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First published online 23 May 2006
doi: 10.1242/jcs.02969
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Research Article |

1 Sir William Dunn School of Pathology, University of Oxford, South Parks Road, Oxford, OX1 3RE, UK
2 Biomedical Sciences Unit, Department of Biological Sciences, Lancaster University, Lancaster, LA1 4YQ, UK
Author for correspondence (e-mail: keith.gull{at}path.ox.ac.uk)
Accepted 2 March 2006
| Summary |
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-tubulin,
-tubulin, Parkin co-regulated gene product (PACRG) or the paraflagellar rod protein PFR2] allowed a dissection of the mechanisms for central pair constraint. This demonstrated that orientation is independent of flagellum attachment and beating, but is influenced by constraints along its length and is entirely dependent on correct positioning at the basal plate.
Key words: Axoneme, Flagellum, Central pair, Paraflagellar rod, Delta-tubulin, Basal body
| Introduction |
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From even such a basic description of axonemal action, one can recognise the need for three kinds of regulation of dynein activation to create a productive axonemal beat. (1) Asymmetric activation: given the symmetry of the outer doublets, the dynein arms on one side must be active while those on the other are quiescent to generate a productive bend. (2) Co-operative activation: a bend of any significant length requires the simultaneous (in)activation of many dynein molecules along that stretch of axoneme. (3) Dynamic activation: the propagation of a bend along the axoneme requires that dyneins in a particular axonemal region be coherently switched between active and inactive states.
One model for the regulation of dynein activation is based on evidence that indicates a core role for the CP microtubules. In this model, signal from an asymmetric CP apparatus (consisting of the two singlet microtubules C1 and C2, and associated structures such as the CP bridge and CP projections) is transduced through the radial spokes, to a doublet-associated regulatory complex, which modifies dynein activity through the action of dynein-associated protein kinases and phosphatases (Porter and Sale, 2000
; Smith and Yang, 2004
). In support of this model, asymmetric contacts between radial spokes and CP projections have been seen in molluscan gill cilia (Warner and Satir, 1974
), suggesting that radial spokes might detach and reattach to the CP projections during microtubule sliding. In Chlamydomonas, CP mutants are paralysed yet retain the ability to undergo microtubule sliding; their defect is therefore not in the force-generating mechanism but in a control system that regulates force generation (reviewed by Smith and Lefebvre, 1997
).
The CP signal transduction model is particularly significant in the context of CP rotation. It has been postulated that rotation of an asymmetric CP could, through interactions between one set of CP projections and radial spokes, regulate the activation of dyneins on successive doublets around the axoneme. By contrast, associations made by the opposing CP microtubule would ensure that dyneins on the opposite side of the axoneme were inactive. The CP would then act as a distributor of regulatory signals around the axoneme.
CP rotation during beating of flagella and cilia has been demonstrated by thin-section electron microscopy in Chlamydomonas and Paramecium. Longitudinal sectioning through Chlamydomonas flagella that were fixed during active swimming revealed the orientation of the CP microtubules as being parallel to the bend plane within the principal bend, then gradually twisting through 180° until once again being parallel to the bend plane within the reverse bend (Mitchell, 2003
). In fact, the CP assumes a natural left-handed helix when separated from the rest of the axoneme by detergent (Goodenough and Heuser, 1985
) and protease treatments (Hosokawa and Miki-Noumura, 1987
). It thus seems reasonable to accept that the CP is also helical inside the native axoneme, and that rotation of the helix through successive waves allows the CP to reorient such that one of the two CP microtubules is always at the outside of the bend. However, this CP rotation is independent of both dynein activation (as it also occurs in curved, quiescent flagella) and also spoke head association (as it occurs during bend propagation in spoke head mutants) (Mitchell and Nakatsugawa, 2004
). Hence, CP orientation is a passive response to bend formation in Chlamydomonas.
CP orientation in Paramecium cilia is also parallel to the plane of bending, and it rotates 360° during each beat cycle in forward and backward swimming (Omoto and Kung, 1979
; Omoto and Kung, 1980
). Serial thin sections also revealed the CP left-handed twist along the Paramecium axoneme. In contrast to Chlamydomonas quiescent flagella, paralysed paramecia do not show CP rotation (Omoto and Kung, 1980
). Therefore, rather than being a passive process, rotation of the CP in Paramecium might be an active one, initiated or constantly driven by a mechanism other than bending, with the orientation of the CP determining the pattern of active sliding around the axoneme (Omoto and Kung, 1980
). It is worth noting that both Chlamydomonas and Paramecium rotate their CP tubules clockwise when viewed from the base of the flagellum.
