Difference between revisions of "DNA Melting Thermodynamics"
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− | + | [[Category:20.309]] | |
+ | [[Category:DNA Melting Lab]] | ||
+ | {{Template:20.309}} | ||
− | + | ==DNA in solution== | |
− | {{LecturePoint| | + | {{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|At | + | {{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>}} |
− | + | ||
− | </math>}} | + | [[Image:DNA_strands_in_solution.gif]] |
+ | |||
+ | {{LecturePoint|The forward reaction in which two ssDNA oligos combine to form dsDNA is called annealing. The reverse process is called thermal denaturation or melting.}} | ||
+ | |||
+ | {{LecturePoint|At low temperatures, dsDNA is favored. As the temperature rises, dsDNA increasingly separates into its component ssDNA oligos. (Think about why with respect to enthalpic and entropic considerations.)}} | ||
+ | |||
+ | {{LecturePoint|The melting temperature, <math>\left . T_m \right .</math>, is defined to be the point where half of the dsDNA is denatured.}} | ||
+ | |||
+ | {{LecturePoint|Short sequences of about 10-40 base pairs (such as those used in the DNA Melting lab) tend to denature all at once. 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 lower temperatures than G-C rich ones.<ref>Breslauer et al., [http://www.pnas.org/content/83/11/3746.full.pdf Predicting DNA duplex stability from the base sequence] PNAS 83: 3746, 1986</ref>}} | ||
+ | |||
+ | Several web tools are available to predict the melting temprature. (See, for example, [http://mfold.rna.albany.edu/?q=DINAMelt/Hybrid2 DINA Melt] or [http://www.basic.northwestern.edu/biotools/oligocalc.html Oligocalc].) | ||
+ | |||
+ | [[Image:AT_Pairing.png]][[Image:GC_Pairing.png]] | ||
+ | |||
+ | ==Fundamental equilibrium relationships== | ||
+ | |||
+ | {{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 . K_{eq} \right .</math> is a function of temperature. We can equate the fundamental definition of the standard free energy change with its relationship to the equilibrium constant in solution:}} | ||
+ | :<math> | ||
+ | \begin{align} | ||
+ | \Delta G^{\circ} & = \Delta H^{\circ} - T \Delta S^{\circ}\\ | ||
+ | & = -R T \ln K_{eq}\\ | ||
+ | \end{align} | ||
+ | </math> | ||
+ | :where | ||
+ | ::<math>\Delta G^{\circ}</math> is the standard change in free energy | ||
+ | ::<math>\Delta H^{\circ}</math> is the standard enthalpy change | ||
+ | ::<math>\left . T \right .</math> is the temperature | ||
+ | ::<math>\Delta S^{\circ}</math> is the standard entropy change | ||
+ | ::<math>\left . R \right .</math> is the [http://en.wikipedia.org/wiki/Gas_constant gas constant] | ||
+ | |||
+ | {{LecturePoint|Solving for <math>\left . K \right .</math>:}} | ||
+ | :<math> | ||
+ | K_{eq} = e^\left [\frac{\Delta S^{\circ}}{R} - \frac{\Delta H^{\circ}}{R T} \right ] \quad (1) | ||
+ | </math> | ||
+ | |||
+ | Note that the above equation can be differentiated with respect to temperature to yield the (perhaps once!) familiar van't Hoff equation. | ||
+ | |||
+ | ==Theoretical relation of dsDNA fraction and thermodynamic parameters== | ||
+ | |||
+ | {{LecturePoint|In the lab, the fraction of dsDNA will be measured with a fluorescent dye that preferentially binds to dsDNA. As such, it will be useful to derive an equation that relates the fraction of dsDNA to temperature and the thermodynamic parameters.}} | ||
{{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 . 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} | |
− | + | ||
− | + | ||
− | + | ||
</math> | </math> | ||
− | == | + | {{LecturePoint|At the melting point, <math>f = \frac{1}{2}</math> by definition and thus <math>K_{eq} = \frac {4}{C_T}</math>.}} |
+ | |||
+ | {{LecturePoint|Substituting for ''K<sub>eq</sub>'' 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} \quad (2) |
− | \Delta | + | |
− | + | ||
− | + | ||
</math> | </math> | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | + | {{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(2f/C_T(1-f)^2)} | |
