Skip to main content
NCBI home page
NIHPA Author Manuscripts logoLink to NIHPA Author Manuscripts
. Author manuscript; available in PMC: 2012 Sep 7.
Published in final edited form as: Structure. 2011 Sep 7;19(9):1211–1218. doi: 10.1016/j.str.2011.07.017

Symmetry-restrained flexible fitting for symmetric EM maps

Kwok-Yan Chan 1,2, James Gumbart 1,2,*, Ryan McGreevy 2, Jean M Watermeyer 3, B Trevor Sewell 3, Klaus Schulten 1,2,**
PMCID: PMC3412758  NIHMSID: NIHMS317172  PMID: 21893283

Summary

Many large biological macromolecules have inherent structural symmetry, being composed of a few distinct subunits, repeated in a symmetric array. These complexes are often not amenable to traditional high-resolution structural determination methods, but can be imaged in functionally relevant states using cryo-electron microscopy (cryo-EM). A number of methods for fitting atomic-scale structures into cryo-EM maps have been developed, including the molecular dynamics flexible fitting (MDFF) method. However, quality and resolution of the cryo-EM map are the major determinants of a method’s success. In order to incorporate knowledge of structural symmetry into the fitting procedure, we developed the symmetry-restrained MDFF method. The new method adds to the cryo-EM map-derived potential further restraints on the allowed conformations of a complex during fitting, thereby improving the quality of the resultant structure. The benefit of using symmetry-based restraints during fitting, particularly for medium to low-resolution data, is demonstrated for three different systems.

Introduction

Structural information is essential for understanding function and mechanism of biological systems. The state of the art in macromolecular structure determination permits the structural characterization of such molecules at multiple resolutions. X-ray crystallography, the predominant high-resolution technique, provides structural data at the atomic scale, but often requires non-physiological conditions to achieve crystallization. Furthermore, certain macromolecular assemblies, such as the ribosome or those with quaternary helical structure, are difficult to crystallize or cannot be crystallized at all. On the other hand, cryo-electron microscopy (cryo-EM) can capture images of molecules in functional states and for large systems, but typically at lower resolution (3.3–30 Å) than X-ray crystallography (Agirrezabala et al., 2008; Frauenfeld et al., 2011; Qian et al., 2005; Seidelt et al., 2009; Zhang et al., 2010b).

The resolution gap between X-ray crystallography and cryo-EM can be bridged by using hybrid computational methods to combine the two sources of experimental data. An early approach is the so-called rigid-body docking, which treats X-ray structures as rigid and searches for an orientation that best fits the cryo-EM map (Wriggers and Chacón, 2001). Even though this approach provides an excellent first approximation to the fit, it lacks the flexibility needed to deal with molecules that exhibit relative motions between individual parts in different functional states. Hence, a new class of flexible fitting methods has been developed, in which the X-ray structures or homology models have more degrees of freedom to deform and fit the cryo-EM map. An early attempt at flexible fitting involved dividing the molecule into different parts and then docking each rigid part into the density independently (Volkmann et al., 2000). Another early approach utilized real-space refinement, a technique developed for X-ray crystallography, to fit the structure into the map (Chapman, 1995; Chen et al., 2001). Other means of flexible fitting include (1) matching reduced-complexity representations of the structure and the density map to deform the structure (Wriggers et al., 1999), (2) altering the structure along the low frequency normal modes to increase the correspondence to the density map (Suhre et al., 2006-Tama et al., 2004b), (3) fitting comparative models based on different sequence-structure alignments and loop conformations of different components (Topf et al., 2006), (4) using a deformable elastic network and restraints from EM data to morph the structure (Schröder et al., 2007), and (5) refining segments of protein homology models locally within EM data before global refinement of the whole protein (Zhu et al., 2010). A recently developed flexible fitting method, molecular dynamics flexible fitting (MDFF), employs molecular dynamics (MD) simulations to perform flexible fitting by incorporating the EM data as an external potential in conventional MD simulations (Trabuco et al., 2008).

In any of these flexible fitting methods, the quality of the final fitted structure is strongly related to the resolution of the map and the quality of the starting model. A lower resolution map does not represent as much information as a higher resolution map and, hence, allows more degrees of freedom to participate a fitting procedure. Moreover, the accuracy of a structure fitted to a low-resolution map maybe limited by structural variability in the dataset, which reduces the useful structural information one can extract. One way to improve the quality and accuracy of structures fitted into EM data, especially in case of low-resolution maps, is to incorporate additional knowledge about the structure in the fitting procedure. For example, many biological systems are symmetric in nature, such as poliovirus exhibiting icosahedral symmetry (Hogle et al., 1985) and potassium channels exhibiting four-fold symmetry (Doyle et al., 1998). Point group symmetry and helical symmetry also occur frequently in biological systems. This symmetry is functionally important, e.g., for increasing structural stability (André et al., 2008; Hoang et al., 2004; Wolynes, 1996), for cooperative or allosteric functions (Blundell and Srinivasan, 1996; Goodsell and Olson, 2000), and for folding (Frauenfelder et al., 1991; Nymeyer et al., 1998). Indeed, symmetry information of biomolecules has already been incorporated in structure prediction tools like Rosetta (André et al., 2007). Protocols to impose restraints enforcing symmetry during MD simulations have also been developed and have been used in the refinement of homology models of potassium channels (Anishkin et al., 2010). Symmetry information can be extracted even from low-resolution EM data, and symmetric averaging is commonly used to improve the signal-to-noise ratio of a dataset. By integrating the same symmetry information into flexible fitting protocols to ensure a symmetric fitted structure, one can effectively reduce the number of available degrees of freedom in the fitting problem and, hence, improve the quality of fitted models. A similar idea has been employed in normal mode-based fitting of virus capsids in which only modes obeying the icosahedral symmetry were retained (Tama et al., 2004b).

