Full cyclic coordinate descent: solving the protein loop closure problem in Cα space
 Wouter Boomsma^{1} and
 Thomas Hamelryck^{1}Email author
DOI: 10.1186/147121056159
© Boomsma and Hamelryck; licensee BioMed Central Ltd. 2005
Received: 25 April 2005
Accepted: 28 June 2005
Published: 28 June 2005
Abstract
Background
Various forms of the socalled loop closure problem are crucial to protein structure prediction methods. Given an N and a Cterminal end, the problem consists of finding a suitable segment of a certain length that bridges the ends seamlessly.
In homology modelling, the problem arises in predicting loop regions. In de novo protein structure prediction, the problem is encountered when implementing local moves for Markov Chain Monte Carlo simulations.
Most loop closure algorithms keep the bond angles fixed or semifixed, and only vary the dihedral angles. This is appropriate for a fullatom protein backbone, since the bond angles can be considered as fixed, while the (φ, ψ) dihedral angles are variable. However, many de novo structure prediction methods use protein models that only consist of Cα atoms, or otherwise do not make use of all backbone atoms. These methods require a method that alters both bond and dihedral angles, since the pseudo bond angle between three consecutive Cα atoms also varies considerably.
Results
Here we present a method that solves the loop closure problem for Cα only protein models. We developed a variant of Cyclic Coordinate Descent (CCD), an inverse kinematics method from the field of robotics, which was recently applied to the loop closure problem. Since the method alters both bond and dihedral angles, which is equivalent to applying a full rotation matrix, we call our method Full CCD (FCDD). FCCD replaces CCD's vectorbased optimization of a rotation around an axis with a singular value decompositionbased optimization of a general rotation matrix. The method is easy to implement and numerically stable.
Conclusion
We tested the method's performance on sets of random protein Cα segments between 5 and 30 amino acids long, and a number of loops of length 4, 8 and 12. FCCD is fast, has a high success rate and readily generates conformations close to those of real loops. The presence of constraints on the angles only has a small effect on the performance. A reference implementation of FCCD in Python is available as supplementary information.
Background
Many protein structure prediction methods require an algorithm that is capable of constructing a new conformation for a short segment of the protein, without affecting the rest of the molecule. In other words, a protein fragment needs to be generated that seamlessly closes the gap between two given, fixed end points. This problem is generally called the loop closure problem, and was introduced in a classic paper by Go and Scheraga more than 30 years ago [1]. It has been the continued subject of intensive research over many years due to its high practical importance in structure prediction.
The loop closure problem arises in at least two different structure prediction contexts. In homology modelling, it is often necessary to rebuild certain loops that differ between the protein being modelled and the template protein [2]. The modelled loop needs to bridge the gap between the end points of the template's loop.
In de novo prediction, local resampling or local moves can be considered as a variant of the loop closure problem. Typically, the conformation of a protein segment needs to be changed without affecting the rest of the protein as a sampling step in a Markov Chain Monte Carlo (MCMC) procedure [3]. In both homology and de novo structure prediction, the problem is however essentially the same.
The classic article by Go and Scheraga [1] describes an analytical solution to finding all possible solutions for a protein backbone of three residues. In this case, the degrees of freedom (DOF) comprise six dihedral angles, ie. the backbone's (φ, ψ) angles. Another approach is to use a fragment library derived from the set of solved protein structures, and look for fragments or combinations of fragments that bridge the given fixed ends [4–6]. More recently, the loop closure problem has been tackled using algorithms borrowed from the field of robotics, in particular inverse kinematics methods [7–9]. Still other methods use various Monte Carlo chain perturbation approaches, often combined with analytical methods [10, 11, 3, 12]. A good overview of loop closure methods and references can be found in Kolodny et al. (2005) [6].
Most methods assume that one is working with a fullatom protein backbone with fixed bond angles and bond lengths, so the DOF consist solely of the backbone's (φ, ψ) angles. However, in many cases not all the atoms of the protein backbone are present in the model. In particular, a large class of structure prediction, design and in silico folding methods makes use of drastically simplified models of protein structure [13, 14].
