Fundamental equation of thermodynamics for protein-ligand binding

For the internal energy and every thermodynamic potential that can be defined by a Legendre transform, there is a fundamental equation that contains all the thermodynamic information about a system. For a system involving the binding of molecular oxygen and hydrogen ions by a protein, fundamental equations are given for the Gibbs energy G, the transformed Gibbs energy G' at specified pH, and the further transformed Gibbs energy G" at specified pH and specified concentration of molecular oxygen. The Maxwell equations for these various Gibbs energies are important because they provide the connection with experimentally determined properties and increase our understanding of these properties. Measurements of the average number of oxygen molecules bound as a function of T, pH and concentration of molecular oxygen make it possible to calculate Delta(f)G"(o) of the reactant. Maxwell equations make it possible to calculate the average number of hydrogen ions bound, Delta(f)S"(o), Delta(f)H"(o) and their partial derivatives. These relations are illustrated with numerical calculations on a simple reaction system.     

An assessment of protein-ligand binding site polarizability

Electronic polarizability, an important physical property of biomolecules, is currently ignored in most biomolecular calculations. Yet, it is widely believed that polarization could account for a substantial fraction of the total nonbonded energy of a system. This belief is supported by studies of small complexes in vacuum. This perception is driving the development of a new class of polarizable force fields for biomolecular calculations. However, the quantification of this term for protein-ligand complexes has never been attempted. Here we explore the polarizable nature of protein-ligand complexes in order to evaluate the importance of this effect. We introduce two indexes describing the polarizability of protein binding sites. These we apply to a large range of pharmaceutically relevant complexes. We offer a recommendation of particular complexes as test systems with which to determine the effects of polarizability and as test cases with which to test the new generation of force fields. Additionally, we provide a tabulation of the amino acid composition of these binding sites and show that composition can be specific for certain classes of proteins. We also show that the relative abundance of some amino acids is different in binding sites than elsewhere in a protein's structure.     

Determination of protein-ligand binding constants at equilibrium in biological samples.

Protein-ligand complexes can be separated functionally into two classes. "Specific" binding is characterized, in relative terms, by a high affinity for the ligand and a low binding capacity. "Non-specific" binding is characterized by a low affinity and a very large capacity. The calculation of equilibrium binding constants for any specific protein-ligand interaction requires the exact determination of the unbound ligand concentration and the specifically bound ligand concentration. These determinations usually require corrections for the contribution of non-specific binding. The use of two correction terms, kn and f, is proposed: kn is the product of the affinity constant k times the number of binding sites n of the non-specific components, while f is the fraction of the non-specific binding included in the experimental estimates of bound ligand. Several theoretical solutions using these terms are proposed for the calculation of specific binding constants. The practical choice of the correction factor may be different when the simultaneous measurement of the affinity constant and maximum number of binding sites, or when only the latter, is desired. In the case of complex binding systesm containing more than one specific component, the individual constants can be determined by non-graphical methods, using computer-aided iterative statistical calculations. A complete solution is given for a system containing two specific plus non-specific interactions and actual experiments are reported for steroid hormone-receptro complexes.     

A new measure of cooperativity in protein-ligand binding

An allosteric binding system consisting of a single ligand and a nondissociating macromolecule having multiple binding sites can be represented by a binding polynomial. Various properties of the binding process can be obtained by analyzing the coefficients of the binding polynomial and such functions as the binding curve and the Hill plot. The Hill plot has an asymptote of unit slope at each end and the departure of the slope from unity at any point can be used to measure the effective interaction free energy at that point. Of particular interest in detecting and measuring cooperativity are extrema of the Hill slope and its value at the half-saturation point. If the binding polynomial is symmetric, then there is an extremum of the Hill slope at the half-saturation point. This value, the Hill coefficient, is a convenient measure of cooperativity. The purpose of this paper is to express the Hill coefficient for symmetric binding polynomials in terms of the roots of the polynomial and to give an interpretation of cooperativity in terms of the geometric pattern of the roots in the complex plane. This interpretation is then applied to the binding polynomials for the MWC (Monod-Wyman-Changeux) and KNF (Koshland-Nemethy-Filmer) models.     

Estimating protein-ligand binding free energy: atomic solvation parameters for partition coefficient and solvation free energy calculation

Solvation energy calculation is one of the main difficulties for the estimation of protein-ligand binding free energy and the correct scoring in docking studies. We have developed a new solvation energy estimation method for protein-ligand binding based on atomic solvation parameter (ASP), which has been shown to improve the power of protein-ligand binding free energy predictions. The ASP set, designed to handle both proteins and organic compounds and derived from experimental n-octanol/water partition coefficient (log P) data, contains 100 atom types (united model that treats hydrogen atoms implicitly) or 119 atom types (all-atom model that treats hydrogen atoms explicitly). By using this unified ASP set, an algorithm was developed for solvation energy calculation and was further integrated into a score function for predicting protein-ligand binding affinity. The score function reproduced the absolute binding free energies of a test set of 50 protein-ligand complexes with a standard error of 8.31 kJ/mol. As a byproduct, a conformation-dependent log P calculation algorithm named ASPLOGP was also implemented. The predictive results of ASPLOGP for a test set of 138 compounds were r = 0.968, s = 0.344 for the all-atom model and r = 0.962, s = 0.367 for the united model, which were better than previous conformation-dependent approaches and comparable to fragmental and atom-based methods. ASPLOGP also gave good predictive results for small peptides. The score function based on the ASP model can be applied widely in protein-ligand interaction studies and structure-based drug design.     

