9780521516358book.pdf

It can be seen that whenKsis numerically large, the equilibrium is in favour of

unbound E and S, i.e. of non-binding, whilst whenKsis numerically small, the

equilibrium is in favour of the formation of ES, i.e. of binding. ThusKsis inversely

proportional to the affinity of the enzyme for its substrate.

The conversion of ES to product (P) can bemost simply represented by the irreversible

equation:

ES!kþ^2 EþP

wherekþ 2 is the first-order rate constant for the reaction.

In some cases the conversion of ES to E and P may involve several stages and may

not necessarily be essentially irreversible. The rate constantkþ 2 is generally smaller

than bothkþ 1 andk 1 and in some cases very much smaller. In general, therefore,

the conversion of ES to products is the rate-limiting step such that the concentration

of ES is essentially constant but not necessarily the equilibrium concentration. Under

these conditions the Michaelis constant,Km, is given by:

Km¼

kþ 2 þk 1

kþ 1 ¼Ksþ

kþ 2

kþ 1 ð^15 :^3 Þ

It is evident that under these circumstances,Kmmust be numerically larger than

Ksand only whenkþ 2 is very small doKmandKsapproximately equal each other.

The relationship between these two constants is further complicated by the fact that

for some enzyme reactions two products are formed sequentially, each controlled by

different rate constants:

EþS^ !

kþ 2

ES!P 1 þEAk!þ^3 EþP 2

where P 1 and P 2 are products, and A is a metabolic product of S that is further

metabolised to P 2. In such circumstances it can be shown that:

Km¼Ks

kþ 3

kþ 2 þkþ 3 ð^15 :^4 Þ

so thatKmis numerically smaller thanKs. It is obvious therefore that care must be

taken in the interpretation of the significance ofKmrelative toKs. Only when the

complete reaction mechanism is known can the mathematical relationship between

KmandKsbe fully appreciated and any statement made about the relationship

betweenKmand the affinity of the enzyme for its substrate.

Although the Michaelis–Menten equation can be used to calculateKmandVmax,

its use is subject to the difficulty of experimentally measuring initial rates at

high substrate concentrations and hence of extrapolating the hyperbolic curve to give

an accurate value ofVmax. Linear transformations of the Michaelis–Menten equation

are therefore commonly used alternatives. The most popular of these is the

Lineweaver–Burk equationobtained by taking the reciprocal of the Michaelis–Menten

equation:

1

 0

¼

Km

Vmax



1

½SŠ

þ

1

Vmax

ð 15 : 5 Þ

587 15.2 Enzyme steady-state kinetics