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## Introduction

The purpose of this set of notes is to guide the student in reading the original literature on the analysis of three dimensional structures by electron microscopy, with emphasis on helical structures.

See also the lecture by David DeRosier: “A Tutorial on Helical Structures”

A good place to start is the classic paper by Crowther *et al.* 1970 [cite source=doi]10.1098/rspa.1970.0119[/cite] (download: CrowtherDeRosierKlug.pdf). This paper deals with the general problem of reconstructing images in Fourier space. The student may wish to refer to some of the earlier work of these authors cited in this paper where it is mentioned in the text, although this should generally not be necessary.

In reading through this paper and those discussed later, it is important not to get bogged down. If a point is not clear, make a note and carry on. Many points will only become apparent once you have an overall picture.

## Important Points In The Paper

- The electron micrograph is a projection of the density in the specimen along one axis and thus the Fourier transform (FT) of the image is the central section of the FT of the object perpendicular to that axis.
- The three-dimensional structure of the object can only be reconstructed from its FT if the sample points of the FT are regularly spaced.
- Information about the 3D FT of the object can be built up from the FT of a sequence of images made at different angles of tilt about a single axis. However, it is practically impossible to sample the whole of space in this way and thus it is necessary to interpolate in order to obtain regularly spaced data points.
- Much of the discussion is about the optimum method for interpolation and need not concern the student for the present. However, an important general principle emerges, namely, the equations are considerably simplified by expressing the problem in a coordinate system appropriate to the symmetry of the experiment. In this case these are cylindrical polar coordinates.
The derivation in section 4.1 of the paper shows how the Fourier transform can be expressed in terms of a set of

*measurable functions*G_{n}(R, Z) and how these can be transformed to give the function describing the object, ρ(x, y, z). The general case given in the paper is analogous to the case given by Klug, Crick and Wyckoff (1958) [cite source=doi]10.1107/S0365110X58000517[/cite] (download: KlugCrick.pdf) for objects which are periodic in z. As these are of immediate interest to us, their treatment will be expanded here.If the density is periodic in ƒ and z (with periodicity c) we may express it as a two dimensional Fourier series

where the g

_{n,l}(R) are complex functions of r defined byThe Fourier transform of ρ in cylindrical polar coordinates is

Substituting the Fourier series expansion we obtain

Because of the periodicity in z the transform will be non-zero when Z = 1/c; thus on any one layer l, we have

This can be simplified using the integral definition of the Bessel function:

where G

_{n,l}(R) is defined as the underlined part. The significance of this result is explained in the paper.In order to obtain our density function we need to know g

_{n,l}(R). This can be obtained from G_{n,l}(R) by reverse transformation because the orthogonality of Bessel functions, namelyThus

- If the object has any intrinsic symmetry then the Fourier transform of the image contains information about more than one central section and the number of views required to sample three dimensional space to the required resolution is reduced. The point is made that in the case of a particle with helical symmetry only one view may be necessary.
- Section 6 of the paper concerns itself with the number of views needed for reconstruction and works through the reconstruction of an icosohedral particle. This together with section 7 and 8 and the Appendix make interesting reading but are not of immediate concern.
**Note**: The comments made about the magnitude of the computational problem do not apply to today’s computers.

As may be readily appreciated from the foregoing, the key to reconstructing the object lies in identifying the functions in G_{n,l}(R) the Fourier transform (FT) of the image. Our aim is to show that because of the high symmetry of a helical object, many of these functions are available in the FT of a single image and identifying them may therefore allow reconstruction to a limited resolution. To do this we need to relate the FT of a helical object to that of the general periodic object discussed earlier.

The FT of a helix was first described by Cochran et al. 1952 [cite source=doi]10.1107/S0365110X52001635[/cite] (download: CochranCrickVand.pdf). They develop their ideas in a semi-intuitive manner by first transforming a continuous helix and then developing the case for a helix made up of discrete atoms. A more conventional and perhaps more generally useful, although less reliable derivation is given in an appendix to a paper by S. Tanaka and S. Naya, 1969 [cite source=doi]10.1143/JPSJ.26.982[/cite] (download: Tanaka_Naya_1968.pdf).

