Simple Reaction-diffusion Model Research Articles

Spatiotemporal pattern in somitogenesis: A non-Turing scenario with wave propagation

Living organisms maintain their lives under far-from-equilibrium conditions by creating a rich variety of spatiotemporal structures in a self-organized manner, such as temporal rhythms, switching phenomena, and development of the body. In this paper, we focus on the dynamical process of morphogens in somitogenesis in mice where propagation of the gene expression level plays an essential role in creating the spatially periodic patterns of the vertebral columns. We present a simple discrete reaction-diffusion model which includes neighboring interaction through an activator, but not diffusion of an inhibitor. We can produce stationary periodic patterns by introducing the effect of spatial discreteness to the field. Based on the present model, we discuss the underlying physical principles that are independent of the details of biomolecular reactions. We also discuss the framework of spatial discreteness based on the reaction-diffusion model in relation to a cellular array, by comparison with an actual experimental observation.

Scaling from mycelial growth to infection dynamics: a reaction diffusion approach

Numerous models have been proposed for the dynamics of fungal growth, and also for the dynamics of infection. Few models, however, have combined the mechanistic interpretation of mycelial growth with epidemiological models for the transmission of infection. Many of the mechanistic models seek to include considerable biological detail, which necessarily leads to a proliferation of state variables and parameters. Including such models within an epidemiological framework makes interpretation of underpinning processes difficult. A simple reaction diffusion model for the growth and spread of fungal mycelium is introduced and analysed, scaling from the small-scale parameters for mycelial dynamics to the large-scale properties of the colony. By coupling the output to a parsimonious epidemiological model for the dynamics of primary infection, we analyse the sensitivity of the probability of successful infection of a host to the colony dynamics associated with local bulking-up, extension, growth and nutrient consumption by the mycelium. In particular we identify optimal trade-offs in bulking-up versus dispersal in controlling infection dynamics.

Violation of the action-reaction principle and self-forces induced by nonequilibrium fluctuations

We show that the extension of Casimir-like forces to fluctuating fluids driven out of equilibrium can exhibit two interrelated phenomena forbidden at equilibrium: self-forces can be induced on single asymmetric objects and the action-reaction principle between two objects can be violated. These effects originate in asymmetric restrictions imposed by the objects' boundaries on the fluid's fluctuations. They are not ruled out by the second law of thermodynamics since the fluid is in a nonequilibrium state. Considering a simple reaction-diffusion model for the fluid, we explicitly calculate the self-force induced on a deformed circle. We also show that the action-reaction principle does not apply for the internal Casimir forces between a circle and a plate. Their sum, instead of vanishing, provides the self-force on the circle-plate assembly.

Turbulent transport and its effect on the dead zone in protoplanetary discs

Protostellar accretion discs have cool, dense midplanes where externally originating ionisation sources such as X-rays or cosmic rays are unable to penetrate. This suggests that for a wide range of radii, MHD turbulence can only be sustained in the surface layers where the ionisation fraction is sufficiently high. A dead zone is expected to exist near the midplane, such that active accretion only occurs near the upper and lower disc surfaces. Recent work, however, suggests that under suitable conditions the dead zone may be enlivened by turbulent transport of ions from the surface layers into the dense interior. In this paper we present a suite of simulations that examine where, and under which conditions, a dead zone can be enlivened by turbulent mixing. We use three-dimensional, multifluid shearing box MHD simulations, which include vertical stratification, ionisation chemistry, ohmic resistivity, and ionisation due to X-rays from the central protostar. We compare the results of the MHD simulations with a simple reaction-diffusion model. The simulations show that in the absence of gas-phase heavy metals, such as magnesium, turbulent mixing has essentially no effect on the dead zone. The addition of a relatively low abundance of magnesium, however, increases the recombination time and allows turbulent mixing of ions to enliven the dead zone completely beyond a distance of 5 AU from the central star, for our particular disc model. During the late stages of protoplanetary disc evolution, when small grains have been depleted and the disc surface density has decreased below its high initial value, the structure of the dead zone may be significantly altered by the action of turbulent transport.