We present here a method to incorporate the symmetry information extracted from EM data into MDFF. The symmetry relationship among subunits is converted into harmonic restraints, which are then applied to maintain the structure’s symmetry during MDFF simulations. The benefit derived from symmetry restraints is most apparent for low-resolution EM data as high-resolution data already contains sufficient information to maintain the symmetry during MDFF simulations. We applied the symmetry restraints to three different systems for which experimental EM data are available, namely the GroEL-GroES complex, a nitrilase from Rhodococcus rhodochrous J1, and a chaperonin from the archaeon Methanococcus maripaludis (Mm-cpn). These test cases represent data having three different symmetries and a range of resolutions, and serve to illustrate the benefits of applying symmetry restraints during MDFF.

Methods

Molecular dynamics flexible fitting (MDFF)

The MDFF method is based on molecular dynamics simulation (Trabuco et al., 2008) and has already been used successfully in numerous applications (Trabuco et al., 2011). Starting from an initial atomic structure, usually an X-ray crystal structure, an MD simulation is performed. In this simulation, two additional energy terms are added to the standard MD force field potential UMD. The first term, UEM, is derived from the EM density, and is used to apply forces proportional to the gradient of the density map to the atoms. This drives the model towards high density regions of the map and, hence, effectively guides the structure to the conformational state represented by the EM density. The stereochemical correctness of the structure is ensured by the standard MD force field. In addition, a second term, USS, which preserves the secondary structure of the system by imposing harmonic restraints on specific dihedrals, is included to avoid overfitting, i.e., to avoid unphysical distortions caused by the UEM term when a structure is too closely fitted to the EM map. A practical guide for carrying out MDFF simulations has been published previously (Trabuco et al., 2009). Since MDFF is based on MD simulations, thermal fluctuations in the fitting simulations can lead to multiple solutions that fit a map. Hence, different symmetric units can arrive at different conformations even for a symmetric map.

Symmetry restraints

The symmetry information of the cryo-EM data can be incorporated into an MDFF simulation through an additional potential to maintain a symmetric structure during the simulations. Since it is a common practice to symmetrize EM data for symmetric molecules during the 3D EM reconstruction process (Egelman, 2000; Ludtke et al., 1999; van Heel et al., 1996), the type of symmetry and its parameters, e.g., helical rise and twist for a helically symmetric system, can be obtained from this process.

A new potential energy term USR is built according to the symmetry information as follows. Let Ri(t) be the set of coordinates for atoms of the i-th symmetric unit at time t during the simulation, where Ri(t) for different units are related by a predefined symmetry. The symmetry information is used to generate, for each unit i, a transformation U↔i, which superimposes the coordinates Ri of unit i onto the first unit. Averaging over these superimposed symmetric units, the coordinates of an average structure, Ravg(t)=〈Ui↔Ri(t)〉, can be calculated. Using the inverse transformation Ui−1↔ for each unit, the average structure is transformed backwards from the first unit to the respective i-th unit; the resulting set of backward-transformed coordinates, Ri′(t)=Ui−1↔Ravg(t)=Ui−1↔<Ui↔Ri(t)> are now perfectly symmetric. The deviation between Ri′(t) and Ri(t) is measured by their root mean square distance (RMSDi), defined by

RMSDi(t)=<∣Ri(t)−Ri′(t)∣2>=<∣Ri(t)−Ui−1↔<Ui↔Ri(t)>∣2> (1)

where the average is taken over a chosen subset of atoms in the i-th unit. A high RMSDi indicates a large deviation from the ideal symmetric structure.

To minimize RMSDi between each unit and its corresponding unit in the average structure, the potential energy term USR mentioned above is added, given by

USR=12k(t)∑i[RMSDi(t)]2 (2)

where the sum is taken over all units. This potential term defines forces applied to the chosen subset of atoms in order to minimize RMSDi(t) during the simulations and, hence, guide the system towards a symmetric structure. The factor k(t) with unit energy/length2 controls the strength of the applied forces and can either be set to a constant or be linearly increased over time. A k(t), linearly increasing up to a finite value, is recommended so that during the early steps of fitting, the structure has sufficient flexibility to explore the conformational space represented by the EM data before converging to a symmetric structure as the symmetry restraint forces increase. The subset of atoms within a unit on which the RMSD calculation is based and to which forces are applied is typically chosen to be the Cα atoms for proteins instead of all atoms, so that the structure will not be over-restrained.

Simulated systems and molecular dynamics protocols

All systems utilized in the example applications were first rigid-body docked into the EM map using Colores from the Situs package (Wriggers et al., 1999), and then solvated in a box of TIP3P (Jorgensen et al., 1983) water molecules, using 10 Å padding in all directions. Counter ions Na+ and Cl− were added to neutralize the systems. The simulations were performed with a development version of NAMD 2.8 (Phillips et al., 2005) using the CHARMM27 force field with CMAP corrections (MacKerell et al., 1998; MacKerell Jr. et al., 2004). Only water and ions were allowed to equilibrate for the first 500 ps by constraining the protein with harmonic restraints, followed by equilibration of side chains as the protein backbone remained constrained. Next, MDFF simulations were performed for 5 ns in each case, coupling only non-hydrogen atoms of the protein to the UEM potential with a grid scaling of 0.3, which controls the balance between UMD and UEM (Trabuco et al., 2008). In addition to secondary structure restraints, restraints were used to maintain the correct chirality at all chiral centers and to keep peptide bonds in the trans-configuration. Finally 3000 steps of energy minimization, in the presence of UEM with grid scaling of 10, were performed to increase the stability of the resulting structure by removing the thermal deviations in the systems. For simulations using symmetry restraints, the force constant k was increased linearly from 0 to 10 (kcal/mol)/Å2 during the 5-ns MDFF simulations and forces were applied only to Cα atoms. All simulations were carried out in the NVT ensemble, using the following parameters: constant temperature at 300 K was maintained using Langevin dynamics with a damping constant of 5 ps−1; long-range electrostatic forces were computed using the particle-mesh Ewald summation method with a grid spacing of 1 Å; the RESPA multiple-time-stepping algorithm (Grubmüller et al., 1991; Tuckerman et al., 1992) was employed with an integration time step of 1 fs, short-range forces evaluated every 2 time steps, and long-range electrostatics evaluated every 4 time steps.

