Net rate constant method

In 1975, Cleland introduced the net rate constant method to simplify the treatment of simple enzyme kinetic mechanisms that do not involve branched pathways (Cleland, 1975). This method can be successfully applied to obtain rate laws for isotope exchange. Since the net rate constant method allows one to obtain Vmax/ M and Ku in terms of the individual rate constants, this method has the greatest value for fee characterization of isotope effects on VmxJKu and ATm (Huang, 1979 Punch Allison, 2000). 

Let us start to illustrate fee method wife fee sinplest Uni Uni monosubstrate mechanism  

To start wife, reaction (4.8) is first converted to a series of unidirectional rate constants (indicated by primes). The steady-state flux through each step is given by 

one has to express the net rate constants in terms of the actual rate constants. For any irreversible step, the net rate constant and the actual rate constant is identical thus. 

Moving leftward, we express as the real forward rate constant, fco multiplied by 

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Net Rate Constant Method for Deriving Enzyme Equations... 

NET RATE CONSTANT METHOD FOR DERIVING ENZYME EQUATIONS... 

NET RATE CONSTANT METHOD Simulation, analysis, and modeling program, SAAM... 

The partitioning is simply calculated by using the net rate constant k[ in equation 3.62. The rate of formation of F relative to that of P is thus equal to k[/kP. Cle-land has shown how the net rate constant method may be applied to more complex problems.18... 

At the time that the previous chapter in Volume 11 was written, the method of King and Altman (7) was the method of choice for deriving steady-state rate equations for enzymic reactions, and this is still true for any mechanism involving branched reaction pathways. The best description of this method may be found in Mahler and Cordes (8). A useful advance was made in 1975 with the introduction of the net rate constant method (9), and because it is the simplest method to use for any nonbranched mechanism, as well as for equations for isotopic exchange, positional isotopic exchange, isotope partitioning, etc., we shall present it here. 

The equations for any other exchange reaction, and indeed for any other mechanism, can be derived similarly. However, in most cases, the rate equations for isotope exchange are derived by the method of King and Altman and various extensions of the same also, the rate equations may be derived efficiently by the net rate constant method (Chapter 4). In most cases, the rate equations for isotope exchange are far too complicated to permit the determination of the usual kinetic constants. Nevertheless, ffiere are a number of simplifying assumptions which will permit the derivation of manageable rate equations in specific cases (Boyer, 1959 Fromm eta/., 1964 Darvey, 1973). 

Equations for the initial rate of isotope exchange in Ping Pong mechanisms may be derived by one of the methods described in Chapter 4 the usual procedure will be to apply the King-Altman method or the net rate constant method (Cleland, 1975) (Table 4). 

Another means is available for studying the exchange kinetics of second-order reactions—we can adjust a reactant concentration. This may permit the study of reactions having very large second-order rate constants. Suppose the rate equation is V = A caCb = kobs A = t Ca, soAtcb = t For the experimental measurement let us say that we wish t to be about 10 s. We can achieve this by adjusting Cb so that the product kc 10 s for example, if A = 10 M s , we require Cb = 10 M. This method is possible, because there is no net reaction in the NMR study of chemical exchange. 

Table 4-1 lists six combinations of rate constants for which an RCS can be defined and two others lacking one. A method has been presented for exploring the concept of the RCS by means of reactant fluxes.11 Consider the case k < (k- + k2), such that the steady-state approximation is valid. One defines an excess rate , for each step i as the difference between the forward rate of that step and the net forward rate v/. Thus, for Step 1,... 

The breakdown of the diffusion theory of bulk ion recombination in high-mobility systems has been clearly demonstrated by the results of the computer simulations by Tachiya [39]. In his method, it was assumed that the electron motion may be described by the Smoluchowski equation only at distances from the cation, which are much larger than the electron mean free path. At shorter distances, individual trajectories of electrons were simulated, and the probability that an electron recombines with the positive ion before separating again to a large distance from the cation was determined. The value of the recombination rate constant was calculated by matching the net inward current of electrons... 

Initial-rate methods have several fundamental advantages. Since the amount of product formed during the period of measurement is small, the reverse reaction decreases the overall net rate inappreciably. Complications from slower side reactions usually are minimal. The concentrations of reactants change but little, and pseudozero-order kinetics are obeyed. For reactions whose rates are in the useful range, measurements of initial rate should be more precise than measurements of rates at later times because the rate is highest at the beginning, where the slope of the curve is steepest and the relative signal-to-noise ratio is most favorable. Initial-rate methods permit the use of reactions whose formation constants may be too small for equilibrium methods. 

The steady-state approximation is a more general method for solving reaction mechanisms. The net rate of formation of any intermediate in the reaction mechanism is set equal to 0. An intermediate is assumed to attain its steady-state concentration instantaneously, decaying slowly as reactants are consumed. An expression is obtained for the steady-state concentration of each intermediate in terms of the rate constants of elementary reactions and the concentrations of reactants and products. The rate law for an elementary step that leads directly to product formation is usually chosen. The concentrations of all intermediates are removed from the chosen rate law, and a final rate law for the formation of product that reflects the concentrations of reactants and products is obtained. 

To study the addition of 2-hydroxy-2-propyl radicals to alkenes, Batchelor and Fischer employed a ketone as the source of these radicals, which then develop multiplet polarizations from F-pairs the normal reaction product is isopropanol. The addition of an alkene scavenges some part of the radicals and thus decreases the isopropanol polarizations. Time-resolved CIDNP experiments yield the addition rate. The advantage of analysing multiplet CIDNP instead of net CIDNP is that relaxation effects are avoided. The method was reported to be applicable for scavenging rate constants between and 10 M s . ... 

Radical Additions. Batchelor and Fischer [109] measured absolute rate constants kaM for addition of 2-hydroxy-2-propyl radicals 21 to alkenes by a new method, using time-resolved CIDNP. Photoexcited 2,4-dihydroxy-2,4-dimethylpentan-3-one 20 served as a source of these radicals. Chart XI summarizes the relevant steps of the underlying reaction scheme in a simplified manner. Their key idea was to evaluate not the net polarizations, which may be strongly distorted by nuclear spin relaxation in the radicals, but the multiplet polarizations, which had previously [110] been shown to be uninfluenced by relaxation at the paramagnetic stage provided that the... 

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