The heat capacity paradox of ligand binding proteins: reconciling the microscopic and macroscopic world

Differential scanning microcalorimetry (DSC) is a superb method for the analysis of protein energetics. However, the relative simplicity of application has led astray many to assume that a proper analysis of the data was possible without a sound knowledge of the underlying statistical thermodynamic principles. In this study, the question is addressed of how to calculate properly the heat capacity signal of a protein in the presence of high affinity ligands. It is shown that the signal corresponds neither to grand canonic nor to canonic heat capacity. Statistical thermodynamic model calculations result only in the observed macroscopic heat capacity signal, if the protein in the calorimetric cell is assumed to form a grand canonic ensemble (T, p, mu controlled) which is, however, heated under constraints typical for a canonic ensemble (T, p, N controlled). As a consequence, the microscopic statistical thermodynamic heat capacity must be carefully distinguished from the macroscopically observable thermodynamic heat capacity in those cases where proteins unfold in the presence of high affinity ligands.     

Grand canonical ensemble Monte Carlo simulation of the dCpG/proflavine crystal hydrate.

The grand canonical ensemble Monte Carlo molecular simulation method is used to investigate hydration patterns in the crystal hydrate structure of the dCpG/proflavine intercalated complex. The objective of this study is to show by example that the recently advocated grand canonical ensemble simulation is a computationally efficient method for determining the positions of the hydrating water molecules in protein and nucleic acid structures. A detailed molecular simulation convergence analysis and an analogous comparison of the theoretical results with experiments clearly show that the grand ensemble simulations can be far more advantageous than the comparable canonical ensemble simulations.     

An efficient algorithm in the grand canonical ensemble: constructing adsorption isotherms

A new computational method is presented that efficiently describes open thermodynamic systems within the grand canonical ensemble formalism. The method is based on the j-walking algorithm, which circumvent sampling difficulties by coupling random walkers in different thermodynamic states. By imposing detailed balance, a new acceptance probability is derived and applied to the construction of adsorption isotherms for atomic monolayers. The method converges much faster than the standard grand canonical Monte Carlo method and permits the construction of accurate adsorption isotherms and the identification of phase transitions occurring in the adsorbed material.     

Grand Molecular Dynamics: A Method for Open Systems

We present a new molecular dynamics method for studying the dynamics of open systems. The method couples a classical system to a chemical potential reservior. In the formulation, following the extended system dynamics approach, we introduce a variable, v to represent the coupling to the chemical potential reservoir. The new variable governs the dynamics of the variation of number of particles in the system. The number of particles is determined by taking the integer part of v. The fractional part of the new variable is used to scale the potential energy and the kinetic energy of an additional particle: i.e., we introduce a fractional particle. We give the ansatz Lagrangians and equations of motion for both the isothermal and the adiabatic forms of grand molecular dynamics. The averages calculated over the trajectories generated by these equations of motion represent the classical grand canonical ensemble (μVT) and the constant chemical potential adiabatic ensemble (μVL) averages, respectively. The microcanonical phase space densities of the adiabatic and isothermal forms the molecular dynamics method are shown to be equivalent to adiabatic constant chemical potential ensemble, and grand canonical ensemble partition functions. We also discuss the extension to multi-component systems, molecular fluids, ionic solutions and the problems and solutions associated with the implementation of the method. The statistical expressions for thermodynamic functions such as specific heat; adiabatic bulk modulus, Grüneissen parameter and number fluctuations are derived. These expressions are used to analyse trajectories of constant chemical potential systems.     

Molecular Dynamics Simulations With First Order Coupling to a Bath of Constant Chemical Potential

Abstract

In molecular dynamics simulations the temperature or pressure can be controlled by applying a weak first-order coupling to a bath of constant temperature or pressure. This weak coupling technique to control system properties using a first-order relaxation equation is analyzed from a statistical mechanics point of view. It is shown, how the weak coupling scheme can be generalized and applied to a bath of contstant chemical potential. The presented method, to which in the following will be referred to as chemical potential weak coupling, is applied and tested on a Lennard-Jones fluid. The thermodynamic quantities known from the literature are accuratly reproduced.

