DNA Melting Thermodynamics

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20.309: Biological Instrumentation and Measurement

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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' $

DNA strands in solution.gif

$ \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.[1]

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

AT Pairing.pngGC Pairing.png

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

References

  1. Breslauer et al., Predicting DNA duplex stability from the base sequence PNAS 83: 3746, 1986