71 research outputs found
Mode transitions in a model reaction-diffusion system driven by domain growth and noise
Pattern formation in many biological systems takes place during growth of the underlying domain. We study a specific example of a reaction–diffusion (Turing) model in which peak splitting, driven by domain growth, generates a sequence of patterns. We have previously shown that the pattern sequences which are presented when the domain growth rate is sufficiently rapid exhibit a mode-doubling phenomenon. Such pattern sequences afford reliable selection of certain final patterns, thus addressing the robustness problem inherent of the Turing mechanism. At slower domain growth rates this regular mode doubling breaks down in the presence of small perturbations to the dynamics. In this paper we examine the breaking down of the mode doubling sequence and consider the implications of this behaviour in increasing the range of reliably selectable final patterns
Square Patterns and Quasi-patterns in Weakly Damped Faraday Waves
Pattern formation in parametric surface waves is studied in the limit of weak
viscous dissipation. A set of quasi-potential equations (QPEs) is introduced
that admits a closed representation in terms of surface variables alone. A
multiscale expansion of the QPEs reveals the importance of triad resonant
interactions, and the saturating effect of the driving force leading to a
gradient amplitude equation. Minimization of the associated Lyapunov function
yields standing wave patterns of square symmetry for capillary waves, and
hexagonal patterns and a sequence of quasi-patterns for mixed capillary-gravity
waves. Numerical integration of the QPEs reveals a quasi-pattern of eight-fold
symmetry in the range of parameters predicted by the multiscale expansion.Comment: RevTeX, 11 pages, 8 figure
Amplitude equations near pattern forming instabilities for strongly driven ferromagnets
A transversally driven isotropic ferromagnet being under the influence of a
static external and an uniaxial internal anisotropy field is studied. We
consider the dissipative Landau-Lifshitz equation as the fundamental equation
of motion and treat it in ~dimensions. The stability of the spatially
homogeneous magnetizations against inhomogeneous perturbations is analyzed.
Subsequently the dynamics above threshold is described via amplitude equations
and the dependence of their coefficients on the physical parameters of the
system is determined explicitly. We find soft- and hard-mode instabilities,
transitions between sub- and supercritical behaviour, various bifurcations of
higher codimension, and present a series of explicit bifurcation diagrams. The
analysis of the codimension-2 point where the soft- and hard-mode instabilities
coincide leads to a system of two coupled Ginzburg-Landau equations.Comment: LATeX, 25 pages, submitted to Z.Phys.B figures available via
[email protected] in /pub/publications/frank/zpb_95
(postscript, plain or gziped
Stable two-dimensional solitary pulses in linearly coupled dissipative Kadomtsev-Petviashvili equations
A two-dimensional (2D) generalization of the stabilized Kuramoto -
Sivashinsky (KS) system is presented. It is based on the Kadomtsev-Petviashvili
(KP) equation including dissipation of the generic (Newell -- Whitehead --
Segel, NWS) type and gain. The system directly applies to the description of
gravity-capillary waves on the surface of a liquid layer flowing down an
inclined plane, with a surfactant diffusing along the layer's surface.
Actually, the model is quite general, offering a simple way to stabilize
nonlinear waves in media combining the weakly-2D dispersion of the KP type with
gain and NWS dissipation. Parallel to this, another model is introduced, whose
dissipative terms are isotropic, rather than of the NWS type. Both models
include an additional linear equation of the advection-diffusion type, linearly
coupled to the main KP-NWS equation. The extra equation provides for stability
of the zero background in the system, opening a way to the existence of stable
localized pulses. The consideration is focused on the case when the dispersive
part of the system of the KP-I type, admitting the existence of 2D localized
pulses. Treating the dissipation and gain as small perturbations and making use
of the balance equation for the field momentum, we find that the equilibrium
between the gain and losses may select two 2D solitons, from their continuous
family existing in the conservative counterpart of the model (the latter family
is found in an exact analytical form). The selected soliton with the larger
amplitude is expected to be stable. Direct simulations completely corroborate
the analytical predictions.Comment: a latex text file and 16 eps files with figures; Physical Review E,
in pres
The influence of gene expression time delays on Gierer-Meinhardt pattern formation systems
There are numerous examples of morphogen gradients controlling long range signalling in developmental and cellular systems. The prospect of two such interacting morphogens instigating long range self-organisation in biological systems via a Turing bifurcation has been explored, postulated, or implicated in the context of numerous developmental processes. However, modelling investigations of cellular systems typically neglect the influence of gene expression on such dynamics, even though transcription and translation are observed to be important in morphogenetic systems. In particular, the influence of gene expression on a large class of Turing bifurcation models, namely those with pure kinetics such as the Gierer–Meinhardt system, is unexplored. Our investigations demonstrate that the behaviour of the Gierer–Meinhardt model profoundly changes on the inclusion of gene expression dynamics and is sensitive to the sub-cellular details of gene expression. Features such as concentration blow up, morphogen oscillations and radical sensitivities to the duration of gene expression are observed and, at best, severely restrict the possible parameter spaces for feasible biological behaviour. These results also indicate that the behaviour of Turing pattern formation systems on the inclusion of gene expression time delays may provide a means of distinguishing between possible forms of interaction kinetics. Finally, this study also emphasises that sub-cellular and gene expression dynamics should not be simply neglected in models of long range biological pattern formation via morphogens
A Problem Solving Environment for Modelling Stony Coral Morphogenesis
Apart from experimental and theoretical approaches, computer simulation is an important tool in testing hypotheses about stony coral growth. However, the construction and use of such simulation tools needs extensive computational skills and knowledge that is not available to most research biologists. Problem solving environments (PSEs) aim to provide a framework that hides implementation details and allows the user to formulate and analyse a problem in the language of the subject area. We have developed a prototypical PSE to study the morphogenesis of corals using a multi-model approach. In this paper we describe the design and implementation of this PSE, in which simulations of the coral's shape and its environment have been combined. We will discuss the relevance of our results for the future development of PSEs for studying biological growth and morphogenesis
How growth affects the fate of cellular substrates.
Cellular metabolites frequently have more than a single function in the cell. For example they may be sources of energy as well as building blocks for several macromolecules. The relative cellular needs for these different functions depend on environmental and intracellular factors. The intermediary products of phosphorylation of pyruvate by mitochondria, for example, are used for growth, while the released ATP is used for both growth and maintenance. Since maintenance has priority over growth, and maintenance is proportional to a cell's mass, a cell's need for ATP vs. building blocks depends on the growth rate, and hence on substrate availability. We show how the concept of Synthesising Units (SUs) in linear and cyclic pathways takes care of the correct variation of the ATP/building block ratio in the context of the Dynamic Energy Budget (DEB) theory. This can only be achieved by an interaction between subsequent SUs in transferring metabolites. Apart from this interaction we also needed an essential feature of the performance of the pathway in the DEB context: the relative amount of enzymes varies with the growth rate in a special way. We solved an important consistency problem between the DEB model at the whole-cell level and a model for pathway dynamics. We observe that alternative whole-cell models, such as the Marr-Pirt model, that keep the relative amount of enzymes constant, and hence independent of the growth rate, will have problems in explaining how pathways can meet cells' growth-dependent needs for building blocks vs. ATP. © 2004 Society for Mathematical Biology. Published by Elsevier Ltd. All rights reserved
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