This question has been bugging me for a while now. I roughly understand how the Higgs mechanism gives elementary particles their rest mass and I also understand that gravity couples to all forms of energy, including binding energy in a nucleus or atom. I also know most of the mass of a system (such as an atom) comes from binding energy between its components, not from the rest masses of those components. What I don't understand is how binding energy affects the inertial mass of the system. Why is the inertial mass not different from the gravitational mass, with the inertial mass being equal to the rest masses of the components and the gravitational mass equal to the total energy of the system? Do the disturbances in the electroweak and/or strong fields associated with binding energy couple to the Higgs field somehow so that the total inertial mass of the system is affected?