DNA Melting Thermodynamics

DNA solution

• Consider a solution of complementary DNA oligonucleotides $ \left . A \right . $ and $ \left . A' \right . $.
• The oligos combine by the reaction $ 1 A + 1 A' \Leftrightarrow 1 AA' $
• The concentration of the unpaired oligos are denoted by $ \left [ A \right ] $ and $ \left [ A' \right ] $. $ \left [ AA' \right ] $ is the concentration of paired oligos.
• At equilibrium, the concentrations of the reaction products are related by: $ K = \frac{\left [ AA' \right ]}{\left [ A \right ] \left [ A' \right ]} $(eq. 1)
• $ \left . C_T \right . $ is the total concentration of single stranded DNA. $ C_T = \left [ A \right ] + \left [ A' \right ] + 2 \left [ AA' \right ] $
• $ \left . f \right . $ is the fraction of DNA that is double stranded $ f = \frac{2 \left [ AA' \right ]}{C_T} $
• Solving for $ \left . K \right . $ in terms of $ \left . f \right . $:

$ \begin{align} K & = \frac{\left [ AA' \right ]}{\left ( \frac{1}{2} C_T - \left [ AA' \right ] \right ) ^ 2} = \frac{\left [ AA' \right ]}{C_T^2 \left ( \frac{1}{2} - \frac{\left [ AA' \right ]}{C_T} \right ) ^ 2} = \frac{\frac{2 \left [ AA' \right ]}{C_T}}{2 C_T \left ( \frac{1}{2} - \frac{1}{2}\frac{2 \left [ AA' \right ]}{C_T} \right ) ^ 2} \\ & = \frac{f}{2 C_T \left ( \frac{1}{2} - \frac{1}{2} f \right ) ^2} \end{align} $

Free energy

$ \begin{align} \Delta G & = \Delta H - T \Delta S \quad (1)\\ & = -R T \ln K \quad (2)\\ \end{align} $

where

$ \Delta G $ is the change in free energy

$ \Delta H $ is the enthalpy change

T is the absolute temperature

$ \Delta S $ is the entropy change

R is the gas constant

K is the dissociation constant

Let $ C_T \quad $ be the total concentration of ssDNA.

$ \begin{align} C_{ss} & = \left [ A \right ] = \left [ A' \right ] \quad (3) \\ C_{ds} & = \left [ AA' \right ] \quad (4) \\ \end{align} $