- Email: [email protected]
An algorithm for straightening images of curved filamentous structures
Ultramicroscopy North-Holland,
367
19 (1986) 367-374 Amsterdam
AN ALGORITHM FOR STRAIGHTENING IMAGES OF CURVED FILAMENTOUS STRUCTURES E.H. EGELMAN Department Received
of MolecularBiophysics and Biochemistry,
23 September
1985; received
Yale University, New’ Haven, Connecticut
0651 I, USA
in final form 20 May 1986
Flexible filamentous structures are characterized by variable curvature in electron micrographs. direct application of three-dimensional reconstruction methods to helical filaments. An algorithm of such filaments is presented based upon an assumption of a normal mode of bending.
1. Introduction Every real structure, from a collagen fibril to the Empire State Building, has a finite flexibility. This flexibility is most frequently quantified in biology in terms of a correlation or persistence length [1,2], which has the dimensions of length (see section 2). A structure is said to be rigid when the correlation length is many times longer than the segment of interest. Conversely, a flexible structure is one whose correlation length is of comparable magnitude or smaller than the segment of interest. Flexibility manifests itself in electron micrographs of filamentous structures by variable curvature along a filament. Because these variations in curvature are statistical, one may often search through a field of filaments to find regions which can be well approximated as straight rigid rods. For instance, F-actin has a correlation length of about 60,000 A [3], and it is therefore quite easy to find regions of about 2000 A in length which are sufficiently straight to be suitable for three-dimensional helical reconstruction [4]. On the other hand, even large populations of more flexible polymers may not yield enough specimens which are sufficiently straight to be candidates for further analysis. It is shown in this paper that if the curvature of the structure being examined can be approximated by a normal mode of bending, the bending which 0304-3991/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
This frequently prevents the for straightening the images
is observed may be easily corrected, and the structure reconstructed as if it were a rigid rod. This method has been used in the three-dimensional reconstructions of the very flexible helical polymer formed by the E. coli recA protein [5,6]. Since the structure of the recA filament has not been previously determined, the efficacy of this method cannot be proved by its application to those filaments. Further, recA filaments are so flexible that it is virtually impossible to find enough fortuitously straight segments to compare the naturally straight filaments with those which have been computationally straightened. Therefore, relatively rigid Tobacco Mosaic Virus specimens are used in this paper as a test object for the correction of bending.
2. Theory Flexibility may be parameterized in terms of either a characterististic resistance to bending, a (with dimensions erg cm) or in terms of a correlation length, b (with dimensions of cm). The two are related by: b = a/kT.
The correlation length may be defined in several different ways. One method involves the change in angle of the filament axis. If the angle between B.V.
368
E. H. Egelman / Algorithm for stratghtening
two tangents to a filament, separated by a contour length L, is +, the following statistical relationship will hold [l]: (cos +) = exp( - L/b),
dz’ = (1 + x/R)
dz,
where x is the coordinate of the point at which dz’ is located. The relative extension or compression is therefore: (dz’ - dz)/dz
where h is the correlation length. The axial repeat in a helix, the resolution desired, and the amount of noise present all determine the minimum length of a filament needed for a helical reconstruction. When this length is of a magnitude comparable to the correlation length, the curvature of the filament usually prevents such a reconstruction from being made. It is important to realize that significant curvature can still occur when the correlation length is four or five times greater than the lengths of interest. When one examines filament segments which are one quarter of the correlation length the characteristic deviations in angle between the two ends of such a segment will be of the order of 40’ (for a rigid rod the angular deviation would be O’, while a completely random coil would have an expectation value of 900). It is necessary to consider how a protein filament is being deformed by the random bending which it is undergoing. The simplest manner is that of the normal mode of a homogeneous rod. In a normal mode of bending a rod is stretched at some points and compressed at others. Lines on the inside surface of the bent rod are compressed, while those along the outside surface are extended. Between the region of compression and extension there is a neutral surface, which undergoes neither extension nor compression. An elastic homogeneous rod undergoes a normal mode of bending when the amount of deformation which occurs is below the elastic limit. A key element in the normal mode description is that the cross-sectional area taken perpendicular to the filament axis remains invariant during the bending. Further, rotations of that cross-section are precluded. We will consider a rod in the xz plane, where the z-axis is parallel to the axis of the undeformed rod [7]. A length element dz along the rod becomes dz’. For an element on the neutral surface, dz = dz’. If R is the radius of curvature of a bend, then: (1)
mages
= x/R
(2)
and can be seen to be linearly related to the distance from the neutral surface. It can be shown that the neutral surface must pass through the centers of mass of the cross-sections of the rod [7]. If a structure is undergoing a normal mode of bending, the projection of that structure in the plane of bending can be straightened by means of a transformation from a curvilinear coordinate system attached to the filament to a Cartesian coordinate system. Such a transformation will expand area elements which were previously on the inside of a bend and compress those which were on the outside, with the amount of extension or compression linearly related to the distance of the element from the neutral surface, thereby correcting for the effects of the bending. The resulting projection of the transformed filament will thus be identical to that of an undeformed straight filament. Although the real bending of a protein polymer, a structure which is neither homogeneous nor isotropic, will be more complicated than the description just given, it is reasonable to believe that the normal mode analysis may provide a first-order approximation for small amounts of bending.
