Subject Michaelis-Menten systems

The quasi-steady approximation, which was introduced in Section 3.1.3.2 and justified on the basis of rapid cycle kinetics in Section 4.1.1, forms the basis of the study of enzyme mechanisms, a field with deep historical roots in the subject of biochemistry. In later chapters of this book, our studies make use of this approximation in building models of biochemical systems. Yet there remains something unsatisfying about the approximation. We have seen in Section 3.1.3.2 that the approximation is not perfect. Particularly during short-time transients, the quasisteady approximation deviates significantly from the full kinetics of the Michaelis-Menten system described by Equations (3.25)-(3.27). Here we mathematically analyze the short timescale kinetics of the Michaelis-Menten system and reveal that a different quasi-steady approximation can be used to simplify the short-time kinetics. 

Carrier-mediated passage of a molecular entity across a membrane (or other barrier). Facilitated transport follows saturation kinetics ie, the rate of transport at elevated concentrations of the transportable substrate reaches a maximum that reflects the concentration of carriers/transporters. In this respect, the kinetics resemble the Michaelis-Menten behavior of enzyme-catalyzed reactions. Facilitated diffusion systems are often stereo-specific, and they are subject to competitive inhibition. Facilitated transport systems are also distinguished from active transport systems which work against a concentration barrier and require a source of free energy. Simple diffusion often occurs in parallel to facilitated diffusion, and one must correct facilitated transport for the basal rate. This is usually evident when a plot of transport rate versus substrate concentration reaches a limiting nonzero rate at saturating substrate While the term passive transport has been used synonymously with facilitated transport, others have suggested that this term may be confused with or mistaken for simple diffusion. See Membrane Transport Kinetics... 

BSA (p <0.0001). The least-squares mean clearance for a patient with a BSA of 1.83 m2 at 2.3 mg/m2 was 62 L/h, but was 30 L/h at 62.2 mg/m2. Decreasing clearance with increasing dose is consistent with Michaelis-Menten elimination kinetics. Between-subject variability was moderate at approximately 30%. Tasidotin did not show any major renal elimination, with only ca. 13% of the dose being found in the urine as unchanged drug. In Study 103, the least-squares mean tasidotin renal clearance was approximately 4.3 L/h (about 13% of total systemic clearance), with a between-subject variability of approximately 51%. Given a glo-... 

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. 

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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