In this work we deal with the design of the robust feedback control ofwastewater treatmentsystem, namely the activated sludge process. This problem is formulated by anonlinearordinary differential system. On one hand, we develop a robust analysis when thespecific growthfunction of the bacterium μ is not well known. On the other hand, when alsothe substrate concentrationin the feed stream s in is unknown, we provide an observer of system andpropose a designof robust feedback control in term of recycle rate, in order to keep thepollutant concentration lowerthan an allowed maximum level s d .

Two-dimensional inviscid channel flow of an incompressible fluid is considered. It isshown that if the flow is steady and features no horizontal stagnation, then the flow mustnecessarily be a parallel shear flow.

The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vriesequations over a slowly varying random bottom is rigorously studied. One motivation forstudying such a system is better understanding the unidirectional motion of interactingsurface and internal waves for a fluid system that is formed of two immiscible layers. Itwas shown recently by Craig-Guyenne-Sulem [1] thatin the regime where the internal wave has a large amplitude and a long wavelength, thedynamics of the surface of the fluid is described by the Schrödinger equation, while thatof the internal wave is described by the Korteweg-de Vries equation. The purpose of thisletter is to show that in the presence of a slowly varying random bottom, the coupledwaves evolve adiabatically over a long time scale. The analysis covers the cases when thesurface wave is a stable bound state or a long-lived metastable state.

In this paper, we propose a computational model to investigate the coupling between cell’s adhesions and actin fibres and how this coupling affects cell shape and stability. To accomplish that, we take into account the successive stages of adhesion maturation from adhesion precursors to focal complexes and ultimately to focal adhesions, as well as the actin fibres evolution from growing filaments, to bundles and finally contractile stress fibres.

We use substrates with discrete patterns of adhesive patches, whose inter-patches distance can be modulated in order to control the location of the adhesions and the resulting fibres architecture. We then investigate the emergence of stable cell morphologies as a function of the inter-patches distance, for two different cell phenotypes generated from the model. Force generated by the stress fibres on the focal adhesions and specifically the influence of the cell contractility are also investigated.

Our results suggest that adhesion lifetime and fibre growing rate are the key parameters in the emergence of stable cell morphologies and the limiting factors for the magnitude of the mean tension force from the fibres on the focal adhesions.

In this work we discuss and analyze spiking patterns in a generic mathematical model oftwo coupled non-identical nonlinear oscillators supplied with a spike-timing dependentplasticity (STDP) mechanism. Spiking patterns in the system are shown to converge to aphase-locked state in a broad range of parameters. Precision of the phase locking, i.e.the amplitude of relative phase deviations from a given reference, depends on the naturalfrequencies of oscillators and, additionally, on parameters of the STDP law. Thesedeviations can be optimized by appropriate tuning of gains (i.e. sensitivity tospike-timing mismatches) of the STDP mechanisms. The deviations, however, can not be madearbitrarily small neither by mere tuning of STDP gains nor by adjusting synaptic weights.Thus if accurate phase-locking in the system is required then an additional tuningmechanism is generally needed. We found that adding a very simple adaptation dynamics inthe form of slow fluctuations of the base line in the STDP mechanism enables accuratephase tuning in the system with arbitrary high precision. The scheme applies to systems inwhich individual oscillators operate in the oscillatory mode. If the dynamics ofoscillators becomes bistable then relative phase may fail to converge to a given valuegiving rise to the emergence of complex spiking sequences.

In this paper, an SIS (susceptible-infected-susceptible)-type epidemic propagation isstudied on a special class of 3-regular graphs, called modified cycle graphs. The modifiedcycle graph is constructed from a cycle graph with N nodes by connecting nodei to thenode i +d in a way that every node has exactly three links.Monte-Carlo simulations show that the propagation process depends on the value ofd in anon-monotone way. A new theoretical model is developed to explain this phenomenon. Thisreveals a new relation between the spreading process and the average path length in thegraph.

In various biological systems and small scale technological applications particles transiently bind to a cylindrical surface. Upon unbinding the particles diffuse in the vicinal bulk before rebinding to the surface. Such bulk-mediated excursions give rise to an effective surface translation, for which we here derive and discuss the dynamic equations, including additional surface diffusion. We discuss the time evolution of the number of surface-bound particles, the effective surface mean squared displacement, and the surface propagator. In particular, we observe sub- and superdiffusive regimes. A plateau of the surface mean-squared displacement reflects a stalling of the surface diffusion at longer times. Finally, the corresponding first passage problem for the cylindrical geometry is analysed.

