It has been clear since the early days of structural biology in the late 1950s that proteins and other biomolecules are continually changing shape, and that these changes have an important influence on both the structure and function of the molecules. X-ray diffraction can provide detailed information about the structure of a protein, but only limited information about how its structure fluctuates over time. Detailed information about the dynamic behaviour of proteins is essential for a proper understanding of a variety of processes, including catalysis, ligand binding and protein–protein interactions, and could also prove useful in drug design.

Currently most of the X-ray crystal structures in the Protein Data Bank are ‘snap-shots’ with limited or no information about protein dynamics. However, X-ray diffraction patterns are affected by the dynamics of the protein, and also by distortions of the crystal lattice, so three-dimensional (3D) models of proteins ought to take these phenomena into account. Molecular-dynamics (MD) computer simulations transform 3D structures into 4D ‘molecular movies’ by predicting the movement of individual atoms.

Combining MD simulations with crystallographic data has the potential to produce more realistic ensemble models of proteins in which the atomic fluctuations are represented by multiple structures within the ensemble. Moreover, in addition to improved structural information, this process—which is called ensemble refinement—can provide dynamical information about the protein. Earlier attempts to do this ran into problems because the number of model parameters needed was greater than the number of observed data points. Burnley et al. now overcome this problem by modelling local molecular vibrations with MD simulations and, at the same time, using a course-grain model to describe global disorder of longer length scales.

Ensemble refinement of high-resolution X-ray diffraction datasets for 20 different proteins from the Protein Data Bank produced a better fit to the data than single structures for all 20 proteins. Ensemble refinement also revealed that 3 of the 20 proteins had a ‘molten core’, rather than the well-ordered residues core found in most proteins: this is likely to be important in various biological functions including ligand binding, filament formation and enzymatic function. Burnley et al. also showed that a HIV enzyme underwent an order–disorder transition that is likely to influence how this enzyme works, and that similar transitions might influence the interactions between the small-molecule drug Imatinib (also known as Gleevec) and the enzymes it targets. Ensemble refinement could be applied to the majority of crystallography data currently being collected, or collected in the past, so further insights into the properties and interactions of a variety of proteins and other biomolecules can be expected.

Since the dawn of structural biology there have been experimental observations of dynamic motion in proteins and other biomolecules (Linderstrøm-Lang and Schellman, 1959). Multiple biophysical methods have firmly established that such atomic ‘wigglings and jigglings’ (Feynman et al., 1963) play an inherent role in both protein structure and function; and, in conjunction with high-resolution structures insights into dynamics aid the understanding of biomolecular functions in catalysis, ligand or drug binding and macromolecular interactions. Presently X-ray diffraction and NMR spectroscopy are the primary source of data for high-resolution protein structures. Whereas microscopy methods may provide information regarding long-range conformational changes, NMR characterizes fluctuations at atomic detail. However, due to the challenging nature of such experiments the number of dynamics studies is relatively sparse in contrast with the wealth of structural information available in the Protein Data Bank (PDB) (Berman et al., 2000). The majority of entries in the PDB derived from X-ray diffraction data are presented as static, single, structures, although there is often extensive disorder resulting from protein dynamics and crystal-lattice distortions. Extracting atomic fluctuations from these diffraction data would dramatically increase the scope for dynamics studies of biomolecules and potentially reveal atomic details of structure–function–dynamic mechanisms that have previously been obscured.

The diffraction data of proteins are affected by multiple sources of disorder, notably arising from atomic vibrations, concerted motions of protein domains and inter-molecular lattice distortions. Structural models of proteins should account for both anisotropic and anharmonic distributions around the mean atomic positions to reproduce the observed Bragg intensities accurately (Vitkup et al., 2002; Furnham et al., 2006). However, explicit modelling of such distributions in macromolecules using current methods requires extensive parameterization inappropriate for the diffraction quality of a typical protein crystal. Multi-conformer structures represent both anisotropic and anharmonic disorder, but despite numerous attempts at automating the inclusion of minor conformations (DePristo et al., 2004; Levin et al., 2007; Terwilliger et al., 2007; Korostelev et al., 2009; van den Bedem et al., 2009; Lang et al., 2010), 95% of all protein residues in the Protein Data Bank (PDB) (Berman et al., 2000) derived from diffraction data are modelled with a single conformation (Lang et al., 2010). As opposed to multiple discrete models, a MD simulation with time-averaged restraints (Gros et al., 1990) results in a population of structures in which the individual models are interrelated by a Boltzmann-weighted energy function. This method introduced by Torda et al. (1989) and implemented in macromolecular crystallography by Gros et al. (1990), showed a reduction in R-value. However, cross-validation introduced subsequently (Brünger, 1992) revealed chronically over-fitted models (Burling and Brunger, 1994; Clarage and Phillips, 1994; Schiffer et al., 1995).

