# Calculating fraction of protein unfolded from spectroscopy data

This is a question from a homework pset of previous year. We are given the absorbance at 222nm of both a wild type and mutant protein at different temperatures. We are then asked to calculate various thermodynamic quantities from these data. The first thing it asked for is the Tm of both wt and mutant. To get the Tm, I need the fraction of protein unfolded, since Tm is defined as the temperature at which fraction unfolded is 0.5. However, I am confused as to how to calculate fraction unfolded. In the answer key, the professor used a equation, f=2*(A222-0.5), without any explanation. Can anyone explain to me where does this equation come from?

Edit: Sorry, I forgot to tell you that f stands for fraction unfolded.

Thank you very much Zeyuan

I've plotted the data for ease of answering, though it is not necessary if you know what to look for in the data.

As expected, the melting curves are sigmoidal with horizontal asymptotes at approximately 0.5 and 1. You should be able to see this in the raw data as well: the absorbances hover around 0.5 before rising rapidly and then plateauing around 1.

You want the fraction of unfolded protein as a function of absorbance, so you need to normalize this data to the range of [0,1]. There's a straightforward formula for this which, put into the terms of your question, would be:

$$f = \frac{A_{222} - min(A_{222})}{max(A_{222}) - min(A_{222})}$$

$$f = \frac{A_{222} - 0.5}{1 - 0.5}$$

$$f = \frac{A_{222} - 0.5}{0.5} = 2(A_{222} - 0.5)$$

Intuitively, you might think about it as follows: at low temperatures, when the protein is folded, absorbance is 0.5. Consider this a blank or baseline measurement which gives the absorbance of the solution when no protein is unfolded. Since you are only interested in the signal from unfolded protein, you can subtract this value (0.5) from the rest of the data.