Methods using normal modes to detect cryo-EM data heterogeneity and interpret it in terms of continuous

macromolecular conformational changes

Slavica Jonic

Sorbonne Université, IMPMC CNRS UMR 7590, 75005 Paris, France

Heterogeneity problem in single particle analysis

2

Difficult analysis because of a low signal-to-noise ratio Two different images: Different orientations of the same object? Different conformations (same orientation)?

Difficulties

Different objects, different projections

One object, different projections

Projections

J. Dubochet

Ribosome: protein synthesis Virus maturation

RNA polymerase: transcription

Single particle heterogeneity due to conformational changes

600 Å

3 With permission from F. Tama

Large-scale conformational changes

Functional motions

Time scale: > ms

Computational methods to study conformational variability

4

- Multireference

- Maximum likelihood

- Statistical analysis

- Manifold embedding

- Hybrid electron microscopy normal

mode analysis (HEMNMA)

Jonic, Curr Opin Struct Biol 2017

Normal mode analysis (NMA)

With permission from F. Tama & O. Miyashita

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Low frequencies Collective motions

High frequencies Localized motions

Dynamics of a macromolecular complex can be described as a linear combination of independent harmonic oscillators Harmonic approximation of the potential energy function around a minimum energy conformation

NMA with pairwise Hookean potential

6

2

0

, ,,

2a b a b a b

CE r r r r

Ep E r

a, r

b

ra , b

0 R

c

Tirion, Phys Rev Lett 1996 :

Rc : Radius of interaction between atoms Arbitrary interaction cut-off distance Determines number of interacting atom pairs Best results for small Rc (also, faster)

C : Bond strength (spring stiffness constant)

Referred to as Elastic Network Model

- Atoms connected via elastic springs

- Given structure is considered to be at the energy minimum (no minimization)

Hessian: 2nd derivative of the potential

orr

ji

p

rr

E

2

H

7

Eigenvalue problem

LHAAt

0

0

2

2

2

1

L

Eigenvector = normal mode

Eigenvalue = frequency2 21

aaA

Size of the system: 3Nx3N

Diagonalization of the second derivative of the potential (Eigenvalue problem)

n

nnjj

ij

ii

rrji

p

pqrrrr

rr

ErE

o

2200

2

2

1)()(

2

1)(

Coordinate change

Adapted from Tama et al., with permission

Coarse grained models such as only 1 point (bead) per residue can be used

Coarse-grained Elastic Network Model

With permission from Tama et al. 8

Hessian diagonalization: Transition from Cartesian to Rotation-Translation-Block (RTB) coordinates

– rotation + translation of block => new basis

– expression of Hessian in this new basis

Hb = PTHP

H

3N

bH

6nB

RTB method: - Atomic structure is represented with a set of blocks (1 or several residues per block)

- NMA is performed in the space defined by rotation and translation of blocks

Good approximation of the low-frequency normal modes

Tama et al., Proteins: Struc Funct Genet 2000; Durand et al., Biopolymers 1994

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Nixi

3,...,1,0

Njjjj

aaa321

,...,,a : Normal mode j (with frequency j )

jq : Amplitude of the atomic displacement along mode j

Normal-mode displacement of the atomic coordinates from this state:

Modeling conformational changes using normal modes

To determine qj, possible conformational states have to be confronted with experimental data such as EM images or density maps

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0

3

1

i

N

j

jijixqax

q

Can be considered as a possible conformational state (qj is unknown and cannot be determined by NMA)

Minimum-energy conformation with N atoms (3N atomic coordinates):

Tomato Bushy Stunt Virus (TBSV): swelling described by mode 28

11

Mode 28 Mode 80 Mode 107

EM swollen

X-ray compact

A

B

C

Ca2+ ions

Aramayo et al., Biochim Biophys Acta 2005

Displacements along normal modes overlap with experimentally observed conformational changes

12

• NMA has been used for flexible 3D-to-3D fitting of atomic structures into EM density maps

• Since 2004, interest in combining NMA and EM image analysis (Brink et al., Structure 2004)

• Hybrid Electron Microscopy Normal Mode Analysis – HEMNMA (Jin et al., Structure 2014)

First method that automatically determines conformations from images using normal modes

Based on iterative elastic projection matching

Allows processing a large number of images to determine all actual confirmations and evaluate

their pertinence

Difficult to achieve with discrete-state EM methods or X-ray crystallography

Combining Normal Mode Analysis and Electron Microscopy image analysis

Conformational distribution Points: images Close points: similar conformations

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• Hybrid approach (combines model and experimental data)

NMA gives normal mode vectors but not their linear combination

(possible motions of unknown amplitudes)

Elastic 3D-to-2D alignment gives the coefficients of the linear combination

(contribution of simple motions to the complex motion)

