4,481 research outputs found
Analysis of Reaction Network Systems Using Tropical Geometry
We discuss a novel analysis method for reaction network systems with
polynomial or rational rate functions. This method is based on computing
tropical equilibrations defined by the equality of at least two dominant
monomials of opposite signs in the differential equations of each dynamic
variable. In algebraic geometry, the tropical equilibration problem is
tantamount to finding tropical prevarieties, that are finite intersections of
tropical hypersurfaces. Tropical equilibrations with the same set of dominant
monomials define a branch or equivalence class. Minimal branches are
particularly interesting as they describe the simplest states of the reaction
network. We provide a method to compute the number of minimal branches and to
find representative tropical equilibrations for each branch.Comment: Proceedings Computer Algebra in Scientific Computing CASC 201
Chemical reaction systems with toric steady states
Mass-action chemical reaction systems are frequently used in Computational
Biology. The corresponding polynomial dynamical systems are often large
(consisting of tens or even hundreds of ordinary differential equations) and
poorly parametrized (due to noisy measurement data and a small number of data
points and repetitions). Therefore, it is often difficult to establish the
existence of (positive) steady states or to determine whether more complicated
phenomena such as multistationarity exist. If, however, the steady state ideal
of the system is a binomial ideal, then we show that these questions can be
answered easily. The focus of this work is on systems with this property, and
we say that such systems have toric steady states. Our main result gives
sufficient conditions for a chemical reaction system to have toric steady
states. Furthermore, we analyze the capacity of such a system to exhibit
positive steady states and multistationarity. Examples of systems with toric
steady states include weakly-reversible zero-deficiency chemical reaction
systems. An important application of our work concerns the networks that
describe the multisite phosphorylation of a protein by a kinase/phosphatase
pair in a sequential and distributive mechanism
Toric dynamical systems
Toric dynamical systems are known as complex balancing mass action systems in
the mathematical chemistry literature, where many of their remarkable
properties have been established. They include as special cases all deficiency
zero systems and all detailed balancing systems. One feature is that the steady
state locus of a toric dynamical system is a toric variety, which has a unique
point within each invariant polyhedron. We develop the basic theory of toric
dynamical systems in the context of computational algebraic geometry and show
that the associated moduli space is also a toric variety. It is conjectured
that the complex balancing state is a global attractor. We prove this for
detailed balancing systems whose invariant polyhedron is two-dimensional and
bounded.Comment: We include the proof of our Conjecture 5 (now Lemma 5) and add some
reference
Dynamic p-enrichment schemes for multicomponent reactive flows
We present a family of p-enrichment schemes. These schemes may be separated
into two basic classes: the first, called \emph{fixed tolerance schemes}, rely
on setting global scalar tolerances on the local regularity of the solution,
and the second, called \emph{dioristic schemes}, rely on time-evolving bounds
on the local variation in the solution. Each class of -enrichment scheme is
further divided into two basic types. The first type (the Type I schemes)
enrich along lines of maximal variation, striving to enhance stable solutions
in "areas of highest interest." The second type (the Type II schemes) enrich
along lines of maximal regularity in order to maximize the stability of the
enrichment process. Each of these schemes are tested over a pair of model
problems arising in coastal hydrology. The first is a contaminant transport
model, which addresses a declinature problem for a contaminant plume with
respect to a bay inlet setting. The second is a multicomponent chemically
reactive flow model of estuary eutrophication arising in the Gulf of Mexico.Comment: 29 pages, 7 figures, 3 table
Translated Chemical Reaction Networks
Many biochemical and industrial applications involve complicated networks of
simultaneously occurring chemical reactions. Under the assumption of mass
action kinetics, the dynamics of these chemical reaction networks are governed
by systems of polynomial ordinary differential equations. The steady states of
these mass action systems have been analysed via a variety of techniques,
including elementary flux mode analysis, algebraic techniques (e.g. Groebner
bases), and deficiency theory. In this paper, we present a novel method for
characterizing the steady states of mass action systems. Our method explicitly
links a network's capacity to permit a particular class of steady states,
called toric steady states, to topological properties of a related network
called a translated chemical reaction network. These networks share their
reaction stoichiometries with their source network but are permitted to have
different complex stoichiometries and different network topologies. We apply
the results to examples drawn from the biochemical literature
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