A probabilistic model for fitting MWC polynomials in protein-ligand binding

Given a binding polynomial in Adair form, A(x) = 1 + beta 1 x + ... + beta n x n, beta i greater than or equal to 0, a basic problem is to determine a method of fitting a model polynomial to A(x) and a quantitative measure of the goodness of fit. This paper presents such a method for fitting Monod-Wyman-Changeux (MWC) model polynomials when A(x) is of degree three or four. The method of fitting is based on the property that the zeros of an MWC polynomial of any degree lie on a circle in the complex plane. The parameters in the MWC model are determined so that if possible this circle coincides with the circle on which lie the zeros of A(x). The measure of goodness of fit is provided by a probabilistic model which gives the probability that a binding polynomial has its zeros on a circle on which lie the zeros of an MWC polynomial and if so, the probability that the juxtaposition of the two sets of zeros can occur by chance alone.     

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

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

A new measure of cooperativity in protein-ligand binding

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

Properties of chemically modified Ni(II)-Fe(II) hybrid hemoglobins. Ni(II) protoporphyrin IX as a model for a permanent deoxy-heme

Chemical modifications, NES-Cys(beta 93), des-Arg(alpha 141), and both modifications on the same molecule, were made to Ni-Fe hybrid hemoglobins, and their effect on individual subunits was investigated by measuring oxygen equilibrium curves, the Fe(II)-N epsilon (His F8) stretching Raman lines, and light-absorption spectra. The oxygen equilibrium properties indicated that modified Ni-Fe hybrid hemoglobins remain good models for the corresponding deoxy ferrous hemoglobins, although K1, the dissociation equilibrium constant for the first oxygen to bind to hemoglobin, was decreased by the chemical modifications. Resonance Raman spectra of deoxy alpha 2 (Fe) beta 2 (Ni) and light-absorption spectra of deoxy alpha 2 (Ni) beta 2 (Fe), revealed that the state of alpha hemes in both hybrid hemoglobins underwent a transition from a deoxy-like state to an oxy-like state caused by these chemical modifications when K1 was about 3 mm Hg (1 mm Hg approximately 133.3 Pa). On the other hand, the state of beta hemes in hybrid hemoglobins was little affected, when K1 was larger than 1 mm Hg. Modified alpha 2 (Fe) beta 2 (Ni) gave a Hill coefficient greater than unity with a maximum of 1.4 when K1 was about 4 mm Hg. The two-state model predicts that the K1 value at the maximum Hill coefficient should be much larger than this value. For oxygen binding to unmodified alpha 2 (Ni) beta 2 (Fe), oxygen equilibrium data suggested no structural change, while the spectral data showed a structural change around Ni(II) protoporphyrin IX in the alpha subunits. A similar situation was encountered with modified alpha 2 (Ni) beta 2 (Fe), although K1 was decreased as a result of the structural changes induced by the modifications.     

Variable Subunit Contact and Cooperativity of Hemoglobins

Tertiary structures of proteins are conserved better than their primary structures during evolution. Quaternary structures or subunit organizations, however, are not always conserved. A typical case is found in hemoglobin family. Although human, Scapharca, and Urechis have tetrameric hemoglobins, their subunit contacts are completely different from each other. We report here that only one or two amino acid replacements are enough to create a new contact between subunits. Such a small number of chance replacements is expected during the evolution of hemoglobins. This result explains why different modes of subunit interaction evolved in animal hemoglobins. In contrast, certain interactions between subunits are necessary for cooperative oxygen binding. Cooperative oxygen binding is observed often in dimeric and tetrameric hemoglobins. Conformational change of a subunit induced by the first oxygen binding to the heme group is transmitted through the subunit contacts and increases the affinity of the second oxygen. The tetrameric hemoglobins from humans and Scapharca have cooperativity in spite of their different modes of subunit contact, but the one from Urechis does not. The relationship between cooperativity and the mode of subunit contacts is not clear. We compared the atomic interactions at the subunit contact surface of cooperative and non-cooperative tetrameric hemoglobins. We show that heme-contact modules M3–M6 play a key role in the subunit contacts responsible for cooperativity. A module was defined as a contiguous peptide segment having compact conformation and its average length is about 15 amino acid residues. We show that the cooperative hemoglobins have interactins involving at least two pairs of modules among the four heme-contact modules at subunit contact. Received: 12 January 2001 / Accepted: 3 April 2001     

