Hopf bifurcation in a gene regulatory network model: Molecular movement causes oscillations

Gene regulatory networks, i.e. DNA segments in a cell which interact with each other indirectly through their RNA and protein products, lie at the heart of many important intracellular signal transduction processes. In this paper we analyse a mathematical model of a canonical gene regulatory network consisting of a single negative feedback loop between a protein and its mRNA (e.g. the Hes1 transcription factor system). The model consists of two partial differential equations describing the spatio-temporal interactions between the protein and its mRNA in a 1-dimensional domain. Such intracellular negative feedback systems are known to exhibit oscillatory behaviour and this is the case for our model, shown initially via computational simulations. In order to investigate this behaviour more deeply, we next solve our system using Greens functions and then undertake a linearized stability analysis of the steady states of the model. Our results show that the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. This shows that the spatial movement of the mRNA and protein molecules alone is sufficient to cause the oscillations. This has implications for transcription factors such as p53, NF-$kappa$B and heat shock proteins which are involved in regulating important cellular processes such as inflammation, meiosis, apoptosis and the heat shock response, and are linked to diseases such as arthritis and cancer

Two-scale convergence for locally-periodic microstructures and homogenization of plywood structures

The introduced notion of locally-periodic two-scale convergence allows to average a wider range of microstructures, compared to the periodic one. The compactness theorem for the locally-periodic two-scale convergence and the characterisation of the limit for a sequence bounded in $H^1(\Omega)$ are proven. The underlying analysis comprises the approximation of functions, which periodicity with respect to the fast variable depends on the slow variable, by locally-periodic functions, periodic in subdomains smaller than the considered domain, but larger than the size of microscopic structures. The developed theory is applied to derive macroscopic equations for a linear elasticity problem defined in domains with plywood structures.Comment: 22 pages, 4 figure

From individual-based mechanical models of multicellular systems to free-boundary problems

In this paper we present an individual-based mechanical model that describes the dynamics of two contiguous cell populations with different proliferative and mechanical characteristics. An off-lattice modelling approach is considered whereby: (i) every cell is identified by the position of its centre; (ii) mechanical interactions between cells are described via generic nonlinear force laws; and (iii) cell proliferation is contact inhibited. We formally show that the continuum counterpart of this discrete model is given by a free-boundary problem for the cell densities. The results of the derivation demonstrate how the parameters of continuum mechanical models of multicellular systems can be related to biophysical cell properties. We prove an existence result for the free-boundary problem and construct travelling-wave solutions. Numerical simulations are performed in the case where the cellular interaction forces are described by the celebrated Johnson-Kendalli-Roberts model of elastic contact, which has been previously used to model cell-cell interactions. The results obtained indicate excellent agreement between the simulation results for the individual-based model, the numerical solutions of the corresponding free-boundary problem and the travelling-wave analysis

Locally periodic unfolding method and two-scale convergence on surfaces of locally periodic microstructures

In this paper we generalize the periodic unfolding method and the notion of two-scale convergence on surfaces of periodic microstructures to locally periodic situations. The methods that we introduce allow us to consider a wide range of non-periodic microstructures, especially to derive macroscopic equations for problems posed in domains with perforations distributed non-periodically. Using the methods of locally periodic two-scale convergence (l-t-s) on oscillating surfaces and the locally periodic (l-p) boundary unfolding operator, we are able to analyze differential equations defined on boundaries of non-periodic microstructures and consider non-homogeneous Neumann conditions on the boundaries of perforations, distributed non-periodically

BADANIE WPŁYWU UŁAMKOWEJ LICZBY SZCZELIN BIEGUNÓW NA GENERACJĘ TURBINY WIATROWEJ PRZY UŻYCIU ULEPSZONEGO ALGORYTMU OPTYMALIZACJI CĘTKOWANEJ HIENY

