  We explore the direct to indirect band gap transitions in MX$_2$ (M= Mo/W, X=
S/Se) transition metal dichalcogenides heterobilayers for different system
compositions, strains, and twist angles based on first principles density
functional theory calculations within the G$_0$W$_0$ approximation. The
obtained band gaps that typically range between 1.4$-$2.0 eV are
direct/indirect for different/same chalcogen atom systems and can often be
induced through expansive/compressive biaxial strains of a few percent. A
direct to indirect gap transition is verified for heterobilayers upon
application of a finite 16$^{\circ}$ twist that weakens interlayer coupling.
The large inter-layer exciton binding energies of the order of $\sim$~250~meV
estimated by solving the Bethe-Salpeter equation suggest these systems are
amenable to be studied through infrared and Raman spectroscopy.
