2.1 Deriving refinement forces

Our algorithm to refine all-atom models into a cryo-EM density map ρ with molecular dynamics is based on initially roughly aligning density map and target structure, generating a model density from coordinates , and then determining a fitting potential U_fit based on a comparison of the generated model density and the target cryo-EM density (Fig 1). This potential is used to derive fitting forces, which are then combined with the force field potential U_ff based on a heuristic balance between the density-derived forces and force field determined by a force constant k.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Atomic structure models are refined into a cryo-EM density using biasing forces that maximize similarity between model and map.

A refinement/simulation is initialized with an atomic model (orange) and a density map (blue). A model density is generated in each voxel (grey boxes). Voxel-wise similarity scores between model density and cryo-EM density are akin to a noise model (light blue curve). The gradient of the similarity score determines the fitting forces (blue arrows). Together with a molecular dynamics force field (red arrows), the fitting forces enable model coordinate updates (dark orange) that make the model more similar to the density under force field constraints. New model densities are generated iteratively from the updated model in each time step of the simulation until acceptable convergence is reached.

https://doi.org/10.1371/journal.pcbi.1011255.g001

The combined driving forces are determined by the total potential energy, (1)

Assuming that a single configuration of atoms gives rise to the observed cryo-EM density, Bayes’ theorem quantifies the probability density that the given model describes the cryo-EM data as [17] (2)

Boltzmann inversion at temperature T connects the left-hand sides of Eqs (1) and (2) [21] where c is an arbitrary potential energy offset (3) (4)

In this formalism, the force field provides the prior that would have determined the model without any additional observations, while the fitting potential provides the conditional probability that a particular given structure yields a target density .

Splitting the fitting potential into a coordinate and density independent force-constant k and a similarity score reveals the similarity to previous approaches [8, 12]. We find that applied density forces imply a similarity score between the model structure and target density, and vice-versa. With this ansatz, Eq (4) relates similarity measures like cross-correlation or inner-product to their implicit assumptions about the likelihood function above, and in turn enables the construction of new similarity scores that drive refinement procedures depending on the assumptions about the underlying measurement process.

2.2 Maximum likelihood model yields negative relative entropy as similarity score

To derive a new refinement potential from the likelihood of measuring a density given the structure, we assume that cryo-EM densities are linked to atom-electron scattering probabilities, where electron-atom interaction leads to a phase shift in electrons. With this assumption, two steps are necessary to calculate the density likelihood from coordinates. First, an electron-scattering probability density ρ^s is created from a given structure. Second, this density is compared to a given measured density. Two further assumptions enable the derivation of new similarity scores.

For the per-voxel scattering probability, we present two different sets of assumptions, leading each to a refinement potential in their own right. Many more assumptions may be laid out; here we choose to present those that go beyond previous modeling, yet are well-treatable in model complexity and integratable into a molecular dynamics framework where forces have to be calculated numerically stable and fast. Therefore, we choose not to integrate the full image formation from the microscope detector to three-dimensional density but rather start the modeling process with a three-dimensional density and some additional assumptions on what each voxel represents. By presenting two different approaches to what voxel values in a density present, it should become even clearer how to adapt our modeling to more complex models.

In the first model we assume that an incident electron will scatter at exactly one voxel and that the probability at which voxel to scatter is proportional to the density values. This requires ∑_v ρ = 1, which we achieve via rescaling the density values. The free rescaling of cryo-EM density values is motivated from the normalization to unity variance around the particle region during image processing [22]. Introducing the expected number of density interactions, s, the scattering process is described via a Dirichlet distribution, (5)

With some approximations (S1 Appendix), we find (6) This newfound potential relates to the traditionally used refinement potentials by defining the similarity score (7)

Using this definition we observe that the scaling parameter s and force constant are related via k = k_BTs.

