I POSTED THIS ABOUT A MONTH AGO. SINCE I DID NOT RECEIVE IT, I WONDER IF
IT GOT THROUGH. IF IT IS A DUPLICATION, PLEASE EXCUSE ME.
Some weeks ago, Athiel Cornish-Bowden responded to a question posed to him;
namely "How does channelling affect the summation relationships?" Athiel
then asked for comments or other explanations. Mine follows.
Unfortunately, email does not lend itself to mathematical notation.
Nonetheless,
let SUM(i) mean the sum for all species of enzyme involved in the
pathway
=8F(F) mean the partial derivative of the quantity within the
parentheses, F
J mean the flux through the pathway
Ei mean the "ith" enzyme in the pathway - for channeling within a
pathway it is assumed that E(i) and E(i+1) interact and the product of Ei
catalysis binds to the active site of E(i+1) without coming into
equilibrium with like molecules in the bulk medium
I assume that the "Summation Theorem" refers to eqn (a) below.
SUM(i) [=8F(J)/J divided by =8F(Ei)/Ei] =3D 1.
eqn(a)
This is a specific case of the more general case which goes by the name of
Euler's Theorem on homogeneous equations. The value of the right hand side
of the equation is the "order" of the equation. For unimolecular chemical
reactions,the order is "one"; that is, the flux through the pathway is
proportional to the enzyme concentration. So, if, for simplicity, there
are two enzymes in the "pathway", if you double the concentration of both,
then the flux doubles. This is the expected result if substrate one (S-1)
binds to enzyme 1 (E-1) and releases product (P-1) which is identical to
S-2, S-2 comes off of E-1 and mixes with the bulk pool and then *** some
one particular S-2, but not neessarily the one produced in the prior
catalytic step *** binds to E-2 and produces P-2. While asking your pardon
for the simplification required by the limits of my skill in making
diagrams for email, the above is meant to convey
S-1 + E-1 ---> P-1 -----> diffuses into the pool of like molecules
P-1 + E-2 ---> P-2
The important point is that reactions involving both E-1 and E-2 are first
order reactions.
However, when intermediates are channeled - by means of the presumptive
interaction of the enzymes - interaction of the enzymes is bimolecular. In
this case, the sum would not be expected to be one. If the flux were
totally through a channeled pathway (an unrealistic scenario for a system
in which the enzymes are interacting in a transient manner), the sum in a
two enzyme systemcan vary from 0.5 to 2 (see, for example, Fig 5, page 105
in Kholodenko et al in Agius & Sherratt 1997 Chnnelling in Intermediary
Metabolism, Portland Press Research Monographs.) Perhaps Kholodenko can be
prevailed upon to provide intuitive reasons which lead to such a result and
the expectations for a pathway including more than two enzymes. I trust
his intuition much more than I do my own.
Daniel H. Kohl, Professor telephone: (314) 935-5387
Department of Biology FAX: (314) 935--4432
Washington University email: kohl at biodec.wustl.edu
St. Louis, MO 63130
Daniel H. Kohl, Professor telephone: (314) 935-5387
Department of Biology FAX: (314) 935--4432
Washington University email: kohl at biodec.wustl.edu
St. Louis, MO 63130
