Conformational sampling and structure prediction of multiple interacting loops in soluble and β-barrel membrane proteins using multi-loop distance-guided chain-growth Monte Carlo method

Abstract

Motivation: Loops in proteins are often involved in biochemical functions. Their irregularity and flexibility make experimental structure determination and computational modeling challenging. Most current loop modeling methods focus on modeling single loops. In protein structure prediction, multiple loops often need to be modeled simultaneously. As interactions among loops in spatial proximity can be rather complex, sampling the conformations of multiple interacting loops is a challenging task.

Results: In this study, we report a new method called multi-loop Distance-guided Sequential chain-Growth Monte Carlo (M-DiSGro) for prediction of the conformations of multiple interacting loops in proteins. Our method achieves an average RMSD of 1.93 Å for lowest energy conformations of 36 pairs of interacting protein loops with the total length ranging from 12 to 24 residues. We further constructed a data set containing proteins with 2, 3 and 4 interacting loops. For the most challenging target proteins with four loops, the average RMSD of the lowest energy conformations is 2.35 Å. Our method is also tested for predicting multiple loops in β-barrel membrane proteins. For outer-membrane protein G, the lowest energy conformation has a RMSD of 2.62 Å for the three extracellular interacting loops with a total length of 34 residues (12, 12 and 10 residues in each loop).

Availability and implementation: The software is freely available at: tanto.bioe.uic.edu/m-DiSGro.

Contact: jinfeng@stat.fsu.edu or jliang@uic.edu

Supplementary information: Supplementary data are available at Bioinformatics online.

1 Introduction

Protein loops are key structural regions involved in recognition and binding of small molecules and proteins. In structure prediction, accurately predicted loop regions can provide valuable information for understanding the function and dynamics of the proteins. Prediction of loop structures, or loop modeling, has received considerable attention in the past (Fiser et al., 2000; Jacobson et al., 2004; Michalsky et al., 2003; Subramani and Floudas, 2012; van Vlijmen and Karplus, 1997; Zhang et al., 2007a; Zhao et al., 2011; Zhu et al., 2006;). Among these, database search methods can make accurate loop prediction if templates of loop fragments to a query protein are found at a high level of confidence (Choi and Deane, 2010; Michalsky et al., 2003). With the rapid expansion of the PDB, fragment-based approaches are very effective for modeling loop structures (Fernandez-Fuentes et al., 2006a; Fernandez-Fuentes and Fiser, 2006b; Fernandez-Fuentes et al., 2010). Nevertheless, fragment based loop modeling approaches are blind to the local environment of loops, and currently are not applicable for modeling multiple loops that are interacting with each other. In such cases, it is necessary to use template-free methods to predict loop structures. Recent advances have enabled template-free prediction of loops with high accuracy (Canutescu and Dunbrack, 2003; Coutsias et al., 2004; Fiser et al., 2000; Jacobson et al., 2004; Tang et al., 2014; Zhao et al., 2011; Zhu et al., 2006).

Most previous loop modeling studies have focused on predicting structure of a single loop. In reality, multiple loops often need to be modeled for a particular protein target, where loops in spatial proximity can interact in complex ways (Housset et al., 1991; Zaccardi et al., 2014). An example of interacting loops can be found in the bovine pancreatic trypsin inhibitor (BPTI, pdb 6pti), where loop A (residues 7–16) interacts with loop B (35–46). A single substitution in loop B (Y35G, pdb 8pti) results in substantial changes to the conformations of both loop A and loop B (Housset et al., 1991). In this case, the conformations of the two loops are stabilized by their interactions.

Interacting loops are also found in membrane proteins. For example, in the β-barrel assembly machinery A protein, the extracellular loops interact with one another, and form a dome over the top of the β-barrel domain (Noinaj et al., 2013). Given that the transmembrane strands of many β-barrel membrane proteins can be predicted (Naveed et al., 2012), accurate loop structure prediction will facilitate the prediction of full structures of this important class of membrane proteins.