3. Method An electron micrograph is digitized using a scanning densitometer and displayed on a rastergraphics device. The image is intentionally oversampled with respect to the intrinsic resolution of the image by at least a factor of two (if the image has useful detail extending out to 20 A resolution, the image is scanned so that the resulting raster is at most 5 A/pixel). It is first necessary to define the intersection of the neutral surface with the plane of bending. This line will be the filament axis, given the constraint that the neutral surface must pass through the centers of mass of every cross-section. If the filament axis were given only by a collection of user-defined points connected in
E. H. Egelman
/ Algorithm for straightening
a continuous way by linear segments, the resulting curve would not be continuously differentiable and the coordinate transformation could not be made. Therefore, a cubic spline function, whose first and second derivative is continuous [8], is statistically fit to an ensemble of user-defined points which are taken along the filament axis. The cubic spline function is: f(z)
=a,+u,z+a*Z~+a,z’+
Cb,(z-
?z,,:, (3)
with (L_fl,)+=
{;-nI
;I:
;:I/> ’
J'
where t is the axial position along the undeformed filament, f(z) defines the transverse location of the deformed filament axis in the xz plane, and the n,‘s are the knots of the spline. The coefficients and knots are determined by minimizing a residual R which contains two components: (a) the discontinuities of the third derivatives of the spline function; (b) the error between the user-defined points and the spline function at those points:
369
rmuges
Computation time is minimal for a reasonable number of points (of the order of 2 s on a VAX 11/750 for 20 user-defined points over a filament length of about 400 pixels) and this procedure is performed interactively: the resulting spline curve is displayed superimposed on the filament (fig. 2) and the positions of the experimental points as well as their weights may be varied until the desired spline curve is achieved. Once the spline function is determined, the image can be re-sampled in the new curvilinear coordinate system determined by the spline. Let us describe the original array of discretely sampled optical densities (containing the curved filawhere the filament axis runs apment) by O,.,, proximately in j (along constant i). The continuous spline function is used to generate a value +, and d, for each value of j (the raster frame), where + is the angle of the tangent to the filament
.
.
.
.
.
.
.
.
.
.
.
flO.iy, 1,’
/
I
j'+l
i’=-v, .
lg+(n,)-g-b,)\ +
R= :
,j= I
&(x,-f(z,))‘l
c .
.
.
.
i+l,
j+l .
,:I
1-l
\
(4) with
‘1’
;
i’=-1,
j'
I
.
1,
.
.dj-/
.
. i
g( n,) = d3f(n,)/dz’,
X.
/I L.i
. .
‘,
and where z,, x, and w, are the N user-defined points for the filament axis and their associated weights, t is an arbitrary weighting factor, the n,‘s are the locations of the M knots, and g+(n,) and g_(n,) indicate the third derivative determined by approaching n, from the positive and negative sides, respectively. Obviously, by increasing the number of knots the data points may be fit exactly, but at the expense of increasing the sum of the third derivative discontinuities at the knots. The weighting factor t controls the smoothness of the curve versus the goodness of fit to the experimental points. The Harwell method is used for this procedure.
/I’
.
.
/
.
\
. //i
.
.
.
.
.
.
.
