Plants & Human Affairs (BIOL106)  -  Stephen G. Saupe, Ph.D.; Biology Department, College of St. Benedict/St. John's University, Collegeville, MN 56321; ssaupe@csbsju.edu; http://www.employees.csbsju.edu/ssaupe

Surface-to-Volume Ratios in Biology

Introduction: 
    The purpose of this lab is to introduce you to the biological importance of surface-to-volume ratios (abbreviated S/V).  Surface area (SA), which is expressed in squared units (e.g., mm², �m²), is the amount of an object that is directly exposed to the environment.  For a cell, it would represent the area of the plasma membrane and for a person it would represent the amount of skin.  Volume is a rough measure of the size of a structure and the amount of space it occupies.  Volume is expressed by cubic units (e.g., mm³, cm³ = milliliters).  The surface-to-volume ratio (S/V) refers to the amount of surface a structure has relative to its size; or stated in a slightly more gruesome manner, S/V ratio is the amount of "skin" compared to the amount of "guts."  To calculate the S/V ratio, simply divide the surface area by the volume.   

    The reason that surface-to-volume ratios are important is because a cell or organism continuously exchanges materials, such as food, waste, water, and heat, with its environment.  Depending on the circumstances, it may be advantageous to have a small S/V while at other times a large S/V is an advantage.  Thus, optimizing S/V ratios has been a driving force in the evolution of all organisms.  Since S/V is a function of both size and shape, these have also been under strong evolutionary pressure.   

    To begin our studies we will examine the effects of both size and shape on surface-to-volume ratios.  Then, we will use this information to answer fundamental questions about cell size and metabolic rate.  Finally, we will apply our experiences to a variety of biological situations. 

Exercise 1.  The Relationship between Volume, SA, and S/V Ratio

In this exercise we will explore the mathematical relationship between volume, surface area, and the S/V ratio.  Consider a cube that is one unit on a side.  If it increases in size, obviously both the surface area and volume will increase.  But, by how much?  And, how will this affect the surface-to-volume ratio?  In this exercise we will calculate the surface area, volume and s/v ratio for a series of cubes, graph the results, and then attempt to answer these questions and others.  First, let�s make a few predictions. 

Hypothesis 1.1:  As a cube gets larger, its S/V ratio will (select one:   decrease / remain the same / increase).

 Hypothesis 1.2:  As a cube gets larger, the surface area of the cube will increase by (select one:  twice / the square of / the cube of) the linear dimension. 

Hypothesis 1.3:  As a cube gets larger, the volume of the cube will increase by (select one:  three times / the square of / the cube of) the linear dimension. 

Method:  Complete Table 1 for a series of cubes of varying size (equations):

Exercise 2. S/V Ratios In Flattened Objects
    In this exercise we will explore how flattening an object impacts the surface to volume ratio. Consider a cube made out of clay that is 8 x 8 x 8 mm on a side. Then, imagine that we can flatten this cube making it thinner and thinner while maintaining the original volume. What will happen to the surface area and s/v ratio as the cube is flattened?  Let�s make a hypothesis: 

Hypothesis 2.1:  As the cube is flattened, the s/v ratio will (select one: increase / decrease / remain the same).

 Method:  Complete Table 2.  To calculate the volume of a rectangle, multiply length by width by height.  To calculate the surface area of a rectangle you will need to calculate the surface area of each side and then add these values together.

 
Exercise 3.  S/V Ratios In Elongated Objects
    In this exercise we will explore how elongating an object impacts the surface to volume ratio. Once again we�ll start with a cube of clay that is 8 x 8 x 8 mm on a side. Now, imagine that we pull on the ends to make it longer and longer while maintaining the original volume. What will happen to the surface area and s/v ratio as the box is flattened?

            Hypothesis 3.1:  As the cube is elongated the s/v ratio will (select one:  increase / decrease / remain the same).

Method:  Complete Table 3.  Do your data support your hypothesis?

Exercise 4. Shape and S/V Ratio

In this exercise we will explore the impact of shape on surface to volume ratios.  Consider three objects � a sphere (or ball), a cube and a long skinny box (or filament) � that have the same volume.  Which object will have the smallest S/V ratio?  Which object is in contact with a greater portion of its environment?  One way to answer this latter question is to calculate the amount or volume of the environment that is within 1.0 mm of the object.  This is important since all organisms/structures exchange materials with their environment and the exchange occurs between the organism and the environment directly in contact with the object. 

