Cellular and Molecular Biology 46 (5), 949-965 (2000)
1
ON THE RIGIDITY OF THE CYTOSKELETON:
Are MAPs crosslinkers or spacers of microtubules?
Alexander Marx, Jan Pless, Eva-Maria Mandelkow, and Eckhard Mandelkow
Max-Planck-Unit for Structural Molecular Biology, c/o DESY, Notkestraße 85, D-22607 Hamburg
Abstract
Microtubules are fibers of the cytoskeleton involved in mitosis, intracellular transport,
motility, and other functions. They contain microtubule-associated proteins (MAPs) bound
to their surface which stabilize microtubules and promote their assembly. There has been a
debate on additional functions of MAPs, e.g. whether MAPs crosslink microtubules and
thus increase their rigidity, or whether they act as spacers between them. We have studied
the packing of microtubules in the presence of MAPs by solution X-ray scattering using
synchrotron radiation. Microtubules free in solution produce a scattering pattern typical of
an isolated hollow cylinder, whereas tightly packed microtubules generate a pattern
dominated by interparticle interference. The interference patterns are interpreted in terms
of the Hosemann paracrystal concept, adapted for arrays of parallel fibers with hexagonal
arrangement in the plane perpendicular to the fiber axes (Briki et al., 1998). Microtubules
without MAPs can rapidly and efficiently be compressed by centrifugation, as judged by
the transition from a "free MT" to a "packed MT" X-ray scattering pattern. MAPs make
the microtubule array highly resistant to packing, even at high centrifugal forces. This
emphasizes the role of MAPs as spacers of microtubules rather than crosslinkers. A
possible function is to keep the microtubule tracks free for the approach of motor proteins
carrying vesicle or organelle cargoes along microtubules.
Key words:
microtubules, MAP, X-ray scattering, Hosemann paracrystal
Abbreviations:
MAP = microtubule-associated protein
C4S = microtubule protein with about 20 w-% MAPs
PC-T = tubulin purified by phosphocellulose chromatography
2
Introduction
Microtubules are hollow cylindrical protein fibers of about 25 nm in diameter, found in many
eucaryotic cells. They form part of the cytoskeleton and are responsible for diverse functions like
the definition of cell shape, cell polarity, organization of cellular compartments, exo- and
endocytosis etc. They are involved in many transport processes in the cell, both directly because
of their intrinsic dynamics, and indirectly as they serve as tracks for motor proteins like kinesin
and dynein. Microtubules are formed by polymerization of the globular protein tubulin, a 55 kD
protein that usually occurs as a heterodimer of two variants of tubulin, called α- and β-tubulin.
When tubulin is purified from cell extracts, a number of proteins known as microtubuleassociated proteins ( MAPs ) co-purify with tubulin through several cycles of assembly and
disassembly. The function of these MAPs are not yet known in detail. MAP2 and tau, which form
the major part of the MAPs in brain tissue, are involved in stabilization of microtubules and
probably regulate the mechanical properties of the microtubules (Matus, 1994; Mickey &
Howard, 1995; Illenberger et al., 1996; Felgner et al. 1997), their organization into larger
assemblies (Chen et al., 1992; Umeyama et al., 1993), and their interaction with motor proteins
(Ebneth et al., 1998; Trinczek et al., 1999) and other microtubule-related proteins (Schoenfeld &
Obar, 1994). An intact microtubule network is essential for the cell, and dysregulation of MAPmicrotubule interactions can have severe effects. Hyperphosphorylation of tau, for instance, may
lead to the breakdown of the microtubule network and is probably one step in the etiology of
Alzheimer's disease (Mandelkow & Mandelkow, 1993; Haass & Mandelkow, 1999).
The physical properties of microtubules can be approached either from the single microtubule
perspective by using microscopic methods like optical force trapping and similar methods
(Mizushima-Sugano et al., 1983; Gittes et al., 1993; Venier et al., 1994; Kurachi et al., 1995;
Kurz & Williams, 1995; Felgner et al., 1997), or from the perspective of the microtubule network
by studying macroscopic samples with large numbers of microtubules. In following this line,
"material properties" like the viscosity of microtubule solutions (Buxbaum et al., 1987; Sato et
al., 1988; Wagner et al., 1999) or the packing volume of sedimented microtubules have been
studied (Brown and Berlin, 1985; Black, 1987). Here we present an approach that combines
aspects of both, the macroscopic and the microscopic approaches. We used small-angle X-ray
scattering to obtain structural information on the nanometer scale from microtubule pellets
prepared by centrifugation of microtubule solutions in the presence and absence of MAPs.
Microtubules without MAPs can easily be packed into bundles by moderate centrifugal forces,
while the presence of MAPs strongly hinders bundling of microtubules. This shows that smallangle X-ray scattering of microtubule pellets can be used for studying the interactions between
microtubules in statistically significant ensembles. Thus, small-angle scattering of sedimented
microtubules could provide a link between the properties of single microtubules and their
collective behavior.
3
Materials and Methods
Microtubule protein ( "C4S protein" ) containing tubulin with about 20 wt-% of microtubuleassociated proteins ( MAPs ) and phosphocellulose-purified tubulin ( "PC tubulin" ) without
MAPs were prepared from pig brain as described by Mandelkow et al. (1985). High-molecular
weight MAP2 was isolated from C4S protein by boiling for 15 min, followed by a clearing spin
and Mono-S column FPLC chromatography (Pharmacia) as described (Wille et al., 1992).
Dephosphorylated MAP2 was prepared by incubation of MAP2 with alkaline phosphatase.
Microtubules were polymerized in standard reassembly buffer (100 mM Pipes at pH 6.9, 1 mM
GTP, 1 mM MgSO4, 1 mM DTT) by incubation with 10−20 µm taxol at 37 °C for 20 min.
Protein concentrations were adjusted to about 20 mg/ml. Solution scattering of microtubules at
this concentration is not visibly affected by interference effects between adjacent microtubules
and are considered as sufficiently "dilute". Microtubule pellets were prepared by centrifugation of
microtubule solutions in the Beckmann TLA100.2 rotor for various times between 5 min and 60
min. The g-values quoted are nominal values calculated with the maximum radius, representative
of the centrifugal forces that act on the pellet during the last stages of centrifugation. The pellets
were transferred into the cell used for X-ray scattering either with a spatula or by means of a thin
mylar foil ( 14 µm ) that was placed at the bottom of the centrifuge tube before centrifugation.
Scattering of the mylar foil is negligible compared to the scattering of the pellets.
Small-angle scattering patterns were measured on instrument X33 of the EMBL Outstation at the
DESY synchrotron laboratory HASYLAB in Hamburg (Koch & Bordas, 1983). Microtubule
solutions and pellets were placed into an X-ray cell with ≈50 µm mica windows. The solution
scattering patterns were corrected for background scattering by subtracting the scattering
intensity of the buffer solution without protein. The pellet scattering was used without any
correction. Reciprocal spacings were calibrated with respect to the periodicity of collagen from
rat tendon.
