Parametric pattern selection in a reaction-diffusion model.
Ontology highlight
ABSTRACT:
SUBMITTER: Stich M
PROVIDER: S-EPMC3812206 | biostudies-literature | 2013
REPOSITORIES: biostudies-literature
Publications
Parametric pattern selection in a reaction-diffusion model.
Stich Michael M Ghoshal Gourab G Pérez-Mercader Juan J
PloS one 20131029 10
We compare spot patterns generated by Turing mechanisms with those generated by replication cascades, in a model one-dimensional reaction-diffusion system. We determine the stability region of spot solutions in parameter space as a function of a natural control parameter (feed-rate) where degenerate patterns with different numbers of spots coexist for a fixed feed-rate. While it is possible to generate identical patterns via both mechanisms, we show that replication cascades lead to a wider choi ...[more]
Similar Datasets
Project description:Ordering and self-organization are critical in determining the dynamics of reaction-diffusion systems. Here we show a unique pattern formation mechanism, dictated by the coupling of thermodynamic instability and kinetic anisotropy. Intrinsically different from the physical origin of Turing instability and patterning, the ordered patterns we obtained are caused by the interplay of the instability from uphill diffusion, the symmetry breaking from anisotropic diffusion, and the reactions. To understand the formation of the void/gas bubble superlattices in crystals under irradiation, we establish a general theoretical framework to predict the symmetry selection of superlattice structures associated with anisotropic diffusion. Through analytical study and phase field simulations, we found that the symmetry of a superlattice is determined by the coupling of diffusion anisotropy and the reaction rate, which indicates a new type of bifurcation phenomenon. Our discovery suggests a means for designing target experiments to tailor different microstructural patterns.
| S-EPMC6534577 | biostudies-literature
Project description:A fundamental question in developmental biology is how spatial patterns are self-organized from homogeneous structures. In 1952, Turing proposed the reaction-diffusion model in order to explain this issue. Experimental evidence of reaction-diffusion patterns in living organisms was first provided by the pigmentation pattern on the skin of fishes in 1995. However, whether or not this mechanism plays an essential role in developmental events of living organisms remains elusive. Here we show that a reaction-diffusion model can successfully explain the shoot apical meristem (SAM) development of plants. SAM of plants resides in the top of each shoot and consists of a central zone (CZ) and a surrounding peripheral zone (PZ). SAM contains stem cells and continuously produces new organs throughout the lifespan. Molecular genetic studies using Arabidopsis thaliana revealed that the formation and maintenance of the SAM are essentially regulated by the feedback interaction between WUSHCEL (WUS) and CLAVATA (CLV). We developed a mathematical model of the SAM based on a reaction-diffusion dynamics of the WUS-CLV interaction, incorporating cell division and the spatial restriction of the dynamics. Our model explains the various SAM patterns observed in plants, for example, homeostatic control of SAM size in the wild type, enlarged or fasciated SAM in clv mutants, and initiation of ectopic secondary meristems from an initial flattened SAM in wus mutant. In addition, the model is supported by comparing its prediction with the expression pattern of WUS in the wus mutant. Furthermore, the model can account for many experimental results including reorganization processes caused by the CZ ablation and by incision through the meristem center. We thus conclude that the reaction-diffusion dynamics is probably indispensable for the SAM development of plants.
| S-EPMC3066213 | biostudies-literature
Project description:Given a reaction-diffusion system interacting via a complex network, we propose two different techniques to modify the network topology while preserving its dynamical behaviour. In the region of parameters where the homogeneous solution gets spontaneously destabilized, perturbations grow along the unstable directions made available across the networks of connections, yielding irregular spatio-temporal patterns. We exploit the spectral properties of the Laplacian operator associated to the graph in order to modify its topology, while preserving the unstable manifold of the underlying equilibrium. The new network is isodynamic to the former, meaning that it reproduces the dynamical response (pattern) to a perturbation, as displayed by the original system. The first method acts directly on the eigenmodes, thus resulting in a general redistribution of link weights which, in some cases, can completely change the structure of the original network. The second method uses localization properties of the eigenvectors to identify and randomize a subnetwork that is mostly embedded only into the stable manifold. We test both techniques on different network topologies using the Ginzburg-Landau system as a reference model. Whereas the correlation between patterns on isodynamic networks generated via the first recipe is larger, the second method allows for a finer control at the level of single nodes. This work opens up a new perspective on the multiple possibilities for identifying the family of discrete supports that instigate equivalent dynamical responses on a multispecies reaction-diffusion system.
