Ab Initio Protein Structure Assembly Using Continuous Structure Fragments and Optimized Knowledge-based Force Field

Abstract

Ab initio protein folding is one of the major unsolved problems in computational biology due to the difficulties in force field design and conformational search. We developed a novel program, QUARK, for template-free protein structure prediction. Query sequences are first broken into fragments of 1–20 residues where multiple fragment structures are retrieved at each position from unrelated experimental structures. Full-length structure models are then assembled from fragments using replica-exchange Monte Carlo simulations, which are guided by a composite knowledge-based force field. A number of novel energy terms and Monte Carlo movements are introduced and the particular contributions to enhancing the efficiency of both force field and search engine are analyzed in detail. QUARK prediction procedure is depicted and tested on the structure modeling of 145 non-homologous proteins. Although no global templates are used and all fragments from experimental structures with template modeling score (TM-score) >0.5 are excluded, QUARK can successfully construct 3D models of correct folds in 1/3 cases of short proteins up to 100 residues. In the ninth community-wide Critical Assessment of protein Structure Prediction (CASP9) experiment, QUARK server outperformed the second and third best servers by 18% and 47% based on the cumulative Z-score of global distance test-total (GDT-TS) scores in the free modeling (FM) category. Although ab initio protein folding remains a significant challenge, these data demonstrate new progress towards the solution of the most important problem in the field.

INTRODUCTION

Despite significant effort, we still have very limited ability to fold proteins by ab initio approaches, i.e. to predict 3D structures of protein sequences without using template structures from other experimentally solved proteins. Successful cases have been witnessed only on small proteins with length below 100 residues, and with a root mean squared deviation (RMSD) typically above 2–5 Å.^1–6 The difficulty of ab initio protein structure prediction is twofold. First, we lack decent force fields to accurately describe the atomic interactions which can be used to guide the protein folding simulations. Apparently, force fields with an incorrectly located global minimum will undoubtedly misfold the target proteins. Although the physics-based atomic force fields^7,8 can provide a reasonable description of protein atomic interactions in many aspects, the implementation requests atomic-level representation which is often too slow to fold a protein structure from scratch. The knowledge-based potentials, which are often in reduced forms and derived from statistical regularities of structures in the Protein Data Bank (PDB),⁹ have shown power in both protein fold recognition and structure assembly simulations,^10,11 where appropriate selections of reference states and structural features are proven to be of critical importance.¹²

Second, given the force fields, we have difficulties in efficiently identifying the global energy minimum which is supposed to be the protein native state in thermodynamic hypothesis assumption,¹³ because most of the composite force fields are characterized with numerous local energy minima which can easily trap the folding simulations. One way of speeding up the computational search process is to reduce the size of the search space. For example, in TOUCHSTONE,⁴ the authors constrained the conformational change of protein structure on a lattice system. In Rosetta,¹⁴ TASSER¹⁵ and I-TASSER,¹⁶ fragment sequences have the structures copied from PDB templates which are kept rigid during the simulation. Rosetta also keeps the bond lengths and bond angles fixed which further decreases the degrees of freedom. These techniques can help to significantly reduce the search space because of the constraint of conformational movements. Nevertheless, it is essential to have the resolution of conformational representations not limited by the constraints. In TOUCHSTONE, the program implemented a grid size of 0.87 Å which resulted in an average resolution of 0.5 Å in RMSD. In Rosetta and TASSER programs, since the fragments are off-lattice, the conformation should have no resolution limit if the fragment structures are ideally selected.

Another way of increasing the efficiency of conformational search, which is also associated with the reduction in the size of the search space, is to reduce the level of protein structure representation. For example, in UNRES,¹⁷ a protein residue is represented by three units of Cα atom, side-chain ellipsoid, and peptide group. In I-TASSER,^16,18 each residue is specified by two units of Cα atom and side-chain center of mass. These reductions of structure representation can dramatically reduce the total number of conformations needed for searching. However, although the reduced models have the gain in conformational search, they may suffer a lower accuracy of energy force field design. Finally, a central theme in protein conformational search is the appropriate design of conformational updating and optimization algorithms, with examples including Monte Carlo and molecule dynamics simulations, which will essentially decide the efficiency of the overall conformational search.

In this work, we develop a new algorithm, QUARK, for ab initio protein structure prediction, with the focus on the elaborate design of both the force field and the search engine. To facilitate the force field development and search engine design, QUARK takes a semi-reduced model to represent protein residues by the full backbone atoms and the side-chain center of mass. For a query sequence, it first predicts a variety of carefully selected structural features by Neural Network (NN). The global fold is then generated by replica-exchange Monte Carlo (REMC) simulations by assembling the small fragments as generated by gapless threading through template library, an idea borrowed from Rosetta and I-TASSER; but different from Rosetta and I-TASSER which have the fragments in either 3/9-mer or from threading alignments, the fragments in QUARK have continuously multiple sizes from 1 to 20 residues. Meanwhile, in contrast to the pure fragment substitutions as taken by Rosetta and the fixed fragment rotation as taken by I-TASSER, QUARK simulations contain composite movements of free chain constructions and fragment substitutions between decoy and fragment structures. These techniques have significantly increased the structural flexibility and the efficiency of conformational search while taking the advantage of the reduction of the conformational search due to fragment assembly.

We then conducted a systematical test and analysis of QUARK in ab initio structure prediction on the basis of 145 small to medium sized globular proteins, on the control with other top ab initio modeling methods. Since these proteins are taken from the PDB, we made a series of stringent filters to rule out homologous information from the template library. We also tested the method in the ninth Critical Assessment of protein Structure Prediction (CASP9) experiment. Although the blind CASP experiment has a much smaller test set of free modeling (FM) targets, some being non-globular, it offers a valuable opportunity to objectively benchmark the method in comparison with all other state-of-the-art programs in the field.

Materials and Methods

Construction of benchmarking datasets

Since ab initio folding methods are designed to predict protein structures without using templates, it is important to establish a stringent benchmarking protein set which completely excludes global topology information of templates. This is essential to train, test and evaluate the method developments, especially in the situation where most of the top-performing methods in the field were designed to exploit fragment structures from experimental structures to assemble the models of target protein sequences.^5,14 The importance and difficulty of ab initio benchmarking were also reflected in the community-wide blind CASP experiments,^19,20 where the portion of free modeling targets has been consistently decreasing due to the difficulties in collecting individual protein domains which have different folds from the existing proteins in the PDB. Even among the limited FM targets, there are an increasing portion of proteins which lack of globular compact shapes, due to the fact that they are often isolated from a small region of interaction complexes, which can prevent them from being useful assessment targets for ab initio folding methods.

We first obtain a non-redundant set of 6,023 high-resolution experimental structures from the PISCES server,²¹ which are culled from the whole PDB based on the identity cutoff 25%, resolution cutoff 1.8 Å and R-factor cutoff 0.25. All the statistical potentials used in the QUARK simulation are derived from this protein set. We also use the same template library for retrieving fragment structures of various lengths for each query sequence. Since the purpose of ab initio prediction is to handle the targets which have no homologous templates hit by threading algorithms and are inaccurate to predict by template-based methods, protein sequences we choose for training and test are the “Hard” targets classified by the LOcal MEta-Threading Server (LOMETS).²² We first run LOMETS for all the sequences in the PISCES list, where homologous templates with sequence identities larger than 30% to each query sequence, are excluded from the threading template library. In total, 665 sequences are considered as “Hard” targets by LOMETS, where a “Hard” target means none of the threading algorithms detects a template with the Z-score higher than the given cutoff. The Z-score cutoffs of the nine threading algorithms in LOMETS are determined by minimizing the false positive and false negative rate.²² We manually check the native structures of the 665 protein chains and exclude the targets which have obvious broken chains or incompact shapes. The remaining list contains 413 proteins. In order to quickly train the various parameters during the design of energy terms and movements, we randomly select 88 small globular proteins as the training set from the remaining list of “Hard” targets. The lengths of protein chains in the training set are in the range of 70 and 100 amino acids. To test the modeling accuracy of the QUARK method, 145 globular proteins are randomly selected as the test set from the remaining list, which is further divided into two subsets: 51 small proteins (70 to 100 amino acids) and 94 medium sized proteins (100 to 150 amino acids). The lists of the training and test proteins are available at http://zhanglab.ccmb.med.umich.edu/QUARK/list.txt.

Since two movements in QUARK use fragments from experimental structures and one energy term is derived from the structural fragments, to avoid the potential influence from homologous templates, we use three filters to exclude homologous templates when generating the position-specific fragments. First, all the template proteins whose sequence identities to the target sequence are >30% are removed. Second, we run MUSTER threading program through the QUARK template library. All the templates with a TM-score >0.3 in the MUSTER alignment are removed from the QUARK template library. Third, we run TM-align²³ to superimpose each template to the target structure. All the templates with a TM-score >0.5 will be removed. We have observed that non-homologous templates, if they have global structural similarity to the target, will have considerable impact to the fragment-based assembly results. In the third filter, all structurally similar templates are removed even they have low sequence homology to the target sequence.

