Overview of molecular forces: omega dihedral angle energy dependence angle
Oliver Smart
(c) O.S. Smart 1995, all rights reserved
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Potential energy curve for the omega dihedral angle


This graph shows the potential energy for a H-N-C-O dihedral angle the Pitzer potential:

E_dihe= sum_(i=1)^n (V_n/2)*[1+cos{n.phi-gamma}]

The parameters used are from the AMBER potential energy function:

n    V_n (in kcal/mol)   gamma (in degrees)
1	  1.3	            0.
2	  5.0	          180.
3         0.0               - 
As you can see only of two of the three terms have a non-zero barrier height. This leads to an effective functional form:

 0.65[1+cos(phi)] + 2.5[1+cos(2phi-180)]

where gamma is the H-N-C-O dihedral angle in degrees. The first term leads to an energy dependence shown by the light green line in the graph and the second leads to the purple/pink line. Added together they lead to the thick black line - giving the potential energy dependence for the dihedral angle change. By the use of a three term Fourier series complex energetic behaviour can be modelled.

The practical consequence of these terms is to ensure that the peptide bond is planar during an energy minimization or molecular dynamics run. Note that the trans configuration for the peptide bond is favoured over the cis form by 1.3 kcal/mol. The dashed red lines indicates an energy of 0.29 kcal/mol which is equal to 1/2RT at a temperature of 300K. This is the energy that an individual degree of freedom can expect at this temperature: one can expect that it would be most unlikely for an individual peptide to surmount the energy barrier of over 5 kcal/mol to flip from a cis to trans conformation. Experimentally it is most unusual to find a peptide other than a proline in a cis conformation.

Advanced material: Other terms effecting the omega dihedral angle

The treatment given above over simplifies the potential energy's dependence on the omega dihedral angle on two counts:
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