First you have to consider the condition in which you intend to use the enzyme (mainly substrate and enzyme concentrations). For instance, if substrate available is saturating for the enzyme, improving kcat alone is enough as velocity will be Vm which is only dependent on kcat (and [E]0) whereas if substrate is limiting, both Km and kcat might be optimized as velocity will depend on kcat/Km (and [E]0).
its a bit broad answer but based on some GPCR and active research in such area, thermostability is important, and therefore maybe obtaining strong protein for different binding and stability assays is great. In addition, maybe for enzymes if they can be stable and active at a broad temperature and different conditions/buffers, that would be great! Very useful for various applications too.
What really matters for improved enzyme kinetics is the product formation; in other words, either you increase the enzyme Kcat or decrease its Km over the desired substrate. Assuming you have a very good stablished assay for products detection, you could use Rounds of directed evolution.
For library generation, the targeting of positions for variability introduction is crucial. I recomend the use of covariance (i.e. Statistica Coupling Analysis, SCA) combined with structural data for targeting positions. Moreover, the combined mutation in previous described neutral and/or beneficial mutations is a good start for variability introduction (see http://www.ncbi.nlm.nih.gov/pubmed/25723163).
You want to measure the catalytic efficiency (kcat/Km), but if you are evolving the activity, you may also want to track the promiscuity of the enzyme (i.e. Affinity for different substrates).
I agree with Richard Hall, you should look for the increase in the Kcat/Km value. However, if you want to use enzyme with very low concentration of substrate, it would be good idea to decrease the Km value rather than increase in the Kcat value. If the substrate concentration is not the issue then improving Kcat would be a good idea.
I agree with Dominique, that the concentration (or availability if you will) of substrate should be considered for the decision. The ultimate goal is to increase kcat/Km.
Of course increasing the thermostability never hurts as that will make the enzyme more durable. And checking for the substrate promiscuity is also good think t omake sure your enzyme is specific enough (or not, if you would be interested in more substrates).
Thanks all for brilliant suggestions. Suppose, the enzyme works on substrate and a cosubstrate. The km values are generally 10 times more for substrate than cosubstrate. Now, the reaction condition permit me to use saturated concentrations of substrate but limiting concentrations of cosubstrate (may be much below the km).
In that case focus on the Km for the co-substrate for sure. The next step is to improve the v_lim and as the last priority Km for substrate.
Of course during the engineering you may see these changes as well (in the Km for substrate or v_lim), but as you will probably have more clones, choose primarily those with improved Km for the co-substrate or v_lim.