Representation of protein conformations

Protein conformations in QUARK are represented by a semi-reduced model, where each residue is specified by full backbone atoms plus the side-chain center of mass (SC), i.e. N, Cα, C, O, Cβ, H and SC (see Figure 1A). Three backbone atoms N, Cα and C are flexible, which have 9 degrees of freedom. Bond lengths and bond angles between these atoms are not fixed, but are restricted in the physically allowed range. The other four off-backbone atoms/units (O, Cβ, H and SC) are added based on their relative positions to the three backbone atoms.

Figure 1.

(A) Representation of QUARK semi-reduced model where a protein chain conformation is specified by the full backbone atoms and the side-chain center of mass (SC). (B) Conformations of N_i, Cα_i and C_i represented in the Cartesian system. (C) Conformations of N_i, Cα_i and C_i represented in the torsion-angle system, where each atom position is determined by the distance, inner angle and torsion angle relationship to the previous three atoms.

Unlike the O, Cβ, H atoms, which can be determined uniquely based on the backbone atoms, the virtual atom SC has uncertainty because of the various side-chain rotamer conformations. We calculate the averaged SC positions in term of 20 different amino acids and backbone (φ, ψ) torsion-angle pairs that are divided into 72 bins with an interval π/36, summarized from the 6,023 high-resolution experimental structures. The SC position of each residue is therefore decided based on the residue type and the backbone torsion angles, which also has been implemented in ModRefiner²⁴ for main-chain energy minimization.

During the simulation, we represent each reduced model in two coordinate systems: Cartesian system and torsion-angle system. In the Cartesian system, backbone atoms are represented by their 3D coordinates while in the torsion-angle system, they are represented by bond lengths, bond angles and torsion angles (see Figure 1B and 1C respectively). Some movements only change 3D coordinates of the atoms which can be easily modified in the Cartesian system. Other movements only change the bond length, bond angle or torsion angle, which can be changed conveniently in the torsion-angle system. If the coordinates are changed in one system, the new coordinates in the other system are updated correspondingly since the two systems are exchangeable. The two different coordinate systems are also useful in the calculation of different energy terms after each movement. For example, pair-wise energy terms rely on the 3D coordinates of atoms while torsion-angle term is based on the backbone torsion angles.

Flowchart of QUARK prediction procedure

QUARK ab initio structure prediction procedure can be divided into three steps, which are shown in Figure 2. The first step is for multiple feature predictions and fragment generation starting from one query sequence. The second step is structural constructions using replica-exchange Monte Carlo simulation based on the semi-reduced protein model. The third step is for decoy structure clustering and full-atomic refinement. Here we first give the outline of the steps and then describe the details of force field design and Monte Carlo movements.

Figure 2.

Flowchart of the QUARK ab initio prediction.

Feature prediction and fragment generation

Given the amino acid sequence, multiple sequence alignment (MSA) is generated by PSI-BLAST²⁵ through a non-redundant sequence database. Secondary structure (SS) types are then predicted by Protein Secondary Structure prediction program PSSpred (http://zhanglab.ccmb.med.umich.edu/PSSpred/) based on multiple Neural Network trainings of sequence profiles calculated from the MSA. Solvent accessibility, real-value φ and ψ angles, β-turn positions are predicted by separate Neural Networks based on the checkpoint file by PSI-BLAST and secondary structure types predicted by PSSpred. The architectures of the four back-propagation Neural Networks²⁶ are presented in Table S1 of the Supporting Information, where features of one residue consist of 20 frequencies in the checkpoint file and 3 probabilities of SS types. Sequence profile from MSA, predicted secondary structure types, as well as the predicted solvent accessibility and real-value torsion angles are then used to generate structural fragments for each segment of the query sequence. Top 200 fragments for each segment are generated by a gapless threading method using a scoring function close to that in the MUSTER²⁷ threading program. The position-specific fragments are used by the fragment substitution movement to replace existing ones in the decoy structure during the simulation. We find that using fragments with continuous lengths from 1 to 20 amino acids for the assembly simulation can achieve better modeling result than that with discrete lengths since the former provide smoother conformational change and more adequate sampling. Fragments with lengths larger than 20 cannot further improve the result since long fragments can be hardly inserted into the well-packed decoy structure during the simulation.

Two structural features will be derived from the top collected fragments with length 10, since we find that 10-mer fragments can be most accurately retrieved by the scoring function by comparing their conformations with the native fragment structures. First, (φ, ψ) torsion-angle pairs at each residue position are clustered by standard clustering algorithms and at most 30 torsion-angle pairs are selected for each residue position. The reduced number of torsion angles along with their associated bond lengths and bond angles constitute a look-up table, which will be efficiently used in one local movement during the simulation. The second important feature extracted from fragments is the distance profile, which is a histogram distribution of pair-wise distances extracted from unrelated experimental structures based on the occurrence of fragments at different positions but from the same templates. The derivation and usage of distance profile will be described in the following section.

The predicted solvent accessibility is also used in the energy term which is represented as the difference between the predicted value and the actual value of the structural decoy. The predicted three-state SS types will guide the simulation to generate decoy structures with the similar SS types. We don’t restrict the decoy to have exactly the same SS types as the PSSpred prediction. If one template fragment is successfully placed into the decoy by the fragment substitution movement, this segment will have the same SS types as the fragment structure, rather than the PSSpred prediction. The predicted probabilities of β-turn positions will be used to guide one movement for β-turn formation.

Replica-exchange Monte Carlo simulation

There are in total 40 replicas implemented in the replica-exchange Monte Carlo simulation. Since the average energy in low-temperature replicas gets saturated in near 100 cycles for most of our training proteins, around 200 cycles are run for each protein by default. However, the simulation will be terminated if the variation of the average energy of the 10 low-temperature replicas is smaller than 10⁻⁴ times of the average energy.

The initial structure for each replica is constructed by randomly connecting the randomly selected fragments with different lengths. We run 10 different REMC simulations with different starting random numbers. The Lehmer random number generator²⁸ is used for random number generation, which has 256 different streams with a long period (2.15E9) in each stream. In our benchmark test, the template modeling score (TM-score)²⁹ of the first model clustered from 10 REMC simulations is on average 15% better than that from one simulation. However, there are no notable differences when more than 10 simulations are implemented.

Decoy clustering and full-atomic refinement

5,000 decoys randomly selected from the last 150 cycles of the 10 low-temperature replicas in the 10 REMC simulations are gathered and clustered by the revised SPICKER program.³⁰ The mean and standard deviation of RMSD for all pairs of decoys are pre-calculated in the new version of SPICKER since different targets may have different distributions of QUARK decoy structures. The minimum and maximum RMSD cutoffs in the new SPICKER algorithm are then automatically adjusted based on the mean and standard deviation. Five largest cluster centers are selected as the representative predicted models, which are the decoy conformations closest to the cluster centroids. Since the QUARK decoy conformation only contains backbone heavy atoms, the final full-atomic structure is constructed by ModRefiner, which was designed to add the missing atoms in the reduced models and refine the physical quality of both backbone and side-chain atomic structures simultaneously.

Design of force fields

The total energy of the QUARK force field is the sum of the eleven terms:

$$E_{\mathit{\text{tot}}}=E_{\mathit{\text{prm}}}+w_1{E}_{\mathit{\text{prs}}}+w_2{E}_{\mathit{\text{ev}}}+w_3{E}_{\mathit{\text{hb}}}+w_4{E}_{\mathit{\text{sa}}}+w_5{E}_{\mathit{\text{dh}}}+w_6{E}_{\mathit{\text{dp}}}+w_7{E}_{\mathit{\text{rg}}}+w_8{E}_{\mathit{\text{bab}}}+w_9{E}_{\mathit{\text{hp}}}+w_{10}{E}_{\mathit{\text{bp}}}$$ (1)

where w₁=0.10, w₂=0.03, w₃=0.03, w₄=4.00, w₅=0.40, w₆=0.60, w₇=1.00, w₈=1.00, w₉=0.05 and w₁₀=0.10, are the weighting factors to balance the energy terms, which were tuned based on the modeling accuracy of the small proteins in the training set. We use a super-dimensional grid-searching strategy to decide those weighting factors, which is the same as that by MUSTER. The pair-wise energy term E_prm is considered as the base energy term which has the weight equal to 1. The other weights are set to zero at the beginning and gradually increased until they cannot improve the TM-score of the best model in the top 5 cluster centers any more. We determine those ten weights in the first round in the order of w₁, w₂ to w₁₀ and then refine them in random orders for several iterations. In each round, we use big intervals to update each weight in the beginning and then try small ones when the weight is approaching the best. The parameterization procedure by this strategy we believe is close to the optimum, which can lead to better modeling accuracy than using the one-round order-dependent parameterization.

The eleven energy terms can be categorized into three levels, i.e. the atomic-level terms (E_prm, E_prs, and E_ev), the residue-level terms (E_hb, E_sa, E_dh, and E_dp), and the topology-level terms (E_rg, E_bab, E_hp, and E_bp). All the energy terms are knowledge-based, in the sense that they are derived from the statistics of experimental structures, although most of them have direct physical sources. In the following, we describe in more detail of the energy terms, with the emphases on the new design of hydrogen-bonding and solvent accessibility potentials and the novel concept of distance profile potential.

