Guy has raised some matters, and the last word has obviously not been
said on the matter of partial vs. total differentials. I am a bit
pressed for time, and my partner in crime Johann is away teaching
biotechnologists the ins and outs of modelling. So, what follows is
only an initial response to some of the matters raised.
First, I would like to attempt to raise David's 60% conviction by
another few percent, secretly hoping of course that we'll eventually
get him up to the 100% mark. He wrote
> Let me give a simple example where I think the total/partial
> distinction matters - the derivation of the Heinrich & Rapoprt
> equation for a flux control coefficient in terms of the elasticities
> and the concentration control coefficients. Thus we are considering
> the flux at a step j, and how this is influenced by a change in the
> rate at a step i. vj = f(Vj, S1....Sn)
>> Then
> dvj/dvi = dvj/dvi + (cvj/cS1)(dS1/dvi) + ... (cvj/cSn)(dSn/dvi)
There is a problem here. If steady-state flux Jj is written as a
function of local rate functions Jj = Jj(v1,.....,vn), where in turn
vi = vi(S,P,parameters), then the total differential is (as already
given in our previous posting) written as
dJj = (cJj/cv1)dv1 + .... + (cJj/cvn)dvn
David's example starts out by assuming that only vi is changed
leaving the other local rates unperturbed. This means
that the above reduces to
dJj = (cJj/cvi)dvi or dJj/dvi = cJj/cvi
so that what David starts out with is already by definition a
partial derivative.
I would write the Heinrich and Rapoport equation as
cJj/cvi = cvj/cvi + (cvj/cS1)(cS1/cvi) + ... (cvj/cSn)(cSn/cvi)
By the way, David's is in my opinion an excellent example of why one
should make a distinction (conceptually and symbolically) between a
steady-state flux J and a local rate v (remember that David said he
thought this unnecessary - or something to that effect - I cannot
find his message so I apologise if I misrepresent his comment). At
first look
dvj/dvi = dvj/dvi + .......
looks impossible, until one realises that the LHS dvj actually
refers to change in steady-state flux (dJj) while the RHS dvj
refers to a change in local rate.
David raised another issue, exemplified by my version of the H&R
equation given above. In it occurs (cvj/cS1)(cS1/cvi), which looks as
if it should be one. This is primarily a problem of symbolism,
because the two derivatives are not inverses of each other. The first
is a partial derivative of the local rate function vj with all other
entities that affect the rate kept constant; the second is a partial
derivative of the implicit steady-state function for S1 with respect
to only vi. This difference should be made explicit by giving the
appropriate subscripts to the partial derivatives, not by calling the
one a partial and the other a total derivative (in our opinion, at
least).
If we do make the total/partial distinction in our definitions
of the MCA coefficients we run into a further problem that hasn't
been raised yet: what is an elasticity coefficient at one level (and,
in David's view clearly - or should I say 40% :-) - a partial
derivative), becomes a response coefficient at another level, namely
when the enzyme mechanism now becomes the system (and therefore a
total derivative). This is unsatisfactory. When partial
derivatives with clear indications of what changes and what remains
contstant are used this problem never arises.
Best regards
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
