Determining Protein Structures from NOESY Distance Constraints by Semidefinite Programming

Abstract

Contemporary practical methods for protein nuclear magnetic resonance (NMR) structure determination use molecular dynamics coupled with a simulated annealing schedule. The objective of these methods is to minimize the error of deviating from the nuclear overhauser effect (NOE) distance constraints. However, the corresponding objective function is highly nonconvex and, consequently, difficult to optimize. Euclidean distance matrix (EDM) methods based on semidefinite programming (SDP) provide a natural framework for these problems. However, the high complexity of SDP solvers and the often noisy distance constraints provide major challenges to this approach. The main contribution of this article is a new SDP formulation for the EDM approach that overcomes these two difficulties. We model the protein as a set of intersecting two- and three-dimensional cliques. Then, we adapt and extend a technique called semidefinite facial reduction to reduce the SDP problem size to approximately one quarter of the size of the original problem. The reduced SDP problem can be solved approximately 100 times faster, and it is also more resistant to numerical problems from erroneous and inexact distance bounds.

1. Introduction

Computing three-dimensional 3D protein structures from their amino acid sequences is one of the most widely studied problems in bioinformatics. Knowing the structure of the protein is key to understanding its physical, chemical, and biological properties. The protein nuclear magnetic resonance (NMR) method is fundamentally different from the X-ray method: It is not a “microscope with atomic resolution”; rather, it provides a network of distance measurements between spatially proximate hydrogen atoms (Güntert, 1998). As a result, the NMR method relies heavily on complex computational algorithms. The existing methods for protein NMR can be categorized into four major groups: (i) methods based on Euclidean distance matrix completion (EDMC) (Braun et al., 1981; Havel and Wüthrich, 1984; Biswas et al., 2008; Leung and Toh, 2009); (ii) methods based on molecular dynamics and simulated annealing (Nilges et al., 1988; Brünger, 1993; Güntert et al., 1997; Schwieters et al., 2003; Güntert, 2004); (iii) methods based on local/global optimization (Braun and Go, 1985; Moré and Wu, 1997; Williams et al., 2001); and (iv) methods originating from sequence-based protein structure prediction algorithms (Shen et al., 2008; Raman et al., 2010; Alipanahi et al., 2011).

In the early years of protein NMR, many EDMC-based methods directly worked on the corresponding Euclidean distance matrix (EDM). The first method to use EDMC for protein NMR was developed by Braun et al. (1981). Other notable methods include EMBED (Havel et al., 1983) and DISGEO (Havel and Wüthrich, 1984). These methods faced two major drawbacks: randomly guessing the unknown distances is ineffective and after several iterations of distance correction, the error in the distances tended to become large (Güntert, 1998); and there was no way to control the embedding dimensionality, which also tended to become large.

A major breakthrough came that combined simulated annealing with molecular dynamics (MD) simulation. Nilges et al. (1998) made some improvements in the MD-based protein NMR structure determination. Instead of an empirical energy function, they proposed a simple geometrical energy function based on the NOE restraints that penalized large violations, and they also combined simulated annealing (SA) with MD. These methods were able to search the massive conformation space without being trapped in one of numerous local minima. The XPLOR method (Brünger, 1993; Schwieters et al., 2003, 2006) was one of the first successful and widely adapted methods that was built on the molecular dynamics simulation package CHARMM (Brooks et al., 1983). The number of degrees of freedom in torsion angle space is nearly 10 times smaller than in Cartesian coordinates space, while being equivalent under mild assumptions. The torsion angle dynamics algorithm implemented in the program CYANA (Güntert, 2004), and previously in the program DYANA (Güntert et al., 1997), is one of the fastest and most widely used methods.

1.1. Gram matrix methods

The EDMC methods are based on using the Gram matrix, or the matrix of inner products. This has many advantages. For example, (i) the Gram matrix and Euclidean distance matrix (EDM) are linearly related; (ii) instead of enforcing all of the triangle inequality constraints, it is sufficient to enforce that the Gram matrix is positive semidefinite; and (iii) the embedding dimension and the rank of the Gram matrix are directly related.

