Coupling Free Energy Research Articles

In vivoand in vitroStudies of TrpR-DNA Interactions

The binding of tryptophan repressor (TrpR) to its operators was examined quantitatively using in vitroand in vivomethods. DNA sequence requirements for 1:1 and tandem 2 :1 (TrpR : DNA) binding in various sequence contexts were studied. The results indicate that the optimal half-site sequence for recognition by one helix-turn-helix motif of one TrpR dimer is 3′ CNTGA 5′ 5′ GNACT 3′, consistent with contacts observed by X-ray diffraction analysis of cocrystalline 1:1 and 2 :1 complexes. Half-sites can be paired to form a palindrome either by direct abutment, forming the nucleation site for a tandem 2 :1 complex, or with an 8-base-pair spacer, forming a 1:1 target. Dimethylsulfate (DMS) methylation-protection footprinting in vitroof 1:1 and 2 :1 complexes formed sequentially on the two unequal half-site pairs of the trpEDCBA operator from Serratia marcescensindicated an obligate hierarchy of site occupancy, with one half-site pair serving as the nucleation site for tandem binding. DMS footprinting of Escherichia colioperators in vivoshowed that, over a wide range of intracellular TrpR concentration, the trpEDCBA operator is occupied by three repressor dimers, aroH is occupied by two dimers, and the 1:1 binding mode is used on the trpR operator. The coexistence of these distinct occupancy states implies that changes in protein concentration affect only the fractional occupancy of each operator rather than the binding mode, which is determined by the number of half-site sequences present in the operator region. Cooperativity of tandem complex formation measured by gel retardation using a symmetrized synthetic operator containing identical, optimal sites spaced as in natural operators was found to be modest, implying a maximum coupling free energy of ∼−2 kcal/mol. On other sequences the apparent degree of cooperativity, as well as the apparent affinity, varied with sequence and sequence context in a manner consistent with the structural models and which suggests compensation between affinity and cooperativity as a mechanism that allows tolerance of operator sequence variation.

Allosteric effects of carbamoyl phosphate synthetase from Escherichia coli are entropy-driven.

When catalyzing the formation of MgATP and carbamate from MgADP and carbamoyl phosphate, Escherichia coli carbamoyl phosphate synthetase (CPS) binds MgADP with a large negative change in heat capacity. The magnitude of this heat capacity change is not appreciably altered by the presence of a saturating concentration of either the allosteric activator ornithine or the inhibitor UMP despite the substantial and opposing effects these ligands have on the binding affinity for MgADP. By contrast, no detectable change in heat capacity is associated with the thermodynamic coupling between MgADP and either ornithine or UMP. The sign of the apparently constant enthalpic and entropic contributions to the coupling free energy for each of these ligands is opposite that of the coupling free energy, indicating that the observed allosteric phenomenology is in net opposed by the enthalpy of the interaction and instead arises from a change in entropy of the system. IMP produces only a very small allosteric effect as indicated by a near-zero value for the MgADP-IMP coupling free energy. However, the enthalpic and entropic contributions are individually larger in absolute value for the IMP coupling than for those pertaining to the other allosteric ligands, and entropy dominates the coupling free energy above 36 degrees C, causing IMP to become an activator at high temperature. In addition, the sign of the coupling enthalpy and entropy for IMP has the same sign as the coupling enthalpy and entropy produced by ornithine, suggesting that IMP and ornithine may similarly influence the enzyme at a molecular level despite binding to different allosteric sites on the enzyme. The data are consistent with a model in which the actions of the allosteric ligands arise primarily from changes in the conformational degeneracy introduced by each ligand. With this model, one can also rationalize the failure of these allosteric ligands to substantially influence kcat.

Release of fibrinopeptides by the slow and fast forms of thrombin.

