Esse é um trabalho teórico e experimental em que princípios básicos de mecânica estatística são utilizados para entender a dinâmica de micro e nano esferas acopladas direta e indiretamente a células aderentes vivas, objetivando a caracterização mecânica das mesmas. Dentre esses princípios básicos estão inclusos, principalmente, conceitos relacionados à difusão. Na difusão clássica, tem-se uma dependência linear do deslocamento quadrático médio com o tempo. Caso contrário, quando o expoente é diferente de um, tem-se o que se chama de difusão anômala. Caso seja maior que um, o processo é superdifusivo, e se menor que um, subdifusivo. Para se estudar o comportamento mecânico de sistemas complexos pode-se usar micro e nanoesferas como elementos de análise. Essas esferas são dispostas no material a ser estudado, e observando a sua dinâmica é possível caracterizar o processo que conduziu essa dinâmica e consequentemente inferir propriedades físicas do material. Nesse trabalho aplicam-se técnicas de rastreamento de partículas, microscópicas e nanoscópicas, para estudar propriedades dinâmica de células, especialmente difusibilidade, remodelação da estrutura celular e campos de força. Para isso foram utilizadas duas técnicas experimentais de rastreamento de micro e nanoesferas e modelos fenomenológicos e de mecânica estatística. Essas propriedades dinâmicas tem uma grande semelhança com materiais vítreos moles. Nesse contexto, certas funções celulares, como divisão, contração, difusão, requerem que as células apresentem fluidez similarmente a um líquido, enquanto que para outras funções, como manter a sua estrutura celular, elas devam ter uma aparência mais rígida. Essas características assemelham-se a um material vítreo, onde desordem e metastabilidade são características subjacentes de suas funções mecânicas. Os resultados experimentais apresentados aqui evidenciam essa metaestabilidade na forma de anomalias e correlações temporais dos vários dados coletados. Também explicamos os dados experimentais encontrados em termos das atividades metabólicas e a remodelação ativa do citoesqueleto. Mostra-se também os dados obtidos para músculo cardíaco em plena atividade pulsátil. Os resultados aqui obtidos têm aplicações diretas em pesquisa básica e clínica. / This is a theoretical and experimental work in which basic principles of statistical mechanics are used to understand the dynamics of micro and nano spheres attached directly or indirectly to living adherent cells, with the aim of the mechanical characterization of them. Among these basic principles, mainly concepts related to diffusion are included. In classical diffusion, there is a linear dependence of the mean squared displacement in time. Otherwise, when the exponent is diferent than one , there is what is called anomalous diffusion. If it is bigger than one, the process is superdiffusive, and if it is smaller than one, subdiffusive. To study the mechanical behavior of complex systems,micro and nanospheres can be used as analysis elements. These spheres are arranged in the material to be studied, and from observation of the dynamics is possible to characterize the leading process of this dynamic and therefore infer physical properties of the material. In this work, particle tracking techniques, for microscopic and nanoscopic spheres, are applied to study dynamic properties of cells, especially diffusivity, remodeling of the cell structure and force fields. For that we used two experimental techniques of tracking of micro and nanospheres, and phenomenological and statistical mechanics models. These dynamic properties have a great similarity to soft glassy materials. In this context, certain cellular functions such as division, contraction, diffusion, require that cells present fluidity similarly to a liquid, while for other functions, such as keeping the cellular structure, they should have a stiffer appearance. These characteristics resemble a glassy material, where disorder and metastability are underlying characteristics of their mechanical functions. The experimental results presented here show this metastability as anomalies and temporal correlations of the various data collected. We also explain the experimental data found in terms of metabolic activity and the active remodeling of the cytoskeleton. Also data obtained for heart muscle in full pulsatile activity is showed. The results obtained have direct applications in basic and clinical research.

