Fluid model of epithelial morphogenesis: Oscillations and structuring.

We have revised a mathematical model of epithelial morphogenesis by Belintsev et al. (1987) (BB model) taking into account the oscillatory nature of morphogenesis and stability analysis (Cherdantsev, 2014). Following the BB model in considering the feedback control of cell shape changes by mechanical forces, we modify it to represent epithelial surface movements observed in different types of Metazoan gastrulation. Basing on these observations, we argue that the epithelial surface movement is that of an incompressible fluid supplemented by a positive feedback between the movement and spreading of the surface flow. Dipole interactions between sources and sinks of surface energy provide a single mechanism both of short-ranged and long-ranged regulation of collective cell and surface movements whose basic variables are the space averaged epithelial surface curvature and lateral pressure within the epithelial surface flux negatively related to its velocity. The short-ranged activation means a movement of the surface up to the lateral pressure gradient under non-linear feedback control of the surface flexure. The break of this feedback with equalization of the surface curvature is sufficient for a self-restriction of the movement spreading. Owing to bistable interdependence between the lateral pressure and epithelial surface curvature, we get a generic oscillatory contour in which the same region oscillates being alternately a sink and source of the surface flow. The opposite phase oscillations of the lateral pressure and curvature allow for both directional propulsion of the surface through the same region and spatial differentiation based on parametric differences between the large-scaled regions that correspond to sources and sinks of the surface.

Morphomechanical reactions and mechanically stressed structures in amphibian embryos, as related to gastrulation and axial organs formation.

Reactions of embryonic tissues to a distributed and concentrated stretching are described and compared with the mechanics of the normal gastrulation movements. A role of mechanically stressed dynamic cell structures in the gastrulation, demarcation of notochord borders and in providing proportionality of the axial rudiments is demonstrated. A morphomechanical scheme of amphibian gastrulation is presented.

Intra-individual variation and evolution of modular structure in Draba plants.

We studied the evolution of quantitative traits related to shoot system architecture in a large genus Draba (Brassicaceae) making emphasis on the dynamics of relationship between individual and intra-individual variation. The results suggest that selection leading to origin of different life forms arises mainly from a necessity of moderation of the non-adaptive contest between the egoistic plant modules, taking care of self-reproduction of their own. We separated two evolutionary trends, one leading to the formation of short-lived monocarpic, and the other to long-lived polycarpic forms from the short-lived polycarpic plants. The first trend concerns with transformation of the innovation shoots into the axillary inflorescences by shortening of their vegetative developmental phase, while the second one - with individuation of the plant modules owing to acquisition of the capacity of rooting and separating from the mother plant. In both trends, the turning points of the evolution are those of originating of the negative for individual plants interactions between the plant modules being indirect non-adaptive consequences of the previous adaptive evolution and initiating selection for rebuilding of the plant modular structure. The difference between selection operating on intra-individual and individual variations is that, in the first case, combining of the characters of different individuals is infeasible. This leaves no choice for the evolution but to change the developmental mechanisms. In the case considered in this work, this is a change in shoot architecture using the material afforded by the natural variability of developmental pathways of the plant modules.

Generic oscillation patterns of the developing systems and their role in the origin and evolution of ontogeny.

The role of generic oscillation patterns in embryonic development on a macroscopic scale is discussed in terms of active shell model. These self-oscillations include periodic changes in both the mean shape of the shell surface and its spatial variance. They lead to origination of a universal oscillatory contour in the form of a non-linear dependence of the average rudiment's curvature upon the curvature variance. The alternation of high and low levels of the variance makes it possible to pursue the developmental dynamics irrespective to the spatiotemporal order of development and characters subject to selection and genetic control. Spatially homogeneous and heterogeneous states can alternate in both time and space being the parametric modifications of the same self-organization dynamics, which is a precondition of transforming of the oscillations into spatial differences between the parts of the embryo and then into successive stages of their formation. This process can be explained as a "retrograde developmental evolution", which means the late evolutionary appearance of the earlier developmental stages. The developing system progressively retreats from the initial self-organization threshold replacing the self-oscillatory dynamics by a linear succession of stages in which the earlier developmental stages appear in the evolution after the later ones. It follows that ontogeny is neither the cause, nor the effect of phylogeny: the phenotype development can be subject to directional change under the constancy of the phenotype itself and, vice versa, the developmental evolution can generate new phenotypes in the absence of the external environmental trends of their evolution.

Morphogenetic origin of natural variation.