Conventional methods for modeling interacting loops are to generate single loops sequentially one loop after another. However, these methods are prone to be trapped at local energy minima formed by intra-loop interactions, since it cannot effectively model the complex interactions among the loops in spatial proximity. To resolve this issue, several methods have been developed that can model two interacting loops (Danielson and Lill, 2010; Rosenbach and Rosenfeld, 1995). In Rosenbach and Rosenfeld (1995), a method for simultaneous prediction of the structure of multiple loops based on the bond-scaling-relaxation loop-closure algorithm was reported. For two loops of lengths of five to seven residues in spatial proximity, their results showed that more accurate predictions can be made than modeling individual loop sequentially. Danielson and Lill (2010) developed the CorLps method, in which an energetically feasible ensemble of individual loops are generated using the loopyMod method (Soto et al., 2008) disregarding the presence of other loops. Although successful in predicting a number of interacting loop pairs, this method does not work well when both loops have ≥9 residues. Methods have also been developed to model loops in certain protein classes (e.g. antibodies, Sellers et al., 2010), using information specific to these classes of proteins. To the best of our knowledge, none of these existing methods can effectively predict three or more interacting loops.

In this study, we describe the multi-loop Distance-guided chain-Growth Monte Carlo method (m-DiSGro) for simultaneously modeling of two or more interacting loops. Based on chain-growth Monte Carlo sampling, which has been applied to a number of studies of proteins (Liang et al., 2002; Lin et al., 2011; Tang et al., 2014; Zhang and Liu, 2006; Zhang et al., 2007b), m-DiSGro simultaneously constructs multiple loops, and does not require completeness of one loop before growing another loop. This sampling strategy randomizes the order of the residues being sampled to generate more diverse loop conformations, making it less likely to over-sample conformations in certain local energy minima. Consequently, inter-loop interactions are taken into account more effectively compared with sampling one loop at a time. In addition, loop growth is guided by empirical end-to-end distance functions and backbone dihedral angle distributions, allowing effective exploration of low-energy conformational space (Liu, 2008; Liu and Chen, 1998; Tang et al., 2014; Wong, 2013; Zhang et al., 2007a). To improve the sampling of long loops, we further introduce a strategy of regrowing loops using fragments.

This article is organized as follows. We first describe the m-DiSGro method in detail. We then present results for loop prediction of soluble proteins using two different test data sets, followed by results on predicting multiple interacting loops in β-barrel membrane proteins. We show that M-DiSGro has significant advantages in modeling native-like multi-loop compared with CorLps reported in Danielson and Lill (2010). The performance is further improved when loop fragments are used.

2 Methods

2.1 Multi-loops distance-guided chain-growth Monte Carlo (m-DiSGro)

Based on our previous DiSGro method for sampling single loops, our task in this study is to model n ≥ 2 loops in spatial proximity in protein structures (Tang et al., 2014). m-DiSGro simultaneously samples conformations of the n loops. At each step of the chain growth process, a residue i in a chosen loop p is added upon completion of the previous residues in all of the n loops. Here loop p is chosen randomly from the n loops, regardless of loop lengths. The newly added residue is represented by three consecutive backbone atoms during the growth process: C atom of residue i, N atom of residue i + 1, and CA atom of residue i + 1 (Fig. 1 ). The coordinates of the three atoms, C_i, $N_{i+1}$ and $C{A}_{i+1}$, are denoted as $x_{\text{C,i}},x_{\text{N},i+1}$ and $x_{\text{CA},i+1}$, respectively. Here $x_{\text{C,i}}$ and $x_{\text{N},i+1}$ are determined by sampling the dihedral angles $(\varphi ,\psi )$. We describe this sampling process in detail in Section ‘Sampling backbone atoms’. The ω dihedral angles that determine the coordinate of CA atoms are sampled from a normal distribution with a mean of 180° and SD of 4°. Side-chains are built upon completion of backbone placement of all loops. The generated conformations of the loops are scored and ranked by our atom-based distance-dependent empirical potential function. Our potential function is an empirical function following (Li et al., 2003; Miyazawa and Jernigan, 1996; Pokarowski et al., 2005; Zhang et al., 2005; Zhou and Zhou, 2002). Explicit water molecules are not considered, and there is no specific solvation energy calculation beyond what is implicit in the empirical potential function. Details of side-chain construction and the atomic potential function are described in Tang et al. (2014).

Fig. 1.

