Exercise sheet 3

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MA311exam 1998 pdf

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MA311exam 2000

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MA160exam00pdf

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A simple theoretical model for the intensification of tropical cyclones and polar lows

A simple theoretical model for the intensification of tropical cyclones and polar lows is developed using a minimal set of physical assumptions. These disturbances are assumed to be balanced systems intensifying through the WISHE (Wind-Induced Surface Heat Exchange) intensification mechanism, driven by surface fluxes of heat and moisture into an atmosphere which is neutral to moist convection. The equation set is linearized about a resting basic state and solved as an initial-value problem. A system is predicted to intensify with an exponential perturbation growth rate scaled by the radial gradient of an efficiency parameter which crudely represents the effects of unsaturated processes. The form of this efficiency parameter is assumed to be defined by initial conditions, dependent on the nature of a pre-existing vortex required to precondition the atmosphere to a state in which the vortex can intensify. Evaluation of the simple model using a primitive-equation, nonlinear numerical model provides support for the prediction of exponential perturbation growth. Good agreement is found between the simple and numerical models for the sensitivities of the measured growth rate to various parameters, including surface roughness, the rate of transfer of heat and moisture from the ocean surface, and the scale for the growing vortex.

A mathematical model of the in vitro keratinocyte response to chromium and nickel exposure

A mathematical model describing the main mechanistic processes involved in keratinocyte response to chromium and nickel has been developed and compared to experimental in vitro data. Accounting for the interactions between the metal ions and the keratinocytes, the law of mass action was used to generate ordinary differential equations which predict the time evolution and ion concentration dependency of keratinocyte viability, the amount of metal associated with the keratinocytes and the release of cytokines by the keratinocytes. Good agreement between model predictions and existing experimental data of these endpoints was observed, supporting the use of this model to explore physiochemical parameters that influence the toxicological response of keratinocytes to these two metals.

Mathematical model for low density lipoprotein (LDL) endocytosis by hepatocytes

Individuals with elevated levels of plasma low density lipoprotein (LDL) cholesterol (LDL-C) are considered to be at risk of developing coronary heart disease. LDL particles are removed from the blood by a process known as receptor-mediated endocytosis, which occurs mainly in the liver. A series of classical experiments delineated the major steps in the endocytotic process; apolipoprotein B-100 present on LDL particles binds to a specific receptor (LDL receptor, LDL-R) in specialized areas of the cell surface called clathrin-coated pits. The pit comprising the LDL-LDL-R complex is internalized forming a cytoplasmic endosome. Fusion of the endosome with a lysosome leads to degradation of the LDL into its constituent parts (that is, cholesterol, fatty acids, and amino acids), which are released for reuse by the cell, or are excreted. In this paper, we formulate a mathematical model of LDL endocytosis, consisting of a system of ordinary differential equations. We validate our model against existing in vitro experimental data, and we use it to explore differences in system behavior when a single bolus of extracellular LDL is supplied to cells, compared to when a continuous supply of LDL particles is available. Whereas the former situation is common to in vitro experimental systems, the latter better reflects the in vivo situation. We use asymptotic analysis and numerical simulations to study the longtime behavior of model solutions. The implications of model-derived insights for experimental design are discussed.

Mathematical modelling of competitive LDL/VLDL binding and uptake by hepatocytes

Elevated levels of low-density-lipoprotein cholesterol (LDL-C) in the plasma are a well-established risk factor for the development of coronary heart disease. Plasma LDL-C levels are in part determined by the rate at which LDL particles are removed from the bloodstream by hepatic uptake. The uptake of LDL by mammalian liver cells occurs mainly via receptor-mediated endocytosis, a process which entails the binding of these particles to specific receptors in specialised areas of the cell surface, the subsequent internalization of the receptor-lipoprotein complex, and ultimately the degradation and release of the ingested lipoproteins' constituent parts. We formulate a mathematical model to study the binding and internalization (endocytosis) of LDL and VLDL particles by hepatocytes in culture. The system of ordinary differential equations, which includes a cholesterol-dependent pit production term representing feedback regulation of surface receptors in response to intracellular cholesterol levels, is analysed using numerical simulations and steady-state analysis. Our numerical results show good agreement with in vitro experimental data describing LDL uptake by cultured hepatocytes following delivery of a single bolus of lipoprotein. Our model is adapted in order to reflect the in vivo situation, in which lipoproteins are continuously delivered to the hepatocyte. In this case, our model suggests that the competition between the LDL and VLDL particles for binding to the pits on the cell surface affects the intracellular cholesterol concentration. In particular, we predict that when there is continuous delivery of low levels of lipoproteins to the cell surface, more VLDL than LDL occupies the pit, since VLDL are better competitors for receptor binding. VLDL have a cholesterol content comparable to LDL particles; however, due to the larger size of VLDL, one pit-bound VLDL particle blocks binding of several LDLs, and there is a resultant drop in the intracellular cholesterol level. When there is continuous delivery of lipoprotein at high levels to the hepatocytes, VLDL particles still out-compete LDL particles for receptor binding, and consequently more VLDL than LDL particles occupy the pit. Although the maximum intracellular cholesterol level is similar for high and low levels of lipoprotein delivery, the maximum is reached more rapidly when the lipoprotein delivery rates are high. The implications of these results for the design of in vitro experiments is discussed.

