Pattern formation in morphogenesis

We first treat the Gierer-Meinhardt equations by linear stability analysis to determine the critical parameter, at which the homogeneous distributions of activator and inhibitor concentrations become unstable. We find two types of instabilities: one leading to spatial pattern formation and another one leading to temporal oscillations. We consider the case where two instabilities are present. Using the method of generalized Ginzburg-Landau equations introduced earlier we then analyze the nonlinear equations. As we are mainly interested in spatial pattern formation on a sphere we consider the problem under an appropriate constraint. Combining the two occurring solutions we find patterns well-known in biology, such as a gradient system and temporal oscillations.     

Pattern formation: fruiting body morphogenesis in Myxococcus xanthus

When Myxococcus xanthus cells are exposed to starvation, they respond with dramatic behavioral changes. The expansive swarming behavior stops and the cells begin to aggregate into multicellular fruiting bodies. The cell-surface-associated C-signal has been identified as the signal that induces aggregation. Recently, several of the components in the C-signal transduction pathway have been identified and behavioral analyses are beginning to reveal how the C-signal modulates cell behavior. Together, these findings provide a framework for understanding how a cell-surface-associated morphogen induces pattern formation.     

Pattern formation in reaction-diffusion models with nonuniform domain growth

Recent examples of biological pattern formation where a pattern changes qualitatively as the underlying domain grows have given rise to renewed interest in the reaction-diffusion (Turing) model for pattern formation. Several authors have now reported studies showing that with the addition of domain growth the Turing model can generate sequences of patterns consistent with experimental observations. These studies demonstrate the tendency for the symmetrical splitting or insertion of concentration peaks in response to domain growth. This process has also been suggested as a mechanism for reliable pattern selection. However, thus far authors have only considered the restricted case where growth is uniform throughout the domain. In this paper we generalize our recent results for reaction-diffusion pattern formation on growing domains to consider the effects of spatially nonuniform growth. The purpose is twofold: firstly to demonstrate that the addition of weak spatial heterogeneity does not significantly alter pattern selection from the uniform case, but secondly that sufficiently strong nonuniformity, for example where only a restricted part of the domain is growing, can give rise to sequences of patterns not seen for the uniform case, giving a further mechanism for controlling pattern selection. A framework for modelling is presented in which domain expansion and boundary (apical) growth are unified in a consistent manner. The results have implications for all reaction-diffusion type models subject to underlying domain growth.     

Pattern formation: fruiting body morphogenesis in Myxococcus xanthus

When Myxococcus xanthus cells are exposed to starvation, they respond with dramatic behavioral changes. The expansive swarming behavior stops and the cells begin to aggregate into multicellular fruiting bodies. The cell-surface-associated C-signal has been identified as the signal that induces aggregation. Recently, several of the components in the C-signal transduction pathway have been identified and behavioral analyses are beginning to reveal how the C-signal modulates cell behavior. Together, these findings provide a framework for understanding how a cell-surface-associated morphogen induces pattern formation.     

A reaction-diffusion theory of morphogenesis with inherent pattern invariance under scale variations

In the framework of reaction-diffusion theory we deal with the problem of pattern regulation in morphogenesis. A generic model is proposed where the kinetic terms follow constraints imposed by scale invariance considerations. These constraints allow a class of kinetic schemes to be formulated so that, starting with an initially homogeneous morphogen distribution in the field, a stable gradient is established of the form: S(chi,L) = Lpf(chi/L). Here L is the length of the morphogenetic field, chi is the position variable and f(chi/L) is some monotonic function of the relative distance. With this distribution a scale invariant gradient can be constructed which leads to pattern regulation. A linear stability analysis of the model permits the definition of the parameter values enabling the system to abandon the homogeneous state spontaneously. Simulations of the evolution of the system towards its final stable state result in approximate pattern invariance for different field lengths. The accuracy of this invariance is in agreement with some recent quantitative experimental findings in both developing and regenerating systems.     

A reaction-diffusion system of a predator-prey-mutualist model

Mutualism is part of many significant processes in nature. Mutualistic benefits arising from modification of predator-prey interactions involve interactions of at least three species. In this paper we investigate the Homogeneous Neumann problem and Dirichlet problem for a reaction-diffusion system of three species—a predator, a mutualist-prey, and a mutualist. The existence, uniqueness, and boundedness of the solution are established by means of the comparison principle and the monotonicity method. For the Neumann problem, we analyze the constant equilibrium solutions and their stability. For the Dirichlet problem, we prove the global asymptotic stability of the trivial equilibrium solution. Specifically, we study the existence and the asymptotic behavior of two nonconstant equilibrium solutions. The main method used in studying of the stability is the spectral analysis to the linearized operators. The O.D.E. problem for the same model was proposed and studied in [13]. Through our results, we can see the influences of the diffusion mechanism and the different boundary value conditions upon the asymptotic behavior of the populations.     

A model of network formation by Physarum plasmodium: Interplay between cell mobility and morphogenesis

The plasmodium of Physarum polycephalum has attracted much attention due its intelligent adaptive behavior. In this study, we constructed a model of the organism and attempted to simulate its locomotion and morphogenetic behavior. By modifying our previous model, we were able to get closer to the actual behavior. We also compared the behavior of the model with that of the real organism, demonstrating remarkable similarity between the two.     

