2011/01/04 by David F. Anderson, Anderson, David F. · 3 citations
Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #Gene Regulatory Network Analysis #Origins and Evolution of Life #Protein Structure and Dynamics
paper · pdf · doi:10.48550/arxiv.1101.0761
This paper is concerned with the dynamical properties of deterministically\nmodeled chemical reaction systems. Specifically, this paper provides a proof of\nthe Global Attractor Conjecture in the setting where the underlying reaction\ndiagram consists of a single linkage class, or connected component. The\nconjecture dates back to the early 1970s and is the most well known and\nimportant open problem in the field of chemical reaction network theory. The\nresolution of the conjecture has important biological and mathematical\nimplications in both the deterministic and stochastic settings. One of our main\nanalytical tools, which is introduced here, will be a method for partitioning\nthe relevant monomials of the dynamical system along sequences of trajectory\npoints into classes with comparable growths. We will use this method to\nconclude that if a trajectory converges to the boundary, then a whole family of\nLyapunov functions decrease along the trajectory. This will allow us to\novercome the fact that the usual Lyapunov functions of chemical reaction\nnetwork theory are bounded on the boundary of the positive orthant, which has\nbeen the technical sticking point to a proof of the Global Attractor Conjecture\nin the past.\n