Molecular Dynamics Flexible Fitting

Fitted ribosome structure
Recent advances in cryo-electron microscopy (cryo-EM) single-particle 3-D reconstruction are revolutionizing structural biology, with several macromolecular complexes being imaged at subnanometer resolution. These advances have driven great interest towards the development of methods to fit atomic structures into cryo-EM maps to obtain "quasi-atomic" models that conform to the EM data. In several cases, atomic structures of individual components of such complexes have been obtained by X-ray crystallography; when combined with cryo-EM images of the entire complexes, complete models can be generated, significantly aiding in the interpretation of the experimental data. Moreover, cryo-EM makes it possible to image macromolecular complexes in different functional and conformational states, while X-ray crystallography can only capture those that lend themselves to crystal packing. Thus, the combination of the two sources of data offers a unique opportunity to obtain high-resolution models of the complexes in different functional states.



We have developed a new method called molecular dynamics flexible fitting (MDFF) to perform flexible fitting of atomic structures into EM maps using MD simulations with NAMD. The method consists of adding external forces proportional to the gradient of the EM map into an MD simulation of the atomic structure, while the stereochemical quality of the structure is preserved by the MD force field. The external forces are defined on a grid, making use of the GridForces feature recently implemented into NAMD. A significant advantage of this flexible-fitting method is that all internal features present in the EM map are used in the fitting process, while the model remains fully flexible.

Incorporating EM data into simulation

Volume slice of an EM map of the ribosome

Volume slice of a 6.7-Angstrom EM map of the ribosome. The arrows show the gradient of the external field that directs the structure into the density.
The method incorporates the EM density map as a potential so that high density areas in the map correspond to energy minima, so that the atoms in the structure are subject to forces proportional to the gradient of the EM map. The potential is defined on a 3-D grid by


where

Here wj corresponds to a per-atom weigh, typically set to the atomic mass, ξ is a force scaling, Φ(r) is the EM density at position r, Φmax is the maximum value of the EM density map, and Φthr is a density threshold. The purpose of the density threshold is to remove from the EM data the solvent contribution (see figure below). An atom placed in this external potential feels a force of





Density thresholding

(A) Density histogram of a 12.8-Angstrom ribosome map showing two distinct peaks pertaining to the solvent and macromolecule. (B) 2-D slice of the density. (C) 2-D slice of the density after thresholding, i.e., after removing the solvent contribution.

Preventing overfitting

A common concern to flexible fitting techniques is overfitting, i.e., structural distortions introduced in the atomic structure in the fitting process. The term overfitting is commonly used in many fields referring to the problem of fitting too many adjustable parameters to a limited ammount of data. In flexible fitting methods, since all degrees of freedom in the atomic structure are allowed to change, it is possible that unphysical deformations are introduced in the atomic structure in order to obtain a better fit. The problem is likely worse in the case of experimental cryo-EM maps, as opposed to noise-free simulated maps used to validade the methods and more objectively assess overfitting; experimental maps contain noise due to image aquisition and reconstruction algorithms, as well as conformational variability of the sample.

In order to prevent overfitting, MDFF introduces harmonic restraints to preserve the secondary structure of proteins and nucleic acids. For proteins, the φ and ψ dihedral angles of residues which are part of helices and β sheets are restrained to their initial position. For nucleic acids, restraints are imposed to seven dihedral angles, as well as two interatomic distances between base pairs, as illustrated in this figure:

Harmonic restraints are imposed to preserve secondary structure of proteins (left) and nucleic acids (right).

Validation: Crystal structures of the same biomolecule captured in two conformational states

With the aim to validate the MDFF method and estimate its accuracy, we apply it to cases where crystal structures in two different conformational states are available. A noise-free simulated map is generated from conformation I, and conformation II is fitted into the simulated map using MDFF. The figure shows such fitting for the E. coli 16S RNA (PDB 2AVY, 2AW7), and the acetyl-CoA synthase (PDB 1OAO). Watch movies of these fittings: E. coli 16S (4.3Mb); Acetyl-CoA Synthase (8.5Mb).

