Balanced Allocation on Graphs: A Random Walk Approach

In this paper we propose algorithms for allocating $n$ sequential balls into $n$ bins that are interconnected as a $d$-regular $n$-vertex graph $G$, where $d\ge3$ can be any integer.Let $l$ be a given positive integer. In each round $t$, $1\le t\le n$, ball $t$ picks a node of $G$ uniformly at random and performs a non-backtracking random walk of length $l$ from the chosen node.Then it allocates itself on one of the visited nodes with minimum load (ties are broken uniformly at random). Suppose that $G$ has a sufficiently large girth and $d=\omega(\log n)$. Then we establish an upper bound for the maximum number of balls at any bin after allocating $n$ balls by the algorithm, called {\it maximum load}, in terms of $l$ with high probability. We also show that the upper bound is at most an $O(\log\log n)$ factor above the lower bound that is proved for the algorithm. In particular, we show that if we set $l=\lfloor(\log n)^{\frac{1+\epsilon}{2}}\rfloor$, for every constant $\epsilon\in (0, 1)$, and $G$ has girth at least $\omega(l)$, then the maximum load attained by the algorithm is bounded by $O(1/\epsilon)$ with high probability.Finally, we slightly modify the algorithm to have similar results for balanced allocation on $d$-regular graph with $d\in[3, O(\log n)]$ and sufficiently large girth

Sub-quadratic time for Riemann-Roch spaces. The case of smooth divisors over nodal plane projective curves

International audienceWe revisit the seminal Brill-Noether algorithm in the rather generic situation of smooth divisors over a nodal plane projective curve. Our approach takes advantage of fast algorithms for polynomials and structured matrices. We reach sub-quadratic time for computing a basis of a Riemann-Roch space. This improves upon previously known complexity bounds

Holstein polarons in a strong electric field: delocalized and stretched states

The coherent dynamics of a Holstein polaron in strong electric fields is considered under different regimes. Using analytical and numerical analysis, we show that even for small hopping constant and weak electron-phonon interaction, the original discrete Wannier-Stark (WS) ladder electronic states are each replaced by a semi-continuous band if a resonance condition is satisfied between the phonon frequency and the ladder spacing. In this regime, the original localized WS states can become {\em delocalized}, yielding both `tunneling' and `stretched' polarons. The transport properties of such a system would exhibit a modulation of the phonon replicas in typical tunneling experiments. The modulation will reflect the complex spectra with nearly-fractal structure of the semi-continuous band. In the off-resonance regime, the WS ladder is strongly deformed, although the states are still localized to a degree which depends on the detuning: Both the spacing between the levels in the deformed ladder and the localization length of the resulting eigenfunctions can be adjusted by the applied electric field. We also discuss the regime beyond small hopping constant and weak coupling, and find an interesting mapping to that limit via the Lang-Firsov transformation, which allows one to extend the region of validity of the analysis.Comment: 10 pages, 13 figures, submitted to PR

Nuclear target search at the single molecule level: protein interactions define the exploration landscape

Gene regulation relies on highly mobile transcription factors (TFs) exploring the nucleoplasm in search of their targets. Our view of the nucleus has evolved from that of an isotropic and homogenous reactor to that of a highly organized yet very dynamic organelle. However important questions remain on how these regulatory factors explore the nuclear environment in search of their DNA or protein targets, and how their exploration strategy affects the kinetics of transcriptional regulation. We implemented a single-molecule tracking assay to determine the TFs dynamics using photoactivatable tags in human cells. We investigated the mobility of several nuclear proteins, including the transcription factor c-Myc and the elongation factor P-TEFb. We found that, while their diffusion speed was comparable, these proteins largely differed in terms of their exploration geometry. We discovered that c-Myc is a global explorer diffusing in the nucleus without spatial constraints. In contrast, the positive transcription elongation factor P-TEFb is a local explorer that oversamples its environment, constrained by a fractal nuclear architecture. Consequently, each c-Myc molecule is equally available for all nuclear sites while P-TEFb reaches its targets in a position-dependent manner. We also measured the mobility of a P-TEFb mutant in which the interaction with the CTD of the RNA Pol II was truncated. In this case, the single-molecule experiments suggested a global exploration of the P-TEFb mutant, consistent with free diffusion. Our observations are in line with a model in which the exploration geometry of TFs is constrained by their interactions and not by exclusion properties. Our findings have strong implications on how proteins react in the nucleus and how their function can be regulated in space and time

APM_GUI: analyzing particle movement on the cell membrane and determining confinement

