Discontinuous Galerkin methods for the p-biharmonic equation from a discrete variational perspective

We study discontinuous Galerkin approximations of the p-biharmonic equation for p∈(1,∞) from a variational perspective. We propose a discrete variational formulation of the problem based on an appropriate definition of a finite element Hessian and study convergence of the method (without rates) using a semicontinuity argument. We also present numerical experiments aimed at testing the robustness of the method.

Minimal regulation of platelet activity by PECAM-1.

PECAM-1 is a member of the superfamily of immunoglobulins (Ig) and is expressed on platelets at moderate level. PECAM-1 has been reported to have contrasting effects on platelet activation by the collagen receptor GPVI and the integrin, alphaIIbbeta3, even though both receptors signal through Src-kinase regulation of PLCgamma2. The present study compares the role of PECAM-1 on platelet activation by these two receptors and by the lectin receptor, CLEC-2, which also signals via PLCgamma2. Studies using PECAM-1 knockout-mice and cross-linking of PECAM-1 using specific antibodies demonstrated a minor inhibitory role on platelet responses to the above three receptors and also under some conditions to the G-protein agonist thrombin. The degree of inhibition was considerably less than that produced by PGI2, which elevates cAMP. There was no significant difference in thrombus formation on collagen in PECAM-1-/- platelets relative to litter-matched controls. The very weak inhibitory effect of PECAM-1 on platelet activation relative to that of PGI2 indicate that the Ig-receptor is not a major regulator of platelet activation. PECAM-1 has been reported to have contrasting effects on platelet activation. The present study demonstrates a very mild or negligible effect on platelet activation in response to stimulation by a variety of agonists, thereby questioning the physiological role of the immunoglobulin receptor as a major regulator of platelet activation.

Relative importance of transpiration rate and leaf morphological traits for the regulation of leaf temperature

Urban greening solutions such as green roofs help improve residents’ thermal comfort and building insulation. However, not all plants provide the same level of cooling. This is partially due to differences in plant structure and function, including different mechanisms that plants employ to regulate leaf temperature. Ranking of multiple leaf/plant traits involved in the regulation of leaf temperature (and, consequently, plants’ cooling ‘service’) is not well understood. We therefore investigated the relative importance of water loss, leaf colour, thickness and extent of pubescence for the regulation of leaf temperature, in the context of species for semi-extensive green roofs. Leaf temperature were measured with an infrared imaging camera in a range of contrasting genotypes within three plant genera (Heuchera, Salvia and Sempervivum). In three glasshouse experiments (each evaluating three or four genotypes of each genera) we varied water availability to the plants and assessed how leaf temperature altered depending on water loss and specific leaf traits. Greatest reductions in leaf temperature were closely associated with higher water loss. Additionally, in non-succulents (Heuchera, Salvia), lighter leaf colour and longer hair length (on pubescent leaves) both contributed to reduced leaf temperature. However, in succulent Sempervivum, colour/pubescence made no significant contribution; leaf thickness and water loss rate were the key regulating factors. We propose that this can lead to different plant types having significantly different potentials for cooling. We suggest that maintaining transpirational water loss by sustainable irrigation and selecting urban plants with favourable morphological traits is the key to maximising thermal benefits provided by applications such as green roofs.

Regulation of mitogen-activated protein kinase cascade in adult rat heart preparations in vitro.

