Improved packing of protein side chains with parallel ant colonies
- Lijun Quan†^{1},
- Qiang Lü†^{1, 2}Email author,
- Haiou Li^{1},
- Xiaoyan Xia^{1, 2} and
- Hongjie Wu^{1, 2, 3}
https://doi.org/10.1186/1471-2105-15-S12-S5
© Quan et al.; licensee BioMed Central Ltd. 2014
Published: 6 November 2014
Abstract
Introduction
The accurate packing of protein side chains is important for many computational biology problems, such as ab initio protein structure prediction, homology modelling, and protein design and ligand docking applications. Many of existing solutions are modelled as a computational optimisation problem. As well as the design of search algorithms, most solutions suffer from an inaccurate energy function for judging whether a prediction is good or bad. Even if the search has found the lowest energy, there is no certainty of obtaining the protein structures with correct side chains.
Methods
We present a side-chain modelling method, pacoPacker, which uses a parallel ant colony optimisation strategy based on sharing a single pheromone matrix. This parallel approach combines different sources of energy functions and generates protein side-chain conformations with the lowest energies jointly determined by the various energy functions. We further optimised the selected rotamers to construct subrotamer by rotamer minimisation, which reasonably improved the discreteness of the rotamer library.
Results
We focused on improving the accuracy of side-chain conformation prediction. For a testing set of 442 proteins, 87.19% of ${\mathcal{X}}_{1}$ and 77.11% of ${\mathcal{X}}_{12}$ angles were predicted correctly within 40° of the X-ray positions. We compared the accuracy of pacoPacker with state-of-the-art methods, such as CIS-RR and SCWRL4. We analysed the results from different perspectives, in terms of protein chain and individual residues. In this comprehensive benchmark testing, 51.5% of proteins within a length of 400 amino acids predicted by pacoPacker were superior to the results of CIS-RR and SCWRL4 simultaneously. Finally, we also showed the advantage of using the subrotamers strategy. All results confirmed that our parallel approach is competitive to state-of-the-art solutions for packing side chains.
Conclusions
This parallel approach combines various sources of searching intelligence and energy functions to pack protein side chains. It provides a frame-work for combining different inaccuracy/usefulness objective functions by designing parallel heuristic search algorithms.
Keywords
Introduction
The accurate packing of side chains plays a very important role in modelling protein structures. In ab initio structure prediction, the goal is to choose a rotamer for each position so that the molecule is close to the natural structure. In homology modelling, the goal is to predict the structure of a protein that is homologous to another of a known structure [1, 2]. In protein design, the goal is to find an amino acids sequence that will fold into a particular backbone [3]. In flexible ligand docking, the goal is to display a structural change ranging from large movements of entire domains to small side-chain rearrangements in the binding site [4–6]. Based on Anfinsen's hypothesis [7], the problem of packing side chains is usually mapped into a combinatorial optimisation problem and can be solved in a number of ways. However, a fixed backbone, an energy function and a possible rotamer set are always foundations of this widely studied formulation. All the current existing algorithms for the side-chain problem can be divided into two categories, heuristic and deterministic.
The side-chain problems have been proven as non-deterministic polynomial-time hard (NP-hard) [8–10]. Even when an approximate solution is sought within O(cnR) from the optimum, where c is a constant, n is the number of residues and R is the average number of rotamers per residue [11, 12], the packing side chains cannot be solved successfully. Computational complexity analysis suggests that any global optimisation algorithms for this problem may, in the worst case, run in exponential time [11]. When they converge, dead-end elimination (DEE) algorithms [13, 14] are designed to find the global minimum energy. Heuristics are not guaranteed to find a global minimum, but they almost always find a low-energy conformation in a reasonable time [15]. Therefore, heuristic algorithms become a natural choice for tackling the side-chain modelling problem. Traditionally, all heuristic approaches solve such side-chain problems as a single-objective optimisation Problem (SOP), using Monte Carlo (MC) [16], Ant Colony (AC) [17], and Simulated Annealing (SA) [18]. Some of the heuristic methods combine multiple strategies, such as a combination of DEE and the A^{ * } algorithm [19], and combination of SA and MC [20–22]. The common feature of these heuristic approaches is that they all use an optimisation based on a single objective function.
