arXiv subjects
Publications and source records attributed to Patrick Allen.
Some of the multiplicity-freeness results in ``Modular Gelfand pairs and multiplicity-free representations'' are stated in overly broad generality. We provide counterexamples and partial corrections.
Nanodiscs are binary discoidal complexes of a phospholipid bilayer circumscribed by belt-like helical scaffold proteins. Using coarse-grained and all-atom molecular dynamics simulations, we explore the stability, size, and structure of nanodiscs formed between the N-terminal domain of apolipoprotein E3 (apoE3-NT) and variable number of 1,2-dimyristoyl-sn-glycero-3-phosphocholine (DMPC) molecules. We study both parallel and antiparallel double-belt configurations, consisting of four proteins per nanodisc. Our simulations predict nanodiscs containing between 240 and 420 DMPC molecules to be stable. The antiparallel configurations exhibit an average of 1.6 times more amino acid interactions between protein chains and 2 times more ionic contacts, compared to the parallel configuration. With one exception, DMPC order parameters are consistently larger in the antiparallel configuration than in the parallel one. In most cases, the root mean square deviation of the positions of the protein backbone atoms is smaller in the antiparallel configuration. We further report nanodisc size, thickness, radius of gyration, and solvent accessible surface area. Combining all investigated parameters, we hypothesize the antiparallel protein configuration leading to more stable and more rigid nanodiscs than the parallel one.
Let a and b be positive integers with a \leq b. An (a,b)-triple is a set {x,ax+d,bx+ 2d}, where x,d \geq 1. Define T(a,b;r) to be the least positive integer n such that any r-coloring of {1,2...,n} contains a monochromatic (a,b)-triple. Earlier results gave an upper bound on T(a,b;2) that is a fourth degree polynomial in b and a, and a quadratic lower bound. A new upper bound for T(a,b;2) is given that is a quadratic. Additionally, lower bounds are given for the case in which a = b, updated tables are provided, and open questions are presented.