Helices

(Modified 2015 Feb 08)

Gizmo's Introduction to Helical Image Processing in Electron Microscopy

Gizmo locates the n=6 diffraction maximum.

Outline

• • What is a helix?

  • The helical lattice

  • The Fourier transform of a helix

  • Image filtering

  • Image averaging

  • Flowchart of image processing

    • • Digitization and spline-straightening

      • Fourier transformation and layer-line collection

      • Alignment and averaging

      • 3-D reconstruction and visualization

  • References

What is a helix?

A helix is a repeating structure whose asymmetric units related by a translation along (Dz, the subunit rise) and rotation about (Df, the subunit rotation) the helical axis.

A hypothetical helix of bacterial flagellar motors

A helical particle can be polar; that is, rotating the helix 180 degrees gives rise to a distinguishable structure. In the figure above, upon 180-degree rotation, the bell-shaped structure would open downward instead of upward.

For a sufficiently long helical particle, rotation and translation of a helix by an integral multiple of the subunit rise and rotation gives rise to a superimposable structure. This feature is of key importance in generating a 3-D reconstruction (below). Strictly speaking, labelling one subunit (as in the figure above) breaks the perfect symmetry of the helix, as does having a helix of finite length.

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The helical lattice

A helical particle can be thought of as a two-dimensional lattice rolled up into a tube. Here, we have cut and unrolled the helix into its corresponding 2-D lattice. The cut is parallel to the helical axis.

The horizontal axis in the figure above corresponds to an angular coordinate about the rotation axis; with the helix cut and flattened, the distance traversed by this axis is 2pr, where r is the radius of the helix. Distance along the helical axis is defined as the z coordinate.

Like any 2-D lattice, a helical lattice can be described by families of lattice lines, equally-spaced parallel lines that go through every point in the lattice. Two pieces of information are needed to describe a lattice family: 1) the repeat distance along the helical axis between adjacent lattice lines and 2) the number of lattice lines needed to pass through every point in the lattice (the order n). Keep in mind that lattice lines wrap around from one side to the other.

The n=-5 lattice family

In the figure above, five lattice lines are needed to pass through every lattice point, hence the order n of 5.. The negative sign is the convention used to denote lattice lines that follow a lefthanded spiral. (For a lefthanded spiral, if your fingers follow the spiral's path, the thumb on a normal hand should point in the direction of motion along the helical axis. Most hardware screws are righthanded.)

Equivalent information can be carried by a vector relating adjacent lattice lines. The length of the vector will be the distance between lattice lines, and the direction is orthogonal to that of the lattice lines.

While two parameters are sufficient to define a lattice family, more information is needed to define the actual lattice. In other words, for the figure showing the n=-5 family above, only a series of stripes is defined, and not the actual lattice points. As with any 2-D lattice, one pair of orthogonal units are necessary and sufficient to regenerate the entire lattice. For the helix shown here, we will use the n=-5 and n=-11 lattice families.

The lattice defined by the n=-5 and n=-11 lattice families

Thus, the intersections of the n=-5 and n=-11 lattice lines define the 2-D lattice.

It will be useful later on to express the repeat distance in units of reciprocal distance, which we will call Z. If, for example, z for the n=-5 lattice family is 40 Angstroms, then Z will simply be 1/40 reciprocal Angstroms.

From two sets of lattice parameters, we can obtain the descriptive parameters for every other lattice family. We accomplish this by adding linear combinations of the unit lattice parameters:

.

Let n=-11 is the first lattice family with Z=1/200, and n=-5 is the second lattice family with Z=1/40. If the coordinate coefficients h and k are (-1,1), then

thus giving us the n=6 family with repeat distance 50. The veracity of this calculation can be confirmed geometrically.

The n=6 lattice family

The vector addition of the linear combination of +1 times the n=-5 unit vector plus -1 times the n=-11 unit vector gives the same answer in perhaps the more intuitive way. Note that, with a positive order, the n=+6 lattice lines are righthanded.

Now that we are able to describe the repeating features in the helical lattice, we can explain the appearance of the Fourier transform.