Before attempting either of these, the student should be familiar with the properties of helices.

## Continuous Helix

A helix is the locus of a point which satisfies the equations

r = constant

z = Pφ/2π

where r, φ, and z are **cylindrical polar coordinates**. The constant r is called the **radius of the helix**.

The helix thus defined lies along 0z and intercepts the plane z = 0 at φ = 0.

The helix is periodic in z. The section between z = 0 and z = P being the same as that between z = P and z = 2P, and so on.

Continuous helices are **enantiomorphic** since their reflection about z = 0, for example, produces a helix which cannot be superimposed on the original.

A helix which corresponds to a positive value of P is called **right handed** and one which corresponds to a negative value of P is called **left handed**. The angle that the tangent to the helix makes with the plane z = 0 is called the pitch angle.

## Radial Projections

On the left is an illustration of a radial projection of a right handed continuous helix.

An extremely useful representation of the helix is obtained as follows: Features of the helix are projected along radial lines onto a cylindrical surface running parallel to the z-axis, the surface is then cut parallel to the z-axis, starting at φ = 0 and opened out flat (see the illustration on the right).

Conventionally the outside surface of the cylinder faces upwards and thus lines of positive slope correspond to right handed helices and lines of negative slope correspond to left handed helices.

## Discontinuous Helix

A discontinuous helix is an infinite set of points which lie on a continuous helix and are separated by a constant axial translation, h.

The pattern of points will repeat itself after a distance c along the axis, i.e., after t=c/P turns of the helix of u=c/h points.

We can therefore describe the helix as having u equivalent points in t complete turns, where u and t are obtained by expressing h/P as a rational function.

If P is positive, then both u and t are positive. If P is negative, then u is negative.The description of a discontinuous helix in terms of h, P, and u is not unique. However, they are usually chosen such that the **unit of twist**. T=2pt/u is a minimum. This is called the basic helix.

The figure on the right shows the radial projection of a right handed discontinuous helix with u=18 and t=5.

## Symmetry of Discontinuous Helical Structure

For any helical structure there will be a set of operations which map the helix onto itself, these operations forming a group.

It is generally possible to select a line within the structure which will be unaffected by any of the symmetry operations. We select this line to be the z- axis.

We can use the radial projection to explore the symmetry possibilities of helices. If we lay out the radial projections of the helix side by side in register we will form an infinite two dimensional plane group. It turns out that if the subunits located at the helix points are enantiomorphous then only two plane groups are possible, p1 and p211.

Apart from the screw displacement, ** s**, which generates the helix by translating a point a distance h parallel to the z-axis together with a rotation of 2πu/t about the z- axis we have the possibility of one or both of the following symmetry operations:

** r**: a rotation of 2π/N

about the z axis, where N is an integer > 1

** 2**: a twofold axis about a line passing through the z-axis and perpendicular to it

The student is now in a position to read Cochran, Crick and Vand. Their most important result is that reflections from a regular helix obey its reflection rule, which in our notation can be expressed

l = tn + um

where l is the layer line number, n is the order of the Bessel function and m is any integer which satisfies the above.

The following illustration from Frazer and MacRea will help to illustrate the convolution described in Section 3.

Equation 8 enables us to identify G_{n,l}(R) i.e.

which is independent of Φ. Making the identification enables us to write the value of the layer line in the form of equation

We have thus achieved our objective in relating the diffraction pattern of a helix to the FT of a general periodic object expressed in cylindrical polar coordinates.

An important omission from this paper is the effect of symmetry. This is discussed by Klug, Crick and Wyckoff (1958) [cite source=doi]10.1107/S0365110X58000517[/cite] or may be easily derived by the student using the formalism of Tanaka and Naya (1969) [cite source=doi]10.1143/JPSJ.26.982[/cite].

