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Department of Computational Biology, School of Medicine and Department of Environmental and Occupational Health, Graduate School of Public Health, University of Pittsburgh, Pittsburgh, Pennsylvania
Correspondence: Address reprint requests to D. M. Zuckerman, Tel.: 412-648-3335; E-mail: dmz{at}ccbb.pitt.edu.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTHEORY AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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50 ns) may not be converged for this small—but highly flexible—system. |
INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTHEORY AND METHODSRESULTSDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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), enzymes (2,3), signaling proteins (4–6), or almost any other kind. Different experiments have enabled observation of protein fluctuations over a huge range of timescales, from picoseconds (7) to microseconds (5) to milliseconds (3,6,8) to seconds and longer (9). Naturally, simulations aim to observe conformational fluctuations as well. A gap remains, however, between the timescale of many biologically important motions (microseconds to seconds), and the timescale accessible to atomically detailed simulation (nanoseconds). To put it another way, some problems are simply not possible to study computationally, since it is so far impossible to run a simulation that is long enough.
For those problems that are at the very edge of being feasible, we would like to know whether we have indeed sampled enough to draw quantitative conclusions. These problems include the calculation of free energies of binding (10,11), ab initio protein folding (12,13), simulation of flexible peptides (14), and conformational changes (15).
Convergence assessment is also crucial for rigorous tests of simulation protocols and empirical force fields; see, e.g., Zaman et al. (16). Many algorithms propose to improve the sampling of conformation space, but quantitative estimation of this type of efficiency is difficult, except in simple cases (17). In the case of force-field validation, it is important to know whether systematic errors are a consequence of the force field, or are due to undersampling.
The observed convergence of a simulation depends on how convergence is defined and measured. It is therefore important to consider what sort of quantity is to be calculated from the simulation, and choose an appropriate way to assess the adequacy of the simulation trajectory (or trajectories). Many relatively simple methods are commonly used, such as measuring distance from the starting structure as a function of simulation time, and calculation of various autocorrelation functions (16,18). Other, more sophisticated methods are based on principal components (19,20) or calculation of energy-based ergodic measures (21).
Many applications, however, require a thorough and equilibrated sampling of the space of structures. All of the methods just listed are only related indirectly to structural sampling. There are many examples of groups of structures that are very close in energy, but very dissimilar structurally. In such cases, we might expect energy-based methods to be insensitive to the relative populations of the different structural groups. It is therefore of interest to develop methods that are more directly related to the sampling of different structures, and see how such methods compare to more traditional techniques.
Daura and co-workers (22,23) previously considered convergence assessment by counting structural clusters, based upon a cutoff in the root mean-square deviation (RMSD) metric. The authors assess the convergence of a simulation by considering the number of clusters as a function of time. Convergence is deemed sufficient when the curve plateaus. This is surely a better measure than simpler, historically used methods, such as RMSD from the starting structure or the running average energy. However, it is worth noting that long after the curve of number of clusters versus time plateaus, the relative populations of the clusters may still be changing. Indeed, an important conformational substate that has been visited just once will appear as a cluster, but its relative population will certainly not have equilibrated.
The method of Daura et al. (22) also suffers from the need to store the entire matrix of pairwise distances. For a trajectory of length N, the memory needed scales as N2, rendering the method impractical for long trajectories. At least two groups have developed methods that rely on nonhierarchical clustering schemes, and therefore require memory that is only linear in N. Karpen et al. (24) developed a method that optimizes the clusters based on distance from the cluster center, with distances measured in dihedral angle space. Elmer and Pande (25) have optimized clusters subject to a constraint on the number of clusters, with distance defined by the atom-atom distance root mean-square deviation (26,27).
In this article, we address systematically the measurement of sampling quality. Our method classifies (or bins) a trajectory based upon the distances between a set of reference structures and each structure in the trajectory. Our method is unique in that it not only builds clusters of structures, it also compares the cluster populations. By comparing different fragments of the trajectory to one another, convergence of the simulation is judged by the relative populations of the clusters. We believe the key to assessing convergence is tracking relative bin populations. Our approach can be directly applied to comparing the efficiency of different sampling methods.
