Difference between revisions of "DNA Melting Thermodynamics"
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==DNA solution== | ==DNA solution== | ||
− | {{LecturePoint|Consider a solution containing equal quantities of complementary single stranded DNA oligonucleotides <math>\left . A \right .</math> and <math>\left . A' \right .</math>.}} | + | {{LecturePoint|Consider a solution containing equal quantities of complementary single stranded DNA (ssDNA) oligonucleotides <math>\left . A \right .</math> and <math>\left . A' \right .</math>.}} |
− | {{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>}} | + | {{LecturePoint|Some of the strands combine to form double stranded DNA (dsDNA). The reaction is governed by the equation <math>1 A + 1 A' \Leftrightarrow 1 A \cdot A'</math>}} |
− | {{LecturePoint| | + | ==Equilibrium concentrations of ssDNA and dsDNA depend on temperature== |
− | K = \frac{\left [ A \cdot A' \right ]}{\left [ A \right ] \left [ A' \right ]} | + | |
− | </math>}} | + | {{LecturePoint|The concentrations of the reaction products are related by the equilibrium constant: <math>K = \frac{\left [ A \cdot A' \right ]}{\left [ A \right ] \left [ A' \right ]}</math>}} |
+ | |||
+ | {{LecturePoint|The value of <math>\left . K \right .</math> is a function of temperature. According to the van't Hoff equation:}} | ||
+ | :<math> | ||
+ | \begin{align} | ||
+ | \Delta G & = \Delta H - T \Delta S\\ | ||
+ | & = -R T \ln K\\ | ||
+ | \end{align} | ||
+ | </math> | ||
+ | :where | ||
+ | ::<math>\Delta G</math> is the change in free energy | ||
+ | ::<math>\Delta H</math> is the enthalpy change | ||
+ | ::T is the absolute temperature | ||
+ | ::<math>\Delta S</math> is the entropy change | ||
+ | ::R is the [http://en.wikipedia.org/wiki/Gas_constant gas constant] | ||
+ | |||
+ | {{LecturePoint|Solving for <math>\left . K \right .</math>:}} | ||
+ | :<math> | ||
+ | K = \exp{\frac{\Delta S}{R} - \frac{\Delta H}{R T}} | ||
+ | </math> | ||
+ | |||
+ | {LecturePoint|At low temperatures, dsDNA is favored. As the temperature increases, more of the strands separate into their component ssDNA oligos.}} | ||
+ | |||
+ | {{LecturePoint|The transformation from dsDNA to dsDNA is called denaturation or melting.}} | ||
+ | |||
+ | {{LecturePoint|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.}} | ||
+ | |||
+ | {{LecturePoint|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 a lower temperature than G-C rich ones.}} | ||
+ | |||
+ | ==Fraction of dsDNA as a function of temperature== | ||
{{LecturePoint|Let <math>\left . C_{SS} \right .</math> represent the concentration of either single stranded oligonucleotide: <math>C_{SS} = {\left [ A \right ] = \left [ A' \right ]}</math>.}} | {{LecturePoint|Let <math>\left . C_{SS} \right .</math> represent the concentration of either single stranded oligonucleotide: <math>C_{SS} = {\left [ A \right ] = \left [ A' \right ]}</math>.}} | ||
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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>}} | {{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>}} | ||
− | {{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 total DNA that is double stranded}} |
− | <math> | + | :<math> |
− | f = \frac{2 \ | + | f = \frac{2 C_{DS}}{C_T} = \frac{C_T - 2 C_{SS}}{C_T} = 1 - 2 \frac{C_{SS}}{C_T} |
− | </math>}} | + | </math> |
+ | |||
+ | {{LecturePoint|Therefore, <math>C_{SS} = \frac{(1 - f)C_T}{2}</math>}} | ||
− | {{LecturePoint| | + | {{LecturePoint|Now we can solve for <math>\left . f \right .</math>: |
:<math> | :<math> | ||
\begin{align} | \begin{align} | ||
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==Free energy== | ==Free energy== | ||
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Let <math>C_T \quad</math> be the total concentration of ssDNA. | Let <math>C_T \quad</math> be the total concentration of ssDNA. |
Revision as of 17:19, 9 April 2008
Contents
DNA solution
$ \bullet $ | Consider a solution containing equal quantities of complementary single stranded DNA (ssDNA) oligonucleotides $ \left . A \right . $ and $ \left . A' \right . $. |
$ \bullet $ | Some of the strands combine to form double stranded DNA (dsDNA). The reaction is governed by the equation $ 1 A + 1 A' \Leftrightarrow 1 A \cdot A' $ |
Equilibrium concentrations of ssDNA and dsDNA depend on temperature
$ \bullet $ | The concentrations of the reaction products are related by the equilibrium constant: $ K = \frac{\left [ A \cdot A' \right ]}{\left [ A \right ] \left [ A' \right ]} $ |
$ \bullet $ | The value of $ \left . K \right . $ is a function of temperature. According to the van't Hoff equation: |
- $ \begin{align} \Delta G & = \Delta H - T \Delta S\\ & = -R T \ln K\\ \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
$ \bullet $ | Solving for $ \left . K \right . $: |
- $ K = \exp{\frac{\Delta S}{R} - \frac{\Delta H}{R T}} $
{LecturePoint|At low temperatures, dsDNA is favored. As the temperature increases, more of the strands separate into their component ssDNA oligos.}}
$ \bullet $ | The transformation from dsDNA to dsDNA is called denaturation or melting. |
$ \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 a lower temperature than G-C rich ones. |
Fraction of dsDNA as a function of temperature
$ \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} $ |
{{LecturePoint|Now we can solve for $ \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
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} $