---
title: "Enzymes and Biochemical Catalysis"
book: "University Biology — Year 1"
subject: biology
language: en
chapter: 13
exercises: 12
source: https://one-course.com/books/biology/3/en/chapter/13-enzymes-and-biochemical-catalysis
---

# Chapter 13 — Enzymes and Biochemical Catalysis

Pour hydrogen peroxide on a cut potato and it froths: a single molecule of catalase in the potato’s [cells](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) splits forty million molecules of peroxide every second, a reaction that, left to itself, would take years. The peroxide was going to decompose anyway — the reaction is downhill — but not in any useful time. Every reaction of the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) is like this: thermodynamics says which way it can go, and an *[enzyme](#def-b1-enzymes-enzyme)* says whether it goes now. This chapter describes what an [enzyme](#def-b1-enzymes-enzyme) does to a reaction, the kinetics by which [enzymes](#def-b1-enzymes-enzyme) are measured and compared, the ways they are inhibited, and the ways the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) switches them on and off.

## 13.1 Catalysis

**Definition 13.1 (Enzyme, substrate, active site).**

An *enzyme* is a [protein](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-peptide) (rarely an [RNA](https://one-course.com/books/biology/3/en/chapter/11-nucleotides-and-nucleic-acids#def-b1-nucleic-acids-chain)) that *catalyses* a reaction: it increases the rate without being consumed and without changing the equilibrium. The molecules it acts on are its *substrates*; they bind in the *active site*, a cleft a few residues line, where the chemistry happens. Enzymes are *specific* — for one substrate or a family, for one bond, for one stereoisomer — and they are named for their substrate and reaction with the suffix *-ase* (lactase, [DNA](https://one-course.com/books/biology/3/en/chapter/11-nucleotides-and-nucleic-acids#def-b1-nucleic-acids-chain) polymerase, succinate dehydrogenase), in six classes: oxidoreductases, transferases, hydrolases, lyases, isomerases, ligases. Many need a [non-protein](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-peptide) *cofactor*: a metal ion (Zn, Mg, Fe), or an organic *coenzyme* — $\mathrm{NAD^+}$, FAD, coenzyme A, most of them made from vitamins, which is what vitamins are for.

**Proposition 13.2 (What an enzyme changes and what it does not).**

A reaction passes through a *[transition state](#prop-b1-enzymes-whatitdoes)*, a strained arrangement of the atoms higher in [free energy](https://one-course.com/books/biology/3/en/chapter/8-water-and-small-biomolecules#prop-b1-water-small-molecules-gibbs) than reactants or products by the *[activation energy](#prop-b1-enzymes-whatitdoes)* $E_a$; only the molecules that thermal agitation carries over this barrier react, in a fraction proportional to $e^{-E_a/RT}$. An [enzyme](#def-b1-enzymes-enzyme) binds the [transition state](#prop-b1-enzymes-whatitdoes) more tightly than the [substrate](#def-b1-enzymes-enzyme) — its [active site](#def-b1-enzymes-enzyme) is complementary to the strained form — and so lowers $E_a$: lowering it by $34\,\mathrm{kJ}/\mathrm{mol}$ (the worth of a few [hydrogen bonds](https://one-course.com/books/biology/3/en/chapter/8-water-and-small-biomolecules#def-b1-water-small-molecules-water)) at $37\,{}^{\circ}\mathrm{C}$ multiplies the rate by $e^{34\,000/2580} \approx
10^6$. It does not change $\Delta G$ or the equilibrium constant, which depend only on the reactants and products: an [enzyme](#def-b1-enzymes-enzyme) speeds the forward and backward reactions equally and brings the system to the same equilibrium faster.

![Free energy along a reaction. The enzyme (red) lowers the barrier of the transition state and leaves the difference between substrate and product — and hence the equilibrium — untouched.](https://one-course.com/images/onecourse/chapters/biology-3/b1-enzymes/fig-fb632ae417d9.svg)

*[Free energy](https://one-course.com/books/biology/3/en/chapter/8-water-and-small-biomolecules#prop-b1-water-small-molecules-gibbs) along a reaction. The [enzyme](#def-b1-enzymes-enzyme) (red) lowers the barrier of the [transition state](#prop-b1-enzymes-whatitdoes) and [leaves](https://one-course.com/books/biology/3/en/chapter/3-functional-organization-of-a-flowering-plant#def-b1-flowering-plant-organization-organs) the difference between [substrate](#def-b1-enzymes-enzyme) and product — and hence the equilibrium — untouched.*

![Hydrogen peroxide on a cut potato: catalase in the cells splits it into water and oxygen forty million times per second per enzyme molecule, and the oxygen froths out.](https://one-course.com/images/onecourse/chapters/biology-3/b1-enzymes/img-7938d5cc44f7.jpg)

*Hydrogen peroxide on a cut potato: catalase in the [cells](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) splits it into water and oxygen forty million times per second per [enzyme](#def-b1-enzymes-enzyme) molecule, and the oxygen froths out.*

**Proposition 13.3 (How the active site works).**

An [active site](#def-b1-enzymes-enzyme) accelerates its reaction by several means at once: it *binds and orients* the [substrates](#def-b1-enzymes-enzyme) so that the reacting groups meet in the right geometry, replacing a rare collision by a certain one; it *strains* the [substrate](#def-b1-enzymes-enzyme) toward the [transition state](#prop-b1-enzymes-whatitdoes) (*induced fit*: the [enzyme](#def-b1-enzymes-enzyme) closes around the [substrate](#def-b1-enzymes-enzyme), and the fit is best for the [transition state](#prop-b1-enzymes-whatitdoes)); its [side chains](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-aminoacid) act as *acids and bases*, handing protons to and from the [substrate](#def-b1-enzymes-enzyme) at the right moment (histidine, with its p$K_a$ near 7, does this in many [enzymes](#def-b1-enzymes-enzyme)); some form a transient *[covalent bond](https://one-course.com/books/biology/3/en/chapter/8-water-and-small-biomolecules#def-b1-water-small-molecules-bonds)* with the [substrate](#def-b1-enzymes-enzyme) (serine proteases); and its metal ions polarise bonds and stabilise charges. Water is largely excluded from the site, so that charges and [hydrogen bonds](https://one-course.com/books/biology/3/en/chapter/8-water-and-small-biomolecules#def-b1-water-small-molecules-water) act at full strength.

