Some proteins contain locally knotted structures. Many algorithms have been developed in order to detect local knotting in protein conformations. In some cases these algorithms are used to rule out computationally generated structures containing local knots as knotted proteins are rare. However, there are several types of proteins which contain local knots. I will give an overview of knotted proteins, the various methods used to define a local knot in a protein, and their potential significance.

The molecular basis of life rests on the activity of biological macro-molecules, mostly nucleic acids and proteins. A perhaps surprising finding that crystallized over the last handful of decades is that geometric reasoning plays a major role in our attempt to understand these activities. In my presentations, I will explore the connection between the biological activities of proteins and geometry, using a representation of molecules as a union of balls. I will cover three topics: (1) the geometry of biomolecular solvation, (2) understanding electrostatics using implicit solvent models, and (3), designing protein shape descriptors.

Part 1: Hydrophobicity. The structure of a biomolecule is greatly influenced by its environment in the cell, which mainly consists of water. Explicit representation of the solvent that includes individual water molecules are costly and cumbersome. It is therefore highly desirable to develop implicit solvent models that are nevertheless accurate. In such models, hydrophobicity is expressed as a weighted sum of atomic accessible surface areas. I will show how these surface areas can be computed from the dual complex, a filtering of the weighted Delaunay triangulation of the centers of the atoms.

Electrostatics plays an important role in stabilizing a molecule. In an implicit solvent model, the electric field generated by a molecule is obtained as a solution of the Poisson-Boltzmann equation, a second order elliptic equation to be solved over the whole space within and around the molecule. Analytical solutions of this equation are not available for large molecules. Numerical solutions are usually obtained using finite element methods on regular meshes. These meshes however are not adequate to represent accurately the surface of the molecule that serves as interface between the interior of the molecule and the solvent. I will discuss the application of tetrahedral meshes for solving the Poisson-Boltzmann equation, based on the meshing of the skin surface of the molecule. The skin surface is a smooth, differentiable surface of the molecule.

As the number of proteins for which a high resolution structure is known grow, it is important to classify them. A classification of protein structure would be useful for example to derive structural signatures for the protein functions. Classification requires a measure of protein structure similarity: while there are tools available to align and superpose protein structures, these tools are usually slow and not practical for large scale comparisons. In this talk, I will discuss the development of protein shape descriptors that allow fast detection of similarity between protein structures.

Over the last three decades computer simulation have been able to bring atomic motion to structural biology. Such motion is not seen in experimental structural studies but is relatively easily studied by applying law of motions to models of the proteins and nucleic acids. By bringing molecular to life in this way, simulation complements experimental work making it is much easily to understand how proteins biological macromolecules function. After introducing molecular structure and the fundamental forces that stabilize it, we consider molecular motion and protein folding.

Lecture 1 will introduce fundamental forces between atoms and consider how these forces give rise to the stable protein structures.

Lecture 2 will describe the complementary methods of simulation: molecular dynamics and normal mode dynamic. We will show how they help understand the stability and the nature of protein motion.

Lecture 3 will consider the protein folding both in terms of predicting the most stable structure and simulating the actual folding pathways.

The lecture will be a basic introduction to multigrid techniques. It will cover some background on stationary iterative methods. The two main components of linear multigrid algorithms: smoothing and coarse-grid correction will be introduced. A two grid algorithm will be introduced that then leads to the description of the multilevel Vand W-cycles. A brief description of algebraic multigrid methods will be followed by a description of the Full Approximation Scheme (FAS) for nonlinear problems. Time permitting, the generalization of these algorithms to handle grids with local refinement will also be outlined.

Solved protein structures from PDB depict a static picture, but proteins are flexible. We are interested in understanding how they move near the native conformation, or between two given conformations, without resorting to heavy-duty molecular dynamics techniques. Geometric simulations focus on motions of constrained structures behaving much like mechanical devices, without concern for certain forces (such as electrostatic or hydrophobic interactions). The idea is to isolate specific problems (pertaining to maintenance of geometric distance and angle constraints, or to collisions), and develop the mathematical and computational tools for addressing them efficiently. We will describe static flexibility analysis tools pioneered in the FIRST software, first-generation geometric simulation as done in FRODA, and recent methods aiming at speeding them up.