Even though s can be estimated from a Bayes’ approach with a conditional posterior estimate on s with a prior p(s), to obtain the likelihood to observe a density, given coordinates (see S1 Appendix), we choose a different approach for the following reason. Contributions to this conditional posterior are exponentially weighted with S_I, so that a good estimate for the parameter depends on the ability to create a sufficiently representative distribution of similar to the structure. In most cases, however, we expect that we need density-guided simulations to force molecules away from initial configurations over significant energy barriers to be able to sample the relevant configurations that contribute to the estimate of s. To overcome these issues, we heuristically scale s with a protocol described below. This allows us to generate a trajectory with structures with ever-increasing likelihood of good structure-to-map overlap, but admittedly at the cost of not sampling from a proper posterior distribution.

In this Dirichlet distribution-based picture the reported density is treated as a probability density, requiring the removal of negative values and normalization to unity. The resulting potential of this modeling approach is proportional to the Kullback-Leibler divergence between simulated and experimental density with a free scaling parameter. This potential in turn can be seen as an inner-product based potential where density is replaced by its logarithm.

In an alternative picture, reported cryo-EM densities at each voxel are proportional to interaction counts rρ_v of N incident electrons, where the scaling factor r is undetermined as mentioned above. We assume that measured scattering probabilities per voxel are independent of other voxel values. This does not exclude spatial correlation between density data but states that the scattering process in one voxel does not influence the electron interaction in other voxels. With this result, it suffices to define a probability distribution at each voxel.

This picture assumes that vitreous ice does not contribute to the scattering, which is commonly achieved by shifting the offset of cryo-EM densities so that water density is represented with voxel values that fluctuate around zero. Only accounting for positive density, we describe this scattering interaction process by a Poisson distribution with parameter . While it is theoretically possible to expand the model to include noise fluctuations and negative densities, we omit this for the sake of reducing model complexity.

With these assumptions (detailed algebraic transformations in S1 Appendix), (8)

In analogy to above, the unknown scaling factor r would be treated as a scale parameter and may be estimated from a conditional posterior distribution, but is left as a free parameter to be heuristically scaled here. We obtain a similarity score between simulated model density and cryo-EM density proportional to the negative relative entropy, or Kullback-Leibler divergence after normalizing ρ such that ∑_v r₁ρ_v = 1, with r = r₁r₂, (9)

Similarly to above, we find that with this similarity score definition, force constant and r₂ are related by k = r₂k_BT (see S1 Appendix).

The newly derived relative-entropy-based similarity score has a domain of [−∞, 0] with perfect agreement at zero. Due to the term, it differs prominently from established similarity scores like cross-correlation [11] and inner-product (formulated as a force following the gradient of a smoothed inverted density which is equivalent in this approach [12]; see S1 Appendix). In contrast, the relative-entropy based score receives the largest contribution from voxels where cryo-EM data has no corresponding model density data.

This leads to a different behavior from established similarity scores with local minima for locally good agreement with cryo-EM data while the relative-entropy based potential will only have minima where there is good global agreement between structure and density. As a consequence, the relative-entropy based density potential is expected to perform better in situations where other potentials cannot escape local minima, at the cost of higher sensitivity to noise in the data, and especially additional density data that is not accounted for in the atomic model.

2.3 The potential energy landscapes based on relative entropy are smooth

The proposed relative entropy density-to-density similarity measure has favorable properties in one-dimensional model refinement of one and two particles to a reference density (Fig 2). Both newly derived potentials have the same analytical form for our one-dimensional model case.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Similarity score determines ruggedness of the effective refinement potential energy landscape, also when balancing it with structural bias.

