DNA Melting Thermodynamics

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20.309: Biological Instrumentation and Measurement

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DNA in solution

<math>\bullet</math> Consider a solution containing equal quantities of complementary single stranded DNA (ssDNA) oligonucleotides <math>\left . A \right .</math> and <math>\left . A' \right .</math>.
<math>\bullet</math> Complementary ssDNA strands bond to form double stranded DNA (dsDNA). The reaction is governed by the equation <math>1 A + 1 A' \Leftrightarrow 1 A \cdot A'</math>

DNA strands in solution.gif

<math>\bullet</math> The forward reaction in which two ssDNA oligos combine to form dsDNA is called annealing. The reverse process is called thermal denaturation or melting.
<math>\bullet</math> 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.)
<math>\bullet</math> The melting temperature, <math>\left . T_m \right .</math>, is defined to be the point where half of the dsDNA is denatured.
<math>\bullet</math> 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.
<math>\bullet</math> 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

<math>\bullet</math> {{{1}}}
<math>\bullet</math> The value of <math>\left . K_{eq} \right .</math> 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:
<math>

\begin{align} \Delta G^{\circ} & = \Delta H^{\circ} - T \Delta S^{\circ}\\ & = -R T \ln K_{eq}\\ \end{align} </math>

where
<math>\Delta G^{\circ}</math> is the standard change in free energy
<math>\Delta H^{\circ}</math> is the standard enthalpy change
<math>\left . T \right .</math> is the temperature
<math>\Delta S^{\circ}</math> is the standard entropy change
<math>\left . R \right .</math> is the gas constant
<math>\bullet</math> Solving for <math>\left . K \right .</math>:
<math>

K_{eq} = e^\left [\frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} \right ] \quad (1) </math>

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

<math>\bullet</math> 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.
<math>\bullet</math> {{{1}}}
<math>\bullet</math> {{{1}}}
<math>\bullet</math> {{{1}}}
<math>\bullet</math> Let <math>\left . f \right .</math> be the fraction of total DNA that is double stranded
<math>

f = \frac{2 C_{DS}}{C_T} = \frac{C_T - 2 C_{SS}}{C_T} = 1 - 2 \frac{C_{SS}}{C_T} </math>

<math>\bullet</math> {{{1}}}
<math>\bullet</math> Now we can solve for <math>\left . K \right .</math> in terms of <math>\left . f \right .</math> and <math>\left . C_T \right .</math>:
<math>

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}

</math>

<math>\bullet</math> {{{1}}}
<math>\bullet</math> Substituting for Keq from equation 1:
<math>

e^\left [\frac{\Delta S}{R} - \frac{\Delta H}{R T} \right ] = \frac{2 f}{(1 - f)^2 C_T} \quad (2) </math>

<math>\bullet</math> Taking the log of both sides and solving for <math>\left . T \right .</math>:
<math>

T(f) = \frac{\Delta H^{\circ}}{\Delta S^{\circ}-R \ln(2f/C_T(1-f)^2)} </math>

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

<math>\bullet</math> For simulating DNA melting experiments, it will be convenient to have an expression for <math>\left . f \right .</math> in terms of <math>\left . T \right .</math>. Unfortunately, this gets pretty yucky. On the bright side, Matlab and Python are good at calculating yuck.
<math>\bullet</math> 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):
<math>

f = \frac{1 + C_T K_{eq} - \sqrt{1 + 2 C_T K_{eq}}}{C_T K_{eq}} </math>

<math>\bullet</math> Substituting from equation 1 gives the desired result.
<math>

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 ]} </math>

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