Plot substrate concentration ([S]) on the x-axis, and reaction rate (v) on the y-axis.

Using you favourite software, fit the resulting curve into Michaelis–Menten kinetics.

v = Vmax*[S]/(Km+[S])

That way you can find the values of Vmax and Km.

Once you have Vmax, it's simple to get kcat, since Vmax = kcat * [Enzyme].

Thus, dividing Vmax by the enzyme concentration will give you kcat.

Read the Wikipedia page for more details:

https://en.wikipedia.org/wiki/Michaelis%E2%80%93Menten_kinetics

As for Ki, plot inhibitor concentration ([I]) on the x-axis, and reaction rate (v) on the y-axis for each substrate concentration.

Ki is where the lines intersect.

See the bottom of the following link:

http://www.ucl.ac.uk/~ucbcdab/enzass/inhibition.htm

The complication is due to the fact that the inhibitors are covalent. The inhibition data can't be treated as if the inhibitor was a simple competitive inhibitor according to Michaelis-Menten kinetics, because the inhibitor is not in rapid equilibrium with the enzyme. If you were to graph progress curves (absorbance versus time) of the enzyme reaction with various inhibitor concentrations, starting the reactions by adding the enzyme to the mixture of nitrocefin and inhibitor, you would see that the initial rates are all the same, but diverge with time. The potency of inhibition increases with time. This is unlike what you would see with an ordinary non-covalent competitive inhibitor, for which the progress curves would diverge from the very beginning and the potency would be the same at all times.

The potency of a covalent inhibitor should not be expressed simply by Ki, since this ignores the time-dependence. Such an inhibitor is characterized by the second order acylation rate constant kinact/Ki. Additionally, since the enzyme is capable of slowly hydrolyzing and releasing the inhibitor, the kinetic characterization should also include a rate constant for dissociation of the inhibitor, koff.

For the uninhibited reaction, it should be possible to use standard Michaelis-Menten kinetics, as long as the enzyme concentration is very low compared to the substrate concentration.

It seems you are able to calculate the Km for the b-lactamase and the substrate nitocefin.  Are you asking for the ki and kcat for the non-beta lactam-based inhibitor ZINC that binds through H-bonding and hydrophobic interaction or are you having to deal with the irreversible inhibitors such as Tazobactam?  It might help to study the inhibition kinetics of a simpler system such as PMSF sulfonylation of serine proteases as opposed to a slightly labile acylation mechanism.

thank you for the reply. 

Arthur Wing Hang Li- i think this is for simple Michaelis–Menten kinetics, which is not possible as said by Adam B Shapiro. as per his suggestion i have to calculate K intact and K off to finish the remaining calculation to find the Ki.

Thanks Dr. Adam B Shapiro it was really a great advice for me. can you please give me an advice.

1. what is initial rate of f nitrocefin hydrolysis? what is the time period considered for the initial rate of hydrolysis.(like time 0 to time 30 sec or 1 min), is there any specific time i have to take into consideration.

2. i am not sure of the calculation with acylation-deacylation mechanism since it asks for k+2, k+3 and K'. how will i derive these values and from which graph can i derive these value.

Thanks for you question on this Dr. Grant Shimamoto, we are working on the beta-lactamase II from Bacillus cereus 569/H/9 and we are working on an structure unknown inhibitor. Is it ok to do the kinetics in the same way or should i chnage any method for this?

It's clearer what you are asking.  I assume your unknown inhibitor is working through an irreversible mechanism even though your attached publication was directed at a non-beta lactam reversible inhibitor, correct?  You may not be allowed to disclose this information but if the assumption is correct then you need to proceed with the same kinetic arguments.  

Initial rate is a crucial and basic concept in Michaelis-Menten kinetics. It is the rate measured at the beginning of the reaction before the substrate begins to become depleted and/or the product begins to accumulate, causing the rate to slow down. It is measured from the start of the reaction (time = 0) to a later time before which the rate begins to slow. It is the slope of the line tangent to the beginning of reaction progress curve. Only initial rates should be used to measure Km and Vmax kinetic parameters for nitrocefin with your enzyme.

Your second question is harder to answer briefly and without diagrams. There are a few different ways to measure these parameters. Also, depending on the circumstances, it may or may not be possible to get a measurement for k+2 and Ki separately.

k+2 in the paper you attached is the rate constant for the formation of the covalent acyl-enzyme. Ki is the equilibrium binding constant of the noncovalent enzyme-inhibitor complex. In cases where the inhibitor binds weakly (Ki is large) and the covalent reaction is relatively fast, it usually is not possible to measure k+2 and Ki, but it is possible to measure the ratio k+2/Ki.

k+3 is the rate constant for deacylation of the enzyme, releasing a chemically altered (typically, hydrolyzed) form of the inhibitor. This can also be measured a few different ways. In some cases, k+3 is very small, requiring experiments lasting several hours.

However you analyze the data for k+2 and Ki, it will consist of a series of nitrocefin reaction progress curves collected at several concentrations of the inhibitor. It is crucial that the reactions be initiated by adding the enzyme to a mixture of nitrocefin and the inhibitor. In other words, the enzyme should not be premixed with the inhibitor, or with nitrocefin.

Another crucial point for some ways of analyzing the data is that the uninhibited reaction progress curve is linear, not gradually slowing down. However, there is another way of analyzing the data that does not impose this requirement.

For the classical method of analysis, there is a paper by Morrison and Walsh that describes the method and gives the equation. I think of it as the kobs method because the analysis involves a plot of a parameter called kobs versus inhibitor concentration. kobs comes from an exponential function that describes the shape of the progress curve in the presence of a slow-binding or covalent inhibitor. This method requires a linear progress curve for the uninhibited reaction.

http://www.ncbi.nlm.nih.gov/pubmed/3281418

It maybe difficult to get this paper, but you may be able to find other papers that use the same method.

The other method involves numerical integration of a set of differential equations. This sounds scary, but there are computer programs that take care of the math for you. Two such programs are DynaFit from BioKin and Global Kinetic Explorer from Kintek. This method does not require a linear progress curve for the uninhibited reaction. It can also give a value for k+3 if k+3 is not too small to measure.

k+3 can also be measured in a recovery experiment. You start by making the acylated enzyme complex in a concentrated form, then remove the excess inhibitor (in some cases you can just dilute it to a low enough concentration that it doesn't matter anymore). Then you measure the initial rate of the reaction with nitrocefin after various amounts of time have elapsed. You make a graph of rate versus elapsed time and fit it to an exponential function to obtain the value of koff.

If it is the case that your inhibitor is not slow-binding or covalent but is simply in rapid equilibrium, then you can forget all of the above (except the part about initial rates) and use steady-state kinetics for competitive inhibitors to get a measurement of Ki.