However, CP rotation is not ubiquitous in axonemal beating. In both ctenophore comb plate cilia and echinoderm sperm flagella, the CP apparatus has a fixed orientation relative to the position of active sliding (Sale, 1986
; Tamm and Tamm, 1981
). For most organisms, it is not yet clear if the CP rotates. This is largely a result of the difficulties in ultra-structural analyses of imposing an absolute frame of reference onto the ninefold symmetric doublets of the axoneme, making it problematic to perceive a change in the axis of the CP relative to the doublets. This situation is considerably eased in kinetoplastid protozoa owing to the presence in the flagellum of a characteristic extra-axonemal structure - the paraflagellar rod (PFR). Here, we use this structure as an external reference to address the orientation of the CP in the flagella of several kinetoplastid species. We show that there is no CP rotation and use a simple geometric correction technique to produce an accurate estimate of the angle of the CP axis in Trypanosoma brucei from a large number of axonemal cross-sections. We also present a first examination of the causes of this constraint using RNA interference (RNAi) to deplete particular flagellar and basal body proteins specifically.
| Results |
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To determine the CP orientation in the kinetoplastid flagella, we analysed 40 randomly selected thin-section transmission electron micrographs showing cross-sections of axonemes from motile procyclic-form T. brucei. Most micrographs showing near `transverse' sections through an axoneme cannot be directly superimposed because there is inevitably a slight (and often a substantial) discrepancy between the angle of the section and the true transverse plane. The result is an apparent deformation of the axoneme (Fig. 1A). To compensate for axonemal cross-sections not being true perpendicular sections, we used a mathematical transformation that moves the data into the transverse plane (Fig. 1B; detailed in Materials and Methods). Fig. 1C shows how this transformation can be used to correct an elliptical axonemal cross-section to a circular one representative of a true transverse section. The circularity of the transformed dataset can be demonstrated by ninefold rotation of the corrected image, showing superposition of the doublets of the axoneme (Fig. 1D).
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We should note that this ellipse-correction procedure would also erroneously `correct' any symmetric deformation of the axoneme in a true transverse section caused by bending forces. However, since the transformation is applied only once, it cannot correct for both a symmetric deformation and a glancing section. Since there is very little eccentricity in the transformed dataset (Fig. 1C), whereas the original cross-sections were often highly elliptical (eccentricities up to 0.8; i.e. the ellipse long axis is nearly twice that of the short axis), it is unlikely that deformation of the axoneme as a result of beating is large in these micrographs when compared with eccentricity as a result of sectional plane. Bending waves can be clearly seen in the fixed flagella when interrogated by light microscopy (data not shown).
Following transformation of the data from the micrographs, the positions of all of the microtubules are extremely constant from axoneme to axoneme (Fig. 2A) and fall on the points predicted on the basis of perfect ninefold symmetry. Using the PFR as an external reference to define the doublet number (by being attached to doublets 4-7), the axis of the CP also has a very constant orientation at 92.7±0.5° (mean ± s.e.m.) to the axis defined by the centre of the axoneme and doublet 1 (Fig. 2A,B). Thus, the orientation of the CP does not vary with respect to the nine outer doublets. Since this dataset comprises cross-sections through axonemes of many different cells and at different positions along the cell, this provides strong evidence for a fixed, specific orientation for the CP in T. brucei, precluding CP rotation in this species. We therefore propose that the mechanism by which the CP might rotate with respect to the outer nine doublet microtubules in some species does not operate in trypanosomes.
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The same geometric correction demonstrated in Fig. 1 for procyclic-form T. brucei axonemes was performed on bloodstream-form T. brucei cells. These two life-cycle stages move very differently from one another, yet the CP of each was found to be fixed in the same orientation (Fig. 3A,B). The same is also true for axonemes of epimastigote-form T. cruzi semi-attached flagella (Fig. 3C), as well as Leishmania major, Herpetomonas megaseliae and Crithidia fasciculata free flagella (Fig. 3D-F). Once more, the position of the CP was invariant and had the same orientation as that seen in T. brucei.
