Lowest frequency normal modes

Nonnal mode analysis was first applied to proteins in the early 1980s [1-3]. Much of the literature on normal mode analysis of biological molecules concerns the prediction of functionally relevant motions. In these studies it is always assumed that the soft normal modes, i.e., those with the lowest frequencies and largest fluctuations, are the ones that are functionally relevant. The ultimate justification for this assumption must come from comparisons to experimental data. Several studies have been made in which the predictions of a normal mode analysis have been compared to functional transitions derived from two X-ray conformers [4-7]. These smdies do indeed suggest that the low frequency normal modes are functionally relevant, but in no case has it been found that the lowest frequency normal mode corresponds exactly to a functional mode. Indeed, one would not expect this to be the case. 

Combining all these techniques suggests that, provided minimization has been achieved, a number of the lowest frequency normal modes of any protein can be accurately determined. 

Figure 6 The calculation of the effectiveforce in the Dimer method. A pair of images, spaced apart by a small distance, on the order q/ 0.1A is rotated to minimize the energy. This gives the direction of the lowest frequency normal mode. The component of the force in the direction of the dimer is then inverted and the minimization of this effectiveforce leads to convergence to a saddle point. No reference is made to the final state.

According to the direct comparison of normal mode vectors, MNM is found to preserve the lowest frequency normal modes. More importantly, MNM outperforms all other normal mode methods in predicting experimental ADPs measured by x-ray crystallography. This is in part due to the elimination of structural shifts during initial energy minimization. 

In normal mode based refinement, the dynamical part of the structure factor is expanded in terms of the normal modes which have been calculated from a minimum-energy structure. Using a subset of the lowest frequency normal modes as a basis set. 

The conventional theory is completed by equating AB to a saddle point on the potential surface between A + B and C + D and by equating s to the imaginary-frequency normal mode of the saddle point. The appropriate saddle point is the highest-energy point on the lowest-energy path from reactants to products. 

If the masses are displaced in an arbiPary way or arbiPary initial velocities are given to them, the motion is asynchronous, a complex mixture of synchronous and antisynchronous motion. But the point here is that even this complex motion can be broken down into two normal modes. In this example, the synchronous mode of motion has a lower frequency than the antisynchronous mode. This is generally Pue in systems with many modes of motion, the mode of motion with the highest symmePy has the lowest frequency. 

Normal mode analysis exists as one of the two main simulation techniques used to probe the large-scale internal dynamics of biological molecules. It has a direct connection to the experimental techniques of infrared and Raman spectroscopy, and the process of comparing these experimental results with the results of normal mode analysis continues. However, these experimental techniques are not yet able to access directly the lowest frequency modes of motion that are thought to relate to the functional motions in proteins or other large biological molecules. It is these modes, with frequencies of the order of 1 cm , that mainly concern this chapter. 

Laux (71) assigns the lowest frequency mode to the methine stretch weakly coupled to the antisymmetric stretch of the adjacent methylene. In addition to the coupled oscillator considerations, normal coordinate and FPC model VCD calculations have also been carried out both for the fragment and for some ring molecules (65, 67-69, 71). 

We shall call the frequency at which t = —2em and t" — 0 the Frohlich frequency coF the corresponding normal mode—the mode of uniform polarization—is sometimes called the Frohlich mode. In his excellent book on dielectrics, Frohlich (1949) obtained an expression for the frequency of polarization oscillation due to lattice vibrations in small dielectric crystals. His expression, based on a one-oscillator Lorentz model, is similar to (12.20). The frequency that Frohlich derived occurs where t = —2tm. Although he did not explicitly point out this condition, the frequency at which (12.6) is satisfied has generally become known as the Frohlich frequency. The oscillation mode associated with it, which is in fact the lowest-order surface mode, has likewise become known as the Frohlich mode. Whether or not Frohlich s name should be attached to these quantities could be debated we shall not do so, however. It is sufficient for us to have convenient labels without worrying about completely justifying them. 

This means that, in the harmonic approximation and to lowest order in h, the classical transition rate is multiplied by a factor depending only on the sums of squares of the normal mode frequencies at the saddle point and minimum ... 

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