Discreteness-Induced Slow Relaxation in Reversible Catalytic Reaction Networks

Slowing down of the relaxation of the fluctuations around equilibrium is investigated both by stochastic simulations and by analysis of Master equation of reversible reaction networks consisting of resources and the corresponding products that work as catalysts. As the number of molecules $N$ is decreased, the relaxation time to equilibrium is prolonged due to the deficiency of catalysts, as demonstrated by the amplification compared to that by the continuum limit. This amplification ratio of the relaxation time is represented by a scaling function as $h = N \exp(-\beta V)$, and it becomes prominent as $N$ becomes less than a critical value $h \sim 1$, where $\beta$ is the inverse temperature and $V$ is the energy gap between a product and a resource

Synchronization of Coupled Nonidentical Genetic Oscillators

The study on the collective dynamics of synchronization among genetic oscillators is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels. Genetic oscillators are biochemical networks, which can generally be modelled as nonlinear dynamic systems. We show in this paper that many genetic oscillators can be transformed into Lur'e form by exploiting the special structure of biological systems. By using control theory approach, we provide a theoretical method for analyzing the synchronization of coupled nonidentical genetic oscillators. Sufficient conditions for the synchronization as well as the estimation of the bound of the synchronization error are also obtained. To demonstrate the effectiveness of our theoretical results, a population of genetic oscillators based on the Goodwin model are adopted as numerical examples.Comment: 16 pages, 3 figure

From ballistic to Brownian vortex motion in complex oscillatory media

We show that the breaking of the rotation symmetry of spiral waves in two-dimensional complex (period-doubled or chaotic) oscillatory media by synchronization defect lines (SDL) is accompanied by an intrinsic drift of the pattern. Single vortex motion changes from ballistic flights at a well-defined angle from the SDL to Brownian-like diffusion when the turbulent character of the medium increases. It gives rise, in non-turbulent multi-spiral regimes, to a novel ``vortex liquid''.Comment: 5 pages, 4 figure

Dynamical order, disorder and propagating defects in homogeneous system of relaxation oscillators

Reaction-diffusion (RD) mechanisms in chemical and biological systems can yield a variety of patterns that may be functionally important. We show that diffusive coupling through the inactivating component in a generic model of coupled relaxation oscillators give rise to a wide range of spatio-temporal phenomena. Apart from analytically explaining the genesis of anti-phase synchronization and spatially patterned oscillatory death regimes in the model system, we report the existence of a chimera state, characterized by spatial co-occurrence of patches with distinct dynamics. We also observe propagating phase defects in both one- and two-dimensional media resembling persistent structures in cellular automata, whose interactions may be used for computation in RD systems.Comment: 6 pages, 4 figure

Oscillations in the expression of a self-repressed gene induced by a slow transcriptional dynamics

We revisit the dynamics of a gene repressed by its own protein in the case where the transcription rate does not adapt instantaneously to protein concentration but is a dynamical variable. We derive analytical criteria for the appearance of sustained oscillations and find that they require degradation mechanisms much less nonlinear than for infinitely fast regulation. Deterministic predictions are also compared with stochastic simulations of this minimal genetic oscillator

Oscillations in the expression of a self-repressed gene induced by a slow transcriptional dynamics

We revisit the dynamics of a gene repressed by its own protein in the case where the transcription rate does not adapt instantaneously to protein concentration but is a dynamical variable. We derive analytical criteria for the appearance of sustained oscillations and find that they require degradation mechanisms much less nonlinear than for infinitely fast regulation. Deterministic predictions are also compared with stochastic simulations of this minimal genetic oscillator

Impact of Interdisciplinary Undergraduate Research in Mathematics and Biology on the Development of a New Course Integrating Five STEM Disciplines

Funded by innovative programs at the National Science Foundation and the Howard Hughes Medical Institute, University of Richmond faculty in biology, chemistry, mathematics, physics, and computer science teamed up to offer first- and second-year students the opportunity to contribute to vibrant, interdisciplinary research projects. The result was not only good science but also good science that motivated and informed course development. Here, we describe four recent undergraduate research projects involving students and faculty in biology, physics, mathematics, and computer science and how each contributed in significant ways to the conception and implementation of our new Integrated Quantitative Science course, a course for first-year students that integrates the material in the first course of the major in each of biology, chemistry, mathematics, computer science, and physics

Turbulence near cyclic fold bifurcations in birhythmic media

We show that at the onset of a cyclic fold bifurcation, a birhythmic medium composed of glycolytic oscillators displays turbulent dynamics. By computing the largest Lyapunov exponent, the spatial correlation function, and the average transient lifetime, we classify it as a weak turbulence with transient nature. Virtual heterogeneities generating unstable fast oscillations are the mechanism of the transient turbulence. In the presence of wavenumber instability, unstable oscillations can be reinjected leading to stationary turbulence. We also find similar turbulence in a cell cycle model. These findings suggest that weak turbulence may be universal in biochemical birhythmic media exhibiting cyclic fold bifurcations.Comment: 14 pages 10 figure

Control of birhythmicity : A self-feedback approach

The authors thankfully acknowledge the insightful suggestions by the anonymous referees. DB acknowledges CSIR, New Delhi, India. TB acknowledges Science and Engineering Research Board (Department of Science and Technology, India) [Grant No. SB/FTP/PS-05/2013]. D.B. acknowledges Haradhan Kundu, Department of Mathematics, University of Burdwan, for his useful suggestions regarding computations.Peer reviewedPublisher PD