Piecewise-smooth stochastic systems are widely used in engineering science. However, the theory of these systems is only in its infancy. In this thesis, we take as an example the Brownian motion with dry friction to illustrate dynamical properties of these systems with respect to three interesting topics: (i) weak-noise approximations, (ii) first-passage time (FPT) problems and (iii) functionals of stochastic processes. Firstly, we investigate the validity and accuracy of weak-noise approximations for piecewise-smooth stochastic differential equations (SDEs), taking as an illustrative example the Brownian motion with pure dry friction. For this model, we show that the weak-noise approximation of the path integral correctly reproduces the known propagator of the SDE at lowest order in the noise power, as well as the main features of the exact propagator with higher-order corrections, provided that the singularity of the path integral is treated with some heuristics. We also consider a smooth regularisation of this piecewise-constant SDE and study to what extent this regularisation can rectify some of the problems encountered in the non-smooth case. Secondly, we provide analytic solutions to the FPT problem of the Brownian motion with dry friction. For the pure dry friction case, we find a phase transition phenomenon in the spectrum which relates to the position of the exit point and affects the tail of the FPT distribution. For the model with dry and viscous friction, we evaluate quantitatively the impact of the corresponding stick-slip transition and of the transition to ballistic exit. We also derive analytically the distributions of the maximum velocity till the FPT for the dry friction model. Thirdly, we generalise the so-called backward Fokker-Planck technique and obtain a recursive ordinary differential equation for the moments of functionals in the Laplace space. We then apply the developed results to analyse the local time, the occupation time and the displacement of the dry friction model. Finally, we conclude this thesis and state some related unsolved problems.

Lam, Yu Fung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 40-43). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Methodology --- p.6 / Chapter 2.1 --- Brownian Motion --- p.6 / Chapter 2.2 --- Reflected Brownian Motion --- p.8 / Chapter 2.3 --- Partially Reflected Brownian Motion --- p.11 / Chapter 3 --- Numerical Analysis --- p.15 / Chapter 3.1 --- Comparison of RBM and PRBM --- p.15 / Chapter 3.2 --- Initial state far from the boundaries --- p.17 / Chapter 3.3 --- Initial state close to a boundary --- p.17 / Chapter 4 --- A Study of the USD/HKD Exchange Rate --- p.23 / Chapter 4.1 --- Data Description --- p.23 / Chapter 4.2 --- Testing for the Mean-reverting Property --- p.25 / Chapter 4.3 --- Testing for Decreasing Volatility near the Boundaries --- p.27 / Chapter 4.4 --- Estimation Results --- p.27 / Chapter 5 --- Conclusion --- p.31 / Chapter A --- Derivation of MLE estimator --- p.33 / Chapter B --- Numerical Laplace Inversion --- p.35 / Chapter C --- Augmented Dickey-Fuller Test --- p.39 / Bibliography --- p.40

Every cell contains several millions of diffusing and reacting biological molecules. The interactions between these molecules ultimately manifest themselves in all aspects of life, from the smallest bacterium to the largest whale. One of the greatest open scientific challenges is to understand how the microscopic chemistry determines the macroscopic biology. Key to this challenge is the development of mathematical and computational models of biochemistry with molecule-level detail, but which are sufficiently coarse to enable the study of large systems at the cell or organism scale. Two such models are in common usage: the reaction-diffusion master equation, and Brownian dynamics. These models are utterly different in both their history and in their approaches to chemical reactions and diffusion, but they both seek to address the same reaction-diffusion question. Here we make an in-depth study into the physical validity of these models under various biological conditions, determining when they can reliably be used. Taking each model in turn, we propose modifications to the models to better model the realities of the cellular environment, and to enable more efficient computational implementations. We use the models to make predictions about how and why cells behave the way they do, from mechanisms of self-organisation to noise reduction. We conclude that both models are extremely powerful tools for clarifying the details of the mysterious relationship between chemistry and biology.