Some analytical results about a simple reaction-diffusion system for morphogenesis

Summary The reaction-diffusion system considered involves only one nonlinear term and is a gradient system. In a bifurcation analysis for the equilibrium states, the global existence of infinitely many solution branches can be shown by the method of Ljusternik-Schnirelmann. Their stability is studied. Using a Ljapunov functional it can be shown that the solutions of the time-dependent system converge to the equilibrium states.     

Pattern formation in a generalized chemotactic model

Many models have been proposed for spatial pattern formation in embryology and analyzed for the standard case of zero-flux boundary conditions. However, relatively little attention has been paid to the role of boundary conditions on the form of the final pattern. Here we investigate, numerically, the effect of nonstandard boundary conditions on a model pattern generator, which we choose to be of a cell-chemotactic type. We specifically focus on the role of boundary conditions and the effects of scale and aspect ratio, and study the spatiotemporal dynamics of pattern formation. We illustrate the properties of the model by application to the spatiotemporal sequence of skeletal development.     

Plant morphogenesis: A geometrical model for the ramification

A geometrical model is proposed that describes the emergence of a primordium at the shoot apex in Dicotyledons. It is based on recent fundamental results on plant morphogenesis, viz.:

the emergence is preceded by the reorganization of the microtubules of the cortical cytoskeleton, leading to a new orientation of the synthesis of the cell wall microfibrils;

the resulting global stress is related to the general orientation of the cell growth;

The model sums up the continuous interactions that link the microtubules, the microfibrils and the cell growth axis. The paper tries to answer three essential questions:

Why does the principal stem shifts its growth direction after each lateral emergence?

Why do the three axes involved in any ramification (namely the old and the new principal stems and the lateral emergence) exhibit a plane configuration whereas this is an essentially three dimensional phenomenon?

Does phyllotaxis exclusively depend upon the local emergence of a primordium?

An interactive procedure between empirical botanical knowledge and the mathematical model leads to an insight of the compatibility mechanisms that link the various microtubules and microfibrils networks, and the apical dome restoration. A geometrical formalism allows a redefinition of both the “generating centre” and the “organizing centre”, and their field effect.     

A simple model for early morphogenesis

Despite the large amount of knowledge which continues to accumulate about early developmental events, very little is known about the processes which control them. Part of the problem may lie in that workers applying different approaches and techniques have different points of view and appear to be reluctant to read each others' literature. My aim in this paper is not to give a generative, formal model for early development, but rather to suggest several connecting strands between the physiological, biochemical, cell biological and experimental embryological approaches which may stimulate new research in fields between those already exploited.     

A Geometro-mechanical model for pulsatile morphogenesis

A model is proposed which imitates the morphogenesis of several species of the lower invertebrate animals, the hydroid polyps and permits the derivation of the geometry (surface curvature) of each developmental stage from that of the preceding stage. The model is based upon two experimentally verified assumptions. First, neighbouring cells are assumed to compress each other laterally in a regular and species-specific pulsatile manner. It is this pressure, and/or an active cell reaction to it, which changes the curvature of a cell layer. Secondly, cell layers are assumed to have quasi-elastic properties tending to smooth out their curvature. With our model, the different pulsatile patterns of cell-cell pressure are reproduced and the elasticity parameters are modulated. As a result, within a large zone of parameter values (a so-called "morphogenetic zone", MZ) realistic shapes of the rudiments are reproduced. The main principles of the model can also be used for interpreting the morphogenesis of other groups of animals. A suggested model emphasizes the self-organizing properties of a "stressed geometry" of embryonic rudiments.     

Parameter estimation and inference for stochastic reaction-diffusion systems: application to morphogenesis in D. melanogaster

Background  

Reaction-diffusion systems are frequently used in systems biology to model developmental and signalling processes. In many applications, count numbers of the diffusing molecular species are very low, leading to the need to explicitly model the inherent variability using stochastic methods. Despite their importance and frequent use, parameter estimation for both deterministic and stochastic reaction-diffusion systems is still a challenging problem.     

Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis

Summary We first perform a linear stability analysis of the Gierer-Meinhardt model to determine the critical parameters where the homogeneous distribution of activator and inhibitor concentrations becomes unstable. There are two kinds of instabilities, namely, one leading to spatial patterns and another one leading to temporal oscillations. Focussing our attention on spatial pattern formation we solve the corresponding nonlinear equations by means of our previously introduced method of generalized Ginzburg-Landau equations. We explicitly consider the two-dimensional case and find both rolls and hexagon-like structures. The impact of different boundary conditions on the resulting patterns is also discussed. The occurrence of the new patterns has all the features of nonequilibrium phase transitions.     

A reaction-diffusion model for interference in meiotic crossing over

One crossover point between a pair of homologous chromosomes in meiosis appears to interfere with occurrence of another in the neighborhood. It has been revealed that Drosophila and Neurospora, in spite of their large difference in the frequency of crossover points, show very similar plots of coincidence-a measure of the interference-against the genetic distance of the interval, defined as one-half the average number of crossover points within the interval. We here propose a simple reaction-diffusion model, where a "randomly walking" precursor becomes immobilized and matures into a crossover point. The interference is caused by pair-annihilation of the random walkers due to their collision and by annihilation of a random walker due to its collision with an immobilized point. This model has two parameters-the initial density of the random walkers and the rate of its processing into a crossover point. We show numerically that, as the former increases and/or the latter decreases, plotted curves of the coincidence vs. the genetic distance converge on a unique curve. Thus, our model explains the similarity between Drosophila and Neurospora without parameter values adjusted finely, although it is not a "genetic model" but is a "physical model," specifying explicitly what happens physically.