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250 million years ago (Douzery et al., 2004
Release of CP constraint in mutants
We next asked what factors were controlling the fixed CP orientation seen in trypanosomes. We used double-stranded RNAi to deplete particular flagellar and basal body proteins to gain insight into the mechanism determining CP orientation in T. brucei. The first of these was
-tubulin - a protein associated with basal bodies. In Chlamydomonas, disruption of the gene encoding
-tubulin results in loss of the C-tubule of the basal body microtubules (Dutcher and Trabuco, 1998
; Fromherz et al., 2003
). In Paramecium, silencing of the gene encoding
-tubulin similarly results in loss of the C-tubule, but additionally loss of B- and even A-tubules was observed (Garreau de Loubresse et al., 2001
). Since the radial symmetry of the axoneme is born from the ninefold symmetry of the basal body, with the nine doublets of the axoneme extending from the A and B tubules of the triplet basal body, there is a prediction that disruption of the basal body configuration will be transferred along the axoneme.
We generated a T. brucei cell line for the inducible knockdown of
-tubulin by RNAi. Prior to induction of
-tubulin RNAi, trypanosomes were morphologically normal, and showed normal growth and motility. Induction of
-tubulin RNAi resulted in a reduction in growth rate after 48 hours (Fig. 4A). Several characteristic phenotypes were noted in trypanosome populations following induction of
-tubulin RNAi that varied in severity as a function of time post-induction (see below). The earliest phenotypic effect we observed, at >24 hours post-induction, was the loss of flagellar motility in trypanosomes that otherwise appeared morphologically normal (Fig. 4B). To confirm the efficacy of
-tubulin RNAi, we raised a monoclonal antibody, LAZ1, against T. brucei
-tubulin (see Materials and Methods) and used it to monitor the levels of
-tubulin protein post-induction. Immunoblotting confirmed that, following induction of RNAi, levels of
-tubulin were progressively reduced such that, by 48 hours post-induction,
-tubulin protein is barely detectable (Fig. 4C). By contrast, levels of ß-tubulin - which acts as both loading and specificity control in these RNAi experiments - were unaffected during the time course of the experiment (Fig. 4C).
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-tubulin function. Distal from the basal bodies, ultra-structural analysis of flagella revealed disruption of the 9+2 arrangement by loss of outer doublets (Fig. 4F,G). At later time-points post-induction (>96 hours), more-severe defects were observed (with several doublets missing from the axoneme); however, at 48 hours post-induction, the majority of disrupted axonemes had lost only one doublet and in some instances only the B-tubule of a single doublet was absent. Interestingly, there was a significant preference for the disappearance of outer doublets 8 and 9 (
2 test, P<0.001). Despite this loss of outer doublets, no loss of the CP occurred, indicating that CP nucleation is unimpaired in these cells.
Some axonemes in the induced
-tubulin RNAi cell line display the canonical 9+2 arrangement. These most probably represent axonemes constructed before
-tubulin levels had dropped below a crucial level and, accordingly, their abundance decreases with increasing time post-induction (data not shown). In these axonemes, as for the non-induced cells, the CP was orientated as seen for wild-type T. brucei (Fig. 4E). However, loss of one outer doublet microtubule, or even just the B-tubule of a doublet, results in a total loss of correct CP orientation with the CP now being orientated apparently randomly around the central axis of the axoneme (Fig. 4F,G and Fig. 5D).
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-tubulin, also seen at the basal body and centriole region in a wide range of eukaryotes (Nigg, 2004
-tubulin localises to the basal plate in Chlamydomonas, where it plays a role in CP nucleation (Silflow et al., 1999
-tubulin depletion in T. brucei - induction of
-tubulin RNAi produces immotile flagella that lack the CP (McKean et al., 2003
-tubulin RNAi induction, there are a small number of cells (
9%) in which only one of the two CP microtubules is lost. We analysed these axonemes with respect to orientation of the remaining microtubule and found that, in these 9+1 axonemes, the single central microtubule had a random distribution (Fig. 5A). Hence, although
-tubulin RNAi does not affect the integrity of the nine outer doublets, the radial symmetry of the 9+1 arrangement is not sufficient for constraining the CP position. Interestingly, the remaining, mis-orientated central microtubule is found at the same distance from the centre of the axoneme, as it would be had its partner still been present, and does not collapse inwards. This argues against a model in which the CP push against the radial spokes to create a force that is transduced out to the doublets.
