If a protein can be induced to crystallize, its molecular structu...

Question

If a protein can be induced to crystallize, its molecular structure 
can be determined by X-ray crystallography. Protein crystals, 
though solid, contain a large amount of water molecules 
along with the protein. The protein chicken egg-white lysozyme, 
for instance, crystallizes with a unit cell having angles of 
90° and with edge lengths of 7.9 * 103 pm, 7.9 * 103 pm, 
and 3.8 * 103 pm. There are eight molecules in the unit 
cell. If the lysozyme molecule has a molecular weight of 
1.44 * 104 and a density of 1.35 g>cm3, what percent of the 
unit cell is occupied by the protein?

Accepted Answer

Hi everybody. This problem reads by using X ray crystallography. The molecular structure of a protein may be determined if it can be made to crystallize despite being solid protein crystals also contain a significant quantity of water molecules, for instance, a protein extracted from a fruit crystallizes with a unit cell that has 90 degree angles and edges that are 6.2 times 10 to the third PICO meters and 2.1 times 10 to the third people meters long. The unit cell has eight molecules. What portion of the unit cell is made up of protein? If the molecule has a molecular weight of 1.36 times 10 to the fourth and a density of 1.32 g per cubic centimeter. Okay, so for this problem, the question that we want to answer is what portion of the unit cell is made up of protein? Okay. And what we're going to need to recall to solve this problem is percent occupied. Unit cell volume is equal to the total protein volume over the unit cell volume Times 100%. So this is what we're going to refer to in order to solve this problem. So let's first start off by calculating our unit cell volume. Okay, so the unit cell volume. This is equal to the edges. We're gonna multiply the edges given. So we're given three edges and these edges are given in PICO meters. We're going to want to convert this two centimeters. Okay, because our density is given in centimeters. So let's go ahead And right here we're going to want to go from PICO meters two centimeters and let's go ahead and start there. So two of our edges are the same. So we have 6.2 times 10 to the 3rd and 6.2 times 10 to the third. So let's go ahead and convert this two centimeters. So 6.2 times 10 to the third, PICO meters. Okay, we want to go from PICO meters two m first and then from meters we can go two centimeters and one PICO meter. There is 10 to the -12 m. Alright, so our units for PICO meters cancel and now we have units of meters. Now we want to go from meters two centimeters in one m. There is 100 centimeters. So now our units of meters cancel and we're left with centimeters. So when we do this calculation, what we're going to get is 6. times 10 to the negative seven centimeters. Okay, now let's go ahead in because two of the edges are the same. Let's do the last edge. Let's convert this from PICO meters. Two centimeters. Okay, so we have 2.1 times 10 to the third PICO meters. And we're going to do the same thing to go from PICO meters. Two centimeters in one PICO meter. There is 10 to the -12 m. Okay, our PICO meters cancel and one m there is 100 centimeters. All right, so our units four m canceled. So let's do the math. And when we do that, what we're going to get is 2.1 times 10 to the negative seven centimeters. So now we have all of our edges and centimeters. So let's go ahead and move this down. And we're gonna plug those values in. Our unit cell volume is going to equal 6.2 times 10 to the negative seven centimeters times 6.2 times 10 to the negative seven centimeters times 2. times 10 to the negative seven centimeters. Okay, So our unit cell volume then Equals 8.072, 4 times 10 To the negative 20 cubic cm. Because we're multiplying 3cm by each other. Okay, So now we calculated our unit cell volume which is the denominator of our equation. Alright, So now we need to calculate the total protein volume. Okay, So let's go ahead and do that next. So, our total protein volume. What we're going to need in order to solve for a total protein volume is we're going to need to calculate our total protein mass first so that we can go from mass two volume. Alright, so in the problem we're told that There is 1.36 times 10 to the fourth units. Okay, so let's go ahead and and use that as our starting point. All right. So we're going to now calculate our total protein mass. Okay, So this is we're going to start off with our 1. times 10 to the four. And this is units. Okay, so we want to go from units 2g. Because we're looking for mass here. Alright, So the conversion that we're going to use is first we're going to go from units two kg. Alright, in one unit There is 1. 054 Times 10 to the negative 27 kilograms. So our unit's canceled. And now we're in kilograms. And we want to go from kilograms to graham's next. Okay, So in one kg There is 1000 g. All right. So now our total protein mass is equal to 2. times 10 To the - g. All right. So now that we know the g, we can calculate the total protein volume. All right. And we're going to use the density to do this. So now we can calculate total protein volume. Okay, So we're gonna start off with the mass that we have. Alright, so we have to point to this is the value we just calculated 2.2583 times 10 to the negative 20g. And now we're going to use the density that was given. All right. So, the density that was given is 1. grams per cubic centimeter. Alright, So our grams cancel. And now we're left with cubic centimeter, which is volume. Alright, So, let's do this calculation. And when we do that, what we get for a total protein volume is 1.7109 times 10 to the negative 20 cubic centimeters. All right. So now we have all the information that we need. So let's go ahead and go back to our our equation up at the top. All right. So our percent occupied unit cell volume is equal to total protein volume over unit cell volume Times 100. So let's go ahead and solve for that. Alright, so we'll go ahead and write it at the bottom. Our percent occupied unit cell volume equals. So we said the total protein volume is this value here? So we have 1. times 10 to the negative 20 cubic centimeters. Alright, so that's our total protein volume. And this is going to be divided by The unit cell volume, which we calculated up above our unit cell volume is the edges all multiplied by each other. So we got this value. Okay, so eight 0724 times to the -20. Okay, so let's go ahead and write back here. So 8.0724 times 10 To the negative cubic cm. And remember this is going to be multiplied by 100%. All right, so our cubic centimeters is going to cancel. And what we're going to get as our final answer Is 21.19%. So this is the portion of the unit cell that is made up of protein. If the molecule has a molecular weight of 1.3, 6 times 10 to the four. Okay, so that is it for this problem. I hope this was helpful.

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