To understand the roles that proteins and other large molecules play inside cells, it is important to determine their structures. One of the techniques that researchers can use to do this is called cryo-electron microscopy (cryo-EM), which rapidly freezes molecules to fix them in position before imaging them in fine detail.

The cryo-EM images are like maps that show the approximate position of atoms. These images must then be processed in order to build a three-dimensional model of the protein that shows how its atoms are arranged relative to each other. One computational approach called Molecular Dynamics Flexible Fitting (MDFF) works by flexibly fitting possible atomic structures into cryo-EM maps. Although this approach works well with relatively undetailed (or ‘low resolution’) cryo-EM images, it struggles to handle the high-resolution cryo-EM maps now being generated.

Singharoy, Teo, McGreevy et al. have now developed two MDFF methods – called cascade MDFF and resolution exchange MDFF – that help to resolve atomic models of biological molecules from cryo-EM images. Each method can refine poorly guessed models into ones that are consistent with the high-resolution experimental images. The refinement is achieved by interpreting a range of images that starts with a ‘fuzzy’ image. The contrast of the image is then progressively improved until an image is produced that has a resolution that is good enough to almost distinguish individual atoms.

The method works because each cryo-EM image shows not just one, but a collection of atomic structures that the molecule can take on, with the fuzzier parts of the image representing the more flexible parts of the molecule. By taking into account this flexibility, the large-scale features of the protein structure can be determined first from the fuzzier images, and increasing the contrast of the images allows smaller-scale refinements to be made to the structure.

The MDFF tools have been designed to be easy to use and are available to researchers at low cost through cloud computing platforms. They can now be used to unravel the structure of many different proteins and protein complexes including those involved in photosynthesis, respiration and protein synthesis.

Cryo-electron microscopy (cryo-EM) has evolved into one of the most effective structure determination tools in modern day structural biology, achieving in recent years resolutions rivalling those of X-ray crystallography or NMR spectroscopy (Cheng, 2015). Furthermore, cryo-EM based structure determination overcomes two major bottlenecks faced in traditional X-ray crystallography, namely, the arduous task of preparing well-ordered crystals of macromolecules (Unger, 2002), and the more fundamental problem with capturing these molecules in unphysiological states as a result of crystal contacts (Neutze et al., 2015). Consequently, cryo-EM provides a natural way of resolving the structures of large macromolecular complexes.

Historically, computational methods were required to bridge the resolution gap between crystallography and cryo-EM to produce atomic-resolution models of biomolecular complexes. Various real-space refinement methods that combine crystallographic structures and cryo-EM densities for structure determination have been developed, including DireX (Schröder et al., 2007), Flex-EM (Topf et al., 2008), Rosetta (DiMaio et al., 2015), FRODA (Jolley et al., 2008), Phenix real space refinement (Afonine et al., 2013), and Molecular Dynamics Flexible Fitting (MDFF) (Trabuco et al., 2008, 2009; McGreevy et al., 2016).

MDFF, in particular, has proven to be an extremely successful refinement method as evidenced by its numerous applications (Goh et al., 2015; McGreevy et al., 2016) ranging from the intricate ribosomal machinery (Villa et al., 2009; Trabuco et al., 2011; Frauenfeld et al., 2011; Wickles et al., 2014) to a host of non-enveloped viruses (Zhao et al., 2013). So far this success has been limited to structure determination from typically low-resolution cryo-EM maps in the 7–25 Å range which, indeed, represented the state-of-the-art at the time of MDFF’s inception (Trabuco et al., 2008). However, seminal advances in detection hardware and programs over the past three years (Li et al., 2013; Milazzo et al., 2011) have enabled now the routine availability of high-resolution (<5 Å) EM maps for a range of biological systems including ion channels (Liao et al., 2013), enzymes (Bartesaghi et al., 2014, 2015), membrane fusion machinery (Zhao et al., 2015), and key functional components of the ribosome (Fischer et al., 2015; Brown et al., 2015).

High-resolution maps pose an imminent challenge to the traditional map-guided structure determination methods as the maps now characterize near-atomic scale features, the interpretation of which requires extremely precise structure building and validation protocols (DiMaio et al., 2015). For example, conformation of the protein sidechains, which are more flexible than the backbone, are now discernible within the maps and, thus, require precise modeling of the dihedral angles up to C_β atoms while also respecting the map boundaries (Barad et al., 2015).

In order to produce atomic models with correct backbone and sidechain geometries, as well as minimal potential energy, structure determination tools must be augmented with chemically accurate force fields and exhaustive search algorithms respecting density constraints. Inspired by crystallographic modeling techniques, where such structure-building requirements have already been addressed for the resolution of 3–5 Å diffraction data (DiMaio et al., 2013; Murshudov et al., 2011; McGreevy et al., 2014), tools such as Rosetta have introduced Monte Carlo simulation-based segment building and refinement protocols with heuristic force fields (DiMaio et al., 2015), to handle high-resolution EM maps. Other notable automated model-building tools that can be used for the refinement of high-resolution EM maps include Buccaneer (Cowtan, 2006), ARP/wARP (Langer et al., 2008), and Moulder (Topf et al., 2006).

Driven by a vision to extend the capabilities of flexible fitting approaches (Topf et al., 2008; Trabuco et al., 2008; Tama et al., 2004; Suhre et al., 2006; Kovacs et al., 2008; Wu et al., 2013) for addressing high-resolution maps, two new MDFF methods are introduced here. These methods, denoted cascade MDFF (cMDFF) and resolution exchange MDFF (ReMDFF), augment the traditional MDFF method (Trabuco et al., 2008, 2009; McGreevy et al., 2016) (called direct MDFF henceforth) with enhanced conformational sampling techniques, namely simulated annealing (Brünger, 1988) and replica exchange molecular dynamics (Sugita and Okamoto, 1999). The central idea behind the techniques introduced is to fit a search model sequentially to a series of maps of progressively higher resolutions, ending with the original experimental resolution; all but the last in the series are computationally blurred lower-resolution derivatives of the original map, so that larger-scale features of the structure are determined first by fitting to the blurred densities, and smaller-scale refinements are performed subsequently during the fitting to higher-resolution densities. Altogether, this treatment enables a richer conformational sampling of the model within the map than direct MDFF, thereby allowing accurate modeling of the global and local structural features from the map; a similar treatment has previously been employed to increase the radius of convergence of MDFF protocols, but with crystallographic data (Singharoy et al., 2015).

