Various disorders including pseudoxanthoma elasticum (PXE) and generalized arterial calcification of infancy (GACI), which are caused by inactivating mutations in ABCC6 and ENPP1, respectively, present with extensive tissue calcification due to reduced plasma pyrophosphate (PPi). However, it has always been assumed that the bioavailability of orally administered PPi is negligible. Here, we demonstrate increased PPi concentration in the circulation of humans after oral PPi administration. Furthermore, in mouse models of PXE and GACI, oral PPi provided via drinking water attenuated their ectopic calcification phenotype. Noticeably, provision of drinking water with 0.3 mM PPi to mice heterozygous for inactivating mutations in Enpp1 during pregnancy robustly inhibited ectopic calcification in their Enpp1 −/− offspring. Our work shows that orally administered PPi is readily absorbed in humans and mice and inhibits connective tissue calcification in mouse models of PXE and GACI. PPi, which is recognized as safe by the FDA, therefore not only has great potential as an effective and extremely low‐cost treatment for these currently intractable genetic disorders, but also in other conditions involving connective tissue calcification.

No abstract

Since 1997 a considerable effort has been spent to study the mixing time of switch Markov chains on the realizations of graphic degree sequences of simple graphs. Several results were proved on rapidly mixing Markov chains on unconstrained, bipartite, and directed sequences, using different mechanisms. The aim of this paper is to unify these approaches. We will illustrate the strength of the unified method by showing that on any P -stable family of unconstrained/bipartite/directed degree sequences the switch Markov chain is rapidly mixing. This is a common generalization of every known result that shows the rapid mixing nature of the switch Markov chain on a region of degree sequences. Two applications of this general result will be presented. One is an almost uniform sampler for power-law degree sequences with exponent γ > 1 + √ 3. The other one shows that the switch Markov chain on the degree sequence of an Erds-Rnyi random graph G(n, p) is asymptotically almost surely rapidly mixing if p is bounded away from 0 and 1 by at least 5 log n n−1 .

No abstract

Since 1997 a considerable effort has been spent on the study of the swap (switch) Markov chains on graphic degree sequences. All of these results assume some kind of regularity in the corresponding degree sequences. Recently, Greenhill and Sfragara published a breakthrough paper about irregular normal and directed degree sequences for which rapid mixing of the swap Markov chain is proved. In this paper we present two groups of results. An example from the first group is the following theorem: let be a directed degree sequence on n vertices. Denote by Δ the maximum value among all in- and out-degrees and denote by the number of edges in the realization. Assume furthermore that . Then the swap Markov chain on the realizations of is rapidly mixing. This result is a slight improvement on one of the results of Greenhill and Sfragara. An example from the second group is the following: let d be a bipartite degree sequence on the vertex set U ⊎ V, and let 0 < c1 ≤ c2 < |U| and 0 < d1 ≤ d2 < |V| be integers, where c1 ≤ d(v) ≤ c2: ∀v ∈ V and d1 ≤ d(u) ≤ d2: ∀u ∈ U. Furthermore assume that (c2 − c1 − 1)(d2 − d1 − 1) < max{c1(|V| − d2), d1(|U| − c2)}. Then the swap Markov chain on the realizations of d is rapidly mixing. A straightforward application of this latter result shows that when a random bipartite or directed graph is generated under the Erdős—Rényi G(n, p) model with mild assumptions on n and p then the degree sequence of the generated graph has, with high probability, a rapidly mixing swap Markov chain on its realizations.

No abstract

We affirmatively answer and generalize the question of Kubicka, Kubicki and Lehel [7] concerning the path-pairability of high-dimensional complete grid graphs. As an intriguing by-product of our result we significantly improve the estimate of the necessary maximum degree in path-pairable graphs, a question originally raised and studied by Faudree, Gyárfás, and Lehel [4].

We prove that every simply connected orthogonal polygon of n vertices can be partitioned into 3n+4 16 (simply connected) orthogonal polygons of at most 8 vertices. It yields a new and shorter proof of the theorem of A. Aggarwal that 3n+4 16 mobile guards are sufficient to control the interior of an n-vertex orthogonal polygon. Moreover, we strengthen this result by requiring combinatorial guards (visibility is only needed at the endpoints of patrols) and prohibiting intersecting patrols. This yields positive answers to two questions of O'Rourke [7, Section 3.4]. Our result is also a further example of the "metatheorem" that (orthogonal) art gallery theorems are based on partition theorems.