**Example 13.4 (Three enzymes).**

Lysozyme, in tears and egg white, strains a sugar ring of the bacterial wall into the shape of the [transition state](#prop-b1-enzymes-whatitdoes) and cuts the chain: $10^8$-fold acceleration. Carbonic anhydrase, in red [cells](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell), holds a zinc ion that turns water into a hydroxide poised to attack $\mathrm{CO_2}$: a million molecules a second, the fastest [enzyme](#def-b1-enzymes-enzyme) after catalase. Chymotrypsin, in the pancreatic juice, uses a serine made reactive by a histidine to cut [proteins](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-peptide) after aromatic residues: a hundred per second, with a covalent intermediate.

## 13.2 Enzyme kinetics

**Definition 13.5 (Initial rate, saturation).**

The *rate* $v$ of an [enzyme](#def-b1-enzymes-enzyme) reaction is the amount of product formed per unit time ($\mathrm{mol}/\mathrm{s}$, or units of [enzyme](#def-b1-enzymes-enzyme): $1\,\text{µ}\mathrm{mol}/\mathrm{min}$). It is measured as the *initial rate* $v_0$, before the [substrate](#def-b1-enzymes-enzyme) is depleted or the product accumulates. At a fixed [enzyme](#def-b1-enzymes-enzyme) concentration $v_0$ rises with the [substrate](#def-b1-enzymes-enzyme) concentration $[S]$ and then *saturates* at a maximum $V_{\max}$: the [enzyme](#def-b1-enzymes-enzyme) is fully occupied and can go no faster.

**Theorem 13.6 (Michaelis–Menten).**

For an [enzyme](#def-b1-enzymes-enzyme) $E$ that binds its [substrate](#def-b1-enzymes-enzyme) reversibly and converts it,

$$
E + S \underset{k_{-1}}{\overset{k_1}{\rightleftharpoons}} ES
\overset{k_{\text{cat}}}{\longrightarrow} E + P ,
$$

the initial rate at [substrate](#def-b1-enzymes-enzyme) concentration $[S]$ is

$$
v_0 = \frac{V_{\max}\,[S]}{K_m + [S]}, \qquad V_{\max} = k_{\text{cat}}\,[E]_{\text{tot}},
\qquad K_m = \frac{k_{-1} + k_{\text{cat}}}{k_1} .
$$

$K_m$, the *[Michaelis constant](#thm-b1-enzymes-mm)*, is the [substrate](#def-b1-enzymes-enzyme) concentration at which the rate is half its maximum, and measures how much [substrate](#def-b1-enzymes-enzyme) the [enzyme](#def-b1-enzymes-enzyme) needs; $k_{\text{cat}}$, the *turnover number*, is the number of [substrate](#def-b1-enzymes-enzyme) molecules one [enzyme](#def-b1-enzymes-enzyme) converts per second when saturated; the ratio $k_{\text{cat}}/K_m$ is the *catalytic efficiency*, the rate constant at low [substrate](#def-b1-enzymes-enzyme), bounded above by the rate at which [substrate](#def-b1-enzymes-enzyme) can reach the [enzyme](#def-b1-enzymes-enzyme) by diffusion, about $10^8$–$10^9$ $\mathrm{L}\,\mathrm{mol}^{-1}\,\mathrm{s}^{-1}$.

**Proof.** Assume a *steady state* in which $ES$ is formed as fast as it disappears (valid after the first milliseconds, while $[S] \gg [E]$): $k_1[E][S] = (k_{-1} + k_{\text{cat}})[ES]$, so $[E][S] = K_m[ES]$ with $K_m$ as defined. Conservation gives $[E] = [E]_{\text{tot}} -
[ES]$; substituting, $([E]_{\text{tot}} - [ES])[S] = K_m[ES]$, hence $[ES] = [E]_{\text{tot}}[S]/(K_m + [S])$. The rate is $v_0 = k_{\text{cat}}[ES]$, which gives the formula; at $[S] \gg K_m$, $v_0 \to k_{\text{cat}}[E]_{\text{tot}} = V_{\max}$, and at $[S] = K_m$, $v_0 = V_{\max}/2$. ∎

![Left: the Michaelis–Menten hyperbola for K_m = 2\, mmol/ L and V_ = 1\, µ mol/ min, with the six measured points of the weekend problem. Right: the same data as a Lineweaver–Burk plot, 1/v_0 against 1/(S): a straight line whose intercepts give V_ and K_m.](https://one-course.com/images/onecourse/chapters/biology-3/b1-enzymes/fig-fcc855eda471.svg)

![Left: the Michaelis–Menten hyperbola for K_m = 2\, mmol/ L and V_ = 1\, µ mol/ min, with the six measured points of the weekend problem. Right: the same data as a Lineweaver–Burk plot, 1/v_0 against 1/(S): a straight line whose intercepts give V_ and K_m.](https://one-course.com/images/onecourse/chapters/biology-3/b1-enzymes/fig-128694e7a7f3.svg)

*Left: the Michaelis–Menten hyperbola for $K_m = 2\,\mathrm{mmol}/\mathrm{L}$ and $V_{\max} = 1\,\text{µ}\mathrm{mol}/\mathrm{min}$, with the six measured points of the weekend problem. Right: the same data as a Lineweaver–Burk plot, $1/v_0$ against $1/[S]$: a straight line whose intercepts give $V_{\max}$ and $K_m$.*

**Method 13.7 (Measuring KmK_mKm​ and Vmax⁡V_{\max}Vmax​).**

1. Prepare a series of [substrate](#def-b1-enzymes-enzyme) concentrations spanning $K_m/4$ to $10K_m$ , with the same [enzyme](#def-b1-enzymes-enzyme) concentration; follow the product (colour, absorbance, gas) for the first minute and take the slope as $v_0$ .
2. Plot $v_0$ against $[S]$ : the hyperbola gives a rough $V_{\max}$ (the plateau) and $K_m$ (the concentration at half of it).
3. For precision, invert: $1/v_0 = (K_m/V_{\max})(1/[S]) +  1/V_{\max}$ ( *Lineweaver–Burk* ). The line’s intercept on the $y$ axis is $1/V_{\max}$ , on the $x$ axis $-1/K_m$ , and its slope $K_m/V_{\max}$ .
4. Divide $V_{\max}$ by the [enzyme](#def-b1-enzymes-enzyme) concentration to get $k_{\text{cat}}$ ; divide by $K_m$ for the efficiency. Repeat with an [inhibitor](#def-b1-enzymes-inhibitors) to see which parameter it changes.

