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Chapter 10

Combined Modelling Techniques

Simulando discitur

(By simulation learning is acquired)

- anon

Introduction: Integrating with other Modelling Methods

Simulating nanoscale objects can be a powerful technique for understanding some of the mechanisms involved in cell function. It can be even more useful when combined with other computer-based techniques, such as 3-D modelling and image processing.

This chapter looks at how the nanosimulator can be used to improve modelling of cytoskeletal structures in plant cells, such as actin and tubulin filaments, and more complex structures such as plasmodesmata. A number of modelling techniques are used; image processing is done on raw electron micrographs, 3-D models are constructed based on these micrographs and other experimental data, and the resulting models are inserted into the nanosimulator as static structures, around which other objects (such as stain particles and antibodies) move and interact. The results of these simulations are taken from the nanosimulator and processed to produce 'Virtual Electron Micrographs' (VEMs) using physically reasonable models. These can then be compared with the original micrographs, to provide a check on the modelled data.

Part of the material in this chapter has been presented at the Third International Workshop on Basic and Applied Research in Plasmodesmal Biology, Zichron-Yakov, Israel 1996 under the title "Computer Assisted Analysis and Modelling of Plasmodesmata" (Betts, C., Conway, D., Radford, J., White, R.), and as "Virtual Electron Microscopy" (Betts, C., White, R.), at the 15th Australian Conference for Electron Microscopy, Hobart, Tasmania.

Electron Micrographs and Image Processing

This section presents some of the raw material biological modellers work with. These 'Electron Micrographs' (or simply 'micrographs') are produced by a Transmission Electron Microscope (TEM), and show images of actin filaments and microtubules, as well as those of plasmodesmata. The images are taken from a variety of sources. Some were made using stained samples embedded in a thin sheet of resin, or sprayed onto a coated grid, or set in vitreous ice. All were taken at very high magnification, at close to the limit of a transmission electron microscope (other types of electron microscope offer higher magnification, but are unsuitable for many types of biological specimens).

Actin and Tubulin

These images show some standard micrographs of actin and tubulin:

Figure 10.1 shows a typical micrograph of negatively stained actin from a coated grid. Note the regularly spaced lines, caused by the helical nature of actin filaments. (1)

Figure 10.2 shows a high magnification image of a microtubule in ice. Note the regularities, again caused by the structure's helical nature. (2)

Figure 10.3 shows another image of two microtubules, showing how microtubules appear within a cell. (3) The details are more difficult to see, and the ends are not visible in this image.



Plasmodesmata, the connectors between plant cells described in Chapter 2, are hard to examine under an electron microscope, due to difficulties in specimen preparation (primarily because it is difficult to cleave a plasmodesmata cleanly; usually the cleavage plane goes over or under the cylindrical structure). The following images show a transverse section and a longitudinal section.

Figure 10.4 shows a plasmodesma cross section. Notice the dark inner 'desmotubule', the (middle ring) plasma membrane, and the outer ring (probably caused by the opening funnel of the plasmodesmata). (4)

Figure 10.5 shows the longitudinal section of a plasmodesma. The desmotubule can be seen as the thick line running down the centre of the plasmodesma, while the plasma membrane can be seen as the thick outer lines (5). The lines bracketing the image at both ends are the cell walls of the two cells between which the plasmodesma runs.

Image Processing

These raw micrographs, combined with other experimental results, are used by biologists in determining structure. However, due to practical difficulties with instruments, they are often far from perfect, as can be seen in the above images. At high magnification, a TEM micrograph becomes grainy and blurred, and there may be additional problems with contrast and photographic development.

Using computers it is possible to 'post-process' such images (as described in Chapter 3), to reduce the severity of these effects. For example, by stretching the grey-scale histogram of an image, the contrast can be improved, and by using an 'sharpening' filter, colour gradients can be intensified. To illustrate, this corrections were applied to figure 10.5 using the standard commercial PC package 'Paintshop Pro 5.0'), slightly improving its quality:

Histogram equalisation (summarised in Chapter 4) is a way of increasing the contrast of an image on a global basis, which can be very useful if an entire image is underexposed or overexposed. However, this technique often fails if the image contains both light and dark regions, but the individual regions are under- or over-exposed. It becomes more useful to do 'local histogram equalization (6), where each pixel is adjusted relative to its local area.

In order to enhance images of plasmodesmata such as figure 10.4, which feature very dark areas, especially in the desmotubule, a simple local histogram equalization algorithm was written as a plug-in to the 'Image Explorer' suite of image manipulation tools supplied by SGI. Applying this local histogram technique on figure 10.4 produced the following image:

Local histogram equalization may make it harder to recognize global details (such as the concentric rings of the plasma membrane). However it brings into better relief the fine detail of objects embedded within the desmotubule, and within the plasma membrane, as well as improving the definition of the lines leading from the desmotubule to the plasma membrane (Fig. 10.7). In fact, some details which are barely visible in the first image become clearly visible in this image, such as the hypothesized linkage strut on the lower right of the desmotubule.