The release of fibrinopeptides A and B by the slow and fast forms of thrombin was studied over the temperature range from 5 to 45 degrees C and the salt concentration range from 100 to 800 mM. The sequential mechanism for the release of fibrinopeptides originally proposed by Shafer was found to be obeyed under all conditions examined. The origin of preferential binding of fibrinogen and fibrin I to the fast form of thrombin in the transition state is in the second-order rate constant for association, k(l). In the case of fibrinogen, the values of k(l) for interaction with the fast and slow forms at 25 degrees C are 19 +/- 4 and 2.5 +/- 0.3 microM(-1) s(-1), with an activation energy of about 10 kcal/mol in both forms. In the case of fibrin I, the analogous values of k(l) are 9.1 +/- 0.7 and 2.5 +/- 0.2 microM(-1) s(-1), and the activation energy is about 4.5 kcal/mol in both forms. The mechanism of recognition of fibrinogen and fibrin I by thrombin entails a diffusion-controlled step with a small energy barrier. Analysis of the temperature dependence of the coupling free energy for allosteric switching indicates that the preferential interaction of fibrinogen and fibrin I with the fast form of thrombin in the transition state is entropy-driven, signaling a contribution of the hydrophobic effect to the slow-->fast transition. The salt dependence of the release of fibrinopeptides shows a constant coefficient Gamma(salt) = d ln(k(cat)/K(m))/d ln [salt] in the concentration range examined. Interestingly, the value of Gamma(salt) is independent of the salt used (NaCl, ChCl, or NaF) and is -1.5 +/- 0.1 for fibrinopeptide A and -2.5 +/- 0.1 for fibrinopeptide B. Hence, Gamma(salt) reflects predominantly the electrostatic contribution to the formation of the transition state, with a larger contribution seen in the interaction of thrombin with fibrin I. It is concluded that the interaction of thrombin with fibrinogen and fibrin I, leading to the release of fibrinopeptides A and B, is driven by electrostatic forces that presumably favor the correct preorientation of the enzyme and the substrate to form a productive complex in the transition state. This electrostatic-steering effect, also reported for thrombin-hirudin interaction, leads to a diffusion-controlled encounter with a very small energy barrier. Once the complex is formed, the enzyme switches to the fast form as a result of entropic factors presumably linked to water release from a more extended surface of recognition. While the release of fibrinopeptides as a function of salt concentration was being studied, an important observation was made on the role of Cl- in the formation of the fibrin clot. This anion drastically and specifically reduces the thickness of fibrin fibers, as judged by the 10-fold decrease in the equilibrium turbidity of clots developed in NaCl as compared to the turbidity of clots developed in NaF. Hence, the transition from a "coarse" to a "fine" clot induced by an increase in ionic strength as first described by Ferry is, instead, due to the specific binding of Cl- to intermediates in the ensuing polymerization. In fact, no change in the clotting curve is observed when the ionic strength is changed with NaF.

Thermodynamic Investigation of Hirudin Binding to the Slow and Fast Forms of Thrombin: Evidence for Folding Transitions in the Inhibitor and Protease Coupled to Binding

Temperature dependent studies of the interaction of the clotting enzyme thrombin with the potent natural inhibitor hirudin reveal a large negative heat capacity change of −1.7(±0.2) kcal/mol per K associated with the formation of the thrombin-hirudin complex, independent of the allosteric state of the enzyme. Binding of N-terminal fragments of hirudin (hir 1–49cand hir 1–43) is characterized by heat capacity changes of −1.2(±0.1) and −0.9(±0.1) kcal/mol per K, respectively. The magnitude of these heat capacity changes is unprecedented for protease-inhibitor interactions. A thermodynamic analysis based on observed heat capacity and entropy changes predicts that binding is accompanied by substantial coupled folding transitions in both hirudin and thrombin. In the absence of a structure of free thrombin, analysis of differences in the predicted number of residues which fold upon binding hirudin and its fragments leads to the following structural model: three surface loops in thrombin (W60d, W148 and fibrinogen binding loops) are disordered in the free state and fold upon formation of the thrombin-hirudin complex. Molecular dynamics simulations, run over a time scale of 5 ps, are consistent with the hypothesis of large scale coupled folding transitions in both hirudin and thrombin upon formation of the complex. Comparison of the thermodynamics for the interaction of hirudin with the slow and fast forms of thrombin allows dissection of the coupling free energy for allosteric switching. The coupling free energy for the slow→fast transition increases linearly, in absolute value, with temperature. The coupling enthalpy and entropy terms for hirudin were found to be ΔH c o=12(±1) kcal/mol andΔS c o=47(±4) cal/mol per K. Preferential interaction with the fast form is therefore due to the balance of two opposite forces, both quite large in magnitude. The contribution of enthalpic effects opposes the slow→fast transition and stabilizes binding to the slow form. The contribution of entropic effects favors the slow→fast transition and stabilizes binding to the fast form. In the physiological temperature range the entropic effects prevail and result in preferential binding of hirudin to the fast form. The region of thrombin recognizing the N-terminal domain of hirudin contains most of the residues that are energetically linked to the slow→fast transition. This region is part of the "allosteric core" of thrombin and includes the W60d loop, shaping the specificity site S2, and the Na +binding loop connecting the last two β-strands of the B chain.

