A modification in an existing water uptake and solute transport numerical model was implemented in order to allow the model to simulate solute uptake by the roots. The convection-dispersion equation (CDE) was solved numerically, using a complete implicit scheme, considering a transient state for water and solute fluxes and a soil solute concentration dependent boundary for the uptake at the root surface, based on the Michaelis- Menten (MM) equation. Additionally, a linear approximation was developed for the MM equation such that the CDE has a linear and a non-linear solution. A radial geometry was assumed, considering a single root with its surface acting as the uptake boundary and the outer boundary being the half distance between neighboring roots, a function of root density. The proposed solute transport model includes active and passive solute uptake and predicts solute concentration as a function of time and distance from the root surface. It also estimates the relative transpiration of the plant, on its turn directly affecting water and solute uptake and related to water and osmotic stress status of the plant. Performed simulations show that the linear and non-linear solutions result in significantly different solute uptake predictions when the soil solute concentration is below a limiting value (Clim). This reduction in uptake at low concentrations may result in a further reduction in the relative transpiration. The contributions of active and passive uptake vary with parameters related to the ion species, the plant, the atmosphere and the soil hydraulic properties. The model showed a good agreement with an analytical model that uses a linear concentration dependent equation as boundary condition for uptake at the root surface. The advantage of the numerical model is it allows simulation of transient solute and water uptake and, therefore, can be used in a wider range of situations. Simulation with different scenarios and comparison with experimental results are needed to verify model performance and possibly suggest improvements. / Uma modificação em um modelo existente de extração de água e transporte de solutos foi realizada com o objetivo de incluir nele a possibilidade de simular a extração de soluto pelas raízes. Uma solução numérica para a equação de convecção-dispersão (ECD), que utiliza um esquema de resolução completamente implícito, foi elaborada e considera o fluxo transiente de água e solutos com uma condição de contorno à superfície da raiz de extração de soluto dependente de sua concentração no solo, baseada na equação de Michaelis- Menten (MM). Uma aproximação linear para a equação de MM foi implementada de tal forma que a ECD tem uma solução linear e outra não-linear. O modelo considera uma raiz singular com geometria radial sendo sua superfície a condição de contorno (limite) de extração e sendo o limite extremo a meia-distância entre raízes vizinhas, função da densidade radicular. O modelo de transporte de soluto proposto inclui extração de soluto ativa e passiva e prediz a concentração de soluto como uma função do tempo e da distância à superfície da raiz, além de estimar a transpiração relativa da planta, que por sua vez afeta a extração de água e solutos e é relacionado com a condição de estresse da planta. Simulações mostram que as soluções linear e não-linear resultam em predições de extração de solutos significativamente diferentes quando a concentração de solutos no solo está abaixo de um valor limitante (Clim). A redução da extração em baixas concentrações pode resultar em uma redução adicional na transpiração relativa. As contribuições ativa e passiva da extração de solutos variam com parâmetros relacionados à espécie de íon, à planta, à atmosfera e às propriedades hidráulicas do solo. O modelo apresentou uma boa concordância com um modelo analítico que aplica uma condição de contorno linear, à superfície da raiz, de extração de solutos dependente da concentração no solo. A vantagem do modelo numérico sobre o analítico é que ele permite simular fluxos transientes de água e solutos, sendo, portanto, possível simular uma maior gama de situações. Se faz necessário simulações com diferentes cenários e comparações com dados experimentais para se verificar a performance do modelo e, possivelmente, sugerir melhorias.

