What does it mean to perform in vitro experiments with mutant bacteria?
Answer clarified based on comment by @DurgaDatta


  What would it mean to say the enzyme activity was tested in vitro on the mutant? When we say the response of the a single gene knock out mutant, the cell should be intact except for lacing one gene, which would mean that the response is in vivo.


The knockouts were performed inside the E. coli cells and, as you noted, the flux measurements were in vivo (in the cell) inside of the chemostat culture. They identified two mutants, Crp and Cya, that yielded much higher biomass. On page 2326, the authors write, 


  In the Cya (encoding adenylate cyclase) and Crp mutants that lack the cAMP-CRP complex, in vivo glyoxylate shunt activity was essentially abolished....


I am not familiar with these genes but the implication is that the wild type regulators have the cAMP-CRP complex. Because this complex is missing from the mutants, the enzymes no longer function the same. However, it would be possible that some type of interaction of the mutants with other enzymes caused the observed result in vivo. They therefore verified the result by extracting the Crp and Cya mutant enzymes from the cells and assaying their enzyme activity in isolation from the other enzymes (in vitro). The technique is explained in the answer by @dgg32.

In your comment to that answer, you ask.


  what difference does it make whether the enzyme was taken from wild-type or the mutant one, since it is the same enzyme?


It is the same enzyme in name but not in function. The mutant enzyme lacks the cAMP-CRP complex so it cannot perform the same function as the wild type. In vitro, they first showed that mutant enzyme activity was altered relative to the wild type.  They then added the wild type enzyme to the in vitro test and restored normal wild type activities:


  To verify that the observed changes were not due to polar effects of the still present marker gene or secondary-site mutations, we complemented the Crp mutant with the plasmid-based Crp gene and cultivated the complemented mutant under the same chemostat conditions. In vitro enzyme activities demonstrated that both the isocitrate lyase and isocitrate dehydrogenase activities were restored to the wild-type level


Yes, this was performed inside the chemostat cultures but only with the enzymes, not the living E. coli cells.