Comparative least-squares analysis of hemoglobin oxygen equilibrium curves

The oxygen-binding properties of hemoglobin have been studied at 600 microM protein concentration with organic phosphate, and analyzed by a series of different nonlinear least-squares analysis methods to determine whether reports of negligibly small values of the third overall Adair parameter, A3, are consequences of the data or a product of the data analysis. Data from other laboratories were analyzed as well. The single most important factor in creating a measurement that yields a small A3 is the use of equally weighted fitting in the Adair equation, while end-weighted fitting generally yields a larger A3. Endpoint extrapolation is ruled out as a major cause of abnormal A3 values. Monte Carlo simulations of the 600 microM results suggest that, if a small A3 were present, end weighting is at least as sensitive to a small A3 as equal weighting. We conclude that equally weighted fitting of the tetrameric Adair equation is unable to resolve the upper asymptote of the oxygen-binding data, resulting in an unusually small value for A3.     

Information theory and the analysis of ligand-binding data

The phenomenological principles of information theory are used in the analysis of ligand-binding phenomena in biological macromolecules. Information maps are constructed to visualize regions of ligand chemical potential with maximum amount of information and to devise suitable experimental strategies therefrom. Extensive simulation studies and analysis of experimental data also point out the properties of information used as a weighting procedure in nonlinear least-squares analyses.     

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.     

Thermodynamic aspects of the co-operativity in four-step oxygenation equilibria of haemoglobin

Accurate oxygen equilibrium curves of human haemoglobin (concentration, 600 μm as haem) were determined by an automatic recording method (Imai et al., 1970) under a variety of conditions combining six different temperatures with seven sets of solute conditions, producing wide-ranging structural constraints on haemoglobin. The heat and entropy change of oxygenation for four individual steps (ΔH_tand ΔS_i, i = 1 to 4) were evaluated by a least-squares method directly from each set of six equilibrium curves without knowing the values of the four equilibrium constants k_t. As shown in previous studies with dilute haemoglobin solutions (Imai &; Tyuma, 1973; Imai &; Yonetani, 1975b) ΔH_i depended strongly on i; small amounts of heat were liberated at oxygenation steps involving the release of H⁺ and anions such as Cl^?, 2,3-diphosphoglycerate, and inositol hexaphosphate, while large amounts of heat were liberated on the oxygenation of the R state or highly constrained T state, from which no or few non-haem ligands are released. The observed amounts of heat, when corrected for the heat of H⁺ and anion release associated with oxygenation, became uniform to a good approximation, indicating that the intrinsic heat of haem oxygenation is essentially equal for the four oxygenation steps, and a large part of the non-uniformity of ΔH_i may be ascribed to the oxygen-linked release of the non-haem ligands. ΔS_i exhibited similar behaviour. The relation, k₁ ? k₂ ? k₃ ? k₄ which usually holds under physiological conditions, is a consequence of the presence of an enthalpy-entropy compensation process at the first three steps and its absence at the fourth step. The compensation temperature was around 300 K. The origin of the co-operativity cannot be specified as either an enthalpic or entropic effect. In the presence of 0.1 m-Cl^? and 2 mm-2,3-diphosphoglycerate, the T to R transition at any oxygenation step is an endothermic process and haemoglobin gains entropy on the transition. The deoxy T structure is stabilised by the enthalpy term, while the oxy R structure is stabilised by the entropy term, so that the T to R transition occurs at a stop where the entropy contribution exceeds the enthalpy contribution. The present study shows that the oxygen-linked binding of non-haem ligands is very important in co-operative oxygen binding by haemoglobin, as predicted by Perutz, (1970).     

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?     

Spreadsheet Method for Evaluation of Biochemical Reaction Rate Coefficients and Their Uncertainties by Weighted Nonlinear Least-Squares Analysis of the Integrated Monod Equation

