Biol Cell (1991) 71, 161-174
161
© Elsevier, Paris
O r i g i n a l article
New data on the microtubule surface lattice Denis C h r ~ t i e n , R i c h a r d H W a d e * Laboratoire de Biologie Structurale, CEA and CNR$ URA 1333, DBMS/DSV, CEN-G 85)(, 38041 Grenoble Cedex, France (Received 3 December 1990; accepted 4 March 1991)
Summary - The in vitro polymerisation of tubulin is a remarkable example of protein self-assemblyin that several closely related microtubule structures coexist on the polymerisation plateau. UnfLxedand unstained in vitro assembled microtubules were observed in vitreous ice by cryo-electron microscopy. New results are reported that considerably extend previous observations [47]. In ice, microtubule images have a distinctive contrast related to the number and skew of the protof'daments. The microtubules observed have from twelve to seventeen protofilaments. Comparison with thin sections of pelleted material allows a direct identification of images from microtubules with thirteen, fourteen and fifteen protofdaments. A surface lattice accommodation mechanism, previously proposed to explain how variable numbers of protofflaments can be incorporated into the basic thirteen protofilament structure, is described in detail. Our new experimental results are shown to be in overall agreement with the theoretical predictions. Only thirteen protofilament microtubules have unskewed protof'flaments,this was confLrmedby observations on axoneme fragments. The results imply that the microtubule surface lattice is based on a mixed packing which combines features of the standard A and B lattices. mierotubules I cryo.electron microscopy I mierolubule structure I microtubule polymorphism
Introduction
Microtubules are one of the filamentary structures commonly found in the cytoplasm of eukaryotic cells. They are an important component of the cytoskeleton and are also involved in the complex architectures of cilia and flagella and in the highly labile mitotic spindle [18]. To understand how these cellular components work, one necessity is to have a detailed model of the organisation of microtubules themselves. What is known at present about the structure of microtubules stems mainly from: i) transverse views obtained by electron microscopy of thin sections of fixed material prepared in the presence of tannic acid [23, 25, 43]; ii) longitudinal views observed by electron microscopy in negative stain [4]; iii) structural studies of tubulin in two dimensional zinc induced crystalline arrays [3, 14, 15, 41]; iv) X-ray fibre diffraction of oriented specimens [6, 28]. As shown in figure 1, the current textbook model of a microtubule deriv~edfrom these investigations is a hollow tube about 240 A in diameter built from a basic motif which is the tubulin heterodimer [2, 18]. The 0t and subunits of this dimer show a considerable degree of sequence homology [24, 40] and have molecular masses [Mr] in the range 50000 to 55 000. Heterodimers stacked head to tail form protofilaments, thirteen of which are incorporated in the microtubule wall. Neighbouring protofiluments have a slight longitudinal shift one along the other so that a line drawn through equivalent subunits in adjacent protofilaments follows a helical path. Of the many helical families that can describe theemicrotubule surface lattice the most familiar is the 120 Apitch left-handed three-start helix set [2, 4] (see figure I). There are two standard models of the tubulin dimer packing. These are known as the A and B lattices from the terminology used
* Correspondence and reprints
for flagellar outer doublets in which the complete A tubule and the incomplete B tubule are supposed to have these packings [4]. The protofdament juxtapositions in the A lattice ensure that the • and the/3 subunits alternate along each of the three start helices, as shown in figure lb. In the B lattice the two subunits make up alternate helices, but this leads to a packing discontinuity along a 'seam' between two of the protofilaments (fig lc). For this reason the A lattice stacking is usually the favoured model [2, 18]. There has been some evidence indicating that the packing may incorporate a mixture of the stacking sequences of both the A and the B lattices [26, 30]. In a previous study of frozen.hydrated microtubules it was found that the protofilaments had a variable twist around the axis of the tubule, the so-called supertwist [29]. If this twist should turn out to be an inherent feature of all microtubules, including those with thirteen protofllaments, the established models of complex microtubule structures such as axoneme outer doublets would need to be seriously revised. Microtubules have been frequently investigated using thin tannic acid stained sections of material from many different organisms [5, 11, 16, 19-21, 31-33, 36, 37, 43-45, 49, 50], and also using in vitro assembled microtubules [1, 7, 8, 10, 27, 34, 35]; protofdament numbers ranging from 8 to 19 are reported in these references. Unfortunately, thin sections give no direct indication of how the microtubule surface lattice can accommodate these different protofilament numbers. This has remained an unanswered problem in microtubule structure research. Previous findings using cryoelectron microscopy, reviewed in [17, 39], show clearly that in vitreous ice biGlogical structures are usually well preserved and maintain their overall shape. Our observations of frozen-hydrated microtubule structures, reported here and previously [46-48], indicate much better shape preservation than for negatively s~zined microtubules, which are almost always flattened, and often distorted or sprit open. In vitreous ice, distortions occur when the ice film is very thin and when microtubules interact at crossing points. Usually, frozen-
162
D Chr~tien, RH Wade
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Lattice accommodation explains how the basic thirteen protofilament microtubule structure can adapt so as to allow tubules to be built with a wide range of protofilament numbers. Since it is now known that an extensive range of protofilament numbers exists in vivo [5, 16, 19-21, 31-33, 36, 37, 43-45, 49, 50], structural polymorphism may well turn out to have a functional significance and so be an important aspect of microtubule design.