The temperature and chemical potential weak coupling methods aim to sample the canonical and grand canonical ensembles respectively. By analyzing the fluctuations in energy and number of particles, the tight relation between the ensembles and the distributions obtained from the weak coupling simulations is demonstrated. The influence of the choice of the coupling parameters on the quality of the approximation of the ensemble distribution is discussed.     

A New Monte Carlo Method for Direct Calculation of the Critical Size and the Formation Work of a Microdrop

New Monte Carlo procedures in open ensembles are proposed. Non-stationary Markov chain procedure in the μl;pT - ensemble provides a direct estimation for the critical size of a condensation nucleus at given p and T. A stationary procedure in the μlpT ensemble with two allowed particle numbers n and n + 1 provides the direct way to calculate the chemical potential and Gibbs free energy of a cluster; in the grand canonical (μlVT) ensemble the same approach gives μl and the Helmholtz free energy. The same procedures are readily applicable to periodic systems representing bulk phases.     

Revolutions in RNA secondary structure prediction

RNA structure formation is hierarchical and, therefore, secondary structure, the sum of canonical base-pairs, can generally be predicted without knowledge of the three-dimensional structure. Secondary structure prediction algorithms evolved from predicting a single, lowest free energy structure to their current state where statistics can be determined from the thermodynamic ensemble. This article reviews the free energy minimization technique and the salient revolutions in the dynamic programming algorithm methods for secondary structure prediction. Emphasis is placed on highlighting the recently developed method, which statistically samples structures from the complete Boltzmann ensemble.     

Thermodynamics of the denaturation of lysozyme in alcohol--water mixtures.

The thermal denaturation of lysozyme was studied at pH 2 in aqueous mixtures of methanol, ethanol, and 1-propanol by high sensitivity differential scanning calorimetry (DSC). The most obvious effect of alcohols was the lowering of Td, the temperature of denaturation, increasingly with higher alcohol concentration and longer alkyl chain. Both the calorimetric and van't Hoff enthalpies of denaturation initially increased and then decreased with increasing alcohol concentration, the ratio of the two enthalpies being nearly unity, 1.007 +/- 0.011, indicating the validity of the two-state approximation for the unfolding of lysozyme in these solvent systems. The reversibility of the denaturation was demonstrated by the reversibility of the DSC curves and the complete recovery of enzymic activity on cooling. The changes in heat capacity on unfolding decreased with increasing alcohol concentration for each alcohol. Experimentally determined values of denaturation temperature and of entropy and heat capacity changes were used to derive the additional thermodynamic parameters delta G degrees and delta S degrees for denaturation as a function of temperature for each alcohol--water mixture. Comparison of the thermodynamic parameters with those reported [Pfeil, W., & Privalov, P.L. (1976) Biophys. Chem. 4, 23--50] in aqueous solution at various values of pH and guanidine hydrochloride concentration showed that these latter changes have no effect on the heat capacity changes, whereas the addition of alcohols causes a sharp decrease.     

Free energies of binding of polychlorinated biphenyls to the estrogen receptor from a single simulation

Relative free energies of binding to the ligand-binding domain of the estrogen receptor have been calculated for a series of 17 hydroxylated polychlorinated biphenyls. Because traditional thermodynamic integration or perturbation approaches are hardly feasible for these numbers of compounds, the one-step perturbation approach is applied and is shown to yield accurate results based on only two 2-ns molecular dynamics simulations of an unphysical, judiciously chosen, reference state. The mean absolute difference between the calculated and experimental binding free energies for the 17 compounds is 3.4 kJ/mol, which illustrates the accuracy of the GROMOS biomolecular force field used. Excluding the three largest ligands from the comparison reduces the deviation to 2.0 kJ/mol (i.e., < k(B)T). Apart from the relative free energy, structural information about the binding mode and binding orientation for every compound can also be extracted from the simulation, showing that a ligand bound to its receptor cannot be represented by a single conformation, but it samples an ensemble of different orientations.     