Theory
Interpretation of small angle scattering patterns
The scattering of microtubules in solutions can be described by the single particle scattering
provided that their orientations and relative positions can be considered as uncorrelated and
random. Absence of correlations implies that the scattering intensity of the solution is the sum of
the scattering intensities of the individual particles, and randomness allows calculation of this
sum by averaging the scattering intensity of a single particle over all possible orientations. In
dilute solutions where the interactions can be neglected, both conditions are fulfilled because of
the independence of the particles. For microtubule solutions at medium concentrations the
4
assumption of randomness and absence of correlations can still be a good approximation.
However, at high concentrations and in microtubule pellets this assumption is usually not valid
and interference effects have to be taken into account.
(1) Single particle scattering
Treatment of the pellet scattering requires a realistic description of the structure and the relative
arrangement of the microtubules. To render the problem amenable to an analytical treatment we
make the assumption that the microtubules can be described as hollow cylinders of infinite length
with inner and outer radii ri and ro, and that these radii are the same for all microtubules. The
structure factor of a hollow cylinder of infinite length, centered around the z-axis and with unit
density between ri and ro is
F 1 ( s ) = Fmt ( s ) δ ( sz )
with
Fmt ( s ) = 2 π
ro
∫rJ
r
i
0
1
( 2 π r s ) dr = ( ro J 1 ( 2 π ro s ) − ri J 1 ( 2 π ri s ))
s
for any vector s of the reciprocal space. Here, s is the magnitude of s, δ is the Dirac delta
function, and J0 and J1 are the Bessel functions of order zero and one. The single particle
structure factor F1( s ) vanishes for any s with nonzero component sz along the z-axis. The
scattering intensity Imt( s ) of a dilute solution of microtubules is proportional to the average of
I1( s ) = |F1( s )| 2 over all vectors s of magnitude s, where s corresponds to the "Bragg
spacing", s = 2 sin( θ ) / λ, with 2 θ scattering angle, λ wavelength.
Imt ( s ) ∝
1
( Fmt ( s ) ) 2
s
The factor 1/s results from averaging of the delta function. Thus, s Imt( s ) is proportional to the
square of the radial part of the structure factor of a hollow cylinder. Figure 1 b shows the
function s Imt( s ) for hollow cylinders with parameters ri = 9.5 nm, ro = 14 nm, and Fig. 1 c the
corresponding function determined from the X-ray scattering intensity Isol( s ) of a solution of
microtubules.
The approximation of microtubules by hollow cylinders is reasonable only for small scattering
angles, below the range that is dominated by periodicities of the tubulin lattice. The axial repeat
of the tubulin lattice is effectively about 4 nm since α- and β-tubulin subunits are normally
indistinguishable by small-angle scattering. The lateral repeat distance between protofilaments is
about 5 nm. Thus, the approximation should be valid for Bragg spacings below ≈0.2 nm-1, which
is consistent with Fig. 1 b and c.
5
(2) Scattering by arrays of parallel microtubules
In pellets obtained by centrifugation of microtubule solutions, the waves scattered by individual
microtubules are phase-correlated, leading to interference effects. Considering microtubules as
hollow cylinders of infinite length considerably simplifies the treatment of interference: Since the
microtubule structure factor is restricted to the plane perpendicular to the microtubule axis, only
those microtubules that are perpendicular to the scattering vector contribute to the total scattered
intensity and, thus, can interfere with each other. Therefore, it suffices to consider the
interference of subsets of microtubules the axes of which are perpendicular to a common vector
(the scattering vector).
From all possible arrangements of microtubules with axes perpendicular to a common vector, say
t, we consider the special case of microtubules with axes parallel to each other. There are two
main reasons for believing that such arrangements of parallel microtubules dominate the
interference effects of concentrated solutions and pellets of microtubules. First, interference
effects increase with the number of particles and the degree of order between the particles that
interfere with each other. For long cylindrical particles it is hard to imagine that other than
parallel arrangements could comprise an appreciable number of particles with high positional
correlation − provided that there are no special interactions that enforce a special arrangement
like, for instance, a cholesteric liquid crystalline state. Furthermore, if all types of arrangements
are present in the sample, the subsets of parallel cylinders will predominate since parallel
alignment allows interference in any direction perpendicular to the common axis, while in the
general case, interference occurs only in two directions ( ± t ). Secondly, it is a common feature
of rod-like molecules in solution to assume a parallel orientation at high concentrations. In thin
sections of microtubule pellets, large areas of parallel fibers have been observed and their
formation has been interpreted in terms of an isotropic - nematic phase transition (Murphy &
Borisy, 1975; Kim et al., 1979; Brown & Berlin, 1985, Somers & Engelborghs, 1990).
Thus, the most relevant subsets of microtubules in pellets and highly concentrated solutions are
arrays of parallel microtubules. The scattering of such an array is defined by the shape of a single
microtubule ( a hollow cylinder of infinite length; assumed to be parallel to z ) and the points of
intersection of the cylinder axes with the plane perpendicular to the axes ( xy-plane ). The traces
of the microtubule axes in the xy-plane define a two-dimensional array of points that completely
determine the inter-microtubular interference effects for the subset of microtubules under
consideration (Figure 1b). This subset of microtubules can be treated as the convolution of the
two-dimensional array of points and the shape of a single microtubule. Thus, from the
convolution theorem of the Fourier transform, the total structure factor of the subset of
microtubules F is the product of the Fourier transform Z of the two-dimensional array of points
and the form factor F1 of a single microtubule. Since F1 is zero outside the xy-plane, the same is
true for the total structure factor F . Thus, we can restrict the analysis to the equatorial part ( the
part living on the xy-plane ) of the structure factor.
F ( sx , sy ) = F 1 ( sx , sy ) Z ( sx , sy )
Assuming that the sample contains many subsets of parallel microtubules without correlation
between the orientation of the subsets, the total intensity scattered by the sample is the average of
| F |2 ,
6
F ( sx , sy )
2
= F 1 ( sx , sy )
2
Z ( sx , s y )
2
= I 1 ( sx , sy ) IIF ( sx , sy ) ,
over all orientations. Here,
IIF ( sx , sy ) = Z ( sx , sy )
2
is the interparticle interference function. Averaging over all sets of microtubules with axes
parallel to z replaces the interference function by its azimuthal average, IIFav( s ). ( I1 has
rotational symmetry, I1( sx, sy ) = I1( s ) = ( Fmt( s ) )2. ) Averaging over all orientations of the
microtubule axes leads to the factor 1/s as in the case of the single microtubule scattering. Thus,
the scattering of a large number of randomly oriented arrays of parallel microtubules is given by
1
( Fmt ( s ) ) 2 IIFav ( s ) ∝ Imt ( s ) IIFav ( s ) .
s
This expression is the expected scattering intensity for a microtubule pellet idealized by random
subsets of parallel cylinders. Figure 1 j shows the scattering intensity ( multiplied by s ) that was
determined experimentally for a pellet of microtubules. The interference function calculated as
the ratio of the pellet scattering (Fig. 1j) and the scattering of microtubules in solution (Fig. 1 c)
is shown in Fig. 1 h. In general, the scattering of random subsets of parallel microtubules in real
samples will be superposed to a background of those microtubules not taken into account by
considering arrays of parallel microtubules.