A protein structure might for example be represented by a chain of Cα atoms or a chain of virtual atoms at the centers of mass of the side chain atoms [15]. In these models, there is obviously no fullatom model of the protein's backbone available.
Most inverse kinematics approaches assume that the DOF consist only of dihedral angles, and keep the bond angles fixed or semifixed. Hence, they cannot be readily applied to the Cαonly case without restricting the search space unnecessarily. In principle, fragment library based methods would apply, but here the problem of data sparsity arises [17, 18]. Often, no suitable fragments can be found if the number of residues between the fixed ends becomes too high.
In order to solve the loop closure problem in Cα space, we extend a particularly attractive approach that was recently introduced by Canutescu & Dunbrack [8]. The algorithm is called Cyclic Coordinate Descent (CCD), and like many other loop closure algorithms it derives from the field of robotics [19]. As pointed out by Canutescu & Dunbrack, the CCD algorithm is meant as a black box method that generates plausible protein segments that bridge two given, fixed endpoints. The final choice is typically made based upon the occurrence of steric clashes, applicable constraints (for example side chain conformations) and evaluation of the energy.
The CCD algorithm does not directly generate conformations that bridge a given gap, but alters the dihedral angles of a given starting segment that already overlaps at the Nterminus such that it also closes at the Cterminus. The starting segment can be generated in many ways, for example by using a fragment library derived from real structures or by constructing random artificial fragments with reasonable conformations. Surprisingly, most protein loops can be closed efficiently by CCD starting from artificial loops constructed with random (φ, ψ) dihedral angles [8].
The CCD algorithm alters the (φ, ψ) dihedral angles for every residue in the segment in an iterative way. In each step, the RMSD between the chain end and the overlap is minimized by optimizing one dihedral angle. Because only one dihedral angle is optimized at a time, the optimal rotation can be calculated efficiently using simple vector arithmetic.
The list of advantages of CCD is impressive: it is conceptually simple and easy to implement, computationally fast, very flexible (ie. capable of incorporating various restraints and/or constraints) and numerically stable. Therefore, we decided to adopt the CCD algorithm for use with Cαonly models. Here, we describe a new version of CCD that optimizes both dihedral angles and bond angles, while maintaining all the advantages of the CCD method. We call our method Full Cyclic Coordinate Descent (FCCD), where "Full" indicates that both dihedral angles and bond angles are optimized, while only the bond lengths remain fixed. At the heart of the FCCD method lies a procedure to superimpose point sets with minimal Root Mean Square Deviation (RMSD), based on singular value decomposition. As is the case for the CCD algorithm, FCCD is not a modelling method in itself. Rather, it can be used as a method to generate possible conformations that can be evaluted using some kind of energy function.
To test the algorithm, we selected random segments from a protein structure database, and evaluated the efficiency of closing the corresponding gaps starting from artificial segments with proteinlike (θ, τ) angles. We show that FCCD is both fast and successful in solving the loop closure problem, even in the presence of angle constraints. Conformations close to those of real protein loops are readily generated. Finally, we discuss possible applications of the FCCD algorithm, and mention some possible disadvantages.
Results and discussion
Overview of the FCCD algorithm
The fixed segment is a list of Cα vector positions that specifies the gap that needs to be bridged. Only the first and last three Cα positions, with corresponding vectors (f_{0}, f_{1}, f_{2}) and (f_{N3}, f_{N2}, f_{N1}) are relevant. We will call these two sets of vectors the N and Cterminal overlaps, respectively. The moving segment is a list of Cα position vectors that will be manipulated by the FCCD algorithm to bridge the gap. The closed segment is the moving segment after its pseudo bond angles and pseudo dihedral angles were adjusted to bridge the N and Cterminal overlaps of the fixed segment. The vectors describing the positions of the Cα atoms in a segment of N residues are labelled from 0 to N  1.
Initially, the first three vectors of the moving loop coincide with the first three vectors of the fixed segment, while the last three vectors are conceivably reasonably close to the last three vectors of the fixed loop. This last condition is however not very critical. The moving segment can be obtained using any algorithm that generates plausible Cα fragments, including deriving them from real protein structures. The fixed segment is typically derived from a real protein of interest, or a model in an MCMC simulation.
