---
title: "Cells: The Common Unit of Life"
book: "High School Biology"
subject: biology
language: en
chapter: 2
exercises: 15
source: https://one-course.com/books/biology/2/en/chapter/2-cells-the-common-unit-of-life
---

# Chapter 2 — Cells: The Common Unit of Life

Scrape the inside of your cheek with a cotton bud, smear it on a slide, add a drop of blue dye, and look through a microscope at four hundred times: a scatter of flat, irregular tiles, each with a darker spot near its centre. Peel the thin skin from inside an onion scale and look again: neat rectangular bricks, each with its own spot. Pond water shows oval creatures rowing past with a fringe of hairs. The three samples have nothing in common to the naked eye. Under the lens they are built from the same object — the [cell](#def-g10-cells-common-unit-cell) — and this chapter is about what that object is, how it is seen, and why every living thing is made of it.

## 2.1 The cell theory

**Definition 2.1 (Cell).**

A *cell* is the smallest unit of a living thing that is itself alive: a volume of watery content, the *cytoplasm*, enclosed by a thin *plasma membrane* that separates it from its surroundings while letting selected substances cross. A cell takes in matter and energy, transforms them, grows, and divides into two cells.

**Proposition 2.2 (The cell theory).**

Three statements, established by microscopy in the nineteenth century, form the [cell](#def-g10-cells-common-unit-cell) theory:

1. every living thing is made of one or more [cells](#def-g10-cells-common-unit-cell) ;
2. the [cell](#def-g10-cells-common-unit-cell) is the basic unit of structure and function of living things — nothing smaller is alive on its own;
3. every [cell](#def-g10-cells-common-unit-cell) comes from a pre-existing [cell](#def-g10-cells-common-unit-cell) , by division.

**Evidence.** Statement 1 rests on the systematic examination of plants (Schleiden, 1838) and animals (Schwann, 1839) with improved microscopes: every tissue examined, however different in appearance, resolved into [cells](#def-g10-cells-common-unit-cell). Statement 3 was established by Virchow (1855) and by the observation of dividing [cells](#def-g10-cells-common-unit-cell) in growing tissues; the alternative — [cells](#def-g10-cells-common-unit-cell) forming spontaneously from fluid — was never observed, and Pasteur’s experiments on microbes ([Chapter 18](https://one-course.com/books/biology/2/en/chapter/18-bacteria-and-antibiotic-resistance#ch-g11-antibiotic-resistance) returns to them) closed the question. Statement 2 follows from the first and from the fact that isolated [cells](#def-g10-cells-common-unit-cell), kept in a suitable liquid, go on living while any fragment of a [cell](#def-g10-cells-common-unit-cell) does not. ∎

**Example 2.3 (One cell or many).**

A yeast, a paramecium or a bacterium is a single [cell](#def-g10-cells-common-unit-cell) that does everything — feeds, moves, divides. An oak or a human is a cooperative of [cells](#def-g10-cells-common-unit-cell) of many kinds, each doing a share of the work: a human body contains some $3 \times 10^{13}$ [cells](#def-g10-cells-common-unit-cell), of roughly two hundred kinds. Between the two lie colonies and simple multicellular organisms, but there is no living thing built of anything other than [cells](#def-g10-cells-common-unit-cell).

## 2.2 Seeing cells: scales and microscopes

**Definition 2.4 (Magnification and resolution).**

The *magnification* of an instrument is the ratio of the size of the image to the size of the object. Its *resolution* is the smallest distance between two points that it still shows as separate. Magnifying an image beyond what the resolution justifies enlarges the blur, not the detail.

**Proposition 2.5 (Two instruments, two scales).**

The light microscope magnifies up to about $\times 1500$ with a [resolution](#def-g10-cells-common-unit-magnification) of about $0.2\,\text{µ}\mathrm{m}$: it shows [cells](#def-g10-cells-common-unit-cell), nuclei, [chloroplasts](#def-g10-cells-common-unit-organelle) and bacteria, in colour and alive. The electron microscope uses a beam of electrons instead of light: [resolution](#def-g10-cells-common-unit-magnification) below $1\,\mathrm{nm}$, [magnification](#def-g10-cells-common-unit-magnification) up to $\times 10^6$; it reveals the inner structure of the [cell](#def-g10-cells-common-unit-cell) — membranes, [mitochondria](#def-g10-cells-common-unit-organelle), [ribosomes](#def-g10-cells-common-unit-organelle), even large molecules — but on a sample that is dead, dried and cut into slices thinner than a micrometre.

