Effects of stochastic time-delayed feedback on a dynamical system modeling a chemical oscillator.

We examine how stochastic time-delayed negative feedback affects the dynamical behavior of a model oscillatory reaction. We apply constant and stochastic time-delayed negative feedbacks to a point Field-Körös-Noyes photosensitive oscillator and compare their effects. Negative feedback is applied in the form of simulated inhibitory electromagnetic radiation with an intensity proportional to the concentration of oxidized light-sensitive catalyst in the oscillator. We first characterize the system under nondelayed inhibitory feedback; then we explore and compare the effects of constant (deterministic) versus stochastic time-delayed feedback. We find that the oscillatory amplitude, frequency, and waveform are essentially preserved when low-dispersion stochastic delayed feedback is used, whereas small but measurable changes appear when a large dispersion is applied.

Steady state equivalence among autocatalytic peroxidase-oxidase reactions.

Peroxidase-oxidase is an enzymatic reaction that can exhibit dynamical scenarios such as bistability, sustained oscillations, and Shilnikov chaos. In this work, we apply the chemical reaction network theory approach to find kinetic constants such that the associated mass action kinetics ordinary differential equations induced by three four dimensional structurally different enzymatic reaction systems can support the same steady states for several chemical species despite differences in their chemical nature.

Extended source model for diffusive coupling.

Motivated by the prevailing approach to diffusion coupling phenomena which considers point-like diffusing sources, we derived an analogous expression for the concentration rate of change of diffusively coupled extended containers. The proposed equation, together with expressions based on solutions to the diffusion equation, is intended to be applied to the numerical solution of systems exclusively composed of ordinary differential equations, however is able to account for effects due the finite size of the coupled sources.

Vesicle Motion during Sustained Exocytosis in Chromaffin Cells: Numerical Model Based on Amperometric Measurements.

Chromaffin cells release catecholamines by exocytosis, a process that includes vesicle docking, priming and fusion. Although all these steps have been intensively studied, some aspects of their mechanisms, particularly those regarding vesicle transport to the active sites situated at the membrane, are still unclear. In this work, we show that it is possible to extract information on vesicle motion in Chromaffin cells from the combination of Langevin simulations and amperometric measurements. We developed a numerical model based on Langevin simulations of vesicle motion towards the cell membrane and on the statistical analysis of vesicle arrival times. We also performed amperometric experiments in bovine-adrenal Chromaffin cells under Ba2+ stimulation to capture neurotransmitter releases during sustained exocytosis. In the sustained phase, each amperometric peak can be related to a single release from a new vesicle arriving at the active site. The amperometric signal can then be mapped into a spike-series of release events. We normalized the spike-series resulting from the current peaks using a time-rescaling transformation, thus making signals coming from different cells comparable. We discuss why the obtained spike-series may contain information about the motion of all vesicles leading to release of catecholamines. We show that the release statistics in our experiments considerably deviate from Poisson processes. Moreover, the interspike-time probability is reasonably well described by two-parameter gamma distributions. In order to interpret this result we computed the vesicles' arrival statistics from our Langevin simulations. As expected, assuming purely diffusive vesicle motion we obtain Poisson statistics. However, if we assume that all vesicles are guided toward the membrane by an attractive harmonic potential, simulations also lead to gamma distributions of the interspike-time probability, in remarkably good agreement with experiment. We also show that including the fusion-time statistics in our model does not produce any significant changes on the results. These findings indicate that the motion of the whole ensemble of vesicles towards the membrane is directed and reflected in the amperometric signals. Our results confirm the conclusions of previous imaging studies performed on single vesicles that vesicles' motion underneath plasma membranes is not purely random, but biased towards the membrane.

Monitoring lipase/esterase activity by stopped flow in a sequential injection analysis system using p-nitrophenyl butyrate.

Lipases and esterases are biocatalysts used at the laboratory and industrial level. To obtain the maximum yield in a bioprocess, it is important to measure key variables, such as enzymatic activity. The conventional method for monitoring hydrolytic activity is to take out a sample from the bioreactor to be analyzed off-line at the laboratory. The disadvantage of this approach is the long time required to recover the information from the process, hindering the possibility to develop control systems. New strategies to monitor lipase/esterase activity are necessary. In this context and in the first approach, we proposed a lab-made sequential injection analysis system to analyze off-line samples from shake flasks. Lipase/esterase activity was determined using p-nitrophenyl butyrate as the substrate. The sequential injection analysis allowed us to measure the hydrolytic activity from a sample without dilution in a linear range from 0.05-1.60 U/mL, with the capability to reach sample dilutions up to 1000 times, a sampling frequency of five samples/h, with a kinetic reaction of 5 min and a relative standard deviation of 8.75%. The results are promising to monitor lipase/esterase activity in real time, in which optimization and control strategies can be designed.

