Mitochondria not only govern energy production, but are also involved in crucial cellular signalling processes. They are one of the most important organelles determining the Ca(2+) regulatory pathway in the cell. Several mathematical models explaining these mechanisms were constructed, but only few of them describe interplay between calcium concentrations in endoplasmic reticulum (ER), cytoplasm and mitochondria. Experiments measuring calcium concentrations in mitochondria and ER suggested the existence of cytosolic microdomains with locally elevated calcium concentration in the nearest vicinity of the outer mitochondrial membrane. These intermediate physical connections between ER and mitochondria are called MAM (mitochondria-associated ER membrane) complexes. We propose a model with a direct calcium flow from ER to mitochondria, which may be justified by the existence of MAMs, and perform detailed numerical analysis of the effect of this flow on the type and shape of calcium oscillations. The model is partially based on the Marhl et al model. We have numerically found that the stable oscillations exist for a considerable set of parameter values. However, for some parameter sets the oscillations disappear and the trajectories of the model tend to a steady state with very high calcium level in mitochondria. This can be interpreted as an early step in an apoptotic pathway.

SUMMARYIn this paper, we prove the existence and uniqueness of a global solution for 2-D micropolar fluid equation with periodic boundary conditions. Then we restrict ourselves to the autonomous case and show the existence of a global attractor.

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In this paper we consider the solutions of micropolar fluid equations in space dimension two with periodic boundary condition. We show that the strong solutions are analytic in time with values in an appropriate Gevrey class of function, provided that external forces and moments are time-independent and are in a Gevrey class.

B and Mast cells are activated by the aggregation of the immune receptors. Motivated by this phenomena we consider a simple spatially extended model of mutual interaction of kinases and membrane receptors. It is assumed that kinase activates membrane receptors and in turn the kinase molecules bound to the active receptors are activated by transphosphorylation. Such a type of interaction implies positive feedback and may lead to bistability. In this study we apply the Steklov eigenproblem theory to analyze the linearized model and find exact solutions in the case of non-uniformly distributed membrane receptors. This approach allows us to determine the critical value of receptor dephosphorylation rate at which cell activation (by arbitrary small perturbation of the inactive state) is possible. We found that cell sensitivity grows with decreasing kinase diffusion and increasing anisotropy of the receptor distribution. Moreover, these two effects are cooperating. We showed that the cell activity can be abruptly triggered by the formation of the receptor aggregate. Since the considered activation mechanism is not based on receptor crosslinking by polyvalent antigens, the proposed model can also explain B cell activation due to receptor aggregation following binding of monovalent antigens presented on the antigen presenting cell.

Abstract. Mitochondria are one of the most important organelles determining Ca 2+ regulatory pathway in the cell. Recent experiments suggested the existence of cytosolic microdomains with locally elevated calcium concentration (CMDs) in the nearest vicinity of the outer mitochondrial membrane (OMM). These intermediate physical connections between endoplasmic reticulum (ER) and mitochodria are called MAM (mitochondria-associated ER membrane) complexes.The aim of this paper is to take into account the direct calcium flow from ER to mitochondria implied by the existence of MAMs and perform detailed numerical analysis of the influence of this flow on the type and shape of calcium oscillations. Depending on the permeability of MAMs interface and ER channels, different patterns of oscillations appear (simple, bursting and chaotic). For some parameters the oscillatory pattern disappear and the system tends to a steady state with extremely high calcium level in mitochondria, which can be interpreted as a crucial point at the beginning of an apoptotic pathway.

We consider the model, proposed by Dawidowicz and Zalasiński, describing the interactions between the heterotrophic and autotrophic organisms coexisting in a terrestrial environment with available oxygen. We modify this model by assuming intraspecific competition between heterotrophic organisms. Moreover, we introduce a diffusion of both types of organisms and oxygen. The basic properties of the extended model are examined and illustrated by numerical simulations.

Communicated by M. LachowiczThis paper is devoted to obtain ladder inequalities for 2D micropolar fluid equations on a periodic domain Q = (0,L) 2 . The ladder inequalities are differential inequalities that connect the evolution of L 2 norms of derivatives of order N with the evolution of the L 2 norms of derivatives of other (usually lower) order. Moreover, we find (with slight assumption on external fields) long-time upper bounds on the L 2 norms of derivatives of every order, which implies that a global attractor is made up from C ∞ functions.