One-dimensional Reaction-diffusion Model Research Articles

2D effects enhance precision of gradient-based tissue patterning

Robust embryonic development requires pattern formation with high spatial accuracy. In epithelial tissues that are patterned by morphogen gradients, the emerging patterns achieve levels of precision that have recently been explained by a simple one-dimensional reaction-diffusion model with kinetic noise. Here, we show that patterning precision is even greater if transverse diffusion effects are at play in such tissues. The positional error, a measure for spatial patterning accuracy, decreases in wider tissues but then saturates beyond a width of about ten cells. This demonstrates that the precision of gradient-based patterning in two- or higher-dimensional systems can be even greater than predicted by 1D models, and further attests to the potential of noisy morphogen gradients for high-precision tissue patterning.

Stability and dynamics of spike-type solutions to delayed Gierer-Meinhardt equations

<p style='text-indent:20px;'>For a specific set of parameters, we analyze the stability of a one-spike equilibrium solution to the one-dimensional Gierer-Meinhardt reaction-diffusion model with delay in the components of the reaction-kinetics terms. Assuming slow activator diffusivity, we consider instabilities due to Hopf bifurcation in both the spike position and the spike profile for increasing values of the time-delay parameter <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula>. Using method of matched asymptotic expansions it is shown that the model can be reduced to a system of ordinary differential equations representing the position of the slowly evolving spike solution. The reduced evolution equations for the one-spike solution undergoes a Hopf bifurcation in the spike position in two cases: when the negative feedback of the activator equation is delayed, and when delay is in both the negative feedback of the activator equation and the non-linear production term of the inhibitor equation. Instabilities in the spike profile are also considered, and it is shown that the spike solution is unstable as <inline-formula><tex-math id="M2">\begin{document}$ T $\end{document}</tex-math></inline-formula> is increased beyond a critical Hopf bifurcation value <inline-formula><tex-math id="M3">\begin{document}$ T_H $\end{document}</tex-math></inline-formula>, and this occurs for the same cases as in the spike position analysis. In all cases, the instability in the profile is triggered before the positional instability. If however the degradation of activator is delayed, we find stable positional oscillations can occur in this system.</p>

Propagation and blocking in a two-patch reaction-diffusion model

This paper is concerned with propagation phenomena for the solutions of the Cauchy problem associated with a two-patch one-dimensional reaction-diffusion model. It is assumed that each patch has a relatively well-defined structure which is considered as homogeneous. A coupling interface condition between the two patches is involved. We first study the spreading properties of solutions in the case when the per capita growth rate in each patch is maximal at low densities, a configuration which we call the KPP-KPP case, and which turns out to have some analogies with the homogeneous KPP equation in the whole line. Then, in the KPP-bistable case, we provide various conditions under which the solutions show different dynamics in the bistable patch, that is, blocking, virtual blocking (propagation with speed zero), or spreading with positive speed. Moreover, when propagation occurs with positive speed, a global stability result is proved. Finally, the analysis in the KPP-bistable frame is extended to the bistable-bistable case.

Coupled modeling of peatlands carbon cycle and carbon dioxide emission from their peat deposits

COmplex Model of BOg LAndscapes (COMBOLA) is a set of dynamic models of carbon and nitrogen turnover, net ecosystem exchange, water balance, heat and water transport, generation and transfer of CO2 and CH4 in a peat deposit on a number of time scales. Combined modeling of carbon turnover and carbon dioxide emission from peat deposits on an annual time scale is carried out as part of COMBOLA for several bog landscapes in Western-Siberian south taiga. One-dimensional vertical reaction-diffusion model for carbon dioxide emission from peat deposit is combined with compartment carbon cycle model allowing estimate both bog type vegetation changes and greenhouse gases balance under probable climate change or perturbations initiated by human activities. Water table level (WTL) separating aerobic peat layer from the anaerobic one is calculated by means of the whole peat layer depth and its water balance relations. Minimal statistical information is used for calibrating temperature dependencies together with coefficients of heat and water transfer equations. The stationary states of carbon cycle are accompanied by steady average profiles of carbon dioxide concentration and can loose stability under human or climatic perturbations. The model dynamics is simulated with several climatic projections of the IPSL atmospheric circulation model for 21st century with anthropogenic scenarios RCP-2.6 and RCP-8.5.

Critical Role of Layer Thickness in Frontal Polymerization.

