Steady state velocity equation

A critical feature of the random ternary complex mechanism is that for either substrate the dissociation constant from the binary enzyme complex may be different from that of the ternary enzyme complex. For example, the Ks value for AX dissociation from the E AX complex will have a value of K v<. The affinity of AX for the enzyme may, however, be modulated by the presence of the other substrate B, so that the dissociation constant for AX from the ternary E.AX.B complex may now be c/Xax, where a is a constant that defines the degree of positive or negative regulation of the affinity of AX for the enzyme by the other substrate. The overall steady state velocity equation for this type of mechanism is given by Equation (2.15) ... 

A second ternary complex reaction mechanism is one in which there is a compulsory order to the substrate binding sequence. Reactions that conform to this mechanism are referred to as bi-bi compulsory ordered ternary complex reactions (Figure 2.13). In this type of mechanism, productive catalysis only occurs when the second substrate binds subsequent to the first substrate. In many cases, the second substrate has very low affinity for the free enzyme, and significantly greater affinity for the binary complex between the enzyme and the first substrate. Thus, for all practical purposes, the second substrate cannot bind to the enzyme unless the first substrate is already bound. In other cases, the second substrate can bind to the free enzyme, but this binding event leads to a nonproductive binary complex that does not participate in catalysis. The formation of such a nonproductive binary complex would deplete the population of free enzyme available to participate in catalysis, and would thus be inhibitory (one example of a phenomenon known as substrate inhibition see Copeland, 2000, for further details). When substrate-inhibition is not significant, the overall steady state velocity equation for a mechanism of this type, in which AX binds prior to B, is given by Equation (2.16) ... 

An inhibitor that binds exclusively to the free enzyme (i.e., for which a = °°) is said to be competitive because the binding of the inhibitor and the substrate to the enzyme are mutually exclusive hence these inhibitors compete with the substrate for the pool of free enzyme molecules. Referring back to the relationships between the steady state kinetic constants and the steps in catalysis (Figure 2.8), one would expect inhibitors that conform to this mechanism to affect the apparent value of KM (which relates to formation of the enzyme-substrate complex) and VmJKM, but not the value of Vmax (which relates to the chemical steps subsequent to ES complex formation). The presence of a competitive inhibitor thus influences the steady state velocity equation as described by Equation (3.1) ... 

If a single particle is falling freely under gravity in an infinitely dilute suspension, it will accelerate until it reaches a steady-state velocity. This final velocity is known as the terminal settling velocity (t/t) and represents the maximum useful superficial velocity achievable in a fluidised bed. Thus, the contained particles will be elutriated from the column if the superficial velocity is above Ut, the value of which can be predicted using the Stokes equation... 

Figure 3.3 Substrate titration of steady state velocity for an enzyme in the presence of a competitive inhibitor at varying concentrations. (A) Untransformed data (B) data as in (A) plotted on a semilog scale (C) data as in (A) plotted in double reciprocal form. For all three plots the data are fit to Equation (3.1).

In the mechanism illustrated by scheme B, significant inhibition is only realized after equilibrium is achieved. Hence the value of vs (in Equations 6.1 and 6.2) would not be expected to vary with inhibitor concentration, and should in fact be similar to the initial velocity value in the absence of inhibitor (i.e., v, = v0, where v0 is the steady state velocity in the absence of inhibitor). This invariance of v, with inhibitor concentration is a distinguishing feature of the mechanism summarized in scheme B (Morrison, 1982). The value of vs, on the other hand, should vary with inhibitor concentration according to a standard isotherm equation (Figure 6.5). Thus the IC50 (which is equivalent to Kfv) of a slow binding inhibitor that conforms to the mechanism of scheme B can be determined from a plot of vjv0 as a function of [/]. 

These practical approaches are by no means mutually exclusive, and attempts should be made to combine as many of these as possible to improve ones ability to experimentally measure the K-pp of tight binding inhibitors. Thus one should always work at the lowest enzyme concentration possible, and drive the substrate concentration as high as possible, when dealing with competitive inhibitors. A long preincubation step should be used before activity measurements, or the progress curves should be fitted to Equation (6.2) so that accurate determinations of the steady state velocity at each inhibitor concentration can be obtained. Finally, the concentration-response data should be fitted to Morrison s quadratic equation to obtain good estimates of the value of Arfpp. 

Yang and Schulz also formulated a treatment of coupled enzyme reaction kinetics that does not assume an irreversible first reaction. The validity of their theory is confirmed by a model system consisting of enoyl-CoA hydratase (EC 4.2.1.17) and 3-hydroxyacyl-CoA dehydrogenase (EC 1.1.1.35) with 2,4-decadienoyl coenzyme A as a substrate. Unlike the conventional theory, their approach was found to be indispensible for coupled enzyme systems characterized by a first reaction with a small equilibrium constant and/or wherein the coupling enzyme concentration is higher than that of the intermediate. Equations based on their theory can allow one to calculate steady-state velocities of coupled enzyme reactions and to predict the time course of coupled enzyme reactions during the pre-steady state. 

In other words, when the magnitude of the left-hand term is negligible, only the incompressible steady-state continuity equation V V = 0 remains. For isothermal, nonreacting flow, it is only when velocity variations are responsible for density variations where compressibility effects are important. 

This equation indicates that the steady state velocity of the fluid is a linear function of the pressure. 

