On surface geometry coupled to morphogen

An expression is derived for both the Gauss and the Mean curvature of a surface, in terms of three simple cell parameters. The surface is thought of as composed of a single-cell thick sheet of cells joined laterally. The three cellular parameters involved are the ratios of (linear) basal to apical dimension in two orthogonal directions, S1 and S2, and the cell thickness "h". These three parameters may be envisioned as functions of a morphogen or morphogens which vary from point to point over the (middle) surface. As an example, the "reaction-diffusion" equations which are often used to describe pattern-formation in early development can be seen as possible candidates for these morphogens, when the resultant surface deformations are given when the dependence of the three cellular parameters are specified as a function of morphogen concentration. The coupling back of the surface deformations to the set of reaction-diffusion equations is simply given, and is through the dependence on geometry of the Laplacian operator which enters these equations.     

The interaction of surface geometry with morphogens

Expressions are given for the Gauss and Mean curvatures of a surface of thickness h. The two curvatures, (K and H), which are given at each point of the middle surface, are adequate to describe the surface. The sheet thickness varies with position in the middle surface bisecting the apical and basal surfaces. The definitions of K and H are in terms of radii of curvature, but such radii are not appropriate variables for determining how morphogens in the surface may couple to the geometry. More suitable expressions are developed here. Two important geometrical constraints must be satisfied, namely the famous Gauss-Bonnet theorem, and an inequality stemming from the definition of the two curvatures. It is argued that these constraints are of great usefulness in determining the form of the coupling of morphogens to the geometry. In particular, when two key morphogens suffice to determine surface geometry, explicit expressions are suggested to determine both Gauss (K) and Mean (H) curvatures as functions of invariant morphogen densities.     

Regulative feedback in pattern formation: towards a general relativistic theory of positional information

Positional specification by morphogen gradients is traditionally viewed as a two-step process. A gradient is formed and then interpreted, providing a spatial metric independent of the target tissue, similar to the concept of space in classical mechanics. However, the formation and interpretation of gradients are coupled, dynamic processes. We introduce a conceptual framework for positional specification in which cellular activity feeds back on positional information encoded by gradients, analogous to the feedback between mass-energy distribution and the geometry of space-time in Einstein's general theory of relativity. We discuss how such general relativistic positional information (GRPI) can guide systems-level approaches to pattern formation.     

Coalescence of interacting cell populations

We analyse the coalescence of invasive cell populations by studying both the temporal and steady behaviour of a system of coupled reaction-diffusion equations. This problem is relevant to recent experimental observations of the dynamics of opposingly directed invasion waves of cells. Two cell types, u and v, are considered with the cell motility governed by linear or nonlinear diffusion. The cells proliferate logistically so that the long-term total cell density, u+v approaches a carrying capacity. The steady-state solutions for u and v are denoted u(s) and v(s). The steady solutions are spatially invariant and satisfy u(s)+v(s)=1. However, this expression is underdetermined so the relative proportion of each cell type u(s) and v(s) cannot be determined a priori. Various properties of this model are studied, such as how the relative proportion of u(s) and v(s) depends on the relative motility and relative proliferation rates. The model is analysed using a combination of numerical simulations and a comparison principle. This investigation unearths some novel outcomes regarding the role of overcrowding and cell death in this type of cell migration assay. These observations have relevance to experimental design and interpretation regarding the identification and parameterisation of mechanisms involved in cell invasion.     

A Model for Selection of Eyespots on Butterfly Wings

Unsolved Problem

The development of eyespots on the wing surface of butterflies of the family Nympalidae is one of the most studied examples of biological pattern formation.However, little is known about the mechanism that determines the number and precise locations of eyespots on the wing. Eyespots develop around signaling centers, called foci, that are located equidistant from wing veins along the midline of a wing cell (an area bounded by veins). A fundamental question that remains unsolved is, why a certain wing cell develops an eyespot, while other wing cells do not.