Results

We applied MDFF with and without symmetry restraints to three exemplary applications, namely GroEL-GroES complexes, nitrilase from R. rhodochrous J1 and Mm-cpn (see Figure 1). For all three cases, we calculated along the fitting trajectory the average RMSD between symmetric 1 units, defined by <RMSDi(t)>=1N∑iRMSDi(t), where N is the number of symmetric units. As shown in Figures 2A,B and D, in all three examples, the average RMSD between symmetric units for fitting with symmetry restraints applied is lower than that for fitting without symmetry restraints. The lower RMSD demonstrates that the symmetry restraints work as intended.

Figure 1.

Figure 1

Open in a new tab

Symmetries of the example applications, showing side and end-on views of each. (A, B) Seven-fold symmetric GroEL-GroES complex (C7 symmetry). (C, D) Helically symmetric nitrilase of R. rhodochrous J1 (D1S4.9 symmetry). (E, F) Sixteen-fold symmetric units of Mm-cpn (D8 symmetry).

Figure 2.

Figure 2

Open in a new tab

Average Cα RMSD values along the fitting trajectories of (A) GroEL-GroES complex, (B) Nitrilase of R. rhodochrous J1, (C) Mm-cpn closed state (4.3 Å) and (D) Mm-cpn open state (8 Å). The average RMSD value is defined through <RMSDi(t)>=1N∑iRMSDi(t) where N is the number of symmetric units. Red and black curves in (A)-(D) correspond to fitting with and without symmetry restraints, respectively. In (D), solid lines represent fitting using a homology model, while dashed lines represent fitting using a crystal structure with (blue) and without (green) symmetry restraints.

Example 1: GroEL-GroES complexes

The first example involves MDFF for the GroEL-GroES complex. The Escherichia coli chaperonin GroEL, together with the lid-like co-chaperonin GroES, form a molecular machine that assists the folding of many proteins with the help of ATP binding and hydrolysis. Upon ATP hydrolysis, the GroEL-GroES complex undergoes conformational changes necessary to carry out its function of mediating protein folding (Roseman et al., 1996). Crystal structures of an analogue of the ATP-bound state (Chaudhry et al., 2003) and a post-hydrolysis ADP-bound state (Xu et al., 1997) are available, but appear to be identical, failing to resolve the structural and functional differences between the two states. In contrast, both states have been captured in cryo-EM maps that have successfully resolved potentially important structural differences between them (Ranson et al., 2006), suggesting that crystal packing may have favored non-physiological conformations in the published crystal structures.

GroEL comprises two back-to-back seven-membered rings and, together with the co-chaperonin GroES, forms a complex with a seven-fold rotational symmetry, as illustrated in Figures 1A and B. We applied MDFF to fit an ADP-bound structure (PDB code 2C7D), which was modelled from cryo-EM data (Ranson et al., 2006), into the EM map of an ATP-bound complex, both with and without symmetry restraints. The target cryo-EM data used here is an intermediate resolution (7.7 Å) map of the complex in an ATP-bound state (Ranson et al., 2006), obtained from the EM data bank (EMD-1180).

To validate the structure fitted with symmetry restraints, a comparison was made to an available structure (PDB code 2C7C) modelled from the same map used here. The RMSD of the full GroEL-GroES complex, compared to the published structure, decreases from 2.21 Å to 1.97 Å (fitted without symmetry restraints) and 1.93 Å (fitted with symmetry restraints) during the fitting. From Figure 3 it can be seen that GroES (RMSD 1.73 Å, Figure 3A), the GroEL cis-ring (RMSD 1.15 Å, Figure 3B) and the GroEL trans-ring (RMSD 2.51 Å, Figure 3C) of the fitted structure are all very similar to the EM-modelled structure. Furthermore, the chief observable structural difference between the published ATP- and ADP-bound structures modelled from EM data, namely a shorter inter-strand distance between β-sheets in adjacent subunits of the trans-ring in the ATP-bound state (Ranson et al., 2006), is also captured in our fitted structure. However, the separation between the β-sheets observed in our fitted structure of the ATP-bound state is wider. The wider inter-strand distance agrees better with the map than the published structure, as shown in Figure 3D. Such a difference can be attributed to the fact that the trans-ring of the published structure was obtained by rigid-body docking of three domains to the map, which failed to capture intra-domain motions.

Figure 3.

Figure 3

Open in a new tab

Comparison of symmetry-restrained fitted structures (blue) to known structures (red) of the corresponding proteins. (A) GroES (RMSD 1.73 Å). (B) GroEL cis-ring (RMSD 1.15 Å). (C) GroEL trans-ring (RMSD 2.51 Å). (D) β-sheet region (residues 37–51 of the GroEL trans-ring subunit) of the symmetry-restrained fitted structure (blue) and the published structure (red) in the ATP-bound state inside the EM map. (E,F) Apical domain of the Mm-cpn lidless open state. Both a homology model (E, RMSD 5.31 Å) and a crystal structure (F, RMSD 2.34 Å) of a closed state were used for fitting.

