Physical chemical studies of short-chain lecithin homologues. I. Influence of the chain length of the fatty acid ester and of electrolytes on the critical micelle concentration

The critical micelle concentration (CMC) of four synthetic phosphatidylcholines (containing two hexanoyl, heptanoyl, octanoyl or nonanoyl residues respectively) in aqueous solutions have been determined by surface tension measurements. The dependence of the CMC on the chain length is discussed on the basis of the mass action model for micelle formation. For the three higher homologues a contribution of 1.08 kT per CH₂ group to the standard free energy of micellisation is found. The change in this free energy in going from the dihexanoyl- to the diheptanoyllecithin is somewhat larger (1.2 kT per CH₂ group).The influence of high concentrations (several moles per liter) of simple electrolytes on the CMC is interpreted as a salting-out of nonpolar solutes in water. Contrary to expectations the effects of NaCl and Lil on the CMC of dioctanoyllecithin are not additive.     

Determination of the Equilibrium Constants of Associating Protein Systems: VII. Thermodynamic Analysis of Linear and Helical Association

The expression of free energy change for linear and helical association based on Casper's definition as applied to Oosawa and Kasai's model is described. Redefinition of the thermodynamic quantity ζ indicates that it is dependent on Δf_B, the free energy change in the formation of B-bonds, rather than mean distortion energy as previously believed.     

Resolvability of free energy changes for oxygen binding and subunit association by human hemoglobin.

Probability distributions of the free energy changes for oxygen binding, subunit association, and quaternary enhancement by human hemoglobin were obtained from Monte Carlo simulations performed on two independent sets of variable protein concentration equilibrium oxygen-binding data. Uncertainties in unliganded and fully liganded dimer to tetramer association free energy changes (0 delta G'2 and 4 delta G'2) were accounted for in the simulations. Distributions of the dimer to tetramer association free energy changes for forming singly and triply liganded tetramers (1 delta G'2 and 3 delta G'2) are well defined and quite symmetric, whereas that for forming doubly liganded tetramers (2 delta G'2) is poorly defined and highly asymmetric. The distribution of the dimer stepwise oxygen-binding free-energy change (delta g'2i) is well defined and quite symmetric as are those of the tetramer stepwise oxygen-binding free-energy changes for binding the first and last oxygens to tetramers (delta g'41 and delta g'44). Distributions of the intermediate tetramer stepwise oxygen-binding free-energy changes (delta g'42 and delta g'43) are poorly defined and highly asymmetric, but are compensatory in that their sum (delta g'4[2 + 3]) is again well defined and nearly symmetric. Distributions of the free energy changes corresponding to the tetramer product Adair oxygen binding constants (delta G'4i) are well defined and quite symmetric for i = 1, 3, 4 but not for i = 2. The distribution of delta g'44 - delta g'2i (the quaternary enhancement free energy change) is relatively narrow, nearly symmetric, and confined to the negative free-energy domain. This suggests that the quaternary enhancement free energy change (a) may be resolved with good confidence from this data and (b) is finite and negative under the conditions of these experiments. Our results also suggest two different four-state combinatorial switch models that provide accurate characterization of hemoglobin's functional behavior.     

Mechanisms of the multiphasic kinetics in the folding and unfolding of globular proteins

The multiphasic kinetics of the protein folding and unfolding processes are examined for a “cluster model” with only two thermodynamically stable macroscopic states, native (N) and denatured (D), which are essentially distributions of microscopic states. The simplest kinetic schemes consistent with the model are: N-(fast) → I-(slow) → D for unfolding and N ← (fast)-D₂ ← (slow)-D₁ for refolding. The fast phase during the unfolding process can be visualized as the redistribution of the native population N to I within its free energy valley. Then, this population crosses over the free energy barrier to the denatured state D in the slow phase. Therefore, the macrostate I is a kinetic intermediate which is not stable at equilibrium. For the refolding process, the initial equilibrium distribution of the denatured state D appears to be separated into D₁ and D₂ in the final condition because of the change in position of the free energy barrier. The fast refolding species D₂ is due to the “leak” from the broadly distributed D state, while the rest is the slow refolding species D₁, which must overpass the free energy barrier to reach N. At an early stage of the folding process the amino acid chain is considered to be composed of several locally ordered regions, which we call clusters, connected by random coil chain parts. Thus, the denatured state contains different sizes and distributions of clusters depending on the external condition. A later stage of the folding process is the association of smaller clusters. The native state is expressed by a maximum-size cluster with possible fluctuation sites reflecting this association. A general discussion is given of the correlation between the kinetics and thermodynamics of proteins from the overall shape of the free energy function. The cluster model provides a conceptual link between the folding kinetics and the structural patterns of globular proteins derived from the X-ray crystallographic data.     