(1) Backbone atomic pair-wise potential

This base energy term accounts for the distance-dependent contact preferences between the backbone atoms (N, Cα, C, O and Cβ):³¹

$$E_{\mathit{\text{prm}}}(i,j,r_{\mathit{\text{ij}}})=-\mathit{\text{RT}}\text{log}(\frac{N_{\mathit{\text{obs}}}(i,j,r_{\mathit{\text{ij}}})}{r_{\mathit{\text{ij}}}^{\alpha }{N}_{\mathit{\text{obs}}}(i,j,r_{\mathit{\text{cut}}})})$$ (2)

where R is the gas constant, T is the temperature, r_ij is the distance between the ith and jth atoms, r_cut=15 Å is the short-range cutoff distance. We have used a formula similar to the DFIRE³² method with α=1.61, but N_obs(i,j,r_ij) which is the observed number of pairs between atoms i and j with distance r_ij, was recalculated on our own from the high-resolution experimental structures. In Figure 3A, we show illustrative curves for three Cα pairs between ASP-ASP, ASP-ARG and ARG-ARG with distance from 0 to 15 Å.

Figure 3.

Illustrations of distance-specific contact potentials for three atom pairs. (A) Pair-wise backbone Cα-Cα potential. (B) Pair-wise SC-SC potential. (C) Excluded volume potential.

(2) Side-chain center pair-wise potentials

We extend the pair-wise atomic potential to that between the virtual atom SC in one residue and the real/virtual atoms N, Cα, C, O, Cβ, SC in another residue and derive the side-chain pair-wise potential E_prs:

$$E_{\mathit{\text{prs}}}(i,j,r_{\mathit{\text{ij}}})=-\mathit{\text{RT}}\text{log}(\frac{N{\text{'}}_{\mathit{\text{obs}}}(i,j,r_{\mathit{\text{ij}}})}{r_{\mathit{\text{ij}}}^{\alpha \text{'}}N{\text{'}}_{\mathit{\text{obs}}}(i,j,r_{\mathit{\text{cut}}})})$$ (3)

For the experimental structures in the template library, we first get the side-chain center SC for each residue, and then calculate the observed number N’_obs(i,j,r_ij) between SC and SC or other backbone atoms. Here, α’=1.40 is determined based on the DFIRE method. Three curves between SC and SC of the same residue pairs are shown in Figure 3B. Since arginine has positively charged side-chain while aspartic acid side-chain is negatively charged, there are apparently more short-range contact pairs between ASP-ARG due to the Coulomb interactions (see line with dot in Figure 3B). From backbone atom pairs in Figure 3A, we cannot distinguish the charge properties of different residues.

(3) Excluded volume

The excluded volume interaction is expressed by:

$$E_{\mathit{\text{ev}}}(i,j,r_{\mathit{\text{ij}}})=\{\begin{array}{cc}{(\mathit{\text{vdw}}(a_i)+\mathit{\text{vdw}}(a_j))}^2-{r_{\mathit{\text{ij}}}}^2\hfill & \text{if }{r}_{\mathit{\text{ij}}}<\mathit{\text{vdw}}(a_i)+\mathit{\text{vdw}}(a_j)\hfill \\ 0\hfill & \text{else}\hfill \end{array}$$ (4)

where a_i is the atom type of the ith atom and vdw(a_i) is its van der Waals radius. This term is used to avoid the over-compactness of the structural model and to reduce the search space by eliminating physically not allowed conformations. Here the penalty score for every pair of atoms is in a quadratic form. Three example curves between different atom pairs are shown in Figure 3C. Since side-chain center is not a real atom and its position is approximately added based on the backbone torsion angles, we define a clash between SC and any other atom when r_ij <1 Å. Although the decoy will tend to contain side-chain clashes if we directly add side-chain atoms based on its backbone structure, we can easily remove those clashes by our refinement program ModRefiner later. Using a big distance cutoff r_ij for side-chain center definitely will help to build the full-atomic model easily, but we found that it would worsen the backbone modeling accuracy. This is because the conformational search space is narrowed if the simulation forbids any overlap between the inaccurate side-chain centers.

(4) Hydrogen-bonding

Hydrogen bonds, especially those between backbone atoms, are one of the major driven forces to form regular secondary structures and stabilize the global topology of protein structures. In the semi-reduce model, we only consider the backbone hydrogen bonds which are between N-H in one residue and O=C in another residue (See Figure S1A). In α-helix, the hydrogen bond (H-bond) is between O=C of residue i and N-H of residue i+4, while in β-sheet, it can be between any pair of residues i and j. Here, we select four geometric features to gauge the bonding, i.e. the distance between O_i and H_j, D(O_i,H_j); the inner angle between C_i, O_i, and H_j, A(C_i,O_i,H_j); the inner angle between O_i, H_j, and N_j, A(O_i,H_j,N_j); and the torsion angle between C_i, O_i, H_j, and N_j, T(C_i,O_i,H_j,N_j).

In Table I, we list the mean and standard deviation of the four features in the four types of H-bonds calculated from the high-resolution experimental structures whose secondary structure types defined by DSSP.³³ In α-helix, there is no real hydrogen bond between O=C of residue i and N-H of residue i+3 (See Figure S1B). However, the four features between them have even less deviations than that between the residues with real hydrogen bond, when we compare the second and third rows in Table I. We therefore add this in our simulation as an additional restraint of H-bond in α-helix. In the last two rows of Table I, the standard deviations of torsion angles in β-sheets are much higher, which means torsion angle doesn’t form a conserved pattern for characterizing β-pairs in β-sheets.

Table I.

Mean and standard deviation of four H-bond features in α-helix and β-sheet structures.

Acceptor i, donor j		D(Oi,Hj) (Å)	A(Ci,Oi,Hj) (°)	A(Oi,Hj,Nj) (°)	T(Ci,Oi,Hj,Nj) (°)
T1	Helix, j=i+4	2.00/0.53	147/10.58	159/11.25	160/25.36
T2	Helix, j=i+3	2.85/0.32	89/7.70	111/8.98	−160/7.93
T3	Parallel	2.00/0.30	155/11.77	164/11.29	180/68.96
T4	Antiparallel	2.00/0.26	151/12.38	163/11.02	−168/69.17

The energy term for a single backbone hydrogen bond here is given by:

$$\begin{array}{cc}{E}_{\mathit{\text{hb}}}(i,j,T_k)={\displaystyle \sum _{l=1}^{n_k}}\frac{{(f_l(i,j)-{\mu }_{\mathit{\text{kl}}})}^2}{2{\delta }_{\mathit{\text{kl}}}^2},\hfill & n_k=\{\begin{array}{cc}4\hfill & k=1,2\hfill \\ 3\hfill & k=3,4\hfill \end{array}\hfill \end{array}$$ (5)

where T_k denotes the kth type of H-bond restraint, f_l(i,j) is the lth feature calculated from decoy structures, and μ_kl and δ_kl are the mean and standard deviation of the lth feature of type k H-bond in the Table I. This energy term is a continuous function of the four geometric parameters. When the energy value is lower, the probability of existing a hydrogen bond between residues i and j will be higher.

In the H-bonding network of experimental structures, hydrogen bonds are arranged continuously in order to form a segment of α-helix or a β-sheet. If only residues i and i+4 form a hydrogen bond and all their neighboring residues don’t form similar hydrogen bonds, it is only an α-turn rather than an α-helix. Therefore, we use double hydrogen-bonding energy terms in Eq. 5 to evaluate the stability of secondary structures of α-helix and β-sheet, which also can avoid forming discontinuous H-bonding network for the decoys during the simulation. In the decoy structure, we count residues i to i+3 as a helical region only when E_hb(i,i+4,T₁)+E_hb(i,i+3,T₂) is lower than a cutoff 16.12, i.e. a 4-mer segment forms a helical region if both interactions between i and i+3, i+4 are well maintained. Based on the high-resolution structures, we consider the continuous helical region output by DSSP as the correct assignment. The cutoff value is then determined by making the coverage and accuracy of the predicted helical regions approximately equal to each other.

In a parallel β-sheet, we consider residues i and j as a β-pair only if there are two neighboring hydrogen bonds between them, i.e. E_hb(i-1,j,T₃)+E_hb(j,i+1,T₃) or E_hb(j-1,i,T₃)+E_hb(i,j+1,T₃) is lower than a cutoff 19.11. Similarly, in an antiparallel β-sheet, E_hb(i,j,T₄)+E_hb(j,i,T₄) or E_hb(j-1,i+1,T₄)+E_hb(i-1,j+1,T₄) should be smaller than a cutoff 20.72. Those four kinds of β-pairs between residues i and j in parallel and antiparallel β-sheets are illustrated in Figure S1C-F separately. We also use the DSSP output to extract the correct β-pairs. The two cutoff values for determining the β-pairs in parallel and antiparallel β-sheets are decided by making the total number of predicted β-pairs approximately equal to that of the native β-pairs. After the calculation of hydrogen bonding energy potential, the three-state SS types of the decoy structure can be assigned, which will then affect the calculation of other related energy terms such as backbone torsion angle, helix-packing and strand-packing etc.

To test the accuracy of the hydrogen-bonding energy for secondary structure assignment, we selected a non-redundant set of 3,881 experimental structures from PDBselect³⁴ and used the SS types calculated by DSSP as the correct assignment. Their eight states are grouped into three states (helix, strand, coil). We only extract the N, Cα, C backbone atoms from each experimental structure and add H and O atoms uniquely based on their relative positions to the backbone atoms. We assign four consecutive residues as helices and two residues as strands if their hydrogen-bonding energy values are lower than the three cutoffs. The assignment results of helix and strand are listed in Table II, where the accuracy is defined as the number of correct assigned residues divided by the total assigned residues and the coverage is the number of correct assigned residues divided by the total residues which are assigned as helix or strand by DSSP.