The cMDFF and ReMDFF methods are demonstrated for structure analysis based on 3.2-Å and 3.4-Å resolution maps of β-galactosidase (Bartesaghi et al., 2014) and the TRPV1 channel (Liao et al., 2013), respectively. The two methods were found to resolve atomic structures with accuracy greater than that of direct MDFF and comparable to that of Rosetta, even with poor choices of search models. The accuracy is evaluated in terms of the quality of fit measured through global and local cross-correlations (GCC and LCC), integrated Fourier shell coefficients (iFSC), and EMRinger scores (Barad et al., 2015), as well as in terms of the quality of structural integrity measures like MolProbity (Chen et al., 2010).

In the second part of the present study we establish that structural flexibility, as measured by root mean square fluctuations or RMSF within the MDFF simulation, provides an ensemble-based indicator of local and overall resolution of a map offering, thus, a quality measure of an EM map based on the inherent dynamics of the imaged macromolecule. In line with this new finding, RMSF values are shown to provide a physical basis for the determination of optimal sharpening B-factors that maximize the signal-to-noise ratio within a map. These B-factors are determined at three different levels of model description: whole-system, per-domain, and per-residue.

Finally, use of the ReMDFF method on cloud computing platforms is discussed. Cloud computing is now a highly suitable approach for computational biology and can be employed for large-scale scientific computing, data analysis, and visualization tasks. For example, Amazon Web Services has been previously demonstrated to be a low-cost cloud computing platform for processing cryo-EM data (Cianfrocco et al., 2015). We demonstrate now the usage of Amazon Web Services, highlighting the platform’s capability for rapidly fitting structures to EM density with ReMDFF. The web-interface makes it readily possible for experimental groups around the world to deploy MDFF in an easy and economical way, bypassing the need for their own staff, software, and hardware resources.

In the following section, we first describe the methodological advances achieved within cMDFF and ReMDFF for the resolution of sub-5 Å maps. Search model preparation, refinement, and structure validation protocols based on these advances are subsequently demonstrated for five exemplary protein complexes that were chosen based on the availability of high-resolution (3–5 Å) EM maps and atomic structures. Finally, the performance of ReMDFF on Amazon’s cloud computing platforms is described, demonstrating that our MDFF software offers an efficient web-based resource for structure determination from EM maps.

Simulation concept

In direct MDFF, an initial atomic structure is subjected to an MD simulation with an additional potential energy term $V_{\text{EM}}$ that is proportional to the sign-inverse of the EM map. Through $V_{\text{EM}}$, steering forces locally guide atoms towards high-density regions, thereby fitting the structure to the map (see Materials and methods).

The equilibrium structure obtained in the simulation represents a global minimum in $V_{\text{EM}}$. For maps in the low resolution range (6–15 Å), this global minimum is broad, accomodating an ensemble of conformations defined by the overall shape of the macromolecule (Trabuco et al., 2008, 2011). In contrast, at the mid-resolution range of 4–6 Å, densities corresponding to the backbones become discernible, and at sub-4 Å resolutions, even sidechains can be resolved. At such high resolutions, ${\displaystyle V_{\mathrm{E}\mathrm{M}}}$ now features multiple proximal local minima which correspond to recurring spatial patterns within a macromolecule, such as helices aligned in parallel or strands in a β-sheet. As shown in Figure 1, the energy barriers separating these local minima are typically twice as high as those in the case of low-resolution maps. The existence of such potential minima in high-resolution maps exposes MDFF to a long-known weakness of traditional MD-based algorithms, namely entrapment of the fitted structure within undesired local minima instead of reaching the global minimum of $V_{\text{EM}}$. Not unexpectedly, therefore, direct MDFF yields structurally poor or functionally irrelevant models with high-resolution EM maps (Figure 2—figure supplement 1) (DiMaio et al., 2015).

Figure 1

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To mitigate this weakness in direct MDFF, we introduce cMDFF and ReMDFF. In cMDFF, the structure is fitted, in a series of MDFF simulations, to maps of gradually increasing resolution. First, the experimental map is smoothened by applying a series of Gaussian blurs with increasing half-widths, $\sigma$, to obtain a set of theoretical maps with gradually decreasing resolution; $\sigma$ = 0 Å corresponds to the experimental map, and ${\displaystyle \sigma >0}$ Å corresponds to a smoothened one (see Materials and methods). Illustrated in Figure 1, the density-dependent potential derived from the smoothest map (i.e. the one with the largest $\sigma$ value) features a clear global minimum representing the large-scale structural features of the protein. Second, a search model is fitted to this map, allowing resolution of these large-scale features. Third, the resulting structure is employed as the search model for fitting to the next higher-resolution map in the series. These fitting and search model refreshment steps are repeated through the series of maps in order of decreasing $\sigma$, until the structure is finally fitted to the experimental map (see Video 1 for TRPV1 refinement. An additional virtual reality version can be found at https://www.youtube.com/watch?v=UwwVC6C9tw0).

Video 1

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The gradual increase in map resolution over the course of the simulations allows the structure to explore a greater conformational space than in direct MDFF. The structure thus avoids entrapment within local minima of the density-dependent potential and is accurately fitted to the near-atomic density features of the experimental map while also resolving the larger scale features.

In ReMDFF, the cascade scheme is infused with a greater degree of automation. Multiple MDFF simulation replica are run in parallel with each replica fitting a model to a map of a specific resolution. Based on a Metropolis formula analogous to that of conventional replica exchange molecular dynamics simulations (Sugita and Okamoto, 1999), but now derived in terms of density and not temperature (see Materials and methods), the models are exchanged at regular time intervals between maps from neighboring pairs of replica. Stepwise improvements in fit occur during exchanges between a poorly fitted model at a high resolution with a well-fitted model at a lower resolution. This well-fitted model is further refined against the high-resolution map until convergence is reached, and exchange between the chosen resolutions ceases. Further details are described in Materials and methods.

