (DB18C6) and alkaline earth metal cations, i.e., Mg2+, Ca2+, and Sr2+, as a concrete example. Because the diameter of the three metal cations is smaller than that of the inner cavity of DB18C6, the values of KA, ka, and kd were analytically determined by assuming the stoichiometry of 1:1 between DB18C6 and the metal cation. They reflected the influence of the difference in the size between the inner cavity of DB18C6 and the metal cations on the inclusion complex formation. It seems that the moment analysis method based on HPLC separation is effective for the multifaceted analysis of chemical reactions because some characteristics of the method are different from those of other conventional methods. It must contribute to the dissemination of an opportunity for the study of chemical reactions to many researchers because of the versatility of HPLC.

Use of DNA forceps to measure receptor-ligand dissociation equilibrium constants in a single-molecule competition assay.

The ability of biophysicists to decipher the behavior of individual biomolecules has steadily improved over the past thirty years. However, it still remains unclear how an ensemble of data acquired at the single-molecule level compares with the data acquired on an ensemble of the same molecules. We here propose an assay to tackle this question in the context of dissociation equilibrium constant measurements. A sensor is built by engrafting a receptor and a ligand onto a flexible dsDNA scaffold and mounting this assembly on magnetic tweezers. This way, looking at the position of the magnetic bead enables one to determine in real-time if the two molecular partners are associated or not. Next, to quantify the affinity of the scrutinized single-receptor for a given competitor, various amounts of the latter molecule are introduced in solution and the equilibrium response of the sensor is monitored throughout the titration protocol. Proofs of concept are established for the binding of three rapamycin analogs to the FKBP12 cis-trans prolyl isomerase. For each of these drugs the mean affinity constant obtained on a ten of individual receptors agrees with the one previously determined in a bulk assay. Furthermore, experimental contingencies are sufficient to explain the dispersion observed over the single-molecule values.

Extracting thermodynamic properties from van ’t Hoff plots with emphasis on temperature-sensing ion channels

ABSTRACT Transient receptor potential (TRP) ion channels are among the most well-studied classes of temperature-sensing molecules. Yet, the molecular mechanism and thermodynamic basis for the temperature sensitivity of TRP channels remains to this day poorly understood. One hypothesis is that the temperature-sensing mechanism can simply be described by a difference in heat capacity between the closed and open channel states. While such a two-state model may be simplistic it nonetheless has descriptive value, in the sense that it can be used to compare overall temperature sensitivity between different channels and mutants. Here, we introduce a mathematical framework based on the two-state model to reliably extract temperature-dependent thermodynamic potentials and heat capacities from measurements of equilibrium constants at different temperatures. Our framework is implemented in an open-source data analysis package that provides a straightforward way to fit both linear and nonlinear van ’t Hoff plots, thus avoiding some of the previous, potentially erroneous, assumptions when extracting thermodynamic variables from TRP channel electrophysiology data.

Determination of binding constants by ultrafast affinity extraction: Theoretical and experimental studies of optimum conditions for analysis

Ultrafast affinity extraction (UAE) is a form of microscale affinity HPLC that can be employed to quickly measure equilibrium constants for solute-binding agent interactions in solution. This study used chromatographic and equilibrium theory with universal plots to examine the general conditions that are needed in UAE to obtain accurate, precise, and robust measurements of equilibrium constants for such interactions. The predicted results were compared to those obtained by UAE in studies that examined the binding of various drugs with two transport proteins: human serum albumin and α1-acid glycoprotein. The most precise and robust conditions for these binding studies occurred for systems with intermediate values for their equilibrium free fraction for the solute (F0 ≈ 0.20–0.80). These trends showed good agreement with those seen in prior studies using UAE. It was further determined how the apparent free fraction of a solute was related to the dissociation rate of this solute, the time allowed for solute dissociation during UAE, and the equilibrium free fraction for the solute. These results also agreed with experimental results, as obtained for the binding of warfarin and gliclazide with human serum albumin. The final section examined how a change in the apparent free fraction, as caused by solute dissociation, affected the accuracy of an equilibrium constant that was measured by UAE. In addition, theoretical plots were generated to allow the selection of conditions for UAE that provided a given level of accuracy during the measurement of an equilibrium constant. The equations created and trends identified for UAE were general ones that can be extended in future work to other solutes and binding agents.

