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 $F$-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 $R$ 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 $R$. 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

Identifying parameter regions for multistationarity

Mathematical modelling has become an established tool for studying the dynamics of biological systems. Current applications range from building models that reproduce quantitative data to identifying systems with predefined qualitative features, such as switching behaviour, bistability or oscillations. Mathematically, the latter question amounts to identifying parameter values associated with a given qualitative feature. We introduce a procedure to partition the parameter space of a parameterized system of ordinary differential equations into regions for which the system has a unique or multiple equilibria. The procedure is based on the computation of the Brouwer degree, and it creates a multivariate polynomial with parameter depending coefficients. The signs of the coefficients determine parameter regions with and without multistationarity. A particular strength of the procedure is the avoidance of numerical analysis and parameter sampling. The procedure consists of a number of steps. Each of these steps might be addressed algorithmically using various computer programs and available software, or manually. We demonstrate our procedure on several models of gene transcription and cell signalling, and show that in many cases we obtain a complete partitioning of the parameter space with respect to multistationarity.Comment: In this version the paper has been substantially rewritten and reorganised. Theorem 1 has been reformulated and Corollary 1 adde