The FCCD algorithm changes the pseudo bond angles and pseudo dihedral angles of the moving loop in such a way that the RMSD between the last three vectors of the moving loop (m_{N3}, m_{N2}, m_{N1}) and the last three vectors of the fixed loop (f_{N3}, f_{N2}, f_{N1}) is minimized, thereby seamlessly closing the gap.
Note that we assume that the last three vectors of the moving and fixed segments can be superimposed with an RMSD of 0.0 Å (see Figure 2). In other words, the first and last pseudo bond angles in both segments are equal. It is however perfectly possible to use segments with different pseudo bond angles at these positions. Since the final possible minimum RMSD will be obviously greater than 0 in this case, the RMSD threshold needs to be adjusted accordingly.
The algorithm proceeds in an iterative way. In each iteration, a vector m_{ i }in the moving segment is chosen that will serve as a center of rotation. This chosen center of rotation will be called the pivot throughout this article. Then, the rotation matrix that rotates (m_{N3}, m_{N2}, m_{N1}) on (f_{N3}, f_{N2}, f_{N1}) around the pivot and resulting in minimum RMSD is determined, and applied to all the vectors m_{ j }downstream i (with i <j <N). In the next iteration, a new pivot is chosen, and the procedure is repeated. The vectors in the chain can be traversed linearly, or they can be chosen at random in each iteration. The difference between FCCD and CCD is that the latter applies a general rotation to the chain using an atom in the chain as a pivot, while the former only applies a rotation around a single axis. The process is stopped when the RMSD falls below a given threshold.
Finding the optimal (with respect to the RMSD) rotation matrix corresponds to finding one optimal pseudo bond angle and pseudo dihedral angle pair. We define θ_{ i }as the bond angle of the vectors m_{i1}, m_{ i }, m_{i+1}and τ_{ i }as the dihedral angle of the vectors m_{i2}, m_{i1}, m_{ i }, m_{i+1}(see Figure 1 and [16]). These definitions have the intuitive interpretation that altering (θ_{ i }, τ_{ i }) changes the positions of all Cα's downstream from position i. Conversely, using pivot m_{ i }and applying a rotation matrix to all the positions downstream from position i corresponds to changing pseudo bond angle θ_{ i } and pseudo dihedral angle τ_{ i }.
For a segment of N Cα's (with N > 3), the pseudo angles range from θ_{1} to θ_{N2}and the pseudo dihedrals range from τ_{2} to τ_{N2}. Since the first and last bond angles of the moving segment are fixed, the pivot points range from position 2 to position N  3 (with N > 4). The pseudo bond angle and pseudo dihedral angle pairs thus range from (θ_{2}, τ_{2}) to (θ_{N3}, τ_{N3}).
Finding the optimal rotation matrix with respect to the RMSD of the Cterminal overlaps can be efficiently solved using singular value decomposition, as described in detail in the following section.
Finding the optimal rotation
In this section we discuss solving the following subproblem arising in the FCCD algorithm: given a chosen pivot point i in the moving segment, find the optimal (θ_{ i }, τ_{ i }) pair that minimizes the RMSD between the last three Cα vectors in the moving segment and the last three Cα vectors in the fixed segment. Recall that the (θ_{ i }, τ_{ i }) pair at position i corresponds to the pseudo bond angles and pseudo dihedral angles defined by vectors m_{i1}, m_{ i }, m_{i+1}and m_{i2}, m_{i1}, m_{ i }, m_{i+1}respectively.
Finding the optimal (θ_{ i }, τ_{ i }) pair simply corresponds to finding the optimal rotation matrix using Cα position i as the center of rotation (see Figure 2). This reformulated problem can be solved by a variant of a well known algorithm to superimpose two point sets with minimum RMSD which makes use of singular value decomposition [20, 21]. Below, we describe this adapted version of the algorithm.