The regulation of mitogen-activated protein kinase (MAPK) and MAPK kinase (MEK) was studied in freshly isolated adult rat heart preparations. In contrast to the situation in ventricular myocytes cultured from neonatal rat hearts, stimulation of MAPK activity by 1 mumol/L phorbol 12-myristate 13-acetate (PMA) was not consistently detectable in crude extracts. After fast protein liquid chromatography, MAPK isoforms p42MAPK and p44MAPK and two peaks of MEK were shown to be activated > 10-fold in perfused hearts or ventricular myocytes exposed to 1 mumol/L PMA for 5 minutes. The identities of MAPK or MEK were confirmed by immunoblotting and, for MAPK, by the "in-gel" myelin basic protein phosphorylation assay. In retrogradely perfused hearts, high coronary perfusion pressure (120 mm Hg for 5 minutes), norepinephrine (50 mumol/L for 5 minutes), or isoproterenol (50 mumol/L for 5 minutes) stimulated MAPK and MEK approximately 2- to 5-fold. In isolated myocytes, endothelin 1 (100 nmol/L for 5 minutes) also stimulated MAPK, but stimulation by norepinephrine or isoproterenol was difficult to detect. Immunoblotting showed that the relative abundances of MAPK and MEK protein in ventricles declined to < 20% of their postpartal abundances after 50 days. This may explain the difficulties encountered in assaying the activity of MAPK in crude extracts from adult hearts. We conclude that potentially hypertrophic agonists and interventions stimulate the MAPK cascade in adult rats and suggest that the MAPK cascade may be an important intracellular signaling pathway in this response.

Regulation of phospholipases C and D in rat ventricular myocytes: stimulation by endothelin-1, bradykinin and phenylephrine.

The physiological activator of protein kinase C (PKC), diacylglycerol, is formed by hydrolysis of phosphoinositides (PI) by phospholipase C (PLC) or phosphatidylcholine by phospholipase D (PLD). We have measured activation of these phospholipases by endothelin-1 (ET-1), bradykinin (BK), or phenylephrine (PE) in ventricular myocytes cultured from neonatal rat. The stimulation of PI hydrolysis after 10 min by 0.1 microM ET-1 (about 12-fold) was much greater than for BK or PE (each about four-fold), and did not correlate with translocation of nPKC delta or nPKC epsilon (Clerk A. Bogoyevitch MA. Andersson MB. Sugden PH, 1994. J Biol Chem 269: 32848-32857: Clerk A, Gillespie-Brown J, Fuller SJ, Sugden PH, 1996. Biochem J 317: 109-118). However, ET-1 and BK stimulated a similar rapid increase in [3H]InsP, formation (< 30 s), which was much greater than that seen with PE. This early phase correlated with PKC translocation. Acute or chronic exposure to 12-O-tetradecanoylphorbol-13-acetate (TPA) or treatment with Ro-31-8220 showed that the stimulation of PI hydrolysis by PE, but not ET-1 or BK, was inhibited by activation of PKC. Furthermore, ET-1 and BK heterologously desensitized the stimulation of PI hydrolysis by PE, ET-1 or BK homologously uncoupled their own receptors from [3H]InsP3 formation, but there was no evidence of heterologous desensitization with these two agonists. Anomalously, chronic exposure to TPA increased the stimulation of PI hydrolysis by BK, but this probably resulted from an increase in BK receptor density. PLD was also rapidly activated by TPA. ET-1, BK or PE. Experiments with Ro-31-8220 showed that the stimulation of PLD by ET-1 and BK was mediated through activation of PKC. We discuss the characteristics of the activation of PI hydrolysis and PLD by ET-1, BK, and PE with respect to the translocation of PKC.

Regulation of the ERK subgroup of MAP kinase cascades through G protein-coupled receptors.

The extracellularly-responsive kinase (ERK) subfamily of mitogen-activated protein kinases (MAPKs) has been implicated in the regulation of cell growth and differentiation. Activation of ERKs involves a two-step protein kinase cascade lying upstream from ERK, in which the Raf family are the MAPK kinase kinases and the MEK1/MEK2 isoforms are the MAPK kinases. The linear sequence of Raf --> MEK --> ERK constitutes the ERK cascade. Although the ERK cascade is activated through growth factor-regulated receptor protein tyrosine kinases, they are also modulated through G protein-coupled receptors (GPCRs). All four G protein subfamilies (Gq/11 Gi/o, Gs and G12/13) influence the activation state of ERKs. In this review, we describe the ERK cascade and characteristics of its activation through GPCRs. We also discuss the identity of the intervening steps that may couple agonist binding at GPCRs to activation of the ERK cascade.

