**Synopsis:** The kinetic behaviour of enzymes is described by the **Michaelis-Menten equation**, and the two characteristic constants associated with this equation, **V _{max} and K_{M}**. Every enzyme has specific values for these constants, which must be measured experimentally. Linear plots like the

**Lineweaver-Burk**plot provide the simplest means of fitting potentially error prone experimental values to the Michaelis-Menten equation.

Inhibitors control enzyme activity by reversibly decreasing the enzyme activity. Different mechanisms of inhibition depend on the relationship between inhibitor and substrate, and can be distinguished by observing how inhibitor affects K_{M} or V_{max} of the enzyme.

**Competitive** inhibition increases K_{M} with no effect on V_{max}.

**Non-competitive** inhibition decreases V_{max} with no effect on K_{M}.

LABORATORY-RELATED CONTENT

The information discussed in this section is related to **Lab 3.**

The kinetic properties of enzymes may be characterized by **measuring reaction rates for a series of different substrate concentrations.** Each rate is measured should be an initial velocity, either by taking the slope of a progress curve ([S] or [P] plotted versus time) at zero time, or by allowing the reaction to proceed for a very brief time and measuring extent of reaction.

The reaction should be linear over the short time interval.

A different progress curve is obtained for each initial substrate concentration, which is indicated by where the curve starts on the [S] axis **(Figure 11.1)**. A slope is measured for each curve at t = 0. These slopes are the initial rates, v_{o} **used in the Michaelis-Menten plot.**

LABORATORY-RELATED CONTENT

The information discussed in this section is related to **Lab 3.**

Initial rates taken from the progress curves are then plotted as a function of [S]. The graph follows a characteristic **hyperbolic** shape that matches the **Michaelis-Menten** equation **(right).**

**V _{max}** is the limiting maximum rate as

**[S] tends to infinity (Figure 11.2)**.

Once V_{max} has been determined we find the point on the curve where **v _{o} = ½ V_{max}**; the concentration

**[S]**at this point gives the value of

**K**.

_{M}**K _{M} and V_{max} tell you about the enzyme’s properties as a catalyst.**

**V**_{max }indicates catalytic rate when 100% of enzyme is occupied by substrate, higher V_{max} means faster reaction, better catalyst

**V _{max}** is not a true constant – it is only constant if the

**same amount of enzyme**is used for each rate measurement. V

_{max}is proportional to the concentration of enzyme present:

**V _{max} =k_{2}[E]_{total}**

From **V _{max}**, we can calculate the true constant

*k*

_{2}, the rate constant for the catalytic step of the two step enzyme reaction.

**(sometimes written as**

*k*_{2}**) is the**

*k*_{cat}**turnover number**of the enzyme

**(Figure 11.3)**.

**K _{M}** indicates the

**affinity of the substrate**for the enzyme.

**Low K**means_{M}**high affinity**, the enzyme binds this substrate**more strongly****High K**means_{M}**low affinity**, the enzyme binds this substrate**more weakly**

**An enzyme that recognizes different substrates will have a different K _{M} for each substrate.** To measure

**K**of substrate

_{M}**B**, keep

**[A] high and constant**, measure rate as a function of varying

**[B]**.

**Measuring V _{max} and K_{M} can be tricky**

Unfortunately it’s not as easy as it seems to estimate V_{max} by inspection of the hyperbolic plot, since the curve keeps creeping up even at very high [S]. Most people underestimate V_{max} by 10-20% when using this method **(Figure 11.4).** If the estimate of V_{max} is bad, the estimate of ½ V_{max} and K_{M} is also affected.

Real experimental data also tends to scatter off the theoretical curve due to measurement errors, making graphing the correct curve even more difficult **(Figure 11.5)**.

LABORATORY-RELATED CONTENT

The information discussed in this section is related to **Lab 3.**

Instead, the Michaelis-Menten equation may be rewritten so it plots as a **straight line**, which is much easier for graphing experimental data. This is known as a **linear transformation**.