Another method for solving the side-chain problem was by using the theory of decomposing the underlining residue relationship. One such method is SCWRL [23–25, 15], which is widely used because of its speed, accuracy and ease of use. SCWRL3 decomposes original residue graphs to connected subgraphs, which cannot be disconnected by the removal of a single vertex. They find the global minimal energy conformation for the residues in these subgraphs [25]. The authors who proposed the SCWRL methods also observed that residues with a single rotamer or a single neighbour can be eliminated from the residue graph. Then SCWRL4 [15] transfers the original residue graphs to a tree for speeding up the solver. However, in the tightly packed environments of protein interiors, these methods will inherently lead to atomic clashes and hinder the prediction accuracy. Therefore, a new method, CIS-RR, performs clash detection-guided iterative searches (CIS) of side-chain rotamers whilst continuously optimising side-chain conformations using a conjugate gradients method [26].
In general, methods for predicting side chains seem to be limited not by the quality of search algorithms, but also by the quality of the energy functions employed [23]. An energy function typically consists of a combination of weighted energy terms. It is not hard to find different approaches, which develope distinctive kinds of energy functions. For example, SCWRL3 use an energy function based on logarithmic probabilities of rotamers and a simple repulsive steric energy term [25]. However, SCWRL4 also uses a short-range, soft van der Waals interaction potential between atoms rather than the linear repulsive-only function used in SCWRL3, as well as an anisotropic hydrogen bond function similar to that used in Rosetta [15, 27]. The energy function of CIS-RR is also a modified the energy function of SCWRL3. The first improvement is to add attractive energy and weights to the van der Waals potential. The second improvement is to penalise the drifting of side chain dihedral angles away from the nearest rotamer library values for the original rotamer term. The existence of different energy functions implies that all energy functions are inaccurate in a universal sense (inaccuracy), but each of them is very useful in some specific sense (usefulness). This hypothesis is referred to as the inaccuracy/usefulness property [28]. The approaches based on SOP all use a single inaccuracy energy function to model side chains, so the results are sometimes inaccurate in a quantitative sense for discriminating native or near-native conformations.
In this study, a novel approach is proposed to assemble the usefulness and decrease the inaccuracy of different energy functions. We believe that it is more reasonable to model packing side chains as a multi-objective optimisation problem (MOP). Different energy functions should be combined to the best possible extent. As this idea has been successfully applied to de novo prediction of protein backbone [28, 29], we also used parallel ant colony optimisation based on SHOP (SHaring One Pheromone matrix) [30]. Our parallel strategy is not for speeding up the predictor, but can be used to hybridise the usefulness of different energy functions. All energy functions can be adopted by an individual colony. In this way, we can avoid the sensitivity of the optimised parameters of energy functions, so we expect to obtain better generality of our predictor. This parallel strategy has been validated experimentally.
Methods
We propose a novel parallel ant colony optimisation (ACO) metaheuristic frame-work for packing protein side chains by single-heuristic multi-objective algorithms (SHMO) to reduce the inaccuracy of a single energy. We denote a heuristic algorithm by h and different energy functions by ε = {E_{1} , . . . , E_{ k } }, where the number of threads amount to k. This type of algorithm is generally denoted by ${\prod}_{h}\left({E}_{i}|\text{\Theta}\right)$ where $\text{\Theta}$ refers to the control parameters in terms of heuristic search algorithms and can usually be tuned empirically before starting, or adaptively during the algorithm [28]. In the pacoPacker algorithm, h adopts ACO, and $\text{\Theta}$ contains two variables, private and public. To be more specific, all ant colonies share one common pheromone matrix T as a public variable, and each ant colony has a private variable including heuristic matrix H_{ i } and two other parameters, α_{ i } and β_{ i } . A = {α_{1} , . . . , α_{ k } }, determines the importance of the pheromone and B = {β_{1} , . . . , β_{ k } }, determines the importance of the heuristic matrix H = {H_{1} , . . . , H_{ k } }. This paper's method can be described as ${\prod}_{AC}\left({E}_{i}|{\alpha}_{i},{\beta}_{i},{H}_{i},T\right)$. The Rosetta3.4 platform [31] is quite mature and supports the object-oriented paradigm, therefore pacoPacker uses Rosetta3.4 for building rotamer libraries, constructing interaction graphs, and scoring structures. Using Rosetta3.4 and OpenMP [32], our scheme is easy to implement.