From top to bottom: a One-dimensional refinement of a single particle (black circle) towards a Gaussian-shaped density (gray) with inner-product (purple), cross-correlation (ochre), relative-entropy swapped (dark blue) and relative-entropy (green) as similarity scores. b Expanded model with two particles (black circles, x₁ smaller and x₂ larger) with two amplitude peaks in a one-dimensional density and target distribution (gray), and the resulting two-dimensional effective potential energy landscapes for inner-product, cross-correlation, swapped relative-entropy and relative-entropy similarity measures. c Combination of the similarity measure and force field contribution to the potential energy landscape, exemplified by a harmonic bond that keeps particles at half the distance between the Gaussian centers. For all relative weights of the contributions of the refinement potential and bond potential energy landscape (ratio 1:2 upper panel, 2:1 middle panel, as illustrated by the scale on the left), the relative entropy similarity score produces smooth landscapes with minima at the positions that are expected from the model input.

https://doi.org/10.1371/journal.pcbi.1011255.g002

In contrast to cross-correlation and inner product similarity measures that have a steep and sudden onset for refinement forces in one dimension, the relative-entropy similarity score has a harmonic shape with long-ranged interactions that allow for efficient minimization. Using relative-entropy, the particle to be refined is attracted by a harmonic spring-like potential to the best-fitting position; far away from the minimum forces are large, but their magnitude decreases monotonically as the minimum is approached. Inner-product and cross-correlation based fitting potentials, however, exert almost no force on the particle outside the Gaussian spread width, while exerting a suddenly increasing force when moving closer to the Gaussian center, and are only insignificant very close to the minimum.

For the refinement of two particles, this advantage is only maintained for one of the newly derived potentials, where the relative-entropy-based potential energy landscape is less rugged and has fewer pronounced features and minima than the corresponding landscapes for the inner-product and cross-correlation based potentials (Fig 2). Only a single diagonal barrier is found in the relative-entropy-based potential landscape, corresponding to a “swapping” of particle positions, which alleviates the search for a global minimum. The inner-product-based free energy landscape has its minimum at a configuration where both particles are at the same position at the highest density. This issue can be alleviated in practical applications through a force-field prior that would enforce a minimum distance between the atoms (e.g. through van der Waals interactions). The swapped relative entropy potential on the other hand exhibits behavior similar to the inner-product, with similar minima but an overall smoother energy landscape.

To model the influence of a force field, the two particles were connected with a harmonic bond with increasing influence. The balance between density-based forces and bond strongly determines the shape of the resulting energy landscape, but here too relative entropy provides a smoother landscape less sensitive to the specific relative weight of refinement and bond potentials.

2.4 Adaptive force scaling reduces work exerted during refinement and allows for comparison of density-based potentials

To enable convergence to high similarity between structure and densities without distorting secondary structure, we employ an adaptive force scaling heuristic. Established protocols where the force constant has to be determined manually require an iterative trial-and-error approach. We address this by introducing an adaptive force-scaling as depicted in Fig 3a to automatically balance force-field and density-based forces during the refinement.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Adaptive scaling of contributions from force-field and cryo-EM density data overcomes potential energy barriers without excessive work input.

a Adaptive force scaling heuristically balances force-field and density influence during refinement simulations. b Particle in energy landscape where density similarity increases from left to right along the black curve. For the upper leg alternative, the similarity decreases despite biasing forces (burgundy arrow), which causes the bias strength to be increased. Conversely, in a scenario where the similarity remains high (lower leg), the biasing force will gradually be reduced to allow the system to better sample the local landscape. c Brownian diffusion in a potential with fixed (grey) and adaptive (burgundy) biasing forces, respectively. The constant biasing force is scaled such that both force-adding schemes yield the same average mean first passage time moving from left to right. The relative-entropy approach leads to significantly lower exerted work on the system (area under the grey and burgundy curves, respectively), which reduces perturbation of the dynamics of the system.

https://doi.org/10.1371/journal.pcbi.1011255.g003

Cryo-EM refinement simulations are non-equilibrium simulations with the aim to drive a system from an initial model state to a final state that is as similar to the cryo-EM density as possible while avoiding structural distortions that result e.g. from unphysical paths. To avoid or at least reduce the latter during refinement, a heuristic protocol has been devised that aims to minimize work exerted on the system while still requiring as little time as possible for the refinement.

