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. I also work on problems related to RNA structural genomics. In particular, apply mathematical and computational biology methods to study RNA 3D structure patterns (motifs), and design models to predict RNA 3D structures.

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.

In collaboration with Tamar Schlick, I have focused on studying RNA structural genomics. In particular, we applied mathematical and computational biology methods to understand 3D structure patterns (motifs) of RNA molecules. By using network analysis and statistics, we studied complex interaction networks formed by a number of known RNA tertiary motifs, showing the existence of higher order motifs built by a combination of smaller sub-motifs. We studied and classified RNA junctions into families, and showed that RNA junctions are composed of recurrent helical configurations. Interestingly, helical elements in junctions are found to align in parallel and perpendicular configurations. We also discovered new RNA motifs, and showed that larger junctions are composed of smaller sub-junction elements. By defining 2D/3D motif restraints and guidelines for the prediction of RNA helical conformations, this analysis can ultimately help in the difficult task of RNA 3D structure prediction.

 

Classification of RNA four-way junctions into nine families according to their coaxial stacking properties and flexible helical arms.

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.

Diamond lattice structure with its corresponding writhe formula. The choice of the vectors are the unit orthogonal vectors (1,0,0), (0,1,0), and (0,0,1).

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.

Red line demarcates calcarine sulcus on the brain surface.