After fitting, free equilibration of both the symmetry-restrained and unrestrained fitted structures was carried out. From the 10-ns trajectory of both simulations, the backbone RMSD of the entire complex with respect to the first frame (i.e., the final fitted structure) was calculated. In Figure 4A, it can be seen that the simulation starting from a structure fitted with symmetry restraints equilibrates faster and has a lower backbone RMSD than the one starting from a structure fitted without symmetry restraints. The shorter time needed to equilibrate and the lower RMSD value (Figure 4A) demonstrate that a more stable structure has been obtained using the symmetry restraints during the fitting. The stability of the fitted structure is particularly important if the system is to be subjected to further MD simulations.

Figure 4.

Figure 4

Open in a new tab

(A) Backbone RMSD of the GroEL-GroES complex with respect to the fitted structure during free equilibration after fitting. Red and black curves correspond to starting structures which were fitted with and without symmetry restraints, respectively. (B) Two-turn-helix fitted structure of R. rhodochrous J1 nitrilase. Red and blue structures were fitted with and without symmetry restraints, respectively. The first and last dimers of the structure fitted without symmetry restraints are pulled towards the density of adjacent dimers and are distorted as a consequence, while these dimers remain intact in case of structures fitted with symmetry restraints.

Example 2: Nitrilase

The second example involves a nitrilase from R. rhodochrous J1, a member of the superfamily of nitrilases, amidases, acyl transferase and N-carbamoyl-D-amino acid amidohydrolases. Nitrilases convert nitriles to the corresponding carboxylic acids and ammonia; an oligomerization of individual protein dimers into spiral homo-oligomers is important for their enzymatic function (Jandhyala et al., 2003; Thuku et al., 2009). In particular, the nitrilase from R. rhodochrous J1 was found to be inactive in its dimer form, but active in its helical-fiber form (Thuku et al., 2007). Negative stain and cryo-EM have been used to resolve the 3D structure of these helical fibers for native nitrilases as well as mutants (Thuku et al., 2009; Woodwarda et al., 2008), in order to understand the relationship between function and spiral quaternary structure formation in the nitrilase family.

We applied MDFF to fit a two-turn-helix model of R. rhodochrous J1 nitrilase, consisting of nine dimers, into a low resolution (18 Å) negative stain EM map of a long helix fiber (EMD-1313) (Thuku et al., 2007). The atomic structure of the dimer of R. rhodochrous J1 nitrilase used is a homology model based on crystal structures of four nitrilase homologues (PDB codes: 2PLQ, 3HKX, 1J31, 2VHH (Thuku, personal communication)). The helical symmetry of the fiber is parametrized by the helical twist (azimuthal rotation around the helical axis) and the helical rise (rise along the helical axis) between adjacent dimers. The helical twist and rise of the published map were −73.65° and 15.8 Å, respectively, with a D1S4.9 symmetry (Thuku et al., 2007).

Because only two helical turns were used for the atomic model, while the map covered six turns, the first and the last dimers at the edge of the model were attracted by the adjacent, empty density. Indeed, in the fitting without symmetry restraints, certain parts of the first and last dimers were pulled into this density, leading to structural distortions, as illustrated in the blue structure in Figure 4B. However, in the fitting with symmetry restraints, both the first and last dimers maintained their dimer form inside their corresponding density envelope (red structure in Figure 4B). The difference in dimer behavior shows that the symmetry restraints prevent edge-distortion effects, which arise whenever one is fitting only a part of the full system into the map. Segmenting the maps and removing the density where no dimers are placed may be a possible way to avoid edge-distortion effects, but map segmentation is prone to human error and proper cutting of boundaries is not always achieved. Although this problem can be addressed more directly by doing a fitting for the full system, there are occasions in which fitting of only a portion is desirable. In the J1 nitrilase system, a two-turn helix model is sufficient for determining how inter-dimer interactions give rise to the stability of a spiral structure; fitting more dimers into the long helical EM data would be computationally costly and unnecessary. This strategy may be appropriate to other symmetries such as virus capsids, which have icosahedral symmetry and for which full-system fitting would be computationally prohibitive.

Example 3: Mm-cpn

The last example features Mm-cpn, an archaeal group II chaperonin, which mediates protein folding. Mm-cpn is composed of two back-to-back eight-membered rings, but unlike the GroEL-GroES complexes, the rings are related by a two-fold axis giving rise to 16 symmetric units in the system. To carry out its function, a built-in lid closes upon ATP hydrolysis, the latter inducing structural rearrangements which have been illuminated by cryo-EM (Zhang et al., 2010a).

We applied MDFF to model the structures represented by two distinct maps, one at medium resolution (8 Å) of a lidless Mm-cpn in the open state (EMD-5140) and one at high resolution (4.3 Å) of a full Mm-cpn in the closed state (EMD-5137) (Zhang et al., 2010a). For the medium-resolution lidless open-state map, MDFF was used to fit a lidless closed-state homology model, built from the sequence of a lidless Mm-cpn and a full closed-state structure (PDB code 3LOS), into the EM data. For the high-resolution full closed-state map, MDFF was used to refine and improve the full closed-state structure (PDB code 3LOS).

The Mm-cpn example showcases the usefulness of symmetry restraints in dealing with lower (8 Å) resolution EM data. As shown in Figure 2C, fitting with and without symmetry restraints gives a similar average RMSD for Cα atoms among the symmetric units in the 4.3-Å-resolution map. However, for the lower resolution (8 Å) open-state map, the symmetry restraints are required to preserve the symmetry of the system. For high-resolution data, the conformational space allowed by the map is more confined than for low-resolution maps and, hence, during the fitting to them, all units converge to the same conformation in the absence of symmetry restraints. In contrast, there are significantly more possible conformations for a symmetric unit fitted to a lower resolution map, causing different units’ conformations to diverge during the fitting and, thus, giving a non-symmetric structure when fitted without the symmetry restraints.