![An enzyme assay: the product absorbs light, and the spectrophotometer records its appearance second by second; the initial slope is v_0.](https://one-course.com/images/onecourse/chapters/biology-3/b1-enzymes/img-1729b017d143.jpg)

*An [enzyme](#def-b1-enzymes-enzyme) assay: the product absorbs light, and the spectrophotometer records its appearance second by second; the initial slope is $v_0$.*

**Example 13.8 (Enzymes compared).**

| [enzyme](#def-b1-enzymes-enzyme) | $K_m$ ($\mathrm{mol}/\mathrm{L}$) | $k_{\text{cat}}$ ($\mathrm{s}^{-1}$) | $k_{\text{cat}}/K_m$ ($\mathrm{L}\,\mathrm{mol}^{-1}\,\mathrm{s}^{-1}$) |
| --- | --- | --- | --- |
| catalase | $2.5 \times 10^{-2}$ | $4 \times 10^{7}$ | $1.6 \times 10^{9}$ |
| carbonic anhydrase | $1.2 \times 10^{-2}$ | $1 \times 10^{6}$ | $8 \times 10^{7}$ |
| chymotrypsin | $1.5 \times 10^{-2}$ | 100 | $7 \times 10^{3}$ |
| lysozyme | $6 \times 10^{-6}$ | 0.5 | $8 \times 10^{4}$ |
| [DNA](https://one-course.com/books/biology/3/en/chapter/11-nucleotides-and-nucleic-acids#def-b1-nucleic-acids-chain) polymerase | $1 \times 10^{-5}$ | 15 | $1.5 \times 10^{6}$ |

Catalase and carbonic anhydrase work at the diffusion limit: every collision with [substrate](#def-b1-enzymes-enzyme) is productive. Chymotrypsin is slow but needs to be — it cuts a [peptide bond](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-peptide), a far harder job than splitting peroxide.

## 13.3 Inhibition

**Definition 13.9 (Inhibitors).**

An *inhibitor* lowers the rate of an [enzyme](#def-b1-enzymes-enzyme). *Irreversible* inhibitors bind covalently and destroy the [enzyme](#def-b1-enzymes-enzyme)’s activity for good (nerve agents on acetylcholinesterase, penicillin on the [enzyme](#def-b1-enzymes-enzyme) that cross-links the bacterial wall, aspirin on the [enzyme](#def-b1-enzymes-enzyme) that makes prostaglandins). *Reversible* inhibitors bind by weak bonds and can be washed away; by where they bind:

- *competitive* : the inhibitor resembles the [substrate](#def-b1-enzymes-enzyme) and occupies the [active site](#def-b1-enzymes-enzyme) ; it raises the apparent $K_m$ (by the factor $1 + [I]/K_i$ ) and [leaves](https://one-course.com/books/biology/3/en/chapter/3-functional-organization-of-a-flowering-plant#def-b1-flowering-plant-organization-organs) $V_{\max}$ unchanged — enough [substrate](#def-b1-enzymes-enzyme) displaces it;
- *non-competitive* : the inhibitor binds elsewhere and spoils the catalysis whether or not [substrate](#def-b1-enzymes-enzyme) is bound; it lowers $V_{\max}$ and [leaves](https://one-course.com/books/biology/3/en/chapter/3-functional-organization-of-a-flowering-plant#def-b1-flowering-plant-organization-organs) $K_m$ unchanged — no amount of [substrate](#def-b1-enzymes-enzyme) overcomes it;
- *uncompetitive* : the inhibitor binds only the $ES$ complex; both $V_{\max}$ and $K_m$ fall.

$K_i$, the dissociation constant of the inhibitor, measures its potency: the lower, the stronger.

![Lineweaver–Burk plots with the two common inhibitors. A competitive inhibitor pivots the line about the y intercept (V_ unchanged, K_m raised); a non-competitive one pivots it about the x intercept (K_m unchanged, V_ lowered).](https://one-course.com/images/onecourse/chapters/biology-3/b1-enzymes/fig-4a1825ba7e80.svg)

*Lineweaver–Burk plots with the two common [inhibitors](#def-b1-enzymes-inhibitors). A competitive [inhibitor](#def-b1-enzymes-inhibitors) pivots the line about the $y$ intercept ($V_{\max}$ unchanged, $K_m$ raised); a non-competitive one pivots it about the $x$ intercept ($K_m$ unchanged, $V_{\max}$ lowered).*

**Example 13.10 (Competition as medicine).**

Methanol is harmless until the liver’s alcohol dehydrogenase turns it into formaldehyde and formic acid, which blind and kill. The treatment is ethanol: a competing [substrate](#def-b1-enzymes-enzyme) with a lower $K_m$, which keeps the [enzyme](#def-b1-enzymes-enzyme) busy while the methanol is excreted unchanged. Statins are competitive [inhibitors](#def-b1-enzymes-inhibitors), resembling the [transition state](#prop-b1-enzymes-whatitdoes), of the [enzyme](#def-b1-enzymes-enzyme) that commits carbon to [cholesterol](https://one-course.com/books/biology/3/en/chapter/9-lipids#def-b1-lipids-amphiphilic); sulfonamides resemble the [substrate](#def-b1-enzymes-enzyme) of a bacterial [enzyme](#def-b1-enzymes-enzyme) that makes folate, which humans do not make and so do not miss.