'N
R
Elements of image charaeterisation (,)
(',)
240 ), A
lattice
|e) O lattice
Fig 1. (a) Showinga side view of a simplehelix of pitch P, repeated motifs are separated verticallyby the rise R. (b) and (c) Thirteen protofilamentmicrotubulescan be consideredas three simple helices stacked together with a verticalspacing a, each helix having a pitch P = 3a. The structural motifs are the • and ~ tubulin subunits representedhere by shaded and unshaded circlesrespectively to show the two standard packing models. An indication of the accepted dimensions is given: (b) the A lattice model, (c) the B lattice model, in this case there is a discontinuity in the subunit packing along the three-starthelices,producing a "seam" between two of the protofilaments.
hydrated microtubules give highly characteristic images when examined by cryo-electron microscopy [47] and yield considerably more detailed information about the microtubule surface lattice than negatively stained specimens or thin sections. The results in vitreous ice show that individual microtubule images can be classed into groups according to their distinctive contrasts. Each group corresponds to a different number of constituent protofi!aments. Our interpretation of the images has allowed us to propose a simple mechanism whereby the basic surface lattice associated with the 13 protofilament structure can adapt so as to accommodate different protofilament numbers [47]. The mechanism involves a protofilament skew which enables the helical surface lattice to be geometrically conti. nuous whatever the number of protofilaments. This accommodation model can be used to predict the main features of the image contrast for microtubules with any number of protofilaments. The two essential features of the predicted, and experimentally observed, microtubule images are their distinctive contrast profiles and their regular longitudinal contrast variations. Consequently, images of frozen-hydrated samples give a direct indication of the number of protofilaments in each individual microtubule. What is more important, the analysis and understanding of the image contrast offers a considerable amount of detailed information about the surface lattice organisation. This type of information is unobtainable by any other method, and leads to a new insight concerning the specificity and precision of the lateral inter-protofilament contacts. Finally, we note that positive identification of each microtubule type is an essential preliminary to any attempt at three-dimensional reconstruction. • In the present article we present new experimental data covering microtubules with from 12 to 17 protofilaments prepared from both 3X and PC tubulin. This completes the image characterisation of the main microtubule types observed in frozen-hydrated samples [47, 48]. We describe the theoretic~ aspects of our surface lattice accommodation model and discuss these in the light of experiment.
