Silvina Ponce Dawson

Antimonotonicity: inevitable reversals of period-doubling cascades

Abstract In many common nonlinear dynamical systems depending on a parameter, it is shown that periodic orbit creating cascades must be accompanied by periodic orbit annihilating cascades as the parameter is varied. Moreover, reversals from a periodic orbit creating cascade to a periodic orbit annihilating one must occur infinitely often in the vicinity of certain common parameter values. It is also demonstrated that these inevitable reversals are indeed observable in specific chaotic systems.

Turing patterns inside cells

Concentration gradients inside cells are involved in key processes such as cell division and morphogenesis. Here we show that a model of the enzymatic step catalized by phosphofructokinase (PFK), a step which is responsible for the appearance of homogeneous oscillations in the glycolytic pathway, displays Turing patterns with an intrinsic length-scale that is smaller than a typical cell size. All the parameter values are fully consistent with classic experiments on glycolytic oscillations and equal diffusion coefficients are assumed for ATP and ADP. We identify the enzyme concentration and the glycolytic flux as the possible regulators of the pattern. To the best of our knowledge, this is the first closed example of Turing pattern formation in a model of a vital step of the cell metabolism, with a built-in mechanism for changing the diffusion length of the reactants, and with parameter values that are compatible with experiments. Turing patterns inside cells could provide a check-point that combines mechanical and biochemical information to trigger events during the cell division process.

Hindered cytoplasmic diffusion of inositol trisphosphate restricts its cellular range of action

Slow diffusion of IP3 in mammalian cells results in a local rather than a global signal. A new paradigm for IP3 signaling Receptors that activate phospholipase C generate the second messenger inositol trisphosphate (IP3). IP3 stimulates calcium release from the endoplasmic and sarcoplasmic reticulum, thereby shaping calcium signals in cells. Dickinson et al. triggered the focal release of IP3 in animal cells and measured calcium “puffs”—intense, localized increases in calcium released from the endoplasmic reticulum. They found that IP3 diffuses much more slowly within cells than had been originally measured in vitro using oocyte cytoplasmic extracts. Thus, rather than functioning as a global cellular signal, IP3 can produce local signals, which increases the complexity of information that can be encoded by cells in response to stimuli that activate receptors that generate this second messenger. The range of action of intracellular messengers is determined by their rates of diffusion and degradation. Previous measurements in oocyte cytoplasmic extracts indicated that the Ca2+-liberating second messenger inositol trisphosphate (IP3) diffuses with a coefficient (~280 μm2 s−1) similar to that in water, corresponding to a range of action of ~25 μm. Consequently, IP3 is generally considered a “global” cellular messenger. We reexamined this issue by measuring local IP3-evoked Ca2+ puffs to monitor IP3 diffusing from spot photorelease in neuroblastoma cells. Fitting these data by numerical simulations yielded a diffusion coefficient (≤10 μm2 s−1) about 30-fold slower than that previously reported. We propose that diffusion of IP3 in mammalian cells is hindered by binding to immobile, functionally inactive receptors that were diluted in oocyte extracts. The predicted range of action of IP3 (<5 μm) is thus smaller than the size of typical mammalian cells, indicating that IP3 should better be considered as a local rather than a global cellular messenger.

Intra-Cluster Percolation of Calcium Signals

Calcium signals are involved in a large variety of physiological processes. Their versatility relies on the diversity of spatio-temporal behaviors that the calcium concentration can display. Calcium entry through inositol 1,4,5-trisphosphate (IP) receptors (IPRs) is a key component that participates in both local signals such as “puffs” and in global waves. IPRs are usually organized in clusters on the membrane of the endoplasmic reticulum and their spatial distribution has important effects on the resulting signal. Recent high resolution observations [1] of Ca puffs offer a window to study intra-cluster organization. The experiments give the distribution of the number of IPRs that open during each puff without much processing. Here we present a simple model with which we interpret the experimental distribution in terms of two stochastic processes: IP binding and unbinding and Ca-mediated inter-channel coupling. Depending on the parameters of the system, the distribution may be dominated by one or the other process. The transition between both extreme cases is similar to a percolation process. We show how, from an analysis of the experimental distribution, information can be obtained on the relative weight of the two processes. The largest distance over which Ca-mediated coupling acts and the density of IP-bound IPRs of the cluster can also be estimated. The approach allows us to infer properties of the interactions among the channels of the cluster from statistical information on their emergent collective behavior.

