Helixplorer-1 help

Helixplorer-1 helps with the determination of helical symmetry parameters. This tool supposes you have at your disposal the diffraction pattern (Fourier transform) of a helical structure. If you only dispose of a cryo-EM 2D class average image, you can use Helixplorer-0 to calculate its power spectrum (PS).

Helixplorer-1 draws the theoretically expected pattern calculated from an ideal helix on the top of your experimental diffraction pattern (Fig.1). It then calculates a similarity score based on the average PS pixels in the areas where the pattern is ("the pattern" here means the peaks of a set of Bessel functions). The obtained scores are then plotted for all tested helical symmetry parameters as 1D or 2D graphs that you can click on (Fig.1, bottom right panel).

A single click changes the current helical symmetry parameter(s) (pitch, rise, radius, etc) to the one(s) defined by the clicked position. Click-and-drag there will zoom-in at a particular region of the parameter space and a right-click will zoom-out. By playing with the "Search symmetry & ranges" menu (Fig.1, the top right) one can explore the helical symmetry parameter space and search for the best agreement between experimental and theoretical patterns.

Hopefully, the plots will show a maximum score (at least locally) for the true helical symmetry parameters of your structure. Of course, there will be ambiguities and the maximum score will not always be unique but Helixplorer-1 should help with selecting a manageable subset of possibilities.

To understand the concepts behind helical symmetry determination in depth requires reading these papers. However, Helixplorer-1 can also be used as a pedagogical tool allowing abstract concepts and mathematical relationships to be visualized.

Electron cryo-microscopy (cryo-EM) images of macromolecules can be combined through 2D classification methods into 2D class averages in order to increase the signal to noise ratio. These class averages represent 2D projections of a 3D object (Fig.2). As helices are elongated structures, in an electron microscopy grid, they usually lay parallel to the grid so most of the images represent projections perpendicular to the viewing direction.

Recent developments in software and data acquisition allow the calculation of high-resolution 2D class averages where even secondary structure elements might become visible. For the study of helical assemblies by cryo-EM, one of the most important steps is the determination of the helical symmetry parameters through the analysis of 2D class averages of helical segments.

Fig.2: 2D class average of a helical structure made from thousands of noisy cryo-EM images.

The Fourier slice theorem says that the Fourier transform (FT) of a 2D projection of a 3D object is equal to a central slice through the 3D Fourier transform of that object. As class averages are calculated in respect to an arbitrary (x,y) origin/center, it is better to use the absolute square of the Fourier transform, also called the power spectrum (PS) which is invariant with respect to the origin/center position.

In the particular case of helical structures, their 3D Fourier transforms (FTs) are composed of discrete parallel layers stacked perpendicularly to the helix axis direction. When a central 2D slice is extracted from the 3D FT, the layers appear as a set of parallel lines (Fig.3). It is the thoughtful analysis of the so called layer-lines that allows the determination of the parameters which define the kind of helical symmetry present in the 3D structure. The layer-lines are mathematically composed of a series of Bessel functions with their peaks running along the horizontal direction.

Fig.3: DNA helix PS taken in 1952 by Rosalind Franklin using X-rays. The meridian (red), the equator (blue) and some layer-lines (black) are overlaid. Based on this picture, Watson and Crick proposed a 3D model for the structure of DNA (Fig.4).

Fig.4: Helical structure of DNA proposed by Watson and Crick in 1953.

The parameters that define a given kind of helical symmetry are: the axial rise p which is the vertical distance between two consecutive sub-units (blue dots in Fig.5), the pitch P that tells the vertical distance covered by one turn of the helix and the radius R. Other secondary parameters include: the helix thickness and the helix axis tilt (also called out-of-plane angle that, ideally, should be zero when the 2D class average represent helical segments laying perpendicular to the imaging direction).

Fig.5: 2D projection of a 3D Helix with radius R composed of discrete sub-units (blue dots) separated vertically by p units of distance and completing one turn every P units of distance.

Helixplorer-1 is a web application that helps to understand how the helical parameters affect the PS and at the same time it helps with the process of indexation (or symmetry parameters determination) of a particular helical structure. This is done through the calculation of scores based on a PS image file that is loaded by the user. Helixplorer-1 can also simulate the effects of having some spread in the helical axis direction (e.g. fibres) and the use of a given wavelength (e.g. X-rays) through Ewald's sphere simulation (fibre diffraction). The possibility of displaying indexes to produce the so called (n,l) plots (ref.1) are both useful and enlightening when one starts analysing helical structures. The possibility of varying parameters like helix radius, thickness, out-of-plane angle, etc. allows one to quickly gain insights about the effects of the above mentioned parameters on the Fourier transform of a helix.

To compare a given set of parameter to the PS loaded by the user Helixplorer-1 calculates a score. The score is simply the average of the PS values at the positions where the theoretical PS is expected to have its maxima. Although very simple, the score is robust and easy to calculate. By producing 1D and 2D plots of the score as a function of the various parameters, the user can estimate the correspondence between the experimental data and some candidate set of parameters.

The power spectrum loaded by the user (and any other information like the symmetry parameters obtained) are *NOT* uploaded to any server. The code is written in javascript and it runs exclusively on the client browser. So this tool provides total anonymity concerning the scientific data being analyzed.

The "Search symmetry & ranges" menu (Fig.1, top right side) allows the user to test a number of parameters at once by creating 1D and 2D plots that will indicate the set of symmetry parameters that best fit the helix PS supplied by the user. The loaded PS image must be adjusted to make sure that the "meridian" and "equatorial" lines are well centered and that observed layer-lines are symmetric in respect to the central meridian. To this goal, the user will be exposed to alternating images of the PS and its vertical mirrored version in order to make adjustments and to spot anomalies (e.g. horizontally asymmetric layer-lines).

