Two methods for fitting a rectangular hyperbola to data from several animals.

Two methods are described for fitting the Michaelis-Menten equation to sets of data with a common michaelis constant but different maximum velocities. One of them uses the method of least squares, and the other is based on the direct plot of Eisenthal & Cornish-Bowden [Biochem. J. (1974) 139, 715-720].     

Improved non-parametric statistical methods for the estimation of Michaelis-Menten kinetic parameters by the direct linear plot.

The theoretical basis for the direct linear plot [Eisenthal & Cornish-Bowden (1974) Biochem. J. 139, 715-720], a non-parametric statistical method for the analysis of data-fitting the Michaelis-Menten equation, was reinvestigated in order to accommodate additional experimental designs and to provide estimates of precision more directly comparable with those obtained by parametric statistical methods. Methods are given for calculating upper and lower confidence limits for the estimated parameters, for accommodating replicate measurements and for comparing the results of two separate experiments. Factors that influence the proper design of experiments are discussed.     

Use of the direct linear plot to estimate binding constants for protein-ligand interactions.

The direct linear plot (Eisenthal and Cornish-Bowden[1974] Biochem. J. $\text{139}$, 715–720) for the determination of enzyme kinetic constants has been assessed as a means of describing specific steroid-protein interactions. In the rat uterine cytoplasmic estrogen receptor system, determination of the equilibrium dissociation constant (K_D) and of the total number of ligand-binding sites (B_max) has been made, and the results are in good agreement with those obtained by Scatchard and Lineweaver-Burk plot analyses. The usefulness of the direct linear plot lies in the speed and simplicity with which it can be constructed and interpreted.     

Application of topological methods in enzyme kinetics.

A criticism [Cornish-Bowden (1976) Biochem. J. 159, 167] of an algebraic method for deriving steady-state rate equations [Indge & Childs (1976) Biochem. J. 155, 567-570] is theoretically founded.     

Mechanistic origin of the sigmoidal rate behaviour of rat liver hexokinase D (''glucokinase'').

Two recent proposals to account for the kinetic co-operativity of hexokinase D ('glucokinase') from rat liver are examined. A model in which the deviations from Michaelis-Menten kinetics result from a random order of binding of the substrates [Pettersson (1986) Biochem. J. 233, 347-350] accounts satisfactorily for the behaviour as a function of glucose concentrations, but it also predicts observable substrate inhibition by MgATP, which is in fact not observed. An alternative proposal in which the deviations arise from recycling of an enzyme-MgADP complex [Pettersson (1986) Eur. J. Biochem. 154, 167-170] also accounts satisfactorily for some of the data, but the required enzyme-MgADP complex could not be detected in isotope-exchange measurements. Thus the mnemonical mechanism proposed originally [Storer & Cornish-Bowden (1977) Biochem. J. 165, 61-69], which explains the deviations in terms of a relatively slow interconversion between two forms of free enzyme, remains the most parsimonious explanation of the behavior of hexokinase D.     

Validity of the jack-knife technique for analysing enzyme kinetic data.

An observation by Duggleby [Biochem. J. (1979) 181, 255-256] that estimates of kinetic parameters by the jack-knife technique [Cornish-Bowden & Wong (1978) Biochem. J. 175, 969--976] are sometimes outside the range of estimates from which they are calculated has been examined. No significant correlation has been found between the occurrence of this behaviour and the actual quality of the estimates.     

A comment on the ''jack-knife'' technique for analysing enzyme-kinetic data.

The use of the 'jack-knife' technique in the analysis of enzyme-kinetic data [Cornish-Bowden & Wong (1978] Biochem. J. 175, 969--976) is examined. The method can give parameter estimates that appear to be incorrect.     

Kinetic evidence for a ''mnemonical'' mechanism for rat liver glucokinase.

Inhibition studies of glucokinase were carried out with the products of the reaction, glucose 6-phosphate and MgADP-, as well as with ADP3-, Mg2+ and ATP4-. The results of these, together with those of kinetic studies of the uninhibited reaction described previously [Storer & Cornish-Bowden (1976) Biochem. J. 159, 7-14], indicate that the enzyme obeys a 'mnemonical' mechanism. This implies that the co-operativity observed with glucose as substrate arises because glucose binds differentially to two forms of the free enzyme that are not in equilibrium under steady-state conditions. The mechanism predicts the decrease in glucose co-operativity observed at low concentrations of MgATP2-. The product-inhibition results suggest that glucose 6-phosphate is released first and that it is possibly displaced by MgATP2- in a concerted reaction.     

