Pattern formation and morphogenesis: A reaction-diffusion model

The problem of cellular differentiation and consequent pattern generation during embryonic development has been mathematically investigated with the help of a reaction-diffusion model. It is by now a well-recognized fact that diffusion of micromolecules (through intercellular gap junctions), which is dependent on the spatial parameter (r), serve the purpose of ‘positional information’ for differentiation. Based on this principle the present model has been constructed by coupling the Goodwin-type equations for RNA and protein synthesis with the diffusion process. The homogeneous Goodwin system can exhibit stable periodic solution if the value of the cooperativity as measured by the Hill coefficient (ρ) is greater than 8, which is not biologically realistic. In the present work it has been observed that inclusion of a negative cross-diffusion can drive the system into local instability for any value of ρ and thus a time-periodic spatial solution is possible around the unstable local equilibrium, eventually leading to a definite pattern formation. Inclusion of a negative cross-diffusion thus makes the system biologically realistic. The cross-diffusion can also give rise to a stationary wave-like dissipative structure.     

Assessment of rabbit hemorrhagic disease in controlling the population of red fox: A measure to preserve endangered species in Australia

Predator's management requires a detailed understanding of the ecological circumstances associated with predation. Predation by foxes has been a significant contributor to the Australian native animal reduction. This paper mainly focuses on the dissemination of rabbit hemorrhagic disease in the rabbit population and its subsequences on red fox (Vulpes vulpes) population, by qualitative and quantitative analyses of a designed eco-epidemiological model with simple law of mass action and sigmoid functional response.Existence of solution has been analyzed and shown to be uniformly bounded. The basic reproduction number (R₀) is obtained and the occurrence of a backward bifurcation at R₀ = 1 is shown to be possible using central manifold theory. Global stability of endemic equilibrium is established by geometric approach. Criteria for diffusion-driven ecological instability caused by local random movements of European rabbits and red fox are obtained. Detailed analyses of Turing patterns formation selected by reaction-diffusion system under zero flux boundary conditions are presented. We found that transmission rate, self and cross-diffusion coefficients have appreciable influence on spatial spread of epidemics. Numerical simulation results confirm the analytical finding and generate patterns which indicate that population of red foxes might be controlled if rabbit hemorrhagic disease (RHD) is introduced into the rabbit population and thus ecological balance can be maintained.     

Turing instabilities in general systems

We present necessary and sufficient conditions on the stability matrix of a general n(≥2)-dimensional reaction-diffusion system which guarantee that its uniform steady state can undergo a Turing bifurcation. The necessary (kinetic) condition, requiring that the system be composed of an unstable (or activator) and a stable (or inhibitor) subsystem, and the sufficient condition of sufficiently rapid inhibitor diffusion relative to the activator subsystem are established in three theorems which form the core of our results. Given the possibility that the unstable (activator) subsystem involves several species (dimensions), we present a classification of the analytically deduced Turing bifurcations into p (1 ≤p≤ (n− 1)) different classes. For n = 3 dimensions we illustrate numerically that two types of steady Turing pattern arise in one spatial dimension in a generic reaction-diffusion system. The results confirm the validity of an earlier conjecture [12] and they also characterise the class of so-called strongly stable matrices for which only necessary conditions have been known before [23, 24]. One of the main consequences of the present work is that biological morphogens, which have so far been expected to be single chemical species [1–9], may instead be composed of two or more interacting species forming an unstable subsystem. Received: 21 September 1999 / Revised version: 21 June 2000 / Published online: 24 November 2000     

强耦合竞争-竞争-互惠模型古典解的整体存在性

应用能量估计方法和bootstrap技巧证明了一类强耦合反应扩散方程系统在任意维空间中古典解的整体存在性,该系统是竞争种群含自扩散和交错扩散,互惠种群仅含自扩散的竞争-竞争-互惠模型．     

Competitive exclusion and persistence in models of resource and sexual competition

 We study a combined mathematical model of resource and sexual competition. The population dynamics in this model is analyzed through a coupled system of reaction-diffusion equations. It is shown that strong sexual competition and low birth rate lead to competitive exclusion of the biological species. If sexual competition is weak, then the persistence of the species is possible, depending on the initial density functions and the growth rates of the species. When sexual competition affects both species, persistence and competitive exclusion results are also obtained in terms of the ecological data in the model. Received 1 November 1995; received in revised form 13 January 1996     

