Flux Summation Theorem

The Flux Summation Theorem states that the sum of all the flux control coefficients of any pathway is equal to unity ... 

As mentioned, there are other summation relationships that can be derived in a similar way. For example, in arriving at the result above, we deduced that a 1% increase in the activity of every enzyme would produce not only a 1% change in flux but also a zero change in the concentration of any metabolite. It follows from this sort of consideration that if we define concentration control coefficients for any particular metabolite, they will sum to zero. These other relationships have, in general, received less attention than the flux summation theorem, which is now regarded as being fundamental to the understanding of metabolic control. 

One of the many important consequences of these summation theorems is that, in the case where all flux control coefficients are positive, all coefficients have values between 0 and 1. In this case, the reaction for which C,1 is greatest represents the reaction to which the flux Jj is most sensitive. In the limit that one flux control coefficient has a value close to 1 and all others have values close to 0, we can say that there exists a rate-limiting step a change in activity of the rate-limiting enzyme would be expected to elicit a proportional change in the flux Jj. 

According to the summation theorem of Metabolic Control Analysis (MCA) (Kacser and Burns 1973 Heinrich and Rapoport 1974 Kell and Westerhoff 1986 Cornish-Bowden et al. 1995 Fell 1996 Heinrich and Schuster 1996), changes in the concentrations of individual enzymes within a metabolic pathway tend to have little effect on metabolic fluxes and on the phenotype under most conditions (Thatcher et al. 1998). However, changes in individual enzyme concentrations could affect the size of metabolite pools even when the alterations in flux are minimal. Because of mass aetion effeets, the change in a metabolic pool could lead to a shift of other eoupled metabolites and change the flux at the nonequilibrium reactions in the pathway. 

The flux control coefficient, C., is a measure of how the flux / changes in response to small perturbations in the concentration or activity of enzyme i. The magnitude of the control coefficient is a measure of how important a particular enzyme is in the determination of the steady-state flux. The summation theorem... 

Since the sum of all the control strengths for a given flux is unity, there may be a temptation to use the values as a measure of the percentage control of the flux at each step. Thus, if were found to be 0.3 so that would be 0.7, according to the summation theorem, it might be concluded that 30% of the control of J is exerted at Ej and 70% at E2. Such reasoning is erroneous because it does not take into account the... 

The developed control theory for metabolic systems allows inferring of, for example, the effects of local changes, like the properties of an enzyme on global properties as the flux through the system. Furthermore, general global properties of the systems were captured by summation and connectivity theorems, see [5] for a comprehensive review. 

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