Computation of entropy contribution to protein-ligand binding free energy

The entropy contribution ΔS to protein-ligand binding free energy is studied for nine protein-lipid complexes. The entropy effect from the loss of the translational/rotational degrees of freedom (ΔS _tr) is calculated using the ideal gas approach. The change in the vibrational entropy (ΔS _vib) is calculated using the effective quantum oscillator approach with frequencies derived from the coordinate covariance matrix, so the inharmonic effects are taken into account. The change in the entropy of solvation (ΔS _solv) is considered using the binomial cell model (developed by the authors) for the hydrophobic effect. The entropy contribution from loss of conformations that are available for the free ligand (ΔS _conf) is also estimated. It is revealed that the negative in view of binding term ΔS _tr is only partly compensated by increasing of ΔS _vib, so T(ΔS _tr + ΔS _vib + ΔS _conf) < 0 for all complexes under investigation, but taking into account ΔS _solv leads to significantly increased ΔS. For all complexes except biotin-streptavidin, the results are found to be in reasonable agreement with experimental data. Published in Russian in Biokhimiya, 2007, Vol. 72, No. 7, pp. 963–973.     

Estimation of growth parameters from multiple-recapture data

This article develops a method for analysis of growth data with multiple recaptures when the initial ages for all individuals are unknown. The existing approaches either impute the initial ages or model them as random effects. Assumptions about the initial age are not verifiable because all the initial ages are unknown. We present an alternative approach that treats all the lengths including the length at first capture as correlated repeated measures for each individual. Optimal estimating equations are developed using the generalized estimating equations approach that only requires the first two moment assumptions. Explicit expressions for estimation of both mean growth parameters and variance components are given to minimize the computational complexity. Simulation studies indicate that the proposed method works well. Two real data sets are analyzed for illustration, one from whelks (Dicathais aegaota) and the other from southern rock lobster (Jasus edwardsii) in South Australia.     

Use of the direct linear plot to estimate binding constants for protein-ligand interactions.

The direct linear plot (Eisenthal and Cornish-Bowden[1974] Biochem. J. $\text{139}$, 715–720) for the determination of enzyme kinetic constants has been assessed as a means of describing specific steroid-protein interactions. In the rat uterine cytoplasmic estrogen receptor system, determination of the equilibrium dissociation constant (K_D) and of the total number of ligand-binding sites (B_max) has been made, and the results are in good agreement with those obtained by Scatchard and Lineweaver-Burk plot analyses. The usefulness of the direct linear plot lies in the speed and simplicity with which it can be constructed and interpreted.     

The salt-dependence of a protein-ligand interaction: ion-protein binding energetics

Using the binding of a nucleotide inhibitor (guanosine-3'-monophosphate) to a ribonuclease (ribonuclease Sa) as a model system, we show that the salt-dependence of the interaction arises due to specific ion binding at the site of nucleotide binding. The presence of specific ion-protein binding is concluded from a combination of differential scanning calorimetry and NMR data. Isothermal titration calorimetry data are then fit to determine the energetic profile (enthalpy, entropy, and heat capacity) for both the ion-protein and nucleotide-protein interactions. The results provide insight into the energetics of charge-charge interactions, and have implications for the interpretation of an observed salt-dependence. Further, the presence of specific ion-binding leads to a system behavior as a function of temperature that is drastically different from that predicted from Poisson-Boltzmann calculations.     

Estimation of drug-protein binding parameters on assuming the validity of thermodynamic equilibrium

This contribution focuses the reader's attention on the pitfalls usually emerging during the phase of evaluation of experimental data of drug-protein binding studies. To overcome the occurrence of problem(s) apparently defying solution, the concept of "affinity spectra" is recommended to be implemented for data evaluation. A (general) "binding study protocol" is also suggested, which can prevent the formation of inadequate conclusions and the generation of unrealistic drug-protein binding parameters.     