A convenient method for evaluation of biochemical reaction rate coefficients and their uncertainties is described. The motivation for developing this method was the complexity of existing statistical methods for analysis of biochemical rate equations, as well as the shortcomings of linear approaches, such as Lineweaver-Burk plots. The nonlinear least-squares method provides accurate estimates of the rate coefficients and their uncertainties from experimental data. Linearized methods that involve inversion of data are unreliable since several important assumptions of linear regression are violated. Furthermore, when linearized methods are used, there is no basis for calculation of the uncertainties in the rate coefficients. Uncertainty estimates are crucial to studies involving comparisons of rates for different organisms or environmental conditions. The spreadsheet method uses weighted least-squares analysis to determine the best-fit values of the rate coefficients for the integrated Monod equation. Although the integrated Monod equation is an implicit expression of substrate concentration, weighted least-squares analysis can be employed to calculate approximate differences in substrate concentration between model predictions and data. An iterative search routine in a spreadsheet program is utilized to search for the best-fit values of the coefficients by minimizing the sum of squared weighted errors. The uncertainties in the best-fit values of the rate coefficients are calculated by an approximate method that can also be implemented in a spreadsheet. The uncertainty method can be used to calculate single-parameter (coefficient) confidence intervals, degrees of correlation between parameters, and joint confidence regions for two or more parameters. Example sets of calculations are presented for acetate utilization by a methanogenic mixed culture and trichloroethylene cometabolism by a methane-oxidizing mixed culture. An additional advantage of application of this method to the integrated Monod equation compared with application of linearized methods is the economy of obtaining rate coefficients from a single batch experiment or a few batch experiments rather than having to obtain large numbers of initial rate measurements. However, when initial rate measurements are used, this method can still be used with greater reliability than linearized approaches.     

Scanning molecular sieve chromatography of interacting protein systems: nonlinear least-squares analysis of small zone experiments

A simple routine for nonlinear least-squares analysis is applied to small zone scanning data, where calibration of the gel column requires the use of fully characterized markers of known molecular size. Application of nonlinear least-squares analysis eliminates the difficulty encountered because of scarcity of calibrating markers for gels whose porosities span certain size ranges by providing a sensitive measure of changes in molecular size of the interacting protein system, in order to determine the interaction parameters, size heterogeneity, and the centroid position for time difference chromatographic experiments.     

A fast and reliable computational method for estimating population genetic parameters

The estimation of ancestral and current effective population sizes in expanding populations is a fundamental problem in population genetics. Recently it has become possible to scan entire genomes of several individuals within a population. These genomic data sets can be used to estimate basic population parameters such as the effective population size and population growth rate. Full-data-likelihood methods potentially offer a powerful statistical framework for inferring population genetic parameters. However, for large data sets, computationally intensive methods based upon full-likelihood estimates may encounter difficulties. First, the computational method may be prohibitively slow or difficult to implement for large data. Second, estimation bias may markedly affect the accuracy and reliability of parameter estimates, as suggested from past work on coalescent methods. To address these problems, a fast and computationally efficient least-squares method for estimating population parameters from genomic data is presented here. Instead of modeling genomic data using a full likelihood, this new approach uses an analogous function, in which the full data are replaced with a vector of summary statistics. Furthermore, these least-squares estimators may show significantly less estimation bias for growth rate and genetic diversity than a corresponding maximum-likelihood estimator for the same coalescent process. The least-squares statistics also scale up to genome-sized data sets with many nucleotides and loci. These results demonstrate that least-squares statistics will likely prove useful for nonlinear parameter estimation when the underlying population genomic processes have complex evolutionary dynamics involving interactions between mutation, selection, demography, and recombination.     

Nonlinear estimation of Monod growth kinetic parameters from a single substrate depletion curve

Monod growth kinetic parameters were estimated by fitting sigmoidal substrate depletion data to the integrated Monod equation, using nonlinear least-squares analysis. When the initial substrate concentration was in the mixed-order region, nonlinear estimation of simulated data sets containing known measurement errors provided accurate estimates of the mu max, Ks, and Y values used to create these data. Nonlinear regression analysis of sigmoidal substrate depletion data was also evaluated for H2-limited batch growth of Desulfovibrio sp. strain G11. The integrated Monod equation can be more convenient for the estimation of growth kinetic parameters, particularly for gaseous substrates, but it must be recognized that the estimates of mu max, Ks, and Y obtained may be influenced by the growth rate history of the inoculum.     

Nonlinear estimation of Monod growth kinetic parameters from a single substrate depletion curve.

Monod growth kinetic parameters were estimated by fitting sigmoidal substrate depletion data to the integrated Monod equation, using nonlinear least-squares analysis. When the initial substrate concentration was in the mixed-order region, nonlinear estimation of simulated data sets containing known measurement errors provided accurate estimates of the mu max, Ks, and Y values used to create these data. Nonlinear regression analysis of sigmoidal substrate depletion data was also evaluated for H2-limited batch growth of Desulfovibrio sp. strain G11. The integrated Monod equation can be more convenient for the estimation of growth kinetic parameters, particularly for gaseous substrates, but it must be recognized that the estimates of mu max, Ks, and Y obtained may be influenced by the growth rate history of the inoculum.     