Conformational changes in the chicken receptor for endocytosis of glycoproteins. Modulation of ligand-binding activity by Ca2+ and pH

Limited proteolysis, gel filtration, and circular dichroism have been used to identify at least three distinct conformational states of a proteolytic fragment containing the ligand-binding domain of the chicken receptor for endocytosis of glycoproteins. Differences in the ligand-binding activity of intact receptor brought about by changing Ca2+ concentrations and pH values can be correlated with different physical states of the binding domain present under similar conditions. An active, ligand-binding state can be detected at either pH 7.8 or 5.4, but 10-fold higher concentrations of Ca2+ are required to stabilize this state at the lower pH. In all cases, the dependence on Ca2+ concentration is second-order, suggesting that two Ca2+ ions are bound to each domain. These studies demonstrate an interdependence between the effects of Ca2+ concentration and pH on both ligand-binding activity and receptor conformation, which is important to consider when describing the binding and dissociation of ligand during endocytosis.     

Thermodynamic stability of carbonic anhydrase: measurements of binding affinity and stoichiometry using ThermoFluor

ThermoFluor (a miniaturized high-throughput protein stability assay) was used to analyze the linkage between protein thermal stability and ligand binding. Equilibrium binding ligands increase protein thermal stability by an amount proportional to the concentration and affinity of the ligand. Binding constants (K(b)) were measured by examining the systematic effect of ligand concentration on protein stability. The precise ligand effects depend on the thermodynamics of protein stability: in particular, the unfolding enthalpy. An extension of current theoretical treatments was developed for tight binding inhibitors, where ligand effect on T(m) can also reveal binding stoichiometry. A thermodynamic analysis of carbonic anhydrase by differential scanning calorimetry (DSC) enabled a dissection of the Gibbs free energy of stability into enthalpic and entropic components. Under certain conditions, thermal stability increased by over 30 degrees C; the heat capacity of protein unfolding was estimated from the dependence of calorimetric enthalpy on T(m). The binding affinity of six sulfonamide inhibitors to two isozymes (human type 1 and bovine type 2) was analyzed by both ThermoFluor and isothermal titration calorimetry (ITC), resulting in a good correlation in the rank ordering of ligand affinity. This combined investigation by ThermoFluor, ITC, and DSC provides a detailed picture of the linkage between ligand binding and protein stability. The systematic effect of ligands on stability is shown to be a general tool to measure affinity.     

Folding energetics of ligand binding proteins II. Cooperative binding of Ca2+ to annexin I

The calcium binding properties of annexin I as observed by thermodynamic DSC studies have been compared to the structural information obtained from X-ray investigation. The calorimetric experiment permitted to evaluate both the reaction scheme - including binding of ligand and conformational changes - and the energetics of each reaction step. According to published X-ray data Annexin I has six calcium binding sites, three medium-affinity type II and three low-affinity type III sites.The present study shows that at 37 degrees C annexin I binds in a Hill type fashion simultaneously two calcium ions in a first step with medium affinity at a concentration of 0.6 mM and another three Ca(2+) ions again cooperatively at 30 mM with low affinity. Therefore it can be concluded that only two medium-affinity type II binding sites are available. The third site, that should be accessible in principle appears to be masked presumably due to the presence of the N terminus. In view of the large calcium concentration needed for saturation of the binding sites, annexin I may be expected to be Ca(2+) free in vivo unless other processes such as membrane interaction occur simultaneously. This assumption is consistent with the finding, that the affinity of annexins to calcium is usually markedly increased by the presence of lipids.     

Direct Determination of Fluid Phase Equilibria by Simulation in the Gibbs Ensemble: A Review

This paper provides an extensive review of the literature on the Gibbs ensemble Monte Carlo method for direct determination of phase coexistence in fluids. The Gibbs ensemble technique is based on performing a simulation in two distinct regions in a way that ensures that the conditions of phase coexistence are satisfied in a statistical sense. Contrary to most other available techniques for this purpose, such as thermodynamic integration, grand canonical Monte Carlo or Widom test particle insertions, the Gibbs ensemble technique involves only a single simulation per coexistence point. A significant body of literature now exists on the method, its theoretical foundations, and proposed modifications for efficient determination of equilibria involving dense fluids and complex intermolecular potentials. Some practical aspects of Gibbs ensemble simulation are also discussed in this review. Applications of the technique to date range from studies of simple model potentials (for example Lennard–Jones, square-well or Yukawa fluids) to calculations of equilibria in mixtures with components described by realistic potentials. We conclude by discussing the limitations of the technique and potential future applications.     

