Cells and Animal Sizes and Shapes: Surface Area (SA), Volume (V) and SA:V Rat

  
Cells and Animal Sizes and Shapes: Surface Area (SA), Volume (V) and SA:V Ratios
Metabolism: Get It In, Move It Out
I. Introduction:
Most cells have size range from 5 µm to 10 µm (5-10 micrometers, microns). The largest body cell is the egg (100 μm) and the smallest is the red blood cell (4-5 μm). Cells come in 4 general shapes: 1) Squamous (flattened, like thick cardboard) (L ≥ W > H) 2) Cuboidal (cube-ish) (Side, S, all equal) 3) Columnar (like upright columns) (L > W ≥ H) and 
4) Spherical (like marbles). These shapes are approximations because most cells lack perfect geometry or symmetry.
Cells must be able to bring in the nutrients (ex. proteins, sugars, ions, O2) across their membranes and send out (across the membrane) their metabolic waste products (ex. CO2, ammonia, NH3). Therefore, there is a limit as to how large a cell can be, because the volume of a cell is a cubic function (V = S3) but the area of a cell is a squared function 
(A = S2). So, as a cells gets larger its volume grows proportionally greater than its area, and their SA to V ratio decreases, which means they cannot get nutrients in fast enough, or move waste out fast enough, to accommodate a very large volume, and mass.
The rate of metabolism is a function of a cell’s volume and mass (larger cells require more to function). But the rate of material exchange across the membrane is a function of its surface area (membranes with greater SA can move materials in and out of a cell faster). So, as cells get larger, they need more to function, but their membrane SA proportionally is smaller and they risk dying (for lack of input of nutrients, or output of wastes). Therefore, cells can only be so large, and they will undergo mitosis (divide to make a daughter cell) in order to have the necessary SA : V ratio.
In anatomy and physiology, the SA to V ratio is directly seen in the design and structure of tissues and organs that have high rates (big demands) on processing large volumes of energy, nutrients and materials. Wherever, there is this high demand the SA:V ratio is high because a lot of materials need to be processed and moved into, or out of cells, tissues and organs.
In marine biology, SA:V ratio is seen in the structure of larger animals, with complex organ systems. Protozoans have no specialized organs for breathing, digestion and excretion because their high SA:V ratio allows them to get in, and out, of the body all required nutrients by diffusion alone. Many simple multicellular animals (ex. sponges, flat worms, jelly fish) also don’t have specialized organ systems because they too have high SA:V ratios. But, as animals get larger, thicker and higher volume, their SA:V ratio decreases and diffusion alone cannot get in, and out, the nutrients they need. So, these phyla (ex. arthropods and vertebrates) have specialized organ systems to get in, and out, their nutrients.
II. Purpose: A) To calculate SA and V quantities for various cell shapes and the SA : V ratio B) To understand and apply the concept of SA : V ratios in various physiological processes and anatomical structures.
III. Equations:
A. For squamous-shaped cells: SA = 2 (L x W) + 4 (W x H) V = L x W x H
B. For columnar-shaped cells: SA = 4 (L x W) + 2 (W x H) V = L x W x H
C. For cuboidal-shaped cells: SA = 6 x S2 V = S3
D. For spherical-shaped cells (π = 3.14), r = radius: SA = 4 x π x r2 V = 1.33 x π x r3
IV. Data Tables:
A. Squamous-shaped Cells (L ≥ W > H)
  
L (μm)
5
10
15
20
5
5
10
 
W (μm)
1
1
1
1
2
2
2
 
H (μm)
1
1
1
1
1
2
1
 
SA (μm2)
 
V (μm3)
 
SA : V (0.1) 
B. Columnar-shaped Cells (H > L = W)
  
L (μm)
5
7.5
10
12.5
15
 
W (μm)
2
2
2
4
4
 
H (μm)
2
2
2
4
4
 
SA (μm2)
 
V (μm3)
 
SA : V (0.1)
C. Cuboid-shaped Cells
  
S (μm)
5
10
15
20
 
SA (μm2)
 
V (μm3)
 
SA : V (0.1)
D. Spherical-shaped Cells
  
r (μm)
5
10
15
20
 
SA (μm2)
 
V (μm3)
 
SA : V (0.1)
V. Conclusion Questions
1. Watch this video: https://www.youtube.com/watch?v=huKUJsqik2I
Summarize in a couple of sentences what you learned.
2. Why is the sphere the worst shape for SA : V ratio (minimum SA : V ratio)? (search Google, and look at Table D)
3. Watch this video: https://www.youtube.com/watch?v=CNkP4rycLbI
A) What did the Agar Cube demonstration show (in terms of SA : V ratios and rate of absorption)?
B) Why do Elephants have such big ears (what does it facilitate)?
C) Why do Flatworms not need specialized organs (ex. heart and lungs) for gas and nutrient exchange?
D) How does being huge allow Whales not to lose too much heat in cold ocean waters?
E) What is a behavioral adaptation that humans do with their arms when it gets cold?
F) What did the agar-cube (cut with ridges) show about SA : V ratios, and how it modeled a villi?
4. Watch this video: https://www.youtube.com/watch?v=wuXSEOKNxN8
A) What did the 2 paper tube demonstration show you? (ex. which dimension creates greater volume, bigger radius or length)?
B) How does Allen’s Rule explain why Eskimos living by the N pole are shorter and rounder, while Africans living by the equator are taller and thinner (in terms of heat exchange and SA : V ratios).