We compared our symmetry-restrained fitted model of the Mm-cpn lidless open state to a recently released crystal structure (PDB code 3KFK (Pereira et al., 2010)), finding that the apical domain exhibits noticeable structural differences between them (RMSD 5.32 Å, see Figure 3E). Such differences may be due to the inaccuracy of the initial homology model of a lidless closed-state. To eliminate this source of inaccuracy, we performed fittings for the lidless open state map using a different starting structure, a crystal structure of the lidless closed state (PDB code 3KFE (Pereira et al., 2010)). As shown in Figure 3F, a structure in better agreement with the open-state crystal structure (RMSD 2.34 Å) was obtained. The improvement in the fitted model demonstrates the importance of using a good starting structure for fitting. Furthermore, the structure obtained with symmetry-restrained fitting is in better agreement with the crystal structure (RMSD 2.34 Å) compared to fitting without symmetry restraints (RMSD 2.83 Å), showing again that symmetry-restrained fitting can improve the quality of fitted structures.

Discussion

We have developed a new type of restraint that maintains the symmetry of a system during MDFF simulations by utilizing symmetry information extracted from cryo-EM data. Application to Mm-cpn demonstrates that when employing MDFF without symmetry restraints, the symmetry is distorted more heavily during fitting of lower resolution data, namely 8 Å vs. 4.3 Å data, because for each symmetric unit, more structural variability is allowed by a lower resolution map. Hence, symmetry restraints should be employed when one is modeling low to medium resolution symmetric EM data by MDFF. However, even when using symmetry restraints, the quality of the starting structure plays a significant role in the fitting, exemplified by Mm-cpn (see Figures 3E,F).

As shown in the free equilibration simulations of the GroEL-GroES complex, a more stable structure, i.e., one more suitable for further MD simulations, can be obtained with the application of symmetry restraints during fitting. The finding is in agreement with previous simulations of homology models of potassium channels refined by symmetry-restrained MD simulations (Anishkin et al., 2010), which is an example of an alternative use of symmetry restraints in MD simulations other than MDFF. Another benefit of using symmetry restraints is the prevention of edge-distortion effects, which arise when one is fitting only a portion of the system into the map, as illustrated above for R. rhodochrous J1. The benefits of using symmetry restraints demonstrated for the three example systems are also applicable to biological complexes possessing other types of symmetry, such as the icosahedral symmetry of virus capsids. The generalization is straightforward as the transformations arising in Equations (1,2) can be defined also for other symmetry types.

In symmetry-restrained fitting simulations, thermal noise tends to drive the system away from symmetry whereas the restraints counteract this tendency, limiting conformations to those that remain close to the symmetric structure. At physiological temperature, small deviations from symmetry arise naturally, especially at the side-chain level. Hence, in general it is not recommended to apply symmetry restraints to all atoms, which can produce a structure that adheres to the symmetry perfectly, but may also be non-physiological due to overly restraining it, akin to the over-fitting problem for cryo-EM maps (Trabuco et al., 2008). Instead, applying symmetry restraints to Cα atoms only can avoid over-restraining while still maintaining the secondary and tertiary structural symmetry of the system. The balance between thermal fluctuations and symmetry is controlled by the force constant k(t), which is recommended to be increased linearly over the fitting simulation so that the system can explore more conformational space as allowed by the map before converging to a symmetric structure. A maximum force constant of 10 (kcal/mol)/Å2 was chosen in our examples, from which deviations of atomic positions from the average will be Δx∼kBT/k=0.25Å only, thus maintaining the symmetry well. Indeed, in our examples, a linearly increasing k(t), reaching 10 (kcal/mol)/Å2, proved to be sufficient to guide the system to a nearly symmetric structure (see convergence of average RMSD in Figures 2) within a simulation time of 5 ns.

There are alternatives to symmetry restraints that can give a symmetric fitted structure. For example, one may fit only one symmetric unit into a segmented map and then generate the whole structure using the symmetry transformations. Although this will result in a perfectly symmetric structure, fitting in the presence of other symmetric units should be preferred as this will produce a more reliable structure at the interfaces between different units. Indeed, clashes may arise when one attempts to generate the whole structure from only a single fitted subunit, while in symmetry-restrained fitting the MD force field prevents such clashes. One may also average the individual structures of all symmetric units at the end of the fitting protocol, performing a so-called “instantaneous symmetrization”. This will give a perfectly symmetric structure, but the average of units in different conformations is not likely to be physically realistic or biological relevant. In contrast, the symmetry restraints will bias all units towards the same conformation during the fitting. The advantage of using symmetry restraints during the simulations over “instantaneous symmetrization” has also been demonstrated previously using symmetry-restrained MD simulations of tetrameric potassium channels (Anishkin et al., 2010).

Symmetry is common and often functionally important for biological systems, but there are cases where symmetry breaking arises during function. An example occurs during viral entry into the cell, which starts with binding of the icosahedrally symmetric virus capsid to receptors on the cell membrane, inducing then a conformational transition of the capsid for injection of the genome into the cell (Hogle, 2002); clearly, upon binding to the receptor, the symmetry of the virus capsid is broken. Naturally, symmetry-restrained fitting does not capture highly asymmetric regions in a system given that the map itself is symmetric, but subsequent MD simulations without any restraints applied permit the development of asymmetry. Root-mean-square fluctuations (RMSF) of atoms during simulations, which measure the deviation of atoms from their average positions, can reveal regions that are more flexible and, hence, more likely to become asymmetric.

It is common for multiple symmetries to co-exist in a biological system; for example, the family of nitrilases exhibits a two-fold symmetry in addition to the helical symmetry. Therefore, extending symmetry restraints to handle multiple symmetries in a single simulation is desirable and, indeed, is already under development. A tutorial on applying symmetry restraints during an MDFF simulation has been developed and is available at http://www.ks.uiuc.edu/Training/Tutorials/.

Highlight.

