Synchronization of Metabolic Oscillations:Two Cells and Ensembles of Adsorbed Cells

We treat synchronization of metabolic oscillations in two cellsand in ensembles of cells adsorbed at the liquid-solid interface.(i) Synchronization of oscillations in two cells is assumedto occur via perturbation of the metabolite concentration nearone cell due to the metabolite diffusion flux from another cell.This direct channel of synchronization may be important ifthe distance between two cells is comparable with the cell diameter.The corresponding coupling coefficient is found to be proportionalto the metabolite diffusion coefficient and inversely proportionalto the cell radius and the distance between the cells.(ii) In the case of ensembles of adsorbed cells, synchronizationof oscillations is considered to be indirect, i.e., to occur viathe metabolite concentration formed outside the cells nearthe interface due to metabolite diffusion from the cells. We havederived a general integral equation relating the metaboliteconcentration near the interface with concentrations inside the cells.PACS: 82.37.-j, 82.40.Bj     

Modelling and Analysis of Planar Cell Polarity

Planar cell polarity (PCP) occurs in the epithelia of many animals and can lead to the alignment of hairs, bristles, and feathers. Here, we present two approaches to modelling this phenomenon. The aim is to discover the basic mechanisms that drive PCP, while keeping the models mathematically tractable. We present a feedback and diffusion model, in which adjacent cell sides of neighbouring cells are coupled by a negative feedback loop and diffusion acts within the cell. This approach can give rise to polarity, but also to period two patterns. Polarisation arises via an instability provided a sufficiently strong feedback and sufficiently weak diffusion. Moreover, we discuss a conservative model in which proteins within a cell are redistributed depending on the amount of proteins in the neighbouring cells, coupled with intracellular diffusion. In this case, polarity can arise from weakly polarised initial conditions or via a wave provided the diffusion is weak enough. Both models can overcome small anomalies in the initial conditions. Furthermore, the range of the effects of groups of cells with different properties than the surrounding cells depends on the strength of the initial global cue and the intracellular diffusion.     

Intra-membrane ligand diffusion and cell shape modulate juxtacrine patterning

A key problem in developmental biology is how pattern and planar polarity are transmitted in epithelial structures. Examples include Drosophila neuronal differentiation, ommatidia formation in the compound eye, and wing hair polarization. A key component for the generation of such patterns is direct cell-cell signalling by transmembrane ligands, called juxtacrine signalling. Previous models for this mode of communication have considered homogeneous distributions in the cell membrane, and the role of polarity has been largely ignored. In this paper we determine the role of inhomogeneous protein and receptor distributions in juxtacrine signalling. We explicitly include individual membrane segments, diffusive transport of proteins and receptors between these segments, and production terms with a combination of local and global responses to ligand binding. Our analysis shows that intra-membrane ligand transport is vital for the generation of long wavelength patterns. Moreover, with no ligand transport, there is no pattern formation for lateral induction, a process in which receptor activation up-regulates ligand production. Biased production of ligand also modulates patterning bifurcations and predicted wavelengths. In addition, biased ligand and receptor trafficking can lead to regular polarity across a lattice, in which each cell has the same orientation-directly analogous to patterns of hairs in the Drosophila wing. We confirm the trends in pattern wavelengths previously observed for patterns with cellular homogeneity-lateral inhibition tends to give short-range patterns, while lateral induction can give patterns with much longer wavelengths. Moreover, the original model can be recovered if intra-membrane bound receptor diffusion is included and rapid equilibriation between the sides is considered. Finally, we consider the role of irregular cell shapes and waves in such networks, including wave propagation past clones of non-signalling cells.     

Toward a biophysical theory of organogenesis: Birefringence observations on regenerating leaves in the succulent,Graptopetalum paraguayense E. Walther

Polarity shifts occur during organogenesis. The histological criterion for polarity is the direction of cell division. The biophysical criterion is the orientation of reinforcing cellulose microfibrils which lie normal to the organ axis and which determine the preferred growth direction. Using cell pattern to deduce cell lineage, and polarized light to study cellulose alignment, both aspects of polarity were examined in the epidermis of regenerating G. paraguayense. In this system new leaves and a stem arise from parallel cell files on a mature leaf. Large (90°) shifts in polarity occur in regions of the epidermis to give the new organs radial symmetry in the surface plane (files radiating from a pole). Study of the shifts in the epidermis showed that, during certain stages, shifts in the division direction are accompanied by shifts in the cellulose deposition direction, as expected. The new cellulose orientation is parallel to the new cross wall. During normal organ extension, however, shifts in division direction do not bring on changes in cellulose pattern. Thus the coupling between the two kinds of polarity is facultative. This variable relation is used in a biophysical model which can account for the reorganization of cell file pattern and cellulose reinforcement pattern into the radial symmetry of the new organ.     

