Inversion formulae for ellipsoid of revolution macromolecular shape functions.

Explicit relations linking ellipsoid of revolution shape parameters with the axial ration (a/b) have been available for a long time. It is not possible to directly invert these functions to give a/b as a function of the shape parameter, so in the past graphical or tabular interpolation methods have been necessary. By the use of simple unconstrained polynomial expansions this restriction is no longer necessary. For the two salient hydration-dependent shape functions (P from the frictional ratio and v from the intrinsic viscosity) and the four salient hydration-independent shape functions (beta and R, both from sedimentation and viscosity measurements, and Lambda and Pi from combining viscosity measurements with respectively fluorescence depolarization or thermodynamic second virial coefficient measurements) the polynomial expansion coefficients (to no more than seventh order) are given for the ranges 1.1 < a/b < 2.0, 2 < 2/b < 10, and 10 < a/b < 100. The polynomial coefficients give an excellent fit to the real data values, except for four uninteresting cases (where the shape function is too insensitive for practical use).