Table of Contents

1. Introduction

1.1 Objectives of the Course
1.2 Generalities on DNA
1.2.1 The B form of the DNA double-helix
1.2.2 Some other DNA forms
1.2.3 The length scales of DNA
1.3 DNA cyclization as a specific modelling issue

Introduction

Introduction

2. The Special Cosserat Theory of Idealized Rods

2.1 Introduction
2.2 Kinematics
2.2.1 The reference state and the adapted framing
2.2.2 The inextensible, unshearable rod
2.3 Balance Laws
2.4 Constitutive Relations
2.5 Equilibrium Conditions
2.6 The planar rod example
2.7 The discrete strut
2.8 The continuous limit of the discrete strut

2.8.1 A little bifurcation theory for the continuous problem
2.8.2 The collocation discretization
2.8.3 VBM and AUTO

Cosserat idealized Rods

Cosserat idealized Rods

3. One-dimensional Calculus of Variations

3.1 First and Second Variations 
3.2 Euler-Lagrange Equations 
3.3 Natural Boundary Conditions 
3.4 Corner Conditions 
3.5 Isoperimetric Constraints 
3.6 Pointwise Constraints 
3.7 Conjugate Point Theory 
3.8 Hamiltonian Formulation

Calculus of Variations

Calculus of Variations

4. Cosserat Rods as Calculus of Variations problems

4.1 The planar case examples (in parallel to section 2.7) 
4.2 The 3D case
4.3 Quaternions

Cosserat Rods as CoV

Cosserat Rods as CoV

5. Bifurcation Theory

5.0 Introduction 
5.1 Basic Problem 
5.2 The planar strut 

5.2.1 Bifurcation Analysis 
5.2.2 Stability Analysis

5.3 The 3D strut 

5.3.1 Linearization 
5.3.2 Simplifications 
5.3.3 Bifurcation points 
5.3.4 Interpretations K1!=K2 
5.3.5 Interpretations K1=K2 
5.3.6 Symmetry breaking

 Bifurcation Theory

 Bifurcation Theory

6. Experimental Approaches to DNA structure and dynamics

6.1 X-Ray diffraction and Crystallography
6.2 Gel electrophoresis and the discovery of DNA bending
6.3 Cyclization probability
6.4 The biology of DNA bending, an example

Experimental Approaches

Experimental Approaches

7. Hamiltonian formulation

7.1 The kinematics of rods
7.2 Force and Moment Balance laws
7.3 Constitutive Relations
7.4 Representation of directors via Euler parameters (quaternions)
7.5 Variational formulations
7.6 Hamiltonian formulations

7.6.1 Discussion of the Hamilton Formulations
7.6.2 Analysis of Integrals and Symmetries

Hamiltonian Formulation

All you ever want to know about quaternions

Hamiltonian Formulation

All you ever want to know about quaternions

8. Framed Curves

8.1 Adapted Framing of a Curve

8.1.1 Frenet-Serret Frame
8.1.2 Natural Frame
8.1.3 Why work with adapted frames

Framed curves

Framed curves

9. Link, Twist and Writhe and DNA Supercoiling

9.1 The Link Integral
9.2 The Writhe Integral
9.3 The Twist
9.4 The Calugareanu-White-Fuller Theorem "Lk=Tw+Wr"

9.4.1 The Self-Linking of a Closed Curve
9.4.2 The Natural frame

9.5 Some Interesting Implications for DNA

9.5.1 The Writhe Framing

Link, Twist and Writhe and DNA supercoiling

Link, Twist and Writhe and DNA supercoiling

10. Figures

Figures (7.9 MB)

Figures (zipped 7.8 MB)

Figures (33 MB)

Figures (zipped 9 MB)

10. Statistical Mechanics of Polymer Chains

see hardcopy handouts given in class

11. Stochastic Differential & Fokker-Plank Equations

see hardcopy handouts given in class

12. Monte-Carlo Methods