Multiple Subunit Fitting into a Low-Resolution Density Map of a Macromolecular Complex Using a Gaussian Mixture Model

Abstract

Recently, electron microscopy measurement of single particles has enabled us to reconstruct a low-resolution 3D density map of large biomolecular complexes. If structures of the complex subunits can be solved by x-ray crystallography at atomic resolution, fitting these models into the 3D density map can generate an atomic resolution model of the entire large complex. The fitting of multiple subunits, however, generally requires large computational costs; therefore, development of an efficient algorithm is required. We developed a fast fitting program, “gmfit”, which employs a Gaussian mixture model (GMM) to represent approximated shapes of the 3D density map and the atomic models. A GMM is a distribution function composed by adding together several 3D Gaussian density functions. Because our model analytically provides an integral of a product of two distribution functions, it enables us to quickly calculate the fitness of the density map and the atomic models. Using the integral, two types of potential energy function are introduced: the attraction potential energy between a 3D density map and each subunit, and the repulsion potential energy between subunits. The restraint energy for symmetry is also employed to build symmetrical origomeric complexes. To find the optimal configuration of subunits, we randomly generated initial configurations of subunit models, and performed a steepest-descent method using forces and torques of the three potential energies. Comparison between an original density map and its GMM showed that the required number of Gaussian distribution functions for a given accuracy depended on both resolution and molecular size. We then performed test fitting calculations for simulated low-resolution density maps of atomic models of homodimer, trimer, and hexamer, using different search parameters. The results indicated that our method was able to rebuild atomic models of a complex even for maps of 30 Å resolution if sufficient numbers (eight or more) of Gaussian distribution functions were employed for each subunit, and the symmetric restraints were assigned for complexes with more than three subunits. As a more realistic test, we tried to build an atomic model of the GroEL/ES complex by fitting 21-subunit atomic models into the 3D density map obtained by cryoelectron microscopy using the C7 symmetric restraints. A model with low root mean-square deviations (14.7 Å) was obtained as the lowest-energy model, showing that our fitting method was reasonably accurate. Inclusion of other restraints from biological and biochemical experiments could further enhance the accuracy.

Integrative structure and functional anatomy of a nuclear pore complex

“…The Cryo-ET density restraint corresponded to the cross-correlation between the Gaussian Mixture Model (GMM) representation of most Nups and the GMM representation of the Cryo-ET density map 103,122–124 (Fig. 3; Extended Data Figs.…”

PRISM-EM: template interface-based modelling of multi-protein complexes guided by cryo-electron microscopy density maps

“…After the segmentation process, subunits can be docked individually. Other tools such as MultiFit (Lasker et al, 2009;Tjioe et al, 2011), GMFit (Kawabata, 2008) and ATTRACT-EM (de Vries & Zacharias, 2012) go a step further and perform multiple docking. They fit multiple protein structures simultaneously based on molecular-docking and molecular-fitting approaches; ATTRACT-EM also refines the models after docking structures into the density map.…”

Improved metrics for comparing structures of macromolecular assemblies determined by 3D electron-microscopy

“…This is required as GMfit is sensitive to the number of GDFs included in the Gaussian mixture model (GMM) approximating the volume density (Kawabata, 2008). With the resulting GMMs, we used GMfit to generate 100 volume alignments for each map pair in the test dataset.…”