Fig. 2

An illustrative example of spherical coordinate system application. The example gene contains 3 isoforms, and their splicing ratios in a sample are p ₁, p ₂, p ₃ with the constraint ∑ ³ _i = 1 p _i  = 1. We let \( {q}_i=\sqrt{p_i},\ i=1,2,3 \) and present this sample as a point (the red point in the plot) in the 3-dimensional rectangular coordinate system. We then apply spherical coordinate system in which this sample can also be regarded as a point (ρ, ϕ₁, ϕ₂) in a spherical coordinate system. Because of the constraint that ∑ ³ _i = 1 p _i  = 1, the spherical coordinates of this sample is (1, ϕ₁, ϕ₂), in which ϕ₁ and ϕ₂ are independent from each other