Practice: Molecules of life

1. Atoms and bonds

RememberEach atom makes a set number of covalent bonds: H 1, O 2, N 3, C 4, S 2, and P 5 in a phosphate group. A double bond counts as 2. In a formula such as $\mathrm{C_6H_{12}O_6}$ , the number after each symbol counts its atoms, and no number means 1.

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How many covalent bonds does a sulfur atom usually make? Count a double bond as 2.

Glycine has the formula $\mathrm{C_{2}H_{5}NO_{2}}$ . How many nitrogen atoms are in one molecule?

Sulfur makes 2 bonds and hydrogen makes 1. A sulfur atom is joined to one carbon atom by a single bond. Its other bond goes to hydrogen. How many hydrogen atoms are attached to the sulfur?

2. Formulas and molecular mass

RememberMolecular mass = the sum of (number of atoms × atomic mass). Atomic masses in daltons: H 1.008, C 12.011, N 14.007, O 15.999, P 30.974, S 32.06. A mass of M Da means M grams per mole, and a mole is $6.022 \times 10^{23}$ molecules.

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What is the molecular mass of ATP, $\mathrm{C_{10}H_{16}N_{5}O_{13}P_{3}}$ ? Use these atomic masses, in daltons (Da): C 12.011, H 1.008, N 14.007, O 15.999, P 30.974.

Glycine, $\mathrm{C_{2}H_{5}NO_{2}}$ , has a molecular mass of 75.067 Da. Nitrogen has an atomic mass of 14.007 Da. What percent of the mass of glycine is nitrogen?

One milliliter of water weighs about 1 g. Water has a molecular mass of 18.015 Da, so a mole of it weighs 18.015 g. How many water molecules is that? One mole is $6.022 \times 10^{23}$ molecules.

A liter of pure water weighs 1,000 g. Water, $\mathrm{H_2O}$ , is made of hydrogen (1.008 Da) and oxygen (15.999 Da). What is the concentration of water, in moles per liter (M)?

A yeast cell has a volume of about 40 µm³ and about the density of water, so it weighs about 40 pg (picograms). About 70% of a cell's mass is water, and water has a molecular mass of 18.015 Da. About how many water molecules are in the cell? One mole is $6.022 \times 10^{23}$ molecules.

3. Energy and kT

Remember $kT = R \times T$ , with R = 0.008314 kJ/(mol·K) and T in kelvin (°C + 273.15). At 37 °C, kT is about 2.58 kJ/mol. A state that costs E more energy is rarer by the Boltzmann factor $e^{-E/kT}$ , and $e^{-x} = 10^{-x/2.303}$ .

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Hydrogen bonds take about 5 to 20 kJ/mol to break. Suppose one takes 20 kJ/mol. At 37 °C, kT is 2.58 kJ/mol. How many kT is that?

A change in a molecule's shape costs 5 kJ/mol of energy. At 37 °C, kT is 2.58 kJ/mol. What is the Boltzmann factor, $e^{-E/kT}$ , for this change?

Two shapes of a molecule differ in energy by 10 kJ/mol. At 37 °C, kT is 2.58 kJ/mol. How many times more likely is the lower-energy shape? Use ratio $= e^{\Delta E / kT}$ , where $\Delta E$ is the energy gap.

Breaking a bond costs 10 kJ/mol. kT is 2.58 kJ/mol at 37 °C and 3.06 kJ/mol at 95 °C. How many times bigger is the bond's Boltzmann factor, $e^{-E/kT}$ , at 95 °C than at 37 °C?

4. DNA and RNA

RememberA pairs with T (with U in RNA), and G pairs with C. A–T pairs have 2 hydrogen bonds and G–C pairs have 3. The two strands run opposite ways, so to write a partner strand 5′ to 3′, pair each letter and then read backward. Each base pair adds 0.34 nm, and the helix turns once every 10.5 base pairs.

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A DNA strand reads $5'\text{-}\texttt{TACGATA}\text{-}3'$ . Write the RNA strand that pairs with it, letter by letter from left to right. In RNA, U takes the place of T: DNA's A pairs with U, T with A, G with C, and C with G.

What percent of the bases in $5'\text{-}\texttt{TTAGTCACATTACCTATCTG}\text{-}3'$ are G or C? This is the strand's G–C content.

An RNA strand, $5'\text{-}\texttt{CAGUUACGUUU}\text{-}3'$ , was made by pairing with a DNA strand. Write that DNA strand from 5′ to 3′. RNA's A pairs with DNA's T, U pairs with A, G with C, and C with G.

