System Of Nonlinear Reaction-diffusion Equations Research Articles

Oscillations in Neuronal Activity: A Neuron-Centered Spatiotemporal Model of the Unfolded Protein Response in Prion Diseases.

Many neurodegenerative diseases (NDs) are characterized by the slow spatial spread of toxic protein species in the brain. The toxic proteins can induce neuronal stress, triggering the Unfolded Protein Response (UPR), which slows or stops protein translation and can indirectly reduce the toxic load. However, the UPR may also trigger processes leading to apoptotic cell death and the UPR is implicated in the progression of several NDs. In this paper, we develop a novel mathematical model to describe the spatiotemporal dynamics of the UPR mechanism for prion diseases. Our model is centered around a single neuron, with representative proteins P (healthy) and S (toxic) interacting with heterodimer dynamics (S interacts with P to form two S's). The model takes the form of a coupled system of nonlinear reaction-diffusion equations with a delayed, nonlinear flux for P (delay from the UPR). Through the delay, we find parameter regimes that exhibit oscillations in the P- and S-protein levels. We find that oscillations are more pronounced when the S-clearance rate and S-diffusivity are small in comparison to the P-clearance rate and P-diffusivity, respectively. The oscillations become more pronounced as delays in initiating the UPR increase. We also consider quasi-realistic clinical parameters to understand how possible drug therapies can alter the course of a prion disease. We find that decreasing the production of P, decreasing the recruitment rate, increasing the diffusivity of S, increasing the UPR S-threshold, and increasing the S clearance rate appear to be the most powerful modifications to reduce the mean UPR intensity and potentially moderate the disease progression.

MODEL TO STUDY INTERDEPENDENT CALCIUM AND IP3 DISTRIBUTION REGULATING NFAT PRODUCTION IN T LYMPHOCYTE

The cooperative system of calcium (Ca[Formula: see text]) and inositol 1,4,5-trisphosphate (IP3) plays a crucial role in maintaining structure and functions of T lymphocyte cells including the production of nuclear factor of activated T cells (NFAT) as an immunological response. NFAT regulates several mechanisms and dysregulation in this process can lead to various disorders like inflammatory bowel diseases, etc. In this study, the two-way feedback model of Ca[Formula: see text] and IP3 dynamics regulating NFAT production in T lymphocytes has been constructed. A system of nonlinear reaction-diffusion equations for calcium and IP3 for a one-dimensional case has been employed in this proposed model. The numerical simulation is performed using finite element and the Crank–Nicholson method. The effects of the parameters like Ryanodine Receptor (RyR), source amplitude and buffers on active NFAT production have been analyzed. The novel insights about regulatory and dysregulation events and mechanisms of Ca[Formula: see text], IP3 and NFAT are concluded from these results.

Mathematical modelling of a mono-enzyme dual amperometric biosensor for enzyme-catalyzed reactions using homotopy analysis and Akbari-Ganji methods

The mathematical model is described in this article as a mono-enzyme dual biosensor. This model involves coupling the enzyme-catalyzed reactions under steady-state conditions for a reaction-diffusion equation describing the substrate and product. Pseudo-experimental responses to chemical mixtures were produced using a computer model of a biocatalytic amperometric biosensor that used a mono-enzyme-catalyzed (nonspecific) competitive conversion of two substrates.The non-linear system of reaction-diffusion equations into an approximate analytical expression is solved by employing the Homotopy Analysis Method (HAM) and Akbari-Ganji Method (AGM) are used. These expressions are compared with numerical simulations carried out using MATLAB. Further, the biosensor's resistance, sensitivity, and current density were analytically expressed.

Steady state concentration in an amperometric biosensor using mathematical modeling

This paper presents an analytical solution to the model. By solving the system of nonlinear reaction-diffusion equations using Akbari–Ganji’s method (AGM), we can derive the semi-analytical expressions for substrate and oxidized and reduced mediator concentrations. These models provide information about enzyme electrode mechanisms and their kinetics. These modeling results can be helpful for sensor design, optimization, and determining how the electrode will react. Further, we studied the effects of many physical and biological parameters on model prediction. The mathematical equations, which are nonlinear partial differential equations, are solved using AGM. Also, we derived the semi-analytical results for concentrations for saturated and unsaturated kinetics. It is verified that the proposed solution is validated by comparing it with numerical solutions.

