Precision determination and Adair scheme analysis of oxygen equilibrium curves of concentrated hemoglobin solution. A strict examination of Adair constant evaluation methods

To examine the validity of the recent finding by Gill et al. (S.J. Gill, E. Di Cera, M.L. Doyle, G.A. Bishop and C.H. Robert, Biochemistry 26 (1987) 3995) that the third overall Adair constant (A3) for human hemoglobin tetramers (Hb A) is too small to be determined and therefore that the contribution of the triply ligated species in the oxygenation process is negligibly small, highly accurate oxygen equilibrium curves for concentrated pure Hb A solutions were determined with an automatic oxygenation apparatus and analyzed by a least-squares curve-fitting method with various options. The present results indicate that an appropriate choice of weighting for data points is the key to the correct evaluation of the Adair constants and the present experimental data cannot accommodate the Adair scheme with A3 = 0, giving distinctly positive values for A3. Several criteria for correct determination of the Adair constants are presented.     

Weighting functions and parameter resolvability for oxygenation data subject to error in the independent variable.

Parameter resolvability and bias has been investigated for weighted nonlinear regression of data where the independent variable is subject to instrumental uncertainty. The specific example of cooperative oxygenation of hemoglobin was studied, where fractional saturation is determined spectrophotometrically and the oxygen activity is measured with a Clark polarographic electrode. For this system the instrumental uncertainty in the oxygen electrode was measured directly and the influence of the uncertainties on resolution of oxygen binding parameters was determined by Monte Carlo simulations. Four weighting functions were tested for their ability to minimize parameter uncertainty and bias: (1) uniform weighting; (2) "propagated weighting" whereby uncertainties in the independent variable are propagated into and added to uncertainties of the dependent variable; (3) Hill plot transform, or "end weighting"; and (4) maximum likelihood analysis, where deviations between fitting function and data are minimized as weighted horizontal and vertical distance vectors. Results of the Monte Carlo simulations favor the use of either uniform weighting, propagated weighting, or maximum likelihood weighting methods. Use of the Hill transform as a weighting function produced poorer parameter resolvability and inaccurate representation of the data in general. Bias error was negligible for all weighting functions.     

Statistical analysis of data pertaining to complex state systems by stepwise regression with reformulated parameters; Application to spectroscopically monitored hemoglobin oxygen binding data

A method is described for the statistical analysis of data pertaining to complex state systems, based on the concept of reformulating the parameters describing the system as a hierarchy of interactions, and this method demonstrated on the analysis of spectroscopically monitored hemoglobin oxygen binding data [K. Imai, Biophys. Chem. 37 (1990) 197-210]. The concept of reformulation was first extended to state parameters other than ΔG°s, such as the extinction coefficients (εs) associated with different ligation states during hemoglobin oxygen binding. The reformulated parameters are incrementally allowed to vary in the data fitting procedure, and the statistical significance of the added parameters tested by F and Kolmogorov-Smirnov tests. The result of this method is the minimal set of statistically significant parameters required to describe the data. The hierarchical nature of reformulated parameters allows the physical significance of the subset of statistically significant parameters to be discussed even when all reformulated terms may not be statistically significant. Applying this method to hemoglobin oxygen binding data with the reformulated Adair model demonstrated that at least two, and at most three, of the four reformulated Adair constants are statistically significant. A reformulated square model was found to give a statistically indistinguishable fit from the Adair model, with the statistically significant thermodynamic terms essentially those proposed by Linus Pauling in 1935. A change in Δ ε with subsequent oxygen binding events was found to be significant in both models. These results are consistent with a model for hemoglobin oxygen binding where a subunit changes its conformation upon oxygen binding, and affects the conformation of adjacent subunits.     

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.     

The linkage between oxygenation and subunit dissociation in human hemoglobin. Consequences for the analysis of oxygenation curves.

For human hemoglobin, a pronounced dependence of oxygenation curves upon protein concentration can be demonstrated experimentally in the range between 10(-4) and 2 X 10(-6) M heme. The effects of such protein concentration dependence upon analysis of saturation curves have been explored using a model-independent linkage analysis which incorporates the dissociation of tetramers to dimers. We have carried out stimulations of oxygenation curves representing a variety of energy distributions designed to cover a wide range of values which are relevant to known hemoglobin systems and experimental conditions. The resulting simulated oxygenation curves were analyzed by least-squares minimization procedures in terms of the tetramer binding isotherm to yield the four apparent Adair constants. These derived constants were compared with the originally assumed values used in the simulation in order to assess the extent to which their values may be altered by the presence of dimer. For each energy distribution the analysis has been carried out over a wide range of protein concentration. We have found that the presence of even small amounts of dimer that are necessarily present at the low protein concentrations commonly employed may have a devastating effect upon the reliability of Adair constant determinations. In addition to these simulated cases, we have analyzed two sets of highly precise experimental data from the literature in order to assess the degree to which constants obtained may have been influenced by the presence of dimer.     