The steady state velocity of water flow thus obtained is plotted as a function of the applied pressure in Fig. 5. The relationship between the applied pressure and the velocity is linear as expected. The friction coefficient, f= l.OxlO11 Pas cm-2, is obtained from the slope of the straight line in Fig. 5 using the following equation... 

A numerical analysis using FlumeCAD was made, solving the incompressible Navier-Stokes equation for the velocity and pressure fields [70], The steady-state velocity field was then used in the coupled solution of three species transport equations (two reagents and one product). Further details are given in [70],... 

For a steady-state case, equations (7.29) and (7.30) allow one at small velocity gradients to obtain a solution as an expansion in series in powers of velocity gradient. Up to the first-order terms with respect to velocity gradients, equation (7.30) immediately gives... 

As momentum is a vector, the steady-state momentum equation above is a vector equation and, in general, needs to be applied in each of the three directions (x, y, z) in which momentum (i.e. velocity) has a... 

The essential feature of equation (2) is the nonuniqueness of its solution. In the range G < G0, two different values of the steady-state velocity of wave propagation correspond to the same value of G. Let us compare the characteristics of the two autowave regimes. For simplicity, we shall consider the range G G0, in which equation (3) is soluble for U. The expression for U corresponding to the smaller velocity mode has the form... 

The original scheme utilized Gaussian isokinetic thermostats, whereas in Eqs. [217] we have replaced it with a Nose-Hoover thermostat. In this equation, the true local streaming velocity is given by iyy, + Usi(q,). In principle, there are no restrictions on u i, so the steady state velocity can be of any form hence, Evans and Morriss refer to Eqs. [217] as profile-unbiased thermostats (PUT). The PUT scheme requires only a reasonable prescription for determining the true local streaming velocity. 

We begin, again, by nondimensionalizing the governing equations. The characteristic length scale is the tube radius R, and an appropriate choice for the characteristic velocity scale would seem to be GR2 / p., as this quantity is proportional to the magnitude of the final steady-state velocities [see, e.g., (3-44)] ... 

The governing equation to determine the temperature distribution in the tube is the thermal energy equation, (2-110). To solve this equation, we need to know the form of the velocity distribution in the tube. We have already seen that the steady-state velocity profile for an isothermal fluid, far downstream from the entrance to the tube, is the Poiseuille flow solution given by (3-44). In the present problem, however, the temperature must depend on both r and z, and hence the viscosity (which depends on the temperature) will also depend on position. The dependence on z is due to the fact that heat is added for all z > 0, and thus the temperature must continue to increase with the increase of z. The dependence on r is due to the fact that there must be a nonzero conductive heat flux in the fluid at the tube wall to match the prescribed heat flux through the wall, and thus the temperature must have a nonzero r derivative. It follows that the velocity field will generally differ from Poiseuille flow. 

By equating the drag force F of the sphere with the difference wa gAp between the gravity and buoyancy forces, one can estimate the steady-state velocity of relative motion of phases (the velocity at which a spherical drop falls or rises) as... 

In the case of laboratory soil column studies, uniformly packed columns with constant water contents and flow velocities are used (steady-state conditions). Equation 5 reduces for such cases to ... 

The steady-state rate equation for the random mechanism will also simplify to the form of Eq. (1) if the relative values of the velocity constants are such that net reaction is largely confined to one of the alternative pathways from reactants to products, of course. It is important, however, that dissociation of the coenzymes from the reactant ternary complexes need not be excluded. Thus, considering the reaction from left to right in Eq. (13), if k-2 k-i, then product dissociation will be effectively confined to the upper pathway this condition can be demonstrated by isotope exchange experiments (Section II,C). Further, if kakiB kik-3 -f- kikiA, then the rate of net reaction through EB will be small compared with that through EA 39). The rate equation is then the same as that for the simple ordered mechanism, except that a is now a function of the dissociation constant for A from the ternary complex, k-i/ki, as well as fci (Table I). Thus, Eqs. (5), (6), and (7) do not hold instead, l/4> < fci and ab/ a b < fc-i, and this mechanism can account for anomalous maximum rate relations. In contrast to the ordered mechanism with isomeric complexes, however, the same values for these two functions of kinetic coefficients would not be expected if an alterna-... 

When the particle moves with a steady state velocity, u0, the FE force is balanced by viscous force, Fn, given by the Stokes equation (eq. (V.7)) ... 

The solution of the hydrodynamic equations requires modeling the system, writing the equations in the appropriate coordinate system (linear, cylindrical, etc.), specifying the boundary conditions, and usually, solving the problem numerically. In electrochemical problems, only the steady-state velocity profile is of interest therefore (9.2.9) is solved for dwldt = 0. 

The initial acceleration of enzyme reactions can be observed by a study of the rate of appearance of the final product during the short time interval between mixing of enzyme and substrate and the attainment of the steady-state concentrations of all the intermediate compounds. Apart from the final steady-state velocity, this method can, in principle, give information about the kinetics of two reaction steps. In the first place, the second-order constant ki which characterizes the initial enzyme-substrate combination can be determined when [ S]o, the initial substrate concentration, is sufficiently small to make this step rate-determining during the pre-steady-state period. Kinetic equations for the evaluation of rate constants from pre-steady-state data have recently been derived (4). Under suitable conditions ki can be evaluated from... 

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