Hypothesis 4.1.  The shape with the largest S/V ratio will be the (select one: cube / sphere / filament)

Hypothesis 4.2.  The shape that is exposed to the greatest amount of its environment is the (select one: cube / sphere / filament)

Method:

1.      Calculate volume, surface area and S/V ratio for each of three different-shaped objects in Table 4. 

2.      To calculate the volume of environment within 1.0 mm of each object, imagine that the object is surrounded by a larger object that is 1.0 mm larger on all sides.  Calculate the volume of this larger object and then subtract the volume of the smaller shape.

Exercise 5. Why Are Cells Small?

          The typical eukaryotic cell is rather small, approximately 100 μm in diameter.  With a few exceptions, the cells of all organisms are about this size.  Thus, the reason that an elephant is bigger than a mouse is because they have more cells, not larger ones.  So, why are the cells so small? 

In this exercise we will use a cylinder of agar to serve as a model cell. The agar has an acid-sensitive dye called bromocresol green incorporated into it.  The dye turns from blue to yellow (clear) in the presence of acid.  To simulate the cell feeding or obtaining a critical nutrient from its environment, we will place the agar cylinder in a container of vinegar and monitor the color of the cylinder.  The uptake of acid, and hence colorless areas of the cell model, will represent the uptake of food/nutrients by the cell. From this, we can calculate the percent of each cell that was fed during the incubation period.   

Hypothesis 5.1:  The cell model in which the greatest percentage of its volume will get fed is the (select one:  small / large) one. 

Hypothesis 5.2:  The cell model in the greatest danger of starving is the (select one:  small / large) one. 
 

Methods:

1.      Obtain 2 agar cylinders (cell models), one small and one large.  Wearing gloves, measure the length and diameter of each cell and record your data in Table 5.1.

2.      Using a plastic spoon, carefully place each cylinder in a container containing vinegar (be careful).

3.      Allow the cylinders to sit in the vinegar for a few minutes until most of the blue color is gone from the smallest cylinder. Then, remove both models with the plastic spoon and place them on a piece of paper towel.

4.      Measure the length and diameter of the colored areas remaining in each model.  Record these data in table 5.

5.      Complete the calculations.

6.      What do you conclude about your hypotheses?

Exercise 6. Why Do Mice Have Greater Metabolic Rates Than Elephants?
   
    It is well known that there is an inverse relationship between body size and metabolic rate. That is, the larger the animal the lower its basal rate of metabolism.  Therefore, mice have a greater metabolic rate than elephants.  The purpose of this exercise is to determine the reason for this relationship. We will use beakers of water to represent organisms with the same shape but different size.  First let's make a few predictions: 
 

Hypothesis 6.1.  The (select one:  small / large) beaker will loose the most total heat.
 

Hypothesis 6.2.  The (select one:  small / large ) beaker will loose the heat at the greatest rate.

Hypothesis 6.3.  The (select one:  small / large ) beaker would require the most total heat input to maintain a constant temperature.

Hypothesis 6.4.  The (select one:  small / large ) beaker would require the greatest total heat input relative to its size to maintain a constant temperature.

Method:

1. Obtain an aluminum pan in which 30 and 100 mL beakers are glued to the bottom.  Pipet 8 mL of water into the small beaker.  Then using a ruler, measure in centimeters the height and width of the water column in the beaker and record your data in Table 6 (rows 1 & 2).
    
2. Using a graduate cylinder, put 80 mL of water into the medium-sized beaker and measure the height and width of the water column.  Record your data.
    
3. Empty the water from the beakers and then pack ice around the empty containers in the pan to a depth of about two inches. Then add cold water to bring the water level up to the top of the ice.
    
4. Measure 80 mL of hot water from the hot plate (be careful!) with a graduate cylinder and place it into the 100 mL beaker.
    
5. Place the thermometer in the hot water. At first the temperature will rise. When it begins to fall again, measure the temperature. If it is below 40 C, start again with hotter water.
    
6. When the temperature reaches 40 C, record the time. Exactly two minutes later record the temperature again. Record your data in Table 6, row 9.
    
7. Repeat steps 4 - 6 except pipet 8 mL of water into the smaller container. 
    

Data & Analysis:   To analyze our data, complete the following steps.

1. Calculate the radius of each container (row 3)
    
2. Calculate the surface area of the water column in each beaker using the equation for a cylinder (row 5).  Although we could use our measurements of water column height and width to calculate the volume of each container (row 4), we don�t need to do so since we�ve already measured the volume of water in each with a graduate cylinder. 
    
