The Collatz tree as a Hilbert hotel:a proof of the 3x + 1 conjecture

The Collatz conjecture maintains that each natural number $n$ is a node in the Collatz tree with a root path to $1$ generated by iterations of the Collatz function, which connects even $n$ to $n/2$ and odd $n$ to $3n+1$. The conjecture holds if a binary subtree $T_{\ge 0}$ exists with as nodes $\textit{all}$ branching numbers constituting the congruence classes ${\left[4,16\right]}_{18}$ with density $2/18$. For each branching number, the $\textit{upward}$ and $\textit{rightward}$ function generate arrows to two branching numbers in the directions $2n$ and $(n-1)/3$. In the automorphism graph of tree $T_{\ge 0}$. The iterates and conjugates of these two functions build outer paths from tree $T_{\ge 0}$ at its root node to two infinite sequences of nodes containing nested subtrees and disjoint cotrees. The nodes and conjugate inner arrows within each subtree and within each cotree correspond one-to-one to the nodes and arrows within tree $T_{\ge 0}$. The cumulative density of the congruence classes of numbers in disjoint cotrees of $T_{\ge 0}$ amounts to the density $2/18$ of all branching numbers, thus proving the conjecture.Comment: Collatz paths between branching classes $[4,16]_18$ and partitioning of the binary tree $T_{ge 0}$ into disjoint cotrees now validate in v5 the density proofs in previous versions (v1 31 Aug 2000

Evidence that a positive feedback loop drives centrosome maturation in fly embryos.

Centrosomes are formed when mother centrioles recruit pericentriolar material (PCM) around themselves. The PCM expands dramatically as cells prepare to enter mitosis (a process termed centrosome maturation), but it is unclear how this expansion is achieved. In flies, Spd-2 and Cnn are thought to form a scaffold around the mother centriole that recruits other components of the mitotic PCM, and the Polo-dependent phosphorylation of Cnn at the centrosome is crucial for scaffold assembly. Here, we show that, like Cnn, Spd-2 is specifically phosphorylated at centrosomes. This phosphorylation appears to create multiple phosphorylated S-S/T(p) motifs that allow Spd-2 to recruit Polo to the expanding scaffold. If the ability of Spd-2 to recruit Polo is impaired, the scaffold is initially assembled around the mother centriole, but it cannot expand outwards, and centrosome maturation fails. Our findings suggest that interactions between Spd-2, Polo and Cnn form a positive feedback loop that drives the dramatic expansion of the mitotic PCM in fly embryos

The Collatz tree as a Hilbert hotel: a proof of the 3x + 1 conjecture

The yet unproven Collatz conjecture maintains that repeatedly connecting even numbers n to n/2, and odd n to 3n + 1, connects all natural numbers by a unique root path to the Collatz tree with 1 as its root. The Collatz tree proves to be a Hilbert hotel. Numbers divisible by 2 or 3 depart. An infinite binary tree remains with one upward and one rightward child per number. Rightward numbers, and infinitely many generations of their upward descendants, each with a well-defined root path, depart thereafter. The Collatz tree is a Hilbert hotel because still higher upward descendants keep descending to all unoccupied nodes. The density of already departed numbers comes nevertheless arbitrarily close to 100% of the natural numbers. The latter proves the Collatz conjecture

Cytoplasmic division cycles without the nucleus and mitotic CDK/cyclin complexes

Cytoplasmic divisions are thought to rely on nuclear divisions and mitotic signals. We demonstrate in Drosophila embryos that cytoplasm can divide repeatedly without nuclei and mitotic CDK/cyclin complexes. Cdk1 normally slows an otherwise faster cytoplasmic division cycle, coupling it with nuclear divisions, and when uncoupled, cytoplasm starts dividing before mitosis. In developing embryos where CDK/cyclin activity can license mitotic microtubule (MT) organizers like the spindle, cytoplasmic divisions can occur without the centrosome, a principal organizer of interphase MTs. However, centrosomes become essential in the absence of CDK/cyclin activity, implying that the cytoplasm can employ either the centrosome-based interphase or CDK/cyclin-dependent mitotic MTs to facilitate its divisions. Finally, we present evidence that autonomous cytoplasmic divisions occur during unperturbed fly embryogenesis and that they may help extrude mitotically stalled nuclei during blastoderm formation. We postulate that cytoplasmic divisions occur in cycles governed by a yet-to-be-uncovered clock mechanism autonomous from CDK/cyclin complexes

Autonomous clocks that regulate organelle biogenesis, cytoskeletal organization, and intracellular dynamics