## 13.4 Regulation

**Definition 13.11 (Allosteric enzymes).**

An *allosteric enzyme* has several subunits and, besides its [active sites](#def-b1-enzymes-enzyme), *regulatory sites* where effectors bind: *activators* shift it toward its active conformation, *[inhibitors](#def-b1-enzymes-inhibitors)* toward the inactive one. Its rate against $[S]$ is sigmoid, not hyperbolic ([cooperativity](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-allostery) among the [active sites](#def-b1-enzymes-enzyme), as for haemoglobin, [Chapter 12](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#ch-b1-proteins)), so that a small change of [substrate](#def-b1-enzymes-enzyme) near the steep part changes the rate greatly; effectors shift the curve sideways (changing the [substrate](#def-b1-enzymes-enzyme) concentration needed) or up and down (changing the maximal rate). [Allosteric](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-allostery) [enzymes](#def-b1-enzymes-enzyme) stand at the branch points of metabolism, and the effectors are the pathway’s own products and the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell)’s energy signals ([ATP](https://one-course.com/books/biology/3/en/chapter/8-water-and-small-biomolecules#def-b1-water-small-molecules-atp), ADP, AMP).

![An allosteric enzyme. The sigmoid curve makes the rate sensitive to substrate near the midpoint; an activator shifts it left (more active at a given (S)), an inhibitor right. Near (S) = 3 the rate can swing from a tenth to nine tenths of maximum.](https://one-course.com/images/onecourse/chapters/biology-3/b1-enzymes/fig-0bb8ee6cbc43.svg)

*An [allosteric enzyme](#def-b1-enzymes-allosteric). The sigmoid curve makes the rate sensitive to [substrate](#def-b1-enzymes-enzyme) near the midpoint; an activator shifts it left (more active at a given $[S]$), an [inhibitor](#def-b1-enzymes-inhibitors) right. Near $[S] = 3$ the rate can swing from a tenth to nine tenths of maximum.*

**Proposition 13.12 (Four ways the cell controls an enzyme).**

1. *Feedback inhibition* : the end product of a pathway is an [allosteric](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-allostery) [inhibitor](#def-b1-enzymes-inhibitors) of its first committed [enzyme](#def-b1-enzymes-enzyme) , so that the pathway runs only as fast as the product is used (isoleucine on threonine deaminase; [ATP](https://one-course.com/books/biology/3/en/chapter/8-water-and-small-biomolecules#def-b1-water-small-molecules-atp) on phosphofructokinase, [Chapter 15](https://one-course.com/books/biology/3/en/chapter/15-cellular-respiration-and-fermentation#ch-b1-respiration-fermentation) ). Response time: milliseconds.
2. *Covalent modification* : a kinase attaches a phosphate to a serine, threonine or tyrosine of the [enzyme](#def-b1-enzymes-enzyme) , a phosphatase removes it, and the two forms differ in activity ( [glycogen](https://one-course.com/books/biology/3/en/chapter/10-carbohydrates#def-b1-carbohydrates-polysaccharide) phosphorylase is switched on by phosphorylation, [glycogen](https://one-course.com/books/biology/3/en/chapter/10-carbohydrates#def-b1-carbohydrates-polysaccharide) synthase off, by the same hormonal signal). Seconds to minutes; reversible; amplifiable in cascades.
3. *Proteolytic activation* : some [enzymes](#def-b1-enzymes-enzyme) are made as inactive precursors ( *zymogens* : trypsinogen, pepsinogen, the clotting factors) and switched on by cutting off a peptide, once and for all, where and when they are wanted.
4. *Amount* : the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) makes more or less of the [enzyme](#def-b1-enzymes-enzyme) by controlling its gene ( [Chapter 20](https://one-course.com/books/biology/3/en/chapter/20-control-of-gene-expression#ch-b1-expression-control) ) and degrades it faster or slower. Minutes to hours.

**Example 13.13 (Isoenzymes).**

Lactate dehydrogenase exists in five forms, tetramers of two subunit types in all combinations; the heart’s form has a low $K_m$ for lactate and is inhibited by pyruvate (it oxidises lactate to feed the Krebs cycle), the muscle’s form has a high $V_{\max}$ and tolerates pyruvate (it makes lactate in a sprint). The pattern of forms in the blood reveals which [organ](https://one-course.com/books/biology/3/en/chapter/2-functional-organization-of-a-mammal#def-b1-mammal-organization-organ) has been damaged — a heart attack releases the heart’s.

## 13.5 Temperature and pH

**Proposition 13.14 (Enzymes and their environment).**

The rate of an [enzyme](#def-b1-enzymes-enzyme) reaction roughly doubles for every $10\,{}^{\circ}\mathrm{C}$ ($Q_{10} \approx 2$) as thermal energy carries more molecules over the barrier, until the [enzyme](#def-b1-enzymes-enzyme) begins to unfold ($40\text{ to }60\,{}^{\circ}\mathrm{C}$ for most; $100\,{}^{\circ}\mathrm{C}$ for the [enzymes](#def-b1-enzymes-enzyme) of hot-spring bacteria) and the rate collapses. Each [enzyme](#def-b1-enzymes-enzyme) has a *pH optimum*, where the ionisation of its catalytic residues and of its [substrate](#def-b1-enzymes-enzyme) is right: pepsin, in the stomach, near pH 2; trypsin, in the intestine, near 8; most intracellular [enzymes](#def-b1-enzymes-enzyme) near 7. Away from the optimum the rate falls, and far from it the [protein](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-peptide) denatures. Temperature and pH act on the [protein](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-peptide); the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) keeps both within the range where its thousand [enzymes](#def-b1-enzymes-enzyme) all work.

**Example 13.15 (A fever and a hot spring).**

A fever of $40\,{}^{\circ}\mathrm{C}$ speeds every reaction of the body by a quarter and begins to unfold the most fragile [proteins](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-peptide); at $42\,{}^{\circ}\mathrm{C}$ the brain’s fail. The [DNA](https://one-course.com/books/biology/3/en/chapter/11-nucleotides-and-nucleic-acids#def-b1-nucleic-acids-chain) polymerase of a bacterium from a $75\,{}^{\circ}\mathrm{C}$ spring survives $95\,{}^{\circ}\mathrm{C}$, which is why it can be cycled through the melting of [DNA](https://one-course.com/books/biology/3/en/chapter/11-nucleotides-and-nucleic-acids#def-b1-nucleic-acids-chain) thousands of times and became the [enzyme](#def-b1-enzymes-enzyme) of the polymerase chain reaction (the Year 3 volume): thermostability is a property of the sequence, not of the chemistry catalysed.

## 13.6 Exercises

**Exercise 13.1 ★.**

What does an [enzyme](#def-b1-enzymes-enzyme) change in a reaction, and what does it leave unchanged? Illustrate with the energy profile.

