  Gravitational contributions to the running of gauge couplings are calculated
by using different regularization schemes. As the $\beta$ function concerns
counter-terms of dimension four, only quadratic divergences from the
gravitational contributions need to be investigated. A consistent result is
obtained by using a symmetry-preserving loop regularization with string-mode
regulators which can appropriately treat the quadratic divergences and preserve
non-abelian gauge symmetry. The harmonic gauge condition for gravity is used in
both diagrammatical and background field calculations, the resulting
gravitational corrections to the $\beta$ function are found to be nonzero,
which is different from previous results presented in the existing literatures.
