3D Reconstruction of 2D Crystals Xiangyan Zeng

Fort Valley State University 1005 State University Dr.

Fort Valley, GA 31030 1-478-825-6528

zengx@fvsu.edu

James Ervin Glover Fort Valley State University 1005 State University Dr.

Fort Valley, GA 31030 1-478-825-6286

gloverj@fvsu.edu

Henning Stahlberg University Basel

CH-4058 Basel, Switzerland 41-61-387-3262

henning.stahlberg@unibas.ch

Owen Hughes Eon Corporation

707 4th St., Suite 305 Davis, CA 95616 1-530-756-6903

owen@eonresearch.com

ABSTRACTHigh Resolution three-dimensional (3D) reconstruction of several proteins has been achieved from two-dimensional (2D) crystals by electron crystallography structure determination. However, for badly ordered 2D crystals, especially non-flat crystals, Fourier-filtering based methods fail, while single particle processing approaches can produce reconstructions of superior resolution by aligning particles in 3D space. We have investigated a single particle processing approach combined with the crystallographic method to generate images centered on the unit cells of 2D crystal images. The implemented software uses the predictive lattice node tracking in 2dx/MRC software to extract particles from the microscope images. These particles are then subjected to a local contrast transfer function (CTF) correction. The tilt geometry obtained in the 2dx software is used to initialize the Euler angles, which along with translations are then refined by a single particle processing approach. Finally, iterative transform algorithms, namely the error reduction algorithm and the hybrid input-output algorithm, are applied to retrieve missing information in the previously obtained 3D reconstruction. Compared with conventional single particle processing for randomly oriented particles, the required computational costs are greatly reduced as the 2D crystals restrict the parameter search space. Preliminary results from a 3D reconstruction of the membrane protein GlpF suggest that the iterative transform process improves 3D resolution.

Categories and Subject DescriptorsI.4.5 [Image Processing and Computer Vision]: Reconstruction – transform methods; I.5.4 [Pattern Recognition]: Applications-computer vision

General Terms

Algorithms, Design, Performance, Experimentation

Keywords3D reconstruction; electron crystallography; single particle processing; error reduction algorithm; hybrid input-output algorithm

1. INTRODUCTIONMembrane proteins represent the main target for drugs that are currently under pharmaceutical development. Protein structure based drug design is one of the most powerful methods of drug development. However, structures of membrane proteins have been extremely difficult to produce. Challenges range from difficulties in production of active purified membrane proteins to difficulties in growing crystals for structure determination. For crystallized samples, cryo-electron microscopy (cryo-EM) has developed popularity in the study of membrane protein structures. Cryo-EM possesses the merit in studying biological specimens that samples are in close to their “native” environment at cryogenic temperatures and thus eliminating undesired conformational changes.

Structure determination by cryo-EM so far has been slow, mainly due to the inefficiencies in data collection and data processing. Computer image processing has greatly advanced the speed and resolution of determining the 3D structure of membrane proteins from 2D crystals. However, challenges remain in image processing for membrane protein structure reconstruction from high-resolution electron microscopy images. First, most biological specimens are extremely radiation sensitive, so they must be imaged with low-dose techniques. Consequently, cryo-EM images have extremely low signal to noise ratio (SNR). The low SNR can only be overcome by averaging a large number of images of individual particles. 2D electron crystallography technique, where the particles to be imaged must be crystallized so that all the individual particles are oriented in exactly the same direction, allows a kind of averaging which can provide high-resolution information about the specimen [1]. This process requires near perfect crystals which in many cases are not amenable. In recent years there has been increasing interest in using single particle cryo-EM where many identical particles of the same structure are imaged in random orientations, and then a

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single composite three-dimensional model is assembled from tens of thousands of those images [2]. Averaging of aligned projections in the same orientation is used to increase the SNR and thus the effective resolution. Second, the range of tilt angles for which projected images of two-dimensionally periodic specimens can be obtained in electron microscopy is limited both by technical aspects and by the limitation of object thickness. The lack of a full set of projections causes missing components in the reciprocal space data for the object. Fourier reconstruction algorithms allows for interpolation in Fourier space. However, it has been realized that isotropic resolution may be drastically reduced[3] Therefore, it is common practice not to interpolate the data but set them to zero, which yields one of the solutions that are compatible with the observed data and assume no constraints on the data. In a summary, the Fourier components which are either missing or deteriorated by noise cause resolution loss.