Porous Gold Structures Templated by Porous Silicon

In this work, we describe the synthesis of porous Au structures by galvanic displacement reaction using porous Si exhibiting arrays of parallel pores with diameters near as template material. Two metal ions were used as Au precursors: tetrachloroaurate(III) and bis(ethylenediamine)Au(III). Fluoride was employed to etch the Si oxides that form as a result of the displacement reaction. The porous metal layer that forms with tetrachloroaurate(III) is an agglomerate of spherical particles with ca. diameter. Metallization with bis(ethylenediamine)Au(III) preserves the morphology of the original porous Si layer, creating a porous Au layer containing an array of parallel pores with ca. diameter. The displacement process is under diffusion control for both reagents. The expansion rate of the porous metal layer into the substrate agrees well with a simple one-dimensional reaction-diffusion model entailing the displacement of Si in a large extent. The electrochemical behavior of the synthesized porous Au layers was studied by voltammetry.

Control of the boiling crisis: analysis of a model system

Controlling the transition between the low (nucleate) and high temperature (film) regimes of boiling is a serious challenge for a number of technological applications. Based on the theoretical analysis of a simplified reaction-diffusion model, it has recently been shown [A. Pumir, V.V. Barelko, Chaos 12, 610 (2002)] that the transition towards the dangerous situation where the high temperature phase tends to invade the whole system requires a higher power in a periodically spatially modulated system than in an homogeneous system. We show here that the transition mechanisms between the various boiling regimes depend on the ratio between the periodicity length along the wire and the characteristic thermal diffusion length. We analyse theoretically a simple experimental setup aimed at testing these ideas. The heater consists of a thin wire, with an applied electric current, with alternatively low resistance and high resistance sections. We determine the gain in stability for a set of realistic values of the parameters.

Light-Controlled Polymerization Kinetics for the Photostabilization of Cholesteric Fingerprint Rolls

We use a focused ultraviolet (UV) laser beam to photopolymerize localized polymer networks in a cholesteric liquid-crystalline host and to thereby stabilize small, light-diffracting “fingerprint” domains within a uniform, planar-aligned sample. As we have previously described, these “microspot” gratings are electrically switchable [1] and can be produced with diameters ranging from a few tens to hundreds of microns. Our focus in this article is to model the growth of the stabilized domain using a simple reaction-diffusion model of the photopolymerization process and to compare this model with data for networks formed both from isotropic and mesogenic monomers. In spite of an initial spatial asymmetry observed in the mesogenic case, both types of light-controlled networks develop in accordance with the fast photoreactivity/weak diffusion limit of the model.

Simple and complex spatiotemporal structures in a glycolytic allosteric enzyme model

Pattern formation in glycolysis is studied with a classical reaction-diffusion allosteric enzyme model. It is found that, similar to recent experimental reports in the yeast extracts, a small magnitude local perturbation can induce transient target waves in a two dimensional oscillatory medium. An above threshold stimulation generates target waves which eventually evolve into spatiotemporal chaos upon collisions with the boundary or other wave activities. Detailed simulation studies show that the studied simple glycolytic reaction-diffusion model can support three types of spatiotemporal behaviors which are independent of the boundary conditions: (1) a spatially uniform stable steady state, (2) periodic global oscillations and (3) spatiotemporal chaos.

Generating function for particle-number probability distribution in directed percolation

We derive a generic expression for the generating function (GF) of the particle-number probability distribution (PNPD) for a simple reaction diffusion model that belongs to the directed percolation universality class. Starting with a single particle on a lattice, we show that the GF of the PNPD can be written as an infinite series of cumulants taken at zero momentum. This series can be summed up into a complete form at the level of a mean-field approximation. Using the renormalization group techniques, we determine logarithmic corrections for the GF at the upper critical dimension. We also find the critical scaling form for the PNPD and check its universality numerically in one dimension. The critical scaling function is found to be universal up to two non-universal metric factors.

A simple two-component reaction-diffusion system showing rich dynamic behavior: spatially homogeneous aspects and selected bifurcation scenarios

We present a simple reaction-diffusion model for two variables. The model was originally designed to have a stable localized solution for a range of parameters as a consequence of the coexistence of a stable limit cycle and a stable fixed point. We classify the spatially homogeneous solutions of the model. In addition we describe several bifurcation scenarios for particle-like solutions as a function of two of the parameters.

