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De Witt Sumners's Faculty Page

De Witt L. Sumners
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Robert O. Lawton Distinguished Professor
Ph.D., University of Cambridge, England, 1967
B.S., Louisiana State University, USA, 1963

Detailed description of research

Summary of Research Interests

  • Molecular Biology
  • Polymer Conformations
  • Human Brain Project

Statement of Research Interests

I am interested in the applications of topology to molecular biology and polymer configuration, both in theory development and computational simulation. Another interest is the mathematical analysis of human brain functional data.

Molecular Biology

The DNA of all organisms has a complex and fascinating topology. It can be viewed as two very long, closed curves that are intertwined millions of times, linked to other closed curves, tied into knots, and subjected to four or five successive orders of coiling to convert it into a compact form for information storage. For information retrieval and cell viability, some geometric and topological features must be introduced, and others quickly removed. Some enzymes maintain the proper geometry and topology by passing one strand of DNA through another via an enzyme-bridged transient break in the DNA; this enzyme action plays a crucial role in cell metabolism, including segregation of daughter chromosomes at the termination of replication and in maintaining proper in vivo (in the cell) DNA topology. Other enzymes break the DNA apart and recombine the ends by exchanging them. These enzymes regulate the expression of specific genes, mediate viral integration into and excision from the host genome, mediate transposition and repair of DNA, and generate antibody and genetic diversity. These enzymes perform incredible feats of topology at the molecular level; the description and quantization of such enzyme action absolutely requires the language and computational machinery of topology.

The long-range goal of this project is to develop a complete set of experimentally observable topological parameters with which to describe and compute enzyme mechanism and the structure of the active enzyme-DNA synaptic intermediate. One of the important unsolved problems in biology is the three-dimensional structure of proteins, DNA and active protein-DNA complexes in solution (in the cell), and the relationship between structure and function. It is the 3-dimensional shape in solution which is biologically important, but difficult to determine. The topological approach to enzymology is an indirect method in which the descriptive and analytical powers of topology and geometry are employed in an effort to infer the structure of active enzyme-DNA complexes in vitro (in a test tube) and in vivo. In the topological approach to enzymology experimental protocol, molecular biologists react circular DNA substrate with enzyme and capture enzyme signature in the form of changes in the geometry (supercoiling) and topology (knotting and linking) of the circular substrate. The mathematical problem is then to deduce enzyme mechanism and synaptic complex structure from these observations.

Polymer Conformations

Polymers in dilute solution can be modeled by means of self-avoiding walks on a lattice, the lattice spacing serving to simulate volume exclusion. Topological entanglement (knotting and linking) restricts the number of configurations available to a macromolecule, and is thus a measure of configurational entropy. A linear polymer can be modeled as a self-avoiding walk (SAW) on the simple cubic lattice; a ring polymer can be modeled as a self-avoiding polygon (SAP) on that same lattice. Microscopic topological entanglement of polymer strands is believed to effect macroscopic physical characteristics of polymer systems, such as the stress-strain curve, rubber elasticity, and various phase change phenomena. Physical properties of semicrystalline polymers are believed to strongly depend on entanglement of polymer strands in the amorphous region. Knots in linear polymers may be trapped as "tight knots" by the crystallization procedure. The dependence of knotting probability on chain thickness can be exploited in a cyclization reaction on linear DNA to determine the effect of salt concentration on chain diameter due to Coulomb shielding. Mathematical models include discrete models on regular lattices, and continuum models in 3-space. In the asymptotic regime (lengths going to infinity) on the simple cubic lattice and in the continuum,one can prove that almost all sufficiently long chains are knotted, almost all sufficiently long circles are chiral, and that the topological entanglement complexity (measure in many ways) goes to infinity at least linearly with the length. For short chains, the pivot algorithm is known to be ergodic on self-avoiding walks and polygons in the simple cubic lattice, and it provides a computationally efficient method for computing entanglement statistics for short chains. Energetic terms can be introduced into the in the Metropolis Monte Carlo computations for knot probability to simulate both solvent quality and Coulomb shielding, producing knot probability curves that qualitatively agree with laboratory random knotting results and other Monte Carlo models (the worm-like model) which include volume exclusion.

Human Brain Project

As a member of an interdisciplinary Human Brain Project research team, I am interested in the mathematical analysis and visualization of human brain functional data. We use cerebral blood flow as a marker for neural activity, obtaining data on blood flow using the modalities of positron emission tomography (PET) and functional magnetic resonance imaging (fMRI). My student Ivo Dinov and I are investigating the use of fractal and wavelet encoding of brain architecture (high resolution magnetic resonance imaging (MRI) scans) and 3-D images of activation foci produced by PET and fMRI scans. The signal-to-noise ratio is low in these human brain functional modalities, and there are serious difficulties inherent in comparing functional data across scans, subjects, groups and modalities. We intend to use these new encoding algorithms, plus other geometrical and topological ideas to aid the group effort in the study of human brain functional data.