3. Calculate the surface/volume ratio (row 6)
    
4. Calculate the initial and final heat content (rows 8 and 10) by multiplying the temperature (rows 7 and 9) by the volume of water (row 4) by the specific heat of water (4.2 joules/cm³ C). [Note - Specific heat is an indication of the amount of heat it takes to change the temperature of a substance.  Joules are a standard energy unit.] Thus, this equation is:
        Heat content (joules) = temp. (C) x vol. (cm³) x specific heat of water (4.2 joules/cm³ C)
    
5. Calculate the total heat loss during the two-minute experiment (row 11) by subtracting the final heat content (row 10) from the initial heat content (row 8).
    
6. Calculate the percent of heat lost by each container (row 12) by the following equation:  (initial heat content - final heat content)/initial heat content  x 100
    
7. Calculate the relative heat loss (joules lost/cm³) for each volume (row 13) by dividing the total heat loss (row 11) by the volume of the vessel (row 4).
    
8. Calculate the relative heat loss (joules lost/cm³) for each volume (row 13) by dividing the total heat loss by the volume of the vessel (row 4).
    
9. Calculate the percent heat lost (row 14).
   

Exercise 7.  Biological Applications of S/V
    Although these exercises are relatively simple, they provide the basis to explain many biological observations.  Listed below are a just a few questions that can be answered using S/V reasoning. 

1. Explain the advantages/disadvantages of block vs. cube ice.
2. Describe the scientific inaccuracy in the episode of Goldilocks and the bowls of porridge.
3. Explain why lungs, gills and intestines have the shape they do.
4. Explain why plants are essentially a cluster of filaments, whereas animals are blobs. In other words, why is a thin, elongated rectangle a good model for a plant, but a square a good model for an animal?
5. Describe the shape of a radiator? Explain why it has this shape.
6. Medieval churches were often built in the shape of a crucifix. Explain why.
7. The earth is geologically active (has a molten core) but the moon is apparently no longer geologically active. Explain why using S/V ratios.
8. Shrews have a reputation for being "mean". In other words, they must feed constantly. Explain why.
9. Why are there few small animals in the arctic?
10. Explain how S/V ratios relate to the form of plants that have evolved in mesic, xeric and hydric environments.
11. Explain why the cells of the spongy layer of a plant leaf are irregularly shaped.
12. Explain why leaves are thin and flat.
13. Offer an explanation why elephants have large, flat ears?
14. Explain why shrews are voracious feeders.
15. Why are cells small?
16. Explain why roots have "hairs."
17. Explain why the shape of animals is basically "spherical", whereas plants and fungi are "filamentous".
    Explain why the rate of cell growth slows as a cell gets larger.
18. Explain why cells divide when they get large.
19. Explain why small animals have a higher metabolic rate than large animals
    

  References:

• Barrett, D. 1983. Body size and temperature: an extended approach. J. Biol. Educ. 71:78.
• Cohen, A, AB Moreh, R Chayoth. 1999. Hands-on Method for Teaching the Concept of the Ratio Between Surface Area & Volume. The American Biology Teacher 61: 691 - 696.
• Diamond, Jared. 1989. How cats survive falls from New York skyscrapers. Natural History, August, pp 20 - 26.
• Haldane, J.B.S. On Being the Right Size.
• Gould, S. 1977. Size and shape. In Ever Since Darwin. Norton, NY.
• Moog, F. (1949) Gulliver was a bad biologist. Scientific American 18(5): 52 - 56 (November)
• Stanek, Jr., J. A. (1983) Why don't cells grow larger? American Biology Teacher 45:393-395.
• Flannery, M (1986) Why is size so important? American Biology Teacher 51: 122 - 125.
• Morowitz, H (19  ) Of mammals great and small.  In The Wine of Life.  St. Martin's Press.
• Went, FW (1968) The size of man.  Scientific American 56: 400 - 413.
• Built to scale.  (1999) Science News 156: 249 - 251.
  

Useful Equations:

Supplies Required:

• Cell Exercise:  Small tray with 2-3 paper towels, spoon, jar of vinegar, ruler, pair of gloves (1/group); cell models (1 of each size per group; 0.04% bromocresol green, 2% agar)
  
• Heat/Metabolism Demo:  2 pans with 8 and 80 ml beakers glued into the bottom; 2 thermometers; hot plate; flask with water; 100 ml graduate cylinder; 10 ml pipet and pump; ice; hot pad/glove

Last updated:  11/10/2005 / � Copyright  by SG Saupe