ReMDFF has advantages over cMDFF both in terms of efficiency and automation as it can take advantage of modern parallel computing hardware and the powerful and adaptive replica exchange interface of NAMD (Jiang et al., 2014). Nonetheless, as presented in the following, both cMDFF and ReMDFF outperform direct MDFF in quality and speed across a range of high-resolution examples.

Search model preparation and refinement

In an initial proof-of-principle computation, cMDFF and ReMDFF were applied to fit a structure of carbon monoxide dehydrogenase to a 3-Å synthetic density map. The same techniques were subsequently applied to obtain refined structures of two more protein systems, namely TRPV1 (Liao et al., 2013) and β-galactosidase (Bartesaghi et al., 2014), for which experimental densities of 3.4 Å and 3.2 Å resolution respectively are available. In each case, a direct MDFF simulation was also performed for the purpose of comparison. The MDFF-derived structures were then subjected to a model validation analysis, to evaluate the quality of the models with established protocols in the cryo-EM field. Additional examples chosen for this analysis include γ-secretase at 4.5 Å (Lu et al., 2014) and 3.4-Å (Bai et al., 2015), β-galactosidase at 2.2-Å (Bartesaghi et al., 2015) and the proteasome at 3.3 Å (Li et al., 2013) resolution. Comparisons between the direct and advanced MDFF protocols, and wherever possible, with other available fitting techniques, such as Rosetta (DiMaio et al., 2015), elucidate the general pros and cons of the flexible fitting strategy.

Proof of principle

The performance of cMDFF and ReMDFF was evaluated on a test system, carbon monoxide dehydrogenase, which exhibits a closed and an open conformation (Darnault et al., 2003). Both these conformations have been crystallized, and are reported respectively in chains C and D of the PDB entry 1OAO. For our demonstration, the closed conformation (1OAO:chain C) was used as the search model, while the open one (1OAO:chain D) was the target.

First, a 3-Å resolution synthetic density map was constructed in Phenix (Adams et al., 2010), employing phases from the 1OAO structure and the associated diffraction data truncated at 3 Å. This map was then masked about chain D to yield a high-resolution envelope characterizing the open conformation. Assuming that the crystallographic model provides an accurate benchmark, the corresponding map for chain D determined here represents the best possible density data at 3 Å resolution that is experimentally attainable for the open conformation. Finally, through direct MDFF, cMDFF and ReMDFF, the search model constructed from chain C was fitted into this density to derive an atomic structure representing the open conformation. Both the cMDFF and ReMDFF refinements were performed for a set of six maps with $\sigma$ values ranging from 5 to 0 Å at constant decrements of 1 Å.

Accuracy of the fitting protocols was evaluated by comparing the fitted chain C structures with the crystallographically reported target chain D model. Direct MDFF of the 3 Å synthetic map performed for 2 ns converged to a structure with an RMSD of 7 Å relative to the target model. In sharp contrast, the cMDFF- and ReMDFF-generated structures are within 1.7 Å and 1 Å RMSD of the target (see the inset of Figure 1). It is also noted that fitting to the lowest-resolution (i.e. one with $\sigma$ = 5 Å) brings about an immediate decrease in RMSD from the target generating structures that are within 2 Å RMSD. Fitting of the structure to subsequent high-resolution maps brought the RMSD down to 1.0 Å.

The results demonstrate that the new protocols are capable of attaining well-fit structures where direct MDFF does not. In particular, one can think of the new protocols as extending the radius of convergence to at least 7 Å, rendering the fitting procedures less dependent on the quality of the starting structure.

Refinement of β-galactosidase

In a second test case, a search model was fitted into the 3.2 Å map (Bartesaghi et al., 2014) of β-galactosidase employing direct MDFF, cMDFF, and ReMDFF. Noting that the radius of convergence of the proposed MDFF protocols was at least 7 Å for the aforementioned test case, the initial search model was prepared such that it had an RMSD of 7 Å from the reported structure. This model was obtained by applying to the reported structure (obtained by de novo modeling within the EM map [Bartesaghi et al., 2014]) a high temperature MD protocol described in Appendix 1, Section 3 and, subsequently, choosing from the collection of trajectory structures one of RMSD 7 Å from the reported structure and with the lowest GCC with respect to the reported map (Bartesaghi et al., 2014) (Figure 2—figure supplement 2a).

Summarized in Table 1, the fitting results, in terms of quality of fit as well as model quality, are significantly better for cMDFF and ReMDFF than for direct MDFF: (i) RMSD of the fitted structure with respect to the reported de novo model is 0.7 Å and 0.9 Å for cMDFF and ReMDFF respectively, much lower than the 3.7 Å RMSD attained with direct MDFF (Figure 2—figure supplement 3a); (ii) EMRinger scores for cMDFF and ReMDFF are 3.16 and 3.45 respectively, higher than the 1.91 obtained for direct MDFF, implying accurate fitting of sidechains into the density; (iii) MolProbity scores are consistently small for all the flexible fitting techniques in part due to fewer, less severe steric clashes and fewer Ramachandran outliers (further detailed in Table 2); (iv) integrated FSC (iFSC2, corresponding to the range 3.4–10 Å on the FSC plot obtained as per Appendix 1 - Section 6), considered a more stringent measure of model quality than CC (DiMaio et al., 2015), attained higher values of 5.22 Å and 4.66 Å for cMDFF and ReMDFF, respectively, than 2.74 Å for direct MDFF. iFSC1, evaluated at the lower resolution range of 5–10 Å improves from 2.11 Å for direct MDFF to 4.22 Å and 3.76 Å for cMDFF and ReMDFF, respectively, showing a trend similar to that of iFSC2 corresponding to the high-resolution range; and (v) GCCs improved from an initial value of 0.48 to 0.56, 0.67 and 0.67 for direct, cMDFF, and ReMDFF protocols respectively. Similarly, typical residue LCC values improved from about 0 to greater than 0.80 (Figure 2—figure supplement 4a and Figure 2—figure supplement 5). Overall, cMDFF and ReMDFF refinements produce structures that interpret the 3.2-Å β-galactosidase map much more accurately than direct MDFF does. Figure 2a shows, visually, how the cMDFF-derived structure differs from the direct MDFF structure in terms of fit. In judging the RMSD values to the target model the reader is reminded that equilibrium MD simulations of a single structure at room temperature typically exhibit RMSD values relative to the initial structure or the average structure of about 3 Å; the same is true for β-galactosidase (see Figure 2—figure supplement 6 in Appendix 1 - Section 7). Consequently, an RMSD of 0.7 Å of the cMDFF/ReMDFF-fitted model relative to the target implies a high-quality refinement. The high quality of this refinement is further supported by visualizations of accurate sidechain placements within the density, shown in Figure 2—figure supplement 7.