Here, we develop new algorithms to address the aforementioned issues. First, we combine the single particle processing and the crystallography method for 3D reconstruction of 2D crystals. Local fluctuations of the unit-cells or large-scale warping of the crystals can be in the plane (two translational displacements and one in-plane rotation) or out of the plane of 2d crystals (two three-dimensional rotations). Unit cells or particles are extracted from the micrograph, and their displacements and rotations are determined by a reference-based single particle processing approach. The initial volume is obtained from the crystallographic method. Combing the crystallographic method with single particle processing can lead to an improvement over the separate application of either. Furthermore, we also show how the contrast transfer function (CTF) of the electron microscope can be included in the processing. Second, motivated by the error reduction algorithm and hybrid input output (HIO) algorithm widely used in phase retrieval problems [4][5][6], we propose to apply iterative transform algorithms for restoring missing information in the 3D structures

The paper is organized as follows: in section 2, we introduce the 3D reconstruction of 2D crystals. Section 3 gives the error reduction algorithm and the HIO algorithm for restoring missing data in 3D reconstruction. The reconstruction results of a synthetic object and the membrane protein GlpF are shown in section 4. The concluding remarks are given in the final section.

2. ELECTRON CRYSTALLOGRAPHY AND 3D RECONSTRUCTION 2.1 Single Particle Processing for 2D Crystals Electron crystallography studies the structure of membrane proteins in the membrane-inserted state. Two-dimensional (2D) crystallization in the membrane is induced by adjusting the protein-lipid-protein interactions via the manipulation of PH and other parameters. Once 2D crystals are obtained, they are imaged in the frozen hydrated state in a transmission electron microscope from different orientation. Computer image processing of recorded electron microscopy images is then used to reconstruct the 3D structure of the protein. Projection theorem states that the Fourier transform of a 2D projection furnishes a central section of a 3D object’s Fourier transform. This suggests that reconstruction can be achieved by filling the 3D Fourier space with data on 2D central planes that are derived from the recorded projections by 2D Fourier transformation.

Crystals provide higher SNR than isolated molecules by merging a large number of particles in identical orientations. The information contained in the image of a perfect two-dimensional crystal is easily extracted in Fourier space because all of the information about the average structure is contained in the relatively small number of diffraction spots in the Fourier transform of the image. 3D reconstruction is straightforward. The information extracted from different crystal images is filled into the 3D Fourier space that ultimately can be used to build the object in real space.

However, real crystals and images are imperfect. These imperfections arise from distortions of both the specimen and the image. For instance, the effect of the contrast transfer function (CTF) of electron microscope can degrade the image quality severely. Another common distortion is that crystals are “bending”. These errors must be corrected before the data are used in 3D structure reconstruction. The few available software packages for 2D crystal image processing include MRC and 2dx [7][8].

In addition, 2D crystallography so far is based on the assumption that the crystal is on a flat sample support so that one image shows the proteins under precisely the same tilt angle throughout the entire micrograph. In reality, however, 2D crystals are rarely flat. When the sample support is not perfectly flat, each particle in the crystal may have slightly different orientations. This lack of flatness is especially critical for tilted samples. In this case, applying the same tilt geometry to the entire image yields low-resolution 3D reconstruction.

We here use a variable tilt-geometry method for 2D crystal image processing. This is a single particle processing method in that unit cells in the crystals are treated like single particles. Approximate tilt geometry of the crystal image and positions of unit cells are obtained using 2dx software. Particles are then extracted from the image and form a stack of single particles. An initial 3D structure is obtained from these particles using Fourier reconstruction method. This structure suffers resolution loss due to the unflatness problem, but can be used as an initial reference. The ultimate 3D structure is obtained through the following iterative refinement process, which determines the local variations in the crystal tilt geometry and improves 3D resolution.