With similarity S as reaction coordinate when driving the system from an initial fit S_start to S_end, the exerted work from the density-guided simulation potential is only determined by the variation of the force-constant as the system progresses: (10)

Eq (10) shows that any protocol that decreases k if the ΔS > 0 and conversely increases k if ΔS < 0 decreases is guaranteed to exert less work to reach the same similarity score than keeping the force constant fixed at the final value of the adaptive scaling protocol, given S_start < S_end. Fig 3b shows how adaptive-force scaling implements this behaviour, starting from a low force constant k.

A one-dimensional Brownian diffusion model system (Fig 3c) is used to test the performance of the concrete scaling protocol as described in the methods section of this paper. In this model, the similarity score simply increases with increasing particle coordinate value. Biasing the system towards increasing coordinate values with adaptive force scaling in contrast to a constant force allows for the particle to reach a given coordinate value in the same average first-passage time at a much lower average work input. Without any coupling of the free energy landscape to the adaptive force scaling protocol other than through the particle trajectory, the adaptive force scaling increases the force just sufficiently to allow overcoming energy barriers but then reduces it again.

Instead of setting a limit to the adaptive force scaling heuristic, we chose to continue the force-scaling until the limits of the molecular dynamics integration algorithm are reached, with the benefit that trajectories are created with structures that represent a wide range of balances between stereo-chemistry and data.

Adaptive force scaling further enables the comparison of relative entropy to other established density-based potentials in simulations with cryo-EM data because it disentangles the effect of the force constant choice from the choice of refinement potential. To carry out this comparison on cryo-EM data with our newly derived similarity score within our framework, a model density generation protocol is required which is shown below.

2.5 Deriving an optimal model density generation for cryo-EM data refinement

To evaluate similarities between structural models and cryo-EM densities, a model electron scattering probability density is generated from atom positions. Two dominant effects are convoluted when modeling electron scattering probabilities: The scattering cross-section of each atom and their thermal motion. Both are approximated with Gaussian functions of amplitude A and width σ. The scattering cross sections determine A (S1 Table). For convenience, we approximate scattering amplitudes by unity for all heavy atoms and zero for hydrogens. The magnitude of thermal fluctuation of atoms at cryogenic temperatures determines the spread width σ.

In practice, these limitations to the model resolution are superseded by the finite performance of the measurement instrument and the reconstruction process where structural heterogeneity, detector pixel size, microscope lenses, and particle alignment limit the resolution. We do not account for structural heterogeneity, because it is an ensemble effect. A connection between the approach presented here and an ensemble model may be made though by employing a probability distribution instead of in Eq (2) and leveraging ensemble simulations [23]. Other resolution-limiting effects are approximated by additional convolution of the generated maps with a Gaussian kernel. Rather than aiming to reproduce the same blur as in the experimental map, we strive to preserve as much information as possible from the physical model.

A balance between information loss due to under-sampling on the grid on the one hand and information loss due to coarse blurring is found where the Gaussian width at half maximum height equals the resolution. The maximum representable resolution on a grid corresponds to twice the Nyqvist frequency δ (corresponding to the pixel and voxel size) so that the Gaussian width σ is approximated in refinement simulations from the highest local resolution or, where that data is not applicable, from the voxel-size, (11)

For computational efficiency Gaussian spreading is truncated at 4σ for all simulations in this publication, accounting for more than 99.8% of the density (Fig A in S1 Appendix). The small limitation on the maximal distance between the initial model structure and the cryo-EM density through this cutoff has proven to be irrelevant for all practical purposes, as density-based forces will “pull” structures into densities as soon as there is minimal overlap between model density and cryo-EM density, which can easily be achieved with an initial alignment. Interestingly, this approach results in a smaller Gaussian spreading width than previously applied ones that aim to reproduce a density map with the same overall resolution as the experimental cryo-EM density. As a result, it maintains as much structural information as possible in the model density while still reducing the computational costs.