**Proof.** *Admitted at this level.* ∎

![Sizes on a logarithmic scale: each step is ten times larger. The naked eye resolves about 0.1\, mm; the light microscope reaches 0.2\, µ m and sees whole cells and bacteria; the electron microscope resolves nanometres and shows the parts inside.](https://one-course.com/images/onecourse/chapters/biology-2/g10-cells-common-unit/fig-b003263f931c.svg)

*Sizes on a logarithmic scale: each step is ten times larger. The naked eye resolves about $0.1\,\mathrm{mm}$; the light microscope reaches $0.2\,\text{µ}\mathrm{m}$ and sees whole [cells](#def-g10-cells-common-unit-cell) and bacteria; the electron microscope resolves nanometres and shows the parts inside.*

**Method 2.6 (Measuring an object on a micrograph).**

A micrograph carries either a [magnification](#def-g10-cells-common-unit-magnification) or a scale bar.

1. With a scale bar: measure the bar on the page (say $12\,\mathrm{mm}$ for a bar labelled $10\,\text{µ}\mathrm{m}$ ) and the object ( $30\,\mathrm{mm}$ ); the real size is $30 \times 10/12 =  25\,\text{µ}\mathrm{m}$ .
2. With a [magnification](#def-g10-cells-common-unit-magnification) : real size $=$ measured size divided by the [magnification](#def-g10-cells-common-unit-magnification) ; a [cell](#def-g10-cells-common-unit-cell) drawn $20\,\mathrm{mm}$ long at $\times 400$ is $20/400 = 0.05\,\mathrm{mm} = 50\,\text{µ}\mathrm{m}$ .
3. Always state the unit, and convert: $1\,\mathrm{mm} =  1000\,\text{µ}\mathrm{m}$ , $1\,\text{µ}\mathrm{m} = 1000\,\mathrm{nm}$ .

**Example 2.7 (Cheek cells).**

A cheek [cell](#def-g10-cells-common-unit-cell) drawn $24\,\mathrm{mm}$ wide from a $\times 400$ image is $24/400 = 0.06\,\mathrm{mm} = 60\,\text{µ}\mathrm{m}$ across, and its [nucleus](#def-g10-cells-common-unit-organelle), drawn $3\,\mathrm{mm}$, is about $8\,\text{µ}\mathrm{m}$. A red blood [cell](#def-g10-cells-common-unit-cell) is $7\,\text{µ}\mathrm{m}$; the bacteria of the mouth, $1\,\text{µ}\mathrm{m}$ long, appear as barely resolved dots at the same [magnification](#def-g10-cells-common-unit-magnification).

## 2.3 Inside the cell

**Definition 2.8 (Organelle).**

An *organelle* is a structure inside a [cell](#def-g10-cells-common-unit-cell), enclosed by its own membrane, that performs a specific task. The principal ones are:

- the *nucleus* , which contains the chromosomes — the DNA of [Chapter 3](https://one-course.com/books/biology/2/en/chapter/3-dna-a-universal-genetic-molecule#ch-g10-universal-dna) — and controls the [cell](#def-g10-cells-common-unit-cell) ’s activity;
- the *mitochondria* , some $1\,\text{µ}\mathrm{m}$ long, where organic molecules are broken down with oxygen to release usable energy (cellular respiration, [Chapter 4](https://one-course.com/books/biology/2/en/chapter/4-cell-metabolism#ch-g10-cell-metabolism) );
- in plant [cells](#def-g10-cells-common-unit-cell) , the *chloroplasts* , green with chlorophyll, where photosynthesis takes place, and a large water-filled *vacuole* that keeps the [cell](#def-g10-cells-common-unit-cell) taut.

Outside the membrane, a plant [cell](#def-g10-cells-common-unit-cell) is boxed in a rigid *cell wall* of cellulose; animal [cells](#def-g10-cells-common-unit-cell) have none. The [cytoplasm](#def-g10-cells-common-unit-cell) of every [cell](#def-g10-cells-common-unit-cell) also contains millions of *ribosomes*, particles of about $25\,\mathrm{nm}$ that assemble [proteins](https://one-course.com/books/biology/2/en/chapter/1-the-chemical-makeup-of-living-things#def-g10-chemistry-of-life-families), visible only with the electron microscope.

