When dealing with open boundary problems with a bias, the full system Fermi-level is somewhat ill-defined. Consider that the left electrode is in equilibrium at its $\mu_L= e_F + V/2$ and the right at $\mu_R = e_F - V/2$. This means that there are already two different Fermi levels and that the Fermi levels are well-defined far away in the electrodes (that is the whole basis of NEGF in transiesta), but the full systems Fermi level is not.
In the same way it becomes difficult to assess how band gaps behave under bias. An increasing band gap (in terms of transmission) may be due to various things such as, two neighbouring gaps overlapping, destructive interference, or some other physical behaviour. Similarly an applied bias may also close transmission band gaps (consider a semi-conductor and the threshold voltage).
So, there is no simple relation between applied bias and the band gap in a semi-conducting material between two metallic electrodes.
With an open boundary system there is also no such thing as HOMO and LUMO. The electrodes should be metallic, and hence the electronic spectrum is extremely closely positioned. Any molecule sandwiched between these electrodes will have their spectrum spread out due to the coupling to the electrodes. I would go as far as say, the HOMO and LUMO does not exist for molecules with high coupling strength to the electrodes.
For weakly coupled molecules, one might argue that one can see the HOMO and LUMO positions by examining projected DOS and/or transmission peaks resembling the characteristics of the lone molecules HOMO/LUMO. This is commonly done but it shouldn't be confused with proper HOMO/LUMO states of a molecule.
In any case one should be careful and clear about the interpretations in such analysis.
Please do note that TranSiesta relies on using metallic electrodes, using semi-conductors as electrodes is not recommended.