Fig. 1. The discrete raster points i, j on which the original image is sampled are indicated by the filled squares. The continuous spline function (approximating the axis of a curved filament) is shown by the curve which runs through this lattice. A discrete grid i’, j’ is affixed to this curve along lines which are normal to the curve. Points on this new grid, indicated by closed circles, will have real values X, J in the old raster frame. The angle between the tangent to the curve and the vertical axis at the grid level i is indicated by O,, while the lateral displacement of the curve at the grid level 1 is indicated by d,.
370
E.H. Egelmun
/ Algorithm
and d is the lateral position of the filament axis. The coordinate transformation is done by evaluating the density along j’ (the axial distance along the curved filament, at discrete integer points) which will be slightly greater than j because of path length differences due to curvature. For every discrete point i’, j’ (where i’ = 0 corresponds to the filament axis) along the curved filament, the continuous coordinates X, y in the original raster frame are determined by a coordinate rotation through an angle +L, with the center of the rotation being on the filament axis. Fig. 1 shows the geometry of this rotation:
(5a)
x=d,+i’cos(&),
y=
[i,cos(&)]
-i’
(5b)
sin($,),
where: k=NearestInteger
[i,cos(,,j
This definition for k involves a negligible error when the sum is rounded-off to an integer. since both the curvature and the rate of change of curvature are very small with respect to the integer raster. The coordinates x, y will not in general fall on the original discrete sample points, i, j. Therefore. the optical density 0(x, y) at real point x, y on the original raster is calculated from the density Oi, on the discrete samples i, ,j using a bilinear interpolation: 0(x.
y)
= [(I -Anx)O,,,+Ax +[(I
-Ado,.,+,
O,,,.,] t-Ax
(1 -A.v) O,,,.,,,]
AY.
(6)
where i and j are the nearest integers smaller than x and y, respectively, and Ax = x - i, A y = J =,j. This density is now used to generate a straightened image on the new raster, i’, j’: O,‘,~,, = b(i’, The weighting to compensate is reciprocally
j’)
0(x.
y).
(7)
factor h(i’, j’) can be introduced for extension and compression and related to the average area of the
for stmghtemng
rmuges
four trapezoids which share the vertex i’, j’. However, for negatively stained images the assumption can be made that there are only two densities present, that of protein and that of heavy metalstain. and compression and extension will not affect those densities. In this case, the weighting factor is set to 1 for all points.
4. Application Tobaco Mosaic Virus (TMV) is a rod-like structure approximately 200 r\ in diameter and 3000 A in length. It appears in general to be quite rigid, but one may search though a large field of viruses to find those which display significant curvature. Statistical arguments may be used to show the probability of finding a given amount of curvature within a population of rigid viruses, but this would require a knowledge of the rigidity of TMV, which has not been measured. Nevertheless, order-ofmagnitude calculations suggest that the number of curved viruses which are found is consistent with a homogeneous population of viruses with energies of bending populating a Maxwell-Boltzmann distribution. One may assume that all the viruses are flexing by small amounts in time, and the electron micrograph has simply given us one “frozen” instant in time. It is reasonable, therefore. to assume that the curved filaments have the same structure as the nominally straight filaments, and do not represent an artefact of specimen preparation. Fig. 2 shows the application of this procedure to two curved viruses, while fig. 3 displays a comparison between the computed Fourier transforms of the curved and straightened viruses. Whereas the first TMV layer line at l/69 A (which arises from the 16-start helices in TMV) is non-existent or very weak in the transforms of the curved specimens, a clear layer line is seen after straightening. The magnitude of the improvement may be seen quantitatively in fig. 4. where the layer line intensities before and after straightening are compared. The strong maximum on the first layer line is at a radial spacing of about l/36 .&. which means that this feature is about l/30 A from the origin. It therefore is reasonable to con-
E.H. Egelman
Fig. 2. Two curved TMV specimens location of the filament axis, and discontinuous due to the finite raster all given a weight of 1. A user may better to the perceived filament axis.
