Polygalacturonic acid (PGA) was hydrolyzed by polygalacturonases (PGs) purified from six fungi. The oligogalacturonide products were analyzed by HPAEC-PAD (high performance anion exchange chromatography-pulsed amperimetric detection) to assess their relative amounts and degrees of polymerization. The abilities of the fungal PGs to reduce the viscosity of a solution of PGA were also determined. The potential abilities of four polygalacturonase-inhibiting proteins (PGIPs) from three plant species to inhibit or to modify the hydrolytic activity of the fungal PGs were determined by colorimetric and HPAEC-PAD analyses, respectively. Normalized activities of the different PGs acting upon the same substrate resulted in one of two distinct oligogalacturonide profiles. Viscometric analysis of the effect of PGs on the same substrate also supports two distinct patterns of cleavage. A wide range of susceptibility of the various PGs to inhibition by PGIPs was observed. The four PGs that were inhibited by all PGIPs tested exhibited an endo/exo mode of substrate cleavage, while the three PGs that were resistant to inhibition by one or more of the PGIPs proceed by a classic endo pattern of cleavage.

We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad-Yogananda by taking into account the class number of the CM order and the splitting of the prime in the CM field. In many cases we can show that our refined bounds are optimal or asymptotically optimal. We also derive asymptotic upper and lower bounds on the least degree of a CM-point on X 1 (N ). Upon comparison to bounds for the least degree for which there exist infinitely many rational points on X 1 (N ), we deduce that, for sufficiently large N , X 1 (N ) will have a rational CM point of degree smaller than the degrees of at least all but finitely many non-CM points.

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