Search space
For an amino-acid sequence t with n length of residues, its side chains are packed with the lowest free energy. Let the rotamer library for t be R = {R_{1} , . . . , R_{ n }}, where the rotamer set is ${R}_{i}=\left\{{r}_{1},...,{r}_{{m}_{i}}\right\}$ for each residue i in t, the number of rotamers belonging to R_{ i } amount to m_{ i }, and different rotamer sets have a different quantity of rotamers. Rotamers were read from Dunbrack backbone dependent rotamer library (2010 version), such that frequencies and dihedral angles varied with the backbone dihedral angles Φ and ψ [33].
Energy function
Score function and ACO parameters.
Thread ID | Score function | Score terms | α | β | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
A | B | C | D | E | F | G | H | I | G | K | L | M | N | O | P | Q | ||||
1 | standard | 0.8 | 0.44 | 0.65 | 0.004 | 0.49 | 0.56 | 1.17 | 1.17 | 1.17 | 1.1 | 0.5 | 2 | 5 | 5 | 1 | 0 | 0 | 3 | 1 |
2 | score12 | 0.8 | 0.44 | 0.65 | 0.004 | 0.49 | 0.56 | 1.17 | 0.585 | 1.17 | 1.1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 1 |
3 | score12 full | 0.8 | 0.44 | 0.65 | 0.004 | 0.49 | 0.56 | 1.17 | 0.585 | 1.17 | 1.1 | 0.5 | 2 | 5 | 5 | 1 | 0 | 0 | 1 | 2 |
4 | score12minpack | 0.8 | 0.44 | 0.65 | 0.004 | 0.49 | 0.56 | 1.17 | 0.585 | 1.17 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 1 | 3 |
5 | score13 | 0.6921 | 0.1754 | 0.5253 | -0.00764 | 0.53 | 0.63 | 1.322 | 0.336 | 2 | 1.883 | 0.5 | 2 | 5 | 5 | 1 | 0.571 | 0 | 2 | 1 |
7 | score13 | 0.6921 | 0.1754 | 0.5253 | -0.00764 | 0.53 | 0.63 | 1.322 | 0.336 | 2 | 1.883 | 0.5 | 2 | 5 | 5 | 1 | 0.571 | 0 | 1 | 1 |
8 | pack no hb env dep | 0.8 | 0.1 | 0.65 | 0.004 | 0.49 | 0.56 | 1.17 | 1.17 | 1.17 | 3.1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 3 | 1 |
6 | RosettaHoles score | The RosettaHoles scores are based on packing information about a cavity ball and the local region surrounding it, most importantly the contact surface area of atoms surrounding the cavity with respect to a sequence of probe radii. | 1 | 2 |
Score terms.
Score term | Label | Description |
---|---|---|
fa_atr | A | lennard-jones attractive |
fa_rep | B | lennard-jones repulsive |
fa_sol | C | lazaridis-jarplus solvation energy |
fa_intra_rep | D | lennard-jones repulsive between atoms in the same residue |
fa_pair | E | pairwise electrostatics term derived from statistics on the pdb database |
fa_dun | F | internal energy of sidechain rotamers as derived from Dunbrack's statistics |
hbond_lr_bb | G | long range (beta or loop) backbone-backbone hydrogen bonds |
hbond_sr_bb | H | short range (helix) backbone-backbone hbonds |
hbond_bb_sc | I | sidechain-backbone hydrogen bond energy |
hbond_sc | J | sidechain-sidechain hydrogen bond energy |
dslf_ss_dst | K | distance score in current disulfide |
dslf_cs_ang | L | csangles score in current disulfide |
dslf_ss_dih | M | dihedral score in current disulfide |
dslf_ca_dih | N | Cα dihedral score in current disulfide |
pro_close | O | proline ring closure energy |
envsmooth | P | Statistically derived fullatom environment potential |
atom_pair_constraint | Q | Harmonic constraints between atoms involved in Watson-Crick base pairs specified by the user in the params file |
Implementation of the algorithm
- 1.