Lewis Acids and Electron-Withdrawing Ligands Accelerate CO Coordination to Dinuclear CuI Compounds.

A series of dinuclear molecular copper complexes were prepared and used to model the binding and Lewis acid stabilization of CO in heterogeneous copper CO2 reduction electrocatalysts. Experimental studies (including measurement of rate and equilibrium constants) and electronic structure calculations suggest that the key kinetic barrier for CO binding may be a σ-interaction between CuI and the incoming CO ligand. The rate of CO coordination can be increased upon the addition of Lewis acids or electron-withdrawing substituents on the ligand backbone. Conversely, Keq for CO coordination can be increased by adding electron density to the metal centers of the compound, consistent with stronger π-backbonding. Finally, the electrochemically measured kinetic results were mapped onto an electrochemical zone diagram to illustrate how these system changes enabled access to each zone.

Recommendations for performing measurements of apparent equilibrium constants of enzyme-catalyzed reactions and for reporting the results of these measurements.

The measurement of values of apparent equilibrium constants K' for enzyme-catalyzed reactions involve a substantial number of critical details, neglect of which could lead to systematic errors. Here, interferences, impurities in the substances used, and failure to achieve equilibrium are matters of substantial consequence. Careful reporting of results is of great importance if the results are to have archival value. Thus, attention must be paid to the identification of the substances, specification of the reaction(s), the conditions of reaction, the definition of the equilibrium constant(s) and standard states, the use of standard nomenclature, symbols, and units, and uncertainties. This document contains a general discussion of various aspects of these equilibrium measurements as well as STRENDA (Standards for Reporting Enzymology Data) recommendations regarding the measurements and the reporting of results.

Building a molecular model for ion channel and transporter stability in membranes through a partnership of experiments and computation.

Ion channels and transporters are the molecular gatekeepers of biology, governing the passage of atoms and molecules in and out of cells. They provide essential nutrients for metabolism, eliminate waste, enable cell-to-cell communication and store the potential energy that fuels life. Yet, despite their overwhelming importance, we still lack a physical and molecular understanding of why membrane transport proteins form stable structures that enable their specific functions within the oil-filled environment of the lipid bilayer. Our lack of understanding here is well rationalized by the inherent challenge of measuring equilibrium reactions of membrane protein assembly in membranes, compounded by the complexity of the reaction solvent as a structured lipid bilayer. However, through a combination of novel experimental single-molecule microscopy approaches coupled with coarse-grained and all-atom computational modeling, we can now measure equilibrium constants of protein association in membranes and dissect out the important molecular contributions from the protein and surrounding membrane. Here, I present our results on two systems where reversible binding within membranes can be quantitatively assessed - the homodimeric CLC-ec1 chloride/proton antiporter and the dual-topology homodimeric Fluc fluoride ion channel. By combining experimental and computational studies, we identify a driving force for protein association that arises from state-dependent perturbation of the membrane structure and a mechanism for tuning dimerization stability by altering solvation energetics through preferential solvation. These results provide generalizable mechanisms that drive membrane protein assembly in membranes that are expected to apply to all ion channels and transporters.

A Distinctive Pattern for Substituent Effects on Transition Metal Centers: Enhanced Electron-Donating Capacity of Cationic Palladium Species

A Distinctive Pattern for Substituent Effects on Transition Metal Centers: Enhanced Electron-Donating Capacity of Cationic Palladium Species

The time-zero HSQC method improves the linear free energy relationship of a polypeptide chain through the accurate measurement of residue-specific equilibrium constants.

