Loop entropy

<h2 id="wang-uhlenbeck-entropy">Wang-Uhlenbeck entropy</h2>
The loop entropy formula becomes more complicated with multiples loops, but may be determined for a Gaussian polymer using a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> method developed by Wang and Uhlenbeck. Let there be 
 
 
 
 M
 
 
 {\displaystyle M}
 
 contacts among the residues, which define 
 
 
 
 M
 
 
 {\displaystyle M}
 
 loops of the polymers. The Wang-Uhlenbeck matrix 
 
 
 
 
 W
 
 
 
 {\displaystyle \mathbf {W} }
 
 is an 
 
 
 
 M
 ×
 M
 
 
 {\displaystyle M\times M}
 
 symmetric, real matrix whose elements 
 
 
 
 
 W
 
 i
 j
 
 
 
 
 {\displaystyle W_{ij}}
 
 equal the number of common residues between loops 
 
 
 
 i
 
 
 {\displaystyle i}
 
 and 
 
 
 
 j
 
 
 {\displaystyle j}
 
. The entropy of making the specified contacts equals

Δ
        S
        =
        α
        
          k
          
            B
          
        
        ln
        ⁡
        det
        
          W
        
      
    
    {\displaystyle \Delta S=\alpha k_{B}\ln \det \mathbf {W} }

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. <a href="/facts/Ribonuclease_A/P3CCAR4F">ribonuclease A</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

W
        
         
        
          
            =
            
              
              
                d
                e
                f
              
            
          
        
        
          
            [
            
              
                
                  58
                
                
                
                  26
                
              
              
                
                  26
                
                
                
                  52
                
              
            
            ]
          
        
      
    
    {\displaystyle \mathbf {W} \ {\overset {\underset {\mathrm {def} }{}}{=}}{\begin{bmatrix}58&&26\\26&&52\end{bmatrix}}}

whose <a href="/facts/Determinant/eGYqJ2TF">determinant</a> is 2340. Taking the logarithm and multiplying by the constants 
 
 
 
 α
 
 k
 
 B
 
 
 
 
 {\displaystyle \alpha k_{B}}
 
 gives the entropy.

<ul><li>Wang, M. C., & Uhlenbeck, G. E. (1945). On the theory of the Brownian motion II. Reviews of Modern Physics, 17(2-3), 323.<a href="http://216.92.172.113/courses/phys39/simulations/Uhlenbeck%20Brownian%20Motion%20Rev%20Mod%20Phys%201945.pdf">[1]</a></li></ul>

Loop entropy open-in-new

Loop entropy