According to "The Calculus with Analytic Geometry" by Letihold (3rd edn.,
pp 930-931) if u = f(x,y) and x and y are functions of both time, t, AND
other variables, then
del u/del t = (del u/del x)*(del x/del t) + (del u/del y)*(del y/del t)
where 'del' is the partial derivative operator. Thus, though the chain
rule is applied, the derivative of u is written as a partial because x
and y, and hence u, depend are functions of other variables as well.
*However*, if x and y are functions of t *only* then one would write
du/dt = (del u/del x)*(dx/dt) + (del u/del y)*(dy/dt)
So, according to this text, the issue is not determined solely by the use
of the chain rule, but by whether there is, ultimately, a single variable
(vs. multiple variables) on which the system depends. For pathway fluxes,
we think of the flux as being a function of all the V_max's, so by this
definition we "should" use the partial derivative for control coeefs.
The problem for us, is that, we typically have
V_i = f(x, y; V_i_max)
and we need to distinguish between the derivative of V_i w.r.t V_i_max
when x and y are held constant (the elasticity) and when x and y vary.
By the text, the control coeff. is the partial derivative, and THERE IS
NO formal derivative notation for the elasticity.
I think that we can get around this by the following construct: if
we wish to consider the control of the pathway fluxes by V_j_max, then
we first restrict the set of equations to set values of V_i_max for
i not equal to j (and all other parameters are restricted to set values
as well). Then the steady state fluxes and concentrations are a function
of precisely one variabl, V_j_max, and the use of 'del' for elasticity
and 'd' for contol coeff. is formally correct.
To be completely formal in doing this, we should then symbolicly
represent the restriction of the model to the set {all params}\{param of
interest}, but that would get to be very cumbersome. I think, though,
that we could satisfy the mathematical purists by saying that this
restriction is *understood* whenever we write "dV/dV_j_max" or
"del V/delV_j_max".
Paul Schlosser