  1. Structural symmetry information can be incorporated into flexible fitting.

  2. Symmetry-based restraints improve quality of fitted structures.

  3. Benefits are most pronounced for lower resolution data.

Acknowledgments

This work was supported by grants from the National Institutes of Health (P41-RR005969) and the National Science Foundation (PHY0822613). All-atom MD simulation discussed here were performed using the package NAMD (Phillips et al., 2005), with computer time provided by NCSA and TACC via TeraGrid Request Allocation Committee grant MCA93S028. The authors thank Jim Phillips and Leonardo Trabuco for helpful discussions. Molecular images in this article were rendered using the molecular visualization software VMD (Humphrey et al., 1996). James Gumbart was supported during completion of this report by an Argonne National Laboratory Director’s Postdoctoral Fellowship.

Footnotes

Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

References

  1. Agirrezabala X, Lei J, Brunelle JL, Ortiz-Meoz RF, Green R, Frank J. Visualization of the hybrid state of tRNA binding promoted by spontaneous ratcheting of the ribosome. Mol Cell. 2008;32:190–197. doi: 10.1016/j.molcel.2008.10.001. [DOI] [PMC free article] [PubMed] [Google Scholar]
  2. André I, Bradley P, Wang C, Baker D. Prediction of the structure of symmetrical protein assemblies. Proc Natl Acad Sci USA. 2007;104:17656–17661. doi: 10.1073/pnas.0702626104. [DOI] [PMC free article] [PubMed] [Google Scholar]
  3. André I, Strauss CEM, Kaplan DB, Bradley P, Baker D. Emergence of symmetry in homooligomeric biological assemblies. Proc Natl Acad Sci USA. 2008;105:16148–16152. doi: 10.1073/pnas.0807576105. [DOI] [PMC free article] [PubMed] [Google Scholar]
  4. Anishkin A, Milac AL, Guy HR. Symmetry-restrained molecular dynamics simulations improve homology models of potassium channels. Proteins: Struct, Func, Bioinf. 2010;78:932–949. doi: 10.1002/prot.22618. [DOI] [PMC free article] [PubMed] [Google Scholar]
  5. Blundell TL, Srinivasan N. Symmetry. stability and dynamics of multidomain and multicomponent protein systems. Proc Natl Acad Sci USA. 1996;93:14243–14248. doi: 10.1073/pnas.93.25.14243. [DOI] [PMC free article] [PubMed] [Google Scholar]
  6. Chapman MS. Restrained real-space macromolecular atomic refinement using a new resolution-dependent electron-density function. Acta Cryst A. 1995;51:69–80. [Google Scholar]
  7. Chaudhry C, Farr GW, Todd MJ, Rye HS, Brunger AT, ams PDA, Horwich AL, Sigier PB. Role of the γ-phosphate of ATP in triggering protein folding by GroEL-GroES: function, structure and energetics. EMBO J. 2003;22:4877–4887. doi: 10.1093/emboj/cdg477. [DOI] [PMC free article] [PubMed] [Google Scholar]
  8. Chen LF, Blanc E, Chapman MS, Taylor KA. Real space refinement of acto-myosin structures from sectioned muscle. J Struct Biol. 2001;133:221–232. doi: 10.1006/jsbi.2000.4321. [DOI] [PubMed] [Google Scholar]
  9. Doyle DA, Cabral JM, Pfuetzer RA, Kuo A, Gulbis JM, Cohen SL, Chait BT, MacKinnon R. The structure of the potassium channel: molecular basis of K+ conduction and selectivity. Science. 1998;280:69–77. doi: 10.1126/science.280.5360.69. [DOI] [PubMed] [Google Scholar]
  10. Egelman EH. A robust algorithm for the reconstruction of helical filaments using single-particle methods. Ultramicroscopy. 2000;85:225–234. doi: 10.1016/s0304-3991(00)00062-0. [DOI] [PubMed] [Google Scholar]
  11. Frauenfeld J, Gumbart J, van der Sluis EO, Funes S, Gartmann M, Beatrix B, Mielke T, Berninghausen O, Becker T, Schulten K, Beckmann R. Cryo-EM structure of the ribosome-SecYE complex in the membrane environment. Nat Struct Mol Biol. 2011;18:614–621. doi: 10.1038/nsmb.2026. [DOI] [PMC free article] [PubMed] [Google Scholar]
  12. Frauenfelder H, Sligar SG, Wolynes PG. The Energy Landscapes and Motions of Proteins. Science. 1991;254:1598–1603. doi: 10.1126/science.1749933. [DOI] [PubMed] [Google Scholar]
  13. Goodsell DS, Olson AJ. Structural symmetry and protein function. Annu Rev Biophys Biomol Struct. 2000;29:105–153. doi: 10.1146/annurev.biophys.29.1.105. [DOI] [PubMed] [Google Scholar]
  14. Grubmüller H, Heller H, Windemuth A, Schulten K. Generalized Verlet Algorithm for Efficient Molecular Dynamics Simulations with Long-Range Interactions. Mol Sim. 1991;6:121–142. [Google Scholar]
  15. Hoang TX, Trovato A, Seno F, Banavar JR, Martian A. Geometry and symmetry presculpt the free-energy landscape of proteins. Proc Natl Acad Sci USA. 2004;101:7960–7964. doi: 10.1073/pnas.0402525101. [DOI] [PMC free article] [PubMed] [Google Scholar]
