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|Complementary ssDNA strands bond 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.}} | {{LecturePoint|The forward reaction where two ssDNA oligos combine to form dsDNA is called annealing. The reverse process is called thermal denaturation or melting.}} | ||
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{{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>}} | {{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>}} | ||
+ | [[Image:ATvsCG.jpg]] | ||
− | == | + | ==Expression for dsDNA fraction== |
{{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|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:}} |
+ | |||
:<math> | :<math> | ||
e^\left [\frac{\Delta S}{R} - \frac{\Delta H}{R T} \right ] = \frac{2 f}{(1 - f)^2 C_T} | 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 | + | {{LecturePoint|Taking the log of both sides and solving for <math>\left . T \right .</math>,}} |
:<math> | :<math> | ||
T(f) = \frac{\Delta H^{\circ}}{\Delta S^{\circ}-R \ln | T(f) = \frac{\Delta H^{\circ}}{\Delta S^{\circ}-R \ln | ||
(2f/C_T(1-f)^2)} | (2f/C_T(1-f)^2)} | ||
+ | </math> | ||
+ | |||
+ | {{LecturePoint|For simulating DNA melting, it would be nice to have an expression for <math>\left . f \right .</math> in terms of <math>\left . T \right .</math>. Unfortunately, this gets pretty yucky. On the bright side, Matlab and Python are good at calculating yuck.}} | ||
+ | |||
+ | {{LecturePoint|Using the quadratic formula and eliminating the nonphysical root:}} | ||
+ | :<math> | ||
+ | f = \frac{1 + C_T K_{eq} - \sqrt{1 + 2 C_T K_{eq}}}{C_T K_{eq}} | ||
+ | </math> | ||
+ | |||
+ | {{LecturePoint|Substituting from equation 1 gives the desired result.}} | ||
+ | :<math> | ||
+ | f = \frac{1 + C_T e^\left [\frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} \right ] - \sqrt{1 + 2 C_T e^\left [\frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} \right ]}}{C_T e^\left [\frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} \right ]} | ||
</math> | </math> |
Revision as of 19:56, 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 $ | Complementary ssDNA strands bond 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] |
Expression for dsDNA fraction
$ \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 solving for $ \left . T \right . $, |
- $ T(f) = \frac{\Delta H^{\circ}}{\Delta S^{\circ}-R \ln (2f/C_T(1-f)^2)} $
$ \bullet $ | For simulating DNA melting, it would be nice to have an expression for $ \left . f \right . $ in terms of $ \left . T \right . $. Unfortunately, this gets pretty yucky. On the bright side, Matlab and Python are good at calculating yuck. |
$ \bullet $ | Using the quadratic formula and eliminating the nonphysical root: |
- $ f = \frac{1 + C_T K_{eq} - \sqrt{1 + 2 C_T K_{eq}}}{C_T K_{eq}} $
$ \bullet $ | Substituting from equation 1 gives the desired result. |
- $ f = \frac{1 + C_T e^\left [\frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} \right ] - \sqrt{1 + 2 C_T e^\left [\frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} \right ]}}{C_T e^\left [\frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} \right ]} $
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