Unlike the basal body proteins
- and
-tubulin, the T. brucei Parkin co-regulated gene products TbPACRG A and TbPACRG B are related proteins that appear to have a role in doublet morphogenesis and locate along the length of the flagellum (Dawe et al., 2005
). Depletion of both homologues simultaneously (although not separately) by RNAi in T. brucei causes flagellar paralysis and leads to premature termination of outer doublets - axoneme formation is initially normal, but the number of missing doublets increases towards the distal tip of the flagellum. The resulting disrupted axonemes look superficially like
-tubulin mutants but the defects are more severe and, importantly, do not originate in the basal body. Fig. 5B shows the orientation of the CP in axonemes with variable numbers of outer doublets (X+2) resulting from PACRG ablation. Disruption of the axoneme again produced a perturbation of CP orientation. However, despite producing a more severe axonemal disruption than that of
-tubulin RNAi, 55% of the X+2 axonemes analysed maintained the CP in a fixed, wild-type orientation. These data indicate a differential effect of axonemal disruptions originating at or near the basal plate compared with those that occur along the length of the flagellum.
Since the above mutants all result in flagellar motility defects, it was possible that CP orientation might be entrained by the intrinsic beating of the flagellum. We addressed this issue by analysing micrographs from a previously described RNAi against PFR2 in T. brucei (Bastin et al., 2000
; Bastin et al., 1998
). In these cells, interference in the building of a normal PFR structure leads to cell paralysis apparently without disruption of the axoneme itself. However, sufficient residual PFR structure remains for it still to be used as an external reference to assign CP orientation, and examination of axonemal cross-sections reveals that the CP remains fixed (Fig. 5C).
| Discussion |
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Relative to an unmoving external reference, an invariable CP orientation can only be achieved if the CP is fixed along its length. However, if the CP is not seen in a constant position, there are three possible explanations - one of which involves an unmoving CP, whereas the other two require CP movement. First, the CP orientation might be invariant (i.e. unmoving) along the length of an individual axoneme, but variant between axonemes as a result of differences in the initial orientation of CPs at their origins. Second, the CP might be fixed at its basal end, but twist around its longitudinal axis during flagellar beating. Since the basal end is unmoving, this model requires a reverse twist of the CP during the recovery stroke to avoid super-coiling of the CP. The net CP movement is thus an oscillation. Last, the CP might be movable at the base and also along its length - allowing the CP to rotate relative to an unmoving external reference during the beat.
Here, we report that CP orientation is fixed within kinetoplastid flagella and hence the CP does not move during flagellar beating. We used RNAi to investigate how CP orientation is defined and whether it can be freed from constraint. The fact that the CP is fixed in free Leishmania flagella suggests that CP orientation is independent of flagellar attachment to the trypanosomatid cell body. This is strengthened by our analysis of the T. cruzi epimastigote flagellum, which is only associated with the cell body in its proximal segments. Yet again, the fixed CP orientation is maintained throughout the attached and free segments of this flagellum.
The RNAi mutants described here all result in forms of flagellar paralysis. However, the orientation of the CP is not being entrained by flagellar beating. This can be demonstrated by the previously described RNAi against PFR2, which also results in immotility in T. brucei (Bastin et al., 1998
), but a careful examination of axonemal cross-sections reveals that the CP remains fixed. Since CP orientation is independent of flagellar attachment and also beating, it must be defined by interactions occurring solely within the flagellum. Such definition might occur by either: (1) determination of CP orientation at its origin at the basal plate, which is then passively inherited distally along the flagellum; or (2) physical constraint of CP orientation by proteins existing along the length of the axoneme. Our data suggest that the former is an absolute requirement for correct CP orientation; mis-orientation of the CP can be prevented only when a native basal body configuration is present in the cell, and CP mis-orientation does not seem able to be corrected even when the ultra-structure of the outer doublets appears normal (see
-tubulin RNAi). However, there does seem to be a partial dependence on constraint by the axoneme because, in cells containing normal basal bodies but with severely disrupted axonemal configurations, correct CP orientation is sometimes lost. Hence, the mechanism by which the kinetoplastid CP is constrained - when it is free to move in species such as Chlamydomonas and Paramecium - acts predominantly at the birth of the CP at the distal-end of the basal body, but is also to a lesser degree constrained by physical contacts made by the CP along the length of the flagellum. Although we cannot entirely rule out movement of the CP in trypanosome RNAi mutants, the removal of the CP constraint in trypanosomes most probably does not result in CP rotation or oscillation, but in a mis-orientation that is invariant within an individual axoneme. This can be seen in the fact that all the RNAi mutants described are paralysed, and hence cannot be moving the CP in response to flagellar beating. Whether or not the CP in such axonemes would move in response to an externally enforced beat remains to be seen.