Models for the morphogenesis of the molluscan shell

We give a review of current theories of morphogenesis of both the general coiling and the ornamentation of molluscan shells. These two aspects of shell growth are closely connected, as ornamentation is primarily due to local perturbations of the general apertural growth field controlling coiling. Also, a new, generalized, free-form apertural growth map model is presented in this paper, illustrating some aspects of the regulation of logarithmic spiral growth. This model is used to simulate the formation of megastriae in ammonoids. We emphasize the importance of damaged specimens and how they regenerated, as illustrated with examples from ammonoids. The phenomenon of ornamental compensation can be explained by a mechanism involving a pre-pattern in the mantle. However, simple reaction-diffusion models for ornamental pattern formation should be regarded only as useful abstractions.

Morphogen Gradient from a Noisy Source

We investigate the effect of time-dependent noise on the shape of a morphogen gradient in a developing embryo. Perturbation theory is used to calculate the deviations from deterministic behavior in a simple reaction-diffusion model of robust gradient formation, and the results are confirmed by numerical simulation. It is shown that such deviations can disrupt robustness for sufficiently high noise levels, and the implications of these findings for more complex models of gradient-shaping pathways are discussed.

Spatio-temporal dynamics of plant invasions: Linking pattern to process

:Biological invasions are characterized by remarkable spatio-temporal dynamics, with many species having extended their distribution range from within a single region to much of the globe within the last century. The comparative analysis of the spatio-temporal dynamics of over 100 taxa from studies undertaken worldwide provides the basis of a critical assessment of current knowledge. At the scale of single habitats, simple reaction-diffusion models may be accurate enough to predict the spread of new invaders without recourse to complex life history parameterization. Average rates of local spread reported for invasive species in the literature range from 2 m·y-1 to a maximum of 370 m·y-1. Average rates of long-distance dispersal are at least two orders of magnitude greater than estimates of local dispersal, with a maximum of 167 km·y-1. While local-scale studies do pick up dispersal events of several kilometres, study sites are rarely sufficiently large or monitored for long enough to characterize these events accurately. Long-distance dispersal events may occur during periods of negligible population increase and appear to bear little relationship to population size. At regional scales, invasive species rarely move across the landscape as a continuous front and both local and long-distance dispersal determine spatial patterns. At these larger spatial scales, both local and long-distance dispersal require parameterization, and this has been achieved through spatially explicit individual-based simulation models using two or more dispersal functions. It is doubtful whether a single estimate of spread encapsulates the spatio-temporal dynamics of invasive species at this scale. Thus, estimates of spread drawn from successive distribution maps will tend to be biased towards long-distance dispersal events. The frequency and distribution of introduction events play a key role in invasion trajectories, and the stochastic nature of such events may explain why the longer a species has been introduced into a region the greater the likelihood that it becomes invasive. However, cumulative counts of localities or samples only provide one perspective on the invasion process and need to be associated with spatial information to depict spread more realistically. This review highlights that monitoring of invasive species must be approached from a hierarchical perspective with data gathered at more than one spatial scale. Such an approach will improve predictions and integrate landscape attributes into invasion dynamics.

OSCILLATING LOCALIZED STRUCTURES IN REACTION–DIFFUSION SYSTEMS

Oscillating localized structures are studied for a simple reaction–diffusion model from an analytical point of view. The result is a particle solution which acts as a source of traveling waves. The analytical expressions obtained are in good agreement with direct numerical simulations.

Coexistence of stable particle and hole solutions for fixed parameter values in a simple reaction diffusion system.

We present a simple autocatalytic reaction-diffusion model for two variables, which shows for fixed parameter values the simultaneous stable coexistence of particle solutions as well of two types of hole solutions. The associated spatially homogeneous system is characterized by the coexistence of one stable fixed point and a stable limit cycle solution. We compare our results to other dissipative systems which have for fixed parameters either stable particle or stable hole solutions including the quintic complex Ginzburg-Landau equation and the envelope equation for optical bistability as well as other reaction-diffusion models.

Analytical approach to localized structures in a simple reaction-diffusion system.

