Lesson 1.7 · Unit 1 · NGSS crosscutting concept 3
Scale and measurement, from the molecule to the biosphere
This course will ask you to move between a water molecule and the Pacific Ocean in a single lesson, and the moves only work if the sizes mean something to you. Biology spans about eighteen orders of magnitude, and almost every important constraint in the subject comes from something being too big or too small for a process to work.
- An order of magnitude is a factor of ten. Two things differing by three orders of magnitude differ by a factor of a thousand.
- The prefixes you need: milli is one thousandth, micro is one millionth, nano is one billionth. A millimeter is 1000 micrometers; a micrometer is 1000 nanometers.
- The working sizes: a water molecule is a fraction of a nanometer, DNA is about 2 nanometers across, a typical bacterium is 1 to 2 micrometers long, a typical animal cell is 10 to 100 micrometers, and the smallest thing an unaided eye resolves is about 100 micrometers.
- Surface area and volume do not scale together. Double a cube's side and its surface area rises fourfold while its volume rises eightfold, so the ratio of surface to volume halves.
- That ratio governs exchange. Everything entering or leaving a cell crosses its surface, while everything needing supply is in its volume, which is why cells are small and why unit 5 returns to this.
- Always carry the units. A number without units is not a measurement, and most arithmetic mistakes in this course are unit mistakes.
Where students lose marks: writing that a cell is "small" or a molecule "very small". These are not measurements. Give a figure with a unit, or a comparison with a factor: "a bacterium is roughly ten times shorter than a typical animal cell".
The question. Why can a cell not simply grow larger when it needs more resources? Work the arithmetic on cubes, which behave like cells for this purpose.
Step one: take a cube of side 1 cm. Surface area is 6 faces times 1 times 1, which is 6 square centimeters. Volume is 1 times 1 times 1, which is 1 cubic centimeter.
Step two: form the ratio. 6 divided by 1 is 6. For every unit of volume there are 6 units of surface.
Step three: double the side to 2 cm. Surface area is 6 times 2 times 2, which is 24 square centimeters. Volume is 2 times 2 times 2, which is 8 cubic centimeters. The ratio is 24 divided by 8, which is 3.
Step four: double again to 4 cm. Surface area is 6 times 4 times 4, which is 96 square centimeters. Volume is 4 times 4 times 4, which is 64 cubic centimeters. The ratio is 96 divided by 64, which is 1.5.
Step five: read the pattern. Each doubling of the side halves the surface to volume ratio: 6, then 3, then 1.5. Surface grows with the square of the side while volume grows with the cube.
Step six: connect it to the cell. Oxygen, glucose and waste all cross the membrane, whose area is the surface. The demand for them is set by the living material inside, which is the volume. As a cell grows, demand outruns supply, and beyond some size the center cannot be served.
Step seven: state the consequences you will meet later. Cells divide rather than enlarge; cells specialized for absorption are folded or projecting to raise area without raising volume; and large organisms need transport systems because they cannot rely on diffusion across an outer surface.
Questions 1 to 5 are recall. Questions 6 to 10 ask for reasoning.
- How many nanometers are in one micrometer?
Show the full solution
1000
- Give the approximate length of a typical bacterium.
Show the full solution
About 1 to 2 micrometers
- A cube has a side of 3 cm. Find its surface area, volume and ratio.
Show the full solution
Surface area 54 square centimeters, volume 27 cubic centimeters, ratio 2
- What happens to the surface to volume ratio as an object gets larger?
Show the full solution
It decreases
- Why is "a cell is very small" not an acceptable answer?
Show the full solution
It is not a measurement; give a figure with units or a comparison with a factor
- A typical animal cell is about 20 micrometers across and DNA is about 2
nanometers wide. How many orders of magnitude separate them?
Show the full solution
Convert to the same unit first. 20 micrometers is 20000 nanometers, and DNA is 2 nanometers, so the cell is 10000 times wider. Ten thousand is ten to the fourth power, so they differ by four orders of magnitude. Doing the conversion before the division is the whole of the method here, and mixing micrometers with nanometers is the commonest way to get an answer wrong by a factor of a thousand. Four orders of magnitude, a factor of 10000
- Explain why the small intestine is lined with millions of projections.
Show the full solution
Absorption happens across a surface, and the quantity of food to be absorbed is set by the volume of the gut contents, so a smooth tube of that diameter would present far too little area for the demand. Folding the lining into villi and covering those in microvilli raises the surface area enormously without increasing the volume of the organ, which is the same trick as increasing a cube's surface without increasing its bulk. The ratio, not the absolute area, is what the problem is about. It raises surface area without raising volume, improving the exchange ratio
- Why do very large organisms need circulatory systems while a single-celled one
does not?
Show the full solution
A single-celled organism has a high surface to volume ratio and a very short distance from its membrane to any point inside, so diffusion alone delivers oxygen and removes waste fast enough. As body size rises, the ratio falls and the distances grow, and diffusion becomes hopelessly slow over centimeters rather than micrometers. A circulatory system solves both problems at once by moving fluid in bulk to within diffusion distance of every cell, which is why size and transport systems appear together across the animal kingdom. Falling surface to volume ratio and longer diffusion distances make diffusion alone insufficient
- Two cells are cubes of side 10 and 20 micrometers. Which meets its needs more
easily, and by what factor do their ratios differ?
Show the full solution
The smaller one. For side 10 micrometers the surface area is 600 and the volume 1000 in consistent units, giving a ratio of 0.6. For side 20 the surface is 2400 and the volume 8000, giving 0.3. The ratios differ by a factor of two, so the smaller cell has twice as much membrane serving each unit of its contents and can supply its interior more easily, which is the general reason cells divide rather than continuing to grow. The smaller cell, with twice the surface per unit volume
- Why does this course insist on units at every step?
Show the full solution
Because a number in biology is meaningless without them and because carrying them catches mistakes automatically. Writing 20 when the figure is 20 micrometers and comparing it with 2 nanometers produces an answer wrong by a factor of a thousand, and the error is invisible until the units are restored. Units also make an answer checkable by someone else, which matters when a result is going to be used, and they force you to notice when a calculation has produced something in the wrong kind of quantity altogether. Units give a number meaning and catch conversion errors automatically