Here, we present an ensemble-refinement method that restricts the number of structures modelled and thereby prevents over-fitting of the data. We model large-scale motions, attributable to, for example, lattice distortions, by an underlying global disorder model. This approach allows MD simulations to sample local atomic fluctuations only, without the need for sampling large-scale global disorder. We show that the method yields reproducible ensembles with improved fit to the X-ray data, as validated by cross validation, R_free (Brünger, 1992), and stereochemical analyses. Analyses of the ensembles show that detailed features are observed indicating atomic fluctuations that may be relevant for the biological function of the macromolecules.

We have shown that far more structural information can be reliably extracted from protein diffraction data than is achieved to date by traditional single-structure modelling methods. Our ensemble refinement method samples distributions that reflect structural details of protein dynamics. The resulting ensemble models provide a more comprehensive description of the molecules and allow interpretation of the molecular function in terms of both the three-dimensional arrangements of the protein residues and their flexibilities. Moreover, ensemble models minimize the risk of structural over-interpretation associated with the seemingly rigid single-structure models. We found comparative analyses of protein molecules in different states to be very useful for identifying detailed changes in structural dynamics that may be mechanistically relevant for the molecular function.

Partitioning large-scale disorder into a global model separates intermolecular variations of protein packing in the crystal from the detailed intra-molecular atomic fluctuations. Effectively, the X-ray gradient dictates the MD sampling to yield featureless, (mF_obs−DF_model)exp[iφ_model], electron-density difference maps, while the global disorder model is accounted for by taking B_TLS into account when computing the atomic densities. In this way, the ensemble of structures is generated to model the anisotropic and anharmonic electron-density distributions precisely, while being restrained by the bonded and non-bonded energy terms used in the MD simulation. The separation of global disorder and local atomic fluctuations contrasts the original approach by Gros et al. (1990), where the MD sampling had to account for both the large scale global disorder and local fluctuations leading to very long relaxation times τ_x of 16 ps. In the current work much shorter relaxation times of 0.25–2 ps can be used, thereby limiting potential over-fitting markedly. The method is applicable to data with a wide range of upper resolution limits. We see marked improvements in R_free for datasets ranging from 1.5 to 2.6-Å upper-resolution limit. A detailed interpretation of the ensembles is allowed, supported by the very high local correlations between independent ensemble refinements. However, at lower resolution limits and for highly disordered loops the local correlation between independent runs drops and detailed interpretation is not feasible. Thus, even though the number of independent parameters in an ensemble model is not clearly defined (and therefore the parameter-to-observation ratio is unclear), the gain in R_free and the very high local correlations between independent runs indicate a high reliability of the ensemble models. However, the method is not a panacea for highly disordered protein regions. In the absence of ordered conformations for a certain region of the protein (as implicitly defined by the diffraction data) the ensemble refinement will sample diverse conformations in order to prevent the build up of negative peaks in the electron-density difference map. In other words, if the data ‘says’ that a region is disordered, ensemble refinement will generate diverse conformations for that region. Furthermore, dataset pathologies caused, for example, by radiation damage may have confounded effects that obscure the dynamics inherent to the protein molecule. Thus, perhaps somewhat counter-intuitively, this modelling method that accounts for inherent protein dynamics does not help to resolve structural details of disordered regions, but is particularly suited to resolve dynamical fluctuations in ordered parts of the protein structure.

Ensemble refinement of 20 protein datasets highlighted global dynamics features of protein molecules. Surprisingly, in some cases the ensembles indicated the existence of folded protein structures that display molten cores. Most likely, such molten cores may indicate intermediates of protein molecules that function in larger complexes (such as PAK pilin), or alternatively these molten cores support dynamical fluctuations that are needed for ligand binding and enzyme functioning (as for birch pollen allergen and proline isomerase respectively). Furthermore, the ensembles show details of specific order–disorder transitions, or conformational exchanges, between active site and core residues (as for HIV protease in the unbound and bound state) that suggest a mechanism of entropy compensation to support the enzymatic activity. The difference in dynamics observed between the ensembles of proline isomerase at cryo and ambient temperatures indicates that flash freezing of a crystal anneals local conformational fluctuations and thereby removes protein dynamics that may be functionally relevant.

