Molecular basis of complete C4 deficiency. A study of three patients.

The highly polymorphic fourth component of human complement (C4) is usually encoded by two genes, C4A and C4B, adjacent to the 21-hydroxylase (21-OH) genes and is also remarkable by the high frequency of the null alleles, C4A*Q0 and C4B*Q0. Complete C4 deficiency is exceptional because this condition appears only in homozygotes for the very rare double-null haplotype C4AQ0,BQ0. This condition in most cases gives rise to systemic lupus erythematosus and an increased susceptibility to infections. The molecular basis for complete C4 deficiency has not yet been established. Therefore we studied the DNA of three previously described C4 deficient patients belonging to unrelated families by restriction fragment length polymorphism analysis using C4 and 21-OH probes. These studies revealed a deletion of the C4B and 21-OHA genes in two patients and no deletion at all in the third patient. Therefore, complete C4 deficiency as a result of homozygosity for the C4AQ0, BQ0 haplotype is not a consequence of a deletion of the C4 genes. The molecular basis of this genetic abnormality is certainly very complex and may vary also from one case to another.

Hopf hypersurfaces in pseudo-Riemannian complex and para-complex space forms

The study of real hypersurfaces in pseudo-Riemannian complex space forms and para-complex space forms, which are the pseudo-Riemannian generalizations of the complex space forms, is addressed. It is proved that there are no umbilic hypersurfaces, nor real hypersurfaces with parallel shape operator in such spaces. Denoting by J be the complex or para-complex structure of a pseudo-complex or para-complex space form respectively, a non-degenerate hypersurface of such space with unit normal vector field N is said to be Hopf if the tangent vector field JN is a principal direction. It is proved that if a hypersurface is Hopf, then the corresponding principal curvature (the Hopf curvature) is constant. It is also observed that in some cases a Hopf hypersurface must be, locally, a tube over a complex (or para-complex) submanifold, thus generalizing previous results of Cecil, Ryan and Montiel.