Selected research projects
Modeling oxygen transport in surgical tissue transfer:
Part of my recent work has focused on mathematical models of oxygen transport in tissue in the context of reconstructive microsurgery. This clinical technique is used to transfer large amounts of tissue from one location on a patient to another in order to restore physical deformities caused by trauma, tumors, or congenital abnormalities. The trend in this field is to transfer tissue using increasingly smaller blood vessels, which decreases problems associated with tissue harvest but increases the possibility that blood supply to the transferred tissue may not be adequate for healing. Mathematical modeling research in this area helps surgeons understand the relationship between tissue volume and blood vessel diameter to ensure success in these delicate operations. I have been an integral part of a research team consisting of applied mathematicians, medical doctors and bioengineers, focusing on the development of mathematical models that can be used to predict successful tissue transfer based on blood vessel diameter, tissue volume, and oxygen delivery. Part of this work has appeared in the Proceedings of the National Academy of Sciences.
Evaluating triggers and enhancers of tau fibrillization in neurodegenerative diseases:
Alzheimer's disease is characterized in part by the aggregation of tau protein into filamentous inclusions. The mechanism of tau filament formation and its modulation by mutation and post-translational modification is of fundamental importance for the control of the disease. In recent work with Dr. Jeff Kuret of the Center for Molecular Neurobiology at the Ohio State University, we investigated the fibrillization of recombinant full-length four-repeat human tau as a function of time and submicromolar tau concentrations using a combination of electron microscopy assays and mathematical modeling methods. The resulting experimental data were fit to a homogeneous nucleation model with rate constant constraints established from filament dissociation rate, critical concentration, and mass-per-unit length measurements. Results indicated for the first time that once assembly-competent protein conformations were attained, the rate-limiting step in the fibrillization pathway was tau dimer formation. Various aspects of this work have appeared in the Journal of Biological Chemistry.
Spatial stochastic modeling of intracellular signaling pathways:
There are numerous sources of stochasticity and heterogeneity in biological systems, and these can have important consequences for understanding the overall system behavior. Intrinsic noise is commonly found in many intracellular signaling pathways. This noise can arise due to low abundance of molecular species, randomness in certain key processes (e.g. binding and unbinding of transcription factors to promoter sites), stochasticity in production processes (transcription, translation) and degradation events. In addition to being inherently stochastic, intracellular signal transduction is inherently spatial.
In a collaborative project with Mark Chaplain at the University of Dundee in Scotland, we seek to investigate and classify the stochastic spatial dynamics of various types of intracellular signaling pathways. In recently published work, we have developed a spatial stochastic model of the Hes1 pathway that yields results in close agreement with experimental studies of Hes1 oscillations observed in mouse embryonic stem cells. Computational investigations of the model suggest that intrinsic noise is the main driving force for the heterogeneity observed in stem cell differentiation responses under the same environmental conditions. Part of this work has appeared in the Journal of the Royal Society Interface.
Random walk distances in data clustering and applications:
Clustering data into groups of similarity is well recognized as an important step in many diverse applications. Well known clustering methods, dating to the 70's and 80's, include the K-means algorithm and its generalization, the Fuzzy C-means (FCM) scheme, and hierarchical tree decompositions of various sorts. More recently, spectral techniques have been employed to much success. However, with the inundation of many types of data sets into virtually every arena of science, it makes sense to introduce new clustering techniques which emphasize geometric aspects of the data, the lack of which has been somewhat of a drawback in most previous algorithms.
In recent work with Sunder Sethuraman of the University of Arizona, we have considered a slate of random walk distances arising in the context of several weighted graphs formed from the data set, in a comprehensive generalized FCM framework, which allow to assign fuzzy variables to data points which respect in many ways their geometry. Our method groups together data which are in a sense well connected as in spectral clustering, but also assigns to them membership values as in FCM.
The effectiveness and robustness of our method has been demonstrated on several standard synthetic benchmarks and other standard data sets such as the Iris and the Yale face data sets. Part of this work has appeared in Advances in Data Analysis and Classification.