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Nucleic Acids Research Advance Access originally published online on June 13, 2008
Nucleic Acids Research 2008 36(Web Server issue):W216-W222; doi:10.1093/nar/gkn367
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Nucleic Acids Research, 2008, Vol. 36, No. suppl_2 W216-W222
© 2008 The Author(s)
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/2.0/uk/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Articles

LocalMove: computing on-lattice fits for biopolymers

Y. Ponty, R. Istrate, E. Porcelli and P. Clote*

Department of Biology, Boston College, Chestnut Hill, MA 02467, USA

*To whom correspondence should be addressed. Tel: +1 617 552 1332; Fax: +1 617 552 2011; Email: clote{at}bc.edu

Received February 29, 2008. Revised May 21, 2008. Accepted May 22, 2008.


   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
Given an input Protein Data Bank file (PDB) for a protein or RNA molecule, LocalMove is a web server that determines an on-lattice representation for the input biomolecule. The web server implements a Markov Chain Monte-Carlo algorithm with simulated annealing to compute an approximate fit for either the coarse-grain model or backbone model on either the cubic or face-centered cubic lattice. LocalMove returns a PDB file as output, as well as dynamic movie of 3D images of intermediate conformations during the computation. The LocalMove server is publicly available at http://bioinformatics.bc.edu/clotelab/localmove/.


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 
Predicting the structure of biopolymers is one of the most important and well-studied computational problems of the 20th century—a problem, that despite enormous advances, remains only partially solved. In an effort to minimize the number of conformations to be explored, coarse-grain lattice models (beads on a string) have been studied by many authors (1–4), while coarse-grain off-lattice models have been used in discrete molecular dynamics (5). In this article, we present the LocalMove web server, which implements a Markov Chain Monte-Carlo (MCMC) algorithm to compute an approximate cubic or face-centered cubic lattice fit of either the coarse-grain or backbone model for an input Protein Data Bank (PDB) (6) file for a protein or RNA molecule.

Finding a self-avoiding walk on the cubic lattice that minimizes the coordinate root mean square deviation (given sequencesFormula and Formula of 3D points, the coordinate root mean square deviation, denoted rms or cRMS, is Formula ) with the original PDB file, after normalization to ensure unit distance between successive monomers, is known to be NP-complete (7). Thus various heuristic approaches (8–13) have been proposed to approximately solve this problem, including Hopfield nets, self-consistent field optimization, integer programming (the application of integer programming (13) provides an optimal, not just approximate, solution, however with exponential run time), etc. Unfortunately, none of these methods is publicly available, so that LocalMove is the only publicly available tool for on-lattice fit of biopolymers, allowing users to postprocess certain threading energies (aka knowledge-based potentials) for structure classification and prediction.

The method LocalMove, presented in this article, performs a Monte-Carlo exploration of the on-lattice conformational landscape through a sequence of local moves, which generalize the single-monomer end and corner moves, and the 2-monomer crankshaft moves used in ref. (14) for the cubic lattice. At each step, a measure of similarity, distance root mean square deviation [dRMS (distance root mean square deviation (dRMS) between two conformations Formula and Formula is defined by

Formula
where D(P) = (di,j) and D(Q) = (ei,j) are the corresponding distance matrices, where di,j = ||pi – pj|| and ei,j = ||qi – qj||)] is evaluated and the candidate move is either accepted or stochastically rejected, according to the Metropolis criterion. Different levels of representation are supported by LocalMove, scaling from the coarse-grain monomer model (C{alpha} for amino acids, C1' for RNA nucleotides or alternatively nucleotide centers of mass), to all backbone atoms. LocalMove supports the cubic and face-centered cubic (FCC) lattices. Various termination conditions can be defined for the walk.

There appears to be little data on the quality, in terms of coordinate root mean square deviation, cRMS, of on-lattice fits, an exception being the data of Reva et al. (15) for approximate cubic lattice fits of C{alpha}-atom traces of proteins from a small representative sample. See Tables 1 and 2 for a comparison of LocalMove with the method of Reva et al. (15,16) on the only published data of on-lattice fits that we could find.


   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS
 DISCUSSION
 REFERENCES
 
LocalMove addresses the problem of finding the best on-lattice fit for the coarse-grain model or backbone model for proteins and RNA, with a number of parameter choices for the user. Lattice type can be either the cubic or FCC lattice, described later.

