A Unified Framework for Hopsets

Given an undirected graph G = (V,E), an (?,?)-hopset is a graph H = (V,E\u27), so that adding its edges to G guarantees every pair has an ?-approximate shortest path that has at most ? edges (hops), that is, d_G(u,v) ? d_{G?H}^(?)(u,v) ? ?? d_G(u,v). Given the usefulness of hopsets for fundamental algorithmic tasks, several different algorithms and techniques were developed for their construction, for various regimes of the stretch parameter ?. In this work we devise a single algorithm that can attain all state-of-the-art hopsets for general graphs, by choosing the appropriate input parameters. In fact, in some cases it also improves upon the previous best results. We also show a lower bound on our algorithm. In [Ben-Levy and Parter, 2020], given a parameter k, a (O(k^?),O(k^{1-?}))-hopset of size O?(n^{1+1/k}) was shown for any n-vertex graph and parameter 0 < ? < 1, and they asked whether this result is best possible. We resolve this open problem, showing that any (?,?)-hopset of size O(n^{1+1/k}) must have ??? ? ?(k)

Modeling Nanoscale Transport Systems

Mathematical formulation and physical models are the foundation of scientific understanding and technological advancement. Our ability to design experiments effectively is heavily dependent on our physical understanding of the system under investigation, and careful mathematical analysis is required in order to effectively progress from scientific concepts towards viable technologies. With increasing system complexity, the focus of mathematical formulation has shifted from simple, elegant models which rely on basic physical concepts to tailored, increasingly complex solutions using high-powered simulations and numerical solutions. While these methods may provide insights into specific systems, adapting these models to different systems is generally difficult, even when the systems under question operate according to the same physical laws. This is especially evident in nanobiotechnology, where the complexity of the systems studied has given rise to experiment-driven focus. Our aim is to focus on the mathematical modeling of transport processes in nanoscale systems, and to construct generalized, conceptual models for three model systems, which in turn could be applied to many biological and engineered systems. The three model systems we use - enzyme cascades, coupled molecular motors and self-assembling molecular shuttles provide a broad basis for generalized transport systems in nanoscale systems. These systems combine diffusive and active transport, as well as diverse assembly conditions and multi-scale systems with size scales spanning nano- to millimeter sizes and system complexity ranging from isolated two-component systems to multimolecular, highly-coupled systems. By applying and adapting these basic models to increasingly complex systems, we can both understand the physics behind nanoscale systems, as well as design these systems with increased robustness, scalability and repeatability

Detecting stars at the galactic centre via synchrotron emission

Stars orbiting within 1\arcsec of the supermassive black hole in the Galactic Centre, Sgr A*, are notoriously difficult to detect due to obscuration by gas and dust. We show that some stars orbiting this region may be detectable via synchrotron emission. In such instances, a bow shock forms around the star and accelerates the electrons. We calculate that around the 10 GHz band (radio) and at 1014 Hz (infrared) the luminosity of a star orbiting the black hole is comparable to the luminosity of Sgr A*. The strength of the synchrotron emission depends on a number of factors including the star's orbital velocity. Thus, the ideal time to observe the synchrotron flux is when the star is at pericenter. The star S2 will be \sim 0.015\arcsec from Sgr A* in 2018, and is an excellent target to test our predictions.Astronom

Origins of Activity Enhancement in Enzyme Cascades on Scaffolds

The concept of “metabolic channeling” as a result of rapid transfer of freely diffusing intermediate substrates between two enzymes on nanoscale scaffolds is examined using simulations and mathematical models. The increase in direct substrate transfer due to the proximity of the two enzymes provides an initial but temporary boost to the throughput of the cascade and loses importance as product molecules of enzyme 1 (substrate molecules of enzyme 2) accumulate in the surrounding container. The characteristic time scale at which this boost is significant is given by the ratio of container volume to the product of substrate diffusion constant and interenzyme distance and is on the order of milliseconds to seconds in some experimental systems. However, the attachment of a large number of enzyme pairs to a scaffold provides an increased number of local “targets”, extending the characteristic time. If substrate molecules for enzyme 2 are sequestered by an alternative reaction in the container, a scaffold can result in a permanent boost to cascade throughput with a magnitude given by the ratio of the above-defined time scale to the lifetime of the substrate molecule in the container. Finally, a weak attractive interaction between substrate molecules and the scaffold creates a “virtual compartment” and substantially accelerates initial throughput. If intermediate substrates can diffuse freely, placing individual enzyme pairs on scaffolds is only beneficial in large cells, unconfined extracellular spaces or in systems with sequestering reactions

Velocity Fluctuations in Kinesin‑1 Gliding Motility Assays Originate in Motor Attachment Geometry Variations

Motor proteins such as myosin and kinesin play a major role in cellular cargo transport, muscle contraction, cell division, and engineered nanodevices. Quantifying the collective behavior of coupled motors is critical to our understanding of these systems. An excellent model system is the gliding motility assay, where hundreds of surface-adhered motors propel one cytoskeletal filament such as an actin filament or a microtubule. The filament motion can be observed using fluorescence microscopy, revealing fluctuations in gliding velocity. These velocity fluctuations have been previously quantified by a motional diffusion coefficient, which Sekimoto and Tawada explained as arising from the addition and removal of motors from the linear array of motors propelling the filament as it advances, assuming that different motors are not equally efficient in their force generation. A computational model of kinesin head diffusion and binding to the microtubule allowed us to quantify the heterogeneity of motor efficiency arising from the combination of anharmonic tail stiffness and varying attachment geometries assuming random motor locations on the surface and an absence of coordination between motors. Knowledge of the heterogeneity allows the calculation of the proportionality constant between the motional diffusion coefficient and the motor density. The calculated value (0.3) is within a standard error of our measurements of the motional diffusion coefficient on surfaces with varying motor densities calibrated by landing rate experiments. This allowed us to quantify the loss in efficiency of coupled molecular motors arising from heterogeneity in the attachment geometry