A theoretical model for the prediction of sequence-dependent nucleosome thermodynamic stability

A theoretical model for predicting nucleosome thermodynamic stability in terms of DNA sequence is advanced. The model is based on a statistical mechanical approach, which allows the calculation of the canonical ensemble free energy involved in the competitive nucleosome reconstitution. It is based on the hypothesis that nucleosome stability mainly depends on the bending and twisting elastic energy to transform the DNA intrinsic superstructure into the nucleosomal structure. The ensemble average free energy is calculated starting from the intrinsic curvature, obtained by integrating the dinucleotide step deviations from the canonical B-DNA and expressed in terms of a Fourier series, in the framework of first-order elasticity. The sequence-dependent DNA flexibility is evaluated from the differential double helix thermodynamic stability. A large number of free-energy experimental data, obtained in different laboratories by competitive nucleosome reconstitution assays, are successfully compared to the theoretical results. They support the hypothesis that the stacking energies are the major factor in DNA rigidity and could be a measure of DNA stiffness. A dual role of DNA intrinsic curvature and flexibility emerges in the determination of nucleosome stability. The difference between the experimental and theoretical (elastic) nucleosome-reconstitution free energy for the whole pool of investigated DNAs suggests a significant role for the curvature-dependent DNA hydration and counterion interactions, which appear to destabilize nucleosomes in highly curved DNAs. This model represents an attempt to clarify the main features of the nucleosome thermodynamic stability in terms of physical-chemical parameters and suggests that in molecular systems with a large degree of complexity, the average molecular properties dominate over the local features, as in a statistical ensemble.     

Use of binding site neighbor-effect parameters to evaluate the interactions between adjacent ligands on a linear lattice. Effects on ligand-lattice association.

A method using binding site "neighbor-effect" parameters (NEPs) is introduced to evaluate the effects of interaction between adjacent ligands on their binding to an infinite linear lattice. Binding site overlap is also taken into account. This enables the conditional probability approach of McGhee & von Hippel to be extended to more complex situations. The general equation for the isotherm is v/LF = SFKF, where v is the ratio of bound ligands to lattice residues, LF is the free ligand concentration, SF is the fraction of binding sites that are free, and KF is the average association constant of a free site. Solutions are derived for three cases: symmetric ligands, and asymmetric ligands on isotropic or anisotropic lattices. For symmetric ligands there is one NEP, E, which is the ratio of the average binding affinity of a free site if the status of the lattice residue neighboring one end of the site is unspecified (left to chance) to the affinity when this residue is free (holding the other neighbor constant). Thus KF is KE2, where K is the affinity of an isolated site. If a site is n residues long, SF is f ffn-1, where f = 1 - nv is the fraction of residues that are free and ff is the conditional probability that a free residue is bordered on a given side by another free residue. The expression for ff is 1/(1 + x/E), where x is v/f, E is (1 - x + [(1 - x)2 + 4x omega]1/2)/2, and omega is the co-operativity parameter. The binding of asymmetric ligands to an isotropic lattice is described by two NEPs; the last case involves four NEPs and a bound ligand orientation parameter. For each case, the expected length distribution of clusters of bound ligands can be calculated as a function of v. When Scatchard plots with the same intercepts and initial slope are compared, it is found that ligand asymmetry lowers the isotherm (relative to the corresponding symmetric ligand isotherm), whereas lattice anisotrophy raises it.     