Table II.

Accuracy and coverage of secondary structure assignment using Cα trace and backbone structure.

Methods	α-helix		β-strand
	Accuracy	Coverage	Accuracy	Coverage
Cα only	94.9%	94.3%	89.2%	87.1%
NHOC	97.3%	97.5%	98.7%	93.7%

As a control, we also implement an algorithm which determines the residue SS type based on the geometry of Cα atoms only, i.e. a set of 6 pair-wise distances of neighboring residues were trained to decide the SS types. From the table, the NHOC-based method here is much more accurate than the method based on Cα only, which demonstrates the benefit from adding other backbone atoms in our model. Both the accuracy and coverage of the NHOC assignments are high (>97%) for α-helix assignment. The accuracy of β-strand determination is also high (>98%) but the coverage is slightly lower (93.7%), partly because some β-pairs such as β-bridges, which only have isolated hydrogen bonds, are neglected by the method here, which evaluates two consecutive H-bonds at a time.

(5) Solvent accessibility

Solvent accessibility (SA) is an important attribution of different amino acids, which potentially determines the relative positions of the residues in the global structure. For a given full-atomic structure, the SA of each residue can be accurately calculated using, e.g. EDTSurf.³⁵ However, the calculation can be time consuming. One challenging issue in the molecular simulation is to design an algorithm that can quickly and yet reliably calculate the SA value for each residue, especially in reduced models.

The energy accounting for the residue-specific solvent accessibility here is written as:

$$E_{\mathit{\text{sa}}}={\displaystyle \sum _{i=1}^L}|s_i-s_i^E|$$ (6)

where L is the sequence length, s_i^E is the expected solvent accessibility of the ith residue, which is predicted by the back-propagation Neural Network trained from the checkpoint file by PSI-BLAST and secondary structure types by DSSP. s_i is the solvent accessibility of the ith residue in the decoy structure, which is calculated from the reduced model by:

$$s_i=1-w{\displaystyle \sum _{d(G_i,G_j)<9\AA }}\frac{A_{\mathit{\text{aa}}(j)}}{d^2(G_i,G_j)}$$ (7)

where A_aa is the pre-calculated maximum solvent accessible surface area for amino acid aa. The weight w here equals to 0.007. G_i is the geometric center of the ith residue which is calculated from the coordinates of N, Cα, C, O, Cβ and SC atoms. d(G_i,G_j) is the distance between the ith and jth residue centers. For each residue i, we check all its neighbors which may bury part of its surface, as shown in Figure 4. The buried part contributed by one of its neighbors is proportional to this neighbor’s maximum surface area and in inverse ratio to the square of their distance. We only consider the neighboring residues in a sphere with a radius 9 Å since too distant residues have no contribution to the burying of the target residue. s_i defined in Eq. 7 has the value in the range of [0, 1] while 0 means it is completely buried and 1 means it completely exposes in solution.

Figure 4.

Illustration of solvent accessibility estimation for a residue i based on the distance to its neighboring residues js.

To evaluate the accuracy of the SA model, we compare the SA values calculated by Eq. 7 with the accurate SA values calculated by EDTSurf based on the full-atomic experimental structures of 145 proteins in our test set. As seen in Table III, the relative error is 4.32% if we use the actual geometric centers of residues to calculate the pair-wise distances in Eq. 7. The average CPU time for one test protein is 1.0 millisecond (ms), which is 6,000 times faster than EDTSurf. Since QUARK uses a reduced model with a side-chain center (SC) representing each residue, if we calculate the residue distance based on the center of N, Cα, C, O and the estimated SC, the SA error is only slightly increased (5.10%). These data demonstrate the feasibility of the equation to quickly and reliably calculate SA values.

Table III.

Accuracy of solvent accessibility estimation and prediction.

Methods	Relative error
Distance based method using full-atomic model	4.32%
Distance based method using reduced model	5.10%
Distance based method using side-chain center	7.52%
Ellipsoid based method by I-TASSER	8.74%
Sequence-based Neural Network prediction	9.94%

We attempted to use the side-chain center instead of the geometric center in the formula, which however resulted in a bigger error 7.52%. This is because the geometric center of one residue is more accurate in determining the contacts with its neighbors for SA estimation. We also implemented another faster algorithm for solvent accessibility estimation, which was used in TOUCHSTONE and I-TASSER. The algorithm first builds the bounding ellipsoid for the entire Cα trace structure based on its three principal axes. The solvent accessibility of each residue is then calculated by the square of the ratio of the distance between the residue and the protein structure center to the distance from the protein center to the surface of the ellipsoid which passes through the residue. The average error for this method is 8.74%; but the computation of the model is slightly faster with average CPU time 0.8 ms for one test protein, since it avoids the calculation of pair-wise distances. In the QUARK simulation, we choose the distance-based method due to the better balance between the accuracy and speed. For real-value SA prediction, solvent accessibility values of training proteins input for Neural Network training are calculated by EDTSurf. In the last row of Table III, the average error between the sequence-based NN prediction and the native solvent accessibility is 9.94% for the test proteins, which means more effort still should be made to improve the sequence-based prediction.

As an illustration, we show the SA results for 1bgfA from three resources in Figure S2. The blue curve shows the solvent accessibilities of all the residues calculated by EDTSurf from the full-atomic experimental structure. The green curve is the SA predicted by Neural Network from amino acid sequence, which is generally consistent with the experimental structure. The estimation by using geometric centers from native full-atomic structure and Eq. 7 is shown in the red curve. We can see that the average error of the structure-based estimation is much lower than that from the sequence-based NN prediction.

(6) Backbone torsion potential

The dihedral-angle potential is calculated as:

$$E_{\mathit{\text{dh}}}=-{\displaystyle \sum _{i=2}^{L-1}}\text{log}(P({\varphi }_i,{\psi }_i|\mathit{\text{aa}}(i),\mathit{\text{ss}}(i)))$$ (8)

where φ_i and ψ_i are the torsion-angle pair of the ith residue; P(φ,ψ|aa,ss) is the conditional probability of φ and ψ at the residue type aa and the secondary structure type ss, which are calculated from the high-resolution experimental structures. For this purpose, 60 (=20×3) Ramachandran plots³⁶ should be generated for 20 amino acids and 3 secondary structure types. In Figure S3, we illustrate four energy spectra converted from the Ramachandran plots. As we can see from Figures S3A–C, the energy spectra for the same residue are different according to different SS types. Although torsion angles are highly conserved in helix, there is still some difference between different residues by comparing the blue regions in Figures S3A and S3D.

(7) Fragment-based distance profile

The distance profile energy term for each decoy is written as:

$$E_{\mathit{\text{dp}}}=-{\displaystyle \sum _{(i,j)\in S_{\mathit{\text{dp}}}}}\text{log}(N_{i,j}(d_{\mathit{\text{ij}}}))$$ (9)

where d_ij is the distance between the ith and jth Cα atoms in the decoy structure. N_i,j(d) is the distance profile for residue i and j extracted from the 10-mer fragment structures, with d divided from 0 to 9 Å in the interval 0.5 Å. S_dp is the set of residue pairs which have distance profiles.

We have already generated top 200 fragment structures for each segment of the query sequence, by gapless threading of the query segment sequence through the template library. In the fragment file, we also record the template name and residue indexes for each selected fragment structure. We then check all the fragments in different positions if they come from the same template. If there are two residues in two different fragment structures (one aligned with residue i and another with residue j in the query sequence) which come from the same template structure, we can directly calculate their Cα distance d_kl (assuming the indexes of the two residues are k and l in the template), since they are in the same coordinate system. If the distance is less than 9 Å, then we consider residue i and j in the query sequence may also have the same distance. A histogram of d_ij, N_i,j(d_ij), will be constructed for all the (i, j) residue pairs by comparing every two fragment structures in two different positions.

Not all the residue pairs are concerned in the distance profile because many are false positive pairs. We filter the residue pairs whose distance profiles are monotonically increasing functions, since we can not distinguish whether those residue pairs have short-range contacts or not from their distributions. We only count the residue pairs which have the peaks of the histograms below 9 Å from their distance distributions. Those residue pairs constitute the set S_dp. We choose fragments with length 10 to extract distance profile information since 10-mer fragments can lead to the highest accuracy of distance restraint prediction.

The concept of distance profile is different from the traditional distance restraint energy term, where only one expected distance is assigned to each residue pair, which is usually the average distance extracted from threading alignments or sequence-based predictions. The average distance can be incorrect if multiple distances appear with high frequencies. The distance profile term designed here includes frequencies of all the distance bins. Therefore all the distances, which have high probabilities, will be appropriately considered in Eq. 9. The best distance will be eventually selected by the simulations with the competitions with other energy terms.

