Difference between revisions of "DNA Melting Thermodynamics"

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{{LecturePoint|<math>\left . C_T \right .</math> is the total concentration of single stranded DNA.  
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{{LecturePoint|Let <math>C_{SS} = {\left [ A \right ] = \left [ A' \right ]}</math>.}}
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\begin{align}
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{{LecturePoint|Let <math>C_{DS} = {\left [ A \cdot A' \right ]}</math>}}
C_T & = \left [ A \right ] + \left [ A' \right ] + 2 \left [ AA' \right ] \\
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    & = 2 \left [ A \right ] + 2 \left [ AA' \right ]
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{{LecturePoint|<math>\left . C_T \right .</math> is the total concentration of DNA strands. <math>\left . C_T = 2 C_{SS} + 2 C_{DS}\right .</math>}}
\end{align}
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</math>}}
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{{LecturePoint|Let <math>\left . f \right .</math> be the fraction of DNA that is double stranded  
 
{{LecturePoint|Let <math>\left . f \right .</math> be the fraction of DNA that is double stranded  

Revision as of 18:04, 8 April 2008

DNA solution

$ \bullet $ Consider a solution containing equal quantities of complementary single stranded DNA oligonucleotides $ \left . A \right . $ and $ \left . A' \right . $.
$ \bullet $ Some of the strands combine to form double stranded DNA. The reaction is governed by the equation $ 1 A + 1 A' \Leftrightarrow 1 A \cdot A' $
$ \bullet $ At equilibrium, the concentrations of the reaction products are governed by the relation: $ K = \frac{\left [ A \cdot A' \right ]}{\left [ A \right ] \left [ A' \right ]} $
$ \bullet $ Let $ C_{SS} = {\left [ A \right ] = \left [ A' \right ]} $.
$ \bullet $ Let $ 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 DNA that is double stranded

$ f = \frac{2 \left [ A\cdot A' \right ]}{C_T} $

$ \bullet $ 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} $