Nested cooperativity in the ATPase activity of the oligomeric chaperonin GroEL

Initial rates of ATP hydrolysis by wild-type GroEL were measured as a function of ATP concentration from 0 to 0.8 mM. Two allosteric transitions are observed: one at relatively low ATP concentrations (< or = 100 microM) and the second at higher concentrations of ATP with respective midpoints of about 16 and 160 microM. Two allosteric transitions were previously observed also in the case of the Arg-196-->Ala GroEL mutant [Yifrach, O., & Horovitz, A. (1994) J. Mol. Biol. 243, 397-401]. On the basis of these observations a mathematical model for nested cooperativity in ATP hydrolysis by GroEL is developed in which there are two levels of allostery: one within each ring and the second between rings. In the first level, each hepatameric ring is in equilibrium between the T and R states, in accordance with the Monod-Wyman-Changeux (MWC) model of cooperativity [Monod et al. (1965) J. Mol. Biol. 12, 88-118]. A second level of allostery is between the rings of the GroEL particle which undergoes sequential Koshland-Némethy-Filmer (KNF)-type transitions from the TT state via the TR state to the RR state [Koshland et al. (1966) Biochemistry 5, 365-385]. Using our model, we estimate the values of the Hill coefficient for the negative cooperativity between rings in wild-type GroEL and the Arg-196-->Ala mutant to be 0.003 (+/- 0.001) and 0.07 (+/- 0.02), respectively. The inter-ring coupling free energies in wild-type GroEL and the Arg-196-->Ala mutant are -7.5 (+/- 0.4) and -3.9 (+/- 0.3) kcal mol-1, respectively.

Linear-Response Calculations of Electron-Phonon Coupling Parameters and Free Energies of Defects

AbstractLinear-response theory provides an efficient approach for calculating the vibrational properties of solids. Moreover, because the use of supercells is eliminated, points with little or no symmetry in the Brillouin zone can be handled. This allows accurate determinations of quantities such as real-space force constants and electron-phonon coupling parameters. We present highly converged calculations of the spectral function α2F(ω) and the average electron-phonon coupling for Al, Pb, and Li. We also present results for the free energy of vacancy formation in Al calculated within the harmonic approximation.

Kinetic and standard thermodynamic parameters of photoinduced charge separation reactions in a bichromophoric anthracene-containing molecule from time-resolved fluorescence measurements in different solvents and solvent mixtures and at different temperatures

The determination of kinetic and standard thermodynamic parameters (reorganization energies, electronic coupling elements, free energy, enthalpy and entropy changes) of intramolecular photoinduced charge separation reactions in ω-(10-Phenyl-9-anthryl)propiophenone from the analysis of biexponential fluorescence decay curves exhibits distinct differences between the electronic factors in aprotic (benzene, acetonitrile, and mixtures thereof) and protic (methanol, ethanol) solvents and an unexpectedly small variation of the standard free energy change with solvent polarity.

Coupling of calcium to the interaction of troponin I with troponin C from cardiac muscle.

The interaction of troponin I (CTnI) with troponin C (CTnC) from bovine cardiac muscle was studied using CTnC modified at Cys35 and Cys84 with the fluorescent probe 2-[(4'-iodoacetamido)-anilino]naphthalene-6-sulfonic acid (CTnCIAANS). The association constant for complex formation between the two proteins was determined at 20 degrees C in 0.4 M KCl, 1 mM DTT, 1 mM EGTA, and 25 mM MOPS, pH 7.2. In the presence of EGTA, Mg2+, and Ca2+ these constants were 1.46 x 10(7), 4.1 x 10(7), and 12.7 x 10(7) M-1, respectively, with corresponding free energy values of -9.62, -10.23, and -10.88 kcal mol-1. The CTnI-CTnCIAANS complex was stabilized by -0.61 kcal when the two Ca/Mg sites of CTnCIAANS were saturated with Mg2+ and by -1.26 kcal when all three Ca2+ sites were occupied by Ca2+. These results suggest that calcium activation in cardiac muscle may be accompanied by a coupling free energy of -0.65 kcal. This value is a factor of 4 smaller than the value previously determined, using a similar method, for the (troponin I).(troponin C) complex from skeletal muscle [Wang, C.-K., & Cheung, H.C. (1985) Biophys. J.48, 727-739]. Since CTnC has only one Ca(2+)-specific site and troponin C from skeletal muscle has two such sites, the present result is a factor of 2 smaller than that for the skeletal complex on the basis of a single specific site. Phosphorylation of CTnI by 3',5'-cyclic AMP-dependent protein kinase resulted in a decrease of the association constants by a factor of 2.5-3.5.(ABSTRACT TRUNCATED AT 250 WORDS)

