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is not self-starting, the first steps are taken with an order 1 method, usually the steepest descent method.

4. Minimization Protocol

The foregoing discussion highlights the fact that the different minimization algorithms have relative strengths and weaknesses. As an example, Figure 5 compares the results of minimizing the same protein with the steepest descent (SD) and conjugated gradients (CG) methods. It is evident that although initially SD reduces the energy faster than CG, in the long run the latter outperforms the former. A similar result is obtained when CG is replaced by ABNR. A detailed comparison between the various minimization algorithms applied to a peptide and a protein is given in Ref. 25.

To optimize the minimization procedure it is usually best to combine several algorithms into a single minimization protocol, taking advantage of their relative strengths. A good minimization scheme will usually start with SD and then use CG or ANBR to finish the job. The number of steps to be used in each phase depends on the goal of the minimization and on the character of the system. When high quality minimization is required, the minimization can be completed with Newton–Raphson. The termination criterion is usually defined in terms of the gradient RMS (GRMS), which is defined as the root-mean-square of all 3N gradients (or forces).

B. Simulated Annealing

Simulated annealing is a popular method that is often used for global optimization, i.e., finding the global minimum of a potential energy surface. The method takes its name from the natural annealing process in which a glass or a metal is first heated and then slowly cooled into a stable low energy state. The key factor in this process is slow cooling. If the cooling is done too fast, the materials will end up in unstable brittle states. Alterna-

Figure 5 A comparison of steepest descent (SD) minimization and conjugated gradients (CG) minimization of the same protein.

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tively phrased, heating up the system shakes and rattles the molecule around the energy landscape, infusing it with thermal energy, analogous to kT, enabling it to jump out of its initial local minimum. The gradual cooling that is subsequently applied decreases the amplitude of these shakes, bringing the details of the energy surface back into focus and causing the system to slowly settle down to a lower energy minimum. Figure 3b is a schematic representation of the simulated annealing process. In fact, we see that simulated annealing bridges between the high temperature conformational sampling simulations that are insensitive to the energy barriers (Fig. 1b) and the low temperature situations, which are sensitive to the details of the energy landscape (Fig. 1a).

Simulated annealing can be easily implemented in both molecular dynamics and Monte Carlo simulations. In molecular dynamics, the temperature is controlled through coupling to a heat bath (Chapter 3); with simulated annealing, the temperature of the bath is decreasing gradually. In Monte Carlo the trial move is accepted or rejected according to a temperature-dependent probability of the Metropolis type [Eq. (1)]. In simulated annealing MC, the temperature used in the acceptance probability is gradually decreased. It should be noted that it is not necessary to anneal all the way to 0 K, because once the kinetic energy kT gets below the characteristic barrier height, a significant change cannot occur. Thus, many simulated annealing protocols cool to room temperature (or somewhat below) and are followed by a local minimization algorithm to remove the excess energy. Specific implementations vary in cooling schedules, initial temperatures, the possibility of repeated heating ‘‘spikes,’’ etc. A detailed account of the method can be found in Ref. 26.

Although simulated annealing is often considered a global optimization method, this is not the case when biomolecules are concerned. It can be shown that in systems characterized by a broad distribution of energy scales, a simulated annealing trajectory (either MD or MC) will have to be extremely long before it is able to find the global minimum. Since the energy landscape of proteins is broadly distributed and rough, this means that simulated annealing is an inefficient and infeasible strategy for protein folding. Nonetheless, even with biomolecules, simulated annealing remains a very useful method for local optimization. Its advantage is that, unlike direct minimization, which takes the molecule only as far as the nearest local minimum, simulated annealing is able to locate lower local minima further away from the initial conformation. An example of the application of this method can be found in Ref. 27. In the context of conformational analysis, simulated annealing is often used in conjunction with a high temperature sampling simulation. Each of the molecular structures generated by the high temperature simulation is first annealed back to room temperature before it is included in the conformational sample and subjected to further analysis [28].

IV. CONFORMATIONAL ANALYSIS

To extract the conformational properties of the molecule that is being studied, the conformational ensemble that was sampled and optimized must be analyzed. The analysis may focus on global properties, attempting to characterize features such as overall flexibility or to identify common trends in the conformation set. Alternatively, it may be used to identify a smaller subset of characteristic low energy conformations, which may be used to direct future drug development efforts. It should be stressed that the different conformational analysis tools can be applied to any collection of molecular conformations. These

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may be generated by the above sampling techniques but can also have an experimental origin, such as NMR models or different X-ray structures of the same molecule (or analogous molecules).