Design of protein-ligand binding based on the molecular-mechanics energy model

While the molecular-mechanics field has standardized on a few potential energy functions, computational protein design efforts are based on potentials that are unique to individual laboratories. Here we show that a standard molecular-mechanics potential energy function without any modifications can be used to engineer protein-ligand binding. A molecular-mechanics potential is used to reconstruct the coordinates of various binding sites with an average root-mean-square error of 0.61 Å and to reproduce known ligand-induced side-chain conformational shifts. Within a series of 34 mutants, the calculation can always distinguish between weak (K_d > 1 mM) and tight (K_d < 10 μM) binding sequences. Starting from partial coordinates of the ribose-binding protein lacking the ligand and the 10 primary contact residues, the molecular-mechanics potential is used to redesign a ribose-binding site. Out of a search space of 2 × 10¹² sequences, the calculation selects a point mutant of the native protein as the top solution (experimental K_d = 17 μM) and the native protein as the second best solution (experimental K_d = 210 nM). The quality of the predictions depends on the accuracy of the generalized Born electrostatics model, treatment of protonation equilibria, high-resolution rotamer sampling, a final local energy minimization step, and explicit modeling of the bound, unbound, and unfolded states. The application of unmodified molecular-mechanics potentials to protein design links two fields in a mutually beneficial way. Design provides a new avenue for testing molecular-mechanics energy functions, and future improvements in these energy functions will presumably lead to more accurate design results.     

The relationship between zeros and factors of binding polynomials and cooperativity in protein-ligand binding

Cooperativity in the protein-ligand binding process is discussed in terms of the zeros of the binding polynomial and the corresponding possible factorizations of the binding polynomial into polynomials having non-negative coefficients. Particular attention is paid to the case in which the real parts of all zeros are negative (Hurwitz polynomial) and the case in which the binding polynomial admits no positive factorization (positive irreducible polynomial). Such factorizations are then interpreted as site linkage patterns and related to cooperativity. The possible combinations of zeros of the binding polynomials for the MWC and KNF tetrahedral, square and linear models are determined and the corresponding factorization and linkage patterns analyzed. An application and interpretation are then made for data obtained from Trout I hemoglobin.     

Ionization of clusters. III. Evaluation of interaction parameters from binding data

Interactions between ionizable groups on the same molecule modulate the binding of protons to an extent where the binding constants may be shifted by orders of magnitude. The first two papers of this series discussed the family of carboxylic acids, pairwise isotropic interactions, and evaluation of single site binding data. This paper presents an extended group of hypothetical binding isotherms. Procedures are illustrated for deriving interaction parameters from binding data. The interaction parameters for about 25 representative compounds with two and three interacting ionizable groups are evaluated and tabulated.     

Estimation of parameters of allometric equations.

Accurate parameter estimation of allometric equations is a question of considerable interest. Various techniques that address this problem exist. In this paper it is assumed that the measured values are normally distributed and a maximum likelihood estimation approach is used. The computations involved in this procedure are reducible to relatively simple forms, and an efficient numerical algorithm is used. A listing of the computer program is included as an appendix.     

Estimation of parameters in a multi-affinity-state model for haemoglobin from oxygen binding data in whole blood and in concentrated haemoglobin solutions.

The reduced fragment of pancreatic trypsin inhibitor lacking the six C-terminal residues, which is produced by cyanogen bromide cleavage, formed a seemingly random mixture of disulphide bonds under refolding conditions where normal pancreatic trypsin inhibitor refolds correctly and quantitatively. This illustrates the importance of the C-terminal residues in folding of the normal protein, the uniqueness of the normal folded conformation, and the apparently central role in protein folding of long-range interactions between residues distant in the primary structure.The intact polypeptide chain of reduced pancreatic trypsin inhibitor in which the methionine residue normally at position 52 had been converted to homoserine refolded slightly less readily than the normal reduced compound. This was observed to be due to an altered spectrum of single-disulphide intermediates: the normally predominant intermediate with the 30–51 disulphide bond was less stable by about 0.8 kcal/mol relative to the other normal single-disulphide intermediates. The other steps in refolding appeared to be normal, although the refolded protein was observed to be susceptible to an unexplained reaction with iodoacetate.     

A regionalizable statistical model of intersecting regions in protein-ligand binding cavities

Finding elements of proteins that influence ligand binding specificity is an essential aspect of research in many fields. To assist in this effort, this paper presents two statistical models, based on the same theoretical foundation, for evaluating structural similarity among binding cavities. The first model specializes in the "unified" comparison of whole cavities, enabling the selection of cavities that are too dissimilar to have similar binding specificity. The second model enables a "regionalized" comparison of cavities within a user-defined region, enabling the selection of cavities that are too dissimilar to bind the same molecular fragments in the given region. We applied these models to analyze the ligand binding cavities of the serine protease and enolase superfamilies. Next, we observed that our unified model correctly separated sets of cavities with identical binding preferences from other sets with varying binding preferences, and that our regionalized model correctly distinguished cavity regions that are too dissimilar to bind similar molecular fragments in the user-defined region. These observations point to applications of statistical modeling that can be used to examine and, more importantly, identify influential structural similarities within binding site structure in order to better detect influences on protein-ligand binding specificity.