A quantitative treatment of the kinetics of the folding transition of ribonuclease A.

New experimental data and a quantitative theoretical treatment are given for the kinetics of the thermal folding transition of ribonuclease A at pH 3.0. A three-species mechanism is used as a starting point for the analysis: U1 (slow) in equilibrium U2(fast) in equilibrium N, where U1 and U2 are two forms of the unfolded enzyme with markedly different rates of refolding and N is the native enzyme. This mechanism is based on certain facts established in previous studies of refolding. The kinetics of unfolding and refolding show two phases a fast phase and a slow phase, over a range of temperatures extending above the transition midpoint, Tm. The three-species mechanism can be used in this range. At higher temperatures a new much faster kinetic phase is also observed corresponding to the transient formation of a new intermediate (I). Although the general solution for a four-species mechanism is complex it is not difficult to extend the three-species analysis for the special case found here, in which the fast reaction (I in equilibrium N) is well separated from the other two reactions. At temperatures below the transition zone the slow phase of refolding becomes kinetically complex. No attempt has been made to extend the analysis to include this effect. The basic test of the three-state analysis is the prediction as a function of temperature of alpha2, the relative amplitude of the fast phase, both for unfolding and refolding. At temperatures above Tm for which the three-state analysis must be extended to include the new intermediate I, a crresponding quanitity alpha2(cor) is predicted and compared with measured values. Data used in the three-state prediction are values of tau2 and tau1, the time constants of the fast and slow kinetic phases, plus a single value of alpha2 measured when tau2 and tau1 are well separated. The observed and predicted values of alpha2 agree within experimental error. The analysis predicts correctly that, for these experiments, alpha2 should have the same value in unfolding as in refolding in the final conditions. The analysis also predicts satisfactorily the equilibrium transition curve from kinetic data alone. Four striking properties of the kinetics are explained or correlated by the analysis: (a) the drop in alpha2 to a minimum near Tm as well as the delayed rise in alpha2 above Tm;(b) the vanishing of alpha1 above the transition zone; (c) the sharp drop in tau1 inside the transition zone followed by a partial leveling off outside this zone; and (d) the passage of tau2 through a maximum near Tm. Through a comparison of observed and predicted values of alpha2, the analysis also rules out the alternative three-species mechanism U1 (slow) in equilibrium N (fast) in equilibrium U2. Finally, the temperature dependence of the amplitude for the fast reaction (I in equilibrium N) is discussed; the behavior of I is like that of U2 and I may be an unfolded species populated at equilibrium...     

Equilibrium and kinetics of the unfolding of alpha-lactalbumin by guanidine hydrochloride (IV): dependence of the N equilibrium A transconformation on the temperature.

The reversible unfolding from the native (N) state to the acid (A) state of alpha-lactalbumin by guanidine-HCl (0.8-2.0 M) was studied at 10-35 degrees C by means of difference spectra and pH-jump measurements. At each temperature, all points plotted as the logarithmic equilibrium constant log KNA of the N equilibrium A process against pH could fall on a curve independent of the denaturant concentration by shifting each point along the log KNA axis, where the shift factor f did not depend on temperature. Moreover, by shifting the points at each temperature along the log (KNA/f) axis, a master curve, independent of both temperature and the denaturant concentration, could be obtained for the pH-dependence of log KNA. From the dependence of the logarithmic rate constants on pH, master curves independent of both temperature and the denaturant concentration could also be made for the N leads to A and the A leads to A processes, where A mean the activated state. The results show the two-state character of the N equilibrium A process. The enthalpy changes and the differences in heat capacity for the N equilibrium A, N equilibrium A and A equilibrium A processes were determined from the accurate measurements of temperature dependence of the unfolding at pH 4.3 and 1.0 M guanidine-HCl. The results show that the disruption of hydrophobic interaction is caused mainly in the A leads to A process, while most of the changes in the pK values of the ionizing groups are caused in the N leads to A process.     