Influence of MgADP on phosphofructokinase from Escherichia coli. Elucidation of coupling interactions with both substrates.

A comprehensive assessment is presented of the mutual influence that MgADP, MgATP, and fructose 6-phosphate (Fru-6-P) have on each other's binding to phosphofructokinase (PFK) from E. coli. When virtually any combination of these ligands binds to PFK it produces a significant perturbation in the intrinsic tryptophan fluorescence intensity and/or polarization which not only provides a means to follow binding in titration experiments but which also underscores the fact that more than two different enzyme conformations result from the binding of these ligands. When MgATP is saturating, the binding of MgADP to the allosteric site increases the affinity the enzyme subsequently displays for Fru-6-P. However, in the absence of MgATP, MgADP can bind to both the allosteric site and the nucleotide portion of the active site, with the latter antagonizing the binding of Fru-6-P to an extent that leads to an overall inhibition of Fru-6-P binding by MgADP. MgADP binding at the allosteric site also inhibits the binding of MgATP, indicating that under many circumstances MgADP should be more properly viewed as an inhibitor rather than an activator of E. coli PFK. After quantifying all of the 20 dissociation constants and 11 coupling parameters between ligand pairs pertinent to this three-ligand system, the more significant coupling parameters have been further characterized by examining their variation with temperature to establish the apparent enthalpy and entropy contributions to the corresponding coupling free energies. For both activating and inhibitory couplings, the enthalpy and entropy terms have the same sign as the coupling free energy.(ABSTRACT TRUNCATED AT 250 WORDS)

Influence of substrates and MgADP on the time-resolved intrinsic fluorescence of phosphofructokinase from Escherichia coli. Correlation of tryptophan dynamics to coupling entropy.

The influence of that MgADP and the substrate ligands MgATP and fructose 6-phosphate (Fru-6-P) have on the structure of E. coli phosphofructokinase (PFK) in the vicinity of the single tryptophan that exists in each subunit has been examined by employing both steady-state and time-resolved measurements of the tryptophan fluorescence. The accessibility of the tryptophan to iodide quenching is over 1 order of magnitude less than experienced by N-acetyltryptophanamide in solution but varies nonetheless with the state of ligation. Most, but not all, of these changes correlate with changes in the degree of local motion available to the tryptophan side chain as determined by steady-state and time-resolved polarization measurements. When the data obtained from differential polarization experiments are fit to a model in which the motion of the tryptophan side chain is able to move with high frequency within a cone of limited amplitude as part of an otherwise slowly tumbling spherical protein, it was found that ligands primarily affect the amplitude of the available local motion. By interpreting these effects with reference to the disproportionation equilibria which define the negative coupling free energy between MgADP and Fru-6-P and the positive coupling free energy between MgADP and MgATP, it is apparent that changes in the local motion amplitudes correlate with the sign of the component coupling entropy previously determined from van't Hoff analyses (Johnson & Reinhart, 1994).(ABSTRACT TRUNCATED AT 250 WORDS)

Comparison of the Inhibition by Phospho(enol)Pyruvate and Phosphoglycolate of Phosphofructokinase from B. stearothermophilus

