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

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

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

Presence of the calcineurin-NFAT pathway and the impact of cyclosporin A on FGF23 treatment.

<p>A) Transcript analysis of laser capture microdissected parathyroid glands from 3- week-old wild-type mice revealed presence of NFATC1-4 and the subunits for calcineurin A and B (CnA and CnB). Each bar represents one hybridization probe. Data is presented as mean (± SEM). B) Immunostaining for NFATC2 showed a partial nuclear translocation in <i>PTH-KL^−/−</i> mice, and cytoplasmatic localization in wild-type mice after treatment with FGF23, indicating activation of the calcineurin-NFAT pathway in parathyroids of <i>PTH-KL^−/−</i> mice. In addition, MCIP1, a facilitator of calcineurin signaling, was markedly higher in <i>PTH-KL^−/−</i> compared to wild-type mice after treatment with FGF23 (10× and 40× magnification). C) Pretreatment of mice with the calcineurin inhibitor CsA followed by a single intravenous FGF23 injection. Wild-type mice had a preserved responsiveness to FGF23 as evidenced by a significant reduction in serum PTH 15 minutes after FGF23 injection (p<0.05) whereas FGF23-mediated inhibition of serum PTH was blunted in <i>PTH-KL^−/−</i> mice. Figure depicts pooled data from two independent experiments, using 60 mg/kg and 150 mg/kg CsA respectively, with similar results. Total n = 10 for wild-type and 7 for <i>PTH-KL^−/−</i> mice. *p<0.05. D) Relative change in PTH level in wild-type and <i>PTH-KL^−/−</i> mice 15 min after a single FGF23 injection, with and without pretreatment of CsA. E) Thyro-parathyroid explants were cultured in serum-free medium and exposed to FGF23 for 2 h with or without pre-treatment by CsA. Wild-type and <i>PTH-KL^−/−</i> mice responded with a similar decrease in PTH when exposed to FGF23. In contrast, in explants pre-treated with CsA (0.83 µM) for 2 h the response to FGF23 treatment was unaltered in wild-type mice but completely blunted in <i>PTH-KL^−/−</i> mice. N = 5–10 for all groups. **p<0.01.</p

Long term effects of FGF23 on induced contractions or relaxations.

<p>Long term effects of FGF23 or vehicle (DMSO 0.17% v/v) on phenylephrine (A), thromboxane A2 (TXA₂) analog U46619 (B), acetylcholine (C) and sodium nitroprusside (D) induced contractions or relaxations, respectively. Data are shown as means ± SEM. Solid black lines indicate responses in the presence of vehicle and gray lines indicate responses in the presence of FGF23.</p

Vascular Egr-1 response to FGF23 injection.

<p>Transcript levels of Egr-1 were analyzed in aorta and kidneys from wild-type mice injected with 0.9% NaCl (n = 3) or 0.15 mg/kg FGF23 (n = 4). In contrast to the kidney, no distinct rise in Egr-1 mRNA was seen in aorta 30 min after an FGF23 injection. Data are presented as fold change, with NaCl set to 1.</p