Origin-Dependence of Dipole Moments of Charged Proteins: Theoretical Foundations and Implications, Revisited

Abstract

Electric dipole moments are widely employed in structural biology and computational chemistry as global descriptors of macromolecular charge distribution, contributing to the understanding of protein interactions, solvation, and orientation in external fields. However, for systems bearing a nonzero net charge, the dipole moment becomes explicitly dependent on the choice of coordinates origin, a consequence grounded in classical electrostatics and sometimes overlooked in structural analyses. This origin-dependence is particularly relevant in biological systems, as proteins are typically charged at physiological pH which differs from their isoelectric points (pI's). Moreover, coordinate manipulations such as centering and alignment are routinely performed during molecular dynamics simulations, docking, and structural comparisons, potentially altering the calculated dipole moment of charged systems. This study reviews the theory of the changes in the dipole moment of charged macromolecules accompanying displacements of the origin of the coordinates system. The theory is illustrated by numerical examples on representative proteins. Using the classical expression ${\overrightarrow{\mu }}^{\prime }=\overrightarrow{\mu }-Q\overrightarrow{a}$, we demonstrate that displacements of the order of a protein's radius of gyration or larger can induce dipoles several hundreds to thousands of debyes. We examine this effect across a range of proteins with varying sizes and identify trends correlating the extent of origin-induced changes with molecular size. These examples highlight the need for standardization in defining coordinate systems in dipole-related analyses. The quantum mechanical status of the dipole moment operator is discussed clarifying that only neutral systems satisfy Dirac's criteria for a true “observable”. Altogether, theory, numerical benchmarks, practical guidelines, and pedagogical insights are presented for reliably calculating and interpreting dipole moments of charged biological macromolecules.