2.6 Refinement against noise-free data

To separate additional noise effects in experimental data and possible limitations in the above model, we first assess refinement with ideal data where a small straight helix model system [14] has been refined against a synthetically generated target density of the same helix in a kinked configuration. As illustrated in Fig 4a, adaptive force scaling and relative-entropy as similarity score efficiently fit the helix into the synthetic cryo-EM density [24].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Refinement into noise-free data with adaptive force scaling.

a Aligned (left) and unaligned (right) starting conformations (black sticks) of a helix subject to refinement simulation into a synthetically generated cryo-EM density (gray mesh). b RMSD per residue of the final refined models starting from the aligned conformation compared to the ground truth model underlying the synthetic density map. Each replicate (n=7) is colored by the similarity measure used, inner-product (purple), cross-correlation (ochre), relative-entropy swapped (dark blue), and relative-entropy (green). c RMSD per residue of the final refined models starting from the unaligned conformation compared to the ground truth model underlying the synthetic density map. Each replicate (n=7) is colored by the similarity measure used, inner-product (purple), cross-correlation (ochre), relative-entropy swapped (dark blue), and relative-entropy (green).

https://doi.org/10.1371/journal.pcbi.1011255.g004

The combination of adaptive force scaling and different similarity scores achieved a consistent global fit when the helix was aligned to the density, with some fluctuations of the results (Fig 3c) due to the stochastic nature of molecular dynamics simulations. All four similarity measures lead to qualitatively similar evolution of the adaptive force constant (S1 Fig; the absolute value has no meaning since it is merely a relative factor describing the balance between force field and density fitting). Simulations starting from both the aligned and unaligned starting positions occasionally get stuck in local minima, which can result in bad fits. The average total RMSD of all replicates was lowest for relative-entropy starting from the aligned position and further improved when the helix was initially unaligned (S2 and S3 Tables). The relative-entropy based potential shows markedly better results for the unaligned refinement and achieved a fit with less 1Å RMSD in 6 out of 7 replicates while inner-product, cross-correlation, and relative-entropy swapped based potentials in some instances completely fail to align the helix (S2 and S3 Figs). For a single helix this is a slightly artificial case, but in a large structure undergoing significant transitions, it will be common for some secondary structure elements to not overlap with the target density in an initial phase of the refinement.

2.7 Refinement against experimental cryo-EM data

Experimental cryo-EM densities of aldolase and a GroEl subunit were used to test the performance of adaptive-force scaling in combination with different refinement potentials on experimental data with increasing amounts of density that is not accounted for in our model description and noise that cannot be fully accounted for by the model assumptions.

By using adaptive-force scaling refinement of a previously published X-Ray structure of rabbit-muscle-aldolase [25] against a recently published independently determined cryo-EM structure [26], we consistently achieve accurate refinement throughout all potentials with good stereochemistry (S4 and S5 Tables). Fig 5a shows the final models of refinement with a global deviation of less than 1Å heavy-atom root mean square deviation (RMSD) from the deposited model using inner-product and cross-correlation measures and just above 1Å for the relative-entropy based density potential.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Refinement of an all-atom X-Ray aldolase structure (PDB id 6ALD) into experimental cryo-EM density (EMD-21023).

a Final structure models from density-guided simulations using different similarity scores colored by unaligned root mean square coordinate deviation (RMSD) per residue from the manually built model (PDB id 6V20). b Fourier shell correlation of starting structure (gray line), rigid-body fit of the starting model to the target density (blue) as well as refinement results in the last simulation frame (solid lines). The reported cryo-EM map resolution and 0.143 FSC are indicated with grey lines. c Unweighted FSC average over the course of refinement simulation.

https://doi.org/10.1371/journal.pcbi.1011255.g005

The close agreement with the cryo-EM data is reflected in the FSC of the models refined against the density (Fig 5b) being nearly indistinguishable from the deposited model. The relative-entropy-based potential emphasizes agreement with global features at the cost of local resolution (S4 Fig), while still providing good agreement to the cryo-EM density.

