Half sphere exposure used in predicting discontinuous B-cell epitopes. Song et al. have developed an online webserver termed HSEpred to predict half-sphere exposure from protein primary sequences. HSEpred server can achieve the correlation coefficients of 0.72 and 0.68 between the predicted and observed HSE-up and HSE-down measures, respectively, when evaluated on a well-prepared non-homologous protein structure dataset. Moreover, residue contact number (CN) can also be accurately predicted by HSEpred webserver using the summation of the predicted HSE-up and HSE-down values, which has further enlarged the application of this new solvent exposure measure. Recently, Heffernan et al. has developed the most accurate predictor for
N-sphere space and is an example of an -manifold. The volume form of an -sphere of radius is given by where is the Hodge star operator; see for a discussion and proof of this formula in the case . As a result, The space enclosed by an -sphere is called an -ball. An -ball is closed if it includes the -sphere, and it is open if it does not include the -sphere. Specifically: Topologically, an -sphere can be constructed as a one-point compactification of -dimensional Euclidean space. Briefly, the -sphere can be described as , which is -dimensional Euclidean space plus