Stochastic properties of cooperative and non-cooperative enzymatic reactions at nanoescale

Abstract

Many of the different processes of synthesis of nanostructured material frequently use catalysts whose rates and properties are studied under stationary regimes. As far as our understanding goes, the most common approach is the Michaelis-Menten model, or a modified version of it, where the rate of change of the enzyme-substrate complex concentration is slow enough that it can be approximated as if it was zero [1]. Although it is common to see the Michaelis-Menten model being used in diverse circumstances, the validity of the algebraic expression known as the Michaelis-Menten equation (𝑉=𝑉𝑚𝑎𝑥[𝑆]𝐾𝑀+[𝑆] where 𝑉 is the rate of product formation, 𝑉𝑚𝑎𝑥 the maximum rate of product formation, [𝑆] is the substrate concentration and 𝐾𝑀 is known as the Michaelis-Menten constant), has been widely studied [2], [3], and a variety of methods have been used to study the dynamics of the reaction [4]–[6]. From the point of view of stochastic processes, it is worth to mention A. F. Bartholomay [7], who formulated the problem of chemical reactions as a probability density that was a function of time and of the concentrations of substrate and enzyme-substrate complex. He obtained the corresponding master equation and demonstrated that the time evolution of the means of the concentrations match with the non-linear differential equations known in the textbooks of chemical kinetics. Sandra Hasstedt [8] studied the same problem with bivalued variables {0,1}, to indicate the presence or absence of a single enzyme molecule. Arányi & Thöt [9] developed a similar approach for states with zero or one enzyme molecule but an unlimited amount of substrate molecules. After 1990, the advancements in technology and measurement techniques using Raman spectroscopy and methods of photo-physics and photochemistry [10]–[12], have made the study of random fluctuations a necessity. In the XXI century the study of stochastic systems has proliferated [13]–[15] brought to attention to the fact that, in the smaller dimensions inside of the cell, enzymes are subject to random fluctuations due to the Brownian motion, causing random displacements of these, therefore changing the reaction rates. After noting that the number of proteins is also very small, the authors of the last references above questioned the description of reactions based in the continuous flow of matter and proposed a formulation in terms of discrete stochastic equations. Puchaka and Kierzek [16] suggested a method named “maximal time step method” aimed at stochastic simulation of systems composed of metabolic reactions and regulatory processes involving small quantities of molecules. Turner et al. [17] reviewed the efforts intended to include the effects of fluctuations in the structural organization of the cytoplasm and the limited diffusion of molecules due to molecular aggregation, and discussed the advantages of these for the modelling of intracellular reactions. In 2008 Valdus Saks et al. [18] showed that cells have a highly compartmentalized inner structure, thus they are not to be considered as simple bags of protein where enzymes diffuse as a gas. Also, this type of works has inspired specialists to design drugs, who have initiated studies about the required sizes for better substrate processing. Among these, one analyzed pairs of compartments in cyanobacterias, which contain two compartments named 𝛼-carboxysome and 𝛽-carboxysome, with dimensions of the order of nanometers [19]. These elements lead us to maintain our position that the analysis of random fluctuations of substance concentrations are relevant in biochemical systems. While there are many enzyme reactions that can be described by the Michaelis-Menten model, it is better suited for cases that follow two conditions: 1. The number of substrate species that can bind to an enzyme is one. 2. The system does not exhibit cooperativity, therefore the curve of the reaction velocity, as a function of substrate concentration, has a hyperbolic shape. Advancements in fluorescence spectroscopy have allowed tracking single catalytic molecules, providing fundamental knowledge about the reactions being catalyzed. Noticing that the proportions of enzyme molecules fluctuate drastically compared to its mean values, Kumar et al. [20] suggest that a stochastic approach demonstrates the appearance of a cooperative dynamic in the chemical reaction kinetics such that, when the number of enzymes participating in the catalysis are few, there is a combined effect of the enzyme fluctuation that renders the Michaelis-Menten model ineffective. The classical model, they affirm, is useful only in cases where substrate concentration is much greater than enzyme concentration, consequently giving rise to modifications being made to the model to introduce parallel pathway mechanisms, which translate to the addition of reaction rates to the Michaelis-Menten model. Many of such pathways involve allosteric regulation of enzymes by different mechanisms. Allosteric regulation occurs when particles (ions or molecules) usually bind reversibly to active or allosteric sites on the enzyme, activating or inhibiting its catalytic activity; therefore, modulating the kinetic behavior of the ensemble of enzymes. Such particles are often given the name of regulators or modulators, and these can be products formed by the enzymatic activity, or products from other reactions from other processes occurring in a larger scale. Unlike single active site enzymes, such as beta-lactamase or the oxygen carrier myoglobin, which have a hyperbolic binding curve; enzymes with multiple binding sites, such as the oxygen carrier hemoglobin, present a sigmoid curve. This shape results from what is called a cooperative binding behavior and can be understood as follows: An enzyme that cooperatively binds its substrates, at low substrate concentration, will behave as if it had poor affinity to it. But as substrate concentration levels increase and more substrate binds to it, the affinity of the enzyme to its substrate also increases. Said another way: From the perspective of a single, multisite enzyme, the first substrate particle that binds to one of the sites will have a hard time binding to it, but as more sites are occupied, the easier it will be for other substrate particles to bind to other sites of the same enzyme, eventually saturating it. In the case of hemoglobin, it is considered that this behavior occurs because its structure undergoes a conformational change from a lower affinity state (T state) to