Normal mode analyses

We now provide an example in wdiich eigenvalue analysis is of direct interest to a problem from chemical engineering practice. Let us say that we have some sfructme (it could be a molecule or some sohd object) whose state is described by the positional degrees of freedom q and the corresponding velocities q. We have some model for the total potential energy of the system U q) and some model of the total kinetic energy K q, q). We wish to compute the vibrational frequencies of the structure. Such a normal mode analysis problem arises when we wish to compute the IR spectra of a molecule (Allen Beers, 2005). 

using the numerical optimization methods outlined in Chapter 5, we identify a state q that is a local minimum of the potential energy. That is, it has a lower potential energy than any neighboring states, and as it is an extremum, V17 = 0. We wish to describe the system s dynamics when it is perturbed slightly from this minimum energy state, and so define 6 = q q. Expanding U(q) about as a Taylor series, with dU/dqm q = 0, yields 

Defining the Hessian matrix H, containing the second derivatives of U q), 

H which from (3 176) is real symmetric, also must be positive-semidefinite, as for a local minimum q, U(q + 6)-U(q)  

Let us assume for the moment that each degree of freedom has the same effective mass nieff, so that the kinetic energy is 

Force field potential function initial force constants -databanks, literature -ab initio calculations 

Vibrational frequencies identification of compounds identification of conformations assignment of experimental spectra 

Force constants potential function calculation of bond properties via correlations calculation of molecular properties such as -Coriolis coupling -centrifugal distortion constants etc. 

L yields information about the individual displacements during the normal modes) the kinetic and potential energies assume the form 

Multiplying Eq. (9) from the left side by L and taking into accoimt that L L =G. the classical secular equation can be formulated as 

---

Seeley G and Keyes T 1989 Normal-mode analysis of liquid-state dynamios J. Chem. Phys. 91 5581-6... 

The combination is in this case an out-of-phase one (Section I). This biradical was calculated to be at an energy of 39.6 kcal/mol above CHDN (Table ni), and to lie in a real local minimum on the So potential energy surface. A normal mode analysis showed that all frequencies were real. (Compare with the prebenzvalene intermediate, discussed above. The computational finding that these species are bound moieties is difficult to confimi experimentally, as they are highly reactive.)... 

Hayward, S., Kitao, A., Berendsen, H.J.C. Model-free methods to analyze domain motions in proteins from simulation A comparison of normal mode analysis and molecular dynamics simulation of lysozyme. Proteins 27 (1997) 425-437. 

The influence of solvent can be incorporated in an implicit fashion to yield so-called langevin modes. Although NMA has been applied to allosteric proteins previously, the predictive power of normal mode analysis is intrinsically limited to the regime of fast structural fluctuations. Slow conformational transitions are dominantly found in the regime of anharmonic protein motion. 

Amadei et al. 1993] Amadei, A., Linssen, A.B.M., Berendsen, H.J.C. Essential Dynamics of Proteins. Proteins 17 (1993) 412-425 [Balsera et al. 1997] Balsera, M., Stepaniants, S., Izrailev, S., Oono, Y., Schiilten, K. Reconstructing Potential Energy Functions from Simulated Force-Induced Unbinding Processes. Biophys. J. 73 (1997) 1281-1287 [Case 1996] Case, D.A. Normal mode analysis of protein dynamics. Curr. Op. Struct. Biol. 4 (1994) 285-290... 

Hayward et al. 1994] Hayward, S., Kitao, A., Go, N. Harmonic and anharmonic aspects in the dynamics of BPTI A normal mode analysis and principal component analysis. Prot. Sci. 3 (1994) 936-943 [Head-Gordon and Brooks 1991] Head-Gordon, T., Brooks, C.L. Virtual rigid body dynamics. Biopol. 31 (1991) 77-100... 

Steven Hayward, Akio Kitao, and Nobuhiro Go. Harmonic and anharmonic aspects in the dynamics of BPTI A normal mode analysis and principal component analysis. Physica Scripta, 3 936-943, 1994. 

D. A. Case. Normal mode analysis of protein dynamics. Curr. Opin. Struc. Biol., 4 385-290, 1994. 

It is also possible to use normal mode analysis [7] to estimate the difference between the exact and the optimal trajectories. Yet another formula is based on the difference between the optimal and the exact actions 2a w [5[Yeract(t)] (f)]]- The action is computed (of course), employ-... 

Energy minimisation and normal mode analysis have an important role to play in the study of the solid state. Algorithms similar to those discussed above are employed but an extra feature of such systems, at least when they form a perfect lattice, is that it is can be possible to exploit the space group symmetry of the lattice to speed up the calculations. It is also important to properly take the interactions with atoms in neighbouring cells into account. 

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. 

More traditional applications of internal coordinates, notably normal mode analysis and MC calculations, are considered elsewhere in this book. In the recent literature there are excellent discussions of specific applications of internal coordinates, notably in studies of protein folding [4] and energy minimization of nucleic acids [5]. 

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. 

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. 

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. 

II. NORMAL MODE ANALYSIS IN CARTESIAN COORDINATE SPACE... 

This section describes the basic methodology of normal mode analysis. Owing to its long history it has been described in detail in the context of many different fields. However, to aid in understanding subsequent sections of this chapter, it is described here in some detail. 

III. NORMAL MODE ANALYSIS OF LARGE BIOLOGICAL MOLECULES... 

B. Normal Mode Analysis in Dihedral Angle Space... 

Given that one does not need to perform an energy minimization and that the Hessian is very sparse, it is not surprising that the computation time is reported to be at least one order of magnitude less than for a conventional normal mode analysis. 

Normal Mode Analysis of Biological Molecules A. Normal Mode X-Ray Refinement... 

Figure 2 Internal RMSF of residues (average over heavy atoms) determined for human lysozyme by the X-ray normal mode refinement method applied to real X-ray data (heavy curve), m comparison with results from a normal mode analysis on a single isolated lysozyme molecule (lightweight curve). (From Ref. 33.)...

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. 

---