(8) Radius of gyration

The propensity to the radius of gyration is written as:

$$E_{\mathit{\text{rg}}}=\{\begin{array}{cc}0\hfill & r_{\text{min}}\le r\le r_{\text{max}}\hfill \\ {(r_{\text{min}}-r)}^2\hfill & r<r_{\text{min}}\hfill \\ {(r-r_{\text{max}})}^2\hfill & r>r_{\text{max}}\hfill \end{array}$$ (10)

where r is the radius of gyration of the simulated decoy structure, r_min and r_max are the minimum and maximum of estimated radius of gyration. The expected radius gyration was estimated based on both protein length and secondary structure elements. Generally, longer proteins have a larger radius of gyration; α-proteins are relative less tightly packed than α/β-proteins especially when they contain some long helices. The average radius and the minimum radius of proteins with different lengths are shown in Figure S4. The minimum radius fits well the equation r_min=2.316L^0.358 (dash line), which has a Pearson correlation coefficient (PCC) 0.991 with the actual values. The average radius of gyration has bigger fluctuations which are approximately fitted with r_avg=δ+2.316L^0.358 (solid line) where the difference δ between the minimum radius and the average radius is 2.5 Å. If we take r_min=2.316L^0.358 and $r_{\text{max}}=\text{max}\{r_{\text{min}}+3\delta ,0.7\sqrt{3/5}{N}_{\text{max }h}\}$ where N_maxh is the number of residues of the longest helix in the structure, we find that 95% of the experimental structures in the PDB have the radius of gyration within [r_min, r_max], i.e. most of the native states have E_rg=0 in Eq. 10.

(9) Strand-helix-strand packing

Since there are rarely left-handed β-α-β motifs in native structures, we add one energy term to penalize this motif during the simulation:

$$E_{\mathit{\text{bab}}}=\{\begin{array}{cc}{E}_{\mathit{\text{Pen}}}\hfill & \text{left}-\text{handed}\hfill \\ 0\hfill & \text{else}\hfill \end{array}$$ (11)

where the penalty energy E_pen equals to the negative value of the total hydrogen bonding energy between the two β-strands in the motif. Given the structure of each decoy, we first scan all the secondary structure elements which have one helix sequentially between two β-strands. Then we check if the two β-strands form a parallel β-sheet. The left-handedness is determined based on the relative position of the center of the helix to the plane formed by the two β-strands. Since the β-α-β motif has identical energy values to its mirror image for most of the other energy terms, Eq. 11 will help avoid the incorrect mirror image models, which have been most often encountered in ab initio structure folding.⁴

(10) Helix packing

The helix-helix packing energy is written as:

$$E_{\mathit{\text{hp}}}(i,j)=-\text{log}(P(d_{\mathit{\text{ij}}},{\phi }_{\mathit{\text{ij}}}))$$ (12)

where d_ij is the distance between the medial axis of the ith helix and that of the jth helix; φ_ij is the torsion angle of the axis vectors which are oriented from N- to C-terminal. P(d_ij, φ_ij) is the probability distribution calculated from the non-redundant experimental structures, where d_ij is split into 30 bins in [0, 15 Å] and φ_ij into 36 bins in [−180°, 180°]. As shown in Figure S5, most of the helix pairs fall in the region of d_ij~9.5 Å and φ_ij~−160°, −40° or 140°.

(11) Strand packing

The β-pairing energy of the ith and jth residues in two paired strands is written as:

$$E_{\mathit{\text{bp}}}(i,j)=-\text{log}(P(\mathit{\text{aa}}(i),\mathit{\text{aa}}(j),T_{\mathit{\text{ij}}}))$$ (13)

where P(A, B, T) is the probability for amino acids A and B in the sheet type T (parallel or antiparallel), calculated from the high-resolution experimental structures. This energy term is used to emphasize the residue types in β-pairs which haven’t been considered by the hydrogen bonding term. As shown in Figure S6, the distributions of residue types between the residue pairs that form backbone hydrogen bonds are highly uneven. Prolines rarely appear in a β-pair since there is no hydrogen atom associated with the nitrogen. By comparing Figures S6A and S6B, there is a slight difference between the two distributions. There are more β-pairs between glutamic acid and arginine in parallel β-sheets than in antiparallel β-sheets.

Design of conformational movements

Efficient conformational search is another critical component of ab initio protein folding, where the design of conformational movements with high acceptance rates is essential for improving the efficiency of Monte Carlo simulations. QUARK performs the conformational search based on the standard replica-exchange Monte Carlo simulation algorithm.³⁷ It involves two types of conformational movements. The global movement consists of periodically conformational swap between neighboring replicas. The local movements include conformational updates implemented in each replica, which is the major focus of discussion in this section.

We have designed eleven local movements for QUARK, which can be also divided into three levels: residue level (M1–M4), segmental level (M5–M8) and topology level (M9–M11) (see Figure 5). Seven movements in the residue level and segmental level also have been included in the ModRefiner program for structure refinement, which have shown efficiency in removing various structural outliers.

Figure 5.

Illustration of eleven movements designed for the QUARK simulations. New conformations after movements are represented by dash lines.

Residue-level movements

The residue level movements only change the conformation of one residue, but may still result in large conformational changes to the global structure. Movements M1, M2 and M3 randomly change one bond length, bond angle and torsion angle of a randomly selected residue. Movement M4 substitutes these three parameters in the selected residue by the clustered values for this residue which are most frequently occurred in the template fragments at the position. This is equivalent to the fragment substitution movement with fragment length equal to 1. Although M4 can be decomposed into the combinations of M1, M2 and M3, it is more efficient with around twice higher acceptance rate than M3 because the torsion-angle values are clustered from experimental fragment structures. M4 implemented in ModRefiner is a little different, which randomly selects one torsion-angle pair from the allowed region in the Ramachandran plot since it doesn’t use fragment structures.

Segment-level movements

There are four movements which change the conformations of a segment sequence. Movement M5 substitutes one fragment in the decoy by another one randomly selected from the position-specific fragment structures. It is one of the important local movements in QUARK that can help reduce the conformational search space and increase the quality of the local structures. To minimize the conformational change of the surrounding residues and to increase the acceptance rate, a Cyclic Coordinate Descent (CCD)³⁸ movement is followed, which tries to adjust the segment conformation and make it connect with the anchor points of the surrounding chain. However, M5 still can be hardly accepted when the fragment is long and the decoy structure becomes compact after a number of cycles of simulations. Hence, we try to use more long fragments at the commencement of the simulation where the decoy structure hasn’t been well packed. The probability of short fragment substitutions will be gradually increased with the process of the simulations. In Figure S7, we show the fragment length distributions in terms of different cycles of QUARK simulations. In each simulation cycle c, the probability of choosing fragment length l follows the discrete Gaussian distribution:

$$\begin{array}{cc}{P}_{\mathit{\text{fragsub}}}(l,c)=\frac{e^{-\frac{{(20-l-0.1c)}^2}{2{\sigma }^2}}}{{\displaystyle \sum _{l=1}^{20}}{e}^{-\frac{{(20-l-0.1c)}^2}{2{\sigma }^2}}},\hfill & \sigma =3/\sqrt{2},c\le 200\hfill \end{array}$$ (14)

where the average length of the moved fragments decreases with the number of the simulation cycles which is proportional to the simulation time. We found that the acceptance rate of M5 was increased by 1.45 times, compared with that using the uniform distribution.

Movement M6 is a LMProt³⁹ perturbation, which first randomly changes the positions of backbone atoms in a selected segment and then tries to restrict all the bond lengths and bond angles within the physically allowed region. Movement M7 rotates the backbone atoms of a randomly selected segment around the axis connecting the two ending Cα atoms. Movement M8 shifts the residue numbers in a segment forward or backward by one residue, which means the coordinates of each residue are copied from its preceded or followed residue in the segment. We then need to delete the unused coordinates of one residue in one terminal and insert new coordinates of another residue in the other terminal. This movement can easily adjust the β-pairing in two well-aligned β-strands.

Topology-level movements

There are three topological movements which try to form well-packed helix pair, β-pair and β-turn. In movement M9, one helix is moved close to another one. The probability of their distance and torsion-angle distribution is the same as that in the helix-packing energy term. The linkage region between the two helices will be rebuilt to keep the backbone connectivity. In the similarly way, one β-pair is formed in movement M10. Since one β-strand is likely to pair with another one which has similar number of residues to form a β-sheet, we pre-calculated the probability for every pair of residues which may form a β-pair based on their secondary structure types and positions in the SS elements. A pair of residues whose predicted SS types are strands have a higher probability than those with SS types equal to coils and helices. During the random β-pair formation movement M10, we select the residue pair based on those pre-calculated probabilities. The possibilities of forming a β-pair in antiparallel and parallel sheets are 75% and 25%, respectively, based on the observation from experimental structures in the PDB.

The probabilities of β-turn positions are predicted by Neural Network, where the correct β-turn positions in the training structures are assigned by PROMOTIF.^40,41 Movement M11 tries to form a β-turn motif for every 4-mer segment along the query sequence. The number of M11 attempts at each position is proportional to the predicted β-turn probability.

The summary of the acceptance rates of all the 11 movements designed for QUARK simulations are shown in Figure S8A–S8D. Movements like M1, M2, M4, M6 and M7, which change the decoy conformation in a smaller magnitude, often have a higher acceptance rate. On the other hand, movements such as M3, M5, M8, M9, M10 and M11, were designed to change the whole part of the conformation from the selected location to the C-terminal or to change the conformation of one segment, or to form a given motif structure. They often have a much larger magnitude of conformational move and generally have a lower acceptance rate.

