Athel has a good point about keeping the discussion broad. Accordingly,
the message that follows is the reply that I'd already sent to Jannie and
Johann.
Incidentally, I also agrre with Athel about elasticities.
Dear Jannie & Johann,
I was interested to read your nomenclature paper, when I finally got it.
The transfer kept stalling - presumably due to net congestion somewhere.
I think you have done a good job in proposing a logical and clear set of
rules. I've got two comments:
1. I'm happy to abide by these symbols and terms in theoretical papers,
BUT I have got reservations about a second retrospective change in
nomenclature for general purpose literature. For example, having
translated the 'Control of Flux' into 'control coefficient' nomenclature in
1995, I think it would be sad if we needed yet another translation into a
new terminology in 1999. This is not likely to aid communication with
general biochemists who are only aware of the broad outlines of MCA and are
not mathematically sophisticated (i.e. the 99% of them). Can we agree a
'dispensation' to use the 1984 terminology in 'popular' literature, perhaps
with some form of words to indicate that this reflects a traditional usage
and is not the current technically exact usage?
I do not think it is difficult for us who are familiar with the theory to
make these slight adaptations, but I do have a strong feeling that it is
both wrong and counterproductive to convey to the general public that
control coefficients are only exactly definable at a very low mechanistic
level. It may be neat for theoretical treatments, but in a practical sense
it overlooks the fact that there are very few enzymes for which the exact
mechanism is agreed beyond any reasonable level of doubt. It was never the
intention of Henrik Kacser, I am sure, to develop a treatment that could
only be used when the mechanistic minutiae of a metabolic system were all
fully known - quite the reverse. We will have a very hard job persuading
people of the necessity to treat metabolic systems at a system level if our
theory appears to presuppose a level of molecular detail that they never
envisaged needing to know.
Furthermore, changing the amount of enzymically active protein is a
physically realisable modulation. The parallel and specific change of two
independent rate constants on the other hand is a purely mathematical
operation: it cannot be implemented as such. We will have an uphill battle
persuading the rest of the biochemical community of the relevance of a
theory that is based on a purely hypothetical definition, however sound we
know it to be.
2. I think it would be really helpful to address the question of total and
partial derivatives in a systematic way once and for all. The problem is
that I am not sure of my ground here, but I'll state my beliefs:
Control and response coefficients are total derivatives in a mathematical
sense because they represent the total change in a function value to a
parameter, including all the indirect effects via dependent variables that
change in response to the parameter. Therefore they should be defined
with the differential operator 'd'.
Partial response coefficients actually would be partial derivatives
(defined with the partial differential operator), because they do not
represent the total differential - but a measure of an isolated component
of the total response. Eqn (2) then expresses the reasonable statement
that the total derivative is the sum of a specific set of partial derivatives.
Elasticity coefficients are true partial derivatives. By referring to them
as 'local' coefficients compared with the 'system' coefficients like
response and control coefficients, I think we are trying to describe in a
physical way what the mathematician means by a total and partial
derivative. (In this context, I think it is formally wrong to distinguish
between rates, v, and fluxes, J, for this purpose. These are obviously the
same thing, and what we are distinguishing between is partial derivatives
on v and total derivatives on v.)
My understanding then is that the connectivity equation and the Heinrich &
Rapoport equation (CJ = I + Sum CS*elasticity S) represent mathematically
justifiable calculations of the total derivative from the relevant partial
derivatives.
I seem to remember that in some of Christoph Giersch's work, it was
important to distinguish between entities that were true control
coefficients (total derivatives) and certain components (partial control
coefficients - defined for specific, limited conditions and not equal to
the total derivative).
Whether I'm right or wrong, I think we should come to a clear statement
about which of these coefficients should be defined with the differential
operator, and which with the partial differential operator (bearing in mind
that the notation of a partial differential can include an explicit
statement of the dependent variables that are constant).
Of course, I recognise that there is a slight context dependence here.
Notwithstanding that I believe that control coefficients are total
derivatives, I would accept that, in stating a summation theorem as the
total effect on a variable of changing all the steps, the effect of a
single step is a partial derivative in this specific context where the set
of independent parameters is considered simultaneously. However, I don't
think this invalidates my claim about the notation of the definition of a
response or control coefficient to a single specific parameter.
Best wishes,
David