Those details of the image interpretation which have been discussed in some detail elsewhere [47] will only be briefly mentioned here for the sake of completeness. If we assume that the specimen preparation method gives good structural conservation, the observed image intensity will depend on the projection of the cylindrical cross-section structure. This is shown schematically for thirteen and fourteen protofilament structures in figure 2 where the protofilaments are represented by the circular mbtifs. It can be proved that, when the number of motifs Nis even, the projected density, and consequently the image contrast, must be centrosymmetric for all angular orientations of the cross-sections. For odd N t h e projection can be either centrosymmetric, this occurs when the motifs project in exact intercalation, or asymmetric, this is the case for all the other orientations of the cross-sections including the situation where the motifs project in register. Taking into account these symmetry considerations, the projections shown in figure 2 have a certain number of general features which should appear in the images. Along the edges of the projected densities there are contributions from several partially superposed motifs and consequently the images should show strongly contrasted bands along their borders. The contrast in the central regions will depend on whether or not the motifs superpose in projection; in the case of superposition we would expect to see fringes parallel to the microtubule axis, otherwise this region will probably be blurred. If the schematic crosssections are drawn approximately to scale it is possible to confirm by computer modelling such general features as the symmetry of the projected density, the fringe number and the relative widths and intensities of the edge bands [48]. Furthermore, longitudinal contrast variations can be produced by a continuous rotation of the cross-section along the microtubule length. This would correspond to the microtubule having skewed protofilaments. The total width of the projected density will vary as a function of the number of motifs. If the spacing between the motifs around the circular cross-section is 8x, the width W of the projection will be given by:
W
=
[N
~x/~r] + 2 h,
where h is the height by which the motif sticks out beyond the contact radius. If we had in hand a series of images corresponding to different protofilament numbers a plot of W as a function of N should follow a straight line of slope proportional to the inter-motif (protofilament) separation whilst the intercept with the W axis would be 2 h. Other important information comes from the amplitude and the phase distributions along the equator of the computer calculated Fourier transforms of microtubule images. The most significant indication of the parity of N comes from the phase behaviour in the region where the Bessel function components JN and Jo overlap [4], see reference [47] for more details.
New data on the microtubule surface lattice
N.13
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(b) FiB 2. Schematic representation of microtubule cr Jss-sectio,ls showing how the projected structure varies with orientation. The protofilaments are represented by the circular motifs. For microtuhules oriented perpendicular to the electron beam the image contrast will he directly related to the projected structure. (a) N = 13, when the number of protofilaments N is odd, the projection may he either symmetric or asymmetric. (b) N - 14, for Neven the projection is symmetric whatever the cross-section orientation.
The surface lattice accommodation model To get a feeling for the model, consider tile highly idealised situation of figure 3a in which a regula~ lattice is drawn on a cylindrical surface, note that at this stage we no longer make a distinction between the ~ and the/3 subunits of the tubulin heterodimer, the reason for this will become clear later. As shown in figure 3b the cylinder can be mapped onto a plane surface with the 13 columns of unit cells corresponding to the protofilaments and with one of the three-start helices shown shaded. Three helices stacked one above the other are required to completely fill the surface lattice, these correspond to the shallow threestart helix family of the Amos and Klug model [4]. What happens when the microtubule has a different number of protofilaments? The fourteen protofilament case, corresponding to an extra column of unit cells in the two-dimensional representation, is shown in figure 3c. Figure 3d shows that when this fiat surface is mapped back onto a cylinder a geometrical lattice discontinuity is created along AB where the two edges of the sheet join. Intuitively, it seems unlikely that this type of discontinuity will be an intrinsic part of a real microtubule structure, since, de-
Fig 3. (a) Showing the surface lattice of a 13 protofllament microtubule, drawn as if the structure were situated uniquely on the surface of a cylinder, this is equivalent to a radial projection. For convenience, no distinction is made between the 0tand the ~ subunits. One of the three-start, shallow pitch helices, is shown shaded. (b) The cyfinder can he cut open along AB and unrolled so that the inner surface is viewed as a flat sheet. The lattice vectors a, b are indicated. Note that the shaded strips transform into continuous helices if the lattice is mapped back from the flat sheet to the cyfindricai representation. (c) Adding an extra column of unit cells givesthe two-dimensional lattice representation of a 14 protofilament microtubule. (d) Mapping this hack onto a cylinder produces a lattice mismatch along the junction of the two edges of the flat lattice. Equation (1) gives the analytical expression for the geometrical mismatch AN.
pending on the specificity of the lateral binding, it will represent an energetically unfavourable situation due to the serious modification of the lateral contact geometry between tubulin subunits on either side of the discontinuity. Is there some simple way to eliminate this structural discontinuity? Before trying to answer this question we first show how the extent of the lattice mismatch will depend on the number of protofilaments and on the relevafit number of helix starts. We then consider two fairly obvious ways of eliminating the discontinuities. These can be tested against the experimental data. There may be other possibilities.