A mathematical model of crystallization in an emulsion

A mathematical model incorporating many of the important processes at work in the crystallization of emulsions is presented. The model describes nucleation within the discontinuous domain of an emulsion, precipitation in the continuous domain, transport of monomers between the two domains, and formation and subsequent growth of crystals in both domains. The model is formulated as an autonomous system of nonlinear, coupled ordinary differential equations. The description of nucleation and precipitation is based upon the Becker–Döring equations of classical nucleation theory. A particular feature of the model is that the number of particles of all species present is explicitly conserved; this differs from work that employs Arrhenius descriptions of nucleation rate. Since the model includes many physical effects, it is analyzed in stages so that the role of each process may be understood. When precipitation occurs in the continuous domain, the concentration of monomers falls below the equilibrium concentration at the surface of the drops of the discontinuous domain. This leads to a transport of monomers from the drops into the continuous domain that are then incorporated into crystals and nuclei. Since the formation of crystals is irreversible and their subsequent growth inevitable, crystals forming in the continuous domain effectively act as a sink for monomers “sucking” monomers from the drops. In this case, numerical calculations are presented which are consistent with experimental observations. In the case in which critical crystal formation does not occur, the stationary solution is found and a linear stability analysis is performed. Bifurcation diagrams describing the loci of stationary solutions, which may be multiple, are numerically calculated.

Mathematical model for NF-kappaB-driven proliferation of adult neural stem cells

Neural stem cells (NSCs) are early precursors of neuronal and glial cells. NSCs are capable of generating identical progeny through virtually unlimited numbers of cell divisions (cell proliferation), producing daughter cells committed to differentiation. Nuclear factor kappa B (NF-kappaB) is an inducible, ubiquitous transcription factor also expressed in neurones, glia and neural stem cells. Recently, several pieces of evidence have been provided for a central role of NF-kappaB in NSC proliferation control. Here, we propose a novel mathematical model for NF-kappaB-driven proliferation of NSCs. We have been able to reconstruct the molecular pathway of activation and inactivation of NF-kappaB and its influence on cell proliferation by a system of nonlinear ordinary differential equations. Then we use a combination of analytical and numerical techniques to study the model dynamics. The results obtained are illustrated by computer simulations and are, in general, in accordance with biological findings reported by several independent laboratories. The model is able to both explain and predict experimental data. Understanding of proliferation mechanisms in NSCs may provide a novel outlook in both potential use in therapeutic approaches, and basic research as well.

Exploratory semiparametric analysis of two-dimensional diffusions in finance

We examine bivariate extensions of Aït-Sahalia’s approach to the estimation of univariate diffusions. Our message is that extending his idea to a bivariate setting is not straightforward. In higher dimensions, as opposed to the univariate case, the elements of the Itô and Fokker-Planck representations do not coincide; and, even imposing sensible assumptions on the marginal drifts and volatilities is not sufficient to obtain direct generalisations. We develop exploratory estimation and testing procedures, by parametrizing the drifts of both component processes and setting restrictions on the terms of either the Itô or the Fokker-Planck covariance matrices. This may lead to highly nonlinear ordinary differential equations, where the definition of boundary conditions is crucial. For the methods developed, the Fokker-Planck representation seems more tractable than the Itô’s. Questions for further research include the design of regularity conditions on the time series dependence in the data, the kernels actually used and the bandwidths, to obtain asymptotic properties for the estimators proposed. A particular case seems promising: “causal bivariate models” in which only one of the diffusions contributes to the volatility of the other. Hedging strategies which estimate separately the univariate diffusions at stake may thus be improved.

Modelagem dinâmica e simulação computacional de poços de petróleo verticais e direcionais com elevação por bombeio mecânico

LINS, Filipe C. A. et al. Modelagem dinâmica e simulação computacional de poços de petróleo verticais e direcionais com elevação por bombeio mecânico. In: CONGRESSO BRASILEIRO DE PESQUISA E DESENVOLVIMENTO EM PETRÓLEO E GÁS, 5. 2009, Fortaleza, CE. Anais... Fortaleza: CBPDPetro, 2009.

Modelagem e simulação de um sistema de bombeio mecânico em poços direcionais utilizando parâmetros concentrados

This work aims presenting the development of a model and computer simulation of a sucker rod pumping system. This system take into account the well geometry, the flow through the tubing, the dynamic behavior of the rod string and the use of a induction motor model. The rod string were modeled using concentrated parameters, allowing the use of ordinary differential equations systems to simulate it s behavior