The synchronization of superparamagnetic beads driven by a micro-magnetic ratchet.

We present theoretical, numerical, and experimental analyses on the non-linear dynamic behavior of superparamagnetic beads exposed to a periodic array of micro-magnets and an external rotating field. The agreement between theoretical and experimental results revealed that non-linear magnetic forcing dynamics are responsible for transitions between phase-locked orbits, sub-harmonic orbits, and closed orbits, representing different mobility regimes of colloidal beads. These results suggest that the non-linear behavior can be exploited to construct a novel colloidal separation device that can achieve effectively infinite separation resolution for different types of beads, by exploiting minor differences in their bead's properties. We also identify a unique set of initial conditions, which we denote the "devil's gate" which can be used to expeditiously identify the full range of mobility for a given bead type.

UNC-6 (netrin) stabilizes oscillatory clustering of the UNC-40 (DCC) receptor to orient polarity.

The receptor deleted in colorectal cancer (DCC) directs dynamic polarizing activities in animals toward its extracellular ligand netrin. How DCC polarizes toward netrin is poorly understood. By performing live-cell imaging of the DCC orthologue UNC-40 during anchor cell invasion in Caenorhabditis elegans, we have found that UNC-40 clusters, recruits F-actin effectors, and generates F-actin in the absence of UNC-6 (netrin). Time-lapse analyses revealed that UNC-40 clusters assemble, disassemble, and reform at periodic intervals in different regions of the cell membrane. This oscillatory behavior indicates that UNC-40 clusters through a mechanism involving interlinked positive (formation) and negative (disassembly) feedback. We show that endogenous UNC-6 and ectopically provided UNC-6 orient and stabilize UNC-40 clustering. Furthermore, the UNC-40-binding protein MADD-2 (a TRIM family protein) promotes ligand-independent clustering and robust UNC-40 polarization toward UNC-6. Together, our data suggest that UNC-6 (netrin) directs polarized responses by stabilizing UNC-40 clustering. We propose that ligand-independent UNC-40 clustering provides a robust and adaptable mechanism to polarize toward netrin.

Algorithms for Geometric Matching, Clustering, and Covering

With the popularization of GPS-enabled devices such as mobile phones, location data are becoming available at an unprecedented scale. The locations may be collected from many different sources such as vehicles moving around a city, user check-ins in social networks, and geo-tagged micro-blogging photos or messages. Besides the longitude and latitude, each location record may also have a timestamp and additional information such as the name of the location. Time-ordered sequences of these locations form trajectories, which together contain useful high-level information about people's movement patterns.

The first part of this thesis focuses on a few geometric problems motivated by the matching and clustering of trajectories. We first give a new algorithm for computing a matching between a pair of curves under existing models such as dynamic time warping (DTW). The algorithm is more efficient than standard dynamic programming algorithms both theoretically and practically. We then propose a new matching model for trajectories that avoids the drawbacks of existing models. For trajectory clustering, we present an algorithm that computes clusters of subtrajectories, which correspond to common movement patterns. We also consider trajectories of check-ins, and propose a statistical generative model, which identifies check-in clusters as well as the transition patterns between the clusters.

The second part of the thesis considers the problem of covering shortest paths in a road network, motivated by an EV charging station placement problem. More specifically, a subset of vertices in the road network are selected to place charging stations so that every shortest path contains enough charging stations and can be traveled by an EV without draining the battery. We first introduce a general technique for the geometric set cover problem. This technique leads to near-linear-time approximation algorithms, which are the state-of-the-art algorithms for this problem in either running time or approximation ratio. We then use this technique to develop a near-linear-time algorithm for this

shortest-path cover problem.