Asymmetric distribution of cooperativity in the binding cascade of normal human hemoglobin. 2. Stepwise cooperative free energy

Stepwise cooperative free energies and intermediate Hill coefficients are used to assess the presence of noncooperative sequences in the database of binding free energies previously obtained for the eight partially ligated intermediates of human hemoglobin, encompassing a variety of hemesite analog substitutions. This analysis is prompted by the observed noncooperative binding of two ligands to hemoglobins that are partially substituted with Zn2+-heme, an analog of deoxy Fe2+-heme (Holt et al. (2005) Biochemistry 44, XXXXX). The results show that noncooperative binding sequences are observed in all hemesite analog studied to date. The noncooperative binding observed in (alpha2Znbeta2FeO2) and (alpha2FeO2beta2Zn) is therefore not a Zn-specific substitution artifact. One of several binding sequences from singly to triply ligated hemoglobin is also observed to occur with little or no positive cooperativity. These results demonstrate the variability possible among different ligation pathways in a highly cooperative multi-subunit system such as hemoglobin. As a direct consequence of this variability, differences among ligation pathways are not always detectable using cooperativity functions based on statistical distributions, such as the Hill coefficient n(H). The limitations of Hill coefficient analysis in evaluating cooperativity in intermediates of complex systems is contrasted with the utility of the stepwise binding parameters.     

Determination of kinetic constants in the general case of cooperative type or Michaelis enzymes inhibited by their substrate: theoretic analysis

For an enzyme (E) susceptible to substrate (S) inhibition, (S) can bind on one hand to (E) and on the other hand to (ES), leading to the dead-end complexes (SE) and (SES). In the general case where the (E)/(S) interaction obeys the Hill equation, the theoretical maximum velocity VM can be estimated when n not equal to 1, from the determination of velocities v beta at substrate concentrations S beta = Sm beta where Sm is the value corresponding to the actual maximum velocity vm. The Hill coefficient (n) as well as the constants KS, KSE and KSES corresponding to the respective dissociations of the complexes (ES), (SE) and (SES) are then determined from the equation: Ln (v/(VM-v] = nLnS-LnKS(1 + Sn/KSE + S2n/KS KSES) and its two asymptotes.     

Component D of chicken hemoglobin and the hemoglobin of the embryonic Tammar wallaby (Macropus eugenii) self-associate upon deoxygenation: Effect on oxygen binding

Extensive measurements of oxygen binding by some vertebrate hemoglobins (Hbs) have suggested an unusually high degree of cooperativity with reported Hill coefficients, n(H), greater than 4.0. We have reexamined this possibility of "super-cooperativity" with chicken Hb components A (alpha(A) (2)beta(2)) and D (alpha(D) (2)beta(2)). Prior studies have shown that component D but not A self-associates to dimers of tetramers upon deoxygenation. This self-association is reflected in the oxygen equilibrium of Hb D which shows a maximal n(H), greater than 4.0 at approximately 4 mM heme concentration. In contrast, component A has maximal n(H) value below 3. The value of the maximal n(H) for Hb D increases linearly with the fraction of octamer present in the deoxy Hb. We anticipate that deoxygenation-dependent self-association will be shown to be a general property of Hb D from birds and reptiles. Neither oxygen equilibria nor sedimentation measurements show any evidence that components A and D interact to form a complex when deoxygenated. We have also reexamined the oxygen equilibria of Hbs of an embryonic marsupial, the wallaby. The equilibria in red cells have been reported to have Hill coefficients as high as 5-6. Although our oxygen equilibrium measurements of solutions of unfractionated wallaby Hb at a concentration of approximately 1 mM show no n(H) values greater than approximately 3.0, sedimentation velocity measurements provide clear evidence for deoxygenation-dependent self-association.     