Thermodynamic stability of water molecules in the bacteriorhodopsin proton channel: a molecular dynamics free energy perturbation study.

The proton transfer activity of the light-driven proton pump, bacteriorhodopsin (bR) in the photochemical cycle might imply internal water molecules. The free energy of inserting water molecules in specific sites along the bR transmembrane channel has been calculated using molecular dynamics simulations based on a microscopic model. The existence of internal hydration is related to the free energy change on transfer of a water molecule from bulk solvent into a specific binding site. Thermodynamic integration and perturbation methods were used to calculate free energies of hydration for each hydrated model from molecular dynamics simulations of the creation of water molecules into specific protein-binding sites. A rigorous statistical mechanical formulation allowing the calculation of the free energy of transfer of water molecules from the bulk to a protein cavity is used to estimate the probabilities of occupancy in the putative bR proton channel. The channel contains a region lined primarily by nonpolar side-chains. Nevertheless, the results indicate that the transfer of four water molecules from bulk water to this apparently hydrophobic region is thermodynamically permitted. The column forms a continuous hydrogen-bonded chain over 12 A between a proton donor, Asp 96, and the retinal Schiff base acceptor. The presence of two water molecules in direct hydrogen-bonding association with the Schiff base is found to be strongly favorable thermodynamically. The implications of these results for the mechanism of proton transfer in bR are discussed.     

Dimeric procaspase-3 unfolds via a four-state equilibrium process.

We have examined the folding and assembly of a catalytically inactive mutant of procaspase-3, a homodimeric protein that belongs to the caspase family of proteases. The caspase family, and especially caspase-3, is integral to apoptosis. The equilibrium unfolding data demonstrate a plateau between 3 and 5 M urea, consistent with an apparent three-state unfolding process. However, the midpoint of the second transition as well as the amplitude of the plateau are dependent on the protein concentration. Overall, the data are well described by a four-state equilibrium model in which the native dimer undergoes an isomeration to a dimeric intermediate, and the dimeric intermediate dissociates to a monomeric intermediate, which then unfolds. By fitting the four-state model to the experimental data, we have determined the free energy change for the first step of unfolding to be 8.3 +/- 1.3 kcal/mol. The free energy change for the dissociation of the dimeric folding intermediate to two monomeric intermediates is 10.5 +/- 1 kcal/mol. The third step in the unfolding mechanism represents the complete unfolding of the monomeric intermediate, with a free energy change of 7.0 +/- 0.5 kcal/mol. These results show two important points. First, dimerization of procaspase-3 occurs as a result of the association of two monomeric folding intermediates, demonstrating that procaspase-3 dimerization is a folding event. Second, the stability of the dimer contributes significantly to the conformational free energy of the protein (18.8 of 25.8 kcal/mol).     

The mechanism of stabilization of the structure of nuclease-T by binding of ligands.