LocalMove applies the Monte-Carlo algorithm (17,18), where energy is defined as follows. Given a conformation P = p1,..., pn, where each pi isin R3, define the distance matrix D(P) = (di,j), where di,j is the Euclidean distance between pi and pj. Define the dRMS between two conformations P = p1,...,pn and Q' = q1,...,qn by

Formula
where D(P) = (di,j) and D(Q) = (ei,j) are the corresponding distance matrices. To determine approximate on-lattice fit, define the energy E(C) of a given lattice conformation by Formula , where P0 is the normalized conformation of monomers C{alpha} or C1' in the coarse-grain model, or backbone atoms, as depicted in Figure 1 in the backbone model. The off-lattice conformation P0 is normalized so that distance between successive atoms is 1.


Figure 1
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  		Figure 1. Backbones of protein (left) and RNA (right). Note that residue resp. nucleotide positions increase from the bottom of the figure towards the top.

 
In LocalMove, if C' denotes the temporary conformation obtained by replacing a k-monomer segment in the current conformation C, then C' becomes the next configuration, provided that C' is a self-avoiding walk and either E(C') ≤ E(C) or a random real z is less than e– (E(C') – E(C))/RT, i.e. the Metropolis criterion holds. Details and parameter choices for the user are suggested below. Algorithmic details, computational experiments for various parameters and extensive benchmarking will appear in a companion methods paper in preparation.

Models
Backbone representation
For protein, on-lattice models have historically considered the coarse-grain representation where each residue is represented by a single point, yielding the C{alpha}-trace. For proteins, this level of granularity seems reasonable, since the average distance between consecutive C{alpha} carbons in proteins extracted from the Nucleic Acid Database (NDB) (19) yields an average of 3.8 Å with a low SD of 0.04 Å. In the case of RNA, a coarse-grain model is less able to capture the essence of an RNA conformation, since the average distance between successive C4' atoms is 6.1 Å with a SD of 0.46 Å. In the case of RNA, the backbone model thus appears to be a better representative of the conformation than is the coarse-grain model.

While it is beyond the scope of the current article to answer the question of choosing the best representation of biopolymers backbone for general on-lattice applications, we tried to offer the user the choice of a suitable representation. Namely, our algorithm extracts a subset of the atoms in the model/chain of interest, and performs its search for the best fit of this selection. The different levels of representation currently supported by LocalMove are:
Proteins RNA
Full backbone N-C{alpha}-C P-O5' – C5' – C4' – C3' – O3'
Coarse-grain C{alpha} or µ C1', N, P or µ

where in the RNA coarse-grain model, the user can select among the carbon C1'- or nitrogen N-atom, both adjacent to the glycosidic bond, the backbone phosphorus or the center of mass of the nucleotide, denoted by µ.

Lattices
LocalMove supports the cubic and FCC lattice. The latter, well-known to crystallographers as one of the Bravais lattices, has contact number 12, meaning that each lattice point has 12 immediate neighbors; see Figure 2. Covell and Jernigan have shown that the FCC lattice is the most appropriate 3D lattice for fitting protein C{alpha}-atoms as a self-avoiding walk; i.e. cRMS values are smaller for the FCC than for the cubic, body-centered cubic and tetrahedral lattices.


Figure 2
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  		Figure 2. Neighbors of a point under various lattice models: (left) 3D cubic lattice, (middle) 3D FCC, (right) numbers of self-avoiding walks of various sizes on FCC. The FCC lattice can be represented as the set of all integral coordinates (x,y,z), such that Figure 2. If p = (x,y,z) and q = (a,b,c), then p,q are immediate neighbors if Figure 2, and |x – a|,|y – b|,|z – c| ≤ 1. Note that immediate neighbors on the FCC lattice are at Euclidean distance Figure 2 from each other, hence comparisons with PDB data are made after normalization that ensures unit distance between successive monomers.

 
Algorithm
Simulated annealing
LocalMove implements the MCMC algorithm, as well as simulated annealing, and the user can set an initial temperature, terminal threshold temperature and temperature scaling factor c (i.e. temperature is periodically decreased by T = c·T). Alternatively, a greedy descent (no Metropolis step) and a Fixed Metropolis probability strategy are implemented.

Three strategies are implemented in LocalMove to choose an initial self-avoiding configuration: Random, a random 3D self-avoiding walk is generated; Straight line; Rounded (greedy). By rounding, we mean a greedy, iterative procedure to place the next monomer (or atom) of a growing chain on the closest lattice point to the previous monomer (or atom), while guaranteeing a self-avoiding walk. If this strategy does not produce a self-avoiding walk, which sometimes happens, then LocalMove chooses a random self-avoiding walk as the initial on-lattice conformation.