Semidefinite programming (SDP) is a natural choice for formulating the EDMC problem using the Gram matrix. SDP-based EDMC methods have demonstrated great success in solving the sensor network localization (SNL) problem (Doherty et al., 2001; Biswas and Ye, 2004; Biswas et al., 2006; Wang et al., 2008; Kim et al., 2009; Krislock and Wolkowicz, 2010). In the SNL problem, the location of a set of sensors is determined based on given short-range distances between spatially proximate sensors. As a result, the SNL problem is inherently similar to the protein NMR problem. The major obstacle in extending SNL methods to protein NMR is the complexity of SDP solvers. To overcome this limitation, Biswas et al. (2008) proposed DAFGAL, which is built on the idea of divide-and-stitch. Leung and Toh (2009) proposed the DISCO method. It is an extension of DAFGAL that can determine protein molecules with more than 10,000 atoms using a divide-and-conquer technique. The improved methods for partitioning the partial distance matrix and iteratively aligning the solutions of the subproblems boosted the performance of DISCO in comparison to DAFGAL.

1.2. Contributions of the proposed SPROS method

Most of the existing methods make some of the following assumptions: (i) (nearly) exact distances between atoms are known; (ii) distances between any type of nuclei (not just hydrogens) are known; (iii) ignoring the fact that not all hydrogen atoms can be uniquely assigned; and (iv) overlooking the ambiguity in the NOE cross-peak assignments. In order to automate the NMR protein structure determination process, we need a robust structure calculation method that tolerates more errors. In this article, we give a new SDP formulation that does not assume (i–iv) above. Moreover, the new method, called “SPROS” (semidefinite programming-based protein structure determination), models the protein molecule as a set of intersecting two- and three-dimension cliques. We adapt and extend a technique called semidefinite facial reduction, which makes the SDP problem strictly feasible and reduces its size to approximately one quarter the size of the original problem. The reduced problem is more numerically stable and can be solved nearly 100 times faster.

Outline We have divided the presentation of the SPROS method into providing the necessary background, followed by giving a description of techniques used for problem size reduction, and finally, showing the performance of the method on experimentally derived data.

Preliminaries Scalars, vectors, sets, and matrices are shown in lowercase, lowercase bold italic, script, and uppercase italic letters, respectively. We work only on real finite-dimensional Euclidean Spaces and define an inner product operator for these spaces: (i) for the space of real p-dimensional vectors, , and (ii) for the space of real p × q matrices, . The Euclidean distance norm of is defined as . We use the Matlab notation that . For a matrix and an index set is the matrix formed by rows and columns of A indexed by . Finally, we let the space of symmetric p × p matrices.

2. The SPROS Method

2.1. Euclidean distance geometry

Euclidean Distance Matrix A symmetric matrix D is called a Euclidean distance matrix (EDM) if there exists a set of points such that:

(1)

The smallest value of r is called the embedding dimension of D, and is denoted embdim(D). The space of all n × n EDMs is denoted .

The Gram Matrix If we define , then the matrix of inner-products, or Gram matrix, is given by . It immediately follows that , where is the set of symmetric positive semidefinite n × n matrices. The Gram matrix and the Euclidean distance matrix are linearly related:

(2)

where 1 is the all-ones vector of the appropriate size, and diag(G) is the vector formed from the diagonal of G. To go from the EDM to the Gram matrix, we use the Moore-Penrose generalized inverse

(3)

where is the centering matrix, and is the set of symmetric matrices with zero diagonal.

Schoenberg's Theorem Given a matrix D, we can determine if it is an EDM with the following well-known theorem (Schoenberg, 1935):

Theorem 1

A matrix is a Euclidean distance matrix if and only if K^†(D) is positive semidefinite. Moreover, embdim(D)  = rank(K^†(D)), for all .

2.2. The SDP formulation

Semidefinite optimization or, more commonly, semidefinite programming, is a class of convex optimization problems that has attracted much attention in the optimization community and has found numerous applications in different science and engineering fields. Notably, several diverse convex optimization problems can be formulated as SDP problems (Vandenberghe and Boyd, 1996). Current state-of-the-art SDP solvers are based on primal-dual interior-point methods.

Preliminary Problem Formulation There are three types of constraints in our formulation: (i) equality constraints, which are the union of equality constraints preserving bond lengths (), bond angles (), and planarity of the coplanar atoms (), giving ; (ii) upper bounds, which are the union of NOE-derived (), hydrogen bonds (), disulfide and salt bridges (), and torsion angle () upper bounds, giving ; and (iii) lower bounds, which are the union of steric or van der Waals () and torsion angle () lower bounds, giving . We assume the target protein has n atoms, . The preliminary problem formulation is given by:

(4)

where and e_i is the ith column of the identity matrix. The centering constraint K1 = 0 ensures that the embedding of K is centered at the origin. Since both upper bounds and lower bounds may be inaccurate and noisy, non-negative penalized slacks, ζ_ij's and ξ_ij's, are included to prevent infeasibility and manage ambiguous upper bounds. The heuristic rank reduction term, , with γ < 0 in the objective function, produces lower-rank solutions (Weinberger and Saul, 2004).

