**K nowledge of DNA thermodynamics is crucial for the design of PCR probes or any other oligonucleotide based hybridization probe!**

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Since DNA hybridization does not strictly follow the Watson-Crick pairing rules DNA hybridization experiments often require optimization. A DNA oligonucleotide has the potential to pair with many sites on the genome including one or a few mismatches. This can lead to false-positive results. Furthermore, the desired target sites of single-stranded genomic DNA or mRNA are often folded into stable secondary structures that must be unfolded to allow for the oligonucleotide to bind. A very stable fold of the target will cause very little probe DNA to bind. Again, this can lead to a false-negative test. And there are other artifacts encountered with DNA that can lead to false results such as probe folding and probe dimerization. To fully understand these phenomena many researchers have investigated the thermodynamics and kinetics of the hybridization reactions in the past. One prominent researcher in this field is John SantaLucia, Jr. In this blog I am briefly reviewing some aspects of the thermodynamics that govern DNA hybridization.

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**The formation of DNA/DNA duplexes is what makes the hybridization reaction work!**

In a simplistic view the formation of an oligonucleotide duplex via hybridization can be viewed as a forward bimolecular reaction. The reverse reaction is considered to be unimolecular. If enough time is allowed for the reaction to occur the forward and the reverse reaction rates will be equal and equilibrium is achieved. The reaction equilibrium can be described as follows:

where A and B imply strands A and B in the random coil state and AB implies the ordered AB duplex state.

The equilibrium constant, K, for the reaction is given by the law of mass action:

The equilibrium constant is independent of the total strand concentrations, [**A**tot] and [**B**tot]. However, **K** depends strongly on temperature, salt concentrations, pH, the presence of DMSO, and other components that might be present in reaction buffers or environment. The system will respond to re-achieve the equilibrium given by equation 2 if the total concentrations of one or both of the strands are changed. This means that the individual concentrations of [A], [B], and [AB] will change.

This is called “**Le Chatelier’s principle”**, or **Chatelier’s principle. **It can be used to predict the effect of a change in conditions on a chemical equilibrium. The principle can be summarized as:

If a chemical system at equilibrium experiences a change in concentration, temperature, volume, or partial pressure, then the equilibrium will shift to counteract the imposed change and a new equilibrium is established. In the case of the duplex formation reaction, the more of the strand A is added, the more [AB] will increase.

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**Example of how to solve simple equilibrium equations**

If we assume that the equilibrium constant is 1.81 x 10^{6}, and [Atot] = [Btot] = 1 x 10^{-5} M we can illustrate how equation 2 is used. (This example was published earlier by SantaLucia, Jr. See references). Both, [**A**tot] and [**B**tot] can be measured by UV absorbance or other techniques.

Equation 3 is called the ‘**equilibrium equation**,’ and equations 4 and 5 are called the ‘**conservation of** **mass**’ **equations**. This equation system can be solved analytically if it is quadratic or numerically if it is of higher order. However, multi-state equilibriums require the computation of more complex equations.

Substituting equations 4 and 5 into equation 3 gives the following equations:

Equation 6 can be rearranged into the familiar form of the quadratic equation:

As X = [AB] the equation 8 can be solved as:

Note that concentrations must be positive and only one root is positive.

Now if we plug the numbers for K {K = a = 1.81 x10^{6}}, **A**tot {= 1 x 10^{-5}}, and **B**tot {= 1 x 10^{-5}} into equation 9 we get [AB] = 7.91 x 10^{-6} M. Using equations 4 and 5 we get [A] = [B] = 2.09 x 10^{-6} M.

**This result means that 79 % of [A**tot**] can be found in the bound duplex structure, [AB].**

However, when performing these calculations the results should be verified if they are reasonable. In this example the melting temperature **T**_{m} is 43.4 °C, whereas the **K** was determined at 37 °C. Since 37 °C is less than the Tm we should expect that the amount bound should be more than 50 %. As we can see the calculated 79% bound is consistent with that expectation.

As this example shows this approach can be used to compute the primer bound to the target if the equilibrium constant and the total strand concentration are known.

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**The primer bound to the target is the quantity that matters in a hybridization assay!**

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**What is the melting temperature?**

The energy required to break the hydrogen bonds that are holding the duplexes together is a function of the number of hydrogen bonds that are present. Both the length and the G+C content of the sequences control this. G-C interactions involve formation of three (3) hydrogen bonds rather than the two (2) formed by A-T interactions.

The temperature at which 50% of specific duplexes break to form single strands is called the “**melting temperature**,” or **T**_{m}. The **T**_{m} can be measured experimentally by detecting the hyperchromatic shift or calculated from the sequence length and relative number of G and C bases. The hyperchromatic shift is the observation of an increase in the absorption of ultraviolet light of a solution that contains oligonucleotides due to a loss of the ordered secondary structure.

Duplexes formed between strands that vary by as little as one (1) base will consequently have a lower **T**_{m} than completely complementary strands.

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**Can the equilibrium constant be predicted?**

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**What is the influence of the temperature?**

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To solve these questions we need Δ*G*°, Δ*H*°, and Δ*S*° and help from the field of thermodynamics.

The basic concepts in thermodynamics are the concept of “**system**” and the “**surroundings**.” The surroundings of a thermodynamic system can be other thermodynamic systems that can interact with it.

