Reaction-diffusion equation , model

Compared to scroll rings and helices, the geometry of even the simplest knotted filament (the trefoil) is quite complex. Knotted scroll waves are unknown (or unrecognized) in the BZ reaction, and there have been few thorough studies of knotted scroll wave solutions to reaction-diffusion equations modeling excitable media [26,28]. The analytical theory [32] of invariant knotted solutions to the filament equations (15) is not only difficult but also (probably) inapplicable to the invariant knots that have been computed numerically, because the latter are compact structures whose dynamics seem to be dominated by interactions between the closely spaced segments of the knotted filament. 

When applying a mechanistic model, nearly all of the computational effort resides in step (3).109 In most mechanistic models, step (3) is modeled by one-dimensional reaction-diffusion equations of the form... 

As discussed in Section 4.3, the linear-eddy model solves a one-dimensional reaction-diffusion equation for all length scales. Inertial-range fluid-particle interactions are accounted for by a random rearrangement process. This leads to significant computational inefficiency since step (3) is not the rate-controlling step. Simplifications have thus been introduced to avoid this problem (Baldyga and Bourne 1989). 

The behaviour of the simpler autocatalytic models in each of these three class A geometries seems to be qualitatively very similar, so we will concentrate mainly on the infinite slab, j = 0. For the single step process in eqn (9.3) the two reaction-diffusion equations, for the two species concentrations, have the form... 

In a theoretical model, we considered the dynamics of bound water molecules and when they become free by translational and rotational motions. Two coupled reaction-diffusion equations were solved. The two rate constants, kbf and kjb, were introduced to describe the transition from bound (to the surface) to free (from the surface) and the reverse, respectively. We also took into account the effect of the bulk water re-entry into the layer—a feedback mechanism—and the role of orientational order and surface inhomogeneity on the observed decay characteristics. With this in mind, the expressions for the change in density with time were written defining the feedback as follows ... 

A second procedure, based on the vibrational assistance model for calculating the solvent-dynamics-modified rate, is given in Ref. 44. The reaction-diffusion equation, adapted from Eq. (1.2), is, for the case where the back reaction is neglected, given by (2.3). The more complete treatment, where the back reaction (recrossings) is included, is given in Ref. 44 ... 

A recent work has demonstrated that the formulation of reaction-diffusion problems in systems that display slow diffusion within a continuous-time random walk model with a broad waiting time pdf of the form (6) leads to a fractional reaction-diffusion equation that includes a source or sink term in the same additive way as in the Brownian limit [63], With the fractional formulation for single-species slow reaction-diffusion obtained by the authors still being linear, no pattern formation due to Turing instabilities can arise. This is due to the fact that fractional systems of the type (15) are close to Gibbs-Boltzmann thermodynamic equilibrium as shown in the next section. 

The model geometries are shown in Figure 43, and the basic dimensionless promoter (O ) reaction-diffusion equation, governing both phenomena, is... 

An expression for the slow relaxation in the hydration layer has been derived where we model the protein surface as an infinite wall in the x-y plane. The starting point is coupled reaction-diffusion equations that describe the time evolution of bound and free water densities (Nandi and Bagchi, 1997 Pal et al., 2002 Bhattacharyya et al., 2003),... 

A disadvantage of ODE models is that they assume spatially homogeneous systems, an assumption that sometimes may lead to wrong predictions. Although in many cases spatial effects can be incorporated in the function I , there are situations where one may need to take into account diffusion and transport of proteins from one compartment to another. For the purpose, reaction-diffusion equations (RDEs) of the form... 

Basic features of the light-sensitive BZ reaction are reproduced by the Oregonator model given as a system of two coupled reaction-diffusion equations ... 

It should be pointed out that although //is not consumed by the reaction, its value is not locally constant because diffusion during PEB and evaporation (as well as unwanted side reactions involving the acid, i.e., all acid loss mechanisms) may induce local variations in the concentration of H. This condition necessitates the use of reaction-diffusion equations to accurately model this system. However, the assumption that H is constant is not without merit, for it is valid under certain conditions. Besides, it helps to simplify the problem. 

Peskin et al [1993] have proposed the Brownian ratchet theory to describe the active force production. The main component of that theory was the interaction between a rigid protein and a diffusing object in front of it. If the object undergoes a Brownian motion, and the fiber undergoes polymerization, there are rates at which the polymer can push the object and overcome the external resistance. The problem was formulated in terms of a system of reaction-diffusion equations for the probabilities of the polymer to have certain number of monomers. Two limiting cases, fast diffusion and fast polymerization, were treated analytically that resulted in explicit force/velocity relationships. This theory was subsequently extended to elastic objects and to the transient attachment of the filament to the object. The correspondence of these models to recent experimental data is discussed in the article by Mogilner and Oster [2003]. 

The dynamic exchange model employs a hydrodynamic approach wherein the dynamics of the three species in the surface layer is described by a reaction-diffusion equation and the bulk water dynamics is described by a simple diffusion equation. Therefore, in this approach, the interactions are not considered explicitly... 

This type of equation is also encountered in other areas, such as nonlinear waves, nucleation theory, and phase field models of phase transitions, where it is known as the damped nonlinear Klein-Gordon equation, see for example [165, 355, 366]. In the (singular) limit r 0, (2.15) goes to the reaction-diffusion equation (2.3). Front propagation in HRDEs has been studied analytically and numerically in [149, 150, 152, 151, 374]. The use of HRDEs in applications is problematic. Such equations are obtained indeed very much in an ad hoc manner for reacting and dispersing particle systems, and they can be derived neither from phenomenological thermodynamic equations nor from more microscopic equations, see below. 

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