Bond lengths and angles Covalent bonds are very stable, and since their fluctuations cannot be detected in NMR experiments, all bond lengths and angles must be set to ideal values computed from accurate X-ray structures; see Engh and Huber (1991). Bonds length and angle constraints are written in terms of the distance between an atom and its immediate neighbor and an atom and its second nearest neighbor, respectively.

Planarity constraints Proteins contain several coplanar atoms, from H, CO, and N atoms in the peptide planes, and from side chain in moieties found in nine amino acids (Hooft et al., 1996). We have enforced planarity by preserving the distances between all coplanar atoms.

Torsion angle constraints Another source of structural information in protein NMR is the set of torsion angle restraints, defined as . We extend the idea proposed in Sussman (1985) and define upper and lower bounds on the torsion angles based on the distance between the first and the fourth atom in the torsion angle. Thus, for example, we can constrain the Φi angle by constraining the distance between the C_i and C_i−1 atoms.

Penalizing incorrect bounds Let ξ be a vector containing all of the slacks for upper bounds. Since , assuming that all the weights are the same, i.e., w_ij = w, we have , where ∥x∥₁ is the ℓ−1 norm of vector x. The fact that minimizing the ℓ₁-norm finds sparse solutions is a widely known and used heuristic (Boyd and Vandenberghe, 2004). In our problem, ξ_ij = 0 implies no violation; consequently, SPROS tends to find a solution that violates a minimum number of upper bounds.

Pseudo-atoms Not all hydrogens can be uniquely assigned, such as the hydrogens in the methyl groups; therefore, upper bounds involving these hydrogens are ambiguous. To overcome this problem, pseudo-atoms are introduced (Güntert, 1998). Given an ambiguous constraint between one of the hydrogens and atom A, by using the triangle inequality, we modify the constraint as follows:

(5)

where ∥HB_i −QB∥ is the same for i = 1, 2, 3. Pseudo-atoms are named corresponding to the hydrogens they represent; only H is changed to Q and the rightmost number is dropped. For example, in leucine, QD1 represents HD11, HD12, and HD13. We adapt the pseudo-atoms used in CYANA (Güntert, 2004).

Side chain simplification In CYANA, hydrogens that do not participate directly in the structural solution are discarded initially and then added at later stages (Güntert, 2004). We have adapted this approach by discarding hydrogens only if they make our problem smaller. In the side chain simplification process, we temporarily discard (i) all of the methyl hydrogens, (ii) all of the methylene hydrogens, (iii) hydroxyl hydrogens of tyrosine and serine, (iv) amino hydrogens of arginine and threonine, and (v) sulfhydryl hydrogen of cysteine. After the SDP problem is solved, the omitted hydrogen atoms are replaced and remain in all post-processing stages.

Challenges in Solving the SDP Problem Solving the optimization problem in Equation (4) can be challenging: For small- to medium-sized proteins, the number of atoms, n, is 1,000–3,500, and current primal-dual interior-point SDP solvers cannot solve problems with n > 2,000 efficiently. Moreover, the optimization problem in Equation (4) does not satisfy strict feasibility, causing numerical problems; see Wei and Wolkowicz (2010).

It can be observed that the protein contains many small intersecting cliques. For example, peptide planes, or aromatic rings, are two-dimensional (2D) cliques, and tetrahedral carbons form 3D cliques. As we show later, whenever there is a clique in the protein, the corresponding Gram matrix, K, can never be full-rank, which violates strict feasibility. By adapting and extending a technique called semidefinite facial reduction, not only do we obtain an equivalent problem that satisfies strict feasibility, but we also significantly reduce the SDP problem size.