• Each particle image may represent a unique conformation (no 2D classification/averaging for the elastic 3D-to-2D alignment) • It can help identify intermediate conformations and continuous conformational changes

Important to note about HEMNMA

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Set of EM single particle images

Phase flipped

• 3D reference

Obtained by an experimental technique or by modeling

Type

Atomic (all-atoms or coarse-grain)

Density map obtained by EM, SAXS, or simulation from an atomic structure

Single conformation (ideal case) or average of different conformations (e.g.,

obtained by 3D reconstruction from a heterogeneous set of images)

HEMNMA input

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Image analysis using normal modes of a 3D reference: • Each image is analyzed independently from other images • Normal mode flexible fitting of the 3D reference with the image

Deformation of the 3D reference using normal modes to find the model that suits best the given image (via iterative elastic 3D-to-2D alignment )

HEMNMA principle

Low correlation

High correlation

Deformation using normal modes, and rotation and shift

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Results of the elastic 3D-to-2D alignment, for each particle image: • Deformation amplitudes for M normal modes (M elastic geometrical parameters) • 3 Euler angles and 2 in-plane shifts (5 rigid-body geometrical parameters) HEMNMA output : • Image mapping in the space of normal-mode deformation amplitudes • 3D reconstructions from images assigned with similar conformations (similar normal-mode deformation amplitudes) • Animated “trajectories”

HEMNMA output

Semi-automatic tracing of “trajectories” in the map of images

Semi-automatic grouping of images with similar conformations (“clustering”)

Map of images

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3D reference

Iterative elastic 3D-to-2D alignment

Phase-flipped particle images

Selected modes

Orientation and position of each particle Dimension reduction

Normal mode analysis

Type of 3D reference?

Map-to-pseudoatoms conversion

Density map

Atomic

Deformation amplitudes along normal modes for each particle

Identification of trajectories and animation

Clustering and 3D reconstructions

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HEMNMA with DNA polymerase Pol α - B subunit complex

3D reconstructions Animated trajectories

Klinge et al., 2009, Embo J.

Jin et al., Structure 2014

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HEMNMA with Tomato Bushy Stunt Virus Aramayo et al., 2005, Biochim Biophys Acta

Jin et al., Structure 2014

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Mode 9

HEMNMA with 70S ribosome with and without EF-G

With and without EF-G (Frank and Gao, EMDB)

Without EF-G, EMDB (Fu et al., PNAS, 2011)

Jin et al., Structure 2014

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HEMNMA integral graphical interface

Practice Session: HEMNMA within Scipion

http://xmipp.cnb.csic.es http://scipion.cnb.csic.es

Sorzano et al., J Struct Biol 2014

Integrated in Xmipp and since recently in Scipion

Method to represent EM density maps with 3D points: Density approximation method

• Approximation of the EM map density using isotropic (circularly symmetric) 3D Gaussian functions that we refer to as pseudoatoms:

: Isotropic 3D Gaussian functions

• Given target approximation error () and Gaussian standard deviation (), the method computes weights (wi), position (ri) and number of Gaussian functions (N) to achieve:

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3

R, rrf

i

N

i

iKwf rrr

1

ˆ

K

M

j

jj

f

ff

M

e

1

ˆ1

rr

• Centers of Gaussian functions are 3D points for building Elastic Network Model

Jonic & Sorzano, IEEE J Sel Topics Signal Process 2016

• Density approximation method that can have other applications than NMA

f : effective range of values in f (insensitive to outliers) M: number of voxels at which the error is evaluated (possibility to use a mask)

Examples of density approximation

Large and , small N Small and , large N

Original

Jonic & Sorzano, IEEE J Sel Topics Signal Process 2016

Spheres represent 3D Gaussian functions (pseudoatoms)

Larger and , smaller N

Smaller and , larger N

Density-map slices

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Applications of this density approximation method

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- NMA of EM density maps to explore potential conformational changes (3D EM Loupe web server, HEMNMA)

- NMA of EM density maps for analysis of conformational heterogeneity in images (HEMNMA)

- Denoising (out of scope here)

Jonic et al., J Struct Biol 2016

Nogales-Cadenas et al., Nucleic Acids Res 2013

Jin et al., Structure 2014

Speed up image analysis: Use a subset of normal modes

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Collectivity measure: Number of atoms or pseudoatoms affected by the mode (Bruschweiler, J. Chem. Phys.. 1995) - Close to 1 for maximally collective motions - Close to 0 for localized motions

• How to select a subset of normal modes? • Principle: Select highly collective and low frequency modes

