Identification of enzyme inhibitory mechanisms from steady-state kinetics

doi:10.1016/j.biochi.2011.05.031

Biochimie

Volume 93, Issue 9, September 2011, Pages 1623-1629

https://doi.org/10.1016/j.biochi.2011.05.031 Get rights and content

Abstract

Enzyme inhibitors are used in many areas of the life sciences, ranging from basic research to the combat of disease in the clinic. Inhibitors are traditionally characterized by how they affect the steady-state kinetics of enzymes, commonly analyzed on the assumption that enzyme-bound and free substrate molecules are in equilibrium. This assumption, implying that an enzyme-bound substrate molecule has near zero probability to form a product rather than dissociate, is valid only for very inefficient enzymes. When it is relaxed, more complex but also more information-rich steady-state kinetics emerges. Although solutions to the general steady-state kinetics problem exist, they are opaque and have been of limited help to experimentalists.

Here we reformulate the steady-state kinetics of enzyme inhibition in terms of new parameters. These allow for assessment of ambiguities of interpretation due to kinetic scheme degeneracy and provide an intuitively simple way to analyze experimental data. We illustrate the method by concrete examples of how to assess scheme degeneracy and obtain experimental estimates of all available rate and equilibrium constants. We suggest simple, complementary experiments that can remove ambiguities and greatly enhance the accuracy of parameter estimation.

Highlights

► Ambiguous text book identifications of enzyme inhibitory mechanism are resolved. ► Relaxing of the substrate equilibration assumption reveals new inhibition features. ► 3 classes of rapid inhibitors are described of which one is also enzyme activating. ► New variables greatly simplify interpretation of experimental observations. ► The precision of parameter estimations from steady-state is characterized.

Introduction

Enzyme inhibitors of natural origin or human design are ubiquitous in the life sciences as tools for basic research and in commercial applications. They are used in mechanistic studies of individual enzymes, cell biological studies of metabolism and signaling pathways, toxicology and pharmaceutical studies on humans. Today, extensive research is aimed at integrating previously separated levels of biology [1]. There have, for instance, been numerous attempts to clarify the behavior of whole cells from kinetic information of key enzymes and their inhibitors, e.g., [2], [3], [4], [5], [6]. The success of such studies depends on high quality data and realistic modeling of enzyme kinetics in the test-tube as well as in the living cell.

There are established techniques for studies of fast kinetics of molecular ensembles and, more recently also for single molecules [7]. At the same time, steady-state enzyme kinetics remains wide spread and its results are often interpreted in terms of the Michaelis–Menten formalism [8]. Its validity depends on particular assumptions (cf. Supplementary information). When these are fulfilled, the formalism allows for simple, quantitative studies of enzyme efficiency, provides kinetic information of enzyme-binding inhibitors and suggests mechanistic interpretations of inhibitor action.

In textbooks of biochemistry or enzyme kinetics a set of linear plots, often of Lineweaver–Burke (LB) type [9] (Fig. 1E–H) but also of other types [10], [11], [12], [13], [14], is commonly used to discriminate between different sorts of enzyme inhibitors as shown in Fig. 1A–D (e.g., [15], [16]).

We have learnt that plots like the one in Fig. 1E define an inhibitor as competitive, meaning that substrate and inhibitor cannot be simultaneously bound to the enzyme (Fig. 1A). In the non-competitive, mixed and uncompetitive cases inhibitor binding is unaffected by, affected by and dependent on substrate binding to the enzyme, respectively. Thus, it is taught that plots like the ones in Fig. 1F–H define a non-competitive (Fig. 1B), mixed (Fig. 1C) and uncompetitive (Fig. 1D) inhibitor, respectively. However, all plots in Fig. 1F–H are drawn for a non-competitive inhibitor (Fig. 1B). What is the explanation for this apparent discrepancy between textbook wisdom and reality?

