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| '''Loop entropy''' is the entropy lost upon bringing together two residues of a polymer within a prescribed distance. For a single loop, the entropy varies logarithmically with the number of residues <math>N</math> in the loop
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| :<math>
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| \Delta S = \alpha k_{B} \ln N \,
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| </math>
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| where <math>k_{B}</math> is Boltzmann's constant and <math>\alpha</math> is a coefficient that depends on the properties of the polymer. This entropy formula corresponds to a power-law distribution <math>P \sim N^{-\alpha}</math> for the probability of the residues contacting.
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| The loop entropy may also vary with the position of the contacting residues. Residues near the ends of the polymer are more likely to contact (quantitatively, have a lower <math>\alpha</math>) than those in the middle (i.e., far from the ends), primarily due to excluded volume effects.
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| ==Wang-Uhlenbeck entropy==
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| The loop entropy formula becomes more complicated with multiples loops, but may be determined for a Gaussian polymer using a matrix method developed by Wang and Uhlenbeck. Let there be <math>M</math> contacts among the residues,
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| which define <math>M</math> loops of the polymers. The Wang-Uhlenbeck matrix <math>\mathbf{W}</math> is an <math>M \times M</math> symmetric, real matrix whose elements <math>W_{ij}</math> equal the number of common residues between loops <math>i</math> and <math>j</math>. The entropy of making the specified contacts equals
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| :<math>
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| \Delta S = \alpha k_{B} \ln \det \mathbf{W}
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| </math> | |
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| As an example, consider the entropy lost upon making the contacts between residues 26 and 84 and residues 58 and 110 in a polymer (cf. [[ribonuclease A]]). The first and second loops have lengths 58 (=84-26) and 52 (=110-58), respectively, and they have 26 (=84-58) residues in common. The corresponding Wang-Uhlenbeck matrix is
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| :<math> | |
| \mathbf{W}\ \overset{\underset{\mathrm{def}}{}}{=}\begin{bmatrix}
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| 58 && 26 \\
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| 26 && 52
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| \end{bmatrix}
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| </math>
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| whose determinant is 2340. Taking the logarithm and multiplying by the constants <math>\alpha k_{B}</math> gives the entropy.
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| ==References==
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| * Wang MC and Uhlenbeck GE. (1945) ''Rev. Mod. Phys.'', '''17''', 323.
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| {{polymer-stub}}
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| [[Category:Thermodynamic entropy]]
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| [[Category:Polymer physics]]
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The author is known as Wilber Pegues. He is an order clerk and it's something he truly enjoy. My wife and I reside in Mississippi and I love each working day residing right here. One of the things she enjoys most is canoeing and she's been performing it for quite a whilst.
Take a look at my web site: psychics online