Analysis of oxygen binding by hemoglobin on the basis of mean intrinsic thermodynamic quantities

The binding data for oxygenation of human hemoglobin, Hb, at various temperatures and in the absence and presence of 2,3-diphosphoglycerate, DPG, and inositol hexakis phosphate, IHP, were analyzed for extraction of mean intrinsic Gibbs free energy, DeltaGo, enthalpy, DeltaHo, and entropy, DeltaSo, of binding at various partial oxygen pressures. This method of analysis considers all the protein species present such as dimer and tetramer forms which were not considered by Imai et al. (Imai K et al., 1970, Biochim Biophys Acta 200: 189-196), in their analysis which was based on Adair equation. In this regard, the values of Hill equation parameters were estimated with high precision at all points of the binding curve and used for calculation of DeltaGo, DeltaHo and DeltaSo were also calculated by analysis of DeltaGo values at various temperatures using van't Hoff equation. The results represent the enthalpic nature of the cooperativity in Hb oxygenation and the compensation effect of intrinsic entropy. The interpretation of results also to be, into account the decrease of the binding affinity of sites for oxygen in the presence of DPG and IHP without any considerable changes in the site-site interaction (extent of cooperativity). In other words, the interactions between bound ligands, organic phosphates and oxygen, are more due to a decreasing binding affinity and not to the reduction of the cooperative interaction between sites. The results also document the more heterotropic effect of IHP compared to DPG.     

Nonlinearities in mating sounds of American crocodiles

We use nonlinear time series analysis methods to analyze the dynamics of the sound-producing apparatus of the American crocodile (Crocodylus acutus). We capture its dynamics by analyzing a recording of the singing activity during mating time. First, we reconstruct the phase space from the sound recording and thereby reveal that the attractor needs no less than five degrees of freedom to fully evolve in the embedding space, which suggests that a rather complex nonlinear dynamics underlies its existence. Prior to investigating the dynamics more precisely, we test whether the reconstructed attractor satisfies the notions of determinism and stationarity, as a lack of either of these properties would preclude a meaningful further analysis. After positively establishing determinism and stationarity, we proceed by showing that the maximal Lyapunov exponent of the recording is positive, which is a strong indicator for the chaotic behavior of the system, confirming that dynamical nonlinearities are an integral part of the examined sound-producing apparatus. At the end, we discuss that methods of nonlinear time series analysis could yield instructive insights and foster the understanding of vocal communication among certain reptile species.     

A simple computer program with statistical tests for the analysis of enzyme kinetics.

A simple computer program that calculates the kinetic parameters of enzyme reactions is described. Parameters are determined by nonlinear, least-squares regression using either Marquardt-Levenberg or Gauss-Newton algorithms to find the minimum sum of squares. Three types of enzyme reactions can be analyzed: single substrate reactions (Michaelis-Menten and sigmoidal kinetics), enzyme activation at a fixed substrate value or enzyme inhibition at a fixed substrate value. The user can monitor goodness of fit through nonparametric statistical tests (performed automatically by the computer) and through visual examination of the pattern of residuals. The program is unique in providing equations for activator and inhibition analysis as well as in enabling the user to fix some of the parameters before regression analysis. The simplicity of the program makes it extremely useful for quickly determining kinetic parameters during the data-gathering process.     

Bidirectional reaction steps in metabolic networks: II. Flux estimation and statistical analysis

Metabolic carbon labelling experiments enable a large amount of extracellular fluxes and intracellular carbon isotope enrichments to be measured. Since the relation between the measured quantities and the unknown intracellular metabolic fluxes is given by bilinear balance equations, flux determination from this data set requires the numerical solution of a nonlinear inverse problem. To this end, a general algorithm for flux estimation from metabolic carbon labelling experiments based on the least squares approach is developed in this contribution and complemented by appropriate tools for statistical analysis. The linearization technique usually applied for the computation of nonlinear confidence regions is shown to be inappropriate in the case of large exchange fluxes. For this reason a sophisticated compactification transformation technique for nonlinear statistical analysis is developed. Statistical analysis is then performed by computing appropriate statistical quality measures like output sensitivities, parameter sensitivities and the parameter covariance matrix. This allows one to determine the order of magnitude of exchange fluxes in most practical situations. An application study with a large data set from lysine-producing Corynebacterium glutamicum demonstrates the power and limitations of the carbon-labelling technique. It is shown that all intracellular fluxes in central metabolism can be quantitated without assumptions on intracellular energy yields. At the same time several exchange fluxes are determined which is invaluable information for metabolic engineering. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 118-135, 1997.     