Statistical considerations in the estimation of enzyme kinetic parameters by the direct linear plot and other methods

The statistical implications of the direct linear plot for enzyme kinetic data, described in the preceding paper (Eisenthal & Cornish-Bowden, 1974), are discussed for the case of the Michaelis-Menten equation. The plot is shown to lead directly to non-parametric confidence limits for the kinetic parameters, V and K(m), which depend on far less sweeping assumptions about the nature of experimental error than those implicit in the method of least squares. Median estimates of V and K(m) can also be defined, which are shown to be more robust than the least-squares estimates in a wide variety of experimental situations.     

A thermodynamic model for the cooperative functional properties of the tetraheme cytochrome c3 from Desulfovibrio gigas.

A thermodynamic model is presented to describe the redox behaviour of the tetraheme cytochrome c3 from Desulfovibrio gigas. This molecule displays different intrinsic redox potentials for the four hemes and during the redox titration process, interactions among different hemes occur, thus altering the values of redox potentials according to which of the hemes are oxidized [Santos, H., Moura, J.J.G., Moura, I., LeGall, J. & Xavier, A.V. (1984) Eur. J. Biochem. 141, 283-296]. This complex cooperative behaviour [Xavier, A.V. (1986) J. Inorg. Biochem. 28, 239-243] has been analyzed here using an I2H4-interaction network [Cornish-Bowden, A. & Koshland, D.E. Jr (1970) J. Biol. Chem. 245, 6241-6250] coupled to a proton-linked equilibrium between two tertiary structures. Such a formalism, which requires a reduced number of parameters, is able to fully account quantitatively for the pH dependence of the NMR redox-titration curves. The 'redox-Bohr' effect is discussed in terms of the available structure and thermodynamic data and a functional mechanism is proposed.     

Is rat skeletal muscle hexokinase an allosteric enzyme?

A recent report by M. Gregoriou, I. P. Trayer, and A. Cornish-Bowden (1986, Eur. J. Biochem. 161, 171-176) showed that the mechanism for rat skeletal muscle hexokinase contains two allosteric sites: one for ATP and one for glucose 6-phosphate. In this report, we show that the allosteric mechanism is at variance with a large amount of kinetic data for the skeletal muscle hexokinase reaction in the literature. In addition, the allosteric mechanism conflicts with isotope exchange at chemical equilibrium data reported by M. Gregoriou, I. P. Trayer, and A. Cornish-Bowden (1983, Eur. J. Biochem. 134, 283-288).     

Studies on adenosine triphosphate transphosphorylases. Amino acid sequence of rabbit muscle ATP-AMP transphosphorylase

The total amino acid sequence of rabbit muscle adenylate kinase has been determined, and the single polypeptide chain of 194 amino acid residues starts with N-acetylmethionine and ends with leucyllysine at its carboxyl terminus, in agreement with the earlier data on its amino acid composition [Mahowald, T. A., Noltmann, E. A., & Kuby, S. A. (1962) J. Biol. Chem. 237, 1138-1145] and its carboxyl-terminus sequence [Olson, O. E., & Kuby, S. A. (1964) J. Biol. Chem. 239, 460-467]. Elucidation of the primary structure was based on tryptic and chymotryptic cleavages of the performic acid oxidized protein, cyanogen bromide cleavages of the 14C-labeled S-carboxymethylated protein at its five methionine sites (followed by maleylation of peptide fragments), and tryptic cleavages at its 12 arginine sites of the maleylated 14C-labeled S-carboxymethylated protein. Calf muscle myokinase, whose sequence has also been established, differs primarily from the rabbit muscle myokinase's sequence in the following: His-30 is replaced by Gln-30; Lys-56 is replaced by Met-56; Ala-84 and Asp 85 are replaced by Val-84 and Asn-85. A comparison of the four muscle-type adenylate kinases, whose covalent structures have now been determined, viz., rabbit, calf, porcine, and human [for the latter two sequences see Heil, A., Müller, G., Noda, L., Pinder, T., Schirmer, H., Schirmer, I., & Von Zabern, I. (1974) Eur. J. Biochem. 43, 131-144, and Von Zabern, I., Wittmann-Liebold, B., Untucht-Grau, R., Schirmer, R. H., & Pai, E. F. (1976) Eur. J. Biochem. 68, 281-290], demonstrates an extraordinary degree of homology.(ABSTRACT TRUNCATED AT 250 WORDS)     