Let Tzyx ),,(r be the 3D coordinate system affixed to the

molecule, and Tjjj yx ),(r be the coordinate system of 2D

images. By projecting the molecule along a direction )(iD ,defined by three Euler angles i , i , and i , we have

projection )()(i

ip r . With the N particles )( jjf r (j=1, 2, …N) and

the initial reference )(rO , the refinement process determines the tilt geometry for each particle through the following steps:

a). Project the 3D reference )(rO in M directions, which

produce the projections )()(i

iOp r (i=1,2,…M)

b). Determine the tilt geometry for each particle. Calculate the cross correlation function between )( jjf r and all the

projections )()(i

iOp r . The tile geometry is the projection

direction m , m , and m that yields the maximum cross correlation value. The peak coordinates in the cross correlation map represent the displacement from the origin.

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c). Shift the particle to the origin and insert the Fourier components of the particle into the central plane with angles

m , m , and m in the 3D Fourier space. d). Perform inverse Fourier transform and obtain a new 3D

structure )(rO , which is used as the reference for the next iteration.

The above-described approach combines conventional 2D crystal image processing with method of single particle image processing for angular assignment. The main advantage is an opportunity for refining the local tilt geometry, which would otherwise be static across the entire image. However, strong noise in single particles may perturb the angle assignment based on cross correlation and be reflected in the 3D structure. In addition, the tilt geometry may not cover enough range in the Fourier space. Therefore, we apply the iterative transform algorithms to retrieve the missing data in the so obtained 3D structure.

2.2 Contrast Transfer Functions (CTF) The discussion in the previous section does not take into account the CTF of the electron microscope. CTF can degrade the image quality severely and must be corrected before 3D reconstruction to achieve high resolution.

A whitening filter wH is obtained by averaging the power

spectrum along the path of the thon rings

Hw (k)1

1N

F 2( ˜ R )(k i)k1 (k i ) k1 (k )

(1)

where RF ~2 is the power spectrum of Fourier transformation of image )(rf , k1 k i k1 k indicates the pixels k i in reciprocal Fourier space that are on the same Thon ring as pixel k . N is the number of Fourier pixels along one Thon ring that are averaged. CTF correction of an image )(rf can be performed in the Fourier space by:

Fn (R~

)(k) F(R~

)(k)Hw (k)Sign[CTF(k)] (2)

Eq. (2) includes a multiplication by Sign CTF k which equals +1 for CTF 0 and -1 for CTF 0 . The multiplication does not change the normalization of the power spectrum. However, it performs a phase flip according to the CTF, which is given by:

CTF k w1 sin k w2 cos k (3) with

k k2 f k 12

2k2Cs (4)

k is the magnitude of k. The two constants w1 and w2 are given by the percentage of amplitude contrast, W, in the image:

w1 1 W 2

w2 W (5)

Usually, the value for W ranges from 0.07 to 0.14 for proteins embedded in ice, and 0.19 to 0.35 for proteins embedded in stain (uranyl acetate). is the electron wavelength and Cs is the spherical aberration coefficient of the objective lens. The defocus

f is given by f k 1

2 DF1 DF2 DF1 DF2 cos 2 k ast (6)

Where DF1 and DF2 are the two defocus values describing the defocus in two perpendicular directions in an image when astigmatism is present, ast gives the angle between the first

direction (described by DF1 ) and the x-axis, and k is the angle between the direction of the scattering vector k and the x-axis. For a particular image, DF1, DF2 and ast can be determined using a fitting procedure that fits a calculated CTF to the observed Thon rings. Note that in Eqs. (4) and (6) a positive value for the defocus indicates an underfocus.

For tilted 2D crystals, the defocus and therefore the CTF varies across the image with position according to the equation:

cDFyxDF )tan()sin()( 22 (7) Where

cDF is the defocus at the origin taken as the center of the image, x , y are the co-ordinates of the particle relative to the origin, is the tilt angle, is the angle between the line to the particle and the x-axis of the image, and is the angle between the tilt axis and the x-axis.

