Steady-state rate equations

The quantitative description of enzyme kinetics has been developed in great detail by applying the steady-state approximation to all intermediate forms of the enzyme. Some of the kinetic schemes are extremely complex, and even with the aid of the steady-state treatment the algebraic manipulations are formidable. Kineticists have, therefore, developed ingenious schemes for writing down the steady-state rate equations directly from the kinetic scheme without carrying out the intermediate algebra." -" ... 

Chain termination. The chlorination of alkanes by rm-butyl hypochlorite is believed to follow a chain mechanism, but there is a dispute about the termination step.10 Derive the steady-state rate equation for each, making the long-chain approximation. 

Assuming that chain-breaking occurs exclusively by step 6, derive the steady-state rate equation for -r/[H2C204]/c/f. Make the steady-state assumption for Cl and CiO , and assume long chains (that is, [Fe2+] added [C121). The answer will involve R. Repeat the derivation for chain termination with step 7. 

Analytic expressions were obtained for the O and OH adsorption isotherms by solving the steady state rate equations. 

For this mechanism, the steady-state rate equation is ... 

The five unmeasurable quantities ( a, b, v, m, n) can be solved for from the five numbered equations in terms of the four partial pressures (pa, pb, Pm Pn) and seven constants (Ka, kal, ka2, kbl, kb2, km, kn). These results are substituted into the steady state rate equation,... 

The King and Altman Method. King and Altman developed a systematic approach for deriving steady-state rate equations, which has contributed to the advance of enzyme kinetics. The first step of this method is to draw an enclosed geometric figure with each enzyme form as one of the corners. Equation (5), for instance, can be rewritten as ... 

EXAMPLE 1. ONE-SUBSTRATE STEADY-STATE RATE EQUATION. We begin with the simplest two-intermediate example of a one-substrate mechanism ... 

This reaction cycle has more steps than the simple Michaelis-Menten scheme. Nonetheless, the steady-state rate equations describing these reaction cycles have indistinguishable functions, and one cannot determine the number of intermediary steps by steady-state kinetics alone. 

Handling rate equations for complex mechanisms. While steady-state rate equations can be derived easily for the simple cases discussed in the preceding sections, enzymes are often considerably more complex and the derivation of the correct rate equations can be extremely tedious. The topological theory of graphs, widely used in analysis of electrical networks, has been applied to both steady-state and nonsteady-state enzyme kinetics 45-50 The method employs diagrams of the type shown in Eq. 9-50. Here... 

The importance of the simplified schematic methods is apparent when one considers that the steady-state rate equation for Eq. 9-50 would have 6 terms in the numerator and 12 terms in the denominator.51 In the more complex case in which EAB breaks down to two products P and Q with a random order of release, the rate equation contains 672 terms in the denominator. 

We end steady state kinetics with two ideas that should provide insight into why many rate equations have their particular mathematical forms. These ideas point to useful short cuts for quickly noting the effects of additional intermediates on mechanisms, and even for solving certain complicated mechanisms by inspection instead of by analyzing the full steady state rate equations. 

Under these conditions, the steady state rate, Equation (8.158), and the preequilibrium rate are totally distinct and experiment should be able to distinguish... 

For excitation from level 1 to 2 the steady state rate equations become... 

The rate equations. The steady state rate equations for the number density of any rotational state in the state other than the one involved in the laser excitation can be written as... 

Such patterns are described by a three-term steady state rate equation (Equation 2)... 

The steady state rate equation for this ping pong mechanism is Equation 4. 

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