Dynamics of the nonlinear Schrödinger equation in the presence of a constant electric field is studied. Both discrete and continuous limits of the model are considered. For the discrete limit, a probabilistic description of subdiffusion is suggested and a subdiffusive spreading of a wave packet is explained in the framework of a continuous time random walk. In the continuous limit, the biased nonlinear Schrödinger equation is shown to be integrable, and solutions in the form of the Painlevé transcendents are obtained.

Propagation of nonlinear baroclinic Kelvin waves in a rotating column of uniformlystratified fluid under the Boussinesq approximation is investigated. The model isconstrained by the Kelvin’s conjecture saying that the velocity component normal to theinterface between rotating fluid and surrounding medium (e.g. a seashore) is possibly zeroeverywhere in the domain of fluid motion, not only at the boundary. Three classes ofdistinctly different exact solutions for the nonlinear model are obtained. The obtainedsolutions are associated with symmetries of the Boussinesq model. It is shown that oneclass of the obtained solutions can be visualized as rotating whirlpools along which thepressure deviation from the mean state is zero, is positive inside and negative outside ofthe whirlpools. The angular velocity is zero at the center of the whirlpools and it ismonotonically increasing function of radius of the whirlpools.

In this paper we present a fluid-structure interaction model of neuron’s membranedeformation. The membrane-actin is considered as an elastic solid layer, while thecytoplasm is considered as a viscous fluid one. The membrane-actin layer is governed byelasticity equations while the cytoplasm is described by the Navier-Stokes equations. Atthe interface between the cytoplasm and the membrane we consider a match between the solidvelocity displacement and the fluid velocity as well as the mechanical equilibrium. Themembrane, which faces the extracellular medium, is free to move. This will change thegeometry in time. To take into account the deformation of the initial configuration, weuse the Arbitrary Lagrangian Eulerian method in order to take into account the meshdisplacement. The numerical simulations, show the emergence of a filopodium, a typicalstructure in cells undergoing deformation.

Plants and animals have highly ordered structure both in time and in space, and one of the main questions of modern developmental biology is the transformation of genetic information into the regular structure of organism. Any multicellular plant begins its development from the universal unicellular state and acquire own species-specific structure in the course of cell divisions, cell growth and death, according to own developmental program. However the cellular mechanisms of plant development are still unknown. The aim of this work was to elaborate and verify the formalistic approach, which would allow to describe and analyze the large data of cellular architecture obtained from the real plants and to reveal the cellular mechanisms of their morphogenesis. Two multicellular embryos of Calla palustris L. (Araceae) was used as a model for the verification of our approach. The cellular architecture of the embryos was reconstructed from the stack of optical and serial sections in three dimensions and described as graphs of genealogy and space adjacency of cells. In result of the comparative analysis of these graphs, a set of regular cell types and highly conservative pattern of cell divisions during five cell generations were found. This mechanism of cellular development of the embryos could be considered as a developmental program, set of rules or grammars applied to the zygote. Also during the comparative analysis the finite plasticity in cell adjacency was described. The structural equivalence and the same morphogenetic potencies of some cells of the embryos were considered as the space-temporal symmetries. The symmetries were represented as a set of regular cell type permutations in the program of development of the embryo cellular architecture. Two groups of cell type permutations were revealed, each was composed of two elements and could be interpreted as the mirror and rotational space symmetries. The results obtained as well as the developed approach can be used in plant tissue modelling based on the real, large and complex structural data.

In mammalian cells, the p53 pathway regulates the response to a variety of stresses, including oncogene activation, heat and cold shock, and DNA damage. Here we explore a mathematical model of this pathway, composed of a system of partial differential equations. In our model, the p53 pathway is activated by a DNA-compromising event of short duration. As is typical for mathematical models of the p53 pathway, our model contains a negative feedback loop representing interactions between the p53 and Mdm2 proteins. A novel feature of our model is that we combine a spatio-temporal approach with the appearance and repair of DNA damage. We investigate the behaviour of our model through numerical simulations. By ignoring the possibility of DNA repair, we first explore the scenario in which the cell has a very inefficient DNA repair mechanism. We find that spatio-temporal oscillations in p53 and Mdm2 may occur, consistent with experimental data. We then allow p53 to be directly involved in repairing DNA damage, since experimental evidence suggests this can happen. We find that oscillations in p53 and Mdm2 can still occur, but their amplitude damps down quickly as the DNA damage is repaired. Finally, we find that a minor change to the location of the DNA damage can notably change the spatial distribution of p53 within the nucleus. We discuss the biological implications of our results.

