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Research Vignettes

Application of Knot Theory to DNA topology (with Shabnam Beheshti)
Knot Theory is a beautiful branch of pure mathematics that incorporates ideas from differential geometry, algebraic topology, and many others.  Its applications are rich and far-reaching, and it is classically used to explain several phenomena in string theory and quantum mechanics.  More recently, it has been shown to explain how DNA knots and links during supercoiling and homologous recombination events.  This is of particular interest to biologists, as it provides a much more comprehensive model than the limited information yielded by electron microscopy.  In addition, it has been shown that DNA curvature can affect the binding affinity of transcription factors to promotor regions.  This allows the topological properties of DNA to have a (possibly marked) effect on gene regulation.  We are investigating whether mathematics can explain the implications of this effect on biological systems.  (This work was supported by an NSF REU fellowship.)

Computational analysis of protein dynamics and flexibility (with Roland L. Dunbrack, Jr.)
The collaborative effort to expand the Protein Data Bank in the last two decades has done much to facilitate the visualization and measurement of protein structures.  In addition to contributions of previously unsolved amino acid sequences, many proteins now have a number of "redundant" tertiary structures for the same amino acid sequence.  We have used cutting-edge dimensionality reduction algorithms as an analytical tool for identifying the differences in these structures that result from domain flexibility and the underlying molecular dynamics of the protein.  In addition to shedding light on the underlying biophysics of when and how domain flexibility occurs, we hope this information can be used to develop therapeutic targets for disregulated pathways.  (This work was supported by a Fox Chase Cancer Center Undergraduate Summer Research Fellowship.)

Modelling the chemotaxis of macrophages into hypoxic tumours (with Helen M. Byrne)
The presence of a tumour triggers an immune response, whereby circulating monocytes differentiate into macrophages that then infiltrate the tumour.  These tumour-associated macrophages (TAMs) do not consist of a single population; rather, there are subpopulations with a range of phenotypes. While some TAMs have anti-tumour phenotypes, others promote growth and metastasis by recruiting angiogenic factors.  In recent years, methodologies have been proposed to use the behaviour of TAMs for therapeutic purposes.  Namely, macrophages can be infected with an oncolytic virus and injected into the bloodstream.  These "virally loaded" macrophages migrate to primary and secondary tumours via chemotaxis where they release the virus.  While this technique has proven effective in mice (Muthana et al. Cancer Research 2011) there are still a number of unanswered questions.  We have extended existing continuum models of tumour spheroid growth to include the effect of these "virally loaded" macrophages.  In doing so, we hope to better understand how this therapy can be used effectively in combination with more traditional chemotherapy and radiation regimens.

Mechanics and analysis of cortex folding (with Alain Goriely, Dominic Vella, and Peter Stewart)
In recent years a great deal of interest has focused on understanding the mechanical basis for the folding pattern (sulcification) observed in the brains of humans and other higher mammals.  One key difficulty in this area is quantifying the sulcification of human brains and that of analogue systems with qualitatively similar folding patterns.  Our aim is to develop the mathematical and software tools necessary to analyse clinical images and hence quantify these patterns.  Such quantification would allow for comparison between the folding observed in different individuals and may also be applied to images from mechanical analogue systems.  The comparison between analogue mechanical models and brains may allow us to discriminate between the various proposed mechanisms of sulcus formation.