Thermo stability is measured by Differential Scanning Calorimetry and different enzyme assays can be used to measure enzyme activity, say by measuring quantity of enzyme product present in different conditions.
I was wondering if there is something behind these two types of measurements to roughly estimate one of from the other.
As far as I know, no, not really.
Activity measurements are indeed influenced by temperature, and different from direct chemical activities which show an ongoing and exponential increase in speed at increasing temperatures enzyme reactions have a temperature optimum. That is indeed largely caused by loss of conformation at higher temperatures, but that is not the same as thermostability. Normally you don't measure conformation changes or folding, although you can traditionally for instance through measurement of changes in optical rotation. What you do measure is product formation of substrate consumption, sometimes indirectly. Conformation loss is often reversible. You can off course access that by first increasing the temperature and then decreasing it again. Which is indeed a classical biochemical approach. But you need that type of data to estimate stability from activity. Oftentimes irreversible conformation loss also leads simply to precipitation (although not in the presence of high urea concentrations).
A bioinformatics approach to estimate one from the other is thus not very useful. It would be interesting to know in how far direct estimations of stability can be made from the protein structure. If the latter question is what you really wanted to ask the question does indeed have a bioinformatics aspect (I could imagine people doubt that and see it as a pure wetlab question), but I don't know the answer to that part.
I have two data sets with two types of measurement (DSC and activity)(not to mention, there is also some overlap between enzymes in these datasets).The purpose at this stage is to label enzymes as high thermostable or medium or low.
For DSC dataset it would be easy to label, but I don't how can I label the other one !
Activity requires stability, but stability is not sufficient for activity. I'm a bit confused by your statement that you can measure enzyme activity by measuring folding, because again, while an enzyme generally needs to be folded to be active, that is not sufficient- you also need the right arrangement of chemical groups at the active site. I've seen people use enzyme activity assays to estimate the progress of a folding reaction (because, as I've said, most enzymes need to be folded to be active), but I've never seen the converse done- i.e., I've never seen anyone use a folding assay to measure enzyme activity.
Intuitively, I'd expect more stable proteins to have higher enzyme activity, but I don't think you can write an equation that would estimate one from the other. It certainly wouldn't be trivial to do so. If you really want to try, though, the common thread behind both types of measurements is thermodynamics. An explanation of that is a bit beyond an answer on a board like this.
Enzymes need of course be stable, but they often also need to be flexible. A too rigid active center will often not work, and increasing rigidity (for instance through oxidative crosslinking) oftentimes decreases the activity even though the more rigid protein could be more stable.
Yes. But in general, the "best" enzymes (as measured by the improvement of the reaction rate) tend to be quite thermostable, and any induced fit required on the enzyme's part will ding the reaction efficiency, which will show up in assays of enzymatic activity, but not necessarily in a clean way. Again, it all comes back to thermodynamics. I honestly think that the answer to the original question is "maybe, maybe not- but certainly not in a way that can be explained in a simple answer on a discussion board".
Maybe you can check some well studied enzymes from thermophilic and mesophilic organisms and compare their catalytic properties (Kcat, Km etc.) .The BRENDA database is a good place to get enzyme data. I'm sure there must have been quite some study on this topic so, search for the relevant references.
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Strictly-speaking, this is biochemistry, not bioinformatics.