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Flux Space

Definition

Let S denote an m x n stoichiometry matrix, and let v denote a vector of n fluxes. The term flux space refers to the set of flux vectors that satisfy all of the constraints defined for the system Sv = 0.

The Null Space

The set of solutions to the equation Sv = 0 is referred to as the null space, or kernel, of S. The dimension of the null space is n - r, where r is the rank of S. By specifying values for the n - r free variables, all n fluxes in v are uniquely determined.

Flux Constraints

Bounded intervals have the form [α, β], where α and β are both finite real numbers and α ≤ β. An n dimensional vector of bounded intervals forms an n dimensional hyperbox, which is a type of convex polytope. When all n fluxes in v are constrained to lie on bounded intervals, a corresponding n dimensional hyperbox B is created.

Flux Space

The dimensions of B and the null space of S are n and n - r, respectively. Since r fluxes can be represented as linear combinations of the other n - r fluxes, B can be mapped to the null space in a straightforward manner. The convex polytope formed from this mapping is referred to as the flux space, which is usually not a hyperbox. Every point, or vector, on the surface or the interior of the flux space satisfies both the "steady state constraint" and the explicitly defined flux constraints.