# Big and small numbers in biology

## Problem

Biology makes good use of numbers both small and large. Try these questions involvin

You might need to use standard biological data not given in the question.

Of course, as these questions involve estimation there are no definitive 'correct' answers. Just try to make your answers to each part as accurate as seems appropriate in the context of the question.

- Estimate how many of the smallest viruses would fit inside the largest bacterium. What assumptions do you make in the calculation?

- Why might it be misleading to say that the size of a bacterium is 200 microns? How might you provide a more accurate description of the size?

- It has been said that 1g of fertile soil may contain as many as 2500 million bacteria. Do you think that this is a high density of bacteria? Estimate the percentage, by weight, of the soil that comprises bacteria. If the bacteria were evenly spread out, estimate the distance between the bacteria and compare this to the size of the bacteria. Does this surprise you?

- The shape of the earth may be approximated closely by a sphere of radius 6 x10$^{6}$m. In June 2008, its human population, according to the US census bureau , was thought to be 6,673,031,923. If everyone spread out evenly on the surface of the earth, what area of the planet would they each have?

- Humans typically live on land. Readjust your answer to the previous question making use of the fact that about 70% of the surface of the earth is water.

- Compare the previous three parts of the question. Are humans more densely packed than the bacteria in the soil?
*(given that humans live on the surface of the earth and bacteria live inside a volume of soil, you might want to consider how best to measure the 'density')*

- Suppose that fertile land on earth extends, on average, down to about 10cm. Estimate how many cubic mm of fertile soil the earth contains. Estimate the number of bacteria living in the fertile land on earth.

- Question the assumption concerning the average depth of fertile land in the previous question. Would you say that it should be smaller, larger or is about right? You might want to use these suggested data (from here ) that the percentages of earth's land surface can be divided into different types: 20% snow covered, 20%
mountains, 20% dry/desert, 30% good land that can be farmed, 10% land with no topsoil. What other data might you need to make a more accurate assessment?

- There are about 300000 platelets in a cubic mm of human blood. How many platelets might you expect to find in a healthy adult male?

- There are about 4 - 6 million erythrocytes and 1000 - 4500 lymphocytes in a cubic mm of blood. A sample of blood on a slide is 2 microns thick. Would you expect many erythrocytes or lymphocytes to overlap on the microscope image?

*Extension: In mathematics, a bound for a* *measurement gives two numbers between which we know for certain that the real measurement must lie. For example, a (not very good) bound on the height of the members of a class would be 1m < heights < 2m. In the previous questions can you find bounds on the quantities? First suggest
a really rough bound which you would know to be true and then see* *if you can sensibly improve on it.*

## Getting Started

In estimation questions don't be afraid to have a go with a guess at some numbers in the problem and then to refine your estimate after checking it makes some sort of sense.

Although there is no 'right' answer to an estimation, there are good or bad estimations and sensible or over-detailed calculations.

Think how you might make your estimation a good one, and think how it makes sense to ignore certain complexities in the calculation.

## Student Solutions

**QUESTION 1**

**Additional data required**

Size of smallest virus: 20 nm

Size of largest bacterium: 0.75 mm (This bacterium is called *Thiomargarita namibiensis* and is so large that it is visible by the naked eye).

*Reasoning*

We assume that the virus is generally spherical, and that the bacterium is cylindrical, with diameter 0.1 mm.

Therefore, the volume of the smallest virus is $$V_1 = \frac{4}{3} \pi r^3 \approx 3.35 \cdot 10^4 \textrm{nm}^3$$

The volume of the largest bacterium is on the other hand $$V_2 = \pi r^2 h \approx 0.0236 \textrm{mm}^3 = 2.36 \cdot 10^{16} \textrm{nm}^3 $$

So, in principle, we could fit

$$ N = \frac{V_2}{V_1} = 7.04 \cdot 10^{11} \textrm{ viruses} $$

inside a *Thiomargarita namibiensis*.

**QUESTION 2**

A bacterium which is 200mm *long* is probably one of the longest bacteria known (since most species have length between 0.5 and 5 microns).

**QUESTION 3**

**Additional data required**

Average mass of a bacterium: $9.5\cdot 10^{-13} \textrm{g}$

Average volume of a bacterium: $0.6 \mu \textrm{m}^3$

Average density of soil: $1.602 \textrm{g} / \textrm{cm}^3 $

*Reasoning*

2500 million bacteria weigh $m = 2500\cdot 10^6 \cdot 9.5\cdot 10^{-13}\textrm{g} = 2.375 \cdot 10^{-3} \textrm{g} $.

So, the percentage of bacteria contained in 1g of soil is

$$2.375 \cdot 10^{-3} = 0.2375 \textrm{%} $$

Now, 1g of soil has volume $\frac{1}{1.602} = 0.624 \textrm{cm}^3 $. So, if the 2500 million bacteria are evenly spread in the soil, they each get a volume

$$ V = \frac{0.624}{2500 \cdot 10^6} = 2.496 \cdot 10^{-10} \textrm{cm}^3 = 2.496 \cdot 10^2 \mu \textrm{m}^3 $$

around them.

If we consider this volume around them as a sphere, the sphere's radius will be given by

$$ r = \sqrt[3]{\frac{3V}{4\pi}} $$

and so, substituting for V as found above, we obtain $ r = 3.906 \mu \textrm{m} $ as the distance between two neighbouring bacteria.