Oxygen binding by alpha(Fe2+)2beta(Ni2+)2 hemoglobin crystals

Oxygen binding by hemoglobin fixed in the T state either by crystallization or by encapsulation in silica gels is apparently noncooperative. However, cooperativity might be masked by different oxygen affinities of alpha and beta subunits. Metal hybrid hemoglobins, where the noniron metal does not bind oxygen, provide the opportunity to determine the oxygen affinities of alpha and beta hemes separately. Previous studies have characterized the oxygen binding by alpha(Ni2+)2beta(Fe2+)2 crystals. Here, we have determined the three-dimensional (3D) structure and oxygen binding of alpha(Fe2+)2beta(Ni2+)2 crystals grown from polyethylene glycol solutions. Polarized absorption spectra were recorded at different oxygen pressures with light polarized parallel either to the b or c crystal axis by single crystal microspectrophotometry. The oxygen pressures at 50% saturation (p50s) are 95 +/- 3 and 87 +/- 4 Torr along the b and c crystal axes, respectively, and the corresponding Hill coefficients are 0.96 +/- 0.06 and 0.90 +/- 0.03. Analysis of the binding curves, taking into account the different projections of the alpha hemes along the optical directions, indicates that the oxygen affinity of alpha1 hemes is 1.3-fold lower than alpha2 hemes. Inspection of the 3D structure suggests that this inequivalence may arise from packing interactions of the Hb tetramer within the monoclinic crystal lattice. A similar inequivalence was found for the beta subunits of alpha(Ni2+)2beta(Fe2+)2 crystals. The average oxygen affinity of the alpha subunits (p50 = 91 Torr) is about 1.2-fold higher than the beta subunits (p50 = 110 Torr). In the absence of cooperativity, this heterogeneity yields an oxygen binding curve of Hb A with a Hill coefficient of 0.999. Since the binding curves of Hb A crystals exhibit a Hill coefficient very close to unity, these findings indicate that oxygen binding by T-state hemoglobin is noncooperative, in keeping with the Monod, Wyman, and Changeux model.     

Cooperativity,absolute interaction,and algebraic optimization

We consider a measure of cooperativity based on the minimal interaction required to generate an observed titration behavior. We describe the corresponding algebraic optimization problem and show how it can be solved using the nonlinear algebra tool SCIP. Moreover, we compute the minimal interactions and minimal molecules for several binding polynomials that describe the oxygen binding of various hemoglobins under different conditions. We compare their minimal interaction with the maximal slope of the Hill plot, and discuss similarities and discrepancies with a view towards the shapes of the binding curves.

     

Preliminary survey of carnivore hemoglobin compositions

Summary The amino acid differences among 12 chains and 10 chains of carnivore hemoglobins are given. These hemoglobins conform to the substitution rate found for hemoglobins in general. A table compares the differences among known-chain sequences when deduced from tryptic peptide compositions and when taken from actual sequences. Among the carnivore hemoglobins studies, tryptic peptide compositions are about 14 % low in giving the number of sequence differences.     

