What does "dep" mean? I'm looking at some data with substrate concentration on the $x$-axis and $k_\text{dep}$ on the $y$ axis. What does $k_\text{dep}$ mean? I understand that it must be a rate of some kind but what is the significance of the 'dep' part? 

The data I'm looking at can be found in Fig. 9 in this paper (PDF):







Essentially this is a dose response curve for a substrate and enzyme. I suspect $k_\text{dep}$ is simply the rate but why $k_\text{dep}$ and not just $k$?
The "dep" probably stands for "depletion": read the paper you linked to for more context. Often times, subscripts on rate constants are used to refer to which reaction in a system of multiple reactions. You'll see $k_1$ and $k_2$ for rate constants of "reaction 1" and "reaction 2", etc.

I suspect in the paper, they were measuring the depletion of a substrate, and not what the substrate was converted to.  So they wouldn't know if they were measuring the rate of a single reaction $\ce{A-&gt;B}$, or the combined rates of multiple reactions such as $\ce{A-&gt;B}$ and $\ce{A-&gt;C}$.  Thus (probably) the desire to indicate $k_{depletion}$ instead of just $k$.