The rate of unfolding of Nuclease-T at pH 8,20 degrees was determined as a function of concentration of the ligands deoxythymidine 3',5'-diphosphate (pdTp) and Ca2+ on the basis of the rate of exchange between free fragment, Nuclease-T(50-149) and labeled fragment, Nuclease-T-(50-149) incorporated in the structure of nuclease-T (Taniuchi, H. (1973) J. Biol. Chem. 248, 5164-5174). The rate constant of unfolding of unliganded Nuclease-T' was 4.6 times 10-4s-1. Those of Nuclease-T' bound with pdTp, with Ca2+, and with both pdtp and Ca2+ were 9.0 times 10-5, 1.6 times 10-4, and 2.2 times 10-5s-1, respectively. The association constants of pdTp and Ca2+ with Nuclease-T' were found to be 1.0 times 10-4 and 2.0 times 10-2 m-1, respectively. Those of pdTp with Nuclease-T' plus Ca2+ and of Ca2+ with Nuclease-T' plus pdTp were 4 times 10-5 and 1.4 times 10-4M-1, respectively. The calculation of free energy change on the basis of the association constants shows that the magnitude of negative free energy change involved in the binding of either of the two ligands increases by approximately 2 kcal when the other ligand is already bound. There is a correlation between the free energy change and the specifically coupled with the cooperative interacions operating throught the three-dimensional structure resulting in strengthening of the interactions throughtout the structure, including those with the ligands, without a large change in conformation.     

Mutations of an epitope hot-spot residue alter rate limiting steps of antigen-antibody protein-protein associations

The antibodies, HyHEL-10 and HyHEL-26 (H10 and H26, respectively), share over 90% sequence homology and recognize with high affinity the same epitope on hen egg white lysozyme (HEL) but differ in degree of cross-reactivity with mutant lysozymes. The binding kinetics, as measured by BIAcore surface plasmon resonance, of monovalent Fab from both Abs (Fab10 and Fab26) to HEL and mutant lysozymes are best described by a two-step association model consistent with an encounter followed by docking that may include conformational changes. In their complexes with HEL, both Abs make the transition to the docked phase rapidly. For H10, the encounter step is rate limiting, whereas docking is also partially rate limiting for H26. The forward rate constants of H10 are higher than those of H26. The docking equilibrium as well as the overall equilibrium constant are also higher for H10 than for H26. Most of the free energy change of association (Delta G degrees) occurs during the encounter phase (Delta G1) of both Abs. H10 derives a greater amount and proportion of free energy change from the docking phase (Delta G2) than does H26. In the H10--HEL(R21Q) complex, a significant slowing of docking results in lowered affinity, a loss of most of Delta G2, and apparently faster dissociation. Slower encounter and docking cause lowered affinity and a loss of free energy change primarily in the encounter step (Delta G1) of H26 with mutant HEL(R21Q). Overall, in the process of complex formation with lysozyme, the mutations HEL(R21X) affect primarily the docking phase of H10 association and both phases of H26. Our results are consistent with the interpretation that the free energy barriers to conformational rearrangement are highest in H26, especially with mutant antigen.     

Quantitative analysis of mixed association between different protein molecules. Physico-chemical and enzymatic properties of rat-liver glutamate dehydrogenase.

The theoretical and experimental analysis of a reversible association-dissociation equilibrium between different proteins (mixed association) is described. The experiments were performed with glutamate dehydrogenases from beef and rat liver. These enzymes are different, especially with respect to their association behavior. The association constant of rat liver glutamate dehydrogenase has been determined by light-scattering measurements. Its value (1.3 x 10(-4) M(-1)) is much lower than that of the beef liver enzyme, but the difference in the free association energy is only 30%. Association between these two enzymes is observed, also employing light-scattering experiments. Theoretical curves for mixed associating systems have been calculated and by comparison with these curves the mixed association constant could be determined. Since the free association energy of the mixed association is very near to the arithmetic mean between the values for the pure enzymes, the association interactions appear to be additive. The model of an open association with a virial coefficient is also true for the rat enzyme and the mixed association. The ultracentrifuge data are also explained by the same model and yield a similar value for the mixed association constant. Differences in the enzyme kinetics are small, but a somewhat reduced lifetime of the ternary complexes with the coenzymes and with subs-rates or GTP can be concluded for the rat liver enzyme. The circular dichroism measurements indicate no significant difference in the dissociation constants of the nucleotides, but the different amplitudes of the ellipticity indicate small differences in the electrical environment of the active center.     

