# DNA Melting Thermodynamics

20.309: Biological Instrumentation and Measurement

## DNA in solution

 $\bullet$ Consider a solution containing equal quantities of complementary single stranded DNA (ssDNA) oligonucleotides $\left . A \right .$ and $\left . A' \right .$.
 $\bullet$ Complementary ssDNA strands bond to form double stranded DNA (dsDNA). The reaction is governed by the equation $1 A + 1 A' \Leftrightarrow 1 A \cdot A'$
 $\bullet$ The forward reaction in which two ssDNA oligos combine to form dsDNA is called annealing. The reverse process is called thermal denaturation or melting.
 $\bullet$ At low temperatures, dsDNA is favored. As the temperature rises, dsDNA increasingly separates into its component ssDNA oligos. (Think about why with respect to enthalpic and entropic considerations.)
 $\bullet$ The melting temperature, $\left . T_m \right .$, is defined to be the point where half of the dsDNA is denatured.
 $\bullet$ Short sequences of about 10-40 base pairs (such as those used in the DNA Melting lab) tend to denature all at once. Longer sequences may melt in segments.
 $\bullet$ Less energy is required to split the double hydrogen bond of A-T pairs than the triple bond of G-C pairs. Thus, A-T rich sequences tend to melt at lower temperatures than G-C rich ones.

Several web tools are available to predict the melting temprature. (See, for example, DINA Melt or Oligocalc.)

## Fundamental equilibrium relationships

 $\bullet$ The concentrations of the reaction products are related by the equilibrium constant: $K_{eq} = \frac{\left [ A \cdot A' \right ]}{\left [ A \right ] \left [ A' \right ]}$
 $\bullet$ The value of $\left . K_{eq} \right .$ is a function of temperature. We can equate the fundamental definition of the standard free energy change with its relationship to the equilibrium constant in solution:
\begin{align} \Delta G^{\circ} & = \Delta H^{\circ} - T \Delta S^{\circ}\\ & = -R T \ln K_{eq}\\ \end{align}
where
$\Delta G^{\circ}$ is the standard change in free energy
$\Delta H^{\circ}$ is the standard enthalpy change
$\left . T \right .$ is the temperature
$\Delta S^{\circ}$ is the standard entropy change
$\left . R \right .$ is the gas constant
 $\bullet$ Solving for $\left . K \right .$:
$K_{eq} = e^\left [\frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} \right ] \quad (1)$

Note that the above equation can be differentiated with respect to temperature to yield the (perhaps once!) familiar van't Hoff equation.

## Theoretical relation of dsDNA fraction and thermodynamic parameters

 $\bullet$ In the lab, the fraction of dsDNA will be measured with a fluorescent dye that preferentially binds to dsDNA. As such, it will be useful to derive an equation that relates the fraction of dsDNA to temperature and the thermodynamic parameters.
 $\bullet$ Let $\left . C_{SS} \right .$ represent the concentration of either single stranded oligonucleotide: $C_{SS} = {\left [ A \right ] = \left [ A' \right ]}$.
 $\bullet$ Similarly, let $\left . C_{DS} \right .$ be the concentration of double stranded DNA: $C_{DS} = {\left [ A \cdot A' \right ]}$
 $\bullet$ $\left . C_T \right .$ is the total concentration of DNA strands. $\left . C_T = 2 C_{SS} + 2 C_{DS}\right .$
 $\bullet$ Let $\left . f \right .$ be the fraction of total DNA that is double stranded
$f = \frac{2 C_{DS}}{C_T} = \frac{C_T - 2 C_{SS}}{C_T} = 1 - 2 \frac{C_{SS}}{C_T}$
 $\bullet$ Therefore, $C_{SS} = \frac{(1 - f)C_T}{2}$
 $\bullet$ Now we can solve for $\left . K \right .$ in terms of $\left . f \right .$ and $\left . C_T \right .$:
$K_{eq} = \frac{C_{DS}}{C_{SS}^2} = \frac{f C_T / 2}{ [(1 - f) C_T / 2] ^ 2} = \frac{2 f}{(1 - f)^2 C_T}$
 $\bullet$ At the melting point, $f = \frac{1}{2}$ by definition and thus $K_{eq} = \frac {4}{C_T}$.
 $\bullet$ Substituting for Keq from equation 1:
$e^\left [\frac{\Delta S}{R} - \frac{\Delta H}{R T} \right ] = \frac{2 f}{(1 - f)^2 C_T} \quad (2)$
 $\bullet$ Taking the log of both sides and solving for $\left . T \right .$:
$T(f) = \frac{\Delta H^{\circ}}{\Delta S^{\circ}-R \ln(2f/C_T(1-f)^2)}$

We now know temperature as a function of dsDNA fraction for a given total DNA strand concentration and DNA identity (therefore ΔH, etc.).

## Simulating DNA melting for tractable calculation of dsDNA fraction

 $\bullet$ For simulating DNA melting experiments, it will be convenient to have an expression for $\left . f \right .$ in terms of $\left . T \right .$. Unfortunately, this gets pretty yucky. On the bright side, Matlab and Python are good at calculating yuck.
 $\bullet$ Taking the log of both sides of equation 2 (after re-substiting in equation 1 for simplicity) and using the quadratic formula (eliminating the nonphysical root):
$f = \frac{1 + C_T K_{eq} - \sqrt{1 + 2 C_T K_{eq}}}{C_T K_{eq}}$
 $\bullet$ Substituting from equation 1 gives the desired result.
$f = \frac{1 + C_T e^\left [\frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} \right ] - \sqrt{1 + 2 C_T e^\left [\frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} \right ]}}{C_T e^\left [\frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} \right ]}$

See the pages DNA Melting Part 1: Simulating DNA Melting - Basics. And if you're interested in a Python implementation see Python:Simulating DNA Melting