LocalMove performs local k-monomer moves, generalizing the move set of Sali et al. (14). Given a current self-avoiding walk p1,...,pn, LocalMove randomly chooses positions i,j, and replaces the intermediate k-monomer walk pi + 1,...,pj – 1, where k = j – i – 1, by a different k-monomer walk p';i + 1,...,p';j – 1 having the same vector difference. Three types of strategies are proposed regarding self-avoidance: strict, where the self-avoidance of the resulting walk is tested in linear time and the move is rejected if the test is failed; local, where only a subset of points adjacent to the insertion point are tested; none, where self-avoidance is not enforced, depending on the option. The relevant parameters handled by LocalMove for such moves are the local move size, the self-avoidance strategy and the strategy for picking a new local move at random.

LocalMove simulations can be stopped for some of the different following reasons: either a limit temperature is bypassed during the simulated annealing; a distance threshold is reached; the maximal number of steps have been performed or the simulation is stalled for too long, leaving few hope for improvement. In the latter, the required improvement over a user-defined period of time can be either relative or absolute.

Additional features
In addition to the features described earlier, our webserver gives its user the possibility to follow in realtime the lattice fitting process. After the beginning of the lattice fitting process, the user's; browser is redirected to a webpage featuring an experiment player based on the popular JMol. Additionally, an email is sent to the user, featuring an unique identifier for the ongoing experiment. By entering this identifier at any time during or after completion of the experiment, the user can access its results or follow its progress. Results are kept until about 1 week after the end of the experiment, and are then deleted. Even if such is the case, the user is proposed to repeat the experiment, using the same parameters or is allowed to modify them in a prefilled version of the webserver form. This allows for a quick and easy modification of an already run experiment.

Finally, movies can be generated automatically after the lattice fitting process is over. To that purpose, snapshots of the molecule are rendered using PyMol each 500 steps of the Monte-Carlo algorithm, and assembled using FFMpeg.


   RESULTS
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
DISCUSSION
 REFERENCES
 
Preliminary results are given in Tables 1 and 2, to compare LocalMove (greedy strategy, rounded initial conformation, self-avoiding walk test for intermediate conformations) with the method of Reva et al. (15) (optimal parameters A=10, T{approx}0.1 – see p. 7 of ref. (15)). Although the method of Reva et al. is clearly superior for cubic lattice fits, it is not publicly available. In contrast, LocalMove provides acceptable approximate lattice representations for cubic and FCC lattices, for various coarse grain and backbone models of both protein and RNA.

Table 1 and 2 respectively list the best scores and average scores for cubic lattice fits of 17 protein chains of various sizes. Scores for the method of Reva et al. (15) are values of RMS in lattice units, while those of LocalMove are values of cRMS in lattice units—i.e. PDB files are scaled to have distance 1.0 between successive monomers (or atoms) when superimposing structures. RMS, as measured in ref. (15), is approximately the same as cRMS; however, there is a technical difference, explained as follows. In Reva's; method, a cubic orthonormal lattice is projected onto the C{alpha}-trace of a protein, self-consistent field is approximated, followed by dynamic programming. It is unclear from ref. (15), whether the (stochastic) cubic orthonormal lattice is defined from any three randomly chosen orthogonal basis vectors emanating from origin (0,0,0), or whether the origin is randomly chosen as well. In contrast, given the C{alpha} traces p1,...,pn and q1,...,qn, BioPython computes cRMS by superimposing the centers of mass, then computing optimal rotation matrix to return the C{alpha}-trace r1,...,rn obtained by q1,...,qn by the computed translation and rotation. The value of cRMS is then

Formula
To illustrate the stability of our approach, we ran LocalMove on all RNA models/chains found in the NDB (Figure 3). Namely, we fit the backbone atoms O5',P,O3',C5',C4',C3' on the FCC lattice. We rescaled the resulting models and superimposed them with the original NDB backbone data, normalized so that adjacent atoms were at distance Formula , the distance between adjacent lattice points in the FCC lattice. Superimposition was performed using Biopython http://biopython.org/. After removal of 17 spurious values, the cRMS values obtained when superimposing the 1735 on-lattice RNA models/chains on (normalized) backbone data from the NDB, we obtain mean cRMS is 0.554169 with SD of 0.145392. Similarly, we obtained LocalMove fits of backbone atoms (N,C{alpha},C) of monochain proteins from PDBselect25](20), a nonredundant protein database, where pairwise sequence identity is at most 25%. When on-lattice fits were superimposed on original (normalized) backbone off-lattice data from PDBselect25, the cRMS had mean of 0.612181 and SD of 0.161009.