<p>Abstract</p> <p>Background</p> <p>Single-particle tracking is a powerful tool for tracking individual particles with high precision. It provides useful information that allows the study of diffusion properties as well as the dynamics of movement. Changes in particle movement behavior, such as transitions between Brownian motion and temporary confinement, can reveal interesting biophysical interactions. Although useful applications exist to determine the paths of individual particles, only a few software implementations are available to analyze these data, and these implementations are generally not user-friendly and do not have a graphical interface,.</p> <p>Results</p> <p>Here, we present APM_GUI (Analyzing Particle Movement), which is a MatLab-implemented application with a Graphical User Interface. This user-friendly application detects confined movement considering non-random confinement when a particle remains in a region longer than a Brownian diffusant would remain. In addition, APM_GUI exports the results, which allows users to analyze this information using software that they are familiar with.</p> <p>Conclusions</p> <p>APM_GUI provides an open-source tool that quantifies diffusion coefficients and determines whether trajectories have non-random confinements. It also offers a simple and user-friendly tool that can be used by individuals without programming skills.</p

The relative orientation of the TM3 and TM4 domains varies between α1 and α3 glycine receptors

Glycine receptors (GlyRs) are anion-conducting members of the pentameric ligand-gated ion channel family. We previously showed that the dramatic difference in glycine efficacies of α1 and α3 GlyRs is largely attributable to their nonconserved TM4 domains. Because mutation of individual nonconserved TM4 residues had little effect, we concluded that the efficacy difference was a distributed effect of all nonconserved TM4 residues. We therefore hypothesized that the TM4 domains of α1 and α3 GlyRs differ in structure, membrane orientation, and/or molecular dynamic properties. Here we employed voltage-clamp fluorometry to test whether their TM4 domains interact differently with their respective TM3 domains. We found a rhodamine fluorophore covalently attached to a homologous TM4 residue in each receptor interacts differentially with a conserved TM3 residue. We conclude that the α1 and α3 GlyR TM4 domains are orientated differently relative to their TM3 domains. This may underlie their differential ability to influence glycine efficacy

Real-Time Imaging and Quantification of Amyloid-β Peptide Aggregates by Novel Quantum-Dot Nanoprobes

Background: Protein aggregation plays a major role in the pathogenesis of neurodegenerative disorders, such as Alzheimer’s disease. However, direct real-time imaging of protein aggregation, including oligomerization and fibrillization, has never been achieved. Here we demonstrate the preparation of fluorescent semiconductor nanocrystal (quantum dot; QD)-labeled amyloid-b peptide (QDAb) and its advanced applications. Methodology/Principal Findings: The QDAb construct retained Ab oligomer-forming ability, and the sizes of these oligomers could be estimated from the relative fluorescence intensities of the imaged spots. Both QDAb coaggregation with intact Ab42 and insertion into fibrils were detected by fluorescence microscopy. The coaggregation process was observed by real-time 3D imaging using slit-scanning confocal microscopy, which showed a typical sigmoid curve with 1.5 h in the lag-time and 12 h until saturation. Inhibition of coaggregation using an anti-Ab antibody can be observed as 3D images on a microscopic scale. Microglia ingested monomeric QDAb more significantly than oligomeric QDAb, and the ingested QDAb was mainly accumulated in the lysosome. Conclusions/Significance: These data demonstrate that QDAb is a novel nanoprobe for studying Ab oligomerization an

Statistical Aging and Non Ergodicity in the Fluorescence of Single Nanocrystals

The relation between single particle and ensemble measurements is adressed for semiconductor CdSe nanocrystals. We record their fluorescence at the single molecule level and analyse their emission intermittency, which is governed by unusual random processes known as Levy statistics. We report the observation of statistical aging and ergodicity breaking, both related to the occurrence of Levy statistics. Our results show that the behaviour of ensemble quantities, such as the total fluorescence of an ensemble of nanocrystals, can differ from the time averaged individual quantities, and must be interpreted with care.Comment: 4 pages, 3 figure

Pre-formulation and systematic evaluation of amino acid assisted permeability of insulin across in vitro buccal cell layers

The aim of this work was to investigate alternative safe and effective permeation enhancers for buccal peptide delivery. Basic amino acids improved insulin solubility in water while 200 and 400 µg/mL lysine significantly increased insulin solubility in HBSS. Permeability data showed a significant improvement in insulin permeation especially for 10 µg/mL of lysine (p < 0.05) and 10 µg/mL histidine (p < 0.001), 100 µg/mL of glutamic acid (p < 0.05) and 200 µg/mL of glutamic acid and aspartic acid (p < 0.001) without affecting cell integrity; in contrast to sodium deoxycholate which enhanced insulin permeability but was toxic to the cells. It was hypothesized that both amino acids and insulin were ionised at buccal cavity pH and able to form stable ion pairs which penetrated the cells as one entity; while possibly triggering amino acid nutrient transporters on cell surfaces. Evidence of these transport mechanisms was seen with reduction of insulin transport at suboptimal temperatures as well as with basal-to-apical vectoral transport, and confocal imaging of transcellular insulin transport. These results obtained for insulin is the first indication of a possible amino acid mediated transport of insulin via formation of insulin-amino acid neutral complexes by the ion pairing mechanism