We are now ready to tackle the most important paper in the present course, namely that by de Rosier and Moore (1970) [cite source=doi]10.1016/0022-2836(70)90036-7[/cite] (download: DeRosier_Moore.pdf). This paper deals with the technicalities of reconstructing three dimensional images from micrographs of structures with helical symmetry. The paper begins with a summary of the theory we have developed thus far. The student should now ensure that the concepts involved are understood.

The final part of the argument has not been previously encountered, namely:

If the selection rule is such that to the resolution required the Bessel functions on the layer line do not overlap, then the expression for the FT reduces to

This enables two estimates of the function G_{n,l}(R) to be made from the micrograph, corresponding to Φ = 0 and Φ = π. These arise from the near and far sides of the object. Further discussion of this point can be obtained in Klug and deRosier, 1966 [cite source=doi]10.1038/212029a0[/cite].

Then follows an outline of the computer program to be described in the rest of the paper. The student will have the opportunity of using these programs during the course. The programs have been set up in such a way that they request the user to supply all the relevant information. In addition to the programs described in the paper, several programs for image manipulation and display are available.

The problem of indexing (see page 359 of DeRosier and Moore) as not been adequately covered thus far. This is the process of assigning values to n and l and thus enabling us to identify Gn,l(R). This is done by constructing a reciprocal lattice to the lattice corresponding to the radial projection of the helix. If a and b are the vectors describing the lattice of the radial projection, then the reciprocal lattice vectors a* and b* are defined by

a.a* = 1 | a.b* = 0 |

b.b* = 1 | b.a* = 0 |

i.e. a* = 1/(a sin γ), b* = 1/(b sin γ), γ* = 180° – γ

The following illustrations from Klug, Crick and Wyckoff (1958) [cite source=doi]10.1107/S0365110X58000517[/cite] should serve to clarify the construction of the reciprocal lattice and its significance.

The restrictions to the FFT algorithm mentioned on page 363 no longer apply because of the use of a modified algorithm. The new restrictions are that the number of sampling points in X or Y must be even and must not have a prime factor greater than 19.

The points made on page 365 et seq. concerning the distortion of the FT due to the object being tilted out of the plane perpendicular to the microscope axis and the measurement and correction of this distortion are important. The technique involves searching for a minimum in a selected function rather than attempting to solve the problem analytically. In practice the angle of tilt and the position of the helix axis can be determined in a single two dimensional search.

The correction for the shift in origin due to inaccurate initial placement of the helix axis is simply to shift the phase of the FT. The amplitudes of the FT do not vary when the micrograph is translated, i.e. it has the same amplitudes whether the origin of the micrograph is taken at (x, y) or at (x + δx, y + δ y). However, this is not true of the phases; a change of origin by ( δx,δy) causes a phases shift in the transform –2 π( ν_{x} δx + ν_{y} δy), where ν_{x} and ν_{y} are the spatial frequencies of a point (n, m) in the digital FT of dimensions Nx x Ny;

ν_{x} = n/Nx and ν_{y} = m/Ny (see Figure 11 below)

An important omission from the discussion is the effect of shear deformation of the particle on its transform – this results in the layer lines being angled away from the horizontal.

Once the appropriate corrections have been carried out, the reconstruction can proceed without difficulty. The next problem lies in the interpretation of the resulting density map. This is best seen reading through these three reviews: Crowther and Klug (1975) [cite source=doi]10.1146/annurev.bi.44.070175.001113[/cite], Aebi *et al.* (1982) [cite source=doi]10.1016/0304-3991(82)90288-1[/cite] (download: Aebi_1982.pdf) and J. E. Mellema (1980), “3D-Structure of Biological Objects by Electron Microscopy. Methods, Results & Fidelity”, J. Microsc. Spectrosc. Electron., 5, 607-626.

All three are extremely readable and cover many practical points which are not covered above.

The practical work of the course will cover the determination of the structure of the nucleosome core particle (hopefully!). This is described by Klug et al, 1980 [cite source=doi]10.1038/287509a0[/cite]