/ Algorrrhm for strarghtening images
371
are shown in (A) and (C). Data points (12 in (A) and 11 in (C)) were generated to mark the these are shown by small crosses in (A) and (C). The resulting spline function appears size (the continuous function is drawn at the nearest raster point). The starting data points are then change weights of points and/or their positions to force the spline function to conform After correcting for normal mode bending, the resulting straightened viruses are shown in (B) and (D).
elude that for this system, at least, the model for bending presented above is useful at 30 A resolution in recovering a structural detail which would
be lost due to curvature. Unfortunately, due to a lack of resolution, layer lines arising from different helical families are in
312
E. H. Egelmun
/ Algorithm for strarghtenmg
rmqqes
Fig 3. (A)-(D) are the computed Fourier transforms of figs. 2A-2D. respectively. The l/69 P\ layer line (of Bessel order 16) is indl icated by the arrows in (B) and (D). The transform in (B) is quite noisy. yet a symmetrical layer line is seen which is absent in (P i). The : transform in (D) is less noisy, but the layer line is somewhat one-sided. This one-sidedness is common in transforms of straig ,ht TM V particles. and is due to differential staining between the top and bottom surfaces. The meridional intensity in both (B) and (I1) whi ch occurs at about layer line 1.5 arises from random perturbations across the one-start helix. and these perturbations have per iod of every other turn of this helix. This was first described for the Dahlemense strain of TMV by Caspar and Holmes [12], but It was noted that low pH can induce similar perturbations in wild-type. Since uranyl acetate stain ia at about pH 4, these observatio n\ are consistent with what has been previously described.
general not present in our images of TMV (even in those specimens which are fortuitously quite straight). It is therefore not possible to show with TMV that the straightening preserved the full helical symmetry. On the other hand, in the application of this method to recA filaments [5,6] several different helical families are seen in straightened images, whereas the intrinsic curvature presents one from recovering virtually any useful data from
uncorrected images. However, the absence of fortuitously straight filaments prevents a direct demonstration of the utility of this method in that application. This method has also been applied to the paired helical filaments associated with Alzheimer’s disease, and has been useful in recovering detail which would not otherwise be obtainable from these flexible filaments [9,10].
E. H. Egelman
/ Algorithm
for straightening images small
amount
of curvature
373
along
a thin
filament
(R/x is large, where x is the distance from the filament axis and R is the radius of curvature). At some level of resolution all of these assumptions must break down for a real structure. However, if these assumptions are approximately valid, the low resolution structure of the filament can be recovered by this method. Applications of this method to TMV, recA filaments [5,6], and the paired helical filaments of Alzheimer’s disease [9,10] support the notion that the normal mode of bending approximation is useful at low resolution.
Acknowledgements This work was initiated while the author was a NATO Fellow at the MRC Laboratory of Molecular Biology, Cambridge, UK. I would like to thank Dr. Andrzej Stasiak, ETH, Ziirich, for the use of his electron micrographs of TMV. It has come to my attention while revising this paper that a similar method for straightening images of flexible filaments has been described in abstract form by Steven et al. [ll].
Fig. 4. A comparison is made between the first layer line intensity of curved versus straightened TMV particles. (A) is the first layer line from a carefully selected naturally straight TMV specimen, and is used as a control since this particular specimen appeared to be the best preserved of all those examined. The dashed curves in (B) and (C) are from the transforms in figs. 3A and 3C, respectively, and correspond to the curved filaments of figs. 2A and 2C. The solid curves in (B) and (C) are from the transforms of figs. 3B and 3D, and correspond to the straightened filaments of figs. 2B and 2D. All layer line intensities in this figure are an average of the near plus far sides, after searching for the position of the filament axis which minimizes the near-far phase residual
References VI L.D. Landau 14 [31 [41 [51 [61 (71 PI
5. Discussion
and conclusion [91
It is important to stress the assumptions in this algorithm, since these ultimately determine the limits of applicability of the method. First, a normal mode of bending for a homogeneous filament is assumed. This means that torsion is not taking place, and that there is no differential compression or extension due to inhomogeneities in the structure. Second, it is assumed that there is a
UOI [ill
WI
and E.M. Lifshitz, Statistical Physics (Pergamon, Oxford, 1958). CR. Cantor and P.R. Schimmel, Biophysical Chemistry III (Freeman, San Francisco, 1980). F. Oosawa, Biophys. Chem. 11 (1980) 443. J. Trinick, J. Cooper, J. Seymour and E.H. Egelman, J. Microscopy 141 (1986) 349. E.H. Egelman and A. Stasiak, J. Mol. Biol., in press. A. Stasiak and E.H. Egelman, Biophys. J. 49 (1986) 5. L.D. Landau and E.M. Lifshitz, Mechanics (Pergamon. Oxford, 1959). J.H. Ahlberg, E.N. Nilson and J.L. Walsh, The Theory of Splines and Their Applications (Academic Press, New York, 1967). R.A. Crowther, CM. Wischik and M. Stewart, in: Proc. 43rd Annual EMSA Meeting, Louisville, KY, 1985, Ed. G.W. Bailey (San Francisco Press, San Francisco, 1985). R.A. Crowther and CM. Wischik, EMBO J. 4 (1985) 3661. A.C. Steven, R. Stall, P.M. Steinert and B.L. Trus, in: Proc. 43rd Annual EMSA Meeting, Louisville, KY, 1985, Ed. G.W. Bailey (San Francisco Press, San Francisco, 1985). D.L.D. Caspar and K.C. Holmes, J. Mol. Biol. 46 (1969) 99.
