Membranes, thin barriers between compartments, can uctuate. An important example in
nature are membranes of biological cells. Cells, these building blocks of biological systems,
have diverse capabilities and shapes. However, the basic structural elements and their chemical composition of most cells are the same. Fluid sheets (membranes) enclose the cell and its compartments, while networks of fillaments, if present, maintain the cell's shape and help organize its contents. These structural elements can have quite di_erent mechanical properties than macroscopic objects of our everyday life. For example, they are very soft solely thermal uctuations at room temperature can generate gentle undulations of
membranes.Flickering" of red blood cells was already recorded in the late 19th century by Browicz using the light microscope. Today, with phase-contrast microscopy, non-invasive spectral analysis of those thermal uctuations of biological membranes can provide useful information of the membrane properties.Theoretical model for determining elastic properties of biological membranes with analysis of thermal uctuations by Milner and Safran is based on Helfrich model of membrane and includes also an implicit assumption that in the thermal uctuations of phospholipid bilayers, the shape uctuation modes are not correlated with the lateral stretching modes and that the mean-field approximation can be used. Using Randomly triangulated surfaces, we can simulate biological membrane systems in their thermodynamical equilibrium, where the stochastic Metropolis-Hastings algorithm allows us to sample their thermal uctuations. In this thesis, the coarse-grained model of the membrane is implemented in the program written in C programming language, where the membrane is represented by randomly triangulated network. The model takes into account the assumptions by Milner and Safran. The output of the simulator is the bending sti_ness of the membrane Kc which can be compared with the input bending stiffness , to verify if the numerical simulations are in accordance with the theoretical predictions of Milner and Safran. The randomly triangulated surfaces Monte-Carlo simulations can become time consuming for large systems, therefore some sort of parallelization is needed to harvest the capabilities of modern computers. Two approaches were made and compared. The problem proved to be embarrassingly parallelizable and we measured near theoretical max. speedup of the simulations by running multiple instances of the simulators and combining their statistics. Systems based on method of measurement of thermal uctuations with phase-contrast microscopy, interference contrast microscopy and uorescent-interference contrast microscopyare examples of non-invasive determination of the elastic properties of membranes. Our system is based on phase-contrast microscope and illumination apparatus presented in and, including image analysis described in. It basically consists of a phase-contrast microscope, a stroboscopic lighting system and a camera, connected to a computer. The camera and the lighting system was synchronized to allow a precise, blur-less registration of the membrane shape at the given moment, which is then analysed using user friendly software on the computer. The results of the simulations con_rmed the assumptions of Milner and Safran. The measurements of the simulations were behaving accordingly to the prediction of the equation of Milner and Safran, thus we concluded that the numerical simulations of nearly spherical vesicles modelled with triangulated networks can be used to determine the bending sti_- ness. Depending on the resolution of the simulations (the density of the mesh) the di_erence between input and measured bending sti_ness can be well below 10%.
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