velocity dispersion and gas mass should follow $r$=$\sigma^2$/($\pi G \Sigma_{\rm disk}$) (see § \[sec:Sigma\_disk\_Sigma\_clump\]). In Fig. \[fig:scaling\] we therefore overlay contours of constant gas mass in the r–$\sigma$ plane, which suggests that the [*initial*]{} gas masses for the clumps is $M_{\rm
  gas}^{initial}$=2$\pm$1$\times$10$^9$M$_{\odot}$ a factor $\sim $1000$\times$ more massive then the star-forming complexes in local galaxies (e.g. see also @Elmegreen09 [@Genzel11; @Wisnioski11b]). Assuming our gas mass estimates from § \[sec:dynSF\], then these star-forming regions may contain as much as $\sim$10–20% of the cold molecular gas in the disk.

Turning to the relation between size and luminosity of the star-forming regions, it is evident from Fig. \[fig:scaling\] that the star formation densities of the high-redshift clumps higher than those locally. Indeed, local star-forming regions follows a scaling relation $$\log\left(\frac{L}{\rm erg\,s^{-1}}\right)\,=\,(2.91\,\pm\,0.15)\log\left(\frac{r}{\rm kpc}\right)\,+\,(32.1\,\pm\,0.3)
  \label{eqn:Lr}$$ We do not have sufficient number of objects or the dynamic range to measure both the slope and zero-point of the size-luminosity relation in the high-redshift clumps, and so instead we fix the slope of the local relation (which is $L\propto r^{2.91\pm0.15}$) and fit for the zero-point evolution and obtain $$\log\left(\frac{L}{\rm erg\,s^{-1}}\right)\,=\,(2.91\,\pm\,0.15)\log\left(\frac{r}{\rm kpc}\right)\,+\,(33.2\,\pm\,0.4)
  \label{eqn:Lr_hiz}$$ This suggests that high-redshift star-forming regions have luminosities at a fixed size that are on average 