In conclusion, this new method of modelling X-ray diffraction data reveals a wealth of detailed information about the dynamics of biomolecules that complements the high-resolution structural information already available from the crystallographic experiment. In depth understanding of structure–dynamics in biomolecules will enhance our insights into the molecular mechanisms that underlie biological processes.

The method of ensemble refinement was implemented in the Phenix software (Adams et al., 2010). Adaptations and new procedures developed for ensemble refinement are given in section ‘Ensemble refinement methods’. Simulations were performed as described in section ‘Ensemble refinement protocol’. Details of the single-structure re-refinements used for comparison with the ensemble models are given in section ‘Single structure re-refinements’. Validation of the global disorder TLS model, the dependency of ensemble refinement on the starting structure and additional ensemble refinement calculations are given in ‘Additional ensemble refinement calculations’.

Ensemble refinement was performed using phenix.ensemble_refinement, as will be made available in the next release of Phenix.

Ensemble refinement methods

Time-averaged restraints

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The overall model structure factors are calculated as (1), defined by Afonine et al. (2005), incorporating overall anisotropic scaling (Sheriff and Hendrickson, 1987) and bulk solvent contributions (Jiang and Brünger, 1994).

(1) ${\mathit{F}}_{\mathit{model}}=k{e}^{\left(-\frac{{\mathit{h}}^T{\mathit{A}}^{-1}{\mathit{B}}_{\mathit{cart}}{\left({\mathit{A}}^{-1}\right)}^T\mathit{h}}{4}\right)}\left({\mathit{F}}_{calc}+k_{\mathit{sol}}{e}^{\left(-\frac{B_{\mathit{sol}}{s}^2}{4}\right)}{\mathit{F}}_{\mathit{mask}}\right),$

where, k is the overall scale factor, h is the column vector with Miller indices, A is the orthogonalisation matrix, B_cart is the anisotropic scale matrix, F_calc is the structure factors calculated from atomic model, k_sol and B_sol are the parameters for the flat bulk solvent model and F_mask are the structure factors calculated from bulk solvent mask.

In order to restrain the instantaneous structures produced during the MD simulation with time and spatially averaged X-ray data, time-averaged restraints are used (Gros et al., 1990). This produces time-averaged (or rolling-average) structure factors such that (1) becomes (2).

(2) ${\mathit{F}}_{\mathit{model}}^t=k{e}^{\left(-\frac{{\mathit{h}}^{\text{T}}{\mathit{A}}^{-1}{\mathit{B}}_{\mathit{cart}}{\left({\mathit{A}}^{-1}\right)}^T\mathit{h}}{4}\right)}\left({\langle {\mathit{F}}_{\mathit{calc}}\rangle }_t+k_{\mathit{sol}}{e}^{\left(-\frac{B_{\mathit{sol}}{s}^2}{4}\right)}{\langle {\mathit{F}}_{\mathit{mask}}\rangle }_t\right).$

This is a time-dependent memory function, that is a ‘rolling’ average, where the size of the averaging window is controlled by the τ_x parameter (typically 1 ps). This averaging function is updated with the current individual structure every 10 time-steps (∆t) during the simulation and is implemented as in (3).

(3) ${\langle {\mathit{F}}_{\mathit{calc}}\rangle }_t=e^{-\mathit{\Delta t}/{\tau }_x}{\langle {\mathit{F}}_{\mathit{calc}}\rangle }_{t-\mathit{\Delta t}}+\left(1-e^{-\mathit{\Delta t}/{\tau }_x}\right){\mathit{F}}_{\mathit{calc}}^t.$

Dual explicit-bulk solvent model

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Due to the stochastic behaviour of solvent molecules and the number of partially disordered or low occupancy sites, explicitly modelled solvent atoms are repositioned every 250 time-steps. Electron-density and difference density maps are generated using F_model^t, excluding reflections in the free R set. Water oxygen atoms with an electron-density peak >1.0 σ in the 2mF_obs − DF_model map or a peak >3.0 σ in the mF_obs − DF_model map are preserved, otherwise the atom is removed. New water atoms are added for positions which have a 2mF_obs − DF_model peak >1.0 σ and a mF_obs − DF_model peak >3.0 σ, and are between 1.8–3.0 Å in distance to an existing atom. For high-resolution cases these criteria are adjusted to include mF_obs − DF_model map peaks >2.5 σ. Newly positioned atoms are assigned a random, Boltzmann-weighted, velocity. Explicitly modelled solvent atoms contribute to the atomic model (F_calc).

Bulk solvent is modelled using a solvent mask (Afonine et al., 2005). The mask structure factors (F_mask) are averaged in the same manner as the atomic model (F_calc) (4).