EXSY (exchange spectroscopy) NMR provides the residue-specific equilibrium constants, K, and residue-specific kinetic rate constants, k, of a polypeptide chain in a two-state exchange in the slow exchange regime. A linear free energy relationship (LFER) discovered in a log k versus log K plot is considered to be a physicochemical basis for smooth folding and conformational changes of protein molecules. For accurate determination of the thermodynamic and kinetic parameters, the measurement bias arising from state-specific differences in the R1 and R2 relaxation rates of 1H and other nuclei in HSQC and EXSY experiments must be minimized. Here, we showed that the time-zero HSQC acquisition scheme (HSQC0) is effective for this purpose, in combination with a special analytical method (Π analysis) for EXSY. As an example, we applied the HSQC0 + Π method to the two-state exchange of nukacin ISK-1 in an aqueous solution. Nukacin ISK-1 is a 27-residue lantibiotic peptide containing three mono-sulfide linkages. The resultant bias-free residue-based LFER provided valuable insights into the transition state of the topological interconversion of nukacin ISK-1. We found that two amino acid residues were exceptions in the residue-based LFER relationship. We inferred that the two residues could adopt special conformations in the transition state, to allow the threading of some side chains through a ring structure formed by one of the mono-sulfide linkages. In this context, the two residues are a useful target for the manipulation of the physicochemical properties and biological activities of nukacin ISK-1.

A suite of mathematical solutions to describe ternary complex formation and their application to targeted protein degradation by heterobifunctional ligands

Small molecule–induced targeted protein degradation by heterobifunctional ligands or molecular glues represents a new modality in drug development, allowing development of therapeutic agents for targets previously considered undruggable. Successful target engagement requires the formation of a ternary complex (TC) when the ligand brings its target protein in contact with an E3 ubiquitin ligase. Unlike traditional drugs, where target engagement can be described by a simple bimolecular equilibrium equation, similar mathematical tools are currently not available to describe TC formation in a universal manner. This current limitation substantially increases the challenges of developing drugs with targeted protein degradation mechanism. In this article, I provide a full, exact, and universal mathematical description of the TC system at equilibrium for the first time. I have also constructed a comprehensive suite of mathematical tools for quantitative measurement of target engagement and equilibrium constants from experimental data. Mechanistic explanations are provided for many common challenges associated with developing this type of therapeutic agent. Insights from these analyses provide testable hypotheses and grant direction to drug development efforts in this promising area. The mathematical and analytical tools described in this article may also have broader applications in other areas of biology and chemistry in which ternary complexes are observed.

Cavity Hydration and Competitive Binding in Methylated β-Cyclodextrin.

Raman multivariate curve resolution (Raman-MCR) spectroscopy is used to measure the vibrational spectrum of water molecules in the cavity of an aqueous methylated β-cyclodextrin (Me-β-CD), as well as to quantify the competitive expulsion of those waters by benzene. The Me-β-CD cavity is found to contain 5-6 water molecules whose structure is remarkably similar to that of bulk water, although slightly more tetrahedral and with fewer weak hydrogen bonds. The binding constant of benzene to Me-β-CD, obtained using Raman-MCR, is found to be similar to that of benzene to β-CD (previously determined by other means). The competitive displacement of water by benzene in Me-β-CD is quantified by explicitly including the release of cavity-bound water molecules in the measured equilibrium constant.

True equilibrium measurement of transcription factor-DNA binding affinities using automated polarization microscopy

The complex patterns of gene expression in metazoans are controlled by selective binding of transcription factors (TFs) to regulatory DNA. To improve the quantitative understanding of this process, we have developed a novel method that uses fluorescence anisotropy measurements in a controlled delivery system to determine TF-DNA binding energies in solution with high sensitivity and throughput. Owing to its large dynamic range, the method, named high performance fluorescence anisotropy (HiP-FA), allows for reliable quantification of both weak and strong binding; binding specificities are calculated on the basis of equilibrium constant measurements for mutational DNA variants. We determine the binding preference landscapes for 26 TFs and measure high absolute affinities, but mostly lower binding specificities than reported by other methods. The revised binding preferences give rise to improved predictions of in vivo TF occupancy and enhancer expression. Our approach provides a powerful new tool for the systems-biological analysis of gene regulation.