Thermal frontal polymerization (FP) is a chemical process during which a cold monomer-initiator mixture is converted into a hot polymer as a polymerization front propagates in the system due to the interplay between heat diffusion and the exothermicity of the reaction. The theoretical description of FP generally focuses on one-dimensional (1D) reaction-diffusion (RD) models where the effect of heat losses is encoded into an effective parameter in the heat equation. We show here the limits of such 1D models to describe FP under nonadiabatic conditions. To do so, the propagation of a polymerization front is analyzed both analytically and numerically in a rectangular two-dimensional (2D) layer. The layer thickness is shown to control the dynamics of the front and to determine its very existence. We find that for given heat losses, a minimum thickness is required for front propagation as recently observed in FP experiments of 2D thin films on wood. Moreover, when the thickness exceeds a critical value, the front is observed to survive independently of the rate of heat losses. This result cannot be predicted with 1D models where front extinction is always possible. A scaling analysis is proposed to highlight the physical interpretation of such a front survival. The influence of dimensionality on thermal instabilities is also analyzed, with a focus on the differences with the 1D predictions.

Plasma-droplet interaction study to assess transport limitations and the role of ⋅OH, O⋅,H⋅,O2(a 1Δg),O3, He(23 S) and Ar(1s 5) in formate decomposition

Plasmas interacting with liquid microdroplets are gaining momentum due to their ability to significantly enhance the reactivity transfer from the gas phase plasma to the liquid. This is, for example, critically important for efficiently decomposing organic pollutants in water. In this contribution, the role of ⋅OH as well as non-⋅OH-driven chemistry initiated by the activation of small water microdroplets in a controlled environment by diffuse RF glow discharge in He with different gas admixtures (Ar, O2 and humidified He) at atmospheric pressure is quantified. The effect of short-lived radicals such as O⋅ and H⋅ atoms, singlet delta oxygen (O2(a 1Δg)), O3 and metastable atoms of He and Ar, besides ⋅OH radicals, on the decomposition of formate dissolved in droplets was analyzed using detailed plasma diagnostics, droplet characterization and ex situ chemical analysis of the treated droplets. The formate decomposition increased with increasing droplet residence time in the plasma, with ∼70% decomposition occurring within ∼15 ms of the plasma treatment time. The formate oxidation in the droplets is shown to be limited by the gas phase ⋅OH flux at lower H2O concentrations with a significant enhancement in the formate decomposition at the lowest water concentration, attributed to e−/ion-induced reactions. However, the oxidation is diffusion limited in the liquid phase at higher gaseous ⋅OH concentrations. The formate decomposition in He/O2 plasma was similar, although with an order of magnitude higher O⋅ radical density than the ⋅OH density in the corresponding He/H2O plasma. Using a one-dimensional reaction–diffusion model, we showed that O2(a 1Δg) and O3 did not play a significant role and the decomposition was due to O⋅, and possibly ⋅OH generated in the vapor containing droplet-plasma boundary layer.

Controlled plasma–droplet interactions: a quantitative study of OH transfer in plasma–liquid interaction

Plasmas in contact with liquids are a rich source of OH radicals and have been extensively studied in the last decade to leverage the ability to generate chemically reactive species in gas phase plasmas to decompose organics. Multiphase transfer of OH radicals is highly transport limited and to overcome transport limits, the plasma activation of aerosols, small liquid droplets, interspersed in the plasma has been proposed. In this work, we report a combined experimental and modeling study of a controlled plasma–droplet interaction experiment using a diffuse RF glow discharge in He + 0.2% H2O with detailed plasma diagnostics, ex situ analysis of the plasma-induced chemistry in the droplet containing formate, droplet trajectory and size measurements. This enables a quantitative study of the reactivity transfer of OH from the gas phase plasma to the liquid phase and how its diffusion limitations impact formate decomposition in the water droplet. For a droplet with a diameter of 36 μm, we observed 50% reduction in formate concentration in the droplets after plasma treatment for droplet residence times in the plasma of ∼10 ms. These short droplet residence times in the plasma allow in some cases for droplet size reductions of ∼5% in spite gas temperatures of 360 K. A one-dimensional reaction–diffusion model was used to calculate the OH transport and formate oxidation inside the droplet and was able to predict the conversion of formate by plasma in a droplet without any fitting parameters. The model further shows that formate conversion is dominated by near-interfacial reactions with OH radicals and is limited by diffusion of formate in the droplet. The results show that a controlled plasma–micro-droplet reactor as reported in this study might be an excellent tool for detailed quantitative plasma–liquid interaction studies.

Electrochemically Induced pH Change: Time-Resolved Confocal Fluorescence Microscopy Measurements and Comparison with Numerical Model.