Regulation of mitogen-activated protein kinase cascades in the heart.

Using primary cultures of neonatal rat ventricular myocytes and isolated adult rat hearts as models, we have characterized extensively the regulation of MAPKs in the heart. The ERKs are activated primarily by GPCR agonists acting through PKC. These agonists can also activate the JNKs although the mechanism is unclear. Cellular stresses stimulate strong activation of the JNKs, but also cause some stimulation of ERKs. Activation of p38-MAPK has so far only been demonstrated in intact adult hearts subjected to stresses and probably leads to activation of MAPKAPK2. Both cellular stresses and GPCR agonists induce phosphorylation of c-Jun, but only the latter causes upregulation of c-Jun protein.

Regulation of Ras.GTP loading and Ras-Raf association in neonatal rat ventricular myocytes by G protein-coupled receptor agonists and phorbol ester. Activation of the extracellular signal-regulated kinase cascade by phorbol ester is mediated by Ras.

The small G protein Ras has been implicated in hypertrophy of cardiac myocytes. We therefore examined the activation (GTP loading) of Ras by the following hypertrophic agonists: phorbol 12-myristate 13-acetate (PMA), endothelin-1 (ET-1), and phenylephrine (PE). All three increased Ras.GTP loading by 10-15-fold (maximal in 1-2 min), as did bradykinin. Other G protein-coupled receptor agonists (e.g. angiotensin II, carbachol, isoproterenol) were less effective. Activation of Ras by PMA, ET-1, or PE was reduced by inhibition of protein kinase C (PKC), and that induced by ET-1 or PE was partly sensitive to pertussis toxin. 8-(4-Chlorophenylthio)-cAMP (CPT-cAMP) did not inhibit Ras.GTP loading by PMA, ET-1, or PE. The association of Ras with c-Raf protein was increased by PMA, ET-1, or PE, and this was inhibited by CPT-cAMP. However, only PMA and ET-1 increased Ras-associated mitogen-activated protein kinase kinase 1-activating activity, and this was decreased by PKC inhibition, pertussis toxin, and CPT-cAMP. PMA caused the rapid appearance of phosphorylated (activated) extracellular signal-regulated kinase in the nucleus, which was inhibited by a microinjected neutralizing anti-Ras antibody. We conclude that PKC- and Gi-dependent mechanisms mediate the activation of Ras in myocytes and that Ras activation is required for stimulation of extracellular signal-regulated kinase by PMA.

Regulation of bcl-2 family proteins during development and in response to oxidative stress in cardiac myocytes: association with changes in mitochondrial membrane potential.

Cardiac myocyte apoptosis is potentially important in many cardiac disorders. In other cells, Bcl-2 family proteins and mitochondrial dysfunction are probably key regulators of the apoptotic response. In the present study, we characterized the regulation of antiapoptotic (Bcl-2, Bcl-xL) and proapoptotic (Bad, Bax) Bcl-2 family proteins in the rat heart during development and in oxidative stress-induced apoptosis. Bcl-2 and Bcl-xL were expressed at high levels in the neonate, and their expression was sustained during development. In contrast, although Bad and Bax were present at high levels in neonatal hearts, they were barely detectable in adult hearts. We confirmed that H(2)O(2) induced cardiac myocyte cell death, stimulating poly(ADP-ribose) polymerase proteolysis (from 2 hours), caspase-3 proteolysis (from 2 hours), and DNA fragmentation (from 8 hours). In unstimulated neonatal cardiac myocytes, Bcl-2 and Bcl-xL were associated with the mitochondria, but Bad and Bax were predominantly present in a crude cytosolic fraction. Exposure of myocytes to H(2)O(2) stimulated rapid translocation of Bad (<5 minutes) to the mitochondria. This was followed by the subsequent degradation of Bad and Bcl-2 (from approximately 30 minutes). The levels of the mitochondrial membrane marker cytochrome oxidase remained unchanged. H(2)O(2) also induced translocation of cytochrome c from the mitochondria to the cytosol within 15 to 30 minutes, which was indicative of mitochondrial dysfunction. Myocytes exposed to H(2)O(2) showed an early loss of mitochondrial membrane potential (assessed by fluorescence-activated cell sorter analysis) from 15 to 30 minutes, which was partially restored by approximately 1 hour. However, a subsequent irreversible loss of mitochondrial membrane potential occurred that correlated with cell death. These data suggest that the regulation of Bcl-2 and mitochondrial function are important factors in oxidative stress-induced cardiac myocyte apoptosis.