367
19 (1986) 367-374 Amsterdam
AN ALGORITHM FOR STRAIGHTENING IMAGES OF CURVED FILAMENTOUS STRUCTURES E.H. EGELMAN Department Received
of MolecularBiophysics and Biochemistry,
23 September
1985; received
Yale University, New’ Haven, Connecticut
0651 I, USA
in final form 20 May 1986
Flexible filamentous structures are characterized by variable curvature in electron micrographs. direct application of three-dimensional reconstruction methods to helical filaments. An algorithm of such filaments is presented based upon an assumption of a normal mode of bending.
1. Introduction Every real structure, from a collagen fibril to the Empire State Building, has a finite flexibility. This flexibility is most frequently quantified in biology in terms of a correlation or persistence length [1,2], which has the dimensions of length (see section 2). A structure is said to be rigid when the correlation length is many times longer than the segment of interest. Conversely, a flexible structure is one whose correlation length is of comparable magnitude or smaller than the segment of interest. Flexibility manifests itself in electron micrographs of filamentous structures by variable curvature along a filament. Because these variations in curvature are statistical, one may often search through a field of filaments to find regions which can be well approximated as straight rigid rods. For instance, F-actin has a correlation length of about 60,000 A [3], and it is therefore quite easy to find regions of about 2000 A in length which are sufficiently straight to be suitable for three-dimensional helical reconstruction [4]. On the other hand, even large populations of more flexible polymers may not yield enough specimens which are sufficiently straight to be candidates for further analysis. It is shown in this paper that if the curvature of the structure being examined can be approximated by a normal mode of bending, the bending which 0304-3991/86/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
This frequently prevents the for straightening the images
is observed may be easily corrected, and the structure reconstructed as if it were a rigid rod. This method has been used in the three-dimensional reconstructions of the very flexible helical polymer formed by the E. coli recA protein [5,6]. Since the structure of the recA filament has not been previously determined, the efficacy of this method cannot be proved by its application to those filaments. Further, recA filaments are so flexible that it is virtually impossible to find enough fortuitously straight segments to compare the naturally straight filaments with those which have been computationally straightened. Therefore, relatively rigid Tobacco Mosaic Virus specimens are used in this paper as a test object for the correction of bending.
2. Theory Flexibility may be parameterized in terms of either a characterististic resistance to bending, a (with dimensions erg cm) or in terms of a correlation length, b (with dimensions of cm). The two are related by: b = a/kT.
The correlation length may be defined in several different ways. One method involves the change in angle of the filament axis. If the angle between B.V.
