The purity of an enzyme does not affect Km, which is = 1/2 V(max), but affects V(max)

Howdy. There's no general relationship between the Km of an enzyme and the kcat.  A good paper that describes what trends there are can be found here: Biochemistry, 2011, 50 (21), pp 4402–4410. 

With regards to substrate scope, the sensitivity of an enzyme to variation in the substrate can be extreme. Some, like proteases and kinases, are highly tolerant to variation. If you're trying to understand how your catalyst works, then you need to measure those details (ie kcat & Km). On the other hand, if you're trying to develop a tool then a simpler approach is to measure under conditions that are operationally desirable, regardless of any changes in the underlying kinetics. You'll need to figure out what's appropriate for you. Good luck!

KM and kcat are muh more informative and clearly preferred over specific activity, but it is more laboratorious to determine so it depends on what you want to and can do. In addition, if you know KM and kcat you can calculate the specific activity at any desired substrate concentration. 

You do not need to know KM and kcat to determine the specific activity. It is usually determined at high substrate concentration and it will then reflect kcat, but that is not required.

"But if a substrate has low Km, then it will also likely have low specific activity." This statement is incorrect. There is no such relationship.

Specific activity is useful for following the purity of an enzyme during a purification procedure. Specific activity should increase at each step in the process. Specific activity is also useful for comparing different batches of an enzyme preparation. (This is only true if there is only one enzyme in the extract capable of catalyzing the reaction.)

If you are working with a purified enzyme and you are interested in comparing how well it uses different substrates, then you should measure Km and kcat for each substrate and calculate kcat/Km.

If you are working with impure preparations, and you are sure that only one enzyme in the preparation is capable of catalyzing the reaction, then you can compare Kms for different substrates, but not kcats.

Specific activity is not necessarily directly proportional to kcat even though a high substrate concentration relative to the Km is used. There are situations in which adding such a high concentration of the substrate does not result in the Vmax rate due to substrate inhibition. If this is happening, it will become apparent when the Km is measured, as long as high enough substrate concentrations are tested.

Andrew, Anders and Adam have mostly answer the question, just two small additions:

A) Maximum specific activity (which as mentioned by Anders can be obtained form KM and kCAT) may be useful in terms of cell or organism physiology, because you may relate the maximum velocity of crude extracts to the amount of active protein in the biological fluid of interest, i.e. the blood. This parameter is usually reported (in addition to kCAT ) in clinical enzymology.

B) While there is no apparent connection between KM and kCAT, these two parameters are hardly ever as simple as defined in textbooks. The kCAT and KM are complex combinations of rate constants, some of which are shared. In addition, kCAT is dominated by the rate-limiting step of the catalytic mechanism, and  such step might be a product release step, which depends on the affinity of the substrate/product binding pocket. Therefore, there is a thermodynamic and kinetic link between kCAT and KM , which reflects on the well known diffusional upper limit to enzyme catalytic efficiency (also known as specificity constant, given by kCAT/KM). For a perfect-catalyser, any factor increasing kCAT would also reduce the apparent affinity and vice versa.

Thank you all for your answers. Perhaps I can also include the details of my experiment to facilitate your answers. I have a proteins that has a His-tag and I purified them using Ni column. I then chose the maximum concentration I could use for assays based on the absorbance properties of some of the substrates. Some of them were above the read limit of the plate reader so I had to settle for the lower concentration that prohibited this problem. Then I used the purified enzymes to do the assays with a panel of substrates and calculated specific activity based on the data from this assay. Can I use these specific activity values to make conclusions about which enzyme and substrate pair have the best turnover under these identical reaction conditions? Thanks again for your help! 

If you know the KM for each substrate, then you may estimate the level of saturation and, since you should know the concentration of Active sites, you should be able to calculate an estimate for kCAT. If you do not know KM, you do not have enough information. In other words, let us assume the concentration of substrate used was X and KM_X is the KM for that substrate, then MM equation becomes:

v0 = kCAT [ET] X/(KM_X+X),

and

kCAT= (KM_X+X)/(v0 [ET] X).

Actually, I would use not just one measurement, but several measurements at X, 1/2X, 1/4X, and so to reduce the uncertainty of my estimate.

If you do not know KM, tray assaying the activity a many substrate concentrations, from your allowed upper limit, to as low as you still can measure and see is this gives you a curve v0 vs [S] with enough curvature to allow an accurate estimation of KM and VMAX using non-linear regression analysis.

If the above is not possible, try a different assay, where the limitations of substrate concentration are overcome.

If you were using an absorbance plate reader for your measurements, you can extend the range of usable substrate concentrations by reducing the volume of the assay. The shorter path length through the lower volume results in lower absorbance.

Another way to deal with the same problem is to use a wavelength offset from the peak absorbance wavelength.

In both cases, you will have to make a new standard curve for the altered conditions.

Rogelio and Adam, thank you for your follow up answers. They have been thoroughly helpful and I intend to implement them in my experiments. Best wishes to you! 

I do agree with Rogelio and Adams. Follow their advice. They are on spot.

I join the discussion with a long delay and I have the following note: First of all, that the question should make sense, the form of the kinetic equation should be specified. Probably it's about the M-M equation, but it's just a guess. It may as well be Adrews equation. And then KM will not be equal to the substrate concentration, which corresponds to half of the maximum rate. Regards,