The geometry of normal tissue and cancer gene expression manifolds

A recent paper shows that in gene expression space the manifold spanned by normal tissues and the manifold spanned by the corresponding tumors are disjoint. The statement is based on a two-dimensional projection of gene expression data. In the present paper, we show that, for the multi-dimensional vectors defining the centers of cloud samples: 1. The closest tumor to a given normal tissue is the tumor developed in that tissue, 2. Two normal tissues define quasi-orthogonal directions, 3. A tumor may have a projection onto its corresponding normal tissue, but it is quasi-orthogonal to all other normal tissues, and 4. The cancer manifold is roughly obtained by translating the normal tissue manifold along an orthogonal direction defined by a global cancer progression axis. These geometrical properties add a new characterization of normal tissues and tumors and may have biological significance. Indeed, normal tissues at the vertices of a high-dimensional simplex could indicate genotype optimization for given tissue functions, and a way of avoiding errors in embryonary development. On the other hand, the cancer progression axis could define relevant pan-cancer genes and seems to be consistent with the atavistic theory of tumors.