Mathematical Model of Synaptic Long-Term Potentiation as a Bistability in a Chain of Biochemical Reactions with a Positive Feedback.

Nitric oxide (NO) is involved in synaptic long-term potentiation (LTP) by multiple signaling pathways. Here, we show that LTP of synaptic transmission can be explained as a feature of signal transduction-bistable behavior in a chain of biochemical reactions with positive feedback, formed by diffusion of NO to the presynaptic site and facilitating the release of glutamate (Glu). The dynamics of Glu, calcium (Ca2+) and NO is described by a system of nonlinear reaction-diffusion equations with modified Michaelis-Menten (MM) kinetics. Numerical investigation reveals that the chain of biochemical reactions analyzed can exhibit a bistable behavior under physiological conditions when production of Glu is described by MM kinetics and decay of NO is modeled by means of two enzymatic pathways with different kinetic properties. Our finding extends understanding of the role of NO in LTP: a short high-intensity stimulus is "memorized" as a long-lasting elevation of NO concentration. The conclusions obtained by analysis of the chain of biochemical reactions describing LTP can be generalized to other chains of interactions or for creating the logical elements for biological computers.

A robust nonlinear model of steam reformers with applications to energy efficiency

We have developed and implemented a computational methodology to solve for the heat flow and the gas concentrations in a class of industrial catalytic steam reformers. The problem under consideration consists of a nonlinearly coupled system composed of a boundary-value problem and reaction-diffusion partial differential equations. Such reformers have broad industrial applications and consume a large amount of energy and natural resources globally. The methodology is applied to tackle two outstanding questions regarding catalytic processes, for which our methodology delivers efficient answers to the following research questions: (i) How to solve the energy/catalyst investment dichotomy, i.e., locations with relatively inexpensive energy require less catalyst and in the opposite case of high energy costs, more catalyst is needed. (ii) How to express the energy input, which is difficult to ascertain for large burner designs, in terms of easily measurable parameters in real time, such as the input temperatures. These questions are answered by our approach leading to a powerful design tool for catalytic processes in general, with huge potential to mitigate environmental impacts. The complex nonlinear system of reaction-diffusion equations is solved by means of an implicit-explicit method and the nonlinear boundary value problem is solved by an iterative method. The coupling between the two systems is also performed through an iterative method. The convergence of the latter is confirmed by extensive numerical investigations, which are consistent with observed measurements in industrial installations. Furthermore, an analysis of the syngas production with different catalyst density, reactor inlet temperature, furnace energy flame and catalyst length are performed, which confirm that the variation of these parameters can result in a more optimized process. A typical consequence is the following: Fixing all parameters, except the catalyst density and the reactor inlet temperature, we show that an increase of 100 K in the latter allows us to use 3 times less catalyst density to achieve the same target CO concentration. In other words, we have a quantifiable and substantial trade off between energy and catalyst investment.

Stable rotational symmetric schemes for nonlinear reaction-diffusion equations

In the simulation of biological pattern forming, it has been observed that the numerical solution is more sensitive to the spatial mesh resolution than the temporal one. Such a higher sensitivity to the spatial resolution is mainly originated from an inaccurate approximation of diffusion differential operators, which might violate the rotational symmetry to be seriously erroneous in low spatial resolutions. Also, it has been known that the second-order Crank-Nicolson time-stepping procedure may introduce spurious oscillations when the initial data or the source term is nonsmooth and the temporal step size is set relatively large. This article studies 9-point finite difference schemes for the diffusion operator to enhance the rotational symmetry, employs the variable-θ method to achieve a nonoscillatory second-order time-stepping procedure, and adopts an effective relaxation linear solver to solve the algebraic systems efficiently. The variable-θ method is proved to satisfy the maximum principle, which guarantees that the time-stepping procedure is unconditionally stable. When the successive over-relaxation method with an optimal relaxation parameter is adopted for the algebraic solver, the iteration converges in 2-4 iterations in most time steps. The overall algorithm is second-order in accuracy and scalable in efficiency. Various examples are given to show the accuracy and efficiency of the proposed algorithm for the numerical solution of the system of nonlinear reaction-diffusion equations.

