Enzyme kinetics quasi-steady approximation

The solution obtained using this system of equations is plotted as dashed lines in Figure 3.4. The solution based on this quasi-steady state approximation closely matches the solution obtained by solving the full kinetic system of Equations (3.27). The major difference between the two solutions is that the quasi-steady approximation does not account explicitly for enzyme binding. Therefore a + b remains constant in this case, while in the full kinetic system a + b + c remains constant. Since the fraction of reactant A that is bound to the enzyme is small (c/a << 1), the quasi-steady approximation is relatively accurate. 

Obtaining quasi-steady approximations for fluxes through reaction mechanisms, including mechanisms more complex than the simple Michaelis-Menten system studied in this section, is a major component of the study of enzyme kinetics. This topic will be treated in some detail in Chapter 4. 

What is the equilibrium constant for the association of reactant A to the enzyme for the kinetic parameters used in Figure 3.4 How close is the reaction A + E C to equilibrium during the simulation that is illustrated How does the quasi-steady approximation depend on the equilibrium constant for enzyme binding ... 

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. 

Section 4.2 we explore the quasi-steady approximation with somewhat more mathematical rigor. However, before undertaking that analysis, let us analyze the reversible enzyme mechanism studied in Chapter 3 from the perspective of cycle kinetics. 

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. 

The flux expression in Equation (4.16) displays the canonical Michaelis-Menten hyperbolic dependence on substrate concentration [S], We have shown that this dependence can be obtained from either rapid pre-equilibration or the assumption that [S] [E]. The rapid pre-equilibrium approximation was the basis of Michaelis and Menten s original 1913 work on the subject [140], In 1925 Briggs and Haldane [24] introduced the quasi-steady approximation, which follows from [S] 2> [E], (In his text on enzyme kinetics [35], Cornish-Bowden provides a brief historical account of the development of this famous equation, including outlines of the contributions of Henri [80, 81], Van Slyke and Cullen [203], and others, as well as those of Michaelis and Menten, and Briggs and Haldane.)... 

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. 

Specific detail on Michaelis-Menten kinetics, quasi steady-state approximations, competitive and non-competitive inhibitions, substrate inhibition, rate expressions for enzyme catalysis and deactivations, Monod growth kinetics, etc. are not presented in an extensive manner although additional information is available in the work of Vasudevan for the interested leader. " Also note that the notation adopted by Vasudevan is employed throughout this chapter. 

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

Appropriate expressions for the fluxes of each of the reactions in the system must be determined. Typically, biochemical reactions proceed through multiple-step catalytic mechanisms, as described in Chapter 4, and simulations are based on the quasi-steady state approximations for the fluxes through enzyme-catalyzed reactions. (See Section 3.1.3.2 and Chapter 4 for treatments on the kinetics of enzyme catalyzed reactions.)... 

If the enzyme charged to a batch reactor is pristine, some time will be required before equilibrium is reached. This time is usually short compared to the batch reaction time and can be ignored. Furthermore, so > Eo is usually true so that the depletion of substrate to establish the equilibrium is negligible. This means that Michaelis-Menten kinetics can be applied throughout the reaction cycle and the kinetic behavior of a batch reactor will be similar to that of a packed-bed PER, as illustrated in Example 12.4. Simply replace t with tbatch to obtain the approximate result for a batch reactor. This approximation is an example of the quasi-steady hypothesis discussed in Section 2.5.3. 

The validity of the quasi-steady-state approximation, has already been mentioned in Subsection 4.8.7. A detailed analysis of enzyme kinetics is given in Heineken et al. (1967), Walter (1977) and Segel (1984). The strict mathematical basis of the assumption is based on a theorem by Tikhonov (1952). He investigated the assumptions leading to separation of the fast and slow components of the solutions of the system... 

Tzafriri, A.R., Edelman, E.R. The total quasi-steady-state approximation is valid for reversible enzyme kinetics. J. Theor. Biol. 226, 303-313 (2004)... 

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