Cells are dynamic systems that generate and respond to forces through the complex interplay between biochemical and mechanical regulations. Since cellular processes often happen at the molecular level and are challenging to be observed under in vivo conditions due to limitations in optical microscopy, multiple analysis tools have been developed to gain insight into those processes. One of the ways to characterize these mechanical properties is by measuring their persistence length, the average length over which filaments stay straight. There are several approaches in the literature for measuring the persistence length of the filaments, including Fourier analysis of images obtained using fluorescence microscopy. Here, we show how curvature can be used to quantify local deformations of cell shape and cellular components. We develop a novel technique, called curvature analysis, to measure the stiffness of bio-filaments from fluorescent images. We test our predictions with Monte-Carlo generated filaments. We also apply our approach to microtubules and actin filaments obtained from in vitro gliding assay experiments with high densities of non-functional motors. The presented curvature analysis is significantly more accurate compared to existing approaches for small data sets. To study the effect of motors on filament deformations and velocities observed in gliding assays with functional and non-functional motors, we developed Langevin dynamics simulations of on glass and lipid surfaces. We found that generally the gliding velocity increases with an increase in motor density and a decrease in diffusion coefficient, and that motor density and diffusion coefficient have no clear effect on filament curvatures, except at a very low diffusion coefficients. Finally, we provide an ImageJ plugin to make curvature and persistence length measurements more accessible to everyone.

Cells are dynamic systems that generate and respond to forces through the complex interplay between biochemical and mechanical regulations. Since cellular processes often happen at the molecular level and are challenging to be observed under in vivo conditions due to limitations in optical microscopy, multiple analysis tools have been developed to gain insight into those processes. One of the ways to characterize these mechanical properties is by measuring their persistence length, the average length over which filaments stay straight. There are several approaches in the literature for measuring the persistence length of the filaments, including Fourier analysis of images obtained using fluorescence microscopy. Here, we show how curvature can be used to quantify local deformations of cell shape and cellular components. We develop a novel technique, called curvature analysis, to measure the stiffness of bio-filaments from fluorescent images. We test our predictions with Monte-Carlo generated filaments. We also apply our approach to microtubules and actin filaments obtained from in vitro gliding assay experiments with high densities of non-functional motors. The presented curvature analysis is significantly more accurate compared to existing approaches for small data sets. To study the effect of motors on filament deformations and velocities observed in gliding assays with functional and non-functional motors, we developed Langevin dynamics simulations of on glass and lipid surfaces. We found that generally the gliding velocity increases with an increase in motor density and a decrease in diffusion coefficient, and that motor density and diffusion coefficient have no clear effect on filament curvatures, except at a very low diffusion coefficients. Finally, we provide an ImageJ plugin to make curvature and persistence length measurements more accessible to everyone.

Mechanical deformations of nucleic acids (NA) play a very important role in many biological life processes. The bending persistence length of DNA is of specific interest, because so much eukaryotic DNA that stores genetic information is tightly packed inside cell nuclei, even though DNA is considered to be a relatively stiff biopolymer. However, recent experiments suggest that DNA may be more flexible than its persistence length (~ 150 bp or ~ 47 nm) suggests, especially for fragments shorter than 100 bp. It is important to reconcile these two seemingly competing pictures of DNA bending by providing a model that can explain the novel results without discrediting old experiments and the widely-accepted worm-like chain model. Another factor that influences both molecular geometry as well as mechanical properties is the ionic atmosphere surrounding the NA. It is known that multivalent ions with charge of +3e and higher can condense DNA into aggregates at high enough concentration. However, most conventional models cannot explain why RNA and DNA condense at different concentrations. Furthermore, our recent simulation results suggest that even though DNA persistence length decreases with multivalent ion concentration due to increasing electrostatic screening, RNA actually becomes stiffer due to a structural transition from the internal binding of the counterions. / Ph. D.