Statistical analysis of the Michaelis-Menten equation

An application of the method of maximum likelihood (ML) is described for analysing the results of enzyme kinetic experiments in which the Michaelis-Menten equation is obeyed. Accurate approximate solutions to the ML equations for the parameter estimates are presented for the case in which the experimental errors are of constant relative magnitude. Formulae are derived that approximate the standard errors of these estimates. The estimators are shown to be asymptotically unbiased and the standard errors observed in simulated data rapidly approach the theoretical lower bound as the sample size increases. The results of a large-scale Monte Carlo simulation study indicate that for data with a constant coefficient of variation, the present method is superior to other published methods, including the conventional transformations to linearity and the nonparametric technique proposed by Eisenthal and Cornish-Bowden (1974, Biochemical Journal 139, 715-720). Finally, the present results are extended to the analysis of simple receptor binding experiments using the general approach described by Munson and Rodbard (1980, Analytical Biochemistry 107, 220-239).     

Estimation of the initial velocity of enzyme-catalysed reactions by non-linear regression analysis of progress curves.

Most methods for studying the kinetic properties of an enzyme involve the determination of initial velocities. When the reaction progress curve shows significant curvature due to depletion of the substrate, accumulation of inhibitory products or instability of the enzyme, estimation of the initial velocity is a subjective and inexact process. Two methods have been suggested [Cornish-Bowden (1975) Biochem. J. 144, 305-312; Boeker (1982) Biochem J. 203, 117-123] that attempt to eliminate this subjective element. The present study offers a third alternative, which is based on fitting a reparameterized form of the integrated Michaelis-Menten equation to the progress curves by non-linear regression. This method yields estimates and standard errors of the initial velocity and of the time to reach 50% reaction. No prior knowledge of the apparent product concentration at zero time or infinite time is required, since both of these quantities are also estimated from the data. It is shown that this method yields reliable estimates of the initial velocity under a wide range of circumstances, including those where the two previously published methods perform poorly.     

Enzymatic assays of L-ornithine and L-delta 1-pyrroline-5-carboxylate in tissues, and orniathine-load test in human subjects

The efficiencies of estimates obtained from the direct linear plot (A. Cornish-Bowden and R. Eisenthal, 1978, Biochem. Biophys, Acta, 523, 268) are shown to be dependent on the spacing of substrate concentrations. When substrate values are harmonically spaced, the direct linear plot should not be used. The nonparametric confidence limits based on the direct linear plot are accurate in their confidence coefficient, but their efficiencies are shown to be dependent on substrate spacing. Harmonic spacing is, in general, a more efficient experimental design for estimating K_m than arithmetic spacing when the appropriate estimation methods are used. If assumptions about the error structure cannot be made, the best procedure for estimating K_m is to have harmonic spacing of substrate values and use weighted least squares for estimation. The most accurate and precise estimation of enzyme kinetic parameters requires knowledge of the error structure and utilization of the appropriate nonlinear regression.     

Evaluation of distribution-free confidence limits for enzyme kinetic parameters

Monte Carlo experiments have been used to test the robustness of distribution-free confidence limits for the parameters of the Michaelis-Menten equation (Porter & Trager, 1977). When used in conjunction with the modified form of the direct linear plot (Cornish-Bowden & Eisenthal, 1978), they prove to be more robust than least-squares confidence limits. In circumstances where the least-squares assumptions are correct, the distribution-free confidence limits define the parameters somewhat less precisely than the corresponding least-squares confidence limits, but this effect is negligible unless there are eight or fewer observations.     

Glutathione reductase from human erythrocytes. Amino-acid sequence of a major fragment that links the FAD, NADP and interface domains.