Confocal fluorescence microscopy is a proven technique, which can image near-electrode pH changes. For a complete understanding of electrode processes, time-resolved measurements are required, which have not been achieved previously. Here we present the first measurements of time-resolved pH profiles with confocal fluorescence microscopy. The experimental results compare favorably with a one-dimensional reaction–diffusion model; this holds up to the point where the measurements reveal three-dimensionality in the pH distribution. Specific factors affecting the pH measurement such as attenuation of light and the role of dye migration are also discussed in detail. The method is further applied to reveal the buffer effects observed in sulfate-containing electrolytes. The work presented here is paving the way toward the use of confocal fluorescence microscopy in the measurement of 3D time-resolved pH changes in numerous electrochemical settings, for example, in the vicinity of bubbles.

Pyrolysis behaviors dominated by the reaction–diffusion mechanism in the fluorine-free metal–organic decomposition process

In this work, pyrolysis behaviors dominated by the reaction–diffusion mechanism were investigated. And one-dimensional reaction–diffusion model is proposed.

Adomyan Decomposition Method for a Two-Component Nonlocal Reaction-Diffusion Model of the Fisher–Kolmogorov–Petrovsky–Piskunov Type

We consider an approach to constructing approximate analytical solutions for the one-dimensional twocomponent reaction-diffusion model describing the dynamics of population interacting with the active substance surrounding the population. The system of model equations includes the nonlocal generalized Fisher–Kolmogorov–Petrovsky–Piskunov equation for the population density and the diffusion equation for the density of the active substance. Both equations contain additional terms describing the mutual influence of the population and the active substance. To find approximate solutions of the system of model equations, we first use the perturbation method with respect to the small parameter of interaction between the population and the active substance. Then we apply the well-known iterative method developed by G. Adomian to solve equations for terms of perturbation series. In the method proposed, the solution is presented as a series whose terms are determined by the corresponding iterative procedure. In this work, the diffusion operator is taken as the operator for which the inverse operator is expressed in terms of the diffusion propagator. This allows one to find the approximate solutions in the class of functions decreasing at infinity. As an illustration, we consider an example of solving the Cauchy problem for the initial functions of a Gaussian form.

Tracing denitrification in the Canada Basin: N2 loss to the atmosphere on the Chukchi Shelf and benthic inputs in deep waters

The global marine fixed nitrogen budget acts as a strong control on oceanic primary productivity, and the Arctic plays a disproportionately large role in the sink terms for this budget. This paper aims to quantify the impact of nitrogen cycling on the Canada Basin, utilizing two tracers of denitrification: N2/Ar, a dissolved gas tracer, and N*, a nutrient ratio tracer. In the Pacific Winter Water (PWW), which forms in the Chukchi Sea, we observe a disconnect between N2/Ar and N*, where the excess N2 expected from N* observations is far larger than the N2 excess we measure. We show that loss of N2 to the atmosphere through ventilation on the Chukchi Shelf likely accounts for this disparity, highlighting the importance of using N2/Ar as a denitrification tracer only in isolated water masses. We additionally observe increasing N2/Ar and decreasing N* in the old deep waters of the Canada Basin, suggesting benthic denitrification has been operating in the deep sediments over the 500-year age of this water mass. We use a one-dimensional vertical reaction-diffusion model to estimate denitrification rates of 0.0053–0.0130 mmol-N m−2 d−1, or 0.04–0.1 Tg N y−1 integrated over the whole basin, which is about half the rates estimated for other deep basins, in-line with lower remineralization rates in the deep Canada Basin. Further measurements of these tracers in the Arctic, particularly directly in the Chukchi Sea, will help constrain the relative importance of physical vs. biological processes on N2 in this region.

Spatially localized structures in the Gray-Scott model.

Spatially localized structures in the one-dimensional Gray-Scott reaction-diffusion model are studied using a combination of numerical continuation techniques and weakly nonlinear theory, focusing on the regime in which the activator and substrate diffusivities are different but comparable. Localized states arise in three different ways: in a subcritical Turing instability present in this regime, and from folds in the branch of spatially periodic Turing states. They also arise from the fold of spatially uniform states. These three solution branches interconnect in complex ways. We use numerical continuation techniques to explore their global behaviour within a formulation of the model that has been used to describe dryland vegetation patterns on a flat terrain.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.