A comparison between the inhibition by phospho(enol)pyruvate (PEP) versus the inhibition by phosphoglycolate (PG) of phosphofructokinase (PFK) from Bacillus stearothermophilus is presented. Both inhibitors act by decreasing the apparent affinity displayed by the enzyme for its substrate fructose 6-phosphate (Fru-6-P) while having little effect on V max. However, the two ligands differ in both their affinity for the enzyme and their effectiveness at antagonizing the subsequent binding of Fru-6-P. Although PG binds with approximately 10-fold lower affinity, it antagonizes the binding of Fru-6-P 3.5-fold more strongly than does PEP. Moreover, the enthalpy and entropy contributions to the coupling free energy between inhibitor and Fru-6-P, from which these antagonisms derive, reveal even greater differences between the ligands. These data indicate, therefore, that the changes in the structure of PFK from B. stearothermophilus that result from PG binding, which have been determined by X-ray crystallography (T. Schirmer and P. R. Evans, 1990 Nature 343, 140-145), may not be comparable to those that result from PEP binding and consequently do not represent the generic "T-state," as has been presumed.

Temperature-induced inversion of allosteric phenomena.

Two instances, involving the enzymes carbamoyl-phosphate synthetase from Escherichia coli and phosphofructokinase from Bacillus stearothermophilus, respectively, are described in which increasing temperature alone causes the actions of an allosteric ligand to change from inhibition to activation. In neither case are these effects due to a change in the activation energy of the enzyme catalyzed reaction induced by the allosteric ligand. Rather, they are due to temperature-dependent changes in the extent to which the binding of allosteric ligand modifies the affinity of the enzyme for substrate. The data can be readily explained by an analysis of the apparent delta H and delta S components of the coupling free energy, which quantitatively describe the actions of allosteric ligands that act in this manner. These observations underscore the shortcomings of expecting to explain the actions of an allosteric ligand solely by the structural perturbations that accompany the binding of an allosteric ligand such as those often revealed by x-ray crystallography.

MgATP and fructose 6-phosphate interactions with phosphofructokinase from Escherichia coli.

A thermodynamic linked-function analysis is presented of the interactions of MgATP and fructose 6-phosphate (Fru-6-P) with phosphofructokinase (PFK) from Escherichia coli in the absence of allosteric effectors. MgATP and Fru-6-P are shown to bind in random fashion by product inhibition of the back-reaction as well as by the kinetically competent binding of each ligand individually as monitored by the consequent changes in the intrinsic fluorescence of E. coli PFK. When Fru-6-P is saturating, the dissociation of MgATP is sufficiently slow that it cannot achieve a binding equilibrium in the steady state, causing the observed Km (49 microM) to significantly exceed the Kd (1.7 microM) deduced from a thermodynamic linkage analysis. The following features distinguish the interactions of MgATP and Fru-6-P with E. coli PFK: MgATP and Fru-6-P antagonize each other's binding to the enzyme in a saturable manner with an overall apparent coupling free energy equal to +2.5 kcal/mol at 25 degrees C; MgATP induces positive cooperativity in the Fru-6-P binding profile, with the Hill coefficient calculated from the Fru-6-P binding curves reaching a maximum of 3.6 when MgATP is saturating; and MgATP exhibits substrate inhibition at low concentrations of Fru-6-P. Simulations based upon the rate equation pertaining to a two-active-site, two-substrate dimer indicate that these features can all result from two independent couplings: an antagonistic MgATP-Fru-6-P coupling extending at least in part between active sites and a MgATP-induced Fru-6-P-Fru-6-P coupling.(ABSTRACT TRUNCATED AT 250 WORDS)

Interaction of wheat germ protein synthesis initiation factors eIF-3, eIF-(iso)4F, and eIF-4F with mRNA analogs

The interaction of wheat germ eIF-3 with the wheat germ cap-binding proteins eIF-(iso)4F and eIF-4F as a function of pH and ionic strength is described. Direct fluorescence titration experiments are used to measure the equilibrium association constants (Keq) for the binary protein/protein complexes as well as for the interaction of eIF-3 with methylated cap analogues and rabbit alpha-globin mRNA oligonucleotide analogues. The Keq values for ternary eIF-3/eIF-(iso)4F/analogue and eIF-3/eIF-4F/analogue interactions were also measured. The equilibrium binding constants were used to calculate coupling free energies, which provide an estimate of the cooperativity for the interaction of the mRNA analogues, eIF-3, and either eIF-4F or eIF-(iso)4F. These data suggest a mechanism in which the binding of eIF-(iso)4F or eIF-4F to mRNA enhances the subsequent binding of eIF-3 to the message. This may lead to favorable positioning of the complex on the ribosome and thereby enhance translation.

