311,120 research outputs found
Pattern formation of reaction-diffusion system having self-determined flow in the amoeboid organism of Physarum plasmodium
The amoeboid organism, the plasmodium of Physarum polycephalum, behaves on
the basis of spatio-temporal pattern formation by local
contraction-oscillators. This biological system can be regarded as a
reaction-diffusion system which has spatial interaction by active flow of
protoplasmic sol in the cell. Paying attention to the physiological evidence
that the flow is determined by contraction pattern in the plasmodium, a
reaction-diffusion system having self-determined flow arises. Such a coupling
of reaction-diffusion-advection is a characteristic of the biological system,
and is expected to relate with control mechanism of amoeboid behaviours. Hence,
we have studied effects of the self-determined flow on pattern formation of
simple reaction-diffusion systems. By weakly nonlinear analysis near a trivial
solution, the envelope dynamics follows the complex Ginzburg-Landau type
equation just after bifurcation occurs at finite wave number. The flow term
affects the nonlinear term of the equation through the critical wave number
squared. Contrary to this, wave number isn't explicitly effective with lack of
flow or constant flow. Thus, spatial size of pattern is especially important
for regulating pattern formation in the plasmodium. On the other hand, the flow
term is negligible in the vicinity of bifurcation at infinitely small wave
number, and therefore the pattern formation by simple reaction-diffusion will
also hold. A physiological role of pattern formation as above is discussed.Comment: REVTeX, one column, 7 pages, no figur
Global-in-time behavior of the solution to a Gierer-Meinhardt system
Gierer-Meinhardt system is a mathematical model to describe biological pattern formation due to activator and inhibitor. Turing pattern is
expected in the presence of local self-enhancement and long-range inhibition.
The long-time behavior of the solution, however, has not yet been clarified mathematically. In this paper, we study the case when its ODE part takes
periodic-in-time solutions, that is, . Under some additional assumptions on parameters, we show that the solution exists global-in-time and absorbed into one of these ODE orbits. Thus spatial patterns eventually dis- appear if those parameters are in a region without local self-enhancement or long-range inhibition
Billiards with Spatial Memory
Many classes of active matter develop spatial memory by encoding information
in space, leading to complex pattern formation. It has been proposed that
spatial memory can lead to more efficient navigation and collective behaviour
in biological systems and influence the fate of synthetic systems. This raises
important questions about the fundamental properties of dynamical systems with
spatial memory. We present a framework based on mathematical billiards in which
particles remember their past trajectories and react to them. Despite the
simplicity of its fundamental deterministic rules, such a system is strongly
non-ergodic and exhibits highly-intermittent statistics, manifesting in complex
pattern formation. We show how these self-memory-induced complexities emerge
from the temporal change of topology and the consequent chaos in the system. We
study the fundamental properties of these billiards and particularly the
long-time behaviour when the particles are self-trapped in an arrested state.
We exploit numerical simulations of several millions of particles to explore
pattern formation and the corresponding statistics in polygonal billiards of
different geometries. Our work illustrates how the dynamics of a single-body
system can dramatically change when particles feature spatial memory and
provide a scheme to further explore systems with complex memory kernels.Comment: 11 pages, 6 figure
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
Coordinated Spatial Pattern Formation in Biomolecular Communication Networks
This paper proposes a control theoretic framework to model and analyze the self-organized pattern formation of molecular concentrations in biomolecular communication networks, emerging applications in synthetic biology. In biomolecular communication networks, bio-nanomachines, or biological cells, communicate with each other using a cell-to-cell communication mechanism mediated by a diffusible signaling molecule, thereby the dynamics of molecular concentrations are approximately modeled as a reaction-diffusion system with a single diffuser. We first introduce a feedback model representation of the reaction-diffusion system and provide a systematic local stability/instability analysis tool using the root locus of the feedback system. The instability analysis then allows us to analytically derive the conditions for the self-organized spatial pattern formation, or Turing pattern formation, of the bionanomachines. We propose a novel synthetic biocircuit motif called activator-repressor-diffuser system and show that it is one of the minimum biomolecular circuits that admit self-organized patterns over cell population
Cellular pattern formation during Dictyostelium aggregation
The development of multicellularity in the life cycle of Dictyostelium discoideum provides a paradigm model system for biological pattern formation. Previously, mathematical models have shown how a collective pattern of cell communication by waves of the messenger molecule cyclic adenosine 3′5′-monophosphate (cAMP) arises from excitable local cAMP kinetics and cAMP diffusion. Here we derive a model of the actual cell aggregation process by considering the chemotactic cell response to cAMP and its interplay with the cAMP dynamics. Cell density, which previously has been treated as a spatially homogeneous parameter, is a crucial variable of the aggregation model. We find that the coupled dynamics of cell chemotaxis and cAMP reaction-diffusion lead to the break-up of the initially uniform cell layer and to the formation of the striking cell stream morphology which characterizes the aggregation process in situ. By a combination of stability analysis and two-dimensional simulations of the model equations, we show cell streaming to be the consequence of the growth of a small-amplitude pattern in cell density forced by the large-amplitude cAMP waves, thus representing a novel scenario of spatial patterning in a cell chemotaxis system. The instability mechanism is further analysed by means of an analytic caricature of the model, and the condition for chemotaxis-driven instability is found to be very similar to the one obtained for the standard (non-oscillatory) Keller-Segel system. The growing cell stream pattern feeds back into the cAMP dynamics, which can explain in some detail experimental observations on the time evolution of the cAMP wave pattern, and suggests the characterization of the Dictyostelium aggregation field as a self-organized excitable medium
Supramolecular structure in the membrane of Staphylococcus aureus
The fundamental processes of life are organized and based on common basic principles. Molecular organizers, often interacting with the membrane, capitalize on cellular polarity to precisely orientate essential processes. The study of organisms lacking apparent polarity or known cellular organizers (e.g., the bacterium Staphylococcus aureus) may enable the elucidation of the primal organizational drive in biology. How does a cell choose from infinite locations in its membrane? We have discovered a structure in the S. aureus membrane that organizes processes indispensable for life and can arise spontaneously from the geometric constraints of protein complexes on membranes. Building on this finding, the most basic cellular positioning system to optimize biological processes, known molecular coordinators could introduce further levels of complexity.
