Deviations from Michaelis-Menten kinetics. Computation of the probabilities of obtaining complex curves from simple kinetic schemes.

1. It is possible to calculate the intrinsic probability associated with any curve shape that is allowed for rational functions of given degree when the coefficients are independent or dependent random variables with known probability distributions. 2. Computations of such probabilities are described when the coefficients of the rational function are generated according to several probability distribution functions and in particular when rate constants are varied randomly for several simple model mechanisms. 3. It is concluded that each molecular mechanism is associated with a specific set of curve-shape probabilities, and this could be of value in discriminating between model mechanisms. 4. It is shown how a computer program can be used to estimate the probability of new complexities such as extra inflexions and turning points as the degree of rate equations increases. 5. The probability of 3 : 3 rate equations giving 2 : 2 curve shapes is discussed for unrestricted coefficients and also for the substrate-modifier mechanisms. 6. The probability associated with the numerical values of coefficients in rate equations is also calculated for this mechanism, and a possible method for determining the approximate magnitude of product-release steps is given. 7. The computer programs used in the computations have been deposited as Supplement SUP 50113 (21 pages) with the British Library Lending Division, Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies can be obtained on the terms indicated in Biochem, J. (1978) 169, 5.     

Kinetics of galactose uptake by perfused rat livers: applicability of a family of models

Mathematical models for general substrate uptake mechanisms in the liver have been used to describe the kinetics of galactose removal. In this study sets of galactose uptake rates to galactose concentration relations obtained by perfusion of ten livers of 200 g rats were examined. The rate of galactose uptake (ν) was related to the galactose concentration in the sinusoids (calculated as the logarithmic mean of in- and outlet concentration, ?). In all experiments a saturation pattern emerged, but the resulting 1/ν, versus $\text{1/}\text{c}\text{?}$ plots were all markedly convex, discarding simple Michaelis-Menten kinetics. Data were therefore examined in the light of a family of kinetic models, including the following modifications: substrate inhibition, porto-systemic shunting, and allosterism. The kinetic constants were assessed by iterative procedures, aiming at linearization of the double reciprocal plots. The two latter models were found to fit the experimental data by entering a shunting of 61% of the hepatic blood flow, or two active sites, respectively. Since this degree of shunting is improbable the results speak in favour of allosterism. The work gives an example of whole-liver kinetic considerations when simple Michaelis-Menten is insufficient.     

The probability that complex enzyme kinetic curves can be caused by activators or inhibitors

Numerous chemical compounds are known that alter the rate of conversion of substrates into products in enzyme-catalysed reactions by interacting with the enzyme rather than substrates. Where this takes place in such a way that the effect is reversible on removing the compound, say by dialysis, and where the compound is unchanged chemically by the enzyme system, we refer to such a compound as a modifier. So protons, inorganic salts, activators, inhibitors or even specific allosteric effectors would all be modifiers, and any chemically reasonable kinetic scheme that is proposed to account for such effects is referred to as modifier mechanism. Three versions of a modifier mechanism of enzyme action are studied. The implicit representation is 2:2 in [S] (with α₀=0) and 2:2 in [M] (with α₀≠0), and this is a short-hand scheme for the minimum chemical formulation, the explicit one, involving discrete ES and EP species, which is 2:2 in [S] (with α₀=0) and 3:3 in [M] (with α₀≠0). If m extra steps are allowed between interconversion of ES and EP species, the degree of the rate equation remains 2:2 in [S] (with α₀=0), but increases to degree (m+3):(m+3) in modifier (with α₀≠0). It is proved that this increase in degree is genuine and that highly complex v([M]) (i.e. v-versus-[M]) curves can occur. Computation of the probabilities of the five possible double-reciprocal plots in 1/v versus 1/[S] show that all of these formulations of the modifier mechanism give similar probabilities, and these are characteristic for the mechanism and quite distinct from the intrinsic curve-shape probabilities. It is also established that the probabilities of alternative complex v([M]) plots are similar for the various formulations, and again the probabilities of the allowed complex curves for the mechanism are quite distinct from the instrinsic probabilities of the ten possible v([M]) curves for a 2:2 function (with α₀≠0). The computer studies reported lead to several conclusions about the probability of modifiers leading to inhibition or activation or causing changes in v([S]) curve shapes, and suggest that differentiation between model mechanisms may be facilitated by knowledge of the intrinsic curve-shape probabilities for the appropriate degree rational function and the characteristic way that this is altered by specific mechanisms. It is shown that, although in some instances new curve-shape complexities are possible when schemes are considered that allow for interconversion of ES and EP species, these are highly improbable and, for theoretical purposes, schemes formulated with node compression provide good approximations to the more complicated explicit schemes. By node compression we refer to the procedure whereby enzyme kinetic schemes are simplified by replacing sequences of steps such as ESX₁X₂...EP... by a single step... ES/EP... that does not formally recognize the existence of the intermediate species. We show that the modifier mechanism studied is one where this process alters the form of the rate equation.     