Conduct side chains based on the selection equation for each ant.
- 2.
Perform the local search on each odd-numbered iteration ant.
- 3.
Update global best ant s_{ gb }with iteration best ant s_{ ib } if E(s_{ ib }) is lower.
- 4.
Update the pheromone matrix T based on s_{ gb }.
- 5.
If the termination criterion is met, let's return to s_{ gb }, or repeat steps 1 to 5.
In this workflow, each colony terminates when one of the following criteria is met: the colony runs for a specified number of iterations; and there is no energy improvement during the last several iterations. Two important equations, the selection equation and the update pheromone matrix equation are explained below.
q_{0} tunes the bias between the two selection policies. A random probability q will be generated when a rotamer is needed. Once the rotamer is picked, ${r}_{j}^{*}$ is inserted into the protein backbone from the position of residue i.
Our SHMO scheme is simple with the help of OpenMP. The pheromone matrix is extracted from AC, and multiple colonies are run as parallel threads with private variables in each colony to co-evolve with the common pheromone matrix.
Rotamer minimization
Results
The principal idea behind pacoPacker is to make the parallel ant colonies share only one pheromone matrix, which can combine different energies to guide each ant in constructing protein side-chain conformations. We tested pacoPacker by making comparisons with two popular side-chain modelling programs, CIS-RR and SCWRL4. CIS-RR combines a novel clash-detection guided iterative search (CIS) algorithm with continuous torsion space optimisation of rotamers (RR) [26]. SCWRL4 is an improved version of SCWRL3 [25] which uses the new rotamer library, more efficient search algorithms and a soft Vander Waals potential plus hydrogen bonding based scoring function [15]. All these predictors are based on discrete rotamers.
Experimental settings
We performed all the tests on a computer cluster containing 20 nodes with 16-core 1.9 GHz AMD Opteron CPU per node under Linux 2.6.18 and GCC 4.1.2. CIS-RR and SCWRL4 were ran using their default settings to produce one prediction for each test instance. We ran pacoPacker, with eight ant colonies running in parallel, on the same test instances. As all these threads were synchronised to work out eight predictions and each is a nondeterministic approach, different numbers of decoys for each test instance were generated. The number of predictions for each test instance ranged from 2130 ([PDB:1CBN] 46 residues) to 4650 ([PDB:1B9O] 635 residues). We selected the highest accuracy rate of each test instance from pacoPacker to compare with CIS-RR and SCWRL4.
The benchmark instances were taken directly from other research, which contained 442 protein targets with lengths of 46 to 1184 amino acid residues [26, 15]. Because [PDB:2QOL] cannot be predicted by CIS-RR and [PDB:1G8Q] is considered as a missing main chain atom by Rosetta, we excluded them from this benchmark. A fair evaluation is a difficult task, so we used two criteria to assess the accuracy of side chain packing. One was defined as the percentage of correctly predicted ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{12}$ angles within thresholds of 40° and 20° compared with the native structures. The second criterion was the root mean square deviation (RMSD) of the side-chain heavy atoms [34]. Both evaluation methodologies are adapted from third-party software [26, 35], where they consider residues with symmetric terminal groups, or with a possibly flipped terminal group.
Protein chain based evaluation performance
Comparison of pacoPacker, CIS-RR and SCWRL4 in the 442 structure set.