2.3. Cliques in a protein molecule

A protein molecule with ℓ amino acid residues has ℓ + 1 planes in its backbone. Moreover, each amino acid has a different side chain with a different structure; therefore, the number of cliques in each side chain varies (see Appendix Tables 1 and 2 for the number of cliques in each amino acid side chain and other information about the cliques). We assume that the ith residue, r_i, has s_i cliques in its side chain, denoted by . For all amino acids (except glycine and proline), the first side chain clique is formed around the tetrahedral carbon CA, , which intersects with two peptide planes and in two atoms: and . Side chain cliques for all twenty amino acids are listed in Appendix Table 2 (see Appendix B). There is a total of cliques in the distance matrix of any protein. To simplify, let , and . For properties of the cliques in the protein molecule, see Appendix A. See Figure 1 for an example with the Glycine amino acid as the i-th residue, together with its neighboring backbone planes.

Appendix Table 1.

Table Summarizing Properties of Different Amino Acids

			Complete side chains				Simplified side chains
A.A.	p	t	a	s	q	k	a	s	q	k
Ala	7.3	3	11	5	2 (2)	5	8	2	1 (1)	3
Arg	5.2	6	29	23	5 (4)	10	20	14	5 (1)	7
Asn	4.6	4	16	10	3 (2)	6	13	7	3 (1)	5
Asp	5.1	4	13	7	3 (2)	6	10	4	3 (1)	5
Cys	1.8	4	12	6	3 (2)	5	8	2	2 (1)	4
Glu	4.0	5	20	14	4 (3)	8	14	8	4 (1)	6
Gln	6.2	5	17	11	4 (3)	8	11	5	4 (1)	6
Gly	6.9	2	8	2	1 (1)	3	8	2	1 (1)	3
His	2.3	4	18	12	3 (2)	6	15	9	3 (1)	5
Ile	5.8	6	22	16	5 (5)	11	13	7	3 (2)	6
Leu	9.3	6	23	17	5 (5)	11	14	8	3 (2)	6
Lys	5.8	7	27	21	6 (6)	13	12	6	5 (1)	7
Met	2.3	6	20	14	5 (4)	10	11	5	4 (1)	6
Phe	4.1	4	24	18	3 (2)	6	21	15	3 (1)	5
Pro	5.0	1	17	12	1 (1)	3	17	12	1 (1)	3
Ser	7.4	4	12	6	3 (2)	6	8	2	2 (1)	4
Thr	5.8	5	15	9	4 (3)	8	11	5	2 (2)	5
Trp	1.3	4	25	19	3 (2)	6	22	16	3 (1)	5
Tyr	3.3	5	25	19	4 (2)	7	21	15	3 (1)	5
Val	6.5	5	19	13	4 (4)	9	13	7	2 (1)	5
w.a.	-	4.5	18.2	12.2	3.6 (3.0)	7.6	12.8	6.8	2.7 (1.3)	5.0

Column p denotes abundance of amino acids in percentile, t denotes the number of torsion angles (excluding ω), a denotes the total number of atoms and pseudo-atoms, s denotes the total number of atoms and pseudo-atoms in the side chains, q denotes the number of cliques in each side chain (the number in the parentheses is the number of 3D cliques), and k denotes the increase in the SDP matrix size. The values in the reduced column denote the same values in the side chain simplified case. The weighted average (w.a.) of quantity x is computed as , where is the set of twenty amino acids.

Appendix Table 2.

Cliques in the Simplified Side Chains of Amino Acids

A.A.	s′	Side chain cliques
Ala	1
Arg	5
Asn	3
Asp	3
Cys	2
Glu	4
Gln	4
Gly	1
His	3
Ile	3
Leu	3
Lys	5
Met	4
Phe	3
Pro	1
Ser	2
Thr	2
Trp	3
Tyr	3
Val	2

If (s′ is the number of cliques in the simplified side chain) is not listed, it is the same as Lys. 2D cliques are marked by an *.

FIG. 1.

The Glycine amino acid as the i-th residue, with its neighboring backbone planes.

2.4. Algorithm for finding the face of the structure

For t < n and , the set of matrices is a face of (in fact every face of can be described in this way); see, e.g., Ramana et al. (1997). We let face () represent the smallest face containing a subset of ; then we have the important property that if and only if there exists such that . Furthermore, in this case, we have that every can be decomposed as , for some , and the reduced feasible set has a strictly feasible point, giving us a problem that is more numerically stable to solve (problems that are not strictly feasible have a dual optimal set that is unbounded and therefore can be difficult to solve numerically; for more information, see Wei and Wolkowicz 2010). Moreover, if t ≪ n, this results in a significant reduction in the matrix size.