Mathematical modeling of plant cell fate transitions controlled by hormonal signals.

Coordination of fate transition and cell division is crucial to maintain the plant architecture and to achieve efficient production of plant organs. In this paper, we analysed the stem cell dynamics at the shoot apical meristem (SAM) that is one of the plant stem cells locations. We designed a mathematical model to elucidate the impact of hormonal signaling on the fate transition rates between different zones corresponding to slowly dividing stem cells and fast dividing transit amplifying cells. The model is based on a simplified two-dimensional disc geometry of the SAM and accounts for a continuous displacement towards the periphery of cells produced in the central zone. Coupling growth and hormonal signaling results in a nonlinear system of reaction-diffusion equations on a growing domain with the growth rate depending on the model components. The model is tested by simulating perturbations in the level of key transcription factors that maintain SAM homeostasis. The model provides new insights on how the transcription factor HECATE is integrated in the regulatory network that governs stem cell differentiation.

Nonclassical Symmetries of a System of Nonlinear Reaction-Diffusion Equations

We study the conditional symmetry of a system of nonlinear reaction-diffusion equations and establish the existence of operators of conditional symmetry for systems of nonlinear reaction-diffusion equations with any number of independent variables and find these operators in the explicit form. The exact solutions of nonlinear reaction-diffusion equations with exponential nonlinearity are constructed.

RENORMALIZED AND ENTROPY SOLUTIONS OF TUMOR GROWTH MODEL WITH NONLINEAR ACID PRODUCTION

This paper establishes the existence of renormalized and entropy solutions for a system of nonlinear reaction-diffusion equations which describes the tumor growth along with acidification and interaction. Under the assumptions of L1 data and no growth conditions with zero Dirichlet boundary conditions, we prove the existence of renormalized and entropy solutions for the considered mathematical model.

Conditional Symmetry of a System of Nonlinear Reaction-Diffusion Equations

The conditional symmetry of a system of nonlinear reaction-diffusion equations is investigated. It is shown that the operators of conditional symmetry exist for the systems of nonlinear reaction-diffusion equations with an arbitrary number of independent variables. Moreover, these operators are found in the explicit form.

Класифікація галілеєво-інваріантних систем нелінійних рівнянь реакції-дифузії

Класифікація галілеєво-інваріантних систем нелінійних рівнянь реакції-дифузії

A Class of Mathematical Models Describing Processes in Spatially Heterogeneous Biofilm Communities

We present a class of deterministic continuum models for spatially heterogeneous biofilm communities. The prototype is a single-species biofilm growth model, which is formulated as a highly non-linear system of reaction-diffusion equations for the biomass density and the concentration of the growth controlling substrate. While the substrate concentration satisfies a standard semi-linear reaction-diffusion equation the equation for the biomass density comprises two non-linear diffusion effects: a porous medium-type degeneracy and super diffusion. When further biofilm processes are taken into account equations for several substrates and multiple biomass components have to be included in the model. The structure of these multi-component extensions is essentially different from the mono-species case, since the diffusion operator forthe biomass componentsdepends on the total biomass in the system and the equations are strongly coupled. We present the prototype biofilm growth model and give an overview of its multi-component extensions. Moreover, we summarize analytical results that were obtained for these models.

Modeling Cell Movement and Chemotaxis Using Pseudopod-Based Feedback

A computational framework is presented for the simulation of eukaryotic cell migration and chemotaxis. An empirical pattern formation model, based on a system of nonlinear reaction-diffusion equations, is approximated on an evolving cell boundary using an arbitrary Lagrangian Eulerian surface finite element method (ALE-SFEM). The solution state is used to drive a mechanical model of the protrusive and retractive forces exerted on the cell boundary. Movement of the cell is achieved using a level set method. Results are presented for cell migration with and without chemotaxis. The simulated behavior is compared with experimental results of migrating Dictyostelium discoideum cells.

