JHS Hofmeyr wrote:
> Let us now apply the above to the (irreversible Michaelis-Menten)
> rate function v of a step in a reaction network: v =
> v(s,e,kcat,Km) (the rate is a function of only concentrations s, e
> of substrate and enzyme, and of the kinetic constants kcat and Km)
>> The total differential dv is written as:
>> dv = (cv/cs)ds + (cv/ce)de + (cv/ckcat)dkcat + (cv/cKm)dKm (4)
>> The following functional relationships exist: s, e, and kcat are
> independent, but Km depends on kcat by Km = (kcat + k-1)/k1.
>> Using the above mathematics it follows that
>> dv/ds = cv/cs
> dv/de = cv/ce
> dv/dkcat = (cv/cKm)(dKm/dkcat) + (cv/ckcat)
> dv/dKm = cv/cKm
> (note that cv/cKm is defined at constant kcat)
>> Therefore, in some cases the total derivative reduces to the
> partial derivative, whereas in others it does not.
I completely agree with the essence of this post (very nice review of
some simple calculus here). However, for an outsider (being simply
curious what kinds of things are being solved by MCA), the simple
statement "Km depends on kcat by Km = (kcat + k-1)/k1" could lead to
opening a huge can of worms. How might a beginning student understand
that it is o.k. to write
Km = (k2 + k-1)/k1
kcat = k2, and therefore
k2 = kcat, which leads to
Km = (kcat + k-1)/k1 giving a nonzero derivative
dKm/dkat = 1/k1 ,
but it is not o.k. to write
kcat = k2
Km = (k2 + k-1)/k1, and therefore
k2 = Km k1 - k-1, which leads to
kcat = Km k1 - k-1 giving a nonzero derivative
dkcat/dKm = k1 ?
I do know the answer, but I still think it helps if we always maintain a
logical distinction between elementary rate constants (k1, k-1, k2) and
steady state constants (kcat, Km). In the general case of an enzyme
reaction with N-1 reversible plus one (final) irreversible step,
E + S <==> X1 : k1 k-1
X1 <==> X2 : k2 k-2
X3 <==> X4 : k3 k-3
...
XN ---> E + P : kN ,
kcat is always a function of (N-1)/2 - 1 rate constants,
kcat = f(k2,k-2,k3,k-3,...,kN),
while Km is a function of N/2 - 1 rate constants:
Km = g(k1,k-1,k2,k-2,k3,k-3,...,kN)
and kcat/Km is always a function of N/2 rate constants:
kcat/Km = h(k1,k-1,k2,k-2,k3,k-3,...,k[N-1],k-[N-1])
So in a sense, yes, Km 'depends' on kcat because it depends (no
quotation marks) on _all_ elementary rate constants on which kcat also
depends (k2 through kN), and then on some more.
I just thought I'd throw this piece or pedantry in, perhaps for other
people new to MCA who wondered (as I did) what could be meant by "Km
depends on kcat but not vice versa".
-- Petr
_____________________________________________________________
Petr Kuzmic Ph.D. * BioKin Ltd. * Madison, WI 53708-8336, USA
pkuzmic at biokin.com * http://www.biokin.com * 608.256.1269 fax
