A generalization of Koide's formula suggests a nearly massless up quark, contrary to model dependent estimates that suggest an up quark mass of about 40%-60% of the down quark mass (e.g. here), while the formula accurately estimates the conventional value of the d quark mass. There is a case for this in supersymmetric theories as a solution to the "strong CP problem.", although this approach has been questioned. Early lattice QCD simulations in the Standard Model also suggested that the massless up quark solves the strong CP problem.
A masslesss or nearly massless up quark can also be brought to bear to account for neutrino mass.
There is somewhat similar interesting inquiry into massless QCD vacuum energy and its relationship to gravity.
It isn't unusual to model QCD with masses for all the light quarks set at zero for calculational convenience and still have the model produce meaningful results.
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