![Two eukaryotic cells, schematically. Both have a membrane, a cytoplasm with mitochondria (orange) and a nucleus; the plant cell adds a cellulose wall, chloroplasts (green) and a large central vacuole. Neither is drawn to scale: a real nucleus is about a tenth of the cell’s width, and mitochondria are a hundred times smaller than the cell.](https://one-course.com/images/onecourse/chapters/biology-2/g10-cells-common-unit/fig-28ce0fddb6aa.svg)

*Two [eukaryotic cells](#def-g10-cells-common-unit-prokaryote), schematically. Both have a membrane, a [cytoplasm](#def-g10-cells-common-unit-cell) with [mitochondria](#def-g10-cells-common-unit-organelle) (orange) and a [nucleus](#def-g10-cells-common-unit-organelle); the plant [cell](#def-g10-cells-common-unit-cell) adds a cellulose wall, [chloroplasts](#def-g10-cells-common-unit-organelle) (green) and a large central [vacuole](#def-g10-cells-common-unit-organelle). Neither is drawn to scale: a real [nucleus](#def-g10-cells-common-unit-organelle) is about a tenth of the [cell](#def-g10-cells-common-unit-cell)’s width, and [mitochondria](#def-g10-cells-common-unit-organelle) are a hundred times smaller than the [cell](#def-g10-cells-common-unit-cell).*

**Definition 2.9 (Prokaryotes and eukaryotes).**

A [cell](#def-g10-cells-common-unit-cell) whose DNA is enclosed in a [nucleus](#def-g10-cells-common-unit-organelle) is a *eukaryotic cell*: animals, plants, fungi and many single-celled organisms are eukaryotes. A *prokaryotic cell* has no [nucleus](#def-g10-cells-common-unit-organelle) and no membrane-bound [organelles](#def-g10-cells-common-unit-organelle): its DNA lies free in the [cytoplasm](#def-g10-cells-common-unit-cell), its size is around $1\,\text{µ}\mathrm{m}$, and it is surrounded by a wall of its own kind. Bacteria are prokaryotes.

**Example 2.10 (Reading a bacterium).**

An electron micrograph of the gut bacterium *Escherichia coli* shows a rod $2\,\text{µ}\mathrm{m}$ long and $0.5\,\text{µ}\mathrm{m}$ wide: a wall, a membrane just inside it, a granular [cytoplasm](#def-g10-cells-common-unit-cell) packed with [ribosomes](#def-g10-cells-common-unit-organelle), and a paler central region where the single circular DNA molecule lies. No [nucleus](#def-g10-cells-common-unit-organelle), no [mitochondria](#def-g10-cells-common-unit-organelle). Yet the [cell](#def-g10-cells-common-unit-cell) feeds, grows and divides every twenty minutes in a rich broth: everything the [cell](#def-g10-cells-common-unit-cell) theory demands, in a thousandth of the volume of a cheek [cell](#def-g10-cells-common-unit-cell).

![A plant cell cut open: the rigid wall, the thin membrane against it, the nucleus, the green chloroplasts and the vacuole that fills most of the volume.](https://one-course.com/images/onecourse/chapters/biology-2/g10-cells-common-unit/fig-3998cff1bdf1.svg)

*A plant [cell](#def-g10-cells-common-unit-cell) cut open: the rigid wall, the thin membrane against it, the [nucleus](#def-g10-cells-common-unit-organelle), the green [chloroplasts](#def-g10-cells-common-unit-organelle) and the [vacuole](#def-g10-cells-common-unit-organelle) that fills most of the volume.*

## 2.4 Many cells, many shapes: specialisation

**Proposition 2.11 (Specialised cells).**

In a multicellular organism the [cells](#def-g10-cells-common-unit-cell) share the same basic plan and the same DNA, but each kind has a shape and an equipment matched to its task: a neuron extends a fibre up to a metre long to carry signals; a red blood [cell](#def-g10-cells-common-unit-cell) has lost its [nucleus](#def-g10-cells-common-unit-organelle) and is a flattened disc packed with haemoglobin; a muscle fibre is a giant [cell](#def-g10-cells-common-unit-cell) full of contractile [proteins](https://one-course.com/books/biology/2/en/chapter/1-the-chemical-makeup-of-living-things#def-g10-chemistry-of-life-families); a root hair is a thin tube that maximises the surface for absorbing water; a guard [cell](#def-g10-cells-common-unit-cell) of a leaf changes shape to open or close a pore. Specialisation is a matter of which parts of the shared plan are developed.

**Proof.** *Admitted at this level.* ∎

**Example 2.12 (Surface and volume).**

Why are [cells](#def-g10-cells-common-unit-cell) so small? Everything a [cell](#def-g10-cells-common-unit-cell) takes in or gives out crosses its membrane, whose area grows with the square of the size, while the needs grow with the volume, the cube. A cubic [cell](#def-g10-cells-common-unit-cell) of side $10\,\text{µ}\mathrm{m}$ has a surface of $600\,\text{µ}\mathrm{m}^{2}$ for a volume of $1000\,\text{µ}\mathrm{m}^{3}$: $0.6\,\text{µ}\mathrm{m}^{2}$ of membrane per cubic micrometre. At side $100\,\text{µ}\mathrm{m}$ the ratio drops to $0.06\,\text{µ}\mathrm{m}^{2}$: ten times less membrane per unit of content. A large [cell](#def-g10-cells-common-unit-cell) would starve at its centre; the [cells](#def-g10-cells-common-unit-cell) that are large (a frog egg, a muscle fibre) are either mostly inert reserve or elongated so that no point is far from the surface.