By comparing Figures S8A and S8B, the replicas at high temperatures have a higher acceptance rate than that at the low temperatures, which is consistent with the Metropolis criterion.⁴² From Figures S8C and S8D, decoy conformations in the beginning of the simulation have a higher acceptance rate than that in the end of the simulation, since the decoy structures at the start are unpacked and have high energies, which therefore can easily accept new movements. But with the number of cycles increasing, the structures become more compact and harder to accept new movements. The average acceptance rate of all the movements in all the replicas and all the 200 cycles is 8.5%. The proportions of these 11 movements attempted to update the decoy structures are listed in Table S2, which are determined by trial and error with the goal to identify the lower energy conformations in a finite simulation time. Although the acceptance rate of fragment substitution movement M5 is low, it still has a high probability to attempt during the simulation, because the local segment or the global topology of the decoy gets improved significantly once a M5 movement is accepted.

Global movements

QUARK runs Monte Carlo simulations in 40 parallel replicas. Although the simulation at low temperatures will detect conformation of lower energies, it can be easily trapped at local energy basin. The replica swap movement is designed to exploit the high-temperature replica simulations to help the low-temperature replicas jump over low energy basins. It is therefore essential to keep a high acceptance rate for swapping each pair of neighboring replicas.

Each replica runs separately within each cycle, where 30L^1/2 (L is the protein length) movements will be attempted based on the Metropolis criterion. After a running cycle is completed, a swap movement will be attempted between every two adjacent replicas to exchange their decoy conformations. The swap movement also follows the Metropolis rule. Figure S9A shows the average acceptance rates of the swap movements for different replicas, where high-temperature replicas have generally higher swap rates. The minimum swap rate is >75% which indicates that the number of replicas is higher enough to keep sufficient replica exchanges. The trajectories of 5 replicas at low temperatures and 5 replicas at high temperatures are shown in Figure S9B. Indeed, the low-temperature replicas tend to search the low energy basin and the high-temperature ones have higher overall energies with higher fluctuations. The neighboring replicas have overlapped energy ranges to ensure replica exchanges.

The temperature distribution of the 40 replicas is shown in Figure S10, which follows an exponential function for the purpose of keeping approximately equal acceptance rate for the global swap movements. The temperature of the ith replica is given in the following formula:

$$T_i=T_{\text{min}}{(\frac{T_{\text{max}}}{T_{\text{min}}})}^{\frac{40-i}{39}},1\le i\le 40$$ (15)

where T_max=2.4+0.016L, T_min=0.6+0.00067L are the temperatures for the first and last replicas. Temperature range of the replicas is larger for bigger proteins in order to keep a reasonable acceptance rate of movements in all replicas since bigger proteins have usually larger energy fluctuation range.

We have also tried different numbers of replicas when we fix the maximum and minimum temperatures. Basically, the modeling result will be better if we use more replicas. The overall swap rate will be higher when we use a larger number of replicas due to the smaller temperature interval between two adjacent replicas. Since more replicas also take more time for the simulation, we decide to use 40 replicas which can achieve a reasonably short running time, high swap rate and high modeling accuracy.

RESULTS AND DISCUSSION

Impact of force field to quality of final models

An accurate force field for protein folding should have a correlation with the structure similarity of the decoy structures to their native structure, so that the energy function can be used to guide the folding simulations towards the native state. To examine the correlation of the QUARK force field, we generated 5,000 structure decoys which were randomly taken from the 10 low-temperature replicas of 10 different REMC runs for each of the 145 test proteins. Homologous templates were removed base on the three strict filters for fragment generation before the REMC simulations. All the decoy structures are deposited at http://zhanglab.ccmb.med.umich.edu/decoys/decoy3.html.

One important question is how much the structural prediction power of QUARK is influenced by the correlation of the force field to the backbone accuracy as evaluated by TM-score. To examine this, we cluster the structure decoys by SPICKER and plot the TM-score of the first cluster center versus the Pearson correlation coefficient of the energy-to-TM-score in Figure 6. As shown in the figure, PCC of the QUARK energy and the TM-score of the first cluster center is typically in the range of [−0.7, 0.2]; the average PCC is −0.185. There is a general tendency that the well-folded targets with higher TM-score correspond to cases with stronger correlation coefficients between TM-score and energy of the decoys. The Pearson correlation coefficient between the TM-score of the first cluster and the PCC of the energy-to-TM-score is −0.469. It is however quite surprising that there are several exceptional cases where QUARK successfully folds the proteins which do not have strong energy-to-TM-score correlation. There are also cases where QUARK potential has a high correlation with TM-score but the final model has a low TM-score. In the following, we specifically examine these cases.

Figure 6.

TM-score of the first cluster center model versus the Pearson correlation coefficient of the QUARK energy and the TM-score of decoys. 5,000 decoys have been generated for each protein to calculate the Pearson correlation coefficient.

Figure 7A shows a normal case from the arginine repressor of bacillus stearothermophilus (PDB ID 1b4b, Chain A), where the QUARK force field has a strong correlation with the TM-score. This correlation results in a well-folded model of the first cluster with a TM-score = 0.624, which is slightly better than the structure model of the lowest total energy (TM-score=0.611). This represents a typical example of QUARK simulations where better energy force fields lead to better modeling results.

Figure 7.

Three illustrative examples for energy-to-TM-score correlations of 5,000 decoys. The cluster center model (in gray) selected from the decoy set is superimposed onto the native structure (in black). (A) 1b4bA. (B) 2bl7A. (C) 2o42A.

Figure 7B is an example from EntA-im (PDB ID: 2bl7, Chain A). The QUARK energy has no obvious correlation with the TM-score (PCC=−0.116) but QUARK generated a model of correct fold (TM-score=0.527). The major error of this model is at the loop region (53N-67T), where the secondary structure prediction program incorrectly assigned the loop as β-strands. Therefore, QUARK generated an additional β-hairpin structure in this loop region. Nevertheless, the global fold of the model which contains four α-helices is the same as the experimental structure. However, many of other decoys whose global topologies are incorrect have consistent secondary structures with the predictions (including the incorrectly assigned β-hairpin region) and therefore have a low energy in the QUARK simulation; this resulted in the weak energy-to-TM-score correlation coefficient for this example. Despite of the low correlation, the conformation of the correct fold has the largest cluster size with the lowest free-energy, which can therefore be picked up by the SPICKER clustering program. One possible reason for the low correlation but high modeling accuracy is because of the unbalanced Monte Carlo simulation using fragment assembly. Local segments of the generated decoys may be biased towards the fragments retrieved from the template library. Those fragment structures in fact are part of the energy surface implicitly which hasn’t been included in the energy calculation.

Figure 7C is another example of exceptional energy-to-TM-score correlation, which is from the M11L apoptosis inhibitor protein (PDB ID: 2o42, Chain A). Although this protein has a strong correlation (PCC=−0.489), all the decoys are within the low TM-score region. The experimental structure of the protein consists of a pack of seven short helices, but the majority of the QUARK decoys have the topology of a bundle of two long helices, mainly due to the incorrect secondary structure prediction. The QUARK energy of the native structure is −406.124 which is higher than almost all that of the decoys because more loop regions in the native structure make the torsion-angle potential higher than that of the decoys which contain more regular helical regions as predicted by PSSpred. This target represents a typical example of the QUARK simulations which were misguided by the incorrect secondary structure predictions.

In Table IV, we analyze the correlation coefficient of each energy term in Eqs. (1–13) to the TM-score, which are calculated from the average of 145 targets, each of which has 5,000 structural decoys. Due to the penalty effect of E_bab, QUARK simulations did not generate left-handed β-α-β motifs. Hence, there is no apparent correlation coefficient between TM-score and E_bab in this calculation. The excluded volume E_ev and the radius of gyration E_rg are continuous penalty scores, which were designed to roughly control the local conflict and global shape. Those are the two energy terms of the weakest correlations with the TM-score to native.

Table IV.

Average Pearson correlation coefficient between each energy term and TM-score of 5,000 decoy structures of 145 testing proteins.

Energy terms		Correlation with TM-score
Atomic-level	Backbone distance-specific contact (Eprm)	−0.157
	Side-chain distance-specific contact (Eprs)	−0.166
	Excluded volume (Eev)	−0.001
Residue-level	Hydrogen bonding (Ehb)	−0.036
	Solvation (Esa)	−0.137
	Backbone torsion (Edh)	−0.013
	Fragment-based distance profile (Edp)	−0.076
Topology-level	Radius of gyration (Erg)	−0.001
	Strand-helix-strand packing (Ebab)	N/A
	Helix-helix packing (Ehp)	−0.039
	Strand-strand packing (Ebp)	−0.030
	Total energy (Etot)	−0.185

The side-chain distance-specific contact potential E_prs has the highest correlation with TM-score although the side-chain centers are added approximately, which demonstrates the usefulness of this newly-designed energy term. The backbone pair-wise contact potential E_prm and solvation potential E_sa are the other two important potentials which contribute to the average correlation of the total energy.

The absolute correlations of the TM-score with the rest five energy terms are all below 0.1. Despite of the low correlation, we find all of them are necessary to the QUARK ab initio folding simulation and dropping off any of them will result in degraded folding results in our training simulations.