Effects of diffusion on the stability of the equilibrium in multi-species ecological systems

Continuous population distributions that undergo self-diffusion, migrational cross-diffusion and interaction in a region of (1-, 2- or 3-dimensional) space are described dynamically by a governing system of nonlinear reaction-diffusion equations. It is shown that the constants associated with migrational cross-diffusion are ordinarily nonnegative or nonpositive, contingent on the type of species interaction. A simple sign relationship obtains between the latter diffusivity constants and the rate constants for species interaction in the neighborhood of a spatially uniform equilibrium state, and this relationship of signs serves to simplify the general stability theory for the growth or decay of arbitrary perturbations on a spatially uniform equilibrium state. The stability of the equilibrium state is analyzed and discussed in detail for the case of a generic two-species model, where the self-diffusion and migrational cross-diffusion of species act to either stabilize or destabilize the equilibrium, depending essentially on the character of the species interaction and also on the magnitude of the Helmholtz eigenvalues associated with the region and boundary conditions. In particular, for a prey-predator or host-parasite model, self-diffusion usually helps to stabilize the equilibrium state and migrational cross-diffusion can only act as an additional stabilizing influence, as evidenced generally by the experiments on such two-species systems. Sufficient conditions are derived for stability of the equilibrium state in the general case for an arbitrarily large number of interacting species. It is shown that the equilibrium state is always stable if all species undergo significant self-diffusion and the Helmholtz eigenvalues are suitably large.     

Diffusion versus network models as descriptions for the spread of prion diseases in the brain

In this paper we will discuss different modeling approaches for the spread of prion diseases in the brain. Firstly, we will compare reaction-diffusion models with models of epidemic diseases on networks. The solutions of the resulting reaction-diffusion equations exhibit traveling wave behavior on a one-dimensional domain, and the wave speed can be estimated. The models can be tested for diffusion-driven (Turing) instability, which could present a possible mechanism for the formation of plaques. We also show that the reaction-diffusion systems are capable of reproducing experimental data on prion spread in the mouse visual system. Secondly, we study classical epidemic models on networks, and use these models to study the influence of the network topology on the disease progression.     

A nonlinear treatment of the protocell model by a boundary layer approximation

The “protocell” is a mathematical model of a self-maintaining unity based on the dynamics of simple reaction-diffusion processes and a self-controlled dynamics of the surface. In this paper its spatio-temporal behaviour far from the stationary structure is investigated by means of a boundary layer approximation. It is shown in detail how a simplified and mathematically feasible equation can be derived from the original parabolic problem. It turns out that the known instability which is initiated in the linear region around the stationary structure is continued further in the direction to a division by nonlinear dynamics.     

Diffusion driven instability in an inhomogeneous domain

Diffusion driven instability in reaction-diffusion systems has been proposed as a mechanism for pattern formation in numerous embryological and ecological contexts. However, the possible effects of environmental inhomogeneities has received relatively little attention. We consider a general two species reaction-diffusion model in one space dimension, with one diffusion coefficient a step function of the spatial coordinate. We derive the dispersion relation and the solution of the linearized system. We apply our results to Turing-type models for both embryogenesis and predator-prey interactions. In the former case we derive conditions for pattern to be isolated in one part of the domain, and in the latter we introduce the concept of “environmental instability”. Our results suggest that environmental inhomogeneity could be an important regulator of biological pattern formation.     

Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains

By using asymptotic theory, we generalise the Turing diffusively-driven instability conditions for reaction-diffusion systems with slow, isotropic domain growth. There are two fundamental biological differences between the Turing conditions on fixed and growing domains, namely: (i) we need not enforce cross nor pure kinetic conditions and (ii) the restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Our theoretical findings are confirmed and reinforced by numerical simulations for the special cases of isotropic linear, exponential and logistic growth profiles. In particular we illustrate an example of a reaction-diffusion system which cannot exhibit a diffusively-driven instability on a fixed domain but is unstable in the presence of slow growth.     

