We have calculated the twist viscosity and the alignment angle between the director and the stream lines in shear flow of a liquid crystal model system, which forms biaxial nematic liquid crystals, as functions of the density, from the Green-Kubo relations by equilibrium molecular dynamics simulation and by a nonequilibrium molecular dynamics algorithm, where a torque conjugate to the director angular velocity is applied to rotate the director. The model system consists of a soft ellipsoid-string fluid where the ellipsoids interact according a repulsive version of the Gay-Berne potential. Four different length-to-width-to-breadth ratios have been studied. On compression, this system forms discotic or calamitic uniaxial nematic phases depending on the dimensions of the molecules, and on further compression a biaxial nematic phase is formed. In the uniaxial nematic phase there is one twist viscosity and one alignment angle. In the biaxial nematic phase there are three twist viscosities and three alignment angles corresponding to the rotation around the various directors and the different alignments of the directors relative to the stream lines, respectively. It is found that the smallest twist viscosity arises by rotation around the director formed by the long axes, the second smallest one arises by rotation around the director formed by the normals of the broadsides, and the largest one by rotation around the remaining director. The first twist viscosity is rather independent of the density whereas the last two ones increase strongly with density. One finds that there is one stable director alignment relative to the streamlines, namely where the director formed by the long axes is almost parallel to the stream lines and where the director formed by the normals of the broadsides is almost parallel to the shear plane. The relative magnitudes of the components of the twist viscosities span a fairly wide interval so this model should be useful for parameterisation experimental data.