Consequences of neglecting the hemoglobin concentration on the determination of Adair binding constants

Determination of the seven apparent equilibrium constants for the binding of oxygen by hemoglobin from measured fractional saturations

Subunit dissociation has to be taken into account in the determination of the oxygen binding constants of hemoglobin, as described by Ackers and Halvorson in 1974. The seven apparent equilibrium constants for a particular set of conditions can be determined by using extrapolations to determine the fractional saturations YT of tetramer and YD of dimer from measured values of the fractional saturation Y of partially dissociated hemoglobin. Analytical methods are used to show that YT as a function of [O2] for tetramers can be calculated from Y of hemoglobin by linear extrapolation of measured Y values at high [heme] versus [heme]-1/2 to [heme]-1/2 = 0. YD for dimers can be calculated from measured Y values by linear extrapolation of Y versus [heme] to [heme] = 0 if sufficiently low [heme] can be used. These extrapolations have been tested with numerical calculations of Y for a particular hemoglobin as a function of [heme] and [O2] by using the seven apparent equilibrium constants determined by Mills, Johnson, and Ackers in 1976. The proposed procedure also yields the apparent association constant K" for 2TotD = TotT, where TotD is the sum of the dimers and TotT is the sum of the tetramers. This thermodynamic analysis of experimental data to determine the seven apparent equilibrium constants is independent of the model used to interpret the values of the thermodynamic parameters.

Mutagenic dissection of hemoglobin cooperativity: effects of amino acid alteration on subunit assembly of oxy and deoxy tetramers

Free energies of oxygen-linked subunit assembly and cooperative interaction have been determined for 34 molecular species of human hemoglobin, which differ by amino acid alterations as a result of mutation or chemical modification at specific sites. These studies required the development of extensions to our earlier methodology. In combination with previous results they comprise a data base of 60 hemoglobin species, characterized under the same conditions. The data base was analyzed in terms of the five following issues. (1) Range and sensitivity to site modifications. Deoxy tetramers showed greater average energetic response to structural modifications than the oxy species, but the ranges are similar for the two ligation forms. (2) Structural localization of cooperative free energy. Difference free energies of dimer-tetramer assembly (oxy minus deoxy) yielded delta Gc for each hemoglobin, i.e., the free energy used for modulation of oxygen affinity over all four binding steps. A structure-energy map constructed from these results shows that the alpha 1 beta 2 interface is a unique structural location of the noncovalent bonding interactions that are energetically coupled to cooperativity. (3) Relationship of cooperativity to intrinsic binding. Oxygen binding energetics for dissociated dimers of mutants strongly indicates that cooperativity and intrinsic binding are completely decoupled by tetramer to dimer dissociation. (4) Additivity, site-site coupling and adventitious perturbations. All these are exhibited by individual-site modifications of this study. Large nonadditivity may be correlated with global (quaternary) structure change. (5) Residue position vs. chemical nature. Functional response is solely dictated by structural location for a subset of the sites, but varies with side-chain type at other sites. The current data base provides a unique framework for further analyses and modeling of fundamental issues in the structural chemistry of proteins and allosteric mechanisms.

Why, when, and how biochemists should use least squares

One of the most commonly used methods for the analysis of experimental data in the biochemical literature is nonlinear least squares (regression). This group of methods are also commonly misused. The purpose of this article is to review the assumptions inherent in the use of least-squares techniques and how these assumptions govern the ways that least-squares techniques can and should be used. Since these assumptions pertain to the nature of the experimental data to be analyzed they also dictate many aspects of the data collection protocol. The examination of these assumptions includes a discussion of questions like: Why would a biochemist want to use nonlinear least-squares techniques? When is it appropriate for a biochemist to use nonlinear least-squares techniques? What confidence can be assigned to the results of a nonlinear least-squares analysis?

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.

Discontinuous release of heat at successive steps of oxygenation in human and bovine hemoglobin at pH 9.0

We have measured the temperature dependence of the oxygen-binding isotherms of human and bovine hemoglobin at pH 9.0 in 0.1 M borate buffer. In both hemoglobins the ionization of the Bohr protons is finished at this pH; therefore, their heat does not interfere with the measurements. Two sets of curves have been obtained, which have been analyzed by either singular or global procedures for estimating the enthalpy changes of subsequent steps of oxygenation. The data indicate that in human hemoglobin the reaction with oxygen is enthalpy driven for steps 1, 2, and 4 while it is entropy driven for step 3. In bovine hemoglobin this phenomenon is even more evident: steps 2 and 4 are enthalpy driven while steps 1 and 3 are entropy driven. The discontinuous distribution of heat at subsequent steps of oxygenation suggests that the T to R transition in hemoglobin is not a monotonic process and involves conformations with novel characteristics.

Evaluation of Adair binding constants from experimental data obtained with an Imai cell. Statistical weighting functions

One group of laboratories uses an 'even-weighted', or unweighted, nonlinear least-squares method for the analysis of experimental oxygen binding data obtained with an Imai oxygenation apparatus. Another group uses an 'end-weighted' nonlinear least-squares analysis. With end weighting each observation is assigned a statistical weight which is proportional to Y(1-Y), where Y is the fractional saturation. In this work we discuss statistical weighting functions as applied to the Imai oxygenation apparatus and then determined what are the best weighting factors for an actual series of published experiments. Based on these calculations, it is concluded that the 'best' weighting for the Imai oxygenation apparatus is a very small amount of end weighting. Furthermore, the amount of end weighting is so small that even weighting, or no weighting at all, is also appropriate.

Consequences of neglecting the hemoglobin concentration on the determination of Adair binding constants

It is commonly believed that the tetrameric Adair constants for oxygen binding to human hemoglobin can be evaluated from a single oxygenation experiment at 'high' hemoglobin concentration without considering the consequence of the presence of alpha beta dimers. We present examples which demonstrate that this is a very dangerous assumption. Without a knowledge of the complete oxygenation-linked dimer-tetramer association reaction (alpha beta Xi----(alpha beta)2Xj), it is impossible to predict a priori how high of a hemoglobin concentration would be required to make this assumption. Furthermore, without a knowledge of the complete oxygenation-linked dimer-tetramer association reaction, it is impossible to predict a priori the direction and magnitude of the systematic errors which are induced by making this assumption.