Schematic illustration of placing C_i and $N_{i+1}$ atoms. Atom C_i has to be on the circle $Q_C$. The position $x_{\text{C,i}}$ of the C_i atom of residue i is determined by $d_{{\text{C}}_{\text{i}}{\text{,C}}_{\text{t}}}$, which is based on known distance $d_{{\text{CA}}_{\text{i}}{\text{,C}}_{\text{t}}}$ and the conditional distribution of $\pi (d_{{\text{C}}_{\text{i}}{\text{,C}}_{\text{t}}}|d_{{\text{CA}}_{\text{i}}{\text{,C}}_{\text{t}}})$. Once $d_{{\text{C}}_{\text{i}}{\text{,C}}_{\text{t}}}$ is sampled, C_i can be placed on two positions with equal probabilities. Here $x_{\text{C,i}}$ is the selected position of C_i. $C{\prime }_{\text{i}}$ is placed at the position $x_{\text{C}\prime ,\text{i}}$ alternative to $x_{\text{C,i}}$. Similarly, the $N_{i+1}$ atom has to be on the circle $Q_N$ and its position $x_{\text{N},i+1}$ is determined by $d_{{\text{N}}_{i\text{+1}}{\text{,C}}_{\text{t}}}$ in a similar fashion

Unlike the CorLps method, which is based on ensembles of individual loops with complete structures (Danielson and Lill, 2010), m-DiSGro does not require completeness of one loop before growing another loop. It generates multiple loops by randomly selecting one residue to grow from all incomplete loops. All previously built residues in different loops immediately become part of the structural environment and contribute to the calculation of the coordinates of future atoms. When compared with methods based on sampling one whole loop at a time, our method can generate more diverse conformations of multiple loops, and can effectively avoid being trapped at some local energy minima.

2.2 Loop fragment libraries

During the chain growth process, a partial conformation may have to adopt a state with very high (or infinite) energy, making further growth fruitless. For example, it may be impossible to add any atoms due to lack of available space and loops may fail to close. m-DiSGro would have to terminate the whole chain-growth process upon such a failure. This is costly if failure occurs when a significant or the full portion of other loops have been built.

To increase the efficiency of sampling, we develop a fragment-based regrowth method to repair failed loops in m-DiSGro, as the near-native regions can be more efficiently explored by using fragments to generate an adequate number of native-like loop conformations. The overall procedure of m-DiSGro with fragments is outlined in Figure 2 .

Fig. 2.

The flowchart of m-DiSGro using fragments. In this example, three interacting loops (L1–L3) are to be modeled. Loop fragment libraries for them are constructed separately using the single loop method DiSGro (Tang et al., 2014). At each step of the chain growth process, a residue is added to a randomly chosen loop of L1-L3. This is repeated until all loops are completed. Here newly added residues are shown in red online and grey in print. In this example, a residue (red/grey circle) is added to L2 in step 1. Another residue (red/grey circle) is added to L3 in step 2. When a failure occurs (L3 in this example), a loop fragment is drawn from the corresponding library to replace the failed conformation. This process of adding residues is continued until the three missing loops are fully constructed

For a protein with n loops to be sampled, we first build n loop fragment libraries, one for each loop. Conformations in each library are sampled independently in the absence of the other n − 1 loops. All loop fragments are generated using the DiSGro single loop prediction method (Tang et al., 2014). Conformations of the individual loops generated are then ranked by the atom-based distance-dependent empirical potential function described in Tang et al. (2014). For each individual loop, the top-50 energetically favorable loop conformations are retained to form the loop fragment library specific to this loop.

During the chain-growth process, when a failure occurs in loop p which begins at residue s and ends at residue t with a length of ($t-s+1$), a randomly selected fragment f of length ($t-s-3$) from its loop fragment library, is used to replace the residues from s to $(t-4)$ of loop p. As we apply the Coutsias-Seok-Jacobson-Dill (CSJD) analytical closure method (Coutsias et al., 2004) at the residue $(t-2)$ to close the loop, the replacement of residues s to $(t-2)$ will lead to a high occurrence of highly similar loop conformations. In order to introduce the necessary diversity in loop conformations while at the same time achieving high sampling efficiency, we only replace the residues s to $(t-4)$ with residues from fragment f. The multi-loop chain-growth process is then continued until the rest of the residues of all loops, including residues $(t-3)$ to t of loop p, are fully built. This fragment strategy is only used for loops of length ≥5. For short loops of length ≤4, the success rate of completing loops without fragments is already sufficient.