Benchmark results

We have collected 51 small proteins whose lengths are in the range of [70, 100) and 94 medium sized proteins with lengths in the range of [100, 150] as the two test sets. These proteins are “Hard” targets defined by LOMETS since there are no significant template structures detected from the threading template library after removing the templates which have a sequence identity >30% to the target. We generated 5,000 decoys as described in the previous section in 10 parallel jobs for each test protein.

Rosetta^14,43 is one of best established ab initio protein folding methods as demonstrated in the CASP experiments.^44,45 Since both QUARK and Rosetta use fragment assembly, we will mainly use Rosetta as a control to benchmark our method. Although there were different versions of Rosetta programs available, we found in our benchmark that the version 2.3.0 generated models with the highest average TM-score, which is thus used in this study. For each target, we first run the Rosetta script to generate top 200 3/9-mer fragments which are retrieved using features of PSI-BLAST²⁵ checkpoint file and PSIPRED⁴⁶ secondary structure prediction. The template library is version 2006-05-05, which contains 6,025 idealized templates. Homologous templates are removed from their template library for fragment generation using the same filters as described for QUARK. We then run the Rosetta release version in 50 parallel jobs, each of which starts from a different random number and generates 100 decoys. At last, we use the same SPICKER program as in QUARK to cluster the entire 5,000 decoys and get top 5 cluster center models.

Table V shows a summary of ab initio structural prediction results by Rosetta and QUARK separately, where the structural quality of the final models is measured by RMSD,⁴⁷ TM-score,²⁹ global distance test-total (GDT-TS) score,⁴⁸ MaxSub⁴⁹ score and backbone hydrogen-bonding score (HB-score) in comparison to the native structures. Both TM-score and MaxSub⁴⁹ score evaluate the backbone accuracy to native after the optimum superposition, but they have different distance cutoffs and scoring functions. GDT-TS score is a little different to the above two scores, which counts the sum of fractions of residue pairs between model and native with distances below 1, 2, 4, and 8 Å respectively, after optimal structural superimpositions. Since models by both Rosetta and QUARK contain full backbone atoms, it allows us to compare their backbone H-bonding quality directly. Here, HB-score is defined as the number of the correctly predicted hydrogen bonds divided by the total number of the hydrogen bonds appearing in the experimental structure as calculated by HBPLUS.⁵⁰ Based on the first QUARK models of all the 145 targets, the Pearson correlation coefficients between TM-score and GDT-TS, MaxSub scores are extremely high, which are 0.968 and 0.965 separately. It reveals that any one of the three scoring functions is adequate enough to evaluate the backbone accuracy to native. The Pearson correlation coefficients are −0.783 and 0.606 between TM-score and RMSD, HB-score, which means they are not equivalent metrics.

Table V.

Ab initio modeling results by QUARK and Rosetta.

		First (best in top 5) cluster center model
		RMSD	TM-score	GDT-TS	MaxSub	HB-score	Time
51 small proteins with [70–100) residues	Rosetta	10.1Å(8.5Å)	0.350(0.393)	0.381(0.418)	0.291(0.337)	0.442(0.491)	25.0hr
	QUARK	9.1Å(7.7Å)	0.404(0.441)	0.428(0.466)	0.349(0.389)	0.503(0.538)	37.7hr
	QUARK-ha	6.4Å(4.6Å)	0.585(0.667)	0.602(0.691)	0.552(0.635)	0.610(0.681)	37.7hr
94 medium proteins with [100–150] residues	Rosetta	13.0Å(11.5Å)	0.317(0.346)	0.293(0.318)	0.224(0.247)	0.410(0.453)	63.3hr
	QUARK	12.5Å(10.7Å)	0.334(0.374)	0.310(0.342)	0.237(0.268)	0.471(0.504)	79.5hr
	QUARK-ha	8.6Å(6.6Å)	0.491(0.541)	0.439(0.483)	0.362(0.410)	0.449(0.503)	79.5hr

^aQUARK simulation using fragments without removing homologous templates to the query sequence. However, the target proteins themselves, if existing in the library, were excluded.

Both Rosetta and QUARK programs have a slightly better result in folding small proteins than medium sized proteins. Overall, 1/3 of short proteins can be correctly folded by QUARK with TM-score >0.5. A TM-score >0.5 means the model most probably has the same fold as the native structure.⁵¹ However, there are only 10 out of 94 medium sized targets which QUARK can correctly predict. It is probably due to the fact that small proteins have usually simpler topology and smaller conformational search space. Therefore, it is relatively easier for the algorithms to identify the correct fold. By comparing the best in top 5 QUARK models with the first model, the relative TM-score differences are 9% for small proteins and 12% for medium sized proteins, which indicate the ranking by the clustering algorithm is also better for small proteins.

Comparing the two folding algorithms, although QUARK outperforms Rosetta in both sets of proteins, the TM-score difference of the final models is slightly larger for the small proteins (15% in TM-score of the first model) than that for the medium sized proteins (6%). This protein-size dependent difference also exists for the best in top 5 models, where the TM-score differences of final models are 12% and 8%, respectively. QUARK models also have a higher HB-score than the Rosetta models. However, the difference does not depend on the protein lengths. For example, the differences of HB-score of the first models are 14% and 15% for small and medium sized proteins, respectively. For the best in top 5 models, the HB-score differences are 10% and 11% for small and medium sized proteins, respectively, which are still independent of the protein lengths but less significant than that of the first model.

We also compare the average CPU hours per target in the last column of Table V. Both programs are running in the cluster with Linux 64-bit system installed. Each cluster node is equipped with eight 2.27 GHZ Intel E5520 Xeon processors and 24GB memory. Since there are totally 40 replicas in the QUARK REMC simulation and only 1/4 of the 10 low-temperature replicas are randomly selected for clustering, the running time of QUARK is longer than that of Rosetta which runs a single trajectory of simulated annealing algorithm. QUARK is 1.51 times slower than Rosetta for small proteins while the ratio becomes 1.26 for medium sized proteins.

In Figures 8A–C, we present a head-to-head comparison of the QUARK and Rosetta models in regard of RMSD, TM-score and HB-score. According to the RMSD of the best in top 5 models, there are 96 out of 145 targets where QUARK models are better than Rosetta models, while QUARK outperforms Rosetta in 95 and 106 cases, respectively, in terms of TM-score and HB-score. We further conduct the paired Student’s t-test to check their difference. The p-values of the RMSD, TM-score and HB-score between the best in top 5 QUARK and Rosetta models are 1.51E-4, 2.87E-7 and 6.88E-14 separately, which show their differences are statistically significant.

Figure 8.

Comparison of the best in top 5 cluster center models by Rosetta and QUARK on 145 test sequences. (A) RMSD. (B) TM-score. (C) HB-score.

Among all the test proteins, a large portion of them can be modeled with an approximately correct topology by both Rosetta and QUARK. However, QUARK models often have better accuracy than Rosetta models for those targets, as seen in Figure 8B. The reason is probably because of the more accurately designed potentials by QUARK for the low-resolution simulations. In Figure 9, we present four successful examples where QUARK generated improved structural models compared with Rosetta. Figures 9A–B show two α-proteins while targets in Figures 9C–D are α/β-proteins. Target in Figure 9A is the Chain T of SecM-stalled E. coli ribosome complex (PDB ID 2gyb), which has a simple topology of a three-helix bundle. QUARK folds this target with a very high accuracy (RMSD=1.30 Å, TM-score=0.90). Rosetta has the fold approximately correct but breaks the third long helix into two which results in a much lower TM-score (0.68). Figure 9B is from the sporulation inhibitor pXO2-61 of Bacillus anthracis (PDB ID: 1yku), an α-protein of 132 amino acids containing 5 long helices. QUARK model has a TM-score 0.61 where all the five helices are with correct orientations. Again, Rosetta breaks two long helices into short ones and misplaces the orientation of the third helix, which results in a low TM-score (0.46). The high modeling accuracy of QUARK for α-proteins in these examples is mainly due to the joint effect of the side-chain pair-wise potential in Eq. 3 and the helix-packing energy term in Eq. 12.

Figure 9.

Examples of successful modeling results by QUARK in the benchmark set. Experimental structure, QUARK model and Rosetta model are in red, green and blue separately. (A) 2gybT. (B) 1ykuA. (C) 2v94B. (D) 1jo0A.

Target 2v94B in Figure 9C is the mutation of ribosomal protein RPS24. Four long β-strands form an antiparallel β-sheet, which can be split into two β-hairpins. Although both QUARK and Rosetta models have the correct topology, one α-helix and one β-hairpin in the Rosetta model deviate from the positions in the native structure. Target in Figure 9D is a hypothetical protein from Haemophilus influenza (PDB ID: 1jo0), which contains one β-hairpin and one β-α-β motif. QUARK model has a slightly higher TM-score than the Rosetta model mainly because it correctly predicts the two β-strands in the β-α-β motif, which is probably attributed to the strand-packing energy term in Eq. 13.

All the above modeling data have been generated with analogous templates completely excluded. In the last rows of Table V, we also show the QUARK prediction result without removing the fragments from homologous templates, i.e. we released the three filters. Although no global template information was directly used in the simulation, the prediction result with homologous fragments makes the average TM-score increased by more than 45%. This is because the fragments from the homologous templates can be assembled in a more cooperative manner in the fragment substitution movements, where structures from the same (or a few) homologous template(s) will result in lower energy basin because of the seamless matches of all the fragments. Moreover, the distance profile potential which is extracted from all the selected fragments will be more accurate when homologous fragments are included. Meanwhile, those data also highlight the ability of the QUARK algorithm in recognizing global template structures, no matter they are homologous or non-homologous. This advantage can be useful for folding the targets in the category of hard template-based modeling (TBM), where analogous but non-homologous templates exist in the PDB but threading algorithms have difficulties in detecting them. A systematic study on folding these proteins is under progress.

