Let say that we have a metabolic pathway and we want to perform a Metabolic Flux

Question

Let say that we have a metabolic pathway and we want to perform a Metabolic Flux Analysis. We've got 4 fluxes and two constraints. So then the degrees of freedom is equal to 2 (4 fluxes - 2 constraints). What is the result of deleting 2 fluxes? (Degrees of freedom = 0, but what implies? or what does it mean to have -1 degrees of freedom?  

Note: the degrees of freedom gives the number of rates that need to be determined experimentally. r are the rates of metabolites, v are the fluxes, the matrix represents the stoichiometric coefficient matrix of the reactions involved in the metabolic pathways.

Explanation / Answer

DOF= no of variables - no of equations

1. deleting two fluxes the degree of freedom reduces to 0 which means the system is fully defined by the fluxes and constraints and the solution gives the values of rates. So rate need not be found out experimentally or no excess data is required.

2. -1 degrees of freedom means the system is overconstrained or there is more than enough information to solve the system. Happens when experimental data is provided n excess.

Note: this is a general answer and is not related to the given matrix which has 4 variables v1 to v4 and 8 equations.

the solution of which is to check for linear dependence by converting to echelon form and computing DOF.

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