Notions of denseness

The notion of a completely saturated packing [Fejes Toth, Kuperberg and Kuperberg, Highly saturated packings and reduced coverings, Monats. Math. 125 (1998) 127-145] is a sharper version of maximum density, and the analogous notion of a completely reduced covering is a sharper version of minimum density. We define two related notions: uniformly recurrent and weakly recurrent dense packings, and diffusively dominant packings. Every compact domain in Euclidean space has a uniformly recurrent dense packing. If the domain self-nests, such a packing is limit-equivalent to a completely saturated one. Diffusive dominance is yet sharper than complete saturation and leads to a better understanding of n-saturation.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol4/paper9.abs.htm

Effect of acid/base balance on H-ATPase 31 kD subunit mRNA levels in collecting duct cells

Effect of acid/base balance on H-ATPase 31 kD subunit mRNA levels in collecting duct cells. The cortical collecting duct (CCD) adapts to disturbances of acid/base balance by adjusting the direction and magnitude of its HCO3 transport. The molecular events involved in this adaptation are incompletely understood, but it seems that adaptation is accompanied by changes in the activity and intracellular distribution of the vacuolar H-ATPase. The goal of this study was to examine the effects of metabolic acidosis and alkali load on the expression of the mRNA encoding the 31 kD subunit of the vacuolar H-ATPase in rabbit CCD cells. Pairs of rabbits received either a NH4Cl load or a NaHCO3 load for 16 hours, resulting in a urinary pH of 5.53 ± 0.38 and 8.42 ± 0.10, respectively. CCD cells were isolated by immunodissection and mRNA levels of the H-ATPase 31 kD subunit and of β-actin were determined from the same cDNA samples by quantitative RT-PCR. H-ATPase mRNA levels were significantly higher in CCD cells from acidotic than alkali-loaded rabbits (2.51 ± 1.3 vs. 0.65 ± 0.2; P < 0.05). Similar differences in the H-ATPase 31 kD subunit mRNA levels were observed by Northern blotting. β-actin mRNA levels were comparable in CCD cells of the two groups. The distribution of the H-ATPase 31 kD subunit mRNA was determined among the three cell types of the CCD, that is in α- and β-intercalated cells (α-ICC and β-ICC) and principal cells (PC) isolated by fluorescence-activated cell sorting. The level of expression was comparable in α-ICCs and β-ICCs, whereas PCs contained very low levels of H-ATPase mRNA. In both α-ICC and β-ICC the levels of the 31 kD H-ATPase mRNA were significantly higher in acidotic than in alkaliloaded rabbits. These results indicate that in the rabbit CCD changes in acid/base balance not only regulate the subcellular distribution of the vacuolar H-ATPase but also alter its expression, at least at the mRNA level

Highly saturated packings and reduced coverings

We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packing $\cal P$ with congruent replicas of a body $K$ is $n$-saturated if no $n-1$ members of it can be replaced with $n$ replicas of $K$, and it is completely saturated if it is $n$-saturated for each $n\ge 1$. Similarly, a covering $\cal C$ with congruent replicas of a body $K$ is $n$-reduced if no $n$ members of it can be replaced by $n-1$ replicas of $K$ without uncovering a portion of the space, and it is completely reduced if it is $n$-reduced for each $n\ge 1$. We prove that every body $K$ in $d$-dimensional Euclidean or hyperbolic space admits both an $n$-saturated packing and an $n$-reduced covering with replicas of $K$. Under some assumptions on $K\subset \mathbb{E}^d$ (somewhat weaker than convexity), we prove the existence of completely saturated packings and completely reduced coverings, but in general, the problem of existence of completely saturated packings and completely reduced coverings remains unsolved. Also, we investigate some problems related to the the densities of $n$-saturated packings and $n$-reduced coverings. Among other things, we prove that there exists an upper bound for the density of a $d+2$-reduced covering of $\mathbb{E}^d$ with congruent balls, and we produce some density bounds for the $n$-saturated packings and $n$-reduced coverings of the plane with congruent circles

On a strong version of the Kepler conjecture

We raise and investigate the following problem that one can regard as a very close relative of the densest sphere packing problem. If the Euclidean 3-space is partitioned into convex cells each containing a unit ball, how should the shapes of the cells be designed to minimize the average surface area of the cells? In particular, we prove that the average surface area in question is always at least 13.8564... .Comment: 9 page

Dowker-type theorems for hyperconvex discs

A hyperconvex disc of radius r is a planar set with nonempty interior that is the intersection of closed circular discs of radius r . A convex disc-polygon of radius r is a set with nonempty interior that is the intersection of a finite number of closed circular discs of radius r . We prove that the maximum area and perimeter of convex disc- n -gons of radius r contained in a hyperconvex disc of radius r are concave functions of n , and the minimum area and perimeter of disc- n -gons of radius r containing a hyperconvex disc of radius r are convex functions of n . We also consider hyperbolic and spherical versions of these statements

Characterization of a mouse cortical collecting duct cell line

Characterization of a mouse cortical collecting duct cell line. A cortical collecting duct (CCD) cell line has been developed from a mouse transgenic for the early region of simian virus 40, Tg(SV40E)Bri/7. CCDs were microdissected and placed on collagen gels. Monolayers were subsequently subcultured onto permeable collagen membranes and maintained in serum-supplemented medium. One line, designated M-1, retained many characteristics of the CCD, including a typical epithelial appearance and CCD-specific antigens. M-1 cells, when grown in monolayers on permeable supports, exhibited a high transepi-thelial resistance (885.7 ± 109.6 ohms/cm2) and developed a lumen negative transepithelial potential difference (PD) of -45.7 ± 3.5mV. The associated short-circuit current (SCC) averaged 71.8 ± 10.3 µA/cm2, and was reduced by 95% by luminal application of amiloride. The cultured cells responded to arginine vasopressin (AVP) with a significant increase in SCC. M-1 cells generated significant transepithelial solute gradients. After 24 hours incubation, the composition of the luminal (L) and basolateral (B) media (in mM) was: [Na+], L = 106.7 ± 0.9 and B = 127.4 ± 0.4; [K+], L = 8.6 ± 0.6 and B = 2.1 ± 0.3; [Cl], L = 68.6 ± 5.8 and B = 101.8 ± 6.6; [HCO3], L = 15.5 ± 1.5 and B = 8.6 ± 1.2; while pH was 7.16 ± 0.03 at the luminal and 6.94 ± 0.03 at the basolateral side. The formation of these concentration gradients indicates that the CCD cultures absorb Na+ and Cl- and secrete K+. Lactate accumulated predominantly at the basolateral side (L = 7.1 ± 0.44 and B = 17.5 ± 0.52 mM); osmotic concentration was 272 ± 1.4 at the luminal and 290 ±3.0 mOsm/kg at the basolateral side. These data demonstrate that the M-1 cell line retains many phenotypic properties of the CCD epithelium and thus should prove useful as a model in studying mechanisms of ion transport in this segment