![A drop of pond water under the light microscope: a paramecium, a single cell a quarter of a millimetre long, rows past green algae with the beat of thousands of tiny hairs. One cell, and a whole organism.](https://one-course.com/images/onecourse/chapters/biology-2/g10-cells-common-unit/img-30a45e48b3ee.jpg)

*A drop of pond water under the light microscope: a paramecium, a single [cell](#def-g10-cells-common-unit-cell) a quarter of a millimetre long, rows past green algae with the beat of thousands of tiny hairs. One [cell](#def-g10-cells-common-unit-cell), and a whole organism.*

**Remark 2.13 (The cell as an open system).**

A [cell](#def-g10-cells-common-unit-cell) is not a sealed bag. Water, oxygen, glucose and ions cross its membrane continuously, and its wastes leave the same way; a cheek [cell](#def-g10-cells-common-unit-cell) placed in pure water swells and bursts within minutes, and one placed in strong salt water shrivels. The membrane’s control of what enters and leaves is the subject of the Year 1 volume; for now, remember that a [cell](#def-g10-cells-common-unit-cell) lives by exchanging with its surroundings, and dies when the exchanges stop.

## 2.5 Exercises

**Exercise 2.1 ★.**

State the three statements of the [cell](#def-g10-cells-common-unit-cell) theory.

**Solution of Exercise 2.1.**

Every living thing is made of one or more [cells](#def-g10-cells-common-unit-cell); the [cell](#def-g10-cells-common-unit-cell) is the basic unit of structure and function; every [cell](#def-g10-cells-common-unit-cell) comes from a pre-existing [cell](#def-g10-cells-common-unit-cell) by division.

**Exercise 2.2 ★.**

Convert: $0.05\,\mathrm{mm}$ to micrometres; $2500\,\mathrm{nm}$ to micrometres; $7\,\text{µ}\mathrm{m}$ to millimetres.

**Solution of Exercise 2.2.**

$50\,\text{µ}\mathrm{m}$; $2.5\,\text{µ}\mathrm{m}$; $0.007\,\mathrm{mm}$.

**Exercise 2.3 ★.**

A [cell](#def-g10-cells-common-unit-cell) is drawn $36\,\mathrm{mm}$ long from a $\times 600$ image. What is its real length?

**Solution of Exercise 2.3.**

$36/600 = 0.06\,\mathrm{mm} = 60\,\text{µ}\mathrm{m}$.

**Exercise 2.4 ★.**

Name three structures found in a plant [cell](#def-g10-cells-common-unit-cell) but not in an animal [cell](#def-g10-cells-common-unit-cell), and one structure found in every [cell](#def-g10-cells-common-unit-cell).

**Solution of Exercise 2.4.**

Plant only: [cell wall](#def-g10-cells-common-unit-organelle), [chloroplasts](#def-g10-cells-common-unit-organelle), large central [vacuole](#def-g10-cells-common-unit-organelle). Every [cell](#def-g10-cells-common-unit-cell): a [plasma membrane](#def-g10-cells-common-unit-cell) (also [cytoplasm](#def-g10-cells-common-unit-cell) and [ribosomes](#def-g10-cells-common-unit-organelle)).

**Exercise 2.5 ★.**

What is the difference between a prokaryotic and a [eukaryotic cell](#def-g10-cells-common-unit-prokaryote)? Give an example of each.

**Solution of Exercise 2.5.**

A [eukaryotic cell](#def-g10-cells-common-unit-prokaryote) keeps its DNA inside a [nucleus](#def-g10-cells-common-unit-organelle) and has membrane-bound [organelles](#def-g10-cells-common-unit-organelle) (a cheek [cell](#def-g10-cells-common-unit-cell), a yeast); a [prokaryotic cell](#def-g10-cells-common-unit-prokaryote) has neither, its DNA lies free in the [cytoplasm](#def-g10-cells-common-unit-cell), and it is about $1\,\text{µ}\mathrm{m}$ (a bacterium such as *E. coli*).

**Exercise 2.6 ★★.**

On a micrograph a scale bar labelled $5\,\text{µ}\mathrm{m}$ measures $15\,\mathrm{mm}$; a [mitochondrion](#def-g10-cells-common-unit-organelle) measures $6\,\mathrm{mm}$ long. Compute the real length of the [mitochondrion](#def-g10-cells-common-unit-organelle) and the [magnification](#def-g10-cells-common-unit-magnification) of the image.

