6C: Cell Size

INTRODUCTION

Why are cells so small?   One of the reasons is that being smaller, gives certain advantages of geometry.

One important geometric principle that affects cells is the fact that smaller objects have more surface relative to their volume.  This causes a small cell to become effectively more interactive with its surroundings.  A larger proportion of its inner waste heat, dissolved waste molecules, etc., as well as useful products that it exports to elsewhere in the body, can exit it through its outer “skin” (membrane) when it is small. Likewise, it can take in needed biochemical inputs from the outside (like nutrients) more efficiently.  The following calculations will illustrate this.

CONSIDER A SIMPLE CELL

We will be working with a very simplified model of a cell, just to make calculations easy.  We will consider a cell to be the shape of a cube, 10 microns (millionths of a meter) on a side.  (Show all calculations for your work below).

If each side of this cell (cube) were 10 microns in length, what would be its volume?   (length x width x height)

What would be its surface area?  (6 x width x height)

Now imagine this cell generates wastes of some sort, which it is trying to rid itself of by diffusion across its outer membrane.  If it has 2 of these waste molecules (on average) in each cubic micron of its internal space, how many of these waste molecules are inside of it, total?   (2 x the volume calculated above)

Imagine the rate of disposal (by diffusion) of this molecule out of the cell is 1 molecule leaving the cell through each 1 micron x 1 micron square of its outer surface each second.  How many molecules would be disposed of to the outside each second? (1 x the surface area)

How many would be disposed of in three seconds?   (3 x waste molecules disposed of each second)

What percentage of the total number of waste molecules in the cell would this be?  (waste molecules disposed of in three seconds  /  waste molecules total, x 100)

After three seconds, what would be the concentration of the remaining waste molecules, in molecules/cubic micron?  ([total waste molecules – waste molecules disposed of in three seconds]/volume)

CONSIDER A LARGER CELL (OTHERWISE IDENTICAL)

Now, imagine the cell was 20 microns on each side instead of 10 microns (i.e., it’s dimensions are 20x20x20).  Use the same calculation procedures as above.

What would be its volume?

What would be its surface area?

If 2 waste molecules are in each cubic micron of its space, how many waste molecules would it have?

If it disposes of 1 waste molecule across each square micron of its outer membrane per second, what is the total number of waste molecules disposed of each second?

How many per three seconds?

What percentage of the total number of waste molecules in the cell would this be?

After three seconds, what would be the concentration of the remaining waste molecules?

WHICH CELL (the 10x10x10 or the 20x20x20) HAS A HIGHER CONCENTRATION OF WASTES REMAINING AFTER 3 SECONDS OF DISPOSING OF ITS WASTES TO THE OUTSIDE?

How many times larger is this higher concentration than the concentration in the other cell?  (larger concentration / smaller concentration)

CONSIDER A DIFFERENTLY SHAPED CELL

In addition to size, a cell’s shape influences the connectivity of a cell to its surroundings.  A sphere is the shape with the least connectivity to its surroundings- the most compact shape- it has the least surface area for its volume.  A cube is, likewise, quite compact, though less so than a sphere.  The less compact a shape is, the more it interacts with its surroundings, by virtue of having more surface area.

Consider our cubic 10x10x10 cell again.  A cell of this volume could also have a less compact shape, such as  5x5x40.  Using the figures we already used (i.e., the cell contains 2 waste molecules per cubic micron, and disposes of one through each square micron of surface per second), determine whether this new cell would be able to eliminate more waste than did the 10x10x10 cell.  Show your 5x5x40 calculations below.

What is the volume of the 5x5x40 cell?

What is it’s surface area?

How many waste molecules does it possess?

How many does it dispose of per second?

How many have been disposed of in 3 seconds?

How many are remaining?

Does this 5x5x40 cell get rid of all its waste? (yes/no)