Conformational species of gramicidin A in non-polar solvent. A kinetic and thermodynamic treatment in the absence and presence of phosphatidylcholine as studied by high-performance liquid chromatography

A kinetic and thermodynamic study has been carried out to characterize quantitatively the conformational equilibrium of gramicidin A (GA) in tetrahydrofuran at different peptide concentrations in the absence and presence of egg yolk phosphatidylcholine by using size-exclusion high-performance liquid chromatographic analysis. In the absence of lipid, the experimental data fit a simple dimer-monomer equilibrium, the rate and equilibrium constants for the dissociation process being (1.6 +/- 0.7) X 10(-7) s-1 and (8.5 +/- 0.3) X 10(-6) M, respectively. A higher extent of monomerization and a decrease in the time required for reaching equilibrium are detected in the presence of phospholipid, the kinetic and thermodynamic effects depending on both lipid and GA concentrations. In order to account for these observations a cyclic equilibrium mechanism is proposed which is analysed in terms of four conformational species, namely, free monomer, free dimer, lipid-bound monomer and lipid-bound dimer. The results obtained are discussed in relation to recent literature data on lipid-protein interactions.     

A mean-field model of the alkane-saturated lipid bilayer above its phase transition. I. Development of the model

A statistical mechanical model of a bilayer of dipalmitoyl-3-sn-phosphatidylcholine molecules in equilibrium with an aqueous phase saturated with an n-alkane is presented. A mean-field approach developed in previous work on a solventless bilayer (Gruen, Biochim. Biophys. Acta. 595:161--183, 1980) is extended to allow alkane chains to exist in the hydrophobic core of the membrane. As the alkane chains are chemically similar to the lipid chains, much of the analysis follows directly from the solventless model. Novel features of the present model are the inclusion of (a) a term which models the free energy cost of creating space for alkane conformations, (b) a term which constrains the chains to pack evenly in the hydrophobic region of the membrane, and (c) a term which estimates the free energy of mixing of the lipid and alkane molecules in the plane of the bilayer. On uptake of alkane, the dimensions of the bilayer increase. Allowance is made for an increase in thickness and/or an increase in area per lipid. A thermodynamic framework is established which allows evaluation of the free energy of a bilayer of arbitrary dimensions with a view to predicting the equilibrium structure.     

Heme reactivity of hemoglobins. Azide and fluoride binding equilibria of free and mercuriated ferri-gamma chains.

The free gamma chains, isolated from human foetal hemoglobin, are stable when oxidized and thus suitable for ligand binding and subunit equilibrium studies. The metaquo-ferri chains, with cysteine-F9 in the free state II ag gamma SH) possess several properties which are different from those of their p-mercuribenzoate derivative (III aq gammaSHgR); these are: stronger binding of a high-field ligand (N3- minus), altered spin equilibrium and an altered subunit equilibrium. A quantitative assessment of the free energy changes associated with all individual steps involved in changing the metaquo chains to their azide derivatives has been made. The results show that the higher apparent reactivity of III ag gammaSH (compared to IIIaq gammaSHgR) for the azide ion is not solely due to compensatory effects arising from differences of subunit dissociation or of spin equilibrium: other process(es) occurring in the ligand binding site have to be considered.     

PH-dependence of the steady-state rate of a two-step enzymic reaction.