(4) ${\langle {\mathit{F}}_{\mathit{mask}}\rangle }_t=e^{-\mathit{\Delta t}/{\tau }_x}{\langle {\mathit{F}}_{\mathit{mask}}\rangle }_{t-\mathit{\Delta t}}+\left(1-e^{-\mathit{\Delta t}/{\tau }_x}\right){\mathit{F}}_{\mathit{mask}}^t.$

The k_sol and B_sol bulk solvent parameters and B_cart scaling parameters used for the duration of the simulation are calculated from the starting structure as described previously (Afonine et al., 2005), they are re-optimized for the final ensemble.

Constrained target functions

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The overall scale factor, k, is constrained during the simulation. For the maximum-likelihood target function, as shown for acentric reflections, (5) during normalisation (6), the sum of the rolling-average structure factor array (2) is scaled to the sum of the structure factor array from the starting model (F_ref) as shown in (7).

(5) $P_{\mathit{x-ray}}=\frac{2E_{\mathit{obs}}}{1-{\mathrm{\sigma }}_A^2}\text{exp}\left(-\frac{E_{\mathit{obs}}^2+{\mathrm{\sigma }}_A^2{E}_{\mathit{model}}^t}{1-{\mathrm{\sigma }}_A^2}\right)I_0\left(\frac{2E_{\mathit{obs}}{\mathrm{\sigma }}_A{E}_{\mathit{model}}^t}{1-{\mathrm{\sigma }}_A^2}\right),$

(6) $E_{\mathit{model}}^t=\frac{k{\mathit{F}}_{\mathit{model}}^t}{\sqrt{\epsilon {\Sigma }_N}},$

(7) $k=\frac{\underset{\mathit{hkl}}{\Sigma }{\mathit{F}}_{\mathit{ref}}}{\underset{\mathit{hkl}}{\Sigma }{\mathit{F}}_{\mathit{model}}^t},$

where, E_obs and E_model are the normalised structure factors, σ_A is the Sigma-A weighting factor, I₀ is a modified Bessel function of order 0 and ε is the expected intensity factor.

Temperature bath and X-ray weight

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The simulations are performed such that the non-solvent atoms are at a target temperature (T_target) of 300 K, where the simulation is coupled to a velocity-scaled temperature-bath (Berendsen et al., 1984). The temperature bath is set to a value less than 300 K, typically 295–298 K. Because the X-ray restraints are computed from a time-dependent memory function, the X-ray energy term is non-conservative and thus heating occurs. During the equilibration phase the X-ray weight (w_x-ray) is modulated by the temperature of the protein atoms (T_protein) every 10 time-steps (∆t), such that the non-solvent atoms sample consistently at the target temperature (8).

(8) $w_{\mathit{x-ray}}^t=w_{\mathit{x-ray}}^{t-\mathit{\Delta t}}\frac{T_{\mathit{target}}}{T_{\mathit{protein}}}.$

Thus, the thermostat offset controls the X-ray weight in a system independent manner whilst maintaining the target temperature. In the acquisition phase the X-ray weight is fixed to the averaged value used in the equilibration phase.

TLS approximation of the global disorder

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The partitioning of inter-molecular disorder is performed before the start of the simulation using ADPs from the traditionally refined starting structure. TLS groups are assigned per molecule or domain as appropriate to model global packing disorder. For each group, TLS parameters are fitted to the ADPs of the starting structure for all non-solvent, non-hydrogen atoms. The agreement of the isotropic equivalents for the fitted TLS ADPs (B_tls) and the reference ADPs (B_ref) is scored as (9) for all non-solvent, non-hydrogen atoms.

(9) $R_i=\left|B_{\mathit{ref}}^i-B_{\mathit{tls}}^i\right|.$

A percentile of atoms with the poorest fitting ADPs (p_TLS) are excluded from the next round of TLS parameter fitting and repeated until the fitted TLS parameters converge. The converged TLS parameters are then applied to all atoms within that group for the duration of the simulation. Solvent atoms are assigned to the TLS group of the closest non-water atom, this assignment is updated every 250 time-steps. This TLS model produced lower R-values than using the ADP values from the re-refined single structure or using the one overall isotropic B-factor for all atoms in the model (Table 5).