Fluorescence study of the binding of m7GpppG and rabbit globin mRNA to protein synthesis initiation factors 4A, 4E, and 4F

The interactions of protein synthesis initiation factors eIF-4E from human erythrocytes and eIF-4A and eIF-4F from rabbit reticulocytes with the cap analogue m7GpppG and rabbit globin mRNA were investigated. The equilibrium binding constants for the binary complex formation of eIF-4E-eIF-4A, m7GpppG-eIF-4E, m7GpppG-eIF-4F, globin mRNA-eIF-4E, globin mRNA-eIF-4F, and globin mRNA-eIF-4A were measured by direct fluorescence titration experiments. The binding of eIF-4E to globin mRNA was found to be 5.5-fold tighter than its binding to m7GpppG; the binding of eIF-4F for globin mRNA and m7GpppG was similar to that of eIF-4E. Association equilibrium constants were determined for the ternary system mRNA-eIF-4E-eIF-4A; four thermodynamically independent equilibria characterize the system. These equilibrium binding constants were used to calculate coupling free energies, which provided an estimate of the cooperativity of the interaction of eIF-4E, eIF-4A, and mRNA. These coupling energies were all found to be small and positive, indicative of anticooperative binding.

Role of coupling entropy in establishing the nature and magnitude of allosteric response.

The coupling free energy between an allosteric ligand and a substrate, delta Gax, is an explicit measure of the nature as well as the magnitude of impact that an allosteric ligand has on the binding of the substrate ligand to the enzyme, with positive values indicating inhibition and negative values indicating activation. By measuring the variation with temperature of the coupling free energy between the allosteric ligand and the substrate, it is possible to determine the enthalpic and entropic components that give rise to the coupling free energy. We have performed this analysis on two different K-type allosteric systems: the allosteric inhibition of rat liver phosphofructokinase by MgATP, and the allosteric activation of beef heart NAD+-dependent isocitrate dehydrogenase by ADP. In both cases the coupling free energy arises as the net result of opposing enthalpic and entropic components, with the coupling enthalpy (delta Hax) favoring activation and the coupling entropy (delta Sax) favoring inhibition. For phosphofructokinase at 25 degrees C, the absolute value of T delta Sax is greater than the absolute value of delta Hax, and net inhibition of rat liver phosphofructokinase by MgATP is realized. For isocitrate dehydrogenase, delta Hax dominates; however, the net activation is substantially mitigated by the magnitude of T delta Sax. Hence, the coupling entropy plays an important role in establishing both the nature and magnitude of the allosteric response. We hypothesize that the negative coupling entropy arises from the particular constraint placed upon the internal dynamical properties of the enzyme by the simultaneous binding of both allosteric and substrate ligands.

Linked-function origins of cooperativity in a symmetrical dimer.

The thermodynamic origins of substrate binding cooperativity in a dimeric enzyme that can bind one substrate (A) and one allosteric ligand (X) to each of two identical subunits are discussed. It is assumed that maximal activity is not subject to allosteric modification and that the substrates and allosteric ligands achieve binding equilibrium in the steady state. Each uniquely ligated form is assumed to be capable of exhibiting unique binding properties, and only the principles of thermodynamic linkage are used to constrain the system further. The explicit relationship between the Hill coefficient, the concentration of X, and the magnitudes of the relevant coupling free energies and dissociation constants is derived. In the absence of X only the homotropic coupling between substrate sites contributes to a nonhyperbolic substrate saturation profile. An allosteric ligand, X, can alter the cooperativity in two distinct ways, one mechanism being manifested when X is saturating and the only only when X is present at saturating concentrations. By evaluating the concentration of substrate required to produce half-maximal velocity as a function of [X], as well as the Hill coefficients when X is absent and fully saturating, the dissociation and coupling constants most important for understanding the mechanisms of allosteric action in an enzyme of this type can be determined.

Non-additivity in protein-protein interactions

The energy of binding between proteins may be seen as the sum of the contributions of the individual amino acid residues. These contributions are additive when the binding energy, due to different amino acid residues, is independent of the interactions between amino acids in the same polypeptide chain. A measure of non-additivity is the coupling free energy. In this communication it is shown that: (1) the coupling free energy is the sum of intramolecular and intermolecular contributions; and (2), when additivity exists, experimentally determined values for the free energy of transfer of amino acids from water to the hydrophobic protein-protein interface are a very good approximation of their contribution to the energy of binding. Additivity cycles can be useful in determining the precise conditions where this approximation holds.

