BIOCHEMICAL REACTIONS ENZYME KINETICS

The subject of biochemical reactions is very broad, covering both cellular and enzymatic processes. While there are some similarities between enzyme kinetics and the kinetics of cell growth, cell-growth kinetics tend to be much more complex, and are subject to regulation by a wide variety of external agents. The enzymatic production of a species via enzymes in cells is inherently a complex, coupled process, affected by the activity of the enzyme, the quantity of the enzyme, and the quantity and viability of the available cells. In this chapter, we focus solely on the kinetics of enzyme reactions, without considering the source of the enzyme or other cellular processes. For our purpose, we consider the enzyme to be readily available in a relatively pure form, off the shelf, as many enzymes are. 

Reactions with soluble enzymes are generally conducted in batch reactors (Chapter 12) to avoid loss of the catalyst (enzyme), which is usually expensive. If steps are taken to prevent the loss of enzyme, or facilitate its reuse (by entrapment or immobilization onto a support), flow reactors may be used (e.g., CSTR, Chapter 14). More comprehensive treatments of biochemical reactions, from the point of view of both kinetics and reactors, may be found in books by Bailey and Ollis (1986) and by Atkinson and Mavituna (1983). 

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Chapter 10 Biochemical Reactions Enzyme Kinetics SOLUTION... 

This chapter solely reviews tlie kinetics of enzyme reactions, modeling, and simulation of biochemical reactions and scale-up of bioreactors. More comprehensive treatments of biochemical reactions, modeling, and simulation are provided by Bailey and Ollis [2], Bungay [3], Sinclair and Kristiansen [4], Volesky and Votruba [5], and Ingham et al. [6]. 

Each of the processes shown in Figure 2.8 can be described by a Michaelis-Menten type of biochemical reaction, a standard generalized mathematical equation describing the interaction of a substrate with an enzyme. Michaelis and Men ten realized in 1913 that the kinetics of enzyme reactions differed from the kinetics of conventional... 

All enzymatic reactions are initiated by formation of a binary encounter complex between the enzyme and its substrate molecule (or one of its substrate molecules in the case of multiple substrate reactions see Section 2.6 below). Formation of this encounter complex is almost always driven by noncovalent interactions between the enzyme active site and the substrate. Hence the reaction represents a reversible equilibrium that can be described by a pseudo-first-order association rate constant (kon) and a first-order dissociation rate constant (kM) (see Appendix 1 for a refresher on biochemical reaction kinetics) ... 

Different from conventional chemical kinetics, the rates in biochemical reactions networks are usually saturable hyperbolic functions. For an increasing substrate concentration, the rate increases only up to a maximal rate Vm, determined by the turnover number fccat = k2 and the total amount of enzyme Ej. The turnover number ca( measures the number of catalytic events per seconds per enzyme, which can be more than 1000 substrate molecules per second for a large number of enzymes. The constant Km is a measure of the affinity of the enzyme for the substrate, and corresponds to the concentration of S at which the reaction rate equals half the maximal rate. For S most active sites are not occupied. For S >> Km, there is an excess of substrate, that is, the active sites of the enzymes are saturated with substrate. The ratio kc.AJ Km is a measure for the efficiency of an enzyme. In the extreme case, almost every collision between substrate and enzyme leads to product formation (low Km, high fccat). In this case the enzyme is limited by diffusion only, with an upper limit of cat /Km 108 — 109M. v 1. The ratio kc.MJKm can be used to test the rapid... 

The various chemical mechanisms of enzyme action will not be discussed here but an overview of enzyme kinetics is essential to allow a full understanding of metabolic control. Enzymes accelerate biochemical reactions. The precise rate of reaction is influenced by a number of physiological (cellular) factors ... 

Kinetic parameters Vmax and Km give information about the relative speed of biochemical reactions and the ease of interaction between the enzyme and its substrate respectively. Inhibitors may increase Km or decrease Vmax and metabolic control often relies on these effects. 

In the equations describing enzyme kinetics in this chapter, the notation varies a bit from other chapters. Thus v is accepted in the biochemical literature as the symbol for reaction rate while Vmax is used for the maximum rate. Furthermore, for simplification frequently Vmax is truncated to V in complex formulas (see Equations 11.28 and 11.29). Although at first glance inconsistent, these symbols are familiar to students of biochemistry and related areas. The square brackets indicate concentrations. Vmax expresses the upper limit of the rate of the enzyme reaction. It is the product of the rate constant k3, also called the turnover number, and the total enzyme concentration, [E]o. The case u, = Vmax corresponds to complete saturation of all active sites. The other kinetic limit, = (Vmax/KM)[S], corresponds to Km >> [S], in other words Vmax/KM is the first order rate constant found when the substrate concentration approaches zero ... 

ENZYME KINETICS. Most enzyme-catalyzed reactions proceed at rates that are much faster than their nonenzymatic counterparts. Nonetheless, there are a number of conditions that account for basal rates for biochemical reactions in the absence, or apparent absence, of the enzyme under consideration ... 

In the design and operation of various bioreactors, a practical knowledge of physical transfer processes - that is, mass and heat transfer, as described in the relevant previous chapters - are often also required in addition to knowledge of the kinetics of biochemical reactions and of cell kinetics. Some basic concepts on the effects of diffusion inside the particles of catalysts, or of immobilized enzymes or cells, is provided in the following section. 

Enzymes are biological catalysts. Without their presence in a cell, most biochemical reactions would not proceed at the required rate. The physicochemical and biological properties of enzymes have been investigated since the early 1800s. The unrelenting interest in enzymes is due to several factors— their dynamic and essential role in the cell, their extraordinary catalytic power, and their selectivity. Two of these dynamic characteristics will be evaluated in this experiment, namely a kinetic description of enzyme activity and molecular selectivity. 