A modification in an existing water uptake and solute transport numerical model was implemented in order to allow the model to simulate solute uptake by the roots. The convection-dispersion equation (CDE) was solved numerically, using a complete implicit scheme, considering a transient state for water and solute fluxes and a soil solute concentration dependent boundary for the uptake at the root surface, based on the Michaelis- Menten (MM) equation. Additionally, a linear approximation was developed for the MM equation such that the CDE has a linear and a non-linear solution. A radial geometry was assumed, considering a single root with its surface acting as the uptake boundary and the outer boundary being the half distance between neighboring roots, a function of root density. The proposed solute transport model includes active and passive solute uptake and predicts solute concentration as a function of time and distance from the root surface. It also estimates the relative transpiration of the plant, on its turn directly affecting water and solute uptake and related to water and osmotic stress status of the plant. Performed simulations show that the linear and non-linear solutions result in significantly different solute uptake predictions when the soil solute concentration is below a limiting value (Clim). This reduction in uptake at low concentrations may result in a further reduction in the relative transpiration. The contributions of active and passive uptake vary with parameters related to the ion species, the plant, the atmosphere and the soil hydraulic properties. The model showed a good agreement with an analytical model that uses a linear concentration dependent equation as boundary condition for uptake at the root surface. The advantage of the numerical model is it allows simulation of transient solute and water uptake and, therefore, can be used in a wider range of situations. Simulation with different scenarios and comparison with experimental results are needed to verify model performance and possibly suggest improvements. / Uma modificação em um modelo existente de extração de água e transporte de solutos foi realizada com o objetivo de incluir nele a possibilidade de simular a extração de soluto pelas raízes. Uma solução numérica para a equação de convecção-dispersão (ECD), que utiliza um esquema de resolução completamente implícito, foi elaborada e considera o fluxo transiente de água e solutos com uma condição de contorno à superfície da raiz de extração de soluto dependente de sua concentração no solo, baseada na equação de Michaelis- Menten (MM). Uma aproximação linear para a equação de MM foi implementada de tal forma que a ECD tem uma solução linear e outra não-linear. O modelo considera uma raiz singular com geometria radial sendo sua superfície a condição de contorno (limite) de extração e sendo o limite extremo a meia-distância entre raízes vizinhas, função da densidade radicular. O modelo de transporte de soluto proposto inclui extração de soluto ativa e passiva e prediz a concentração de soluto como uma função do tempo e da distância à superfície da raiz, além de estimar a transpiração relativa da planta, que por sua vez afeta a extração de água e solutos e é relacionado com a condição de estresse da planta. Simulações mostram que as soluções linear e não-linear resultam em predições de extração de solutos significativamente diferentes quando a concentração de solutos no solo está abaixo de um valor limitante (Clim). A redução da extração em baixas concentrações pode resultar em uma redução adicional na transpiração relativa. As contribuições ativa e passiva da extração de solutos variam com parâmetros relacionados à espécie de íon, à planta, à atmosfera e às propriedades hidráulicas do solo. O modelo apresentou uma boa concordância com um modelo analítico que aplica uma condição de contorno linear, à superfície da raiz, de extração de solutos dependente da concentração no solo. A vantagem do modelo numérico sobre o analítico é que ele permite simular fluxos transientes de água e solutos, sendo, portanto, possível simular uma maior gama de situações. Se faz necessário simulações com diferentes cenários e comparações com dados experimentais para se verificar a performance do modelo e, possivelmente, sugerir melhorias.

In the first part of this thesis, we will explore an iterative procedure to determine the detailed asymptotic behaviour of solutions of a certain class of nonlinear vector differential equations which approach a nonlinear sink as time tends to infinity. This procedure is indifferent to resonance in the eigenvalues. Some attention will be given to finding approximations to solutions which are themselves flows. Moreover, we will address the writing of one component in terms of another in the case of a planar system. In the second part of this thesis, we will explore the Michaelis-Menten mechanism of a single enzyme-substrate reaction. The focus is an analysis of the planar reduction in phase space or, equivalently, solutions of the scalar reduction. In particular, we will prove the existence and uniqueness of a slow manifold between the horizontal and vertical isoclines. Also, we will determine the concavity of all solutions in the first quadrant. Moreover, we will establish the asymptotic behaviour of all solutions near the origin, which generally is not given by a Taylor series. Finally, we will determine the asymptotic behaviour of the slow manifold at infinity. Additionally, we will study the planar reduction. In particular, we will find non-trivial bounds on the length of the pre-steady-state period, determine the asymptotic behaviour of solutions as time tends to infinity, and determine bounds on the solutions valid for all time. In the third part of this thesis, we explore the (nonlinear) Lindemann mechanism of unimolecular decay. The analysis will be similar to that for the Michaelis-Menten mechanism with an emphasis on the differences. In the fourth and final part of this thesis, we will present some open problems.

In the first part of this thesis, we will explore an iterative procedure to determine the detailed asymptotic behaviour of solutions of a certain class of nonlinear vector differential equations which approach a nonlinear sink as time tends to infinity. This procedure is indifferent to resonance in the eigenvalues. Some attention will be given to finding approximations to solutions which are themselves flows. Moreover, we will address the writing of one component in terms of another in the case of a planar system. In the second part of this thesis, we will explore the Michaelis-Menten mechanism of a single enzyme-substrate reaction. The focus is an analysis of the planar reduction in phase space or, equivalently, solutions of the scalar reduction. In particular, we will prove the existence and uniqueness of a slow manifold between the horizontal and vertical isoclines. Also, we will determine the concavity of all solutions in the first quadrant. Moreover, we will establish the asymptotic behaviour of all solutions near the origin, which generally is not given by a Taylor series. Finally, we will determine the asymptotic behaviour of the slow manifold at infinity. Additionally, we will study the planar reduction. In particular, we will find non-trivial bounds on the length of the pre-steady-state period, determine the asymptotic behaviour of solutions as time tends to infinity, and determine bounds on the solutions valid for all time. In the third part of this thesis, we explore the (nonlinear) Lindemann mechanism of unimolecular decay. The analysis will be similar to that for the Michaelis-Menten mechanism with an emphasis on the differences. In the fourth and final part of this thesis, we will present some open problems.

abstract: Signaling cascades transduce signals received on the cell membrane to the nucleus. While noise filtering, ultra-sensitive switches, and signal amplification have all been shown to be features of such signaling cascades, it is not understood why cascades typically show three or four layers. Using singular perturbation theory, Michaelis-Menten type equations are derived for open enzymatic systems. When these equations are organized into a cascade, it is demonstrated that the output signal as a function of time becomes sigmoidal with the addition of more layers. Furthermore, it is shown that the activation time will speed up to a point, after which more layers become superfluous. It is shown that three layers create a reliable sigmoidal response progress curve from a wide variety of time-dependent signaling inputs arriving at the cell membrane, suggesting that natural selection may have favored signaling cascades as a parsimonious solution to the problem of generating switch-like behavior in a noisy environment. / Dissertation/Thesis / Ph.D. Applied Mathematics 2013

Success of hybrid poplar plantations will rely on the efficient management of nutrients and weeds. Relatively little is known about the root uptake characteristics of hybrid poplar and weeds, their belowground interactions and particularly, the quantitative understanding of nutrient uptake using mechanistic models under weed-competing conditions. Therefore, the objectives of this study were to investigate the effects of dandelion and quackgrass on the growth of hybrid poplar, to establish their root uptake characteristics and to quantify their nutrient uptake using the soil supply and nutrient demand (SSAND) model. In a pot study, hybrid poplar stem height, root collar diameter, shoot and root biomass, root length, and N, P and K uptake significantly decreased in the presence of dandelion and quackgrass weeds. Similar weed competition effects on growth of hybrid poplar were also observed in the field at the Pasture and Alfalfa sites where hybrid poplar was grown with and without weeds for 50, 79 and 100 days. In a hydroponic experiment, Imax values for NH4-N, NO3-N, P and K varied significantly among hybrid poplar seedlings and dandelion and quackgrass weed species and was greatest for dandelion followed by hybrid poplar and then quackgrass. The Km values were lowest for quackgrass compared to the other plant species for all of the nutrients. Simulation results from the SSAND model for the pot study showed that N uptake was underpredicted in hybrid poplar by 58 to 73%, depending upon soil type and weed treatment. Incorporation of N mineralization as a model input improve the hybrid poplar N uptake predictions by 24 and 67% in the Pasture and Alfalfa soil, respectively, when grown without weeds. SSAND model underestimated P uptake by 84-89% and overestimated K uptake by 28 to 59% for hybrid poplar depending upon the soil type and weed treatment. In the field, N uptake by hybrid poplar was in close agreement to measured N uptake in the control treatment. N uptake was greatly underestimated for both hybrid poplar and weeds in the weed treatment. Including changing water content greatly improves the N uptake by hybrid poplar and weeds in weed treatments. Results from this study suggest weed control is an essential practice to establish successful hybrid poplar plantations. Also, SSAND model can be an effective tool for predicting the nutrient uptake under two plant species competing environment if all the processes of nutrient supply are adequately described in the model.