Interaction of hemoglobin and its component alpha and beta chains with band 3 protein

The equilibrium binding of hemoglobin to isolated band 3 protein exhibited positive cooperativity [Hill coefficient = 1.65 +/- 0.1; total number of binding sites at pH 6.6 in 5 mM sodium phosphate buffer = 32 500 +/- 940 pmol/mg; Ka = (3.0 +/- 0.5) X 10(5) M-1]. The binding was reversible and ionic in nature as the bound hemoglobin was readily displaced by KCl, ATP, and 2,3-diphosphoglycerate, the latter two being more effective than KCl on a molar basis. The ratio of the interaction of hemoglobin to band 3 protein per se was 1:1, whereas the band 3 preparation as a whole (protein + lipids) was 3:1. Saturating levels of glyceraldehyde-3-phosphate dehydrogenase blocked only 33% of the total binding sites which were localized at the cytoplasmic segment; the remaining 67% was localized in lipids by their extraction with acetone. Reconstitution of acetone-extracted band 3 with phospholipid liposomes indicated phosphatidylserine as the binding site. The positive cooperativity in binding to acetone-extracted band 3 was increased (Hill constant = 2.1 +/- 0.1) compared to the band 3 preparation. After separation of the alpha and beta chains of hemoglobin, only the alpha chain binds to band 3 with positive cooperativity to an extent of 45-50% of native hemoglobin with similar affinity. The binding capacity of p-(hydroxymercuri)benzoate (HMB) derivatives of hemoglobin and its alpha chain was less than that of native hemoglobin, whereas HMB-beta chain or beta chain did not bind.(ABSTRACT TRUNCATED AT 250 WORDS)     

A flow procedure to determine oxygen binding isotherms for low affinity and easily oxidized hemoglobins.

1. The determination of binding isotherms for low affinity hemoglobins is particularly difficult because of rapid autoxidation. 2. In carp Hb (pH 6 + IHP, 25 degrees C), one-quarter of the hemes are oxidized within 3 min, preventing the accurate determination of even P50 or X. 3. We circumvent this problem by rapidly flowing HbO2, initially at pH 9 (X = 1.4 microM), against a low pH buffer to bring the system rapidly to equilibrium in the low affinity form. Diode-array spectrophotometry allows a complete spectrum to be obtained less than 5 sec, after flow ceases, before significant oxidation has occurred. In tandem with the stopped flow apparatus is an oxygen electrode to measure O2 activity. 4. At 22 degrees C, the half-saturation oxygen activity (X) is 227 microM, and the Hill number is 0.91, for carp Hb (T state) implying significant differences in the O2 affinities of the alpha and beta chain hemes or, two different T states.     

Iodo-alpha-conotoxin MI selectively binds the alpha/delta subunit interface of muscle nicotinic acetylcholine receptors

The embryonic mouse muscle nicotinic acetylcholine receptor (nAChR) is a ligand-gated ion channel formed by alpha1, beta1, delta, and gamma subunits. The receptor contains two ligand binding sites at alpha/delta and alpha/gamma subunit interfaces. [(3)H]Curare preferentially binds the alpha/gamma interface. We describe the synthesis and properties of a high-affinity iodinated ligand that selectively binds the alpha/delta interface. An analogue of alpha-conotoxin MI was synthesized with an iodine attached to Tyr-12 (iodo-alpha-MI). The analogue potently blocks the fetal mouse muscle subtype of nAChR expressed in Xenopus oocytes. It failed, however, to block alpha3beta4, alpha4beta2, or alpha7 nAChRs. Iodo-alpha-MI potently blocks the alpha1beta1delta but not the alpha1beta1gamma subunit combination expressed in Xenopus oocytes indicating selectivity for the alpha/delta subunit interface. Alpha-conotoxin MI was subsequently radioiodinated, and its properties were further evaluated. Saturation experiments indicate that radioiodinated alpha-conotoxin MI binds to TE671 cell homogenates with a Hill slope of 0.95 +/- 0.0094. Kinetic studies indicate that the binding of [(125)I]alpha-conotoxin MI is reversible (k(off) = 0.084 +/- 0.0045 min(-1)); k(on) is 8.5 x 10(7) min(-1) M(-1). The calculated k(d) is 0.98 nM. This potency is approximately 20-fold higher than the unmodified alpha-MI peptide. Unlike [(125)I]alpha-bungarotoxin, [(125)I]alpha-conotoxin MI binding to TE671 cell homogenates is fully displaceable by the small molecule antagonist d-tubocurarine.     

