Note of pedagogical information. Einstein once decried the fact that so many teachers have their non-swimming students jump into the deep end of the pool - right into the math without gaining a clear vision or conception of what is happening. Trying to correct this situation, this author will approach these enzymological studies in a way that first gives the student a conception of what is happening before trying to mathematize it. Thus we shall first do a QUALitative study, and then some QUANTitative work with a lot of precision, math and graphing involved.

The first thing the student needs to understand is the normal pathway executed by lactase, or β-galactosidase. And next the student must understand how a chemical analog works - in this case an analog of lactose called ONPG (ortho-nitrophenyl-β-D-galactopyranoside). These two pathways are shown here:

It should be pointed out that both lactose and ONPG have a galactose moiety on their left halves.

QUALITATIVE WORK

Now let's get our hands wet in lab! This is an extremely simple procedure that should take 15 minutes or less.

1. Begin by dissolving a few crushed specks (about the size of two grains of table sugar) of ONPG powder in about 1 ml of water. This is done first because ONPG does not dissolve as quickly as the other reagents. (Why no precise weights? This is 'qualitative'!)
2. To about 1 ml of water, dissolve a few specks scraped off of a Lactaid^® pill. Because the words lactose and lactase look so similar, hereonafter, lactase will be denoted as "ENZ."
3. To four separate test tubes containing about 1 ml of water each, dissolve pinches of the following sugars: sucrose (negative control), lactose (positive control), galactose and glucose. It is best if enough is added so as to result in saturated solutions - some undissolved sugar remains on the bottoms.
4. Add a drop of the dissolved ENZ to each tube.
5. Add about 4 drops of the ONPG solution to each tube, and await color development.
6. Results: sucrose should have no inhibitory effect and that tube should become yellow quickly; lactose should compete with the ONPG and color development should be very slow since there is a great excess of lactose versus ONPG. If either galactose or glucose also slow the color development, then you know which is a "product-" or "feedback-inhibitor.

QUANTITATIVE WORK

In this exercise, which will take two or more hours to complete, you will do the procedures with precision. In the process you will determine the kinetics of lactase under various concentrations of ONPG and galactose.

Reagents:

• 10 ml of saturated aqueous solution of galactose (GAL), refrigerate.
• 20 ml of freshly-made ONPG solution
  (200 mg ONPG + 20 ml water = approx 0.03 M)
  Subdivide this into several small PLASTIC containers, and FREEZE those you do not need immediately.
• 500 ml of the enzyme lactase (ENZ) made by dissolving a Lactaid^® pill into 500 ml of water.

You must now determine the most useful concentration of ENZ, as you neither want it to do the reaction too quickly, nor have it take hours. You would like to have the unihibited reaction be about a quarter the way to completion in five minutes. Here's how to do this:

1. To find out what "quarter-yellow" means you must first add ingredients together, and allow them to sit for perhaps 30 minutes. To four ml of water, add 1 ml of your concentrated lactase solution, and now add 0.5 ml of ONPG, mix, and allow to sit for 30 minutes. (While this is incubating at room temperature proceed with the next unit, below.) DO NOT ALLOW the pipette to touch the tube. You do not want to contaminate your ONPG solution with any lactase picked up from the test tubes.
2. After 30 minutes (more than enough time for the ENZ to hydrolyze the ONPG to completion), that solution will be defined as "fully yellow." Your new comparison standard for timing the speed of reactions will be a quarter of that fully yellow value: add 16.5 ml (3x5.5 ml) to the tube to dilute it 4-fold. This intensity is thus a "quarter-yellow." Discard all but 5.5 ml of that standard. Cap or cork the STANDARD to prevent evaporation.

You will now make various known dilutions of the ENZ so that one of them will eventually be found to have the right degree of activity for your kinetics work.

The following steps assume that you do not have access to a spectrophotometer. If you are a fortunate [high school] teacher in possession of a spectrophotometer, zero it with a blank (water-filled) tube at 420 nm and have the students take readings every 30 seconds for several minutes until they see they are getting a straight line. Discard the points that are curving away from the line and determine the rate of the reaction in OD units/minute.)

1. I would suggest for a start that you make a set of 2-fold serial dilutions of a stock solution of lactase (see fig to the right). Note that the term "volume" or "vol" means some arbitrary volume. "2 vols" means twice that much; etc.In this, each beaker is one-half the concentration as that in the previous beaker.
   
2. Put 1 ml of these into 5 clean test tubes labelled #1 through #5.
3. Add 4 ml of water to each of the five tubes; mix
4. Start your timer
5. Add 0.5 ml of the ONPG solution to tubes #1 through #5; mix. DO NOT ALLOW the pipette to touch the tube. You do not want to contaminate your ONPG solution with any lactase picked up from the test tubes.
6. Determine how long it took for the tubes #1 through #5 to become a as yellow as your "1/4 yellow standard".
7. Choose the respective enzyme concentration that appropriately gave you a quarter-yellow in about 5 minutes.
8. Hereonafter, your velocities will be determined by how long it took tubes to get to the quarter-yellow value. All tubes will have received indentical amounts of that appropriately determined ENZ dilution.