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Table 1

Structure	RMSD(Å)	EMRinger	iFSC1(Å)	iFSC2(Å)	MolProb.	GCC
de novo (Bartesaghi et al., 2014)	0.0	2.25	4.03	5.00	3.14	0.67
Refined de novo	0.6	4.23	4.19	5.20	1.23	0.68
Initial	7.7	0.24	0.14	0.15	1.49	0.48
Direct MDFF	3.7	2.31	2.11	2.74	1.38	0.56
cMDFF	0.7	3.16	4.22	5.22	1.37	0.67
ReMDFF	0.9	3.45	3.76	4.66	1.13	0.67

Table 2

	de novo (Bartesaghi et al., 2014)	Refined de novo	Initial	Direct MDFF	cMDFF	ReMDFF
Clashscore	53.7	0.0	0.0	0.0	0.0	0.0
Poor rotamers (%)	11.6	3.8	4.2	3.0	4.4	1.37
Favored rotamers (%)	67.4	90.8	87.8	92.1	89.8	95.3
Ramachandran outliers (%)	0.2	0.7	2.7	3.0	1.6	2.7
Ramachandran favored (%)	97.4	95.8	91.1	91.1	94.4	90.9
MolProbity	3.14	1.23	1.49	1.38	1.37	1.13
C$\beta$ deviations (%)	0.0	0.05	4.92	0.18	0.29	0.39
Bad bonds (%)	0.09	0.04	3.61	0.02	0.01	0.03
Bad angles (%)	0.03	0.60	3.98	0.63	0.49	0.37
RMS distance (Å)	0.007 (0.025%)	0.019 (0%)	0.035 (0.237%)	0.022 (0%)	0.019 (0%)	0.021 (0%)
RMS angle (degrees)	1.1 (0.009%)	2.2 (0.009%)	3.6 (1.177%)	2.4 (0.103%)	2.1 (0.018%)	2.3 (0.085%)
Cis prolines (%)	8.06	8.06	6.45	6.45	6.45	8.06
Cis non-prolines (%)	1.15	1.15	0.0	0.0	1.15	0.0

The cMDFF- and ReMDFF-refined structures were found to be comparable in every quality measure in comparison to the reported de novo structure (Bartesaghi et al., 2014); in fact, the overall Molprobity and EMRinger scores are significantly better in cMDFF and ReMDFF. However, a closer look at the Molprobity score (Table 2) reveals that even though cMDFF vastly improves clash score and poor rotamers, it marginally increases the percentage of Ramachandran outliers and Cβ deviations relative to the de novo structure. Nonetheless, both cMDFF and ReMDFF improved structural statistics with respect to the initial model (Table 1, third row) which was intentionally chosen to have a large deviation (RMSD of 7.7 Å) from the de novo structure.

Noting that the quality of MDFF output depends strongly on that of the search model, a second cMDFF simulation was also performed to refine the de novo structure within the reported map. The simulation, labeled ‘refined de novo’ in Table 1, yielded a structure that was superior in all the quality measures considered in comparison to the de novo structure as well as to the structures obtained from the various MDFF fittings of the other, 7.7 Å deviated initial model. A closer look at the Molprobity scores (Table 2) now reveals that not only are clash score and poor rotamers vastly improved, but the Ramachandran outliers and Cβ deviations are also comparable to the de novo structure.

A third cMDFF refinement was further performed with a search model of even lower structural quality (Supplementary file 1E, third row) compared to that of Table 1 (third row). Structural statistics for the refinements from this lower-quality search model are provided in Supplementary file 1F. Comparison of the three cMDFF refinements starting with three search models of varying structural quality reveal that the poorer the secondary structure of the search model, measured in terms of higher percentages of rotamer and Ramachandran outliers, the worse the local structural statistics of the MDFF/cMDFF/ReMDFF-refined model. Surprisingly, large scale map-model validation measures, such as RMSD, GCC, or iFSC values, remain insensitive to such local discrepancies in the refined model. This insensitivity is apparent in the similarity of RMSD, GCC, and iFSC values of the three refined models of β-galactosidase (Table 1 and Supplementary file 1E), which indeed feature very different number of Ramachandran outliers: 0.7% (Table 2-Refined de novo), 1.6% (Table 2-cMDFF) and 7.8% (Supplementary file 1F-cMDFF).

The high-precision of sub-5 Å maps demands correct assignment of secondary structure, which determines backbone geometry directly and sidechain conformations indirectly. In light of the initial model quality dependence of MDFF, it is advisable to begin refinement with an initial model of maximal secondary structure information, yet with minimal tertiary structure. Dependable initial secondary structure is also required since MD simulations have limited capability of recovering the structure of a protein fold if the search model begins with a random coil conformation (Freddolino et al., 2010). To this end, notwithstanding observed cMDFF improvements of the secondary structure (Table 2 columns 4 vs. 6, and also demonstrated for γ-secretase [Supplementary file 1D]), moderate- to high-confidence homology models will serve as the most optimal starting point, as has been successfully shown for MDFF with low-resolution EM (Chan et al., 2011; Noble et al., 2013; Wickles et al., 2014) and crystallographic data (McGreevy et al., 2014; Li et al., 2014). Quantitatively speaking, employing the current example (β-galactosidase) and the following one (TRPV1), it is now shown that a search model which deviates from the target by an RMSD of 7–25 Å featuring Ramachandran outliers of ~3%, poor rotamers of 11–38%, overall Molprobity score of ~4 and EMRinger score of ~0.20 can be accurately refined against a sub-5 Å EM map with cMDFF or ReMDFF.