Quantitative interpretation of vertical profiles of calcium and pH in the coral coelenteron

Abstract Scleratinian corals (hard corals) and their symbiotic algae are the ecological engineers of biodiverse and geological important coral reef habitats. The complex, linked physiological processes that enable the holobiont (coral + algae) to calcify and generate reef structures are consequently of great interest. However, the mechanism of calcification is difficult to study for several reasons including the small spatial scales of the processes and the close coupling between the symbiont and host. In this study, we explore the utility of pH and Ca 2+ microelectrodes for constraining the rates and spatial distribution of photosynthesis, respiration, and calcification. The work focuses on vertical profiles of pH and Ca 2+ through the coelenteron cavity, a semi-isolated compartment of modified seawater amenable to quantitative interpretation. In two studied species, Turbinaria reniformis and Acropora millepora , Ca 2+ concentrations decreased in a roughly linear manner from the mouth to the base of the coelenteron, indicating the primary physiological process affecting Ca 2+ concentration is removal for calcification below the coelenteron. In contrast, the H + concentration remained relatively constant over much of the coelenteron cavity before it increased sharply toward the base of the coelenteron, indicative of proton-pumping from the calcification fluid below. The estimated H + gradient between the coelenteron cavity and the calcification site was >10 times higher than previously predicted. Consequently, the energy required to export protons from the calcifying fluid was estimated to be ~3 times higher than previously calculated. A one-dimensional reaction-diffusion model was used to interpret the pH profile considering the effects of photosynthesis, respiration, and calcification. This model provided a good fit to the observed pH profile and helped to constrain the rates and spatial distribution of these processes. Our modeling results also suggested that adult corals with deeper polyps may be more sensitive to ocean acidification (OA) because of enhanced difficulty to transport H + out of the coelenteron and into the surrounding seawater.

A mathematical model of Noggin and BMP densities in adult neural stem cells

New neurons are generated in the adult hippocampus throughout life by neural stem cells (NSCs) in a dynamic process responsive to external signalling cues. NSCs in the adult hippocampus divide infrequently, and it has been shown that bone morphogenetic protein (BMP) modulates their quiescence. Infusion of Noggin, a BMP antagonist, blocks this signalling. We investigate the balance of BMP and Noggin in this particular niche and qualitatively reproduce experimental results obtained and qualitatively reproduce experimental results with a one-dimensional reaction-diffusion model. We use the model to connect BMP signalling profiles with specific cellular outcomes and to determine whether the transient infusion of BMP leads to a signalling profile which can be reversed by the infusion of Noggin. Additionally, we analyse the role of diffusion in this system for generating signalling profiles with dramatically different cell-fate outcomes and show that diffusion-driven instability is not possible in our system of reaction–diffusion equations.

Dynamics of gradient formation by intracellular shuttling.

A number of important cellular functions rely on the formation of intracellular protein concentration gradients. Experimental studies discovered a number of mechanisms for the formation of such gradients. One of the mechanisms relies on the intracellular shuttling of a protein that interconverts between the two states with different diffusivities, under the action of two enzymes, one of which is localized to the plasma membrane, whereas the second is uniformly distributed in the cytoplasm. Recent work reported an analytical solution for the steady state gradient in this mechanism, obtained in the framework of a one-dimensional reaction-diffusion model. Here, we study the dynamics in this model and derive analytical expressions for the Laplace transforms of the time-dependent concentration profiles in terms of elementary transcendental functions. Inverting these transforms numerically, one can obtain time-dependent concentration profiles of the two forms of the protein.

Bursting regimes in a reaction-diffusion system with action potential-dependent equilibrium.

The equilibrium Nernst potential plays a critical role in neural cell dynamics. A common approximation used in studying electrical dynamics of excitable cells is that the ionic concentrations inside and outside the cell membranes act as charge reservoirs and remain effectively constant during excitation events. Research into brain electrical activity suggests that relaxing this assumption may provide a better understanding of normal and pathophysiological functioning of the brain. In this paper we explore time-dependent ionic concentrations by allowing the ion-specific Nernst potentials to vary with developing transmembrane potential. As a specific implementation, we incorporate the potential-dependent Nernst shift into a one-dimensional Morris-Lecar reaction-diffusion model. Our main findings result from a region in parameter space where self-sustaining oscillations occur without external forcing. Studying the system close to the bifurcation boundary, we explore the vulnerability of the system with respect to external stimulations which disrupt these oscillations and send the system to a stable equilibrium. We also present results for an extended, one-dimensional cable of excitable tissue tuned to this parameter regime and stimulated, giving rise to complex spatiotemporal pattern formation. Potential applications to the emergence of neuronal bursting in similar two-variable systems and to pathophysiological seizure-like activity are discussed.