I (s) ∝
(3) The interparticle interference function
The lateral arrangement of microtubules in dense pellets is likely to be governed by close
packing. Therefore, it seems natural to assume a quasi-hexagonal arrangement of the
microtubules. The averaged interference function, IIFav , for hexagonal arrays of points ( twodimensional crystals of finite size ) randomly dispersed in the xy-plane is shown in Fig. 1 f .
For arrays of large size, IIFav consists of a series of lines, corresponding to the diffraction lines
of a two-dimensional powder. Some of the maxima of IIFav in Fig. 1 f are labeled with the
corresponding Miller indices. The linewidth is inversely proportional to the size of the crystals.
By comparing interference functions for hexagonal arrays of different size with the experimental
result shown in Fig. 1 j, it turns out that the hexagonal arrays must not exceed the second
coordination shell of the center point ( Fig. 1d ) in order to explain the observed width of the
interference maxima by a pure size effect.
A more realistic description of microtubule packing in pellets should include stochastic
deviations from the perfect hexagonal arrangement. We used the concept of the "Hosemann
paracrystal" to calculate interference functions for hexagonal arrays in two dimensions with
disorder of the second kind ( liquid-like disorder with loss of long-range correlation's;
Hosemann, 1950a, 1950b; Guinier, 1963; Vainshtein, 1966 ). This concept has recently been
adapted to two-dimensional paracrystals of hexagonal symmetry ( Briki et al., 1998 ). The
principles of the calculations are described in Appendix A. In short, the Hosemann paracrystal
concept uses a statistical description of the positions of the lattice points: given a lattice point at
the origin of the paracrystal, the positions of the next neighbors along the axes a , b , c of the
7
unit cell ( for the three-dimensional case ) are defined by probability distributions, ha , hb , and
hc ; then the probability distributions for the position of any lattice point with average position
na a + nb b + nc c is given by na × nb × nc −fold convolution of the respective next-neighbordistributions ha, hb, and hc . Thus, the distributions of the lattice points become more and more
"smeared" with increasing distance from the origin.
In our adaptation of the Hosemann paracrystal for the two-dimensional, hexagonal case ( which is
similar to that given by Briki et al., 1998 ), the paracrystal is defined by only two parameters: the
average distance between next neighbors a and a standard deviation σ describing the nextneighbor probability distributions, which are assumed to be two-dimensional Gaussian
distributions with rotational symmetry, identical for all six symmetry related directions. The
standard deviation is a measure of the degree of disorder. In the limit σ → 0 the paracrystal
becomes equivalent to a perfect crystal, while large values of σ correspond to liquid-like
disorder.
Although the ideal paracrystal is infinite by definition, correlations between lattice points and,
thus, interference effects are limited to distances of the order a3 / σ2 . Nevertheless, real
structures could probably better be described by finite paracrystals the size of which might be
smaller than the region of correlations. The interference function of a finite paracrystal is the
convolution of the shape function S and the interference function of the infinite paracrystal
(Guinier, 1963):
S ( sx , sy ) =
1
2
Fshape ( sx , sy ) ,
A
where Fshape( sx, sy ) is the Fourier transform of the function that is 1 within the region of the
finite paracrystal and 0 outside; A is the area of the two-dimensional paracrystal. For a circular
disk of radius R , the shape function is
S ( s x , sy ) =
1
πs
2
J 1 ( 2π R s ) 2 .
Figure 1 g shows the orientationally averaged interference function for finite paracrystals with
circular shape of radius R = 2 a ( a = 35 nm, σ = 0.075 a ), and Fig. 1i the expected scattering
intensity calculated by multiplication of this interference function by the single microtubule
scattering pattern shown in Fig. 1 b .
8
Results
(1) Microtubule bundling: scattering theory versus experiment
The scattering pattern of Fig. 1 j was recorded for a dense pellet obtained by 10 min of
centrifugation at 400,000 g of a solution of microtubules polymerized from tubulin purified by
phosphocellulose chromatography ( PC tubulin ). Ideally, the interference function should be
calculated as the ratio of the pellet scattering and the "single particle scattering". The latter
corresponds, in principle, to the scattering of a dilute solution of microtubules polymerized under
the same conditions. In real experiments, the scattering of dilute solutions usually differs from
the single particle scattering for a number of reasons. The signal of microtubules in dilute
solution is weak and can easily be affected by any kind of disturbing effects. For instance, part of
the protein does not polymerize at all or it polymerizes into aberrant forms of aggregates and,
thus, contributes to the background scattering. Furthermore, the structure of microtubules varies
by bending and flattening, and the number of protofilaments is not identical for all microtubules.
One of the consequences is that the minima of the scattering pattern are filled up, compared to the
ideal patterns. Thus, in the vicinity of the minima, especially of the first minimum, the solution
scattering is a poor estimate of the single particle scattering.
Usually, disturbing effects are less pronounced when microtubules are polymerized in the
presence of MAPs. The best solution scattering pattern we measured, as judged by the form of
the first and second maxima and the depth of the first minimum, was that of a solution of
microtubules polymerized from of PC tubulin with 20% of the MAPs present in C4S protein
( Fig. 1 c ). Even if this pattern is used as an estimate of the single particle scattering, the
interference function is probably underestimated in the vicinity of the first minimum at s =
0.035 nm-1. This would explain why the first maximum of the interference function ( Fig. 1 h ) is
low compared to the theoretical functions of Fig. 1 f and g. Because of the difficulties to obtain
quantitatively reliable estimates of the interference functions, it is obvious that in comparing the
interference functions with model calculations, only the position and the width of the interference
maxima should be considered.
With this in mind, the interference functions in Fig. 1 f and g, derived for hexagonal bundles of
microtubules ( Fig. 1 d ) and for paracrystalline arrays of parallel microtubules ( Fig. 1 e ),
can be regarded as adequate representations of the experimental result. If described by strictly
hexagonal bundles, the number of microtubules forming the bundles must be very limited,
probably of the order of ten, in order to match the width of the interference maxima. If statistical
deviations from the hexagonal arrangement are included, as in the model of the Hosemann
paracrystal of finite size, the interference function ( Fig. 1 g ) seems even more realistic.
