213 research outputs found
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
Canonical active Brownian motion
Active Brownian motion is the complex motion of active Brownian particles.
They are active in the sense that they can transform their internal energy into
energy of motion and thus create complex motion patterns. Theories of active
Brownian motion so far imposed couplings between the internal energy and the
kinetic energy of the system. We investigate how this idea can be naturally
taken further to include also couplings to the potential energy, which finally
leads to a general theory of canonical dissipative systems. Explicit analytical
and numerical studies are done for the motion of one particle in harmonic
external potentials. Apart from stationary solutions, we study non-equilibrium
dynamics and show the existence of various bifurcation phenomena.Comment: 11 pages, 6 figures, a few remarks and references adde
Analyzing Stability of Equilibrium Points in Neural Networks: A General Approach
Networks of coupled neural systems represent an important class of models in
computational neuroscience. In some applications it is required that
equilibrium points in these networks remain stable under parameter variations.
Here we present a general methodology to yield explicit constraints on the
coupling strengths to ensure the stability of the equilibrium point. Two models
of coupled excitatory-inhibitory oscillators are used to illustrate the
approach.Comment: 20 pages, 4 figure
On the existence of Hopf bifurcations in the sequential and distributive double phosphorylation cycle
Protein phosphorylation cycles are important mechanisms of the post
translational modification of a protein and as such an integral part of
intracellular signaling and control. We consider the sequential phosphorylation
and dephosphorylation of a protein at two binding sites. While it is known that
proteins where phosphorylation is processive and dephosphorylation is
distributive admit oscillations (for some value of the rate constants and total
concentrations) it is not known whether or not this is the case if both
phosphorylation and dephosphorylation are distributive. We study four
simplified mass action models of sequential and distributive phosphorylation
and show that for each of those there do not exist rate constants and total
concentrations where a Hopf bifurcation occurs. To arrive at this result we use
convex parameters to parameterize the steady state and Hurwitz matrices
The zero-Hopf bifurcations of a four-dimensional hyperchaotic system
We consider the four-dimensional hyperchaotic system ẋ=a(y-x), y˙=bx+u-y-xz, ż=xy-cz, and u˙=-du-jx+exz, where a, b, c, d, j, and e are real parameters. This system extends the famous Lorenz system to four dimensions and was introduced in Zhou et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 27, 1750021 (2017). We characterize the values of the parameters for which their equilibrium points are zero-Hopf points. Using the averaging theory, we obtain sufficient conditions for the existence of periodic orbits bifurcating from these zero-Hopf equilibria and give some examples to illustrate the conclusions. Moreover, the stability conditions of these periodic orbits are given using the Routh-Hurwitz criterion
Analysis of symmetries in models of multi-strain infections
In mathematical studies of the dynamics of multi-strain diseases caused by antigenically diverse pathogens, there is a substantial interest in analytical insights. Using the example of a generic model of multi-strain diseases with cross-immunity between strains, we show that a significant understanding of the stability of steady states and possible dynamical behaviours can be achieved when the symmetry of interactions between strains is taken into account. Techniques of equivariant bifurcation theory allow one to identify the type of possible symmetry-breaking Hopf bifurcation, as well as to classify different periodic solutions in terms of their spatial and temporal symmetries. The approach is also illustrated on other models of multi-strain diseases, where the same methodology provides a systematic understanding of bifurcation scenarios and periodic behaviours. The results of the analysis are quite generic, and have wider implications for understanding the dynamics of a large class of models of multi-strain diseases
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