The functional reasons for CP rotation in some but not all axonemes have yet to be fully understood. Bray suggested that the CP rotates whenever a ciliary beat is three dimensional - i.e. when the effective and recovery strokes occur in different planes (Bray, 1992
). Paramecium supports this condition since it performs non-planar, asymmetrical waves - during forward or reverse swimming, the cilia beat presents a planar power stroke and a three-dimensional recovery stroke (Omoto and Kung, 1979
). However, Chlamydomonas performs planar waves whether swimming forwards (asymmetrical) or backwards (symmetrical) (Bessen et al., 1980
). In addition, Chlamydomonas flagella are able to propagate planar asymmetrical and symmetrical waveforms despite the lack of CP (Frey et al., 1997
; Wakabayashi et al., 1997
). Another proposal is that cilia and flagella of organisms that rely on rapid changes in waveform, beat frequency and effective stroke orientation require variable CP position, whereas an unchanging orientation is supposed to be a derived simplification in organelles that have a permanent bend plane, such as ctenophore comb plate cilia and many metazoan spermatozoa (Mitchell, 2004
). However, this hypothesis also appears incomplete since trypanosomes - here shown to have a fixed CP orientation - present complex waveforms. The main waveform is a planar, symmetrical wave run from the distal tip to its base (Holwill, 1965a
; Holwill, 1965b
), but trypanosomes can also re-orientate themselves by reversing their flagellar wave (Sugrue et al., 1988
), which becomes asymmetrical and runs from base to tip (Holwill and McGregor, 1976
). Swimming reversal has also been reported in Chlamydomonas (Bessen et al., 1980
) and Paramecium (Naitoh and Kaneko, 1973
), although the mode is distinct in each case (Holwill and McGregor, 1976
). The trypanosome flagellum is not only a dynamic motility organelle, but has also evolved to perform other biological processes, such as participating in cell division (Moreira-Leite et al., 2001
) and moving the parasite through challenging environments during its cell cycle in the vertebrate and invertebrate host. In the invertebrate host, the flagellum is an organelle for attachment to the host epithelia (Vickerman, 1973
; Vickerman and Tetley, 1990
), which ensures transmission to vertebrates. Whether the trypanosome swimming behaviour is not complex enough to require movable CP orientation to differentially regulate specific dynein arms is not yet clear. However, it seems reasonable to suggest that the swimming changes trypanosomes have to deal with might well be precisely regulated but by means other than a variable CP orientation.
This report has brought further insights into the relationship of functional constraints and axoneme integrity in eukaryotic flagella. This raises interesting questions about the molecular mechanisms of such phenomena and one wonders whether differences between axonemes capable of CP rotation versus axonemes bearing fixed CPs could be reflected in flagellar composition. In this regard, flagellar proteome comparisons hold much promise.
| Materials and Methods |
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-tubulin and PACRG (double knockdown of A and B isoforms) have been described previously (Bastin et al., 1998
-tubulin could be induced, a 500 bp fragment of the T. brucei
-tubulin open reading frame was amplified from genomic DNA using the primers 5'-CTCGAGAAGCTTCGCGAGAGTGCGGATGCTGT-3' and 5'-ACTTCTCGAGAGCACTTCCCGC-3' (restriction enzyme recognition sequence underlined). This amplicon was digested with HindIII and XhoI restriction endonucleases and ligated into the vector pZJM (Wang et al., 2000Promastigote-form Leishmania major Friedlin VI strain cells were cultured at 28°C in Medium 199 with Earle's Salt and L-glutamine supplemented with 40 mM HEPES, 10% v/v FCS and 5 µg ml-1 haemin. Promastigote-form Herpetomonas megaseliae and choanomastigote-form Crithidia fasciculata cells were cultured in Brain and Hearth Infusion medium supplemented with 5% v/v FCS. Epimastigote-form Trypanosoma cruzi CL Brenner strain cells were kindly supplied by J. Kelly (London School of Hygiene and Tropical Medicine, London, UK).
