Practice: The cell by the numbers

1. Powers of ten

RememberTo multiply, add the exponents. To divide, subtract them. A negative exponent makes a number small, not negative: $10^{-3} = 0.001$ .

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Write 150,000,000 in scientific notation. It is $1.5 \times 10^{?}$ . What is the exponent?

Write $1.2 \times 10^{6}$ as an ordinary number.

What is $(4 \times 10^{3}) \times (8 \times 10^{2})$ ?

What is $\dfrac{6 \times 10^{3}}{4 \times 10^{10}}$ ?

What is $10^{-9} \times 10^{-8} \times 10^{19}$ ? The answer is a power of ten: type just its exponent.

2. Metric prefixes and units

RememberEach prefix is 1,000 times smaller than the one before: milli $10^{-3}$ , micro $10^{-6}$ , nano $10^{-9}$ , pico $10^{-12}$ , femto $10^{-15}$ . Going to a smaller unit makes the number bigger.

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An E. coli cell is about 2 µm long. How many nanometers is that?

Convert $1.5\ \text{g}$ to µg.

A human hair is about 80 µm thick. An E. coli cell is about 2 µm long. How many E. coli cells, end to end, would span the width of the hair?

3. Logarithms and orders of magnitude

Remember $\log_{10}(x)$ answers "10 to what power gives $x$ ?" Multiplying numbers adds their logs. Worth remembering: $\log_{10}(3) \approx 0.5$ .

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What is $\log_{10}(100{,}000{,}000)$ ?

Using $\log_{10}(3) \approx 0.5$ , estimate $\log_{10}(3 \times 10^{-8})$ .

Suppose $\log_{10}(a) = 4.6$ and $\log_{10}(b) = 0.6$ . What is $\log_{10}(a \times b)$ ?

A human cell is about 20 µm across. A hydrogen atom is about 0.1 nm across. About how many orders of magnitude apart are they? Round to the nearest whole number.

4. Volumes

RememberCube: $V = s^3$ . Sphere: $V = \frac{4}{3}\pi r^3 \approx 4.19\,r^3$ , where the radius $r$ is half the diameter. To convert: $1\ \mu\text{m}^3 = 10^{-15}\ \text{L}$ .

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A cube is $10\ \mu\text{m}$ on each side. What is its volume?

Cell B has the same shape as cell A and 8 times the volume. How many times wider is it?

A cell has a volume of $600\ \mu\text{m}^3$ . What is that in liters?

A human cell is about 20 µm across. Treat it as a sphere. What is its volume?

A spherical cell is $4\ \mu\text{m}$ across. What is its volume in liters?

An influenza virus is about 100 nm across. Treat it as a sphere. What is its volume in µm³?

5. Counting molecules

Remember $N = C \times V \times N_A$ , with $C$ in mol/L, $V$ in liters, and $N_A = 6.022 \times 10^{23}$ . Going the other way: $C = N / (N_A \times V)$ .

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How many molecules are in $1.5 \times 10^{-9}$ moles of a substance?

In syn3A, 1 µM is about 20 molecules. Roughly how many copies of a protein at 2 µM are in one syn3A cell?

A small molecule is at $2\ \text{mM}$ in a human cell, which has a volume of $3{,}000\ \mu\text{m}^3$ . How many molecules of it are in the cell?

A human cell, with a volume of $3{,}000\ \mu\text{m}^3$ , contains $10^{5}$ copies of a protein. What is the protein's concentration in nM?

What concentration does a single molecule make in a yeast cell, which has a volume of $40\ \mu\text{m}^3$ ? Give it in nM.

A protein is at the same concentration in a human cell ( $3{,}000\ \mu\text{m}^3$ ) and in syn3A ( $0.0335\ \mu\text{m}^3$ ). How many times more copies of it are in a human cell?

6. Masses and estimates

RememberCells have about the density of water, 1 g/mL. About 30% of a cell is dry mass, and 55% of that is protein. An amino acid is about 110 Da, and $M$ Da means $M$ grams per mole.

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A cell has a volume of $0.0335\ \mu\text{m}^3$ and about the density of water, 1 g/mL. What is its mass in grams?

A protein has 150 amino acids, each weighing about 110 Da (daltons). What is its mass in daltons?

A protein has 200 amino acids of about 110 Da each. What does one molecule of it weigh, in grams?

Estimate how many proteins are in a bacterium, which has a volume of $0.5\ \mu\text{m}^3$ . Use the lesson's rough numbers: the density of water (1 g/mL), 70% water, protein is 55% of the dry mass, and a typical protein is 400 amino acids of 110 Da each.

7. Rates and times

RememberTime = amount of work ÷ rate. Track the units: amino acids ÷ (amino acids per second) = seconds. There are 60 s in a minute and 3,600 s in an hour.

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A ribosome adds 20 amino acids per second. How many seconds does it take to make a protein 1,200 amino acids long?

Copying E. coli's genome, $4.6 \times 10^{6}$ base pairs, starts at one point and runs in both directions. Suppose each of the two replication forks moves at 800 base pairs per second. How many minutes does the copying take?

A ribosome adds 10 amino acids per second. How many proteins of 300 amino acids can one ribosome make in 105 minutes?

A cell like E. coli must make $3 \times 10^{6}$ proteins, each 300 amino acids long, in the 20 minutes before it divides. One ribosome adds 20 amino acids per second. At least how many ribosomes does the cell need?

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.

What is $\dfrac{10^{-9}}{10^{-2}}$ ? The answer is a power of ten: type just its exponent.

Copying E. coli's genome, $4.6 \times 10^{6}$ base pairs, starts at one point and runs in both directions. Suppose each of the two replication forks moves at 800 base pairs per second. How many minutes does the copying take?

A protein has 1000 amino acids of about 110 Da each. What does one molecule of it weigh, in grams?

An HIV particle is about 120 nm across. Treat it as a sphere. What is its volume in µm³?

What is $\dfrac{2 \times 10^{-1}}{5 \times 10^{-3}}$ ?

A ribosome adds 15 amino acids per second. How many proteins of 500 amino acids can one ribosome make in 20 minutes?

Using $\log_{10}(3) \approx 0.5$ , estimate $\log_{10}(3 \times 10^{1})$ .

A small round bacterium is 0.3 µm across. Treat it as a sphere. What is its volume?