KINETICS

As kinetics use some unit of velocity, we will define velocity as:

"V" = 60 min / (time to become quarter-yellow)

CONTROLS FOR SUGAR EFFECTS
Add ingredients left to right and only when ready to run that particular tube. ONPG* is always added LAST!
For larger amounts of ONPG, put the drops of ONPG into a second clean tube. Start timing when you dump the first tube into the second, and then dump all the contents back into the first tube for good mixing.
SUGAR	ENZ	Water	SUGAR	ONPG*	Time to "quarter-yellow"	60/Time
No Sugar	1 ml	4 ml	none	0.5 ml*	approx 5 min	approx 12/hr
Sucrose	1 ml	0	4.0 ml SUC	0.5 ml*	.	.
Glucose	1 ml	0	4.0 ml GLC	0.5 ml*	.	.
Galactose	1 ml	0	4.0 ml GAL	0.5 ml*	.	.
Lactose	1 ml	0	4.0 ml LAC	0.5 ml*	.	.

Here come the studies that yield data that will reveal what sort of inhibition galactose has on ENZ. In this work, drops of ONPG are used in lieu of milliliters for ease of measurement in labs that do not possess micropipetters. However, if you do have micropipetters, you might consider 16 drops as equivalent to 500 μl, and work down from there. The really neat thing about this is that precision is the key and not what units you use. Just be sure that when you halve 500 μl you use 250 μl, or 16 drops to 8 drops and not 9 or 7 and that your drops are always the same size from the same dropper. The reason for being able to disregard units is that in the subsequent mathematical calculations entailed in Lineweaver-Birk plotting methods all the units cancel out.

SET-UP FOR TEST TUBE ARRAY FOR MAIN TEST
Add ingredients left to right and only when ready to run that particular tube. ONPG is always added LAST!
Expt'l goal	ENZ	Water	GAL	ONPG	Time to "quarter-yellow"	60/Time
.
No GAL	2 ml	4.0 ml	0	16 drops*	.	.
"	2 ml	4.3 ml	0	8 drops*	.	.
"	2 ml	4.5 ml	0	4 drops	.	.
"	2 ml	4.6 ml	0	2 drops	.	.
.
Hi GAL	2 ml	0.0 ml	4 ml	16 drops*	.	.
"	2 ml	0.3 ml	4 ml	8 drops*	.	.
"	2 ml	0.5 ml	4 ml	4 drops	.	.
"	2 ml	0.6 ml	4 ml	2 drops	.	slowest of all!
.
Med GAL	2 ml	2.0 ml	2 ml	16 drops*	.	.
"	2 ml	2.3 ml	2 ml	8 drops*	.	.
"	2 ml	2.5 ml	2 ml	4 drops	.	.
"	2 ml	2.6 ml	2 ml	2 drops	.	.
.
Lo GAL	2 ml	3.0 ml	1 ml	16 drops*	.	.
"	2 ml	3.3 ml	1 ml	8 drops*	.	.
"	2 ml	3.5 ml	1 ml	4 drops	.	.
"	2 ml	3.6 ml	1 ml	2 drops	.	.

THE LINEWEAVER-BIRK GRAPH

The Lineweaver-Birk graph is a "double reciprocal" plot, since it is of 1/V and 1/[ONPG]. Among the several types of "LB" plots one can get for inhibited enzymes, the ones for allosteric (second site) and competitive inhibitions are the most common:

What you need to do to make your LB plot is rather simple:

1. Plot all your 1/V values as merely the time it took to turn half-yellow (the previous column in your data chart). That's simple!
2. Plot all your 1/[ONPG] values as 1/16, 1/8, 1/4 and 1/2. In other words, since units mean nothing, make one hash-mark far out to the right, and then halve that distance, and halve that, and halve that again. Very simple indeed!
3. And all you are looking for is whether the lines converge on the vertical axis, or somewhere to the left of it.

Discussion

If all the lines converge on the vertical axis, it means that at infinite concentration of ONPG, no amount of inhibitor (galactose) will alter that value. This is the mark of a competitive inhibitor.

Consider a case in which you were to add some lead ion into the ENZ solution. Suppose that the lead inactivate half of the ENZ molecules. It would be equivalent that your ENZ concentration was only half of what it was. If you consider this sort of lead inhibitor, it will always affect where the lines cross the vertical axis. Twice as many ENZ molecules working as fast as they can will do twice as much work as half as many ENZ molecules working as fast as they can. (Indeed the LB plot of this system would be parallel lines - one of the other types of LB plots!)

Lost? Perhaps it would help if you consider the supermarket analogy of inhibition.

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