Modulation of the Drag Force Exerted by Microfluidic Flow on Laser-Trapped Particles: A New Method to Assess Surface-Binding Kinetics, Analyte Size, and Solution Viscosity

We have designed and validated a platform capable of assessing minuscule changes of the viscous drag force that different fluid environments exert on individual, micrometer-sized particles. Based on a combination of optical tweezers and a microfluidic device, this platform allows us to expose a laser-trapped particle to a series of microenvironments, created through microfluidic laminar flow, and to monitor biochemical interactions on the surface of the particle. In addition, we can determine the viscosity of the different environments with excellent resolution. We have used this setup to examine the interaction between human IgG and both protein-A as well as protein-G. At saturation, bound IgG added an apparent thickness of 12.6±.77(SD)nm and 16.0±1.4(SD) nm to protein-A- and protein-G-coated beads, respectively. This is in agreement with the expected size of human IgG of ∼10-16 nm dependent on orientation. Moreover, we measured equilibrium constants of KA=4.21x107 M−1 for the protein-A:IgG interaction and KA=7.94x107 M−1 for the protein-G:IgG interaction, in agreement with literature values. Finally, we measured the viscosity of different solutions of bovine serum albumin, human serum, and glycerol with µPa·s resolution. Measurements were performed in a microfluidic flow chamber and in microwells on a glass slide. The microwells allowed us to carry out comparative viscosity measurements with very small volumes of solution, less than 50 µL. The results obtained in each case agree with published models. This platform opens up a new playing field for the study of biochemical surface interactions. Its potential applications include the investigation of biomolecular interactions at the surface of live biological particles, such as individual bacteria, fungal cells, or large viruses, and the determination of the viscosity of costly solutions or small samples.

A model-free method for measuring dimerization free energies of CLC-ec1 in lipid bilayers.

The thermodynamic reasons why membrane proteins form stable complexes inside the hydrophobic lipid bilayer remain poorly understood. This is largely because of a lack of membrane-protein systems amenable for equilibrium studies and a limited number of methods for measuring these reactions. Recently, we reported the equilibrium dimerization of the CLC-ec1 Cl-/H+ transporter in lipid bilayers (Chadda et al. 2016. eLife https://doi.org/10.7554/eLife.17438), which provided a new type of model system for studying protein association in membranes. The measurement was conducted using the subunit-capture approach, involving passive dilution of the protein in large multilamellar vesicles, followed by single-molecule photobleaching analysis of the Poisson distribution describing protein encapsulation into extruded liposomes. To estimate the fraction of dimers (FDimer ) as a function of protein density, the photobleaching distributions for the nonreactive, ideal monomer and dimer species must be known so that random co-capture probabilities can be accounted for. Previously, this was done by simulating the Poisson process of protein reconstitution into a known size distribution of liposomes composed of Escherichia coli polar lipids (EPLs). In the present study, we investigate the dependency of FDimer and ΔG° on the modeling through a comparison of different liposome size distributions (EPL versus 2:1 POPE/POPG). The results show that the estimated FDimer values are comparable, except at higher densities when liposomes become saturated with protein. We then develop empirical controls to directly measure the photobleaching distributions of the nonreactive monomer (CLC-ec1 I201W/I422W) and ideal dimer (WT CLC-ec1 cross-linked by glutaraldehyde or CLC-ec1 R230C/L249C cross-linked by a disulfide bond). The measured equilibrium constants do not depend on the correction method used, indicating the robustness of the subunit-capture approach. This strategy therefore presents a model-free way to quantify protein dimerization in lipid bilayers, offering a simplified strategy in the ongoing effort to characterize equilibrium membrane-protein reactions in membranes.