Key Idea and Model

We illustrate that the key to understanding focus point selection may be in the venation system of the wing disc. Our main hypothesis is that changes in morphogen concentration along the proximal boundary veins of wing cells govern focus point selection. Based on previous studies, we focus on a spatially two-dimensional reaction-diffusion system model posed in the interior of each wing cell that describes the formation of focus points. Using finite element based numerical simulations, we demonstrate that variation in the proximal boundary condition is sufficient to robustly select whether an eyespot focus point forms in otherwise identical wing cells. We also illustrate that this behavior is robust to small perturbations in the parameters and geometry and moderate levels of noise. Hence, we suggest that an anterior-posterior pattern of morphogen concentration along the proximal vein may be the main determinant of the distribution of focus points on the wing surface. In order to complete our model, we propose a two stage reaction-diffusion system model, in which an one-dimensional surface reaction-diffusion system, posed on the proximal vein, generates the morphogen concentrations that act as non-homogeneous Dirichlet (i.e., fixed) boundary conditions for the two-dimensional reaction-diffusion model posed in the wing cells. The two-stage model appears capable of generating focus point distributions observed in nature.

Result

We therefore conclude that changes in the proximal boundary conditions are sufficient to explain the empirically observed distribution of eyespot focus points on the entire wing surface. The model predicts, subject to experimental verification, that the source strength of the activator at the proximal boundary should be lower in wing cells in which focus points form than in those that lack focus points. The model suggests that the number and locations of eyespot foci on the wing disc could be largely controlled by two kinds of gradients along two different directions, that is, the first one is the gradient in spatially varying parameters such as the reaction rate along the anterior-posterior direction on the proximal boundary of the wing cells, and the second one is the gradient in source values of the activator along the veins in the proximal-distal direction of the wing cell.     

Single effective neuron: dendritic coupling effects and stochastic resonance

We consider a model of a neuron coupled with a surrounding dendritic network subject to Langevin noise and a weak periodic modulation. Through an adiabatic elimination procedure, the single-neuron dynamics are extracted from the coupled stochastic differential equations describing the network of dendrodendritic interactions.Our approach yields areduced neuron model whose dynamics may correspond to neurophysiologically realistic behavior for certain ranges of soma and bath parameters. Cooperative effects (e.g., stochastic resonance) arising from the interplay between the noise and modulation are discussed in detail.     

Morphogenetic factors controlling differentiation and dedifferentiation of epidermal cells in the gynoecium of Catharanthus roseus

During postgenital fusion of the distal adaxial surfaces of the two originally separate carpel primordia of Catharanthus roseus (L.) G. Don, approx. 400 epidermal cells undergo rapid dedifferentiation into parenchymatous cells. To characterize the mechanism of the induction of dedifferentiation, various types of both water-permeable and water-impermeable barriers were placed between pre-fusion carpels. Barriers which did not allow the passage of water-soluble agents blocked dedifferentiation. Barriers which allowed passage of water-soluble agents did not block dedifferentiation of the contacting epidermal cells, implicating a diffusible agent or morphogen as the factor responsible for dedifferentiation. Experiments with barriers of known pore size demonstrated that the molecular weight of this morphogen was less than 1000. The two cell walls and thin cuticle present at the site of this postgenital fusion do not block the movement of some substances between the fusing carpels. Tracer studies with tritium-labeled asparagine confirmed that substances can be transported across the fusion plane.Abbreviation 2,4-D 2,4-dichlorophenoxyacetic acid I=Walker (1978b)     

Geometrical aspects of surface morphogenesis

This paper is concerned with the morphogenesis of structures which form thin deformable sheets. A general formalism is presented for the deformation of a sheet in the presence of an isotropic local body stress. This formalism leads to a set of equations, based on the theory of shells, in which corrections are made in the geometry due to large deformations. Under certain conditions the equations may be solved to give the surface metric tensor as a function of the local tension. A numerical example based on a simple "threshold" model is also presented.     

Sampling and efficiency of metric matrix distance geometry: A novel partial metrization algorithm

Summary In this paper, we present a reassessment of the sampling properties of the metric matrix distance geometry algorithm, which is in wide-spread use in the determination of three-dimensional structures from nuclear magnetic resonance (NMR) data. To this end, we compare the conformational space sampled by structures generated with a variety of metric matrix distance geometry protocols. As test systems we use an unconstrained polypeptide, and a small protein (rabbit neutrophil defensin peptide 5) for which only few tertiary distances had been derived from the NMR data, allowing several possible folds of the polypeptide chain. A process called metrization in the preparation of a trial distance matrix has a very large effect on the sampling properties of the algorithm. It is shown that, depending on the metrization protocol used, metric matrix distance geometry can have very good sampling properties'indeed, both for the unconstrained model system and the NMR-structure case. We show that the sampling properties are to a great degree determined by the way in which the first few distances are chosen within their bounds. Further, we present a new protocol (partial metrization) that is computationally more efficient but has the same excellent sampling properties. This novel protocol has been implemented in an expanded new release of the program X-PLOR with distance geometry capabilities.     

