On Saturated kk-Sperner Systems

Given a set $X$, a collection $\mathcal{F}\subseteq\mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. conjectured that, if $|X|$ is sufficiently large with respect to $k$, then the minimum size of a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ is $2^{k-1}$. We disprove this conjecture by showing that there exists $\varepsilon>0$ such that for every $k$ and $|X| \geq n_0(k)$ there exists a saturated $k$-Sperner system $\mathcal{F}\subseteq\mathcal{P}(X)$ with cardinality at most $2^{(1-\varepsilon)k}$. A collection $\mathcal{F}\subseteq \mathcal{P}(X)$ is said to be an oversaturated $k$-Sperner system if, for every $S\in\mathcal{P}(X)\setminus\mathcal{F}$, $\mathcal{F}\cup\{S\}$ contains more chains of length $k+1$ than $\mathcal{F}$. Gerbner et al. proved that, if $|X|\geq k$, then the smallest such collection contains between $2^{k/2-1}$ and $O\left(\frac{\log{k}}{k}2^k\right)$ elements. We show that if $|X|\geq k^2+k$, then the lower bound is best possible, up to a polynomial factor.Comment: 17 page

Saturation in the Hypercube and Bootstrap Percolation

Let $Q_d$ denote the hypercube of dimension $d$. Given $d\geq m$, a spanning subgraph $G$ of $Q_d$ is said to be $(Q_d,Q_m)$-saturated if it does not contain $Q_m$ as a subgraph but adding any edge of $E(Q_d)\setminus E(G)$ creates a copy of $Q_m$ in $G$. Answering a question of Johnson and Pinto, we show that for every fixed $m\geq2$ the minimum number of edges in a $(Q_d,Q_m)$-saturated graph is $\Theta(2^d)$. We also study weak saturation, which is a form of bootstrap percolation. A spanning subgraph of $Q_d$ is said to be weakly $(Q_d,Q_m)$-saturated if the edges of $E(Q_d)\setminus E(G)$ can be added to $G$ one at a time so that each added edge creates a new copy of $Q_m$. Answering another question of Johnson and Pinto, we determine the minimum number of edges in a weakly $(Q_d,Q_m)$-saturated graph for all $d\geq m\geq1$. More generally, we determine the minimum number of edges in a subgraph of the $d$-dimensional grid $P_k^d$ which is weakly saturated with respect to `axis aligned' copies of a smaller grid $P_r^m$. We also study weak saturation of cycles in the grid.Comment: 21 pages, 2 figures. To appear in Combinatorics, Probability and Computin

Bcl-xL-mediated remodeling of rod and cone synaptic mitochondria after postnatal lead exposure: electron microscopy, tomography and oxygen consumption.