However, even in this model, correlations between microtubules do not exceed the second
coordination shell (about 20 microtubules ). This type of bundling is confirmed by thin-section
electron microscopy (Fig. 2). The original micrograph in Fig. 2 a shows an area of several µm2 of
mostly parallel microtubules perpendicular to the plane of section (visible by their circular crosssections). The positions of these microtubules are represented in Fig. 2 b by the traces of their
long axes. The microtubules are not evenly distributed but concentrated in clusters of
9
paracrystalline order.
The scattering pattern predicted according to the paracrystalline model ( Fig. 1 i ) is in good
agreement with the observed pellet scattering. ( This indicates, that a large part of the
imponderables that deteriorate the interference function are due to the estimate of the single
particle scattering by the scattering of microtubule solutions. ) The main effects of microtubule
pelleting are the reduction of the intensity near the center and a shift of the maxima at 0.5 nm-1
and 0.9 nm-1 , as well as narrowing and skewing of these maxima. The shifts are in opposite
directions, decreasing the distance between the maxima, and are due to the "powder diffraction"
lines with indices (11), (20), in combination, and (21) of the interference function. The first
maximum of the interference function (10) coincides with the first minimum of the solution
scattering ( single particle scattering ) and therefore does not show up as a prominent maximum
in the pellet scattering. The parameters used in the models are a = 35 nm for the average distance
between next-neighbor microtubules ( corresponding to the hexagonal lattice constant in the twodimensional crystal ) and σ = 0.075 a for the Hosemann paracrystals. Thus, the distance
between next neighbors exceeds the nominal diameter of the microtubules ( 28 nm in the
cylindrical model ) by 7 nm.
The example shown in Fig. 1 j represents the case of optimal packing of microtubules that we
observed in our experiments. In the following, we discuss some other results obtained under
different conditions. For reasons set forth in the previous paragraphs, we do not consider
interference functions calculated from recorded scattering patterns, but restrict the discussion to
the untreated data.
(2) Sedimentation of PC-T and C4S microtubules: time-dependence
First we compare the effect of sedimentation of microtubules polymerized from purified tubulin
( PC-T pellets, Fig. 3 a ) and from tubulin with MAPs ( C4S pellets, Fig. 3 b ). The scattering of
PC-T pellets shown in Fig. 3 a is equivalent to that shown in Fig. 1 j, demonstrating that pellets
prepared from purified tubulin are well reproducible. The patterns obtained after 5 and 10 min of
centrifugation ( at 400,000 g ) are similarly affected by interference effects. Thus, after 5 min of
centrifugation, the packing of microtubules has already reached the final state. The
reproducibility of the scattering of PC-T pellets is favored by the fact that these pellets are
compact and can easily be handled. This is in contrast to pellets prepared from C4S protein. C4S
pellets are usually less dense and less reproducible, and the scattering patterns can vary
considerably between different experiments, depending on minor changes in sample preparation
and handling. Nevertheless, a general trend can be discerned by comparing results obtained in
series of measurements done with special care taken for identical treatment of all samples.
The scattering pattern of a C4S pellet obtained after 10 min centrifugation is still very similar to
the pattern of a solution of C4S microtubules. After 30 min, first differences appear; they become
only slightly more pronounced after 60 min centrifugation. Two patterns recorded from different
samples are shown for the 30, 45, and 60 min of centrifugation in order to demonstrate the
variability of the samples in this series of experiments. The patterns obtained after prolonged
centrifugation seem to approach the form of the PC-T pattern inasmuch as the first maximum
shifts from 0.05 nm-1 to 0.057 nm-1, both maxima tend to sharpen, and the intensity near the
10
center decreases. However, the patterns obtained by prolonged centrifugation are still far from
PC-T-like. It seems that they are superpositions of the PC-T pattern with the solution scattering,
indicating that only part of the microtubules has reached lateral packing as in PC-T pellets, while
the other part is still uncorrelated and scatters similar to microtubules in dilute solution. Since the
contribution of microtubule bundles is amplified by coherent scattering, the subset of
microtubules that is responsible for interference effects in C4S patterns is probably very small. In
any case, it is obvious that the presence of MAPs strongly hinders parallel packing of
microtubules.
(3) Sedimentation of microtubules with varying concentrations of MAPs
To further analyze the inhibitory effect of MAPs on the rearrangement and packing of
microtubules, a series of pellets was prepared from mixtures of PC tubulin and C4S protein,
varying the relative contents of MAPs. All the pellets were prepared by centrifugation at high
centrifugal force ( 400,000 g ) for 10 min. The scattering patterns are presented in Fig. 4 together
with the patterns of dilute solutions of PC and C4S microtubules. The pellet of pure C4S
microtubules, corresponding to 100% MAPs, does not show any sign of interference effects ( in
this series of measurements ). Deviations from solution scattering are hardly visible at
concentrations above 50% of the maximum MAP concentration and gradually increase with
decreasing MAP contents below about 40%. It seems that the transition from solution-like
scattering to PC-T pellet-like scattering occurs in a limited range of MAP concentrations between
about 40% and 30%. However, with regard to the variability of the pellets prepared with MAPs it
cannot be excluded, that the impression of a step-like transition is incidental.
For a more quantitative description of the transition from solution-like scattering to the scattering
of microtubule paracrystals we chose the scattering intensity near the center (i.e. close to the
beamstop), Icen = I( s = 0.0175 nm-1 ), normalized by the scattering intensity at the position of the
relative maximum in the scattering of well-packed PC-T microtubules, Iiif = I( s = 0.058 nm-1 ).
Normalization is indispensable in order to account for the different numbers of microtubules
contributing to the scattering signal of solutions and pellets of varying densities. Theoretically, it
would be preferable to normalize with scattering intensities at larger s-values where the
interference function approaches unity ( s > 0.15 nm-1 ). However, in this region the scattering
signal is low and normalization would by extremely sensitive to small errors. At s = 0.058 nm-1
the scattering signal is relatively strong since it is close to the first subsidiary maximum which is
characteristic of microtubules in dilute solution, and it remains high even in the interference
patterns of packed microtubules.
The normalized near-center-intensity ( Icen / Iiif ) of the scattering patterns in Figure 4 are shown
in the inset on the top of the figure. With increasing concentrations of MAPs, Icen / Iiif
approaches the value corresponding to solution scattering ( single-particle scattering ). The most
plausible form of the functional dependence between Icen / Iiif and the MAP concentration is a
sigmoidal curve. The scatter of the data points makes it difficult to estimate the width of the
transition region. Both, a smooth and almost linear relationship and a step-like transition would
be compatible with the data. In any case, it is obvious that sufficiently high contents of MAPs
prevent efficient packing of microtubules even at high centrifugal forces.