2.3 Sampling backbone atoms

The procedure to sample backbone atoms is the same as that of the single loop method DiSGro (Tang et al., 2014). Below we give a brief description for completeness. Let C_t be the C-terminal anchor atom in the end residue t of a loop. We describe the sampling procedure for (C_i, O_i, $N_{i+1}$ and $C{A}_{i+1}$) atoms as an example (Fig. 1).

C_i is generated first, followed by $N_{i+1}$. Denote the distance $|x_{\text{C,t}}-x_{\text{CA,i}}|$ between $x_{\text{CA,i}}$ and $x_{\text{C,t}}$ as $d_{{\text{CA}}_{\text{i}}{\text{,C}}_{\text{t}}}$, and the distance $|x_{\text{C,i}}-x_{\text{C,t}}|$ between $x_{\text{C,i}}$ and $x_{\text{C,t}}$ as $d_{{\text{C}}_{\text{i}}{\text{,C}}_{\text{t}}}$. As the bond length $l_{{\text{CA}}_{\text{i}}{\text{,C}}_{\text{i}}}$, and the bond angle ${\theta }_{\text{C,i}}$ are fixed, C_i will be located on a circle $Q_C$ (Fig. 1):

$$\begin{array}{l}{Q}_C=\{x\in {\mathbb{R}}^3|\text{\hspace{0.17em}such\hspace{0.17em}that\hspace{0.17em}}||x-x_{\text{CA,i}}||=l_{{\text{CA}}_{\text{i}}{\text{,C}}_{\text{i}}},\\ \text{\hspace{0.17em}and\hspace{0.17em}}(x-x_{\text{CA,i}})·(x_{\text{CA,i}}-x_{\text{N,i}})=\cos {\theta }_{\text{C,i}}\}.\end{array}$$ (1)

Given a fixed $d_{{\text{C}}_{\text{i}}{\text{,C}}_{\text{t}}}$, C_i can be placed on two positions $x_{\text{C,i}}$ and $x_{\text{C}\prime ,\text{i}}$ on circle $Q_C$, $x_{\text{C,i}}$ and $x_{\text{C}\prime ,\text{i}}$ are labeled as C_i and $C{\prime }_{\text{i}}$, respectively. As the probability for placing C_i on either position is equal, we randomly select one position to place atom C_i.

Sampling from the empirical distributions of $d_{{\text{C}}_{\text{i}}{\text{,C}}_{\text{t}}}$ and mapping back to C_i should encourage the growth of loops to connect to the terminal C_t atom. Further analysis of the empirical distribution of $d_{{\text{C}}_{\text{i}}{\text{,C}}_{\text{t}}}$ given $d_{{\text{CA}}_{\text{i}}{\text{,C}}_{\text{t}}}$ shows that $d_{{\text{CA}}_{\text{i}}{\text{,C}}_{\text{t}}}$ can be very informative for sampling $d_{{\text{C}}_{\text{i}}{\text{,C}}_{\text{t}}}$ in some cases. This leads us to design a strategy of sampling $x_{{\text{C}}_{\text{i}}}$ based on the conditional distribution of $\pi (d_{{\text{C}}_{\text{i}}{\text{,C}}_{\text{t}}}|d_{{\text{CA}}_{\text{i}}{\text{,C}}_{\text{t}}})$. Atom $N_{i+1}$ is generated in a similar way as C_i. Details can be found in Tang et al. (2014).

The trial positions of (C_i, $N_{i+1},{\text{CA}}_{i+1}$) are then subject to a filtering procedure using an empirically derived backbone dihedral angle distribution. One filtered trial is selected according to its probability calculated using an atomic distance-dependent empirical potential function. The coordinate of O_i atom is determined by (N_i, CA_i, C_i). See Tang et al. (2014) for more details.

3 Results

3.1 Test sets

To assess the accuracy of m-DiSGro and facilitate direct comparison with the CorLps method reported in Danielson and Lill (2010), we use their test set, which we name as the CorLps Set. It is obtained from high-resolution (1.53 Å) structure of trypsin (pdb 1utk) (Leiros et al., 2004). It contains 36 pairs of interacting loops, including 7 pairs of 6–6-residue loops, 10 pairs of 6–9-residue loops, 3 pairs of 9–9-residue loops, 9 pairs of 6–12-residue loops, 5 pairs of 9–12-residue loops and 2 pairs of 12–12-residue loops.