**Solution of Exercise 13.1.**

It lowers the [activation energy](#prop-b1-enzymes-whatitdoes) (the height of the [transition state](#prop-b1-enzymes-whatitdoes)), and so the rate, in both directions; it [leaves](https://one-course.com/books/biology/3/en/chapter/3-functional-organization-of-a-flowering-plant#def-b1-flowering-plant-organization-organs) $\Delta G$, the equilibrium constant and the position of equilibrium unchanged. On the profile the peak is lowered, the two ends are not.

**Exercise 13.2 ★.**

Define $K_m$, $V_{\max}$, $k_{\text{cat}}$ and $k_{\text{cat}}/K_m$, with units.

**Solution of Exercise 13.2.**

$K_m$: [substrate](#def-b1-enzymes-enzyme) concentration at half $V_{\max}$ ($\mathrm{mol}/\mathrm{L}$). $V_{\max}$: rate at saturating [substrate](#def-b1-enzymes-enzyme) ($\mathrm{mol}/\mathrm{s}$, or $\text{µ}\mathrm{mol}/\mathrm{min}$). $k_{\text{cat}}$: [substrate](#def-b1-enzymes-enzyme) molecules converted per second per [enzyme](#def-b1-enzymes-enzyme) at saturation ($\mathrm{s}^{-1}$). $k_{\text{cat}}/K_m$: efficiency, the second-order rate constant at low [substrate](#def-b1-enzymes-enzyme) ($\mathrm{L}\,\mathrm{mol}^{-1}\,\mathrm{s}^{-1}$).

**Exercise 13.3 ★.**

From the Lineweaver–Burk figure of the [inhibitors](#def-b1-enzymes-inhibitors), read $V_{\max}$ and $K_m$ for the uninhibited [enzyme](#def-b1-enzymes-enzyme) and for each [inhibitor](#def-b1-enzymes-inhibitors).

**Solution of Exercise 13.3.**

No [inhibitor](#def-b1-enzymes-inhibitors): intercept 1, so $V_{\max} = 1$; $x$ intercept $-0.5$, so $K_m = 2$. Competitive: $V_{\max} = 1$, $x$ intercept $-0.25$, $K_m = 4$. Non-competitive: intercept 2, $V_{\max} = 0.5$; $x$ intercept $-0.5$, $K_m = 2$.

**Exercise 13.4 ★.**

Name the four ways a [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) regulates an [enzyme](#def-b1-enzymes-enzyme)’s activity and give the timescale of each.

**Solution of Exercise 13.4.**

[Allosteric](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-allostery) (feedback) regulation, milliseconds; [covalent modification](#prop-b1-enzymes-control) (phosphorylation), seconds to minutes; proteolytic activation of a zymogen, once, on demand; control of the amount by gene expression and degradation, minutes to hours.

**Exercise 13.5 ★★.**

An [enzyme](#def-b1-enzymes-enzyme) has $K_m = 0.1\,\mathrm{mmol}/\mathrm{L}$ and $V_{\max} =
50\,\text{µ}\mathrm{mol}/\mathrm{min}$. Compute $v_0$ at $[S] = 0.02$, $0.1$, $1$ and $10\,\mathrm{mmol}/\mathrm{L}$. At what $[S]$ is $v_0 = 0.9\,V_{\max}$?

**Solution of Exercise 13.5.**

$v_0 = 50[S]/(0.1 + [S])$: 8.3, 25, 45.5 and $49.5\,\text{µ}\mathrm{mol}/\mathrm{min}$. $0.9 = [S]/(0.1 + [S])$ gives $[S] = 0.9\,K_m/0.1 = 0.9\,\mathrm{mmol}/\mathrm{L}$.

**Exercise 13.6 ★★.**

Lowering $E_a$ by $20\,\mathrm{kJ}/\mathrm{mol}$ at $37\,{}^{\circ}\mathrm{C}$ multiplies the rate by what factor? By how much must $E_a$ fall to gain a factor $10^{10}$?

**Solution of Exercise 13.6.**

$e^{20\,000/2580} = e^{7.75} = 2300$. For $10^{10}$: $\Delta E_a =
RT\ln 10^{10} = 2.58\times 23.0 = 59\,\mathrm{kJ}/\mathrm{mol}$.

**Exercise 13.7 ★★.**

$2\,\mathrm{pmol}$ of carbonic anhydrase in $1\,\mathrm{mL}$ gives $V_{\max} = 120\,\mathrm{mmol}/\mathrm{L}$ per minute. Compute $k_{\text{cat}}$ and, with $K_m = 12\,\mathrm{mmol}/\mathrm{L}$, the efficiency. Compare with the diffusion limit.

**Solution of Exercise 13.7.**

$V_{\max} = 0.12\,\mathrm{mol}/\mathrm{L}$ per minute in $1\,\mathrm{mL}$: $1.2 \times 10^{-4}\,\mathrm{mol}/\mathrm{min}$ $= 2.0 \times 10^{-6}\,\mathrm{mol}/\mathrm{s}$ of product; [enzyme](#def-b1-enzymes-enzyme) $2 \times 10^{-12}\,\mathrm{mol}$: $k_{\text{cat}} = 2\times 10^{-6}/2\times 10^{-12} =
1 \times 10^{6}\,\mathrm{s}^{-1}$. Efficiency $10^6/0.012 = 8 \times 10^{7}\,\mathrm{L}\,\mathrm{mol}^{-1}\,\mathrm{s}^{-1}$, within a factor of ten of the diffusion limit: nearly every encounter with a [substrate](#def-b1-enzymes-enzyme) molecule is productive.

**Exercise 13.8 ★★.**

A competitive [inhibitor](#def-b1-enzymes-inhibitors) at $2\,\mathrm{mmol}/\mathrm{L}$ doubles the apparent $K_m$. Compute $K_i$. What [substrate](#def-b1-enzymes-enzyme) concentration restores the rate to $90\,\%$ of $V_{\max}$ with and without the [inhibitor](#def-b1-enzymes-inhibitors), if $K_m = 1\,\mathrm{mmol}/\mathrm{L}$?

