I think that whether a control coefficient is regarded as a partial or total
derivative would depend on context. If the variable is a metabolite
concentration, and the parameter is one enzyme out of a set of several,
i.e. S = f(E1,E2......), then it is clear that this is a partial derivative.
If the variable is a flux, then the position is more complicated. We
could take the position that the flux at any step in the network is
expressible as a function of a parameter set only, i.e. external pool
concentrations, Km values, enzyme amounts. i.e. J = f(X0, X1, X2....., Km1...,
E1, E2, ......), and therefore a partial derivative expression could be
used. On the other hand, we might say that the flux at a particular step
is expressible as the enzyme rate law of that step, with the (variable)
metabolite concentrations pertaining at steady state. Thus for flux Ji,
Ji = f(Si,Sj,Sk......., Ei). If now we want the control coeff with repect
to enzyme Ej, we have to consider Ji = f(Si(Ej), Sj(Ej), Sk(Ej)....,Ei).
In this case, which I think will lead to relationships such as the
Heinrich-Rapoport equation, the derivative of Ji wrt Ej has the character
of a total derivative, that is expressible in terms of a number of partial
derivatives.
I think we have to acept that this sort of issue is of concern to MC
theorists, particularly if they're trying to communicate with mathematicians,
but will be of no concern at all to MC analysts. Certainly I hope to have
made the issue sufficiently obscure to persuade you that it needn't
arise in the FAQ.
David Fell
daf at bms.brookes.ac.uk
