Bakhtier Vasiev

Vortex initiation in a heterogeneous excitable medium

Abstract We studied numerically the process of vortex initiation in the heterogeneous active medium which is described by a FitzHugh-Nagumo-type model. Vortex initiation results from interaction of two external stimuli with a stepwise inhomogeneity in refractoriness. The influence of distance between the place of stimulation and heterogeneity and geometrical sizes of the heterogeneity on the process of vortex initiation is examined. The drift and interaction of vortices is also studied.

Modeling Chemotactic Cell Sorting during Dictyostelium discoideum Mound Formation

Coordinated cell movement is a major mechanism of the multicellular development of most organisms. The multicellular morphogenesis of the slime mould Dictyostelium discoideum, from single cells into a multicellular fruiting body, results from differential chemotactic cell movement. During aggregation cells differentiate into prestalk and prespore cells that will form the stalk and spores in the fruiting body. These cell types arise in a salt and pepper pattern after what the prestalk cells chemotactically sort out to form a tip. The tip functions as an organizer because it directs the further development. It has been difficult to get a satisfactory formal description of the movement behavior of cells in tissues. Based on our experiments, we consider the aggregate as a drop of a viscous fluid and show that this consideration is very well suited to mathematically describe the motion of cells in the tissue. We show that the transformation of a hemispherical mound into an elongated slug can result from the coordinated chemotactic cell movement in response to scroll waves of the chemoattractant cAMP. The model calculations furthermore show that cell sorting can result from differences in chemotactic cell movement and cAMP relay kinetics between the two cell types. During this process, the faster moving and stronger signaling cells collect on the top of the mound to form a tip. The mound then extends into an elongated slug just as observed in experiments. The model is able to describe cell movement patterns in the complex multicellular morphogenesis of Dictyostelium rather well and we expect that this approach may be useful in the modeling of tissue transformations in other systems.

PROPAGATING WAVES CONTROL DICTYOSTELIUM DISCOIDEUM MORPHOGENESIS

The morphogenesis of Dictyostelium results from the coordinated movement of starving cells to form a multicellular aggregate (mound) which transforms into a motile slug and finally a fruiting body. Cells differentiate in the mound and sort out to form an organised pattern in the slug and fruiting body. During aggregation, cell movement is controlled by propagating waves of the chemo-attractant cAMP. We show that mounds are also organised by propagating waves. Their geometry changes from target or single armed spirals during aggregation to multi-armed spiral waves in the mound. Some mounds develop transiently into rings in which multiple propagating wave fronts can still be seen. We model cell sorting in the mound stage assuming cell type specific differences in cell movement speed and excitability. This sorting feeds back on the wave geometry to generate twisted scroll waves in the slug. Slime mould morphogenesis can be understood in terms of wave propagation directing chemotactic cell movement.

Modeling gastrulation in the chick embryo: formation of the primitive streak.

The body plan of all higher organisms develops during gastrulation. Gastrulation results from the integration of cell proliferation, differentiation and migration of thousands of cells. In the chick embryo gastrulation starts with the formation of the primitive streak, the site of invagination of mesoderm and endoderm cells, from cells overlaying Kollers Sickle. Streak formation is associated with large-scale cell flows that carry the mesoderm cells overlying Kollers sickle into the central midline region of the embryo. We use multi-cell computer simulations to investigate possible mechanisms underlying the formation of the primitive streak in the chick embryo. Our simulations suggest that the formation of the primitive streak employs chemotactic movement of a subpopulation of streak cells, as well as differential adhesion between the mesoderm cells and the other cells in the epiblast. Both chemo-attraction and chemo-repulsion between various combinations of cell types can create a streak. However, only one combination successfully reproduces experimental observations of the manner in which two streaks in the same embryo interact. This finding supports a mechanism in which streak tip cells produce a diffusible morphogen which repels cells in the surrounding epiblast. On the other hand, chemotactic interaction alone does not reproduce the experimental observation that the large-scale vortical cell flows develop simultaneously with streak initiation. In our model the formation of large scale cell flows requires an additional mechanism that coordinates and aligns the motion of neighboring cells.

Becoming Multicellular by Aggregation; The Morphogenesis of the Social Amoebae Dicyostelium discoideum

The organisation and form of most organisms is generated during theirembryonic development and involves precise spatial and temporal controlof cell division, cell death, cell differentiation and cell movement.Differential cell movement is a particularly important mechanism in thegeneration of form. Arguably the best understood mechanism of directedmovement is chemotaxis. Chemotaxis plays a major role in the starvationinduced multicellular development of the social amoebae Dictyostelium.Upon starvation up to 105 individual amoebae aggregate to form afruiting body. In this paper we review the evidence that the movement ofthe cells during all stages of Dictyostelium development is controlled bypropagating waves of cAMP which control the chemotactic movement ofthe cells. We analyse the complex interactions between cell-cell signallingresulting in cAMP waves of various geometries and cell movement whichresults in a redistribution of the signalling sources and therefore changes thegeometry of the waves. We proceed to show how the morphogenesis,including aggregation stream and mound formation, slug formation andmigration, of this relatively simple organism is beginning to be understoodat the level of rules for cell behaviour, which can be tested experimentallyand theoretically by model calculations.