A general method of deriving the best binding site model consistent with experimental binding data

An analysis of binding data is presented which yields the best binding site model consistent with the experimental data. The analysis is applicable to homotropic binding and yields the number of independent sites, number of interacting sites (dimers and tetramers of sites), intrinsic association constants, and degree of interaction. The information is derived from the roots of a binding polynomial constructed by the fitted Adair constants.     

Regression analysis of mean quality-adjusted lifetime with censored data

In clinical trials of chronic diseases such as acquired immunodeficiency syndrome, cancer, or cardiovascular diseases, the concept of quality-adjusted lifetime (QAL) has received more and more attention. In this paper, we consider the problem of how the covariates affect the mean QAL when the data are subject to right censoring. We allow a very general form for the mean model as a function of covariates. Using the idea of inverse probability weighting, we first construct a simple weighted estimating equation for the parameters in our mean model. We then find the form of the most efficient estimating equation, which yields the most efficient estimator for the regression parameters. Since the most efficient estimator depends on the distribution of the health history processes, and thus cannot be estimated nonparametrically, we consider different approaches for improving the efficiency of the simple weighted estimating equation using observed data. The applicability of these methods is demonstrated by both simulation experiments and a data example from a breast cancer clinical trial study.     

Normalization in the fitting of data by iterative methods. Application to tracer kinetics and enzyme kinetics

1. The normalization of biochemical data to weight them appropriately for parameter estimation is considered, with reference particularly to data from tracer kinetics and enzyme kinetics. If the data are in replicate, it is recommended that the sum of squared deviations for each experimental variable at each time or concentration point is divided by the local variance at that point. 2. If there is only one observation for each variable at each sampling point, normalization may still be required if the observations cover more than one order of magnitude, but there is no absolute criterion for judging the effect of the weighting that is produced. The goodness of fit that is produced by minimizing the weighted sum of squares of deviations must be judged subjectively. It is suggested that the goodness of fit may be regarded as satisfactory if the data points are distributed uniformly on either side of the fitted curve. A chi-square test may be used to decide whether the distribution is abnormal. The proportion of the residual variance associated with points on one or other side of the fitted curve may also be taken into account, because this gives an indication of the sensitivity of the residual variance to movement of the curve away from particular data points. These criteria for judging the effect of weighting are only valid if the model equation may reasonably be expected to apply to all the data points. 3. On this basis, normalizing by dividing the deviation for each data point by the experimental observation or by the equivalent value calculated by the model equation may both be shown to produce a consistent bias for numerically small observations, the former biasing the curve towards the smallest observations, the latter tending to produce a curve that is above the numerically smaller data points. It was found that dividing each deviation by the mean of observed and calculated variable appropriate to it produces a weighting that is fairly free from bias as judged by the criteria mentioned above. This normalization factor was tested on published data from both tracer kinetics and enzyme kinetics.     

Nonlinear optical effects in oxygen-binding reactions of hemoglobin A0

Optical spectra have been taken in the Soret band (440-400 nm) under different oxygen partial pressures for hemoglobin (Hb) A0 at pH 7.0, 15 degrees C, 2-3 mM heme, 30 mM inositol hexaphosphate, 0.1 Hepes and 0.1 M NaCl. Application of the matrix method of singular value decomposition (SVD) to the difference spectra for different oxygen pressures shows the presence of at least two distinct optical transitions. From this result one concludes that the optical response to oxygen binding is nonlinear in the Soret band. The degree of nonlinearity has been determined by fitting the data at different wavelengths to the four-step reaction Adair equation with the inclusion of optical parameters that describe the intermediate oxygenated species. It is found that the data are well-represented by two optical parameters at each wavelengths, one which represents the optical change for the addition of the first and second oxygen molecules and the other which corresponds to the change for the addition of the third and fourth oxygen molecules. The ratio of these optical parameters depends only moderately upon wavelength with an average value of 0.8 over the Soret band. Thus, there is an approx. 20% smaller optical response for the first two ligated species than that for the last two ligated species. The overall Adair equilibrium constants are evaluated as follows: beta 1 = 0.081 +/- 0.003 Torr-1, beta 2 = 2.53 x 10(-3) +/- 2.4 x 10(-4) Torr-2, beta 3 = 1.25 x 10(-5) +/- 1.0 x 10(-6) Torr-3, beta 4 = 1.77 x 10(-6) +/- 1.5 x 10(-7) Torr-4.     