Success of hybrid poplar plantations will rely on the efficient management of nutrients and weeds. Relatively little is known about the root uptake characteristics of hybrid poplar and weeds, their belowground interactions and particularly, the quantitative understanding of nutrient uptake using mechanistic models under weed-competing conditions. Therefore, the objectives of this study were to investigate the effects of dandelion and quackgrass on the growth of hybrid poplar, to establish their root uptake characteristics and to quantify their nutrient uptake using the soil supply and nutrient demand (SSAND) model. In a pot study, hybrid poplar stem height, root collar diameter, shoot and root biomass, root length, and N, P and K uptake significantly decreased in the presence of dandelion and quackgrass weeds. Similar weed competition effects on growth of hybrid poplar were also observed in the field at the Pasture and Alfalfa sites where hybrid poplar was grown with and without weeds for 50, 79 and 100 days. In a hydroponic experiment, Imax values for NH4-N, NO3-N, P and K varied significantly among hybrid poplar seedlings and dandelion and quackgrass weed species and was greatest for dandelion followed by hybrid poplar and then quackgrass. The Km values were lowest for quackgrass compared to the other plant species for all of the nutrients. Simulation results from the SSAND model for the pot study showed that N uptake was underpredicted in hybrid poplar by 58 to 73%, depending upon soil type and weed treatment. Incorporation of N mineralization as a model input improve the hybrid poplar N uptake predictions by 24 and 67% in the Pasture and Alfalfa soil, respectively, when grown without weeds. SSAND model underestimated P uptake by 84-89% and overestimated K uptake by 28 to 59% for hybrid poplar depending upon the soil type and weed treatment. In the field, N uptake by hybrid poplar was in close agreement to measured N uptake in the control treatment. N uptake was greatly underestimated for both hybrid poplar and weeds in the weed treatment. Including changing water content greatly improves the N uptake by hybrid poplar and weeds in weed treatments. Results from this study suggest weed control is an essential practice to establish successful hybrid poplar plantations. Also, SSAND model can be an effective tool for predicting the nutrient uptake under two plant species competing environment if all the processes of nutrient supply are adequately described in the model.

Vitamin K is a fat soluble compound which is synthesized by the gut microbiota and produced in many tissues within the body. Considering its role in the liver as a cofactor for gamma carboxylase enzymes, treatment of dark circles and pigments under the eye among others. It is clear that is some circumstances vitamin K has to cross biological barriers, particularly, when the vitamin is produced by microbiota in the intestine or applied topically on skin. Thus it is important to develop methods that allow studies of vitamin K permeability through the skin including its participation in redox reactions and transdermal permeability. Taking into account that transdermal permeability is strongly limited for high molecular weight compounds, i.e., compounds with higher than 500Da, the study was conducted with vitamin K of  lower molecular weight. Specifically vitamin K4 model, i.e., 1,4-dihydroxy-2 naphthoic acid, with molecular weight of 204g/mol. Vitamin K4 is suitable for this kind of study , because it can work as reducing (antioxidant) compound as well as has relatively beneficial physicochemical characteristics for transdermal permeability. Permeability studies were conducted with skin covered oxygen electrode and franz diffusion cell. Data from measurements were analyzed to estimate diffusion coefficients, apparent Michaelis-Menten constants and flux of a vitamin K4 model whilst contribution of different permeability pathways was determined theoretically.