**Lineweaver-Burk or double reciprocal plot**

Take **reciprocals** of both sides of the Michaelis-Menten equation and then cancel terms **(Figure 11.6).**

If we substitute **y for 1 / v _{o}** and

**x for 1 / [S]**, this is now the standard equation for a straight line, with

**slope = K**, and

_{M}/ V_{max}**y intercept = 1 / V**.

_{max}A straight-line plot is much preferred over a curve, particularly when the data is slightly scattered due to experimental error. Slopes and intercepts are relatively easily obtained from a straight-line graph. **The result is known as the Lineweaver-Burk plot, or double reciprocal plot (Figure 11.7).**

The Lineweaver-Burk plot is usually extrapolated to the negative **x**-axis. Although there’s no data with negative **x** values, the negative x-intercept can be used to obtain **K _{M}**.

X-intercept = – (Y-intercept) / slope.

**A hint for correlating axis labels and intercepts:** The y-axis is 1 / v_{o}, so its intercept is 1 / **V _{max}.**The x axis is

**1 / [S],**so its intercept is

**– 1 / K**(the negative sign corrects for the intercept having a negative value). Remember that K

_{M}_{M}is a concentration, the concentration [S] that happens to give 50% of maximum rate or

**0.5 × V**.

_{max}## Various substances can act on enzymes to reduce their activity

**Inactivation** results from a reactive molecule that may form covalent bonds with key amino acids, preventing the enzyme from completing its reaction cycle. Inactivation tends to be **irreversible**. Essentially, an inactivator reduces the quantity of available enzyme irreversibly and in a **stoichiometric** manner:

3 µmol enzyme + 2 µmol inactivator leaves 1 µmol enzyme to continue working.

e.g. **Disopropylfluorophosphate** reacts with *acetylcholinesterase* irreversibly, blocking transmission of nerve impulses. Many so-called nerve gases act this way and many are halogen-phosphorus compounds.

**Reversible inhibition** results from a substance that binds to an enzyme and limits its capacity to catalyze reaction. The binding is **non-covalent and reversible**, and if inhibitor is removed, normal activity is restored. The concentration of inhibitor, like substrate, is typically much higher than enzyme concentration.

Enzymes need to be **regulated** in the course of normal metabolism, i.e. an enzyme that is temporarily not needed is **turned off**. Reversible inhibition can contribute to regulation, since activity can be restored by removing the inhibitor without having to make new enzyme. Enzyme regulation and its consequences are major themes of BIOC*3560.

Many of the substances that we use as drugs acting by inhibiting a key enzyme in the body. For example, **acetylsalicylic acid** (ASA or aspirin) inhibits an enzyme called *cyclooxygenase, *which is responsible for making prostaglandins that stimulate the inflammatory response. When ASA inhibits cyclooxygenase, less prostaglandin is made and inflammation is kept under control. Finding and analyzing properties of enzyme inhibitors is an important aspect of pharmaceutical research.

There are several reversible** inhibition mechanisms**, distinguished by the relationship between inhibitor and the substrate of the enzyme.

**Competitive inhibition**: the enzyme either binds substrate or binds inhibitor, but not both. In other words, the substrate and inhibitor**compete**for occupation of the enzyme molecule.**Non-competitive inhibition**: inhibitor can bind to enzyme whether substrate is also bound or not, i.e. substrate binding has no effect on inhibition.**Uncompetitive inhibition**: opposite to competitive, the inhibitor can**only**bind to the ES complex if substrate bind first. This mode may occur with two-substrate enzymes. We won’t be discussing this mode other than to mention it here.**Mixed inhibition**: is the combination of non-competitive with either of the other mechanisms. The true non-competitive inhibition is rare. In fact, most cases are actually mixed inhibition that closely approximates the non-competitive case.

## 1. Competitive inhibition:

A competitive inhibitor I can only bind to the unoccupied enzyme E, not to the ES complex **(Figure 11.8)**. The quantity of complex EI that forms is governed by the equilibrium constant K_{i}, known as the inhibition constant. If more EI forms, less enzyme is available to form productive ES complex.