First, the Cterminal overlaps of the moving and the fixed segment need to be translated to the new origin that will be used as pivot for the optimal rotation. This new origin is the pivot vector m_{ i } at Cα position i in the moving segment. The new vector coordinates of the moving and the fixed segments are put in two matrices (respectively M and F), with the coordinates of the vectors positioned column wise:
M = [m_{N3} m_{ i } m_{N2} m_{ i } m_{N1} m_{ i }]
F = [f_{N3} m_{ i } f_{N2} m_{ i } f_{N1} m_{ i }]
Then, the correlation matrix Σ is calculated using M and F :
Σ = FM^{ T }
Any real n × m matrix A can be written as the product of an orthogonal n × n matrix U, a diagonal n × m matrix D and an orthogonal m × m matrix V^{ T }[22]. Such a factorization is called a singular value decomposition of A. The positive diagonal elements of D are called the singular values. Hence, Σ can be written as:
Σ = UDV^{ T }
The optimal rotation Γ is then calculated as follows:
Γ = USV^{ T }
The value of the diagonal 3 × 3 matrix S is determined by the product det(U)det(V^{ T }), which is either 1 or 1. If this product is 1 then S = diag(1, 1, 1), else S is the 3 × 3 unit matrix. The matrix S ensures that Γ is always a pure rotation, and not a rotationinversion [21].
In order to apply to all the vectors that are downstream from the pivot point i, these vectors are first translated to the origin of the rotation (ie. pivot point m_{ i }), left multiplied by Γ and finally translated back to the original origin:
where i <j <N.
Adding angle constraints to FCCD
It is straightforward to constrain the (θ, τ) angles to a given probability distribution. For each rotation matrix Γ, the resulting new pseudo bond angles and dihedral angles can easily be calculated. The new angles can for example be accepted or rejected using a simple rejection sampling Monte Carlo scheme, comparing the probabilities of the previous pair (θ^{ prev }, τ^{ prev }) with that of the next pair (θ^{ next }, τ^{ next }). If P (θ^{ next }, τ^{ next }) > P (θ^{ prev }, τ^{ prev }) the change is accepted, otherwise it is accepted with a chance proportional to P (θ^{ next }, τ^{ next }) / P (θ^{ prev }, τ^{ prev }). A similar approach was used by Canutescu & Dunbrack [8], and we describe its performance in combination with FCCD in the following section.
More advanced methods could take the probability of the sequence of angles into account as well, for example using a Hidden Markov Model of the backbone [23]. The pseudo code in Table 3 illustrates accepting/rejecting rotations using an unspecified 'accept' function, whose details will depend on the application.
FCCD's performance
In order to evaluate the general efficiency of the method, we selected random fragments of various sizes from a representative database of protein structures, and used these fragments as fixed segments. Hence, the evaluation described below is not limited to loops, but extends to random protein segments. This is a relevant test, since local moves in a typical MCMC simulation are indeed performed on random segments.
The fixed segments were sampled from a dataset of fold representatives (see Methods). First we selected a random fold representative, and subsequently extracted a random continuous fragment of suitable length. The lengths varied from 10 to 30 with a step size of 5. It should be noted that the length of the segment here refers to the number of Cα atoms between the ends that need to be bridged.
Performance of the FCCD algorithm for various segment lengths. The first and second number in columns 2–4 refer to unconstrained and constrained FCCD, respectively. Columns 2 and 3 respectively show the average time and number of iterations needed for closing a single segment successfully. The percentage of loops successfully closed in under 1000 iterations is shown in the last column.
Segment length  Average time (ms)  Average iterations  % Closed 

5  4.5/51.7  14.0/27.0  99.90/86.50 
10  5.2/28.3  10.5/16.8  99.40/98.20 
15  5.6/28.6  7.8/12.1  99.60/99.40 
20  6.2/27.1  6.3/9.0  99.80/99.40 
25  7.6/31.7  5.5/7.6  99.00/99.90 
30  7.1/31.0  4.4/6.3  99.70/99.40 
A first observation is the effect of the angle constraints. These slow down FCCD with a factor of 10 for small segments (5 residues) and roughly a factor of 5 for larger segments (10 residues or more). Nonetheless FCCD including constraints remains quite speed efficient: small five residue segments are on average closed in about 50 ms, while larger segments (from 10 to 30 residues) are closed considerably faster (on average in about 30 ms). The explanation for this is of course that it is easier to close large segments because they have more DOF. Hence, FCCD, like CCD, is fast and easily handles large segments efficiently.