In terms of efficiency, the ReMDFF protocol exhibits the quickest convergence, arriving at steady state within 0.1 ns of simulation, whereas cMDFF requires around 0.8 ns. Both methods employed eleven maps with Gaussian blurs starting from a width of 5 Å and decreasing in steps of 0.5 Å towards the original reported map. To ensure that the cMDFF procedure did not over-fit the structures, cross-validation using EMRinger and FSC analysis was performed using half-maps from the EMD-5995 entry. iFSC and EMRinger values were found to be almost identical in both direct and cross comparisons. Details are provided in Appendix 1 - Section 11.

In addition to the MDFF simulations described so far for β-galactosidase, other simulations were performed to investigate in more detail the capabilities of MDFF. These simulations are described in Appendix 1 - Section 7. First, it was found that fitting of the C_β atoms to the density is crucial for accurate placement of the sidechains. In agreement with prior EMRringer results (Barad et al., 2015), it is confirmed that MDFF placement of the backbone does not guarantee correct sidechain geometries, even with state-of-the-art CHARMM36 (Klauda et al., 2010) force fields. Second, MD simulation of the cMDFF-fitted β-galactosidase model revealed that this model is indeed an excellent representation of the most probable structures of the thermodynamic ensemble that characterizes the 3.2 Å map.

Refinement of TRPV1

In the third test case, cMDFF and ReMDFF protocols were employed to fit an initial model to the 3.4-Å map (Liao et al., 2013) of the temperature-sensing protein TRPV1. The search model was prepared from the reported de novo structure through an interactive MD protocol described in Appendix 1 - Section 3. The model deviated from the de novo structure by an overall RMSD of 10 Å and locally, by about 25 Å in the vicinity of the ankyrin repeats represented by residues 199 to 430. This degree of deviation is in the ballpark of the lowest resolutions of usable EM maps and, therefore, represents the upper limit of uncertainty between a search model and the fitted structure that MDFF can still reconcile. Having to address an RMSD of 25 Å between search and target models, the present example represents an extreme test case for judging the radius of convergence of the proposed MDFF methods.

Fitting results for TRPV1, described in Appendix 1 - Section 8, were significantly better for cMDFF and ReMDFF than for direct MDFF, but now with much poorer search models than those employed for the β-galactosidase refinements. For example, cMDFF and ReMDFF refinements produced structures that interpret the 3.4-Å TRPV1 map within an RMSD of 2.4 Å and 2.5 Å from the target de novo model, much more accurately than does direct MDFF which converges to structures at an RMSD of 7.9 Å.

The cMDFF- and ReMDFF-obtained structures were found to be better in every overall quality metric in comparison to the de novo structure. Figure 2b illustrates the contrast in fit between the cMDFF and direct MDFF-derived structures. To observe the effect of MDFF on a substantially well-fitted initial structure, a direct MDFF simulation was also performed to refine the de novo structure within the reported map. The simulation yielded a structure that was comparable in all the quality measures considered to the structures obtained from cMDFF and ReMDFF (Appendix 1 - Section 8). However, the TRPV1 fitting results show that cMDFF and ReMDFF can have a radius of convergence as high as 25 Å in RMSD, whereas direct MDFF requires at the outset a well-fitted structure to deliver a satisfactory model. Also, since the number of Ramachandran outliers were minimal in both the 25 Å-deviated and de novo initial models, the cMDFF, ReMDFF, and direct MDFF-refined models exhibited low percentages of Ramachandran outliers, as reflected in Supplementary file 1B.

As was observed already for β-galactosidase, the ReMDFF protocol exhibited the quickest convergence, arriving at steady state within 0.02 ns of simulation, whereas cMDFF required around 0.27 ns. Both methods employed six maps with Gaussian blurs starting from a width of σ = 5 Å and decreasing in steps of 1 Å to the reported σ = 0 Å map. Cross-validation with half-maps was also performed on the cMDFF structure, as per the β-galactosidase simulations, to ensure that it was not over-fitted. As in the case of β-galactosidase, iFSC and EMRinger scores for direct and cross comparisons were similar. FSC analysis results are described in Appendix 1 - Section 11.

A separate set of model validation analyses was performed on the well-resolved TM portion of TRPV1 to pursue a direct comparison of a MDFF refined model with one from Rosetta (Barad et al., 2015). As reported in Appendix 1 - Section 9, MDFF produced results comparable to those of Rosetta when all the heavy atoms are coupled to the density: though the EMRinger score is marginally lower relative to that from Rosetta, the GCC and iFSCs are higher for MDFF; also MDFF provides a marginally higher MolProbity score. Altogether, major discrepencies between Rosetta and direct MDFF that were reported for the high-resolution EM maps (DiMaio et al., 2015) are now absent when employing the cMDFF and ReMDFF protocols, even with poorer choices of search models than those used with Rosetta. Thus, cMDFF and ReMDFF enable flexible fitting techniques to pursue resolution of structures within state-of-the-art maps obtained via cryo-EM.

Model validation

An EM density map represents a thermodynamic ensemble of atomic conformations (Schröder et al., 2007; Brunger et al., 2012; Schröder et al., 2010). Conventionally, however, only a single model representing a best fit to the map is reported. One may ask how statistically representative a single model can be. To quantify the deviation of a fitted model from the rest of an ensemble of simulated molecules, root mean square fluctuation (RMSF) of the model relative to the ensemble-averaged structure was computed during an MDFF refinement simulation employing the protocol described in Methods. In the following, the RMSF of a fitted model is first shown to be indicative of the quality of fit of the model, as well as to represent the degree of natural conformational variation exhibited within the thermodynamic ensemble underlying the map. Second, the RMSF values are found to correlate both locally and globally with the resolution of an EM map, providing an interpretation of map quality based on the inherent (i.e., natural) dynamics of the macromolecule under observation. Finally, RMSF values are also employed to identify optimal B-factor values for the sharpening of a map. Altogether, the results of the present study demonstrate that RMSF of a fitted model during an MDFF refinement provides valuable information on the model.