The geometrical mL~match in the surface lattice The tubulin subunit spacing along each protofilament is almost certainly invari~nt due to the longitudinal bonding between the ~-~ heterodimers. This imposes the strong geometrical constraint that the value of P, the three-start helix pitch, be set to P = 3a. If the protofilaments are parallel to the microtubule axis, there is an integer number of motifs N for each helical turn, N is also the number of protofilaments. Consequently, the value of the
164
D Chr~tien, RH Wade
vertical displacement between adjacent subunits along the helix, ie the rise R, can be found from the relation P = N × R. Of course, in general, a helical structure can have any integer or non-integer number of motifs per turn. In the accepted 120 ~ pitch thirteen protof'dament microtubule model the subunit separation along a protof'dament is a = 40 ~,, and the rise along a three start helix is 9.23 ~,. Figure 3 shows how the introduction of extra protofilamerits produces a mismatch along the basic three-start helices. A convenient way to visualise this is to modify the two dimensional lattice representation of figure 3 so as to juxtapose the left and right edges of the lattice. This is represented in figure 4 which shows the geometrical mismatch at the interface between the first and the N th protofilaments. With reference to figure 4, denote the subunit separation along the protofilaments by a (the unit cell height) and the number of helix starts by S (the standard value will be S --- SO = 3). Consider the shaded, slanting strips that correspond to one of the S start helices. Along protofilament I the second strip is found at a height D = Sa above the first set. The equivalent strip on protofilament N is found at a height D N = S o a N / N o above the same baseline; N O is the number of protofilaments in the reference structure and in the present context N o = 13. The extent of the geometrical mismatch is given by:
AN = (DN-D) = (SoaN/No)-Sa,
(1)
This general expression for the geometrical mismatch holds for all protofilament numbers N and for helix start values
PROTOFILAMENI"S
312 '
161
© Elsevier, Paris
O r i g i n a l article
New data on the microtubule surface lattice Denis C h r ~ t i e n , R i c h a r d H W a d e * Laboratoire de Biologie Structurale, CEA and CNR$ URA 1333, DBMS/DSV, CEN-G 85)(, 38041 Grenoble Cedex, France (Received 3 December 1990; accepted 4 March 1991)
Summary - The in vitro polymerisation of tubulin is a remarkable example of protein self-assemblyin that several closely related microtubule structures coexist on the polymerisation plateau. UnfLxedand unstained in vitro assembled microtubules were observed in vitreous ice by cryo-electron microscopy. New results are reported that considerably extend previous observations [47]. In ice, microtubule images have a distinctive contrast related to the number and skew of the protof'daments. The microtubules observed have from twelve to seventeen protofilaments. Comparison with thin sections of pelleted material allows a direct identification of images from microtubules with thirteen, fourteen and fifteen protofdaments. A surface lattice accommodation mechanism, previously proposed to explain how variable numbers of protofflaments can be incorporated into the basic thirteen protofilament structure, is described in detail. Our new experimental results are shown to be in overall agreement with the theoretical predictions. Only thirteen protofilament microtubules have unskewed protof'flaments,this was confLrmedby observations on axoneme fragments. The results imply that the microtubule surface lattice is based on a mixed packing which combines features of the standard A and B lattices. mierotubules I cryo.electron microscopy I mierolubule structure I microtubule polymorphism
Introduction
Microtubules are one of the filamentary structures commonly found in the cytoplasm of eukaryotic cells. They are an important component of the cytoskeleton and are also involved in the complex architectures of cilia and flagella and in the highly labile mitotic spindle [18]. To understand how these cellular components work, one necessity is to have a detailed model of the organisation of microtubules themselves. What is known at present about the structure of microtubules stems mainly from: i) transverse views obtained by electron microscopy of thin sections of fixed material prepared in the presence of tannic acid [23, 25, 43]; ii) longitudinal views observed by electron microscopy in negative stain [4]; iii) structural studies of