However, since inhibitor I can’t bind to ES, very high substrate concentrations can overcome the inhibitor by forcing the substrate binding equilibrium in the direction of ES, and this brings the enzyme up to its normal V_{max}. Hence the term competitive to describe this inhibition.

**Michaelis Menten hyperbolic plot:** shows the rate v_{o} of the enyme reaction **with a constant concentration [I]** of inhibitor as [S] is varied **(Figure 11.9)**.

The rate rises more gradually when inhibitor is present, but eventually reaches approaches the normal V_{max} when [S] is very high.

**If inhibitor concentration [I] is set equal to K _{i}**, this causes the K

_{M}‘ observed to be

**doubled**relative to uninhibited enzyme.

**Characteristics of a competitive inhibitor:V _{max} is unchanged, observed K_{M}‘ increases.**

- In the presence of inhibitor at concentration [I],
- Remember
**higher K**implies that the enzyme binds its substrate with_{M}**lower affinity**.**i.e. a higher concentration of [S] is needed**before the enzyme can reach 50% of V_{max}. - K
_{i}is equal to the concentration of inhibitor that doubles the observed K_{M}of the enzyme.If we set the inhibitor concentration [I] = K_{i}, then . The term is the**inhibition factor**, and appears in all inhibition equations.

The change in K_{M} is detected by plotting the data in one of the linear forms, e.g. in the **Lineweaver Burk Plot (Figure 11.10) —**

The dashed line represents the activity of the normal uninhibited enzyme, and then the experiment is repeated with a series of different concentrations of I. The other two lines show the effect increasing [I] .

Each line shows the behaviour of the enzyme for a **given value of [I]**. The higher concentration of inhibitor gives a steeper slope to the line. The series of lines pivot on the y intercept, since V_{max} **is not changed for competitive inhibition.** The X-intercept becomes smaller as [I] increases, since **K _{M} increases for competitive inhibition.**

To measure K_{i}, one finds the inhibitor concentration [I] that **doubles the observed K _{M}‘**. This is represented by the middle line in the figure (x-intercept halved means

**K**was doubled.).

_{M}‘## 2. Non-competitive inhibition:

A non-competitive inhibitor I can bind both to unoccupied enzyme E, and to ES complex **(Figure 11.11)**. The EI complex can bind S, but EIS is unable to proceed to give products.

The quantity of complexes EI and EIS that form are governed by the equilibrium constant K_{i}. If more EI or EIS forms, less enzyme is available to form productive ES complex. Since I can also bind to ES, high substrate concentrations do not overcome the inhibitor. Hence the term **non-competitive** to describe this inhibition. True non-competitive inhibition requires K_{i} to be the same at both stages (this is rare). If the K_{i} for I binding to empty E is not the same as for I binding to occupied ES, **mixed inhibition** will be observed.

**Michaelis Menten hyperbolic plot (Figure 11.12):** shows the rate v_{o} of the enzyme reaction **with a constant concentration [I]** of inhibitor as [S] is varied. The rate rises more gradually when inhibitor is present, and levels off at a **lower V’ _{max}**.

**If inhibitor concentration [I] is set equal to K _{i}**, this causes the V’

_{max}observed to be

**halved**relative to uninhibited enzyme.

**Characteristics of a non-competitive inhibitor:** V’_{max} **decreases**, **K _{M} is unchanged**.

**Lineweaver-Burk Plot (Figure 11.13):**

The dashed line represents the activity of the uninhibited enzyme. The other two lines show the effect of added inhibitor I. The series of lines pivot on the** -ve x-intercept**, since **K _{M} is unchanged for non-competitive inhibition.** Y-intercept and slope increase due to the reciprocal dependence on V’

_{max}, which decreases. To measure K

_{i}, one finds the inhibitor concentration [I] that

**halves the observed**V’

_{max}.

**Mixed inhibition** is indicated by decrease in V_{max} coupled to increase in K_{M}. If the change in V_{max} is large and the change in K_{M}‘ is small, then it is a reasonable approximation (and mathematically much simpler) to treat the case as non-competitive.