Validation of MDFF

Examples of fittings of atomic structures into noise-free simulated maps using MDFF. (top) For each system, the target structure from which the simulated maps were generated is shown in gray and the fitted structure in green, with the initial structure on the left and the final on the right. (bottom) Same organization as before, now showing the target map and coloring the fitted structure by RMSD per residue with respect to the target structure.

Naturally, the resolution of the EM density map determines the quality of the quasi-atomic model obtained by flexible fitting: the higher the resolution, the more information is available on the structure, and thus the smaller the uncertainty about the positions of the atoms. Sub-nanometer maps have clear information of secondary structure elements. We use RMSD (with respect to the target structure when know, and to the initial structure when using real data) and cross-correlation coefficients to monitor the progress of the fit during MDFF simulations. The figure below shows the effect of resolution on these measures. It is important to note that when these measures have converged, and thus the fitting is complete, the atomic structure fluctuates around a conformation dictated by the molecular dynamics set up, and the EM map. One can select a set of representative structures that conform to the data available in the EM map. In this way, even though the low resolution of the map results in an indeterminate problem in structure determination, we can explore different conformations that are likely present in the original EM micrographs.

RMSD (top) and cross-correlation coefficient (bottom) between the fitted and target structure for the acetyl-CoA synthase example, using several simulated map resolutions. The regions highlighted in orange correspond to strucutres fluctuating around an equilibrum determined by the EM map and the MD force field. A set of representative structures is shown on the right.

Application: The bacterial ribosome

We have applied the MDFF method to obtain atomic models from cryo-EM maps of the E. coli ribosome in different functional states and imaged at different resolutions. A multi-step protocol was used in which the fitting of the RNA structure converged in the first step, followed by two steps in which the fitting of the ribosomal proteins was improved; in the last step, the remaining ligands - tRNAs and elongation factor Tu (EF-Tu) in the example presented here - were fitted. The figure below provides an overview of the fitting into a 6.7-Angstrom map of the ribosome in complex with tRNAs and EF-Tu.


Watch a movie (7.2 Mb) of the multistep protocol.

In the analysis of a previous 9.0-Angstrom cryo-EM map of the same complex, a model of the A/T-site tRNA was obtained by dividing the molecule into two parts and manually fitting them independently, followed by an optimization. A similar model was now obtained by MDFF, but this time without any assumptions. It can be seen that the anticodon loop of the tRNA bends when it binds to the ribosome, storing elastic energy which is dissipated when the tRNA is accommodated into the A site.


Watch a movie (2.7 Mb) of the fitting of the ternary complex.

      

Conformation of the tRNA in the A/T site. (left) The crystal structure from the free EF-Tu:tRNA:GTP ternary complex (TC) used as a starting point for the fitting is shown in red; the A/T tRNA model obtained by MDFF is shown in blue; the A/T tRNA model previously proposed is shown in green. (right) Comparison between the A/T tRNA model (blue) and a partial crystal structure of the A-site tRNA (red).

As a last example, the figure below shows the different conformations of the GTPase-associated center (GAC) obtained by applying MDFF to ribosome maps in two different functional states: bound to ternary complex (corresponding to the initial step of tRNA delivery to the ribosome) and after tRNA accommodation.

Conformational dynamics of the GTPase-associated center (GAC). Shown are differences in the conformation of the GAC between the TC-bound ribosome (top) and the accommodated ribosome (bottom). Rigid-body docked structures into the corresponding maps, used as initial coordinates for flexible fitting, are shown on the left; flexibly fitted structures are shown on the right.

Publications

Leonardo G. Trabuco*, Elizabeth Villa*, Kakoli Mitra, Joachim Frank, and Klaus Schulten.
Flexible fitting of atomic structures into electron microscopy maps using molecular dynamics. Structure, 16, 673-683, 2008.
* Equal contribution.


Investigators

Page created and maintained by Elizabeth Villa and Leonardo Trabuco.

This website is reproduced in part with permission from Flexible fitting of atomic structures into electron microscopy maps using molecular dynamics. L.G. Trabuco, E. Villa, K. Mitra, J. Frank, and K. Schulten. Structure, 16, 673-683. Copyright 2008 Elsevier.

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