There are many other possibilities for enhancing images. As a further example, a question that has often arisen regarding the large particles that are either embedded in the desmotubule (7), (8), or positioned between the desmotubule and the plasma membrane (9). Specifically it is unclear whether or not these particles are symmetrically arranged.

To help resolve this question, the image (Fig. 10.7) was processed using a polar Fourier transform (again, written as a plug-in for Explorer on the SGI), after which the angular high frequency components were dropped, using a low frequency angular bandpass filter (see Chapter 4 for a brief explanation of polar Fourier transform bandpass filtering).

Figure 10.8 shows the general progression. The original image is converted to a polar Fourier transform, reproduced as a negative for clarity. The higher frequency angular components (angular frequency in these examples is the vertical axis) are trimmed off, and the inverse polar Fourier transform applied to obtain the final image, using only angular low frequency components.

The resulting image shows only the low frequency, i.e. rotationally repeated, components. We can see this in the image on the left, where the circular rings remain visible, but most of the other detail is either lost or reduced in intensity. Features closer to the centre are more accurately reproduced, since they subtend a relatively larger arc, and are thus more accurately described by the truncated Fourier spectrum.

The preceding image does not in fact show any strong evidence of rotational symmetry in the electron opaque (i.e. dark) areas in the desmotubule in the preceding image. Such rotational symmetry should express itself as a repeating rotational pattern in figure 10.9, but no such pattern is discernable. However a more intensive study on a large number of images using this and similar techniques would be needed before reaching any conclusion.


Using Micrographs and Processed Images

When the image, and even the processed image, does not give enough detail to fully determine the structure, other methods of investigation can be used to add further information. Usually however the electron micrograph at least places some general boundaries around the problem, by giving the rough dimensions of the structure, and possibly details about the regularity of the structure. These features, combined with available information on proteins present (derived from other experiments and analyses), can be used to create theoretical models that test the practicality of different arrangements of the proteins, membranes and other molecules making up biological structures.

3-D Models

A useful and long-established means of studying proteins and protein structures is the creation of 3-D models. Such 3-D models have ranged from toothpicks and plasticine to quite sophisticated works of machined wood and metal. Increasingly though, the ease with which computers can help researchers produce and manipulate 3-D models has led to the widespread adoption of computer-assisted modelling by researchers attempting to determine what protein conformations and arrangements are geometrically possible.

A Note on Static 3-D Models

Despite the popularity of molecular modelling programs and 3-D CAD packages, it was not possible to find modelling software that could be easily used for this project, and as a result a simple 3-D modelling package was written from scratch, comprising of a number of separate viewing, scene creation, and processing utilities. This small package was able to read and save object definitions, output data in a variety of forms that could be used by other programs (including the nanosimulator), display results in real time using the Silicon Graphics hardware graphics library, and (most importantly) read in a mathematical description of object placement (especially helices). This last feature allowed the "single line" definition of thousands of separate objects making up a large helix, and was essential in creating some of the models described below.

All these features were available, and implemented to a far greater degree of sophistication, in other pre-existing packages, but unfortunately no single package combined all the features required. In order to maintain compatibility with future work, and the possibility of a better package appearing in the future, the modelling package was written to output data in both povray and wavefront formats, allowing other programs to also use the data.

In this thesis, most of the images are rendered in the 'Povray' raytracer, rather than using screen shots from the computer console, since this allows a great deal more detail to be displayed. To this end, minor details such as camera and illuminant positions are sometimes set manually to better frame the objects, as are material properties, such as whether the objects appear glossy or matte, and whether spheres are displayed as entirely discrete, or whether adjacent spheres are allowed to melt into each other. These finishing touches do not affect the geometry of the model, and are simply done to make the pictures clearer by giving the reader more visual cues to aid interpretation of the 3-D structure represented by the object. The reader should bear in mind that while the 3-D modeller utilities can model a wide variety of objects, such as cylinders, hollow cylinders, rectangular blocks etc., the nanosimulator itself is only capable of manipulating spheres.

Models of Actin and Tubulin

Creating models of actin filaments and microtubules was a straightforward task. Models can easily be created by saving data from a nanosimulator session that had resulted in the growth of actin filaments or microtubules. This is normally done when the output of the nanosimulator is viewed and studied after the program is run.

Another way to create models of actin and tubulin assemblies is to model them as simple helices, twisting strands defined with mathematical precision. This can be an easier approach if it is necessary to precisely position an object. For example, if the object needs to be centred mathematically on the origin, this can still be accomplished with data saved from the nanosimulator, but usually requires the object to be manually re-positioned, since the object will have been moving about randomly within the simulation volume.