Overall, the success rate of FCCD is excellent, and very little affected by constraints. For 5 residue segments, adding constraints diminishes the number of successfully closed segments from 99.9% to 86.5%. This effect is however much less pronounced for larger segments: more than 98% percent of the moving/fixed segment pairs can be successfully closed. In short, FCCD is both speed efficient and has a high success rate, even in the presence of constraints.
Evaluation of FCCD's sampling space
Does FCCD potentially generate realistic protein conformations? FCCD could be used to propose possible conformations that are subsequently evaluated by an energy function. In this context, it is of course imperative to generate realistic conformations. To answer this question, we evaluate FCCD's ability to generate closed segments that are close to real protein loops. We used 30 real loops with lengths of 4, 8 and 12 residues as fixed segments. The loop length refers to the number of residues between the N and Cterminal overlaps.
Minimum RMSD (out of 1000 tries) between a fixed segment derived from a protein structure and a closed segment generated by FCCD. The length of the loops is shown between parentheses in the upper row.
Loop (4)  RMSD  Loop (8)  RMSD  Loop (12)  RMSD 

1dvj, A, 20–23  0.59  1cru, A, 85–92  2.31  1cru, A, 358–369  3.37 
1dys, A, 47–50  0.67  1ctq, A, 144–151  2.22  1ctq, A, 26–37  2.40 
1egu, A, 404–407  0.61  1d8w, A, 334–341  2.04  1d4o, A, 88–99  3.20 
1ej0, A, 74–77  0.61  1ds1, A, 20–27  2.20  1d8w, A, 43–54  2.74 
1i0h, A, 123–126  0.73  1gk8, A, 122–129  2.20  1ds1, A, 282–293  3.16 
1id0, A, 405–408  0.66  1i0h, A, 145–152  2.42  1dys, A, 291–302  2.90 
1qnr, A, 195–198  0.54  1ixh, 106–113  1.98  1egu, A, 508–519  3.06 
1qop, A, 44–47  0.58  1lam, 420–427  2.16  1f74, A, 11–22  3.12 
1tca, 95–98  0.76  1qop, B, 14–21  2.17  1q1w, A, 31–42  3.04 
1thf, D, 121–124  0.56  3chb, D, 51–58  1.97  1qop, A, 175–186  2.97 
Average RMSD  0.63  Average RMSD  2.17  Average RMSD  3.00 
Table 3
maxit = maximum number of iterations 

moving = N × 3 matrix of Cα positions in moving segment 
fixed = N × 3 matrix of Cα positions in fixed segment 
threshold = desired minimum RMSD 
N = length of the segments 
M = 3 × 3 matrix (centered coordinates along columns) 
F = 3 × 3 matrix (centered coordinates along columns) 
S = diag(1, 1, 1) 
repeat maxit: 
# Start iteration over pivots 
for i from 2 to N3: 
pivot = moving[i,:] 
# Make pivot point origin 
for j from 0 to 2: 
M [:,j] = moving [N3+j,:]pivot 
F [:,j] = fixed [N3+j,:]pivot 
# Find the rotation Γ that minimizes RMSD 
Σ = FM^{ T } 
U, D, V^{ T }= svd(Σ) 
# Check for reflection 
if det(U)det(V^{ T })<0: 
U = US 
Γ = UV^{ T } 
# Evaluate and apply rotation 
if accept(Γ): 
# Apply the rotation to the moving segment 
for j from i+1 to N1: 
moving [j,:] = Γ (moving [j,:]pivot)+pivot 
rmsd = calc_rmsd(moving [N3,:], fixed [N3,:]) 
# Stop if RMSD below threshold 
if rmsd<threshold: 
return moving, rmsd 
# Failed: RMSD threshold not reached before maxit 
return 0 
