Steady-State Random Bi Uni mechanism

If the breakdown of the central complex in bisubstrate reactions is not the sole rate-limiting step, than the rate equation becomes quite complex. For example, consider the Steady-State Random Bi Uni system shown below  

The complete rate equation, obtained with the King-Altman method, and after grouping similar terms, is quite complex  

Even in the absence of product P, the rate equation is complex, a rational polynomial of the order 2 2 with respect to both substrates. The reciprocal plots are nonlinear, but the departure from linearity may be very difficult to detect if both routes to EAB are about equally favorable. 

This example clearly shows that completely randomized steady-state bisubstrate reactions wiU produce extremely complex rate equations which are, in most cases, unmanageable and almost useless for practical purposes. Thus, for example, the rate equation for an Ordered Bi Bi mechanism has 12 terms in the denominator (compare Eq. (9.8)). A completely Random Bi Bi mechanism yields an even more comphcated rate equation with 37 new terms in the denominator. Eor this reason, and in such cases, we shah usuahy revert to simplifying assumptions, usually introducing the rapid equilibrium segments in the mechanism in order to reduce the rate equations to manageable forms. 

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Occasionally rate expressions are described as 1/1, 2/1, etc., functions, referring to the maximum power of the substrate concentration in the numerator (N) and denominator (D). For example, consider the case of the steady-state random Bi Uni mechanism. The reciprocal form of the rate expression (at constant [B]) has the general form of 1/v = ( o + a[A] -t da2[A] )/ (na[A] + na2[A] ) where the Rvalues are collections of rate constants. If both the numerator and denominator of this reciprocal form of the rate expression are divided by the substrate concentration raised to the highest power in which it appears (in this case, [A] ), then the numerator has a term in 1/[A] (as well as 1/[A] and 1/[A]°) whereas the denominator has terms in 1/[A] and 1/[A]°. Thus, this rate expression is a 2/1 function. See Multisubstrate Mechanisms... 

Let us take the last example in Table 1, the Steady-State Random Bi Uni mechanism. The number of enzyme species or comers in the basic figure is four and the number of lines is five. Thus,... 

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