May 20, 2018, Sunday
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Applying Convex Analysis to Extreme Pathways

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The analysis conducted in examining the steady-state flux distribution spanning vectors of the stoichiometric-diffusion matrix revealed the need to constrain a subset of the spanning vectors -- namely, the stoichiometric fluxes -- to be strictly positive and examining the effect that such a constraint would have on the diffusion balances, which constituted the other subset of the spanning vectors. This problem is mathematically represented as follows:

{\frac  {\partial \phi }{\partial t}}=(S:D){\textbf  {v}},

where v is further decomposed into

{\textbf  {v}}=(v_{{i}},b_{{j}})^{T},

with v_{i} corresponding to the stoichiometric fluxes and b_{j} corresponding to the diffusion fluxes. v_{i} is constrained such that v_{i}\geq 0, representing the unidirectional requirement for the system to be biologically relevent. b_{j} is unconstrained; that is, -\infty \leq b_{j}\leq \infty .

Convex Polyhedral Cones

As it turns out, Schilling et al. studied a very similar problem, denoting v_{i} as an internal flux that are only considered unidirectionally and are contained strictly within the biological system. b_{j} was denoted as a external flux that could proceed bidirectionally and thus effectively involved diffusion in and out of the system. It was noted that the presence of linear inequalities forced the application of convex analysis concepts in lieu of the traditionally used linear algebra concepts: specifically, the set of solutions to a matrix inequality is given by a convex set, and the solution generated under such constraints produces, geometrically, a convex polyhedral cone in n-dimensional space, where n represents the number of fluxes (both stoichiometric and diffusive) involved in the system.

Conceptually, however, there are some similarities between the theory of spanning vectors of these cones and of linear algebra structures. In particular, extreme rays are vectors that correspond to the edges of the cone, emanating from the origin and are "conically independent", a concept which is approximately equivalent to the concept of basis in linear algebra.

Biologically, these extreme rays correspond to extreme pathways, which essentially give "routes in" and "routes out" of the system.