-tubulin for monoclonal antibody generation
For recombinant
-tubulin expression, the T. brucei
-tubulin open reading frame was amplified from genomic DNA using the primers 5'-CATATGCCTGACAACCGTGTCGTT-3' and 5'-GGATCCTCAAGCCGCTCCGTATGCGGCAGCAGT-3' (restriction enzyme recognition sequence underlined). This amplicon was ligated into the TA cloning site of pGEMT-Easy (Promega), released by NdeI and BamHI digestion, and ligated into the Escherichia coli expression vector pET15b (Novagen). Recombinant
-tubulin was expressed in FB810 (a RecA derivative of BL21) (McKean et al., 2001
) and purified by Ni-affinity chromatography followed by preparative SDS-PAGE. BALB/c mice were immunised with purified recombinant
-tubulin and hybridomas produced as described previously (Woods et al., 1989
).
Protein sample preparation and western blotting
For protein depletion analysis,
-tubulin RNAi mutant cells (2x108) were harvested at various time-points after the addition of tetracycline, washed in PBS (150 mM NaCl, 10 mM sodium phosphate, pH 7.4) containing protease inhibitors, and resuspended in boiling Laemmli sample buffer. SDS-PAGE and immersion transfer to nitrocellulose membrane were performed using standard techniques described elsewhere (Sambrook and Russell, 2001
). For immunoblotting, membranes were blocked with 3% w/v skimmed milk in TTBS (20 mM Tris-HCl, 500 mM NaCl, 0.05% Tween 20, pH 7.5), labelled with either anti-
-tubulin monoclonal antibody, or anti-ß-tubulin monoclonal antibody KMX (Birkett et al., 1985
), developed with horseradish peroxidase (HRP)-conjugated anti-mouse immunoglobulins (Sigma) and detected by the ECL detection system (Amersham).
Sample preparation for electron microscopy
T. brucei, T. cruzi, L. major, H. megaseliae and C. fasciculata samples for thin-section transmission electron microscopy were prepared by fixing cells in culture with 4% formaldehyde and 2% glutaraldehyde for 1 hour at room temperature. Cells were post-fixed in 1% osmium tetroxide in 100 mM sodium phosphate buffer, pH 7 for 1 hour at 4°C, washed several times in double-distilled water, stained en bloc with 1% aqueous uranyl acetate for 16 hours at 4°C, dehydrated through acetone and embedded in epoxy resin.
Geometric correction of electron micrographs
To correct for variations in the angle of sectioning in micrographs collected in this study, the following mathematical correction was applied. Considering the axoneme as a cylinder, any cross-section perpendicular to the longitudinal axis (the z' axis) will depict a perfect circle. However, any `glancing' section, which is not exactly 90° to the z' axis, will show an ellipse whose eccentricity increases as the angle between the section and a true perpendicular section (
) increases (see Fig. 1). Hence, an ellipse in the observed xy plane - formed by taking a section at an angle (90-
)° to the true longitudinal axis z' - can be superimposed onto the true transverse plane, x'y', by the transformation:
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From our `transverse' sections, the positions of the centre of each A tubule of the axonemal outer doublet microtubules and the centre of each of the CP microtubules were mapped by hand using the software NIH Image as an interface. The position of each microtubule in the observed xy plane was thus defined by a point (xi, yi). The origin was defined by the mean xi and yi values for the nine doublets and an estimate of
was made from the eccentricity of the doublet positions:
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i to the y' axis. The x'y' axes were rotated such that the mean of the differences of each
i from its expected value [2
(n-1)/9, where n is the doublet number] was zero. Finally, the microtubule positions were normalised against the mean d' value for the doublets. For most axonemes, minimum and maximum doublet distances provide a good estimate of ellipse angle and eccentricity (see Fig. 2A). However, in situations such as the PACRG mutant cell lines, where there is gross disruption of the outer doublets, this measurement is no longer sufficient. For these axonemes, it is necessary to use information from all remaining doublets, such as fitting the data points to an ellipse. We used the free software LSM (http://www.prz.rzeszow.pl/~janand/) to perform non-linear least-squares fitting of the data. All other calculations follow as above.
The above ellipse corrections are obviously just as applicable to images as they are to individual data points (see Fig. 1C). In this case, we used parameters estimated from non-linear least-squares fitting as above and the software NIH Image (http://rsb.info.nih.gov/nih-image/) with custom macros (available from the authors of this article on request).
| Acknowledgments |
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| Footnotes |
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