Torsion angle functions

Figure 2-108. Derivation of a syrMbolic potential energy function from the torsion angle distribution of a torsion fragment.

Figure 2-108 shows the correspondence between a histogram and the derived empirical energy function for the torsion angle fragment C-N H)-C(H)(H -C. 

Evalii atm g average distances, angles, and torsion angles, pins their deviations, can facilitate understanding of detailed molecular properties and functional characteristics. 

Sketch the form of the potential energy as a function of torsional angle cj) for the torsional vibration in (a) ethane, (b) CH3NO2, (c) 2-fluorophenol, (d) CH2FOH, and... 

To understand the function of a protein at the molecular level, it is important to know its three-dimensional stmcture. The diversity in protein stmcture, as in many other macromolecules, results from the flexibiUty of rotation about single bonds between atoms. Each peptide unit is planar, ie, oJ = 180°, and has two rotational degrees of freedom, specified by the torsion angles ( ) and /, along the polypeptide backbone. The number of torsion angles associated with the side chains, R, varies from residue to residue. The allowed conformations of a protein are those that avoid atomic coUisions between nonbonded atoms. 

The torsional strain is a sinusoidal function of the torsion angle. Torsional strain results from the barrier to rotation about single bonds as described for ethane on p. 56. For molecules with a threefold barrier such as ethane, the form of the torsional barrier is... 

Fig. 3.1. Potential energy as a function of torsion angle for ethane.

As mentioned in Section 2.2.3, the out-of-plane energy may also be described by an improper torsional angle. For the example shown in Figure 2.6, a torsional angle ABCD may be defined, even though there is no bond between C and D. The out-of-plane oop may then be described by an angle for example as a harmonic function... 

Table 2.3 Comparison of functional forms used in common force fields. The torsional energy, [ors is in all cases given as a Fourier series in the torsional angle...

The transformation from a set of Cartesian coordinates to a set of internal coordinates, wluch may for example be distances, angles and torsional angles, is an example of a non-linear transformation. The internal coordinates are connected with the Cartesian coordinates by means of square root and trigonometric functions, not simple linear combinations. A non-linear transformation will affect the convergence properties. This may be illustrate by considering a minimization of a Morse type function (eq. (2.5)) with D = a = ] and x = AR. 

In Figure 2 we have plotted the energy stored in the bond lengths,bond angles, and torsional angle as a function of chain extension for the three values of the constraining force shown in Figure 1. At a value of 7c =... 

Figure 1. The torsional angle, 4>, and bond angle, S, are shown as a function of dj for the four atom chain with values of 7. equal to 5.0 x 10, 5.0 x 10, and 5.0 x 10 kg/mol-ns. (Reproduced with permission from ref. 13. Copyright 1988 Wiley.)...

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