Long-time behavior of the N -finger solution of the Laplacian growth equation

Abstract The non-singular N -finger solutions of the Laplacian Growth Equation, Im [ ovbar | ∂ f ( x , t )∂ t ) (∂ f ( x , t )/∂ x )] = 1, describing the motion of the interface in numerous non-equilibrium processes, such as dendritic growth, flows through porous media, electrodeposition, etc., is analyzed. The motion of the interface is described by N + 1 moving singularities (simple poles) in the upper-half of an auxiliar “mathematical plane”. In the long-time limit these singularities tend to the real axis, following an exponential law. Meanwhile, the physical interface develops at most N separated fingers. In the case of enough separation, each of the gaps between fingers corresponds to one singularity while each finger is locally similar to the Saffman-Taylor one. The analogy with the N -soliton solutions of exactly integrable PDEs, such as Korteweg-de Vries, Nonlinear Schrodinger, and sine-Gordon equations, is discussed. Using the asymptotic properties of the N -finger solution, canonical variables of “action-angle”-type are introduced.

Saltatory and Continuous Calcium Waves and the Rapid Buffering Approximation

Calcium waves propagate inside cells due to a regenerative mechanism known as calcium-induced calcium release. Buffer-mediated calcium diffusion in the cytosol plays a crucial role in the process. However, most models of calcium waves either treat buffers phenomenologically or assume that they are in equilibrium with calcium (the rapid buffering approximation). In this article we address the issue of whether this approximation provides a good description of wave propagation. We first compare the timescales present in the problem, and determine the situations in which the equilibrium hypothesis fails. We then present a series of numerical studies based on the simple fire-diffuse-fire model of wave propagation. We find that the differences between the full and reduced descriptions may lead to errors that are above experimental resolution even for relatively fast buffers in the case of saltatory waves. Conversely, in the case of continuous waves, the approximation may give accurate results even for relatively slow buffers.

Collections of heteroclinic cycles in the Kuramoto-Sivashinsky equation

Abstract We study the Kuramoto-Sivashinky equation with periodic boundary conditions in the case of low-dimensional behavior. We analyze the bifurcations that occur in a six-dimensional (6D) approximation of its inertial manifold. We mainly focus on the attracting and structurally stable heteroclinic connections that arise for these parameter values. We reanalyze the ones that were previously described via a 4D reduction to the center-unstable manifold (Ambruster et al., 1988, 1989). We also find a parameter region for which a manifold of structurally stable heteroclinic cycles exist. The existence of such a manifold is responsible for an intermittent behavior which has some features of unpredictability.

The PIM-simplex method: an extension of the PIM-triple method to saddles with an arbitrary number of expanding directions

Abstract The growing interest in non-attracting chaotic sets of high-dimensional dynamical systems requires the development of numerical techniques for their study. The PIM-triple method [H.E. Nusse, J.A. Yorke, Physica D 36 (1989) 137] is a very good method to obtain trajectories on saddles with one positive Lyapunov exponent. In this paper, we combine the same ideas with an algorithm for finding local extrema of multi-variable functions to develop an extension of the method (the PIM-simplex method ) that is suitable for the study of sets with an arbitrary number of expanding directions.

Towards a global classification of excitable reaction–diffusion systems

Abstract Patterns in reaction–diffusion systems near primary bifurcations can be studied locally and classified by means of amplitude equations. This is not possible for excitable reaction–diffusion systems. In this paper we propose a global classification of two variable excitable reaction–diffusion systems. In particular, we claim that the topology of the underlying two-dimensional homogeneous dynamics organizes the systems behavior. We believe that this classification provides a useful tool for the modeling of any real system whose microscopic details are unknown.

An analysis of unidimensional soliton gas models of magnetohydrodynamic turbulence in the solar wind

Two statistical models of Alfven solitons are compared whose evolution is described by the one-dimensional derivative nonlinear Schroedinger (DNLS) equation, contrasting their predictions with solar wind observations. Both distribution functions give the same mean number of solitons. One of the distribution functions follows an exponential law with soliton energy and the other follows a power law; the latter gives better results than the former. Within these models, the variation of the observed spectra with the heliocentric distance can be explained. This variation is related to the radial dependence of the mean level of modulation instability in the medium. 47 refs.