Regulation of protein kinase B and 4E-BP1 by oxidative stress in cardiac myocytes.

Stimulation of phosphatidylinositol 3'-kinase (PI3K) and protein kinase B (PKB) is implicated in the regulation of protein synthesis in various cells. One mechanism involves PI3K/PKB-dependent phosphorylation of 4E-BP1, which dissociates from eIF4E, allowing initiation of translation from the 7-methylGTP cap of mRNAs. We examined the effects of insulin and H(2)O(2) on this pathway in neonatal cardiac myocytes. Cardiac myocyte protein synthesis was increased by insulin, but was inhibited by H(2)O(2). PI3K inhibitors attenuated basal levels of protein synthesis and inhibited the insulin-induced increase in protein synthesis. Insulin or H(2)O(2) increased the phosphorylation (activation) of PKB through PI3K, but, whereas insulin induced a sustained response, the response to H(2)O(2) was transient. 4E-BP1 was phosphorylated in unstimulated cells, and 4E-BP1 phosphorylation was increased by insulin. H(2)O(2) stimulated dephosphorylation of 4E-BP1 by increasing protein phosphatase (PP1/PP2A) activity. This increased the association of 4E-BP1 with eIF4E, consistent with H(2)O(2) inhibition of protein synthesis. The effects of H(2)O(2) were sufficient to override the stimulation of protein synthesis and 4E-BP1 phosphorylation induced by insulin. These results indicate that PI3K and PKB are important regulators of protein synthesis in cardiac myocytes, but other factors, including phosphatase activity, modulate the overall response.

Regulation of mitogen-activated protein kinases in cardiac myocytes through the small G protein Rac1.

Small guanine nucleotide-binding proteins of the Ras and Rho (Rac, Cdc42, and Rho) families have been implicated in cardiac myocyte hypertrophy, and this may involve the extracellular signal-related kinase (ERK), c-Jun N-terminal kinase (JNK), and/or p38 mitogen-activated protein kinase (MAPK) cascades. In other systems, Rac and Cdc42 have been particularly implicated in the activation of JNKs and p38-MAPKs. We examined the activation of Rho family small G proteins and the regulation of MAPKs through Rac1 in cardiac myocytes. Endothelin 1 and phenylephrine (both hypertrophic agonists) induced rapid activation of endogenous Rac1, and endothelin 1 also promoted significant activation of RhoA. Toxin B (which inactivates Rho family proteins) attenuated the activation of JNKs by hyperosmotic shock or endothelin 1 but had no effect on p38-MAPK activation. Toxin B also inhibited the activation of the ERK cascade by these stimuli. In transfection experiments, dominant-negative N17Rac1 inhibited activation of ERK by endothelin 1, whereas activated V12Rac1 cooperated with c-Raf to activate ERK. Rac1 may stimulate the ERK cascade either by promoting the phosphorylation of c-Raf or by increasing MEK1 and/or -2 association with c-Raf to facilitate MEK1 and/or -2 activation. In cardiac myocytes, toxin B attenuated c-Raf(Ser-338) phosphorylation (50 to 70% inhibition), but this had no effect on c-Raf activity. However, toxin B decreased both the association of MEK1 and/or -2 with c-Raf and c-Raf-associated ERK-activating activity. V12Rac1 cooperated with c-Raf to increase expression of atrial natriuretic factor (ANF), whereas N17Rac1 inhibited endothelin 1-stimulated ANF expression, indicating that the synergy between Rac1 and c-Raf is potentially physiologically important. We conclude that activation of Rac1 by hypertrophic stimuli contributes to the hypertrophic response by modulating the ERK and/or possibly the JNK (but not the p38-MAPK) cascades.