The Face of a Single Clique Here, we solve the Single Clique problem, which is defined as follows: Let D be a partial EDM of a protein. Suppose the first n₁ points form a clique in the protein, such that for , all distances are known. That is, the matrix is completely specified. Moreover, let r₁ = embdim(D₁). We now show how to compute the smallest face containing the feasible set .

Theorem 2

(Single Clique, Krislock and Wolkowicz, 2010). Let the matrix be defined as follows:

be a full column rank matrix such that range(V₁) = range(K^†(D₁));

Then U₁ has full column rank, 1 range(U), and

Computing the V₁ Matrix In Theorem 2, we can find V₁ by computing the eigen decomposition of K^†(D[C₁]) as follows:

(6)

It can be seen that V₁ has full column rank (columns are orthonormal) and also that range(V₁) = range(K^†(D₁)).

2.5. The face of a protein molecule

The protein molecule is made of q cliques, , such that D[C_l] is known, and we have r_l = embdim(D[C_l]), and n_l = |C_l|. Let be the feasible set of the SDP problem. If for each clique , we define , then

(7)

where are the centered symmetric matrices. For , let , where U_l is computed as in Theorem 2. We have (Krislock and Wolkowicz, 2010):

(8)

where is a full column rank matrix that satisfies range.

We now have an efficient method for computing the face of the feasible set . To have better numerical accuracy, we developed a bottom-up algorithm for intersecting subspaces (see Algorithm 1 in the Appendix).

After computing U, we can decompose the Gram matrix as , for . However, by exploiting the centering constraint, K1 = 0, we can reduce the matrix size one more. If has full column rank and satisfies range(V) = null, then we have (Krislock and Wolkowicz, 2010):

(9)

For more details on facial reduction for Euclidean distance matrix completion problems, see Krislock (2010).

Constraints for Preserving the Structure of Cliques If we find a base set of points in each clique such that embdim, then by fixing the distances between points in the base set and fixing the distances between points in and points in , the entire clique is kept rigid. Therefore, we need to fix only the distances between base points (Alipanahi et al., 2012), resulting in a three- to four-fold reduction in the number of equality constraints. We call the reduced set of equality constraints .

2.6. Solving and refining the reduced SDP problem

The SPROS method flowchart is depicted in Appendix Figure 3. In it, we describe the blocks for solving the SDP problem and for refining the solution. From Equation (9), we can formulate the reduced SDP problem as follows:

(10)

where .

Weights and the regularization parameter For each type of upper and lower bound, we define a fixed penalizing weight for violations. For example, for upper bounds (similarly for lower bounds) we have . We set and because upper bounds from hydrogen bonds and disulfide/salt bridges are assumed to be more accurate than are NOE-derived upper bounds. Moreover, the range of torsion angle violations is 10 times smaller than NOE violations.

Let and R be the radius of the protein. Then, the maximum upper bound violation is 2R. Moreover, . Discarding the role of lower bound violations, with the goal of approximately balancing the two terms, a suitable γ is:

(11)

where 0 ≤ ɛ ≤ 1 is the fraction of violated upper bounds. In practice ɛ ≈ 0.01 – 0.30 and works well.

Post-Processing We perform a refinement on the raw structure determined by the SDP solver. For this refinement, we use a BFGS-based quasi-Newton method (Lewis and Overton, 2012) that only requires the value of the objective function and its gradient at each point. Letting X⁽⁰⁾ = X_SDP, we iteratively minimize the following objective function:

(12)

where f (θ) = max(0,θ) and g(θ) = min(0,−θ). We set w_ɛ = 2,  = 1, and . In addition, to balance the regularization term, we set

where −1 ≤ α ≤ 1 is a parameter controlling the regularization. If α < 0, the distances between atoms are maximized, because, after projection, some of the distances have been shortened, this term helps to compensate for that error. However, if α > 0, the distances between atoms are minimized, resulting in better packing of atoms in the protein molecule. In practice, different values for α can be used to generate slightly different structures, thus creating a bundle of structures.

Fixing incorrect chiralities Chirality constraints cannot be enforced using only distances. Consequently, some chiral centers may have the incorrect enantiomer. In this step, SPROS checks the chiral centers and resolves any problems.

Improving the stereochemical quality Williamson and Craven (2009) have described the effectiveness of explicit solvent refinement of NMR structures and suggest that it should be a standard procedure. For protein structures that have regions of high mobility/uncertainty due to few or no NOE observations, we have successfully employed a hybrid protocol from XPLOR-NIH that incorporates thin-layer water refinement (Linge et al., 2003) and a multidimensional torsion angle database (Kuszewski et al., 1996, 1997).