Relative to other metals, little is known about lead toxicity in fish. The use of predictive models such as the biotic ligand model (BLM) has been limited, a situation that is at least partially due to the lack of understanding of lead's acute toxic mechanism and characterization of key binding sites involved in this toxicity. Using the rainbow trout as a model species, the acute toxic mechanism for lead was found to be ionoregulatory disruption. While having no apparent respiratory or acid/base effects, Pb exposure resulted in significant ionoregulatory impacts that affected Ca2+ homeostasis, as well as Na+ and Cl- balance. Active Ca2+ uptake by the gills obeyed typical Michaelis-Menten kinetics, and Pb interacted in a competitive fashion with the uptake process. Exposure to increasing waterborne Pb concentrations resulted in significant increases in Km value while Jmax showed little or no change. A slower, non-competitive interaction occurred after prolonged Pb-exposure, evidenced by a significant reduction of high-affinity Ca2+ -ATPase activity that correlated well with branchial Pb accumulation. Conversely, calcium had a protective effect against branchial Pb accumulation, this relationship being predominately competitive in nature. Voltage-independent calcium channel blockers La3+, Cd, and Zn significantly reduced gill Pb burden while the voltage-dependent, L-type calcium channel blockers, nifedipine and verapamil, did not, suggesting Pb enters fish by a similar mechanism to that of Ca2+. Stimulated stanniocalcin release by CaCl2 injection also significantly reduced branchial Pb accumulation. Based on the evidence presented in this thesis, it is apparent that acute Pb toxicity occurs by ionoregulatory disruption. It is likely that Pb shares a similar uptake pathway as that for Ca2+ and that resulting accumulation results in disruption of Ca2+ influx as well as Na+ and Cl- balance. This study has provided data essential to the characterization of key binding sites involved in Pb toxicity, and ultimately, validates the development and application of predictive models such as the BLM. / Thesis / Master of Science (MSc)

The flavonols are a varied and abundant sub-class of flavonoids that are associated with a number of essential physiological functions in plants and pharmacological activities in animals. The 2-oxoglutarate-dependant dioxygenases(2-ODDs), flavonol synthase (FLS) and flavanone 3-hydroxylase (F3H), are essential for flavonol synthesis. The primary goal of this study has been to gain a deeper understanding of the biochemistry of these enzymes in Arabidopsis. To accomplish this goal, an activity assay employing recombinant protein expression and HPLC as a detection system was developed for F3H and adapted for use with FLS. The assay was employed to establish the biochemical parameters of F3H from Arabidopsis, and to further characterize the F3H mutant allele, <i>tt6</i>(87). Enzymatic activity was demonstrated for F3H enzymes from <i>Ipomoea alba</i> (moonflower), <i>Ipomoea purpurea</i> (common morning glory), <i>Citrus sinensis</i> (sweet orange), and <i>Malus X domestica</i> (newton apple), each of which had previously been identified solely based on sequence homology. Arabidopsis contains six genes with high similarity to <i>FLS</i> from other plant species; however, all other central flavonoid pathway enzymes in Arabidopsis are encoded by single genes. The hypothesis that differential expression of FLS isozymes with varying substrate specificities is responsible for observed tissue-specific differences in flavonol accumulation was tested. Sequence analysis revealed that <i>AtFLS2, 4</i> and <i>6</i> contain premature stop codons that eliminate residues essential for enzyme activity. AtFLS1 was found to have a strong preference for dihydrokaempferol as a substrate. However, no enzyme activity was observed for AtFLS3 or AtFLS5 with a number of different substrates under a variety of reaction conditions. To identify structural elements that may contribute to the observed differences in biochemical activity, homology models for each of the isoforms were generated utilizing Arabidopsis anthocyanin synthase (ANS) as a template. A domain at the N-terminus of AtFLS1 that is missing in the other isozymes was insufficient to convey activity to an AtFLS1/5 chimera. These findings suggest a single catalytically-active form of FLS exists in Arabidopsis. The possibility that the apparently expressed but non-catalytic proteins, AtFLS2, 3, and 5, serve noncatalytic roles in flavonol production were explored by yeast 2-hybrid analysis. / Ph. D.