The multivariate L1-median and associated data depth

This paper gives three related results: (i) a new, simple, fast, monotonically converging algorithm for deriving the L1-median of a data cloud in ℝd, a problem that can be traced to Fermat and has fascinated applied mathematicians for over three centuries; (ii) a new general definition for depth functions, as functions of multivariate medians, so that different definitions of medians will, correspondingly, give rise to different dept functions; and (iii) a simple closed-form formula of the L1-depth function for a given data cloud in ℝd.

Substitution rate calibration of small subunit ribosomal RNA identifies chlorarachniophyte endosymbionts as remnants of green algae.

Chlorarachniophytes are amoeboid algae with chlorophyll a and b containing plastids that are surrounded by four membranes instead of two as in plants and green algae. These extra membranes form important support for the hypothesis that chlorarachniophytes have acquired their plastids by the ingestion of another eukaryotic plastid-containing alga. Chlorarachniophytes also contain a small nucleus-like structure called the nucleomorph situated between the two inner and the two outer membranes surrounding the plastid. This nucleomorph is a remnant of the endosymbiont's nucleus and encodes, among other molecules, small subunit ribosomal RNA. Previous phylogenetic analyses on the basis of this molecule provided unexpected and contradictory evidence for the origin of the chlorarachniophyte endosymbiont. We developed a new method for measuring the substitution rates of the individual nucleotides of small subunit ribosomal RNA. From the resulting substitution rate distribution, we derived an equation that gives a more realistic relationship between sequence dissimilarity and evolutionary distance than equations previously available. Phylogenetic trees constructed on the basis of evolutionary distances computed by this new method clearly situate the chlorarachniophyte nucleomorphs among the green algae. Moreover, this relationship is confirmed by transversion analysis of the Chlorarachnion plastid small subunit ribosomal RNA.