Powers of x: fractional exponents
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UBC Calculus Online Course Notes

How big should a cell be?

On this page, we will use a simple biological model to illustrate the usefulness of the properties of power functions we've been discussing are useful.

Goals:

  • To explore surface area and volume of a simple geometric object and discuss how they change as the object grows in size.
  • To discover an important fact about the ratio of surface area and volume in small and large objects.
  • To draw conclusions about the physical limitations on cell size.
  • To review and apply power functions in a biological context.


Simple properties of a sphere

We will be refering frequently to the Volume, V, and surface area, S of a sphere of radius r:


\begin{eqnarray*} 
V & = & \frac{4 \pi}{3} r^3 \\ 
S & = & 4 \pi r^2 
\end{eqnarray*}


Questions (1):
Suppose the sphere is a balloon, which is being inflated. Let us examine what happens to its volume and surface area.
  • (a) As the radius increases, which of these properties increases faster, the volume or the surface area of the balloon?
  • (b) What is the ratio of the volume to the surface area (V/S). How does this ratio depend on the radius, r? At what rate does the ratio (V/S) increase as the radius increases?
  • (c) Sketch a graph of (V/S) as a function of r.
  • (d) Sometimes it is more conventional to discuss the surface area to volume ratio of a growing object, i.e, to investigate (S/V). Sketch a graph of (S/V) as a function of r.
  • (e) The formulae above tell us the volume and the area of a sphere of a given radius. But suppose we are given either the volume or the surface area and asked to find the radius.
    • (i) Find the radius as a function of the volume
    • (ii) Find the radius as a function of the surface area.
    • (iii) What is the radius of a balloon whose volume is 1 litre? (1 litre = $ 1 \times 10^3 {\rm cm}^3 $ ).
    • What is the radius of a balloon whose surface area is $ 10 {\rm cm}^2 $ ?


Sizes and shapes of typical cells

If you look through a book on Cellular Biology, you will discover that most cells have speciallized shapes that make them uniquely suited to their functions. Few cells are really spherical. Many have long appendages, cylindrical parts, or branch-like structures. But here, we will neglect all these beautiful complexities and look at a simple yeast-like spherical cell. The question we will eventually explore is what determines the size (and shape) of a cell, and why some size limitations exist. Why should animals be made of millions of tiny cells, instead of just a few hundred large ones? Why must the physiological construction of animals beyond a small size include a circulatory system ? All these questions seem extremely complicated, but actually, a little mathematical argument can go a long way in helping us to understand these issues.

To delve into this mystery of size and shape, we will formulate a model. A model is just a representation of reality that simplifies things and represents the most important aspects, while neglecting or idealizing the other aspects. To formulate this model we will make a few assumptions:

Assumptions

  1. The density of a living cell is roughly equal to the density of water, i.e the mass per unit volume is approximately 1 gm/cm3.
  2. The cell is roughly spherical.
  3. The cell must absorb vital substances (oxygen, nutrients)from its environment. It can do so only through its surface.
  4. The rate of absorption is proportional to the surface area of the cell.
  5. The cell consumes (uses up) these substances continually.
  6. The rate of consumption is proportional to the mass of the cell. The bigger the mass of the cell, the more nutrient it must consume per unit time to stay alive.


Questions (2):

  • (a) Discuss the assumptions listed above. Which ones are likely to be realistic? unrealistic?
  • (b) A typical cell has a radius of roughly 10 $ \mu $ . (Note: $ 1 \mu= 10^{-6}m=10^{-4}cm $ ). Find the mass of such a cell. (Express your answer in $ \mu g $ , where $ 1 \mu g= 10^{-6}gm $ .)
  • (c) What would the radius of the cell be if its mass were 1 gm?
  • (d) Express the mass of the cell as a function of the radius. Be sure to include the units.
  • (e) Express the cell radius as a function of its mass.
  • (f) Express the surface area of the cell as a function of its mass.



The best cell size

In order for the cell to survive, the overall rate of consumption of nutrients (or oxygen) should match the overall rate of uptake. In general, for a given type of cell, we may assume that the following intrinsic properties are determined by the type of cell membrane, the rate of metabolism of the cell, etc, and are therefore constant:

A=rate of absorption per unit area per unit time
C=rate of consumption per unit volume per unit time

Now let us see how the size of the cell affects the net consumption and absorption rates which depend on the cell surface area, S, and the cell mass, m. (Remember that the cell mass is proportional to its volume. Since the cell density is roughly $ 1 gm/cm^3 $ , the constant of proportionality is 1):

Net rate of nutrient uptake (Absorption) = A S
Net rate of Consumption = C V = C m

The implications are explored below:



Questions (3):

  • (a) What is the ratio of nutrient uptake to nutrient consumption and how does this ratio depend on the radius r of the cell?
  • (b) How does the ratio depend on the mass of the cell?
  • (c) For what cell size would the uptake rate exactly match the consumption rate?
  • (d) Why is this match necessary for the cell to be viable?
  • (e) What would happen if the cell was smaller? Bigger?
  • (f) What kinds of adjustment(s) could the cell make as it grows to compensate for the changes in the ratio of uptake to consumption?

A careful manipulation of powers in the expressions for surface are and volume will have convinced you that the size of the cell has strong implications on its ability to absorb nutrients quickly enough to feed itself. The restriction on oxygen absorption is even more critical than the replenishment of other substances, such as glucose. For these reasons, cells larger than some maximal size ( roughly 1 mm in diameter) rarely occur. Furthermore, organisms that are bigger than this size cannot rely on simple diffusion to carry oxygen to their parts- they must develop a circulatory system to allow more rapid dispersal of such life-giving substances, or else they perish.


More information about the effect of cell size and shape on the cell and its ability to absorb substances can be found in:

M. LaBarbera and S. Vogel (1982) The design of fluid transport systems in organisms. Am. Sci. 70, 54-60
Steven Vogel (1988) Life's Devices, Princeton University Press, Princeton, NJ
L Edelstein-Keshet (1988) Mathematical Models in Biology, McGraw Hill
There is also an interesting presentation on a similar topic on the Web Page The Shape of Things by Derrick Sugg.