Functional properties of human hemoglobins synthesized from recombinant mutant beta-globins.

The previous and following articles in this issue describe the recombinant synthesis of three mutant beta-globins (beta 1 Val----Ala, beta 1 Val----Met, and the addition mutation beta 1 + Met), their assembly with heme and natural alpha chains into alpha 2 beta 2 tetramers, and their X-ray crystallographic structures. Here we have measured the equilibrium and kinetic allosteric properties of these hemoglobins. Our objective has been to evaluate their utility as surrogates of normal hemoglobin from which further mutants can be made for structure-function studies. The thermodynamic linkages between cooperative oxygenation and dimer-tetramer assembly were determined from global regression analysis of multiple oxygenation isotherms measured over a range of hemoglobin concentration. Oxygen binding to the tetramers was found to be highly cooperative (maximum Hill slopes from 3.1 to 3.2), and similar patterns of O2-linked subunit assembly free energies indicated a common mode of cooperative switching at the alpha 1 beta 2 interface. The dimers were found to exhibit the same noncooperative O2 equilibrium binding properties as normal hemoglobin. The most obvious difference in oxygen equilibria between the mutant recombinant and normal hemoglobins was a slightly lowered O2 affinity. The kinetics of CO binding and O2 dissociation were measured by stopped-flow and flash photolysis techniques. Parallel studies were carried out with the mutant and normal hemoglobins in the presence and absence of organic phosphates to assess their allosteric response to phosphates. In the absence of organic phosphates, the CO-binding and O2 dissociation kinetic properties of the mutant dimers and tetramers were found to be nearly identical to those of normal hemoglobin. However, the effects of organic phosphates on CO-binding kinetic properties of the mutants were not uniform: the beta 1 + Met mutant was found to deviate somewhat from normalcy, while the beta 1 Val----Met mutant reproduced the native allosteric response. Further characterization of the allosteric properties of the beta 1 Val----Met mutant was made by measuring the pH dependence of its overall oxygen affinity by tonometry. Regulation of oxygen affinity by protons was found to be nearly identical to normal hemoglobin from pH 5.8 to 9.3 (0.52 +/- 0.07 protons released per oxygen bound at pH 7.4). The present study demonstrates that the equilibrium and kinetic functional properties of the recombinant beta 1 Val----Met mutant mimic reasonably well those of normal hemoglobin. We conclude that this mutant is well-suited to serve as a surrogate system of normal hemoglobin in the production of mutants for structure-function studies.     

Beta-oxidation as channeled reaction linked to citric acid cycle: evidence from measurements of mitochondrial pyruvate oxidation during fatty acid degradation.

1. The kinetics of mitochondrial mammalian pyruvate dehydrogenase multienzyme complex (PDHC) is studied by the formation of CO2 using tracer amounts of [1-14C]pyruvate. It is found that the Hill plot results in a (pseudo-)cooperativity with a transition of n-1----3 at a pyruvate concentration about Ks. 2. Addition of L-carnitine, octanoate, palmitoyl-CoA or palmitate + L-carnitine + fatty acid-binding protein results in a Hill coefficient of n = 2 following the kinetics of pyruvate oxidation. 3. Addition of fatty acid-binding protein to an assay system oxidizing palmitate in presence of L-carnitine alters the pattern of the kinetics in the Hill plot so that an apparently lower level of L-carnitine is necessary for the reaction course of beta-degradation. 4. It is concluded that beta-degradation is a coordinated, multienzyme-complex based mechanism tightly linked to citric acid cycle and it is proposed that L-carnitine is actively involved into the reaction and not only functioning as carrier-molecule for transmembrane transport.     