Deriving the rate equations for product inhibition patterns in bisubstrate enzyme reactions

In this work, the full rate equations for 17 completely reversible bisubstrate enzyme kinetic mechanisms, with two substrates in the forward and two in the reverse direction, have been presented; among these are rapid equilibrium, steady-state, and mixed steady-state and rapid equilibrium mechanisms. From each rate equation eight product inhibition equations were derived, four for the forward and four for the reverse direction. All the corresponding product inhibition equations were derived in full; thus a total of 17 × 8 = 136 equations, were presented. From these equations a list of product inhibition patterns were constructed and presented in a tabular form, both for the primary plots (intercept effects) and the secondary plots (slope effects).

The purpose of this work is to help investigators in practical work, especially biologists working with enzymes, to choose quickly an appropriate product inhibition pattern for the identification of the kinetic mechanism. The practical application of above product inhibition analysis was illustrated with three examples of yeast alcohol dehydrogenase-catalyzed reactions.     

Activation of the β-galactoside transport system in escherichia coli ML-308 by n-alkanols. Modification of lipid-protein interaction by a change in bilayer fluidity

The net rate of transport of $\text{o-}\text{nitrophenyl}\text{-β-}\text{d}\text{-}\text{galactopyranoside}$ by Escherichia coli ML-308 is increased at temperatures below the apparent phase transition of the lipid bilayer in the presence of n-alkanols. The degree of activation produced is determined both by the concentration and chain length of the n-alkanol used. At low concentrations n-alkanols “activate” transport, but do not cause either cell lysis or denaturation of β-galactosidase.Arrhenius plots of the kinetic constants for transport indicate the K_m shows discontinuity with increasing temperature, while the slope for V changes only gradually. The presence of 10.5 mM n-hexanol increases the value of both K_m and V at low temperature. A comparison of these data to those obtained at a single substrate concentration (1.85 mM o-nitrophenyl-β-d-galactopyranoside) suggests the biphasic behavior of Arrhenius plots at that concentration may be attributed to changes in the K_m for the transport process.     

The effect of palmitoyl-CoA binding to albumin on the apparent kinetic behavior of carnitine palmitoyltransferase I

Substrate saturation plots of carnitine palmitoyltransferase I activity from isolated rat liver mitochondria vs. palmitoyl-CoA concentration in the presence of bovine serum albumin have been reported to yield sigmoidal kinetics. Under identical assay conditions we have confirmed these observations as reflected by nonlinear Lineweaver-Burke plots (1/ν₁ vs. 1|S|) and an average Hill coefficient of n_app. = 1.98 ± 0.09 (Mean±S.E. from four separate experiments). For these determinations the enzyme activity was plotted against the total [palmitoyl-CoA] in the presence of 0.13% bovine serum albumin. Utilizing the total [palmitoyl-CoA] to determine the kinetic properties of carnitine palmitoyltransferase I would be valid only if the relationship between total and free [palmitoyl-CoA] was linear, which is not the case as we have previously shown. When carnitine palmitoyltransferase I substrate saturation kinetics were reanalyzed using the previously determined free [palmitoyl-CoA]'s, the plots revealed a shift to standard hyperbolic kinetics. This observation was confirmed by an average Hill coefficient of n_app. = 1.04 ± 0.10 (Mean±S.E.) and linear Lineweaver-Burke plots. The double-reciprocal plots from these analyses yielded an average S_0.5 of 2.55 ± 0.82 μM(Mean±S.E.) palmitoyl-CoA and V_max of 19.69 ± 5.48 nmol/min per mg protein. These studies clearly demonstrate the importance of defining the free [palmitoyl-CoA] when analyzing the kinetics of carnitine palmitoyltransferase I in the presence of bovine serum albumin.     

