In this article a variational reduction method, how to handle the case of heterogenousdomains for the Transport equation, is presented. This method allows to get rid of therestrictions on the size of time steps due to the thin parts of the domain. In the thinpart of the domain, only a differential problem, with respect to the space variable, is tobe approximated numerically. Numerical results are presented with a simple example. Thevariational reduction method can be extended to thin domains multi-branching in 3dimensions, which is a work in progress.

The paper is devoted to the method of computer simulation of protein interactions takingpart in photosynthetic electron transport reactions. Using this method we have studiedkinetic characteristics of protein-protein complex formation for four pairs of proteinsinvolved in photosynthesis at a variety of ionic strength values. Computer simulationsdescribe non-monotonic dependences of complex formation rates on the ionic strength as theresult of long-range electrostatic interactions. Calculations confirm that the decrease inthe association second order rate constant at low values of the ionic strength is causedby the protein pairs spending more time in “wrong” orientations which do not satisfy thedocking conditions and so do not form the final complex capable of the electrontransfer.

A hybrid model of red blood cell production, where cells are considered as discreteobjects while intra-cellular proteins and extra-cellular biochemical substances aredescribed with continuous models, is proposed. Spatial organization and regulation of redblood cell production (erythropoiesis) are investigated. Normal erythropoiesis issimulated in two dimensions, and the influence on the output of the model of someparameters involved in cell fate (differentiation, self-renewal, and death by apoptosis)is studied.

Linear oscillators are used for modeling a diverse array of natural systems, for instance acoustics, materials science, and chemical spectroscopy. In this paper I describe simple models of structural interactions in biological molecules, known as elastic network models, as a useful topic for undergraduate biology instruction in mathematical modeling. These models use coupled linear oscillators to model the fluctuations of molecular structures around the equilibrium state. I present many learning activities associated with building and understanding these models, ranging from analytical to computational. I provide a number of web resources where students can obtain structural data, perform calculations, and suggest research directions for independent projects.

How does DNA, the molecule containing genetic information, change its three-dimensionalshape during the complex cellular processes of replication, recombination and repair? Thisis one of the core questions in molecular biology which cannot be answered without helpfrom mathematical modeling. Basic concepts of topology and geometry can be introduced inundergraduate teaching to help students understand counterintuitive complex structuraltransformations that occur in every living cell. Topoisomerases, a fascinating class ofenzymes involved in replication, recombination and repair, catalyze a change in DNAtopology through a series of highly coordinated mechanistic steps. Undergraduate biologyand mathematics students can visualize and explore the principles of topoisomerase actionby using easily available materials such as Velcro, ribbons, telephone cords, zippers andtubing. These simple toys can be used as powerful teaching tools to engage students inhands-on exploration with the goal of learning about both the mathematics and the biologyof DNA structure.

Since the release of the Bio 2010 report in 2003, significant emphasis has been placed on initiating changes in the way undergraduate biology and mathematics courses are taught and on creating new educational materials to facilitate those changes. Quantitative approaches, including mathematical models, are now considered critical for the education of the next generation of biologists. In response, mathematics departments across the country have initiated changes to their introductory calculus sequence, adding models, projects, and examples from the life sciences, or offering specialized calculus courses for biology majors. However, calculus-based models and techniques from those courses have been slow to propagate into the traditional general biology courses. And although modern biology has generated exciting opportunities for applications of a broad spectrum of mathematical theories, the impact on the undergraduate mathematics courses outside of the calculus/ordinary differential equations sequence has been minimal at best. Thus, the limited interdisciplinary cross-over between the undergraduate mathematics and biology curricula has remained stubbornly stagnant despite ongoing calls for integrated approaches.

We think that this phenomenon is due primarily to a lack of appropriate non-calculus-based interdisciplinary educational materials rather than inaccessibility of the essential underlying mathematical and biology concepts. Here we present a module that uses Boolean network models of the lac operon regulatory mechanism as an introduction to the conceptual importance of mathematical models and their analysis. No mathematics background beyond high school algebra is necessary to construct the model, which makes the approach particularly appropriate for introductory biology and mathematics courses. Initially the module focuses on modeling via Boolean logic, Boolean algebra, discrete dynamical systems, and directed graphs. The analysis of the model, however, leads to interesting advanced mathematical questions involving polynomial ideals and algebraic varieties that are beyond the mathematical proficiency of most biology students but are of interest in advanced level abstract algebra courses. These questions can also be used to map a path toward further research-level problems. All computations are carried out by computational algebra systems and the advanced mathematical theory implemented by the software can be covered in detail in abstract algebra courses with mathematics students. Biology students and students in lower level mathematic courses, on the other hand, can view the implementation as a “black box” and focus on the interpretation and the implications of the output.

The module is available from the authors for classroom testing and adoption.