Computational Modeling Reveals that a Combination of Chemotaxis and Differential Adhesion Leads to Robust Cell Sorting during Tissue Patterning

Robust tissue patterning is crucial to many processes during development. The "French Flag" model of patterning, whereby naïve cells in a gradient of diffusible morphogen signal adopt different fates due to exposure to different amounts of morphogen concentration, has been the most widely proposed model for tissue patterning. However, recently, using time-lapse experiments, cell sorting has been found to be an alternative model for tissue patterning in the zebrafish neural tube. But it remains unclear what the sorting mechanism is. In this article, we used computational modeling to show that two mechanisms, chemotaxis and differential adhesion, are needed for robust cell sorting. We assessed the performance of each of the two mechanisms by quantifying the fraction of correct sorting, the fraction of stable clusters formed after correct sorting, the time needed to achieve correct sorting, and the size variations of the cells having different fates. We found that chemotaxis and differential adhesion confer different advantages to the sorting process. Chemotaxis leads to high fraction of correct sorting as individual cells will either migrate towards or away from the source depending on its cell type. However after the cells have sorted correctly, there is no interaction among cells of the same type to stabilize the sorted boundaries, leading to cell clusters that are unstable. On the other hand, differential adhesion results in low fraction of correct clusters that are more stable. In the absence of morphogen gradient noise, a combination of both chemotaxis and differential adhesion yields cell sorting that is both accurate and robust. However, in the presence of gradient noise, the simple combination of chemotaxis and differential adhesion is insufficient for cell sorting; instead, chemotaxis coupled with delayed differential adhesion is required to yield optimal sorting.     

Distorted metrics on trees and phylogenetic forests

We study distorted metrics on binary trees in the context of phylogenetic reconstruction. Given a binary tree T on n leaves with a path metric d, consider the pairwise distances {d(u,v)} between leaves. It is well known that these determine the tree and the d length of all edges. Here, we consider distortions d of d such that, for all leaves u and v, it holds that |d(u,v)-dmacr(u,v)|1.....T₀ such that the true tree T may be obtained from that forest by adding alpha-1 edges and alpha-1les2^-Omega(M/g)n. Our distorted metric result implies a reconstruction algorithm of phylogenetic forests with a small number of trees from sequences of length logarithmic in the number of species. The reconstruction algorithm is applicable for the general Markov model. Both the distorted metric result and its applications to phylogeny are almost tight     

Models for cell differentiation and generation of polarity in diffusion-governed morphogenetic fields

Models based on molecular mechanisms are presented for pattern formation in developing organisms. It is assumed that there exists a diffusion governed gradient in the morphogenetic field. It is shown that cellular differentiation and the subsequent pattern formation result from the interaction of the diffusing morphogen with the genetic regulatory mechanism of cells. In a second stage it is shown that starting from a homogeneous distribution of morphogen, polarity can be generated spontaneously in the morphogenetic field giving rise to the establishment of a gradient. The stability of these gradients is demonstrated. The onset of a morphogenetic gradient and pattern formation are combined in a single coherent model. Size invariance and its biological implications are discussed.     

The Dynamics of Root Growth: A Geometric Model

A new model for macroscopic root growth based on a dynamical Riemannian geometry is presented. Assuming that the thickness of the root is much less than its length, the model is restricted to growth in one dimension (1D). We treat 1D tissues as continuous, deformable, growing geometries for sizes larger than 1 mm. The dynamics of the growing root are described by a set of coupled tensor equations for the metric of the tissue and velocity field of material transport in non-Euclidean space. These coupled equations represent a novel feedback mechanism between growth and geometry. We compare 1D numerical simulations of these tissue growth equations to two measures of root growth. First, sectional growth along the simulated root shows an elongation zone common to many species of plant roots. Second, the relative elemental growth rate calculated in silico exhibits spatio-temporal dynamics recently characterized in high-resolution root growth studies but which thus far lack a biological hypothesis to explain them. In our model, these dynamics are a direct consequence of considering growth as both a geometric reaction–diffusion process and expansion due to a distributed source of new materials.     