All life demands the temporal and spatial control of essential biological functions. In bacteria, the recent discovery of coordinating elements provides a framework to begin to explain cell growth and division. Here we present the discovery of a supramolecular structure in the membrane of the coccal bacterium Staphylococcus aureus, which leads to the formation of a large-scale pattern across the entire cell body; this has been unveiled by studying the distribution of essential proteins involved in lipid metabolism (PlsY and CdsA). The organization is found to require MreD, which determines morphology in rod-shaped cells. The distribution of protein complexes can be explained as a spontaneous pattern formation arising from the competition between the energy cost of bending that they impose on the membrane, their entropy of mixing, and the geometric constraints in the system. Our results provide evidence for the existence of a self-organized and nonpercolating molecular scaffold involving MreD as an organizer for optimal cell function and growth based on the intrinsic self-assembling properties of biological molecules
A Selection Criterion for Patterns in Reaction-Diffusion Systems
Alan Turing's work in Morphogenesis has received wide attention during the
past 60 years. The central idea behind his theory is that two chemically
interacting diffusible substances are able to generate stable spatial patterns,
provided certain conditions are met. Turing's proposal has already been
confirmed as a pattern formation mechanism in several chemical and biological
systems and, due to their wide applicability, there is a great deal of interest
in deciphering how to generate specific patterns under controlled conditions.
However, techniques allowing one to predict what kind of spatial structure will
emerge from Turing systems, as well as generalized reaction-diffusion systems,
remain unknown. Here, we consider a generalized reaction diffusion system on a
planar domain and provide an analytic criterion to determine whether spots or
stripes will be formed. It is motivated by the existence of an associated
energy function that allows bringing in the intuition provided by phase
transitions phenomena. This criterion is proved rigorously in some situations,
generalizing well known results for the scalar equation where the pattern
selection process can be understood in terms of a potential. In more complex
settings it is investigated numerically. Our criterion can be applied to
efficiently design Biotechnology and Developmental Biology experiments, or
simplify the analysis of hypothesized morphogenetic models.Comment: 19 pages, 10 figure
Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model
Pattern formation in biological, chemical and physical problems has received
considerable attention, with much attention paid to dissipative systems. For
example, the Ginzburg--Landau equation is a normal form that describes pattern
formation due to the appearance of a single mode of instability in a wide
variety of dissipative problems. In a similar vein, a certain "single-wave
model" arises in many physical contexts that share common pattern forming
behavior. These systems have Hamiltonian structure, and the single-wave model
is a kind of Hamiltonian mean-field theory describing the patterns that form in
phase space. The single-wave model was originally derived in the context of
nonlinear plasma theory, where it describes the behavior near threshold and
subsequent nonlinear evolution of unstable plasma waves. However, the
single-wave model also arises in fluid mechanics, specifically shear-flow and
vortex dynamics, galactic dynamics, the XY and Potts models of condensed matter
physics, and other Hamiltonian theories characterized by mean field
interaction. We demonstrate, by a suitable asymptotic analysis, how the
single-wave model emerges from a large class of nonlinear advection-transport
theories. An essential ingredient for the reduction is that the Hamiltonian
system has a continuous spectrum in the linear stability problem, arising not
from an infinite spatial domain but from singular resonances along curves in
phase space whereat wavespeeds match material speeds (wave-particle resonances
in the plasma problem, or critical levels in fluid problems). The dynamics of
the continuous spectrum is manifest as the phenomenon of Landau damping when
the system is ... Such dynamical phenomena have been rediscovered in different
contexts, which is unsurprising in view of the normal-form character of the
single-wave model
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