A computer aided method using symbolic computations that enables the calculation of thesource terms (Boltzmann) in Grad’s method of moments is presented. The method is extremelypowerful, easy to program and allows the derivation of balance equations to very highmoments (limited only by computer resources). For sake of demonstration the method isapplied to a simple case: the one-dimensional stationary granular gas under gravity. Themethod should find applications in the field of rarefied gases, as well. Questions ofconvergence, closure are beyond the scope of this article.

Waxy Crude Oils (WCO’s) are characterized by the presence of heavy paraffins insufficiently large concentrations. They exhibit quite complex thermodynamical andrheological behaviour and present the peculiar property of giving rise to the formation ofsegregated wax deposits, when temperature falls down the so called WAT, or Wax AppearanceTemperature. In extreme cases, segregated waxes may lead to pipeline occlusion due todeposition on cold walls. In this paper we review the mathematical models formulated todescribe: (i) wax cystallization or thawing in cooling/heating cycles; (ii) the mechanismsof mass transport in saturated non-isothermal solutions; (iii) the experimental deviceused to measure wax solubility and wax diffusivity; (iv) wax deposition in pipelinescarrying a warm, wax-saturated WCO through cold regions; (v) wax deposition accompanied bygelification during the cooling of a WCO under a thermal gradient.

The development of functional-structural plant models has opened interesting perspectivesfor a better understanding of plant growth as well as for potential applications inbreeding or decision aid in farm management. Parameterization of such models is however adifficult issue due to the complexity of the involved biological processes and theinteractions between these processes. The estimation of parameters from experimental databy inverse methods is thus a crucial step. This paper presents some results anddiscussions as first steps towards the construction of a general framework for theparametric estimation of functional-structural plant models. A general family of models ofCarbon allocation formalized as dynamic systems serves as the basis for our study. Anadaptation of the 2-stage Aitken estimator to this family of model is introduced as wellas its numerical implementation, and applied in two different situations: first amorphogenetic model of sugar beet growth with simple plant structure, multi-stage anddetailed observations, and second a tree growth model characterized by sparse observationsand strong interactions between functioning and organogenesis. The proposed estimationmethod appears robust, easy to adapt to a wide variety of models, and generally provides asatisfactory goodness-of-fit. However, it does not allow a proper evaluation of estimationuncertainty. Finally some perspectives opened by the theory of hidden models arediscussed.

Atmospheric flow equations govern the time evolution of chemical concentrations in theatmosphere. When considering gas and particle phases, the underlying partial differentialequations involve advection and diffusion operators, coagulation effects, and evaporationand condensation phenomena between the aerosol particles and the gas phase. Operatorsplitting techniques are generally used in global air quality models. When consideringorganic aerosol particles, the modeling of the thermodynamic equilibrium of each particleleads to the determination of the convex envelope of the energy function. Two strategiesare proposed to address the computation of convex envelopes. The first one is based on aprimal-dual interior-point method, while the second one relies on a variationalformulation, an appropriate augmented Lagrangian, an Uzawa iterative algorithm, and finiteelement techniques. Numerical experiments are presented for chemical systems ofatmospheric interest, in order to compute convex envelopes in various spacedimensions.

We propose a novel approach to introducing hypothesis testing into the biologycurriculum. Instead of telling students the hypothesis and what kind of data to collectfollowed by a rigid recipe of testing the hypothesis with a given test statistic, we askstudents to develop a hypothesis and a mathematical model that describes the nullhypothesis. Simulation of the model under the null hypothesis allows students to comparetheir experimental data to what they would expect under the null hypothesis, thus leadingto a much more intuitive understanding of hypothesis testing. This approach has beentested both in the classroom and in faculty workshops, and we provide some suggestions forimplementations based on our experiences.

Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusionshould be nonlinear if there exist non-diagonal terms. The vast variety of nonlinearmulticomponent diffusion equations should be ordered and special tools are needed toprovide the systematic construction of the nonlinear diffusion equations formulticomponent mixtures with significant interaction between components. We develop anapproach to nonlinear multicomponent diffusion based on the idea of the reaction mechanismborrowed from chemical kinetics.

Chemical kinetics gave rise to very seminal tools for the modeling of processes. This isthe stoichiometric algebra supplemented by the simple kinetic law. The results of thisinvention are now applied in many areas of science, from particle physics to sociology. Inour work we extend the area of applications onto nonlinear multicomponent diffusion.

We demonstrate, how the mechanism based approach to multicomponent diffusion can beincluded into the general thermodynamic framework, and prove the corresponding dissipationinequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementaryprocess cannot have an arbitrary form. For the general kinetic law (the generalized MassAction Law), additional conditions are proved. The cell–jump formalism gives anintuitively clear representation of the elementary transport processes and, at the sametime, produces kinetic finite elements, a tool for numerical simulation.

An exploratory study is performed to investigate the use of a time-dependent discreteadjoint methodology for design optimization of a high-lift wing configuration augmentedwith an active flow control system. The location and blowing parameters associated with aseries of jet actuation orifices are used as design variables. In addition, a geometricparameterization scheme is developed to provide a compact set of design variablesdescribing the wing shape. The scaling of the implementation is studied using severalthousand processors and it is found that asynchronous file operations can greatly improvethe overall performance of the approach in such massively parallel environments. Threedesign examples are presented which seek to maximize the mean value of the liftcoefficient for the coupled system, and results demonstrate improvements as high as 27%relative to the lift obtained with non-optimized actuation. This lift gain is more thanthree times the incremental lift provided by the non-optimized actuation.

