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We analyze a simplified mathematical model of the complete degradation of monochlorophenol. The model takes form of a system of six ordinary differential equations, the dynamics of which can be reduced to the dynamics of a three-dimensional system on the invariant set. We extend the previous analysis by considering multiple substrate inflow. We also focus on the bifurcations occurring in the system and their biological meaning. / Thesis / Master of Science (MSc)

Circadian rhythms are periodic physiological cycles that recur about every 24 hours, by means of which organisms integrate their physiology and behavior to the daily cycle of light and temperature imposed by the rotation of the earth. Circadian derives from the Latin word circa "about" and dies "day". Circadian rhythms have three noteworthy properties. They are endogenous, that is, they persist in the absence of external cues (in an environment of constant light intensity, temperature, etc.). Secondly, they are temperature compensated, that is, the nearly 24 hour period of the endogenous oscillator is remarkably independent of ambient temperature. Finally, they are phase shifted by light. The circadian rhythm can be either advanced or delayed by applying a pulse of light in constant darkness. Consequently, the circadian rhythm will synchronize to a periodic light-dark cycle, provided the period of the driving stimulus is not too far from the period of the endogenous rhythm. A window on the molecular mechanism of 24-hour rhythms was opened by the identification of circadian rhythm mutants and their cognate genes in Drosophila, Neurospora, and now in other organisms. Since Konopka and Benzer first discovered the period mutant in Drosophila in 1971 (Konopka and Benzer, 1971), there have been remarkable developments. Currently, the consensus opinion of molecular geneticists is that the 24-hour period arises from a negative feedback loop controlling the transcription of clock genes. However, a better understanding of this mechanism requires an approach that integrates both mathematical and molecular biology. From the recent discoveries in molecular biology and through a mathematical approach, we propose that the mechanism of circadian rhythm is based upon the combination of both negative and positive feedback. / Master of Science

Submitted by Irene Nascimento (irene.kessia@ufpe.br) on 2016-08-24T17:57:59Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) dissertaçãosuper_final_(1).pdf: 1088878 bytes, checksum: f1d95f7419b99281751c7ea750e47cf8 (MD5) / Made available in DSpace on 2016-08-24T17:57:59Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) dissertaçãosuper_final_(1).pdf: 1088878 bytes, checksum: f1d95f7419b99281751c7ea750e47cf8 (MD5) Previous issue date: 2011-08-31 / CAPES / Nesta dissertação é apresentada uma solução estacionária para um modelo de dinâmica populacional de uma única espécie, considerando a dispersão da população num espaço heterogêneo e um crescimento logístico da população. No primeiro capítulo, para dar ao leitor alguma intimidade com os conceitos apresentados estudamos alguns modelos de dinâmica populacional de uma única espécie. Referimo-nos a uma única população para dizer que não analisamos aqui a interação entre diversas espécies. No segundo capítulo concentra-se a parte substancial do nosso trabalho. Na seção 1 apresentamos o modelo, na seção 2 apresentamos a solução estacionária para o problema e na seção 3 fazemos uma discussão sobre efeitos de isolamento para uma população. / This thesis presents a stationary solution to a model of population dynamics of a single species, considering the dispersion of biological population in a heterogeneous space and a logistic population growth. In the rst chapter, to give the reader some familiarity with the concepts presented study some models of population dynamics of a single species. We refer to a single population to say we do not analyze the interaction between di erent species. The second chapter focuses on the substantial part of our work. In Section 1 presents the problem and the model, section 2 presents the stationary solution to the problem and in Section 3 we make a discussion about isolation e ects on a population

