Open Access

Full cyclic coordinate descent: solving the protein loop closure problem in Cα space

BMC Bioinformatics20056:159

DOI: 10.1186/1471-2105-6-159

Received: 25 April 2005

Accepted: 28 June 2005

Published: 28 June 2005

Abstract

Background

Various forms of the so-called loop closure problem are crucial to protein structure prediction methods. Given an N- and a C-terminal 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 semi-fixed, and only vary the dihedral angles. This is appropriate for a full-atom 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 vector-based optimization of a rotation around an axis with a singular value decomposition-based 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 [46]. More recently, the loop closure problem has been tackled using algorithms borrowed from the field of robotics, in particular inverse kinematics methods [79]. 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 full-atom 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 full-atom model of the protein's backbone available.

In the case of Cα-only models, the structure can be described as a sequence of pseudo bonds, pseudo angles θ and pseudo dihedral angles τ [16]. Here, the term 'pseudo' indicates that the consecutive Cα's are not actually connected by chemical bonds. As in the case of the protein's backbone, the pseudo bond lengths can be considered fixed (typically 3.8 Å). In contrast, the pseudo bond angles between three consecutive Cα atoms are most definitely not fixed, but vary between 1.4 and 2.7 radians. Hence, a Cα-only model of N residues can be represented by a sequence of N - 2 pseudo bond angles θ and N - 3 pseudo dihedral angles τ (Figure 1).
https://static-content.springer.com/image/art%3A10.1186%2F1471-2105-6-159/MediaObjects/12859_2005_Article_484_Fig1_HTML.jpg
Figure 1

A protein segment's Cα trace. The Cα positions are numbered, and the pseudo bond angles θ and pseudo dihedrals τ are indicated. The segment has length 5, and is thus fully described by two pseudo dihedral and three pseudo bond angles.

Most inverse kinematics approaches assume that the DOF consist only of dihedral angles, and keep the bond angles fixed or semi-fixed. Hence, they cannot be readily applied to the Cα-only 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 N-terminus such that it also closes at the C-terminus. 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 Cα-only 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 protein-like (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

Figure 2 illustrates the essence of the FCCD algorithm, and Table 3 provides detailed pseudo code. Here we define some of the terms that will be used throughout the article, and provide a high level overview of the FCCD algorithm.
https://static-content.springer.com/image/art%3A10.1186%2F1471-2105-6-159/MediaObjects/12859_2005_Article_484_Fig2_HTML.jpg
Figure 2

The action of the FCCD algorithm in Cα space. The Cα traces of the moving, fixed and closed segments are shown in red, green and blue, respectively. The Cα atoms are represented as spheres. The labels f0, f1 and f2 indicate the three fixed vectors at the N-terminus that are initially common between the fixed and moving segments. The loop is closed when the three C-terminal vectors of the moving segment (labelled mN-3, mN-2, mN-1) superimpose with an RMSD below the given threshold on the three C-terminal vectors of the fixed segment (labelled (fN-3, fN-2, fN-1). This figure and Figure 3 were made with PyMol http://​www.​pymol.​org.

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 (f0, f1, f2) and (fN-3, fN-2, fN-1) are relevant. We will call these two sets of vectors the N- and C-terminal 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 C-terminal 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 (mN-3, mN-2, mN-1) and the last three vectors of the fixed loop (fN-3, fN-2, fN-1) 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 (mN-3, mN-2, mN-1) on (fN-3, fN-2, fN-1) 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 mi-1, m i , mi+1and τ i as the dihedral angle of the vectors mi-2, mi-1, m i , mi+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 θ1 to θN-2and the pseudo dihedrals range from τ2 to τN-2. 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 (θ2, τ2) to (θN-3, τN-3).

Finding the optimal rotation matrix with respect to the RMSD of the C-terminal 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 mi-1, m i , mi+1and mi-2, mi-1, m i , mi+1respectively.

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 C-terminal 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 = [mN-3- m i | mN-2- m i | mN-1- m i ]

F = [fN-3- m i | fN-2- m i | fN-1- 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 rotation-inversion [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:

https://static-content.springer.com/image/art%3A10.1186%2F1471-2105-6-159/MediaObjects/12859_2005_Article_484_Equa_HTML.gif

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.

The moving segments were generated using random dihedral and bond angles in regions accessible to proteins (see previous section). This was done by sampling the (θ i , τ i ) pairs according to a probability distribution derived from a set of representative protein structures (see Methods). The bond length was fixed at 3.8 Å, in tune with the consensus Cα-Cα distance in protein structures. The last bond angle in the moving segment was chosen equal to the last bond angle in the fixed loop to make a final RMSD of 0.0 Å possible. The RMSD threshold was 0.1 Å. The maximum number of iterations was set to 1000, where one iteration is a sweep over all positions. We ran the FCCD program on 1000 different fixed segments. Table 1 summarizes the results.
Table 1

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 C-terminal overlaps.

FCCD was applied using (θ, τ) constraints and an RMSD threshold of 0.1 Å. The maximum number of iterations was set to 1000. For each loop, we attempted to generate closed segments from 1000 random moving segments within the allowed number of iterations. The moving segments were generated as described in the previous section. For all 30 loop cases, we then identified the closed segment that resembled the input loop best as judged by the RMSD. For the calculation of the RMSD, we included the N-and C-terminal overlaps. The results are shown in Table 2, and the best fitting loops for each loop size are shown in Figure 3.
Table 2

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 N-3:

pivot = moving[i,:]

# Make pivot point origin

for j from 0 to 2:

M [:,j] = moving [N-3+j,:]-pivot

F [:,j] = fixed [N-3+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 N-1:

   moving [j,:] = Γ (moving [j,:]-pivot)+pivot

rmsd = calc_rmsd(moving [N-3,:], fixed [N-3,:])

# Stop if RMSD below threshold

if rmsd<threshold:

   return moving, rmsd

# Failed: RMSD threshold not reached before maxit

return 0

The accept function rejects or accepts the proposed rotation, based on the resulting (θ, τ) pair. The svd function performs singular value decomposition, and calc_rmsd calculates the RMSD between two lists of vectors.

Table 4

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

    
https://static-content.springer.com/image/art%3A10.1186%2F1471-2105-6-159/MediaObjects/12859_2005_Article_484_Fig3_HTML.jpg
Figure 3

Loops generated by FCCD (blue) that are close to real protein loops (green). The loops with lowest RMSD to a given loop of length 4 (top), 8 and 12 (bottom) are shown (loops 1qnr, A, 195–198, 3chb, D, 51–58 and 1ctq, A, 26–37). The N- terminus is at the left hand side.

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 E-value 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 Intra-European 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

(1)
Bioinformatics center, Institute of Molecular Biology and Physiology, University of Copenhagen, Universitetsparken 15

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Copyright

© Boomsma and Hamelryck; licensee BioMed Central Ltd. 2005

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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