368
E. H. Egelman / Algorithm for stratghtening
two tangents to a filament, separated by a contour length L, is +, the following statistical relationship will hold [l]: (cos +) = exp( - L/b),
dz’ = (1 + x/R)
dz,
where x is the coordinate of the point at which dz’ is located. The relative extension or compression is therefore: (dz’ - dz)/dz
where h is the correlation length. The axial repeat in a helix, the resolution desired, and the amount of noise present all determine the minimum length of a filament needed for a helical reconstruction. When this length is of a magnitude comparable to the correlation length, the curvature of the filament usually prevents such a reconstruction from being made. It is important to realize that significant curvature can still occur when the correlation length is four or five times greater than the lengths of interest. When one examines filament segments which are one quarter of the correlation length the characteristic deviations in angle between the two ends of such a segment will be of the order of 40’ (for a rigid rod the angular deviation would be O’, while a completely random coil would have an expectation value of 900). It is necessary to consider how a protein filament is being deformed by the random bending which it is undergoing. The simplest manner is that of the normal mode of a homogeneous rod. In a normal mode of bending a rod is stretched at some points and compressed at others. Lines on the inside surface of the bent rod are compressed, while those along the outside surface are extended. Between the region of compression and extension there is a neutral surface, which undergoes neither extension nor compression. An elastic homogeneous rod undergoes a normal mode of bending when the amount of deformation which occurs is below the elastic limit. A key element in the normal mode description is that the cross-sectional area taken perpendicular to the filament axis remains invariant during the bending. Further, rotations of that cross-section are precluded. We will consider a rod in the xz plane, where the z-axis is parallel to the axis of the undeformed rod [7]. A length element dz along the rod becomes dz’. For an element on the neutral surface, dz = dz’. If R is the radius of curvature of a bend, then: (1)
mages
= x/R
(2)
and can be seen to be linearly related to the distance from the neutral surface. It can be shown that the neutral surface must pass through the centers of mass of the cross-sections of the rod [7]. If a structure is undergoing a normal mode of bending, the projection of that structure in the plane of bending can be straightened by means of a transformation from a curvilinear coordinate system attached to the filament to a Cartesian coordinate system. Such a transformation will expand area elements which were previously on the inside of a bend and compress those which were on the outside, with the amount of extension or compression linearly related to the distance of the element from the neutral surface, thereby correcting for the effects of the bending. The resulting projection of the transformed filament will thus be identical to that of an undeformed straight filament. Although the real bending of a protein polymer, a structure which is neither homogeneous nor isotropic, will be more complicated than the description just given, it is reasonable to believe that the normal mode analysis may provide a first-order approximation for small amounts of bending.
3. Method An electron micrograph is digitized using a scanning densitometer and displayed on a rastergraphics device. The image is intentionally oversampled with respect to the intrinsic resolution of the image by at least a factor of two (if the image has useful detail extending out to 20 A resolution, the image is scanned so that the resulting raster is at most 5 A/pixel). It is first necessary to define the intersection of the neutral surface with the plane of bending. This line will be the filament axis, given the constraint that the neutral surface must pass through the centers of mass of every cross-section. If the filament axis were given only by a collection of user-defined points connected in
E. H. Egelman
/ Algorithm for straightening
a continuous way by linear segments, the resulting curve would not be continuously differentiable and the coordinate transformation could not be made. Therefore, a cubic spline function, whose first and second derivative is continuous [8], is statistically fit to an ensemble of user-defined points which are taken along the filament axis. The cubic spline function is: f(z)
=a,+u,z+a*Z~+a,z’+
Cb,(z-
?z,,:, (3)
with (L_fl,)+=
{;-nI
;I:
;:I/> ’
J'
where t is the axial position along the undeformed filament, f(z) defines the transverse location of the deformed filament axis in the xz plane, and the n,‘s are the knots of the spline. The coefficients and knots are determined by minimizing a residual R which contains two components: (a) the discontinuities of the third derivatives of the spline function; (b) the error between the user-defined points and the spline function at those points:
369
rmuges
Computation time is minimal for a reasonable number of points (of the order of 2 s on a VAX 11/750 for 20 user-defined points over a filament length of about 400 pixels) and this procedure is performed interactively: the resulting spline curve is displayed superimposed on the filament (fig. 2) and the positions of the experimental points as well as their weights may be varied until the desired spline curve is achieved. Once the spline function is determined, the image can be re-sampled in the new curvilinear coordinate system determined by the spline. Let us describe the original array of discretely sampled optical densities (containing the curved filawhere the filament axis runs apment) by O,.,, proximately in j (along constant i). The continuous spline function is used to generate a value +, and d, for each value of j (the raster frame), where + is the angle of the tangent to the filament
.
.
.
.
.
.
.
.
.
.
.
flO.iy, 1,’
/
I
j'+l
i’=-v, .
lg+(n,)-g-b,)\ +
R= :
,j= I
&(x,-f(z,))‘l
c .
.
.
.
i+l,
j+l .
,:I
1-l
\
(4) with
‘1’
;
i’=-1,
j'
I
.
1,
.
.dj-/
.
. i
g( n,) = d3f(n,)/dz’,
X.
