4,345 research outputs found
Intermediates, Catalysts, Persistence, and Boundary Steady States
For dynamical systems arising from chemical reaction networks, persistence is
the property that each species concentration remains positively bounded away
from zero, as long as species concentrations were all positive in the
beginning. We describe two graphical procedures for simplifying reaction
networks without breaking known necessary or sufficient conditions for
persistence, by iteratively removing so-called intermediates and catalysts from
the network. The procedures are easy to apply and, in many cases, lead to
highly simplified network structures, such as monomolecular networks. For
specific classes of reaction networks, we show that these conditions for
persistence are equivalent to one another. Furthermore, they can also be
characterized by easily checkable strong connectivity properties of a related
graph. In particular, this is the case for (conservative) monomolecular
networks, as well as cascades of a large class of post-translational
modification systems (of which the MAPK cascade and the -site futile cycle
are prominent examples). Since one of the aforementioned sufficient conditions
for persistence precludes the existence of boundary steady states, our method
also provides a graphical tool to check for that.Comment: The main result was made more general through a slightly different
approach. Accepted for publication in the Journal of Mathematical Biolog
A network dynamics approach to chemical reaction networks
A crisp survey is given of chemical reaction networks from the perspective of
general nonlinear network dynamics, in particular of consensus dynamics. It is
shown how by starting from the complex-balanced assumption the reaction
dynamics governed by mass action kinetics can be rewritten into a form which
allows for a very simple derivation of a number of key results in chemical
reaction network theory, and which directly relates to the thermodynamics of
the system. Central in this formulation is the definition of a balanced
Laplacian matrix on the graph of chemical complexes together with a resulting
fundamental inequality. This directly leads to the characterization of the set
of equilibria and their stability. Both the form of the dynamics and the
deduced dynamical behavior are very similar to consensus dynamics. The
assumption of complex-balancedness is revisited from the point of view of
Kirchhoff's Matrix Tree theorem, providing a new perspective. Finally, using
the classical idea of extending the graph of chemical complexes by an extra
'zero' complex, a complete steady-state stability analysis of mass action
kinetics reaction networks with constant inflows and mass action outflows is
given.Comment: 18 page
A Projection Argument for Differential Inclusions, with Applications to Persistence of Mass-Action Kinetics
Motivated by questions in mass-action kinetics, we introduce the notion of
vertexical family of differential inclusions. Defined on open hypercubes, these
families are characterized by particular good behavior under projection maps.
The motivating examples are certain families of reaction networks -- including
reversible, weakly reversible, endotactic, and strongly endotactic reaction
networks -- that give rise to vertexical families of mass-action differential
inclusions. We prove that vertexical families are amenable to structural
induction. Consequently, a trajectory of a vertexical family approaches the
boundary if and only if either the trajectory approaches a vertex of the
hypercube, or a trajectory in a lower-dimensional member of the family
approaches the boundary. With this technology, we make progress on the global
attractor conjecture, a central open problem concerning mass-action kinetics
systems. Additionally, we phrase mass-action kinetics as a functor on reaction
networks with variable rates.Comment: v5: published version; v3 and v4: minor additional edits; v2:
contains more general version of main theorem on vertexical families,
including its accompanying corollaries -- some of them new; final section
contains new results relating to prior and future research on persistence of
mass-action systems; improved exposition throughou
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