3. Results

We tested the performance of SPROS on 18 proteins: 15 protein data sets from the database of converted restraints (DOCR) database in the NMR restraints grid (Doreleijers et al., 2003, 2005) and three protein data sets from Donaldson's laboratory at York University, Toronto, Canada. We chose proteins with different sizes and topologies, as listed in Table 1. Finally, the input to the SPROS method is exactly the same as the input to the widely used CYANA method.

Table 1.

Information About the Proteins Used in Testing SPROS

ID	Topo.	Len.	Weight	n	n′	Cliques (2D/3D)				Bound types
1G6J	a+b	76	8.58	1434	405	304 (201/103)	5543	1167	1354	21/29/17/33	31.9 ± 15.3	1291	32	63	0
1B4R	B	80	7.96	1281	346	248 (145/103)	4887	1027	787	26/25/6/43	17.1 ± 10.8	687	30	22	78
2E8O	A	103	11.40	1523	419	317 (212/105)	5846	1214	3157	19/29/26/26	71.4 ± 35.4	3070	24	87	0
1CN7	a/b	104	11.30	1927	532	393 (253/140)	7399	1540	1560	46/24/12/18	23.1 ± 13.4	1418	31	80	62
2KTS	a/b	117	12.85	2075	593	448 (299/149)	7968	1719	2279	22/28/14/36	34.6 ± 17.4	2276	25	0	3
2K49	a+b	118	13.10	2017	574	433 (291/142)	7710	1657	2612	22/27/18/38	40.9 ± 21.1	2374	27	146	92
2K62	B	125	15.10	2328	655	492 (327/165)	8943	1886	2367	21/32/15/32	33.9 ± 18.6	2187	32	180	0
2L3O	A	127	14.30	1867	512	393 (269/124)	7143	1492	1270	24/38/20/18	22.5 ± 12.7	1055	25	156	59
2GJY	a+b	144	15.67	2337	639	474 (302/172)	8919	1875	1710	7/30/19/44	25.0 ± 16.6	1536	29	98	76
2KTE	a/b	152	17.21	2576	717	542 (360/182)	9861	2089	1899	17/31/22/30	24.3 ± 20.8	1669	30	124	106
1XPW	B	153	17.44	2578	723	541 (355/186)	9837	2081	1206	0/31/11/58	17.0 ± 10.8	934	37	210	62
2K7H	a/b	157	16.66	2710	756	563 (363/200)	10452	2196	2768	29/33/13/25	30.3 ± 11.3	2481	19	239	48
2KVP	A	165	17.28	2533	722	535 (344/191)	9703	2094	5204	31/26/23/20	59.2 ± 25.0	4972	22	232	0
2YT0	a+b	176	19.17	2940	828	627 (419/208)	11210	2404	3357	23/28/14/35	34.9 ± 22.3	3237	30	120	0
2L7B	A	307	35.30	5603	1567	1205 (836/369)	21421	4521	4355	10/30/44/16	27.6 ± 14.4	3459	23	408	488
1Z1V	A	80	9.31	1259	362	272 (181/91)	4836	1046	1261	46/24/18/13	28.6 ± 16.3	1189	15	0	72
HACS1	B	87	9.63	1150	315	237 (156/81)	4401	923	828	46/21/5/27	20.2 ± 14.2	828	20	0	36
2LJG	a+b	153	17.03	2343	662	495 (327/168)	9009	1909	1347	40/29/8/22	16.4 ± 11.9	1065	28	204	78

The second, third, and fourth columns list the topologies, sequence lengths, and molecular weight of the proteins; the fifth and sixth columns, n and n′, list the original and reduced SDP matrix sizes, respectively. The seventh column lists the number of cliques in the protein. The eighth and ninth columns, and , list the number of equality constraints in the original and reduced problems, respectively. The 10th column, , lists the total number of upper bounds for each protein. The 11th column, bound types, lists intra-residue, |i − j| = 0, sequential, |i − j| = 1, medium range, 1 < |i − j| ≤ 4, and long range, |i − j| > 4, respectively, in percentile. The 12th column, , lists the average number of upper bounds per residue, together with the standard deviation. The 13th column, , lists the number of NOE-inferred upper bounds. The 14th column, , lists the fraction of pseudo-atoms in the upper bounds in percentile. The last two columns, and , list the number of upper bounds inferred from torsion angle restraints, and hydrogen bonds, disulfide and salt bridges, respectively.