The models presented here were defined using simple mathematical helix definitions, and then rendered using povray. The model of actin in figure 10.10 is the standard model (10), however the microtubule model requires more explanation. Microtubules assemble into a number of different types of helix, and a given microtubule may even have a different structure within different regions (11).

These different structures are categorised by the number of protofilaments, or component longitudinal strands, within the microtubule. The most common variety (and the one modelled in previous examples by the nanosimulator) is the 13-protofilament model, but 14-protofilament microtubules are also quite common, and 12 and 15 protofilament microtubules also occur. As it is equally easy, using a mathematical description, to model any of these, a 12-protofilament microtubule based on the observational work of D.Chretién et al. (12) was chosen. The most obvious difference between these different microtubule types is that only the 13-start microtubule has straight protofilaments - in all other types the protofilaments form slowly twisting helices, as can be seen in figure 10.11 below. A final point to note is that in this particular representation no visual distinction is made between -tubulin and -tubulin.


Proposed Models of Plasmodesmata

Plasmodesmata, the connecting passages between cells in higher plants, were introduced in Chapter 2. The exact structure of plasmodesmata is not known for certain, although a number of similar models exist. A brief summary of some of the proposed models is given below:

The Textbook Model

This model (Fig. 10.12) shows the commonly accepted basic features of plasmodesmata; a tunnel between the cell walls lined with the plasma membrane, with a thin tube of endoplasmic reticulum (E.R.) passing through the centre, connecting, to some extent, the E.R of the adjacent cells. (13)

The Overall Model

The Overall Model (Fig. 10.13) extends the standard model by postulating the existence of a number of globular proteins in the lumen between the desmotubule and the external plasma membrane sheath. (14)


The Ding Model

This model (Fig. 10.14) modifies the standard model by including a role for a number of undetermined globular proteins embedded in the central desmotubule (or 'appressed endoplasmic reticulum protein complex', as it is referred to by the authors), and on the inner side of the plasma membrane sheath that lines the outer walls of the plasmodesmata, rather than in the lumen as above. (15), (16)

The Thomson Model

The Thomson model (Fig. 10.15), derived from observation of Tamarix salt glands, postulates particles in the lumen and embedded in the plasma membrane wall, in order to explain features found in freeze-fracture images. (17)


The White Model

The White model (Fig. 10.18,10.20), based on the discovery of actin within plasmodesmata and TEM images showing "blobs" and "spiral blobs", postulates the existence of filamentous actin strands within the lumen, and raises the possibility of these being twisted around the central desmotubule. (18)

The Radford Model

The Radford model (Fig. 10.19, 10.21) extends the White model to take into account the presence of myosin (an actin-associated motor protein) and TEM images showing apparent "spokes" within plasmodesmata. (19)

3-D-Models of Plasmodesmata

Four of the above models, namely the Ding, Overall, White, and Radford models, were modelled using the 3-D modelling tool. The modelling tool takes a short mathematical description of the models and renders them as a combination of rods, spheres, cylinders, cylindrical sheaths and helices of spheres. It is also capable of outputting their details in a variety of formats, for use by other programs, such as other real time viewers, ray tracers, and the nanosim program.

The fastest way of observing these models is by immediate rendering using the Silicon Graphics system. This also allows real time examination of the model, as the author's 3-D viewing program allows basic 3-D manipulation of viewpoint using a combination of mouse movement for rotation, and keyboard operation for panning the view point in and out.

3-D Computer Models of Plasmodesmata; Real Time Views

The following images (Fig. 10.16 to10.19) are screen shots from the 3-D viewing program, showing the computer models constructed from four of the theories outlined above:

3-D Models of Plasmodesmata; Raytraced Views

The above models can be very useful for real time viewing and examination, when an impression of the three dimensionality of the model can be obtained by real-time manipulations such as rotating the object, and by viewing from multiple angles. However, for still frame images it can be useful to render the objects in more detail using a ray tracer, to produce an image at a far greater resolution. This is important for high quality print reproduction, and it also provides a number of visual cues such as shadows, reflective effects, and improved shading, which may make the object appear more substantial and thus easier for viewers to interpret.

To this end, as part of the modelling program suite a simple conversion program was written to turn the mathematical description of models (such as those above) into povray raytracer files. This allowed the production of higher quality images such as these renditions of the White and Radford models below (Fig. 10.20, 10.21):

Combining Models with the Nanosimulator


The nanosimulator was enhanced to allow it to read in pre-initialised structures, with the (current) limitation that such structures had to be static, and would not be able to move like other objects within the simulation. This was easily enforced within the program by giving such objects a diffusion coefficient of zero.

This allowed the input of models, such as those described above, into the nanosim program, where their interaction with other, motile, particles could be simulated. There are a great many possibilities for work combining pre-modelled structures with the nanosim program, even with the current limitation that such structures must be static.

Some such possibilities include: