On steady-state properties of certain max-plus products

The asymptotic properties of inhomogeneous products in the max-plus algebra context have been investigated. In particular, for products involving matrices with the same unique critical circuit, we have obtained some sufficiency conditions under which the rank of the final product matrix is less than or equal to the length of the critical circuit of the matrices in the product. For a product comprising of matrices with the same unique critical circuit of length l, the asymptotic rank is 1

Tissue-specific deletion of the <i>Klotho</i> gene.

<p><i>Upper left panel</i>. Immunohistochemical staining confirmed successful deletion of Klotho in parathyroid glands of <i>PTH-KL^−/−</i> mice. 20× magnification. <i>Upper right panel</i>. The gross appearance of <i>PTH-KL^−/−</i> mice was normal although females had lower body weight and shorter crown-rump length compared to wild-type littermates. No differences were seen for male <i>PTH-KL^−/−</i> mice. *p<0.05 **<0.01. <i>Lower panel:</i> Serum levels of 1,25(OH)₂D were doubled in <i>PTH-KL^−/−</i> mice compared to their wild-type littermates (p<0.05), while PTH, calcium and FGF23 remained normal.</p

Parathyroid transcriptional data.

<p>Transcriptional analysis of laser microdissected parathyroid tissue revealed differential expression of <i>Cfd</i>, <i>Smad4</i> and <i>Fabp4</i> in <i>PTH-KL^−/−</i> mice (n = 3 for each genotype). *p<0.05 ***p<0.001.</p

Arterial Klotho Expression and FGF23 Effects on Vascular Calcification and Function

<div><p>Recent studies support a role for FGF23 and its co-receptor Klotho in cardiovascular pathology, yet the underlying mechanisms remain largely elusive. Herein, we analyzed the expression of Klotho in mouse arteries and generated a novel mouse model harboring a vascular smooth muscle cell specific deletion of Klotho (<i>Sm22-KL^−/−</i>). Arterial Klotho expression was detected at very low levels with quantitative real-time PCR; Klotho protein levels were undetectable by immunohistochemistry and Western blot. There was no difference in arterial Klotho between <i>Sm22-KL^−/−</i> and wild-type mice, as well as no changes in serum markers of mineral metabolism. Intravenous delivery of FGF23 elicited a rise in renal (0.005; p<0.01) but not arterial Egr-1 expression, a marker of Klotho-dependent FGF23 signaling. Further, the impact of FGF23 on vascular calcification and endothelial response was evaluated in bovine vascular smooth muscle cells (bVSMC) and in a murine <i>ex vivo</i> model of endothelial function, respectively. FGF23 treatment (0.125–2 ng/mL) did not modify calcification in bVSMCs or dilatory, contractile and structural properties in mice arterial specimen <i>ex vivo</i>. Collectively, these results demonstrate that FGF23-Klotho signaling is absent in mouse arteries and that the vascular response was unaffected by FGF23 treatment. Thus, our data do not support Klotho-mediated FGF23 effects in the vasculature although confirmative studies in humans are warranted.</p> </div

Immunohistochemical staining and Western blotting for expression of Klotho in different arteries from <i>Sm22-KL^−/−</i> mice.

<p>(A) Klotho protein was undetectable in the vascular wall of different arteries (aorta, mesenteriac, femoral and lung arteries), as determined by immunohistochemical staining, when compared with a distinct positive staining for Klotho in kidney distal tubule. Upper panels show Klotho staining of aorta with two different antibodies (KM2076 and KM2119). Lower panel shows Klotho staining of mesenterial, femoral and lung arteries, using the same antibodies. All results are shown in 40x magnification. Scale bar, 10 µM. (<b>B</b>) Immunoblot analysis of pooled whole aortic lysates from three wild-type and three <i>Sm22-KL^−/−</i> mice revealed no Klotho protein (using two different antibodies; KM2076 and KM2119), when compared with the positive control of whole kidney extracts from wild-type mice, which showed a strong band for Klotho at 130 kDa.</p

No effect of FGF23 on βGP-induced mineral deposition by bVSMCs.

