Near-optimal mean value estimates for multidimensional Weyl sums

We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed

Novel Roles of GATA4/6 in the Postnatal Heart Identified through Temporally Controlled, Cardiomyocyte-Specific Gene Inactivation by Adeno-Associated Virus Delivery of Cre Recombinase.

GATA4 and GATA6 are central cardiac transcriptional regulators. The postnatal, stage-specific function of the cardiac transcription factors GATA4 and GATA6 have not been evaluated. In part, this is because current Cre-loxP approaches to cardiac gene inactivation require time consuming and costly breeding of Cre-expressing and "floxed" mouse lines, often with limited control of the extent or timing of gene inactivation. We investigated the stage-specific functions of GATA4 and GATA6 in the postnatal heart by using adeno-associated virus serotype 9 to control the timing and extent of gene inactivation by Cre. Systemic delivery of recombinant, adeno-associated virus 9 (AAV9) expressing Cre from the cardiac specific Tnnt2 promoter was well tolerated and selectively and efficiently recombined floxed target genes in cardiomyocytes. AAV9:Tnnt2-Cre efficiently inactivated Gata4 and Gata6. Neonatal Gata4/6 inactivation caused severe, rapidly lethal systolic heart failure. In contrast, Gata4/6 inactivation in adult heart caused only mild systolic dysfunction but severe diastolic dysfunction. Reducing the dose of AAV9:Tnnt2-Cre generated mosaics in which scattered cardiomyocytes lacked Gata4/6. This mosaic knockout revealed that Gata4/6 are required cell autonomously for physiological cardiomyocyte growth. Our results define novel roles of GATA4 and GATA6 in the neonatal and adult heart. Furthermore, our data demonstrate that evaluation of gene function hinges on controlling the timing and extent of gene inactivation. AAV9:Tnnt2-Cre is a powerful tool for controlling these parameters

Mosaic inactivation of <i>Gata4/6</i> in the neonatal heart.

<p><b>(A).</b> Experimental timeline. <b>(B).</b> Titration of AAV9:Tnnt2-Cre in neonatal Rosa26^mTmG mice. GFP marks Cre-recombined cells. The percentage of GFP⁺ cells is indicated. Bar = 20 μm. <b>(C).</b> Relationship of virus dose to ejection fraction (red, left axis) and recombination efficiency (black, right axis). <b>(D-E).</b> Gata4^fl/fl::Gata6^fl/fl::Rosa26^mTmG neonates were treated with low dose AAV9:Tnnt2-Cre. Cardiomyocytes were purified at P7 by magnetic cell sorting, then FACS sorted (D). <i>Gata4</i> and <i>Gata6</i> levels, measured by qRTPCR, confirmed gene knockout in GFP⁺ cells (E). <b>(F).</b> Size of GATA4/6 knockout cardiomyocytes at P7. The cross sectional area of GFP⁺ (recombined) and RFP⁺ (non-recombined) cardiomyocytes in P7 cryosections was compared using the Mann-Whitney test. <b>(G-H).</b> Size of GATA4/6 deficient cardiomyocytes in adult mosaic knockout hearts. Representative images of cardiomyocytes dissociated from adult hearts with mosaic GATA4/6 knockout by AAV9:Tnnt-Cre administered at P1. (G). Bar = 20 μm. Quantitation of GFP⁺ and GFP^- cardiomyocyte size (H) showed that GFP⁺ cardiomyocytes were significantly smaller by Mann-Whitney test. In F and H, black lines represent the median and 1st and 3rd quartiles.</p

Antibodies used in this study.

<p>WB, western blot; IF, immunofluorescent staining.</p><p>Antibodies used in this study.</p

Primers used in this study.

<p>Primers used in this study are indicated (5’-3’).</p><p>Primers used in this study.</p

Cardiac-selective gene inactivation with AAV9-Tnnt2-Cre.

<p>AAV9-Tnnt2-Cre was administered to Rosa26^mTmG neonatal mice. Cre recombination activates GFP expression. <b>(A)</b>. Brightfield and GFP fluorescent signal from the indicated organs. skel. m., skeletal muscle. <b>(B)</b>. Immunofluorescent staining of heart cryosections. Green indicates native membrane-localized GFP fluorescence. Cardiomyocytes were stained by TNNI3 (representative examples indicated by arrowheads). Boxed area in top image is magnified in the bottom image. Arrows indicate non-myocytes. Bar, 100 μm (C, top) and 20 μm (C, bottom).</p

AAV9:Tnnt2-Cre inactivation of <i>Gata4/6</i> in adult heart.

<p><b>(A).</b> Experimental timeline. Virus was administred at 6 weeks of age. After weekly echocardiography, tissue was analyzed at 11 weeks of age. <b>(B).</b> Quantitation of <i>Gata4</i> and <i>Gata6</i> inactivation in ventricular myocardium by qRTPCR. <b>(C).</b> Loss of GATA4 and GATA6 protein with AAV9:Tnnt-Cre but not AAV9:Tnnt2-Luc. Ventricular myocardial protein was analyzed by western blotting. <b>(D).</b> Mild systolic dysfunction following adult stage inactivation of Gata4 and Gata6, as determined by echocardiography. n = 9 (knockout) or 4 (control). *, P<0.05.</p