One central question of neuroscience asks how a neural system can generate the diversity of complex behaviors needed to meet the range of possible demands placed on an organism by an ever changing environment. In many cases, it appears that animals assemble complex behaviors by recombining sets of simpler behaviors known as behavioral primitives. The crawling behavior of the nematode worm Caenorhabditis elegans represents a classic example of such an approach since worms use the simple behaviors of forward and reverse locomotion to assemble more complex behaviors such as search and escape. The relative simplicity and well-described anatomy of the worm nervous system combined with a high degree of genetic tractability make C. elegans an attractive organism with which to study the neural circuits responsible for assembling behavioral primitives into complex behaviors. Unfortunately, difficulty probing the physiological properties of central synapses in C. elegans has left this opportunity largely unfulfilled. In this dissertation we address this challenge by developing techniques that combine whole-cell patch clamp recordings with optical stimulation of neurons. We do this using transgenic worms that express the light-sensitive ion channel Channelrhodopsin-2 (ChR2) in putative pre-synaptic neurons and fluorescent protein reporters in the post-synaptic neurons to be targeted by electrodes. We first apply this new approach to probe C. elegans circuitry in chapter II where we test for connectivity between nociceptive neurons known as ASH required for sensing aversive stimuli, and premotor neurons required for generating backward locomotion, known as AVA. In chapter III we extend our analysis of the C. elegans locomotory circuit to the premotor neurons required for generating forward locomotion, known as AVB. We identify inhibitory synaptic connectivity between ASH and AVB and between the two types of premotor neurons, AVA and AVB. Finally, we use our observations to develop a biophysical model of the locomotory circuit in which switching emerges from the attractor dynamics of the network. Primitive selection in C. elegans may thus represent an accessible system to test kinetic theories of decision making. This dissertation includes previously published co-authored material.

In this thesis, we study the relationship between the geometric shape of a region in the plane, and certain probabilistic information about the behavior of Brownian particles inside the region. The probabilistic information is contained in the function h(r), called the harmonic measure distribution function. Consider a domain Ω in the plane, and fix a basepoint z0. Imagine lining the boundary of this domain with fly paper and releasing a million fireflies at the basepoint z0. The fireflies wander around inside this domain randomly until they hit a wall and get stuck in the fly paper. What fraction of these fireflies are stuck within a distance r of their starting point z0? The answer is given by evaluating our h-function at this distance; that is, it is given by h(r). In more technical terms, the h-function gives the probability of a Brownian first particle hitting the boundary of the domain Ω within a radius r of the basepoint z0. This function is dependent on the shape of the domain Ω, the location of the basepoint z0, and the radius r. The big question to consider is: How much information does the h-function contain about the shape of the domain’s boundary? It is known that an h-function cannot uniquely determine a domain, but is it possible to construct a domain that generates a given hfunction? This is the question we try to answer. We begin by giving some examples of domains with their h-functions, and then some examples of sequences of converging domains whose corresponding h-functions also converge to the h-function. In a specific case, we prove that artichoke domains converge to the wedge domain, and their h-functions also converge. Using another class of approximating domains, circle domains, we outline a method for constructing bounded domains from possible hfunctions f(r). We prove some results about these domains, and we finish with a possible for a proof of the convergence of the sequence of domains constructed.