**Solution of Exercise 13.8.**

$1 + 2/K_i = 2$: $K_i = 2\,\mathrm{mmol}/\mathrm{L}$. For $90\,\%$: $[S] = 9K_m^{\text{app}}$: $9\,\mathrm{mmol}/\mathrm{L}$ without, $18\,\mathrm{mmol}/\mathrm{L}$ with the [inhibitor](#def-b1-enzymes-inhibitors).

**Exercise 13.9 ★★.**

Explain why [feedback inhibition](#prop-b1-enzymes-control) acts on the first committed step of a pathway rather than the last, and why the [inhibitor](#def-b1-enzymes-inhibitors) is usually the end product rather than an intermediate.

**Solution of Exercise 13.9.**

Inhibiting the first committed step stops the whole pathway at once and wastes no intermediates, which would otherwise accumulate; the end product is the signal that matters — its abundance is exactly the information that the pathway is no longer needed — whereas an intermediate’s level says nothing about demand.

**Exercise 13.10 ★★★.**

Trypsinogen is activated in the intestine, not in the pancreas that makes it; a small fraction activated early destroys the pancreas. Explain the logic of zymogens, the role of the [enzyme](#def-b1-enzymes-enzyme) that activates trypsinogen, and why the pancreas also makes a trypsin [inhibitor](#def-b1-enzymes-inhibitors).

**Solution of Exercise 13.10.**

A protease active in the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) that makes it would digest that [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell); made as an inactive zymogen, it is harmless until enteropeptidase, an [enzyme](#def-b1-enzymes-enzyme) of the intestinal lining, cuts trypsinogen to trypsin in the duodenum, where digestion is wanted; trypsin then activates the other zymogens (a cascade). Because a trace of trypsin can start the cascade prematurely, the pancreas packs a specific trypsin [inhibitor](#def-b1-enzymes-inhibitors) into its granules as insurance; failure of the arrangement is pancreatitis.

**Exercise 13.11 ★★★.**

An [allosteric enzyme](#def-b1-enzymes-allosteric) with $n = 3$ and half-saturation at $[S]_{0.5}
= 3$ has its curve shifted to $[S]_{0.5} = 5.5$ by an [inhibitor](#def-b1-enzymes-inhibitors). Compute the rate at $[S] = 3$ with and without the [inhibitor](#def-b1-enzymes-inhibitors), and compare with the change a competitive [inhibitor](#def-b1-enzymes-inhibitors) of the same $K_m$-shift would produce on a Michaelis–Menten [enzyme](#def-b1-enzymes-enzyme). What does [cooperativity](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-allostery) add to regulation?

**Solution of Exercise 13.11.**

Without: $3^3/(3^3 + 3^3) = 0.50$. With: $27/(166 + 27) = 0.14$: the rate falls to $28\,\%$ of its value. A Michaelis–Menten [enzyme](#def-b1-enzymes-enzyme) with $K_m$ raised from 3 to 5.5: $3/6 = 0.50$ to $3/8.5 = 0.35$, i.e. $70\,\%$. [Cooperativity](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-allostery) makes the same shift of the curve more than twice as effective near the working point: a switch rather than a dimmer.

**Exercise 13.12 ★★★.**

“An [enzyme](#def-b1-enzymes-enzyme) is a catalyst that has been told what to do.” Discuss in a paragraph: catalysis versus specificity, regulation as the added information, and the cost of a regulated [enzyme](#def-b1-enzymes-enzyme) compared with a bare catalyst.

**Solution of Exercise 13.12.**

A bare catalyst (a platinum surface, an acid) speeds many reactions indiscriminately; an [enzyme](#def-b1-enzymes-enzyme) speeds one reaction on one [substrate](#def-b1-enzymes-enzyme) — the specificity of its [active site](#def-b1-enzymes-enzyme) is already information about what the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) wants done. Regulation adds a second layer: [allosteric](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-allostery) sites, phosphorylation sites and zymogen peptides tell the [enzyme](#def-b1-enzymes-enzyme) when and where to act, in response to the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell)’s state. The cost is a large [protein](https://one-course.com/books/biology/3/en/chapter/12-amino-acids-and-proteins#def-b1-proteins-peptide) (hundreds of residues for a site of a dozen), slower turnover than the best inorganic catalysts, and the machinery to make, modify and destroy it; the benefit is a chemistry that runs only where and when it is useful, which is what a metabolism is.

## 13.7 Problem: An Enzyme Measured

**Problem 13.1.**

Weekend problem — an esterase assayed at six [substrate](#def-b1-enzymes-enzyme) concentrations, alone and with two [inhibitors](#def-b1-enzymes-inhibitors), its constants extracted and its behaviour in the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) predicted, ending on its $K_m$, $V_{\max}$ and $K_i$

An esterase is assayed in $1.0\,\mathrm{mL}$ at $25\,{}^{\circ}\mathrm{C}$, pH 7.5, with $10\,\mathrm{nmol}/\mathrm{L}$ of [enzyme](#def-b1-enzymes-enzyme). The initial rates ($\text{µ}\mathrm{mol}/\mathrm{min}$) at six [substrate](#def-b1-enzymes-enzyme) concentrations, alone and in the presence of $1.0\,\mathrm{mmol}/\mathrm{L}$ of [inhibitor](#def-b1-enzymes-inhibitors) I or $1.0\,\mathrm{mmol}/\mathrm{L}$ of [inhibitor](#def-b1-enzymes-inhibitors) J:

| $[S]$ ($\mathrm{mmol}/\mathrm{L}$) | 0.5 | 1 | 2 | 5 | 10 | 20 |
| --- | --- | --- | --- | --- | --- | --- |
| $v_0$, no [inhibitor](#def-b1-enzymes-inhibitors) | 0.200 | 0.333 | 0.500 | 0.714 | 0.833 | 0.909 |
| $v_0$, with I | 0.111 | 0.200 | 0.333 | 0.556 | 0.714 | 0.833 |
| $v_0$, with J | 0.100 | 0.167 | 0.250 | 0.357 | 0.417 | 0.455 |

**Part I — The [enzyme](#def-b1-enzymes-enzyme) alone.**

1. Compute $1/[S]$ and $1/v_0$ for the six points without [inhibitor](#def-b1-enzymes-inhibitors) .
2. Show that the points lie on a straight line and find its slope and intercept.
3. Deduce $V_{\max}$ and $K_m$ .
4. Verify with the [Michaelis–Menten equation](#thm-b1-enzymes-mm) at $[S] =  5\,\mathrm{mmol}/\mathrm{L}$ .
5. Compute the amount of [enzyme](#def-b1-enzymes-enzyme) in the assay (moles) and $k_{\text{cat}}$ in $\mathrm{s}^{-1}$ .
6. Compute $k_{\text{cat}}/K_m$ and compare with the diffusion limit.
7. At what [substrate](#def-b1-enzymes-enzyme) concentration is the [enzyme](#def-b1-enzymes-enzyme) working at $95\,\%$ of $V_{\max}$ ?