In Helixplorer-1, the user can vary the helical parameters using the slider bars located at the top left side of the user interface (UI) (Fig.1). By varying them one can see how they influence the positions of the Bessel function maxima J_n that appear as colored horizontal lines at the central part of the UI. Each J_n can be labeled with indexes such as ℓ,m,n according to the nomenclature used in the 1952 paper. The index j (from 1 to 6) enumerates the first (strongest) consecutive maxima of each individual Bessel function.

We can observe that only the J_n's with n=0 touches the meridian line (thanks to a well known property of the Bessel functions). This is an important feature that must be exploited during the search for the parameters associated to a PS image. If the PS image present lines that intersect the meridian, these can be readily associated to the J_n=0's maxima. The meridian-intersecting lines closest to the equator indicate the axial rise p of the helix.

If you observe the asteriks in Fig.3, they indicate the layer-lines crossing the meridian (in red). They are 3.4 Å^-1 appart from the PS center and this is the distance between consecutive DNA nucleotides (the axial rise p in Fig.4).

To load a 2D class average power spectrum (PS) image into the UI, the user must click on the "Load image" button and select "Browse". If you only have 2D class averages, you can calculate its power spectrum by using Helixplorer-0. Often images have an even number of pixels and this makes the definition a central pixel (the Fourier (0,0) offset coefficient) ambiguous. For this reason, sometimes it is necessary to apply a small horizontal shift ΔX (of about 0.5 pixel) in order to have the meridian living precisely at the center of the display. This is easily accomplished thanks to the permanent switching between two versions of the PS image, one being the vertically mirrored version of the other. Likewise, as the class averages are not always perfectly vertical, a small angular rotation γ can also be applied to make the PS image vertical, i.e. with horizontal layer-lines. These settings are found in the "Image adjustments" tab at the top right of the UI (Fig.1).

2D class averages of helical specimens are not guaranteed to represent helical segments laying perfectly perpendicular to the electron microscope beam direction. This means that it may be necessary to take into account a tilt angle θ (called out-of-plane angle). The main impact of non null θ angle is that it can make the J_n=₀'s disappear.

To fully understand the concepts behind helical structures we recommend the thoughtful reading of the following 1952 and 1970 papers on which everything was based on to create this web application:

The structure of synthetic polypeptides. I. The transform of atoms on a helix

A new derivation of the Fourier transform of a helical structure

Zeitschrift fur Kristallographie, Bd. 132, S. 152-156 (1970)

The Electron Microscopy Data Bank (EMDB) is a public repository for electron microscopy maps of macromolecular complexes. It contains lots of meta-data and Helixplorer-0, 1 and 2 can directly access this information from any of the deposited structures through their EMDB-id codes. Helixplorer has a drop-down menu where you can select the helical EMDB entries. This makes easy to learn by looking how each family of biological macromolecules compare in terms of helical parameters and patterns.

Helixplorer-1 follows the rational of the above 1952 and 1970 papers by adopting a "crystallographic" point-of-view where the Fourier transform and the analysis of the layer-lines are at the heart of the indexation procedure.

However, a helix can also be seen as a particular kind of 2D crystal that can be rolled up to form a 3D helix. This alternative approach highlights other aspects of a helix as it explicitly imposes constrains on the characteristics of a lattice so that it can be folded to form a valid 3D helix. Characteristics as crystal packing, bending energy, presence of a seam (as in the case of microtubules), etc. become more evident. This is the approach adopted by Helixplorer-2.

Finally, the goal of Helixplorer-0 is simply allow users to calculate the PS of their 2D class averages and to overlay families of parallel lines on top of them in order to obtain some rough estimates of the radius, pitch, axial rise, etc.

Just as a curiosity, the 1952 paper above was sent for publication the 16th February 1952 before the famous photograph 51 was taken by Franklin and Gosling in May that same year. This means that Francis Crick already had a pretty deep understanding of helical structures before he saw the X-ray diffraction data from Rosalind Franklin.

The 1970's Ramachandran paper is also noteworthy because it get rid of the necessity of having a "true repeat" in the helical mathematical framework. In the approach adopted by the 1952 paper it may seem that only perfectly periodic helices (with some true repeat c) can have layer-lines, which is not true.

To achieve maximum speed, Helixplorer-1 store the positions of the Bessel functions J_n maxima in a table for quick access. These were previously calculated for n from 0 to 400 by searching for zero crossings of the J_n first derivative. The PS values are kept in a Uint8Array and the access to the JavaScript HTML DOM elements are kept to a minimum.

In our GUI, you can make adjustments to various parameters using control sliders: .

For the most accurate adjustments, we highly recommend utilizing keyboard shortcuts listed below. Sometimes, even small changes as little as ±0.001 can significantly impact the results.

Use the Page Up and Page Down keys for coarse tuning, adjusting the parameter by ±1.0. For even larger adjustments of ±10.0, simply add the Shift key to the combination.

To finely adjust the parameter by ±0.01, utilize the Arrow Up ↑ and Arrow Down ↓ keys. If you need medium adjustments of ±0.1, combine the Shift key with the arrow ↑ ↓ keys. Conversely, to achieve even finer adjustments of ±0.001, use the arrow keys ↑ ↓ in conjunction with the Ctrl key.

In summary, follow these key combinations to modify the numeric field corresponding to the parameter you wish to vary:

This page and the information contained on it are intended to facilitate the indexation of helical 3D structures and their understanding for teaching purposes. Others can copy, distribute, display, and make derivative works for non-commercial purposes, as long as credit is given.