A major CNBr fragment of glutathione reductase, peptide Q [Krohne-Ehrich, G., Schirmer, R.H. & Untucht-Grau, R. (1977) Eur. J. Biochem. 80, 65-71], was further fractionated by trypsin, chymotrypsin, thermolysin and clostripain digestion. The peptides were isolated and most of them were sequenced by solid-phase Edman degradation. The whole peptide Q was sequenced N-terminally up to position 51 by the same technique. A total sequence of 128 amino acids (28% of the whole protein) was obtained and could be localized in the electron density map [Schulz, G.E., Schirmer, R.H., Sachsenheimer, W. & Pai, E.F. (1978) Nature (Lond.) 273, 120-124] from position 259-387. This part of the polypeptide links and participates in all three domains of the flavoenzyme.     

The amino acid sequence of Neurospora NADP-specific glutamate dehydrogenase. Peptic and chymotryptic peptides and the complete sequence.

Peptic and chymotryptic peptides were isolated form the NADP-specific glutamate dehydrogenase of Neurospora crassa and substantially sequenced. Out of 452 residues in the polypeptide chain, 265 were recovered in the peptic and 427 in the chymotryptic peptides. Together with the tryptic peptides [Wootton, J. C., Taylor, J. G., Jackson, A. A., Chambers, G. K. & Fincham, J. R. S. (1975) Biochem. J. 149, 749-755], these establish the complete sequence of the chain, including the acid and amide assignments, except for seven places where overlaps are inadequate. These remaining alignments are deduced from information on the CNBr fragments obtained in another laboratory [Blumenthal, K. M., Moon, K. & Smith, E. L. (1975), J. Biol. Chem. 250, 3644-3654]. Further information has been deposited as Supplementary Publication SUP 50054 (17 pages) with the British Library (Lending Division), Boston Spa, Wetherby, W. Yorkshire LS23 7BQ, U.K., from whom copies may be obtained under the terms given in Biochem. J. (1975) 145, 5.     

Evidence against the incorporation into protein of amino acids directly from the membrane transport system in rat heart

The possibility suggested recently [Hider, R.C., Fern, E.B. and London, D.R. (1969) Biochem. J. 114, 171-178; Hider, R.C., Fern, E.B. and London, D.R. (1971) Biochem. J. 121, 817-827; van Venrooij, W.J., Poort, C., Kramer, M.F. and Jansen, M.T. (1972) Eur. J. Biochem. 30, 427-433; and Adamson, L.F., Herington, A.C. and Bornstein, J. (1972) Biochim. Biophys. Acta 282, 352-365] that protein synthesis takes place using amino acids directly from the membrane transport system and not from an intracellular pool has been investigated in rat heart. The tissue was perfused first for 30 min with either [14C]glycine or [14C]leucine and then for a further 30 min with identical medium containing [3H]glycine or [3H]leucine, respectively. After an initial lag, [14C]glycine was incorporated into protein at a linear rate up to 60 min. The [3H]glycine was accumulated into tissue water and incorporated just as readily as the [14C]glycine had been. The rate of total protein synthesis agrees with literature values only if intracellular and not extracellular specific activity values are used in the calculation. Some glycine was converted to serine or threonine. Leucine influx and efflux were very rapid in contrast to the relatively slow exchange reported for incubated tissues [Hider, R.C., Fern, E.B. and London, D.R. (1969) Biochem. J. 114, 171-178; Hider, R.C., Fern, E.B. and London, D.R. (1971) Biochem. J. 121, 817-827; van Venrooij, W.J., Poort, C., Kramer, M.F. and Jansen, M.T. (1972) Eur. J. Biochem. 30, 427-433]. The results are consistent with the existence of an intracellular precursor pool for glycine. Some possible reasons for the discrepancies between this and the other studies are discussed.     

A new subunit of horse liver alcohol dehydrogenase and subunit composition of the polymorphic form.

The most cathodal (on starch-gel electrophoresis), steroid-active band of horse liver alcohol dehydrogenase, whose catalytic properties were shown to be dependent on the livers used as a starting material [Pietruszko (1974) Biochem. Biophys. Res. Commun. 60, 687-694], has been prepared from A-type and S-type horse livers by identical methods. Results presented here show that different isoenzymes are present in these preparations.