A mathematical theory of stochastic microlensing. II. Random images, shear, and the Kac-Rice formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the star's mass. As a consequence, the pdf's of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of global expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars. © 2009 American Institute of Physics.

Augmentation of cardiac contractility mediated by the human beta(3)-adrenergic receptor overexpressed in the hearts of transgenic mice.

BACKGROUND: Stimulation of beta(1)- and beta(2)-adrenergic receptors (ARs) in the heart results in positive inotropy. In contrast, it has been reported that the beta(3)AR is also expressed in the human heart and that its stimulation leads to negative inotropic effects. METHODS AND RESULTS: To better understand the role of beta(3)ARs in cardiac function, we generated transgenic mice with cardiac-specific overexpression of 330 fmol/mg protein of the human beta(3)AR (TGbeta(3) mice). Hemodynamic characterization was performed by cardiac catheterization in closed-chest anesthetized mice, by pressure-volume-loop analysis, and by echocardiography in conscious mice. After propranolol blockade of endogenous beta(1)- and beta(2)ARs, isoproterenol resulted in an increase in contractility in the TGbeta(3) mice (30%), with no effect in wild-type mice. Similarly, stimulation with the selective human beta(3)AR agonist L-755,507 significantly increased contractility in the TGbeta(3) mice (160%), with no effect in wild-type mice, as determined by hemodynamic measurements and by end-systolic pressure-volume relations. The underlying mechanism of the positive inotropy incurred with L-755,507 in the TGbeta(3) mice was investigated in terms of beta(3)AR-G-protein coupling and adenylyl cyclase activation. Stimulation of cardiac membranes from TGbeta(3) mice with L-755,507 resulted in a pertussis toxin-insensitive 1.33-fold increase in [(35)S]GTPgammaS loading and a 1.6-fold increase in adenylyl cyclase activity. CONCLUSIONS: Cardiac overexpression of human beta(3)ARs results in positive inotropy only on stimulation with a beta(3)AR agonist. Overexpressed beta(3)ARs couple to G(s) and activate adenylyl cyclase on agonist stimulation.

In vivo inhibition of elevated myocardial beta-adrenergic receptor kinase activity in hybrid transgenic mice restores normal beta-adrenergic signaling and function.

BACKGROUND: The clinical syndrome of heart failure (HF) is characterized by an impaired cardiac beta-adrenergic receptor (betaAR) system, which is critical in the regulation of myocardial function. Expression of the betaAR kinase (betaARK1), which phosphorylates and uncouples betaARs, is elevated in human HF; this likely contributes to the abnormal betaAR responsiveness that occurs with beta-agonist administration. We previously showed that transgenic mice with increased myocardial betaARK1 expression had impaired cardiac function in vivo and that inhibiting endogenous betaARK1 activity in the heart led to enhanced myocardial function. METHODS AND RESULTS: We created hybrid transgenic mice with cardiac-specific concomitant overexpression of both betaARK1 and an inhibitor of betaARK1 activity to study the feasibility and functional consequences of the inhibition of elevated betaARK1 activity similar to that present in human HF. Transgenic mice with myocardial overexpression of betaARK1 (3 to 5-fold) have a blunted in vivo contractile response to isoproterenol when compared with non-transgenic control mice. In the hybrid transgenic mice, although myocardial betaARK1 levels remained elevated due to transgene expression, in vitro betaARK1 activity returned to control levels and the percentage of betaARs in the high-affinity state increased to normal wild-type levels. Furthermore, the in vivo left ventricular contractile response to betaAR stimulation was restored to normal in the hybrid double-transgenic mice. CONCLUSIONS: Novel hybrid transgenic mice can be created with concomitant cardiac-specific overexpression of 2 independent transgenes with opposing actions. Elevated myocardial betaARK1 in transgenic mouse hearts (to levels seen in human HF) can be inhibited in vivo by a peptide that can prevent agonist-stimulated desensitization of cardiac betaARs. This may represent a novel strategy to improve myocardial function in the setting of compromised heart function.

Enhanced myocardial relaxation in vivo in transgenic mice overexpressing the beta2-adrenergic receptor is associated with reduced phospholamban protein.

To assess the effect of targeted myocardial beta-adrenergic receptor (AR) stimulation on relaxation and phospholamban regulation, we studied the physiological and biochemical alterations associated with overexpression of the human beta2-AR gene in transgenic mice. These mice have an approximately 200-fold increase in beta-AR density and a 2-fold increase in basal adenylyl cyclase activity relative to negative littermate controls. Mice were catheterized with a high fidelity micromanometer and hemodynamic recordings were obtained in vivo. Overexpression of the beta2-AR altered parameters of relaxation. At baseline, LV dP/dt(min) and the time constant of LV pressure isovolumic decay (Tau) in the transgenic mice were significantly shorter compared with controls, indicating markedly enhanced myocardial relaxation. Isoproterenol stimulation resulted in shortening of relaxation velocity in control mice but not in the transgenic mice, indicating maximal relaxation in these animals. Immunoblotting analysis revealed a selective decrease in the amount of phospholamban protein, without a significant change in the content for either sarcoplasmic reticulum Ca2+ ATPase or calsequestrin, in the transgenic hearts compared with controls. This study indicates that myocardial relaxation is both markedly enhanced and maximal in these mice and that conditions associated with chronic beta-AR stimulation can result in a selective reduction of phospholamban protein.