Method | ${\mathcal{X}}_{1}\left(4{0}^{\circ}\right)$ | ${\mathcal{X}}_{1}\left(2{0}^{\circ}\right)$ | ${\mathcal{X}}_{12}\left(4{0}^{\circ}\right)$ | ${\mathcal{X}}_{12}\left(2{0}^{\circ}\right)$ | RMSD (Å) |
---|---|---|---|---|---|
SCWRL4 | 82.80% | 79.61% | 74.98% | 68.21% | 2.07 |
CIS-RR | 84.88% | 82.07% | 77.13% | 70.13% | 1.62 |
pacoPacker | 87.19% | 83.53% | 77.11% | 70.02% | 1.60 |
Individual residues based evaluation performance
Effects of rotamer minimisation
From the results presented in the previous two sections, we show the superiority of ${\mathcal{X}}_{1}$ while the performance of ${\mathcal{X}}_{2}$ is not strong. For example, when compare the number of red crosses on Figure 4(A) with Figure 4(B), pacoPacker has 342 best-performing proteins for ${\mathcal{X}}_{1}$, which is more than the 210 best-performing proteins for ${\mathcal{X}}_{12}$. In addition, Cys, Ser, Thr and Val only on wing ${\mathcal{X}}_{1}$, clearly dominate the area of ${\mathcal{X}}_{1}$. High quality ${\mathcal{X}}_{1}$ is significant for side-chain prediction, because it is a foundation of residue. On the other side, there is still room for improvement of ${\mathcal{X}}_{2}$, so we naturally optimised each rotamer as it was placed (rotamer minimization). An overview of how this method performs is given below.
Discussion
Under the inaccuracy/usefulness property hypothesis, SOP is not an ideal computational model for protein structure prediction [28]. This means that even if the corresponding SOP is completely solved, the SOP answer may not be correct, and in most cases it will not be perfect. PacoPacker proposes a novel hybrid parallel approach to repack protein side chains based on SHOP [28, 30].
Best conformations of pacoPacker distributed on different threads.
ID | Thread0 | Thread1 | Thread2 | Thread3 | Thread4 | Thread5 | Thread6 | Thread7 |
---|---|---|---|---|---|---|---|---|
Quantity | 29 | 31 | 35 | 29 | 33 | 39 | 107 | 139 |
Firstly, from the view of an individual colony, the pheromone matrix accumulates the search experience of ants, which describes which rotamer should be a priori considered as the choice for each residue. Such an experience bias is established by evaluating the conformations found by the previous generation of ants using the corresponding energy function. Then by sharing T , each colony can achieve different search experiences from other colonies asynchronously, and each colony is also directed by their own energy functions to co-evolve towards a better state. The process of sharing one T can accumulate the search experience of all parallel ant colonies and propagate the bias among them. As the pheromone matrix T provides an indeterministic bias for all the running colonies, it may be easier to find better solutions.
The proportion of proteins repacked by pacoPacker with lower RMSD compared with other predictors.
Sequence Length | Number | CW CIS-RR | CW SCWRL4 | CW both |
---|---|---|---|---|
> 500 | 53 | 28.3% | 30.2% | 13.2% |
500~ 400 | 34 | 41.2% | 38.2% | 20.6% |
400~ 300 | 62 | 56.5% | 62.9% | 41.9% |
300~ 200 | 108 | 59.3% | 66.7% | 51.9% |
200~ 100 | 139 | 63.3% | 67.6% | 51.1% |
< 100 | 46 | 76.1% | 76.1% | 63.0% |
Conclusions
In summary, pacoPacker makes each heuristic search work with its own energy function and they complement each other in a qualitative way. Different energy functions train search trajectories to obtain different search intelligences. Our parallel strategy diffuses the intelligence to all the parallel searches by SHOP, so that all ant colonies can share their accumulated hybridised intelligence. Such co-evolvement guided by multiple objective functions simultaneously has an impact on the nature folding procedure of native proteins [28]. The prediction accuracy of packing side chains was improved for most of the proteins, which proves that pacoPacker has feasibility and practical value, but at a cost of increased CPU time. However, an important reason for using pacoPacker is that it does not need training and tuning of the energy function parameters before the predictor can work.
Notes
Declarations
Acknowledgements
The authors acknowledge the support received from Rong Chen for helping with the analysis of the experiments and Caixia Wang for helping with the preparation of the paper. Funder had no role in study design, data collection and analysis, decision to publish, or preparation of the paper.
Declarations
This study was supported by a grant from the National Natural Science Foundation of China (No. 61170125).
This article has been published as part of BMC Bioinformatics Volume 15 Supplement 12, 2014: Selected articles from the IEEE International Conference on Bioinformatics and Biomedicine (BIBM 2013): Bioinformatics. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/15/S12.
Authors’ Affiliations
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