Starting from a classical mechanical model, a set of Langevin equations for the surface diffusion of adsorbed molecules is developed. In contrast to previous work, these Langevin equations take full account of the rotations and internal vibrations of the adsorbed molecule. These equations are then applied to a stiff dimer diffusing in one dimension, and the results compared with previous calculations for the same system. It is shown that the modifications in our new approach give significantly different results than this previous calculation, and therefore must be taken into account in future calculations for systems of this kind. Next a new approximation method is developed by assuming that the motion of the molecule is confined to the lowest energy path between adsorption sites. This method is applicable to an arbitrarily complex molecule, and is complimentary to the first method, in that it can account for deformation of the molecule by the surface but not the internal vibrations of the molecule (whereas the first method accounts for internal vibrations but not deformation). This approximation method is then applied to a flexible dimer in two dimensions (one dimension along the surface and one perpendicular). The results are discussed and compared with those of the stiff dimer in one dimension, explaining and clarifying the difference between our results and those of previous calculations.

<p>Cells express a variety of proteins on their surface that allows them to sample the world. These proteins are embedded in the plasma membrane, a bilayer of lipids that surrounds the cell. Since the lipid and protein dimensions are in the nanometer range, they are subject to thermal agitation by water molecules and show characteristic diffusive motion. The diffusive movement of these proteins plays a critical role in the cell's ability to react to external signals and regulate its internal environment. </p><p>One prominent application of protein diffusion is in the synaptic connection, where is the highly localized concentration of receptors. The receptive dendrite membrane contains many types of receptors that are accumulated to form functional microdomains opposite the presynaptic terminal buttons that release neurotransmitters. Experiments reveal that receptors move from extrasynaptic locations to synaptic locations by lateral diffusion, thereby concentrating receptors at synapses. Two key processes that control synaptic AMPAR numbers are receptor diffusion within the synaptic and extrasynaptic space and interactions between receptors and PSD scaffold proteins. Electron microscopy images suggest that the PSD is highly crowded potentially limiting the ability of receptors to diffuse and interact with scaffold proteins. However, the contribution of macromolecular crowding to receptor retention remains to be tested systematically. </p><p>Here, we combine experimental and computational approaches to test the effect of synaptic steric hindrance on receptor mobility and enrichment. We first investigate how the diffusion is influenced by membrane geometry. The membrane itself can have three-dimensional structure, which means that the actual path length of diffusion can be different from a projected path length. Here, we use a position Langevin equation for diffusion, which incorporates curvature and gradient effects of surfaces. Numeric simulation of the equation allows for the prediction of effective diffusion coefficients over corrugated surfaces.</p><p>In order to examine the distinct contributions of crowding and receptor-scaffold binding, we developed a computational model for AMPA-receptor diffusion in the synaptic and extrasynaptic space, which contains immobile obstacles, representing scaffolding, receptor and adhesion molecules in the PSD. The spatial distribution of scaffold proteins was determined directly from photo-activated localization microscopy measurements that mapped molecular positions with a resolution of ~20 nm. The AMPAR/scaffold association and dissociation rates were adjusted by computer simulations to fit single-particle tracking and fluorescence recovery after photobleaching measurements. The model predicts the recovery curves are influenced mostly by size changes while variation of kinetic rates did not significantly alter receptor residence time or mobility. We also examined the effect of binding, by adding a single synaptic binding motif to a small transmembrane protein, which slows its diffusion within the synapse. These results suggest that both protein size and binding play important roles in retaining surface-diffusing TM proteins within the excitatory synapse and shed light on the biophysical mechanisms that lead to high density of AMPARs at synapses.</p> / Dissertation