A short piece of double-stranded DNA has this sequence on one strand: $5'\text{-}\texttt{ATAAGCAGTACGTA}\text{-}3'$ . A–T pairs are held together by 2 hydrogen bonds, and G–C pairs by 3. How many hydrogen bonds hold the two strands together?

In a sample of double-stranded DNA, 23% of the bases are T. What percent are G or C (the G–C content)?

E. coli's genome is 4,641,652 base pairs long. Each base pair adds 0.34 nm to the length of the double helix. How long is it, stretched out straight, in micrometers?

E. coli's genome is 4,641,652 base pairs long, and each base pair adds 0.34 nm of length. An E. coli cell is about 2 µm long. How many times longer than the cell is its DNA, stretched out straight?

Each DNA base is one of 4 letters, so it carries 2 bits of information. A gene is 1,500 base pairs long. How many bytes does it take to store its sequence? Use 8 bits = 1 byte.

5. Counting possibilities

RememberWith 4 choices at each of $n$ positions there are $4^n$ DNA sequences, and with 20 amino acids there are $20^n$ protein chains. As powers of ten: $4^n = 10^{0.602 n}$ and $20^n = 10^{1.301 n}$ .

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How many different DNA sequences are exactly 2 bases long? Each position can be A, C, G, or T.

How many different DNA sequences are 50 bases long? The answer is $4^{50}$ . Write it as a power of ten, $10^{?}$ , using $\log_{10}(4) = 0.602$ , and type just the exponent.

Fill in the exponent: $4^{10} = 2^{?}$ .

What is the shortest length $n$ for which there are more possible DNA sequences of length $n$ , $4^n$ , than Avogadro's number, $6.022 \times 10^{23}$ ? Use $\log_{10}(4) = 0.602$ .

6. Proteins and membranes

RememberAn amino acid in a protein weighs about 110 Da. A sphere has surface area $4\pi r^2$ . A lipid bilayer is about 4 to 5 nm thick and holds about 5 million lipids per µm², counting both layers (Alberts, Molecular Biology of the Cell).

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A protein is 150 amino acids long. At about 110 Da per amino acid, what is its mass in kDa (kilodaltons)?

A lipid bilayer is about 5 nm thick. An E. coli cell is about 1 µm wide. How many membrane thicknesses would fit across the width of the cell?

Treat a human cell as a sphere 20 µm across. What is the surface area of its membrane, in µm²? Use $A = 4\pi r^2$ .

A lipid bilayer holds about 5 million lipid molecules per µm², counting both layers. A round bacterium is 1 µm across. About how many lipids are in its membrane? Use $A = 4\pi r^2$ .

Mixed review

8 problems from across the lesson, in random order. Here the topic is hidden until you solve each one, so part of the job is working out which tool you need.

Fill in the exponent: $4^{12} = 2^{?}$ .

How many different DNA sequences are 50 bases long? The answer is $4^{50}$ . Write it as a power of ten, $10^{?}$ , using $\log_{10}(4) = 0.602$ , and type just the exponent.

A short piece of double-stranded DNA has this sequence on one strand: $5'\text{-}\texttt{TGGATCAGGTCGG}\text{-}3'$ . A–T pairs are held together by 2 hydrogen bonds, and G–C pairs by 3. How many hydrogen bonds hold the two strands together?

Adenine, $\mathrm{C_{5}H_{5}N_{5}}$ , has a molecular mass of 135.13 Da. Nitrogen has an atomic mass of 14.007 Da. What percent of the mass of adenine is nitrogen?

The DNA double helix makes one full turn every 10.5 base pairs. How many turns are there in a stretch of DNA of 1,050 base pairs?

Breaking a bond costs 50 kJ/mol. kT is 2.58 kJ/mol at 37 °C and 3.06 kJ/mol at 95 °C. How many times bigger is the bond's Boltzmann factor, $e^{-E/kT}$ , at 95 °C than at 37 °C?

What is the shortest length $n$ for which there are more possible DNA sequences of length $n$ , $4^n$ , than a billion, $10^9$ ? Use $\log_{10}(4) = 0.602$ .

A syn3A cell has a volume of about 0.0335 µm³ and about the density of water, so it weighs about 0.0335 pg (picograms). About 70% of a cell's mass is water, and water has a molecular mass of 18.015 Da. About how many water molecules are in the cell? One mole is $6.022 \times 10^{23}$ molecules.