Centrosomin (Cnn) is an integral centrosomal protein in Drosophila with orthologues in several species, including humans. The human orthologue of Cnn is required for brain development with Cnn hypothesised to play a similar role in Drosophila. Control of Cnn incorporation into centrosomes is crucial for controlling asymmetric division in certain types of Drosophila stem cells. FRAP experiments on Cnn show that Cnn recovers in a pe- culiar fashion, which suggest that Cnn may be incorporated closest to the centrioles and then spread radially outward, either diffusively or ad- vectively. The aim of this thesis is to understand the mechanism of Cnn incorporation into the Drosophila centrosomes, to determine the mode of transport of the incorporated Cnn, and to obtain parameter estimates for transport and biochemical reactions. A crucial unknown in the modelling process is the distribution of Cnn receptors. We begin by constructing coupled partial differential equation models with either diffusion or advection as the mechanism for incorpo- rated Cnn transport. The simplest receptor distribution we begin with involves a spherical, infinitesimally thick, impermeable shell. We refine the diffusion models using the insights gained from comparing the model out- put with data (gathered during mitosis) and through careful assessment of the behaviour of the data. We show that a Gaussian receptor distribution is necessary to explain the Cnn FRAP data and that the data cannot be explained by other simpler receptor distributions. We predict the exact form of the receptor distribution through data fitting and present pre- liminary experimental results from our collaborators that suggest that a protein called DSpd2 may show a matching distribution. Not only does this provide strong experimental support for a key prediction from our model, but it also suggests that DSpd2 acts as a Cnn receptor. We also show using the mitosis FRAP data that Cnn does not exhibit appreciable radial movement during mitosis, which precludes the use of these data to distinguish between diffusive and advective transport of Cnn. We use long time Cnn FRAP data gathered during S-phase for this purpose. We fit the S-phase FRAP data using the DSpd2 profiles gath- ered for time points corresponding to the Cnn FRAP experiments. We also use data from FRAP experiments where colchicine is injected into the embryos to destroy microtubules (since microtubules are suspected to play a role in advective transport of Cnn). From the analysis of all these data we show that Cnn is transported in part by advection and in part by diffusion. Thus, we are able to provide the first mechanistic description of the Cnn incorporation process. Further, we estimate parameters from the model fitting and predict how some of the parameters may be altered as nuclei progress from S-phase to mitosis. We also generate testable predic- tions regarding the control of the Cnn incorporation process. We believe that this work will be useful to understand the role of Cnn incorporation in centrosome function, particularly in asymmetrically dividing stem cells.

Parkinson's disease (PD) is an incurable neurodegenerative motor disorder caused by the inexorable loss of dopamine neurones from the substantia nigra pars compacta. Cell loss is characterised by the perturbation of multiple physiological processes (including mitochondrial function, autophagy and dopamine homeostasis) and much of this pathophysiology can be reproduced in vitro using the mitochondrial toxin MPP+ (1-methyl-4-phenylpyridinium). It was hypothesised that MPP+ toxicity could be modelled using protein-protein interaction networks (PPIN) in order to better understand the interplay of systems-level processes that result in eventual cell death in MPP+ models and PD. Initially, MPP+ toxicity was characterised in the human, dopamine-producing cell line BE(2)-M17 and it was confirmed that the neurotoxin resulted in time and dose dependent apoptosis. A radio-label pulse-chase assay was developed and demonstrated that MPP+ induced decreased autophagic flux preceded cell death. Autophagic dysfunction was consistent with lysosome deacidification due to cellular ATP depletion. Pertinent PPINs were sampled from publically available data using a seedlist of proteins with validated roles in MPP+ toxicity. These PPINs were subjected to a series of analyses to identify potential therapeutic targets. Two topological methods based on betweenness centrality were used to identify target proteins predicted to be critical for the crosstalk between mitochondrial dysfunction and autophagy in the context of MPP+ toxicity. Combined knockdown of a subset of target proteins potentiated MPP+ toxicity and the combined resulted in cellular rescue. Neither of these effects was observed following single knockdown/overexpression confirming the need for multiple interventions. Cellular rescue occurred via an autophagic mechanism; prominent autophagosomes were formed and it was hypothesised that these structures allowed for the sequestration of damaged proteins. This thesis demonstrates the value of PPINs as a model for Parkinson's disease, from network creation through target identification to phenotypic benefit.