**Solution of Exercise 2.6.**

Real length $6 \times 5/15 = 2\,\text{µ}\mathrm{m}$. [Magnification](#def-g10-cells-common-unit-magnification): the bar is $15\,\mathrm{mm}$ for $5\,\text{µ}\mathrm{m}$ $= 0.005\,\mathrm{mm}$, so $15/0.005 = 3000$: $\times 3000$.

**Exercise 2.7 ★★.**

Why can a light microscope not show [ribosomes](#def-g10-cells-common-unit-organelle), whatever its [magnification](#def-g10-cells-common-unit-magnification)? Use the two words of [Definition 2.4](#def-g10-cells-common-unit-magnification).

**Solution of Exercise 2.7.**

A [ribosome](#def-g10-cells-common-unit-organelle) ($25\,\mathrm{nm}$) is ten times smaller than the light microscope’s [resolution](#def-g10-cells-common-unit-magnification) ($0.2\,\text{µ}\mathrm{m}$ $= 200\,\mathrm{nm}$): two neighbouring [ribosomes](#def-g10-cells-common-unit-organelle), or a [ribosome](#def-g10-cells-common-unit-organelle) and its surroundings, cannot be told apart. More [magnification](#def-g10-cells-common-unit-magnification) only enlarges the blur.

**Exercise 2.8 ★★.**

Iodine solution is added to a slice of potato under the microscope: the blue-black colour appears in oval grains inside the [cells](#def-g10-cells-common-unit-cell). Using [Chapter 1](https://one-course.com/books/biology/2/en/chapter/1-the-chemical-makeup-of-living-things#ch-g10-chemistry-of-life), say what the grains are and what this shows about where a plant stores its reserves.

**Solution of Exercise 2.8.**

The grains are starch, stored inside the [cells](#def-g10-cells-common-unit-cell) as solid grains (starch is insoluble). A plant’s reserves are kept inside its [cells](#def-g10-cells-common-unit-cell), not in some space between them.

**Exercise 2.9 ★★.**

Onion skin [cells](#def-g10-cells-common-unit-cell) show a [nucleus](#def-g10-cells-common-unit-organelle) and a wall but no [chloroplast](#def-g10-cells-common-unit-organelle), while the [cells](#def-g10-cells-common-unit-cell) of a moss leaf are full of them. Explain the difference from where each tissue lives.

**Solution of Exercise 2.9.**

The onion scale grows underground, in the dark, and stores reserves: [chloroplasts](#def-g10-cells-common-unit-organelle) would be useless there. The moss leaf lives in the light and makes its food by photosynthesis, so its [cells](#def-g10-cells-common-unit-cell) are packed with [chloroplasts](#def-g10-cells-common-unit-organelle). Same plan, different equipment.

**Exercise 2.10 ★★.**

Compute the surface-to-volume ratio of a cubic [cell](#def-g10-cells-common-unit-cell) of side $1\,\text{µ}\mathrm{m}$ (a bacterium) and of side $20\,\text{µ}\mathrm{m}$ (a typical animal [cell](#def-g10-cells-common-unit-cell)). Which one can rely on simple diffusion through its membrane alone?

**Solution of Exercise 2.10.**

Side $1\,\text{µ}\mathrm{m}$: surface $6\,\text{µ}\mathrm{m}^{2}$, volume $1\,\text{µ}\mathrm{m}^{3}$, ratio 6 per micrometre. Side $20\,\text{µ}\mathrm{m}$: surface $2400\,\text{µ}\mathrm{m}^{2}$, volume $8000\,\text{µ}\mathrm{m}^{3}$, ratio 0.3. The bacterium has twenty times more membrane per unit of content and lives by diffusion alone; the larger [cell](#def-g10-cells-common-unit-cell) needs internal transport and [organelles](#def-g10-cells-common-unit-organelle).

**Exercise 2.11 ★★.**

A bacterium in rich broth divides every $20\,\mathrm{min}$. Starting from one [cell](#def-g10-cells-common-unit-cell), how many are there after $2\,\mathrm{h}$? After $4\,\mathrm{h}$?

**Solution of Exercise 2.11.**

$2\,\mathrm{h}$ is 6 divisions: $2^6 = 64$ [cells](#def-g10-cells-common-unit-cell). $4\,\mathrm{h}$ is 12: $2^{12} = 4096$.

**Exercise 2.12 ★★★.**

A red blood [cell](#def-g10-cells-common-unit-cell) has no [nucleus](#def-g10-cells-common-unit-organelle) and no [mitochondria](#def-g10-cells-common-unit-organelle). Does it satisfy [Definition 2.1](#def-g10-cells-common-unit-cell)? Discuss, knowing that it lives about 120 days, cannot divide, and is produced by nucleated [cells](#def-g10-cells-common-unit-cell) in the bone marrow.