<p>(<b>A</b>) bVSMCs were cultured in 6-well plates in 10% FCS-DMEM till 90% confluent, and then incubated in medium containing 5 mM βGP and mock transfected media (control), or with medium containing βGP supplemented with FGF23 in concentrations ranging from 0.125–2 ng/mL. Representative phase contrast images of cells stained with alizarin red on days 10 are shown. The red stain indicates areas of mineralization. Scale bar = 500 µm. (<b>B</b>) Alizarin red dye elution was performed in order to quantify mineralization. Results represent data from day 10. Data are shown as mean ± SEM (n = 3 individual samples).</p

Responsiveness to renal failure and a single FGF23 injection.

<p>A) Mice were challenged with renal failure during 4 weeks by dietary adenine delivery. <i>PTH-KL^−/−</i> (n = 8) and wild-type (n = 6) mice developed uremia and secondary hyperparathyroidism of similar magnitude. B) Mice were injected intravenously with recombinant FGF23 protein at the dose 0.15 mg/kg and blood samples from the tail vein were drawn after 15 minutes. <i>PTH-KL^−/−</i> mice had a similar rapid reduction in PTH followed by a single FGF23 injection indicating a preserved FGF23 responsiveness in <i>PTH-KL^−/−</i> mice despite ablated Klotho expression. Figure shows pooled data from two independent experiments (n = 9 for <i>PTH-KL^−/−</i>; n = 7 for wild-type mice). *p<0.05 C) The impact of Klotho-dependent FGF23 signaling was investigated in parathyroid glands with dual immunofluorescence staining of Klotho and phosphorylated ERK1/2. Importantly, phosphorylated ERK1/2 was strongly induced 15 minutes after a single intravenous FGF23 injection in wild-type mice. In contrast, <i>PTH-KL^−/−</i> mice failed to phosphorylate ERK1/2 specifically in cells lacking Klotho, supporting a disruption of functional Klotho-mediated FGF23 signaling in <i>PTH-KL^−/−</i> mice (10× and 40× magnification).</p

Proposed model of FGF23-Klotho function in parathyroid glands.

<p>FGF23 is an endocrine ligand that in the parathyroid glands either binds to an FGF-receptor and membrane-bound Klotho, to elicit activation of the MAPK-pathway, or acts via a calcineurin-dependent signaling pathway. Both pathways induce a rapid inhibition of PTH secretion within 15 minutes. The MAPK-pathway is likely the dominant pathway in physiology, although their relative contributions are unknown. In addition to mediating MAPK-dependent FGF23 signaling, Klotho may play an intrinsic role in modulating parathyroid transcriptional activity for sustained demands and long-term parathyroid function. Speculatively, the clinical use of calcineurin inhibitors (CNi) that block calcineurin signaling may increase the susceptibility to develop, or aggravate pre-existing, hyperparathyroidism in patients with reduced Klotho level such as in chronic kidney disease and primary hyperparathyroidism.</p

Serum biochemistries in <i>Sm22-KL</i>^−/− and wild-type mice.

<p>At 8 weeks of age <i>Sm22-KL^−/−</i> mice had normal serum biochemistries reflecting mineral metabolism as compared to wild-type controls. Results are displayed as median (range).</p

Klotho gene expression in mouse arteries.

<p>Gene expression of Klotho in kidney extracts, arteries and lung of wild-type mice, <i>Sm22-KL^−/−</i> mice, <i>β-KL^−/−</i> mice, and DCT209 cells. The Klotho transcript levels remained unchanged between wild-type and <i>Sm22-KL^{<b>−/−</b>}</i> mice in all arteries investigated and were 7000 times lower when compared to kidney transcript levels. <i>Sm22-KL^−/−</i> transcript levels remained at same levels or lower when compared with the <i>β-KL^−/−</i> mice and the DCT209 cell line.</p