A model of gradient interpretation based on morphogen binding

We propose a model in which pattern formation is controlled by several concentration gradients of “morphogens” and by allosteric proteins which bind them. In this model, each protein can bind up to two molecules of each morphogen and has an “active state” when one molecule of each morphogen is bound. The concentration of the active state of such a “morphogen binding protein” varies with position in a way that depends on the values given the binding constants. In a contour map of the active state concentration, the contours can have a variety of simple shapes.Simply-shaped regions of cell differentiation can be defined directly by concentration contours of a morphogen binding protein using a threshold-sensing mechanism. More complex shapes may be generated using several proteins and a “winner-take-all” rule according to which each protein specifies some particular sort of cell differentiation and the differentiation of cells in any position is governed by the protein with the highest active state concentration.We present an application of our model to the vertebrate limb skeleton; we use the “winner-take-all” mechanism and thirteen morphogen binding proteins, eleven of which specify cartilage formation. In this model we use one morphogen binding protein to specify the shaft of a typical long bone and one for each epiphysis. Our model is reasonably successful in imitating the in vivo positions and orientations of developing bones and in generating simple, plausible-looking articular surfaces.In addition to the morphogen-binding model we propose a mechanism which could transform morphogen-binding patterns into high-amplitude patterns capable of controlling the activity of structural genes. This “amplifying mechanism” can account for two previously unexplained features of limb skeletal development: the early formation of the diffusely-bounded “scleroblastema” in the limb bud and the center-to-edge gradations in cartilage formation rate which are later seen within individual chondrification foci.A simple modification of the morphogen-binding model provides an explanation for the general anatomical phenomenon of metamerism: The model can account for the formation of inexactly repeating patterns (such as the pattern of the vertebral column) and suggests a mechanism by which such patterns could (1) evolve from exactly repeating patterns, and (2) acquire, in further evolution, a high degree of specialization of the individual repeating units.The most promising approach for testing the morphogen-binding model would appear to involve experiments in which cytoplasm is transferred between cells at various stages of pattern development. Support for the model could also come from the discovery of certain kinds of hereditary limb defects.     

A membrane-carrier mechanism for generating new sources and sinks within gradient-controlled tissues—Its application to regeneration in Oncopeltus

A model is presented that can, in principle, generate new sources and sinks within an existing gradient in the concentration of a morphogen. The novel and crucial feature of the model is that morphogens are transported between cells by membrane-based carrier molecules and not by diffusion. A further aspect of the model is the presence of a second substance within each cell whose concentration is uniform over the tissue; this molecule binds to but is not transported by the carrier and is therefore a competitive inhibitor of the morphogen. The concentration of free inhibitor in a cell determines its fate: if at any time it exceeds some threshold, that cell becomes a morphogen source; if it falls below a second threshold, the cell becomes a sink; in between them, the cell shows no special properties. Provided that differences in morphogen concentration between adjacent cells are not too great, the mechanism is indistinguishable from a normal, diffusion gradient. Examination of the kinetics of the system over a one-dimensional line of cells, however, shows that any stable morphogen difference leads to a carrier imbalance and to a change in the degree of inhibitor binding. If this difference is sufficiently great and if there is morphogen homostasis in each cell, then the free inhibitor concentration in the high morphogen cell may exceed the higher threshold causing it to become a source while the low morphogen cell becomes a sink.A numerical example of the mechanism is given and the results calculated for two-dimensional cellular arrays on either side of a morphogen discontinuity. The predictions match the observations of Wright & Lawrence (1981a, b) on Oncopeltus. These authors showed that, if pieces of epidermis from sufficiently different positions were grafted together in vivo, an ectopic boundary would form with regions of reversed polarity on either side of the join. The ability of the model to explain the regeneration and axial graft observations on hydra is also discussed and some experiments that might test the model are put forward. It is suggested that the significance of the membrane-carrier mechanism in vivo is twofold: first, to interpret the basic segmentation mechanism in embryogenesis by turning its morphogen discontinuities into source-sink pairs and so generating actual boundaries; second, to act as a homeostatic mechanism in later development, thus ensuring the maintenance of boundaries.     