11
(4) Sedimentation at varying centrifugal forces
How important is the centrifugal force applied to the samples ? It is anticipated that sufficiently
weak centrifugation will have no effect at all, and that packing of microtubules will start to
become apparent at g-values that correlate with the concentration of MAPs. We measured the
scattering of pellets prepared with varying centrifugal forces at four different concentrations of
MAPs. The normalized near-center intensities are shown in Figure 5. Again, the scatter of the
data points makes it difficult to deduce the exact form of the transition from solution-like to
interference scattering; the curves are drawn only as a guide. However, the form of the curves is
not completely arbitrary: it is assumed that the curves start at 0 g with the value corresponding
to solution scattering ( interpolated in the case of the 25% C4S series ), and that the tangents at
this point are horizontal. Even though there are large uncertainties in the interpretation of the
data, it is evident that the transition zone shifts to higher g-values with increasing MAP
concentrations. Thus, the higher the concentration of MAPs, the higher the forces required for
efficient packing of microtubules.
(5) Dephosphorylation of MAP2
Finally, we examined the effect of dephosphorylation of MAP2 on the packing of microtubules.
For these experiments, microtubules were polymerized from mixtures of PC tubulin and MAP2
in proportions of 20 wt% MAP2 and 80 wt% tubulin. Three preparations of MAP2 in different
states of phosphorylation were used: The first preparation ( MAP2total ) was partially
phosphorylated MAP2 as obtained from C4S tubulin protein by isolation of the total MAP2
fraction. This preparation contains about 10 mol of endogenous phosphate per mol MAP2
(estimation based on similar preparations of MAP2 from rat brain; Brugg & Matus, 1991). It is a
mixture of two fractions with different degrees of phosphorylation (Hagestedt et al., 1989). The
second preparation ( MAP2fr2 ) contained only the less phosphorylated fraction of MAP2total. The
third preparation ( MAP2dephos ) was obtained by dephosphorylation of the total MAP2 fraction
with alkaline phosphatase. The scattering patterns of pellets obtained by 10 min centrifugation of
microtubules polymerized with each of the MAP2 preparations are shown in Figure 6. MAP2total
has a similar effect on the packing of microtubules as the mixture of MAPs present in C4S
protein. It largely prevents packing of microtubules resulting in solution-like scattering. Selection
of the less phosphorylated fraction of endogenously phosphorylated MAP2 leads to changes in
the scattering pattern even though it remains similar to the scattering of dilute solutions. With
MAP2 dephosphorylated by alkaline phosphatase changes were even more pronounced. These
results indicate that dephosphorylated MAP2 is less inhibitory to microtubule bundling than
phosphorylated MAP2. Thus, it seems that the inhibitory effect of MAPs on microtubule packing
increases with the degree of phosphorylation.
12
Discussion
(1) Method. Small-angle X-ray scattering and the complex structure of microtubule pellets
The packing of microtubules by sedimentation in a centrifuge is a complicated process of
diffusion and reorientation of individual microtubules in the presence of other microtubules of
increasing density. For dispersions of rod-like molecules a discontinuous transition from the
disordered, isotropic phase to a nematic liquid-crystalline phase is expected at some
concentration that depends on the aspect ratio (Onsager, 1949) and the rigidity of the rods (Wulff
& De Rocco, 1971). Since microtubules can be considered as semi-rigid rods, lateral arrangement
of microtubules is thermodynamically favored at high concentrations. Accordingly, large areas of
parallel microtubules have been found in cross-sections of sedimented microtubules and they
have been explained in terms of an isotropic-nematic phase transition (Brown & Berlin, 1985;
Black, 1987; Somers & Engelborghs, 1990). On the other hand, the translational and rotational
diffusion of rod-like molecules is strongly reduced by the presence of other rods and this effect
rapidly increases with the length of the molecules. Thus, kinetically, a disordered, isotropic state
in solutions of rod-like molecules can be stable at high concentrations where thermodynamics
predicts a nematic, anisotropic phase (Edwards & Evans, 1982; Janmey, 1991).
Thus, the state of a microtubule pellet reached after a certain time of sedimentation depends on
the total history of its preparation. Among the parameters that determine the final state are: the
time and speed of centrifugation, the number concentration and length distribution of the
microtubules before and during centrifugation, the buffer conditions ( pH, ionic strength, taxol... )
that may affect the rigidity of and the interaction forces between microtubules, the presence and
nature of MAPs bound to the surface, and probably more. In principle, all these parameters can
be used to study the properties of microtubules and their interactions, just by changing the
parameters and measuring the "response" to these changes. However, any combination of these
parameters may lead to a different arrangement of microtubules, the detailed description of which
would require another set of parameters, including parameters for different scales from the
nanometer scale up to macroscopic distances.
With regard to the possible structural diversity of microtubule pellets it is perhaps surprising that
their small-angle scattering displays any kind of systematic behavior. The reason can be found in
the theoretical analysis of the pellet scattering given above. Only those microtubules contribute to
the scattering signal whose axis are perpendicular to the scattering vector and this implies for
most cases that the scattering intensity is dominated by the scattering of microtubule bundles
(arrays of parallel microtubules). Thus, small-angle scattering of sedimented microtubules works
like a filter, that selects one special feature of the sample structure and discards all the others.
This is at the same time an advantage and a disadvantage of the method. It is an advantage as it
allows interpretation of the scattering results. On the other hand, the method is insensitive to most
of the other structural characteristics of the samples. For instance, the scattering of C4S pellets,
which are certainly not dilute solutions of microtubules, are practically indistinguishable from
solution scattering patterns.
13
For the interpretation of the small-angle scattering patterns in the range s < 0.2 nm-1 , the concept
of finite Hosemann paracrystals in two dimensions seems to provide an adequate representation
of the arrays of parallel microtubules present in the pellets. The advantage of this concept is that
it uses only three parameters, the mean distance between next neighbors, the disorder parameter
σ, and the radius of the paracrystalline regions R, the meaning of which is intuitively clear. With
these parameters, the whole range from the perfect hexagonal arrangement to liquid-like disorder
(in the plane perpendicular to the microtubule axes) can be described.
(2) Application. Effect of MAPs on the packing of microtubules by centrifugal forces
According to the Hosemann paracrystal model, the small-angle scattering of well-packed pellets
of PC microtubules are characterized by coherence regions of radius R ≈ 2 a , corresponding to
"bundles" of not more than about 20 microtubules. This number is surprisingly low in view of
electron-microscopy cross-sections of sedimented microtubules (Brown & Berlin, 1985) and
cellular processes (Chen et al., 1992), both showing wide arrays of parallel microtubules. This
discrepancy can probably be explained by the absence of positional correlation between
paracrystalline regions of radius R. ( This would be in analogy to the mosaicity of threedimensional crystals, a common phenomenon in protein crystallography. ) Thus, the
predominant sub-structures of microtubule pellets which determine the small-angle scattering
could best be described as "bundles of bundles". Because of the small coherence range, the
oscillations of the interference function rapidly fade out with increasing Bragg spacings s , even
for well-ordered PC-T pellets. At s > 0.15 nm-1, the interference function approaches its longdistance limit of 1 . Consequently, at Bragg spacings larger than 0.15 nm-1 , the scattering of
parallel bundles of microtubules approximates that of a single microtubule and can be used for
low-resolution analysis of the internal structure of microtubules (Mandelkow et al., 1977).