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In this thesis we concentrate on instances of the GLE which can be represented in a Markovian form in an extended phase space. We extend previous results on the geometric ergodicity of this class of GLEs using Lyapunov techniques, which allows us to conclude ergodicity for a large class of GLEs relevant to molecular dynamics applications. The main body of this thesis concerns the numerical discretisation of the GLE in the extended phase space representation. We generalise numerical discretisation schemes which have been previously proposed for the underdamped Langevin equation and which are based on a decomposition of the vector field into a Hamiltonian part and a linear SDE. Certain desirable properties regarding the accuracy of configurational averages of these schemes are inherited in the GLE context. We also rigorously prove geometric ergodicity on bounded domains by showing that a uniform minorisation condition and a uniform Lyapunov condition are satisfied for sufficiently small timestep size. We show that the discretisation schemes which we propose behave consistently in the white noise and overdamped limits, hence we provide a family of universal integrators for Langevin dynamics. Finally, we consider multiple-time stepping schemes making use of a decomposition of the fluctuation-dissipation term into a reversible and non-reversible part. These schemes are designed to efficiently integrate instances of the GLE whose Markovian representation involves a high number of auxiliary variables or a configuration dependent fluctuation-dissipation term. We also consider an application of dynamics based on the GLE in the context of large scale Bayesian inference as an extension of previously proposed adaptive thermostat methods. In these methods the gradient of the log posterior density is only evaluated on a subset (minibatch) of the whole dataset, which is randomly selected at each timestep. Incorporating a memory kernel in the adaptive thermostat formulation ensures that time-correlated gradient noise is dissipated in accordance with the fluctuation-dissipation theorem. This allows us to relax the requirement of using i.i.d. minibatches, and explore a variety of minibatch sampling approaches.

Orientador: Roberto E. Lagos Monaco / Banca: José Antonio Roversi / Banca: Bernardo Laks / Resumo: Neste trabalho estudamos o modelo do movimento Browniano de uma partícula carregada sob a ação de campos elétrico e magnético, externos e homogêneos, no formalismo de Langevin. Calculamos a energia cinética média através do teorema da flutuação-dissipação e obtivemos uma expressão para a temperatura efetiva das partículas Brownianas em função da temperatura do reservatório e dos campos externos. Esta temperatura efetiva mostrou-se sempre maior que a temperatura do reservatório, o que explica a expressão "portadores quentes". Estudamos essa temperatura efetiva no regime assintótico, ou seja, no estado estacionário atingido em tempos muito longos (quando comparado com o tempo de colisão) e a utilizamos para escrever as equações de transporte em semicondutores, denominadas equações de Shockley generalizadas sendo que incluem nesse caso também a ação do campo magnético. Uma aplicação direta e relevante foi a modelagem para o já conhecido efeito Gunn para portadores assumidos como Brownianos. A temperatura efetiva calculada por nós no regime transiente permitiu estudar também os efeitos do reservatório na relaxação da temperatura efetiva à temperatura terminal (de não equilíbrio e estacionária). Nossos resultados no que diz respeito ao efeito Gunn, embora seja o modelo mais simples de um portador Browniano, mostrou uma surpreendente concordância com resultados experimentais, sugerindo que modelos mais sofisticados devam incluir os elementos apresentados neste estudo / Abstract: We present a Brownian model for a charged particle in a field of forces, in particular, electric and magnetic external homogeneous fields, within the Langevin formalism. We compute the average kinetic energy via the fluctuation dissipation and obtain an expression for the Brownian particle's effective temperature. The latter is a function of the heat bath temperature and both external fields. This effective temperature is always greater than the heat bath temperature, therefore the expression "hot carriers". This effective temperature, in the asymptotic regime, the stationary state at long times (greater than the collision time), is used to write down the transport equations for semiconductors, namely the generalized Shockley equations, now incorporating the magnetic field effect. A direct and relevant application follows: a model for the well known Gunn effect, assuming a Brownian scheme. In the transient regime the computed effective temperature also allow us to probe some features of the heat bath, as the effective temperature relaxes to its terminal stationary value. As for our results in the Gunn effect model, the simplest of all in a Brownian scheme, we obtain a surprisingly good agreement with experimental data, suggesting that more involved models should include our minimal assumptions / Mestre