Attractors generated from switching unstable dissipative systems.

In this paper, we present a class of 3-D unstable dissipative systems, which are stable in two components but unstable in the other one. This class of systems is motivated by whirls, comprised of switching subsystems, which yield strange attractors from the combination of two unstable "one-spiral" trajectories by means of a switching rule. Each one of these trajectories moves around two hyperbolic saddle equilibrium points. Both theoretical and numerical results are provided for verification and demonstration.

A model for renal arterial branching based on graph theory.

The kidney is one of the most complicated organs in terms of structure and physiology, in part because it is highly vascularized. The renal vascular development occurs through two mechanisms that sometimes overlap: vasculogenesis and angiogenesis. Here, we consider angiogenesis to model the renal arterial tree with the two processes of vascular angiogenesis: sprouting and splitting. We recognize the vessels are not tubes with ends that get glued but physiological factors are relevant into the vascular development. Our contribution integrates the graph theory and physiological information to derive a quantitative model for the vascular tree in the sense that the vertices and edges represent, respectively, a branching point and a vessel. From such a premise, development of the arterial vascular tree of the kidney is mathematically expressed, including physiological processes as the effect of the vascular endothelial growth factor (VEGF) on the vessel length. A definition of the graph is used to visualize the topology of vascular tree in kidney providing physiological information into the edges. Thus, renal arterial branching is modeled as a graph where edges are labeled and oriented.

Pattern formation by two precipitated species during solvent evaporation.

We investigate the pattern formation produced by precipitated species during solvent evaporation through the numerical solution of a set of partial differential equations that account for the mechanisms of evaporation, diffusion, and precipitation. A pattern is formed because solvent evaporation provokes precipitation of species near the border of the system producing ringlike depositions from the edge to the center. Solvent evaporation is modeled as occurs with a liquid drop on a surface. The spacing between rings and its width are constant and roughly constant, respectively. Pattern formation follows the evaporation process inducing trends on pattern formation that are different to those produced by the precipitation of two species in a diffusive front (Liesegang rings). The spatial structure of rings under solvent evaporation is similar to those observed during solvent evaporation on two oppositely charged colloids and is attributable to the competition between precipitation and evaporation processes.

A family of driving forces to suppress chaos in jerk equations: Laplace domain design.

A family of driving forces is discussed in the context of chaos suppression in the Laplace domain. This idea can be attained by increasing the order of the polynomial in the expressions of the driving force to account for the robustness and/or the performance of the closed loop. The motivation arises from the fact that chaotic systems can be controlled by increasing the order of the Laplace controllers even to track arbitrary orbits. However, a larger order in the driving forces can induce an undesirable frequency response, and the control efforts can result in either peaking or large energy accumulation. We overcame these problems by showing that considering the frequency response (interpreted by norms), the closed-loop execution can be improved by designing the feedback suppressor in the Laplace domain. In this manner, the stabilization of the chaotic behavior in jerk-like systems is achieved experimentally. Jerk systems are particularly sensitive to control performance (and robustness issues) because the acceleration time-derivative is involved in their models. Thus, jerky systems are especially helped by a robust control design.

Synchronization of chaotic systems with different order.

The chaotic synchronization of third-order systems and second-order driven oscillator is studied in this paper. Such a problem is related to synchronization of strictly different chaotic systems. We show that dynamical evolution of second-order driven oscillators can be synchronized with the canonical projection of a third-order chaotic system. In this sense, it is said that synchronization is achieved in reduced order. Duffing equation is chosen as slave system whereas Chua oscillator is defined as master system. The synchronization scheme has nonlinear feedback structure. The reduced-order synchronization is attained in a practical sense, i.e., the difference e=x(3)-x(1)(') is close to zero for all time t> or =t(0)> or =0, where t(0) denotes the time of the control activation.

An extension to chaos control via Lie derivatives: Fully linearizable systems.

The technique of using Lie derivatives to control chaos introduced by Kocarev et al. [Chaos, Solitons Fractals 9, 1359-1366 (1998)] is extended in this contribution. Here, by using Lie derivatives in an extended space state, it is proved that chaos can be practically suppressed via feedback in spite of the Lie derivative being ill-posed at the reference. The main idea is to construct a dynamically equivalent system. In this way, the chaotic system can be practically stabilized around any point of singularity x(0). The Lorenz equation is used as an illustrative example to show the application in the chaos control context. (c) 2002 American Institute of Physics.