A. Similarity Measures

A similarity measure is required for quantitative comparison of one structure with another, and as such it must be defined before the analysis can commence. Structural similarity is often measured by a root-mean-square distance (RMSD) between two conformations. In Cartesian coordinates the RMS distance dij between conformation i and conformation j of a given molecule is defined as the minimum of the functional

		N	1/2
dij	1	|rk(i) rk(j) |2	(12)
	N
		k 1

where N is the number of atoms in the summation, k is an index over these atoms, and r(ki), r(kij) are the Cartesian coordinates of atom k in conformations i and j. The minimum value of Eq. (12) is obtained by an optimal superposition of the two structures. The resulting RMS distances are usually compiled into a distance matrix ∆, where the elements ∆ij are the RMS distances between conformations i and j.

Since the summation in Eq. (12) may be on any subset of atoms, it can be finetuned to best suit the problem at hand. The summation may be over the whole molecule, but it is very common to calculate conformational distances based only on non-hydrogen ‘‘heavy’’ atoms or, in the case of proteins, even based on only the backbone Cα atoms. Alternatively, in a study related to drug design one may consider, for example, focusing only on atoms that make up the pharmacophore region or that are otherwise known to be functionally important.

The conformational distance does not have to be defined in Cartesian coordinates. For comparing polypeptide chains it is likely that similarity in dihedral angle space is more important than similarity in Cartesian space. Two conformations of a linear molecule separated by a single low barrier dihedral torsion in the middle of the molecule would still be considered similar on the basis of dihedral space distance but will probably be considered very different on the basis of their distance in Cartesian space. The RMS distance is dihedral angle space differs from Eq. (12) because it has to take into account the 2π periodicity of the torsion angle,

N k 1				1/2
		N
dij	1	min[(θk(j) θk(j))2,(2π θk(i) θk(j))2]		(13)

where N is the number of dihedral angles in the summation and θk(i), θ(kij) are the values of the dihedral angle θk in the two structures. As with the Cartesian distance, any appropriate subset of dihedral angles may be used, ranging from only the backbone φ, ψ angles to a full set that includes all the side chain χ angles.

It is up to the researcher to decide whether to use a Cartesian similarity measure or a dihedral measure and what elements to include in the summation [29]. It should be stressed that while the RMS distances perform well and are often used, there are no restrictions against other similarity measures. For example, similarity measures that emphasize chemical interactions, hydrophobicity, or the relative orientation of large molecular domains rather than local geometry may serve well if appropriately used.

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B. Cluster Analysis

The distance matrix , which holds the relative distances (by whatever similarity measure) between the individual conformations, is rarely informative by itself. For example, when sampling along a molecular dynamics trajectory, the matrix can have a block diagonal form, indicating that the trajectory has moved from one conformational basin to another. Nonetheless, even in this case, the matrix in itself does not give reliable information about the size and shape of the respective basins. In general, the distance matrix requires further processing.

Cluster analysis is a common analytical technique used to group conformations. This approach highlights structural similarity, as defined by the distance measure being used, within a conformational sample. Starting from one selected conformation (often that of the lowest energy), all conformations that are within a given cutoff distance from this structure are grouped together into the first cluster C1. Next, one of the conformations that were not grouped into the first cluster is selected, and a new cluster is formed around it. The process continues until all the conformations in the sample are assigned to a cluster Ci. This process often generates overlapping clusters, that is, clusters with nonzero intersection Ci Cj ≠ 0, The overlapping clusters are typically treated in one of two ways:

(1) Group together the overlapping clusters Ci and Cj to form a single large cluster that is their union Ci Cj (Fig. 6a) or (2) make the overlapping clusters disjoint by removing their intersection Ci Cj from one of the clusters, typically the one that started with a higher energy conformation (Fig. 6b). Since the optimal cutoff distance by which to cluster the conformations is not a priori known, cluster analysis is usually performed hierarchically. Starting with a short cutoff the analysis is repeated again and again, each time with a larger cutoff distance. The results are often represented as a dendogram. More information about the various clustering algorithms can be found in Ref. 30.

In many conformational studies, cluster analysis is used as a way to focus future effort on a small set of characteristic conformations. One conformation, typically the lowest energy one, is picked from each of the highly populated conformational clusters. The resulting small number of distinctly different conformations are then used as starting points for further computational analysis (such as free energy simulations) or as a basis for generating a pharmacological hypothesis used for directing future drug development [18]. It should be noted, however, that conformational clusters generated by the above procedure do not necessarily represent the correct basin structure of the underlying energy landscape.

Figure 6 A schematic representation of two clustering methods, in which each point represents a single molecular conformation and the circles are the similarity cutoff distances used to define the clusters. (a) Three clusters are defined when overlapping clusters are grouped together. (b) Five clusters are defined when the overlaps are removed from one of the overlapping clusters.