Energetics of the binding of calcium and troponin I to troponin C from rabbit skeletal muscle

We determined the free energy of interaction between rabbit skeletal troponin I (TNI) and troponin C (TNC) at 10 degrees and 20 degrees C with fluorescently labeled proteins. The sulfhydryl probe 5-iodoacetamidoeosin (IAE) was attached to cysteine (Cys)-98 of TNC and to Cys-133 of TNI, and each of the labeled proteins was titrated with the other unlabeled protein. The association constant for formation of the complex between labeled TNC (TNC*) and TNI was 6.67 X 10(5) M-1 in 0.3 M KCl, and pH 7.5 at 20 degrees C. In the presence of bound Mg2+, the binding constant increased to 4.58 X 10(7) M-1 and in the presence of excess of Ca2+, the association constant was 5.58 X 10(9) M-1. Very similar association constants were obtained when labeled TNI was titrated with unlabeled TNC. The energetics of Ca2+ binding to TNC* and the complex TNI X TNC* were also determined at 20 degrees C. The two sets of results were used to separately determine the coupling free energy for binding TNI and Mg2+, or Ca2+ to TNC. The results yielded a total coupling free energy of -5.4 kcal. This free energy appeared evenly partitioned into the two species: TNI X TNC(Mg)2 or TNI X TNC(Ca)2, and TNI X TNC(Ca)4. The first two species were each stabilized by -2.6 kcal, with respect to the Ca2+ free TNI X TNC complex, and TNI X TNC(Ca)4 was stabilized by -2.8 kcal, respect to TNI X TNC(Ca)2 or TNI X TNC(Mg)2. The coupling free energy was shown to produce cooperatively complexes formed between TNI and TNC in which the high affinity sites were initially saturated as a function of free Ca2+ to yield TNI X TNC(Ca)4. This saturation occurred in the free Ca2+ concentration range 10(-7) to 10(-5) M. The cooperative strengthening of the linkage between TNI and TNC induced by Ca2+ binding to the Ca2+-specific sites of TNC may have a direct relationship to activation of actomyosin ATPase. The nature of the forces involved in the Ca2+-induced strengthening of the complex is discussed.

Spin-fluctuation theory of strong coupling corrections to the free energy in superfluid 3He

Previous spin-fluctuation theories yielding the strong coupling corrections ..delta..beta-bar/sub i/ to the five coefficients of the fourth-order invariants in the free energy functional are extended. First, the superfluid part of the susceptibility is calculated up to order ..delta../sup 4/for all momenta and frequencies and the contribution arising from p-wave fluctuations of the order parameter is included. Then the frequency sums yielding the ..delta..beta-bar/sub i/ are calculated by taking into account the full momentum and frequency dependence of the superfluid susceptibility and the spin fluctuation propagator. The results for the ..delta..beta-bar/sub i/ are plotted vs. a cutoff q/sub c/ on the momentum integration for spin-fluctuation parameters IequivalentN(0)I=0.75 and I=0.95. The cutoff takes into account in a rough way the effect of additional terms in the free energy functional which were neglected in previous theories. These additional terms are due to the implicit dependence of the superfluid susceptibility on the spin-fluctuation parameter I via the gap parameter ..delta... The gap equation providing the relation between ..delta.. and I is derived in the weak coupling approximation. The cutoffs obtained by fitting the experimental values of the three combinations of ..delta..beta-bar/sub i/(arising from the measured specific heat discontinuities on the melting curve) are comparablemore » to the cutoff obtained from the spin-fluctuation contribution to the weak coupling free energy (q/sub c/=0.3(2k/sub F/) for I=0.75). The corrections due to the momentum and frequency dependence of the superfluid susceptibility and the spinfluctuation propagator are large and point in the direction of better agreement with experiment: The ratio R/sub 1/ =..delta..beta-bar/sub 5//(..delta..beta-bar/sub 2/+..delta..beta-bar/sub 4/) decreases from 2 to about 1.2. It is concluded that spin-fluctuation theory in its present form cannot account quantitatively for the measured specific heat discontinuities on the melting curve.« less