Alexander M. Samsonov

Mechanisms of gap gene expression canalization in the Drosophila blastoderm

BackgroundExtensive variation in early gap gene expression in the Drosophila blastoderm is reduced over time because of gap gene cross regulation. This phenomenon is a manifestation of canalization, the ability of an organism to produce a consistent phenotype despite variations in genotype or environment. The canalization of gap gene expression can be understood as arising from the actions of attractors in the gap gene dynamical system.ResultsIn order to better understand the processes of developmental robustness and canalization in the early Drosophila embryo, we investigated the dynamical effects of varying spatial profiles of Bicoid protein concentration on the formation of the expression border of the gap gene hunchback. At several positions on the anterior-posterior axis of the embryo, we analyzed attractors and their basins of attraction in a dynamical model describing expression of four gap genes with the Bicoid concentration profile accounted as a given input in the model equations. This model was tested against a family of Bicoid gradients obtained from individual embryos. These gradients were normalized by two independent methods, which are based on distinct biological hypotheses and provide different magnitudes for Bicoid spatial variability. We showed how the border formation is dictated by the biological initial conditions (the concentration gradient of maternal Hunchback protein) being attracted to specific attracting sets in a local vicinity of the border. Different types of these attracting sets (point attractors or one dimensional attracting manifolds) define several possible mechanisms of border formation. The hunchback border formation is associated with intersection of the spatial gradient of the maternal Hunchback protein and a boundary between the attraction basins of two different point attractors. We demonstrated how the positional variability for hunchback is related to the corresponding variability of the basin boundaries. The observed reduction in variability of the hunchback gene expression can be accounted for by specific geometrical properties of the basin boundaries.ConclusionWe clarified the mechanisms of gap gene expression canalization in early Drosophila embryos. These mechanisms were specified in the case of hunchback in well defined terms of the dynamical system theory.

Splitting induced generation of soliton trains in layered waveguides

We report first experimental registration of the splitting induced generation of a soliton train from a single incident strain soliton in two- and three-layered elastic waveguides. The origin is in the nonlinear response of the wave to an abrupt change of physical properties of the waveguide. We show a good agreement between our experimental results and theoretical estimates, based on a weakly nonlinear solution for the doubly dispersive (Boussinesq type) equation with piecewise constant coefficients for the waveguide made of a piecewise isotropic nonlinearly elastic material.

Travelling wave solutions for nonlinear dispersive equations with dissipation

It is shown that in order to obtain travelling wave solution many nonlinear dispersive equations with dissipative terms can be reduced by means of elementary transformations to the 1st order Abel o.d.e., and consequently, to the Emden-Fowler equation, if both nonlinear and dissipative terms are polynomials. These reductions can be integrated in closed form in terms of the Weierstrass elliptic function p, containing kink solutions as appropriate limits. Additional conditions for the equation coefficients and wave parameters are established for the wave existence, and their physical meaning is analysed. Some of nonlinear o.d.e., particularly, of higher order, do not provide a reduction to the Abel equation, therefore a different algorithm is proposed to obtain some exact solutions in terms of elliptic p-function.

Comparison of the effect of cyanoacrylate- and polyurethane-based adhesives on a longitudinal strain solitary wave in layered polymethylmethacrylate waveguides

We study the effect of two types of adhesive bonding on the propagation of a localized longitudinal strain wave in two- and three-layered elastic bars. The detectable variation in the decay rate of the wave at relatively long distances of propagation is observed. It is proposed that such variation can be used as an indicator of the type of an interface.

Bulk strain solitary waves in bonded layered polymeric bars with delamination

We report the registration of delamination induced variations in the dynamics of bulk strain solitary waves in layered polymeric bars with the glassy and rubber-like adhesives, for the layers made of the same material. The key phenomenon in a layered structure with the glassy bonding is the delamination caused fission of a single incident soliton into a wave train of solitons, with the detectable increase in the amplitude of the leading solitary wave. The significant feature of bulk strain solitons in structures bonded with the rubber-like adhesive is the generation of radiating solitary waves, whilst co-propagating ripples disappear in the delaminated area. The observed variations may be used for the detection of delamination in lengthy layered structures.

DEEP--differential evolution entirely parallel method for gene regulatory networks

The Differential Evolution Entirely Parallel (DEEP) method is applied to the biological data fitting problem. We introduce a new migration scheme, in which the best member of the branch substitutes the oldest member of the next branch that provides a high speed of the algorithm convergence. We analyze the performance and efficiency of the developed algorithm on a test problem of finding the regulatory interactions within the network of gap genes that control the development of early Drosophila embryo. The parameters of a set of nonlinear differential equations are determined by minimizing the total error between the model behavior and experimental observations. The age of the individuum is defined by the number of iterations this individuum survived without changes. We used a ring topology for the network of computational nodes. The computer codes are available upon request.

Long non-linear strain waves in layered elastic half-space

Elastic strain wave propagation in a thin non-linearly elastic layer superimposed on non-linearly elastic half-space is studied. Two layer-half-space contact models are considered. It is found that the Benjamin-Ono equation can be derived for description of longitudinal non-linear strain waves, when the contact between the layer and the half-space is provided only by means of the normal stresses and displacements. When the full contact problem is considered the more complicated integro-differential equation is derived. It is found that long non-linear periodical and solitary strain waves as well as envelope waves may exist in the first case, while only envelope wave solutions are found to the full contact problem. Linear wave analysis shows that the Korteveg-de Vries equation, often usable, is unlikely to be an adequate model for longitudinal surface strain waves. Application of the results obtained to experiments devoted to superconductivity threshold control in thin metal films as well as to generation of acoustic solitons in layered half-space is discussed.

Evolution of bulk strain solitons in long polymeric waveguides

We have studied the propagation of longitudinal strain solitons in waveguides made of polystyrene and plexiglas (polymethyl methacrylate) and long enough to provide noticeable changes in the wave shape and amplitude. The decay rate for nonlinear waves in these materials is estimated from the experimental data. An anomalously weak attenuation of strain solitons in polymers featuring a high dissipation of linear waves is discovered for the first time.

Model with asymptotically stable dynamics for Drosophila gap gene network

We consider a model of gap gene expression during the early development of Drosophila embryo. Parameter values in the model have been obtained by fitting to experimental patterns under an additional condition that the solution be asymptotically stable at large times. The patterns at the beginning of gastrulation in such solutions are very close to an actual attractor in the model. It is shown that such solutions are more robust to perturbations of concentrations and parameter values in the model.

How gap genes make their domains: An analytical study based on data driven approximations.

We consider a mathematical formulation of the problem of protein production during segment determination in the Drosophila blastoderm, together with some preliminary results of its analytical study. We reformulate the spatial difference equations as a set of nonlinear partial differential equations and obtain their dimensionless form in the continuum limit. Using previous results obtained by the gene circuit method, we find an asymptotic statement of the problem with a small parameter. Some results of the comparison method applied to the model are obtained, and exact stationary upper solutions are derived. They exhibit distinctive features of localized bell-shaped structures. (c) 2001 American Institute of Physics.