An error analysis for two-state protein-folding kinetic parameters and phi-values: progress toward precision by exploring pH dependencies on Leffler plots

The interpretation of φ-values has led to an understanding of the folding transition state ensemble of a variety of proteins. Although the main guidelines and equations for calculating φ are well established, there remains some controversy about the quality of the numerical values obtained. By analyzing a complete set of results from kinetic experiments with the SH3 domain of α-spectrin (Spc-SH3) and applying classical error methods and error-propagation formulas, we evaluated the uncertainties involved in two-state-folding kinetic experimental parameters and the corresponding calculated φ-values. We show that kinetic constants in water and m values can be properly estimated from a judicious weighting of fitting errors and describe some procedures to calculate the errors in Gibbs energies and φ-values from a traditional two-point Leffler analysis. Furthermore, on the basis of general assumptions made with the protein engineering method, we show how to generate multipoint Leffler plots via the analysis of pH dependencies of kinetic parameters. We calculated the definitive φ-values for a collection of single mutations previously designed to characterize the folding transition state of the α-spectrin SH3 domain. The effectiveness of the pH-scanning procedure is also discussed in the context of error analysis. Judging from the magnitudes of the error bars obtained from two-point and multipoint Leffler plots, we conclude that the precision obtained for φ-values should be ∼25%, a reasonable limit that takes into account the propagation of experimental errors.     

Cooperative denaturation kinetics of homogeneous polymers

The kinetics of denaturation of a homogeneous, helical biopolymer with nearest neighbor interactions are described, using a kinetic Ising model in which the configuration of its neighbors dictates the transition probability for a single residue in the chain. The actual kinetics that are simulated using Monte Carlo techniques are compared with the results of analytical kinetic equations for the fraction of helix, <s>, generated using the mean-field approximation. This mean-field rate equation is expanded as a hierarchy of terms that characterize the nature of rate constants for interacting systems. The first term in the expansion is first order in <s> and varies linearly with the interaction energy. Subsequent rate terms involve higher powers of <s> and demonstrate the need for nonlinear equations in systems with larger interaction energies. Both the simulations and the mean-field approximation show an intrinsic induction period for the single-step kinetic process. They also yield an apparent first-order rate constant that changes as the reaction proceeds. However, only the simulated kinetics yield ordered regions of chain and a nonzero, nearest-neighbor correlation function.     

A Strategy to Calculate the Patterns of Nutrient Consumption by Microorganisms Applying a Two-Level Optimisation Principle to Reconstructed Metabolic Networks

Bacterial responses to environmental changes rely on a complex network of biochemical reactions. The properties of the metabolic network determining these responses can be divided into two groups: the stoichiometric properties, given by the stoichiometry matrix, and the kinetic/thermodynamic properties, given by the rate equations of the reaction steps. The stoichiometry matrix represents the maximal metabolic capabilities of the organism, and the regulatory mechanisms based on the rate laws could be considered as being responsible for the administration of these capabilities. Post-genomic reconstruction of metabolic networks provides us with the stoichiometry matrix of particular strains of microorganisms, but the kinetic aspects of in vivo rate laws are still largely unknown. Therefore, the validity of predictions of cellular responses requiring detailed knowledge of the rate equations is difficult to assert. In this paper, we show that by applying optimisation criteria to the core stoichiometric network of the metabolism of Escherichia coli, and including information about reversibility/irreversibility only of the reaction steps, it is possible to calculate bacterial responses to growth media with different amounts of glucose and galactose. The target was the minimisation of the number of active reactions (subject to attaining a growth rate higher than a lower limit) and subsequent maximisation of the growth rate (subject to the number of active reactions being equal to the minimum previously calculated). Using this two-level target, we were able to obtain by calculation four fundamental behaviours found experimentally: inhibition of respiration at high glucose concentrations in aerobic conditions, turning on of respiration when glucose decreases, induction of galactose utilisation when the system is depleted of glucose and simultaneous use of glucose and galactose as carbon sources when both sugars are present in low concentrations. Preliminary results of the coarse pattern of sugar utilisation were also obtained with a genome-scale E. coli reconstructed network, yielding similar qualitative results.     

On the detection of homotropic effects in enzymes of low co-operativity. Application to modified aspartate transcarbamoylase