1. The pH-dependence is considered of a reaction between E and S that proceeds through an intermediate ES under "Briggs-Haldane' conditions, i.e. there is a steady state in ES and [S]o greater than [E]T, where [S]o is the initial concentration of S and [E]T is the total concentration of all forms of E. Reactants and intermediates are assumed to interconvert in three protonic states (E equilibrium ES; EH equilibrium EHS; EH2 equilibrium EH2S), but only EHS provides products by an irreversible reaction whose rate constant is kcat. Protonations are assumed to be so fast that they are all at equilibrium. 2. The rate equation for this model is shown to be v = d[P]/dt = (kcat.[E]T[S]o/A)/[(KmBC/DA) + [S]o], where Km is the usual assembly of rate constants around EHS and A-D are functions of the form (1 + [H]/K1 + K2/[H]), in which K1 and K2 are: in A, the molecular ionization constants of ES; in B, the analogous constants of E; in C and D, apparent ionization constants composed of molecular ionization constants (of E or ES) and assemblies of rate constants. 3. As in earlier treatments of this type of reaction which involve either the assumption that the reactants and intermediate are in equilibrium or the assumption of Peller & Alberty [(1959) J. Am. Chem. Soc. 81, 5907-5914] that only EH and EHS interconvert directly, the pH-dependence of kcat. is determined only by A. 4. The pH-dependence of Km is determined in general by B-C/A-D, but when reactants and intermediate are in equilibrium, C identical to D and this expression simplifies to B/A. 5. The pH-dependence of kcat./Km, i.e. of the rate when [S]o less than Km, is not necessarily a simple bell-shaped curve characterized only by the ionization constants of B, but is a complex curve characterized by D/B-C. 6. Various situations are discussed in which the pH-dependence of kcat./Km is determined by assemblies simpler than D/B-C. The special situation in which a kcat./Km-pH profile provides the molecular pKa values of the intermediate ES complex is delineated.     

Aspartokinase I-homoserine dehydrogenase I of Escherichia coli K12. Concentration-dependent dissociation to dimers in the presence of L-threonine.

The concentration-dependent association-dissociation equilibrium of the bifunctional enzyme aspartokinase I-homoserine dehydrogenase I of Escherichia coli K12 has been investigated at pH 7.6 in the presence of 10 mM L-threonine and 0.1 M KCl by equilibrium gel permeation monitored by a single-photon counting spectrophotometer. The results obtained are consistent with the existence of a dimer-tetramer equilibrium with the association constant of 2.6 X 10(7) M-1 (deltaG0 = -9.9 kcal/mol of dimer). The limiting partition cross-sections estimated by a three-parameter least squares minimization procedure indicate that the molecular radii of the dimer and tetramer are 53.8 A and 70 A, respectively. Both the dimeric and tetrameric forms of the enzyme possess dehydrogenase activity. Treatment of the enzyme with the chaotropic salts, potassium thiocyanate or potassium trichloroacetate, generates a monomeric form that is devoid of dehydrogenase activity. The catalytically inactive monomeric form of the enzyme has a molecular radius between 43 and 45.5 A and a molecular weight of approximately 80,000 as determined by small zone gel chromatography and sedimentation equilibrium studies.     

Null space in the Hodgkin-Huxley Equations. A critical test.

Voltage perturbation methods based upon topological concepts are used to elicit responses from the Hodgkin-Huxley (HH) nonlinear differential equations. These responses present a critical check upon the validity of the HH model for electrical activity across squid axon membrane. It is shown that when a constant current is applied such that a stable equilibrium and rhythmic firing are present, the following predictions are inherent in the HH system of equations: (a) Small instantaneous voltage perturbations to the axon given at points along its firing spike result in phase resetting curves (when new phase versus old phase is plotted) with an average slope of 1. (b) A larger voltage perturbation (from certain points along the firing spike) results in the permanent cessation of periodic firing, with membrane voltage rapidly approaching the equilibrium value. (c) A still larger perturbation yields phase resetting curves with an average slope equal to 0. These predictions, coupled with Tasaki's experimental demonstration that squid axons in excellent condition do give repetitive firing under constant current, provide a critical test of the validity of the HH model.     

Reaction free energy surfaces in myosin-actin-ATP systems.

