Are Infrared Dark Clouds (IRDCs) special in terms of their physical properties (mass, temperature, star-formation activity) or do they behave as any other star-forming molecular cloud? In this letter, we display column density and temperature maps derived from Herschel, and ATLASGAL dust continuum observations of a sample of prominent massive IRDCs, i.e. G11.11-0.12 (the 'snake'), G18.82-0.28, G28.37+0.07, and G28.53-0.25. We disentangle the velocity structure of the clouds using 13CO 1-0 (BU-FCRAO GRS) and 12CO 3-2 (JCMT) data, showing that our IRDCs are embedded in massive giant molecular clouds (GMCs). The probability distribution function of column densities (PDF) for all clouds have a power-law tail for high column densities, independent of their evolutionary stage (G11.11-0.12, G18.82-0.28, and G28.37+0.07 contain (proto)-stars, G28.53-0.25 shows no signs of star-formation), we attribute to self-gravity. This is in contrast with the purely lognormal PDFs reported using near/mid-IR extinction maps. The power-law tail seen for G28.53-0.25 can not stem from radiative feedback processes. Moreover, by comparing the PDF slope and the radial column density profile, we attribute this power-law tail to the effect of a large-scale collapse of the cloud. We are in line with earlier interpretations that see massive IRDCs as the densest regions within GMCs, which form massive stars or clusters. At least some of them are probably the same features as ridges (high-column density regions with N>10^23 cm^-2 over small areas) that were defined for GMCs. Because IRDCs are only confined to the densest (gravity dominated) cloud centers, the PDF constructed from such a clipped image does not represent the (turbulence dominated) low column density regime of the cloud. It it thus not possible to link the column density PDF dispersion to the turbulent structure of the cloud.