486 research outputs found
Power-law Kinetics and Determinant Criteria for the Preclusion of Multistationarity in Networks of Interacting Species
We present determinant criteria for the preclusion of non-degenerate multiple
steady states in networks of interacting species. A network is modeled as a
system of ordinary differential equations in which the form of the species
formation rate function is restricted by the reactions of the network and how
the species influence each reaction. We characterize families of so-called
power-law kinetics for which the associated species formation rate function is
injective within each stoichiometric class and thus the network cannot exhibit
multistationarity. The criterion for power-law kinetics is derived from the
determinant of the Jacobian of the species formation rate function. Using this
characterization we further derive similar determinant criteria applicable to
general sets of kinetics. The criteria are conceptually simple, computationally
tractable and easily implemented. Our approach embraces and extends previous
work on multistationarity, such as work in relation to chemical reaction
networks with dynamics defined by mass-action or non-catalytic kinetics, and
also work based on graphical analysis of the interaction graph associated to
the system. Further, we interpret the criteria in terms of circuits in the
so-called DSR-graphComment: To appear in SIAM Journal on Applied Dynamical System
General theory for stochastic admixture graphs and F-statistics
We provide a general mathematical framework based on the theory of graphical
models to study admixture graphs. Admixture graphs are used to describe the
ancestral relationships between past and present populations, allowing for
population merges and migration events, by means of gene flow. We give various
mathematical properties of admixture graphs with particular focus on properties
of the so-called -statistics. Also the Wright-Fisher model is studied and a
general expression for the loss of heterozygosity is derived
Elimination of Intermediate Species in Multiscale Stochastic Reaction Networks
We study networks of biochemical reactions modelled by continuous-time Markov
processes. Such networks typically contain many molecular species and reactions
and are hard to study analytically as well as by simulation. Particularly, we
are interested in reaction networks with intermediate species such as the
substrate-enzyme complex in the Michaelis-Menten mechanism. These species are
virtually in all real-world networks, they are typically short-lived, degraded
at a fast rate and hard to observe experimentally.
We provide conditions under which the Markov process of a multiscale reaction
network with intermediate species is approximated in finite dimensional
distribution by the Markov process of a simpler reduced reaction network
without intermediate species. We do so by embedding the Markov processes into a
one-parameter family of processes, where reaction rates and species abundances
are scaled in the parameter. Further, we show that there are close links
between these stochastic models and deterministic ODE models of the same
networks
Bounded Coordinate-Descent for Biological Sequence Classification in High Dimensional Predictor Space
We present a framework for discriminative sequence classification where the
learner works directly in the high dimensional predictor space of all
subsequences in the training set. This is possible by employing a new
coordinate-descent algorithm coupled with bounding the magnitude of the
gradient for selecting discriminative subsequences fast. We characterize the
loss functions for which our generic learning algorithm can be applied and
present concrete implementations for logistic regression (binomial
log-likelihood loss) and support vector machines (squared hinge loss).
Application of our algorithm to protein remote homology detection and remote
fold recognition results in performance comparable to that of state-of-the-art
methods (e.g., kernel support vector machines). Unlike state-of-the-art
classifiers, the resulting classification models are simply lists of weighted
discriminative subsequences and can thus be interpreted and related to the
biological problem
Graphical criteria for positive solutions to linear systems
We study linear systems of equations with coefficients in a generic partially
ordered ring and a unique solution, and seek conditions for the solution to
be nonnegative, that is, every component of the solution is a quotient of two
nonnegative elements in . The requirement of a nonnegative solution arises
typically in applications, such as in biology and ecology, where quantities of
interest are concentrations and abundances. We provide novel conditions on a
labeled multidigraph associated with the linear system that guarantee the
solution to be nonnegative. Furthermore, we study a generalization of the first
class of linear systems, where the coefficient matrix has a specific block form
and provide analogous conditions for nonnegativity of the solution, similarly
based on a labeled multidigraph. The latter scenario arises naturally in
chemical reaction network theory, when studying full or partial
parameterizations of the positive part of the steady state variety of a
polynomial dynamical system in the concentrations of the molecular species
Node Balanced Steady States: Unifying and Generalizing Complex and Detailed Balanced Steady States
We introduce a unifying and generalizing framework for complex and detailed
balanced steady states in chemical reaction network theory. To this end, we
generalize the graph commonly used to represent a reaction network.
Specifically, we introduce a graph, called a reaction graph, that has one edge
for each reaction but potentially multiple nodes for each complex. A special
class of steady states, called node balanced steady states, is naturally
associated with such a reaction graph. We show that complex and detailed
balanced steady states are special cases of node balanced steady states by
choosing appropriate reaction graphs. Further, we show that node balanced
steady states have properties analogous to complex balanced steady states, such
as uniqueness and asymptotical stability in each stoichiometric compatibility
class. Moreover, we associate an integer, called the deficiency, to a reaction
graph that gives the number of independent relations in the reaction rate
constants that need to be satisfied for a positive node balanced steady state
to exist.
The set of reaction graphs (modulo isomorphism) is equipped with a partial
order that has the complex balanced reaction graph as minimal element. We
relate this order to the deficiency and to the set of reaction rate constants
for which a positive node balanced steady state exists
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