Brooks Emerick

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.

Using the semi-discrete framework, we investigate the migration of parasitoids between two locations. Typically, hosts are immobile and parasitoids can fly. We consider the idea of redistribution of hosts between yearly updates. This means that each patch has a similar proportion of hosts. This figure shows the stability region for different values of alpha (the proportion of hosts at the first patch) versus the log of the local migration rates. We see that much of the space is stable, indicating that coexistence at both patches occurs. However, there is a sliver of unstable possibilities. The paper is 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 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.

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. 

A minimium cardinality set covering problem is a well-known integer programming problem. Given a random matrix of ones and zeros with fixed dimensions and density, it is possible to determine the the smallest set of columns from that matrix that covers all rows. However, once this minmal set of columns is found, is there any optimal solutions of the same cardinality? The answer is sure, in some cases. In fact, there may be a large number of alternative solutions of the same cardinality. Is there a way to predict this number? We seek an answer to this question using statistics and machine learning. The plot above shows a boxplot of the number of optimal solutions for each minimal cardinality for 10x20 matrices with 20% density.

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. The paper can be found here.