We study quotients of mapping class groups Gamma(g,1) of oriented surfaces with one boundary component by the subgroups I-g,I-1(k) in the Johnson filtrations, and we show that the stable classifying spaces Zx B(Gamma(infinity)/I-infinity(k))+ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces Zx B Gamma(+)(infinity) and Zx B(Gamma(infinity)/I-infinity(1))+ similar or equal to Zx BSp(Z)(+). We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.