The reason for this apparent paradox is the above mentioned (often tacit) textbook assumption that free and enzyme-bound substrate are in equilibrium; a condition fulfilled in Fig. 1F but not in Fig. 1G–H. Fulfillment of this condition means that the K_M-value, (q₁ + k_c)/k₁, for the inhibitor is equal to the dissociation constant, K_I = q₁/k₁. This implies that the rate constant, q₁, of substrate release from the enzyme–substrate complex, ES, is much larger than the rate constant, k_c, of product formation, i.e., q₁ >> k_c. Accordingly, the efficiency, k_cat/K_M, of the enzyme [16] is far below its maximal asymptote, k₁. In some textbooks, this restrictive assumption is stated (e.g., [15], [17], [18]), but in others not (e.g., [16], [19]). When the equilibrium assumption is relaxed, LB plots relating to the schemes in Fig. 1B–C may look very different, depending on the degree to which ES is equilibrated with E + S, as illustrated in Fig. 1F–H.

In Fig. 1G–H ES is not equilibrated with E + S, but the kinetics is of Michaelis–Menten type with linear LB plots. The reason is that in those cases free and enzyme-bound inhibitor are close to equilibrium at all substrate and inhibitor concentrations [18], [20], [21]. In spite of their simplicity, these plots cannot be interpreted unambiguously without additional information as explained below.

The steady-state flows in Fig. 2C are not of Michaelis–Menten type. Their algebraic expressions [22], [23] are opaque and have had little impact on the interpretation of experiments.

The starting point of this work is the identification of a set of parameters, the “Qβ-set”, that allows for an intuitive description of how steady-state flows for product formation are affected by not competitive enzyme inhibitors. We clarify the origin of non-linearity in the LB plots, identify ambiguities of interpretation of experiments due to degeneracy of the scheme in Fig. 2C and show how to extract the maximum possible information from experiments. We illustrate how the general steady-state solutions approach Michaelis–Menten behavior for increasingly rapid enzyme–inhibitor interactions, and use this limiting case to discuss how ambiguities in the assessment of inhibitors as non-competitive, uncompetitive or mixed can be removed. We also explain how a subclass of not competitive enzyme inhibitors can function as activators at low and as inhibitors only at high concentration.

Section snippets

Substrate and inhibitor in equilibrium with enzyme – the text book case

The general scheme in Fig. 2C simplifies to that in Fig. 2A when q₁ >> k_c, so that there is close to zero probability of an enzyme-bound substrate to be transformed to product rather than dissociate from the enzyme. In this limit both substrate and inhibitor are in equilibrium with the enzyme (Fig. 2A). The ratio between the total steady-state rate of product formation, v, and the total enzyme concentration, e₀, is given by [18] $\frac{v}{e_{0}} = \frac{[S]}{\frac{1}{k_{c}} (1 + \frac{[I]}{K_{I 2}}) [S] + \frac{1}{R} (1 + \frac{[I]}{K_{I 1}})}$ Inversion of both sides of Eq.

Discussion

Applications of steady-state kinetics to classify enzyme inhibitors are normally based on textbook formulations, which are tacitly [16], [19] or explicitly [15], [17], [18] based on an assumption of substrate–enzyme equilibration. How to validate this assumption and how to evaluate experimental data when the assumption is false are two rarely discussed topics. A major aim of the present work is to facilitate evaluation of experiments, in which the textbook assumption of substrate–enzyme

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Research paperIdentification of enzyme inhibitory mechanisms from steady-state kinetics

Abstract

Highlights

Introduction

Section snippets

Substrate and inhibitor in equilibrium with enzyme – the text book case

Discussion

FEBS Lett.

Biochimie

Biochim. Biophys. Acta

J. Biol. Chem.

J. Biol. Chem.

J. Biol. Chem.

J. Biol. Chem.

J. Biol. Chem.

J. Biol. Chem.

Drug efflux pump deficiency and drug target resistance masking in growing bacteria

Proc. Natl. Acad. Sci. U S A

Erythromycin resistance by L4/L22 mutations and resistance masking by drug efflux pump deficiency

Embo J.

A metabolic model of human erythrocytes: practical application of the E-cell simulation environment

J. Biomed. Biotechnol.

Impact of genome reduction on bacterial metabolism and its regulation

Science

Single-biomolecule kinetics: the art of studying a single enzyme

Ann. Rev. Anal. Chem.

Research paper
Identification of enzyme inhibitory mechanisms from steady-state kinetics