The biological function and the technological applications of semiflexible polymers, such as DNA, actin filaments and carbon nanotubes, strongly depend on their rigidity. Semiflexible polymers are characterized by their persistence length, the definition of which is the subject of the first part of this thesis. Attractive interactions, that arise e.g.~in the adsorption, the condensation and the bundling of filaments, can change the conformation of a semiflexible polymer. The conformation depends on the relative magnitude of the material parameters and can be influenced by them in a systematic manner. In particular, the morphologies of semiflexible polymer rings, such as circular nanotubes or DNA, which are adsorbed onto substrates with three types of structures, are studied: (i) A topographical channel, (ii) a chemically modified stripe and (iii) a periodic pattern of topographical steps. The results are compared with the condensation of rings by attractive interactions. Furthermore, the bundling of two individual actin filaments, whose ends are anchored, is analyzed. This system geometry is shown to provide a systematic and quantitative method to extract the magnitude of the attraction between the filaments from experimentally observable conformations of the filaments. / Die biologische Funktion und die technologischen Anwendungen semiflexibler Polymere, wie DNA, Aktinfilamente und Nanoröhren aus Kohlenstoff, werden wesentlich von deren Biegesteifigkeit bestimmt. Semiflexible Polymere werden charakterisiert durch ihre Persistenzlänge, mit deren Definition sich der erste Teil dieser Arbeit befasst. Anziehende Wechselwirkungen, wie sie z.B. bei der Adsorption, der Kondensation und der Bündelung von Filamenten auftreten, können die Konformation eines semiflexiblen Polymers verändern. Die Konformation ist dabei abhängig von der relativen Größe der Materialparameter und kann durch diese gezielt beeinflusst werden. Im Einzelnen werden hier die Morphologien semiflexibler Polymerringe, wie z.B. DNA oder ringförmiger Nanoröhren, untersucht, die auf drei verschieden strukturierten Substraten adsorbieren: (i) Ein topographischer Kanal, (ii) ein chemisch modifizierter Streifen und (iii) ein periodisches Muster topographischer Oberflächenstufen. Die Ergebnisse werden mit der Kondensation von Ringen durch anziehende Wechselwirkungen verglichen. Des Weiteren wird die Bündelung zweier Aktinfilamente, deren Enden verankert sind, untersucht. Diese Systemgeometrie liefert eine systematische Methode, um die Stärke der Anziehung zwischen den Filamenten aus experimentell beobachtbaren Konformationen zu berechnen.

Title: Linear semiflexible polyelectrolytes in solutions Author: Petra Bačová Department: Faculty of Science, Charles University in Prague Supervisor: Doc. Ing. Zuzana Limpouchová, CSc. Supervisor's e-mail address: zl@vivien.natur.cuni.cz Consultant: Mgr. Peter Košovan, PhD. Abstract: In this thesis I used the Molecular dynamics simulations for study of charged polymers (polyelectrolytes) and their behaviour in solutions. Wide range of polyelectrolytes are se- miflexible and in contrast to neutral polymers it is possible to influence their stiffness by changing the properties of solution as for example ionic strength. The chain flexibility may be characterized by the persistence length. Thesis explains how to express the persistence length from orientational correlation function which shows the double exponential decay. Two contributions to chain stiffness are discussed and the interest is concentrated around electrostatic persistence length which seems to be scale dependent. An effect of added salt on the chain conformation is studied. Salt is treated implicitly within the Debye-Hückel approximation. The results are confronted with OSF theory and the conclusions of vari- ational calculations of Maghi and Netz. The presented thesis describes the conformational behaviour of polyelectrolytes in salty solutions,...