**Solution of Exercise 2.12.**

It has a membrane, a [cytoplasm](#def-g10-cells-common-unit-cell) and takes in and gives out substances, so it fits most of the definition, but it cannot divide and cannot renew its [proteins](https://one-course.com/books/biology/2/en/chapter/1-the-chemical-makeup-of-living-things#def-g10-chemistry-of-life-families): it is a [cell](#def-g10-cells-common-unit-cell) that has given up part of the programme in exchange for room for haemoglobin, and it is made and replaced by complete [cells](#def-g10-cells-common-unit-cell). The definition describes the general case; the red blood [cell](#def-g10-cells-common-unit-cell) is a specialised, terminal form of it.

**Exercise 2.13 ★★★.**

In 1665 Hooke saw "[cells](#def-g10-cells-common-unit-cell)" in a slice of cork — empty boxes with thick walls. Explain what he actually saw, in the vocabulary of this chapter, and why he could not have seen a [nucleus](#def-g10-cells-common-unit-organelle) in it.

**Solution of Exercise 2.13.**

Cork is dead tissue: Hooke saw the cellulose walls of [cells](#def-g10-cells-common-unit-cell) whose living content had disappeared — empty boxes. A [nucleus](#def-g10-cells-common-unit-organelle) exists only in a living [cell](#def-g10-cells-common-unit-cell)’s [cytoplasm](#def-g10-cells-common-unit-cell), and in any case his microscope’s [resolution](#def-g10-cells-common-unit-magnification) and the absence of stains would have hidden it.

**Exercise 2.14 ★★★.**

Viruses are particles of $20\,\mathrm{nm}$ to $300\,\mathrm{nm}$ that contain [nucleic acid](https://one-course.com/books/biology/2/en/chapter/1-the-chemical-makeup-of-living-things#def-g10-chemistry-of-life-families) and [protein](https://one-course.com/books/biology/2/en/chapter/1-the-chemical-makeup-of-living-things#def-g10-chemistry-of-life-families), cannot grow or divide on their own, and multiply only inside a [cell](#def-g10-cells-common-unit-cell). Are they [cells](#def-g10-cells-common-unit-cell)? Are they alive? Argue from the [cell](#def-g10-cells-common-unit-cell) theory.

**Solution of Exercise 2.14.**

A virus has no [cytoplasm](#def-g10-cells-common-unit-cell), no membrane of its own making, no metabolism, and cannot divide: it fails the definition of a [cell](#def-g10-cells-common-unit-cell). By the [cell](#def-g10-cells-common-unit-cell) theory it is not alive on its own; it is a particle that borrows a [cell](#def-g10-cells-common-unit-cell)’s machinery to make copies of itself. Whether to call it "alive" is a matter of definition; that it is not a [cell](#def-g10-cells-common-unit-cell) is not.

**Exercise 2.15 ★★★.**

A muscle fibre is a single [cell](#def-g10-cells-common-unit-cell) $3\,\mathrm{cm}$ long and $50\,\text{µ}\mathrm{m}$ wide, containing hundreds of nuclei. Compute its volume and compare with that of a cubic $20\,\text{µ}\mathrm{m}$ [cell](#def-g10-cells-common-unit-cell); then explain, from [Example 2.12](#ex-g10-cells-common-unit-surface), why its shape and its many nuclei make such a size workable.

**Solution of Exercise 2.15.**

Volume $\pi \times (25\,\text{µ}\mathrm{m})^2 \times 30000\,\text{µ}\mathrm{m}
\approx 5.9 \times 10^{7}\,\text{µ}\mathrm{m}^{3}$, about 7000 times the $8000\,\text{µ}\mathrm{m}^{3}$ of the cube. Being a thin cylinder, no point of the fibre is more than $25\,\text{µ}\mathrm{m}$ from the membrane, so exchanges remain fast; and each [nucleus](#def-g10-cells-common-unit-organelle) governs only its own stretch of the fibre, so no [nucleus](#def-g10-cells-common-unit-organelle) has to serve a volume larger than an ordinary [cell](#def-g10-cells-common-unit-cell)’s.

## 2.6 Problem: Counting the Cells of a Body

**Problem 2.1.**

Weekend problem — a microscopy session, from the onion skin on the slide to an estimate of the number of cells in a human body

A class spends an afternoon at the microscope. The objectives available give total [magnifications](#def-g10-cells-common-unit-magnification) of $\times 40$, $\times 100$ and $\times 400$; the field of view at $\times 100$ is $1.8\,\mathrm{mm}$ across.

