Mutually Exclusive Spiky Pattern and Segmentation Modeled by the Five-Component Meinhardt--Gierer System.

SIAM Journal of Applied Mathematics 01/2008; 69:419-452. DOI: 10.1137/060673138
Source: DBLP

ABSTRACT We consider the flve-component Meinhardt-Gierer model for mutually exclusive patterns and seg- mentation which was proposed in (11). We prove rigorous results on the existence and stability of mutually exclusive spikes which are located in difierent positions for the two activators. Su-cient conditions for existence and stability are derived, which depend in particular on the relative size of the various difiusion constants. Our main analytical methods are the Liapunov-Schmidt reduction and nonlocal eigenvalue problems. The analytical results are conflrmed by numerical simulations. We analyze the flve-component Meinhardt-Gierer system whose components are two activators and one inhibitor as well as two lateral activators. It has been introduced and very successfully used in various modeling aspects by Meinhardt and Gierer (11). In particular, it can explain the phenomenon of mutual exclusion and handle segmentation in the simplest case of two difierent segments. This model has been reviewed and its many implications have been discussed in detail by Meinhardt in Chapter 12 of (10). The most important features of this system can be highlighted as lateral activation of mutually exclusive states. To each of the local activators a lateral activator is associated in a spatially nonlocal and time-delayed way. The consequence of the presence of the two lateral activators in the system is the possibility to have stable patterns which for the two activators are mutually exclusive, or in other words, the patterns for the two activators are located in difierent positions. It is clear that mutually exclusive patterns are not possible for a three-component system with only two activators and one inhibitor since mutually exclusive patterns for the two activators could destabilize each other in various ways. Therefore the lateral activators are needed. Numerical simulations of mutually exclusive patterns have been performed in (11), (10). Many interesting features have been discovered and explained but those works do not give analytical solutions and they are not mathematically rigorous. To obtain mathematically rigorous results, in this study we show the existence and stability of mutually exclusive spikes in such a system. The overall feedback mechanism of the system can be summarized as follows: Lateral activation is coupled with self-activation and overall inhibition. We will explain this in more detail after the system has been formulated quantitatively. A widespread pattern in biology is segmentation. The mutual exclusion efiect described in this paper is a special case of segmentation where only two difierent segments are present. Examples for biological segmentation are the body segments of insects or the segments of insect legs. The segments usually resemble each other strongly, but on the other hand they are difierent from each other. Segments may for example have an internal polarity which is often visible by bristles or hairs. This internal pattern within a segment depends on the position of the segment within the sequence in its natural state. In some biological cases a good understanding of how segment position and internal structure are related has been obtained. One famous example are surgical experiments on insects, e.g. for cockroach legs. Creating a discontinuity in the normal neighborhood of structures by cutting a leg and pasting one piece to the end of another partial leg creates a discontinuity in the segment structure as some segments are missing their natural neighbors. This forces the emergence of new stable patterns in the cockroach leg such that all segments get back their natural neighbors. However, the resulting pattern can be very difierent from any naturally occurring pattern.

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    ABSTRACT: We study a cooperative consumer chain model which consists of one producer and two consumers. It is an extension of the Schnakenberg model suggested in Gierer and Meinhardt [Kybernetik (Berlin), 12:30-39, 1972] and Schnakenberg (J Theor Biol, 81:389-400, 1979) for which there is only one producer and one consumer. In this consumer chain model there is a middle component which plays a hybrid role: it acts both as consumer and as producer. It is assumed that the producer diffuses much faster than the first consumer and the first consumer much faster than the second consumer. The system also serves as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir. In the small diffusion limit we construct cluster solutions in an interval which have the following properties: The spatial profile of the third component is a spike. The profile for the middle component is that of two partial spikes connected by a thin transition layer. The first component in leading order is given by a Green's function. In this profile multiple scales are involved: The spikes for the middle component are on the small scale, the spike for the third on the very small scale, the width of the transition layer for the middle component is between the small and the very small scale. The first component acts on the large scale. To the best of our knowledge, this type of spiky pattern has never before been studied rigorously. It is shown that, if the feedrates are small enough, there exist two such patterns which differ by their amplitudes.We also study the stability properties of these cluster solutions. We use a rigorous analysis to investigate the linearized operator around cluster solutions which is based on nonlocal eigenvalue problems and rigorous asymptotic analysis. The following result is established: If the time-relaxation constants are small enough, one cluster solution is stable and the other one is unstable. The instability arises through large eigenvalues of order [Formula: see text]. Further, there are small eigenvalues of order [Formula: see text] which do not cause any instabilities. Our approach requires some new ideas: (i) The analysis of the large eigenvalues of order [Formula: see text] leads to a novel system of nonlocal eigenvalue problems with inhomogeneous Robin boundary conditions whose stability properties have been investigated rigorously. (ii) The analysis of the small eigenvalues of order [Formula: see text] needs a careful study of the interaction of two small length scales and is based on a suitable inner/outer expansion with rigorous error analysis. It is found that the order of these small eigenvalues is given by the smallest diffusion constant [Formula: see text].
    Journal of Mathematical Biology 11/2012; · 2.37 Impact Factor


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