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Die vorliegende Arbeit befasst sich mit dem Wachstum von semiflexiblen Polymeren in ungeordneten Medien. Um dieses Wachstum zu simulieren, kommen hoch entwickelte Monte Carlo Algorithmen zum Einsatz. Eine Zahl von Polymeren wird mithilfe eines breadth-first (zuerst in die Breite gehenden) Algorithmus erzeugt. Dies geschieht im zwei- und dreidimensionalen Raum um die erhaltenen Ergebnisse zu validieren und zwischen beiden Dimensionen zu vergleichen. Der verwendete Algorithmus wurde dahingehend modifiziert den Arbeitsaufwand, welcher durch das Umgehen der Hindernisse der Hintergrundunordnung entsteht, zu kompensieren, um die nötige Rechenzeit im Rahmen zu halten und somit dessen Effizienz zu erhöhen. Es kommen verschiedene Typen von Unordnung zum Einsatz, nämlich die Gitterunordnung (einfache und korrelierte Unordnung) sowie kontinuierliche Unordnung. Die Erzeugung dieser ungeordneten Systeme hängt von folgenden Parametern ab, Dichte, Korrelationsstärke und Lennard-Jones-Wechselwirkungen. Die wichtigste untersuchte Größe ist die Persistenzlänge der Polymere, welche aus der Tangenten-Tangentenkorrelation, gegeben durch die Konturvektoren der Polymere, berechnet wird. Weitere Observablen sind der mittlere quadratische Ende-zu-Ende Abstand, die Ende-zu-Ende Abstandsverteilung und die Polymerform. Ziel ist die Gegenüberstellung der Ergebnisse aus zwei und drei Dimensionen bezüglich des Einflusses der verschiedenen Hintergrundunordnungen auf die Renormalisierung der Persistenzlänge der Polymere.:1 Introduction and Motivation . . 1 1.1 Preamble . . . . . . 1 1.2 Objective . . . . . 2 2 Models . . . . . . 5 2.1 Polymers . . . . . . 5 2.1.1 Heisenberg chain model . . . . . . 5 2.1.2 From the Heisenberg chain to the worm like chain 6 2.1.3 Worm like chain model . . . . . . 7 2.1.4 Weakly bending rod 11 2.2 Disorder . . . . . . 12 2.2.1 Lattice disorder . . 13 2.2.2 Continuous disorder 14 2.3 Disorder creation . 14 2.3.1 Lattice disorder . . 14 2.3.2 Continuous disorder 16 3 Simulation methods . . . . . . . 19 3.1 Monte Carlo Method . . . . . . . 19 3.1.1 Monte Carlo simulation concept . 19 3.2 Growth algorithms . 22 3.2.1 Rosenbluth algorithm . . . . . . . 23 3.2.2 Off-lattice growth algorithm . . . 24 3.2.3 Perm algorithm . . 31 4 Data analysis . . . 33 4.1 Error estimation . . 33 4.2 Observables and simulation parameters . . . . . . 35 4.2.1 Mean square end-to-end distance and mean end-to-end distance . 35 4.2.2 End-to-end distance distribution function . . . . . 36 4.2.3 Tangent-tangent correlation function . . . . . . . 37 4.2.4 Persistence length renormalization 40 4.2.5 Polymer shape: prolateness and asphericity . . . 41 5 Numerical results 43 5.1 Simulation parameters . . . . . . . 43 vii Contents 5.1.1 Length scales . . . 43 5.1.2 Disorder . . . . . . 44 5.1.3 Seeds . . . . . . . 44 5.1.4 Histograms . . . . 45 5.1.5 Polymer shape . . . 45 5.2 Free polymer . . . 46 5.2.1 Analysis . . . . . . 46 5.2.2 Comparison of 2D and 3D . . . . 46 5.2.3 Conclusion . . . . . 51 5.3 Lattice disorder . . 53 5.3.1 Analysis . . . . . . 53 5.3.2 Simple lattice . . . 53 5.3.3 Comparison of free polymer and simple lattice disorder . . . . . 63 5.4 Continuous disorder 65 5.4.1 Analysis . . . . . . 65 5.4.2 Comparison of polymers in continuous disorder to free polymers and lattice dis- order 65 5.5 Correlated lattice disorder . . . . . 72 5.5.1 Conclusion . . . . . 78 6 Summary and Outlook . . . . . 79 Danksagung . . . . . . . 81 Bibliography . . . . . . . 