Projections in a normed linear space and a generalization of the pseudo-inverse

The concept of a "projection function" in a finite-dimensional real or complex normed linear space H (the function P_M which carries every element into the closest element of a given subspace M) is set forth and examined.

If dim M = dim H - 1, then P_M is linear. If P_N is linear for all k-dimensional subspaces N, where 1 ≤ k < dim M, then P_M is linear.

The projective bound Q, defined to be the supremum of the operator norm of P_M for all subspaces, is in the range 1 ≤ Q < 2, and these limits are the best possible. For norms with Q = 1, P_M is always linear, and a characterization of those norms is given.

If H also has an inner product (defined independently of the norm), so that a dual norm can be defined, then when P_M is linear its adjoint P_M^H is the projection on (kernel P_M)^⊥ by the dual norm. The projective bounds of a norm and its dual are equal.

The notion of a pseudo-inverse F⁺ of a linear transformation F is extended to non-Euclidean norms. The distance from F to the set of linear transformations G of lower rank (in the sense of the operator norm ∥F - G∥) is c/∥F⁺∥, where c = 1 if the range of F fills its space, and 1 ≤ c < Q otherwise. The norms on both domain and range spaces have Q = 1 if and only if (F⁺)⁺ = F for every F. This condition is also sufficient to prove that we have (F⁺)^H = (F^H)⁺, where the latter pseudo-inverse is taken using dual norms.

In all results, the real and complex cases are handled in a completely parallel fashion.

Gene enrichment using antibodies to DNA/RNA hybrids : mapping the ribosomal DNA of slime mold and rat

A novel method for gene enrichment has been developed and applied to mapping the rRNA genes of two eucaryotic organisms. The method makes use of antibodies to DNA/RNA hybrids prepared by injecting rabbits with the synthetic hybrid poly(rA)•poly(dT). Antibodies which cross-react with non-hybrid nucleic acids were removed from the purified IgG fraction by adsorption on columns of DNA-Sepharose, oligo(dT)-cellulose, and poly(rA)-Sepharose. Subsequent purification of the specific DNA/RNA hybrid antibody was carried out on a column of oligo(dT)-cellulose to which poly(rA) was hybridized. Attachment of these antibodies to CNBr-activated Sepharose produced an affinity resin which specifically binds DNA/RNA hybrids.

In order to map the rDNA of the slime mold Dictyostelium discoideum, R-loops were formed using unsheared nuclear DNA and the 178 and 268 rRNAs of this organism. This mixture was passed through a column containing the affinity resin, and bound molecules containing R- loops were eluted by high salt. This purified rDN A was observed directly in the electron microscope. Evidence was obtained that there is a physical end to Dictyostelium rDN A molecules approximately 10 kilobase pairs (kbp) from the region which codes for the 268 rRNA. This finding is consistent with reports of other investigators that the rRNA genes exist as inverse repeats on extra-chromosomal molecules of DNA unattached to the remainder of the nuclear DNA in this organism.

The same general procedure was used to map the rRNA genes of the rat. Molecules of DNA which contained R-loops formed with the 188 and 288 rRNAs were enriched approximately 150- fold from total genomal rat DNA by two cycles of purification on the affinity column. Electron microscopic measurements of these molecules enabled the construction of an R-loop map of rat rDNA. Eleven of the observed molecules contained three or four R-loops or else two R-loops separated by a long spacer. These observations indicated that the rat rRNA genes are arranged as tandem repeats. The mean length of the repeating units was 37.2 kbp with a standard deviation of 1.3 kbp. These eleven molecules may represent repeating units of exactly the same length within the errors of the measurements, although a certain degree of length heterogeneity cannot be ruled out. If significantly shorter or longer repeating units exist, they are probably much less common than the 37.2 kbp unit.

The last section of the thesis describes the production of antibodies to non-histone chromosomal proteins which have been exposed to the ionic detergent sodium dodecyl sulfate (SDS). The presence of low concentrations of SDS did not seem to affect either production of antibodies or their general specificity. Also, a technique is described for the in situ immunofluorescent detection of protein antigens in polyacrylamide gels.