Structure-based thermodynamic analysis of the dissociation of protein phosphatase-1 catalytic subunit and microcystin-LR docked complexes

The relationship between the structure of a free ligand in solution and the structure of its bound form in a complex is of great importance to the understanding of the energetics and mechanism of molecular recognition and complex formation. In this study, we use a structure-based thermodynamic approach to study the dissociation of the complex between the toxin microcystin-LR (MLR) and the catalytic domain of protein phosphatase-1 (PP-1c) for which the crystal structure of the complex is known. We have calculated the thermodynamic parameters (enthalpy, entropy, heat capacity, and free energy) for the dissociation of the complex from its X-ray structure and found the calculated dissociation constant (4.0 x 10(-11)) to be in excellent agreement with the reported inhibitory constant (3.9 x 10(-11)). We have also calculated the thermodynamic parameters for the dissociation of 47 PP-1c:MLR complexes generated by docking an ensemble of NMR solution structures of MLR onto the crystal structure of PP-1c. In general, we observe that the lower the root-mean-square deviation (RMSD) of the docked complex (compared to the X-ray complex) the closer its free energy of dissociation (deltaGd(o)) is to that calculated from the X-ray complex. On the other hand, we note a significant scatter between the deltaGd(o) and the RMSD of the docked complexes. We have identified a group of seven docked complexes with deltaGd(o) values very close to the one calculated from the X-ray complex but with significantly dissimilar structures. The analysis of the corresponding enthalpy and entropy of dissociation shows a compensation effect suggesting that MLR molecules with significant structural variability can bind PP-1c and that substantial conformational flexibility in the PP-1c:MLR complex may exist in solution.     

Modeling of the Psychophysical Response Curves Using the Grand Canonical Ensemble in Statistical Physics

This paper aims to investigate the peripheral mechanism of taste perception by the use of the grand canonical ensemble in statistical physics. It allows a better understanding of this process and its interpretation at a microscopic level. The experimental part allowed us to obtain psychophysical curves relative to four nutritive sweeteners (sucrose, fructose, glucose, and maltitol). These curves represent the intensity of sweetness as a function of sugar concentration in water. A Sensory Measuring Unit for Recording Flux (SMURF) device is used to obtain intensity–time curves for each of the studied sweeteners. To model these curves we have established the expressions of some models using grand canonical ensemble formalism in statistical physics and applying some simplification approaches. The fitting of the psychophysical data with statistical models by numerical simulation demonstrates a good correlation between the models and the experimental curves. Some physicochemical parameters interfere in the expression of the established models. These parameters are classified in two categories: on one hand, the steric aspect, which is manifested by the density of taste receptor site, and the number of molecules per receptor site and on the other hand, an energetic aspect illustrated by the concentration at half saturation, which gives indirectly the sweet molecule-receptor site energy of interaction.     

Analytical expressions for the homotropic binding of ligand to protein dimers and trimers

Cooperative binding of a ligand to multiple subsites on a protein is a common theme among enzymes and receptors. The analysis of cooperative binding data (either positive or negative) often relies on the assumption that free ligand concentration, L, can be approximated by the total ligand concentration, L(T). When this approximation does not hold, such analyses result in inaccurate estimates of dissociation constants. Presented here are exact analytical expressions for equilibrium concentrations of all enzyme and ligand species (in terms of K(d) values and total concentrations of protein and ligand) for homotropic dimeric and trimeric protein-ligand systems. These equations circumvent the need to approximate L and are provided in Excel worksheets suitable for simulation and least-squares fitting. The equations and worksheets are expanded to treat cases where binding signals vary with distinct site occupancy.     

Fragmentation-tree density representation for crystallographic modelling of bound ligands

The identification and modelling of ligands into macromolecular models is important for understanding molecule's function and for designing inhibitors to modulate its activities. We describe new algorithms for the automated building of ligands into electron density maps in crystal structure determination. Location of the ligand-binding site is achieved by matching numerical shape features describing the ligand to those of density clusters using a "fragmentation-tree" density representation. The ligand molecule is built using two distinct algorithms exploiting free atoms with inter-atomic connectivity and Metropolis-based optimisation of the conformational state of the ligand, producing an ensemble of structures from which the final model is derived. The method was validated on several thousand entries from the Protein Data Bank. In the majority of cases, the ligand-binding site could be correctly located and the ligand model built with a coordinate accuracy of better than 1 ?. We anticipate that the method will be of routine use to anyone modelling ligands, lead compounds or even compound fragments as part of protein functional analyses or drug design efforts.