Difference between revisions of "DNA Melting Thermodynamics"

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e^\left [\frac{\Delta S}{R} - \frac{\Delta H}{R T} \right ] = \frac{2 f}{(1 - f)^2 C_T}
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e^\left [\frac{\Delta S}{R} - \frac{\Delta H}{R T} \right ] = \frac{2 f}{(1 - f)^2 C_T} \quad (2)
 
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{{LecturePoint|For simulating DNA melting, it would be nice 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.}}
 
{{LecturePoint|For simulating DNA melting, it would be nice 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.}}
  
{{LecturePoint|Using the quadratic formula and eliminating the nonphysical root:}}
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{{LecturePoint|Taking the log of both sides of equation 2 and using the quadratic formula (eliminating the nonphysical root):}}
 
:<math>
 
:<math>
 
f = \frac{1 + C_T K_{eq} - \sqrt{1 + 2 C_T K_{eq}}}{C_T K_{eq}}
 
f = \frac{1 + C_T K_{eq} - \sqrt{1 + 2 C_T K_{eq}}}{C_T K_{eq}}

Revision as of 19:59, 9 April 2008

DNA 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 where two ssDNA oligos combine to form dsDNA is called annealing. The reverse process is called thermal denaturation or melting.

Equilibrium concentrations of ssDNA and dsDNA

$ \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. According to the van't Hoff equation:
$ \begin{align} \Delta G^{\circ} & = \Delta H^{\circ} - T \Delta S^{\circ}\\ & = -R T \ln K\\ \end{align} $
where
$ \Delta G^{\circ} $ is the change in free energy
$ \Delta H^{\circ} $ is the enthalpy change
$ \left . T \right . $ is the temperature
$ \Delta S^{\circ} $ is the 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) $
$ \bullet $ At low temperatures, dsDNA is favored. As the temperature increases, more of the strands separate into their component ssDNA oligos.
$ \bullet $ Short sequences of about 10-40 base pairs (such as those used in the DNA Melting lab) tend to denature all at once, while 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]

File:ATvsCG.jpg

Expression for dsDNA fraction

$ \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. $ \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} $ and $ K_{eq} = \frac {4}{C_T} $.
$ \bullet $ Substituting 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)} $
$ \bullet $ For simulating DNA melting, it would be nice 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 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 ]} $


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