**Part II — [Inhibitor](#def-b1-enzymes-inhibitors) I.**

8. Compute $1/v_0$ for the six points with I and find the slope and intercept of their line.
9. Deduce the apparent $V_{\max}$ and $K_m$ with I.
10. Which parameter changed? Classify the [inhibitor](#def-b1-enzymes-inhibitors) .
11. Compute $K_i$ from $K_m^{\text{app}} = K_m(1 + [I]/K_i)$ .
12. At $[S] = 2\,\mathrm{mmol}/\mathrm{L}$ , what concentration of I halves the rate? At $[S] = 20\,\mathrm{mmol}/\mathrm{L}$ ?
13. Propose what I might be, structurally, and where it binds.

**Part III — [Inhibitor](#def-b1-enzymes-inhibitors) J.**

14. Compute the slope and intercept of the line with J and deduce the apparent $V_{\max}$ and $K_m$ .
15. Classify J and compute its $K_i$ from $V_{\max}^{\text{app}} =  V_{\max}/(1 + [J]/K_i)$ .
16. Explain why raising the [substrate](#def-b1-enzymes-enzyme) cannot overcome J.
17. Doubling the [enzyme](#def-b1-enzymes-enzyme) concentration in the presence of J does what to the rate? And in the presence of I?

**Part IV — In the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell).** The [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) holds the [substrate](#def-b1-enzymes-enzyme) at $0.5\,\mathrm{mmol}/\mathrm{L}$ and the [enzyme](#def-b1-enzymes-enzyme) at $10\,\mathrm{nmol}/\mathrm{L}$ in a volume of $1000\,\text{µ}\mathrm{m}^{3}$, at $37\,{}^{\circ}\mathrm{C}$, with $Q_{10} = 2$.

18. Compute the rate in the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) (molecules of product per second in the whole [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) ).
19. The product is needed at $3 \times 10^{6}\,$ molecules per second. By what factor must the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) raise the rate, and name two ways it could.
20. The product is a competitive [inhibitor](#def-b1-enzymes-inhibitors) of the [enzyme](#def-b1-enzymes-enzyme) with $K_i = 0.2\,\mathrm{mmol}/\mathrm{L}$ , and accumulates to $0.4\,\mathrm{mmol}/\mathrm{L}$ . Compute the rate then, and comment on what this achieves.
21. The [enzyme](#def-b1-enzymes-enzyme) is phosphorylated by a kinase, which lowers its $K_m$ to $0.4\,\mathrm{mmol}/\mathrm{L}$ . Compute the rate at $0.5\,\mathrm{mmol}/\mathrm{L}$ of [substrate](#def-b1-enzymes-enzyme) before and after.
22. The [enzyme](#def-b1-enzymes-enzyme) ’s pH optimum is 7.5 and its activity halves at pH 6.5. A [lysosome](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#def-b1-eukaryotic-cell-endomembrane) is at pH 5. Predict, qualitatively, its activity there and explain in terms of ionisable residues.
23. A mutation replaces the histidine of the [active site](#def-b1-enzymes-enzyme) by alanine. Predict the effect on $K_m$ and on $k_{\text{cat}}$ , with a reason for each.
24. Explain why the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) keeps the [substrate](#def-b1-enzymes-enzyme) near $K_m$ rather than far above it.
25. State the result: $K_m$ , $V_{\max}$ , $k_{\text{cat}}$ of the esterase, the type and $K_i$ of each [inhibitor](#def-b1-enzymes-inhibitors) .

