3,577 research outputs found
On proving the absence of oscillations in models of genetic circuits
International audienceUsing computer algebra methods to prove that gene regulatory networks cannot oscillate appears to be easier than expected. We illustrate this claim on a family of models related to historical examples
The Dynamics of Hybrid Metabolic-Genetic Oscillators
The synthetic construction of intracellular circuits is frequently hindered
by a poor knowledge of appropriate kinetics and precise rate parameters. Here,
we use generalized modeling (GM) to study the dynamical behavior of topological
models of a family of hybrid metabolic-genetic circuits known as
"metabolators." Under mild assumptions on the kinetics, we use GM to
analytically prove that all explicit kinetic models which are topologically
analogous to one such circuit, the "core metabolator," cannot undergo Hopf
bifurcations. Then, we examine more detailed models of the metabolator.
Inspired by the experimental observation of a Hopf bifurcation in a
synthetically constructed circuit related to the core metabolator, we apply GM
to identify the critical components of the synthetically constructed
metabolator which must be reintroduced in order to recover the Hopf
bifurcation. Next, we study the dynamics of a re-wired version of the core
metabolator, dubbed the "reverse" metabolator, and show that it exhibits a
substantially richer set of dynamical behaviors, including both local and
global oscillations. Prompted by the observation of relaxation oscillations in
the reverse metabolator, we study the role that a separation of genetic and
metabolic time scales may play in its dynamics, and find that widely separated
time scales promote stability in the circuit. Our results illustrate a generic
pipeline for vetting the potential success of a potential circuit design,
simply by studying the dynamics of the corresponding generalized model
Negative circuits and sustained oscillations in asynchronous automata networks
The biologist Ren\'e Thomas conjectured, twenty years ago, that the presence
of a negative feedback circuit in the interaction graph of a dynamical system
is a necessary condition for this system to produce sustained oscillations. In
this paper, we state and prove this conjecture for asynchronous automata
networks, a class of discrete dynamical systems extensively used to model the
behaviors of gene networks. As a corollary, we obtain the following fixed point
theorem: given a product of finite intervals of integers, and a map
from to itself, if the interaction graph associated with has no
negative circuit, then has at least one fixed point
On the role of differential algebra in biological modeling
Extended abstract of an invited talk at Differential Algebra and related Computer Algebra (Catania, Italy, March 28th, 2008)International audienceDifferential algebra is an algebraic theory for studying systems of polynomial ordinary differential equations (ODE). Among all the methods developed for system modeling in cellular biology, it is particularly related to the well-established approach based on nonlinear ODE. A subtheory of the differential algebra, the differential elimination, has proved to be useful in the parameters estimation problem. It seems however still more promising in the quasi-steady state approximation theory, recent results show
Population Dynamics of Globally Coupled Degrade-and-Fire Oscillators
This paper reports the analysis of the dynamics of a model of pulse-coupled
oscillators with global inhibitory coupling. The model is inspired by
experiments on colonies of bacteria-embedded synthetic genetic circuits. The
total population can be either of finite (arbitrary) size or infinite, and is
represented by a one-dimensional profile. Profiles can be discontinuous,
possibly with infinitely many jumps. Their time evolution is governed by a
singular differential equation. We address the corresponding initial value
problem and characterize the dynamics' main features. In particular, we prove
that trajectory behaviors are asymptotically periodic, with period only
depending on the profile (and on the model parameters). A criterion is obtained
for the existence of the corresponding periodic orbits, which reveals the
existence of a sharp transition as the coupling parameter is increased. The
transition separates a regime where any profile can be obtained in the limit of
large times, to a situation where only trajectories with sufficiently large
groups of synchronized oscillators perdure
A passivity-based stability criterion for a class of interconnected systems and applications to biochemical reaction networks
This paper presents a stability test for a class of interconnected nonlinear
systems motivated by biochemical reaction networks. One of the main results
determines global asymptotic stability of the network from the diagonal
stability of a "dissipativity matrix" which incorporates information about the
passivity properties of the subsystems, the interconnection structure of the
network, and the signs of the interconnection terms. This stability test
encompasses the "secant criterion" for cyclic networks presented in our
previous paper, and extends it to a general interconnection structure
represented by a graph. A second main result allows one to accommodate state
products. This extension makes the new stability criterion applicable to a
broader class of models, even in the case of cyclic systems. The new stability
test is illustrated on a mitogen activated protein kinase (MAPK) cascade model,
and on a branched interconnection structure motivated by metabolic networks.
Finally, another result addresses the robustness of stability in the presence
of diffusion terms in a compartmental system made out of identical systems.Comment: See http://www.math.rutgers.edu/~sontag/PUBDIR/index.html for related
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