A nonlocal eigenvalue problem and the stability of spikes for reaction-diffusion systems with fractional reaction rates

We consider a nonlocal eigenvalue problem which arises in the study of stability of spike solutions for reaction-diffusion systems with fractional reaction rates such as the Sel'kov model, the Gray-Scott system, the hypercycle Eigen and Schuster, angiogenesis, and the generalized Gierer-Meinhardt system. We give some sufficient and explicit conditions for stability by studying the corresponding nonlocal eigenvalue problem in a new range of parameters

Novel Aspects in Pattern Formation Arise from Coupling Turing Reaction-Diffusion and Chemotaxis

Recent experimental studies on primary hair follicle formation and feather bud morphogenesis indicate a coupling between Turing-type diffusion driven instability and chemotactic patterning. Inspired by these findings we develop and analyse a mathematical model that couples chemotaxis to a reaction-diffusion system exhibiting diffusion-driven (Turing) instability. While both systems, reaction-diffusion systems and chemotaxis, can independently generate spatial patterns, we were interested in how the coupling impacts the stability of the system, parameter region for patterning, pattern geometry, as well as the dynamics of pattern formation. We conduct a classical linear stability analysis for different model structures, and confirm our results by numerical analysis of the system. Our results show that the coupling generally increases the robustness of the patterning process by enlarging the pattern region in the parameter space. Concerning time scale and pattern regularity, we find that an increase in the chemosensitivity can speed up the patterning process for parameters inside and outside of the Turing space, but generally reduces spatial regularity of the pattern. Interestingly, our analysis indicates that pattern formation can also occur when neither the Turing nor the chemotaxis system can independently generate pattern. On the other hand, for some parameter settings, the coupling of the two processes can extinguish the pattern formation, rather than reinforce it. These theoretical findings can be used to corroborate the biological findings on morphogenesis and guide future experimental studies. From a mathematical point of view, this work sheds a light on coupling classical pattern formation systems from the parameter space perspective

Models of self-organization in biological development

Bibliography: p. 297-320.In this thesis we thus wish to consider the concept of self-organization as an overall paradigm within which various theoretical approaches to the study of development may be described and evaluated. In the process, an attempt is made to give a fair and reasonably comprehensive overview of leading modelling approaches in developmental biology, with particular reference to self-organization. The work proceeds from a physical or mathematical perspective, but not unduly so - the major mathematical derivations and results are relegated to appendices - and attempts to fill a perceived gap in the extant review literature, in its breadth and attempted impartiality of scope. A characteristic of the present account is its markedly interdisciplinary approach: it seeks to place self-organization models that have been proposed for biological pattern formation and morphogenesis both within the necessary experimentally-derived biological framework, and in the wider physical context of self-organization and the mathematical techniques that may be employed in its study. Hence the thesis begins with appropriate introductory chapters to provide the necessary background, before proceeding to a discussion of the models themselves. It should be noted that the work is structured so as to be read sequentially, from beginning to end; and that the chapters in the main text were designed to be understood essentially independently of the appendices, although frequent references to the latter are given. In view of the vastness of the available information and literature on developmental biology, a working knowledge of embryological principles must be assumed. Consequently, rather than attempting a comprehensive introduction to experimental embryology, chapter 2 presents just a few biological preliminaries, to 'set the scene', outlining some of the major issues that we are dealing with, and sketching an indication of the current status of knowledge and research on development. The chapter is aimed at furnishing the necessary biological, experimental background, in the light of which the rest of the thesis should be read, and which should indeed underpin and motivate any theoretical discussions. We encounter the different hierarchical levels of description in this chapter, as well as some of the model systems whose experimental study has proved most fruitful, some of the concepts of experimental embryology, and a brief reference to some questions that will not be addressed in this work. With chapter 3, we temporarily move away from developmental biology, and consider the wider physical and mathematical concepts related to the study of self-organization. Here we encounter physical and chemical examples of spontaneous structure formation, thermodynamic considerations, and different approaches to the description of complexity. Mathematical approaches to the dynamical study of self-organization are also introduced, with specific reference to reaction-diffusion equations, and we consider some possible chemical and biochemical realizations of self-organizing kinetics. The chapter may be read in conjunction with appendix A, which gives a somewhat more in-depth study of reaction-diffusion equations, their analysis and properties, as an example of the approach to the analysis of self-organizing dynamical systems and mathematically-formulated models. Appendix B contains a more detailed discussion of the Belousov-Zhabotinskii reaction, which provides a vivid chemical paradigm for the concepts of symmetry-breaking and self-organization. Chapter 3 concludes with a brief discussion of a model biological system, the cellular slime mould, which displays rudimentary development and has thus proved amenable to detailed study and modelling. The following two chapters form the core of the thesis, as they contain discussions of the detailed application of theoretical concepts and models, largely based on self-organization, to various developmental situations. We encounter a diversity of models which has arisen largely in the last quarter century, each of which attempts to account for some aspect of biological pattern formation and morphogenesis; an aim of the discussion is to assess the extent of the underlying unity of these models in terms of the self-organization paradigm. In chapter 4 chemical pre-patterns and positional information are considered, without the overt involvement of cells in the patterning. In chapter 5, on the other hand, cellular interactions and activities are explicitly taken into account; this chapter should be read together with appendix C, which contains a brief introduction to the mathematical formulation and analysis of some of the models discussed. The penultimate chapter, 6, considers two other approaches to the study of development; one of these has faded away, while the other is still apparently in the ascendant. The assumptions underlying catastrophe theory, the value of its applications to developmental biology and the reasons for its decline in popularity, are considered. Lastly, discrete approaches, including the recently fashionable cellular automata, are dealt with, and the possible roles of rule-based interactions, such as of the so-called L-systems, and of fractals and chaos are evaluated. Chapter 7 then concludes the thesis with a brief assessment of the value of the self-organization concept to the study of biological development

