3x+13x+1 inverse orbit generating functions almost always have natural boundaries

The $3x+k$ function $T_{k}(n)$ sends $n$ to $(3n+k)/2$ resp. $n/2,$ according as $n$ is odd, resp. even, where $k \equiv \pm 1~(\bmod \, 6)$. The map $T_k(\cdot)$ sends integers to integers, and for $m \ge 1$ let $n \rightarrow m$ mean that $m$ is in the forward orbit of $n$ under iteration of $T_k(\cdot).$ We consider the generating functions $f_{k,m}(z) = \sum_{n>0, n \rightarrow m} z^{n},$ which are holomorphic in the unit disk. We give sufficient conditions on $(k,m)$ for the functions $f_{k, m}(z)$ have the unit circle $\{|z|=1\}$ as a natural boundary to analytic continuation. For the $3x+1$ function these conditions hold for all $m \ge 1$ to show that $f_{1,m}(z)$ has the unit circle as a natural boundary except possibly for $m= 1, 2, 4$ and $8$. The $3x+1$ Conjecture is equivalent to the assertion that $f_{1, m}(z)$ is a rational function of $z$ for the remaining values $m=1,2, 4, 8$.Comment: 15 page

Imaging and energetics of single SSB-ssDNA molecules reveal intramolecular condensation and insight into RecOR function.

Escherichia coli single-stranded DNA (ssDNA) binding protein (SSB) is the defining bacterial member of ssDNA binding proteins essential for DNA maintenance. SSB binds ssDNA with a variable footprint of ∼30-70 nucleotides, reflecting partial or full wrapping of ssDNA around a tetramer of SSB. We directly imaged single molecules of SSB-coated ssDNA using total internal reflection fluorescence (TIRF) microscopy and observed intramolecular condensation of nucleoprotein complexes exceeding expectations based on simple wrapping transitions. We further examined this unexpected property by single-molecule force spectroscopy using magnetic tweezers. In conditions favoring complete wrapping, SSB engages in long-range reversible intramolecular interactions resulting in condensation of the SSB-ssDNA complex. RecO and RecOR, which interact with SSB, further condensed the complex. Our data support the idea that RecOR--and possibly other SSB-interacting proteins-function(s) in part to alter long-range, macroscopic interactions between or throughout nucleoprotein complexes by microscopically altering wrapping and bridging distant sites

Direct imaging of RecA nucleation and growth on single molecules of SSB-coated ssDNA.

Escherichia coli RecA is the defining member of a ubiquitous class of DNA strand-exchange proteins that are essential for homologous recombination, a pathway that maintains genomic integrity by repairing broken DNA. To function, filaments of RecA must nucleate and grow on single-stranded DNA (ssDNA) in direct competition with ssDNA-binding protein (SSB), which rapidly binds and continuously sequesters ssDNA, kinetically blocking RecA assembly. This dynamic self-assembly on a DNA lattice, in competition with another protein, is unique for the RecA family compared to other filament-forming proteins such as actin and tubulin. The complexity of this process has hindered our understanding of RecA filament assembly because ensemble measurements cannot reliably distinguish between the nucleation and growth phases, despite extensive and diverse attempts. Previous single-molecule assays have measured the nucleation and growth of RecA--and its eukaryotic homologue RAD51--on naked double-stranded DNA and ssDNA; however, the template for RecA self-assembly in vivo is SSB-coated ssDNA. Using single-molecule microscopy, here we directly visualize RecA filament assembly on single molecules of SSB-coated ssDNA, simultaneously measuring nucleation and growth. We establish that a dimer of RecA is required for nucleation, followed by growth of the filament through monomer addition, consistent with the finding that nucleation, but not growth, is modulated by nucleotide and magnesium ion cofactors. Filament growth is bidirectional, albeit faster in the 5'→3' direction. Both nucleation and growth are repressed at physiological conditions, highlighting the essential role of recombination mediators in potentiating assembly in vivo. We define a two-step kinetic mechanism in which RecA nucleates on transiently exposed ssDNA during SSB sliding and/or partial dissociation (DNA unwrapping) and then the RecA filament grows. We further demonstrate that the recombination mediator protein pair, RecOR (RecO and RecR), accelerates both RecA nucleation and filament growth, and that the introduction of RecF further stimulates RecA nucleation

