  Models of modified gravity, whereby local tests of gravity are evaded thanks
to a screening mechanism of the chameleon or Damour-Polyakov types, lead to a
spatial variation of the particle masses and the fine structure constant. This
is triggered by the environmental dependence of the value of the scalar field
whose presence modifies gravity. In dense media, the field settles at a density
dependent value while in sparse environments it takes the background
cosmological value. We estimate that the maximal deviation of constants from
their present values is constrained by local tests of gravity, and must be less
than $10^{-6}$.