− | + | ||
− | + | ||
− | + | ||
</math> | </math> | ||
+ | |||
+ | We now know temperature as a function of dsDNA fraction for a given total DNA strand concentration and DNA identity (therefore'' ΔH'', etc.). | ||
+ | |||
+ | ==Simulating DNA melting for tractable calculation of dsDNA fraction== | ||
+ | |||
+ | {{LecturePoint|For simulating DNA melting experiments, it will be convenient 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|Taking the log of both sides of equation 2 (after re-substiting in equation 1 for simplicity) and using the quadratic formula (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> | ||
+ | |||
+ | See the pages [[DNA Melting Part 1: Simulating DNA Melting - Basics]]. And if you're interested in a Python implementation see [[Python:Simulating DNA Melting]] | ||
+ | |||
+ | ==References== | ||
+ | <References /> | ||
+ | |||
+ | {{Template:20.309 bottom}} |
Latest revision as of 21:43, 1 November 2013
DNA in 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 in which two ssDNA oligos combine to form dsDNA is called annealing. The reverse process is called thermal denaturation or melting. |
$ \bullet $ | At low temperatures, dsDNA is favored. As the temperature rises, dsDNA increasingly separates into its component ssDNA oligos. (Think about why with respect to enthalpic and entropic considerations.) |
$ \bullet $ | The melting temperature, $ \left . T_m \right . $, is defined to be the point where half of the dsDNA is denatured. |
$ \bullet $ | Short sequences of about 10-40 base pairs (such as those used in the DNA Melting lab) tend to denature all at once. 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] |
Several web tools are available to predict the melting temprature. (See, for example, DINA Melt or Oligocalc.)
Fundamental equilibrium relationships
$ \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. We can equate the fundamental definition of the standard free energy change with its relationship to the equilibrium constant in solution: |
- $ \begin{align} \Delta G^{\circ} & = \Delta H^{\circ} - T \Delta S^{\circ}\\ & = -R T \ln K_{eq}\\ \end{align} $
- where
- $ \Delta G^{\circ} $ is the standard change in free energy
- $ \Delta H^{\circ} $ is the standard enthalpy change
- $ \left . T \right . $ is the temperature
- $ \Delta S^{\circ} $ is the standard 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) $
Note that the above equation can be differentiated with respect to temperature to yield the (perhaps once!) familiar van't Hoff equation.
Theoretical relation of dsDNA fraction and thermodynamic parameters
$ \bullet $ | In the lab, the fraction of dsDNA will be measured with a fluorescent dye that preferentially binds to dsDNA. As such, it will be useful to derive an equation that relates the fraction of dsDNA to temperature and the thermodynamic parameters. |
$ \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} $ |
$ \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} $ by definition and thus $ K_{eq} = \frac {4}{C_T} $. |
$ \bullet $ | Substituting for Keq from equation 1: |
- $ e^\left [\frac{\Delta S}{R} - \frac{\Delta H}{R T} \right ] = \frac{2 f}{(1 - f)^2 C_T} \quad (2) $
$ \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)} $
We now know temperature as a function of dsDNA fraction for a given total DNA strand concentration and DNA identity (therefore ΔH, etc.).
Simulating DNA melting for tractable calculation of dsDNA fraction
$ \bullet $ | For simulating DNA melting experiments, it will be convenient 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 $ | Taking the log of both sides of equation 2 (after re-substiting in equation 1 for simplicity) and using the quadratic formula (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 ]} $
See the pages DNA Melting Part 1: Simulating DNA Melting - Basics. And if you're interested in a Python implementation see Python:Simulating DNA Melting
References
- ↑ Breslauer et al., Predicting DNA duplex stability from the base sequence PNAS 83: 3746, 1986