Hybrid transgenic mice reveal in vivo specificity of G protein-coupled receptor kinases in the heart.

G protein-coupled receptor kinases (GRKs) phosphorylate activated G protein-coupled receptors, including alpha(1B)-adrenergic receptors (ARs), resulting in desensitization. In vivo analysis of GRK substrate selectivity has been limited. Therefore, we generated hybrid transgenic mice with myocardium-targeted overexpression of 1 of 3 GRKs expressed in the heart (GRK2 [commonly known as the beta-AR kinase 1], GRK3, or GRK5) with concomitant cardiac expression of a constitutively activated mutant (CAM) or wild-type alpha(1B)AR. Transgenic mice with cardiac CAMalpha(1B)AR overexpression had enhanced myocardial alpha(1)AR signaling and elevated heart-to-body weight ratios with ventricular atrial natriuretic factor expression denoting myocardial hypertrophy. Transgenic mouse hearts overexpressing only GRK2, GRK3, or GRK5 had no hypertrophy. In hybrid transgenic mice, enhanced in vivo signaling through CAMalpha(1B)ARs, as measured by myocardial diacylglycerol content, was attenuated by concomitant overexpression of GRK3 but not GRK2 or GRK5. CAMalpha(1B)AR-induced hypertrophy and ventricular atrial natriuretic factor expression were significantly attenuated with either concurrent GRK3 or GRK5 overexpression. Similar GRK selectivity was seen in hybrid transgenic mice with wild-type alpha(1B)AR overexpression concurrently with a GRK. GRK2 overexpression was without effect on any in vivo CAM or wild-type alpha(1B)AR cardiac phenotype, which is in contrast to previously reported in vitro findings. Furthermore, endogenous myocardial alpha(1)AR mitogen-activated protein kinase signaling in single-GRK transgenic mice also exhibited selectivity, as GRK3 and GRK5 desensitized in vivo alpha(1)AR mitogen-activated protein kinase responses that were unaffected by GRK2 overexpression. Thus, these results demonstrate that GRKs differentially interact with alpha(1B)ARs in vivo such that GRK3 desensitizes all alpha(1B)AR signaling, whereas GRK5 has partial effects and, most interestingly, GRK2 has no effect on in vivo alpha(1B)AR signaling in the heart.

Ligand-induced overexpression of a constitutively active beta2-adrenergic receptor: pharmacological creation of a phenotype in transgenic mice.

Transgenic overexpression (40- to 100-fold) of the wild-type human beta2-adrenergic receptor in the hearts of mice leads to a marked increase in cardiac contractility, which is apparently due to the low level of spontaneous (i.e., agonist-independent) activity inherent in the receptor. Here we report that transgenic mice expressing a mutated constitutively active form of the receptor (CAM) show no such phenotype, owing to its modest expression (3-fold above endogenous cardiac beta-adrenergic receptor levels). Surprisingly, treatment of the animals with a variety of beta-adrenergic receptor ligands leads to a 50-fold increase in CAM beta2-adrenergic receptor expression, by stabilizing the CAM beta2-adrenergic receptor protein. Receptor up-regulation leads in turn to marked increases in adenylate cyclase activity, atrial tension determined in vitro, and indices of cardiac contractility determined in vivo. These results illustrate a novel mechanism for regulating physiological responses, i.e., ligand-induced stabilization of a constitutively active but inherently unstable protein.

Receptor-specific in vivo desensitization by the G protein-coupled receptor kinase-5 in transgenic mice.

Transgenic mice were generated with cardiac-specific overexpression of the G protein-coupled receptor kinase-5 (GRK5), a serine/threonine kinase most abundantly expressed in the heart compared with other tissues. Animals overexpressing GRK5 showed marked beta-adrenergic receptor desensitization in both the anesthetized and conscious state compared with nontransgenic control mice, while the contractile response to angiotensin II receptor stimulation was unchanged. In contrast, the angiotensin II-induced rise in contractility was significantly attenuated in transgenic mice overexpressing the beta-adrenergic receptor kinase-1, another member of the GRK family. These data suggest that myocardial overexpression of GRK5 results in selective uncoupling of G protein-coupled receptors and demonstrate that receptor specificity of the GRKs may be important in determining the physiological phenotype.

Myocardial expression of a constitutively active alpha 1B-adrenergic receptor in transgenic mice induces cardiac hypertrophy.

Transgenic mice were generated by using the alpha-myosin heavy chain promoter coupled to the coding sequence of a constitutively active mutant alpha 1B-adrenergic receptor (AR). These transgenic animals demonstrated cardiac-specific expression of this alpha 1-AR with resultant activation of phospholipase C as shown by increased myocardial diacylglycerol content. A phenotype consistent with cardiac hypertrophy developed in adult transgenic mice with increased heart/body weight ratios, myocyte cross-sectional areas, and ventricular atrial natriuretic factor mRNA levels relative to nontransgenic controls. These transgenic animals may provide insight into the biochemical triggers that induce hypertrophy in cardiac disease and serve as a convenient experimental model for studies of this condition.