A mass conserved reaction-diffusion system captures properties of cell polarity

Cell polarity is a general cellular process that can be seen in various cell types such as migrating neutrophils and Dictyostelium cells. The Rho small GTP(guanosine 5'-tri phosphate)ases have been shown to regulate cell polarity; however, its mechanism of emergence has yet to be clarified. We first developed a reaction-diffusion model of the Rho GTPases, which exhibits switch-like reversible response to a gradient of extracellular signals, exclusive accumulation of Cdc42 and Rac, or RhoA at the maximal or minimal intensity of the signal, respectively, and tracking of changes of a signal gradient by the polarized peak. The previous cell polarity models proposed by Subramanian and Narang show similar behaviors to our Rho GTPase model, despite the difference in molecular networks. This led us to compare these models, and we found that these models commonly share instability and a mass conservation of components. Based on these common properties, we developed conceptual models of a mass conserved reaction-diffusion system with diffusion-driven instability. These conceptual models retained similar behaviors of cell polarity in the Rho GTPase model. Using these models, we numerically and analytically found that multiple polarized peaks are unstable, resulting in a single stable peak (uniqueness of axis), and that sensitivity toward changes of a signal gradient is specifically restricted at the polarized peak (localized sensitivity). Although molecular networks may differ from one cell type to another, the behaviors of cell polarity in migrating cells seem similar, suggesting that there should be a fundamental principle. Thus, we propose that a mass conserved reaction-diffusion system with diffusion-driven instability is one of such principles of cell polarity.     

How predation can slow, stop or reverse a prey invasion

Observations on Mount St Helens indicate that the spread of recolonizing lupin plants has been slowed due to the presence of insect herbivores and it is possible that the spread of lupins could be reversed in the future by intense insect herbivory [Fagan, W. F. and J. Bishop (2000). Trophic interactions during primary sucession: herbivores slow a plant reinvasion at Mount St. Helens. Amer. Nat. 155, 238–251]. In this paper we investigate mechanisms by which herbivory can contain the spatial spread of recolonizing plants. Our approach is to analyse a series of predator-prey reaction-diffusion models and spatially coupled ordinary differential equation models to derive conditions under which predation pressure can slow, stall or reverse a spatial invasion of prey. We focus on models where prey disperse more slowly than predators. We comment on the types of functional response which give such solutions, and the circumstances under which the models are appropriate.     

The chemical basis of morphogenesis

It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns onHydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading. Reprinted from thePhilosophical Transactions of the Royal Society (part B), Vol. 237, pp. 37–72 (1953) with the permission of the Royal Society, London.     

On the heterogeneity of reaction-diffusion generated pattern

Several current reaction-diffusion mechanisms have been proposed as models for morphogenesis in the Turing (1952,Phil. Trans. R. Soc. Lond. B 237, 37–72) sense. We introduce and exploit a quantity, we have termed heterogeneity, which allows us to elaborate the differences between the various models with regard to spatial pattern formation. It is shown that this quantity provides a concise view for the comparison of theoretical models with experimental observations. Two model mechanisms are treated explicitly both for linear and for biased diffusion.     

Sexual dimorphism and mating systems: jumping to conclusions

Previous studies have suggested that there is a strong relationship between a high degree of aggressive competition among males for access to fertile females and large body and canine size in males. It has further been suggested that such a relationship among living primates can be used to infer the social organization of extinct primate species from the degree of sexual dimorphism exhibited. Our field studies of patas (Erythrocebus patas) and blue monkeys (Cercopithecus mitis), two species which had previously been characterized as having one-male ‘harem’ group structures, indicate considerable variability in mating systems. We suggest, on the basis of our observations of these species, that factors other than male-male competition (e.g., predation) may also have influenced the degree of dimorphism in primates.     

CNS delivery of l-dopa by a new hybrid glutathione–methionine peptidomimetic prodrug

Parkinson’s disease (PD) is a neurodegenerative disorder associated primarily with loss of dopamine (DA) neurons in the nigrostriatal system. With the aim of increasing the bioavailability of l-dopa (LD) after oral administration and of overcoming the pro-oxidant effect associated with LD therapy, we designed a peptidomimetic LD prodrug (1) able to release the active agent by enzyme catalyzed hydrolysis. The physicochemical properties, as well as the chemical and enzymatic stabilities of the new compound, were evaluated in order to check both its stability in aqueous medium and its sensitivity towards enzymatic cleavage, providing the parent LD drug, in rat and human plasma. The radical scavenging activities of prodrug 1 was tested by using both the DPPH–HPLC and the DMSO competition methods. The results indicate that the replacement of cysteine GSH portion by methionine confers resistance to oxidative degradation in gastric fluid. Prodrug 1 demonstrated to induce sustained delivery of DA in rat striatal tissue with respect to equimolar LD dosages. These results are of significance for prospective therapeutic application of prodrug 1 in pathological events associated with free radical damage and decreasing DA concentration in the brain.     