Measurements of the specific volume of sedimented microtubules (Brown & Berlin, 1985; Black,
1987) revealed a difference between purified microtubules and microtubules with MAPs bound
to the surface. For microtubules without MAPs, a minimum value of ca. 7 µl/mg tubulin was
obtained. In the presence of MAPs, the specific volume increased several times. This was
attributed to the projection domains of MAP2 and tau, that seem to function as spacers between
microtubules. In this regard, the effect of tau is negligible compared to MAP2, reflecting the
difference in size (1669 amino acids for MAP2 vs. 242 amino acids for tau) and, possibly, in
rigidity of the projection domains (Black, 1987; Chen et al., 1992). MAP2 and tau are neuronal
MAPs specifically located in dendrites and axons, respectively. Interestingly, the distance
between microtubules in dendrites and processes of MAP2-transfected SF9 insect cells is
consistently larger than the distances found in microtubule bundles of axons and tau-transfected
cells (Chen et al., 1992).
Assuming a perfect hexagonal arrangement, a specific volume of 7 µl/mg corresponds to a
center-to-center distance of ≈53 nm. This is 1.5 times larger than the mean distance in
paracrystalline regions as observed by small angle X-ray scattering. The value calculated from
the specific volume is an average over the whole sample including ordered and less ordered
domains, while the value obtained by small angle X-ray scattering is determined by the well-
14
ordered regions only.
Nevertheless, our results also show a clear difference between pellets of pure microtubules and
pellets obtained from microtubules with MAPs. While MAP-free microtubules rapidly pack at
high centrifugal forces ( and even at moderate centrifugal forces as can be extrapolated from the
MAP concentration series in Figure 5 ), bundling of microtubules with MAPs bound to the
surface is strongly reduced. At the total MAP concentration of C4S protein ( 100% MAPs )
centrifugation at 400,000 g for one hour has only little effect in terms of interference effects in
the scattering pattern. At reduced MAP concentrations, bundling of microtubules becomes
detectable at centrifugal forces in the range below 400,000 g. The extent of bundling depends on
the duration and the intensity of centrifugation, the exact relationship of these factors still remains
to be elucidated. Furthermore, we found that the influence of MAPs depends on the degree of
phosphorylation, as evidenced by the measurements with partially phosphorylated MAP2:
dephosphorylation of MAP2 enhances the formation of microtubule bundles. It has been shown
that phosphorylation of specific sites in the microtubule binding domain of MAP2 leads to the
dissociation from microtubules (Illenberger et al., 1996). More generally, there is some evidence
that phosphorylation decreases the affinity of MAP2 to microtubules (e.g., Burns & Islam, 1984),
although there is no simple relationship between the extent of phosphorylation and microtubule
binding (Brugg & Matus, 1991). Thus, the effect of MAP2 dephosphorylation on the packing of
microtubules is probably not only a consequence of reduced binding to microtubules but rather
reflects a change in the conformation and the net charge of the projection domain.
How can the inhibitory effect of MAPs on the packing of microtubules be explained ? First, it
should be noted that it cannot be the effect of the sheer volume, since there is enough space
between the microtubules to accommodate the MAPs, even in the tight bundles we observed with
purified microtubules. This is in accordance with the observation that the presence of MAPs per
se, without attachment to microtubules, does not affect the specific volume of microtubule pellets
(Brown & Berlin, 1985). Then, two essentially different ways of action can easily be imagined
that could account for the inhibitory effect of MAPs on the packing of microtubules. (1) MAPs
could prevent reorientation and alignment of microtubules into parallel arrays. (2) MAPs could
act as spacers between microtubules that keep aligned microtubules at large distance to each
other. The former mechanism would imply that MAPs can crosslink microtubules and, thus, can
suppress the isotropic-nematic phase transition expected for rod-like molecules. On the basis of
our small-angle scattering data this cannot be excluded . However, measurements of the
birefringence of tubulin solutions under conditions of microtubule self-assembly revealed that in
the presence of MAPs spontaneous orientation occurs simultaneously with polymerization, while
in the case of purified tubulin, formation of ordered regions takes hours unless it is assisted by
flow-induced orientation (Somers & Engelborghs, 1990). This was explained by more
pronounced repulsive forces between microtubules that are covered with MAPs. Thus, from the
kinetics of spontaneous orientation of microtubules the second mechanism mentioned above
seems more likely. It is also compatible with our results if we assume that the paracrystalline
order of parallel microtubules rapidly approaches zero when the average distance is enlarged by
the presence of MAPs that act as spacers between microtubules. However, the results of our
measurements also show that these spacers are not rigid, but can be compressed or even
overcome by application of sufficiently high centrifugal forces.
15
Conclusion
Small-angle X-ray scattering of microtubule pellets seems to be a promising tool for studying
physical properties of microtubules and microtubule bundles, as well as their modulation by
MAPs and other factors of physiological significance. The model of finite, hexagonal Hosemann
paracrystals in two dimensions proves to be a useful concept for the analysis of the small-angle
scattering patterns. A systematic exploration of the three-dimensional parameter space of this
model has still to be done. It would be important for a careful assessment of the limits of the
method. Even so, it was possible to obtain positive information about the effect of MAPs on the
bundling of microtubules. Pure microtubules are easily packed into tight bundles by moderate
centrifugation. In the presence of MAPs, packing of microtubules is severely hindered. At
increasing concentrations of MAPs, higher and higher centrifugal forces and longer times of
centrifugation are required to overcome the resistance of the MAPs.
Acknowledgments
We thank M. Koch and the staff of the EMBL Outstation in Hamburg for making small-angle Xray scattering instrument X33 available and for numerous helpful discussions.
16
Appendix A
Here we describe the calculation of the interparticle interference function IIF according to the
Hosemann paracrystal model. We use an adaptation of the Hosemann paracrystal concept to the
two-dimensional, hexagonal case, that is analogous to that given by Briki et al. (1998). Since
Briki et al. refer to unpublished results and the formulas they derived differ from ours in several
points we repeat these formulas here in our notation and sketch the way we used to arrive at these
formulas. For the general background of the Hosemann paracrystal concept we refer to the
literature cited in the main text.
