Dear group,
here goes (somwhat late) the common view of Reinhart Heinrich and myself
about the question of logarithmic derivatives in Metabolic Control Analysis.
Of course, dlny/dlnx is equivalent to (x/y)(dy/dx) as long as infinitesimal
changes are considered. However, when introducing the concept of control
coefficients or elasticity coefficients to students, we always start with the
latter definition, written in the form
(x/y) Delta y / Delta x or even better (Delta y/y)/(Delta x/x),
which is applicable also for finite changes Delta y and Delta x. Therefore,
its physical meaning is wider than that of the first definition, and it makes
clearer that relative changes are considered. In the limit Delta y --> 0,
Delta x --> 0 one obtains dlny/dlnx by taking into account that dlny = dy/y
and dlnx = dx/x. Starting with dlny/dlnx is more difficult since the students
would have to know, before, something about the derivative of logarithms.
In other words, dlny/dlnx gains its justification from (x/y)(Delta y/Delta x)
in the limit Delta y --> 0, Delta x --> 0, while the opposite conclusion is
not so straightforward. We agree with the opinion posted to this group
earlier saying that in the definition (x/y) Delta y/ Delta x, there is some
ambiguity in that it must be defined at which state the ratio x/y is taken.
We think the original steady state is the natural reference state and, hence,
the ratio has to be taken at that point.
As I wrote earlier, taking the logarithm of quantities with a physical unit
is no problem as long as differentials are considered.
Stefan Schuster
(Berlin)
