top of page

Current Research Projects


My group is currently working on five main projects. 1) We are developing, simulating and anlyzing a multiscale model to understand how cells within an epithelial tissue communicate over long distances to form patterns. 2) We are using machine learning to identify features of biological patterns that can be used to distinguish modified patterns from nonmodified ones. 3) We are developing a multiscale model of bleb-based chemotaxis to clarify the role of membrane to cortex linker proteins in regulating the size and frequency of blebs as well as understand how cells translocate in confined environments using blebs. 4) We are developing a risk prediction model for men of African ancestry with prostate cancer to reduce overdiagnosis and overtreatment in this population. 5) We are developing efficient exponential time differencing schemes with time adaptivity and dimensional splitting to improve the solution of advection-diffusion-reaction equations. 

Open Positions: 

  1. PhD position in multiscale modeling of pattern formation (Fall 2024, Funded).

  2. PhD position in machine learning for pattern recognition (Fall 2024, Funded).

  3. PhD position in multiscale modeling of cell motility (Fall 2024)

  4. PhD position in computational mathematics (Fall 2024).

  5. Undergraduate research position in mathematical modeling (Summer 2024, Paid)

  6. Undergraduate research position in computational mathematics (Summer 2024, Paid).

  7. Undergraduate/Masters student research in statistical learning (Fall 2024)


Interested students should send me an email at with a CV and brief statement of interest. 


Pattern Formation

Dorsal Thorax of Fruit Fly

Repeating patterns (hairs, intestinal villi, etc) are important for epithelia that sense the environment. Optimizing the organization of these tissues helps them to function normally. In the fuit fly Drosophila melanogaster, the organization of sensory bristles on its dorsal thorax is important for the proper function of its peripheral nervous system (Fig A). Notch signaling is a mechanism whose proper functioning is critical to the organization of sensory bristles. During development, sensory bristols emerge from sensory organ precussor cells (Fig B,  bright white spots).   Until recently, Notch signaling was thought to be activated by purely local cell-cell contacts. This philosophy has however been insufficient in explaining the organization of sensory bristles over large areas of the fly thorax. Recent modeling and experiments have suggested that distant cells can engage in Notch signaling via interacting signaling filopodia (cytonemes). However the precise mechanism by which this signaling occurs is unkown. This project is in collaboration with Dr. Ginger Hunter, a developmental biologist at Clarkson University. It seeks to use mathematical modeling, numerical simulation and analysis of biological data to understand how Notch signals are 1) activated and 2) transported along cytonemes. Our cellullar model for Notch activation and molecular model for Notch transport will be combined within a tissue model of the fly dorsal thorax using the vertex modeling framework. The resulting multiscale model will be used to test the formation of bristle spot patterns via long-range cytoneme interaction and validated using experimental data.


Funding Source: This project is supported by the Joint DMS/NIGMS initiative to support research at the interface of biological and mathematical sciences under Award Number R01GM152810. 


Pattern Recognition

sop pattern

A major challenge for researchers is the ability to quantify and classify complex cell and tissue patterns so that modified patterns can be distinguished from normal ones. Such classification, when performed algorithmically, can help developmental biologists quickly determine whether a patricular gene or protein plays a role in the organization of tissue patterns.  In collaboration with Dr. Ginger Hunter at Clarkson University, we are using Drosophila melanogaster as a model system to formulate a classification system that differentiates between normal and modified sensory organ precursor (SOP) patterns (see Figure above). These SOP patterns develop into sensory bristles which are important for the proper functioning of the fly's peripheral nervous system. The project has two goals. 1) Identify features that are sensitive to changes in a normal SOP pattern. 2) Use these features, within a machine learning algorithm, to distinguish between modified and normal SOP patterns. Currently, we are exploring the effectiveness of density and clustering of SOP patterns as clasification features. Future work will explore features obtained from methods like Topological Data Analysis and Graph Theory. 


Cell Motility

bleb life cycle

Eukaryotic cells such as white blood cells and cancer cells have been observed to move by making spherical, pressure-driven blister-like protrusions of their cell membrane, referred to as blebs. In contrast to the well- known actin driven motility structures such as lamellipodia and pseudopodia which rely heavily on the ability of cells to rapidly polymerize actin, blebs use a build-up in intracellular pressure to push the cell membrane forward by overcoming the adhesive force of membrane-to-cortex linker proteins (Fig A). In order for cells to use blebs for movement, the expanded membrane must be stabilized through the formation of a new cortex beneath the protruded membrane (Fig B). The old cortex (at the initial location of the membrane) must also be fully degraded to permit organelles and larger proteins to flow past the old cell boundary (Fig C).  Little is known about how the reformation and degradation of the cortex is regulated during bleb formation. Other questions remain about the role of membrane-to-cortex linker proteins in regulating bleb size and frequency.  Additionally,  it is not clear how cells coordinate the protrusion of their leading edge and retraction of their rear to translocate when blebbing.  To resolve these questions, we are developing a multiscale model of bleb-based motility that combines mechanical forces generated from interacellular pressure to expand the membrane with the kinetics of actin and myosin during the formation and degradation of the cortex to stabiilze the expanded bleb. The model also couples the contractile force generated from myosin at the rear of the cell with the dynamics of cell-substrate adhesion proteins to achieve translocation.  Decisions about where to bleb and retract are driven by the distribution of protein kinases which responds to the prevailing chemoattractant gredient.   The numerical simulation of the model is being done using the level set method to faciliate an easy extension of the current 2D modeling framework to study motility in more realistic 3D environments. 


Clinical Prediction Models

prostate cancer incidence

Prostate cancer (PCa) is the most common noncutaneous melanoma in men in the United States and the second leading cause of death by cancer in men. Predicting a patient’s risk for prostate cancer is an important first step to identifying and treating the disease. Risk prediction involves biomarkers as predictor variables in a statistical learning model such as logistic regression or support vector machines. In collaboration with Dr. Olorunseun Ogunwobi, cancer biologist at Michigan State University, we are developing a risk prediction model using novel biomarkers discovered in his lab to provide a more accurate assessment of a patients risk for prostate cancer.  We are particularly interested in improving risk prediction for men of African ancestry (moAA) who bear a substantial disease burden (See Pie Chart above) and are typically under represented in efforts to develop risk prediction models. We are exploring the predictive power of generalized linear models, support vector machines and Copula-based generative models. 


Computational Mathematics

Bruselator solution profile

Advection-diffusion-reaction (ADR) equations are mathematical models that describe the spatio-temporal dynamics of many physical and biological processes such as the transport of pollutants in ground water, tumor angiogenesis and the formation of biological patterns (Fig A, B show patterns from Brusselator ADR system). Computational solutions to ADR equations present several difficulties such as stiff reaction kinetics which imposes stability restrictions on the temporal step size of explicit schemes. ADR problems are typically posed in two or three dimensional space which necessitates domain decomposition or dimensional splitting methods to obtain reasonable cpu run times. Here, we are developing efficient exponential time differencing schemes, based on rational approximations of the matrix exponential, which can be coupled to finite difference, finite element and spectral methods to solve ADR problems. Our schemes have dimensional splitting capability to reduce the run time of multidimensional problems. We are currently developing higher order and adaptive time formulations of the scheme to handle very stiff problems.

bottom of page