A model for calculating the Bohr effect in hemoglobin equilibria.

The unusual aspects of the reaction of oxygen with hemoglobin are believed to be due to the free energy of the conformational change in the hemoglobin molecule upon oxygenation. The conformational free energy change due to oxygenation can be estimated in terms of the surface free energy of an emuslion droplet of the same size as the hemoglobin molecule. Calculations on the basis of this model lead to an equilibrium constant that varies with pH as in the acid and alkaline Bohr Effects, and that also varies with the ionic strength. The model used in this paper provides a simple way of estimating the variation of the equilibrium constant of a reaction involving a globular protein where the free energy of conformational changes can be evaluated in terms of surface properties.     

Calculating the free energy of association of transmembrane helices

A large number of experimental studies have been devoted to quantifying the interaction between transmembrane (TM) helices in detergent micelles and, more recently, in bilayers. Theoretical calculation of association free energy of TM helices would be useful for predicting the propensity of given sequences to oligomerize and for understanding the difference between association in micelles and in bilayers. In this article, the theoretical foundation for calculating the standard association free energy of TM helices is laid out and is applied to glycophorin A in both micelles and bilayers. The standard association free energy is decomposed into the effective energy, translational, rotational, and conformational entropy terms. The effective energy of association is obtained by molecular dynamics simulations in an implicit membrane model. The translational and rotational entropy of association is calculated from the probability distribution of the translational and rotational degrees of freedom obtained from the molecular dynamics simulations. The side-chain conformational entropy of association is estimated from the probability distribution obtained by rigid rotation of all side-chain dihedral angles. The calculated standard association free energy of glycophorin A in N-dodecylphosphocholine micelles is in good agreement with the experimental value. The translational entropy cost is larger, whereas the rotational entropy cost is smaller in bilayers than in micelles. The standard association free energy in 1,2-dimyristoyl-sn-glycero-3-phosphocholine bilayers is calculated to be approximately 1.3 kcal/mol more favorable than in N-dodecylphosphocholine micelles, consistent with available experimental data.     

Standardizing the free energy change of transmembrane helix-helix interactions

Side-to-side associations of transmembrane alpha-helices are integral components of the structure and function of helical membrane proteins. A fundamental unknown in the understanding of the chemical principles driving the lateral interactions between transmembrane alpha-helices is the balance of forces arising from the polypeptide sequence versus the hydrophobic solvent. To begin to address this question, a consideration of basic thermodynamic principles has been applied to assess the experimental free energy change associated with transmembrane helix dimerization in micelles. This analysis demonstrates the ability to partition the apparent free energy of transmembrane helix-helix association into two components. The first component is a statistical energy term, which arises from the fact that there are an unequal number of reactants and products. The second component is a standard state free energy change, which informs on the molecular details of the transmembrane helix self-association reaction. The advantage of separating these two energy terms arises from the fact that extrapolation to the standard state free energy change normalizes the statistical energy term so that it applies equivalently in all experimental systems. Accompanying experimental results for the glycophorin A transmembrane alpha-helix dimer measured in micelles are well described by these theoretical components assuming an ideal-dilute solution.     

Thermodynamics of glycophorin A transmembrane helix dimerization in C14 betaine micelles

We have used sedimentation equilibrium analytical ultracentrifugation to measure the free energy change for the glycophorin A transmembrane helix-helix dimerization in C14 betaine micelles. By varying the amount of micellar C14 betaine, we show that the protein association reaction in the micellar C14 phase behaves as an ideal-dilute solution. In this hydrophobic environment, the mole-fraction standard state free energy change for self-association of the SNGpA99 glycophorin A construct is -5.7 (+/-0.3, N=5) kcal mol(-1) at 25 degrees C. Compared with previous results carried out in C(8)E(5) micellar solutions, the free energy of dimerization is 1.3 kcal mol(-1) less favorable in C14 betaine micelles. In contrast, when considered on a per-interface basis, the formation of the glycophorin A transmembrane dimer in C14 betaine micelles may be more favorable than the association of several designed transmembrane peptides.     