RMSF and quality of fit

The relationship between RMSF values and quality of model fit is demonstrated for the cMDFF refinement of β-galactosidase at 3.2 Å resolution. The initial conformation is a poor fit of the map, characterized by low values of GCC, LCC, and iFSC (the row containing ‘initial’ structure in Table 1). Such conformations belong to a diverse ensemble of poorly fit structures, explored by the search model in the early phase of the fitting, that gives rise to high initial RMSF values shown in Figure 4—figure supplement 1. In the ending phase of the refinement, well-fitted structures are obtained with improved GCC, LCC, and iFSC (cMDFF row of Table 1). Owing to the high resolution (3.2 Å) of the EM density map, the population of these well-fitted structures is much smaller than that of partially-fitted structures. Thus, as can be observed in Figure 4—figure supplement 1, the converged ensemble of the well-fitted structures exhibit much smaller fluctuations than the initial one. Low RMSF values for the fitted structure indicate therefore, that (i) the structure has been modeled unambiguously within the map, and (ii) the structure can be regarded as representative of the ensemble underlying the 3.2-Å β-galactosidase map; since per-residue fluctuations about the fitted structure (Figure 4—figure supplement 1) are less than 1 Å, only marginal backbone and sidechain variations within the ensemble arise (Singharoy et al., 2013).

RMSF and quality of map

Apart from representing the quality of fit, RMSF values monitored during an MDFF simulation correlate closely with the overall and local resolution of an EM map. Even though high-resolution cryo-EM data are becoming increasingly obtainable, resolution is not always uniform throughout a map. For example, Figure 3 reflects the variation in local resolution of map regions corresponding to residues of β-galactosidase and TRPV1. Conformational flexibility can cause heterogeneity in the cryo-EM data (Leschziner and Nogales, 2007), producing local resolutions lower than that of the overall map.

Figure 3

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Local resolution analysis (Kucukelbir et al., 2014) can be especially important for determining the parts of a high-resolution map that realistically contain side chain information and the parts that do not, preventing over-interpretation of the latter. MDFF protocols can be adjusted to account for such local variations and better inform the process of model validation. For example, if analysis shows that certain residues reside in low-resolution regions of the density, the per-atom weighting factor applied to the forces derived from the density can be lowered.

The RMSF-resolution correlation is found to hold in the cases of TRPV1 (PDB 3J5P), 2.2-Å β-galactosidase (PDB 5A1A), γ-secretase (PDB 5A63 and 4UPC), and the T20S Proteasome (PDB 3J9I). Generally, the lower the resolution of the map, the higher the corresponding overall RMSF during MDFF simulations. For example, the overall RMSF during MDFF of the 4.5-Å γ-secretase model and map is greater than that of the 3.4-Å model and map which, in turn, is greater than that of the 2.2-Å map and model of β-galactosidase (see RMSF labels on the upper column of Figure 4). The correlation between map resolution and model RMSF extends further to local features within the density. In Figure 4 (upper row), RMSFs of atoms plotted against local resolutions of the corresponding map regions display linear correlation between the two quantities. Again, higher RMSF indicates lower local resolution.

Figure 4 with 4 supplements see all

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The physical basis for considering RMSF values of a group of fitted atoms during an MDFF refinement as an indicator of the map resolution follows from the linear correlation of these values with RMSF of the same set of atoms during unbiased MD (Figure 4, bottom row). Noting that the RMSF value during unbiased MD simulations reflects flexibility (Karplus, 1990), this linear correlation clearly establishes the dependence of RMSF during MDFF refinement on the inherent flexibility of the macromolecule, at least for our four demonstration systems (see Appendix 1 - Section 2 for MD simulation details). Since the flexibility of a molecule during the imaging process contributes to the limiting resolution of the resulting EM density map (Kucukelbir et al., 2014), it is not surprising that overall and local map resolutions correlate with overall and local RMSF values of the best fitted model (Figure 4).

In broad terms, through the present study we establish that RMSF, together with GCC, LCC, EMRinger (Barad et al., 2015), and iFSC, provide a comprehensive set of criteria for evaluating model and map quality on both global and local levels. The added value of RMSF is particularly evident on the local level, where the other measures may not perform as consistently. For example, a high LCC may be the result of a highly flexible structure fitting to a low-resolution region of the map, and not necessarily of a good representation of the local structure. As a result, although multiple low-resolution regions of the model in Figure 4—figure supplement 2a possess similar LCCs, disparate RMSFs of the same regions clearly indicate differences in local quality of the model. Likewise, EMRinger scoring, when applied to small groups of residues, does not correlate with local resolution (Figure 4—figure supplement 2b), and, therefore, is incapable of distinguishing regions of small number of atoms by local model quality. In contrast, RMSF clearly resolves local resolution, and, thus, resolves the map and model quality even with as few as 100 atoms.

RMSF and B-factor determination

High contrast within an EM map allows clear identification of secondary structural elements. However, experimental imaging discrepancies arising from specimen movement and charging, radiation damage, and partial microscope coherence, or computational discrepancies due to inaccurate determination of the single particle order parameters (Wade, 1992; Fernández et al., 2008), introduce fuzziness to the EM map, thus hindering secondary structure identification. B-factor sharpening restores lost contrast by resolving the fuzzy features and, therefore, is a crucial step of map generation that affects map interpretation. Here, we describe the use of RMSF as a physical basis for the determination of optimal B-factors that preserve contrast during map sharpening.