The unweighted FSC average [27] serves as an established similarity score that is not related to the biasing potentials which were used to refine the system (Fig 5b). All underlying potentials appear to lead to refinement simulations that converge in less than 2 ns, as shown in Fig 5c. The less rugged and long-range potential properties of relative-entropy based density forces are reflected in a rapid rigid-body like initial fit to global structural features, while the other potentials show gradual improvements in fit.

The refinement of a GroEl subunit in two different conformations as determined by cryo-EM [28] stretches the limits of the model assumptions of our refinement potential by refining it against a more noisy model with imperfect map-to-model correspondence. Similar to aldolase refinement, adaptive force-scaling allows for rapid and reliable refinement into the model density, as shown in Fig 6 (S6 and S7 Tables). However, the relative-entropy based potential’s propensity to taking all density into account leads to deviations from the published model in regions with density that has no correspondence in the all-atom model (S5 and S6 Figs).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Refinement of an all-atom GroEL cryo-EM structure (PDB id 5W0S state 1) into experimental cryo-EM density (EMD-8750, additional map 3).

a Final structure models from density-guided simulations using different similarity scores colored by unaligned root mean square coordinate deviation (RMSD) per residue from the deposited model (PDB id 5W0S state 3). b Fourier shell correlation of starting structure (gray line), rigid-body fit of the starting model to the target density (blue) as well as refinement results in the last simulation frame (solid lines) deviations of an equilibrium simulation (dotted lines). c Unweighted FSC average over the course of refinement simulation.

https://doi.org/10.1371/journal.pcbi.1011255.g006

2.8 Model rectification by combining force-field and cryo-EM data

To assess performance in larger structural transitions, we repeated the aldolase refinement when starting from initial model structures that have been distorted by heating with partially unfolded secondary structure elements (Fig 7, as described in Methods). Fig 7b shows the final relative-entropy based model of the refinement procedure that achieved 1.13 Å heavy-atom RMSD from the manually built model. Structural details at map resolution match in secondary structure elements. In contrast to refinement of the undistorted X-ray structure, the relative-entropy based potential gains less from the long-rangedness of the potential and the rapid alignment of large-scale features, because structural rearrangements were needed on all length scales. The adaptive force scaling protocol alleviates differences between density-based potentials in refinement speed and allows for refinement with good structural agreement in less than 3 ns (S7 Fig). The adaptive force scaling protocol allows the modeling to be more steered by cryo-EM data and reach model structures that would not have been accessible by modeling using stereochemical information from the force-field alone.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Cryo-EM data rectifies model distortions with density-guided simulations.

a Distorted starting model RMSD with respect to manually built model (PDB id 6V20). b Final model structure after refinement into a cryo-EM density (EMD-21023) using adaptive force scaling and relative-entropy similarity score. c Close-up of structural features of the final simulation model (green lines) and cryo-EM density (gray mesh). d Fourier shell correlation of starting structure (gray line) as well as refinement results in the last simulation frame (solid lines). The reported cryo-EM map resolution and 0.143 FSC value are indicated with grey lines. e Un-weighted FSC average over the course of a refinement simulation.

https://doi.org/10.1371/journal.pcbi.1011255.g007

4.1 Calculating density-based forces

For ease of implementation and computational efficiency the derivative of Eq (4) is decomposed into a similarity measure derivative and a simulated density model derivative, summed over all density voxels v (12)

Though the convolution in eq (12) might be evaluated with possible performance benefits in Fourier space, we have implemented the more straightforward real-space approach.