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Helium-3 surface spin echo spectroscopy (HeSE) has been used to measure the diffusive dynamics of adsorbates on close-packed metal surfaces, namely hydrogen on Cu(111), Pd(111) and Ru(0001), carbon and oxygen on Ru(0001), and oxygen on Cu(111). Chapter 2 reviews the HeSE technique and describes the relevant dynamical models and statistical methods used to interpret data in later chapters. The performance of the ionizing detector is analysed, with a focus on the signal-to-noise ratio. In Chapter 3 expressions for the classical intermediate scattering function (ISF) are introduced for open and closed systems. The effects of corrugation and surface-perpendicular motion on the amplitude of different components in the ISF are modelled analytically and compared with simulation. The exact ISF for a particle on a flat surface, obeying the Generalized Langevin Equation with exponential memory friction, is calculated analytically. In Chapter 4 the analytical ISF is calculated for quantum Brownian motion and for coherent tunneling dynamics in a tight binding system. The bounce method for calculating quantum mechanical hopping rates in dissipative systems is applied to model diffusion of hydrogen on Ru(0001). Chapter 5 presents the first HeSE measurements of carbon and oxygen diffusion. C/Ru(0001) diffusion is assigned to a small carbon cluster. The jump rate has an activation energy $E_{A}=292\pm7\,$meV in the temperature range $550\,\textrm{K}\leq T \leq 1300\,$K. Oxygen diffusion is significantly slower. By comparison of literature data with the new HeSE results, the activation energy for oxygen diffusion at low coverage is estimated as $650\pm10$meV. Oxygen measurements at high coverage $\theta\approx0.22\,$ML are consistent with strong mutual O-O interactions. Surface diffusion is also observed after exposing Cu(111) to oxygen. Chapter 6 presents low-coverage measurements of protium (H) and deuterium (D) diffusion on Ru(0001), Pd(111) and Cu(111). In the quantum activated regime there is evidence for multiple jumps in all three systems, suggesting a low dynamical friction. The measurements on Ru(0001) indicate that the deep tunneling rate is much slower for D than for H.

Sudden cardiac death due to the development of lethal arrhythmias is the dominant cause of mortality in the UK, yet the mechanisms underlying their onset, maintenance and termination are still poorly understood. Therefore biomarkers are used to determine arrhythmic risk within patients and of new drug compounds. In recent years, the magnitude of variations in the length of successive beats, measured over a short period of time, has been shown to be a powerful predictor of arrhythmic risk. This beat-to-beat variability is thought to be the manifestation of the random opening and closing dynamics of individual ion channels that lie within the membrane of cardiac cells. Computational models have become an important tool in understanding the electrophysiology of the heart. However, current state-of-the-art electrophysiology models do not incorporate this intrinsic stochastic behaviour of ion channels. Those that do use computationally costly methods, restricting their use in complex tissue scale simulations, or employ stochastic simulation methods that result in negative numbers of channels and so are inaccurate. Therefore, using current stochastic modelling techniques to investigate the role of stochastic ion channel behaviour in beat-to-beat variability presents difficulties. In this thesis we take a mathematically rigorous and novel approach to develop accurate and computationally efficient models of stochastic ion channel dynamics that can be incorporated into existing electrophysiology models. Two different models of stochastic ion channel behaviour, both based on a system of stochastic differential equations (SDEs), are developed and compared. The first model is based on an existing SDE model from population dynamics called the Wright-Fisher model. The second approach incorporates boundary conditions into the SDE model of ion channel dynamics that is obtained in the limit from the discrete-state Markov chain model, and is called a reflected SDE. Of these two methods, the reflected SDE is found to more accurately capture the stochastic dynamics of the discrete-stateMarkov chain, seen as the ‘gold-standard’ model and also provides substantial computational speed up. Thus the reflected SDE is an accurate and efficient model of stochastic ion channel dynamics and so allows for detailed investigation into beat-to-beat variability using complex computational electrophysiology models. We illustrate the potential power of this method by incorporating it into a state-of-the-art canine cardiac cell electrophsyiology model so as to explore the effects of stochastic ion channel behaviour on beat-to-beat variability. The stochastic models presented in this thesis fulfil an important role in elucidating the effects of stochastic ion channel behaviour on beat-to-beat variability, a potentially important biomarker of arrhythmic risk.

Orientador: José Antonio Souza / Dissertação (mestrado) - Universidade Federal do ABC. Programa de Pós-Graduação em Nanociências e Materiais Avançados, 2013.

The use of hazardous toxic substances is very common in the industrial sector. The substances are often stored in tanks in storage compartments or transported between industrial premises. In case of an accident involving these substances, severe harm can affect both population and the environment. This leaves a demand for an accurate prediction of the substance concentration distribution to mitigate the risks as much as possible and in advance create suitable safety measures. Toxic gases and vapors are often denser than air making it affected by negative buoyancy forces. This will make the gas descend and spread horizontally when reaching the ground. Swedish Defence Research Agency (FOI) carries today a model called LillPello for simulating the dispersion of gases, yet it does not account for the specific case of a dense gas. Therefore, this thesis aims to implement the necessary effects needed to accurately simulate the dispersion of a dense gas. These effects were implemented in Fortran 90 by solving five conservation equations for energy, momentum (vertical and horizontal) and mass. The model was compared against experimental data of a leak of ammonia (NH3). By analyzing the result of the simulations in this thesis, we can conclude that the overall result is satisfactory. We can notice a small concentration underestimation at all measurement points and the model produced a concentration power law coefficient which lands inside the expected range. Two out of the three statistical quantities Geometric Mean (MG), Geometric Variance (VG) and Factor of 2 (FA2) produced values within the ranges of acceptable values. The drawback of the model as it is implemented today is its efficiency, so the main priority for the future of this thesis is to improve this. The model should also be analyzed on more experiments to further validate its accuracy. / Användandet av giftiga ämnen är vanligt inom den industriella sektorn. Ämnena är oftast lagrade i behållare positionerade i lagringsutrymmen eller så transporteras ämnena mellan industrilokaler. I samband med en olycka innehållande dessa substanser kan stora skador drabba både befolkning och miljön. Detta leder till ett behov av att noggrant kunna förutspå koncentrationsfördelningen för att minska riskerna, samt i förväg kunna skapa lämpliga säkerhetsåtgärder. Giftiga gaser och ångor är oftast tyngre än luft vilket gör att gasen blir påverkad av negativ bärkraft. Detta gör att gasen sjunker och sprids horisontalt när den når marken. Totalförsvarets Forskningsinstitut (FOI) besitter idag en modell kallad LillPello som simulerar spridning av gaser, men den hanterar inte det specifika fallet av en tunggas. Därför siktar detta projekt på att, in i LillPello, implementera de nödvändiga effekterna som behövs för att korrekt kunna simulera spridningen av en tunggas. Dessa effekter är implementerad i Fortran 90 genom att lösa fem konserveringsekvationer för energi, momentum (vertikal och horisontell) samt massa. Modellen jämfördes mot data från ett fältexperiment där ammoniak (NH3) släpptes ut. Genom att analysera resultatet från simuleringar kan vi dra slutsatsen att det övergripande resultatet är tillfredsställande. Vi kan notera en underskattning för alla koncentrationsmätningar i simuleringarna och modellen producerade en potenslagsexponent vars värde hamnade innanför den accepterade gränsen. Två utav de tre beräknade statistiska kvantiteterna: Geometriskt medelvärde (MG), Geometrisk varians (VG) och Faktor av 2 (FA2) producerade värden inom de acceptabla gränserna. Största nackdelen med modellen är dess effektivitet och därför är största prioritet för det fortsatta arbetet inom detta projekt att effektivisera implementeringen. Modellen ska även bli vidare analyserad mot fler experiment för att validera dess noggrannhet.