Difference between revisions of "DNA Melting Thermodynamics"
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{{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|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| | + | {{LecturePoint|Hydrogen bonds form between complementary ssDNA strands 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|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== | ==Equilibrium concentrations of ssDNA and dsDNA== | ||
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:<math> | :<math> | ||
\begin{align} | \begin{align} | ||
| − | \Delta G & = \Delta H - T \Delta S\\ | + | \Delta G^{\circ} & = \Delta H^{\circ} - T \Delta S^{\circ}\\ |
& = -R T \ln K\\ | & = -R T \ln K\\ | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
:where | :where | ||
| − | ::<math>\Delta G</math> is the change in free energy | + | ::<math>\Delta G^{\circ}</math> is the change in free energy |
| − | ::<math>\Delta H</math> is the enthalpy change | + | ::<math>\Delta H^{\circ}</math> is the enthalpy change |
| − | ::T is the | + | ::<math>\left . T \right .</math> is the temperature |
| − | ::<math>\Delta S</math> is the entropy change | + | ::<math>\Delta S^{\circ}</math> is the entropy change |
| − | ::R is the [http://en.wikipedia.org/wiki/Gas_constant gas constant] | + | ::<math>\left . R \right .</math>is the [http://en.wikipedia.org/wiki/Gas_constant gas constant] |
{{LecturePoint|Solving for <math>\left . K \right .</math>:}} | {{LecturePoint|Solving for <math>\left . K \right .</math>:}} | ||
:<math> | :<math> | ||
| − | K_{eq} = e^\left [\frac{\Delta S}{R} - \frac{\Delta H}{R T} \right ] \quad (1) | + | K_{eq} = e^\left [\frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} \right ] \quad (1) |
</math> | </math> | ||
{{LecturePoint|At low temperatures, dsDNA is favored. As the temperature increases, more of the strands separate into their component ssDNA oligos.}} | {{LecturePoint|At low temperatures, dsDNA is favored. As the temperature increases, more of the strands separate into their component ssDNA oligos.}} | ||
| − | |||
| − | |||
{{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|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.}} | ||
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= \frac{f C_T / 2}{ [(1 - f) C_T / 2] ^ 2} | = \frac{f C_T / 2}{ [(1 - f) C_T / 2] ^ 2} | ||
= \frac{2 f}{(1 - f)^2 C_T} | = \frac{2 f}{(1 - f)^2 C_T} | ||
| + | </math> | ||
{{LecturePoint|At the melting point, <math>f = \frac{1}{2}</math> and <math>K_{eq} = \frac {4}{C_T}</math>.}} | {{LecturePoint|At the melting point, <math>f = \frac{1}{2}</math> and <math>K_{eq} = \frac {4}{C_T}</math>.}} | ||
| − | |||
| − | |||
{{LecturePoint|Substituting from equation 1,}} | {{LecturePoint|Substituting from equation 1,}} | ||
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</math> | </math> | ||
| − | {{LecturePoint|Taking the log of both sides and applying the quadratic formula gives <math>\left . f \right .</math> as a function of | + | {{LecturePoint|Taking the log of both sides and applying the quadratic formula gives an expression for <math>\left . f \right .</math> as a function of <math>\left . T \right .</math>,}} |
| + | :<math> | ||
| + | T(f) = \frac{\Delta H^{\circ}}{\Delta S^{\circ}-R \ln | ||
| + | (2f/C_T(1-f)^2)} | ||
| + | </math> | ||
Revision as of 18:27, 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 $ | Hydrogen bonds form between complementary ssDNA strands 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] |
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. $ \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} $
| $ \bullet $ | Taking the log of both sides and applying the quadratic formula gives an expression for $ \left . f \right . $ as a function of $ \left . T \right . $, |
- $ T(f) = \frac{\Delta H^{\circ}}{\Delta S^{\circ}-R \ln (2f/C_T(1-f)^2)} $
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