Up-regulation of c-jun mRNA in cardiac myocytes requires the extracellular signal-regulated kinase cascade, but c-Jun N-terminal kinases are required for efficient up-regulation of c-Jun protein.

Cardiac hypertrophy, an important adaptational response, is associated with up-regulation of the immediate early gene, c- jun, which encodes the c-Jun transcription factor. c-Jun may feed back to up-regulate its own transcription and, since the c-Jun N-terminal kinase (JNK) family of mitogen-activated protein kinases (MAPKs) phosphorylate c-Jun(Ser-63/73) to increase its transactivating activity, JNKs are thought to be the principal factors involved in c- jun up-regulation. Hypertrophy in primary cultures of cardiac myocytes is induced by endothelin-1, phenylephrine or PMA, probably through activation of one or more of the MAPK family. These three agonists increased c- jun mRNA with the rank order of potency of PMA approximately endothelin-1>phenylephrine. Up-regulation of c- jun mRNA by endothelin-1 was attenuated by inhibitors of protein kinase C (GF109203X) and the extracellular signal-regulated kinase (ERK) cascade (PD98059 or U0126), but not by inhibitors of the JNK (SP600125) or p38-MAPK (SB203580) cascades. Hyperosmotic shock (0.5 M sorbitol) powerfully activates JNKs, but did not increase c- jun mRNA. These data suggest that ERKs, rather than JNKs, are required for c- jun up-regulation. However, endothelin-1 and phenylephrine induced greater up-regulation of c-Jun protein than PMA and phosphorylation of c-Jun(Ser-63/73) correlated with the level of c-Jun protein. Up-regulation of c-Jun protein by endothelin-1 was attenuated by inhibitors of protein kinase C and the ERK cascade, probably correlating with a primary input of ERKs into transcription. In addition, SP600125 inhibited the phosphorylation of c-Jun(Ser-63/73), attenuated the increase in c-Jun protein induced by endothelin-1 and increased the rate of c-Jun degradation. Thus whereas ERKs are the principal MAPKs required for c- jun transcription, JNKs are necessary to stabilize c-Jun for efficient up-regulation of the protein.

Regulation of gene and protein expression in cardiac myocyte hypertrophy and apoptosis

Considerable efforts have been expended in elucidating the inter-cellular and intra-cellular signaling pathways which elicit cardiac myocyte hypertrophy or apoptosis, and in identifying the changes which are associated with the end-stage of the response. The challenge now is to link the two. Although some of the signaling effects will be the acute modulation of existing protein function, long-term effects which bring about and maintain the hypertrophic state or which culminate in cell death are mediated at the level of gene and protein expression. With the advances in micro-array technology and genome sequencing, it is now possible to obtain a picture of the global gene expression profile in myocytes or in whole heart which dictates the proteins which could be made. This is not the final picture since additional regulation at the level of translation modulates the relative proportions of each protein that can be made from the transcriptome. Even here, further regulation of protein stability and turnover means that ultimately it is still necessary to examine the proteome to determine what may cause the functional changes in a cell. Thus, in order to gain a full picture of events which regulate the response and gain some insight into possible points of intervention for therapy, it is necessary to examine gene expression, mRNA translation and protein expression in concert.

Noether-type discrete conserved quantities arising from a finite element approximation of a variational problem

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether’s first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.