• Objectives of Course
• The problem of DNA mini-circles
• Mathematical Topics that will be Covered
• Computational Topics that will be covered
• Biological Topics that will be covered
• Biological Topics that will not be covered

Naturally, the course is certainly supposed to be of interest to people whose primary motivation is to learn about the mathematical modelling and computation of DNA.
The focus of the course will be on the structural properties of DNA at the length scale of a few hundred, to several hundred base pairs.
Much of our study will be centered on the motif of DNA mini-circles.
 
Less obviously, the course is designed also to be of interest to people who are interested in mathematical modelling generally.
That is the application of DNA is taken as a case study of  the process of gaining understanding through:

• taking experimental data,
• developing a mathematical model,
• analyzing the model,
• computing with the model, and
• comparing computational predictions with further experimental data

A mini-circle is a relatively short, and therefore stiff, piece of DNA in which the double helix has closed on its own tail to form a twisted circle.

The properties of such mini-circles can be measured experimentally, and the mini-circles can even be observed directly using cryo-Electron-Microscopy. They form a very convenient motif for studying structural properties of DNA experimentally.

They also have lots (!) of very interesting mathematical features.
 

Mini-circles will be the concrete problem that we will use as a specific context in which to introduce:

1) general approaches to modelling the structural, or tertiary, properties of DNA and other macro-molecules.
2) Several, rather general, mathematical and computational techniques.

Most of the first semester will focus on ways of describing the minimum energy, equilibrium shapes of mini-circles.

Toward the end of the first semester, and continuing throughout the second semester we will consider models of the dynamics of DNA in a solvent, which will lead us to statistical mechanics theories for polymers, Monte Carlo simulations, and various stochastic partial differential equation models of the Brownian and Langevin dynamics of DNA.
 

In equilibrium problems:

- Non-dimensionalization and scaling
- The One Dimensional Calculus of Variations
- The Hamiltonian formulation of self-adjoint two-point boundary value problems for ordinary differential equations
- Bifurcation theory and the roles of symmetries and integrals
- the theory of the second variation and stability of equilibria
- the role of isoperimetric constraints
- The geometry and topology of Link, Twist and Writhe
- Quaternion parametrization of SO(3) (i.e. proper rotation matrices)

In Statistical Mechanics Problems:
- Maxwell-Boltzmann probability distributions in phase and configuration space
- Expectation values in polymer chain models
 
 

From equilibrium models:

- Numerical methods for two-point boundary value   problems
- Collocation as a Space discretization
- Numerical Parameter continuation
- Numerical Symmetry breaking
- Averaging and fitting continuum constitutive relations to discrete data

From polymer chain statistical mechanics models:
- numerical implementations of Monte Carlo methods

          - All-atom models
          - Wedge-angle models
          - Bead models
          - Polymer chain models (freely jointed,
                             freely rotating, twisted worm-like)

There are many, very important and interesting questions in mathematics, computation and statistics related to sequencing the Human Genome, i.e. determining the list of base pairs that makes up the DNA in humans (or other organisms). There will probably not be time to mention such issues. We will be concentrating on models related to determination of  the three dimensional or tertiary structure of DNA.