Figure 5, Figure 5—figure supplement 3, Figure 4—figure supplement 4 demonstrate the relationship between RMSF and B-factor for the 2.2-Å map of β-galactosidase, 3.4-Å map of TRPV1 and 4.5-Å map of γ-secretase; these three systems were chosen due to their reasonable size and availability of unsharpened data. An initial decrease in the RMSF values is observed with increase in B-factor sharpening of the map. However, the RMSF eventually reaches a minimum before increasing as the B-factor is further increased. Surprisingly, for all the three structures the B-factor corresponding to the minimum RMSF coincides with the one determined by Guinier analysis (Fernández et al., 2008; Rosenthal and Henderson, 2003) to provide maximum contrast. In fact, for TRPV1, B-factors determined from the RMSF minima are found to be higher for the soluble regions than for the transmembrane helices (Figure 5—figure supplement 1, b vs. c), again, in agreement with those B-factors derived from the Guinier analysis. Therefore, the RMSF analysis of maps with varying B-factor sharpening provides an alternate procedure for the determination of the optimal B-factor, which has been traditionally determined by Guinier analysis.

Figure 5 with 4 supplements see all

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The rationale for optimal B-factor selection based on an analysis of RMSF values is the following. As pointed out above and in Figure 4, RMSF of a fitted model is a function of the quality of the corresponding map. Following this argument, the maps in Figure 5, Figure 5—figure supplement 1 and Figure 5—figure supplement 2 that produce the model with the lowest RMSF during fitting represent the density envelopes where atoms can be positioned with the least uncertainty at the given experimental resolutions. The corresponding sharpening B-factor thus provides the optimal contrast for atom placement and secondary structure determination.

Our argument for the RMSF-based selection of B-factors is supported by the well-established quadratic relationship of B-factors with the RMSF of atoms of a structure in an experimental setting, i.e. 8π²/3(RMSF)² (Rosenthal and Henderson, 2003; Liu and Xiong, 2014). This relationship indeed implies that the smaller the average atomic fluctuations, the lower are the B-factors, and by definition (Liu and Xiong, 2014) the higher are the measured structure factor amplitudes, and hence the contrast, of the resulting map in both crystallographic and EM experiments (Liu and Xiong, 2014; Fernández et al., 2008). By analogy, RMSF within MDFF is a measure of these average atomic fluctuations under the experimental setting (Figure 4, bottom row) and, therefore, within a set of maps of varying B-factors, the highest contrast is indicated by the lowest RMSF obtained during fitting of a model. Since Guinier analysis of EM data selects B-factors based on the same atomic displacement-structure factor relationship that justifies the RMSF-based selection of B-factors (Rosenthal and Henderson, 2003), the B-factor selected from our analysis of the lowest RMSF matches excellently with those from the Guinier analysis of EM maps.

The higher contrast of the B-factor sharpened map, over the unsharpened one, is evident from the higher quality of the models derived from the sharpened maps. Presented in Figure 5a, Figure 5—figure supplement 3b, and Figure 5—figure supplement 1a, for the cases of β-galactosidase, γ-secretase, and TRPV1, respectively, are EMRinger scores of models fitted to unsharpened and sharpened maps. These scores indicate clearly that the models derived from maps at B-factor sharpening of −50 (Figure 5a), −100 (Figure 5—figure supplement 3b), and −100 (Figure 5—figure supplement 1a) are more accurate than those derived from the unsharpened ones. Indeed, as further illustrated in Figure 5a, Figure 5—figure supplement 3b, and Figure 5—figure supplement 1, the maximum EMRinger score is attained for the same B-factor that produces the minimum RMSF. Sharpening by B-factors of magnitudes any higher than 50 for β-galactosidase, and 100 for γ-secretase or TRPV1 damages key density features thus reducing the quality of the associated fitted models. In addition, correlation of the RMSF of the fitted structure with that from an unbiased MD of the system is still preserved (Figure 5b and Figure 5—figure supplement 2b) for the B-factor sharpened map, confirming that the dynamical fluctuations of the structure within the sharpened map reflects the inherent dynamics of the system. Altogether, RMSF establishes a unique map-model validation criterion that represents an ensemble view of the fitted structures, while also preventing over-sharpening of the EM maps.

RMSF and per-residue B-factors

The quadratic relationship between RMSF and B-factors can be further employed to determine per-residue B-factors. Presented in Figure 3, the per-residue RMSF${}^2$ of β-galactosidase and TRPV1, derived from MDFF, show excellent agreement with the distribution of local resolution and experimentally reported B-factors: regions with higher RMSF₂ correspond to lower local resolution and higher per-residue B-factors, and vice versa. In fact, the quantitative agreement between the B-factors derived from MDFF through the computation of 8π²/3(RMSF)² (0.02 $<$ (RMSF)²$<$4.43 [Figure 3]) and those reported in the experiments is remarkable (Figure 5—figure supplement 4); a cross-correlation of 55% is found between the data sets, which improves to 60% in the structured regions. Larger discrepencies are observed between the RMSF-based and reported B-factors of TRPV1, particularly in the soluble region (Figure 3b). This discrepency is expected, as for poorly resolved regions MD provides up to six-fold higher B-factors (Kuzmanic et al., 2014); the higher B-factors indeed have been demonstrated to be a more accurate representation of conformational diversity (Kuzmanic et al., 2014). For the TRPV1 example, the highest computed B-factor is 8π²/3*9.25 = 240.5 (Figure 3b) compared to the reported value 53.17. However, the majority of the computed B-factors fluctuatate about the value of 8π²/3*1.73 = 44.98, which is in fair agreement with the reported values of 20–30.

