Petr Kuzmic wrote
> 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 is, of course, absolutely right. We were a bit sloppy.
Fortunately, this is unimportant for our argument in favour of
partial derivatives.
Jannie
Prof. Jannie Hofmeyr Tel: +27 21 808 3039
Dept. of Biochemistry Fax: +27 21 808 3022
University of Stellenbosch e-mail: jhsh at maties.sun.ac.za
Stellenbosch 7600
South Africa
