Reaction-diffusion Waves Research Articles

Infection spreading in tissue as a reaction-diffusion wave.

Viral infection develops in the organism due to virus replication inside infected cells and its transmission from infected to uninfected cells through the extracellular matrix or cell junctions. In this work, we model infection spreading in tissue with a delay reaction-diffusion system of equations for the concentrations of uninfected cells, infected cells and virus. We prove the wave existence, determine its speed of propagation and introduce a simplified one-equation model obtained from the complete model using a quasi-stationary approximation.

A dynamical system framework for a mathematical model of atherosclerosis

Atherosclerosis is a chronic inflammatory disease occurs due to plaque accumulation in the inner artery wall. In atherosclerotic plaque formation monocytes and macrophages play a significant role in controlling the disease dynamics. In the present article, the entire biochemical process of atherosclerotic plaque formation is presented in terms of an autonomous system of nonlinear ordinary differential equations involving concentrations of oxidized low-density lipoprotein (LDL), monocytes, macrophages, and foam cells as the key dependent variables. To observe the capacity of monocytes and macrophages the model has been reduced to a two-dimensional temporal model using quasi steady state approximation theory. Linear stability analysis of the two-dimensional ordinary differential equations (ODEs) model has revealed the stability of the equilibrium points in the system. We have considered both one- and two-parameter bifurcation analysis with respect to parameters associated to the rate at which macrophages phagocytose oxidized LDL and the rate at which LDL enters into the intima. The bifurcation diagrams reveal the oscillating nature of the curves representing concentration of monocytes and macrophages with respect to significant model parameters. We are able to find the threshold values at which the plaque accumulation accelerates in an uncontrollable way. Further to observe the impact of diffusion, a spatiotemporal model has been developed. Numerical investigation of the partial differential equations (PDEs) model reveals the existence of travelling wave in the system which ensures the fact that the development of atherosclerotic plaque formation follows reaction–diffusion wave.

Chemistry does general relativity: reaction-diffusion waves can model gravitational lensing

Gravitational lensing is a general relativistic (GR) phenomenon where a massive object redirects light, deflecting, magnifying, and sometimes multiplying its source. We use reaction-diffusion (RD) Belousov-Zhabotinsky (BZ) chemistry to study this astronomical effect in a table-top experiment. We experimentally observe BZ waves passing through non-planar, quasi-two-dimensional molds and reproduce the waveforms in computer simulations using planar RD waves propagating with variable diffusion. We tune the variable diffusion to match the Schwarzschild-coordinate light speed near a spherical mass so the RD propagation approximates Einstein’s famous light deflection relation. We discuss varying the diffusion or reaction rates with a gel matrix or with illumination, electric field, or temperature gradients.

Artificial cell system as a tool for investigating pattern formation mechanisms of intracellular reaction-diffusion waves

Artificial cell system as a tool for investigating pattern formation mechanisms of intracellular reaction-diffusion waves

Modeling of Viral Infection with Inflammation

Viral infection spreads in cell culture or tissue as a reaction–diffusion wave. It is characterized by three main parameters: virus replication number, Rv, which determines whether infection progresses, wave speed, c, which correlates with the virus virulence, and viral load, J(v), which determines the infection transmission rate. In this work, we study how the inflammation triggered by viral infection influences its progression. We obtain analytical expressions for Rv,c, and J(v) and show how they depend on the intensity of inflammation characterized by one or two parameters. Analytical and numerical results show that inflammation decreases the viral replication number, virus virulence, and infectivity, though there are different cases depending on the parameters of the model.

Inflammation propagation modeled as a reaction–diffusion wave

Inflammation is a physiological process aimed to protect the organism in various diseases and injuries. This work presents a generic inflammation model based on the reaction–diffusion equations for the concentrations of uninflamed cells, inflamed cells, immune cells and the inflammatory cytokines. The analysis of the model shows the existence of three different regimes of inflammation progression depending on the value of a parameter R called the inflammation number. If R>1, then inflammation propagates in cell culture or tissue as a reaction–diffusion wave due to diffusion of inflammatory cytokines produced by inflamed cells. If 0<R<1, then inflammation vanishes and the system converges to the stable inflammation-free equilibrium. Finally if R<0, inflammation also propagates as a reaction–diffusion wave, but the mechanism of propagation is different, it is determined by positive feedback between inflammation and immune response. From the biological point of view, these three regimes correspond to acute inflammation resolved due to the immune response, to the disappearance of inflammatory reaction, and to an auto-immune inflammatory reaction or cytokine storm. We focus on finding the wave speed and other characteristics of inflammation progression by analytical and numerical methods, in order to deduce a qualitative understanding of various inflammatory reactions.