Table 5

PDB	Resolution (Å)	Global isotropic B-factor				Refined ADPs			Fitted TLS ADPs
		Rwork	Rfree	Wilson B-factor (Å2)	Global B-factor (Å2)	Rwork	Rfree	Scale factor	Rwork	Rfree	pTLS
3K0M	1.3	0.117	0.147	12.0	12.0	0.125	0.146	0.9	0.103	0.130	0.3
3K0N	1.4	0.121	0.153	19.1	19.1	0.126	0.153	0.9	0.114	0.133	0.1
1UOY	1.5	0.103	0.148	10.4	9.4	0.107	0.144	0.9	0.101	0.136	0.3
3CA7	1.5	0.129	0.194	16.8	13.4	0.142	0.192	0.9	0.142	0.190	0.5
1X6P	1.6	0.108	0.158	15.9	12.7	0.113	0.152	0.8	0.121	0.150	0.8
1F2F	1.7	0.116	0.184	15.6	14.8	0.123	0.167	0.8	0.126	0.167	0.7
1BV1	2.0	0.125	0.192	22.6	18.1	0.135	0.191	0.8	0.145	0.182	0.6
Mean	-	0.117	0.168	-	-	0.125	0.164	-	0.122	0.155	-

Generation of the final ensemble

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Structure factors for the final ensemble are calculated from the population of collected structure as in (2) where F_calc and F_mask are defined as (10) and (11).

(10) ${\langle {\mathit{F}}_{\mathit{calc}}\rangle }_{\mathit{final}}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{\mathit{F}}_{\mathit{calc}}^i,}$

(11) ${\langle {\mathit{F}}_{\mathit{mask}}\rangle }_{\mathit{final}}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{\mathit{F}}_{\mathit{mask}}^i.}$

The acquisition phase is split into several time blocks, in each of which 250 structures are typically stored. The R-values of all possible contiguous time blocks are calculated and the periods with the lowest R_work are selected. This selection reduces the R_work by 0–1.0% (mean improvement in 0.3%). For the 1YTT dataset with high quality experimental phases, the block selection for lowest R_work corresponds well with the overall map correlation coefficient computed between the experimentally phased map and the map derived from the ensemble model (Figure 13). Next, to reduce the redundancy in the number of structures in the final ensemble (during the simulation thousands of structures are collected), we calculate the smallest number of structures that reproduce the R_free within 0.1%. This is performed by iteratively parsing the stored structures with increasing time spacing (see Figure 1D). The overall and bulk-solvent scale factors are optimised for the final ensemble. The ensembles of structures are stored using the standard PDB format for multiple models, with B-factors listed as computed from the TLS model and overall B-factor scaling contributions.

Figure 13

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Calculation of atomic positional probability

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All atoms comprising the ensemble are assigned a probability (P_i) based on the positional likelihood of atom i in a given model relative to the complete ensemble of models. F_calc electron-density maps are calculated for each model in the ensemble and 〈F_calc〉 electron-density map is calculated for the complete ensemble as (10). P_i is calculated as (12).

(12) $P_i=\frac{{\rho }_{\langle {\mathit{F}}_{\mathit{calc}}\rangle }^i}{{\rho }_{{\mathit{F}}_{\mathit{calc}}}^i}.$

Calculating from an electron-density function allows for non-Gaussian distributions unlike RMSF, which is calculated from mean atomic position. These probabilities aid the visual inspection of the ensemble models and allow the observer to control the level of detail displayed (Figure 14).

Figure 14

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Author details

1. B Tom Burnley

   Crystal and Structural Chemistry, Bijvoet Center for Biomolecular Research, Department of Chemistry, Faculty of Science, Utrecht University, Utrecht, The Netherlands

   Contribution

   BTB: developed the method, programmed the method in PHENIX, analysed the data and wrote the manuscript

   Competing interests

   The authors have declared that no competing interests exist

2. Pavel V Afonine

   Lawrence Berkeley National Laboratory, Berkeley, United States

   Contribution

   PVA: helped program the method in PHENIX and assisted in the writing of the manuscript

   Competing interests

   The authors have declared that no competing interests exist

3. Paul D Adams

   1. Lawrence Berkeley National Laboratory, Berkeley, United States
   2. Department of Bioengineering, University of California Berkeley, Berkeley, United States

   Contribution

   PDA: helped program the method in PHENIX and assisted in the writing of the manuscript

   Competing interests

   The authors have declared that no competing interests exist

4. Piet Gros

   Crystal and Structural Chemistry, Bijvoet Center for Biomolecular Research, Department of Chemistry, Faculty of Science, Utrecht University, Utrecht, The Netherlands

   Contribution

   PG: developed the method, analysed the data and wrote the manuscript

   For correspondence

   p.gros@uu.nl

   Competing interests

   The authors have declared that no competing interests exist

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