Figure 3
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  		Figure 3. (Left) Distribution of cRMS for LocalMove best on-lattice fits for the backbones (O5',P,O3',C5',C4',C3') of 1735 RNA models/chains from the NDB, superimposed with the (normalized) NDB files. Statistics for cRMS: mean is 0.554169, SD is 0.145392, both measured in lattice units. (Right) Distribution of cRMS for LocalMove best on-lattice fits for the backbone (N,C{alpha},C) of 1733 (monochain) proteins from PDBselect25 (20), a nonredundant protein database (pairwise, proteins have at most 25% sequence identity), superimposed with the (normalized) original PDB files. Statistics for cRMS: mean is 0.612181, SD is 0.161009.

 

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  		Table 1. Comparison of best scores out of 100 runs. Scores are RMS for the optimized method of Reva et al. (15) (A = 10,T{approx}0.1,shells1,2), while remaining scores are cRMS using various strategies LocalMove (See text for distinction between RMS and cRMS.)

 

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  		Table 2. Comparison of average scores out of 100 runs, for method of Reva et al. (15) and the four strategies of LocalMove, as explained in Table 1

 
For these experiments with both NDB and PDBselect25, LocalMove was run for one million steps, using the greedy (Monte Carlo with zero probability for Metropolis moves) strategy with at most three-monomer moves. In this case, the greedy strategy attempts to minimize dRMS; i.e. LocalMove accepts a randomly proposed k-monomer move, for k ≤ 3, provided that the dRMS score of the proposed move is lower. The initial on-lattice structure determined by rounding. We allow early termination when relative improvement is <0.01%; i.e. after every 15 000 steps, if the relative difference between best score and that of an ancestor 15 000 steps prior to current step is less than 0.0001, (recall that score means dRMS) then computation terminates. Technically, this means that we compute whether s0 – s1/s0 < 0.0001, where s0 denotes the ancestor score 15 000 steps before and s1 denotes the current move.


   DISCUSSION
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
REFERENCES
 
In this article, we present a new web server, LocalMove, capable of determining approximate on-lattice fits of protein and RNA 3D conformations on the cubic and the FCC lattice (Figure 4). LocalMove returns the PDB file of the approximate on-lattice fit, and interactively displays a dynamic movie of 3D images of intermediate conformations during the computation. In Tables 1 and 2, we benchmark LocalMove against what appears to be the only publicly available data set for previous on-lattice fits. To the best of our knowledge, no other method is publicly available to compute on-lattice fits of protein and RNA molecules. Reva's; method and most of the earlier methods handle only the cubic lattice, known not to be optimal for biopolymer folding, while LocalMove handles cubic and FCC lattices with a variety of coarse grain and backbone models for both protein and RNA.


Figure 4
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  		Figure 4. FCC lattice fits for the full backbone of two ribozymes models [PDB IDs 1SJ3:R (left panel) and 1GID:A (right panel)] superimposed with their original models.

 
We believe that the new server, LocalMove (Figure 5), as well as our previous 3D RNA motif detection server, DIAL, described in Ferrè et al. (21), will contribute to better detection and classification of RNA motifs, essential ultimately for predicting tertiary structure, catalytic sites and function of RNA.


Figure 5
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  		Figure 5. Screen shot of LocalMove web server.

 

   ACKNOWLEDGEMENTS
 
We would like to thank anonymous referees for helpful comments and suggestions. The initial MCMC algorithm LocalMove was designed by P.C. and implemented in C++ for the undergraduate thesis of R.I. and E.P., directed by P.C., who then entirely re-wrote the core of LocalMove, while Y.P. extended the algorithm and built the web server, requiring an interface using Python, JavaScript Jmol and mySQL. Invaluable technical help was provided by staff bioinformatics programmer, Jason Persampieri. LocalMove is somewhat related to the software MoCaPro, written by Sebastian Will under the direction of P.C. at Ludwig-Maximilians-Universität München. The latter software generalized the work of Sali et al. (14); MoCaPro is not publicly available and cannot perform the computations supported by LocalMove.

The research of P.C. and Y.P. was supported by National Science Foundati grant DBI-0543506 with PI P.C. Funding to pay the Open Access publication charges for this article was provided by NSF grant DBI-0543506.

Conflict of interest statement. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.


   Footnotes
 
The authors wish it to be known that, in their opinion, the first two authors should be regarded as joint First Authors


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 REFERENCES
 

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