If we select for consideration any reaction M1 in equilibrium M2 in the myosin-ATPase cycle, the question arises as to the relations between the rate constants for (1) M1 equilibrium M2, (2) AM1 in equilibrium AM2 (A = actin), (3) A + M1 in equilibrium AM1, and (4) A + M2 equilibrium AM2, with actin and myosin either (a) in solution or (b) in the myofilament structure. It is shown here, by means of examples, that a single so-called potential of mean force, W, and structural free energy, Am, suffice to determine the reaction free energy surfaces for all of these transitions (W for the solution case, W + Am for the structured case). In fact, Am is the same for all reactions in the myosin-ATPase cycle. Of course, though indispensable as the starting point and adequate for qualitative understanding, the reaction free energy surface does not provide (without additional theory) the actual values of the rate constants or of the corresponding basic free energy changes in the myosin states involved. These rate constants and free energies are discussed, in a preliminary way, in two other papers.     

Kinetics of the spontaneous transient unfolding of a native protein studied with monoclonal antibodies. Monomer/dimer transition in the tryptophan-synthase beta 2 subunit

Included in a series of monoclonal antibodies obtained after immunization with the native holo beta 2 subunit of tryptophan synthase of Escherichia coli (EC 4.2.1.20), are some that interact preferentially with a denatured state of the antigen (Friguet et al., 1984). A study of the equilibrium and kinetic characteristics of the interaction of one of these antibodies with native apo beta 2 (i.e. free of pyridoxal 5'-phosphate) and with one of its proteolytic domains is reported here. The antibody is shown to interact strongly with the isolated domain in accordance with a simple equilibrium. In the presence of native beta 2, the antibody binds exclusively to the dissociated beta-monomer. The interaction of this antibody with native apo beta 2 is used to determine the equilibrium and kinetic constants of the monomer-dimer equilibrium. The values obtained are 4.5 X 10(-8) M for the equilibrium constant and 7.9 X 10(-3) s-1 for the rate constant of the dissociation of apo beta 2 into beta-monomers.     

Electron transfer between the two photosystems. II. Equilibrium constants

(1) The equilibrium constants for the redox reactions occurring between Photosystem (PS) I donors were measured on chloroplasts, dark-adapted in the presence of sodium ascorbate and 3-(3′,4′-dichlorophenyl)-1,1-dimethylurea (DCMU) and then illuminated by d.c. light. The equilibrium constant for the electron transfer between plastocyanin and P-700 is close to 1 and the overall equilibrium constant between cytochrome f and P-700 is about 2.3. As these equilibrium constants do not depend upon the intensity of the d.c. beam, the low values we measured cannot be due to kinetic limitations. (2) The equilibrium constants were measured also in the absence of DCMU using chloroplasts in oxidizing conditions (ferricyanide or far red illumination) illuminated by a saturating flash. During the course of the reduction of PS I donors by plastoquinol molecules formed by the flash, the equilibrium constants are higher than in the preceding conditions: the value for plastocyanin to P-700 is close to 5, and that for cytochrome f to P-700 is about 25. (3) The variations of these equilibrium constants are tentatively interpreted as being due to mutual electrostatic interactions between cytochrome b and f which are included in the same complex. This model implies that the perturbation of the redox properties of cytochrome f by a positive charge located on cytochrome b is identical to the perturbation of the redox properties of cytochrome b by a positive charge located on cytochrome f.     

Kinetic study of the reduction mechanism for Desulfovibrio gigas cytochrome c3.

The kinetic aspects of the reduction process in cytochrome c3 from Desulfovibrio gigas have been investigated over a wide range of pH values ranging between pH 5.8 and pH 9.8. The data have been analyzed in the framework of an I2H4 interaction network coupled to a proton-linked equilibrium between two tertiary structures (Cornish-Bowden, A. & Koshland, D.E. Jr (1970) J. Biol. Chem. 245, 6241-6250). The kinetic rate constants for the reduction of the four hemes for the two tertiary conformations have been characterized in the framework of the thermodynamic network obtained from the equilibrium analysis (Coletta, M., Catarino, T., LeGall, J.J. & Xavier, A.V. (1991) Eur. J. Biochem. 202, 1101-1106). The intrinsic reduction rate constants determined by reaction with sodium dithionite for two hemes (namely heme 4 and heme 1) are significantly faster than those for the other two heme residues. In view of the equilibrium redox properties, heme 4 (with the fastest reduction rate) may then work as the kinetic electron-capturing site for the electrons from sodium dithionite. The transfer to hemes 2 and 3 then occurs by virtue of their free-energy levels at equilibrium. At our experimental conditions, there is also transfer of electrons to hemes 2 and 3 from heme 1, which is reduced at a slower rate than heme 4, thus contributing to the biphasic kinetics observed for the overall process. The kinetic parameters obtained are discussed in terms of the mechanism proposed for the coupling between the electron and proton transfer, as induced by the heme/heme cooperativity network.     