The interference function IIF( s ) is the Fourier transform of the probability density z(r) of
finding some point of the paracrystalline lattice at position r, given that one point of the
paracrystal is positioned at the origin (Guinier, 1963). In the following we use the term "atom"
for the lattice points of the paracrystalline lattice in order to avoid confusion. The atoms are
referred to the lattice points of a perfect crystal of the underlying symmetry, in our case a twodimensional, infinite crystal of hexagonal symmetry. Since the position of the atoms are not
known exactly, they are described by probability distributions with densities hi( r ) with index i
running over all atoms. Thus, z(r) = Σ hi( r ) where the sum runs over all atoms. In the
Hosemann paracrystal concept, the distributions hi are all constructed from the next-neighbordistributions ha and hb along the axes that span the crystal lattice (in two-dimensions): the
distributions hi are multiple convolutions of the next-neighbor-distributions ha and hb (see
references). With the convolution theorem of the Fourier transform this leads in the expression of
IIF( s ) to multiple products of the transforms Ha( s ) and Hb( s ) of the distributions ha and hb.
We assume that the distributions ha and hb are rotationally symmetric, two-dimensional Gaussian
distributions, which are identical except that they are centered at the ends of the lattice vectors a
and b , respectively. Thus,
 −r2 
1


ha( r + a ) = hb( r + b ) =
exp
2 
2π σ 2
 2σ 
and
(
(
Ha ( s ) = exp − 2 π 2 σ 2 s 2
Hb ( s ) = exp − 2 π 2 σ 2 s 2
) exp ( − 2 π i
) exp ( − 2 π i
s⋅a )
s⋅b )
where σ = σx = σy denotes the standard deviation of the marginal distributions along x and y of
ha and hb (x, y: Cartesian coordinates).
As Briki et al. (1998) pointed out, the usual procedure that is applicable for rectangular lattices
would destroy the hexagonal symmetry of the paracrystal. This is demonstrated in Fig. 7 a. The
distribution of the atom with reference lattice point a + b would be the convolution of ha and hb,
which is broader than the next-neighbor distributions (except in the singular case where ha and hb
degenerate to δ-functions, which corresponds to the perfect hexagonal lattice). This is in conflict
with the assumption of hexagonal symmetry since lattice vector a + b is symmetrically
equivalent to a and b , and therefore, the atoms referenced by a , b , and a + b should have
equivalent distributions.
17
To retain the hexagonal symmetry we define six basic lattice vectors, a1 = a , a2 = a + b ,
a3 = b , a4 = -a1 , a5 = -a2 , a6 = -a3 (Fig. 7 b), and the corresponding next-neighbor
distributions for these directions, all equivalent to ha and hb . Then, in calculating IIF we sum
the contributions of the atoms of each of the six sectors defined by ak and ak+1 ( k = 1, ... 6; with
a7 = a1 ) separately. The procedure is analogous to the usual one except that the distributions of
the atoms belonging to any one of the sectors are calculated from the next-neighbor distributions
corresponding to basic lattice vectors that define this sector. The result is:
 H3
H2 
H3 
H*
 H1
 H2
1
IFF ( s ) = 1 + 2 Re 
 + 2 Re 
 + 2 Re 
−
−
−
−
−
1
H
1
H
1
H
1
H
1
H
1
2 
2
3 
3 1− H *


1


+


 H1 
 H2 
 H3 
+ 2 Re 

 + 2 Re 
 + 2 Re 
 1− H 1 
 1− H 2 
 1− H 3 
with
(
Hk ( s ) = exp − 2 π 2 σ 2 s 2
) exp ( − 2 π i
s ⋅ ak )
( k = 1, 2 , 3 ) .
These formulas are similar in structure to those given by Briki et al., but differ in some points,
part of them seem to be simple typos: (i) The terms corresponding to the Hk 's appear in different
combinations. This is probably the consequence of a different grouping of the lattice points
chosen by Briki and coworkers. It should be noted that their functions H1a , H1b , H1c are
referring to the directions a , b , and c = a5 . (ii) Our expression for IIF contains the unity,
corresponding to the δ-function in z( r ) that describes the distribution of the atom at the origin.
(iii) The phase factors appear without the obligatory imaginary unit in the formulas of Briki et al.,
which is simply a typo. (iv) In the exponential defining the modulus of the H-functions, an
additional factor 2/3 appears. This has the effect of rescaling the standard deviation describing the
distribution of next-neighbors by ( 2/3 )1/2 .
18
Figure Legends
Figure 1
Small angle X-ray scattering of microtubules and microtubule-pellets.
Top row: Solutions scattering of Microtubules.
(a) representation of a single microtubule as hollow cylinder of infinite length with inner and
outer radius ri and ro. (b) theoretical scattering intensity Imt multiplied by s for a large number of
hollow cylinders (ri = 9.5 nm, ro = 14 nm) at random orientation and position. (c) experimental
scattering intensity Isol multiplied by s of a solution of microtubules polymerized from a mixture
of 80% PC tubulin and 20% C4S protein (tubulin with MAPs).
Center row: Interference functions.
(d), (e) Two-dimensional arrays of points representing the traces of the microtubule-axes in three
finite bundles of microtubules perpendicular to the plane and at independent, random orientations
and positions within the plane;
(d) arrays of finite size with perfect hexagonal symmetry corresponding to regular bundles of 19
microtubules; (e) illustration of finite paracrystals in two dimensions according to Hosemann
corresponding to disordered, hexagonal bundles of microtubules. (f), (g) Rotationally averaged
interference functions IIFav corresponding to the models of microtubule bundles illustrated in
(d) and (e). (h) Interference function calculated as the ratio of the experimental scattering patterns
shown in (j) and (c).
Bottom row: Scattering patterns of microtubule-pellets
(i) Scattering pattern of a large number of bundles of microtubules with independent, random
orientations and positions according to model (e), calculated by multiplication of the functions
shown in (b) and (g). (j) Scattering pattern of a pellet of microtubules polymerized from a
solution of PC tubulin and sedimented by centrifugation at 400,000 g for 10 min.
Figure 2
Electron microscopy and optical diffraction of packed microtubules.
(a) Thin-section cryo-electron microscopy image of a microtubule pellet obtained by moderate
centrifugation at 80.000 g. Microtubules were polymerized from microtubule protein with MAPs
and treated by trypsin for proteolytic cleavage of the projection domains prior to centrifugation.
The selected area shows part of a large bundle of parallel microtubules cut perpendicularly to
their long axes. (b) Position of the microtubules marked by the center points of their crosssections. (c) Convolution of the image in (b) with a circle representing the cross-section of a
microtubule. This can be considered as the "filtered image" of the micrograph shown in (a) with
the background of protein not belonging to the bundle of parallel microtubules removed. (d), (e),
19
and (f) optical diffraction images corresponding (a), (b), and (c), respectively. The microtubules
of the large bundle are not evenly distributed, but clustered in quasi-hexagonal, paracrystalline
regions of random size and orientiation. The ring of high intensity in the optical diffraction of (b)
at s = 0.032 nm-1 corresponds to the first maximum of the interference function of quasihexagonal arrays with next-neighbor distance of about 36 nm. In the diffraction pattern of the
micrograph (a) the intensity at the position of the first interference maximum is largely reduced
due to the single-particle structure factor that has a minimum in this region. This effect is much
more pronounced in the diffraction pattern of the filtered image (c) as seen by the dark ring at
0.039 nm-1 in (f).