Establishment of global patterns of planar polarity during growth of the Drosophila wing epithelium

Epithelial tissues develop planar polarity that is reflected in the global alignment of hairs and cilia with respect to the tissue axes. The planar cell polarity (PCP) proteins form asymmetric and polarized domains across epithelial junctions that are aligned locally between cells and orient these external structures. Although feedback mechanisms can polarize PCP proteins intracellularly and locally align polarity between cells, how global PCP patterns are specified is not understood. It has been proposed that the graded distribution of a biasing factor could guide long-range PCP. However, we recently identified epithelial morphogenesis as a mechanism that can reorganize global PCP patterns; in the Drosophila pupal wing, oriented cell divisions and rearrangements reorient PCP from a margin-oriented pattern to one that points distally. Here, we use quantitative image analysis to study how PCP patterns first emerge in the wing. PCP appears during larval growth and is spatially oriented through the activities of three organizer regions that control disc growth and patterning. Flattening morphogen gradients emanating from these regions does not reduce intracellular polarity but distorts growth and alters specific features of the PCP pattern. Thus, PCP may be guided by morphogenesis rather than morphogen gradients.     

Further contributions to the theory of cell polarity and self-regulation

In continuation of a previous study a case of self-regulating polarity is investigated, in which a metabolite neither enters nor leaves the cell, being produced in one part of it and consumed in another. Such a cell possesses a larger degree of autonomy and independence of external conditions than the previously discussed one. Application to cellular locomotion is indicated.     

Choosing sides: establishment of polarity in zygotes of fucoid algae

The acquisition and expression of polarity during early embryogenesis underlies developmental pattern. In many multicellular organisms an initial asymmetric division of the zygote is critical to the determination of different cell fates of the early embryonic cells. Zygotes of the marine fucoid algae are initially apolar and become polarized in response to external cues. This results in an initial asymmetric division of the zygote. Subsequent divisions occur in a highly ordered spatial and temporal pattern. A combination of cell biological and biochemical studies is providing new details, and some controversies concerning the mechanisms by which zygotic polarity is acquired and amplified. Here, we discuss some of the more recent studies that are allowing improved understanding of polarization in this system.     

Small regulatory RNAs may sharpen spatial expression patterns

The precise establishment of gene expression patterns is a crucial step in development. Formation of a sharp boundary between high and low spatial expression domains requires a genetic mechanism that exhibits sensitivity, yet is robust to fluctuations, a demand that may not be easily achieved by morphogens alone. Recently, it has been demonstrated that small RNAs (and, in particular, microRNAs) play many roles in embryonic development. Whereas some RNAs are essential for embryogenesis, others are limited to fine-tuning a predetermined gene expression pattern. Here, we explore the possibility that small RNAs participate in sharpening a gene expression profile that was crudely established by a morphogen. To this end, we study a model in which small RNAs interact with a target gene and diffusively move from cell to cell. Though diffusion generally smoothens spatial expression patterns, we find that intercellular mobility of small RNAs is actually critical in sharpening the interface between target expression domains in a robust manner. This sharpening occurs as small RNAs diffuse into regions of low mRNA expression and eliminate target molecules therein, but cannot affect regions of high mRNA levels. We discuss the applicability of our results, as examples, to the case of leaf polarity establishment in maize and Hox patterning in the early Drosophila embryo. Our findings point out the functional significance of some mechanistic properties, such as mobility of small RNAs and the irreversibility of their interactions. These properties are yet to be established directly for most classes of small RNAs. An indirect yet simple experimental test of the proposed mechanism is suggested in some detail.     

Cell cycling and cell enlargement in developing leaves of Arabidopsis.