The forward model ρ^s is calculated using fast Gaussian spreading as used in [30]; the integral over the three-dimensional Gaussian function over a voxel is approximated by its function value at the voxel center v at little information loss (Fig B in S1 Appendix). Amplitudes of the Gaussian functions [31] have been approximated with unity for all atoms except hydrogen. The explicit terms that follow for S(ρ, ρ^s) and are stated in the S1 Appendix.

4.2 Multiple time-stepping for density-based forces

For computational efficiency, density-based forces are applied only every N_fit steps. The applied force is scaled by N_fit to approximate the same effective Hamiltonian as when applying the forces every step while maintaining time-reversibility and energy conservation [32, 33]. The maximal time step should not exceed the fastest oscillation period of any atom within the map potential divided by π. This oscillation period depends on the choice of reference density, the similarity measure, and the force constant and has thus been estimated heuristically.

4.3 Adaptive force scaling

Adaptive force constant scaling decreases the force-constant when similarity increases by a factor 1 + α, with α > 0, and reversely increases it by a factor 1 + 2α when similarity decreases. The larger increase than decrease factor enforces an increase in similarity over time.

To avoid spurious fast changes in force-constant, similarity decrease and increase are determined by comparing similarity scores of an exponential moving average. The simulation time scale is coupled to the adaptive force scaling protocol by setting , where Δt is the smallest time increment step of the simulation and τ determines the time-scale of the coupling.

This adaptive force scaling protocol ensures a growing influence of the density data in the course of the simulation, eventually dominating the force-field. Simulations with adaptive force scaling are terminated when overall forces on the system are too large to be compatible with the integration time step.

4.4 Comparing refined structures to manually built models

Root mean square deviations (RMSD) of all heavy atom coordinates (excluding hydrogen atoms) used absolute positions without super-position as structures because the cryo-EM density provides the absolute frame of reference. This is an upper bound to RMSD values between refined and manually built models calculated with rotational and translational alignment.

4.5 Comparing refined structures to cryo-EM densities

Fourier shell correlation curves and un-weighted Fourier shell correlation averages [27] were calculated at 4 ps intervals from structures during the trajectories by generating densities from the model structures using a Gaussian σ of 0.45 Å, corresponding to a resolution as defined in EMAN2 [24] to 2Å.

4.6 Map and model preparation before refinement

4.7 Generation of a distorted model

To generate a distorted starting model, the aldolase protein X-ray was heated to 433 K over a period of 5 ns. During heating, the pressure was controlled with the Berendsen barostat, favoring simulation stability over thermodynamic considerations. To disentangle effects from decreasing the temperature and fitting, the distorted structure was subjected to 5 ns of equilibration at 300 K before starting the density-guided simulations with the same protocol as described above.

4.8 Molecular dynamics simulation

All simulations were carried out with GROMACS version 2021.3 [29] and the CHARMM27 force-field [37, 38] in an NPT and NVT ensembles with neutralized all-atom systems in 150 mM NaCl solution. The temperature was regulated with the velocity-rescaling thermostat at a coupling frequency of 0.2 ps to ensure rapid dissipation of excess energy from density-based potential, when structures are very dissimilar from the cryo-EM density, i.e., far from equilibrium. The pressure was controlled with the Parinello-Rahman barostat for aldolase simulations, helix simulations were carried out at constant volume. Aldolase and GroEL were aligned roughly to the density by placing their center of geometry in the center of the cryo-EM density box. Forces from density-guided simulations were applied every N_fit = 10 steps according to the protocol described above [33]. All simulations to refine a structure against a density were carried out with adaptive force-scaling. We used a coupling constant of τ = 4 ps, balancing time to result with time for structures to relax. For aldolase simulations, the Gaussian spread width was determined by using a lower bound on the highest estimated local resolution of 1.83 Å. Spread width for GroEl simulations was set to 1.0455Å, based on the 1.23Å voxel size in the map used for refinement. Periodic boundary conditions are treated as described in the S1 Appendix. All simulation setup parameters and workflows have been made available.