How do cells perceive time? Do cells use temporal information to regulate the production/degradation of their enzymes, membranes, and organelles? Does controlling biological time influence cytoskeletal organization and cellular architecture in ways that confer evolutionary and physiological advantages? Potential answers to these fundamental questions of cell biology have historically revolved around the discussion of 'master' temporal programs, such as the principal cyclin-dependent kinase/cyclin cell division oscillator and the circadian clock. In this review, we provide an overview of the recent evidence supporting an emerging concept of 'autonomous clocks,' which under normal conditions can be entrained by the cell cycle and/or the circadian clock to run at their pace, but can also run independently to serve their functions if/when these major temporal programs are halted/abrupted. We begin the discussion by introducing recent developments in the study of such clocks and their roles at different scales and complexities. We then use current advances to elucidate the logic and molecular architecture of temporal networks that comprise autonomous clocks, providing important clues as to how these clocks may have evolved to run independently and, sometimes at the cost of redundancy, have strongly coupled to run under the full command of the cell cycle and/or the circadian clock. Next, we review a list of important recent findings that have shed new light onto potential hallmarks of autonomous clocks, suggestive of prospective theoretical and experimental approaches to further accelerate their discovery. Finally, we discuss their roles in health and disease, as well as possible therapeutic opportunities that targeting the autonomous clocks may offer

Centriole distal-end proteins CP110 and Cep97 influence centriole cartwheel growth at the proximal end

Centrioles are composed of a central cartwheel tethered to nine-fold symmetric microtubule (MT) blades. The centriole cartwheel and MTs are thought to grow from opposite ends of these organelles, so it is unclear how they coordinate their assembly. We previously showed that in Drosophila embryos an oscillation of Polo-like kinase 4 (Plk4) helps to initiate and time the growth of the cartwheel at the proximal end. Here, in the same model, we show that CP110 and Cep97 form a complex close to the distal-end of the centriole MTs whose levels rise and fall as the new centriole MTs grow, in a manner that appears to be entrained by the core cyclin-dependent kinase (Cdk)–Cyclin oscillator that drives the nuclear divisions in these embryos. These CP110 and Cep97 dynamics, however, do not appear to time the period of centriole MT growth directly. Instead, we find that changing the levels of CP110 and Cep97 appears to alter the Plk4 oscillation and the growth of the cartwheel at the proximal end. These findings reveal an unexpected potential crosstalk between factors normally concentrated at opposite ends of the growing centrioles, which might help to coordinate centriole growth

Sensitive detection and quantification of SARS-CoV-2 in saliva

Abstract Saliva has significant advantages as a test medium for detection of SARS-CoV-2 infection in patients, such as ease of collection, minimal requirement of supplies and trained personnel, and safety. Comprehensive validation in a large cohort of prospectively collected specimens with unknown SARS-CoV-2 status should be performed to evaluate the potential and limitations of saliva-based testing. We developed a saliva-based testing pipeline for detection of SARS-CoV-2 nucleic acids using real-time reverse transcription PCR (RT-PCR) and droplet digital PCR (ddPCR) readouts, and measured samples from 137 outpatients tested at a curbside testing facility and 29 inpatients hospitalized for COVID-19. These measurements were compared to the nasal swab results for each patient performed by a certified microbiology laboratory. We found that our saliva testing positively detects 100% (RT-PCR) and 93.75% (ddPCR) of curbside patients that were identified as SARS-CoV-2 positive by the Emergency Use Authorization (EUA) certified nasal swab testing assay. Quantification of viral loads by ddPCR revealed an extremely wide range, with 1 million-fold difference between individual patients. Our results demonstrate for both community screening and hospital settings that saliva testing reliability is on par with that of the nasal swabs in detecting infected cases, and has potential for higher sensitivity when combined with ddPCR in detecting low-abundance viral loads that evade traditional testing methods

A homeostatic clock sets daughter centriole size in flies

Centrioles are highly structured organelles whose size is remarkably consistent within any given cell type. New centrioles are born when Polo-like kinase 4 (Plk4) recruits Ana2/STIL and Sas-6 to the side of an existing “mother” centriole. These two proteins then assemble into a cartwheel, which grows outwards to form the structural core of a new daughter. Here, we show that in early Drosophila melanogaster embryos, daughter centrioles grow at a linear rate during early S-phase and abruptly stop growing when they reach their correct size in mid- to late S-phase. Unexpectedly, the cartwheel grows from its proximal end, and Plk4 determines both the rate and period of centriole growth: the more active the centriolar Plk4, the faster centrioles grow, but the faster centriolar Plk4 is inactivated and growth ceases. Thus, Plk4 functions as a homeostatic clock, establishing an inverse relationship between growth rate and period to ensure that daughter centrioles grow to the correct size