Polymers in Confined Geometry.pdf

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36 CHAPTER 3. POLYMERS IN CONFINED GEOMETRY the differences between the approaches will not effect the results of calculations, besides minor boundary effects. Therefore the series ansatz is not further investigated. The calculation become very tedious. 3.3.3 Radial distribution function The radial distribution function (RDF) p(R) is another interesting quantity. It is the probability distribution of the end-to-end distance vector R that incorporates more information than only the moments. It is defined as p(R) = 〈δ(R − |R|)〉 . (3.51) All configurations with the specified end-to-end distance |R| = R are counted. For the free polymer this quantity can be calculated—or at least approximated— over the whole stiffness range. The result for the Gaussian chain was shown in eq. (2.5). For other regions confer [7] and [47]. The calculations to obtain the RDF are quite involved for the free polymer. For a confined polymer this becomes even more difficult. The advantage of the polymer in bulk solution is, that the problem can be formulated by exclusively using tangent-vectors describing the segments, independent of the embedding space. By introducing a confining environment a non-local interaction term has to be introduced, since the problem is now dependent on the actual positions of the chain joints, which are given as integrals over the tangents. In Fourier space problems arise due to higher order modes. Therefore we do without an analytical approach and resort to simulation results discussed in section 5.2.5.
Chapter 4 Simulation methods This chapter starts with a review of the Monte-Carlo method in general and how to calculate partition sums in the canonical ensemble. The Metropolis algorithm is introduced as a method to implement the idea of Markov chains in computer programs. In this context the problem of correlated sampling and the importance of the quality of random number generators are discussed. Then we turn to the discussion of these methods in the context of the wormlike chain model in bulk solution and in confined environments. Finally we give a list of the important observables which are obtained in simulations and where they are used. 4.1 Motivation As in most cases in physics there is no way of exploiting all interesting parameter values of a given problem analytically. So one has to resort to simulation techniques [4, 13, 22]. We saw in the chapters 2 and 3 that analytical results are limited to either the very stiff and flexible polymer, or the strongly confined and totally free environment. For problems from biotechnology it is very important to obtain quantitative results particularly in the intermediate regime—where up to now only qualitative scaling arguments were available (cf. section 3.2). The problem of flexible DNA confined to nano-channels with lateral dimensions in the order of the persistence length, as introduced in section 3.1, is just in this complicated parameter regime. Also the properties of semi-flexible F-Actin filaments, with a contour-length of about the persistence length, are in this parameter range. We therefore have to rely on simulation techniques. The Monte-Carlo simulation is the method of choice, since it allows to effectively sample the important conformations of the polymers. The problem of other methods, e.g. molecular dynamics (MD) simulations, is the large difference in time-scales. Typical changes on the monomer level are much faster than conformational changes of the whole 37
Page 1 and 2: Diploma Thesis Polymers in Confined
Page 3: TMTOWTDI—There’s More Than One
Page 7 and 8: Contents 1 Introduction 1 2 Polymer
Page 9 and 10: Chapter 1 Introduction Polymers are
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: 3.3. ANALYTICAL RESULTS 35 x 2 (s/
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-
Page 59 and 60: 4.5. SIMULATION OF CONFINED WORM-LI
Page 61 and 62: 4.6. SAMPLED QUANTITIES AND THEIR R
Page 63 and 64: 4.6. SAMPLED QUANTITIES AND THEIR R
Page 65 and 66: Chapter 5 Simulation results In thi
Page 67 and 68: 5.2. THE HARMONIC POTENTIAL 59 ˆt
Page 69 and 70: 5.2. THE HARMONIC POTENTIAL 61 5.2.
Page 71 and 72: 5.2. THE HARMONIC POTENTIAL 63 beha
Page 73 and 74: 5.2. THE HARMONIC POTENTIAL 65 R 2
Page 75 and 76: 5.2. THE HARMONIC POTENTIAL 67 R 2
Page 77 and 78: 5.2. THE HARMONIC POTENTIAL 69 R
Page 79 and 80: 5.2. THE HARMONIC POTENTIAL 71 p(R)
Page 81 and 82: 5.3. THE HARD WALLS 73 〈Ra〉 1 0
Page 83 and 84: Chapter 6 Conclusion Emerging from
Page 85 and 86: Appendix A Discrete worm-like chain
Page 87 and 88: which, after taking all the derivat
Page 89 and 90: Appendix B Calculation for the conf
Page 91 and 92: B.2. END-TO-END DISTANCE CORRELATIO
Page 93 and 94: B.3. PARALLEL END-TO-END DISTANCE C
Page 95 and 96:
Glossary L filament length lp persi
Page 97 and 98:
Bibliography [1] M. Abramowitz and
Page 99 and 100:
BIBLIOGRAPHY 91 [29] P. L’Ecuyer.
Page 101:
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