2

1

2

logexp1

ij

N

i

ijjAA

N

C

Displacement of atom in the mode j

Normalization factor

Number of atoms

11

2

N

i

ijA

i

Select M normal modes with collectivity above a threshold

Selection of normal modes based on a collectivity threshold

26 Jin et al., Structure 2014

1. Weight the modes according to the increasing frequency from 1 to M 2. Weight the modes according to the decreasing collectivity from 1 to M 3. Sum the two weights for each mode and divide the sum by 2M (normalized weight) 4. Order the modes according to the increasing normalized weight 5. Select L modes with the lowest normalized weights

Selection of normal modes based on scoring and ordering

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Sorzano et al., J Struct Biol 2014

Scoring and ordering normal modes of 70S

Mode 13 from synthetic map 6083 pseudoatoms (σ = 3, = 2%)

Mode 9 from EM map 9411 pseudoatoms (σ = 3, = 2%)

Mode 11 from atomic structure, 10204 Cα and P atoms 28

Synthetic (left) and EM (right) maps, 1283 voxels of size (3 Å)3

Mode 28 from atomic structure, 53700 Cα atoms

Mode 33 from synthetic map 9372 pseudoatoms (σ = 4, = 4%)

Mode 28 from EM map 9475 pseudoatoms (σ = 4, = 4%)

Scoring and ordering normal modes of TBSV

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Synthetic (left) and EM (right) maps, 1283 voxels of size (3.2 Å)3

• 3D density maps of different conformational states obtained with discrete-state methods can be regarded as discrete samples of continuous trajectories

• Can we “extrapolate” continuous trajectories from discrete, unordered samples obtained by discrete-state methods?

Structure Mapping (StructMap) methodology

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StructMap (Sorzano et al., Byophys J 2016)

• Projects a set of 3D density maps onto a low-dimensional

space of their conformational distances

• Positions of the density maps in the low-dimensional distance

space can sometimes give an idea about possible trajectories

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StructMap algorithm

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StructMap with synthetic data

Synthetic data set of RyR1 Used in Practice session

Sanchez Sorzano et al., Biophys J 2016

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StructMap using experimental data with combined conformational and compositional heterogeneity

Behrmann et al., Cell 2015

Focus the analysis on conformational heterogeneity:

Use masks to minimize compositional differences among EM density maps

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StructMap with human 80S ribosome from polysomes (1)

• Mask that suits the shape of 2875 map : Partly removed mass in A and F sites

• Observations in the distance space: Grouping of EM maps in three groups

Rotation of 40S w.r.t. 60S Rolling

Unrolling and different compositions

Sanchez Sorzano et al., Biophys J 2016 Behrmann et al., Cell 2015

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StructMap with human 80S ribosome from polysomes (2)

Other observations in the distance space:

8-like “trajectory” form obtained by connecting points following the sequence of EM density

maps proposed by Behrmann et al., 2015

Sanchez Sorzano et al., Biophys J 2016 Behrmann et al., Cell 2015

Combine StructMap and HEMNMA to “extrapolate” continuous trajectories

Discrete-state methods (here, maximum likelihood)

StructMap method

Density-map mapping onto a low-dimensional space (here, 1D) of distances

Image mapping onto a low-dimensional space (here, 3D) of distances using density map 3 as the reference to elastically align with images

A set of density maps obtained by image analysis

Continuous-state methods (here, HEMNMA)

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• Principle: Analyze images with HEMNMA using normal modes of the reference density maps selected with StructMap

Further reading

• Brink J, Ludtke SJ, Kong Y, Wakil SJ, Ma J, Chiu W: Experimental verification of conformational variation of human fatty acid synthase as predicted by normal mode analysis. Structure 2004, 12:185-191. • Jin Q, Sorzano CO, de la Rosa-Trevin JM, Bilbao-Castro JR, Nunez-Ramirez R, Llorca O, Tama F, Jonic S: Iterative elastic 3D-to-2D alignment method using normal modes for studying structural dynamics of large macromolecular complexes. Structure 2014, 22:496-506. • Sorzano CO, de la Rosa-Trevin JM, Tama F, Jonic S: Hybrid Electron Microscopy Normal Mode Analysis graphical interface and protocol. J Struct Biol 2014, 188:134-141. • Sanchez Sorzano CO, Alvarez-Cabrera AL, Kazemi M, Carazo JM, Jonic S: StructMap: Elastic Distance Analysis of Electron Microscopy Maps for Studying Conformational Changes. Biophys J 2016, 110:1753-1765. • Jonic S, Sorzano CO: Versatility of Approximating Single-Particle Electron Microscopy Density Maps Using Pseudoatoms and Approximation-Accuracy Control. Biomed Res Int 2016, 2016:7060348. • Jonic S: Computational methods for analyzing conformational variability of macromolecular complexes from cryo-electron microscopy images. Curr Opin Struct Biol 2017, 43:114-121.