tubulin in two dimensional zinc induced crystalline arrays [3, 14, 15, 41]; iv) X-ray fibre diffraction of oriented specimens [6, 28]. As shown in figure 1, the current textbook model of a microtubule deriv~edfrom these investigations is a hollow tube about 240 A in diameter built from a basic motif which is the tubulin heterodimer [2, 18]. The 0t and subunits of this dimer show a considerable degree of sequence homology [24, 40] and have molecular masses [Mr] in the range 50000 to 55 000. Heterodimers stacked head to tail form protofilaments, thirteen of which are incorporated in the microtubule wall. Neighbouring protofiluments have a slight longitudinal shift one along the other so that a line drawn through equivalent subunits in adjacent protofilaments follows a helical path. Of the many helical families that can describe theemicrotubule surface lattice the most familiar is the 120 Apitch left-handed three-start helix set [2, 4] (see figure I). There are two standard models of the tubulin dimer packing. These are known as the A and B lattices from the terminology used
* Correspondence and reprints
for flagellar outer doublets in which the complete A tubule and the incomplete B tubule are supposed to have these packings [4]. The protofdament juxtapositions in the A lattice ensure that the • and the/3 subunits alternate along each of the three start helices, as shown in figure lb. In the B lattice the two subunits make up alternate helices, but this leads to a packing discontinuity along a 'seam' between two of the protofilaments (fig lc). For this reason the A lattice stacking is usually the favoured model [2, 18]. There has been some evidence indicating that the packing may incorporate a mixture of the stacking sequences of both the A and the B lattices [26, 30]. In a previous study of frozen.hydrated microtubules it was found that the protofilaments had a variable twist around the axis of the tubule, the so-called supertwist [29]. If this twist should turn out to be an inherent feature of all microtubules, including those with thirteen protofllaments, the established models of complex microtubule structures such as axoneme outer doublets would need to be seriously revised. Microtubules have been frequently investigated using thin tannic acid stained sections of material from many different organisms [5, 11, 16, 19-21, 31-33, 36, 37, 43-45, 49, 50], and also using in vitro assembled microtubules [1, 7, 8, 10, 27, 34, 35]; protofdament numbers ranging from 8 to 19 are reported in these references. Unfortunately, thin sections give no direct indication of how the microtubule surface lattice can accommodate these different protofilament numbers. This has remained an unanswered problem in microtubule structure research. Previous findings using cryoelectron microscopy, reviewed in [17, 39], show clearly that in vitreous ice biGlogical structures are usually well preserved and maintain their overall shape. Our observations of frozen-hydrated microtubule structures, reported here and previously [46-48], indicate much better shape preservation than for negatively s~zined microtubules, which are almost always flattened, and often distorted or sprit open. In vitreous ice, distortions occur when the ice film is very thin and when microtubules interact at crossing points. Usually, frozen-
162
D Chr~tien, RH Wade
%I:AM
UlI.AgFOtTS
;i;!
t
Lattice accommodation explains how the basic thirteen protofilament microtubule structure can adapt so as to allow tubules to be built with a wide range of protofilament numbers. Since it is now known that an extensive range of protofilament numbers exists in vivo [5, 16, 19-21, 31-33, 36, 37, 43-45, 49, 50], structural polymorphism may well turn out to have a functional significance and so be an important aspect of microtubule design.
'N
R
Elements of image charaeterisation (,)
(',)
240 ), A
lattice
|e) O lattice
Fig 1. (a) Showinga side view of a simplehelix of pitch P, repeated motifs are separated verticallyby the rise R. (b) and (c) Thirteen protofilamentmicrotubulescan be consideredas three simple helices stacked together with a verticalspacing a, each helix having a pitch P = 3a. The structural motifs are the • and ~ tubulin subunits representedhere by shaded and unshaded circlesrespectively to show the two standard packing models. An indication of the accepted dimensions is given: (b) the A lattice model, (c) the B lattice model, in this case there is a discontinuity in the subunit packing along the three-starthelices,producing a "seam" between two of the protofilaments.