3. RETRIEVING MISSING DATA BY ITERATIVE TRANSFORM ALGORITHMS In its general form, the missing data problem is to estimate the original form of a signal in a functional space from the measurements of physically related signals and a priori information or constraints. Simulation studies have been undertaken to investigate the retrieval of missing phase, missing cone, and missing wedge using the iterative transform algorithm. It was noted that even small amount of noise dramatically affected the retrieval performance of the algorithms [9]. Noise-free condition is not acceptable in electron microscopy. Therefore, we consider components polluted by noise also as missing data. From a very different point of view, we study the application of the iterative transform algorithms in 3D electron microscopy.

In 3D imaging by electron microscopy, the measurements consist of the available projections f and the original signal is a 3D real space object O. The relationship between them is

))(()( rr OPf iDii (i=1, …, m) (8)

iD denotes the projection orientation, and P is the 2D projection of O onto f. The a-prior knowledge is finite object extent and positive electron densities.

The 3D reconstruction scheme is to restore the original object from the projection data. As in many inverse problems, the

solution is any object ^O that is consistent with the measurements

and satisfies other constraints, i.e.

)()()( rrr OConandfOP iiiD (9)

Here the key point is that the projection orientation iD is not physically measured but obtained by single particle processing. Therefore, it is quite possible that the 3D object obtained in Section 2 is only partially consistent with the measurements. The following algorithms are developed to enhance the consistency with the measurements and to restore the data missing in the measurements.

3.1 ConstraintsBecause of the ill-posed nature of the problem, the constraints are critical for achieving an optima solution. We apply the following constraints in Fourier space and real space.

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3.1.1 Finite Support and Positivity The constraints of objects in real space are their spatial limitation of “finite support”. The object is zero outside a certain volume Vand positive inside.

otherwiseV

O00

)(^ r

r (10)

In out experiments, approximate object extent constraints are sufficient.

3.1.2 Fourier Constraint The Fourier components that belong to “true” signal set are retained over the iterations.

otherwiser

rand

OF

n

n

n

n uuu

u

uu

)(

)(0

)()(

^1

(11)

u is the coordinate system in Fourier space, E is the object in Fourier space, and F is the Fourier transform operation. The signal truth is practically unavailable in single particle processing; we retain the strong Fourier components up to a certain resolution.

3.2 Error Reduction Algorithm The error reduction algorithm directly implements the positivity

constraint and updates a current nO^

as below:

otherwiseEFandVifEF

O nnn

00))(())((

)(11

1

^ rrrr (12)

1F is the inverse Fourier transform.

3.3 Hybrid Input-Output Algorithm A variant of HIO algorithm is proposed in [6], which is given as:

otherwiseEFrO

VifEFO

nnn

nn

))(()*21()(

))(()(

1^

1

1^

r

rrr (13)

is a relaxation parameter.

3.4 Iterative Transform Algorithms for 3D ReconstructionThe iterative transform algorithm starts with the 3D object obtained in Section 2, and iteratively update the data using constraints in Fourier space and real space.

a). Prepare the initial estimate of 3D object by filling its missing Fourier components with random values. Perform inverse Fourier transform to produce the 3D structure in real space.

b). Impose the constraints onto the object, which includes: 1) . Set pixels outside the object boundaries zero. 2) . Update the negative pixels using (12) or (13)

c). Perform Fourier transform of the object. Apply the Fourier constraint described by (11).

d). Perform inverse Fourier transform. Go to step (b).

4. EXPERIMENTAL RESULTS OF 3D RECONSTRUCTIONIn this section, we demonstrate 3D reconstruction of 2D crystals using the method that combines single particle processing and crystallography image processing, and the iterative transform algorithms.

