The hydrogenase works by taking H2's electrons and passing them through the electron transport chain to oxygen to generate a membrane potential (like voltage) that provides energy for the production of ATP (the cell's energy currency, that it uses to power many of its reactions). This can be an important process for crop production, since some crops (legumes) are colonized with bacteria that fix nitrogen for them, but at the time it was unclear whether the uptake hydrogenase in this system was actually helpful for the crop at all. So the authors decided to study the question in an easier system: free-living Azotobacter.
In this study, the strains under investigation were a mutant strain, offspring of the wild-type, that lacked hydrogenase, called MCD103; and another strain derived from MCD103, called MCD503, that did have a hydrogenase because they crossed MCD103 with a plasmid containing wild-type hydrogenase genes, resulting in MCD503, which was the same as MCD103 in every way except the hydrogenase (presumably).
These strains were grown together in continuous culture/chemostats, fixing nitrogen. If one had a growth advantage over the other, it would come to dominate the culture in time. They measured proportions of the strains in two ways: first, by plating them out on agar and using a technique called "scrying" (which seemed to consist of exposing the colonies to H2 in a sealed box with an indicator present, such that those that had hydrogenase would stay white and those that didn't would turn blue-black) and then counting the colonies of each kind. Second, by plating and transferring individual colonies to a 96-well plate, then exposing all of them to radioactive hydrogen (tritium, 3H2) and measuring which ones retained radioactivity (indicative of consuming the hydrogen with their hydrogenase). These sound like rather painful and burdensome procedures; nowadays people would probably just do gene sequencing or transcript analysis to see how many copies of hydrogenase genes/transcripts were present over time. I suppose the techniques in this paper may give a more direct measure though.
So anyway, about what they found. When sucrose (sugar) was the limiting nutrient (that is, when the cells were consuming all the sugar they were given and could've consumed even more), the hydrogenase-positive strain MCD503 came to dominate the culture over time, regardless of the proportions of the strains at the beginning of the experiment. Even when initially there were 99 hydrogenase-negative cells for every one hydrogenase-positive cell, before too long they saw the amount of hydrogen produced falling quickly as hydrogenase activity increased. The domination happened faster at higher dilution rates, which makes sense because faster-growing cells would be able to tolerate these better. When fixed nitrogen (ammonium) was added to the cultures, neither strain dominated the other consistently. So it seems that hydrogenase is important when fixing nitrogen.
When nutrients other than sucrose were limiting, though, the situation was not always the same. When phosphate was limiting, MCD503 still dominated, though not as well as with sucrose limitation. However, when oxygen was limiting, the strain missing its hydrogenase (MCD103) was dominant! Even when MCD503 started as 78% of the cells present, it fell to less than 20% before stabilizing. So it seems that hydrogenase-negative strains can deal with low oxygen better.
When sulfate was limiting, MCD503 declined slowly, but it declined more quickly when iron was limiting (makes some sense because the hydrogenase requires iron in its cofactor).
It was interesting to note, also, that MCD503 (hydrogenase-positive) consumed its own hydrogen but also that produced by MCD103 (hydrogenase-negative). Might've contributed to its faster growth in some conditions, and this is consistent with how the domination slows down as more and more of the population is MCD503 (thus there is less hydrogen produced by its competitor to steal).
Speculations about explanations for the findings, as far as I understand them: carbon-limiting results make sense because H2 oxidation adds to the energy recovery and makes up for some of the limitation.
Phosphate-limiting results make sense because hydrogen could help increase consumption of oxygen to protect nitrogenase and hydrogenase, which are sensitive to it.
Oxygen-limiting results make sense because hydrogen oxidation might take precedence over other kinds in the electron transport chain due to greater affinity.
And sulfate- and iron-limiting results make sense because hydrogenase requires sulfur in addition to iron, so the cells would devote some of their nutrients to this enzyme instead of other, more important ones; while hydrogenase-negative strains wouldn't have this disadvantage.
So it seems that the hydrogenase is helpful in some circumstances and harmful in others. Interesting results.