3.1. Implementation

The SPROS method has been implemented and tested in Matlab 7.13 (apart from the water refinement, which is done by XPLOR-NIH). For solving the SDP problem, we used the SDPT3 method (Tütüncü et al., 2003). For minimizing the post-processing objective function (Eq. 12), we used the BFGS-based quasi-Newton method implementation by Lewis and Overton (2012). All the experiments were carried out on an Ubuntu 11.04 Linux PC with a 2.8 GHz Intel Core i7 Quad-Core processor and 8 GB of memory.

3.2. Determined structures

From the 18 test proteins, 9 of them were calculated with backbone RMSDs less than or equal to 1.0Å, and 16 have backbone RMSDs less than 1.5Å. Detailed analysis of calculated structures is listed in Table 2. The superimposition of the SPROS and reference structures for three of the proteins are depicted in Figure 2. More detailed information about the determined structures can be found in Alipanahi, (2011).

Table 2.

Information About Determined Structures of the Test Proteins

				RMSD			Violations		Ramachandran
ID	ts	tw	tt	Backbone	Heavy atoms	CBd.	0.1Å	1.0Å	Fav.	Alw.	Out.
1G6J	44.5	175.5	241.0	0.68 ± 0.05	0.90 ± 0.05	0	4.96 (0.08 ± 0.07)	0.85 (0)	100	100	0
1B4R	21.4	138.0	179.0	0.85 ± 0.06	1.06 ± 0.06	0	20.92 (13.87 ± 0.62)	6.14 (2.28 ± 0.21)	80.8	93.6	6.4
2E8O	129.8	181.3	340.9	0.58 ± 0.02	0.68 ± 0.01	0	31.33 (31.93 ± 0.14)	9.98 (10.75 ± 0.13)	96.2	100	0
1CN7	75.0	230.1	339.7	1.53 ± 0.11	1.80 ± 0.10	0	10.27 (7.63 ± 0.80)	3.18 (2.11 ± 0.52)	96.1	99.0	1.0
2KTS	116.7	231.0	398.5	0.92 ± 0.06	1.13 ± 0.06	0	25.36 (27.44 ± 0.58)	6.49 (10.36 ± 0.68)	86.1	95.7	4.3
2K49	140.7	240.7	422.7	0.99 ± 0.14	1.24 ± 0.16	0	13.75 (15.79 ± 0.67)	2.80 (4.94 ± 0.46)	93.8	97.3	2.7
2K62	156.1	259.0	464.2	1.40 ± 0.08	1.72 ± 0.08	1	33.74 (42.92 ± 0.95)	10.79 (21.20 ± 1.20)	87.8	95.9	4.1
2L3O	61.7	212.0	310.0	1.28 ± 0.15	1.59 ± 0.15	0	21.53 (19.81 ± 0.58)	7.33 (7.61 ± 0.31)	80.4	92.8	7.2
2GJY	113.7	285.9	455.7	0.99 ± 0.07	1.29 ± 0.09	0	11.67 (8.36 ± 0.59)	0.36 (0.49 ± 0.12)	85.4	92.3	7.7
2KTE	139.9	297.7	503.2	1.39 ± 0.17	1.85 ± 0.16	1	35.55 (31.97 ± 0.46)	11.94 (11.96 ± 0.40)	79.4	90.8	9.2
1XPW	124.8	297.1	489.7	1.30 ± 0.10	1.68 ± 0.10	0	9.74 (0.17 ± 0.09)	1.20 (0.01 ± 0.02)	87.9	97.9	2.1
2K7H	211.7	312.0	591.0	1.24 ± 0.07	1.49 ± 0.07	0	17.60 (16.45 ± 0.30)	4.39 (4.92 ± 0.35)	92.3	96.1	3.9
2KVP	462.0	282.4	814.8	0.94 ± 0.08	1.05 ± 0.09	0	15.15 (17.43 ± 0.29)	4.01 (5.62 ± 0.21)	96.6	100	0
2YT0	292.1	421.5	800.1	0.79 ± 0.05	1.04 ± 0.06	1	29.04 (28.9 ± 0.36)	6.64 (6.60 ± 0.30)	90.5	97.6	2.4
2L7B	1101.1	593.0	1992.1	2.15 ± 0.11	2.55 ± 0.11	3	19.15 (21.72 ± 0.36)	4.23 (4.73 ± 0.23)	79.2	91.6	8.4
1Z1V	30.6	158.8	209.2	1.44 ± 0.17	1.74 ± 0.15	0	3.89 (2.00 ± 0.25)	0.62 (0)	90.9	98.5	1.5
HACS1	17.4	145.0	176.1	1.00 ± 0.07	1.39 ± 0.10	0	20.29 (15.68 ± 0.43)	4.95 (3.73 ± 0.33)	83.6	96.7	3.3
2LJG	94.7	280.4	426.3	1.24 ± 0.09	1.70 ± 0.10	1	28.35 (25.3 ± 0.51)	10.76 (8.91 ± 0.49)	80.6	90.7	9.3