83 A Appendix . . . . . A A.1 Tangent-tangent correlation . . . . A A.1.1 Simple lattice disorder . . . . . . . A A.1.2 Correlated lattice disorder . . . . . C A.2 Mean square end-to-end distance . D A.2.1 Continuous disorder D A.2.2 Correlated lattice disorder . . . . . F A.3 End-to-end probability distribution L A.3.1 Continuous disorder L A.3.2 Correlated lattice disorder . . . . . N A.4 Persistence length renormalization S A.4.1 Correlated lattice disorder . . . . . S / The present thesis reports on the growth of semi-flexible polymers in disordered media. Highly advanced Monte Carlo algorithms are used to simulate the polymer growth. A number of polymers is grown with the help of a breadth-first growing algorithm. This is done in two and three dimensions to validate and compare the results. The algorithm was modified to be able to generate the desired number of polymers in decent computing time, by introducing a special guiding field to handle the additional workload introduced by the obstacles of the background disorder. Different types of disorder are used, lattice disorder (simple lattice disorder and correlated disorder) and continuous disorder. The creation of those disorder configurations is steered by certain parameters as density, correlation strength and interactions governed by the Lennard-Jones-potential. The main value of interest is the persistence length, which is calculated from the tangent-tangent-correlation of the contour vectors of the polymers. Further observables which are investigated are the mean square end-to-end distance, the end-to-end distance distribution and the polymer shape. The goal is to show the influence of the different disorder types on the persistence length renormalization will be shown, with special focus on differences and similarities between two and three dimensions.:1 Introduction and Motivation . . 1 1.1 Preamble . . . . . . 1 1.2 Objective . . . . . 2 2 Models . . . . . . 5 2.1 Polymers . . . . . . 5 2.1.1 Heisenberg chain model . . . . . . 5 2.1.2 From the Heisenberg chain to the worm like chain 6 2.1.3 Worm like chain model . . . . . . 7 2.1.4 Weakly bending rod 11 2.2 Disorder . . . . . . 12 2.2.1 Lattice disorder . . 13 2.2.2 Continuous disorder 14 2.3 Disorder creation . 14 2.3.1 Lattice disorder . . 14 2.3.2 Continuous disorder 16 3 Simulation methods . . . . . . . 19 3.1 Monte Carlo Method . . . . . . . 19 3.1.1 Monte Carlo simulation concept . 19 3.2 Growth algorithms . 22 3.2.1 Rosenbluth algorithm . . . . . . . 23 3.2.2 Off-lattice growth algorithm . . . 24 3.2.3 Perm algorithm . . 31 4 Data analysis . . . 33 4.1 Error estimation . . 33 4.2 Observables and simulation parameters . . . . . . 35 4.2.1 Mean square end-to-end distance and mean end-to-end distance . 35 4.2.2 End-to-end distance distribution function . . . . . 36 4.2.3 Tangent-tangent correlation function . . . . . . . 37 4.2.4 Persistence length renormalization 40 4.2.5 Polymer shape: prolateness and asphericity . . . 41 5 Numerical results 43 5.1 Simulation parameters . . . . . . . 43 vii Contents 5.1.1 Length scales . . . 43 5.1.2 Disorder . . . . . . 44 5.1.3 Seeds . . . . . . . 44 5.1.4 Histograms . . . . 45 5.1.5 Polymer shape . . . 45 5.2 Free polymer . . . 46 5.2.1 Analysis . . . . . . 46 5.2.2 Comparison of 2D and 3D . . . . 46 5.2.3 Conclusion . . . . . 51 5.3 Lattice disorder . . 53 5.3.1 Analysis . . . . . . 53 5.3.2 Simple lattice . . . 53 5.3.3 Comparison of free polymer and simple lattice disorder . . . . . 63 5.4 Continuous disorder 65 5.4.1 Analysis . . . . . . 65 5.4.2 Comparison of polymers in continuous disorder to free polymers and lattice dis- order 65 5.5 Correlated lattice disorder . . . . . 72 5.5.1 Conclusion . . . . . 78 6 Summary and Outlook . . . . . 79 Danksagung . . . . . . . 81 Bibliography . . . . . . . 83 A Appendix . . . . . A A.1 Tangent-tangent correlation . . . . A A.1.1 Simple lattice disorder . . . . . . . A A.1.2 Correlated lattice disorder . . . . . C A.2 Mean square end-to-end distance . D A.2.1 Continuous disorder D A.2.2 Correlated lattice disorder . . . . . F A.3 End-to-end probability distribution L A.3.1 Continuous disorder L A.3.2 Correlated lattice disorder . . . . . N A.4 Persistence length renormalization S A.4.1 Correlated lattice disorder . . . . . S