Statistical analyses of enzyme inhibitor kinetics: a hierarchical model-dependent approach.

Inference of the nature of an enzyme inhibitor and calculations of inhibitor constants from linear plots are prone to be unreliable when experimental error is significant. This may be minimised by a method of statistical analysis using nonlinear regression methods and modelling using the variance ratio test of a hierarchy of models of inhibitor behaviour.     

Statistical kinetics of macromolecular dynamics

Fluctuations in biochemical processes can provide insights into the underlying kinetics beyond what can be gleaned from studies of average rates alone. Historically, analysis of fluctuating transmembrane currents supplied information about ion channel conductance states and lifetimes before single-channel recording techniques emerged. More recently, fluctuation analysis has helped to define mechanochemical pathways and coupling ratios for the motor protein kinesin as well as to probe the contributions of static and dynamic disorder to the kinetics of single enzymes. As growing numbers of assays are developed for enzymatic or folding behaviors of single macromolecules, the range of applications for fluctuation analysis increases. To evaluate specific biochemical models against experimental data, one needs to predict analytically the distribution of times required for completion of each reaction pathway. Unfortunately, using traditional methods, such calculations can be challenging for pathways of even modest complexity. Here, we derive an exact expression for the distribution of completion times for an arbitrary pathway with a finite number of states, using a recursive method to solve algebraically for the appropriate moment-generating function. To facilitate comparisons with experiments on processive motor proteins, we develop a theoretical formalism for the randomness parameter, a dimensionless measure of the variance in motor output. We derive the randomness for motors that take steps of variable sizes or that move on heterogeneous substrates, and then discuss possible applications to enzymes such as RNA polymerase, which transcribes varying DNA sequences, and to myosin V and cytoplasmic dynein, which may advance by variable increments.     

Markov chain methods in enzyme kinetics

An enzyme cluster is a system of enzymes attached to a membrane, whose spatial organization makes it possible for the product of one enzymatic reaction tobe directly available as a substrate of another reaction within the cluster. We show how to model enzyme clusters by Markov chains, and how to compute their overall reaction rate. As a by-product we prove a formula for the number of completed cycles per unit time in a Markov chain.     

The origins of enzyme kinetics

The equation commonly called the Michaelis–Menten equation is sometimes attributed to other authors. However, although Victor Henri had derived the equation from the correct mechanism, and Adrian Brown before him had proposed the idea of enzyme saturation, it was Leonor Michaelis and Maud Menten who showed that this mechanism could also be deduced on the basis of an experimental approach that paid proper attention to pH and spontaneous changes in the product after formation in the enzyme-catalysed reaction. By using initial rates of reaction they avoided the complications due to substrate depletion, product accumulation and progressive inactivation of the enzyme that had made attempts to analyse complete time courses very difficult. Their methodology has remained the standard approach to steady-state enzyme kinetics ever since.     

Selective neutrality and enzyme kinetics

This article appeals to a recent theory of enzyme evolution to show that the properties, neutral or adaptive, which characterize the observed allelic variation in natural populations can be inferred from the functional parameters, substrate specificity, and reaction rate. This study delineates the following relations between activity variables, and the forces—adaptive or neutral—determining allelic variation: (1) Enzymes with broad substrate specificity: The observed polymorphism is adaptive; mutations in this class of enzymes can result in increased fitness of the organism and hence be relevant for positive selection. (2) Enzymes with absolute substrate specificity and diffusion-controlled rates: Observed allelic variation will be absolutely neutral; mutations in this class of enzymes will be either deleterious or have no effect on fitness. (3) Enzymes with absolute or group specificity and nondiffusion-controlled rates: Observed variation will be partially neutral; mutants which are selectively neutral may become advantageous under an appropriate environmental condition or different genetic background. We illustrate each of the relations between kinetic properties and evolutionary states with examples drawn from enzymes whose evolutionary dynamics have been intensively studied. Received: 12 December 1996 / Accepted: 22 April 1997     

The kinetics of enzyme inactivation

The course of hydrolysis of β-glycerophosphate catalyzed by a group of different enzyme extracts, both with and without the addition of Mg, with and without preincubation of the enzyme, has been studied and the results discussed on the basis of a mathematical analysis. In all the extracts, it appears that two distinct and independently acting constituent enzymes—or perhaps “principles” of the same enzyme—are present, one acting much more rapidly but also more rapidly inactivated than the other. Storage in the refrigerator changes markedly the behavior of both constituents, though in different ways. There is evidence that in some cases an enzyme is limited in its hydrolytic “capacity” in the sense that after an enzyme molecule has decomposed a definite number of substrate molecules, it thereafter becomes entirely passive. Further, there is evidence, in the case of one extract, that the roles of catalytically more and less active constituents in the absence of Mg are reversed in its presence. Finally, a damped periodicity is found which indicates the presence of two factors of an unknown sort which influence and are influenced by the inactivation of the enzyme.     

Microscopic diffusion-reaction coupling in steady-state enzyme kinetics

The theory of diffusion-controlled processes is applied to describe the steady state of a reversible enzymatic reaction with special emphasis on the effects of enzyme saturation. A standard macroscopic steady-state treatment requires only that the average diffusional influx of substrate equals the net reaction flux as well as the average diffusional efflux of product. In contrast, the microscopic diffusion-reaction coupling used here takes properly into account the conditional concentration distributions of substrate and product: Only when the enzyme is unoccupied will there be a diffusional association flux; when the enzyme is occupied, the concentration distributions will relax towards their homogeneous bulk values. In this way the relaxation effects of the non-steady state will be constantly reoccurring as the enzyme shifts between unoccupied and occupied states. Thus, one is forced to describe the steady state as the weighted sum of properly time-averaged non-stationary conditional distributions. The consequences of the theory for an appropriate assessment of the parameters obtained in Lineweaver-Burk plots are discussed. In general, our results serve to justify the simpler macroscopic coupling scheme. However, considerable deviations between the standard treatment and our analysis can occur for fast enzymes with an essentially irreversible product release.     

A Markov chain occurring in enzyme kinetics

A certain Markov chain which was encountered by T. L. Hill in the study of the kinetics of a linear array of enzymes is studied. An explicit formula for the steady state probabilities is given and some conjectures raised by T. L. Hill are proved.     

Parameter estimation in approximate enzyme kinetics models

Enzyme kinetic modeling is based on a very specific approximation to a more detailed model; this approximation is often called the Michaelis-Menten approximation. Its success lies not only in an initial concentration difference that is set experimentally, but also in a certain difference among kinetic rate constants. Other differences among these constants lead to additional model approximations, often much simpler than the Michaelis-Menten approximation. Under the conditions where these alternate approximations apply, the Michaelis-Menten approximation would give misleading parameter estimates.