A) An assay. That is to say a protocol where you put substrate and enzyme in a pH buffered solution and incubate at constant temperature for a know period of time. Along the assay or at specific time points along the assay you take a sample and measure either the amount of remaining substrate or the amount of product formed, using any analytical technique that is appropriate to measure the substance (substrate or product) in a quantitative manner. Make sure that during the time sampled, the total amount of substrate consumed does not exceed 5% of the total amount of substrate.
B) The increase in the amount of product, or the decrease in the amount of substrate divided by the time elapsed is a measure of enzyme activity. Is you take several samples along the assay time, plot the amount of product or substrate against time and take the magnitude of the slope as measure of activity.
C) Repeat the activity measurement starting with several initial concentrations of substrate. Make sure you see important changes in activity. If you do not, the range of concentrations chosen might either be too high (too much activity) or too low (almost no activity).
Then refer to the page cited by Steingrimur to do the calculations.
Assuming your enzyme obeys the Michaelis-Menten kinetics, which is the simplest case, you will need to do several reactions in which the initial enzyme concentration is the same for all of them, but not the substrate. The initial substrate concentration must vary. Then, if you need or want, you can plot a graph of the initial rate (at nmol min-1, for instance) versus the substrate concentration (at uM, for instance) (I would recommend at least ten points, doing triplicate for each one). This graph will not give you the Km and Kcat values yet, at least not the precise ones, but will give you a notion of the michaelian behavior. The ideal interval between these points will generally depend on the next step, but you may want a point at a very high substrate concentration to see the saturation behavior.
Km and Kcat can be obtained from these data by using a numerical procedure in which the Michaelis-Menten equation is linearized. There are several ways to linearize this equation, and the most used is the Lineweaver-Burk representation, in which you plot 1/v versus 1/[S]. There are other two representations that are less used, each one with its advantages and disadvantages (it is explained at Steingrimur’s link). So, after choosing a representation, just plot the required parameters using your data and do a linear regression. Then, with the obtained equation coefficients, one can easily calculate the wanted kinetics parameters (what each coefficient gives is explained at that link too).
I am supposing that you know the basics about enzymatic reactions, such as the temperature and pH effects, initial rate definition etc. If you need more details about the experimental procedure, let us know.
Thank you so much Steingrimur Stefansson, Rogelio Rodríguez-Sotres and Gustavo Avelar Molina. I will follow your recommendations.
Gustavo Avelar it will be really useful to know more about the experimental procedure. I will appreciate any good readings you may recommend. Thanks again for your help!
Firstly, I need to know what you already know about the enzyme in study. Is the wild protein (the protein without mutations) biochemically characterized? Do you know its substrate(s) and product(s), or at least which kind of reaction it catalyzes? Is there a known analytical technique to quantitatively measure the product formation or substrate consumption? Are its optimum pH and temperature known, for a known buffer? Has someone ever developed an assay for this enzyme?
For the questions you don't know the answer for sure, try to find them in the literature. If you do not succeed with this, do another question separated from this, or ask here if you think it is more appropriated.
If you want to know more about enzyme kinetics, there are a lot of Biochemistry books that should help you with the basics. I would recommend the book "Lehninger Principles of Biochemistry".
You have already recieved many good suggestions from great expertise. Now, if you are a beginner to enzymology or mutational study, I would suggest, before you test or measure activity and kinetic parameter (Km or Kcat) of a mutant enzme, get yourself well-versed with the standard enzyme assay used widely for your wild-type enzyme (if any reported) and also make a note of its kinetic parameters like optimum pH, assay buffer, required cofactors etc.
To put in simple words for you, at first you need to know the details of the enyme assay used to measure the activity of your wild-type enzyme. To get initial experience of dealing with enzymatic reaction or calculation of kinetic parameters, you can also repeat thier experiments and see if you can obtained similar kinetic parameters for the same or different wild-type enzyme close to the reported values (just for a experience purpose).
Now, to determine the Km experimentally, you need to prepare various concentration of substrate (lowest to highest), keeping all other assay & cofactors concentration constant.
Initiate the assay with addition of the enzyme for a specific period of time (for discontinious assay), stop the reaction and measure the prodcuct formed. You can plot a graph of the initial rate versus the substrate concentration using any kinetic programme (Origin http://www.originlab.com/doc/Tutorials/Fitting-NLFit-Built-in /Graphpad) and may easily obtain Km value here and can further also calculate its Kcat numerically.
Hope, you find it useful, if you still have some issue you may share with us.
Good luck with your research, hope you gain good experience out of this.