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Fourier transform of a helix

The diffraction pattern is the physical manifestation of the mathematical entity known as the Fourier transform. Essentially, the Fourier transform looks for repeating structures. The repeat need not be in the form of a lattice, although a lattice, as a highly regular repeating structure, shows a strong signal. Fourier data are complex numbers, i.e., with real and imaginary components. For many purposes Fourier data are perhaps easier to think about as vector quantities, containing amplitude and phase. The Fourier transform contains all the information present in the real-space image. In other words, Fourier transformation is fully reversible, without any loss of information. Thus, there are some operations that are easier to carry out in Fourier space than in real space.

To understand the Fourier transform (FT) of a helix, I will first describe the FT of the corresponding 2-D lattice, then the 2-D lattice folded over into a flattened tube, and lastly with the helical particle itself.

The Fourier transform of a 2-D lattice is a 2-D lattice. The repeat distance of lattice lines in real space is inversely related to the repeat distance of the lattice lines in Fourier space. For the reciprocal relation between real space and Fourier space, Fourier space is sometimes called reciprocal space.

Fourier transform of a 2-D lattice

For each lattice family in real space, there will be a corresponding diffraction maximum in Fourier space. The vectors describing the lattice lines in real space will point in the same direction, although the lengths will be different (but relatable).

A flattened tube is similar to a 2-D lattice, but has a front and a back similar to the case of a helix. In the flattened tube described here, the left half of the 2-D lattice above is folded behind the right half.

Fourier transform of a flattened tube

The front (black) and back (gray) of the flattened tube give qualitatively identical diffraction patterns. However, as the front and back are flipped horizontally relative to each other, so are the diffraction patterns. By fortuitous accident, the diffraction pattern of the more intense front of the tube is stronger than that of the back of the tube.

To reach a helix, we unflatten the flattened tube. The lateral spacings of lattice points on the front-most and back-most surfaces of the helical particle (i.e., surfaces whose tangents are parallel to the image plane) will be approximately the same as the spacings of the corresponding flattened tube. On the edges of the helix, the lattice points in projection will appear closer together than in the flattened tube.

Fourier transform of a helical particle

The additive contributions from these parts (the front & back vs. the edges) of the helix will result in diffraction maxima like those of a flattened tube but that smear outwards. Diffraction maxima of a helix are called layer-lines.

Comparison of the FT's of a 2-D lattice, a flattened tube, and a helical particle

Note how the lateral symmetries change, but the heights of the diffraction maxima remain unchanged.

Once we know how to find the diffraction maxima corresponding to our two basis lattice families, we can find the diffraction maxima for any lattice families. As shown above, the n=6 lattice family could be described as:

with distance units assumed to be in (reciprocal) Angstroms. Due to the reciprocal nature of the Fourier transform, the repeat distances measured in reciprocal Angstroms add simply, such that the height of the n=6 layer-line is the height of the n=-5 minus the height of the n=-11 layer-line.

A Fourier transform of the hook from the bacterial flagellum

Similarly, the height of the n=1 layer-line can be obtained by subtracting the height of the n=-11 layer-line from twice the height of the n=-5 layer-line.

The horizontal axis in the FT is called the equator, and the vertical axis is known as the meridian. The Fourier transform of a real (non-complex) object has what is called Friedel symmetry; rotating the FT 180 degrees gives a pattern with identical amplitudes but the phases will be flipped.

The utility of the ability to predict the location of layer-lines is that, since we know where signal will lie, we can eliminate the noise that appears elsewhere.

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Image filtering

Images taken on the electron microscope are noisy. We reduce the noise in two ways: by removing data that cannot be signal and by averaging the information from many images together.

On the left in an unfiltered image. On the right is the same image after Fourier filtration. In this case, filtration means that only Fourier data found on calculated layer-lines are used to generate the resulting image.

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Image averaging

Typically, the signal-to-noise ratio improves by the square root of the number of samples taken. In other words, quadrupling the sample size improves the signal-to-noise ratio by a factor of two. This is the case with averaging images.

Going from one image to 16 should improve the signal-to-noise ratio by a factor of four, and going from 16 images to 262 should improve the signal-to-noise ratio by approximately another factor of four.

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Flowchart of image processing

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Digitization & spline-straightening

Digitization involves scanning microscopy data into the computer. In some cases, the microscopy data may hve been recorded directly into a digital format using a CCD camera.

Helical processing methods require that the particle of interest is a perfect helix. Real structures will be imperfect, but some imperfections can be corrected on the computer. A slight curvature is one aberration that can be corrected, by fitting the particle to a spline and then straightening that spline.

