# areal density and mean spacing

The area fraction associated with membrane proteins in the red blood cell (diameter = 8 Âµm) membrane is roughly 23%, while the lipids themselves take up roughly 77% of the membrane area.

a.Approximately how many phospholipids make up the red blood cell membrane?

b.How many membrane proteins would you estimate are in the red blood cell membrane

c.What is their mean spacing?

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

We are given the diameter of a red blood cell as 8 Âµm (so radius of 4 Âµm), and the proportion of the membrane that is lipids vs proteins.

To go from a proportion to an absolute number, we'll need to find out the total area of the red blood cell, as well as the area taken up by each component.

Finding the surface area of the red blood cell is a geometry problem that requires some assumptions as to the shape of the red blood cell. As you may know, in real life red blood cells are disks with indents on either side, which makes them a bit of a complicated shape to get the area from just the radius. If you're interested, there is a paper by Deuling and Helfrich (Biophysical Journal 16(8) pp 861-868) describing the formulas that closely approximate a red blood cell. With those and no small measure of calculus, you can find the surface area.

A more reasonable approach is to make an assumption as to whether the red blood cell more closely approximates a sphere or a disk. I might suggest assuming that the red blood cell is a disk, in which case the surface area, A, is given by:

A = pi * r^2 * 2

Where that last factor of two is to represent the fact that there are two surfaces to the red blood cell, i.e., top and bottom. (For a sphere, the formula would be A = 4 * pi * r^2, a factor of two higher than the disk assumption). Depending on the guidance from your instructor and your own reasoning, a case can be made for "splitting the difference" and just assuming that the red blood cell has an area of A = 3 * pi * r^2.

Whatever method you use for determining the surface area of the blood cell, you'll now have some value for A.

For part (a) you first multiply that by the proportion of the total area taken up by lipids (77%) to get the total area taken up by phosholipids. Then, divide that by the area each phospholipid takes up (call it "p") to get the number of phospholipids.

# phospholipids = A [Âµm^2] * 0.77 / p [Âµm^2/phospholipid]

There is one potential catch here, and that's what's known as packing efficiency. If there is empty space between the lipids, then some of the total surface area will be wasted on empty space, and you'll have fewer total phospholipids than would be calculated here. This can be seen for example when you try to pack circular coins onto a piece of paper -- if you divide the area of the paper by the area of the coins, you won't be able to fit as many coins as that formula would tell you because of the packing efficiency. However, it's entirely possible that phospholipids are not simple circles and do in fact pack perfectly, or just as good, that whoever estimated the figure for area per phospholipid has already taken that into account for you. Either way, you should state your assumptions in how you got to whatever answer you arrive at.

For part (b), the calculation is the same, just replace with the appropriate values for proteins (and the proportion goes from 0.77 to 0.23).

Part (c) is a little trickier: you have to try to estimate how far apart each protein is. Here is one way I can think of to approach that.

First assume that you have the same number of proteins as calculated in part (b), but imagine if the proteins completely covered the surface of the cell. In that case, find out how much area each protein would get (# proteins / A). Let's call that "a" (small area).

a = # proteins/A

Then, assume that the proteins take up a circular area, and find out what the radius of that area would be. Work backwards from the formula for the area of a circle:

a = pi * r^2

r = sqrt(a/pi)

So then the centre of one protein would be 2*r away from the centre of another protein, on average.

You may need to subtract from that distance the diameter of the protein itself, depending on whether "mean spacing" is defined as the distance from the centre of one protein to the centre of another, or from the edge of one to the near edge of the next.

Hope that helps!

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