  16. Hogle JM. Poliovirus Cell Entry: Common Structural Themes in Viral Cell Entry Pathways. Annu Rev Microbiol. 2002;56:677–702. doi: 10.1146/annurev.micro.56.012302.160757. [DOI] [PMC free article] [PubMed] [Google Scholar]
  17. Hogle JM, Chow M, Filman DJ. Three-dimensional Structure of Poliovirus at 2.9 Å Resolution. Science. 1985;229:1358. doi: 10.1126/science.2994218. [DOI] [PubMed] [Google Scholar]
  18. Humphrey W, Dalke A, Schulten K. VMD – Visual Molecular Dynamics. J Mol Graphics. 1996;14:33–38. doi: 10.1016/0263-7855(96)00018-5. [DOI] [PubMed] [Google Scholar]
  19. Jandhyala D, Berman M, Meyers PR, Sewell BT, Wilson RC, Benedik MJ. CynD, the Cyanide Dihydratase from Bacillus pumilus: Gene Cloning and Structural Studies. Appl Environ Microbiol. 2003;69:4794–4805. doi: 10.1128/AEM.69.8.4794-4805.2003. [DOI] [PMC free article] [PubMed] [Google Scholar]
  20. Jorgensen WL, Chandrasekhar J, Madura JD, Impey RW, Klein ML. Comparison of Simple Potential Functions for Simulating Liquid Water. J Chem Phys. 1983;79:926–935. [Google Scholar]
  21. Ludtke SJ, Baldwin PR, Chiu W. EMAN: Semiautomated software for high-resolution single-particle reconstructions. J Struct Biol. 1999;128:82–97. doi: 10.1006/jsbi.1999.4174. [DOI] [PubMed] [Google Scholar]
  22. MacKerell AD, Jr, Bashford D, Bellott M, Dunbrack RL, Jr, Evanseck JD, Field MJ, Fischer S, Gao J, Guo H, Ha S, Joseph D, Kuchnir L, Kuczera K, Lau FTK, Mattos C, Michnick S, Ngo T, Nguyen DT, Prodhom B, Reiher IWE, Roux B, Schlenkrich M, Smith J, Stote R, Straub J, Watanabe M, Wiorkiewicz-Kuczera J, Yin D, Karplus M. All-atom empirical potential for molecular modeling and dynamics studies of proteins. J Phys Chem B. 1998;102:3586–3616. doi: 10.1021/jp973084f. [DOI] [PubMed] [Google Scholar]
  23. MacKerell AD, Jr, Feig M, Brooks CL., III Extending the Treatment of Backbone Energetics in Protein Force Fields: Limitations of Gas-Phase Quantum Mechanics in Reproducing Protein Conformational Distributions in Molecular Dynamics Simulations. J Comp Chem. 2004;25:1400–1415. doi: 10.1002/jcc.20065. [DOI] [PubMed] [Google Scholar]
  24. Nymeyer H, García AE, Onuchic JN. Folding funnels and frustration in off-lattice minimalist protein landscapes. Proc Natl Acad Sci USA. 1998;95:5921–5928. doi: 10.1073/pnas.95.11.5921. [DOI] [PMC free article] [PubMed] [Google Scholar]
  25. Pereira JH, Ralston CY, Douglas NR, Meyer D, Knee KM, Goulet DR, King JA, Frydman J, Adams PD. Crystal Structures of a Group II Chaperonin Reveal the Open and Closed States Associated with the Protein Folding Cycle. J Biol Chem. 2010;285:27958–27966. doi: 10.1074/jbc.M110.125344. [DOI] [PMC free article] [PubMed] [Google Scholar]
  26. Phillips JC, Braun R, Wang W, Gumbart J, Tajkhorshid E, Villa E, Chipot C, Skeel RD, Kale L, Schulten K. Scalable Molecular Dynamics with NAMD. J Comp Chem. 2005;26:1781–1802. doi: 10.1002/jcc.20289. [DOI] [PMC free article] [PubMed] [Google Scholar]
  27. Qian P, Hunter CN, Bullough PA. The 8.5 Å Projection Structure of the Core RC-LH1-PufX Dimer of Rhodobacter sphaeroides. J Mol Biol. 2005;349:948–960. doi: 10.1016/j.jmb.2005.04.032. [DOI] [PubMed] [Google Scholar]
  28. Ranson NA, Clare DK, Farr GW, Houldershaw D, Horwich AL, Saibil HR. Allosteric signalling of ATP hydrolysis in GroEL-GroES complexes. Nat Struct Mol Biol. 2006;13:147–152. doi: 10.1038/nsmb1046. [DOI] [PMC free article] [PubMed] [Google Scholar]
  29. Roseman AM, Chen S, White H, Braig K, Saibil HR. The Chaperonin ATPase Cycle: Mechanism of Allosteric Switching and Movements of Substrate-Binding Domains in GroEL. Cell. 1996;87:241–251. doi: 10.1016/s0092-8674(00)81342-2. [DOI] [PubMed] [Google Scholar]
  30. Schröder GF, Brunger AT, Levitt M. Combining efficient conformational sampling with a deformable elastic network model facilitates structure refinement at low resolution. Structure. 2007;15:1630–1641. doi: 10.1016/j.str.2007.09.021. [DOI] [PMC free article] [PubMed] [Google Scholar]
  31. Seidelt B, Innis CA, Wilson DN, Gartmann M, Armache JP, Villa E, Trabuco LG, Becker T, Mielke T, Schulten K, Steitz TA, Beckmann R. Structural insight into nascent polypeptide chain-mediated translational stalling. Science. 2009;326:1412–1415. doi: 10.1126/science.1177662. [DOI] [PMC free article] [PubMed] [Google Scholar]
  32. Suhre K, Navaza J, Sanejouand YH. NORMA: a tool for flexible fitting of high-resolution protein structures into low-resolution electron-microscopy-derived density maps. Acta Cryst D. 2006;62:1098–1100. doi: 10.1107/S090744490602244X. [DOI] [PubMed] [Google Scholar]