We have also developed a new data set of 2-, 3- and 4-interacting loops, called MultiSet, to assess the effectiveness of m-DiSGro in more complex situations. Proteins in this set are taken from (Fiser et al. (2000) and Soto et al. (2008). In total, MultiSet contains twenty 2-interacting loops, eleven 3-interacting loops and eight 4-interacting loops (see Table 2). We use a strict criterion to select interacting loops: all loops must be spatially close and interact with each other. For example, each of the four loops has to be interacting with the other three loops in a four-loop target. We use the edge simplices from the alpha-shape computed from the protein structures to detect interactions among neighboring loops (Edelsbrunner and Mücke, 1994; Li et al., 2003; Liang et al., 1998). Using alpha-shape eliminates spurious neighbor interactions that are not in physical contact (Ouyang and Liang, 2008; Zhang et al., 2005). Here we use the solvent radius of 0.5 Å following (Singh and Thornton, 1993). In MultiSet, loops have lengths ranging from 4 to 16. To construct a challenging data set of two interacting loops, both loops have to be at least 10-residue long. For three- and four-loop target proteins, only those with at least one long loop (length ≥ 10) are included. Multiple interacting loops are present across different protein families. Based on the classification scheme of loops of the Archdb (Bonet et al., 2013), 42.9% of long loops (≥10) and 48.6% of loops in the MultiSet have already been classified, suggesting modeling multiple loops will be applicable to a large number of proteins.

Table 2.

The average RMSD of the lowest energy conformations of loops in the MultiSet

PDB	Loop 1	Loop 2	Loop 3	Loop 4	REmin (Å)
					w/o frag	w/frag
Four loops
1aoz	106–115	221–225	252–259	507–510	1.35	0.93
1frd	34–49	56–67	74–86	90–94	3.61	4.29
1hnj	109–113	141–150	156–159	273–278	1.52	1.11
1ixh	85–96	106–113	120–130	186–190	2.97	2.89
1nwp	36–48	65–79	84–90	112–121	3.53	2.77
1oth	87–90	141–145	155–168	263–273	4.43	2.75
2dri	8–14	39–42	64–69	89–98	1.38	1.09
2mnr	216–228	239–251	262–275	291–299	3.23	2.96
Average					2.75	2.35
Three loops
1bhe	194–203	216–225	248–252		2.09	1.39
1cgt	136–140	192–204	229–235		2.02	1.03
1ctt	63–73	89–102	251–256		2.99	3.23
1ddt	23–27	65–78	167–174		2.96	2.10
1ed8	198–202	233–239	245–255		1.30	2.26
1el5	79–88	230–242	257–263		2.96	3.16
1nln	44–53	70–75	109–119		2.80	2.47
1nls	11–23	98–102	201–208		2.14	1.94
1oyc	117–123	192–196	245–259		2.50	2.50
1p1m	105–108	277–288	364–369		2.25	2.18
1php	192–200	217–220	317–327		1.52	1.21
Average					2.32	2.13
Two loops
1ads	38–50	256–265			1.39	1.14
1art	159–169	190–201			3.55	5.18
1cb0	60–73	190–199			3.58	2.35
1ctm	77–90	133–144			3.85	2.31
1lcf	100–112	232–247			3.09	2.63
1ms9	306–316	333–342			4.00	3.45
1nar	87–97	192–201			1.93	1.61
1nfp	49–64	89–99			3.60	3.45
1nhq	32–43	50–62			5.73	2.86
1nox	42–51	96–105			1.83	1.63
1ojq	88–99	141–151			2.32	2.17
1pgs	51–61	83–95			3.41	3.35
1rcf	36–48	72–81			2.06	2.24
1srp	41–55	138–150			4.66	3.64
1thg	96–106	163–174			2.26	1.72
2exo	127–139	203–213			3.75	2.96
2olb	77–89	176–190			4.95	3.04
2pia	30–42	99–112			3.36	1.89
4enl	138–147	391–402			1.85	1.43
8acn	444–458	635–644			3.94	3.23
Average					3.26	2.61

R_Emin by m-DiSGro and m-DiSGro with fragment are listed.

Loops are designated by the beginning and the end residue positions.

The test set of loops in β-barrel membrane proteins contains four proteins satisfying the following criteria: (i) no in-plug structure is present; (ii) no incorrect loop structures are included; and (iii) loops are of length 2–17 residues. Our test set contains both extracellular and periplasmic loops, which are modeled separately. This test set is called β-Barrel Set (see Table 3).