Perceptual oscillations in gender classification of faces, contingent on stimulus history

Perception is a proactive ‘‘predictive’’ process, in which the brain takes advantage of past experience to make informed guesses about the world to test against sensory data. Here we demonstrate that in the judgment of the gender of faces, beta rhythms play an important role in communicating perceptual experience. Observers classified in forced choice as male or female, a sequence of face stimuli, which were physically constructed to be male or female or androgynous (equal morph). Classification of the androgynous stimuli oscillated rhythmically between male and female, following a complex waveform comprising 13.5 and 17 Hz. Parsing the trials based on the preceding stimulus showed that responses to androgynous stimuli preceded by male stimuli oscillated reliably at 17 Hz, whereas those preceded by female stimuli oscillated at 13.5 Hz. These results suggest that perceptual priors for face perception from recent perceptual memory are communicated through frequency-coded beta rhythms

Trustworthiness perception is mandatory : Task instructions do not modulate fast periodic visual stimulation trustworthiness responses

Acknowledgments The authors thank Bruno Rossion, Joan Liu-Shuang, Talia Retter, and Amy Dawel for their advice and helpful discussions, and Alex Todorov for providing the face stimuli. Supported by an RTP scholarship from the University of Western Australia to DS, an Australian Research Council (ARC) Discovery Early Career Research Award to CS (DE190101043), and ARC Discovery Projects to CS and RP (DP170104602) and RP (DP140101743).Peer reviewedPublisher PD

Computational Studies on the Platinum(II)-Catalyzed Cycloisomerization of 1,6-Dienes into Bicyclopropanes: A Mechanistic Quandary Evaluated by DFT

The mechanism of the (bis(phosphanylethyl)phosphane)Pt2+ catalyzed cyclo-isomerization reaction of 7-methyl-octa-1,6-diene to form 1-isopropylbicyclo[3.1.0]hexane was studied using computational methods. The cyclopropanation step was found to be the turnover-limiting step. The overall reaction proceeds via both a 5-exo and a 6-endo route. W conformations were shown to facilitate cyclopropanation, but do not have any influence on the rate of the 1,2-hydride shifts

The Panchromatic Hubble Andromeda Treasury XI: The Spatially-Resolved Recent Star Formation History of M31

We measure the recent star formation history (SFH) across M31 using optical images taken with the \texit{Hubble Space Telescope} as part of the Panchromatic Hubble Andromeda Treasury (PHAT). We fit the color-magnitude diagrams in ~9000 regions that are ~100 pc $\times$ 100 pc in projected size, covering a 0.5 square degree area (~380 kpc$^2$, deprojected) in the NE quadrant of M31. We show that the SFHs vary significantly on these small spatial scales but that there are also coherent galaxy-wide fluctuations in the SFH back to ~500 Myr, most notably in M31's 10-kpc star-forming ring. We find that the 10-kpc ring is at least 400 Myr old, showing ongoing star formation over the past ~500 Myr. This indicates the presence of molecular gas in the ring over at least 2 dynamical times at this radius. We also find that the ring's position is constant throughout this time, and is stationary at the level of 1 km/s, although there is evidence for broadening of the ring due to diffusion of stars into the disk. Based on existing models of M31's ring features, the lack of evolution in the ring's position makes a purely collisional ring origin highly unlikely. We find that the global SFR has been fairly constant over the last ~500 Myr, though it does show a small increase at 50 Myr that is 1.3 times the average SFR over the past 100 Myr. During the last ~500 Myr, ~60% of all SF occurs in the 10-kpc ring. Finally, we find that in the past 100 Myr, the average SFR over the PHAT survey area is $0.28\pm0.03$ M$_\odot$ yr$^{-1}$ with an average deprojected intensity of $7.3 \times 10^{-4}$ M$_\odot$ yr$^{-1}$ kpc$^{-2}$, which yields a total SFR of ~0.7 M$_\odot$ yr$^{-1}$ when extrapolated to the entire area of M31's disk. This SFR is consistent with measurements from broadband estimates. [abridged]Comment: 23 pages, 17 figures, 2 tables, accepted for publication in Ap