Statistical thermodynamic analysis of peptide and protein insertion into lipid membranes.

A statistical thermodynamic approach is used to analyze the various contributions to the free energy change associated with the insertion of proteins and protein fragments into lipid bilayers. The partition coefficient that determines the equilibrium distribution of proteins between the membrane and the solution is expressed as the ratio between the partition functions of the protein in the two phases. It is shown that when all of the relevant degrees of freedom (i.e., those that change their character upon insertion into the membrane) can be treated classically, the partition coefficient is fully determined by the ratio of the configurational integrals and thus does not involve any mass-dependent factors, a conclusion that is also valid for related processes such as protein adsorption on a membrane surface or substrate binding to proteins. The partition coefficient, and hence the transfer free energy, depend only on the potential energy of the protein in the membrane. Expressing this potential as a sum of a "static" term, corresponding to the equilibrium (minimal free energy) configuration of the protein in the membrane, and a "dynamical" term representing fluctuations around the equilibrium configuration, we show that the static term contains the "solvation" and "lipid perturbation" contributions to the transfer free energy. The dynamical term is responsible for the "immobilization" free energy, reflecting the loss of translational and rotational entropy of the protein upon incorporation into the membrane. Based on a recent molecular theory of lipid-protein interactions, the lipid perturbation and immobilization contributions are then expressed in terms of the elastic deformation free energy resulting from the perturbation of the lipid environment by the foreign (protein) inclusion. The model is formulated for cylindrically shaped proteins, and numerical estimates are given for the insertion of an alpha-helical peptide into a lipid bilayer. The immobilization free energy is shown to be considerably smaller than in previous estimates of this quantity, and the origin of the difference is discussed in detail.     

Role of bound water in protein-ligand association processes

The role of water molecules on the protein-ligand interface during macromolecular association has been determined. The free energy of association of insulin has been calculated by the methods of molecular mechanics and continual electrostatics (Poisson-Boltzmann model). The previously developed scheme of the decomposition of association free energy onto contributions from individual interactions has been used to calculate intermolecular interactions, the solvation free energy, and the entropies of the process of macromolecular association. An analysis of the calculated oscillation spectra indicated that the presence of water molecules on the protein-protein interface promotes an increase in the contribution of vibration entropy to the free energy of association due to the enhancement of the flexibility of the complex. It was shown that water molecules involved in the formation of protein-water-ligand hydrogen bond network change the balance of forces in the system.     

Thermodynamics of Ca2+ transport through sarcoplasmic reticulum membranes during the transient-state of simulated reactions

The kinetics of a chemical model of Ca2+ transport and coupled ATPase activity in sarcoplasmic reticulum membranes were solved for the transient-state of simulated reactions, using a numerical integration procedure. The simulation conditions reproduced in vitro experiments using either fragmented membranes or vesicles with Ca2+ accumulating ability. The results yielded the concentrations of all the ligands and intermediates of the enzymatic cycle as a function of the reaction time. These results were applied to calculations of several thermodynamic variables: (1) the step by step profile of the standard free energy change of the cycle. (2) The step by profile of the actual free energy change of the cycle, and its evolution with the reaction time. (3) The separate contributions of ATP hydrolysis and Ca2+ transport to the overall free energy change with the reaction. (4) The dependence of the velocity of the free energy change with the reaction time. (5) The efficiency of the transport system, and its change with the reaction time. (6) The separate contributions of the Ca2+ gradient and some enzymatic intermediates as free energy stores. The main findings are: (1) the step by step diagrams of the free energy change calculated from the results of the kinetic analysis better describe the thermodynamic profile of the cycle than previously reported diagrams of the standard free energy and basic free energy changes. The relative contribution of each partial step to the driving force of the whole reactions, as well as their changes upon the advancement of the reactions, are derived from the diagrams. (2) Free energy yielded by ATP hydrolysis is stored by the system, not only as a Ca2+ gradient, but also as enzymatic intermediates of the reaction. The progressive increase of both free energy pools upon the advancement of the reaction is quantitated.     