Numerical solutions of a nonlinear reaction-diffusion system

This paper is concerned with finite difference solutions of a coupled system of nonlinear reaction-diffusion equations. The investigation is devoted to the finite difference system for both the time-dependent problem and its corresponding steady-state problem. The existence and uniqueness of a non-negative finite difference solution and three monotone iterative algorithms for the computation of the solutions are given. It is shown that the time-dependent problem has a unique non-negative solution, whereas the steady-state problem may have multiple non-negative solutions depending on the parameters in the problem. The different non-negative steady-state solutions can be computed from the monotone iterative algorithms by choosing different initial iterations. Also discussed is the asymptotic behaviour of the time-dependent solution in relation to the steady-state solutions. The asymptotic behaviour result gives some conditions ensuring the convergence of the time-dependent solution to a positive or semitrivial non-negative steady-state solution. Numerical results are given to demonstrate the theoretical analysis results.

Modelling Amperometric Biosensors Based on Chemically Modified Electrodes.

The response of an amperometric biosensor based on a chemically modified electrode was modelled numerically. A mathematical model of the biosensor is based on a system of non-linear reaction-diffusion equations. The modelling biosensor comprises two compartments: an enzyme layer and an outer diffusion layer. In order to define the main governing parameters the corresponding dimensionless mathematical model was derived. The digital simulation was carried out using the finite difference technique. The adequacy of the model was evaluated using analytical solutions known for very specific cases of the model parameters. By changing model parameters the output results were numerically analyzed at transition and steady state conditions. The influence of the substrate and mediator concentrations as well as of the thicknesses of the enzyme and diffusion layers on the biosensor response was investigated. Calculations showed complex kinetics of the biosensor response, especially when the biosensor acts under a mixed limitation of the diffusion and the enzyme interaction with the substrate.

Multilevel Additive Schwarz Preconditioners for the Bidomain Reaction-Diffusion System

Multilevel additive Schwarz methods are analyzed and studied numerically for the anisotropic cardiac Bidomain model in three dimensions. This is the most complete model to date of the bioelectrical activity of the heart tissue, consisting of a degenerate parabolic system of nonlinear reaction-diffusion equations coupled with a stiff system of several ordinary differential equations describing the ionic currents through the cellular membrane. Due to the presence of very different scales in both space and time, the numerical discretization of this system by finite elements in space and semi-implicit methods in time produces very ill-conditioned linear systems that must be solved at each time step. The proposed multilevel algorithm employs a hierarchy of nested meshes with overlapping Schwarz preconditioners on each level and is fully additive, hence parallel, within and among levels. Convergence estimates are proved for the resulting multilevel algorithm, showing that its convergence rate is independent of the number of subdomains (scalability), of the mesh sizes of each level and of the number of levels (optimality). Several parallel tests on a Linux cluster confirm the scalability and optimality of the method, as well as its parallel efficiency on both Cartesian and deformed domains in three dimensions.

Modelling a Peroxidase-based Optical Biosensor.

The response of a peroxidase-based optical biosensor was modelled digitally.A mathematical model of the optical biosensor is based on a system of non-linear reaction-diffusion equations. The modelling biosensor comprises two compartments, an enzyme layerand an outer diffusion layer. The digital simulation was carried out using finite differencetechnique. The influence of the substrate concentration as well as of the thickness of both theenzyme and diffusion layers on the biosensor response was investigated. Calculations showedcomplex kinetics of the biosensor response, especially at low concentrations of the peroxidaseand of the hydrogen peroxide.

Group classification of systems of nonlinear reaction-diffusion equations with triangular diffusion matrix

We complete the group classification of systems of two coupled nonlinear reaction-diffusion equations with general diffusion matrix begun in author’s previous works. Namely, all nonequivalent equations with triangular diffusion matrix are classified. In addition, we describe symmetries of diffusion systems with nilpotent diffusion matrix and additional terms with first-order derivatives.

A reaction-diffusion system of $\lambda$ – $\omega$ type Part I: Mathematical analysis

We study two coupled reaction-diffusion equations of the boundary. The equations are close to a supercritical Hopf bifurcation in the reaction kinetics and are model equations for oscillatory reaction-diffusion equations. Global existence, uniqueness and continuous dependence on initial data of strong and weak solutions are proved using the classical Faedo-Galerkin method of Lions [15] and compactness arguments. We also present a complete case study for the application of this method to systems of nonlinear reaction-diffusion equations.