**Part I — The onion.**

1. At $\times 100$ , about 9 onion [cells](#def-g10-cells-common-unit-cell) fit side by side across the field of view. Estimate the width of one [cell](#def-g10-cells-common-unit-cell) , in micrometres.
2. At $\times 400$ the field of view is four times narrower. How wide is it, and how many of these [cells](#def-g10-cells-common-unit-cell) fit across it?
3. A student draws one [cell](#def-g10-cells-common-unit-cell) $28\,\mathrm{mm}$ wide from the $\times 400$ image. What is the [magnification](#def-g10-cells-common-unit-magnification) of the drawing relative to the real [cell](#def-g10-cells-common-unit-cell) ?
4. Iodine stains the [nucleus](#def-g10-cells-common-unit-organelle) yellow-brown; the student measures it at $4\,\mathrm{mm}$ on the drawing. Real diameter?
5. Name the structures the student should label on the drawing, and one structure present that cannot be seen at this [magnification](#def-g10-cells-common-unit-magnification) .

**Part II — The cheek and the pond.**

6. Cheek [cells](#def-g10-cells-common-unit-cell) stained with methylene blue appear as flat polygons about $60\,\text{µ}\mathrm{m}$ across but only $5\,\text{µ}\mathrm{m}$ thick. Why are they flat, given where they come from?
7. A drop of pond water shows a paramecium $240\,\text{µ}\mathrm{m}$ long. Can it be seen with the naked eye? At which [magnification](#def-g10-cells-common-unit-magnification) does it fill a quarter of the field?
8. Tiny moving dots about $1\,\text{µ}\mathrm{m}$ long are visible near the paramecium at $\times 400$ . What are they, most probably, and what would an electron microscope add?
9. The paramecium is a single [cell](#def-g10-cells-common-unit-cell) ; the cheek [cell](#def-g10-cells-common-unit-cell) is one of trillions. In one sentence each, say what "specialised" means for the cheek [cell](#def-g10-cells-common-unit-cell) and why it cannot apply to the paramecium.
10. A student adds a drop of salt water to the cheek [cells](#def-g10-cells-common-unit-cell) and sees them shrivel. Explain with [Remark 2.13](#rem-g10-cells-common-unit-open) .

**Part III — The size of a [cell](#def-g10-cells-common-unit-cell).**

11. Take a typical human [cell](#def-g10-cells-common-unit-cell) as a cube of side $20\,\text{µ}\mathrm{m}$ . Compute its volume in cubic micrometres, then in cubic millimetres ( $1\,\mathrm{mm}^{3} = 10^9\,\text{µ}\mathrm{m}^{3}$ ).
12. Compute its surface area and the ratio surface/volume.
13. Repeat for a bacterium taken as a cube of side $1\,\text{µ}\mathrm{m}$ . By what factor is its ratio larger?
14. [Cells](#def-g10-cells-common-unit-cell) take up a mass roughly equal to that of an equal volume of water ( $1\,\mathrm{g}$ per $\mathrm{cm}^{3}$ ). Estimate the mass of the $20\,\text{µ}\mathrm{m}$ [cell](#def-g10-cells-common-unit-cell) in nanograms ( $1\,\mathrm{g} = 10^9\,\mathrm{ng}$ ).
15. A red blood [cell](#def-g10-cells-common-unit-cell) is a disc $7\,\text{µ}\mathrm{m}$ across and $2\,\text{µ}\mathrm{m}$ thick. Estimate its volume (treat it as a cylinder, $V = \pi r^2 h$ ) and compare with the cube of question 11.

**Part IV — The number of [cells](#def-g10-cells-common-unit-cell) in a body.**

16. A student of mass $60\,\mathrm{kg}$ has a volume of about $60\,\mathrm{L}$ . If the body were made only of $20\,\text{µ}\mathrm{m}$ cubic [cells](#def-g10-cells-common-unit-cell) , how many would it contain?
17. In fact about a third of the body’s volume is fluid between the [cells](#def-g10-cells-common-unit-cell) and other non-cellular material. Correct the estimate.
18. Red blood [cells](#def-g10-cells-common-unit-cell) alone number about $2.5 \times 10^{13}$ . Using the volume of question 15, what volume do they occupy? Is that consistent with a blood volume of about $4.5\,\mathrm{L}$ of which 45% is [cells](#def-g10-cells-common-unit-cell) ?
19. Combine questions 17 and 18: give an order of magnitude for the total number of [cells](#def-g10-cells-common-unit-cell) in the body, and say why a precise count is impossible.
20. State the result: how many [cells](#def-g10-cells-common-unit-cell) , roughly, make a human, and by what process — from which starting number — were they all produced?