Table 3.

The RMSD of the lowest energy conformations, R_Emin by m-DiSGro with fragment using β-Barrel Set

Location	PDB	Loop 1	Loop 2	Loop 3	Loop 4	m-DiSGro
Extra-cellular loops	1p4t	18–23	51–60	95–105	136–139	1.54
	1qj8	14–21	51–57	94–98	132–135	1.26
	1thq	75–81	112–122	143–152		2.08
	2 × 9 k	97–106	138–149	177–188		2.62
	Average					1.88
Peri-plasmic loops	1p4t	37–40	71–78	119–122		0.92
	1qj8	31–37	71–77	115–121		1.71
	1thq	60–66	93–101	133–136		1.83
	2 × 9 k	80–83	123–126	162–165	196–204	0.93
	Average					1.35

3.2 Multi-loop structure prediction on CorLps Set

We first test m-DiSGro using the CorLps Set. The sampled loop conformations are ranked based on their energy values calculated using our atom-based distance-dependent empirical potential function (Tang et al., 2014) instead of the DFIRE empirical potential function as in Danielson and Lill (2010). For each loop pair, the RMSD of the lowest energy conformation to the native structure R_Emin among 10 000 trial conformations is reported. Our results are all reported as global backbone RMSD, calculated using the N, CA, C and O atoms of the backbone. The conformation of a loop is obtained by superimposing its two ends on the flanking secondary structures, while the deviation is calculated using the backbone atoms of the loop structure.

From the results summarized in Table 1, we find that m-DiSGro performs significantly better than CorLps. Compared with CorLps, m-DiSGro has a R_Emin of 1.14 versus 2.35 Å for 6–6-residue loop pairs, 2.00 versus 3.39 Å for 6–9-residue loop pairs, 2.14 versus 4.38 Å for 9–9-residue loop pairs, 2.67 versus 4.43 Å for 6–12-residue loop pairs, 2.94 versus 5.47 Å for 9–12-residue loop pairs and 3.32 versus 6.97 Å for 12–12-residue loop pairs, respectively. Overall, the value of R_Emin is reduced significantly using m-DiSGro compared with CorLps.

Table 1.

Comparison of R_Emin of loop conformations generated by CorLps, m-DiSGro, and m-DiSGro with fragment using the CorLps Set

Lengths of loop pairs	REmin (Å)
	CorLps	s-DiSGro	m-DiSGro	m-DiSGro + frag
6–6 (7)	2.35	1.61	1.14	1.19
6–9 (10)	3.39	2.37	2.00	1.57
9–9 (3)	4.38	3.47	2.14	1.64
6–12 (9)	4.43	4.01	2.67	2.53
9–12 (5)	5.47	4.55	2.94	2.41
12–12 (2)	6.97	4.65	3.32	2.87
all (36)	4.02	3.15	2.22	1.93

R_Emin, average RMSD of the lowest energy conformations compared with native loop conformation resolved experimentally. Lengths of Loop Pairs (X-Y): the pairs of interacting loops with lengths X and Y. The number of loop pairs is listed in parentheses.

When fragments are used, m-DiSGro further improves R_Emin to 1.57 from 2.00 Å for 6–9-residue loop pairs, 1.64 from 2.14 Å for 9–9-residue loop pairs, 2.53 from 2.67 Å for 6–12-residue loop pairs, 2.41 from 2.94 Å for 9–12-residue loop pairs and 2.87 from 3.32 Å for 12–12-residue loop pairs, respectively. Only for 6–6-residue loop pairs, m-DiSGro with fragment has a slightly larger R_Emin of 1.19 Å. These results indicate that using loop fragment library is effective in improving the modeling accuracy when loops are nine residues or longer.

We also report results using a single independent loop modeling method s-DiSGro for comparison. A maximum number of 10  000 loop combinations are generated by initially combining the top-100 ranked single loop conformations of each loop region, similar to CorLps, except single loop conformations are generated here by the DiSGro method (Tang et al., 2014) instead of the loopyMod (Soto et al., 2008) used in CorLps. s-DiSGro performs much worse in modeling multiple interacting loops compared with m-DiSGro, with or without fragments (3.15 versus 1.93/2.22 Å), but shows improved accuracy compared with CorLps method (3.15 versus 4.02 Å).