The stochastic dynamics of chemical reactions can be accurately described by chemicalmaster equations. An approximated time-evolution equation of the Langevin type has beenproposed by Gillespie based on two explicit dynamical conditions. However, whennumerically solve these chemical Langevin equations, we often have a small stopping time–atime point of having an unphysical solution–in the case of low molecular numbers. Thispaper proposes an approach to simulate stochasticities in chemical reactions withdeterministic delay differential equations. We introduce a deterministic Brownian motiondescribed by delay differential equations, and replace the Gaussian noise in the chemicalLangevin equations by the solutions of these deterministic equations. This modificationcan largely increase the stopping time in simulations and regain the accuracy as in thechemical Langevin equations. The novel aspect of the present study is to apply thedeterministic Brownian motion to chemical reactions. It suggests a possible direction ofdeveloping a hybrid method of simulating dynamic behaviours of complex gene regulationnetworks.

This paper demonstrates the development of a simple model of carbon flow during plant growth. The model was developed by six undergraduate students and their instructor as a project in a plant ecophysiology course. The paper describes the structure of the model including the equations that were used to implement it in Excel®, the plant growth experiments that were conducted to obtain information for parameterizing and testing the model, model performance, student responses to the modeling project, and potential uses of the model by other students.

We are concerned with the proof of a generalized solution to an ill-posed variational inequality. This is determined as a solution to an appropriate minimization problem involving a nonconvex functional, treated by an optimal control technique.

Cancer treatment using the antiangiogenic agents targets the evolution of the tumor vasculature. The aim is to significantly reduce supplies of oxygen and nutrients, and thus starve the tumor and induce its regression. In the paper we consider well established family of tumor angiogenesis models together with their recently proposed modification, that increases accuracy in the case of treatment using VEGF antibodies. We consider the optimal control problem of minimizing the tumor volume when the maximal admissible drug dose (the total amount of used drug) and the final level of vascularization are also taken into account. We investigate the solution of that problem for a fixed therapy duration. We show that the optimal strategy consists of the drug-free, full-dose and singular (with intermediate values of the control variable) intervals. Moreover, no bang-bang switch of the control is possible, that is the change from the no-dose to full-dose protocol (or in opposite direction) occurs on the interval with the singular control. For two particular models, proposed by Hahnfeldt et al. and Ergun et al., we provide additional theorems about the optimal control structure. We investigate the optimal controls numerically using the customized software written in MATLAB®, which we make freely available for download. Utilized numerical scheme is based on the composition of the well known gradient and shooting methods.

We construct interfacial solitary structures (spots) generated by a bistable chemicalreaction or a non-equilibrium phase transition in a surfactant film. The structures arestabilized by Marangoni flow that prevents the spread of a state with a higher surfacetension when it is dynamically favorable. In a system without surfactant massconservation, a unique radius of a solitary spot exists within a certain range of valuesof the Marangoni number and of the deviation of chemical potential from the Maxvellconstruction, but multiple spots attract and coalesce. In a conservative system, there isa range of stable spot sizes, but solitary spots may exist only in a limited parametricrange, beyond which multiple spots nucleate. Repeated coalescence and nucleation leads tochaotic dynamics of spots observed computationally in Ref. [1].

Viruses are obligate intracellular parasites that rely on the host cell for expansion.With the development of global analyses techniques like transcriptomics, proteomics andsiRNA library screening of complete cellular gene sets, a large range of host cell factorshave been discovered that either support or restrict virus growth. Here we summarize someof the recent findings and focus our discussion on the hepatitis C virus and the humanimmunodeficiency virus, two major pathogens that threat global health. The identificationof cellular proteins affecting multiple viruses points to the existence of centralregulation nodes that might be exploited for both, a quantitative description ofhost-virus interactions within single infected cells and the development of novel,broad-spectrum antiviral drugs.

Epidemiologic data suggest that schools and daycare facilities likely play a major rolein the dissemination of influenza. Pathogen transmission within such small, inhomogenouslymixed populations is difficult to model using traditional approaches. We developedsimulation based mathematical tool to investigate the effects of social contact networkson pathogen dissemination in a setting analogous to a daycare center or grade school. Herewe show that interventions that decrease mixing within child care facilities, includinglimiting the size of social clusters, reducing the contact frequency between socialclusters, and eliminating large gatherings, could diminish pathogen dissemination.Moreover, these measures may amplify the effectiveness of vaccination or antiviralprophylaxis, even if the vaccine is not uniformly effective or antiviral compliance isincomplete. Similar considerations should apply to other small, imperfectly mixedpopulations, such as offices and schools.