Following trauma, such as an impact injury or stroke, brain tissue can swell. Swelling is the result of water accumulation in the tissue that is driven by pathological changes, such as increased permeability of the capillary walls and osmotic pressure changes within the tissue. Swelling causes increased intracranial pressure and mechanical deformation of the brain tissue, exacerbating the original injury. Furthermore, prolonged local swelling can lead to the spread of damage to the (initially undamaged) surrounding tissue, since compression and increased intracranial pressure may restrict blood flow in this tissue. In this thesis, we develop mathematical models to examine the consequences of pathophysiological damage mechanisms on the swelling, and associated stress and strain, experienced by brain tissue. Mixture theory is used to represent brain tissue as a mixture of elastic solid, fluid and solutes. This modelling approach allows elastic deformations to be coupled with hydrodynamic pressure and osmotic gradients; the consequences of different mechanisms of damage may then be quantified. We consider three particular problems motivated by experimental observations of swelling brain tissue. Firstly, we investigate the swelling of isolated, damaged, brain tissue slices; we show that mechanisms leading to an osmotic pressure difference between the tissue slice and its surroundings can explain experimental observations for swollen tissue slices. Secondly, we use our modelling approach to demonstrate that local changes in capillary permeability can cause significant stresses and strains in the surrounding tissue. Thirdly, we investigate the conditions under which a locally swollen, damaged, region can cause compression of the vasculature within the surrounding tissue, and potentially result in damage propagation. To do this, we propose a coupled model for the oxygen concentration within, and mechanical deformation of, brain tissue. We use our model to assess the impact of treatment strategies on damage propagation through the tissue, and show that performing a craniectomy reduces the extent of propagation.

This thesis is concerned with investigating new methods for the analysis of large-scale dynamical systems using convex optimization. The proposed methodology is based on composite Lyapunov theory and is computationally implemented using polynomial programming techniques. The main result of this work is the development of a system decomposition framework that makes it possible to analyze systems that are of such a scale that traditional methods cannot cope with. We begin by addressing the problem of model invalidation. A barrier certificate method for invalidating models in the presence of uncertain data is presented for both continuous and discrete time models. It is shown how a re-parameterization of the time dependent variables can improve the numerical conditioning of the underlying optimization problem. The main contribution of this thesis is the development of an automated dynamical system decomposition framework that permits us to verify the stability of systems that typically have a state dimension large enough to render traditional computational methods intractable. The underlying idea is to decompose a system into a set of lower order subsystems connected in feedback in such a manner that composite methods for stability verification may be employed. What is unique about the algorithm presented is that it takes into account both dynamics and the topology of the interconnection graph. In the first instance we illustrate the methodology with an ecological network and primal Internet congestion control scheme. The versatility of the decomposition framework is also highlighted when it is shown that when applied to a model of the EGF-MAPK signaling pathway it is capable of identifying biologically relevant subsystems in addition to stability verification. Finally we introduce stability metrics for interconnected dynamical systems based on the theory of dissipativity. We conclude by outlining a clustering based decomposition algorithm that explicitly takes into account the input and output dynamics when determining the system decomposition.

Understanding the perfusion of blood and drugs in tumours is fundamental to foreseeing the efficacy of treatment regimes and predicting tumour growth. In particular, the dependence of these processes on the tumour vascular structure is poorly established. The objective of this thesis is to derive effective equations describing blood, and drug perfusion in vascular tumours, and specifically to determine the dependence of these on the tumour vascular structure. This dependence occurs through the interaction between two different length scales - that which characterizes the structure of the vascular network, and that which characterizes the tumour as a whole. Our method throughout is to use homogenization as a tool to evaluate this interaction. In Chapter 1 we introduce the problem. In Chapter 2 we develop a theoretical model to describe fluid flow in solid tumours through both the vasculature and the interstitium, at a number of length scales. Ultimately we homogenize over a network of capillaries to form a coupled porous medium model in terms of a vascular density. Whereas in Chapter 2 it is necessary to specify the vascular structure to derive the effective equations, in Chapter 3 we employ asymptotic homogenization through multiple scales to derive the coupled equations for an arbitrary periodic vascular network. In Chapter 4, we extend this analysis to account for advective and diffusive transport of anticancer drugs delivered intravenously; we consider a range of reaction properties in the interstitium and boundary conditions on the vascular wall. The models derived in Chapters 2–4 could be applied to any drug type and treatment regime; to demonstrate their potential, we simulate the delivery of vinblastine in dorsal skinfold chambers in Chapter 5 and make quantitative predictions regarding the optimal treatment regime. In the final Chapter we summarize the main results and indicate directions for further work.