  33. Tama F, Miyashita O, Brooks CL., III Flexible Multi-scale Fitting of Atomic Structures into Low-resolution Electron Density Maps with Elastic Network Normal Mode Analysis. J Mol Biol. 2004a;337:985–999. doi: 10.1016/j.jmb.2004.01.048. [DOI] [PubMed] [Google Scholar]
  34. Tama F, Miyashita O, Brooks CL., III Normal mode based flexible fitting of high-resolution structure into low-resolution experimental data from cryo-EM. J Struct Biol. 2004b;147:315–326. doi: 10.1016/j.jsb.2004.03.002. [DOI] [PubMed] [Google Scholar]
  35. Thuku RN, Brady D, Benedik MJ, Sewell BT. Microbial nitrilases: versatile, spiral forming, industrial enzymes. J Appl Microbiol. 2009;106:703–727. doi: 10.1111/j.1365-2672.2008.03941.x. [DOI] [PubMed] [Google Scholar]
  36. Thuku RN, Weber BW, Varsani A, Sewell BT. Post-translational cleavage of recombinantly expressed nitrilase from Rhodococcus rhodochrous J1 yields a stable, active helical form. FEBS J. 2007;274:2099–2108. doi: 10.1111/j.1742-4658.2007.05752.x. [DOI] [PubMed] [Google Scholar]
  37. Topf M, Baker ML, Marc-Renom MA, Chiu W, Sali A. Refinement of protein structures by iterative comparative modeling and cryoEM density fitting. J Mol Biol. 2006;357:1655–1668. doi: 10.1016/j.jmb.2006.01.062. [DOI] [PubMed] [Google Scholar]
  38. Trabuco LG, Schreiner E, Gumbart J, Hsin J, Villa E, Schulten K. Applications of the molecular dynamics flexible fitting method. J Struct Biol. 2011;173:420–427. doi: 10.1016/j.jsb.2010.09.024. [DOI] [PMC free article] [PubMed] [Google Scholar]
  39. Trabuco LG, Villa E, Mitra K, Frank J, Schulten K. Flexible Fitting of Atomic Structures into Electron Microscopy Maps Using Molecular Dynamics. Structure. 2008;16:673–683. doi: 10.1016/j.str.2008.03.005. [DOI] [PMC free article] [PubMed] [Google Scholar]
  40. Trabuco LG, Villa E, Schreiner E, Harrison CB, Schulten K. Molecular Dynamics Flexible Fitting: A practical guide to combine cryo-electron microscopy and X-ray crystallography. Methods. 2009;49:174–180. doi: 10.1016/j.ymeth.2009.04.005. [DOI] [PMC free article] [PubMed] [Google Scholar]
  41. Tuckerman M, Berne BJ, Martyna GJ. Reversible multiple time scale molecular dynamics. J Chem Phys. 1992;97:1990–2001. [Google Scholar]
  42. van Heel M, Harauz G, Oriova EV, Schmidt R, Schatz M. A new generation of the IMAGIC image processing system. J Struct Biol. 1996;116:17–24. doi: 10.1006/jsbi.1996.0004. [DOI] [PubMed] [Google Scholar]
  43. Volkmann N, Hanein D, Ouyang G, Trybus KM, DeRosier DJ, Lowey S. Evidence for cleft closure in actomyosin upon ADP release. Nat Struct Biol. 2000;7:1147–1155. doi: 10.1038/82008. [DOI] [PubMed] [Google Scholar]
  44. Wolynes PG. Symmetry and the energy landscapes of biomolecules. Proc Natl Acad Sci USA. 1996;93:14249–14255. doi: 10.1073/pnas.93.25.14249. [DOI] [PMC free article] [PubMed] [Google Scholar]
  45. Woodwarda J, Webera B, Scheffera M, Benedikc M, Hoengerd A, Sewell B. Helical structure of unidirectionally shadowed metal replicas of cyanide hydratase from Gloeocercospora sorghi. J Struct Biol. 2008;161:111–119. doi: 10.1016/j.jsb.2007.09.019. [DOI] [PubMed] [Google Scholar]
  46. Wriggers W, Chacón P. Modeling tricks and fitting techniques for multiresolution structures. Structure. 2001;9:779–788. doi: 10.1016/s0969-2126(01)00648-7. [DOI] [PubMed] [Google Scholar]
  47. Wriggers W, Milligan RA, McCammon JA. Situs: A package for docking crystal structures into low-resolution maps from electron microscopy. J Struct Biol. 1999;125:185–195. doi: 10.1006/jsbi.1998.4080. [DOI] [PubMed] [Google Scholar]
  48. Xu Z, Horwich AL, Sigler PB. The crystal structure of the asymmetric GroEL-GroES-(ADP)7chaperonin complex. Nature. 1997;388:741–750. doi: 10.1038/41944. [DOI] [PubMed] [Google Scholar]
  49. Zhang J, Baker ML, Schröder GF, Douglas NR, Reissmann S, Jakana J, Dougherty M, Fu CJ, Levitt M, Ludtke SJ, Frydman J, Chiu W. Mechanism of folding chamber closure in a group II chaperonin. Nature. 2010a;463:379–383. doi: 10.1038/nature08701. [DOI] [PMC free article] [PubMed] [Google Scholar]
  50. Zhang X, Jin L, Fang Q, Hui WH, Zhou ZH. 3.3 Å Cryo-EM Structure of a Nonenveloped Virus Reveals a Priming Mechanism for Cell Entry. Cell. 2010b;141:472–482. doi: 10.1016/j.cell.2010.03.041. [DOI] [PMC free article] [PubMed] [Google Scholar]
  51. Zhu J, Cheng L, Fang Q, Zhou ZH, Honig B. Building and refining protein models within cryo-electron microscopy density maps based on homology modeling and multiscale structure refinement. J Mol Biol. 2010;397:835–851. doi: 10.1016/j.jmb.2010.01.041. [DOI] [PMC free article] [PubMed] [Google Scholar]

RESOURCES