CHRISTIAN LAING
 
Research overview

My research combines the areas of knot theory, differential geometry, data mining, and computer science to produce applications to neuroscience and biopolymers such as RNA, DNA, and Proteins. Part of my work has focused on simplifying and generalizing measures of entanglement such as the writhe, the linking number and average crossing number. I applied these results to define and compute a number of geometric shape descriptors to characterize the structure of biopolymers such as proteins and RNA tertiary structures. Analysis of these shape descriptors uses data mining techniques; I have also adapted these concepts to differentiate human brain anatomical characteristics.

RNA tertiary structures

RNA folding is recognized as hierarchical. An RNA sequence forms secondary structural elements (helices and single strands), followed by recurrent tertiary interactions, and then it folds into a native structure. RNA tertiary motifs are recurrent interactions connecting between secondary structural elements. Understanding the role of RNA tertiary motifs in RNA folding will help to understand RNA 3D prediction.
Biopolymer entanglement

Lattice models of self-avoiding random walks have been extensively used to simulate polymerchains with volume exclusion. It is of interest to quantify microscopic entanglement, and if possible, relate it to macroscopic physical properties of the polymer ensemble, such as the stress-strain curve, rubber elasticity, and various phase change phenomena.

The writhe is a measure of non-planarity that can indicate chirality of knots. The presence of knots in a closed circular DNA plasmid can give information about the binding and mechanism of enzymes acting on the DNA molecule. Approximating the writhe of a space curve in general requires choosing a number of planar projections, computing the projected writhe for each of these projections, and averaging the results. For polygons on lattices and space groups, the restricted geometry of the lattice or space group leads to simplification of the writhe calculation, and provides an exact calculation for the writhe.

Polymer structures and characteristic curves on surfaces in nature can take many form circles, open arcs, branched structures, and a mix of these. So it is important to generalize the writhe formulae for linear and branched complexes on lattices or space groups. Important special cases are the calculation of writhe of biopolymers like DNA, RNA, and proteins.
Geometric measures applied to neuroscience

Geometric measures such as the writhe, average crossing number, ropelength and thickness can also be defined in a natural way for oriented graph embeddings and apply to the characterization of sulcus paths along the surface of the human brain.

Brain surfaces can be obtained from triangulated meshes extracted from MRI data. A set of sulcus paths can be traced to describe the anatomy of the brain. Such approach can provide an automatic way to distinguish sulcus paths coming from either the left or right hemisphere, as well as gender.