As an applied mathematician working in the field of mathematical biology, my goal as a researcher is to investigate important questions from complex biological systems using mathematical and computational models. Modeling can provide valuable insights when traditional experiments are either too expensive, or impossible. The main focus of my research has been using mechanical models of microorganisms to study their coordinated behavior and interactions with their environment. These models often require a wide variety of mathematics, in particular partial differential equations and numerical analysis.


Mixing and pumping by pairs of helices in a viscous fluid

Abstract: Here we study the fluid dynamics of a pair of rigid helices rotating at a constant velocity, tethered at their bases, in a viscous fluid. Our computations use a regularized Stokeslet framework, both with and without a bounding plane, so we are able to discern precisely what flow features are unaccounted for in studies that ignore the surface from which the helices emanate. We examine how the spacing and phase difference between identical rotating helices affects their pumping ability, axial thrust, and power requirements. We also find that optimal mixing of the fluid around two helices is achieved when they rotate in opposite phase, and that the mixing is enhanced as the distance between the helices decreases.

Reversals and collisions optimize protein exchange in bacterial swarms (2017)

Abstract: Swarming groups of bacteria coordinate their behavior by self-organizing as a population to move over surfaces in search of nutrients and optimal niches for colonization. Many open questions remain about the cues used by swarming bacteria to achieve this self-organization. While chemical cue signaling known as quorum sensing is well-described, swarming bacteria often act and coordinate on time scales that could not be achieved via these extracellular quorum sensing cues. Here, cell-cell contact-dependent protein exchange is explored as a mechanism of intercellular signaling for the bacterium Myxococcus xanthus. A detailed biologically calibrated computational model is used to study how M. xanthus optimizes the connection rate between cells and maximizes the spread of an extracellular protein within the population. The maximum rate of protein spreading is observed for cells that reverse direction optimally for swarming. Cells that reverse too slowly or too fast fail to spread extracellular protein efficiently. In particular, a specific range of cell reversal frequencies was observed to maximize the cell-cell connection rate and minimize the time of protein spreading. Furthermore, our findings suggest that predesigned motion reversal can be employed to enhance the collective behavior of biological synthetic active systems.

Controlling a Cockroach Infestation (2016)

Abstract: The cockroach is one of the world’s most prolific and resilient pests, with over 3,500 species worldwide. It is important to understand the growth and adaptive mechanisms of cockroach colonies in order to safely control these populations. We present a continuous time, age-structured population model of the Blattella germanica cockroach that includes the application of pesticides and the development of resistant subpopulations. The resulting system of differential equations is then used to optimize treatment strategies using analytical and heuristic optimization techniques. While the model shows that the roach-free equilibrium is always unstable, the strategic application of pesticides can keep populations low, even when a drug-resistant subpopulation develops.

Flow Induced by Bacterial Carpets and Transport of Microscale Loads (2015)

Abstract: In this paper we utilize themethod of regularized Stokeslets to explore flow fields induced by ‘carpets’ of rotating flagella. We model each flagellum as a rigid, rotating helix attached to a wall, and study flows around both a single helix and a small patch of multiple helices. To test our numerical method and gain intuition about flows induced by a single rotating helix, we first perform a numerical time-reversibility experiment. Next, we investigate the hypothesis put forth in (Darnton et al., Biophys J 86, 1863–1870, 2004) that a small number of rotating flagella could produce “whirlpools” and “rivers” a small distance above them. Using our model system, we are able to produce “whirlpools” and “rivers” when the helices are rotating out of phase. Finally, to better understand the transport of microscale loads by flagellated microorganisms, we model a fully coupled helix-vesicle system by placing a finite-sized vesicle held together by elastic springs in fluid near one or two rotating helices. We compare the trajectories of the vesicle and a tracer particle initially placed at the centroid of vesicle and find that the two trajectories can diverge significantly within a short amount of time. Interestingly, the divergent behavior is extremely sensitive to the initial position within the fluid.

Sizing it up: The mechanical feedback hypothesis of organ growth regulation (2014)

Abstract: The question of how the physical dimensions of animal organs are specified has long fascinated both experimentalists and computational scientists working in the field of developmental biology. Research over the last few decades has identified many of the genes and signaling pathways involved in organizing the emergent multi-scale features of growth and homeostasis. However, an integrated model of organ growth regulation is still unrealized due to the numerous feedback control loops found within and between intercellular signaling pathways as well as a lack of understanding of the exact role of mechanotransduction. Here, we review several computational and experimental studies that have investigated the mechanical feedback hypothesis of organ growth control, which postulates that mechanical forces are important for regulating the termination of growth and hence the final physical dimensions of organs. In particular, we highlight selected computational studies that have focused on the regulation of growth of the Drosophila wing imaginal disc. In many ways, these computational and theoretical approaches continue to guide experimental inquiry. We demonstrate using several examples how future progress in dissecting the crosstalk between the genetic and biophysical mechanisms controlling organ growth might depend on the close coupling between computational and experimental approaches, as well as comparison of growth control mechanisms in other systems.