In contrast with the ease of observing heterotropic effects in allosteric enzymes of low co-operativity, the detection of homotropic effects is often difficult. As a consequence, erroneous conclusions about the uncoupling of homotropic and heterotropic effects can result unless sensitive techniques are used for analyzing the kinetic data. Simulations of experiments as well as actual measurements on the allosteric enzyme, aspartate transcarbamoylase, of Escherichia coli and some of its modified forms, were performed in attempts to develop stringent diagnostic procedures for the detection of homotropic effects in enzymes of low co-operativity. The analyses show that direct saturation plots (velocity versus substrate concentration), double reciprocal plots, and Hill plots yield misleading results in that the co-operativity known to be present is not observed. In contrast, Eadie plots (velocity/substrate concentration versus velocity) are much more sensitive in revealing homotropic effects. Since the observed co-operativity depends on both the allosteric equilibrium constant, L, and the number of active sites, n, simulations were performed on the effect of those parameters. The maxima in the Eadie plots increased as L was lowered and conversely the maxima decreased as n was reduced. These changes were confirmed with a mutant aspartate transcarbamoylase which had the same specific activity as the wild-type enzyme and a lower value of L, and also with a hybrid enzyme containing fewer active sites and the same L value. Analogous experiments on nitrated aspartate transcarbamoylase derivatives of decreasing activity showed that Eadie plots were of value in distinguishing between the changes in L and n values resulting from the inactivation. Data from the literature were analyzed in the form of Eadie plots and in all cases homotropic effects were readily detectable for aspartate transcarbamoylase derivatives previously claimed to be devoid of co-operativity.     

Linear n-compartment catenary models: Formulas to describe tracer amount in any compartment and identification of parameters from a concentration-time curve

Resolution of kinetic equations and parameter identification are discussed for n-compartment linear catenary models with elimination allowed from any compartment. For a given input, general formulas are derived to describe the tracer amount in any compartment as a function of the model parameters. Conversely, explicit procedures are given to identify the model parameters when the concentration-time curve is known in one arbitrary compartment, the tracer being injected into the same compartment. In this inverse problem, the solution is not unique: the model transfer rate constants can only be localized in a finite set of intervals.     

The effect of chromosomal polytenization on the rates of RNA synthesis and decay in the salivary glands of the blowfly, Calliphora erythrocephala

The rates of total RNA synthesis and accumulation have been measured in the polytenic salivary gland cells of the blowfly, Calliphora erythrocephala, by three methods: (1) injecting larvae with [2-³H]adenosine and determining its flow into the cellular ATP pool and RNA, (2) measuring the increase in glandular RNA optically, and (3) measuring the rate of flow of ATP out of the cellular pool. The size of the steady-state pool of rapidly turning over RNA and its half-life, were calculated from these kinetic data and, also, by an independent measurement of the steady-state content of nuclear RNA. These parameters were compared at a number of developmental stages which differed in degree of chromosomal polytenization. The results indicate that these polytenic cells synthesize RNA at a rate approximately 10³ times those of other diploid eukaryotic cells. This rate is independent of the increase in chromosomal polyteny that accompanies larval development. Approximately 67% of the newly synthesized salivary gland RNA is an unstable component with an average first-order half-life of 20–25 min. The remainder is a long-lived species with an estimated average first-order half-life of about 30 hr.     

An analysis of learned kin recognition in hymenoptera

A model is developed to explain data on the probability with which a strange conspecific hymenopteran female will be accepted into a group of sisters. The analysis is based on a genetic labeling system (primarily odors) of m loci and n_i equally frequent alleles at the i-th locus (i = 1,…, m). Three recognition mechanisms are considered (viz: genotype recognition; foreign-label rejection; and habituated-label acceptance) where all three mechanisms depend on individuals learning the labels represented in their group. The probability with which non-kin will be accepted into large sibling group is calculated for a number of different labeling systems. These different labeling systems are compared and a comparision is also made between the three recognition mechanisms mentioned above. A general expression is then derived, in terms of the number of loci and alleles in the labeling system and the size of the sibling group, for the probability with which “strang” sisters are accepted into a group of sisters with whom they have had no prior contact. These results are applied to existing data on the primitively eusocial sweat bee Lasioglossum zephyrum and the present indication is that recognition in L. zephyrum can be modeled by foreign-label rejection with a genetic labeling system of four or five loci with eight to ten alleles in total (i.e. each locus will have two or at most three alleles).     

Maintenance of lower proportions of (n − 6) eicosanoid precursors in phospholipids of human plasma in response to added dietary (n − 3) fatty acids