Higher-order tension spline-based numerical technique for time fractional reaction-diffusion wave equation with damping

Higher-order tension spline-based numerical technique for time fractional reaction-diffusion wave equation with damping

Disappearance, division, and route change of excitable reaction-diffusion waves in deformable membranes

Shapes of biomembrane in living cells are regulated by curvature-inducing proteins. However, the effects of membrane deformation on signal transductions such as chemical waves have not been researched adequately. Here, we report that membrane deformation can alter the propagation of excitable reaction-diffusion waves using state-of-the-art simulations. Reaction waves can induce large shape transformations, such as membrane budding and necking, that erase or divide the wave, depending on the curvature generated by the waves, feedback to the wave propagation, and the ratio of the reaction and deformation times. In genus-2 vesicles, wave division occurs at branching points and collided waves disappear together. We demonstrate that the occasional disappearance of the waves can alter the pathway of wave propagation. Our findings suggest that membrane deformation and reaction waves can together regulate signal transductions on biomembranes.

Buoyancy-Driven Chemohydrodynamic Patterns in A + B → Oscillator Two-Layer Stratifications

Chemohydrodynamic patterns due to the interplay of buoyancy-driven instabilities and reaction-diffusion patterns are studied experimentally in a vertical quasi-two-dimensional reactor in which two solutions A and B containing separate reactants of the oscillating Belousov-Zhabotinsky system are placed in contact along a horizontal contact line where excitable or oscillating dynamics can develop. Different types of buoyancy-driven instabilities are selectively induced in the reactive zone depending on the initial density jump between the two layers, controlled here by the bromate salt concentration. Starting from a less dense solution above a denser one, two possible differential diffusion instabilities are triggered depending on whether the fast diffusing sulfuric acid is in the upper or lower solution. Specifically, when the solution containing malonic acid and sulfuric acid is stratified above the one containing the slow-diffusing bromate salt, a diffusive layer convection (DLC) instability is observed with localized convective rolls around the interface. In that case, the reaction-diffusion wave patterns remain localized above the initial contact line, scarcely affected by the flow. If, on the contrary, sulfuric acid diffuses upward because it is initially dissolved in the lower layer, then a double-diffusion (DD) convective mode develops. This triggers fingers across the interface that mix the reactants such that oscillatory dynamics and rippled waves develop throughout the whole reactor. If the denser solution is put on top of the other one, then a fast developing Rayleigh-Taylor (RT) instability induces fast mixing of all reactants such that classical reaction-diffusion waves develop later on in the convectively mixed solutions.

Glycolytic Wave Patterns in a Simple Reaction‐diffusion System with Inhomogeneous Influx: Dynamic Transitions

An inhomogeneous profile of chemostatted species generates a rich variety of patterns in glycolytic waves depicted in a Selkov reaction-diffusion framework here. A key role played by diffusion amplitude and symmetry in the chemostatted species profile in dictating the fate of local spatial dynamics involving periodic, quasiperiodic, and chaotic patterns and transitions among them are investigated systematically. More importantly, various dynamic transitions, including wave propagation direction changes, are illustrated in interesting situations. Besides numerical results, our analytical formulation of the amplitude equation connecting complex Ginzburg-Landau and Lambda-omega representation shed light on the phase dynamics of the system. This systematic study of the glycolytic reaction-diffusion wave is in line with previous experimental results in open spatial reactor and will provide a knowledge about the dynamics that shape and control biological information processing and related phenomena.

Curvature dependences of wave propagation in reaction–diffusion models

Reaction–diffusion waves in multiple spatial dimensions advance at a rate that strongly depends on the curvature of the wavefronts. These waves have important applications in many physical, ecological and biological systems. In this work, we analyse curvature dependences of travelling fronts in a single reaction–diffusion equation with general reaction term. We derive an exact, non-perturbative curvature dependence of the speed of travelling fronts that arises from transverse diffusion occurring parallel to the wavefront. Inward-propagating waves are characterized by three phases: an establishment phase dominated by initial and boundary conditions, a travelling-wave-like phase in which normal velocity matches standard results from singular perturbation theory and a dip-filling phase where the collision and interaction of fronts create additional curvature dependences to their progression rate. We analyse these behaviours and additional curvature dependences using a combination of asymptotic analyses and numerical simulations.

