Difference between revisions of "DNA Melting Thermodynamics"
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{{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|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>}} | ||
| − | ==Equilibrium concentrations of ssDNA and dsDNA | + | ==Equilibrium concentrations of ssDNA and dsDNA== |
| − | {{LecturePoint|The concentrations of the reaction products are related by the equilibrium constant: <math> | + | {{LecturePoint|The concentrations of the reaction products are related by the equilibrium constant: <math>K_{eq} = \frac{\left [ A \cdot A' \right ]}{\left [ A \right ] \left [ A' \right ]}</math>}} |
| − | {{LecturePoint|The value of <math>\left . | + | {{LecturePoint|The value of <math>\left . K_{eq} \right .</math> is a function of temperature. According to the van't Hoff equation:}} |
:<math> | :<math> | ||
\begin{align} | \begin{align} | ||
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{{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) | |
</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|The transformation from dsDNA to | + | {{LecturePoint|The transformation from dsDNA to ssDNA 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|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 | + | {{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 lower temperatures than G-C rich ones.<ref>Breslauer et al., PNAS 83: 3746, 1986</ref>}} |
==Fraction of dsDNA as a function of temperature== | ==Fraction of dsDNA as a function of temperature== | ||
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{{LecturePoint|Similarly, let <math>\left . C_{DS} \right .</math> be the concentration of double stranded DNA: <math>C_{DS} = {\left [ A \cdot A' \right ]}</math>}} | {{LecturePoint|Similarly, let <math>\left . C_{DS} \right .</math> be the concentration of double stranded DNA: <math>C_{DS} = {\left [ A \cdot A' \right ]}</math>}} | ||
| − | {{LecturePoint|<math>\left . C_T \right .</math> is the total concentration of DNA | + | {{LecturePoint|<math>\left . C_T \right .</math> is the total concentration of DNA. <math>\left . C_T = 2 C_{SS} + 2 C_{DS}\right .</math>}} |
{{LecturePoint|Let <math>\left . f \right .</math> be the fraction of total DNA that is double stranded}} | {{LecturePoint|Let <math>\left . f \right .</math> be the fraction of total DNA that is double stranded}} | ||
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{{LecturePoint|Therefore, <math>C_{SS} = \frac{(1 - f)C_T}{2}</math>}} | {{LecturePoint|Therefore, <math>C_{SS} = \frac{(1 - f)C_T}{2}</math>}} | ||
| − | {{LecturePoint|Now we can solve for <math>\left . f \right .</math>: | + | {{LecturePoint|Now we can solve for <math>\left . K \right .</math> in terms of <math>\left . f \right .</math> and <math>\left . C_T \right .</math>:}} |
:<math> | :<math> | ||
| − | + | 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} | |
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | == | + | {{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,}} | |
:<math> | :<math> | ||
| − | \ | + | e^\left [\frac{\Delta S}{R} - \frac{\Delta H}{R T} \right ] = \frac{2 f}{(1 - f)^2 C_T} |
| − | + | ||
| − | + | ||
| − | + | ||
</math> | </math> | ||
| + | |||
| + | {{LecturePoint|Taking the log of both sides and applying the quadratic formula gives <math>\left . f \right .</math> as a function of | ||
Revision as of 18:03, 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 $ | 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
| $ \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 & = \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_{eq} = e^\left [\frac{\Delta S}{R} - \frac{\Delta H}{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 $ | The transformation from dsDNA to ssDNA 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 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} {{LecturePoint|At the melting point, <math>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} $
{{LecturePoint|Taking the log of both sides and applying the quadratic formula gives $ \left . f \right . $ as a function of
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