This paper proposes a quantitative model of the reaction-diffusion type to examine thedistribution of interferon-α (IFNα) in a lymph node(LN). The numerical treatment of the model is based on using an original unstructured meshgeneration software Ani3D and nonlinear finite volume method for diffusion equations. Thestudy results in suggestion that due to the variations in hydraulic conductivity ofvarious zones of the secondary lymphoid organs the spatial stationary distribution ofIFNα is essentially heterogeneous across the organs. Highly protecteddomains such as sinuses, conduits, co-exist with the regions in which where the stationaryconcentration of IFNα is lower by about 100-fold. This is the first studywhere the spatial distribution of soluble immune factors in secondary lymphoid organs ismodelled for a realistic three-dimensional geometry.

We have developed a chemical kinetics simulation that can be used as both an educationaland research tool. The simulator is designed as an accessible, open-source project thatcan be run on a laptop with a student-friendly interface. The application can potentiallybe scaled to run in parallel for large simulations. The simulation has been successfullyused in a classroom setting for teaching basic electrochemical properties. We have shownthat this can be used for simulating fundamental molecular and chemical processes and evensimplified models of predator–prey interactions. By giving the simulated entities spatialextent in the lattice, the particles do not interpenetrate, and clusters of particles canspatially exclude one another. Our simulation demonstrates that spatial inhomogeneityleads to different results than those that are obtained by using standard ordinarydifferential equation models, as previously reported.

Physics based simulation is widely seen as a way of increasing the information aboutaircraft designs earlier in their definition, thus helping with the avoidance ofunanticipated problems as the design is refined. This paper reports on an effort to assessthe automated use of computational fluid dynamics level aerodynamics for the developmentof tables for flight dynamics analysis at the conceptual stage. These tables are then usedto calculate handling qualities measures. The methodological questions addressed area)geometry and mesh treatment for automated analysis from a high level conceptual aircraftdescription and b) sampling and data fusion to allow the timely calculation of large datatables. The test case used to illustrate the approaches is based on a refined designpassenger jet wind tunnel model. This model is reduced to a conceptual description, andthe ability of this geometry to allow calculations relevant to the final design to bedrawn is then examined. Data tables are then generated and handling qualitiescalculated.

We consider a dense granular shear flow in a two-dimensional system. Granular systems(composed of a large number of macroscopic particles) are far from equilibrium due toinelastic collisions between particles: an external driving is needed to maintain themotion of particles. Theoretical description of driven granular media is especiallychallenging for dense granular flows. This paper focuses on a gravity-driven densegranular Poiseuille flow in a channel. A special focus here is on the intriguingphenomenon of fluid-solid coexistence: a solid plug in the center of the system,surrounded by fluid layers. To find and analyze various flow regimes, a multi-scaleapproach is taken. On macro scale, granular hydrodynamics is employed. On micro scale,event-driven molecular dynamics simulations are performed. The entire phase diagram ofparameters is explored, in order to determine which flow regime occurs in various regionsin the parameter space.

Theoretical studies and numerical experiments suggest that unstructured high-ordermethods can provide solutions to otherwise intractable fluid flow problems within complexgeometries. However, it remains the case that existing high-order schemes are generallyless robust and more complex to implement than their low-order counterparts. These issues,in conjunction with difficulties generating high-order meshes, have limited the adoptionof high-order techniques in both academia (where the use of low-order schemes remainswidespread) and industry (where the use of low-order schemes is ubiquitous). In this shortreview, issues that have hitherto prevented the use of high-order methods amongst anon-specialist community are identified, and current efforts to overcome these issues arediscussed. Attention is focused on four areas, namely the generation of unstructuredhigh-order meshes, the development of simple and efficient time integration schemes, th edevelopment of robust and accurate shock capturing algorithms, and finally the developmentof high-order methods that are intuitive and simple to implement. With regards to thisfinal area, particular attention is focused on the recently proposed flux reconstructionapproach, which allows various well known high-order schemes (such as nodal discontinuousGalerkin methods and spectral difference methods) to be cast within a single unifyingframework. It should be noted that due to the experience of the authors the review isdirected somewhat towards aerodynamic applications and compressible flow. However, many ofthe discussions have a wider applicability. Moreover, the tone of the review is intendedto be generally accessible, such that an extended scientific community can gain insightinto factors currently pacing the adoption of unstructured high-order methods.

There are two mathematical models of elastic walls of healthy and atherosclerotic bloodvessels developed and studied. The models are included in a numerical model of globalblood circulation via recovery of the vessel wall state equation. The joint model allowsus to study the impact of arteries atherosclerotic disease of a set of arteries onregional haemodynamics.