**Solution of Problem 13.1.**

**1.** $1/[S]$: 2, 1, 0.5, 0.2, 0.1, 0.05. $1/v_0$: 5.0, 3.0, 2.0, 1.4, 1.2, 1.1. **2.** Each step of $1/[S]$ changes $1/v_0$ in proportion: slope $(5.0 - 1.1)/(2 - 0.05) = 2.0$; intercept $1.0$. **3.** $V_{\max} = 1/1.0 = 1.0\,\text{µ}\mathrm{mol}/\mathrm{min}$; $K_m =
\text{slope}\times V_{\max} = 2.0\,\mathrm{mmol}/\mathrm{L}$. **4.** $1.0\times 5/(2 + 5) = 0.714$: as measured. **5.** $10 \times 10^{-9}\,\mathrm{mol}/\mathrm{L}\times1 \times 10^{-3}\,\mathrm{L} = 1 \times 10^{-11}\,\mathrm{mol}$; $V_{\max} = 1 \times 10^{-6}\,\mathrm{mol}/\mathrm{min} = 1.67 \times 10^{-8}\,\mathrm{mol}/\mathrm{s}$; $k_{\text{cat}} = 1.67\times 10^{-8}/10^{-11} = 1670\,\mathrm{s}^{-1}$. **6.** $1670/(2\times 10^{-3}) = 8.3 \times 10^{5}\,\mathrm{L}\,\mathrm{mol}^{-1}\,\mathrm{s}^{-1}$: a thousand times below the diffusion limit; only one collision in a thousand is productive. **7.** $0.95 = [S]/(2 + [S])$: $[S] = 19\,K_m = 38\,\mathrm{mmol}/\mathrm{L}$. **8.** $1/v_0$: 9.0, 5.0, 3.0, 1.8, 1.4, 1.2. Slope $(9.0 -
1.2)/1.95 = 4.0$; intercept $1.0$. **9.** $V_{\max}^{\text{app}} = 1.0\,\text{µ}\mathrm{mol}/\mathrm{min}$ (unchanged); $K_m^{\text{app}} = 4.0\times 1.0 = 4.0\,\mathrm{mmol}/\mathrm{L}$. **10.** $K_m$ doubled, $V_{\max}$ unchanged: competitive. **11.** $4 = 2(1 + 1/K_i)$: $K_i = 1.0\,\mathrm{mmol}/\mathrm{L}$. **12.** With the [inhibitor](#def-b1-enzymes-inhibitors), $v = V_{\max}[S]/(K_m(1 + [I]/K_i) +
[S])$; halving the rate requires $K_m(1 + [I]/K_i) + [S] = 2(K_m +
[S])$, i.e. $[I] = K_i(K_m + [S])/K_m$: at $2\,\mathrm{mmol}/\mathrm{L}$, $[I] =
1\times 4/2 = 2\,\mathrm{mmol}/\mathrm{L}$; at $20\,\mathrm{mmol}/\mathrm{L}$, $[I] = 22/2 =
11\,\mathrm{mmol}/\mathrm{L}$. The more [substrate](#def-b1-enzymes-enzyme), the more [inhibitor](#def-b1-enzymes-inhibitors) it takes. **13.** A molecule resembling the [substrate](#def-b1-enzymes-enzyme) (or its [transition state](#prop-b1-enzymes-whatitdoes)) — an ester analogue that cannot be hydrolysed — binding in the [active site](#def-b1-enzymes-enzyme). **14.** $1/v_0$: 10.0, 6.0, 4.0, 2.8, 2.4, 2.2; slope $4.0$, intercept $2.0$: $V_{\max}^{\text{app}} = 0.5\,\text{µ}\mathrm{mol}/\mathrm{min}$, $K_m^{\text{app}} = 4.0\times 0.5 = 2.0\,\mathrm{mmol}/\mathrm{L}$. **15.** $V_{\max}$ halved, $K_m$ unchanged: non-competitive; $2 = 1 + 1/K_i$: $K_i = 1.0\,\mathrm{mmol}/\mathrm{L}$. **16.** J binds at a site other than the [active site](#def-b1-enzymes-enzyme), on both free [enzyme](#def-b1-enzymes-enzyme) and $ES$, with the same affinity: [substrate](#def-b1-enzymes-enzyme) does not compete with it, and at any $[S]$ half the [enzyme](#def-b1-enzymes-enzyme) molecules are inactivated. **17.** With J the rate doubles (half of twice as much [enzyme](#def-b1-enzymes-enzyme) is still active); with I it also doubles (the rate is proportional to [enzyme](#def-b1-enzymes-enzyme) at every $[S]$): doubling the [enzyme](#def-b1-enzymes-enzyme) never tells the [inhibitors](#def-b1-enzymes-inhibitors) apart. **18.** Rate at $25\,{}^{\circ}\mathrm{C}$: $V_{\max}[S]/(K_m + [S])$ with $V_{\max} = k_{\text{cat}}[E]$: per litre, $1670\times 10^{-8} =
1.67 \times 10^{-5}\,\mathrm{mol}/\mathrm{L}/\mathrm{s}$; $\times 0.5/2.5 = 3.3 \times 10^{-6}\,\mathrm{mol}/\mathrm{L}/\mathrm{s}$; at $37\,{}^{\circ}\mathrm{C}$, $\times 2^{1.2} = 2.3$: $7.7 \times 10^{-6}\,\mathrm{mol}/\mathrm{L}/\mathrm{s}$; in $1 \times 10^{-12}\,\mathrm{L}$: $7.7 \times 10^{-18}\,\mathrm{mol}/\mathrm{s}$ $= 4.6 \times 10^{6}$ molecules per second. **19.** No rise needed: the rate ($4.6 \times 10^{6}$) exceeds the demand ($3 \times 10^{6}$); if it had to rise, the [cell](https://one-course.com/books/biology/3/en/chapter/5-the-cell-unit-of-life#def-b1-cell-unit-of-life-cell) could raise the [substrate](#def-b1-enzymes-enzyme), or phosphorylate the [enzyme](#def-b1-enzymes-enzyme) to lower its $K_m$, or make more [enzyme](#def-b1-enzymes-enzyme). **20.** $K_m^{\text{app}} = 2(1 + 0.4/0.2) = 6\,\mathrm{mmol}/\mathrm{L}$: rate $\times 0.5/6.5$ instead of $0.5/2.5$: $38\,\%$ of before, $1.8 \times 10^{6}$ per second — the product throttles its own synthesis to below the demand, so it is consumed and its level falls, which releases the [enzyme](#def-b1-enzymes-enzyme): a self-adjusting supply. **21.** Before: $0.5/2.5 = 0.20$ of $V_{\max}$; after: $0.5/0.9 =
0.56$: nearly threefold. **22.** Near zero: at pH 5 the histidine of the [active site](#def-b1-enzymes-enzyme) is protonated and cannot act as a base, and acidic residues of the site are neutralised; the [enzyme](#def-b1-enzymes-enzyme)’s ionisation state, and perhaps its fold, are wrong. It is built for the [cytosol](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#def-b1-eukaryotic-cell-organelle), not the [lysosome](https://one-course.com/books/biology/3/en/chapter/6-functional-organization-of-the-eukaryotic-cell#def-b1-eukaryotic-cell-endomembrane). **23.** $K_m$ changes little (binding is mostly by the rest of the site); $k_{\text{cat}}$ collapses by orders of magnitude, since the histidine was the catalytic base that activated the water or serine. **24.** Near $K_m$ the rate responds to changes of [substrate](#def-b1-enzymes-enzyme) (slope near $V_{\max}/2K_m$); far above it the [enzyme](#def-b1-enzymes-enzyme) is saturated and insensitive, and the excess [substrate](#def-b1-enzymes-enzyme) would be an osmotic and chemical burden. Working near $K_m$ makes the pathway controllable by supply. **25.** $K_m = 2.0\,\mathrm{mmol}/\mathrm{L}$, $V_{\max} = 1.0\,\text{µ}\mathrm{mol}/\mathrm{min}$ ($10\,\mathrm{nmol}/\mathrm{L}$ [enzyme](#def-b1-enzymes-enzyme)), $k_{\text{cat}} = 1670\,\mathrm{s}^{-1}$; I competitive, $K_i = 1.0\,\mathrm{mmol}/\mathrm{L}$; J non-competitive, $K_i =
1.0\,\mathrm{mmol}/\mathrm{L}$.