PERFORMANCE OF THE HEPP MICRO-OSMOMETER

Estimation of the molecular weight of horse serum albumin from the osmotic pressure of solutions containing 0.713 to 5.12 gm. per 100 cc. shows that the Hepp osmometer yields the same values as the standard simple osmometer of Adair. Accuracy and precision of the instrument decrease noticeably at concentrations of albumin less than 0.7 gm. per 100 cc. A determination in duplicate can be carried out with this instrument in less than 2 hours. The instrument is easily operated.     

An Adaptive Location‐Scale Test

Increasing locations are often accompanied by an increase in variability. In this case apparent heteroscedasticity can indicate that there are treatment effects and it is appropriate to consider an alternative involving differences in location as well as in scale. As a location‐scale test the sum of a location and a scale test statistic can be used. However, the power can be raised through weighting the sum. In order to select values for this weighting an adaptive design with an interim analysis is proposed: The data of the first stage are used to calculate the weights and with the second stage's data a weighted location‐scale test is carried out. The p‐values of the two stages are combined through Fisher's combination test. With a Lepage‐type location‐scale test it is illustrated that the resultant adaptive test can be more powerful than the ‘optimum’ test with no interim analysis. The principle to calculate weights, which cannot be reasonably chosen a priori, with the data of the first stage may be useful for other tests which utilize weighted statistics, too. Furthermore, the proposed test is illustrated with an example from experimental ecology.     

Kinetic determination of tight-binding impurities in enzyme inhibitors

A novel rate equation to characterize the dose-response behavior of a moderately potent ("classical") enzyme inhibitor contaminated with a very potent ("tight-binding") impurity is derived. Mathematical properties of the new rate equation show that, for such contaminated materials, experimentally observed I(50) values are ambiguous. The four-parameter logistic equation, conventionally used to determine I(50) values, cannot be used to detect the presence of tight-binding impurities in inhibitor samples. In contrast, fitting the newly derived rate equation to inhibitor dose- response curves can, in favorable cases, reveal whether the unknown material is chemically homogeneous or whether it is contaminated with a tight-binding impurity. The limitations of our method with respect to the detectable range of inhibition constants (both classical and tight-binding) were examined by using Monte-Carlo simulations. To test the new analytical procedure experimentally, we added a small amount (0.02 mole%) of a tight-binding impurity (K(i)=0.065 nM) to an otherwise weak inhibitor of human mast-cell tryptase (K(i)=50.4 microM). The resulting material was treated as "unknown." Our kinetic equation predicts that such adulterated material should show I(50)=0.40 microM, which was identical to the experimentally observed value. The best-fit value of the apparent inhibition constants for the tight-binding inhibitor was K(i)=(0.107+/-0.035)nM, close to the true value of 0.065 nM.     

Marginal analyses of clustered data when cluster size is informative

We propose a new approach to fitting marginal models to clustered data when cluster size is informative. This approach uses a generalized estimating equation (GEE) that is weighted inversely with the cluster size. We show that our approach is asymptotically equivalent to within-cluster resampling (Hoffman, Sen, and Weinberg, 2001, Biometrika 73, 13-22), a computationally intensive approach in which replicate data sets containing a randomly selected observation from each cluster are analyzed, and the resulting estimates averaged. Using simulated data and an example involving dental health, we show the superior performance of our approach compared to unweighted GEE, the equivalence of our approach with WCR for large sample sizes, and the superior performance of our approach compared with WCR when sample sizes are small.     

On the Hill plot of NMR data for titration of protein residues

The Hill plots of NMR titration data for protein residues disclose more clearly than the usual titration curves the presence of multiple weak perturbations originating from other titratable groups, and should be used whenever the conventional curve fitting is poor. For a quantitative interpretation, we derive here expressions for the Hill equation and the Hill coefficient when the titration of the observed group is perturbed by more than one titratable group. When the generalized Hill equation is fitted to the data, values of the interaction parameters between the observed group and the others are extracted provided that there are no mutual interactions between the latter groups. The method is applied to the titration data of two histidyl residues of l-arginine phosphotransferase (E.C. 2.7.3.3.) in the transition state analogue complex (enzyme-Mg²⁺-ADP-NOsk3/–l-Arg). From the Hill plots, interactions with three titratable groups are disclosed for both residues, and the fitting with the Hill equation reveals that they experience perturbations from the same three groups. Microscopic pK values are obtained for all the involved groups, indicating large changes (up to 3 pH units) upon protonation of the interacting groups. As compared to the conventional fitting procedure, the use and fitting of Hill plots yields from NMR data more information on the neighbourhood of enzyme residues and on the changes intervening therein through the steps involved in the catalysis.     