Ultrasonic Monitoring of Biocatalysis in Solutions and Complex Dispersions

The rapidly growing field of chemical catalysis is dependent on analytical methods for non-destructive real-time monitoring of chemical reactions in complex systems such as emulsions, suspensions and gels, where most analytical techniques are limited in their applicability, especially if the media is opaque, or if the reactants/products do not possess optical activity. High-resolution ultrasonic spectroscopy is one of the novel technologies based on measurements of parameters of ultrasonic waves propagating through analyzed samples, which can be utilized for real-time non-invasive monitoring of chemical reactions. It does not require optical transparency, optical markers and is applicable for monitoring of reactions in continuous media and in micro/nano bioreactors (e.g., nanodroplets of microemulsions). The technology enables measurements of concentrations of substrates and products over the whole course of reaction, analysis of time profiles of the degree of polymerization and molar mass of polymers and oligomers, evolutions of reaction rates, evaluation of kinetic mechanisms, measurements of kinetic and equilibrium constants and reaction Gibbs energy. It also provides tools for assessments of various aspects of performance of catalysts/enzymes including inhibition effects, reversible and irreversible thermal deactivation. In addition, ultrasonic scattering effects in dispersions allow real-time monitoring of structural changes in the medium accompanying chemical reactions.

The yeast kinesin-5 Cin8 interacts with the microtubule in a noncanonical manner

Kinesin motors play central roles in establishing and maintaining the mitotic spindle during cell division. Unlike most other kinesins, Cin8, a kinesin-5 motor in Saccharomyces cerevisiae, can move bidirectionally along microtubules, switching directionality according to biochemical conditions, a behavior that remains largely unexplained. To this end, we used biochemical rate and equilibrium constant measurements as well as cryo-electron microscopy methodologies to investigate the microtubule interactions of the Cin8 motor domain. These experiments unexpectedly revealed that, whereas Cin8 ATPase kinetics fell within measured ranges for kinesins (especially kinesin-5 proteins), approximately four motors can bind each αβ-tubulin dimer within the microtubule lattice. This result contrasted with those observations on other known kinesins, which can bind only a single “canonical” site per tubulin dimer. Competition assays with human kinesin-5 (Eg5) only partially abrogated this behavior, indicating that Cin8 binds microtubules not only at the canonical site, but also one or more separate (“noncanonical”) sites. Moreover, we found that deleting the large, class-specific insert in the microtubule-binding loop 8 reverts Cin8 to one motor per αβ-tubulin in the microtubule. The novel microtubule-binding mode of Cin8 identified here provides a potential explanation for Cin8 clustering along microtubules and potentially may contribute to the mechanism for direction reversal.

In-situ evaluation of dye adsorption on TiO2using QCM

We measured the adsorption characteristics of two organic dyes; triphenylamine-cyanoacrylic acid (TPA-C) and phenoxazine (MP13), on TiO2 , directly in a solution based on quartz crystal microbalance (QCM). Monitoring the adsorbed amount as a function of dye concentration and during rinsing allows determination of the equilibrium constant and distinction between chemisorbed and physisorbed dye. The measured equilibrium constants are 0.8 mM-1 for TPA-C and 2.4 mM-1 for MP13. X-ray photoelectron spectroscopy was used to compare dried chemisorbed layers of TPA-C prepared in solution with TPA-C layers prepared via vacuum sublimation; the two preparation methods render similar spectra except a small contribution of water residues (OH) on the solution prepared samples. Quantitative Nanomechanical Mapping Atomic Force Microscopy (QNM-AFM) shows that physisorbed TPA-C layers are easily removed by scanning the tip across the surface. Although not obvious in height images, adhesion images clearly demonstrate removal of the dye.

A Nexus between Theory and Experiment: Non-Empirical Quantum Mechanical Computational Methodology Applied to Cucurbit[n]uril⋅Guest Binding Interactions.

