Cell size & surface-area-to-volume

Everything goes in and out through the surface

A cell is constantly trading with its surroundings. Oxygen and nutrients have to come in; carbon dioxide and other wastes have to go out. All of that traffic passes through the plasma membrane, the cell's outer surface.

So there are two quantities to keep an eye on. The surface area (SA) sets how fast the cell can exchange materials. The volume (V) sets how much cytoplasm is using those materials, and so how much the cell needs. What matters is the balance between them, the surface-area-to-volume ratio (SA:V).

Squares versus cubes

Take a cube with sides of length L. Its surface is six square faces, so SA = 6L². Its volume is L³. Now double the side length. The area goes up 2² = 4 times, but the volume goes up 2³ = 8 times. Volume wins every time you scale up.

• 1 cm cube: SA = 6 cm², V = 1 cm³, SA:V = 6 : 1
• 2 cm cube: SA = 24 cm², V = 8 cm³, SA:V = 3 : 1
• 3 cm cube: SA = 54 cm², V = 27 cm³, SA:V = 2 : 1

Dividing the two formulas gives a shortcut: for a cube, SA:V = 6/L. For a sphere of radius r, SA = 4πr² and V = (4/3)πr³, so SA:V = 3/r. Either way, the ratio is inversely proportional to size: double the size and you halve the ratio.

The second problem: distance

Inside the cell, most molecules move by diffusion, the random jiggling of molecules that spreads them from where they're concentrated to where they're scarce. Diffusion is fast over tiny distances but hopelessly slow over large ones.

The reason is that the time needed grows with the square of the distance: roughly t ≈ x²/(2D), where x is the distance and D is the diffusion coefficient, a measure of how quickly that molecule spreads in that medium. For oxygen in water, D is about 2 × 10⁻⁹ m² per second. Plug in some distances and the pattern jumps out:

• 10 µm (a typical cell): about 0.025 seconds
• 1 mm: about 4 minutes
• 1 cm: about 7 hours

Going 1,000 times farther takes a million times longer. That's why a cell a centimetre across couldn't supply its centre by diffusion alone, and why large animals need a circulatory system to carry oxygen by bulk flow to within a few micrometres of every cell. These figures are rough estimates, but the scaling is the point.

How living things keep SA:V high

Selection has favoured several ways around the problem. You'll meet these again and again in biology:

• Stay small. Prokaryotes are typically 0.1–5 µm across and most eukaryotic cells 10–100 µm. When a cell grows, it eventually divides, and two small cells have a higher combined SA:V than one big one.
• Change shape. Flattening or stretching a cell adds surface without adding volume, and it also shortens the distance from any point to the membrane. A flat cell or a long thin one exchanges much better than a sphere of the same volume.
• Fold the surface. Microvilli, finger-like projections of the plasma membrane, carpet the cells lining your small intestine and greatly increase the area for absorbing food. Root hairs do the same job for water uptake in plants.
• Fold internal membranes. The cristae of the inner mitochondrial membrane pack more membrane, and so more ATP-making machinery, into each mitochondrion.
• Divide the interior. Eukaryotic cells are bigger than prokaryotes partly because their organelles break the cytoplasm into small compartments, each with its own membrane surfaces.
• At the scale of whole organisms, the same logic explains lung alveoli, intestinal villi, gills and the thin, flat shape of leaves, all huge surfaces for exchange packed into a small body.

Key terms

Surface area (SA)

The total area of a cell's outer membrane, which sets how fast it can exchange materials.

Volume (V)

The amount of space inside the cell, which sets how much it needs to exchange.

SA:V ratio

Surface area divided by volume; it falls as an object of a given shape gets bigger.

Diffusion

The net spreading of molecules from where they are more concentrated to where they are less concentrated, caused by random motion.

Diffusion coefficient (D)

How quickly a particular molecule spreads through a particular medium.

Microvilli

Finger-like folds of the plasma membrane that increase surface area, for example in the small intestine.

Cristae

Folds of the inner mitochondrial membrane that increase its area.

Check yourself

Try answering in your head before you open each answer.

• 1.A spherical cell's radius doubles from 5 µm to 10 µm. By what factor do its surface area, volume and SA:V change? What does that mean for its oxygen supply?Show answerHide

  Area rises 4× and volume 8×, so SA:V halves (from 3/5 = 0.6 µm⁻¹ to 3/10 = 0.3 µm⁻¹). Each unit of cytoplasm now has half as much membrane supplying it, and oxygen has twice as far to diffuse to reach the centre, which takes about four times as long.

• 2.A spherical cell divides into two identical spherical daughter cells with the same total volume. Does the total SA:V go up or down? Estimate by how much.Show answerHide

  It goes up. Each daughter has half the volume, so its radius is (½)^(1/3) ≈ 0.79 of the parent's. Since SA:V = 3/r for a sphere, each daughter's SA:V is about 1/0.79 ≈ 1.26 times the parent's. Dividing is one way cells restore a high SA:V.

• 3.Using t ≈ x²/(2D), an oxygen molecule takes about 0.025 s to diffuse 10 µm. Roughly how long would it take to diffuse 100 µm? What does this suggest about the thickness of tissues that get oxygen by diffusion alone?Show answerHide

  Ten times the distance means 10² = 100 times the time, so about 2.5 s. Every further tenfold increase multiplies the time by another 100, so tissues supplied only by diffusion have to stay very thin (or have blood vessels running close to every cell).