Low-resolution structure of the proteolytic fragments of the Rapana venosa hemocyanin in solution

Rapana venosa hemocyanin (Hc) is a giant oxygen-binding protein consisting of different subunits assembled in a hollow cylinder. The polypeptide chain of each subunit is believed to be folded in several oxygen-binding functional units of molecular mass 50 kDa, each containing a binuclear copper active site. Limited proteolysis with alpha-chymotrypsin of native R. venosa hemocyanin allows the separation of three functional proteolytic fragments with molecular masses of approximately 150, 100, and 50 kDa. The functional fragments, purified by combining gel filtration chromatography and ion-exchange FPLC, were analyzed by means of small-angle X-ray scattering (SAXS). The gyration radius of the 50-kDa Rapana Hc fraction (2.4 nm) agrees well with that calculated on the basis of the dimensions determined by X-ray crystallography for one functional unit of Octopus Hc (2.1 nm). Independent shape determination of the 50- and 100-kDa proteolytic fragments yields consistent low-resolution models. Simultaneous fitting of the SAXS data from these fragments provides a higher-resolution model of the 100-kDa species made of two functional units tilted with respect to each other. The model of the 150-kDa proteolytic fragment consistent with the SAXS data displays a linear chain-like aggregation of the 50-kDa functional units. These observations provide valuable information for the reconstruction of the three-dimensional structure of the minimal functional subunit of gastropod hemocyanin in solution. Furthermore, the spatial relationships among the different functional units within the subunit will help in elucidation of the overall quaternary structure of the oligomeric native protein.     

Possibilities and limitations of weighted averaging

A statistical analysis of a weighted averaging procedure for the estimation of small signals buried in noise (Hoke et al. 1984a) is given. The weighting factor used by this method is in inverse proportion to the variance estimated for the noise. It is shown that, compred to conventional averaging, weighted averaging can improve the signal-to-noise ratio to a high extent if the variance of the noise changes as a function of time. On the other hand, uncritical application of the method involves the danger that the signal amplitude is underestimated. How serious this effect is depends on the number of degrees of freedom available for the estimation of the weighting factor. The effect can be neglected, if this number is sufficiently increased by means of an appropriate preprocessing.     

Site—site interaction in the lipid activation of β-hydroxybutyrate dehydrogenase

Kinetic data for the activation of the β-hydroxybutyrate dehydrogenase by long-chain lecithins [(1979) Biochemistry 18, 2420–2429; (1983) J. Biol. Chem. 258, 208–214] are analyzed. A previous kinetic model [(1982) Biochemistry 21, 3899–3908] is shown not to apply. Instead, the use of a two-site Adair equation points to a strongly cooperative interaction between the lecithin binding sites (ΔG, ?2.8 kcal/mol).     

Determination of inhibition constants, I50 values and the type of inhibition for enzyme-catalyzed reactions

A procedure is proposed for determining whether an inhibitor of an enzyme-catalyzed reaction is competitive, noncompetitive, or uncompetitive with respect to the substrate. The method is based on fitting the equation for noncompetitive inhibition to data obtained by measuring the rate of the reaction over a range of substrate and inhibitor concentrations. The results of this fit may suggest that the inhibition may be either competitive or uncompetitive, whereupon the data are reanalyzed using the appropriate equation. Comparison of this second fit with the first using an F test permits a statistical decision to be made on the type of inhibition. The chosen fit yields values and standard errors for the Michaelis-Menten parameters (maximum velocity and Michaelis constant), as well as the inhibition constant(s). From these values it is then possible to predict the I50, and its standard error, at any chosen substrate concentration, thereby facilitating comparison with results obtained with similar inhibitors, for homologous enzymes, or in other laboratories.     

A few remarks on 'combined action of DC and AC magnetic fields on ion motion in a macromolecule'

Zhadin and Barnes [2005:26:323-330] concluded that they solved the differential equation describing combined action of DC and AC magnetic fields on thermal motion of ions in a biological macromolecule and, as a result, a diversity of biological phenomena could be explained. It is shown here that biological phenomena cannot be explained based on this model. Adair [2006:27:332-334] gave several arguments for the statement that the interaction of weak magnetic fields with ions trapped in protein cavities cannot produce detectable biological effects through changing the character of the ion orbits. The arguments are analyzed here and some are shown to be questionable or unjustified. We stress the difference between the conclusion made by Adair and that stated in this article.     

Congruence among Mammalian Mitochondrial Genes

Mitochondrial protein coding genes were combined into a single matrix that included the 13 protein coding genes for 22 mammals, resulting in 11,448 characters each, or more than a quarter of a million base pairs of mitochondrial sequence. This matrix was examined for three separate a priori weighting strategies, including equal weighting, transversion weighting, and codon weighting. The incongruence length difference test obtained a significant value for equally weighted data and for one of our differentially weighted analyses. Thus, in one case, transversion only, weighting did effect previously incongruent data matrices, making them congruent. The tree produced for the various weighting strategies was identical in all of the treatments. These results support the conclusion that a priori weighting schemes should not be used to remove incongruence (presumably one does not want to introduce incongruence), but instead a simultaneous analysis strategy which includes treating all of the data equally weighted should be considered.