Effective generation of right-angled artin groups in mapping class groups

We show that given a collection $$X=\{f_1$$ , ..., $$f_m\}$$ of pure mapping classes on surface S, there is an explicit constant N, depending only X, such their Nth powers $$\{f_1^N$$ $$f_m^N\}$$ generate the expected right-angled Artin subgroup MCG(S). Moreover, we these subgroups are undistorted, and each element pseudo-Anosov largest possible subsurface.