The Gierer-Meinhardt system in various settings.

Tse, Wang Hung.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 75-77).Abstract also in Chinese.Chapter 1 --- Introduction --- p.1Chapter 2 --- On bounded interval with n jumps in inhibitor diffusivity --- p.3Chapter 2.1 --- Introduction --- p.3Chapter 2.2 --- Preliminaries --- p.5Chapter 2.3 --- Review of previous results in the two segment case: interior spike and spike near the jump discontinuity of the diffusion coefficient --- p.7Chapter 2.4 --- The construction and analysis of spiky steady-state solutions --- p.9Chapter 2.5 --- Stability Analysis --- p.10Chapter 2.6 --- Spikes near the jump discontinuity xb of the inhibitor diffusivity --- p.11Chapter 2.7 --- Stability Analysis II: Small Eigenvalues of the Spike near the Jump --- p.16Chapter 2.8 --- Existence of interior spikes for N segments --- p.20Chapter 2.9 --- Existence of a spike near a jump for N segments --- p.24Chapter 2.10 --- Appendix: The Green´ةs function for three segments --- p.25Chapter 3 --- On a compact Riemann surface without boundary --- p.30Chapter 3.1 --- Introduction --- p.30Chapter 3.2 --- Some Preliminaries --- p.35Chapter 3.3 --- Existence --- p.43Chapter 3.4 --- Refinement of Approximate Solution --- p.50Chapter 3.5 --- Stability --- p.52Chapter 3.6 --- Appendix I: Expansion of the Laplace-Beltrami Operator --- p.67Chapter 3.7 --- Appendix II: Some Technical Calculations --- p.7

Concentration-Dependent Domain Evolution in Reaction–Diffusion Systems

Pattern formation has been extensively studied in the context of evolving (time-dependent) domains in recent years, with domain growth implicated in ameliorating problems of pattern robustness and selection, in addition to more realistic modelling in developmental biology. Most work to date has considered prescribed domains evolving as given functions of time, but not the scenario of concentration-dependent dynamics, which is also highly relevant in a developmental setting. Here, we study such concentration-dependent domain evolution for reaction–diffusion systems to elucidate fundamental aspects of these more complex models. We pose a general form of one-dimensional domain evolution and extend this to N-dimensional manifolds under mild constitutive assumptions in lieu of developing a full tissue-mechanical model. In the 1D case, we are able to extend linear stability analysis around homogeneous equilibria, though this is of limited utility in understanding complex pattern dynamics in fast growth regimes. We numerically demonstrate a variety of dynamical behaviours in 1D and 2D planar geometries, giving rise to several new phenomena, especially near regimes of critical bifurcation boundaries such as peak-splitting instabilities. For sufficiently fast growth and contraction, concentration-dependence can have an enormous impact on the nonlinear dynamics of the system both qualitatively and quantitatively. We highlight crucial differences between 1D evolution and higher-dimensional models, explaining obstructions for linear analysis and underscoring the importance of careful constitutive choices in defining domain evolution in higher dimensions. We raise important questions in the modelling and analysis of biological systems, in addition to numerous mathematical questions that appear tractable in the one-dimensional setting, but are vastly more difficult for higher-dimensional models

Dynamics of pulse solutions in Gierer-Meinhardt model with time dependent diffusivity