hydrated microtubules give highly characteristic images when examined by cryo-electron microscopy [47] and yield considerably more detailed information about the microtubule surface lattice than negatively stained specimens or thin sections. The results in vitreous ice show that individual microtubule images can be classed into groups according to their distinctive contrasts. Each group corresponds to a different number of constituent protofi!aments. Our interpretation of the images has allowed us to propose a simple mechanism whereby the basic surface lattice associated with the 13 protofilament structure can adapt so as to accommodate different protofilament numbers [47]. The mechanism involves a protofilament skew which enables the helical surface lattice to be geometrically conti. nuous whatever the number of protofilaments. This accommodation model can be used to predict the main features of the image contrast for microtubules with any number of protofilaments. The two essential features of the predicted, and experimentally observed, microtubule images are their distinctive contrast profiles and their regular longitudinal contrast variations. Consequently, images of frozen-hydrated samples give a direct indication of the number of protofilaments in each individual microtubule. What is more important, the analysis and understanding of the image contrast offers a considerable amount of detailed information about the surface lattice organisation. This type of information is unobtainable by any other method, and leads to a new insight concerning the specificity and precision of the lateral inter-protofilament contacts. Finally, we note that positive identification of each microtubule type is an essential preliminary to any attempt at three-dimensional reconstruction. • In the present article we present new experimental data covering microtubules with from 12 to 17 protofilaments prepared from both 3X and PC tubulin. This completes the image characterisation of the main microtubule types observed in frozen-hydrated samples [47, 48]. We describe the theoretic~ aspects of our surface lattice accommodation model and discuss these in the light of experiment.
Those details of the image interpretation which have been discussed in some detail elsewhere [47] will only be briefly mentioned here for the sake of completeness. If we assume that the specimen preparation method gives good structural conservation, the observed image intensity will depend on the projection of the cylindrical cross-section structure. This is shown schematically for thirteen and fourteen protofilament structures in figure 2 where the protofilaments are represented by the circular mbtifs. It can be proved that, when the number of motifs Nis even, the projected density, and consequently the image contrast, must be centrosymmetric for all angular orientations of the cross-sections. For odd N t h e projection can be either centrosymmetric, this occurs when the motifs project in exact intercalation, or asymmetric, this is the case for all the other orientations of the cross-sections including the situation where the motifs project in register. Taking into account these symmetry considerations, the projections shown in figure 2 have a certain number of general features which should appear in the images. Along the edges of the projected densities there are contributions from several partially superposed motifs and consequently the images should show strongly contrasted bands along their borders. The contrast in the central regions will depend on whether or not the motifs superpose in projection; in the case of superposition we would expect to see fringes parallel to the microtubule axis, otherwise this region will probably be blurred. If the schematic crosssections are drawn approximately to scale it is possible to confirm by computer modelling such general features as the symmetry of the projected density, the fringe number and the relative widths and intensities of the edge bands [48]. Furthermore, longitudinal contrast variations can be produced by a continuous rotation of the cross-section along the microtubule length. This would correspond to the microtubule having skewed protofilaments. The total width of the projected density will vary as a function of the number of motifs. If the spacing between the motifs around the circular cross-section is 8x, the width W of the projection will be given by:
W
=
[N
~x/~r] + 2 h,
where h is the height by which the motif sticks out beyond the contact radius. If we had in hand a series of images corresponding to different protofilament numbers a plot of W as a function of N should follow a straight line of slope proportional to the inter-motif (protofilament) separation whilst the intercept with the W axis would be 2 h. Other important information comes from the amplitude and the phase distributions along the equator of the computer calculated Fourier transforms of microtubule images. The most significant indication of the parity of N comes from the phase behaviour in the region where the Bessel function components JN and Jo overlap [4], see reference [47] for more details.
New data on the microtubule surface lattice
N.13
163 a
...,...-..
*',
i.:"
I I I
%
:, ..,
,o
I I I
0
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-
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,
,
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(a)
I.) JnamsliloJoTq
bbau
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A
•
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..., -
.
,
,
,
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(b) FiB 2. Schematic representation of microtubule cr Jss-sectio,ls showing how the projected structure varies with orientation. The protofilaments are represented by the circular motifs. For microtuhules oriented perpendicular to the electron beam the image contrast will he directly related to the projected structure. (a) N = 13, when the number of protofilaments N is odd, the projection may he either symmetric or asymmetric. (b) N - 14, for Neven the projection is symmetric whatever the cross-section orientation.
