The quasi-steady approximation

We will first look at the solidifying process on a flat, cooled wall, as shown in Fig. 2.37. The wall, with thickness and thermal conductivity Aw will be cooled by a fluid having a temperature i 0, whereby the heat transfer coefficient a is decisive. On the other side of the wall a solidified layer develops, which at 

During the time interval dt the phase interface moves a distance of ds. This releases the fusion enthalpy 

The solidification-time for a flat layer of thickness s is found by integrating (2.216) to be 

For k — oo the temperature at x = 0 has the value i) i) ]. This is the boundary condition in the Stefan problem discussed in 2.3.6.1. From (2.218) we obtain the first term of the exact solution (2.213), corresponding to Ph — oo, and therefore the time t according to (2.214). With finite heat transfer resistance (1 /k) the solidification-time is greater than i it no longer increases proportionally to s2. 

In the same manner the solidification-times for layers on cylindrical (tubes) and spherical surfaces can be calculated. We will derive the result for a layer which develops on the outside 

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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. 

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 ... 

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. 

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 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),... 

The description of the curve s motion can be further simplified when the conditions of the quasi-steady approximation are fulfilled. To formulate them, let us consider the narrow boundary layer of width about Iq near the free end of the propagating curve. Outside this layer, in the external region I > Iq, the curvatures are so small that the curvature dependence of the normal propagation velocity V can be neglected and the motion of the curve can be satisfactorily described in the approximation of the WR model. Moreover, we have seen that small perturbations, originating in this region, cannot influence the processes inside the boundary layer. When Dkc < Vo, the width of the boundary layer is much smaller than the pitch of the spiral. 

The equations of the quasi-steady approximation can be used to describe the process of formation of a steadily rotating spiral from an arbitrary broken initial curve or to consider relaxation to the steady rotation after a finite perturbation. Moreover, using this system of equations, one can easily determine the behaviour of spirals in time-dependent or nonuniform media. 

If the properties of a medium change in time and/or in space, the parameters of the kinematical model (i.e. Vo. D, 7 and Gq or kc = G0/7) become certain functions of the spatial coordinates and/or of time. In the quasi-steady approximation the motion of the tip is influenced only by the dependence of the properties of the medium along its trajectory. It means that the kinematical parameters in the Equations (38)-(41) would represent certain functions of Xo and Yq. When such functions are known, the system of these equations can be solved to determine the trajectory of motion of the tip, and, hence, the behaviour of the entire spiral wave. 

Note that the quasi-steady approximation can be applied for the description of processes in time-dependent or nonuniform excitable media only if the characteristic time T and length L of variation of the medium s properties are not very small. Namely, we should require that T tq, L Iq and L/Vq > To. However, these conditions are not very restrictive since To is much smaller than the rotation period of the spiral wave and Iq is much smaller than the pitch of the spiral. 

If one tries to develop a perturbation theory proceeding directly from the reaction-diffusion equations this meets with serious difficulties. They arise because translation and rotation perturbation modes for the spiral wave are not spatially localized. We bypass such difficulties by using the quasi-steady approximation formulated in the previous section. In this approximation the trajectory of the tip motion can be calculated by solving a system of ordinary differential equations which depend only on the local properties of the medium in the vicinity of the tip. The perturbations which originate outside a small neighbourhood of the end point propagate quickly to the periphery and do not influence the motion of the tip. The evolution of the entire curve can then be calculated in the WR approximation using the known trajectory of the tip motion as a dynamic boundary condition. 

After putting kc(t) = Go t)/j into Equation (38) of the quasi-steady approximation and linearizing this equation, one can find variations of the curvature ko at the free end of the curve which are induced by the modulation of Gq. Substitution of these variations into the linearized equation (39) yields the time dependence of the angle cco- The motion of the tip of the spiral wave can then be calculated from (40) and (41). We find that in the linear... 

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