At 08:49 28/05/98 Jannie wrote:
::>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
>:>> 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),
My fault for not explaining myself clearly. I was following the original
Heinrich & Rapoport papers which did not distinguish in this way between
rates and fluxes. When I first found out about Control Analysis, I found
this less clear than the Kacser & Burns distinction between rates and
fluxes. Nevertheless, a flux Jj can be simply the rate vj of step j.
(Obviously there are other possibilities, such as a measured flux like CO2
evolution that can be a function of the rates of several steps, but that is
not necessary for the argument here). In this case, the way that step vi
affects flux Jj is by its effects on the rate of step vj, and therefore I
am justified in considering the case Jj = vj = f(Vj, S1.....Sn). The
functional dependence of the rates of the other steps on the S1....Sn is
not relevant.
>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.
Obviously so if you start from the assumption that it is obligatory to
always consider the vector of steps as part of the parameter list. I
should have been more rigourous by considering the derivatives with respect
to a parameter pi of the step vi.
My point was that if you wanted the control coefficient of pi (acting
exclusively on step vi) on flux Jj, it is the total derivative of the rate
function vj wrt pi (or vi as I wrote it).
>>I would write the Heinrich and Rapoport equation as
>>cJj/cvi = cvj/cvi + (cvj/cS1)(cS1/cvi) + ... (cvj/cSn)(cSn/cvi)
>A natural choice if you are aiming at deriving the whole set of Heinrich &
Rapoprt equations for each step on each flux using matrix algebra.
>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.
>Well, I did say something to that effect, because 'local rate' is merely a
non-mathematical term for 'partial derivative'. The fact is that 'fluxes'
are just simple linear combinations (involving stoichiometric coefficients)
of the rates of steps, the simplest case being that the flux at step j is
the rate of step j, as I said above. And in that case, when you want the
effect on the flux j, you are mathematically determing the total derivative
of rate vj.
I have no problem with using this as a terminology to disguise from
biochemists that we are making a distinction between partial and total
derivatives. As I said, I found this helpful myself in getting to grips
with Control Analysis. BUT if you are talking about having a consistent
mathematical basis of the definitions, you can't have it on the basis of a
fiction that fluxes do or mean something different from the rates.
>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).
>That is something I agree with 100%. If we build that into your equations
1, 3 & 8, I'll shut up.
>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.
Why? Your words '...now becomes the system...' mean that the function
under consideration has changed. There is no reason to expect the total
derivatives of 2 different systems to be the same.
>When partial
>derivatives with clear indications of what changes and what remains
>contstant are used this problem never arises.
>Again, I accept this, but point out that we are all guilty of not notating
our partial derivatives with this crucial information.
David
=========================================================
David Fell, School of Biological & Molecular Sciences,
Oxford Brookes University, Oxford OX3 0BP, UK
Tel. +44 (0)1865 483247 FAX +44 (0)1865 484017
New research pages at:
http://www.brookes.ac.uk/bms/research/molcell/fell/mca_rg/mca_home.html
