Biological Physics/Absorption

When observing light absorption is defined as the.

Absorbance can also be written in the formula for Beer's Law: $$A = \epsilon cl$$

This formula written out is Absorbance = extinction coefficient × concentration × length of light path through the sample. The extinction coefficient is typically calculated by adding up the extinction coefficients of three amino acids that mimic all proteins(Tryptophan, Tyrosine, and Cytosine). Absorption is essentially a unit-less value due to the cancellation of units within the formula: . We can measure absorption using a spectrophotometer or spectrometer designed to measure absorption.

We can calculate the fraction of unfolded proteins, which is basically calculating the probability of unfolded proteins. In this case, the fraction of unfolded (denatured) protein can be written as follows: $$f_d = \frac{A_{max}-A}{A_{max}-A_{min}}$$ This equates to the change in absorption due to unfolding divided by the total change observed when all protein is denatured. Similarly, the protein fraction that remains folded or in its native state is $$f_n = \frac{A}{A_{max}-A_{min}}$$, which is the absorption at a particular concentration, A, relative to the total absorption change observed. Note the sum of fd and fn is 1. The ratio of denatured to native protein fractions is the equilibrium constant. The equilibrium constant (Keq) can be written as: $$K_{eq} = \frac{f_d}{f_n}$$

Recall that entropy is written as: $$S = k_B * ln(\Omega)$$

So the difference in probability between denatured and natured proteins is written as: $$S = k_{B} * ln(\frac{\Omega _{d}}{\Omega _{total}}) - k_{B} * ln(\frac{\Omega_{n}}{\Omega_{total}})$$

If we move Boltzmann's constant outside of the parentheses and realize that the multiplicity for the total change will cancel, we end up with the formula: $$S = k_B[ln(\Omega _d) - ln(\Omega _n)$$ Given algebraic properties for the natural logarithm, the formula can be re-arranged to appear as: $$S = k_B * ln(\frac{\Omega _d}{\Omega _n})$$

Recall that enthalpy is written as: $$\Delta S *T$$ so that $$\Delta H = k_B * T * ln(\frac{\Omega _d}{\Omega _n})$$

However, this says that it is unfavorable to denatured protein even though there is more denatured protein. To correct for this, we simply add a negative sign so that the resulting formula appears as: $$\Delta H = k_{B} * T * ln(\frac{f_d}{f_n})$$