The first step of spline-fitting is picking points along the helical axis (left-most image above). This process can be automated or done manually. The second step is that these points are fit to a spline (middle image). Lastly, the image is interpolated such that the calculated spline is now straight (right-most image).

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Fourier transformation & layer-line collection

The algorithm used to calculate Fourier transforms is typically the Fast Fourier transform (FFT). Here, it is often optimal to use image dimensions that are a power of two. FFT calculation is very fast on modern computers. However, old-timers reminiscing about how slow computers used to be can be very time-consuming.

For real samples, it is likely that only a subset of layer-lines will be visibile in a given FT. After calculating the FT, one selects the visible layer-lines. If we know the symmetry of the particle of interest and can assign the order n to the visible layer-lines, we can calculate as many layer-lines as we would like.

The information not found on layer-lines is assumed to contain only noise and is thrown out. The remaining operations can be carried out on the few lines of layer-line Fourier data.

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Alignment & averaging

In order to be able to average images together, the particle of interest must be in the same orientation in all images. In principle, helical particles can be relatable by a shift and rotation. In practice however, one can get an approximate match using only a shift for typical helical particles. (For short or sparse helices, this approximation can fail.)

Alignment using only shift parameters allows us to use cross-correlation methods, which are very fast and straightforward. (Methods for matching helices using true true translational and rotational shifts have been implemented recently.) In cross-correlation, an experimental image is aligned to a reference image. The combination of shifts in x and y that give the best match is found. While this sounds like a brute-force search, this calculation is very fast in Fourier space.

In this way, any number of experimental images can be aligned to the reference. The aligned experimental images are lastly added with a straightforward, vector addition of their Fourier data.

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3-D reconstruction and visualization

The Projection Theorem (sometimes known as the Central Section Theorem) states that the (2-D) Fourier transform of a 2-D projection is a slice through the 3-D Fourier transform of the (3-D) object. What is required in order to obtain a 3-D reconstruction, then, is to obtain different views of our particle of interest and fill its 3-D FT.

For a helical particle, its symmetry essentially tells us how the different views look. Above, we learn that one view of a helical particle is equivalent to other views related by a integral multiple of the subunit shift & rotation. Thus, knowing a view in one direction gives us in essence a view along many directions.

A schematic of 3-D reconstruction

The last step in the 3-D reconstruction is Fourier-Bessel inversion. The inputs here are a layer-line data set and knowledge of the helical symmetry. Thus, the purpose of averaging is to improve the signal-to-noise ratio, and the symmetry allows us to fill the 3-D FT with information from a 2-D FT.

Now with a 3-D volume, there exist many means of visualization. Surface maps are straightforward, although internal details will not be seen. Wireframe maps allow a limited degree of transparency and inner detail. Two-dimensional slices (not shown) through the 3-D volume allow a relatively faithful presentation of the data, as it requires no choice of contour level, but the objects are only 2-D.

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References

Egelman EH (1986)

An algorithm for straightening images of curved filamentous structures.

Ultramicroscopy. 19(4):367-73.

This article describes a spline-straightening procedure.

Egelman EH (2000)

A robust algorithm for the reconstruction of helical filaments using single-particle methods.

Ultramicroscopy. 85(4):225-34.

This article describes a methodology for finding the relative shift and angle between a reference and an experimental image, rather than using cross-correlation in x & y.

Jones TA, Zou JY, Cowan SW, Kjeldgaard (1991)

Improved methods for building protein models in electron density maps and the location of errors in these models.

Acta Crystallogr A. 47 ( Pt 2):110-9.

This is the reference for the molecular imaging program O.

Moody MF (1990)

Image analysis of electron micrographs.

In Biophysical Electron Microscopy: Basic Concepts and Modern Techniques (P. W. Hawkes & U. Valdre, eds.) Academic Press, New York pp. 145-287.

This review contains just about everything, as the broad scope of the title suggests.

Owen CH, Morgan DG, DeRosier DJ (1996)

Image analysis of helical objects: the Brandeis Helical Package.

J Struct Biol. 116(1):167-75.

Many of the figures shown here were generated by the Brandeis Helical Package.

Stewart M (1988)

Computer image processing of electron micrographs of biological structures with helical symmetry.

J Electron Microsc Tech. 9(4):325-58.

An excellent review of helical image processing

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