Competition between the (n ? 3) and (n ? 6) types of highly unsaturated fatty acids can diminish the abundance of (n ? 6) eicosanoid precursors in a tissue, which in turn can diminish the intensity of tissue responses that are mediated by (n ? 6) eicosanoids. The mixture of 20- and 22-carbon highly unsaturated fatty acids maintained in the phospholipids of human plasma is related to the dietary intake of 18:2 (n ? 6) and 18:3 (n ?3) by empirical hyperbolic equations in a manner very similar to the relationship reported for laboratory rats (Lands, W.E.M., Morris, A. and Libelt, B. (1990) Lipids 25, 505–516). Analytical results from volunteers ingesting self-selected diets showed an inter-individual variance for the proportion of (n ? 6) eicosanoid precursors in the fatty acids of plasma phospholipids of about 5%, but the variance among multiple samples taken from the same individual throughout the day was less (about 3%), closer to the experimental variance of the analytical procedure (about 1%). The reproducibility of the results makes it likely that analysis of fatty-acid composition of plasma lipids from individuals will prove useful in estimating the diet-related tendency for severe thrombotic, arthritic of other disorders that are mediated by (n ? 6) eicosanoids. Additional constants and terms were included in the equations to account for the effects of 20- and 22-carbon highly unsaturated (n ? 3) fatty acids in the diet. A lower constant for the 20- and 22-carbon (n ? 3) fatty acids compared to that for the 18-carbon (n ? 3) fatty acid in decreasing the ability of dietary 18:2 (n ? 6) to maintain 20:4 (n ? 6) in tissue lipids confirmed the greater competitive effectiveness of the more highly unsaturated n ? 3 fatty acids in the elongation/ desaturation process. Also, a lower constant for direct incorporation of 20-carbon fatty acids of the n ? 6 vs. the n ? 3 type indicated a greater competitive effectiveness of 20:4 (n ? 6) relative to 20:5 (n ? 3) in reesterification after release from tissue lipids. The equations may be used in reverse to estimate the dietary intakes of the (n ? 3) and (n ? 6) fatty acids by using the composition of the fatty acids that had been maintained in plasma lipids.     

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.     

Allosteric and related phenomena: An analysis of sigmoid and non-hyperbolic functions

The graphs of rational polynomial functions are of considerable importance in enzyme kinetics but a full analysis of the turning points and inflexions is only possible for 2 : 2 functions due to the complexity of the first and second derivatives for 3 : 3 and higher degree functions. This paper describes a simple method which can be applied to rational functions of any degree in order to discover the relationship between the coefficients necessary for the curve to have an initial sigmoid inflexion and a final inflexion (which implies a maximum in the graph). This technique is then applied to the current allosteric models. In addition, a full analysis of the 2 : 2 function with a maximum is given together with a method of separating this type of behaviour (partial substrate inhibition) from the 1 : 2 function (dead-end substrate inhibition).     

Deriving the rate equations for product inhibition patterns in bisubstrate enzyme reactions

In this work, the full rate equations for 17 completely reversible bisubstrate enzyme kinetic mechanisms, with two substrates in the forward and two in the reverse direction, have been presented; among these are rapid equilibrium, steady-state, and mixed steady-state and rapid equilibrium mechanisms. From each rate equation eight product inhibition equations were derived, four for the forward and four for the reverse direction. All the corresponding product inhibition equations were derived in full; thus a total of 17 x 8 = 136 equations, were presented. From these equations a list of product inhibition patterns were constructed and presented in a tabular form, both for the primary plots (intercept effects) and the secondary plots (slope effects). The purpose of this work is to help investigators in practical work, especially biologists working with enzymes, to choose quickly an appropriate product inhibition pattern for the identification of the kinetic mechanism. The practical application of above product inhibition analysis was illustrated with three examples of yeast alcohol dehydrogenase-catalyzed reactions.     

Ambidentate H-bonding by heme-bound NO: structural and spectral effects of –O versus –N H-bonding

Resonance Raman studies have uncovered puzzling complexities in the structures of NO adducts of heme proteins. Although CO adducts of heme proteins obey well-behaved anti-correlations between Fe–C and C–O stretching frequencies, which reflect changes in backbonding induced by distal H-bonding residues, the corresponding NO data are scattered. This scatter can be traced to distal influences, since protein-free NO–hemes do show well-behaved anti-correlations. Why do distal effects produce irregularities in νFeN/νNO plots but not in νFeC/νCO plots? We show via density functional theory (DFT) computations on model systems that the response to distal H-bonding differs markedly when the NO acceptor atom is N versus O. Backbonding is augmented by H-bonding to O, but the effect of H-bonding to N is to weaken both N–O and N–Fe bonds. The resulting downward deviation from the νFeN/νNO backbonding line increases with increasing H-bond strength. This effect explains the deviations observed for a series of myoglobin variants, in which the strength of distal H-bonding is modulated by distal pocket residue substitutions. Most of the data follow a positive νFeN/νNO correlation with the same slope as that calculated for H-bonding to N. Such deviations are not observed for CO adducts, because the CO π* orbital is unoccupied, and serves as a delocalized acceptor of H-bonds. H-bonding to N primes NO–heme for reduction to the HNO adduct, a putative intermediate in NO-reducing enzymes.