Difference between revisions of "DNA Melting Thermodynamics"

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==DNA solution==
 
==DNA solution==
  
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{{LecturePoint|Consider a solution of complementary DNA oligonucleotides <math>\left . A \right .</math> and <math>\left . A' \right .</math>.}}
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{{LecturePoint|Consider a solution containing equal quantities of complementary single stranded DNA oligonucleotides <math>\left . A \right .</math> and <math>\left . A' \right .</math>.}}
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{{LecturePoint|The oligos combine by the reaction <math>1 A + 1 A' \Leftrightarrow 1 AA'</math>}}
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{{LecturePoint|The concentration of the unpaired oligos are denoted by <math>\left [ A \right ]</math> and <math>\left [ A' \right ]</math>. <math>\left [ AA' \right ]</math> is the concentration of paired oligos.}}
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{{LecturePoint|Some of the strands combine to form double stranded DNA. The reaction is governed by the equation <math>1 A + 1 A' \Leftrightarrow 1 A \cdot A'</math>}}
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{{LecturePoint|At equilibrium, the concentrations of the reaction products are related by: <math>
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K = \frac{\left [ AA' \right ]}{\left [ A \right ] \left [ A' \right ]}
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{{LecturePoint|At equilibrium, the concentrations of the reaction products are governed by the relation: <math>
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K = \frac{\left [ A \cdot A' \right ]}{\left [ A \right ] \left [ A' \right ]}
 
</math>}}
 
</math>}}
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{{LecturePoint|<math>\left . C_T \right .</math> is the total concentration of single stranded DNA. <math>
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C_T = \left [ A \right ] + \left [ A' \right ] + 2 \left [ AA' \right ]
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{{LecturePoint|<math>\left . C_T \right .</math> is the total concentration of single stranded DNA.  
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:<math>
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\begin{align}
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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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\end{align}
 
</math>}}
 
</math>}}
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{{LecturePoint|<math>\left . f \right .</math> is the fraction of DNA that is double stranded <math>
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f = \frac{2 \left [ AA' \right ]}{C_T}
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{{LecturePoint|Let <math>\left . f \right .</math> be the fraction of DNA that is double stranded  
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<math>
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f = \frac{2 \left [ A\cdot A' \right ]}{C_T}
 
</math>}}
 
</math>}}
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{{LecturePoint|Solving for <math>\left . K \right .</math> in terms of <math>\left . f \right .</math>:
 
{{LecturePoint|Solving for <math>\left . K \right .</math> in terms of <math>\left . f \right .</math>:
 
:<math>
 
:<math>

Revision as of 03:56, 3 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 $ $ \left . C_T \right . $ is the total concentration of single stranded DNA.
$ \begin{align} C_T & = \left [ A \right ] + \left [ A' \right ] + 2 \left [ AA' \right ] \\ & = 2 \left [ A \right ] + 2 \left [ AA' \right ] \end{align} $
$ \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} $
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