Galerkin-Ritz procedures for approximate solutions to systems of reaction-diffusion equations

For any essentially nonlinear system of reaction-diffusion equations of the generic form ∂c_i/∂t=D_i∇²c_i+Q_i(c,x,t) supplemented with Robin type boundary conditions over the surface of a closed bounded three-dimensional region, it is demonstrated that all solutions for the concentration distributionn-tuple function c=(c ₁(x,t),...,c _n (x,t)) satisfy a differential variational condition. Approximate solutions to the reaction-diffusion intial-value boundary-value problem are obtainable by employing this variational condition in conjunction with a Galerkin-Ritz procedure. It is shown that the dynamical evolution from a prescribed initial concentrationn-tuple function to a final steady-state solution can be determined to desired accuracy by such an approximation method. The variational condition also admits a systematic Galerkin-Ritz procedure for obtaining approximate solutions to the multi-equation elliptic boundary-value problem for steady-state distributions c=−c(x). Other systems of phenomenological (non-Lagrangian) field equations can be treated by Galerkin-Ritz procedures based on analogues of the differential variational condition presented here. The method is applied to derive approximate nonconstant steady-state solutions for ann-species symbiosis model.     

Parameter domains for instability of uniform states in systems with many delays

 We present a computational method for determining regions in parameter space corresponding to linear instability of a spatially uniform steady state solution of any system of two coupled reaction-diffusion equations containing up to four delay terms. At each point in parameter space the required stability properties of the linearised system are found using mainly the Principle of the Argument. The method is first developed for perturbations of a particular wavenumber, and then generalised to allow arbitrary perturbations. Each delay term in the system may be of either a fixed or a distributed type, and spatio-temporal delays are also allowed. Received 19 September 1995; received in revised form 4 September 1996     

Modelling the radiotherapy effect in the reaction-diffusion equation

PurposeIn recent years, the reaction-diffusion (Fisher-Kolmogorov) equation has received much attention from the oncology research community due to its ability to describe the infiltrating nature of glioblastoma multiforme and its extraordinary resistance to any type of therapy. However, in a number of previous papers in the literature on applications of this equation, the term (R) expressing the ‘External Radiotherapy effect’ was incorrectly derived. In this note we derive an analytical expression for this term in the correct form to be included in the reaction-diffusion equation.MethodsThe R term has been derived starting from the Linear-Quadratic theory of cell killing by ionizing radiation. The correct definition of R was adopted and the basic principles of differential calculus applied.ResultsThe compatibility of the R term derived here with the reaction-diffusion equation was demonstrated. Referring to a typical glioblastoma tumour, we have compared the results obtained using our expression for the R term with the ‘incorrect’ expression proposed by other authors.     

Excitation of a magnetospheric MHD cavity by Kelvin-Helmholtz instability

The Kelvin-Helmholtz instability is investigated analytically by using a one-dimensional nonuniform model of the Earth’s magnetosphere and the adjacent solar wind region. Its properties are shown to be essentially governed by the presence of an MHD cavity that arises in the magnetosphere because of the non-uniformity of the latter and also because of the jump in the parameters of the medium at the magnetopause (the outer boundary of the magnetosphere). System oscillations constitute a discrete spectrum of eigenmodes, which are determined by the wave vector k _t along the tangential discontinuity and also by the mode number n = 0, 1, 2, …, playing the role of the wavenumber along a coordinate normal to the magnetopause. Analytic expressions are obtained for the frequency and instability growth rate of each eigenmode and for the functions describing its spatial structure. All these quantities depend parametrically on the solar wind velocity V _W , or more precisely, on the Doppler frequency shift ω _W = k _t · V _W . For each eigenmode, there is a lower instability threshold depending on the parameter ω _W and a sharp maximum in the growth rate at the eigenfrequency of the magnetospheric cavity. For ω _W values below the threshold, the properties of an eigenmode are highly sensitive to the type of solar wind nonuniformity. Three cases are considered: a uniform solar wind and solar winds in which the speed of sound increases or decreases away from the magnetopause.