The surface lattice accommodation model To get a feeling for the model, consider tile highly idealised situation of figure 3a in which a regula~ lattice is drawn on a cylindrical surface, note that at this stage we no longer make a distinction between the ~ and the/3 subunits of the tubulin heterodimer, the reason for this will become clear later. As shown in figure 3b the cylinder can be mapped onto a plane surface with the 13 columns of unit cells corresponding to the protofilaments and with one of the three-start helices shown shaded. Three helices stacked one above the other are required to completely fill the surface lattice, these correspond to the shallow threestart helix family of the Amos and Klug model [4]. What happens when the microtubule has a different number of protofilaments? The fourteen protofilament case, corresponding to an extra column of unit cells in the two-dimensional representation, is shown in figure 3c. Figure 3d shows that when this fiat surface is mapped back onto a cylinder a geometrical lattice discontinuity is created along AB where the two edges of the sheet join. Intuitively, it seems unlikely that this type of discontinuity will be an intrinsic part of a real microtubule structure, since, de-
Fig 3. (a) Showing the surface lattice of a 13 protofllament microtubule, drawn as if the structure were situated uniquely on the surface of a cylinder, this is equivalent to a radial projection. For convenience, no distinction is made between the 0tand the ~ subunits. One of the three-start, shallow pitch helices, is shown shaded. (b) The cyfinder can he cut open along AB and unrolled so that the inner surface is viewed as a flat sheet. The lattice vectors a, b are indicated. Note that the shaded strips transform into continuous helices if the lattice is mapped back from the flat sheet to the cyfindricai representation. (c) Adding an extra column of unit cells givesthe two-dimensional lattice representation of a 14 protofilament microtubule. (d) Mapping this hack onto a cylinder produces a lattice mismatch along the junction of the two edges of the flat lattice. Equation (1) gives the analytical expression for the geometrical mismatch AN.
pending on the specificity of the lateral binding, it will represent an energetically unfavourable situation due to the serious modification of the lateral contact geometry between tubulin subunits on either side of the discontinuity. Is there some simple way to eliminate this structural discontinuity? Before trying to answer this question we first show how the extent of the lattice mismatch will depend on the number of protofilaments and on the relevafit number of helix starts. We then consider two fairly obvious ways of eliminating the discontinuities. These can be tested against the experimental data. There may be other possibilities.
The geometrical mL~match in the surface lattice The tubulin subunit spacing along each protofilament is almost certainly invari~nt due to the longitudinal bonding between the ~-~ heterodimers. This imposes the strong geometrical constraint that the value of P, the three-start helix pitch, be set to P = 3a. If the protofilaments are parallel to the microtubule axis, there is an integer number of motifs N for each helical turn, N is also the number of protofilaments. Consequently, the value of the
164
D Chr~tien, RH Wade
vertical displacement between adjacent subunits along the helix, ie the rise R, can be found from the relation P = N × R. Of course, in general, a helical structure can have any integer or non-integer number of motifs per turn. In the accepted 120 ~ pitch thirteen protof'dament microtubule model the subunit separation along a protof'dament is a = 40 ~,, and the rise along a three start helix is 9.23 ~,. Figure 3 shows how the introduction of extra protofilamerits produces a mismatch along the basic three-start helices. A convenient way to visualise this is to modify the two dimensional lattice representation of figure 3 so as to juxtapose the left and right edges of the lattice. This is represented in figure 4 which shows the geometrical mismatch at the interface between the first and the N th protofilaments. With reference to figure 4, denote the subunit separation along the protofilaments by a (the unit cell height) and the number of helix starts by S (the standard value will be S --- SO = 3). Consider the shaded, slanting strips that correspond to one of the S start helices. Along protofilament I the second strip is found at a height D = Sa above the first set. The equivalent strip on protofilament N is found at a height D N = S o a N / N o above the same baseline; N O is the number of protofilaments in the reference structure and in the present context N o = 13. The extent of the geometrical mismatch is given by:
AN = (DN-D) = (SoaN/No)-Sa,
(1)
This general expression for the geometrical mismatch holds for all protofilament numbers N and for helix start values
PROTOFILAMENI"S
312 '