Dispersive processes with a time dependent diffusivity appear in a plethora of physical systems. Most often a solution is attained for a predefined form of diffusion coefficient D(t). Here existence of pulse solutions with an arbitrary time dependence thereof is proved for the Gierer-Meinhardt model with three types of transport: regular diffusion, sub-diffusion and L´evy flights. Admission of a solution of the classical pulse shape, but for an unencumbered form of D(t) is a valuable property that allows to study phenomena of the ilk observed in various ostensibly unrelated applications. Closed form solutions are obtained for some pulse constellations. Transitions between periods of nearly constant diffusivities trigger respective cross-over between counterpart solutions known for a constant diffusivity, thereupon exhibiting otherwise unattainable behaviour, qualitatively reconstructing observable evolution peculiarities of tagged molecular structures, such as essential slowing down or speeding up during various stages of motion, inexplicable with a single constant diffusion coefficient.Peer reviewedanomalous diffusionnon-linear pattern formationreaction - diffusionGierer-Meinhardt modelpulse solutiontransient diffusion regime

Establishment of an Institute for Fusion Studies. Technical progress report, November 1, 1994--October 31, 1995

The Institute for Fusion Studies is a national center for theoretical fusion plasma physics research. Its purposes are to (1) conduct research on theoretical questions concerning the achievement of controlled fusion energy by means of magnetic confinement--including both fundamental problems of long-range significance, as well as shorter-term issues; (2) serve as a national and international center for information exchange by hosting exchange visits, conferences, and workshops; and (3) train students and postdoctoral research personnel for the fusion energy program and plasma physics research areas. During FY 1995, a number of significant scientific advances were achieved at the IFS, both in long-range fundamental problems as well as in near-term strategic issues, consistent with the Institute`s mandate. Examples of these achievements include, for example, tokamak edge physics, analytical and computational studies of ion-temperature-gradient-driven turbulent transport, alpha-particle-excited toroidal Alfven eigenmode nonlinear behavior, sophisticated simulations for the Numerical Tokamak Project, and a variety of non-tokamak and non-fusion basic plasma physics applications. Many of these projects were done in collaboration with scientists from other institutions. Research discoveries are briefly described in this report

Computational Study of Acid-Catalyzed Reactions in Zeolites

Aldol condensation is a very important reaction in organic synthesis because it leads to the formation of C-C bonds. Because of that, the use of different catalysts and in particular the use of zeolites for the catalysis of this reaction has been previously studied. The first step of aldol condensation is the the acid- catalyzed keto-enol tautomerization of the aldehyde or ketone. In this thesis, we study all the possible locations of BA sites in the vicinity of each of the defined inequivalent T site positions in FER and MOR zeolites and establish the most stable location of proton siting at force field and periodic DFT level. The reactions involving the small carbonyl compounds, acetaldehyde and acetone, are studied at the specific acid site locations in both zeolites in order to discover which are the active sites that can stabilize the reactants and therefore how the existing catalyst can be improved. Besides the H-bonding and other interactions, the confinement effect are of equal importance in the determination of factors that influence the reactivity of these complexes. In order to understand the keto-enol tautomeric mechanism in zeolites and identify the transition states, constrained geometry optimizations were performed of acetaldehyde in FER and MOR using periodic DFT. From this we determined the formation of enol product via and an one and two-step concerted mechanism. The calculations reveal that C-β deprotonation is the kinetic bottleneck for enol formation. The concerted mechanism was performed at each of the inequivalent T position in both zeolites. We found that proton transfer is a consequence of a cooperation between the acid site and its total environment acting on the molecular adsorbate. The adsorption and stability of the intermediates is dependent upon the heterogeneity of acid sites and their local geometry, the pore channels and cavities and interactions such as dispersive and H-bonding that do not reflect, and are often independent of, acid strength. Thirdly, we studied the keto-enol tautomerization of acetaldehyde in FER and MOR in the presence of a single water or methanol molecular and found the activation barriers to reduce further with an increase in stability of the adsorbed enol form with larger reverse barriers in the larger pores of FER. We establish the importance of the specific role of the H-bonding using these solvent molecules. In the smaller pores of MOR, the presence of a solvent supresses the catalytic interconversion due to steric repulsion. Lastly, we explored the location of monovalent alkali ions in FER and found that the most stable location of the cations is in the FER cavity and dependent upon the Al position. We studied the effect of hydration on the mobility of the cesium ion in the cavities of FER using static DFT calculations supplemented with ab initio molecular dynamics. We estabished the cesium ion prefers to coordinate with the framework oxygens of the zeolite rather than oxygen atoms of the water molecules as well as the position of the cesium ion is affected by the Al siting. The coordination number of cesium is ~ 10 with the ion interacting with only 1-3 water molecules. In addition, we identified a self-organization of water molecules across the channels forming a H-bonding network