Polymers in Confined Geometry.pdf

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iv CONTENTS 4.4.2 Self-avoiding chains . . . . . . . . . . . . . . . . . . . . . . 51 4.5 Simulation of confined worm-like chains . . . . . . . . . . . . . . . 51 4.5.1 Harmonic potential . . . . . . . . . . . . . . . . . . . . . . 52 4.5.2 Hard-wall tubes . . . . . . . . . . . . . . . . . . . . . . . . 53 4.6 Sampled quantities and their relevance . . . . . . . . . . . . . . . 53 5 Simulation results 57 5.1 The unconfined chain . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2 The harmonic potential . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2.1 Undulations . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.2.2 Tangent-tangent correlation . . . . . . . . . . . . . . . . . 61 5.2.3 Scaling plot . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.2.4 End-to-end distances . . . . . . . . . . . . . . . . . . . . . 69 5.2.5 Radial distribution function . . . . . . . . . . . . . . . . . 70 5.3 The hard walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6 Conclusion 75 A Discrete worm-like chain correlations 77 B Calculation for the confined polymer 81 B.1 Undulation correlations . . . . . . . . . . . . . . . . . . . . . . . . 81 B.2 End-to-end distance correlation . . . . . . . . . . . . . . . . . . . 82 B.3 Parallel end-to-end distance correlation . . . . . . . . . . . . . . . 84 Glossary 87 Bibliography 89
Chapter 1 Introduction Polymers are one of the main constituents of cells. The cytoskeleton is formed by a set of protein filaments (microtubules, actin filaments, intermediate filaments). It provides mechanic stability and a railway network for intracellular transport. Genetic information is also stored in a macromolecule, deoxyribonucleic acid (DNA). As the basic ‘cookbook’ of life it provides the recipe for sequencing all the other biopolymers. DNA is not only carrier of genetic information, but its mechanic properties (such as bending and torsional stiffness) are also essential for how this information is processed. For example some processes in gene expression are regulated by loop formation and other conformational changes of DNA. These few examples are supposed to give a flavor of the manifold role biopolymers play in cellular processes (cf. [3, 5, 6, 14]). Biopolymers differ from synthetic polymers in one important aspect. They are stiff on length scales relevant for the biophysical processes they are involved in. This stiffness is characterized in terms of the persistence length which measures the length scale (along the backbone) over which correlations of the tangents (describing the polymer configuration) decay. For DNA the persistence length is approximately lp ≈ 50 nm. This is much larger than a typical microscopic length scale of DNA, e.g. the diameter h ≈ 2 nm: lp ≫ h semi-flexible polymer. F-Actin is even stiffer, with lp ≈ 17 nm and h ≈ 7 nm. This unique feature gives rise to a multitude of interesting phenomena genuinely different from those found for their synthetic cousins (such as polyethylene), where lp is comparable to the microscopic scale h (cf. [17, 31]). Our focus in this thesis will be on biopolymers in confined geometry. This is motivated by rapid growing interest in observing and manipulating singe polymer chains in biotechnological applications using micro- and nanofluidic devices. Using optical tweezers, it is possible to exert small forces on polymers. By 1
Page 1 and 2: Diploma Thesis Polymers in Confined
Page 3: TMTOWTDI—There’s More Than One
Page 7: Contents 1 Introduction 1 2 Polymer
Page 11 and 12: (a) λ-DNA (b) T2-DNA Figure 1.2: A
Page 13 and 14: Chapter 2 Polymer models This chapt
Page 15 and 16: 2.2. IDEAL PHANTOM CHAIN 7 configur
Page 17 and 18: 2.3. SEMI-FLEXIBLE POLYMERS 9 In th
Page 19 and 20: 2.3. SEMI-FLEXIBLE POLYMERS 11 to i
Page 21 and 22: 2.3. SEMI-FLEXIBLE POLYMERS 13 for
Page 23 and 24: 2.5. REAL CHAINS 15 no overhangs al
Page 25 and 26: 2.5. REAL CHAINS 17 Figure 2.5: Sna
Page 27 and 28: 2.5. REAL CHAINS 19 ln R cases intr
Page 29 and 30: Chapter 3 Polymers in confined geom
Page 31 and 32: 3.2. SCALING RELATIONS 23 using eq.
Page 33 and 34: 3.2. SCALING RELATIONS 25 b a d R |
Page 35 and 36: 3.3. ANALYTICAL RESULTS 27 The mean
Page 37 and 38: 3.3. ANALYTICAL RESULTS 29 From thi
Page 39 and 40: 3.3. ANALYTICAL RESULTS 31 a consta
Page 41 and 42: 3.3. ANALYTICAL RESULTS 33 the proj
Page 43 and 44: 3.3. ANALYTICAL RESULTS 35 x 2 (s/
Page 45 and 46: Chapter 4 Simulation methods This c
Page 47 and 48: 4.3. SAMPLING THE CANONICAL ENSEMBL
Page 55 and 56: 4.4. SIMULATION OF UNCONFINED WORM-
Page 57 and 58: 4.4. SIMULATION OF UNCONFINED WORM-
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4.5. SIMULATION OF CONFINED WORM-LI
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4.6. SAMPLED QUANTITIES AND THEIR R
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4.6. SAMPLED QUANTITIES AND THEIR R
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Chapter 5 Simulation results In thi
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5.2. THE HARMONIC POTENTIAL 59 ˆt
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5.2. THE HARMONIC POTENTIAL 61 5.2.
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5.2. THE HARMONIC POTENTIAL 63 beha
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5.2. THE HARMONIC POTENTIAL 65 R 2
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5.2. THE HARMONIC POTENTIAL 67 R 2
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5.2. THE HARMONIC POTENTIAL 69 R
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5.2. THE HARMONIC POTENTIAL 71 p(R)
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5.3. THE HARD WALLS 73 〈Ra〉 1 0
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Chapter 6 Conclusion Emerging from
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Appendix A Discrete worm-like chain
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which, after taking all the derivat
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Appendix B Calculation for the conf
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B.2. END-TO-END DISTANCE CORRELATIO
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B.3. PARALLEL END-TO-END DISTANCE C
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Glossary L filament length lp persi
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Bibliography [1] M. Abramowitz and
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BIBLIOGRAPHY 91 [29] P. L’Ecuyer.
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