The growth and the density dependence of the low temperature mobility of a series of two-dimensional electron systems confined to un-intentionally doped, low extended defect density InAs quantum wells with Al$_{1-x}$Ga$_{x}$Sb barriers are reported. The electron mobility limiting scattering mechanisms were determined by utilizing dual-gated devices to study the dependence of mobility on carrier density and electric field independently. Analysis of the possible scattering mechanisms indicate the mobility was limited primarily by rough interfaces in narrow quantum wells and a combination of alloy disorder and interface roughness in wide wells at high carrier density within the first occupied electronic sub-band. At low carrier density the functional dependence of the mobility on carrier density provided evidence of coulombic scattering from charged defects. A gate-tuned electron mobility exceeding 750,000 cm$^{2}$/Vs was achieved at a sample temperature of 2 K.