Normal mode technique

G. Zhang and T. Schlick. LIN A new algorithm combining implicit integration and normal mode techniques for molecular dynamics. J. Comp. Chem., 14 1212-1233, 1993. 

Combining ImpUcit Integration and Normal Mode Techniques for Molecular Dynamics. 

The technique of normal mode analysis has been described as a relatively simple procedure for obtaining an exact solution to the approximate equations of motion for a chemical system. Despite its severe approximation (that the dynamics of a system can be represented by the sum of harmonic terms that are only strictly valid for small displacements), the normal mode technique has proven to perform well at predicting many experimentally observed properties. The preceding applications have illustrated the variety of ways in which normal modes can serve to define the dynamic structure and eneiget-ics of small molecules, proteins, and nucleic acids and to aid in the interpretation and refinement of experimental data. This technique is likely to see increased use in the future. 

Following a brief discussion of normal mode (NM) techniques, we will describe the LIN method and summarize the results obtained. 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

A particular advantage of the low-mode search is that it can be applied to botli cyclic ajic acyclic molecules without any need for special ring closure treatments. As the low-mod> search proceeds a series of conformations is generated which themselves can act as starting points for normal mode analysis and deformation. In a sense, the approach is a system ati( one, bounded by the number of low-frequency modes that are selected. An extension of th( technique involves searching random mixtures of the low-frequency eigenvectors using Monte Carlo procedure. 

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. 

In the following, the method itself is introduced, as are the various techniques used to perform normal mode analysis on large molecules. The method of normal mode refinement is described, as is the place of normal mode analysis in efforts to characterize the namre of a protein s conformational energy surface. 

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. 

A number of studies have compared normal mode analysis predictions with results from more realistic simulation techniques or experiments. These studies shed light on the nature of the conformational energy surface and the effect of solvent. 

Various techniques exist that make possible a normal mode analysis of all but the largest molecules. These techniques include methods that are based on perturbation methods, reduced basis representations, and the application of group theory for symmetrical oligomeric molecular assemblies. Approximate methods that can reduce the computational load by an order of magnitude also hold the promise of producing reasonable approximations to the methods using conventional force fields. 

Evidence exists that some of the softest normal modes can be associated with experimentally determined functional motions, and most studies apply normal mode analysis to this purpose. Owing to the veracity of the concept of the normal mode important subspace, normal mode analysis can be used in structural refinement methods to gain dynamic information that is beyond the capability of conventional refinement techniques. 

One way to do so is to look at the normal mode corresponding to the imaginary frequency and determine whether the displacements that compose it tend to lead in the directions of the structures that you think the transition structure connects. The symmetry of the normal mode is also relevant in some cases (see the following example). Animating the vibrations with a chemical visualization package is often very useful. Another, more accurate way to determine what reactants and products the transition structure coimects is to perform an IRC calculation to follow the reaction path and thereby determine the reactants and products explicity this technique is discussed in Chapter 8. 

The entropy difference A5tot between the HS and the LS states of an iron(II) SCO complex is the driving force for thermally induced spin transition [97], About one quarter of AStot is due to the multiplicity of the HS state, whereas the remaining three quarters are due to a shift of vibrational frequencies upon SCO. The part that arises from the spin multiplicity can easily be calculated. However, the vibrational contribution AS ib is less readily accessible, either experimentally or theoretically, because the vibrational spectrum of a SCO complex, such as [Fe(phen)2(NCS)2] (with 147 normal modes for the free molecule) is rather complex. Therefore, a reasonably complete assignment of modes can be achieved only by a combination of complementary spectroscopic techniques in conjunction with appropriate calculations. 

Molecular mechanics calculations are an attempt to understand the physical properties of molecular systems based upon an assumed knowledge of the way in which the energy of such systems varies as a function of the coordinates of the component atoms. While this term is most closely associated with the conformational energy analyses of small organic molecules pioneered by Allinger (1), in their more general applications molecular mechanics calculations include energy minimization studies, normal mode calculations, molecular dynamics (MD) and Monte Carlo simulations, reaction path analysis, and a number of related techniques (2). Molecular mechanics... 

Projection operators are a technique for constructing linear combinations of basis functions that transform according to irreducible representations of a group. Projection operators can be used to form molecular orbitals from a basis set of atomic orbitals, or to form normal modes of vibration from a basis of displacement vectors. With projection operators we can revisit a number of topics considered previously but which can now be treated in a uniform way. 

Furthermore, it may be seen that for all the normal modes of relaxation, including the most rapid, the freely jointed chain model and the Rouse model are identical if we set n = N + 1 that is, the relaxation time xp of the pth normal mode of a freely-jointed chain is the same as that of a Rouse marcromolecule composed of N + 1 subchains, each of mean square end-to-end length b2. Moreover, for the special choice a = 0, Eq. (10) is true for arbitrarily large departures from equilibrium. We thus seem to have confirmed analytically the discovery of Verdier24 that quite short chains executing a stochastic process described by Eqs. (1) and (3) on a simple cubic lattice display Rouse relaxation behavior. Of course, Verdier s Monte Carlo technique permits study of excluded volume effects, quite beyond the range of our present efforts. 

The determination of the normal modes and their frequencies, however, depends upon solving the secular equation, a 3N X3N determinant. This rapidly becomes nontrivial as N increases. Methods do exist which somewhat simplify the computational problem. Thus, if the molecule has symmetry, the 3Ar X 3N determinant can be resolved into sub-determinants of lower order, each of which involves only normal frequencies of a given symmetry class. These determinants are of course easier to solve. (We will return shortly to the subject of symmetry considerations since they not only aid in the solution of the secular equation, but they permit the determination — without any other information about the molecule — of many characteristics of the normal modes, such as their number, activity in the infrared and Raman spectra, possibilities of interaction, and so on.) In addition, special techniques have been developed for facilitating the setting up and solving of the secular equation [Wilson, Decius, and Cross (245)]. Even these, however, become prohibitive for the large N encountered in complex molecules such as high polymers. 

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