The average R_Emin of 36 interacting loop pairs are 1.93 Å (m-disgro with fragment), 2.22 Å (m-DiSGro without fragment), 3.15 Å (s-DiSGro) and 4.02 Å (CorLps). Overall, m-DiSGro with fragment shows significantly improved accuracy in modeling multiple interacting loops.

m-DiSGro is also much faster than CorLps. The reported computational time of CorLps is about 900 cpu minutes for two 12-residue interacting loops on a single core of a Intel Xeon 2.66 GHz quad-core machine (Danielson and Lill, 2010). The computation cost for m-DiSGro without fragment is only 14 cpu minutes on a single 2 GHz AMD Opteron processor. m-DiSGro with fragment takes a slightly longer time of 17 min, as loop fragment libraries specific to the loops need to be constructed on the fly.

3.3 Multi-loop structure prediction on MultiSet

We then test m-DiSGro on the MultiSet. Results are summarized in Table 2.

For the eight proteins with four-interacting loops, the average R_Emin by m-DiSGro with and without fragment are 2.35 and 2.75 Å, respectively. m-DiSGro with fragment is close to the accuracy level of sub-angstrom (1.04 Å) for three target proteins containing only one loop with more than 10 residues. The remaining five target proteins are challenging, as they have at least two long loops with length ≥10. For these five target proteins, m-DiSGro with fragment has an R_Emin of 3.13 Å. Of these five, three of them have three long loops. For example, the structure of mandelate racemase (pdb 2mnr) has four loops of length 13, 13, 14 and 9, respectively. The R_Emin by m-disgro with fragment is 2.96 Å. Another example is azurin (pdb 1nwp), which has four loops of length 7, 10, 13 and 15, respectively. It is challenging to model a 15-residue loop in an inexact environment when three other interacting loops also need to be modeled as well. The total length of these loops is 45 residues. As can be seen from Figure 3 a, m-DiSGro with fragment achieves good accuracy for 1nwp, with an R_Emin of 2.77 Å.

Fig. 3.

Examples of prediction results of interacting loops of soluble proteins and β-barrel membrane proteins using m-DiSGro with fragment. The lowest energy modeled loops by m-DiSGro with fragment is in red/dark. The native loops are in white/light. (a) Four interacting loops of azurin (pdb 1nwp, residues 36–48, 65–79, 84–90 and 112–121). The total length of the four loops is 45. The R_Emin is 2.77 Å. (b) Three extracellular loops of OmpG protein (pdb 2 × 9 k, residues 97–106, 138–149 and 177–188). The total length of the three loops is 34. The R_Emin of this three-loop target is 2.62 Å. There exists a short α-helical secondary structure element of four residues which is correctly predicted

For the eleven 3-loop target proteins, R_Emin by m-DiSGro with and without fragment are 2.13 and 2.32 Å, respectively. For the seven 3-loop target proteins with only one loop longer than 10 residues, m-DiSGro with fragment has an average R_Emin of 1.86 Å. The R_Emin of the rest of the three-loop target proteins with at least two long loops are 2.52 Å by m-DiSGro with fragment.

For the twenty 2-loop target proteins, R_Emin values are 2.61 and 3.26 Å with and without using fragments, respectively. This two-loop data set is challenging, as both loops have lengths ≥10-residue. Among the three- and four-loop target proteins, short loops can be modeled fairly easily with high accuracy, which contribute to the overall improved RMSD. In thirteen of the twenty 2-loops target proteins, m-DiSGro with fragment achieves good accuracy of R_Emin ≤ 3 Å, with six of them of RMSD ≤ 2 Å. Even when there is a loop of length ≥ 15, m-DiSGro with fragment also gives good R_Emin. An example is lactoferrin (pdb 1lcf), which has a 16- and a 13-residue loops. The RMSD of the lowest energy conformation is 2.63 Å.

Overall, m-DiSGro with fragment achieves good accuracy in modeling multi-loops in the MultiSet. It also achieves an improved accuracy compared with m-DiSGro without fragment, with an average R_Emin of 2.35 versus 2.75 Å for eight 4-loop, 2.13 versus 2.32 Å for eleven 3-loop and 2.61 versus 3.26 Å for twenty 2-loop target proteins.

