A versatile equation to describe reversible enzyme inhibition and activation kinetics: Modeling β-galactosidase and butyrylcholinesterase

Abstract

Current treatments for Alzheimer's disease involve inhibiting cholinesterases. Conversely, cholinesterase stimulation may be deleterious. Homocysteine is a known risk factor for Alzheimer's and vascular diseases and its active metabolite, homocysteine thiolactone, stimulates butyrylcholinesterase. Considering the opposing effects on butyrylcholinesterase of homocysteine thiolactone and cholinesterase inhibitors, understanding how these molecules alter this enzyme may provide new insights in the management of dementia. Butyrylcholinesterase does not strictly adhere to Michaelis–Menten parameters since, at higher substrate concentrations, enzyme activation occurs. The substrate activation equation for butyrylcholinesterase does not describe the effects of inhibitors or non-substrate activators. To address this, global data fitting was used to generate a flexible equation based on Michaelis–Menten principles. This methodology was first tested to model complexities encountered in inhibition by imidazole of β-galactosidase, an enzyme that obeys Michaelis–Menten kinetics. The resulting equation was sufficiently flexible to permit expansion for modeling activation or inhibition of butyrylcholinesterase, while accounting for substrate activation of this enzyme. This versatile equation suggests that both the inhibitor and non-substrate activator examined here have little effect on the substrate-activated form of butyrylcholinesterase. Given that butyrylcholinesterase inhibition can antagonize stimulation of this enzyme by homocysteine thiolactone, cholinesterase inhibition may have a role in treating Alzheimer and vascular diseases related to hyperhomocysteinemia.

Introduction

Neurodegenerative disorders such as Alzheimer's disease (AD), exhibit changes in the activities of cholinesterases that control levels of the cholinergic neurotransmitter acetylcholine [1], [2], [3]. Low levels of acetylcholine lead to cognitive and behavioural dysfunction and current symptomatic treatment for AD involves the use of drugs that inhibit cholinesterases [4]. Conversely, the stimulation of cholinesterase activity may be considered deleterious towards the integrity of the cholinergic system [5]. A number of lines of evidence have indicated that butyrylcholinesterase (BuChE, EC 3.1.1.8) may be an important enzyme in the functioning of the cholinergic system [6], [7], [8], [9], [10]. BuChE hydrolysis of choline esters has been found to be stimulated by the presence of homocysteine thiolactone, and this metabolite of homocysteine may be the active derivative of the amino acid that makes the latter a risk factor for dementia and vascular diseases [11], [12], [13], [14].

Considering the opposing effects on BuChE produced by homocysteine thiolactone and cholinesterase inhibitors [4], [5], [15], it is imperative to have a better understanding of how these molecules interact with this enzyme because of their potential importance in the management of disorders such as dementia.

Cholinesterases, although usually described using Michaelis–Menten kinetic parameters, do not strictly adhere to this model, which assumes that, at fixed enzyme concentration, catalysis occurs at a single maximum reaction rate [16]. This is not the case for cholinesterases because, at elevated substrate concentrations, substrate binding to an allosteric site on the enzyme results in modification of the catalytic rate. In acetylcholinesterase (EC 3.1.1.7), this allosteric binding results in inhibition [17]. For BuChE, allosteric substrate binding results in stimulation of catalytic activity [18]. The dependence of catalytic reaction rate on substrate-induced changes in cholinesterase activity presents a problem when modeling the effect of reversible inhibitors or stimulators on these enzymes.

Although an equation to describe the substrate activation of BuChE has been described (Eq. (1), Table 1) [19], this equation does not lend itself to the description of additional non-substrate reversible inhibition or activation events that affect this cholinesterase. The effect of inhibitors on enzymes that obey Michaelis–Menten kinetics has been extensively investigated. However, even with such enzymes, anomalous modes of inhibition occur that are not adequately described by conventional equations. The inhibition of β-galactosidase by imidazole represents such a case, and this system was employed to test the use of a global fitting methodology to develop equations to describe experimental data provided by enzymes with unusual kinetic behaviour.

