The discrete dynamics of symmetric competition in the plane

We consider the generalized Lotka-Volterra two-species system xn + 1 = xn exp(r1(1 - xn) - s1yn) yn + 1 = yn exp(r2(1 - yn) - s2xn) originally proposed by R. M. May as a model for competitive interaction. In the symmetric case that r1 = r2 and s1 = s2, a region of ultimate confinement is found and the dynamics therein are described in some detail. The bifurcations of periodic points of low period are studied, and a cascade of period-doubling bifurcations is indicated. Within the confinement region, a parameter region is determined for the stable Hopf bifurcation of a pair of symmetrically placed period-two points, which imposes a second component of oscillation near the stable cycles. It is suggested that the symmetric competitive model contains much of the dynamical complexity to be expected in any discrete two-dimensional competitive model.

Hormone-stimulated polyphosphoinositide breakdown in rat liver plasma membranes. Roles of guanine nucleotides and calcium

Calcium-sensitive inositide release in a purified rat liver plasma membrane preparation is increased by calcium-mobilizing hormones in the presence of guanine nucleotides. Vasopressin-stimulated inositide release is evident in the presence of GTP or its nonhydrolyzable analogs guanyl-5'-yl imidodiphosphate and guanosine 5'-(3-O-thio)triphosphate (GTP gamma S). The stimulation of inositide release by (-)-epinephrine (alpha 1), angiotensin II, or vasopressin in the presence of either 1 microM or 10 microM GTP gamma S correlates with the number of receptors present for each hormone. The guanine nucleotide and hormonal stimulation is evident on both inositol trisphosphate production and phosphatidylinositol bisphosphate degradation. Ethylene glycol bis(beta-aminoethyl ether)-N,N,N',N'-tetraacetic acid (1 mM) completely abolishes stimulation by guanine nucleotides and hormone. Prior treatment of plasma membranes with cholera toxin or islet activating protein or prior injection of animals with islet activating protein does not affect stimulation of inositide release by GTP gamma S or GTP gamma S plus vasopressin. Stimulation by GTP gamma S is dependent upon magnesium and is inhibitable by guanosine 5'-(2-O-thio) diphosphate. Inositide release from the plasma membrane exhibits half-maximal stimulation by calcium at approximately 100 nM free calcium in the presence of 1.5 mM MgCl2 and at approximately 10 microM free calcium in the presence of 10 mM MgCl2. Addition of guanine nucleotides decreases the requirement for calcium and also increases the activity at saturating calcium. The results presented suggest that calcium-mobilizing hormones stimulate polyphosphoinositide breakdown in rat liver plasma membranes through a novel guanine nucleotide binding protein.

Regulation of a liver plasma membrane phosphoinositide phosphodiesterase by guanine nucleotides and calcium

Rat liver plasma membranes are enriched in a Ca2+-dependent phosphodiesterase active on phosphatidylinositol 4,5-P2 and phosphatidylinositol 4-P, but not phosphatidylinositol. Inositol-P3 is the first product of the reaction, but is rapidly degraded. Micromolar concentrations of GTP and its nonhydrolyzable analogues stimulate the reaction, whereas GDP, GMP and other nucleoside triphosphates are inactive. GTP and its analogues decrease the requirement of the reaction for Ca2+ and also increase its activity at saturating Ca2+. These results support the hypothesis that guanine nucleotides and a guanine nucleotide binding regulatory protein are involved in coupling the receptors for Ca2+-mediated agonists to the breakdown of plasma membrane phosphatidylinositol 4,5-P2.