Realistic protein-protein association rates from a simple diffusional model neglecting long-range interactions, free energy barriers, and landscape ruggedness

We develop a simple but rigorous model of protein-protein association kinetics based on diffusional association on free energy landscapes obtained by sampling configurations within and surrounding the native complex binding funnels. Guided by results obtained on exactly solvable model problems, we transform the problem of diffusion in a potential into free diffusion in the presence of an absorbing zone spanning the entrance to the binding funnel. The free diffusion problem is solved using a recently derived analytic expression for the rate of association of asymmetrically oriented molecules. Despite the required high steric specificity and the absence of long-range attractive interactions, the computed rates are typically on the order of 10(4)-10(6) M(-1) sec(-1), several orders of magnitude higher than rates obtained using a purely probabilistic model in which the association rate for free diffusion of uniformly reactive molecules is multiplied by the probability of a correct alignment of the two partners in a random collision. As the association rates of many protein-protein complexes are also in the 10(5)-10(6) M(-1) sec(-1) range, our results suggest that free energy barriers arising from desolvation and/or side-chain freezing during complex formation or increased ruggedness within the binding funnel, which are completely neglected in our simple diffusional model, do not contribute significantly to the dynamics of protein-protein association. The transparent physical interpretation of our approach that computes association rates directly from the size and geometry of protein-protein binding funnels makes it a useful complement to Brownian dynamics simulations.     

Thyroxine-protein interactions. Binding constants for interaction of thyroxine analogues with the thyroxine binding site on human thyroxine-binding globulin.

The binding constants for interaction of various thryoxine analogues with the thyroxine binding site on human thyroxine-binding globulin have been determined. Equilibrium dialysis, at pH 7.4 and 37 degrees C, was used to measure the competitive effects of different iodothyronine compounds on the binding of 125I-labeled thyroxine to highly purified thyroxine-binding globulin. Relative to L-thyroxine, K = 6 . 10(9) M-1, the association constants of some important analogues were D-thyroxine, 1.04 . 10(9) M-1, 3,5-diiodo-3'-isopropyl-L-thyronine, 4.9 . 10(8) M-1; L-triiodothyronine, 3.3 . 10(8) M-1, 3,3',5'-DL-triiodothyronine (reverse triiodothyronine), 3.1. 10(8) M-1; tetraiodothyropropionic acid, 2.7 . 10(8) M-1; tetraiodothyroacetic acid, 2.6 . 10(8) M-1; 3', 5'- diiodo-DL-thyronine, 8.3 . 10(7) M-1; and 3,5-diiodo-DL-thyronine, 7.1 . 10(7) M-1. Calculation of the deltaG0 values for binding of the analogues indicates that a major contribution to the free energy favoring binding is made by the alanine side chain of thyroxine. A change in configuration of the alpha-amino group from the L to D form causes an unfavorable change of 1 kcal/mol in the free energy of binding. Removal of the alpha-amino group as in tetraiodothyropropionic acid causes an unfavorable change of 1.9 kcal/mol in the free energy of binding. With regard to ring substituents, the results indicate that the two inner 3,5-iodines make about the same contribution to binding as the two outer 3', 5'-iodines.     

Quantitative model for binary measurements of protein-protein interactions.

We develop a stochastic model for quantifying the binary measurements of protein-protein interactions. A key concept in the model is the binary response function (BRF) which represents the conditional probability of successfully detecting a protein-protein interaction with a given number of the protein complexes. A popular form of the BRF is introduced and the effect of the sharpness (Hill's coefficient) of this function is studied. Our model is motivated by the recently developed yeast two-hybrid method for measuring protein-protein interaction networks. We suggest that the same phenomenological BRF can also be applied to the mass spectroscopic measurement of protein-protein interactions. Based on the model, we investigate the contributions to the network topology of protein-protein interactions from (i) the distribution of protein binary association free energy, and from (ii) the cellular protein abundance. It is concluded that the association constants among different protein pairs cannot be totally independent. It is also shown that not only the association constants but also the protein abundance could be a factor in producing the power-law degree distribution of protein-protein interaction networks.