The use of computational simulation to study the dynamics and interactions of macromolecules has become an important tool in the field of biochemistry. A common method to perform these simulations is to use all-atom explicit-solvent molecular dynamics (MD). However, due to the limitations in computational power currently available, this method is not practical for simulating large-scale biomolecular systems on long timescales. An alternative is to perform implicit-solvent Brownian dynamics (BD) simulations using a coarse grained (CG) model that allows for increased computational efficiency. However, if simulations using the CG model are not realistic, then the gain in computational efficiency from using a CG model is not worthwhile. This thesis describes the derivation of a set of bonded and nonbonded CG potential functions for use in implicit-solvent BD simulations of proteins derived from all-atom explicit-solvent MD simulations of amino acids. To determine which force field and water model to use in the MD simulations, Chapter II describes 1 Μs all-atom explicit-solvent MD simulations of glycine, asparagine, phenylalanine, and valine solutions at 50, 100, 200 and 300 mg/ml concentrations performed using eight different force field and water model combinations. To evaluate the accuracy of the force fields at high solute concentrations, the density, viscosity, and dielectric increments of the four amino acids were calculated from the simulations and compared to experimental results. Additionally, the change in the strength of hydrophobic and electrostatic interactions with increasing solute concentration was calculated for each force field and water model combination. As a result of this study, the Amber ff99SB-ILDN force field and TIP4P-Ew explicit-solvent water model were chosen for all subsequent MD simulations. Chapter III describes the derivation of CG bonded potential functions from 1 Μs all-atom explicit-solvent MD simulations of each of the twenty amino acids, including a separate simulation for protonated histidine. The angle and dihedral probability distributions sampled during the MD simulations were used to optimize the bonded potential functions using the iterative Boltzmann inversion (IBI) method. Chapter IV describes the derivation of CG nonbonded potential functions from 1 Μs all-atom explicit-solvent MD simulations of every possible pairing of the amino acids (231 different systems). The radial distribution functions calculated from these MD simulations were used to optimize a set of nonbonded CG potential functions using the IBI method. The optimized set of bonded and nonbonded potential functions, which is termed COFFDROP (COarse-grained Force Field for Dynamic Representation Of Proteins), quantitatively reproduced all of the calculated MD distributions. To determine if COFFDROP would be useful for simulations of bimolecular systems, Chapter V describes the testing of the transferability of the force field. First, COFFDROP was used to simulate concentrated amino acid solutions. The clustering of the solutes in these simulations was directly compared with results from corresponding all-atom explicit-solvent MD simulations and found to be in excellent agreement. Next, BD simulations of 9.2 mM solutions of the small protein villin headpiece were performed. The proteins aggregated during these simulations, which is in agreement with results from MD simulation but in disagreement with experiment. After scaling the strength of COFFDROP's nonbonded potential functions by a factor of 0.8 and rerunning the BD simulations, the amount of aggregation was comparable to experimental observations. Based on these results, COFFDROP is likely to be applicable in CG BD simulations of large, highly concentrated, biomolecular systems.

No description available.

<p>Lévy flights and Lévy walks are two mathematical models used to describe anomalous diffusion(i.e. those having mean square displacements nonlinearly related to time (as opposed to Brownian motion)). Lévy flights follow probability distributions p(|<strong>r</strong>|) yielding infinite mean square displacements since some rare steps are very long. Lévy walks, however, have coupled space-time probability distributions penalising very long steps. Both Lévy flights and Lévy walks are dominated by a few long steps, but most steps are much, much smaller. The semi-experimental part ofthis work dealt with how fluorescent probes moved in systems of cationic starch and latex/solutions of dodecyl trimethyl ammonium bromide, respectively. Visually, no Lévy walks couldbe detected. However, mathematical regression suggested enhanced diffusion and subdiffusion. Moreover, time-dependent diffusion coefficients were calculated. Also examined was how Microsoft Excel could be used to generate normal diffusion as well as anomalous diffusion.</p> / <p>Lévyflygningar och Lévypromenader är matematiska modeller som används för att beskriva anomal diffusion (i.e. dessa då medelvärdet av kvadratförflyttningarna är icke-linjärt relaterat tilltiden (till skillnad från Brownsk rörelse)). Lévyflygningar följer sannolikhetsfördelningar p(|<strong>r</strong>|)som ger oändliga medelkvadratförflyttningar eftersom vissa steg är väldigt långa. Lévypromenader,å andra sidan, har kopplade rum-tid-sannolikhetsfördelningar som kraftigt reducerar demycket långa stegen. Både Lévyflygningar och -promenader domineras av ett fåtal långa steg ävenom de flesta steg är mycket, mycket mindre. Den semiexperimentella delen av detta arbetestuderade hur fluorescerande prober rör sig i katjonisk stärkelse respektive latex/lösningar avdodecyltrimetylammoniumbromid. Inga Lévypromenader kunde ses. Emellertid taladematematisk regression för att superdiffusion och subdiffusion förelåg. Tidsberoende diffusionskoefficienter beräknades också. I detta arbete undersöktes även hur Microsoft Excel kan användas för att generera både normal och anomal diffusion.</p>

by Ioannis Karatzas and Steven E. Shreve. / "June 1985." This report constitutes the first three chapters of a book to be published by Springer-Verlag. / Includes bibliography. / ARO Grant DAAG-29-84-K-005