My research interests lie in mathematical modeling and analysis of biological phenomena. My research efforts to date can be divided into two major components -- multi-scale modeling of colon tissue cell dynamics and colorectal cancer initiation, and semi-discrete modeling of host-parasitoid population dynamics. My professional goal is to continue to apply mathematical techniques to biological problems, with an emphasis on collaborative research and undergraduate mentorship. Below are several figures that summarize some past and present research projects.
The colonic crypt is a small fold or invagination in the epithelial layer of the colon, which is responsible for consistent cell renewal. The normal crypt cycle is a mechanism by which the crypt itself divides to form two identical crypts. This process is characterized by an initial growth of the crypt followed by a budding at the base. The crypt bifurcation elongates and extends upwards until it reaches the lumen, finishing the crypt fission phase. The crypt cycle is a continuous process responsible for epithelial maintenance. To model this, we consider the crypt as a stack of cell rings such that each ring divides into two identical rings in a cascade-type fashion that begins with one cell at the base. The evolution of each cell-ring-division is modeled using the Level Set Method. The video above is the result of our working model.
The colonic crypt is a small finger-like invagination in the epithelial lining of the colon (right). Cells of different phenotypes make up the tissue of the crypt walls. Stem cells reside at the base and continually create daughter cells that migrate up the crypt until they are terminally differentiated and sloughed away near the top. Our model uses a degree of stemness (left) as a measure of cell-age and defines cell movement using non-linear diffusion. At steady-state, an abundance of proliferating cells (middle, blue curve) are kept above the stemness threshold and in the proliferating zone (left). Below this threshold, cells are differentiated (middle, red curve) and exit through the top. Each figure represents the steady-state output of the model under normal circumstances (no mutation). The paper is here.
The Wnt Pathway is a cascade of chemical reactions within a cell that ultimately determines the destruction or accumulation of a proliferation promoting protein called beta-catenin. This figure shows a single trajectory in the phase space of a system of nonlinear differential equations that describes the interactions of three key proteins involved the Wnt pathway: beta-catenin, Axin, and APC. When the concentration of Axin is relatively small, the three concentrations become periodic in time. This indicates that a cell switches, periodically, between a proliferative and quiescent state. The paper is here.
This project considered a system of linear ODEs to model the degradation of a peptide substrate reporter for protein kinase B (VI-B) in five different cell cultures from data provided by the Chemistry department. After solving these equations one by one, we find the best fit parameters that match the data using least squares tools in Matlab with an iterative approach. Drawing from the histogram plots of parameter distributions, we conclude that the most popular values which yield the smallest residuals are best fit parameters. The figure shows the best fit of our model to the given data. A poster detailing the full problem is here. The paper can be found here.
Using the semi-discrete framework, we address the tendency of parasitoids to feed on host larvae. This figure compares the stability region of the non-zero equilibrium of the semi-discrete model with and without host-feeding when coupled with a density dependent host-mortality. The non-zero equilibrium corresponds to the system that yields coexistence between the host and the parasitoid, which is observed in nature. Host-feeding burdens the parasitoid population since the no-parasitoid equilibrium stability region is larger. Hence, host-feeding causes an inefficiency in the parasitoid's yearly reproductive habits, which yields a higher population of hosts per generation. Find the paper here.
Using the traditional discrete structure of host-parasitoid models, we assume that the parasitoid has a different attack rate for each host, which means each host has variable risk. The risk, x, is assumed to be independent of local host density and distributed according to the continuous probability distribution, p(x). We implement the variability in existing host-parasitoid models to search for the conditions of coexistence of the host larvae population and two parasitoid population, which have different time frames of attack and different rates of attack. The figure shows a stable trajectory in which all three populations coexist. An investigation of the stability region for coexistence is still underway. A poster detailing the model is here.
Using the semi-discrete framework, we investigate the migration of parasitoids between two locations. Typically, hosts are immobile and parasitoids can fly. This figure shows the surface of the third, most restrictive Jury condition for evaluating the stability region in (alpha, beta) space, where alpha and beta are the proportions of host and parasitoids at one location, respectively. The surface lies above the zero plane, indicating that the model is stable for all values of alpha and beta. However, we can see that for some values of alpha and beta, the surface nearly touches the zero plane, which means the system is `less stable' for those values.
Using the semi-discrete framework, we incorporate an infected host-feeding scenario, which assumes that the parasitoids feed on already infected hosts. Essentially, this means a parasitoid does not know a host has already been oviposited, resulting in a loss of potential adult parasitoids via host-feeding. This, interestingly, results in an oscillatory behavior for a critical value of R (viable eggs per adult host), which produces chaotic behavior for even larger values of R. This figure shows a period doubling bifurcation in the parameter R for the parasitoid population.