Implications of diffusion and time-varying morphogen gradients for the dynamic positioning and precision of bistable gene expression boundaries

The earliest models for how morphogen gradients guide embryonic patterning failed to account for experimental observations of temporal refinement in gene expression domains. Following theoretical and experimental work in this area, dynamic positional information has emerged as a conceptual framework to discuss how cells process spatiotemporal inputs into downstream patterns. Here, we show that diffusion determines the mathematical means by which bistable gene expression boundaries shift over time, and therefore how cells interpret positional information conferred from morphogen concentration. First, we introduce a metric for assessing reproducibility in boundary placement or precision in systems where gene products do not diffuse, but where morphogen concentrations are permitted to change in time. We show that the dynamics of the gradient affect the sensitivity of the final pattern to variation in initial conditions, with slower gradients reducing the sensitivity. Second, we allow gene products to diffuse and consider gene expression boundaries as propagating wavefronts with velocity modulated by local morphogen concentration. We harness this perspective to approximate a PDE model as an ODE that captures the position of the boundary in time, and demonstrate the approach with a preexisting model for Hunchback patterning in fruit fly embryos. We then propose a design that employs antiparallel morphogen gradients to achieve accurate boundary placement that is robust to scaling. Throughout our work we draw attention to tradeoffs among initial conditions, boundary positioning, and the relative timescales of network and gradient evolution. We conclude by suggesting that mathematical theory should serve to clarify not just our quantitative, but also our intuitive understanding of patterning processes.     

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.     

On the role of glypicans in the process of morphogen gradient formation

Glypicans are cell surface molecules that influence signaling and gradient formation of secreted morphogens and growth factors. Several distinct functions have been ascribed to glypicans including acting as co-receptors for signaling proteins. Recent data show that glypicans are also necessary for morphogen propagation in the tissue. In the present study, a model describing the interaction of a morphogen with glypicans is formulated, analyzed and compared with measurements of the effect of glypican Dally-like (Dlp) overexpression on Wingless (Wg) morphogen signaling in Drosophila melanogaster wing imaginal discs. The model explains the opposing effect that Dlp overexpression has on Wg signaling in the distal and proximal regions of the disc and makes a number of quantitative predictions for further experiments. In particular, our model suggests that Dlp acts by allowing Wg to diffuse on cell surface while protecting it from loss and degradation, and that Dlp rather than acting as Wg co-receptor competes with receptors for morphogen binding.     

Cell balance equation for chemotactic bacteria with a biphasic tumbling frequency

Alts three-dimensional cell balance equation characterizing the chemotactic bacteria was analyzed under the presence of one-dimensional spatial chemoattractant gradients. Our work differs from that of others who have developed rather general models for chemotaxis in the use of a non-smooth anisotropic tumbling frequency function that responds biphasically to the combined temporal and spatial chemoattractant gradients. General three-dimensional expressions for the bacterial transport parameters were derived for chemotactic bacteria, followed by a perturbation analysis under the planar geometry. The bacterial random motility and chemotaxis were summarized by a motility tensor and a chemotactic velocity vector, respectively. The consequence of invoking the diffusion-approximation assumption and using intrinsic one-dimensional models with modified cellular swimming speeds was investigated by numerical simulations. Characterizing the bacterial random orientation after tumbles by a turn angle probability distribution function, we found that only the first-order angular moment of this turn angle probability distribution is important in influencing the bacterial long-term transport. Mathematics Subject Classification (2000):60G05, 60J60, 82A70     

Scale-invariant pattern in the alga Micrasterias

Spatial structures arise in a variety of different physical, chemical and biological systems. A striking example is found during morphogenesis in the single-celled alga Micrasterias, where cell extensions called lobes branch repeatedly to produce a highly regular, apparently self-similar pattern. Lobe outgrowth in Micrasterias is thought to be controlled by the local accumulation of growth determinants at the lobe tips. These tip-growth sites undergo successive spatial bifurcations, leading to the recursively branched, final cell form. I have tested for scale invariance of this form, by measuring the distribution of tips as a function of position along the cell perimeter in mature Micrasterias cells of four different species. This tip distribution should reflect the steady-state distribution of growth determinants at the end of the spatial bifurcation process. For each cell measured, the distribution of tips resembled a Cantor set with three levels of constant, nested scaling. Significantly, roughly the same scale factor (3.0) was found at each scaling level in individual cells, and among different cells in each of the four species measured. These data suggest that scaling by this factor is intrinsic to the pattern formation process in Micrasterias.