/I L.i
. .
‘,
and where z,, x, and w, are the N user-defined points for the filament axis and their associated weights, t is an arbitrary weighting factor, the n,‘s are the locations of the M knots, and g+(n,) and g_(n,) indicate the third derivative determined by approaching n, from the positive and negative sides, respectively. Obviously, by increasing the number of knots the data points may be fit exactly, but at the expense of increasing the sum of the third derivative discontinuities at the knots. The weighting factor t controls the smoothness of the curve versus the goodness of fit to the experimental points. The Harwell method is used for this procedure.
/I’
.
.
/
.
\
. //i
.
.
.
.
.
.
.
Fig. 1. The discrete raster points i, j on which the original image is sampled are indicated by the filled squares. The continuous spline function (approximating the axis of a curved filament) is shown by the curve which runs through this lattice. A discrete grid i’, j’ is affixed to this curve along lines which are normal to the curve. Points on this new grid, indicated by closed circles, will have real values X, J in the old raster frame. The angle between the tangent to the curve and the vertical axis at the grid level i is indicated by O,, while the lateral displacement of the curve at the grid level 1 is indicated by d,.
370
E.H. Egelmun
/ Algorithm
and d is the lateral position of the filament axis. The coordinate transformation is done by evaluating the density along j’ (the axial distance along the curved filament, at discrete integer points) which will be slightly greater than j because of path length differences due to curvature. For every discrete point i’, j’ (where i’ = 0 corresponds to the filament axis) along the curved filament, the continuous coordinates X, y in the original raster frame are determined by a coordinate rotation through an angle +L, with the center of the rotation being on the filament axis. Fig. 1 shows the geometry of this rotation:
(5a)
x=d,+i’cos(&),
y=
[i,cos(&)]
-i’
(5b)
sin($,),
where: k=NearestInteger
[i,cos(,,j
This definition for k involves a negligible error when the sum is rounded-off to an integer. since both the curvature and the rate of change of curvature are very small with respect to the integer raster. The coordinates x, y will not in general fall on the original discrete sample points, i, j. Therefore. the optical density 0(x, y) at real point x, y on the original raster is calculated from the density Oi, on the discrete samples i, ,j using a bilinear interpolation: 0(x.
y)
= [(I -Anx)O,,,+Ax +[(I
-Ado,.,+,
O,,,.,] t-Ax
(1 -A.v) O,,,.,,,]
AY.
(6)
where i and j are the nearest integers smaller than x and y, respectively, and Ax = x - i, A y = J =,j. This density is now used to generate a straightened image on the new raster, i’, j’: O,‘,~,, = b(i’, The weighting to compensate is reciprocally
j’)
0(x.
y).
(7)
factor h(i’, j’) can be introduced for extension and compression and related to the average area of the
for stmghtemng
rmuges
four trapezoids which share the vertex i’, j’. However, for negatively stained images the assumption can be made that there are only two densities present, that of protein and that of heavy metalstain. and compression and extension will not affect those densities. In this case, the weighting factor is set to 1 for all points.
4. Application Tobaco Mosaic Virus (TMV) is a rod-like structure approximately 200 r\ in diameter and 3000 A in length. It appears in general to be quite rigid, but one may search though a large field of viruses to find those which display significant curvature. Statistical arguments may be used to show the probability of finding a given amount of curvature within a population of rigid viruses, but this would require a knowledge of the rigidity of TMV, which has not been measured. Nevertheless, order-ofmagnitude calculations suggest that the number of curved viruses which are found is consistent with a homogeneous population of viruses with energies of bending populating a Maxwell-Boltzmann distribution. One may assume that all the viruses are flexing by small amounts in time, and the electron micrograph has simply given us one “frozen” instant in time. It is reasonable, therefore. to assume that the curved filaments have the same structure as the nominally straight filaments, and do not represent an artefact of specimen preparation. Fig. 2 shows the application of this procedure to two curved viruses, while fig. 3 displays a comparison between the computed Fourier transforms of the curved and straightened viruses. Whereas the first TMV layer line at l/69 A (which arises from the 16-start helices in TMV) is non-existent or very weak in the transforms of the curved specimens, a clear layer line is seen after straightening. The magnitude of the improvement may be seen quantitatively in fig. 4. where the layer line intensities before and after straightening are compared. The strong maximum on the first layer line is at a radial spacing of about l/36 .&. which means that this feature is about l/30 A from the origin. It therefore is reasonable to con-
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Fig. 2. Two curved TMV specimens location of the filament axis, and discontinuous due to the finite raster all given a weight of 1. A user may better to the perceived filament axis.
