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

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

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

Robustness of circadian clocks to daylight fluctuations: hints from the picoeucaryote Ostreococcus tauri

The development of systemic approaches in biology has put emphasis on identifying genetic modules whose behavior can be modeled accurately so as to gain insight into their structure and function. However most gene circuits in a cell are under control of external signals and thus quantitative agreement between experimental data and a mathematical model is difficult. Circadian biology has been one notable exception: quantitative models of the internal clock that orchestrates biological processes over the 24-hour diurnal cycle have been constructed for a few organisms, from cyanobacteria to plants and mammals. In most cases, a complex architecture with interlocked feedback loops has been evidenced. Here we present first modeling results for the circadian clock of the green unicellular alga Ostreococcus tauri. Two plant-like clock genes have been shown to play a central role in Ostreococcus clock. We find that their expression time profiles can be accurately reproduced by a minimal model of a two-gene transcriptional feedback loop. Remarkably, best adjustment of data recorded under light/dark alternation is obtained when assuming that the oscillator is not coupled to the diurnal cycle. This suggests that coupling to light is confined to specific time intervals and has no dynamical effect when the oscillator is entrained by the diurnal cycle. This intringuing property may reflect a strategy to minimize the impact of fluctuations in daylight intensity on the core circadian oscillator, a type of perturbation that has been rarely considered when assessing the robustness of circadian clocks

Equivalence of the Siegert-pseudostate and Lagrange-mesh R-matrix methods

Siegert pseudostates are purely outgoing states at some fixed point expanded over a finite basis. With discretized variables, they provide an accurate description of scattering in the s wave for short-range potentials with few basis states. The R-matrix method combined with a Lagrange basis, i.e. functions which vanish at all points of a mesh but one, leads to simple mesh-like equations which also allow an accurate description of scattering. These methods are shown to be exactly equivalent for any basis size, with or without discretization. The comparison of their assumptions shows how to accurately derive poles of the scattering matrix in the R-matrix formalism and suggests how to extend the Siegert-pseudostate method to higher partial waves. The different concepts are illustrated with the Bargmann potential and with the centrifugal potential. A simplification of the R-matrix treatment can usefully be extended to the Siegert-pseudostate method.Comment: 19 pages, 1 figur

The Dynamics of Zeroth-Order Ultrasensitivity: A Critical Phenomenon in Cell Biology

It is well known since the pioneering work of Goldbeter and Koshland [Proc. Natl. Acad. Sci. USA, vol. 78, pp. 6840-6844 (1981)] that cellular phosphorylation- dephosphorylation cycle (PdPC), catalyzed by kinase and phosphatase under saturated condition with zeroth order enzyme kinetics, exhibits ultrasensitivity, sharp transition. We analyse the dynamics aspects of the zeroth order PdPC kinetics and show a critical slowdown akin to the phase transition in condensed matter physics. We demonstrate that an extremely simple, though somewhat mathematically "singular" model is a faithful representation of the ultrasentivity phenomenon. The simplified mathematical model will be valuable, as a component, in developing complex cellular signaling network theory as well as having a pedagogic value.Comment: 8 pages, 3 figure

Threshold responses to morphogen gradients by zero-order ultrasensitivity

Translating a graded morphogen distribution into tight response borders is central to all developmental processes. Yet, the molecular mechanisms generating such behavior are poorly understood. During patterning of the Drosophila embryonic ventral ectoderm, a graded mitogen-activated protein kinase (MAPK) activation is converted into an all-or-none degradation switch of the Yan transcriptional repressor. Replacing the cardinal phosphorylated amino acid of Yan by a phosphomimetic residue allowed its degradation in a MAPK-independent manner, consistent with Yan phosphorylation being the critical event in generating the switch. Several alternative threshold mechanisms that could, in principle, be realized by this phosphorylation, including first order, cooperativity, positive feedback and zero-order ultrasensitivity, were analyzed. We found that they can be distinguished by their kinetics and steady-state responses to Yan overexpression. In agreement with the predictions for zero-order kinetics, an increase in Yan levels did not shift the degradation border, but significantly elevated the time required to reach steady state. We propose that a reversible loop of Yan phosphorylation implements a zero-order ultrasensitivity-like threshold mechanism, with the capacity to form sharp thresholds that are independent of the level of Yan

Competing Patterns of Signaling Activity in Dictyostelium discoideum

Quantitative experiments are described on spatio-temporal patterns of coherent chemical signaling activity in populations of {\it Dictyostelium discoideum} amoebae. We observe competition between spontaneously firing centers and rotating spiral waves that depends strongly on the overall cell density. At low densities, no complete spirals appear and chemotactic aggregation is driven by periodic concentric waves, whereas at high densities the firing centers seen at early times nucleate and are apparently entrained by spiral waves whose cores ultimately serve as aggregation centers. Possible mechanisms for these observations are discussed.Comment: 10 pages, RevTeX, 4 ps figures, accepted in PR

The statistical mechanics of complex signaling networks : nerve growth factor signaling

It is becoming increasingly appreciated that the signal transduction systems used by eukaryotic cells to achieve a variety of essential responses represent highly complex networks rather than simple linear pathways. While significant effort is being made to experimentally measure the rate constants for individual steps in these signaling networks, many of the parameters required to describe the behavior of these systems remain unknown, or at best, estimates. With these goals and caveats in mind, we use methods of statistical mechanics to extract useful predictions for complex cellular signaling networks. To establish the usefulness of our approach, we have applied our methods towards modeling the nerve growth factor (NGF)-induced differentiation of neuronal cells. Using our approach, we are able to extract predictions that are highly specific and accurate, thereby enabling us to predict the influence of specific signaling modules in determining the integrated cellular response to the two growth factors. We show that extracting biologically relevant predictions from complex signaling models appears to be possible even in the absence of measurements of all the individual rate constants. Our methods also raise some interesting insights into the design and possible evolution of cellular systems, highlighting an inherent property of these systems wherein particular ''soft'' combinations of parameters can be varied over wide ranges without impacting the final output and demonstrating that a few ''stiff'' parameter combinations center around the paramount regulatory steps of the network. We refer to this property -- which is distinct from robustness -- as ''sloppiness.''Comment: 24 pages, 10 EPS figures, 1 GIF (makes 5 multi-panel figs + caption for GIF), IOP style; supp. info/figs. included as brown_supp.pd

Vortex Glass and Vortex Liquid in Oscillatory Media

We study the disordered, multi-spiral solutions of two-dimensional homogeneous oscillatory media for parameter values at which the single spiral/vortex solution is fully stable. In the framework of the complex Ginzburg-Landau (CGLE) equation, we show that these states, heretofore believed to be static, actually evolve on ultra-slow timescales. This is achieved via a reduction of the CGLE to the evolution of the sole vortex position and phase coordinates. This true defect-mediated turbulence occurs in two distinct phases, a vortex liquid characterized by normal diffusion of individual spirals, and a slowly relaxing, intermittent, ``vortex glass''.Comment: 4 pages, 2 figures, submitted to Physical Review Letter