Abstract
Implementation of Einstein's law connecting charge, diffusion coefficient, and mobility to interpret experimental data on proteins from single molecule electrophoresis through nanopores faces serious difficulties. The protein charge and diffusion coefficient, inferred with the Einstein law, can be orders of magnitude smaller than their bare values depending on the electrolyte concentration, pore diameter, chemical nature of the pore wall, and the externally applied voltage. The main contributors to the discrepancies are the coupled dynamics of the protein and its counterion cloud, confinement effects inside the pore, and the protein-pore-surface interaction. We have addressed these ingredients by harking on classical theories of electrophoresis of macroions and hydrodynamics inside pores, and deriving new results for pore-protein interactions. Putting together various components, we present approximate analytical formulas for the effective charge, diffusion coefficient, and mobility of a protein in the context of single molecule electrophoresis experiments. For the omnipresent pore-protein interactions, nonlinear dependence of the velocity of protein on voltage sets in readily and analytical formulas for this effect are presented. The derived formulas enable the determination of the bare charge and size of a protein from the experimentally measured apparent values.