A computational model for BMP movement in sea urchin embryos

Bone morphogen proteins (BMPs) are distributed along a dorsal-ventral (DV) gradient in many developing embryos. The spatial distribution of this signaling ligand is critical for correct DV axis specification. In various species, BMP expression is spatially localized, and BMP gradient formation relies on BMP transport, which in turn requires interactions with the extracellular proteins Short gastrulation/Chordin (Chd) and Twisted gastrulation (Tsg). These binding interactions promote BMP movement and concomitantly inhibit BMP signaling. The protease Tolloid (Tld) cleaves Chd, which releases BMP from the complex and permits it to bind the BMP receptor and signal.In sea urchin embryos, BMP is produced in the ventral ectoderm, but signals in the dorsal ectoderm. The transport of BMP from the ventral ectoderm to the dorsal ectoderm in sea urchin embryos is not understood. Therefore, using information from a series of experiments, we adapt the mathematical model of Mizutani et al. (2005) and embed it as the reaction part of a one-dimensional reaction–diffusion model. We use it to study aspects of this transport process in sea urchin embryos. We demonstrate that the receptor-bound BMP concentration exhibits dorsally centered peaks of the same type as those observed experimentally when the ternary transport complex (Chd-Tsg-BMP) forms relatively quickly and BMP receptor binding is relatively slow. Similarly, dorsally centered peaks are created when the diffusivities of BMP, Chd, and Chd-Tsg are relatively low and that of Chd-Tsg-BMP is relatively high, and the model dynamics also suggest that Tld is a principal regulator of the system. At the end of this paper, we briefly compare the observed dynamics in the sea urchin model to a version that applies to the fly embryo, and we find that the same conditions can account for BMP transport in the two types of embryos only if Tld levels are reduced in sea urchin compared to fly.

Measuring the Impact of Lipid Interactions on the Mobility and Localization of Synaptic Proteins in Live Synapses

The presynaptic protein complexin can both promote and inhibit fusion through interactions between its central helix and the SNARE complex. A poorly conserved C-terminal domain (CTD) is also required for inhibition of spontaneous fusion. We found that the CTD binds lipids through a novel protein motif and directs complexin onto synaptic vesicles where it can efficiently engage the SNAREs and inhibit spontaneous fusion. Using in vivo dynamic imaging approaches in C. elegans, we observed that complexin is sequestered within presynaptic terminals through its CTD while its escape rate out of the synapse depends sensitively on synaptic activity. Complexin exhibits reduced mobility in synaptic boutons compared to neighboring axonal regions and its mobility is enhanced when synaptic vesicles were removed, consistent with their role in capture and retention of complexin. Finally, several common lipid-interacting protein motifs were imaged at synapses in the presence and absence of synaptic activity, and the impact of disrupting these lipid-binding domains was quantified. Simple one-dimensional reaction diffusion models were used to quantify the dynamics of protein exchange between en passant synapses.

Spread of infectious diseases in a hyperbolic reaction-diffusion susceptible-infected-removed model

A one-dimensional hyperbolic reaction-diffusion model of epidemics is developed to describe the dynamics of diseases spread occurring in an environment where three kinds of individuals mutually interact: the susceptibles, the infectives, and the removed. It is assumed that the disease is transmitted from the infected population to the susceptible one according to a nonlinear convex incidence rate. The model, based upon the framework of extended thermodynamics, removes the unphysical feature of instantaneous diffusive effects, which is typical of parabolic models. Linear stability analyses are performed to study the nature of the equilibrium states against uniform and nonuniform perturbations. Emphasis is given to the occurrence of Hopf and Turing bifurcations, which break the temporal and the spatial symmetry of the system, respectively. The existence of traveling wave solutions connecting two steady states is also discussed. The governing equations are also integrated numerically to validate the analytical results and to characterize the spatiotemporal evolution of diseases.

Ecological systems, nonlinear boundary conditions, and Σ-shaped bifurcation curves

We examine a one-dimensional reaction diffusion model with a weak Allee growth rate that appears in population dynamics. We combine grazing with a certain nonlinear boundary condition that models negative density dependent dispersal on the boundary and analyze the effects on the steady states. In particular, we study the bifurcation curve of positive steady states as the grazing parameter is varied. Our results are acquired through the adaptation of a quadrature method and Mathematica computations. Specifically, we computationally ascertain the existence of6-shaped bifurcation curves with several positive steady states for a certain range of the grazing parameter.