Electrophoretic mobility equation for protein with molecular shape and charge multipole effects

We derive a simple formula for the free solution electrophoretic mobility of protein by including both molecular shape and charge distribution effects. The molecular shape of protein is described by a deformed sphere model, while the charge distribution is represented in terms of net charge, charge dipole, and charge quadrupole. The deformed sphere model approximates the radial coordinate of the protein surface as a simple quadratic equation based on the atomic coordinate data. Charge dipole does not affect the mobility of protein. Combined with the quadratic coefficients of the surface equation, charge quadrupole affects the mobility. When the charge quadrupole contribution is negligible, the mobility equation simplifies to the Henry equation in which the sphere radius is replaced with the hydrodynamic radius of protein. The deformed sphere model predicts correctly the hydrodynamic radius of protein from the atomic coordinate data. The hydrodynamic radius is not the radius of sphere of equal volume but the effective radius that correlates with the translational diffusivity of protein. To illustrate the utility of our mobility equation we study the electrophoresis of lysozyme and compare our results with previously published works.