3-sphere two 3-balls. This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres. After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean
Half sphere exposure as the number of Cα atoms in the upper half (containing the pseudo-Cβ atom) and analogously HSE-down is defined as the number of Cα atoms in the opposite sphere. If only Cα atoms are available (as is the case for many simplified representations of protein structure), a related measure, called HSEα, can be used. HSEα uses a pseudo-Cβ instead of the real Cβ atom for its calculation. The position of this pseudo-Cβ atom (pCβ) is derived from the positions of preceding Cα and the following Cα. The Cα-pCβ vector is calculated by adding the Cα-Cα and Cα-Cα vectors. HSE is