The second, third, and fourth columns list SDP time, water refinement time, and total time, respectively. For the backbone and heavy atom RMSD columns, the mean and standard deviation between the determined structure and the reference structures is reported (backbone RMSDs less than 1.5Å are shown in bold). The seventh column, CBd, lists the number of residues with “CB deviations” larger than 0.25Å computed by MolProbity, as defined by Chen et al. 2010). The eighth and ninth columns list the percentage of upper bound violations larger than 0.1Å and 1.0Å, respectively (the numbers for the reference structures are in parentheses). The last three columns list the percentage of residues with favorable and allowed backbone torsion angles and outliers, respectively.

FIG. 2.

Superimposition of structures determined by SPROS in blue and the reference structures in red.

To further assess the performance of SPROS, we compared the SPROS and reference structures for 1G6J, Ubiquitin, and 2GJY, PTB domain of Tensin, with their corresponding X-ray structures, 1UBQ and 1WVH, respectively. For 1G6J, the backbone (heavy atoms) RMSDs for SPROS and the reference structures are 0.42Å (0.57Å) and 0.73 ± 0.04Å (0.98 ± 0.04Å), respectively. For 2GJY, the backbone (heavy atoms) RMSDs for SPROS and the reference structures are 0.88Å (1.15Å) and 0.89 ± 0.08Å (1.21 ± 0.06Å), respectively.

3.3. Discussion

The SPROS method was tested on 18 experimentally-derived protein NMR data sets of sequence lengths ranging from 76 to 307 (weights ranging from 8 to 35 KDa). Calculation times were in the order of a few minutes per structure. Accurate results were obtained for all of the data sets, although with some variability in precision. The best attribute of the SPROS method is its tolerance for, and efficiency at, managing many incorrect distance constraints (that are typically defined as upper bounds).

The reduction methodology developed for SPROS is an ideal choice for protein-ligand docking. If the side chains participating at the interaction surface are only declared to be flexible, it has the effect of reducing the SDP matrix size to less than 100. Calculations under these specific parameters can be achieved in a few seconds, thereby making SPROS a worthwhile choice for automated, high-throughput screening.

Our final goal is a fully automated system for NMR protein structure determination, from peak picking (Alipanahi et al., 2009) to resonance assignment (Alipanahi et al., 2011) to protein structure determination. An automated system without the laborious human intervention will have to tolerate more errors than usual. This was the initial motivation of designing SPROS. The key is to tolerate more errors. Thus, we are working toward incorporating an adaptive violation weight mechanism to identify the most significant outliers in the set of distance restraints automatically.

4. Appendix

A. Properties of cliques

Let , and . Let r_i = embdim. The following properties hold for cliques in a protein molecule:

• 1. , given |i − i′| > 1.

• 2. , given i′ ≠ i, i + 1.

• 3. , given i′ ≠ i.

• 4. |C_i| ≥ r_i + 1.

• 5. 3 ≤ |C_i| ≤ 16.

• 6. .

• 7.  such that .

• 8. If , then .

• 9. .

C. Efficient subspace intersection algorithm

D. SPROS flow chart

FIG. 3.

SPROS method flowchart.

Acknowledgments

This work was supported in part by NSERC grant OGP0046506, NSERC grant RGPIN327487-09, NSERC grant 238934, an NSERC Discovery Accelerator Award, Canada Research Chair program, an NSERC collaborative grant, David R. Cheriton graduate scholarship, OCRiT, Premier's Discovery Award, and Killam Prize. Nathan Krislock is currently a PIMS Postdoctoral Fellow at the University of British Columbia.

Author Disclosure Statement

No competing financial interests exist.