In the next section, we present a detailed description of the algorithm and discuss possible choices of metric. We then demonstrate the method on simulations of met-enkephalin, a structurally diverse peptide.
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THEORY AND METHODS |
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Histogram construction
We generate the set of reference structures and corresponding histogram from a trajectory in the following simple way (our choice for measuring conformational distance will be discussed below):
Such a partitioning guarantees a set of clusters whose centers are at least dc apart. Furthermore, for a trajectory of N frames, the number of reference structures, M, and therefore the memory needed to store the resulting M x N matrix of distances, is controlled by dc. For physically reasonable cutoffs (e.g., 
Å RMSD), the number of reference structures is at least an order-of-magnitude smaller than the number of frames in the trajectory. The memory requirements are therefore manageable, since the computation of pairwise distances scales as N log N.
There is nothing in principle that prevents the use of a more carefully chosen set of reference structures with our classification scheme. For example, we may consider a set of structures that correspond to minima of the potential energy surface. The cutoff might then be chosen to be the smallest observed distance between any pair of the minimum energy structures, and the set of reference structures so determined could be augmented by the random selection of more reference structures to span the whole trajectory.
However, we expect that the purely random selection used here will naturally include the lowest free-energy substates, since these are the most populated. In either case, any set of reference structures defines a unique histogram for any trajectory.
Trajectory analysis
Once we have a set of reference structures, we may easily compare two different trajectories classified by the same set of reference structures, by comparing the populations of the various bins as observed in the two trajectories: given a (normalized) population pi(1) for cluster i in the first trajectory, and pi(2) in the second, the difference in the populations 
Pi = |pi(1) – pi(2)| measures the convergence of substate i's population between the two trajectories.
Note that the two trajectories just discussed may be two different pieces of the same simulation. In this way, we may self-consistently assess the convergence of a continuous simulation, by looking to see whether the relative populations of the most populated substates are changing with time. Of course, this cannot answer affirmatively that a simulation has converged (no method can do so); however, it may answer negatively. In fact, we will see later that our method indicates that structural convergence may be much slower than previously appreciated.
Our approach should also be applicable to some types of noncontinuous trajectories, such as those generated by multiple starts (e.g., (28)) or parallel exchange protocols (e.g., (29,30)). For multiple independent trajectories, one can compare the two histograms generated from 1), the first halves and 2), the second halves of all simulations. If converged, these two histograms should agree. One could also compare histograms generated by grouping entire trajectories into distinct sets. For a parallel exchange simulation, where the ensemble is built from a set of continuous trajectories, histograms from the first and second halves of the simulation can be compared.
The comparison of histograms clearly will not be appropriate when ensembles are generated in a fully decorrelated way. For instance, starting from a single long trajectory, one could generate two ensembles by randomly selecting structures, or perhaps by selecting structures at two different fixed time intervals. So long as the number of structures in each ensemble greatly exceeds the number of reference structures used for classification, it is hard to see how such histograms could be significantly different. In such cases, dynamical correlations have been explicitly discarded, and the histograms can only differ statistically.
Structural metrics
Many different metrics have been used to measure distance between conformations. The choice depends on both physical and mathematical considerations. For example, dihedral angle-based metrics are well suited to capture local structural information (24), but are not sensitive to more global rearrangements of the molecule. Least-squares superposition followed by calculation of the average positional fluctuation per atom (RMSD) is quite popular, but the problem of optimizing the superposition can be both subtle and time-consuming for large, multidomain proteins (31). In addition, RMSD does not satisfy a triangle inequality (32). This is not an issue for the algorithm presented here, but is a consideration for more sophisticated clustering methods (25). We will use RMSD to measure distance here, though we note that distance root-mean-square deviation (or sometimes, distance-matrix error) (26,27) may be appropriate when RMSD is not.
