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Exchange surfaces: 1. The mathematics of surface exchange area to volume ratio in cells, organs and whole multicellular organisms

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There are various sections to work through, after 1 they can be read and studied in any order.

Sub-index of biology notes on exchange surfaces

1. A mathematical 'extra' on surface area to volume ratio calculations and implications for exchange surfaces in living organisms - cells, organs or whole organisms

The greater the surface area the greater the possible rate of material transfer.

The most compact shape to give the lowest surface area/volume ratio is a sphere, but that's not very practical for the working of many specialised cells, tissues or organs - but very good for single-celled organisms!

However, systems in living organisms that involve transfer of substances, do need as large a surface area as possible within the volume the 'system' occupies.

To this end, many organs have evolved to give the maximum surface area as possible within the volume the 'system' occupies.

A bit of area/volume maths to illustrate this idea with cubes of various sizes (6 faces):

A 1 cm cube has a volume of 1 cm3 (1 x 1 x 1), a total surface are of 6 x 1 x 1 = 6 cm2

So the surface area / volume ratio = 6 / 1 = 6.0 cm-1  (6 : 1 ratio)

A 2 cm cube has a volume of 8 cm3 (2 x 2 x 2), a total surface are of 6 x 2 x 2 = 24 cm2

So the surface area / volume ratio = 24 / 8 = 3.0 cm-1  (3 : 1 ratio)

A 3 cm cube has a volume of 27 cm3 (3 x 3 x 3), a total surface are of 6 x 3 x 3 = 54 cm2

So the surface area / volume ratio = 54 / 27 = 2.0 cm-1  (2 : 1 ratio)

A 4 cm cube has a volume of 64 cm3 (4 x 4 x 4), a total surface area of 6 x 4 x 4 = 96 cm2

So the surface area / volume ratio = 96 / 54 = 1.5 cm-1  (1.5 : 1 ratio)

You can see clearly that the smaller (thinner etc.) the 'system' or parts of the 'system' the greater the surface to volume ratio - potentially increasing the rate of transfer of substances.

Good examples of this are the millions of tiny air sacs (alveoli) in the lungs and the thin multi-layered sections of gills in fishes - both of which are to do with animal respiration.

Another good example is the fine and numerous villi in the intestine where their large surface area is very efficient for absorbing nutrients from absorbed food.

The villi can be envisaged as tall thin rectangular blocks in shape to maximise surface area.

An extra calculation based on a volume of 8 units. to make the point about villi.

A 2 x 2 x 2 block has a surface to volume ratio of 3 : 1 (see above).

A 1 x 2 x 4 block has a surface area to volume ratio of 3.5 : 1 (see adaptations)

0.1 x 0.1 x 800 block has a surface area of (2 x 0.12) + (4 x 0.1 x 800) = 0.02 + 320 = 320.02 = ~320 (3 sf)

This gives a surface to volume ratio of 320 / 8 = 40 : 1, much higher than the blocks above, over 10 x higher in fact.

Just think about the very fine capillaries in the blood system too.

In any surface area : volume calculations, make sure all measurements and calculations are quoted with the same length units!


The implications of these calculations for transfer of substances

This is the mathematics behind why for small cells in single or multicellular organisms, the transfer of nutrients, oxygen and waste products, diffusion rates are high - substances can be moved quickly in and out of cells.

As the volume of a cell increases, the distance from the outer cell membrane through the cytoplasm to the centre of the cell increases.

This slows down the rate of exchange of substances in or out of the cell from or to the environment.

Cells larger than 1 mm in diameter may not be viable because the rate of diffusion is too slow to supply nutrients and oxygen sustain the cell's life-supporting biochemistry.

Multicellular organisms, with many layers of cells, tend to have a smaller surface to volume ratio and therefore need specialised organ systems with large surface areas for the efficient transfer of substances and also thermal energy to avoid heating.

Because multicellular organisms have many layers of cells, this increases the time needed for nutrients and oxygen to diffuse in and reach the inner cells.

Therefore the cells of the outer layers would tend to use up the resources first and faster, depriving inner cells life-supporting resources.

Therefore adaptations have evolved to enable complex multicellular organisms to overcome this problem.

Examples of surface : volume ratio in various organisms

(based on the same length units)

Single cell bacterium 6 x 106 : 1

single celled amoeba 6 x 104 : 1

fly 6 x 102 : 1dog 6 : 1

whale 0.06 : 1

You can see there is quite a contrast between microscopic single celled and large multicellular organisms!

This mathematical 'extra' was 'adapted' from the pages

Structural adaptations of plants and animals

and note that surface area is an important variable in Fick's Law of diffusion

and appropriate to points in Diffusion, osmosis and active transport

Summary of learning objectives and key words or phrases

Be able to calculate the surface area to volume ratio for an exchange surface in cells or organs.

Be able to calculate the surface area to volume ratio for whole multicellular animal organisms.



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