A quantity A quantity/volume
mg, mole M (mol/L), M
g, mol mM, mg/mL
units units/mL
Quantities such as milligrams (mg), micromoles (mol), and units refer to amounts. Concentration is the amount per volume, so that molar (M), micromolar (M), milligrams per milliliter (mg/ml), and units per milliliter (units/ml) are concentrations. A unit is the amount of enzyme that will catalyze the conversion of 1 mol of substrate to product in 1 min under a given set of conditions.
S, P, AND E (SUBSTRATE, PRODUCT, ENZYME)
Enzyme (E) converts substrate(s) (S) to product(s) (P) and acceler- ates the rate.1Enzymes are named by a systematic set of rules that nobody follows. The only given is that enzyme names end in -ase and may have something in them that may say something about the type of reaction they catalyze—such as chymotrypsin, pepsin, and enterokinase (all proteases).
8 Enzyme Kinetics • 97 •
The concentrations of substrate and product are invariably in molar units (M; this includes mM, M, etc.), but enzyme concentrations may be given in molar (M), milligrams per milliliter (mg/mL), or units/mL. The amount of enzyme you have can be expressed in molecules, mil- ligrams, nanomoles (nmol), or units. A unit of enzyme is the amount of enzyme that will catalyze the formation of 1 mol of product per minute under specifically defined conditions. A unit is an amount, not a con- centration.
Units of enzyme can be converted to milligrams of enzyme if you know a conversion factor called the specific activity. Specific activity is the amount of enzyme activity per milligram of protein (micromoles of product formed per minute per milligram of protein, or units per mil- ligram). For a given pure enzyme under a defined set of conditions, the specific activity is a constant; however, different enzymes have different specific activities. To convert units of enzyme to milligrams of enzyme, divide by the specific activity: units/(units/milligram) milligrams. Spe- cific activity is often used as a rough criterion of purity, since in crude mixtures very few of the milligrams of protein will actually be the enzyme of interest. There may be a large number of units of activity, but there will also be a large amount of protein, most of which is not the enzyme. As the enzyme becomes more pure, you’ll get the same units but with less protein, and the specific activity will increase. When the protein is pure, the specific activity will reach a constant value.
Enzyme concentrations in milligrams per milliliter can be converted to molar units by dividing by the molecular weight (in mg/mmol).2
m m g g /m /m m L ol m m m L ol M
ACTIVE SITE
The active site is the special place, cavity, crevice, chasm, cleft, or hole that binds and then magically transforms the substrate to the product. The kinetic behavior of enzymes is a direct consequence of the protein’s having a limited number (often 1) of specific active sites.
2
• 98 • Basic Concepts in Biochemistry
Most of enzyme kinetics (and mechanism) revolves around the active site. As we’ll see later, saturation kinetics is one of the direct con- sequences of an active site.
How fast a given amount of substrate is converted to a product depends on how much enzyme is present. By measuring how much prod- uct is formed in a given time, the amount of enzyme present can be deter- mined. An assay requires that you have some way to determine the concentration of product or substrate at a given time after starting the reaction. If the product and substrate have different UV or visible spec- tra, fluorescence spectra, and so forth, the progress of the reaction can be followed by measuring the change in the spectrum with time. If there are no convenient spectral changes, physical separation of substrate and product may be necessary. For example, with a radioactive substrate, the appearance of radioactivity in the product can be used to follow product formation.
There are a number of interchangeable words for velocity: the change in substrate or product concentration per time; rate; just plain v (for velocity, often written in italics to convince you it’s special); activ- ity; or the calculus equivalent, the first derivative of the product or sub- strate concentration with respect to time, d[P]/dt or d[S]/dt (the minus means it’s going away). Regardless of confusion, velocity (by any of its names) is just how fast you’re going. Rather than miles per hour, enzyme velocity is measured in molar per minute (M/min) or more usually in micromolar per minute (M/min).
VELOCITY
Velocity—rate, v, activity, d[P]/dt, d[S]/dt—is how fast an en- zyme converts substrate to product, the amount of substrate con- sumed, or product formed per unit time. Units are micromoles per minute (mol/min) units.
ASSAY
An assay is the act of measuring how fast a given (or unknown) amount of enzyme will convert substrate to product—the act of measuring a velocity.
8 Enzyme Kinetics • 99 •
Figure 8-1 shows what happens when enzyme is added to a solution of substrate. In the absence of enzyme, product appearance is slow, and there is only a small change in product concentration with time (low rate). After enzyme is added, the substrate is converted to product at a much faster rate. To measure velocity, you have to actually measure two things: product (or substrate) concentration and time. You need a device to measure concentration and a clock. Velocity is the slope of a plot of product (or substrate) concentration (or amount) against time.
Velocity can be expressed in a number of different units. The most common is micromolar per minute (M/min); however, because the velocity depends on the amount of enzyme used in the assay, the veloc- ity is often normalized for the amount of enzyme present by expressing the activity in units of micromoles per minute per milligram of enzyme [mol/(min mg)]. This is called a specific activity. You may be won- dering (or not) where the volume went—after all, product concentration is measured in molar units (M; mol/L). Well, it’s really still there, but it
[Pr oduct] ( mol/mL)
∆
[Product]∆
time mL min mol velocity = velocity = slope time (min).
冸
冹
Figure 8-1The VELOCITY of product formation (or substrate disappearance) is defined as the change in product concentration per unit time. It is the slope of a plot of product concentration against time. The velocity of product formation is the same as the velocity of substrate disappearance (except that substrate goes away,
• 100 • Basic Concepts in Biochemistry
canceled out, and you don’t see it because both the product and enzyme are expressed in concentration units; both are per milliliter. Let’s do a numerical example.
Adding 0.1 g of fumarase to a solution of 5 mM fumarate (sub- strate) in a final volume of 1.0 mL results in the formation of 0.024 mol of malate (product) per minute. At the beginning of the reaction, the con- centration of product is zero. If we wait 10 min. 10 0.024 mol of malate will be made, or 0.24 mol. In a volume of 1 mL, 0.24 mol rep- resents a concentration of 0.24 mM. So, over a period of 10 min, the con- centration of malate went from 0 to 0.24 mM while the concentration of fumerate went from 5 to 4.76 mM (5 mM 0.24 mM). The spe- cific activity of the enzyme would be [0.024 mol/(min mL)]/(0.1 103 mg fumarase/mL) or 240 mol/(min mg).
To make matters worse, velocity is often reporting using the change in the amount of product per time (mol/min). To actually determine the concentration, you need to know the volume. The key unit that always shows up somewhere with velocities and never cancels out is the per time part; the rest can usually be sorted out, depending on whether you’re dealing with amounts or concentrations.
As substrate is consumed, the substrate concentration falls and the reaction may get slower. As product is made, the reaction may slow down if the product is an inhibitor of the enzyme. Some enzymes are unstable and die as you’re assaying them. All these things may cause the veloc- ity to change with time. If the velocity is constant with time, the plot of product against time is a straight line; however, if velocity changes with time (the slope changes with time), this plot is curved (Fig. 8-2).
Usually enzyme activities are measured under conditions under which only a tiny bit of the substrate is converted to product (like 1 to 5 per- cent)3 This means that the actual concentration of substrate will be very