**Solution of Problem 2.1.**

**1.** $1.8/9 = 0.2\,\mathrm{mm} = 200\,\text{µ}\mathrm{m}$.

**2.** $1.8/4 = 0.45\,\mathrm{mm}$; about two [cells](#def-g10-cells-common-unit-cell) fit across it.

**3.** $28/0.2 = 140$: the drawing is $\times 140$.

**4.** $4/140 \approx 0.029\,\mathrm{mm}$, about $30\,\text{µ}\mathrm{m}$.

**5.** [Cell wall](#def-g10-cells-common-unit-organelle), [plasma membrane](#def-g10-cells-common-unit-cell) (pressed against the wall), [cytoplasm](#def-g10-cells-common-unit-cell), [nucleus](#def-g10-cells-common-unit-organelle), [vacuole](#def-g10-cells-common-unit-organelle). Present but invisible at $\times 400$: the [ribosomes](#def-g10-cells-common-unit-organelle) (and the [mitochondria](#def-g10-cells-common-unit-organelle) are at the limit).

**6.** They come from the surface of a lining that is several [cells](#def-g10-cells-common-unit-cell) thick and constantly rubbed: the outermost [cells](#def-g10-cells-common-unit-cell) are flattened and shed, which is why a cotton bud collects them.

**7.** $0.24\,\mathrm{mm}$ is just above the naked eye’s $0.1\,\mathrm{mm}$ limit: a barely visible speck. At $\times 100$ it spans $0.24/1.8 \approx 13\%$ of the field, at $\times 400$ more than half; no available objective gives exactly a quarter.

**8.** Bacteria. The electron microscope would show their wall, membrane, [ribosomes](#def-g10-cells-common-unit-organelle) and DNA region, and the absence of a [nucleus](#def-g10-cells-common-unit-organelle).

**9.** The cheek [cell](#def-g10-cells-common-unit-cell) does one job (protecting a surface) in a body where other [cells](#def-g10-cells-common-unit-cell) feed it, oxygenate it and coordinate it; the paramecium has nobody else and must feed, move, sense and divide as a single [cell](#def-g10-cells-common-unit-cell).

**10.** The outside is now saltier than the [cytoplasm](#def-g10-cells-common-unit-cell), so water leaves the [cell](#def-g10-cells-common-unit-cell) across the membrane; the [cell](#def-g10-cells-common-unit-cell) loses volume and shrivels.

**11.** $20^3 = 8000\,\text{µ}\mathrm{m}^{3} = 8 \times 10^{-6}\,\mathrm{mm}^{3}$.

**12.** $6 \times 20^2 = 2400\,\text{µ}\mathrm{m}^{2}$; ratio $2400/8000 = 0.3$ per micrometre.

**13.** Surface $6\,\text{µ}\mathrm{m}^{2}$, volume $1\,\text{µ}\mathrm{m}^{3}$, ratio 6: twenty times larger.

**14.** $8000\,\text{µ}\mathrm{m}^{3} = 8 \times 10^{-9}\,\mathrm{cm}^{3}$, hence about $8 \times 10^{-9}\,\mathrm{g} = 8\,\mathrm{ng}$.

**15.** $\pi \times 3.5^2 \times 2 \approx 77\,\text{µ}\mathrm{m}^{3}$, about a hundredth of the cube.

**16.** $60\,\mathrm{L} = 6 \times 10^{16}\,\text{µ}\mathrm{m}^{3}$; divided by 8000: about $7.5 \times 10^{12}$ [cells](#def-g10-cells-common-unit-cell).

**17.** Two thirds of that: about $5 \times 10^{12}$.

**18.** $2.5 \times 10^{13} \times 77 \approx 1.9 \times 10^{15}\,\text{µ}\mathrm{m}^{3} \approx 1.9\,\mathrm{L}$. Blood [cells](#def-g10-cells-common-unit-cell): $0.45 \times 4.5 \approx
2.0\,\mathrm{L}$ — consistent.

**19.** About $5 \times 10^{12}$ large [cells](#def-g10-cells-common-unit-cell) plus $2.5 \times 10^{13}$ red [cells](#def-g10-cells-common-unit-cell): some $3 \times 10^{13}$, a few tens of trillions. [Cell](#def-g10-cells-common-unit-cell) sizes span a factor of a hundred and the mix of kinds varies, so any count is an estimate to within a factor of two or three.

**20.** Roughly thirty trillion ($3 \times 10^{13}$) [cells](#def-g10-cells-common-unit-cell), all produced by successive [cell](#def-g10-cells-common-unit-cell) divisions from a single starting [cell](#def-g10-cells-common-unit-cell), the fertilised egg.