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Elementary Vectors and Conformal Sums in Polyhedral Geometry and their Relevance for Metabolic Pathway Analysis

Overview of attention for article published in Frontiers in Genetics, May 2016
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Title
Elementary Vectors and Conformal Sums in Polyhedral Geometry and their Relevance for Metabolic Pathway Analysis
Published in
Frontiers in Genetics, May 2016
DOI 10.3389/fgene.2016.00090
Pubmed ID
Authors

Stefan Müller, Georg Regensburger

Abstract

A fundamental result in metabolic pathway analysis states that every flux mode can be decomposed into a sum of elementary modes. However, only a decomposition without cancelations is biochemically meaningful, since a reversible reaction cannot have different directions in the contributing elementary modes. This essential requirement has been largely overlooked by the metabolic pathway community. Indeed, every flux mode can be decomposed into elementary modes without cancelations. The result is an immediate consequence of a theorem by Rockafellar which states that every element of a linear subspace is a conformal sum (a sum without cancelations) of elementary vectors (support-minimal vectors). In this work, we extend the theorem, first to "subspace cones" and then to general polyhedral cones and polyhedra. Thereby, we refine Minkowski's and Carathéodory's theorems, two fundamental results in polyhedral geometry. We note that, in general, elementary vectors need not be support-minimal; in fact, they are conformally non-decomposable and form a unique minimal set of conformal generators. Our treatment is mathematically rigorous, but suitable for systems biologists, since we give self-contained proofs for our results and use concepts motivated by metabolic pathway analysis. In particular, we study cones defined by linear subspaces and nonnegativity conditions - like the flux cone - and use them to analyze general polyhedral cones and polyhedra. Finally, we review applications of elementary vectors and conformal sums in metabolic pathway analysis.

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Mendeley readers

The data shown below were compiled from readership statistics for 16 Mendeley readers of this research output. Click here to see the associated Mendeley record.

Geographical breakdown

Country Count As %
Unknown 16 100%

Demographic breakdown

Readers by professional status Count As %
Student > Ph. D. Student 5 31%
Student > Master 3 19%
Researcher 2 13%
Student > Bachelor 1 6%
Other 1 6%
Other 1 6%
Unknown 3 19%
Readers by discipline Count As %
Biochemistry, Genetics and Molecular Biology 4 25%
Mathematics 3 19%
Agricultural and Biological Sciences 2 13%
Computer Science 1 6%
Medicine and Dentistry 1 6%
Other 0 0%
Unknown 5 31%
Attention Score in Context

Attention Score in Context

This research output has an Altmetric Attention Score of 1. This is our high-level measure of the quality and quantity of online attention that it has received. This Attention Score, as well as the ranking and number of research outputs shown below, was calculated when the research output was last mentioned on 24 May 2016.
All research outputs
#18,345,702
of 23,577,761 outputs
Outputs from Frontiers in Genetics
#6,355
of 12,603 outputs
Outputs of similar age
#238,703
of 336,100 outputs
Outputs of similar age from Frontiers in Genetics
#50
of 69 outputs
Altmetric has tracked 23,577,761 research outputs across all sources so far. This one is in the 19th percentile – i.e., 19% of other outputs scored the same or lower than it.
So far Altmetric has tracked 12,603 research outputs from this source. They receive a mean Attention Score of 3.7. This one is in the 41st percentile – i.e., 41% of its peers scored the same or lower than it.
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We're also able to compare this research output to 69 others from the same source and published within six weeks on either side of this one. This one is in the 21st percentile – i.e., 21% of its contemporaries scored the same or lower than it.