Experiment 3:  Inhibition of ß-galactosidase

Theory

  Inhibitors are substances that inhibit enzyme activity. Such substances are of great economic interest as pharmaceuticals and herbicides, in addition to the scholary interest of figuring out how enzymes work, of course. Inhibition can be either reversible or not; in reversible inhibition, subcategories can be made to futher define the binding models.

  For the different forms of reversible inhibition, we can use a modified version of the Michaelis-Menten equation. For simplification purposes, we can calculate an apparent constant. The inhibition constant Ki is the dissociation constant for the EI and ESI complexes and serves as a measure for the strength/effectiveness of the inhibitor itself: the smaller the K_i, the stronger the inhibitor.

  An overiew of reversible inhibition would include reviewing equilibrium principles, inhibitor binding models, MM theory, and LWB plots.

  From the apparent Michaelis constant K_M^app in the LWB plot below, we can find the inhibition constant Ki and Ki'. Ki can also be found from a certain point of some sort on a Dixon diagram (one that plots 1/v vs [I] (see figure 1).

Figure 1. The inhibition constant Ki can be directly calculated from a Dixon diagram.]

  A very easily determined parameter for the strength of an inhibitor is the IC₅₀ value. This reveals at which [I] the reaction is 1/2 (-Vmax), where -Vmax is the pint of maximum inhibition (lowest rate). The IC₅₀ is borrowed from common usage in pharmacology in the dose response plot (see figure 2). The IC₅₀ is quite popular for level analysis of chemically similar substances, for investigating entire cell systems, for indicating the bioavailability of given materials, and for characterizing inhibitor efficiency in vivo but in any case, they give no indication as to the type of inhibition occurring. If we can determine this, we can find the K_i.

Figure 2. Dose response plot used to calculate IC₅₀.

  Irreversible inhibitors mostly bind the target enzyme covalently. The term "irreversible" is technically incorrect; it is reversible, but at such a slow rate that significant damage to the system arises. These types of inhibitors are used in herbicides (eg, phosphoric acid esters), in pharmaceuticals (eg, lactam anibiotics), and in investigating enzyme mechanisms (eg, TPCK or tosyl-L-phenylalanine chloromethyl ketone which involves His57 and chymotrypsin somehow). Darker uses of these include chemical warfare applications (eg, Sarin, a nerve gas). E.coli ß-gal can be competitively inhibited with galactose or isopropylthiogalactopyranoside among others. Due to kinetic differences between the types of inhibition, we can determine the type and characteristic parameters of inhibition in this experiment by running the enzyme test from yesterday (expt 2) in the presence of different [I].

Workup

  Measure the rates for 4 different [I], independent of [S]. Use [S] from yesterday so that you can put your measured values directly into the LWB plot you've already drawn up.

1. Make up the following dilutions from the NaK phosphate buffer stock solution (42 mM or 10 mg/mL): 1:2, 1:5, 1:10, 1:20
2. Using the smallest [S] from yesterday, make sure each run contains 50 µL inhibitor dilution; total volume should be the same as yesterday: 2.5 mL
3. Lather, rinse, repeat with the 2nd smallest [S].
4. Fill in your experimental values from expt 2 in your LWB plot (already mentioned).
5. Calculate the K_M^app from the LWB plot and the K_I from the K_M you found yesterday.
6. Plug your data into a Dixon plot and read off the K_I. Approximate a meaningful error value.
7. Using semi-log paper, draw a dose response curve for the lowest [S]. Make sure you use v_o from the v_i/v_o ratio, NOT v_max!  v_o is the rate of reaction WITHOUT inhibitor (from expt 2). Read off your IC₅₀ value.
8. In the writeup, include all data used in the v vs. [S] graph from expt 2 and determine from the K_I value (calculated from IC₅₀) whether we did in fact witness competitive inhibition kinetics.

Procedure

1. Make up a NaK-phospate buffer (pH 6.8) dilution (check with TA for ratio) from the ß-gal stock solution.
2. Measure the rate of o-NPG hydrolysis at different substrate concentrations. Calculate the rates at max and min [S]. Pipette the following mixture into 2 cuvettes  (x = 0.04 mL bzw. 0.02 mL):

   2.4 - x 0.05 x

   mL mL mL

   NaK phosphate buffer, pH 6.8, 50 mM MgCl2 solution - 50 mM o-NPG, 5 mg/mL in H2O

3. Incubate at 30°C for 5 minutes, then add 0.05 mL of the ß-gal dilution, mix well, and measure the DA immediately with the photometer (l = 405 nm) over a span of 2 minutes. Don't forget to turn the recorder on! From the paper output, we can find the change of extinction per minute!
4. Using the Beer-Lambert law, find the rate of traction in M/min from the DE/min found in the last point.
5. Plot your data in an LWB diagram.
6. Run 4 more activity tests in which [S] lies at equidistant points between the extremes (from step 2).
7. From the LWB plot, find K_M and v_max.
8. Calculate k_cat for the E. coli ß-gal from v_max and the molar [enzyme] in the test solution.
9. Include the following in the protocol:  v vs. [S] plot, a [S]/v vs. [S] plot (Eisenthal), and a v vs. [S] plot (Cornish-Bowden) - include all kinetic paramters associated with each diagram! From that last diagram, the Cornish-Bowden, estimate the error for K_M and v_max. Compare with the values as seen on the diagrams.

Questions

1. Describe a few factors that majorly contribute to the catalytic effects of enzymes. What are some advantages and disadvantages that enzymes have over in/organic catalysts?
2. What simple assumptions does the Michaelis-Menten model make to describe enzyme kinetics? Why are they made?
3. What would happen if we were to simultaneously use both D- and L-amino acids in protein synthesis in the presence of a biocatalyst?
4. In the experiement, we measure the ß-gal initiated hydrolysis of o-NPG with a photometer. How would you measure the hydrolysis if we used lactose instead? Give an example of at least one activity test.
5. One of the conditions for the MM equation is a turnover < 10%.
   • Why do we make this assumption?
   • Roughly calculate how long hydrolysis can be observed at the smallest possible [S], keeping the amount of o-NPG hydrolyzed abouve 10%. Using this, calculate the rate of reaction.
   • In part (b), we approximated the rate at a constant [S]. What would you give as a mathematically correct decription of (b)? (Tip: Set up a differential equation and solve!)
6. Under physiological conditions, why is k_cat/K_M a better measure for catalytic activity than K_M?
7. To verify the specificity of a protease, different N-acetyl amino acid esters were used. The amino acids were varied, and the following values were drawn. Order the substrate by increasing substrate specificity and explain in your own words.

   KM (M)		kcat (s-1)
   Gly Leu  Ala Ile Val	0.2 1*10-4 6*10-3 3*10-4 4*10-4	0.0001 0.1  0.05 0.5 0.5

8. Calculate the relative rates (v_o/v_max) given the following parameters and assuming MM kinetics:
   1. [S] = 0.2 * K_M, [S] = 0.8 * K_M, [S] = 10 * K_M
   2. Calculate the values of [S] (as a multiple of K_M) at which we would hit 10% and 99% of v_max.
9. What problems would we run into with the introduction of coupled enzyme tests?
10. Running an enzyme test, we used a [S] = 2.5*10^-5 M. After 6 minutes, half of the substrate was used up. K_M = 5*10^-3M
    1. Determine the order of the reation under these conditions.
    2. Determine the applicable reaction velocity constants.
    3. v_max = __________
    4. How high is the [product]?
11. Draw a v vs. [S] plot for competitive, purely non-competitive, and mixed inhibition, and compare to an uninhibited reaction. (Draw in such a way that the differences are obvious.)

    Given:  V_S reactions with two isoenzymes (I and II):

    Enzyme	Vmax (U/mL)	KM (mmol/L)
    I II	10 1	9.8 0.2

    • Which isoenzyme has higher activity at [S] = 0.2, assuming MM kinetic conditions?
    • Isoenzyme II is inhibited competitively by Inhibitor I (K_I = 0.1 mmol/L) while Isoenzyme I is unaffected. Which isoenzyme has the larger activity at [S] =0.2 mmol/L and [I] = 1.0 mmol/L?
12. Calculate the free enthalpy for the EI bond (T = 298K) for two competitive inhibitors if K_I1 = 10nM and K_I2 = 1µM = 1000nM.
13. You know that 2 ß-lactam antibiotics are efficient.
    1. What's the underlying mechanism of increasing bacterial resistance to this type of antibiotic?
    2. Name at least one other type of inhibitory antibiotic and its target enzyme. How would you explain resistance against this antibiotic?
14. Graphically determine the type of inhibition occurring in an reaction with the following charateristics:

    [S] (mM)	vo (mM/min) without inhibitor	vo (mM/min) with inhibitor
    1 2 4 8 12	1.3 2.0 2.8 3.6 4.0	0.8 1.2 1.7 2.2 2.4

15. "Suicide inhibitors" are special irreversible inhibitors. What's so special about them? Give an example from the literature.
16. We want to run an activity test to determine the presence of a competitive inhibitor. Calculate the rate of reaction.

    Given:

    [S] KM [I] KI vmax without inhibitor

    2.0*10-4 M 2.5*10-4 M 2.5*10-3 M >2.5*10-3 M 55 mM/min

Things to know:

• how to derive the formulas for competitive, non-competitive, and uncompetitive inhibition
• determination of K_M, vmax, and K_I