83 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
A survey of methods for deciding whether a reaction network is multistationary
Which reaction networks, when taken with mass-action kinetics, have the
capacity for multiple steady states? There is no complete answer to this
question, but over the last 40 years various criteria have been developed that
can answer this question in certain cases. This work surveys these
developments, with an emphasis on recent results that connect the capacity for
multistationarity of one network to that of another. In this latter setting, we
consider a network that is embedded in a larger network , which means
that is obtained from by removing some subsets of chemical species and
reactions. This embedding relation is a significant generalization of the
subnetwork relation. For arbitrary networks, it is not true that if is
embedded in , then the steady states of lift to . Nonetheless, this
does hold for certain classes of networks; one such class is that of fully open
networks. This motivates the search for embedding-minimal multistationary
networks: those networks which admit multiple steady states but no proper,
embedded networks admit multiple steady states. We present results about such
minimal networks, including several new constructions of infinite families of
these networks
Finding the positive feedback loops underlying multi-stationarity
Bistability is ubiquitous in biological systems. For example, bistability is
found in many reaction networks that involve the control and execution of
important biological functions, such as signalling processes. Positive feedback
loops, composed of species and reactions, are necessary for bistability, and
generally for multi-stationarity, to occur. These loops are therefore often
used to illustrate and pinpoint the parts of a multi-stationary network that
are relevant (`responsible') for the observed multi-stationarity. However
positive feedback loops are generally abundant in reaction networks but not all
of them are important for subsequent interpretation of the network's dynamics.
We present an automated procedure to determine the relevant positive feedback
loops of a multi-stationary reaction network. The procedure only reports the
loops that are relevant for multi-stationarity (that is, when broken
multi-stationarity disappears) and not all positive feedback loops of the
network. We show that the relevant positive feedback loops must be understood
in the context of the network (one loop might be relevant for one network, but
cannot create multi-stationarity in another). Finally, we demonstrate the
procedure by applying it to several examples of signaling processes, including
a ubiquitination and an apoptosis network, and to models extracted from the
Biomodels database.
We have developed and implemented an automated procedure to find relevant
positive feedback loops in reaction networks. The results of the procedure are
useful for interpretation and summary of the network's dynamics.Comment: 16 pages, 4 figure
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
Injectivity, multiple zeros, and multistationarity in reaction networks
Polynomial dynamical systems are widely used to model and study real
phenomena. In biochemistry, they are the preferred choice for modelling the
concentration of chemical species in reaction networks with mass-action
kinetics. These systems are typically parameterised by many (unknown)
parameters. A goal is to understand how properties of the dynamical systems
depend on the parameters. Qualitative properties relating to the behaviour of a
dynamical system are locally inferred from the system at steady state. Here we
focus on steady states that are the positive solutions to a parameterised
system of generalised polynomial equations. In recent years, methods from
computational algebra have been developed to understand these solutions, but
our knowledge is limited: for example, we cannot efficiently decide how many
positive solutions the system has as a function of the parameters. Even
deciding whether there is one or more solutions is non-trivial. We present a
new method, based on so-called injectivity, to preclude or assert that multiple
positive solutions exist. The results apply to generalised polynomials and
variables can be restricted to the linear, parameter-independent first
integrals of the dynamical system. The method has been tested in a wide range
of systems.Comment: Final version, Proceedings of the Royal Society
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