Time variation of proton-electron mass ratio and fine structure constant with runaway dilaton

Recent astrophysical observations indicate that the proton-electron mass ratio and the fine structure constant have gone through nontrivial time evolution. We discuss their time variation in the context of a dilaton runaway scenario with gauge coupling unification at the string scale $M_{\rm s}$. We show that the choice of adjustable parameters allows them to fit the same order magnitude of both variations and their (opposite) signs in such a scenario.