SABMark identifiers of the 236 structures used as fold representatives
1ew6a_  1ail__  1l1la_  1kid__  1n8yc1  1gzhb1  1e5da1  1ep3b2  1ihoa_  1m0wa1 

1dhs__  1gpua2  2lefa_  1nsta_  1eaf__  1iiba_  1d5ra2  1foha3  1gpua3  1crza2 
3pvia_  1i6pa_  1e4ft1  1kx5d_  2pth__  1lu9a2  1dkla_  1fsga_  1m2oa3  2dpma_ 
1ajsa_  1fxoa_  3tgl__  1bx4a_  1mtyg_  1duvg2  1qopb_  1iata_  1k2yx2  1f0ka_ 
1ayl_1  1toaa_  8abp__  1nh8a1  1bi5a2  2mhr__  1a2pa_  3lzt__  1dkia_  1e7la2 
1bf4a_  1bb8__  1kpf__  1mu5a2  1lfda_  1gpea2  1jqca_  1a2va2  1jfma_  1ll7a2 
1cjxa1  1lo7a_  1fm0e_  1fs1b2  1o0wa2  1dtja_  1k0ra3  1evsa_  1jpdx2  1qd1a1 
1d5ya3  1h3fa2  1iq0a3  1tig__  1xxaa_  1ck9a_  1gyxa_  1e5qa2  1ivsa2  1qbea_ 
3grs_3  1f08a_  1c7ka_  1lkka_  1dq3a3  1uox_1  12asa_  1bob__  1m4ja_  1dv5a_ 
1f5ma_  1k2ea_  1ei1a2  1jdw__  1ln1a_  2pola2  1f0ia1  1rl6a1  1fvia2  1j7la_ 
1is2a1  1e8ga2  1qr0a1  2dnja_  1kuua_  1qh5a_  1ii7a_  1b8pa2  1j7na3  1chua3 
1f00i3  1grj_1  1nkd__  1mwxa3  1jp4a_  1ih7a2  1eula2  1gnla_  1maz__  2por__ 
4htci_  1es7b_  1tocr1  1d1la_  1fd3a_  1i8na_  1h8pa1  4sgbi_  1fltv_  1quba1 
1d4va3  1tpg_2  1iuaa_  1fv5a_  1mdya_  1zmec1  1fjgn_  1eska_  1i50i2  1fbva4 
1dmc__  1e53a_  1ezvb1  1jeqa1  1k3ea_  1rec__  1lm5a_  1k82a1  1jaja_  1m0ka_ 
1c0va_  1kqfc_  1ocrk_  1h67a_  2cpga_  1ljra1  1brwa1  1hs7a_  2cbla2  1jmxa2 
1hyp__  1cuk_2  1ecwa_  1l9la_  1g7da_  1jkw_1  1dgna_  1iqpa1  1pa2a_  1ko9a1 
1f1za1  1ks9a1  2sqca2  1d2ta_  1h3la_  1wer__  1b3ua_  1n1ba2  1poc__  1e79i_ 
1m1qa_  1enwa_  1g4ma1  1e5ba_  1qhoa2  1kv7a2  1l4ia2  1c8da_  1amm_1  1ca1_2 
1phm_2  1d7pm_  1jjcb2  1flca1  1gr3a_  1mjsa_  1a8d_1  1lf6a2  1fqta_  1jb0e_ 
1jh2a_  1lcya1  1mgqa_  1hcia1  1b3qa2  1jlxa1  1dar_1  1exma2  1ejea_  1agja_ 
1e79d2  2rspa_  1h0ha1  1gtra1  2erl__  1btn__  1lf7a_  1jmxa5  1crua_  1m1xa4 
1hx0a1  1goia1  1ciy_2  1daba_  3tdt__  1gg3a1  1pmi__  1bdo__  1h3ia2  1gppa_ 
1f39a_  1k6wa1  1jqna_  1lu9a1  1m6ia1  1o94a3 
It is clear that FCCD readily generates closed segments that are reasonably close to the real loops, with an average RMSD of about 0.6, 2.2 and 3.0 Å for loops of 4, 8 and 12 residues, respectively. The highest minimum RMSD values for these loop lengths are 0.76, 2.42 and 3.37 Å, respectively, indicating that FCCD in general can come up with a reasonably close conformation. Using more initial moving segments will obviously increase the chance of encountering a close conformation. Additionally, one can also expect an even better performance with a more refined way to constrain the (θ, τ) angles.
Conclusion
In this article, we introduce an algorithm that solves the loop closure problem for Cα only protein models. The method is conceptually similar to the CCD loop closure method introduced by Canutescu and Dunbrack [8], but optimizes dihedral and bond angles simultaneously, while the former method only optimizes one angle at a time. At the heart of the method lies a modified algorithm to superimpose point sets with minimum RMSD, based on singular value decomposition [20, 21].