Controlling the Periodicity of a Reaction-Diffusion Wave in Artificial Cells by a Two-Way Energy Supplier.

Reaction-diffusion (RD) waves, which are dynamic self-organization structures generated by nanosize molecules, are a fundamental mechanism from patterning in nano- and micromaterials to spatiotemporal regulations in living cells, such as cell division and motility. Although the periods of RD waves are the critical element for these functions, the development of a system to control their period is challenging because RD waves result from nonlinear physical dynamics under far-from-equilibrium conditions. Here, we developed an artificial cell system with tunable period of an RD-driven wave (Min protein wave), which determines a cell division site plane in living bacterial cells. The developed system is based on our finding that Min waves are generated by energy consumption of either ATP or dATP, and the period of the wave is different between these two energy suppliers. We showed that the Min-wave period was modulated linearly by the mixing ratio of ATP and dATP and that it was also possible to estimate the mixing ratio of ATP and dATP from the period. Our findings illuminated a previously unidentified principle to control the dissipative dynamics of biomolecules and, simultaneously, built an important framework to construct molecular robots with spatiotemporal units.

Reaction-Diffusion Patterning of DNA-Based Artificial Cells.

Biological cells display complex internal architectures with distinct micro environments that establish the chemical heterogeneity needed to sustain cellular functions. The continued efforts to create advanced cell mimics, namely, artificial cells, demands strategies for constructing similarly heterogeneous structures with localized functionalities. Here, we introduce a platform for constructing membraneless artificial cells from the self-assembly of synthetic DNA nanostructures in which internal domains can be established thanks to prescribed reaction–diffusion waves. The method, rationalized through numerical modeling, enables the formation of up to five distinct concentric environments in which functional moieties can be localized. As a proof-of-concept, we apply this platform to build DNA-based artificial cells in which a prototypical nucleus synthesizes fluorescent RNA aptamers that then accumulate in a surrounding storage shell, thus demonstrating the spatial segregation of functionalities reminiscent of that observed in biological cells.

Excitable reaction-diffusion waves of curvature-inducing proteins on deformable membrane tubes.

Living cells employ excitable reaction-diffusion waves for internal cellular functions, in which curvature-inducing proteins are often involved. However, the role of their mechanochemical coupling is not well understood. Here, we report the membrane deformation induced by the excitable reaction-diffusion waves of curvature-inducing proteins and the alternation in the waves due to the deformation, using a coarse-grained simulation of tubular membranes with a modified FitzHugh-Nagumo model. Protein-propagating waves deform tubular membranes and large deformations induce budding and erase waves. The wave speed and shape are determined by a combination of membrane deformation and spatial distribution of the curvature-inducing protein. Waves are also undulated in the azimuthal direction depending on the condition. Rotationally symmetric waves locally deform the tubes into a symmetric shape but maintain a straight shape on average. Our simulation method can be applied to other chemical reaction models and used to investigate various biomembrane phenomena.

Virus replication and competition in a cell culture: Application to the SARS-CoV-2 variants

Viral replication in a cell culture is described by a delay reaction–diffusion system. It is shown that infection spreads in cell culture as a reaction–diffusion wave, for which the speed of propagation and viral load can be determined both analytically and numerically. Competition of two virus variants in the same cell culture is studied, and it is shown that the variant with larger individual wave speed out-competes another one, and eliminates it. This approach is applied to the Delta and Omicron variants of the SARS-CoV-2 infection in the cultures of human epithelial and lung cells, allowing characterization of infectivity and virulence of each variant, and their comparison.

Infection spreading in cell culture as a reaction-diffusion wave

Infection spreading in cell culture occurs due to virus replication in infected cells and its random motion in the extracellular space. Multiplicity of infection experiments in cell cultures are conventionally used for the characterization of viral infection by the number of viral plaques and the rate of their growth. We describe this process with a delay reaction-diffusion system of equations for the concentrations of uninfected cells, infected cells, virus, and interferon. Time delay corresponds to the duration of viral replication inside infected cells. We show that infection propagates in cell culture as a reaction-diffusion wave, we determine the wave speed and prove its existence. Next, we carry out numerical simulations and identify three stages of infection progression: infection decay during time delay due to virus replication, explosive growth of viral load when infected cells begin to reproduce it, and finally, wave-like infection progression in cell culture characterized by a constant or slowly growing total viral load. The modelling results are in agreement with the experimental data for the coronavirus infection in a culture of epithelial cells and for some other experiments. The presence of interferon produced by infected cells decreases the viral load but does not change the speed of infection progression in cell culture. In the 2D modelling, the total viral load grows faster than in the 1D case due to the increase of plaque perimeter.