Cell cycling plays an important role in plant development, including: (1) organ morphogenesis, (2) cell proliferation within tissues, and (3) cell differentiation. In this study we use a cyclin::beta-glucuronidase reporter construct to characterize spatial and temporal patterns of cell cycling at each of these levels during wild-type development in the model genetic organism Arabidopsis thaliana (Columbia). We show that a key morphogenetic event in leaf development, blade formation, is highly correlated with localized cell cycling at the primordium margin. However, tissue layers are established by a more diffuse distribution of cycling cells that does not directly involve the marginal zone. During leaf expansion, tissue proliferation shows a strong longitudinal gradient, with basiplastic polarity. Tissue layers differ in pattern of proliferative cell divisions: cell cycling of palisade mesophyll precursors is prolonged in comparison to that of pavement cells of the adjacent epidermal layers, and cells exit the cycle at different characteristic sizes. Cell divisions directly related to formation of stomates and of vascular tissue from their respective precursors occur throughout the period of leaf extension, so that differing tissue patterns reflect superposition of cycling related to cell differentiation on more general tissue proliferation. Our results indicate that cell cycling related to leaf morphogenesis, tissue-specific patterns of cell proliferation, and cell differentiation occurs concurrently during leaf development and suggest that unique regulatory pathways may operate at each level.     

Mathematical Analysis of Spontaneous Emergence of Cell Polarity

Cell polarization, in which intracellular substances are asymmetrically distributed, enables cells to carry out specialized functions. While cell polarity is often induced by intracellular or extracellular spatial cues, spontaneous polarization (the so-called symmetry breaking) may also occur in the absence of spatial cues. Many computational models have been used to investigate the mechanisms of symmetry breaking, and it was proved that spontaneous polarization occurs when the lateral diffusion of inactive signaling molecules is much faster than that of active signaling molecules. This conclusion leaves an important question of how, as observed in many biological systems, cell polarity emerges when active and inactive membrane-bound molecules diffuse at similar rates while cycling between cytoplasm and membrane takes place. The recent studies of Rätz and Röger showed that, when the cytosolic and membrane diffusion are very different, spontaneous polarization is possible even if the membrane-bound species diffuse at the same rate. In this paper, we formulate a two-equation non-local reaction-diffusion model with general forms of positive feedback. We apply Turing stability analysis to identify parameter conditions for achieving cell polarization. Our results show that spontaneous polarization can be achieved within some parameter ranges even when active and inactive signaling molecules diffuse at similar rates. In addition, different forms of positive feedback are explored to show that a non-local molecule-mediated feedback is important for sharping the localization as well as giving rise to fast dynamics to achieve robust polarization.     

Cell polarity and epithelial oncogenesis

Pathologists have long recognized that tumour formation in epithelia leads to disruption of normal epithelial cell polarity. Despite this, few studies have taken advantage of new information on the biogenesis of cell polarity to analyse the process of epithelial oncogenesis. Recent studies of epithelial cell lines now indicate that the pattern of breakdown of polarity during oncogenesis may reflect the way in which normal epithelial cells achieve polarity. These results suggest not only a novel way to study the development of polarity in vitro, but also new ideas for the early detection of cancer.     

Stochastic amplification of spatial modes in a system with one diffusing species

The problem of pattern formation in a generic two species reaction–diffusion model is studied, under the hypothesis that only one species can diffuse. For such a system, the classical Turing instability cannot take place. At variance, by working in the generalized setting of a stochastic formulation to the inspected problem, spatially organized patterns can develop, seeded by finite size corrections. General conditions are given for the stochastic patterns to occur. The predictions of the theory are tested for a specific case study.     

Formation of spacing pattern and morphogenesis of chick feather buds is regulated by cytoskeletal structures

Chick feather buds develop sequentially in a hexagonal array. Each feather bud develops with anterior posterior polarity, which is thought to develop in response to signals derived from specialized regions of mesenchymal condensation and epithelial thickening. These developmental processes are performed by cellular mechanisms, such as cell proliferation and migration, which occur during chick feather bud development. In order to understand the mechanisms regulating the formation of mesenchymal condensation and their role in feather bud development, we explanted chick dorsal skin at stage HH29+ with cytochalasin D, which inhibits cytoskeletal formation. We show that the aggregation of mesenchymal cells can be prevented by cytochalasin D treatment in a concentration-dependent manner. Subsequently, cytochalasin D disrupts the spacing pattern and inhibits feather bud axis formation as well. In addition, expression patterns of Bmp-4 and Msx-2, key molecules for early feather bud development, were disturbed by cytochalasin D treatment. Our results fully indicate that both the cytoskeletal structure and cell activity via gene regulation are of fundamental importance in mesenchymal condensation leading to proper morphogenesis of feather bud and spacing pattern formation.     