3.4 Multi-loop structure prediction of β-Barrel membrane proteins

We also assess m-DiSGro in predicting loop conformations of β-barrel membrane proteins. Results are summarized in Table 3. For extracellular loops, the average RMSD of the lowest energy loop conformations R_Emin of the four proteins by m-DiSGro with fragment is 1.88 Å. Among these, 2 of the 4 extracellular loops in the Neisserial surface protein A, NspA (pdb 1p4t) are longer than 10 residues (Vandeputte-Rutten et al., 2003). The predicted extracellular loop structures have an R_Emin of 1.54 Å. The outer membrane protein G, OmpG (pdb 2 × 9 k) has three long extracellular loops of length 10, 12 and 12, respectively. The R_Emin of this three-loop target protein is 2.62 Å. Noticeably, there is a short helix in the middle of the 12-residue (138–149) loop. m-DiSGro with fragment successfully predicts this short helical secondary structural element without any explicit additional secondary structure information (Fig. 3b).

The periplasmic loops are usually short turns connecting β-strands. Most of them have lengths <10-residue. Prediction of these periplasmic loops is not as challenging as prediction of extracellular loops. The average RMSD R_Emin of the lowest energy periplasmic loop conformations of the four proteins by m-DiSGro with fragment is 1.35 Å. For the four interacting loops at the periplasmic end of OmpG, R_Emin is only 0.93 Å.

4 Conclusion and Discussion

In this study, we present a multi-loop chain-growth Monte Carlo method (m-DiSGro) for modeling interacting loops. Our method samples multiple loops by growing one residue at a randomly selected loop in each step. The calculation of the positions of newly added atoms is determined by the coordinates of previous placed atoms in all loops. This method is capable of exploring large conformational space of multi-loop effectively, without frequently being trapped in narrow or dead space. With further incorporation of loop fragment libraries, even failed loops can be efficiently regrown. m-DiSGro has significant advantages in predicting multiple interacting loops over previous methods, such as the CorLps method (Danielson and Lill, 2010), as interacting information among loops are taken into account during the simultaneous chain-growth process.

m-DiSGro can model multiple interacting loops simultaneously and efficiently, especially for those target proteins with long loops. For a set of 36 interacting loop pairs, m-DiSGro with and without fragment have better average R_Emin of 1.93 and 2.22 Å, compared with CorLps of 4.02 Å. For 12–12-residue loop pairs, the R_Emin by m-DiSGro with fragment is 2.87 Å, and the average computing time is only 17 min. Furthermore, our method is the first that considers three- and four-interacting loops explicitly. It successfully predicts these loop structures, with an R_Emin of 2.13 Å for three-loop target proteins, and 2.35 Å for four-loop target proteins. Preliminary results on building loops in β-barrel membrane proteins suggest that our method can be extended to model multiple loops of other complex membrane proteins.

Our study shows that by randomizing the order of residues to be sampled, we can generate more diverse sets of loop conformations, which helps to increase the overall sampling effectiveness. However, it is possible that there exist specific orders by which loop residues are to be sampled that are especially effective for certain proteins. For example, residues in more constrained environment may need to be sampled first. One approach towards this is to look-ahead and further sample a number of states that can be grown into in the next step for each loop, and then select one state to place the next residue according to either energy or other specific bias criterion. Those sampled but unused states can be reused in future steps. This approach has been successfully applied to side chain conformations of proteins (Zhang and Liu, 2006) and reaction network sampling (Cao and Liang, 2013).

There are several directions for further improvement. The single loop method DiSGro is an important component of m-DiSGro. DiSGro is also the limiting step for the accuracy of modeled multiple interacting loops, especially when there are long loops. We envision that this can be improved by sampling dipeptide segment instead of sampling individual residue currently implemented in DiSGro. It will likely lead to improved sampling efficiency further and enable longer loops to be modeled (Zhao et al., 2011). Our work is also related to modeling loops in a flexible environment (Sellers et al., 2008; Subramani and Floudas, 2012), as other loops can be regarded as the flexible environment of the loop currently being modeled. Furthermore, the atom-based distance-dependent empirical potential function taken from Ref (Tang et al., 2014), can be improved by using nonlinear kernel for training (Hu et al., 2004), or by optimization using rapid iterations through a physical convergence function (Thomas and Dill, 1996).

Funding

This work was supported by National Institutes of Health (GM079804 and 1R21GM101552), NSF (DBI 1062328 and MCB-1415589) and the Chicago Biomedical Consortium with support from the Searle Funds at The Chicago Community Trust.

Conflict of Interest: none declared.