Electric current control of spiral wave dynamics

The control of spiral wave parameters by electric current was investigated in the Belousov-Zhabotinsky (BZ) reaction. It was found that the wavelength and the period of spiral waves increase by a factor of up to 3 with electric current (both dc and ac). Using this procedure spiral waves with a period higher than the period of medium bulk oscillations were observed. It was also found that hysteresis phenomena occur in the system considered.

Coordination of Cell Differentiation and Migration in Mathematical Models of Caudal Embryonic Axis Extension

Vertebrate embryos display a predominant head-to-tail body axis whose formation is associated with the progressive development of post-cranial structures from a pool of caudal undifferentiated cells. This involves the maintenance of active FGF signaling in this caudal region as a consequence of the restricted production of the secreted factor FGF8. FGF8 is transcribed specifically in the caudal precursor region and is down-regulated as cells differentiate and the embryo extends caudally. We are interested in understanding the progressive down-regulation of FGF8 and its coordination with the caudal movement of cells which is also known to be FGF-signaling dependent. Our study is performed using mathematical modeling and computer simulations. We use an individual-based hybrid model as well as a caricature continuous model for the simulation of experimental observations (ours and those known from the literature) in order to examine possible mechanisms that drive differentiation and cell movement during the axis elongation. Using these models we have identified a possible gene regulatory network involving self-repression of a caudal morphogen coupled to directional domain movement that may account for progressive down-regulation of FGF8 and conservation of the FGF8 domain of expression. Furthermore, we have shown that chemotaxis driven by molecules, such as FGF8 secreted in the stem zone, could underlie the migration of the caudal precursor zone and, therefore, embryonic axis extension. These mechanisms may also be at play in other developmental processes displaying a similar mode of axis extension coupled to cell differentiation.

A mathematical model of mortality dynamics across the lifespan combining heterogeneity and stochastic effects

The mortality patterns in human populations reflect biological, social and medical factors affecting our lives, and mathematical modelling is an important tool for the analysis of these patterns. It is known that the mortality rate in all human populations increases with age after sexual maturity. This increase is predominantly exponential and satisfies the Gompertz equation. Although the exponential growth of mortality rates is observed over a wide range of ages, it excludes early- and late-life intervals. In this work we accept the fact that the mortality rate is an exponential function of age and analyse possible mechanisms underlying the deviations from the exponential law across the human lifespan. We consider the effect of heterogeneity as well as stochastic factors in altering the exponential law and compare our results to publicly available age-dependent mortality data for Swedish and US populations. In a model of heterogeneous populations we study how differences in parameters of the Gompertz equation describing different subpopulations account for mortality dynamics at different ages. Particularly, we show that the mortality data on Swedish populations can be reproduced fairly well by a model comprising four subpopulations. We then analyse the influence of stochastic effects on the mortality dynamics to show that they play a role only at early and late ages, when only a few individuals contribute to mortality. We conclude that the deviations from exponential law at young ages can be explained by heterogeneity, namely by the presence of a subpopulation with high initial mortality rate presumably due to congenital defects, while those for old ages can be viewed as fluctuations and explained by stochastic effects.

The drift of a vortex in an inhomogeneous system of two coupled fibers

Abstract We studied the behavior of a vortex in an excitable medium having a stepped inhomogeneity, which is represented by a system of two coupled fibers. Numerical experiments were performed and analytical expressions were obtained for the determination of vortex drift velocity as a function of the parameters in the FitzHugh-Nagumo model.

Scaling of morphogenetic patterns in reaction-diffusion systems.

Development of multicellular organisms is commonly associated with the response of individual cells to concentrations of chemical substances called morphogens. Concentration fields of morphogens form a basis for biological patterning and ensure its properties including ability to scale with the size of the organism. While mechanisms underlying the formation of morphogen gradients are reasonably well understood, little is known about processes responsible for their scaling. Here, we perform a formal analysis of scaling for chemical patterns forming in continuous systems. We introduce a quantity representing the sensitivity of systems to changes in their size and use it to analyse scaling properties of patterns forming in a few different systems. Particularly, we consider how scaling properties of morphogen gradients forming in diffusion-decay systems depend on boundary conditions and how the scaling can be improved by passive modulation of morphogens or active transport in the system. We also analyse scaling of morphogenetic signal caused by two opposing gradients and consider scaling properties of patterns forming in activator–inhibitor systems. We conclude with a few possible mechanisms which allow scaling of morphogenetic patterns.