Research

Some topics I'm currently interested in.

Mechanical Interactions in Cellular Systems

The self-assembly and aggregation of proteins and other macromolecules is a vital component in many cellular processes. For example, BAR and ENTH proteins are recruited to specially shape lipid bilayers at the microscopic level, which form the coatings of cellular parts, such as the endoplasmic reticulum, but are also necessary for endo- and exocytosis - the process by which bubbles carrying waste and nutrients are created and absorbed in the cell. Another example involves the neutralization of diseased cellular bodies by the immune system via the recruitment of white blood cells. One of the steps in this process requires the formation and the organization of molecular tethers that bridge the gap between white blood-cells and diseased cells; a vital mechanism in the extraction of diseased molecules called antigens which trigger the creation of killer T-cells necessary to eliminate the diseased threat.

I'm interested in devising models of membrane shaping and the immune system which involve purely mechanical interactions; a contribution that is increasingly being shown as highly nontrivial in experiments. A simplified model of cylindrical cross-sectional shaping I've done has shown that proteins are capable of pattern formation typical of chemical systems whilst, in the 2D setting, simulations of proteins with mechanical quadrupole-like properties form a network aggregate that may act as a prelude for global changes to the membrane geometry; very complex phenomenon arising from the enforcing of "simple" mechanical principles!

Nonlinear dynamics, chaos, and applications thereof

Differential equations which are nonlinear appear in remarkably diverse systems of a physical, biological, and abstract nature. Typically, such systems do not allow an exact explicit solution to be written, but there are other ways to determine the nature of the problem and its solutions. Mathematically, the solutions are described as flows along a manifold in the space of dependent variables, and its derivatives, of the equation. By studying the properties and structures of these flows, we can obtain important information about the solutions, such as where this system will tend to under some asymptotic limit, and about the mathematical structure of the system and its stability under perturbations imposed on it.

Unlike linear systems, nonlinear systems can exhibit chaos of which there are two types: dissipative, for example the Lorenz equation with its strange attractor, and Hamiltonian, which features no attractors at all. The latter interests me because of its mathematical structure; systems that are Hamiltonian have trajectories which are confined to an invariant manifold, or, more physically, it is a statement that the total energy of the system stays constant for all time. The invariant structure lends the system to be more analytically intractable. Using theories pioneered by Kolmogorov, Arnold, and Moser as well as Melnikov, one can determine criterion for the onset of chaos, which I've applied to the rigid body rotations of celestial bodies. Currently, I'm studying under what perturbations an integrable system (i.e. a system where the number of degrees of freedom is half of the conserved quantities of motion) will exhibit chaos. Such a result would have significant implications for empirical measurements of even "well-behaved" nonlinear systems.

Deformable rotators

In outer space, numerous bodies rotate, such as pulsars, planets both solid and liquid, asteroids, and cosmic dust granules. All such bodies are deformable - a stress acting on the material will produce a strain - however, this leads to a deceptively simple question: What happens to the rotation of such objects when they are allowed to bend, compress, or stretch? For a special subset of celestial bodies which exhibit non-principal axis (NPA) tumbling, their precession - similar to that of a spinning top - relaxes as a result of inelastic stresses; an object rotating around its longest principal axis will eventually rotate around its shortest axis as a result of its deformability. Recent work I've published suggests that, by using an approach more consistent with the field of continuum mechanics, the mechanism of precession decay occurs on a much wider range of timescales than could be predicted by previous studies involving phenomenological assertions.

An even more interesting question involves determining what occurs after the object has reached a minimal energy rotating state around the shortest principle axis. Accepted theoretical models predict that the rotator will continue moving in exactly the same manner forever; a result which contradicts our physical intuition that the object should eventually come to stop! Currently, I'm devising a model which features the effects of non-trivial deformations, rather than supposing the body is close to rigid, to obtain insight into the interactions between the kinematics of the rotator and its changing geometry as a result of inertial forces.

Vortex dynamics

Vortices are regions in the fluid where the flow rotates around an axis. Though they are conceptually simple objects in 3D flow, the evolution of a single vortex filament, let alone a multiple number of them, exhibits highly non-trivial dynamics. Such dynamics still constitute an important question in fluid dynamics, in particular, the onset of turbulence in the inertial or high Reynolds number regime. However, there are also important applications from fields as wide-ranging as aerospace engineering, studying the vortices shed from wing-tips, to the motion of hurricanes and cyclones in climate science.

My interest involves looking at the dynamics of multiple vortices and extending the single vortex filament solutions seen in the Biot-Savart and Local Induction Approximation regimes, the work of which has culminated in a paper published (with nifty simulations that can be seen here). Currently, I am looking at developing a simplified model for quantum vortex filaments with the goal of studying vortex entanglement and the onset of quantum turbulence.

Page updated

Google Sites

Report abuse