PAR polarity: From complexity to design principles

The par-titioning-defective or PAR proteins comprise the core of an essential cell polarity network that underlies polarization in a wide variety of cell types and developmental contexts. The output of this network in nearly every case is the establishment of opposing and complementary membrane domains that define a cell?s polarity axis. Yet, behind this simple pattern is a complex system of interactions, regulation and dynamic behaviors. How these various parts combine to generate polarized patterns of protein localization in cells is only beginning to become clear. This review, part of the Special Issue on Cell Polarity, aims to highlight several emerging themes and design principles that underlie the process of cell polarization by components of the PAR network.     

Symmetrically dividing cell specific division axes alteration observed in proteasome depleted C. elegans embryo

A fertilised Caenorhabditis elegans embryo shows an invariable pattern of cell division and forms a multicellular body where each cell locates to a defined position. Mitotic spindle orientation is determined by several preceding events including the migration of duplicated centrosomes on a nucleus and the rotation of nuclear-centrosome complex. Cell polarity is the dominant force driving nuclear-centrosome rotation and setting the mitotic spindle axis in parallel with the polarity axis during asymmetric cell division. It is reasonable that there is no nuclear-centrosome rotation in symmetrically dividing blastomeres, but the mechanism(s) which suppress rotation in these cells have been proposed because the rotations occur in some polarity defect embryos. Here we show the nuclear-centrosome rotation can be induced by depletion of RPN-2, a regulatory subunit of the proteasome. In these embryos, cell polarity is established normally and both asymmetrically and symmetrically dividing cells are generated through asymmetric cell divisions. The nuclear-centrosome rotations occurred normally in the asymmetrically dividing cell lineage, but also induced in symmetrically dividing daughter cells. Interestingly, we identified RPN-2 as a binding protein of PKC-3, one of critical elements for establishing cell polarity during early asymmetric cell divisions. In addition to asymmetrically dividing cells, PKC-3 is also expressed in symmetrically dividing cells and a role to suppress nuclear-centrosome rotation has been anticipated. Our data suggest that the expression of RPN-2 is involved in the mechanism to suppress nuclear-centrosome rotation in symmetrically dividing cells and it may work in cooperation with PKC-3.     

Turing bifurcations with a temporally varying diffusion coefficient

This paper is concerned with the possibility of Turing bifurcations in a reaction-diffusion system in which the diffusion coefficient of one species varies periodically in time. This problem was introduced and investigated numerically by Timm and Okubo (J. Math. Biol. 30, 307, 1992) in the context of predator-prey interactions in plankton populations. Here, I consider the simple case in which the temporal variation in diffusivity has a square-tooth form, alternating between two constant values, with a period that is long compared with the time scale of the kinetics. The analysis is valid for any set of reaction kinetics. I derive explicit expressions for the Floquet multipliers that determine the stability of the steady state, and thereby obtain the conditions for diffusion driven instability to occur. These conditions imply that, depending on the kinetics, the homogeneous equilibrium may be either more or less stable than when the diffusion coefficient is a constant equal to the mean of the variable diffusivity. I go on to consider the form of the solution when diffusion driven instability does occur, and I use perturbation theory to determine the effect of a small temporal variation in the diffusion coefficient on the spatial wavelength of the pattern that results from diffusion driven instability.     

Stability of polarity in the epidermis of a beetle, Tenebrio molitor L.