Labeling two structures by a and b, the traditional root mean-square deviation (RMSD) is defined to be the minimum of the root mean-square average of interatomic distances over all possible translations and rotations of xb—namely,
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It is clear that the choice of dc, together with the choice of metric, determines the resolution of the histogram. Reducing dc increases the number of reference structures, and reduces the size of the bins. How is dc chosen? There is no general answer, and a suitable cutoff will depend on the problem under investigation.
The typical RMSD between a pair of structures will depend on the size of the molecule, its flexibility, and the conditions of the simulation. If the magnitude of some important conformational change is known in advance, then this information will guide the selection of an appropriate cutoff. If not, then a series of histograms should be constructed at several values of dc. The behavior of the histogram as a function of dc will give a sense of the appropriate value, as we will see below.
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RESULTS |
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The trajectories analyzed in this section were generated by Langevin dynamics simulations, as implemented in the Tinker v. 4.2.2 simulation package (33). The temperature was 298 K, the friction constant was 5 ps–1, and solvation was treated by the GB/SA method (34). Two 100-ns trajectories were generated, each starting from the PDB structure 1plw, model 1. The trajectories will be referred to as "plw-a" and "plw-b". Coordinates were written every 10 ps, for a total of 104 frames per trajectory.
Previous methods: RMSD analysis and cluster counting
An often-used indicator of equilibration is the RMSD from the starting structure (see Fig. 1 A). Such plots are motivated by the recognition that the starting structure (e.g., a crystal structure) may not be representative of the protein under the simulation conditions—solvent, force field, and temperature. This is the case in Fig. 1 A—the computation was performed with an implicit water model, while the experimental structure was determined in the presence of bicelles (35). The system fails to settle down to a relatively constant distance from the starting structure—rather, it is moving between various substates, some nearer and some farther from the starting structure. Although this is not surprising for a peptide renowned for its floppy character, it also indicates that this method cannot determine when the peptide simulation has converged. Indeed, Fig. 1 A can tell us little about the convergence of the simulation, only that it spends most of its time more than 2.0 Å from the starting structure.
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), albeit with clusters built by the procedure described in Theory and Methods (Histogram Construction). The premise is that convergence is achieved when the number of clusters no longer increases, as this means that the simulation has visited every substate. This analysis suggests that convergence is observed by 7 ns, and the curve has the comforting appearance of saturation. However, Fig. 1 B is insensitive to the relative populations of the clusters. To illustrate the problem, consider a simple potential, with two smooth wells separated by a high barrier. By simple cluster counting, a simulation will be converged as soon as it has crossed the barrier once. It is clear, however, that many crossings will be required before the populations of the two states have equilibrated. We will address this question using our ensemble-based method. We find, in fact, that the relative populations of the clusters continue to change, long after their number has equilibrated.
Ensemble-based assessment of trajectories
The use of our systematic approach is much more revealing. We first discuss the selection of an appropriate cutoff. We then demonstrate two different applications of the ensemble-based comparison of trajectories—a comparison between a trajectory and a gold-standard ensemble, and a self-consistent convergence analysis of a single trajectory.
Reference structure generation and cutoff selection
A compound trajectory was formed from trajectories plw-a and plw-b, by discarding the first nanosecond of each trajectory and concatenating the two into a single, 198-ns trajectory ("plw-ab"). We then generated a set of reference structures for the compound trajectory, as described earlier: a structure is picked at random, and it is temporarily discarded along with every structure within a predefined cutoff distance, dc. The process is repeated on the remaining structures until the trajectory has been exhausted. The result is a set of reference structures that are separated from one another by at least the predefined cutoff distance. Lowering the cutoff (making the reference structures more similar) increases the resolution of the clustering, and increases the number reference structures (see Table 1). Although RMSD is system-size-dependent (36), for a particular system the cutoff defines a resolution.