We studied individual pathways of gastrulation in two related amphibian species making an emphasis on the developmental dynamics of normal variation in the geometry of gastrulation movements. Analyzing the variation dynamics, we show that the linear succession of developmental stages is a secondary phenomenon disguising self-oscillations that lie at the heart of the dorsal blastopore lip morphogenesis. Characteristic features of the equations derived to describe the oscillations are, first, their dependence only on the movement geometry and, second, including of the dynamics of spatial variance directly into the movement equations, making it clear that the reasons for variability of morphogenesis are the same that for morphogenesis itself. The equations describing morphogenetic oscillations are mathematically similar to those describing natural selection in that the system tends to minimize its variance, individual or within-individual one, but the spatially uniform state turns to be unstable. Comparing of the dynamics of natural developmental variation in gastrulation in two frog species shows that, depending on the mechanics and geometry mass cell movements, different types of gastrulation movements have different proportions of the between- to within-individual differences, which strongly influences the choice of characters subject to evolution. Instead of being a source of constraints imposed on externally guided evolutionary trends, morphogenesis becomes a driving force of the adaptively silent, but directional evolution of the developing systems, which seems to be the only possible way of originating of the evolutionary novelties, both in evolution and ontogeny of the biological structures.

Morphogenesis of active shells.

We consider the active shell as a single-cell or epithelial sheet surface that, sharing basic properties of stretched elastic shells, is capable of active planar movement owing to recruiting of the new surface elements. As model examples of their morphogenesis, we consider the growth and differentiation of single-cell hairs (trichomes) in plants of the genus Draba, and the epiboly and formation of the dorsoventral polarity in loach. The essential feature of the active shell behavior at both cellular and supracellular levels is regular deviating from the spatially homogeneous form, which is a primary cause of originating of the active mechanical stresses inside the shell in addition to its passive stretching by the intrinsic forces. Analyzing the quantitative morphological data, we derive the equations in which the temporal self-oscillations and spatial differentiation are distinguishable only at the parametric level depending on the proportion of active to passive stresses. In contrast to the ordinary activator-inhibitor systems, the self-oscillation dynamics is principally non-local and, consequently, one-parametric, the shell surface curvature being an analog of the inhibitor, while its spatial variance being an analog of the activator of shaping. Analyzing variability and evolution of the hair cell branching, we argue that the linear ontogeny (succession of the developmental stages) is a secondary evolutionary phenomenon originating from cyclic self-organizing algorithms of the active shell shaping.

Geometry and mechanics of teleost gastrulation and the formation of primary embryonic axes.

Examination of normal shaping dynamics and immediate and long-term responses to blastoderm cutting in zebrafish and loach embryos prior to the onset of gastrulation and during the course of epiboly revealed that anteroposterior (AP) and dorsoventral (DV) polarity formation is connected with shaping of the blastoderm circumferential region, which stretches along and shrinks across its movement axes and originates the non-isotropic fields of tensile stresses. Based on data from cutting experiments and quantitative morphology, we reconstructed the movement-shaping patterns of epiboly and embryonic shield formation. We revealed that AP and DV axes originate as a mass cell movement subject to the movement-shaping equivalence principle, which means the spatial series of differently shaped areas corresponding to the time succession of the same area shaping. Maintenance of the main body axes in orthogonal orientation depends on the mechanical equilibrium principle allowing for converting shape asymmetry into that of tensile stresses and vice versa. The causal relationship between the main movement-shaping axes and that of embryonic polarity was proved in cutting experiments in which the DV axis direction was subject to rearrangement so as to adjust to the new direction of mass cell movement axes induced by healing the wound in the blastoderm circumferential region.

The dynamic geometry of mass cell movements in animal morphogenesis.

There is an infinite number of interactions between morphogenetic processes of different time and space scales. How do these unfold in a regular series of mass morphogenetic movements to produce a basically simple and reproducible structure? I present a new morphogenetic concept -- the spatial unfolding (SU) of cell movements, whose definition rests on the correspondence between the continuous spatial series of cell shapes and the succession of changes in the shape of a single cell moving in an epithelial sheet whose shape is also subject to change. The change in the shape of moving cells is the only measure of their translocation both in space and time. The SU provides a morpho-dynamics description of mass cell movements which is completely independent of both an external coordinate system and external forces. The cell geometry of SU allows us to derive the future embryonic form from the actual one by a movement-shaping algorithm operating on the basis of positive and negative geometric feedbacks between the cell movement in the epithelial sheet plane and the epithelial sheet shaping, the feedback system providing a geometric alternative to Turing's self-organization via reaction-diffusion systems. Putting together histological, quantitative morphological and experimental data permits us to isolate four SU, each acting in morphogenesis as an irreducible whole, which seem to include all real examples of epithelial morphogenesis in multicellular animals, from Coelenterates to Chordates.