Impact of global interaction and symmetry on pattern selection and bifurcation

Spherical Caps in Cell Polarization

Intracellular symmetry breaking plays a key role in wide range of biological processes, both in single cells and in multicellular organisms. An important class of symmetry-breaking mechanisms relies on the cytoplasm/membrane redistribution of proteins that can autocatalytically promote their own recruitment to the plasma membrane. We present an analytical construction and a comprehensive parametric analysis of stable localized patterns in a reaction-diffusion model of such a mechanism in a spherical cell. The constructed patterns take the form of high-concentration patches localized into spherical caps, similar to the patterns observed in the studies of symmetry breaking in single cells and early embryos.

Localized structures in a nonlinear wave equation stabilized by negative global feedback: one-dimensional and quasi-two-dimensional kinks

We study the evolution of fronts in a nonlinear wave equation with global feedback. This equation generalizes the Klein-Gordon and sine-Gordon equations. Extending previous work, we describe the derivation of an equation governing the front motion, which is strongly nonlinear, and, for the two-dimensional case, generalizes the damped Born-Infeld equation. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the classical case (with no global feedback), leading in most cases to a localized solution; i.e., the stabilization of one phase inside the other. The nature of the localized solution depends on the strength of the global feedback as well as on other parameters of the model.

Asymptotic solutions of stationary patterns in convection-reaction-diffusion systems

We study and map the possible stationary patterns that emerge in a convection-reaction-diffusion (CRD) system using a learning polynomial kinetics. We classify the patterns according to the kinetic model (oscillatory, bistable, or intermediate), the instability nature of the bounded system (convective or absolute), the applied boundary conditions and the system length. This analysis presents a unifying approach to various pattern-inducing mechanisms such as DIFICI (differential flow induced chemical instability), which predicts moving patterns in systems with wide difference of convective rates, and differential capacity patterns, which predicts stationary patterns in cross-flow reactors with a large heat capacity. Previous studies of CRD systems have considered only oscillatory kinetics. Nonlinear analysis, which follows the front motion by approximating its velocity, accounts for the stability of the stationary, whether spatially periodic or other, patterns. The most dominant state is the large-amplitude stationary spatially periodic pattern. With oscillatory kinetics these emerge in the convectively unstable domain above the amplification threshold. The domain of absolute instability, which is determined analytically for unbounded systems, is divided in the bounded system into two subdomains with moving DIFICI waves or stationary patterns. With bistable kinetics the large-amplitude stationary patterns can be sustained only within a narrow subdomain but other stationary patterns, that incorporate several fronts upstream and an "almost homogeneous" tail downstream, can be sustained as well. With intermediate kinetics the large-amplitude axisymmetric stationary patterns may coexist with small-amplitude stationary nonaxisymmetric patterns.

Theory of electrochemical pattern formation

The spatial coupling in electrochemical systems is mediated by ion migration under the influence of the electric field. Since field effects spread very rapidly, every point of an electrode can communicate with every other one practically instantaneously through migration coupling. Based on mathematical potential theory we present the derivation of a generally applicable reaction-migration equation, which describes the coupling via an integral over the whole electrode area. The corresponding coupling function depends only on the geometry of the electrode setup and has been computed for commonly used electrode shapes (such as ring, disk, ribbon or rectangle). The pattern formation observed in electrochemical systems in the bistable, excitable and oscillatory regime can be reproduced in computer simulations, and the types of patterns occurring under different geometries can be rationalized. (c) 2002 American Institute of Physics.

Using weighted global control for stabilizing patterned states

A general method to control a desired patterned state in reaction-diffusion processes is presented. Weighted global control is aimed to keep weighted spatially averaged properties of state variable at preset values. It is shown that weighted global control creates a stable direction in the global space of system states and affects system dynamics globally. We apply it for a specific two-component reaction-diffusion system and show that the desired pattern is attainable for a wide range of the control parameters. (c) 1999 American Institute of Physics.