The algorithm is fast, numerically stable and leads to a solution for the great majority of loop closure problems studied here. Importantly, the method remains efficient even in the presence of constraints on the dihedral and bond angles. FCCD readily handles large gaps, and potentially generates realistic conformations. Compared to other loop closure methods, FCCD is surprisingly easy to implement provided a function is available to calculate the singular value decomposition of a matrix.
A possible disadvantage is that FCCD has a tendency to induce large changes to the pseudo angles at the start of the moving segment while angles near the end are less affected, which is also the case for CCD [8]. This can for example be avoided by selecting the pivot points in a random fashion, or by limiting the allowed change in the angles per iteration. Occasionally the method gets stuck, which can be avoided by incorporating stochastic changes away from the encountered local minimum. One can also simply try again with a new random moving segment. We believe that CCD and FCCD despite these disadvantages are among the most efficient loop closure algorithms currently available.
The FCCD algorithm proposed here has great potential for use in structure prediction methods that only make use of Cα atoms, or that otherwise do not include all backbone atoms [15, 13, 14]. FCCD could be used for example to implement local moves in a MCMC procedure. The moving segments could be derived from a fragment database or generated from a probabilistic model of the protein backbone. The latter model could range from a primitive probability distribution over allowed (θ, τ) angle pairs like we used here to a Hidden Markov Model that also models the sequence of (θ, τ) angle pairs.
We are planning to use the FCCD algorithm in combination with a sophisticated probabilistic model of the protein's backbone, which will steer both the generation of the initial moving loop and the acceptance/rejection of the angles. The performance of FCCD in this context will be the subject of a future publication.
Methods
Implementation
The FCCD algorithm was implemented in C, using the LAPACK [24] function dgesvd for the calculation of the singular value decomposition. Handling PDB files and calculating the (θ, τ) angles [16] was done using Biopython's Bio.PDB module [25]. We used a 2.5 GHz Pentium processor to calculate the benchmarks. A reference implementation of FCCD in Python is available as supplementary information.
Structure databases
For the calculation of the (θ, τ) probability distribution and the generation of random protein fragments, we used the SABMark 1.63 Twilight Zone database [26]. SABMark Twilight Zone contains 2230 high quality protein structures, divided over 236 different folds. All protein pairs have a BLAST Evalue below 1, and thus presumably belong to different superfamilies. A dataset of fold representatives was generated by selecting a single structure at random for each fold (see Table 4).
The loops used to evaluate FCCD's sampling space were derived from Canutescu & Dunbrack [8]. We shifted two loops (1d8w, A, 46–57 and 1qop, A, 178–189) by three residues to ensure that all loops had three flanking residues on each side.
Calculation of the (θ, τ) probability distribution
The bond angle θ was subdivided in 18 bins and the dihedral angle τ in 36 bins, in both cases starting at 0 degrees and with a bin width of 10 degrees. All (θ, τ) angles were extracted from all structures in the SABMark Twilight Zone database that consisted of a polypeptide chain without breaks. In total, 257534 angle pairs were extracted. Each such (θ, τ) angle pair was assigned to a bin pair, and the number of angle pairs assigned to each bin pair was stored in a 18 × 36 count matrix. Finally, the normalized count matrix was used to assign a probability to any given (θ, τ) angle pair.
List of abbreviations
 • CCD:

Cyclic Coordinate Descent
 • DOF:

Degrees Of Freedom
 • FCCD:

Full Cyclic Coordinate Descent
 • MCMC:

Markov Chain Monte Carlo
 • RMSD:

Root Mean Square Deviation
Declarations
Acknowledgements
Wouter Boomsma is supported by the Lundbeckfond http://www.lundbeckfonden.dk/. Thomas Hamelryck is supported by a Marie Curie IntraEuropean Fellowship within the 6th European Community Framework Programme. We acknowledge encouragement and support from Prof. Anders Krogh, Bioinformatics Center, Institute of Molecular Biology and Physiology, University of Copenhagen.
Authors’ Affiliations
References
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