CASP9 blind test

There are 26 domains/targets in the ninth community-wide CASP experiment (CASP9), which were categorized by the assessors as free modeling targets. Most of these targets have no structural similarity to any proteins in the PDB while few targets (e.g. T0537, T0550-D1, T0571-D1 and T0624) have weakly homologous templates which cannot be found by threading algorithms or threading alignments are not satisfactory.⁵² Structural models of all the server and human groups were assessed based on the GDT-TS score and then normalized according to their mean and standard deviation to get their corresponding Z-scores. A positively high Z-score means the modeling accuracy is much better than the average result. A summary of the top 20 groups, ranked by the Z-score of GDT-TS score, from the automated server prediction section is listed in Table VI, the data of which were taken from the FM assessor’s website (http://prodata.swmed.edu/CASP9/evaluation/domainscore_sum/human_server-best-Z.html). As shown in the table, models generated by QUARK server has a total Z-score=40.6 in this blind test, which is about 18% higher than the second best server, and 47% higher than the third best server from other groups.

Table VI.

Cumulative Z-score of GDT-TS score of the top 20 servers on the 26 FM targets in CASP9 ranked based on the best in top 5 models (Data was taken from http://prodata.swmed.edu/CASP9/evaluation/domainscore_sum/human_server-best-Z.html, The Zhang-Server prediction from our lab using a different method is not listed here).

Server name	Z-score of GDT-TS score
QUARK	40.5894
BAKER-ROSETTASERVER	34.3905
MULTICOM-CLUSTER	27.6390
CHUNK-TASSER	27.4773
MULTICOM-REFINE	25.8908
RAPTORX-MSA	25.0199
RAPTORX	24.9208
RAPTORX-BOOST	24.7337
PRO-SP3-TASSER	24.5677
MULTICOM-NOVEL	24.3617
MULTICOM-CONSTRUCT	24.2406
PRDOS2	20.4793
GWS	19.7966
PHYRE2	19.2882
JIANG_ASSEMBLY	19.1613
MUFOLD-MD	18.7891
GSMETASERVER	17.8072
ZHOU-SPARKS-X	16.4345
PCOMB	15.8549
MUFOLD-SERVER	15.0945

Overall, QUARK server correctly predicted 6 out of 26 targets, where QUARK models have TM-score >0.5 to their native structures. Five of them are small α-proteins and one is a small α/β-protein. If we use TM-score cutoff 0.4, we can have additionally 4 meaningful predictions for medium sized targets. In Figure 10A–C, we show three successful examples of QUARK modeling results relative to other CASP9 predictors in the FM category, which belong to the categories of α, α/β and β proteins. First, target T0547 is a big protein with 611 amino acids, which is divided into four domains. The third domain T0547-D3 was considered as a FM domain since there were no good threading alignments for this domain. The native structure shown in Figure 10A contains two long helices and one short helix. QUARK server correctly folded the domain with a TM-score=0.653. Comparing the cartoon representations of the QUARK model and the native structure, the two long helices and the coil region between them in the model are nearly identical to that in the native structure. The RMSD in this region is 1.26 Å to the native structure, although the overall RMSD of the model is quite high (5.88 Å).

Figure 10.

Examples of QUARK modeling results in CASP9. Experimental structure and QUARK model are in red and green separately. (A) T0547-D3. (B) T0618-D1. (C) T0624-D1. (D) T0529-D1.

Target T0618-D1 in Figure 10B is mainly an α-protein but contains a short β-hairpin. Our secondary structure prediction program predicted it as a whole α-protein, which led the final QUARK models to contain only helices. Nevertheless, one QUARK model had the core region of the helix bundle correctly constructed, which resulted in a TM-score=0.478 and GDT-TS score=41.77. The modeling error was mainly due to the spatial shift of the third helix which stemmed from the incorrect secondary structure prediction in the region of β-sheet.

T0624-D1 is a small β-protein which contains seven β-strands. The C-terminal strand contacts with the N-terminal strand and another strand in the middle of the sequence to form an antiparallel β-sheet in Figure 10C. The QUARK model had the orders of the strands correctly packed but there was a relatively severe twist of the N-terminal 3-strand domain relative to the other 3-strand domain. Nevertheless, it had a relatively high GDT-TS score of 44.56. The major reason for the successful modeling of this target is that most of the β-strands are regularly packed with low sequence separations in the native structure where QUARK has the advantage in folding such proteins of simple β-hairpin topology as observed in our benchmark test.

Despite the successes in CASP9, the ability of ab initio folding by QUARK is still quite limited as indicated by the overall CASP9 assessments.² Compared with the encouraging performance of the ab intio predictions in early CASP experiments (e.g. CASP3-4),^53,54 one reason for the apparently low GDT-TS/TM-scores in CASP9 is due to the lack of targets with regular globular topology. Many FM targets have the structures dominated by irregular shapes such as a super-long α-helix which cannot be folded on their own (e.g. T0616-D1 and T0629-D2). Another reason for the obviously low modeling scores is related to the limitation of current methods on handling large proteins. Figure 10D shows an example from T0529-D1 which is 339 residues long, containing 12 short α-helices and 4 β-sheets in the experimental structure. Such topological complexity raises tremendous challenges to the conformational sampling and force field design for the current ab initio simulations. The QUARK model is among those of the highest TM-score which is however still close to random (0.17). These data highlight the fact that there are so far no methods which can fold proteins with more than 200 amino acids without using templates.

CONCLUSIONS

We have developed a new algorithm, QUARK, for ab initio protein structure prediction. The protein conformations are specified by the full-atom of backbone (N, Cα, C, O, Cβ, H) and side-chain center of mass (SC). Such a semi-reduced model facilitates the design of atomic-level force fields such as H-bonding, van der Waals, backbone torsion-angle and atomic pair-wise interactions, which cannot be implemented by the conventional reduced models^4,17 that simplify each residue by two or three virtual points. The conformations are represented in both Cartesian and torsion-angle coordinate systems, which significantly facilitate the conformational movements and energy calculations. Since the ability of ab initio protein structure predictions is mainly barred by the inaccurate force field and limited conformational search power, we present an effort to attack both aspects of the problem by the development and benchmarking of the QUARK algorithm.

Folding a protein structure by ab initio modeling essentially requires force fields to guide both local structural packing and global topology assembly. There have been a number of well developed energy terms for local secondary structure constructions. But the more important short-range interactions between residues far apart in the sequence are generally lacked. The energy force field of QUARK covers three levels of structural packing: atom-, residue- and topology-level energy terms. Especially, it includes several topology-level terms to account for the motif packing which are essential for assembling the global topology of protein structures in the template-free protein folding. Another helpful term is the fragment-based distance profile, where distance contacts between residue pairs with both short and long sequence separations can be derived from unrelated experimental structures based on the cooperative occurrence of fragment motifs in the same templates. This is different from the traditional contact predictions from homologous templates^55,56 or machine-learning,^57,58 but provides comparable accuracy for free-modeling targets. The other energy terms from traditional considerations including solvent accessibility, hydrogen-bonding, and pair-wide atomic interactions have been reconstructed and validated which help enhance both local and global structural packing.

To speed up the conformational search, eleven local structural movements have been designed with the major focus on increasing the average acceptance rate. Although the fragment substitution and rotation movements were borrowed from Rosetta and I-TASSER programs, which are essential for reducing the size of the conformational search space, the composite design of local movements helps increase the flexibility and efficiency of the conformational search significantly.

Based on the benchmark test, we showed that more than 1/3 of short proteins (<100 residues) could be successfully folded by QUARK with a TM-score >0.5 even after a stringent exclusion of all experimental structures which have any sequence or structure similarity to the target. For proteins with lengths >100 residues, the successful rate is lower but there are still 31% targets for which QUARK constructed reasonable structural folds with a TM-score >0.4. The average TM-score of the models by QUARK on the 145 benchmark proteins is 10% higher than that by Rosetta, one of the state-of-the-art algorithms in ab initio protein structure prediction. The QUARK method was also tested in the recent community-wide CASP9 experiment. The total Z-score of the GDT-TS score is 18% higher than the second best program and 47% higher than the third best program from other groups. These data demonstrated new progress in the field of ab initio protein structure predictions.

Nevertheless, we witnessed several cases where global energy minima have not been reached in current simulations. Some cases were due to the complexity of the structural topology such as β-proteins of complicated strand arrangement. We are working on enumerating all types of typical β-topologies as QUARK starting conformations which hopefully can partially address this issue. The other cases were due to the incorrect local and global restraints which trap the simulations to specific topologies, which are caused apparently by the combined effect of both force field and search engine. Therefore, continuous efforts in both aspects of force field development and conformational search improvement are still required.

Abbreviations

NN

Neural Network

REMC

replica-exchange Monte Carlo

RMSD

Root Mean Squared Deviation

SA

solvent accessibility

SS

secondary structure