Figure 3
Effect of the time of centrifugation on the bundling of microtubules.
(a) Scattering patterns, s I( s ) vs. s, of two pellets obtained by 5 and 10 min centrifugation at
400,000 g of a solution of microtubules polymerized from purified tubulin ( PC-T) , shown with
the pattern of a dilute solution of PC-T microtubules (sol). Interference of the microtubule
scattering indicating alignment and packing of microtubules leads to the depression of the central
scattering and to a shift and sharpening of the of maxima at about 0.5 nm-1 and 0.9 nm-1.
Pelleting of PC-T microtubules is fast and efficient compared to C4S microtubules.
(b) The same as in (a) for pellets of microtubules polymerized in the presence of MAPs ( C4S
microtubules ), centrifuged at 400,000 g for times between 10 min and 60 min. Interference
effects, even after prolonged centrifugation, are less pronounced, compared to PC-T pellets. For
30, 45, and 60 min of centrifugation two scattering patterns recorded from different samples are
shown.
Figure 4
Effect of the concentration of MAPs on the bundling of microtubules.
Scattering patterns, s I( s ) vs. s, of pellets of microtubules polymerized from tubulin solutions
with variant amounts of MAPs. The tubulin solutions were mixtures of PC tubulin and C4S
protein at different proportions, such that the percentages indicated are the relative concentrations
of MAPs compared to the maximum concentration present in undiluted C4S protein
(corresponding to 100% ). All pellets were obtained by 10 min of centrifugation at 400,000 g.
Scattering patterns of dilute solutions of pure PC-T and C4S microtubules are shown in order to
facilitate recognition of the interference effects.
Inset: normalized near-center-scattering intensity as a function of the C4S concentration (for
details see legend to Fig. 5); diamonds correspond to the pellet scattering patterns, circles to
solution scattering patterns. The white circles correspond to solution scattering patterns not
shown in the main part of this figure.
20
Figure 5
Effect of varying centrifugal force on the packing of microtubules in the presence of MAPs.
Icen / Iiif : normalized near-center-scattering intensity of microtubule solutions ( circles ) and
microtubule pellets ( diamonds ) as measured by the ratio of the intensities at s = 0.0175 nm-1
( Icen ) close to the beamstop, and s = 0.058 nm-1 ( Iiif ) at the maximum of the interference
pattern of well-packed PC-T pellets ( Fig. 3 a ). The percentages indicated are the relative
concentrations of MAPs compared to the maximum concentration present in undiluted C4S
protein. All pellets were obtained by 10 min of centrifugation with the exception of the 75,000 g
pellets with 20% and 30% C4S, which have been centrifuged for 20 min and 90 min ( two pellets
for each concentration ). The 400,000 g value in the top panel ( white diamond ) corresponds to
the 12% C4S pellet of the C4S concentration series shown in Figure 4. With increasing MAP
concentration, the transition from a solution-like scattering pattern to the interference pattern of
packed microtubules requires increasing centrifugal forces.
Figure 6
Effect of phosphorylation of MAP2 on the bundling of microtubules.
Scattering patterns, s I( s ) vs. s, of pellets of microtubules polymerized from tubulin solutions
mixed with three different preparations of MAP2. "MAP2total": endogenously phosphorylated
MAP2 as separated from the MAPs co-purifying with C4S protein. "MAP2fr2": one of two
fraction of the "MAP2" preparation that is less phosphorylated than the other. "MAP2dephos":
dephosphorylated MAP2 obtained by treatment with alkaline phosphatase. All pellets were
obtained by 10 min of centrifugation at 400,000 g. The amount of MAP2 in the tubulin solutions
was 20-wt%, comparable to the total amount of MAPs in C4S protein. The inhibitory effect of
MAP2 on the bundling of microtubules increases with the degree of phosphorylation.
Figure 7
(Appendix A)
Probability distributions of some atoms around the center.
(a) Using only next-neighbor-distributions at a and b to construct all other distributions leads to
loss of hexagonal symmetry.
(b) To preserve the symmetry, the distributions at a1 to a6 are used equivalently as basic
distributions; the distributions of the other atoms are calculated separately for each sector
(defined by a1 to a6, dashed lines) by using the corresponding pair of basic distributions.
21
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Wille, H., Mandelkow, E.-M., Dingus, J., Vallee, R.B., Binder, L.I., Mandelkow, E. (1992)
Domain structure and antiparallel dimers of microtubule-associated protein 2 (MAP2). Journal of
Structural Biology 108, 49-61
Wulff, A., De Rocco, A.G. (1971) Statistical mechanics for long semiflexible molecules: a model
for the nematic mesophase. J. Chem. Phys. 55, 12-27
Figure 1
ro
ri
e
d
a
s Imt / [arb. units]
IIFav
IIFav
s I / [arb. units]
0.00
0
0
1
2
3
4
5
0.00
0.00
0
1
2
3
4
5
0
10
0.05
0.05
11
0.05
20
0.10
0.10
30
31
s / [nm-1]
0.10
s / [nm-1]
21
s / [nm-1]
40
0.15
0.15
0.15
i
g
f
b
0.20
0.20
0.20
s Isol / [arb. units]
IIFpellet
s Ipellet / [arb. units]
0
0
1
2
0
0.00
0.00
0.00
0.05
0.05
0.05
0.10
s / [nm-1]
0.10
s / [nm-1]
0.10
s / [nm-1]
0.15
0.15
0.15
j
h
c
0.20
0.20
0.20
Figure 2
e
d
500 nm
b
a
f
c
5
s I / [arb. units]
a
3
2
sol
1
5'
0
10'
0.00
b
PC-T
4
0.05
0.10
0.15
s I / [arb. units]
8
0.20
0.25
C4S
6
sol
10'
4
20'
30'
2
45'
60'
0
0.00
0.05
0.10
0.15
0.20
0.25
s / [nm-1]
Figure 3
20
Icen / Iiif
16
14
10
0
0
s I / [arb. units]
12
50
% C4S
100
PC-T solution
10
PC-T
8
12%
25%
6
33%
4
38%
50%
78%
C4S
2
C4S solution
0
0.00
0.05
0.10
0.15
0.20
0.25
s / [nm-1]
Figure 4
10 % C4S
10
Icen / Iiif
0
20 % C4S
10
0
25 % C4S
10
0
30 % C4S
10
0
0
100
200
300
400
g-value / 1000
Figure 5
s I / [arb. units]
5
4
3
2
MAP2total
1
MAP2fr2
0
MAP2dephos
0.00
0.05
0.10
0.15
0.20
0.25
s / [nm-1]
Figure 6
a
b
a
b
a3
a4
a5
a2
a1
a6
Figure 7