4.1 3D Reconstruction of a Synthetic Object Before we proceed to the 3D reconstruction of 2D crystals, we analyze the performance of the iterative transform algorithms using a synthetic binary object as shown in Figure 1, which consists of four cylinders with an inner radius of 5 pixels, an outer radius of 10 pixels, and a height of 5 pixels. The object was padded into an 80x80x80 cube. In order to simulate the circumstances in 3D reconstruction of electron micrographs, we generated 1200 projections in six different orientation categories.

The tilt-angles are less than 60 . 200 projections in each category simulate the unit cells in a 2D crystal and have slightly different tile geometry oscillating around the crystal tilt geometry. Gaussian noise was added to all the projections and the SNR was 1/16.

A 3D structure was obtained by the reconstruction method described in section 2. The error reduction algorithm and the HIO algorithm were then applied to this 3D structure. In both methods, the support boundary V is 60x60x40. Figure 2 shows the central sections in the direction perpendicular to z-axis of the object and the 3D structures. The two methods were able to retrieve part of the missing data, both in low frequency components. It is also noted that the error reduction algorithms may get stuck in local optima, where the HIO algorithm highly depends on the selection of the relaxation parameter . We used a fixed parameter over the iterations, which probably is not in favor of this method. Further study on how to select this parameter is under planning. The absolute difference between the reconstructed real space object and the original object is calculated within the object boundary 50x50x10. The results are shown in Figure 3. The absolute difference of the HIO algorithm quickly declines in the first several iterations and slightly goes up after that.

4.2 3D Reconstruction of Protein GlpFThe 3D structure of GlpF, the glycerol facilitator of Escherichia coli, was built from 2D crystal data. Images were recorded at various tilt-angles up to 60 [10][11], from which we selected six for the following experiment.

We first processed each images using software 2dx to obtain tilt geometry, and parameters for correcting the contrast transfer function. The approximate coordinates of unit cells are also obtained, which are then used to extract 80x80 particles form the CTF corrected 2D crystals. All the particles are padded into 120x120 squares and merged, and the initial 3D structure is shown in Figure 4. This structure is the equivalent result of the crystallography method used in 2dx. The single particle processing algorithm in Section 2 is applied to refine the initial structure. Figure 5 shows the top view and the side view of the refined 3D structure. The iterative transform algorithms were then applied to this structure. The support volume is 80x80x60. The HIO algorithm was not stable and easily converged to wrong structures. This coincides with the unstable convergence observed

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in the first experiment. The 3D structure obtained using the error reduction algorithm is shown in Figure 6. Although we have no quantitative measurement to evaluate the performance, the differences between these two structures are visible.

Figure 3. A convergence plot showing the absolute difference between the original object and the 3D structures restored using the error reduction algorithm and the HIO algorithm

Figure 4. Initial 3D Structure of GlpF (side view)

Figure 1. Synthetic object with four cylinders

A B

C D

Figure 2. Central sections in Fourier space of (A) original object, (B) 3D structure obtained using the single particle processing method, and 3D structures restored using (C) the error reduction algorithm, (D) the HIO algorithm

A B

Figure 5. 3D Structure of GlpF obtained using the single particle processing method, (A) side view, (B) top view

Figure 6. 3D Structure of GlpF restored using the error reduction algorithm, (A) side view, (B) top view

A B

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5. CONCLUSIONSIn this paper, we study 3D structure determination of membrane proteins from 2D crystals. A single particle processing approach is combined with crystallography technique to achieve high resolution. The iterative transform algorithms are applied to retrieve missing data caused by noise interruption and limited projections. Experiments were carried out on synthetic data and electron microscopy images. The results quantitatively or visually demonstrate the effectiveness of the algorithms in 3D reconstruction of 2D crystals.

Future work is to explore new criterion other than cross correlation for angle and translation refinement. Developing global optimization algorithms to overcome local optima is also under planning. It is an open question that to what extent the iterative transform algorithms have restored the missing data caused by noise and that by missing projections. This is also an interesting issue and deserves further study.

6. ACKNOWLEDGMENTSThis work is supported by the National Institutes of Health, grant no. R43-GM083921-01.

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