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Although it seems obvious that the most revealing cutoff will be system-specific, our histograms are more robust than they first appear. Because reference structures are chosen arbitrarily, the divisions between bins will not reflect basins of the landscape. In other words, many, if not most, bins can be expected to include a number of full and partial local basins. Thus, a lack of convergence in a macroscopic bin, at least in principle, can report on more local, microscopic states. Further, because our approach is so inexpensive compared to the simulation itself, more than one binning of configuration space can and should be considered; see Self-Referential Convergence Assessment and Fig. 4.
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Though we do not pursue it here, we note that the tail of the distribution—where half of all the bins account for only 10% of the population—might contain some very interesting structures. Indeed, at the very end of the tail are found bins that sometimes contain a single structure. Might some of these low population bins represent transition states? For now, we set this question aside, and focus instead on convergence assessment.
Comparing trajectories to a gold-standard ensemble
In some applications, we want to compare a trajectory to a gold-standard ensemble. For example, the gold-standard might be the ensemble sampled by a long molecular dynamics simulation, and we may wish to check the ensemble produced by a new simulation protocol against the long molecular dynamics trajectory.
For met-enkephalin, we use our histogram approach to illustrate, in Fig. 3, the evolution of convergence in two long (99 ns) trajectories. The compound trajectory (198 ns) is taken as a gold-standard, from which reference structures are calculated using a cutoff dc = 3.0 Å. We can then assess the convergence of portions of the trajectory against this full ensemble (see Fig. 3, A–D).
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Fig. 3 presents a picture of a very conformationally diverse peptide, especially given the large cutoff (dc = 3.0 Å) used. The first three substates contain only 52% of the observed structures, while the first nine account for 74%. Indeed, the (experimentally determined) starting structure is located in the second-most populated bin.
We also analyzed the entire set of NMR model structures. These were determined in the presence of bicelles, as it was hypothesized that interaction of the peptide with the cell membrane induces a shift in the conformational distribution (35). We classified the entire set of 80 NMR structures against our set of reference structures. The overwhelming majority of the NMR structures, 75%, were nearest to reference structure 23, the second-least populated bin in our simulation. The next largest group of NMR structures (15 of 80) were nearest to McCallum et al. (2), which held a comparable portion of the simulation trajectory. The remaining five NMR structures were scattered among four different bins. While not conclusive, the comparison between our simulation data and the NMR structures supports the hypothesis that binding to the membrane induces a shift in the distribution of met-enkephalin conformers, relative to the distribution observed in water. Such conformational diversity is not surprising for a peptide, which is known to be a promiscuous neurotransmitter by virtue of its flexibility (35,37,38). However, it will be interesting to revisit the issue in the study of a protein.
Self-referential convergence assessment
We want to assess convergence without the use of a gold-standard. Our previous analysis (Fig. 3) might be used to compare simulation protocols; ensembles from a new protocol may be compared to a gold-standard ensemble. (Here, the gold-standard is the 198-ns compound trajectory.) However, it is not useful as a means of assessing the convergence of a single simulation. After all, given only a 4-ns trajectory, one would like an assessment without reference to the answer.
Fig. 4 therefore demonstrates a purely self-referential scheme for on-the-fly analysis of a continuous trajectory. Fig. 4 A compares, for example, the first two-nanoseconds to the second two-nanoseconds. The series of plots in Fig. 4 shows that the populations of the clusters are still changing significantly, even between the first and second 50-nanoseconds. Presuming we had run only a single 100-ns simulation, we could make Fig. 4 C, and describe the convergence by saying, at a resolution of 3.0 Å RMSD, considering bins containing 75% of the structures, six of nine bins have not yet converged to within 1/2 kBT. Note the contrast with Fig. 1 B, where it appears convergence is reached after just 7 ns. This contrast is all the more striking considering that dc = 3.0 Å is a rather conservative choice. At a higher resolution (smaller dc), the observed convergence is worse.
To test whether our analysis is sensitive to the (random) selection of reference structures, Fig. 4 shows two independent sets of reference structures. There is little difference in the results. Both classifications indicate that >50 ns are required for convergence when dc = 3.0 Å.