PurposePostnatal lead exposure produces rod-selective and Bax-mediated apoptosis, decreased scotopic electroretinograms (ERGs), and scotopic and mesopic vision deficits in humans and/or experimental animals. Rod, but not cone, inner segment mitochondria were considered the primary site of action. However, photoreceptor synaptic mitochondria were not examined. Thus, our experiments investigated the structural and functional effects of environmentally relevant postnatal lead exposure on rod spherule and cone pedicle mitochondria and whether Bcl-xL overexpression provided neuroprotection.MethodsC57BL/6N mice pups were exposed to lead only during lactation via dams drinking water containing lead acetate. The blood [Pb] at weaning was 20.6±4.7 µg/dl, which decreased to the control value by 2 months. To assess synaptic mitochondrial structural differences and vulnerability to lead exposure, wild-type and transgenic mice overexpressing Bcl-xL in photoreceptors were used. Electron microscopy, three-dimensional electron tomography, and retinal and photoreceptor synaptic terminal oxygen consumption (QO(2)) studies were conducted in adult control, Bcl-xL, lead, and Bcl-xL/lead mice.ResultsThe spherule and pedicle mitochondria in lead-treated mice were swollen, and the cristae structure was markedly changed. In the lead-treated mice, the mitochondrial cristae surface area and volume (abundance: measure correlated with ATP (ATP) synthesis) were decreased in the spherules and increased in the pedicles. Pedicles also had an increased number of crista segments per volume. In the lead-treated mice, the number of segments/crista and fraction of cristae with multiple segments (branching) similarly increased in spherule and pedicle mitochondria. Lead-induced remodeling of spherule mitochondria produced smaller cristae with more branching, whereas pedicle mitochondria had larger cristae with more branching and increased crista junction (CJ) diameter. Lead decreased dark- and light-adapted photoreceptor and dark-adapted photoreceptor synaptic terminal QO(2). Bcl-xL partially blocked many of the lead-induced alterations relative to controls. However, spherules still had partially decreased abundance, whereas pedicles still had increased branching, increased crista segments per volume, and increased crista junction diameter. Moreover, photoreceptor and synaptic QO(2) were only partially recovered.ConclusionsThese findings reveal cellular and compartmental specific differences in the structure and vulnerability of rod and cone inner segment and synaptic mitochondria to postnatal lead exposure. Spherule and pedicle mitochondria in lead-exposed mice displayed complex and distinguishing patterns of cristae and matrix damage and remodeling consistent with studies showing that synaptic mitochondria are more sensitive to Ca(2+) overload, oxidative stress, and ATP loss than non-synaptic mitochondria. The lead-induced decreases in QO(2) likely resulted from the decreased spherule cristae abundance and smaller cristae, perhaps due to Bax-mediated effects as they occurred in apoptotic rod inner segments. The increase in pedicle cristae abundance and CJ diameter could have resulted from increased Drp1-mediated fission, as small mitochondrial fragments were observed. The mechanisms of Bcl-xL-mediated remodeling might occur via interaction with formation of CJ protein 1 (Fcj1), whereas the partial protection of synaptic QO(2) might result from the enhanced efficiency of energy metabolism via Bcl-xL's direct interaction with the F1F0 ATP synthase and/or regulation of cellular redox status. These lead-induced alterations in photoreceptor synaptic terminal mitochondria likely underlie the persistent scotopic and mesopic deficits in lead-exposed children, workers, and experimental animals. Our findings stress the clinical and scientific importance of examining synaptic dysfunction following injury or disease during development, and developing therapeutic treatments that prevent synaptic degeneration in retinal and neurodegenerative disorders even when apoptosis is blocked

Borrowing from historical control data in a Bayesian time-to-event model with flexible baseline hazard function

There is currently a focus on statistical methods which can use historical trial information to help accelerate the discovery, development and delivery of medicine. Bayesian methods can be constructed so that the borrowing is "dynamic" in the sense that the similarity of the data helps to determine how much information is used. In the time to event setting with one historical data set, a popular model for a range of baseline hazards is the piecewise exponential model where the time points are fixed and a borrowing structure is imposed on the model. Although convenient for implementation this approach effects the borrowing capability of the model. We propose a Bayesian model which allows the time points to vary and a dependency to be placed between the baseline hazards. This serves to smooth the posterior baseline hazard improving both model estimation and borrowing characteristics. We explore a variety of prior structures for the borrowing within our proposed model and assess their performance against established approaches. We demonstrate that this leads to improved type I error in the presence of prior data conflict and increased power. We have developed accompanying software which is freely available and enables easy implementation of the approach.Comment: Sampler for regression coefficients (beta and beta_0) updated on page 5 of the Supplementary Materia

On saturated k-Sperner systems

Given a set X , a collection F ⊆ P (X) is said to be k-Sperner if it does not contain a chain of length k + 1 under set inclusion and it is saturated if it is maximal with respect to this property. Gerbner et al. [11] conjectured that, if |X| is sufficiently large with respect to k, then the minimum size of a saturated k-Sperner system F ⊆ P (X) is 2k-1 . We disprove this conjecture by showing that there exists ε > 0 such that for every k and |X| > n0 (k) there exists a saturated k-Sperner system F ⊆P (X) with cardinality at most 2 (1- ε)k. A collection F ⊆ P (X) is said to be an oversaturated k-Sperner system if, for every S∈P (X) \ F, F∪{ S } contains more chains of length k +1 than F. Gerbner et al. [11] proved that, if |X| > k, then the smallest such collection contains between 2k/2-1 and O (log k k 2 k) elements. We show that if |X| > k2 + k, then the lower bound is best possible, up to a polynomial factor