Quasi-steady approximation

Hence, under the quasi-steady approximation, the movement of the species is dictated by a macroscopic convection-diffusion-reaction equation with an instantaneous adsorption/desorption source term. A notable consequence of the three-scale approach is the double-averaging representation for the partition coefficient A which is defined as... 

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 ... 

The quasi-steady approximation is generally valid when the amount that enzyme complex concentrations change is less than the amount that reactant concentrations change over the timescale of interest. This is true, for example, in Section 3.1.3.2 as long as dc/dt <reactant concentrations are large compared to enzyme concentrations (a condition that is by no means universally true in vivo) is not necessarily required to apply the approximation. 

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. 

The derivation of the Michaelis-Menten equation in the previous section differs from the standard treatment of the subject found in most textbooks in that the quasi-steady approximation is justified based on the argument that the catalytic cycle kinetics is rapid compared to the overall biochemical reactant kinetics. In... 

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.)... 

Invoking the quasi-steady approximation, the steady state flux expression for this system can be shown to be 3... 

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. 

The quasi-steady approximation is strictly valid when the rate of change of enzyme-bound intermediate concentrations is small compared to the rate of change of reactant concentrations. This is the case either when a given reaction remains in an approximately steady state (reactant concentrations remain nearly constant) or when total reactant concentrations are significantly higher than total enzyme concentration, as illustrated in Figure 3.4. 

As an example of a flux expression derived from the quasi-steady approximation consider the reversible Michaelis-Menten flux arrived at in Section 3.1.3.2 ... 

Next we introduce the quasi-steady approximation, which yields ... 

For the subcritical pressure range of interest, gas-phase heat and mass diffusion rates are of the order of 10 -1 cm /sec while the liquid-phase heat transfer rate is of the order of 10" cm /sec, and the liquid surface area regression rate is approximately 10" -10" ctn /sec. Inasmuch as the gas-phase transfer rates are much faster than all of the liquid-phase transfer rates, gas-phase heat and mass transfer can be represented as quasi-steady processes. The validity of this quasi-steady approximation has been substantiated by the numerical study of Hubbard et al. (10). Furthermore, Law and Sirignano (6) have demonstrated that effects caused by the hquid surface regression during the droplet heating period are negligible relative to the liquid-phase heat conduction rate. 

Integral Equation Solutions. As a consequence of the quasi-steady approximation for gas-phase transport processes, a rigorous simultaneous solution of the governing differential equations is not necessary. This mathematical simplification permits independent analytical solution of each of the ordinary and partial differential equations for selected boundary conditions. Matching of the remaining boundary condition can be accomplished by an iterative numerical analysis of the solutions to the governing differential equations. 

Neglecting the heat stored in the solidified body corresponds, due to c = 0, to the limiting case of Ph —> 00. This is the so-called quasi-steady approximation, which gives from (2.213), a solidification time... 

Fig. 2.37 Temperature profile in the solidification of a flat layer under the assumption of the quasi-steady approximation...

As a comparison with the exact solution of the Stefan problem shows, the quasisteady approximation discussed in the last section only holds for sufficiently large values of the phase transition number, around Ph > 7. There are no exact solutions for solidification problems with finite overall heat transfer resistances to the cooling liquid or for problems involving cylindrical or spherical geometry, and therefore we have to rely on the quasi-steady approximation. An improvement to this approach in which the heat stored in the solidified layer is at least approximately considered, is desired and was given in different investigations. 

The functions i(x, t) and Si(t) can be recursively determined from the exact formulation of the problem with the heat conduction equation and its associated boundary conditions. Thereby o(x,t) and so(t) correspond to the quasi-steady approximation with Ph — oo. 

The term (3 /g v dv)/dt represents the accumulation of material in the cluster size range below the critical particle size range u. In homogeneous nucleation theoiy (Chapter 10), this term vanishes there is a steady state for this portion of the distribution in which material is removed as fast as it is supplied. (This is actually true only as a quasi-steady approximation.) The second term on the right-hand side of (11.18) can be written as follows ... 

The quasi-steady approximation requires the assumption that the dissolved oxygen concentration varies linearly across the zone of internal oxidation. Therefore, the oxygen flux through the internal oxidation zone (lOZ) is given by Pick s first law as Equation (5.3),... 

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