The observed ensembles and corresponding convergence depend on both the metric used and the value of dc. (This is of course true of any clustering algorithm.) It is therefore important to report this information along with any statements about the convergence of a particular simulation. Indeed, lowering the cutoff, and hence increasing the resolution of the classification, is bound to reduce the observed level of convergence. Instead of Fig. 4, in which each panel is a different length of the trajectory, we could have plotted the same trajectory length at different resolutions. At a high enough resolution, we will always find some substates that are under- or overpopulated. In other words, since all trajectories are finite, a physically acceptable value of dc must be chosen.
While the choice of dc is somewhat ad hoc in the present implementation, plots like those in Fig. 4 still can provide valuable, quantitative information. For example, imagine that we wish to calculate the free energy difference between two experimentally known conformations that differ by 3.0 Å RMSD. In this case, Fig. 4 suggests that we cannot expect an accuracy better than 1/2 kBT. Perhaps more importantly, any fixed choice of cutoff can be useful in comparing different simulation methods—even if the difficult question of absolute convergence is not addressed.
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DISCUSSION |
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The results for met-enkephalin indicate that it takes quite some time (>50 ns) for the relative populations of the various substates to equilibrate, even with a fairly promiscuous cutoff (3.0 Å RMSD) that partitions the trajectory into relatively few bins. Because we can expect that many transitions into and out of each substate will be required to equilibrate their relative populations, a simple cluster-counting approach (Fig. 1 B) will present a deceptively optimistic picture of convergence. To carefully assess convergence of a simulation, we must therefore compare the populations of the various substates from different fragments of the trajectory. A simple, fast way to carry out such a comparison is provided by the ensemble method described above. A higher level of rigor can be achieved by comparing multiple pairs of independent blocks of the trajectory.
It must be stressed that—though our method may provide an unambiguous negative answer to the question, Is the simulation converged?—it may only provide a provisionally positive answer. A longer simulation may well reveal longer timescale phenomena, parts of structure space not yet visited.
Our approach should be useful, in its present form, as a means to assess the relative efficiencies of two simulation methods. (The cutoff dc can always be reduced enough to suggest poorer convergence of at least one of the trajectories analyzed.) Many algorithms have recently generated broad interest by virtue of their potential to enhance the sampling of biomolecular conformation space. Some of these algorithms, notably the various parallel exchange simulations (39), invest considerable CPU time in pursuit of this goal. It is therefore important to ask whether these methods are in fact worth the extra expense, i.e., Does running the algorithm in question increase the quantity: (observed conformational sampling)/(total CPU time)?
In particular, these parallel exchange algorithms should be compared to 1), single, parallelized trajectories, as are possible with NAMD (40), for example; and 2), multiple independent trajectories as suggested by Caves et al. (28). The CPU time is easy enough to quantify, and we hope the present report will aid in evaluating the quality of sampling.
In the future, we will study trajectories of larger proteins, to develop criteria for determining cutoffs in larger systems. On the one hand, the upper bound on RMSD distance between any pair of structures increases with the size of the protein. On the other hand, larger proteins may not be as structurally diverse as small, floppy peptides—at least on the timescale currently accessible to simulation. Work already underway on a G-protein coupled receptor should shed light on these issues (A. Grossfield, personal communication, 2006). Furthermore, the approach should already be able to compare different simulation methods in large systems. The systems that may be treated with our method are not limited to proteins, or even single chains. Indeed, the method is immediately applicable for analyzing simulations of polymers, nucleic acids, or macromolecular complexes.
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ACKNOWLEDGEMENTS |
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Submitted on February 8, 2006; accepted for publication March 30, 2006.
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  F. M. Ytreberg and D. M. Zuckerman A black-box re-weighting analysis can correct flawed simulation data PNAS, June 10, 2008; 105(23): 7982 - 7987. [Abstract] [Full Text] [PDF]
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