The mathematical description of enzyme catalysis is based on the formation of an enzyme–substrate complex (ES) and the subsequent conversion of substrate into products [16]. The Michaelis–Menten equation (Eq. (2), Table 1), which describes the rate of product formation, is defined by the steady state substrate affinity constant (K_m), and the maximum velocity for substrate conversion to product (V_max) [20]. The substrate affinity constant (K_m) represents the equilibrium constant for breakdown of the ES complex into enzyme and substrate, hence, the larger the numerical value, the lower the affinity. The term maximum velocity (V_max) could imply that an enzyme may operate at multiple catalytic rates. Instead, V_max is known to be proportional to the concentration of the enzyme present [16]. At constant enzyme concentration, the observed reaction rate is dependent on the proportion of the enzyme population in the catalytically-active ES state. The Michaelis–Menten equation is a measure of the amount of the enzyme population in this state and is directly dependent on the substrate affinity constant (K_m) and the substrate concentration. Increasing substrate concentration produces a hyperbolic rise in the reaction rate by changing the proportion of the enzyme population found in the ES state from 0 to 100%, when V_max is achieved.

For many enzymes, the rate of the product formation can be affected by the presence of molecules that either inhibit or activate the enzyme [21]. Reversible enzyme inhibitors are molecules that transiently interact with the enzyme and decrease the catalytic conversion of substrate to product [22], [23], [24]. This interaction is characterized by an increase in the K_m, a decrease in the V_max, or a change in both K_m and V_max, depending on the type of inhibition. Differences in the way inhibitors interfere with catalysis led to their earlier categorization as competitive or non-competitive inhibitors [22], [23], [24]. In competitive inhibition (Eq. (3), Table 1), substrate and inhibitor compete for the same site on the enzyme. In the competitive model, inhibition decreases the affinity of the enzyme for substrate, producing a linear increase in the K_m value, which, theoretically, can increase to infinity in response to increasing inhibitor concentration. In contrast, for non-competitive inhibition, the inhibitor binds to a site on the enzyme distinct from that for substrate binding and does not alter K_m. The non-competitive inhibition model reflects a decrease in the V_max of the enzyme-catalyzed reaction and produces a hyperbolic decrease in enzyme activity with increasing inhibitor concentration, independent of substrate concentration. The non-competitive inhibition equation (Eq. (4), Table 1) implies that the degree of enzyme inhibition is directly proportional to the percent of the enzyme population associated with the non-competitive inhibitor, and that this fraction of the enzyme population has zero catalytic activity.

The Michaelis–Menten-derived equations for competitive (Eq. (3)) and non-competitive (Eq. (4)) inhibitors represent the most basic models for enzyme–inhibitor interactions that affect either K_m (Eq. (3)) or V_max (Eq. (4)) [22], [23], [24]. However, many inhibitors are found to alter enzyme catalysis in ways not readily described by Eqs. (3) and (4). For example, the mixed non-competitive equation (Eq. (5), Table 1) has been developed to model inhibitors that alter both K_m and V_max, while the partial competitive Eq. (6) describes an inhibitor that binds to the same site as the substrate but does not completely block substrate access to the catalytic site [22], [23], [24]. Other equations have also been generated to deal with other exceptions to the two basic modes of inhibition (i.e. Eqs. (3), (4)) [22], [23], [24], [25]. This partitioning of expressions to describe each particular form of inhibition has led to an increasing number of equations for enzyme inhibition.

While equations for inhibition are numerous, equations that describe enzyme stimulation have received much less attention [25], [26]. However, the kinetic description of inhibition and stimulation are known to be closely related [24], prompting an examination of the inhibitory term associated with competitive inhibition Eq. (3) and non-competitive inhibition Eq. (4). This examination suggested an expandable equation, based on the mass action binding of modifiers to an enzyme, that allowed for the mathematical description of both inhibition and stimulation events. Furthermore, this basic equation was sufficiently flexible to model the additional complex substrate activation phenomenon exhibited by BuChE. Global fitting of observed kinetic data to this versatile equation led to the observation that both the inhibitor galantamine, and the activator homocysteine thiolactone, exert their modifying effects primarily on the non-substrate-activated form of BuChE.