Overall, B-factors of cryo-EM maps are typically calculated by Guinier analysis for maps with resolutions better than ~10 Å (Fernández et al., 2008; Rosenthal and Henderson, 2003). Using a mask around specific regions of interest, estimation of local B-factors for different parts of a map is also possible. However, this method is limited to large domains of macromolecular complexes due to problems associated with tight masking of cryo-EM maps. For maps with resolutions better than ~3 Å, local B-factors could be estimated and refined in X-ray crystallography programs. However, most highresolution cryo-EM maps have resolutions between 3 to 5 Å (Liao et al., 2013; Bartesaghi et al., 2014; Zhao et al., 2015; Fischer et al., 2015). In this resolution range, B-factors could be estimated from MDFF and used as prior information to improve model building and refinement. For resolutions better than 3 Å, MDFF-derived values may serve as initial estimates of B-factors. Furthermore, many cryo-EM maps show local variations in resolution, complicating the model building process. A possible solution may be to combine local B-factor refinement for highresolution regions of a map and B-factors derived from MDFF for lower resolution regions. A resolution-dependent weighting scheme could be incorporated to combine the different values for optimal performance, which may help improve the accuracy of atomic models derived from high-resolution cryo-EM maps.

Flexible fitting methods have facilitated structure determination from low-resolution EM maps for more than a decade (Tama et al., 2004; Suhre et al., 2006; Velazquez-Muriel et al., 2006; Orzechowski and Tama, 2008; Topf et al., 2008; Kovacs et al., 2008; Lopéz-Blanco and Chacón, 2013; Wu et al., 2013) and continue to be the methods of choice for resolving molecular systems with atomic resolution. MDFF, in particular, has been a front-runner among methods that have facilitated the discovery of some of the most complicated structures in modern day structural biology (Hsin et al., 2009; Sener et al., 2009; Gumbart et al., 2011; Frauenfeld et al., 2011; Zhao et al., 2013; Wickles et al., 2014).

cMDFF and ReMDFF, the new variants of MDFF introduced above, offer now accurate fitting of atomic-level structures within sub-5 Å EM maps, a feat thus far inaccessible to direct MDFF. These new methods extend the radius of convergence of MDFF to at least 25 Å, fitting models to maps of resolutions as high as 3.2 Å. This radius of convergence is at least twice that reported for Rosetta refinements of the 20S proteasome (DiMaio et al., 2015). Such a broad radius of convergence will allow the refinement of extremely poorly guessed initial models with MDFF, as demonstrated in the cases of β-galactosidase and TRPV1 reported here.

ReMDFF simulations involving the so-called replica-exchange molecular dynamics converge quickly using a small number of replicas and are thus amenable to cloud computing applications. Running ReMDFF on the cloud lowers greatly the barrier to usage of the method, providing a cost-effective and practical solution to fitting structures to high-resolution cryo-EM densities for researchers who neither own nor can administer their own advanced computer hardware.

The accuracy of structures refined by cMDFF and ReMDFF has been confirmed by standard error analysis protocols, both in terms of quality of fit as well as in terms of the quality of the model. The results clearly show that sidechain refinements through MDFF produce accurate placement of C_α and C_β atoms and modeling of the associated dihedrals.

Beyond the standard error analysis protocols, which apply to single static structures, the quality of both the fit and the model can be evaluated by ensemble-based measures. An example demonstrated in this study pertains to the use of RMSF during MDFF refinement for simultaneously evaluating the quality of model, map, and fit. Furthermore, utilizing the fact that the inherent flexibility of a macromolecule is a key determinant of the achievable resolution of the corresponding map, RMSF values have been employed to identify the optimal amount of sharpening for a given map offering the highest contrast, as established typically through Guinier analysis. The RMSF computations in MDFF provide a viable means of determining per-residue B-factors.

Altogether, interpretation of a map as being representative of an ensemble rather than a single model brings to light new ways of model validation. EM maps, including those at high-resolution, typically do not have a uniform local resolution (Kucukelbir et al., 2014) and contain low-resolution regions, such as flexible exterior or transmembrane segments (Leschziner and Nogales, 2007). The sub-5 Å EM maps have also been able to resolve proteins in multiple conformations (Matthies et al., 2016). Both these type of maps will continue to benefit from accurate ensemble-based flexible fitting techniques for the foreseeable future. MDFF provides a natural method to model flexible regions; de novo models constructed for one conformational state can be flexibley fitted into the density of the other state(s), thus avoiding the arduous task of model construction for all the conformational states for capturing a conformational transition process with cryo-EM.

In the following section, we first outline the methodology underlying direct MDFF, cMDFF, and ReMDFF, along with specific protocols applied for the refinement of β-galactosidase and TRPV1 models. Second, we describe the computations of local and global root mean square fluctuation (RMSF) values, that are utilized for the evaluation of map-model quality. Finally, details on the computational implementation of all the three MDFF protocols are provided.

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Author details

1. Abhishek Singharoy

   Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, United States

   Contribution

   AS, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   Contributed equally with

   Ivan Teo and Ryan McGreevy

   For correspondence

   singharo@illinois.edu

   Competing interests

   The authors declare that no competing interests exist.

2. Ivan Teo

   1. Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, United States
   2. Department of Physics, University of Illinois at Urbana-Champaign, Urbana, United States

   Contribution

   IT, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   Contributed equally with

   Abhishek Singharoy and Ryan McGreevy

   Competing interests

   The authors declare that no competing interests exist.

3. Ryan McGreevy

   Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, United States

   Contribution

   RM, Conception and design, Acquisition of data, Analysis and interpretation of data, Drafting or revising the article

   Contributed equally with

   Abhishek Singharoy and Ivan Teo

   Competing interests

   The authors declare that no competing interests exist.

4. John E Stone

   Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, United States

   Contribution

   JES, Conception and design, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

5. Jianhua Zhao

   Department of Biochemistry and Biophysics, University of California San Francisco School of Medicine, San Francisco, United States

   Contribution

   JZ, Analysis and interpretation of data, Drafting or revising the article

   Competing interests

   The authors declare that no competing interests exist.

6. Klaus Schulten

   1. Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, United States
   2. Department of Physics, University of Illinois at Urbana-Champaign, Urbana, United States

   Contribution

   KS, Computational simulation technologies, Conception and design, Analysis and interpretation of data, Drafting or revising the article

   For correspondence

   kschulte@ks.uiuc.edu

   Competing interests

   The authors declare that no competing interests exist.

   "This ORCID iD identifies the author of this article:" 0000-0001-7192-9632

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