a higher affinity state (R state). It is the substrate the one responsible with regulating the oxygenation process. This kind of regulation is called homoalloesteric. Another mechanism of enzymatic regulation is known as heteroallosteric that activate or inhibit enzyme function, by binding and shifting the conformational state of the structure from T to R (in the case of an activator) or R to T (in the case of an inhibitor). In the hemoglobin, these particles are the 𝐶𝑂2, biphosphoglycerate (BPG), and H+ ions that regulate the liberation of the oxygen in tissues that require the payload carried by the protein. In our aim to develop a model capable of simulating cooperative binding, we opted to focus on studying the oxygenation process of hemoglobin since its dissociation curve (ODC) presents the sigmoid shape that is characteristic to this type of binding behavior, as well as being a system that has been well studied in the medical sciences. In medical practice, the oxygen dissociation curve of the hemoglobin (ODC) is used to determine the oxygenation capacity of a living being. This is a curve on a plane where the 𝑂2 saturation of an Hb is measured in respect to the partial pressure of oxygen, 𝑃𝑂2 [21], [22]. There exist a variety of algebraic expression that attempt to parametrize the ODC [23]–[26] and one of the simpler ones is the Hill equation [27], where two parameters are enough to determine its geometry. Although there have been attempts in past research, spanning over a century, to understand the phenomenon of oxygenation of Hb, there still exist aspects of it that remain obscure. As can be seen in physical statistics textbooks, this law rests on the hypothesis of equilibrium [28], but as we will see here, this is not a sufficient description because it does not consider how it reaches equilibrium if the oxygenation process, which consists in the penetration of 𝑂2 into the erythrocyte, is a non-equilibrium phenomenon. The relaxation process towards equilibrium has been studied for decades. Using modified diffusion equations, W. Moll [29] studied in 1968 the rate of 𝑂2 transportation from the instant it binds to the Hb to when it is released to the tissues. With a similar approach, Baxley et al. [30] in 1983 studied 𝑂2 transport through capillary vessels. Clark et al. [31] also analyzed the release of 𝑂2 considering that the diffusion coefficient is different inside the carrier cell from the diffusion near the cell-membrane. A general review, updated in 1989, was written by Popel [32], who discusses at length the kinetics of Hb and 𝑂2 in the transport process of blood flowing through different geometries. Fischer et al. [33] developed a dissolution model of a bubble of 𝑂2 immersed in blood, trying to understand the dynamics of the diffusion process of this gas in blood. The rise in computational techniques in the last four decades has allowed scientists to study the transport phenomenon using numerical methods. An example is the work by Hyakutake and Kishimoto [34], who studied oxygen carriers based in Hb, approaching diffusion equations with various parameters, among them the 𝑃50 value, diffusion constant 𝐷, length of a blood vessel and 𝐶𝑂2 concentration. A great deal of interest has been given to understanding how 𝑂2 molecules diffuse through liquid membranes to reach Hb molecules, or membranes similar to biological tissues. This is with the express goal of designing external and internal respirators or oxygen concentrators aimed to help patients with respiratory deficiencies [35]–[38]. In a line of research close to our work, recently Scrima et al. [39] studied the out of equilibrium oxygen-hemoglobin dissociation curve (ODC). It is an experimental approach where, among other results, they obtain the sigmoidal shape of the curve, as well as values of equilibrium constants for the four oxygenation states, and also identify a source of sequential cooperativity and of conformational cooperativity. Moreover, Agliari et al. [40], [41] have developed a very interesting connection between three apparently dissimilar fields of knowledge: enzyme kinetics, ferromagnetism and neural networks. Particularly, they have used Hopfield neural networks at the thermodynamic limit, which theoretically consist in an infinite number of neurons, and have established various connections between these three phenomena. Their approach consists in using a McCulloch-Pitts neuron as a model for a binding site, so the output of this neuron is +1 if the binding site is occupied and −1 if it is vacant. If the network consists of 𝑁 neurons, this can also be understood as an Ising spin model system of 𝑁 spins [42]. This work is divided into two chapters: The first chapter is an application that expands our previous work [43] in the extension of the Michaelis-Menten model. We use our extended version to study the hydrolization of rifampicin (a member of the penicillin family of antibiotics) by beta-lactamase enzymes. There are two new contributions: 1. After studying the formalism of stochastic velocities, this approach was used to analyze the quasi-stationary state of the reaction process, as well as its entropy. 2. One of the results previously found was the decrease in entropy in purely theoretical simulated reactions. We replicate this result using experimental data for the simulations, as well as propose an explanation of this phenomenon based on the formalism stated in the previous point. We suggest that a decrease in entropy is possible due to work being done on the system, also propose that the source of this energy comes from the normal vibration modes of the enzyme, which have well defined vibration frequencies. This is a relevant result, because the specificity of enzyme catalysis could be related to a process of resonance between the active site of the enzyme and the molecular structure of the substrate. The approach developed here is useful for any number of participating enzymes and substrate molecules, provided that the initial number of enzymes is smaller than the initial number of molecules; therefore, it is applicable to nano-scaled systems. The second chapter is a new approach to the simulation of enzymatic reactions in reduced spaces that present cooperative behavior. This model is based in the use of finite Hopfield neural networks, taking as type case the oxygenation of a cluster of hemoglobin molecules. Our results are valid for a finite number of participating oxygen molecules, therefore it is useful for micro and nano devices that could be placed inside living organisms that are suffering from diminished oxygenation capacity.