Why and when channelling can decrease pool size at constant net flux in a simple dynamic channel.

Cornish-Bowden and Cárdenas (Cornish-Bowden, A. and Cárdenas M.L. (1993) Eur. J. Biochem. 213, 87-92) have suggested that simulation results peviously published by us (Mendes, P., Kell, D.B. and Westerhoff, H.V. (1992) Eur. J. Biochem. 204, 255-266) which had demonstrated that large reductions of intermediate pool sizes could be accompanied by increasing channel flux in a model metabolic pathway, were an artefact of changes in the pathway's overall flux of the order of 0.0075%, or of inappropriate alterations of enzyme activities. They also asserted to prove that the "channelling of an intermediate cannot affect its free concentration at constant net flux". We consider the co-response of the intermediate metabolite concentration ('pool') and the channel flux to changes in kinetic (or thermodynamic) parameters. Both by analytical proofs and by numerical examples we show that this co-response can be positive, negative or null, depending on the parameter change. In particular, we prove that there is always a number of ways of changing parameters such that the intermediate metabolite concentration decreases with increasing channel flux, whether the total flux varies or is constant. We also show that increased stability of the (dynamic) enzyme-intermediate-enzyme complex, as well as a single parameter change that similarly displays no cross-over effects, can lead to decreased intermediate metabolite concentration and increased channel flux at constant total flux. In general, a non-zero co-response of the intermediate metabolite concentration ('pool') and the channel flux to changes in kinetic (or other) parameters is the rule rather than the exception. More specifically: (i) The algebraic analysis ('general proof') given in Cornish-Bowden and Cárdenas (1993) contains the constraint that the elasticities of various steps to the modulation parameters which were used to vary the channel flux at constant net flux were unity. This is an unfortunate and unnecessary constraint which, when lifted, means that the concentration of the pool in the general case can indeed change at constant net flux. A 'simplified proof' given in Cornish-Bowden and Cárdenas (1993) also fails, due in addition to the consequent failure to include mass conservation relations for some of the enzymes. (ii) In the systems studied by Cornish-Bowden and Cárdenas (1993), flux is properly to be considered as a variable (since it varies during the transition to the steady state), and not a parameter, and as such cannot per se affect the magnitude of other variables in the steady state. (iii) By relaxing the constraint referred to in (i), above, and by making dual modulations (i.e., of more than one parameter at once) which are different from those carried out in Cornish-Bowden and Cárdenas (1993) we find many instances in which channelling (described by a parameter p) does significantly affect the concentration of the pool intermediate C at constant total flux. (iv) In the same pathways, but in which the flux is held constant by setting it via a zero-order flux-generating reaction, the addition of a channel is also able to significantly to modulate the size of the pool at constant total flux. Our results show that the effectiveness of channelling in decreasing a pool, even at constant flux, is very much a reality.