| S-EPMC6212431 | biostudies-literature
Project description:BackgroundAlan 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. Ever since, extensive work on several kinds of pattern-generating reaction diffusion systems has been done. Nevertheless, prediction of specific patterns is far from being straightforward, and a great deal of interest in deciphering how to generate specific patterns under controlled conditions prevails.ResultsTechniques allowing one to predict what kind of spatial structure will emerge from reaction-diffusion systems remain unknown. In response to this need, 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. Our criterion is motivated by the existence of an associated energy function that allows bringing in the intuition provided by phase transitions phenomena.ConclusionsOur 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 work constitutes a first step towards rigorous pattern prediction in arbitrary geometries/conditions. Advances in this direction are highly applicable to the efficient design of Biotechnology and Developmental Biology experiments, as well as in simplifying the analysis of morphogenetic models.
| S-EPMC3925790 | biostudies-literature
Project description:Shapes of biological membranes are dynamically regulated in living cells. Although membrane shape deformation by proteins at thermal equilibrium has been extensively studied, nonequilibrium dynamics have been much less explored. Recently, chemical reaction propagation has been experimentally observed in plasma membranes. Thus, it is important to understand how the reaction-diffusion dynamics are modified on deformable curved membranes. Here, we investigated nonequilibrium pattern formation on vesicles induced by mechanochemical feedback between membrane deformation and chemical reactions, using dynamically triangulated membrane simulations combined with the Brusselator model. We found that membrane deformation changes stable patterns relative to those that occur on a non-deformable curved surface, as determined by linear stability analysis. We further found that budding and multi-spindle shapes are induced by Turing patterns, and we also observed the transition from oscillation patterns to stable spot patterns. Our results demonstrate the importance of mechanochemical feedback in pattern formation on deforming membranes.
| S-EPMC7659017 | biostudies-literature
Project description:The hair bundle, an apical specialization of the hair cell composed of several rows of regularly organized stereocilia and a kinocilium, is essential for mechanotransduction in the ear. Its precise organization allows the hair bundle to convert mechanical stimuli to electrical signals; mutations that alter the bundle's morphology often cause deafness. However, little is known about the proteins involved in the process of morphogenesis and how the structure of the bundle arises through interactions between these molecules. We present a mathematical model based on simple reaction-diffusion mechanisms that can reproduce the shape and organization of the hair bundle. This model suggests that the boundary of the cell and the kinocilium act as signaling centers that establish the bundle's shape. The interaction of two proteins forms a hexagonal Turing pattern--a periodic modulation of the concentrations of the morphogens, sustained by local activation and long-range inhibition of the reactants--that sets a blueprint for the location of the stereocilia. Finally we use this model to predict how different alterations to the system might impact the shape and organization of the hair bundle.
| S-EPMC4217459 | biostudies-literature
Project description:Shapes and patterns observed in internal organs and tissues are reproducibly and robustly produced over a long distance (up to millimeters in length). The most fundamental remaining question is how these long geometries of shape and pattern form arise from the genetic message. Recent studies have demonstrated that extracellular matrix (ECM) critically participates as a structural foundation on which cells can organize and communicate. ECMs may be a key to understanding the underlying mechanisms of long-distance patterning and morphogenesis. However, previous studies in this field mainly focused on the complexes and interaction of cells and ECM. This paper pays particular attention to ECM and demonstrates that collagen, a major ECM component, natively possesses the reproducible and definite patterning ability reaching centimeter-scale length. The macroscopic pattern consists of striped transparent layers. The observation under crossed Nicols demonstrates that the layers consist of alternately arranged polarized and unpolarized parts. Confocal fluorescence microscopy studies revealed that the polarized and unpolarized segments include collagen-rich and -poor regions, respectively. The patterning process was proposed based on the Liesegang banding formation, which are mineral precipitation bands formed in hydrogel matrixes. These findings will give hints to the questions about long-distance cell alignment and provide new clues to artificially control cell placement over micron size in the field of regenerative medicine.
