Use of the F test for determining the degree of enzyme-kinetic and ligand-binding data. A Monte Carlo simulation study

1. Initial-rate data were simulated for 13 representative enzyme mechanisms with the use of several distributions of rate constants in order to locate conditions leading to v([S]) curves in physiological ranges of substrate concentration. 2. In all, 420 sets of such v([S]) curves were generated with the use of several choices of substrate concentration range (two, three or four orders of magnitude), number of experimental points (10, 15 or 20), error on v (5-10%) and standard deviation on v (5-9%) in order to simulate experimental results in a number of possible ways. 3. Curve-fitting was carried out to rational functions of degree 1:1, 2:2, . . ., 5:5 until there was no statistically significant decrease in the sum of weighted squared residuals as judged by the F test at 95% and 99% confidence levels. 4. It was checked whether the non-linear regression program had located a good minimum in the sum of squares by also fitting the data with the correct values of parameters as starting estimates. 5. A similar procedure was adopted with 110 sets of binding data simulated for 11 models, and the F test was used to see if fractional-saturation data generated by a binding polynomial of order n could be adequately fitted by one of order m, m less than n. 6. From the 530 simulations the F test was successful in fixing the correct degree with a probability of 0.62 at the 95% confidence level, but this fell with increase in degree as follows: 1:1 (0.98), 2:2 (0.71), 3:3 (0.43) and 4:4 (0.34), the first numbers being the degree of the rate equation and those in parentheses referring to the 95% confidence level. 7. It made little difference whether the 95% or the 99% confidence level was consulted, as there were very few borderline cases. 8. The chance of detecting deviations from Michaelis-Menten kinetics, i.e. terms of at least second-order in a rate equation of degree n:n, n greater than 1, was estimated to be about 0.8. 9. The probability of the F test leading to a spurious result due to error in the data was found to be about 0.04. 10. The probability with which 4:4 mechanisms can lead to v([S]) plots with no, one, two or three turning points was computed, and it was established that there is a small but finite chance that the increase in degree that occurs in some mechanisms when ES in equilibrium EP interconversions are explicitly allowed for can be detected by the F test.     

Inhibition of enzymes by metal ion-chelating reagents. Theory and new graphical methods of study

1. The mechanism of inhibition of enzymes by metal ion-chelating reagents is discussed and equations derived. 2. Two distinct mechanisms are postulated and graphical methods are given for differentiating between them. 3. Where the metal ion is actually removed from the enzyme to form a co-ordination complex in solution, a procedure is described for obtaining the stability constant for metal-enzyme interaction, the number of metal ions involved and the stoicheiometry of metal ion-ligand interaction.     

Steady-state kinetic studies of the negative co-operativity and flip-flop mechanism for Escherichia coli alkaline phosphatase.

1. A study of variations in experimental error of velocity measurement with substrate concentration for alkaline phosphatase reveals that the standard error is not constant or strictly proportional to velocity, but obeys a more complex dependence. 2. By using an approach based on error estimates at each individual substrate concentration, we show that the double-reciprocal plots in general are curved, necessitating a high-degree rate equation. The curves are analysed according to a recent classification of possible curve shapes for the 3:3 function, which is shown to be the lowest-degree rate equation satisfying the experimental data. 4. Other workers have supposed the enzyme to follow Michaelis-Menten kinetics, and it is shown that this assumption is approximately true at low temperatures in the absence of phosphate. 5. A study of the effects of phosphate concentration, pH and temperature on the kinetics shows that there is a gradual alteration in curve shape with these experimental variables, resulting in an apparent reduction in degree under certain special conditions, and particularly at low temperature. 6. It is shown that the steady-state kinetics do not require a flip-flop or half-of-sites reactivity mechanism as claimed, and a mechanism is proposed, a rate equation calculated and an analysis attempted. 7. An analysis of the product-inhibition effects for a linked two-sited Uni Bi enzyme is given. Alterations of asymptotic double-reciprocal slopes and limiting (1/nu) intercepts with products is discussed, and it is shown how the theory of product inhibition can be extended to complex kinetic situations to extract information as to molecular mechanism. 8. Deviations from Michaelis-Menten kinetics are expressed in terms of the magnitude of the appropriate Sylvester resultants.     

The determination of positive and negative co-operativity with allosteric enzymes and the interpretation of sigmoid curves and non-linear double reciprocal plots for the MWC and KNF models

(i) It is proved that only four independent constants can ever be obtained by extrapolation procedures applied to non-hyperbolic steady-state or binding data, (ii) Analysis of the algebraic graphs $\text{y}\text{x}$, $\text{(1/y)}\text{(1/x)}$, $\text{y}\text{(}\text{y}\text{x}\text{)}$ and ($\text{x}\text{y}\text{)/x}$ is shown to require a knowledge of the sign of six curve shape determinants. In each case, the sign is a necessary and sufficient condition for a specific curve shape feature, (iii) The precise graphical effect of positive and negative co-operativity then requires the definition of two reference curves, the osculating hyperbola at zero substrate concentration, OH(0), and the osculating hyperbola at infinite substrate concentration OH(∞). These are better first order approximations than the Hill equation, (iv) Rules for determining unambiguously the sign of initial, final and overall co-operativity coefficients by inspection of non-hyperbolic binding curves are then possible, (v) These rules require that saturation data for:

$\text{y=}\text{∑}\text{i=1}\text{n}\text{a}{}_{\mathrm{i}}\text{x}{}^{\mathrm{i}}\text{∑}\text{i=0}\text{n}\text{β}{}_{\mathrm{i}}\text{x}{}^{\mathrm{i}}$

be fitted by computer for low concentrations to the hyperbola:

$\text{OH}\text{(o)=}\text{(}\text{-a}{}_1{}^2\text{ψ}{}_{11}{}^{20}\text{)x}\text{[(}\text{-a}{}_1\text{β}{}_0\text{ψ}{}_{11}{}^{20}\text{)+x]}$

while regression of high substrate concentration data is to:

$\text{OH}\text{(∞)=}\text{(}\text{a}{}_{\mathrm{n}}\text{β}{}_{\mathrm{n}}\text{)x}\text{[(}\text{φ}{}_{\mathrm{n,n-1}}\text{a}{}_{\mathrm{n}}\text{β}{}_{\mathrm{n}}\text{)+x]}$

. Comparisons of the best fit pseudo-kinetic constants then gives the type of co-operativity present in an unambiguous way with no assumptions as to molecular mechanism, (vi) These rules are then applied to the MWC and KNF allosteric models of ligand binding and the constraints necessary for specific curve shape effects are given, (vii) The graphical expression of positive or negative final co-operativity depends only on events at high substrate concentration but overall and initial co-operativities produce specific geometric effects depending upon the difference between behaviour of saturation data at both extremes of concentration, (viii) This apparent anomaly is explained by a discussion of the relationships between the osculating hyperbolae, the theoretical parent hyperbola and the Hill plot asymptotes.     

Human placental diamine oxidase. Improved purification and characterization of a copper- and manganese-containing amine oxidase with novel substrate specificity.

1. Isoelectric focusing studies of human placental diamine oxidase showed the pI value of the active enzyme to be 6.5. This information was used in modifying the enzyme purification by incorporating column chromatography on DEAE-Sephadex with ionic strength and pH gradient elution and this, together with affinity chromatography on concanavalin A--Sepharose, gave a highly purified preparation, with a specific activity of 7.0 units/mg. 2. The enzyme gave the expected stoicheiometry with p-dimethylaminomethylbenzylamine as substrate (Keq. 2700) and also oxidized [8-arginine]vasopressin, [8-lysine]vasopressin, collagen and tropocollagen. Polyacrylamide gel slices showed identical migration of diamine-oxidizing and [8-lysine]vasopressin-oxidizing activity. 3. The molecular weight, determined by ultracentrifugation, sodium dodecyl sulphate/polyacrylamide-gel electrophoresis, variable polyacrylamide-gel electrophoresis and Sephadex G-200 column chromatography, was estimated to be approx. 70000. 4. E.s.r. spectroscopy showed that copper and manganese were present in the purified enzyme. This result was confirmed by atomic absorption spectroscopy, which indicated a stoicheiometry for copper and manganese of approx. 1.0 and 1.2g-atom respectively/70000mol.wt. unit. 5. The e.s.r. spectral intensity did not decrease nor did the spectral line shape change when excess of p-dimethylaminomethylbenzylamine was added to the enzyme. 6. Addition of K13CN to the enzyme eliminated the copper e.s.r. signal without affecting the manganese signal. 7. The placental enzyme therefore appears to differ from other amine oxidases in terms of its metal cofactor requirement, molecular weight and substrate specificity, and possible roles in vivo for this enzyme are discussed.     

Brucella abortus strain RB51 vaccination in elk. I. Efficacy of reduced dosage

Bovine brucellosis is a serious zoonotic disease affecting some populations of Rocky Mountain elk (Cervus elaphus nelsoni) and bison (Bison bison) in the Greater Yellowstone Area, USA. The fear that elk and/or bison may spread Brucella abortus to livestock has prompted efforts to reduce or eliminate the disease in wildlife. Brucella abortus strain RB51 (RB51) vaccine has recently been approved for use in cattle. Unlike strain 19 vaccine, RB51 does not cause false positive reactions on standard brucellosis serologic tests. If effective, it may become the vaccine of choice for wildlife. In February 1995, 45 serologically negative female elk calves were trapped and taken to the Sybille Wildlife Research and Conservation Education Unit near Wheatland, Wyoming, USA. In May 1995, 16 of these elk calves were hand-vaccinated with 1 x 10(9) colony forming units (CFU) of RB51, 16 were vaccinated with 1 x 10(8) CFU RB51 by biobullet, and 13 were given a saline placebo. The elk were bred in fall of 1996 and they were challenged with 1 x 10(7) CFU of B. abortus strain 2308 by intraconjunctival inoculation in March 1997. Thirteen (100%) control elk aborted, 14 (88%) hand-vaccinated elk aborted, and 12 (75%) biobullet vaccinated elk aborted or produced nonviable calves. These results suggest that a single dose of 1 x 10(8) to 1 x 10(9) CFU RB51 does not provide significant protection against B. abortus induced abortion in elk. However, the vaccine appears to be safe at this dose and additional study may reveal a more effective RB51 vaccine regimen for elk.