**QUESTION 4**

Recall that the surface area of a sphere is given by $ S = 4\pi r^2 $. Then, the surface area of the Earth is $$ S = 4\pi \cdot (6\cdot10^{6})^2 \textrm{m}^2 \approx \ 4.53 \cdot 10^{14} \textrm {m}^2 $$

Hence, if the humans were evenly spread out on it, each would have an area of

$$ A = \frac{4.53 \cdot 10^{14} \textrm {m}^2}{6,673,031,923} \approx 6.78 \cdot 10^{4} \textrm{m}^2 $$

around them.

**QUESTION 5**

Now, if we only restrict to the land, we have only 30% of the area found above. Hence, each human would have an area of

$$ A = 0.3\cdot (6.78 \cdot 10^{4} \textrm{m}^2) = 2.03 \cdot 10^{4} \textrm{m}^2 $$

around them.

**QUESTION 6**

A reasonable way to measure if humans or bacteria are more densely packed, is to evaluate how many bacteria "fit" into one's bacterium "personal space" and compare that to the corresponding fact for humans.

We found that a bacterium has a space $V_1 = 2.496 \cdot 10^2 \mu \textrm{m} $ around it, whereas its average volume is $V_2 = 0.6 \mu \textrm{m}^3$. Hence, the number of bacteria fitting inside is

$$ N = \frac{V_1}{V_2} = 416 $$

Now, we will perform a similar calculation for humans, noting that each needs approximately $0.5 \textrm{2}$ around them when standing. Then, the number of human's fitting inside one's "personal space" is

$$ N = \frac{2.03 \cdot 10^4}{0.5} = 40600 $$

Hence, the number of humans fitting inside the personal space is much higher than the corresponding fact for bacteria, we conclude that we humans are much less packed than bacteria in fertile soil.

**QUESTION 7**

*Additional data required*

Total fertile (arable and agricultural) land surface area: 62.6 million km $^2$

*Reasoning*

In total, we see that the volume of fertile soil is $$ V = 62.6 \cdot 10^6 \cdot 10^6 \textrm{m}^2 \cdot 10^{-2} \textrm{m} = 62.6 \cdot 10^{10} \textrm{m}^3 = 62.6 \cdot 10^{19} \textrm{mm}^3 $$

So, if we assume that we can find one bacterium in every $ 2.496 \cdot 10^2 \mu \textrm{m} $, we estimate that the total number of baceria in the fertile soil of the Earth is around

$$ N = \frac{62.6 \cdot 10^{28}}{2.496 \cdot 10^2} = 2.5 \cdot 10^{27} $$

**QUESTION 8**

In general, we would expect the depth of fertile land to be much more than 10cm, since we would hope that this extends at least as much as a plant's roots do. This reasoning is correct, and indeed fertile soil does extend to the depth of the plants' roots (which can be up to several meters). However, most of the bacteria are contained in the *topsoil*, which
is on average the first 10 cm below the ground. So our estimation about the depth is actually quite good.

**QUESTION 9**

*Additional data required*

Average blood content in an adult male: 5.2 litres

*Reasoning*

Since there are 5.2 litres = 5.2$\times$10$^6$ mm$^3$ of blood in the human body, we calculate that in the average adult male there are about

$$ 300,000 \times 5.2 \times 10^6 = 1.52 \times 10^{12} \textrm{ platelets} $$

**QUESTION 10**

*Additional data required*

Diameter of an erythrocyte or a lymphocyte: 7 $\mu \textrm{m}$ (they have roughly the same size)

*Reasoning*

First of all we need to make an assumption about the other dimensions of the sample. Given that a microscope slide is 7.5 cm long and 2.5 cm wide, it is fairly reasonable to assume that a circular drop of blood in the centre of the slide has radius about 1cm.

Therefore, the drop occupies a volume of

$$ V = \pi r^2h=\pi \times 1\textrm{cm}^2 \times 2 \mu \textrm {m} = 2\pi \times 10^{-1} \textrm {mm}^3 \approx 0.63 \textrm{mm}^3 $$

So, we expect that in this small sample there are about 2.5 - 4 million erythrocytes and 600 - 2000 lymphocytes (since we are given the density per cubic millimetre).

Each erythrocyte or lymphocyte should occupy an area of $$A_1 = \pi r^2 \approx 38.5 \mu \textrm{m}^2 = 38.5 \times 10^{-8} \textrm{cm}^2$$.

Thus, the about 3 million cells, should occupy an area of $A = 3 \times 10^6 \times A_1 = 1.15 \textrm{cm}^2$.

The area of the drop is, however relatively larger, at $A = \pi r^2 \approx 3.14 \textrm{cm}^2 $, so the overlaps shouldn't be too many, as there is enough space around each cell.

## Teachers' Resources

### Why do this problem ?

Practice with the use of numbers is a crucial biological skill. These interesting questions will allow you to practice these skills whilst developing awareness of orders of magnitude in scientific contexts.### Possible approach

There are several parts to this question. The individual pieces could be used as starters or filler activities for students who finish classwork early. Enthusiastic students might work through them in their own time. Since there is no absolutely 'correct' answer to many of these questions, they might productively be used for discussion: students create their own answers and then explain them to the rest of the class. Does the class agree? Disagree? Is there an obvious best 'collective' answer?### Key questions

- What assumptions will you need to make in this question?

- How accurate do you think you answer is?

- What order of magnitude checks could you make to test that your answer is sensible?