A training set of eleven X-ray structures determined for biomimetic complexes between cucurbit[n]uril (CB[7 or 8]) hosts and adamantane-/diamantane ammonium/aminium guests were studied with DFT-D3 quantum mechanical computational methods to afford ΔGcalcd binding energies. A novel feature of this work is that the fidelity of the BLYP-D3/def2-TZVPP choice of DFT functional was proven by comparison with more accurate methods. For the first time, the CB[n]⋅guest complex binding energy subcomponents [for example, ΔEdispersion , ΔEelectrostatic , ΔGsolvation , binding entropy (-TΔS), and induced fit Edeformation(host) , Edeformation(guest) ] were calculated. Only a few weeks of computation time per complex were required by using this protocol. The deformation (stiffness) and solvation properties (with emphasis on cavity desolvation) of cucurbit[n]uril (n=5, 6, 7, 8) isolated host molecules were also explored by means of the DFT-D3 method. A high ρ2 =0.84 correlation coefficient between ΔGexptl and ΔGcalcd was achieved without any scaling of the calculated terms (at 298 K). This linear dependence was utilized for ΔGcalcd predictions of new complexes. The nature of binding, including the role of high energy water molecules, was also studied. The utility of introduction of tethered [-(CH2 )n NH3 ]+ amino loops attached to N,N-dimethyl-adamantane-1-amine and N,N,N',N'-tetramethyl diamantane-4,9-diamine skeletons (both from an experimental and a theoretical perspective) is presented here as a promising tool for the achievement of new ultra-high binding guests to CB[7] hosts. Predictions of not yet measured equilibrium constants are presented herein.

Stepwise formation of H3O(+)(H2O)n in an ion drift tube: Empirical effective temperature of association/dissociation reaction equilibrium in an electric field.

We measured equilibrium constants for H3O(+)(H2O)n-1 + H2O↔H3O(+)(H2O)n (n = 4-9) reactions taking place in an ion drift tube with various applied electric fields at gas temperatures of 238-330 K. The zero-field reaction equilibrium constants were determined by extrapolation of those obtained at non-zero electric fields. From the zero-field reaction equilibrium constants, the standard enthalpy and entropy changes, ΔHn,n-1 (0) and ΔSn,n-1 (0), of stepwise association for n = 4-8 were derived and were in reasonable agreement with those measured in previous studies. We also examined the electric field dependence of the reaction equilibrium constants at non-zero electric fields for n = 4-8. An effective temperature for the reaction equilibrium constants at non-zero electric field was empirically obtained using a parameter describing the electric field dependence of the reaction equilibrium constants. Furthermore, the size dependence of the parameter was thought to reflect the evolution of the hydrogen-bond structure of H3O(+)(H2O)n with the cluster size. The reflection of structural information in the electric field dependence of the reaction equilibria is particularly noteworthy.

The Effect of Fluorescent Protein Tags on Phosphoglycerate Kinase Stability Is Nonadditive.

It is frequently assumed that fluorescent protein tags used in biological imaging experiments are minimally perturbing to their host protein. As in-cell experiments become more quantitative and measure rates and equilibrium constants, rather than just "on-off" activity or the presence of a protein, it becomes more important to understand such perturbations. One criterion for a protein modification to be a perturbation is additivity of two perturbations (a linear effect on the protein free energy). Here we show that adding fluorescent protein tags to a host protein in vitro has a large nonadditive effect on its folding free energy. We compare an unlabeled, three singly labeled, and a doubly labeled enzyme (phosphoglycerate kinase). We propose two mechanisms for nonadditivity. In the "quinary interaction" mechanism, two tags interact transiently with one another, relieving the host protein from unfavorable tag-protein interactions. In the "crowding" mechanism, adding two tags provides the minimal crowding necessary to overcome destabilizing interactions of individual tags with the host protein. Both of these mechanisms affect protein stability in cells; we show here that they must also be considered for tagged proteins used for reference in vitro.