I am trying to separate the highlighted area from this object. I have pressed B and then P to separate which works to an extent but it does not separate the whole area. I have rotated the object in the second picture. You can clearly see that only one side of this area has been selected. I basically tried to fix this by 'coloring in' the section by constantly pressing B and drawing little boxes over the black areas however, my molecules still look really patchy when I drag it away from its natural position and the framework underneath the molecules were also not selected.
The easiest way to select all the vertices in part of a dense mesh, is to switch to wireframe shading (simply press Z or from the 3D view header Viewport Shading > Wireframe) then use a selection tool like border select, to select the region of interest.
With either method (wireframe or Limit Selection to Visible) you have disabled the depth culling on the selection, so you can select all the geometry, even the pieces "hiding" behind others.
With the piece selected the way you want then it is a simple matter to separate it P > Selection.