Enzymatic ester hydrolysis is a common and widespread biochemical reaction. Since simple procedures are available to follow the kinetics of hydrolytic reactions, great efforts have been made during the last years to explain this form of catalysis in chemical terms, i.e., in analogy to known non-enzymatic reactions, and to define the components of the active sites. The ultimate aim of this research is the synthesis of an artificial enzyme with the same substrate specificity and comparable speeds of reaction as the natural catalyst. 

Enzymes are biocatalysts, as such they facilitate rates of biochemical reactions. Some of the important characteristics of enzymes are summarized. Enzyme kinetics is a detailed stepwise study of enzyme catalysis as affected by enzyme concentration, substrate concentrations, and environmental factors such as temperature, pH, and so on. Two general approaches to treat initial rate enzyme kinetics, quasi-equilibrium and steady-state, are discussed. Cleland s nomenclature is presented. Computer search for enzyme data via the Internet and analysis of kinetic data with Leonora are described. 

Coulson, R.A. (1993). The flow theory of enzyme kinetics role of solid geometry in the control of reaction velocity in live animals. Inti. J. Biochem. 25 1445-1474. 

Part II of this book represents the bulk of the material on the analysis and modeling of biochemical systems. Concepts covered include biochemical reaction kinetics and kinetics of enzyme-mediated reactions simulation and analysis of biochemical systems including non-equilibrium open systems, metabolic networks, and phosphorylation cascades transport processes including membrane transport and electrophysiological systems. Part III covers the specialized topics of spatially distributed transport modeling and blood-tissue solute exchange, constraint-based analysis of large-scale biochemical networks, protein-protein interactions, and stochastic systems. 

Thermodynamic concepts are useful to apply to the study of enzyme-mediated enzyme kinetics. Through a variety of reaction mechanisms, specific enzymes catalyze specific biochemical reactions to turn over faster than they would without the enzyme present. Making use of the fact that enzymes are not able to alter the overall thermodynamics (free energy, etc.) of a chemical reaction, we can develop sets of mathematical constraints that apply to the kinetic constants of enzyme reaction mechanism. 

There is almost no biochemical reaction in a cell that is not catalyzed by an enzyme. (An enzyme is a specialized protein that increases the flux of a biochemical reaction by facilitating a mechanism [or mechanisms] for the reaction to proceed more rapidly than it would without the enzyme.) While the concept of an enzyme-mediated kinetic mechanism for a biochemical reaction was introduced in the previous chapter, this chapter explores the action of enzymes in greater detail than we have seen so far. Specifically, catalytic cycles associated with enzyme mechanisms are examined non-equilibrium steady state and transient kinetics of enzyme-mediated reactions are studied an asymptotic analysis of the fast and slow timescales of the Michaelis-Menten mechanism is presented and the concepts of cooperativity and hysteresis in enzyme kinetics are introduced. 

While the majority of these concepts are introduced and illustrated based on single-substrate single-product Michaelis-Menten-like reaction mechanisms, the final section details examples of mechanisms for multi-substrate multi-product reactions. Such mechanisms are the backbone for the simulation and analysis of biochemical systems, from small-scale systems of Chapter 5 to the large-scale simulations considered in Chapter 6. Hence we are about to embark on an entire chapter devoted to the theory of enzyme kinetics. Yet before delving into the subject, it is worthwhile to point out that the entire theory of enzymes is based on the simplification that proteins acting as enzymes may be effectively represented as existing in a finite number of discrete states (substrate-bound states and/or distinct conformational states). These states are assumed to inter-convert based on the law of mass action. The set of states for an enzyme and associated biochemical reaction is known as an enzyme mechanism. In this chapter we will explore how the kinetics of a given enzyme mechanism depend on the concentrations of reactants and enzyme states and the values of the mass action rate constants associated with the mechanism. 

Since the catalytic cycle operates with relatively rapid kinetics, E and ES will obtain a steady state governed by Equations (4.2) and (4.3) and the quasi-steady state concentrations of enzyme and complex will change rapidly in response to relatively slow changes in [S]. Thus the quasi-steady approximation is justified based on a difference in timescales between the catalytic cycle kinetics and the overall rate of change of biochemical reactions. 

Often the key entity one is interested in obtaining in modeling enzyme kinetics is the analytical expression for the turnover flux in quasi-steady state. Equations (4.12) and (4.38) are examples. These expressions are sometimes called Michaelis-Menten rate laws. Such expressions can be used in simulation of cellular biochemical systems, as is the subject of Chapters 5, 6, and 7 of this book. However, one must keep in mind that, as we have seen, these rates represent approximations that result from simplifications of the kinetic mechanisms. We typically use the approximate Michaelis-Menten-type flux expressions rather than the full system of equations in simulations for several reasons. First, often the quasi-steady rate constants (such as Ks and K in Equation (4.38)) are available from experimental data while the mass-action rate constants (k+i, k-i, etc.) are not. In fact, it is possible for different enzymes with different detailed mechanisms to yield the same Michaelis-Menten rate expression, as we shall see below. Second, in metabolic reaction networks (for example), reactions operate near steady state in vivo. Kinetic transitions from one in vivo steady state to another may not involve the sort of extreme shifts in enzyme binding that have been illustrated in Figure 4.7. Therefore the quasi-steady approximation (or equivalently the approximation of rapid enzyme turnover) tends to be reasonable for the simulation of in vivo systems. 

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