A method for analyzing temporal patterns of variability of a time series from Poincare plots

The Poincaré plot is a popular two-dimensional, time series analysis tool because of its intuitive display of dynamic system behavior. Poincaré plots have been used to visualize heart rate and respiratory pattern variabilities. However, conventional quantitative analysis relies primarily on statistical measurements of the cumulative distribution of points, making it difficult to interpret irregular or complex plots. Moreover, the plots are constructed to reflect highly correlated regions of the time series, reducing the amount of nonlinear information that is presented and thereby hiding potentially relevant features. We propose temporal Poincaré variability (TPV), a novel analysis methodology that uses standard techniques to quantify the temporal distribution of points and to detect nonlinear sources responsible for physiological variability. In addition, the analysis is applied across multiple time delays, yielding a richer insight into system dynamics than the traditional circle return plot. The method is applied to data sets of R-R intervals and to synthetic point process data extracted from the Lorenz time series. The results demonstrate that TPV complements the traditional analysis and can be applied more generally, including Poincaré plots with multiple clusters, and more consistently than the conventional measures and can address questions regarding potential structure underlying the variability of a data set.     

New Weighting Methods for Phylogenetic Tree Reconstruction Using Multiple Loci

Efficient determination of evolutionary distances is important for the correct reconstruction of phylogenetic trees. The performance of the pooled distance required for reconstructing a phylogenetic tree can be improved by applying large weights to appropriate distances for reconstructing phylogenetic trees and small weights to inappropriate distances. We developed two weighting methods, the modified Tajima–Takezaki method and the modified least-squares method, for reconstructing phylogenetic trees from multiple loci. By computer simulations, we found that both of the new methods were more efficient in reconstructing correct topologies than the no-weight method. Hence, we reconstructed hominoid phylogenetic trees from mitochondrial DNA using our new methods, and found that the levels of bootstrap support were significantly increased by the modified Tajima–Takezaki and by the modified least-squares method.     

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.     

The Usefulness of the Sting Apparatus in Phylogenetic Reconstructions in Vespids,with Emphasis on the Epiponini: More Support for the Single Origin of Eusociality in the Vespidae

This study aimed at testing the utility of characters derived from chitinous structures of the sting apparatus for elucidating relationships among the genera of Epiponini. The characters were obtained from the spiracular and quadrate plates, gonostylus, and sting. The data matrix was analyzed using parsimony with equal and implied weighting. Sting characters were also optimized on the tree of Wenzel & Carpenter (1994). Consensus of analysis using equal weights parsimony resulted in a tree with low resolution, but the use of implied weighting improved the results and a consensus tree with a better resolution was obtained. Implied weighting analysis showed an interesting result with Vespinae and Epiponini (the taxa that present the highest degree of sociality) together in a clade. The overall uniformity in morphology of sting apparatus and a possible influence of sociality on morphology could explain these results. The evolution of some characters is discussed.     

Studies on cobalt myoglobins and hemoglobins. Preparation of isolated chains containing cobaltous protoporphyrin IX and characterization of their equilibrium and kinetic properties of oxygenation and EPR spectra.

Human hemoglobin containing cobalt protoporphyrin IX or cobalt hemoglobin has been separated into two functionally active alpha and beta subunits using a new method of subunit separation, in which the -SH groups of the isolated subunits were successfully regenerated by treatment with dithiothreitol in the presence of catalase. Oxygen equilibria of the isolated subunit chains were examined over a wide range of temperature using Imai's polarographic method (Imai, K., Morimoto, H., Kotani, M., Watari, H., and Kuroda, M. (1970) Biochim. Biophys. Acta 200, 189-196). Kinetic properties of their reversible oxygenation were investigated by the temperature jump relaxation method at 16 degrees. Electron paramagnetic resonance characteristics of the molecules in both deoxy and oxy states were studies at 77K. The oxygen affinity of the individual regenerated chains was higher than that of the tetrameric cobalt hemoglobin and was independent of pH. The enthalpy changes of the oxygenation have been determined as -13.8 kcal/mol and -16.8 kcal/mol for the alpha and beta chains, respectively. The rates of oxygenation were similar to those reported for iron hemoglobin chains, whereas those of deoxygenation were about 10(2) times larger. The effects of metal substitution on oxygenation properties of the isolated chains were correlated with the results obtained previously on cobalt hemoglobin and cobalt myoglobin. The EPR spectrum of the oxy alpha chain showed a distinctly narrowed hyperfine structure in comparison with that of the oxy beta chain, indicating that the environment around the paramagnetic center (the bound oxygen) is different between these chains. In the deoxy form, EPR spectra of alpha and beta chains were indistinguishable. These observations suggest that one of the inequivalences between alpha and beta chains might exist near the distal histidine group.