Nonequilibrium dynamics of a fluid vesicle: Turing patterns and traveling waves

In living cells, the shapes of biomembranes are dynamically regulated by proteins. We briefly review our recent studies on the membrane dynamics coupled with reaction-diffusion dynamics of curvature-inducing proteins. We simulated a vesicle in a fluid phase using a dynamically triangulated membrane model combined with a modified Brusselator model. The mechanochemical feedback of curvature-inducing proteins changes the condition of Turing patterns relative to those that occur on a non-deformable curved surface. Budding and multi-spindle shapes are also stabilized by Turing patterns. The speed of traveling waves increases or decreases with increasing local membrane curvature depending on the condition. Moreover, the reaction-diffusion waves can induce the self-oscillation of vesicle shapes.

Space and Genotype-Dependent Virus Distribution during Infection Progression

The paper is devoted to a nonlocal reaction-diffusion equation describing the development of viral infection in tissue, taking into account virus distribution in the space of genotypes, the antiviral immune response, and natural genotype-dependent virus death. It is shown that infection propagates as a reaction-diffusion wave. In some particular cases, the 2D problem can be reduced to a 1D problem by separation of variables, allowing for proof of wave existence and stability. In general, this reduction provides an approximation of the 2D problem by a 1D problem. The analysis of the reduced problem allows us to determine how viral load and virulence depend on genotype distribution, the strength of the immune response, and the level of immunity.

Initial value problem for the $$(2+1)$$-dimensional time-fractional generalized convection–reaction–diffusion wave equation: invariant subspaces and exact solutions

This work investigates how we can extend the invariant subspace method to $$(2+1)$$ -dimensional time-fractional non-linear PDEs. More precisely, the systematic study has been provided for constructing the various dimensions of the invariant subspaces for the $$(2+1)$$ -dimensional time-fractional generalized convection–reaction–diffusion wave equation along with the initial conditions for the first time. Additionally, the special types of the above-mentioned equation are discussed through this method separately such as reaction–diffusion wave equation, convection–diffusion wave equation and diffusion wave equation. Moreover, we explain how to derive variety of exact solutions for the underlying equation along with initial conditions using the obtained invariant subspaces. Finally, we extend this method to $$(2+1)$$ -dimensional time-fractional non-linear PDEs with time delay. Also, the effectiveness and applicability of the method have been illustrated through the $$(2+1)$$ -dimensional time-fractional cubic non-linear convection–reaction–diffusion wave equation with time delay. In addition, we observe that the obtained exact solutions can be viewed as the combinations of the Mittag-Leffler function and polynomial, exponential and trigonometric type functions.

Virus-Host Interactions Shape Viral Dispersal Giving Rise to Distinct Classes of Traveling Waves in Spatial Expansions

Reaction-diffusion waves have long been used to describe the growth and spread of populations undergoing a spatial range expansion. Such waves are generally classed as either pulled, where the dynamics are driven by the very tip of the front and stochastic fluctuations are high, or pushed, where cooperation in growth or dispersal results in a bulk-driven wave in which fluctuations are suppressed. These concepts have been well studied experimentally in populations where the cooperation leads to a density-dependent growth rate. By contrast, relatively little is known about experimental populations that exhibit density-dependent dispersal. Using bacteriophage T7 as a test organism, we present novel experimental measurements that demonstrate that the diffusion of phage T7, in a lawn of host E. coli, is hindered by steric interactions with host bacteria cells. The coupling between host density, phage dispersal and cell lysis caused by viral infection results in an effective density-dependent diffusion coefficient akin to cooperative behavior. Using a system of reaction-diffusion equations, we show that this effect can result in a transition from a pulled to pushed expansion. Moreover, we find that a second, independent density-dependent effect on phage dispersal spontaneously emerges as a result of the viral incubation period, during which phage is trapped inside the host unable to disperse. Additional stochastic agent-based simulations reveal that lysis time dramatically affects the rate of diversity loss in viral expansions. Taken together, our results indicate both that bacteriophage can be used as a controllable laboratory population to investigate the impact of density-dependent dispersal on evolution, and that the genetic diversity and adaptability of expanding viral populations could be much greater than is currently assumed.