The design of machines with permanent magnets is actively developing day by day and is often used in wind energy. The main advantages of such variable speed drives are high efficiency, high power density and torque density. When designing a wind generator with two rotors and permanent magnets, it is necessary to solve such a problem as the correct choice of the number of poles and slots to increase efficiency and minimize the cost of the machine. In this work, an improved spotted hyena optimization algorithm is used to obtain the optimal combination of slots and poles. This optimization algorithm makes it possible to obtain the number of fractional slots per pole and evaluate the operating efficiency of a wind generator with a double rotor and ferrite magnets. At the first stage of machine design, various combinations of slots are installed. Next, the optimal combination is selected from various slot-pole combinations, taking into account the Enhanced Spotted Hyena Optimization (ESHO) algorithm, in which a multi-objective function is configured. Accordingly, the multi-objectives are the integration of reverse electromotive force, output torque, gear torque, flux linkage, torque ripple along with losses. Analysis of the results obtained shows that the proposed algorithm for determining the optimal slot combination is more efficient than other slot combinations. It has also been found that the choice of slot and pole combination is critical to the efficient operation of permanent magnet machines.Projektowanie maszyn z magnesami trwałymi aktywnie rozwija się z dnia na dzień i jest często wykorzystywane w energetyce wiatrowej. Głównymi zaletami takich napędów o zmiennej prędkości są wysoka sprawność, wysoka gęstość mocy i gęstość momentu obrotowego. Podczas projektowania generatora wiatrowego z dwoma wirnikami i magnesami trwałymi konieczne jest rozwiązanie takiego problemu, jak prawidłowy dobór liczby biegunów i szczelin w celu zwiększenia wydajności i zminimalizowania kosztów maszyny. W niniejszej pracy zastosowano ulepszony algorytm optymalizacji hieny plamistej w celu uzyskania optymalnej kombinacji szczelin i biegunów. Ten algorytm optymalizacji umożliwia uzyskanie liczby ułamkowych szczelin na biegun i ocenę wydajności operacyjnej generatora wiatrowego z podwójnym wirnikiem i magnesami ferrytowymi. Na pierwszym etapie projektowania maszyny instalowane są różne kombinacje szczelin. Następnie wybierana jest optymalna kombinacja spośród różnych kombinacji szczelin i biegunów, biorąc pod uwagę algorytm Enhanced Spotted Hyena Optimization (ESHO) (ulepszony algorytm optymalizacjihieny cętkowanej hieny), w którym skonfigurowana jest funkcja wielocelowa. W związku z tym, celami wielozadaniowymi są integracja odwrotnej siły elektromotorycznej, wyjściowego momentu obrotowego, momentu obrotowego przekładni, połączenia strumienia, tętnienia momentu obrotowego wraz ze stratami. Analiza uzyskanych wyników pokazuje, że proponowany algorytm określania optymalnej kombinacji szczelin jest bardziej wydajny niż inne kombinacje szczelin. Stwierdzono również, że wybór kombinacji szczelin i biegunów ma kluczowe znaczenie dla wydajnej pracy maszyn z magnesami trwałymi

Preface. Bifurcations and Pattern Formation in Biological Applications

In the preface we present a short overview of articles included in the issue "Bifurcations and pattern formation in biological applications" of the journal Mathematical Modelling of Natural Phenomena

Periodic Homogenization and Material Symmetry in Linear Elasticity

Here homogenization theory is used to establish a connection between the symmetries of a periodic elastic structure associated with the microscopic properties of an elastic material and the material symmetries of the effective, macroscopic elasticity tensor. Previous results of this type exist but here more general symmetries on the microscale are considered. Using an explicit example, we show that it is possible for a material to be fully anisotropic on the microscale and yet the symmetry group on the macroscale can contain elements other than plus or minus the identity. Another example demon- strates that not all material symmetries of the macroscopic elastic tensor are generated by symmetries of the periodic elastic structure.Comment: 18 pages, 5 figure

Micromechanics of root development in soil

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Homogenization of Biomechanical Models for Plant Tissues

In this paper homogenization of a mathematical model for plant tissue biomechanics is presented. The microscopic model constitutes a strongly coupled system of reaction-diffusion-convection equations for chemical processes in plant cells, the equations of poroelasticity for elastic deformations of plant cell walls and middle lamella, and Stokes equations for fluid flow inside the cells. The chemical process in cells and the elastic properties of cell walls and middle lamella are coupled because elastic moduli depend on densities involved in chemical reactions, whereas chemical reactions depend on mechanical stresses. Using homogenization techniques we derive rigorously a macroscopic model for plant biomechanics. To pass to the limit in the nonlinear reaction terms, which depend on elastic strain, we prove the strong two-scale convergence of the displacement gradient and velocity field