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Three-dimensional Gaussian functions have been shown useful in representing electron microscopy (EM) density maps for studying macromolecular structure and dynamics. Methods that require setting a desired number of Gaussian functions or a maximum number of iterations may result in suboptimal representations of the structure. An alternative is to set a desired error of approximation of the given EM map and then optimize the number of Gaussian functions to achieve this approximation error. In this article, we review different applications of such an approach that uses spherical Gaussian functions of fixed standard deviation, referred to as pseudoatoms. Some of these applications use EM-map normal mode analysis (NMA) with elastic network model (ENM) (applications such as predicting conformational changes of macromolecular complexes or exploring actual conformational changes by normal-mode-based analysis of experimental data) while some other do not use NMA (denoising of EM density maps). In applications based on NMA and ENM, the advantage of using pseudoatoms in EM-map coarse-grain models is that the ENM springs are easily assigned among neighboring grains thanks to their spherical shape and uniformed size. EM-map denoising based on the map coarse-graining was so far only shown using pseudoatoms as grains.

Single-particle analysis is an electron microscopy (EM) technique that allows determining the structure at near-atomic resolutions for a large range of macromolecular complexes [

EM-map representations with a reduced number of points or with a set of 3D Gaussian functions have been shown useful in studying macromolecular structure and dynamics [

An alternative is to set a desired (target) error of approximation of the given EM map and then optimize the number of Gaussian functions, their position, and their weights to achieve the target approximation error, as in the approach that we introduced in [

The advantage of using pseudoatoms in applications based on NMA and ENM, with respect to other types of grains (Table

Comparison of Vector Quantization, Gaussian Mixture Model, and our pseudoatomic model.

Vector Quantization | Gaussian Mixture Model | Our pseudoatomic model | |
---|---|---|---|

Grain | 3D point (codebook vector) | 3D Gaussian distribution function | 3D radial basis function (isotropic Gaussian distribution function) |

Grain geometry | Spherical | Ellipsoidal | Spherical |

Algorithm | Self-organizing map (SOM) | Maximum likelihood method using the expectation maximization algorithm | Iterative adding and removing of pseudoatoms and gradient descent refinement |

Goal of algorithm | Minimize the mean-square deviation of the codebook vectors from the corresponding 3D data | Find the model with the maximum likelihood function | Find the model with the minimum number of grains for the given error of density approximation |

Number of grains | Fixed | Fixed | Adjustable |

Grain weight | Adjustable | Adjustable | Adjustable |

Grain position | Adjustable | Adjustable | Adjustable |

Grain size | Adjustable | Adjustable | Fixed |

Application of elastic network model | Easy | Difficult | Easy |

In this article, we review the mentioned applications of this EM-map approximation method while only briefly reminding the method. For algorithmic details (e.g., related to adding/removing grains), the reader is addressed to [

We start this section with a brief background on the approach for converting EM maps in sets of pseudoatoms. Then, we provide a brief reminder on NMA that is used in applications of the conversion approach to studying conformational changes of macromolecular complexes.

A function

Different pseudoatom representations can be obtained with this approach by varying the Gaussian-function standard deviation (

Approximation of density maps to a desired level of accuracy (

NMA models complex motions by linear combinations of harmonic oscillations around a minimum-energy conformation. NMA is often based on the standard ENM [

Nodes of the elastic network model are 3D point particles. Each node is connected, via harmonic springs, with other nodes within a sphere of a given radius (the radius is referred to as interaction cutoff distance). In our approach, the coordinates of nodes of the ENM are the center coordinates of pseudoatoms with which the given EM density map is represented. Given

A displacement of nodes of the ENM along normal modes modifies the given conformation, which is used in simulations of structural flexibility (Figure

Normal modes of pseudoatomic structures from EM density maps for simulations of structural flexibility. (a, b) EM density map of a compact conformation of Tomato Bushy Stunt Virus from [

As explained in Background, NMA of intermediate-resolution EM maps provides normal modes that can be used to simulate (predict) different conformational states of the same complex. Normal modes of EM maps are calculated based on the EM-map coarse-grained representations with pseudoatoms. Both EM-map coarse-graining and NMA can be performed using

Coarse-grain representations of EM density maps and normal modes of those coarse-grain representations can be used to analyze experimental EM data. More precisely, conformations actually present in EM data can be interpreted using simulated conformations (NMA-based simulations). In the framework of conformational heterogeneity analysis, normal modes were used for image analysis first in [

Given an EM density map,

Exploring actual conformational changes by image analysis using normal modes of Gaussian-based (pseudoatomic) representation of EM density maps (analysis with HEMNMA). (a) EM density maps of three conformations of a DNA polymerase Pol

It should be noted that

Exploring actual conformational changes by analyzing a set of EM density maps using normal modes of Gaussian-based (pseudoatomic) representation of these maps (analysis with StructMap). (a) States along the elongation cycle of 80S ribosome, from which eleven states were determined by EM in [

In the context of revealing sequences of conformational changes,

Cryo-EM images have a low signal-to-noise ratio. Thus, a large number of such images must be averaged in 3D space to reduce noise. However, even high-resolution 3D reconstructed EM maps may still contain significant amounts of noise. We have shown that pseudoatom representations of EM density maps can be used for EM-map denoising [

The method was used to denoise EM density maps of several complexes obtained at subnanometer resolutions by single-particle analysis or subtomogram averaging (beta-galactosidase, ribosome, and empty and full virus particles) [

Denoising of EM density maps based on their approximation to a desired level of accuracy (

The method for EM-map approximation using Gaussian functions of standard deviation

In this approach, the EM-map approximation is a weighted sum of Gaussian functions of given standard deviation

The target approximation error control allows different applications of this method. We could learn from experiments how to choose this target approximation error to suit these different applications. For instance, in EM-map denoising applications, target approximation errors smaller than

Advantages of structural representations with pseudoatoms have recently been explored in the context of 3D reconstruction. For instance, a Bayesian approach to ab initio, low resolution 3D reconstruction proposed in [

The authors declare that they have no competing interests.

This research was partially supported by the CNRS (France) and the CSIC (Spain) [Projet International de Coopération Scientifique, PICS 2011]; the French National Research Agency ANR [ANR-11-BSV8-010-04]; the European Social Fund and the Ministerio de Educación y Ciencia [“Ramón y Cajal” fellowship to COSS]; the Spanish Ministry of Economy and Competitiveness [AIC–A–2011–0638 and BIO2013-44647-R]; and the Comunidad de Madrid [CAM S2010/BMD-2305]. HPC resources from GENCI (France) [Grants to project 072174 (2011–2016)] are also gratefully acknowledged.

_{s}-corrected cryo-EM