Persistence of the Jordan center in Random Growing Trees

The Jordan center of a graph is defined as a vertex whose maximum distance to other nodes in the graph is minimal, and it finds applications in facility location and source detection problems. We study properties of the Jordan Center in the case of random growing trees. In particular, we consider a regular tree graph on which an infection starts from a root node and then spreads along the edges of the graph according to various random spread models. For the Independent Cascade (IC) model and the discrete Susceptible Infected (SI) model, both of which are discrete time models, we show that as the infected subgraph grows with time, the Jordan center persists on a single vertex after a finite number of timesteps. Finally, we also study the continuous time version of the SI model and bound the maximum distance between the Jordan center and the root node at any time.Comment: 28 pages, 14 figure

Structure, function and regulation of integral membrane transport proteins

Integral membrane transport proteins are essential for the transport of a wide variety of substrates such as ions, drugs and metabolites across the membranes of microorganisms. They are found in diverse locations such as outer and inner membranes of archaeal, prokaryotic and eukaryotic cells, as well as the outer layers of the membrane-bound organelles, including chloroplasts, endoplasmic reticulum and mitochondria. These proteins are capable of facilitating the transport of substrates both from outside the cell to inside, and vice versa, and they can be very specific, thereby playing a significant role in substrate selection. Integral membrane transport proteins also play important roles in regulation of transport, defense against drugs and antibiotics, and hemostasis. Thus, understanding the structural framework and detailed functional mechanisms of these proteins will contribute to our knowledge of biological processes. In this study, we have investigated the structure, function and regulation of three different types of integral membrane transport proteins. Chapter 2 focuses on elucidating the structure and function of the transcriptional regulator Rv0678 of Mycobacterium tuberculosis, which negatively regulates the expression of the MmpS4-MmpL4 system, which plays a key role in the biosynthesis and transport of lipids from the cytoplasm towards the periplasm. Lipids play an important role in cell wall remodeling and permeability, thereby functioning in bacterial defense against antibiotics. We crystallized Rv0678 and identified the ligand bound to the protein as 2-stearoylglycerol. We used the structure and functional studies to elucidate a possible DNA binding mechanism for the protein. Chapter 3 discusses the crystal structure of the outer membrane channel, CmeC, of CmeABC efflux system of Campylobacter jejuni. CmeABC is a multidrug efflux system that pumps out quinolones and bile acid derivatives. The crystal structure revealed the structural framework of the channel and captured it in closed conformation. Chapter 4 focuses on understanding the structural basis of nitrite transport into the chloroplast in Chlamydomonas reinhardtii. Formate-nitrite transporter family proteins, NAR1.1 and NAR1.5, were crystallized using hanging drop vapor diffusion. Stopped flow light scattering experiments were carried out to identify nitrite as the substrate for both proteins. Site-directed mutagenesis revealed key residues that form the channel and allowed us to propose a possible nitrite transport mechanism

Autobidders with Budget and ROI Constraints: Efficiency, Regret, and Pacing Dynamics

We study a game between autobidding algorithms that compete in an online advertising platform. Each autobidder is tasked with maximizing its advertiser's total value over multiple rounds of a repeated auction, subject to budget and/or return-on-investment constraints. We propose a gradient-based learning algorithm that is guaranteed to satisfy all constraints and achieves vanishing individual regret. Our algorithm uses only bandit feedback and can be used with the first- or second-price auction, as well as with any "intermediate" auction format. Our main result is that when these autobidders play against each other, the resulting expected liquid welfare over all rounds is at least half of the expected optimal liquid welfare achieved by any allocation. This holds whether or not the bidding dynamics converges to an equilibrium and regardless of the correlation structure between advertiser valuations