We study from an analytical point of view a simple reaction-diffusion model, which admits stable oscillating localized structures as a consequence of the coexistence between a stable limit cycle and a stable fixed point. Using a generalized matching approach we are able to find approximate analytical expressions for localized oscillating structures in this reaction-diffusion model capturing all the essential ingredients of these breathing particlelike solutions.

Preparation of InAs by annealing of two-layer In–As electrodeposits

In this work, we investigate the galvanostatic electrodeposition of In from aqueous solutions on electrodeposited amorphous arsenic substrates and the temporal variation of the composition of the In–As two-layer subjected to annealing. Surface morphology, composition and crystallographic structure of the deposits before and after annealing were analysed by scanning electron microscopy (SEM), laser profilometry, X-ray energy dispersive spectroscopy (EDX) and X-ray diffraction (XRD). The amorphous arsenic film was structurally characterised, determining the radial distribution function (RDF) and the main interatomic distances. The annealing procedure consisted in holding the samples in vacuo at 60, 100 and 140 °C for increasing times. The results showed the formation of InAs, owing to reaction-diffusion going on at the In–As interface. The values of the In diffusion coefficient in the range 60–140 °C (from 4×10 −21 to 2.2×10 −18 m 2 s −1) were evaluated from XRD data with a simple reaction-diffusion model. The activation energy (90 kJ mol −1) and the pre-exponential factor (4.5×10 −7 m 2 s −1) were also estimated. The results were discussed in connection with those previously obtained regarding the indium diffusion process into amorphous antimony with formation of InSb.

Small-scale invasion pattern, spread rate, and lag-phase behavior of Rhamnus frangula L.

We tested the ability of the simple reaction–diffusion spread model to quantify the rate of spread of Rhamnus frangula L., a non-indigenous shrub, at the scale of a single forest stand in southeastern New Hampshire. Cut stems of R. frangula were collected in a Cartesian grid pattern within the invaded stand and aged. A contour map of the invasion pattern was produced using kriging. The empirical invasion rate was calculated as the linear distance per year since initial invasion. The intrinsic rate of increase and diffusion coefficient were estimated in the field and the model prediction calculated. Observed mean rate of linear expansion was measured to be 6.7 m per year with a standard deviation of 1.24 m per year, while the predicted linear asymptotic rate of spread was calculated be 6.3 m per year. Thus, the simple reaction–diffusion model yielded a very accurate prediction of actual spread rate. There may have been a lag-phase in the invasion at this site as the spread rate was found to be slower in the early stages of invasion compared to later stages. At the scale of single stands, the simple reaction–diffusion model may be accurate enough to predict the spread of new invaders without recourse to complex life-history spread rate models. Additionally, if lag-phases in spread rate occur with regular frequency, then attempts to retrospectively predict which species may be invasive will be hampered by incorrectly classifying a future invader as currently non-invasive.

Evolution: geometrical and dynamical aspects.

We develop a possible axiomatic approach to the evolution theory that has been previously discussed in Freguglia [2002]. The axioms synthesize the fundamental ideas of evolution theory and allow a geometrical and dynamical interpretation of the generation law. Using the axioms we derive a simple reaction-diffusion model which introduces the species as self-organized stationary distribution of a finite population and simulates the evolution of a phenotypic character under the effect of an external perturbing action. The dynamical properties of the model are briefly presented using numerical simulations.

A reaction-diffusion model for interference in meiotic crossing over.

One crossover point between a pair of homologous chromosomes in meiosis appears to interfere with occurrence of another in the neighborhood. It has been revealed that Drosophila and Neurospora, in spite of their large difference in the frequency of crossover points, show very similar plots of coincidence-a measure of the interference-against the genetic distance of the interval, defined as one-half the average number of crossover points within the interval. We here propose a simple reaction-diffusion model, where a "randomly walking" precursor becomes immobilized and matures into a crossover point. The interference is caused by pair-annihilation of the random walkers due to their collision and by annihilation of a random walker due to its collision with an immobilized point. This model has two parameters-the initial density of the random walkers and the rate of its processing into a crossover point. We show numerically that, as the former increases and/or the latter decreases, plotted curves of the coincidence vs. the genetic distance converge on a unique curve. Thus, our model explains the similarity between Drosophila and Neurospora without parameter values adjusted finely, although it is not a "genetic model" but is a "physical model," specifying explicitly what happens physically.