Structures of Invertebrate Hemoglobins

Hemoglobin is widely distributed among the invertebrates. Intracellularhemoglobins consist of relatively small molecules with mol wtsof 15–17,000 or dimeric, tetrameric or octameric aggregatesof 15–17,000 mol wt subunits. Sequence homology is presentbut not extensive in those pigments which have been studiedand the characteristic myoglobin fold of vertebrate hemoglobinoccurs in at least two invertebrate hemoglobins. The wide arrayof aggregation states among invertebrate hemoglobins providessome simple models for understanding homotropic functional propertiesexhibited by many of these pigments. Polymeric extracellularhemoglobins are present in annelids molluscs crustacean arthropodsand nematodes. Annelid extracellular hemoglobins and chlorocruorinsconsist of 3 x 10⁶ mol wt two-tiered hexagonal arrays of submultipleswhich in turn are based on polypeptide chain subunits of molwt 14–16 000. Molluscan extracellular hemoglobins areconstructed from a different subunit arrangement. In the planorbidsnail and clam extracellular hemoglobins the subunits appearto be 175 000 and 300 000 mol wt linear series of 15–17000 dalton oxygen binding domains respectively. Planorbid snailnative hemoglobin presents circular structures 200 Å indiameter in the electron microscope with 10-fold symmetry inat least one view, and clam extracellular hemoglobins are huge345 by 1200 Å rodlike structures. Crustacean extracellularhemoglobins are also polymeric pigments and at least in a fewspecies appear to have subunits which are tandemly linked oxygenbinding domains. The polymeric hemoglobins of nematodes havemolecular weights of about 330 000. The subunit molecular weightand heme content suggest a value of 40,000 daltons which setthe nematode pigments apart from all other hemoglobins so farstudied. An overview of invertebrate hemoglobin structures andsome of the questions they pose are presented in this paper.     

beta-Adrenergic receptors of brain cells. Membrane integrity implies apparent positive cooperativity and higher affinity

Beta-Adrenergic receptors were studied in intact cells of chick, rat and mouse embryo brain in primary cultures, by the specific binding of [3H]dihydro-L-alprenolol ([3H]DHA). The results were compared to the receptor binding of broken cell preparations derived from the cell cultures or from the forebrain tissues used for the preparation of the cultures. Detailed analysis of [3H]DHA binding to living chick brain cells revealed a high-affinity, stereoselective, beta-adrenergic-type binding site. Equilibrium measurements indicated the apparent positive cooperativity of the binding reaction. By direct fitting of the Hill equation to the measured data, values of Bmax = 12.01 fmol/10(6) cells (7200 sites/cell), Kd = 60.23 pM and the Hill coefficient n = 2.78 were found. The apparent cooperative character of the binding was confirmed by the kinetics of competition with L-alprenolol, resulting in maximum curves at low ligand concentrations. The rate constants of the binding reaction were estimated as k+ = 8.31 X 10(7) M-1 X min-1 and k- = 0.28 min-1 from the association results, and k- = 0.24 min-1 from the dissociation data. The association kinetics supported the cooperativity of the binding, providing a Hill coefficient n = 1.76; Kd, as (k-/k+)1/n was found to be 101 pM. Analysis of the equilibrium binding of [3H]DHA to rat and mouse living brain cells resulted in values of Bmax = 13.04 fmol/10(6) cells (7800 sites/cell), Kd = 43.85 pM and n = 2.52, and Bmax = 8.08 fmol/10(6) cells (4800 sites/cell), Kd = 46.70 pM and n = 1.63, respectively, confirming the apparent cooperativity of the beta-receptor in mammalian objects, too. The [3H]DHA equilibrium binding to broken cell preparations of either chick, rat or mouse brain cultures or forebrain tissues was found to be non-cooperative, with a Hill coefficient n = 1, Kd in the range 1-2 nM, and a Bmax of 10(3) - 10(4) sites/cell. Our findings demonstrate that cell disruption causes marked changes in the kinetics of the beta-receptor binding and in the affinity of the binding site, although the number of receptors remains unchanged.