An elastic model of the large-scale structure of duplex DNA.

A general model for the large-scale, time-independent structure of duplex DNA is developed based on elastic considerations. The general conditions of elastic equilibrium are given. These equations are solved for the equilibrium shape of stressed duplex DNA, based on the assumption that the double helix behaves mechanically as a symmetric, linearly elastic rod. It is shown that, in general, two orders of superhelicity will arise at equilibrium. Several possible applications of this approach to the supercoiling of closed circular DNA are described.     

The binding of substrates and a product of the enzymatic reaction to glutathione S-transferase A.

The binding of substrates and a product to glutathione S-transferase A from rat liver was studied by use of equilibrium dialysis and equilibrium partition in a two-phase system. The radioactive substrates glutathione and bromosulfophthalein as well as a product of glutathione and 3,4-dichloro-1-nitrobenzene, S-(2-chloro-4-nitrophenyl)glutathione, gave hyperbolic binding isotherms with a stoichiometry of 2 mol per mol of enzyme (i.e. 1 molecule per subunit). Glutathione (and glutathione disulfide) had an equilibrium (dissociation) constant for the binding of about 10 microM, whereas bromosulfophthalein and the product had equilibrium constants of about 0.5 microM. All ligands showed the same binding stoichiometry, and competition experiments involving unlabeled ligands indicated that glutathione and the glutathione derivatives were binding to the same site. Low affinity sites appeared to exist in addition to the specific high affinity sites (one per subunit) for all ligands tested. The binding studies are fully consistent with a steady state random kinetic mechanism for the enzyme.     

Temperature dependence of the repulsive pressure between phosphatidylcholine bilayers.

Bilayer structure and interbilayer repulsive pressure were measured from 5 to 50 degrees C by the osmotic stress/x-ray diffraction method for both gel and liquid crystalline phase lipid bilayers. For gel phase dibehenoylphosphatidylcholine (DBPC) the bilayer thickness and pressure-distance relations were nearly temperature-independent, and at full hydration the equilibrium fluid spacing increased approximately 1 A, from 10 A at 5 degrees C to 11 A at 50 degrees C. In contrast, for liquid crystalline phase egg phosphatidylcholine (EPC), the bilayer thickness, equilibrium fluid spacing, and pressure-distance relation were all markedly temperature-dependent. As the temperature was increased from 5 to 50 degrees C the EPC bilayer thickness decreased approximately 4 A, and the equilibrium fluid spacing increased from 14 to 21 A. Over this temperature range there was little change in the pressure-distance relation for fluid spacings less than approximately 10 A, but a substantial increase in the total pressure for fluid spacings greater than 10 A. These data show that for both gel and liquid crystalline bilayers there is a short-range repulsive pressure that is nearly temperature-independent, whereas for liquid crystalline bilayers there is also a longer-range pressure that increases with temperature. From analysis of the energetics of dehydration we argue that the temperature-independent short-range pressure is consistent with a hydration pressure due to polarization or electrostriction of water molecules by the phosphorylcholine moiety. For the liquid crystalline phase, the 7 A increase in equilibrium fluid spacing with increasing temperature can be predicted by an increase in the undulation pressure as a consequence of a temperature-dependent decrease in bilayer bending modulus.     

The inhibitors thapsigargin and 2,5-di(tert-butyl)-1,4-benzohydroquinone favour the E2 form of the Ca2+,Mg(2+)-ATPase.