For PCR, the “**system”** can be defined as the contents of the test tube that contains the nucleic acid strands, solvent, buffer, salts, and all the other chemicals. The “**surrounding**” can be defined as the **rest of the entire universe**. However, we do not need to consider what is going on in the whole universe. We only need to focus on the system and determine the changes in certain properties of the system alone. We need to determine Δ*G*°* _{T}*, Δ

*H*°, and Δ

*S*°, whether a process is spontaneous and we need to determine the

**equilibrium**.

For the process to reach equilibrium heat is released from the system to the surroundings when strands change from the random coil to the duplex state. At constant pressure this change is called the change in enthalpy, Δ*H*. Δ*H*° indicates the energy values given for the idealized “**standard state**”. This simply means that the energy change refers to the amount of energy that would be released if a scientist could prepare each species in 1 M concentration, mix them, and then allow them to come to equilibrium. However, this is a non-equilibrium condition. The more heat is released from the reaction system to the surroundings the more disorder is produced in the surroundings. Because of the **second law** **of thermodynamics** the heat the reaction system produces the more the reaction is favored.

**Note that a duplex is more ordered than a random coil because of conformational entropy!**

**Entropy** is the amount of additional information needed to specify the exact physical state of a system. This is the modern microscopic interpretation of entropy in statistical mechanics. In other words, entropy is an expression of disorder or randomness that increases in the system or surroundings.

The amount of order in the system changes during the hybridization reaction. Furthermore, solvent molecules and counterions bind differently to duplexes and random coils. These effects are accounted for in the change of the entropy of the system, the Δ*S*°. The Gibb’s free energy change for going from random coil to duplex is expressed as:

Where *T* is the Kelvin temperature, is given in kcal/mol, and is given in cal/mol K. For more accurate calculations the change in heat capacity, ΔC_{p}, may also need to be considered.

**The Gibbs energy** (∆G) or **free enthalpy** is a thermodynamic potential or chemical potential that is minimized when a system reaches equilibrium at constant pressure and temperature.

The relationship between the Gibb’s free energy change, the temperature T and the equilibrium constant at the temperature T is:

Where R is the gas constant (= 1.9872 cal/ml K).

A **good rule of thumb** for estimating the qualitative behavior of ΔG°T is:

At 25°C – every increment of change of a unit of – 1.4 kcal/mol in Δ*G*°_{25 }results in a change in the equilibrium constant by a multiplicative factor of 10.

A change in Δ*G*°_{25 }of – 4.2 kcal/mol equals 3 x (– 1.4 kcal/mol) = 100.

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**The nearest-neighbor model**

Given Δ*H*° and Δ*S*°, we can compute the concentration distribution for all species at all temperature. Applying the “**nearest-neighbor** (NN)” model allows accurately predicting these from the strand sequences. The nearest-neighbor model assumes that a DNA helix is a string of interactions between ‘neighboring’ base pairs. Santa Lucia has published empirical equations that allow the NN model to be extended to include salt dependence, terminal dangling ends, and all possible internal and terminal mismatches as detailed in many recent papers.

Multiple softwares for this type of calculations have been developed. The links to a selected few are listed below:

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**DNA-MFOLD** server: http://mfold.rna.albany.edu/?q=mfold/dna-folding-form

**Hyther** server: http://ozone3.chem.wayne.edu/cgi-bin/login/login/showLoginPage.cgi

**The UNA Fold** web server: http://www.bioinfo.rpi.edu/applications/

**Visual OMP**: http://www.dnasoftware.com/product/visual-omp

**De Novo DNA** https://www.denovodna.com/software/

**Oligo Calc**: Oligonucleotide Properties Calculator: http://www.basic.northwestern.edu/biotools/OligoCalc.html

**Software to download:**

**MeltDNA**: http://sourceforge.net/projects/meltdna/

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**References**

M. **Mandel** and J. Marmur (1968). “Use of Ultravialet Absorbance-Temperature Profile for Determining the Guanine plus Cytosine Content of DNA”. *Methods in Enzymology*. Methods in Enzymology **12** (2): 198–206. doi:10.1016/0076-6879(67)12133-2. ISBN 978-0-12-181856-2.

R.M. **Myers**, T. Maniatis, and L.S. Lerman (1987). “Detection and Localization of Single Base Changes by Denaturing Gradient Gel Electrophoresis”. *Methods in Enzymology*. Methods in Enzymology **155**: 501–527. doi:10.1016/0076-6879(87)55033-9. ISBN 978-0-12-182056-5. PMID 3431470.

**Breslauer**, K.J. et al.; Frank, R; Blöcker, H; Marky, LA (1986). “Predicting DNA Duplex Stability from the Base Sequence”. *Proc. Natl. Acad. Sci. USA.* **83** (11): 3746–3750. doi:10.1073/pnas.83.11.3746. PMC 323600. PMID 3459152. (pdf)

Rychlik, W.; Spencer, W. J.; Rhoads, R. E. (1990). “Optimization of the annealing temperature for DNA amplification in vitro”. *Nucleic Acids Res.* **18** (21): 6409–6412. doi:10.1093/nar/18.21.6409. PMC 332522. PMID 2243783.

**Owczarzy** R., Vallone P.M., Gallo F.J., Paner T.M., Lane M.J. and Benight A.S (1997). “Predicting sequence-dependent melting stability of short duplex DNA oligomers”. *Biopolymers* **44** (3): 217–239. doi:10.1002/(SICI)1097-0282(1997)44:3<217::AID-BIP3>3.0.CO;2-Y. PMID 9591477. (pdf)

John SantaLucia Jr. (1998). “A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics”. *Proc. Natl. Acad. Sci. USA* **95** (4): 1460–5. doi:10.1073/pnas.95.4.1460. PMC 19045. PMID 9465037.

John SantaLucia Jr.; Donald Hicks (June 2004). “The thermodynamics of DNA structural motifs”. *Annual Review of Biophysics and Biomolecular Structure* **33**: 415–440. doi:10.1146/annurev.biophys.32.110601.141800. PMID 15139820.

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Categories: Bioanalysis, BNAs, DNA, DNA Analysis, DNA Hybridization, DNA Thermodynamics, Genetics, Genome, Hybridization, Thermodynamics

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