Beer's Law

"The deeper the pot, the darker the brew." - Beer's Law as described by my former university chemistry teacher.

Lately I've been doing a lot of work with UV/VIS, and I've been trying to find extinction coefficients for some relatively complex proteins. Turns out there really aren't known extinction coefficients for complex proteins (at least not the ones I'm interested in), but I had fun playing with the Beer-Lambert equation, so I thought I'd share:

A = ε c L

-where A is the absorbance of the solution (usually measured at 280nm for protein), ε is the extinction coefficient, c is the protein concentration in the same units as ε, and L is the path length in cm (= 1cm when using 1cm cuvettes). Solving for c and taking out L since it's equal to 1, we're left with:

c = A / ε 

What a sweet little equation! Right? Well, it's not exactly that simple. Usually I just take coefficients for granted and without a second thought. This time, I wanted to know where ε comes from. To calculate ε - you need to know how many tyrosine, tryptophan, and cysteine residues are in your protein of interest, what the molar absorptivity is for each of the above residues, then take the weighted sum. Viola! The extinction coefficient.

For a well-known, commonly used protein - bovine serum albumin (BSA), the extinction coefficient is 43,824 M⁻¹cm⁻¹. Using this coefficient in the equation above will yield a protein concentration in moles/L (M), which isn't super practical. Thankfully, someone brilliant thought up the idea to calculate the 280nm absorbance of a 1% solution (equivalent to 10mg/mL) of BSA. Since 1% is equal to 10mg/mL, we end up with this equation:

c mg/mL = (A / ε percent)10

To calculate ε percent, we need to find the relationship between ε molar  and ε percent. Prepare to head down the rabbit hole with me for a little while...
Here's our original equation, specific to molar concentration:

c molar = A / ε molar

Converting this to percent concentration looks like this:

c percent =       A        x  molar mass  x      100mL       x       1L       
                 ε molar                                          1g                1000mL

Simplify:

c percent =    (A) (molar mass) 
                     (ε molar) (10)   

And then, looking back to our original equation we already know that c percent = A / ε percent, so:

      A        =   (A) (molar mass) 
  ε percent              (ε molar) (10)   

Solving for  ε percent and simplifying:

ε percent =    (ε molar) (10)   
                  molar mass

Now for the fun part! We can finally calculate ε percent for BSA, and calculate the concentration from the absorbance at 280nm.

ε percent BSA =    (43,824) (10)    = 6.6      Hooray!
                               66,463

From our equation for protein concentration mg/mL, way up the page - the protein concentration for a solution of BSA with an A280 of 0.165 is: drumroll please!

c mg/mL = (A / ε percent)10
c mg/mL = (0.165 / 6.6)10 = 0.25mg/mL

There you have it, Beer's Law and the Beer-Lambert equation in many forms. :)

Reference: ThermoPierce Extinction Coefficients PDF

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Response to J.S. Butcher's comment below!

1/26/13
I'd be happy to try to answer your questions! Whew, I was really worried that post would be incredibly boring... :)

Mmmmkay, here we go!

Absorbance is a measurement of the amount of light absorbed at a given wavelength, and is unitless (when solved for A, all the other units cancel each other out).
The extinction coefficient is the potential for a mole of protein molecules to absorb light at the given wavelength. [I'm not sure why it was named the "extinction" coefficient, but you can use "molar absorptivity" interchangeably. Molar absorptivity might be a better term as the definition is in the name.]

So, absorbance isn't measured in nanometers, rather you can shine a certain wavelength (in nanometers) of light through your solution to see how much is absorbed. For example, you could shine red light (~650nm) through it and see what happens. You can even perform a scan, say from 200nm - 800nm, and compare the absorbances across that range to determine at which wavelength your solution absorbs the most.

280nm is usually the chosen wavelength for proteins because we know that tyrosine, tryptophan, and cysteine (to a lesser extent) will most strongly absorb light at that wavelength. That's also the reason that the tyrosine, tryptophan, and cysteine residues are used to determine the extinction coefficient - the absorbance of the protein solution at 280nm is directly related to the number of these residues present in the solution. I'm sure different amino acids absorb at other wavelengths, and this has probably been explored. But I think tyr, trp, and cys provide the most reliable "sample" of the protein, I guess you could say.

Your explanation of the Beer-Lambert equation is pretty much spot-on. I'd say that ε is a measure of light-absorbing "stuff" in a unit of protein, and c is the concentration of that unit of protein in the solution. I like to imagine little floating particles, but depending on what protein you have, your concentration of protein may stay the same, but each particle may be more transparent or less transparent. That, is ε.

I'll explain why ε has such funky units. Any unit to the -1 power, just means that it's in the bottom, under a division sign. Here's a visual - when we solve the Beer-Lambert equation for ε, we get:

ε =     A     
        c L

So we have cm on the bottom, and molar concentration (M) on the bottom, and A on top, which is unitless. So you can also think of ε as the absorptivity per M, per cm (and n represents the value for a particular protein, 43,824 for BSA):

ε =      n     
       M  cm

To answer your question about cuvettes - we do use quartz cuvettes in the lab, they look like glass, but I think they chip differently, more like a super-compressed salt or something. Quartz cuvettes are only required at lower wavelengths, usually below 380nm (UV range). We also use plastic, or semi-methacrylate cuvettes for some applications because they're suitable in the visible range (380-780nm). Glass is also fine at that range, but they're probably more expensive and definitely not as easy to dispose of.

And finally, to answer your question as to the practical reason for running UV/Vis on proteins - it's a very quick and easy way to determine the precise protein concentration of a solution (if you have a purified protein). Let's say you want 100mL of a 1mg/mL solution of BSA. You go and weigh out 100mg of BSA, then add it to a 100mL volumetric flask, and dilute it to the 100mL line with water. Well, you couldn't have weighed 100.00000mg, and even if you had, the purity of the protein is usually somewhere around 95-98%. There might be moisture in it or something else to begin with. Volumetric flasks, if they are Class A certified, have tolerances of their own depending on the size, for example the tolerance for a 100mL flask might be +/- 0.25mL. There are just a lot of places for error to happen, but luckily, you just pop your final solution into the UV/Vis at 280nm, calculate with the Beer-Lambert equation for concentration in mg/mL, and there you go!

Now, trying to determine the protein concentration of an unknown, complex composition of proteins in a solution is another story, which I will get to in a future post. Oooh, la la ;)