Steady-State Random

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... 

Fromm and Rudolph have discussed the practical limitations on interpreting product inhibition experiments. The table below illustrates the distinctive kinetic patterns observed with bisubstrate enzymes in the absence or presence of abortive complex formation. It should also be noted that the random mechanisms in this table (and in similar tables in other texts) are usually for rapid equilibrium random mechanism schemes. Steady-state random mechanisms will contain squared terms in the product concentrations in the overall rate expression. The presence of these terms would predict nonhnearity in product inhibition studies. This nonlin-earity might not be obvious under standard initial rate protocols, but products that would be competitive in rapid equilibrium systems might appear to be noncompetitive in steady-state random schemes , depending on the relative magnitude of those squared terms. See Abortive Complex... 

EB][A]/[EAB], and = [EA][B]/[EAB]. The steady-state random Bi Bi rate expression is a more complex equation containing additional terms of [A] [B] and [A][B] in the numerator and [A], [B], [A] [B], and [A][B] in the denominator. Rudolph and Fromm" have looked at the effect of the magnitude of these other terms on initial rate and product inhibition studies. See Multisubstrate Mechanisms... 

This brings us to the final mechanism we need to consider for a 2-substrate reaction, namely a random-order mechanism. We have assumed that we would be alerted to the possibility of a steady-state random-order mechanism by non-linear primary or secondary plots, but it is possible to get linear kinetics with a random-order mechanism. If we make the assumption that the further reaction of the ternary complex EAB is much slower than the network of reactions connecting E to EAB via EA and EB, then there are only 4 kinetically significant complexes and their concentrations are related to one another by substrate concentrations and dissociation constants. This is the rapid-equilibrium random-order mechanism, and the assumption made is analogous to the Michaelis-Menten equilibrium assumption for a 1-substrate mechanism. 

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,... 

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 upper-part of Table 3 shows the rapid equilibrium mechanisms and the lower-part the steady-state mechanisms. At the end of the table is the Steady-State Random Bi Bi mechanism that is included for comparison. The steady-state random case in practice gives the same patterns as rapid equilibrium ones one can usually teU the difference only by differential rates of isotopic exchange or the measurement of stickiness. 

Steady-State Random. Rate constants k., 4, fcg, and ky less than or not much larger than VJE. ... 

Finite but unequal isotope effects on the two substrate V/K values suggests a Steady-State Random kinetic mechanism. The smaller of the V/K isotope effects reflects the stickier substrate. In reaction (17.61), it means that the rate constant fcg is substantial, and that the central complex is able to dissociate back to EA as well as to EB. 

Equal isotope effect on the two V/K values suggests one of several possibilities, including an Equilibrium Ordered mechanism with or without a dead-end EB complex, a Rapid Equilibrium Random mechanism, or a Steady-State Random mechanism in which the rates of release of A and B from the central complex are equal. 

Based on isotope effects only, it is not possible to distinguish the Rapid Equilibrium Ordered from the Rapid Equilibrium Random mechanism. However, the first mechanism gives a distinctive initial velocity pattern that intersects on the ordinate with B as the varied substrate. To teU the difference between the Rapid Equilibrium Random and the Steady-State Random mechanism will require other methods, such as the isotope trapping method (Rose et al, 1974), or isotopic exchange. 

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