Cell polarity in the insect epidermis may be coupled to the orientation of anisometric cuticle components. Rotation of squares of sternite epidermis in the larva results in a corresponding rotation in the highly ordered orientation of cuticle fibers in the adult “crossply” cuticle. The patterns of fiber orientation resulting from graft rotation can be explained by the presence of an axial gradient of positional information.The polarity of the rotated tissue is, however, not fixed. Interaction between the polarity of the graft and host tissue may result in a partial shift of graft polarity toward the axial polarity of the host tissue. This interaction is apparently restricted to a limited period of the cell cycle: cell division. In Tenebrio, the sternite epidermis proliferates only once during metamorphosis, 140–90 hr before pupation. Rotational grafts performed before, during, and after this period present a graded series of “relaxation” patterns in fiber orientation in the graft area. Maximal graft repolarization coincides with maximal cell division on the sternite. The epidermal gradient, or cell response to the gradient, appears to be nonlinear along the segment.If no cell division intervenes between graft rotation and fiber deposition, graft polarity remains stable. This stability necessitates a “memory” component in the epidermis. It is suggested that periodic assessment of tissue polarity occurs concomitant with a particular process of cell division.     

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.     

The Turing-Child energy field as a driver of early mammalian development

The equivalence of the early mammalian cells, of importance in assisted reproductive technologies (ART), is considered. It is suggested that this controversial topic can be settled by finding whether the cells are distinguished by the Turing-Child (TC) field, as expressed for example by patterns of mitochondrial activity. The division of the pronuclear embryo is driven by a symmetrical bipolar TC pattern whose experimental shape and chemical nature is predicted by TC theory. This bipolar pattern drives the subsequent cell divisions too, and according to present experimental results all cells are equivalent until compaction since they are not distinguished by the TC field in normal development. Interphase cells exhibit homogeneous mitochondrial activity, or perinuclear, or perinuclear and cortical activity, and these patterns too and the rotational symmetry observed are predicted by TC theory. The first differentiation, into an inner mass cell and the trophectoderm, as well as the formation of cell polarity in the trophectoderm are considered. It is suggested that these two events are driven by a peripheral spherical shell of high energy metabolism in the morula; such a shell is predicted by TC theory in a compacted multicellular sphere whose cells are connected by gap junctions. The experimental patterns of mitochondrial activity in unfertilized oocytes exhibit rotational symmetry or polarity. The shape and the chemical nature of these patterns also are predicted and explained by TC theory in a sphere. The change in the spatial pattern of mitochondrial activity with development is attributed to a change in the spatial pattern of mitochondrial activity and not to physical translocation of mitochondria. The experimental finding that these spatial patterns of mitochondrial activity are observed only in live and not in dead biological material is explained by the TC pattern being biology's unique and universal dissipative structure that requires ongoing specific biochemical reactions and energy dissipation.     

A theory of biological pattern formation

One of the elementary processes in morphogenesis is the formation of a spatial pattern of tissue structures, starting from almost homogeneous tissue. It will be shown that relatively simple molecular mechanisms based on auto- and cross catalysis can account for a primary pattern of morphogens to determine pattern formation of the tissue. The theory is based on short range activation, long range inhibition, and a distinction between activator and inhibitor concentrations on one hand, and the densities of their sources on the other. While source density is expected to change slowly, e.g. as an effect of cell differentiation, the concentration of activators and inhibitors can change rapidly to establish the primary pattern; this results from auto- and cross catalytic effects on the sources, spreading by diffusion or other mechanisms, and degradation.Employing an approximative equation, a criterium is derived for models, which lead to a striking pattern, starting from an even distribution of morphogens, and assuming a shallow source gradient. The polarity of the pattern depends on the direction of the source gradient, but can be rather independent of other features of source distribution. Models are proposed which explain size regulation (constant proportion of the parts of the pattern irrespective of total size). Depending on the choice of constants, aperiodic patterns, implying a one-to-one correlation between morphogen concentration and position in the tissue, or nearly periodic patterns can be obtained. The theory can be applied not only to multicellular tissues, but also to intracellular differentiation, e.g. of polar cells.The theory permits various molecular interpretations. One of the simplest models involves bimolecular activation and monomolecular inhibition. Source gradients may be substituted by, or added to, sink gradients, e.g. of degrading enzymes. Inhibitors can be substituted by substances required for, and depleted by activation.Sources may be either synthesizing systems or particulate structures releasing activators and inhibitors.Calculations by computer are presented to exemplify the main features of the theory proposed. The theory is applied to quantitative data on hydra — a suitable one-dimensional model for pattern formation — and is shown to account for activation and inhibition of secondary head formation.