2,5-Di(tert-butyl)-1,4-benzohydroquinone has been shown to inhibit the Ca2+,M(2+)-ATPase of sarcoplasmic reticulum with an affinity of 0.4 microM. It has been shown to shift the E2-E1 equilibrium for the ATPase towards E2, as shown previously for the inhibitor thapsigargin. The shift towards E2 results in a decrease in affinity for Ca2+, as also observed for thapsigargin. A marked decrease in the rate of the E2-E1 transition is observed for both BHQ and thapsigargin. A decrease in the equilibrium level of phosphorylation by Pi and of the steady-state level of phosphorylation by ATP are consistent with a decrease in the equilibrium constant for phosphorylation by Pi and an increase in the rate of dephosphorylation.     

Modelling the activation,opening, inactivation and reopening of the voltage-gated sodium channel.

A model of the voltage-gated sodium channel is put forward suggesting that the four S4 voltage-sensors behave as screw-helices making a series of discrete transitions that carry one elementary charge for each notch of the screw helix. After the channel has been activated by the first two steps R in equilibrium with P in equilibrium with A in all four domains, followed by a voltage-independent rearrangement, it is opened by a third cooperative step A in equilibrium with B in domains I, II and III in conjunction with hydration. Inactivation is a voltage-dependent process controlled by the third step A in equilibrium with I in sensor IVS4, and the closing of the channel is brought about its dehydration. From the inactivated steady state the channel may be reopened by a fourth step, I in equilibrium with C in sensor IVS4 and rehydration. The computed kinetics of the model are shown to conform closely with those observed experimentally.     

Heme models. I. Solution behavior of a water soluble iron porphyrin.

A well-behaved water soluble iron-porphyrin system, meso-tetra-(4-carboxyphenyl) porphinato iron (III) was synthesized. Its solution behavior is described using visable and electron paramagnetic resonance (EPR) spectroscopy. The complex exists in solution as three distinct forms of bridged dimers, oxo, hydroxo and aquo, with the following pK's: oxo + H+ in equilibrium hydroxo, pK = 9.58; hydroxo + H+ in equilibrium aquo, pK = 6.72. In the presence of excess imidazole the second pK is found to be 7.05. Detailed analysis of the interaction of the hydroxo-bridged form with imidazole is presented. It is found that one dimer unit simultaneously binds two imidazole molecules, with an over-all equilibrium constant log Keq = -1.22. EPR spectra are presented for the various forms of iron-porphyrin discussed.     

Properties of solution of the diffusion-reaction equation. II

Two theorems relating to properties of the solutions of the equations of continuity for the concentrations of the chemical species in a diffusion-reaction system are proved. The theorems concern boundary conditions under which the flux of a specified species can be guaranteed to be directed into the reaction region and the circumstances under which any two of the conditions (i) stationarity, (ii) flux equilibrium, and (iii) chemical equilibrium, imply the third. Application of these theorems to apparent active transport and to the properties of the differential equations for specific activities in a distributed tracer system are noted.     

Studies on the function of two adjacent N6,N6-dimethyladenosines near the 3' end of 16S ribosomal RNA of Escherichia coli. IV. The effect of the methylgroups on ribosomal subunit interaction.

The effect of the presence or absence of the methylgroups of the m2(6)Am2(6)A sequence near the 3' end of 16S rRNA of Escherichia coli on the interaction of the ribosomal subunits has been studied, using wild-type (methylated) and mutant (unmethylated) ribosomes. Subunit exchange experiments and competitive association experiments show a strong preference of the 50S subunit for association with methylated 30S subunits. The results indicate that the equilibrium constant of the reaction 70S in equilibrium with 30S + 50S is dependent on the methylgroups; mutant 30S.50S couples are less stable than wild-type 30S.50S couples. It is postulated that the methylgroups also stimulate the interaction between 30S subunits and initiation factor IF-3.