| S-EPMC8756805 | biostudies-literature
Project description:Prior to receiving visual stimuli, spontaneous, correlated activity in the retina, called retinal waves, drives activity-dependent developmental programs. Early-stage waves mediated by acetylcholine (ACh) manifest as slow, spreading bursts of action potentials. They are believed to be initiated by the spontaneous firing of Starburst Amacrine Cells (SACs), whose dense, recurrent connectivity then propagates this activity laterally. Their inter-wave interval and shifting wave boundaries are the result of the slow after-hyperpolarization of the SACs creating an evolving mosaic of recruitable and refractory cells, which can and cannot participate in waves, respectively. Recent evidence suggests that cholinergic waves may be modulated by the extracellular concentration of ACh. Here, we construct a simplified, biophysically consistent, reaction-diffusion model of cholinergic retinal waves capable of recapitulating wave dynamics observed in mice retina recordings. The dense, recurrent connectivity of SACs is modeled through local, excitatory coupling occurring via the volume release and diffusion of ACh. In addition to simulation, we are thus able to use non-linear wave theory to connect wave features to underlying physiological parameters, making the model useful in determining appropriate pharmacological manipulations to experimentally produce waves of a prescribed spatiotemporal character. The model is used to determine how ACh mediated connectivity may modulate wave activity, and how parameters such as the spontaneous activation rate and sAHP refractory period contribute to critical wave size variability.
| S-EPMC4256014 | biostudies-literature
Project description:The development of an adaptive trait simulator package for inferring trait evolution along a phylogenetic tree is shown. Stochastic processes of the continuous type are broadly applied to modeling trait evolution when the evolutionary relationship among species and traits of study interest are present. By including several popular stochastic processes, evolutionary information embedded in a dataset can be revealed. The highlights of the method include: 1.The implementation of the popular Cox-Ingersol-Ross process for modeling rate evolution within the package to prevent rates from becoming negative and thus is potentially a useful extension to study adaptive trait evolution in randomly evolved environment.2.The established trait simulator approach along with approximate Bayesian computation procedure provides a feasible statistical inference without model likelihood.3.The procedure proposed for trait simulator along phylogenetic tree can be applied to all established models of trait evolution in literature, thus providing users an alternative option to analyze their data.
| S-EPMC7341453 | biostudies-literature
Project description:We have developed and implemented a novel mathematical model for simulating transients in surface pH (pH(S)) and intracellular pH (pH(i)) caused by the influx of carbon dioxide (CO(2)) into a Xenopus oocyte. These transients are important tools for studying gas channels. We assume that the oocyte is a sphere surrounded by a thin layer of unstirred fluid, the extracellular unconvected fluid (EUF), which is in turn surrounded by the well-stirred bulk extracellular fluid (BECF) that represents an infinite reservoir for all solutes. Here, we assume that the oocyte plasma membrane is permeable only to CO(2). In both the EUF and intracellular space, solute concentrations can change because of diffusion and reactions. The reactions are the slow equilibration of the CO(2) hydration-dehydration reactions and competing equilibria among carbonic acid (H(2)CO(3))/bicarbonate (HCO(3)(-)) and a multitude of non-CO(2)/HCO(3)(-) buffers. Mathematically, the model is described by a coupled system of reaction-diffusion equations that-assuming spherical radial symmetry-we solved using the method of lines with appropriate stiff solvers. In agreement with experimental data [Musa-Aziz et al. 2009, PNAS 106 5406-5411], the model predicts that exposing the cell to extracellular 1.5% CO(2)/10 mM HCO(3)(-) (pH 7.50) causes pH(i) to fall and pH(S) to rise rapidly to a peak and then decay. Moreover, the model provides insights into the competition between diffusion and reaction processes when we change the width of the EUF, membrane permeability to CO(2), native extra- and intracellular carbonic anhydrase-like activities, the non-CO(2)/HCO(3)(-) (intrinsic) intracellular buffering power, or mobility of intrinsic intracellular buffers.
| S-EPMC3471386 | biostudies-literature