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are shown in (A) and (C). Data points (12 in (A) and 11 in (C)) were generated to mark the these are shown by small crosses in (A) and (C). The resulting spline function appears size (the continuous function is drawn at the nearest raster point). The starting data points are then change weights of points and/or their positions to force the spline function to conform After correcting for normal mode bending, the resulting straightened viruses are shown in (B) and (D).
elude that for this system, at least, the model for bending presented above is useful at 30 A resolution in recovering a structural detail which would
be lost due to curvature. Unfortunately, due to a lack of resolution, layer lines arising from different helical families are in
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Fig 3. (A)-(D) are the computed Fourier transforms of figs. 2A-2D. respectively. The l/69 P\ layer line (of Bessel order 16) is indl icated by the arrows in (B) and (D). The transform in (B) is quite noisy. yet a symmetrical layer line is seen which is absent in (P i). The : transform in (D) is less noisy, but the layer line is somewhat one-sided. This one-sidedness is common in transforms of straig ,ht TM V particles. and is due to differential staining between the top and bottom surfaces. The meridional intensity in both (B) and (I1) whi ch occurs at about layer line 1.5 arises from random perturbations across the one-start helix. and these perturbations have per iod of every other turn of this helix. This was first described for the Dahlemense strain of TMV by Caspar and Holmes [12], but It was noted that low pH can induce similar perturbations in wild-type. Since uranyl acetate stain ia at about pH 4, these observatio n\ are consistent with what has been previously described.
general not present in our images of TMV (even in those specimens which are fortuitously quite straight). It is therefore not possible to show with TMV that the straightening preserved the full helical symmetry. On the other hand, in the application of this method to recA filaments [5,6] several different helical families are seen in straightened images, whereas the intrinsic curvature presents one from recovering virtually any useful data from
uncorrected images. However, the absence of fortuitously straight filaments prevents a direct demonstration of the utility of this method in that application. This method has also been applied to the paired helical filaments associated with Alzheimer’s disease, and has been useful in recovering detail which would not otherwise be obtainable from these flexible filaments [9,10].
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(R/x is large, where x is the distance from the filament axis and R is the radius of curvature). At some level of resolution all of these assumptions must break down for a real structure. However, if these assumptions are approximately valid, the low resolution structure of the filament can be recovered by this method. Applications of this method to TMV, recA filaments [5,6], and the paired helical filaments of Alzheimer’s disease [9,10] support the notion that the normal mode of bending approximation is useful at low resolution.
Acknowledgements This work was initiated while the author was a NATO Fellow at the MRC Laboratory of Molecular Biology, Cambridge, UK. I would like to thank Dr. Andrzej Stasiak, ETH, Ziirich, for the use of his electron micrographs of TMV. It has come to my attention while revising this paper that a similar method for straightening images of flexible filaments has been described in abstract form by Steven et al. [ll].
Fig. 4. A comparison is made between the first layer line intensity of curved versus straightened TMV particles. (A) is the first layer line from a carefully selected naturally straight TMV specimen, and is used as a control since this particular specimen appeared to be the best preserved of all those examined. The dashed curves in (B) and (C) are from the transforms in figs. 3A and 3C, respectively, and correspond to the curved filaments of figs. 2A and 2C. The solid curves in (B) and (C) are from the transforms of figs. 3B and 3D, and correspond to the straightened filaments of figs. 2B and 2D. All layer line intensities in this figure are an average of the near plus far sides, after searching for the position of the filament axis which minimizes the near-far phase residual
References VI L.D. Landau 14 [31 [41 [51 [61 (71 PI
5. Discussion
and conclusion [91
It is important to stress the assumptions in this algorithm, since these ultimately determine the limits of applicability of the method. First, a normal mode of bending for a homogeneous filament is assumed. This means that torsion is not taking place, and that there is no differential compression or extension due to inhomogeneities in the structure. Second, it is assumed that there is a
UOI [ill
WI
and E.M. Lifshitz, Statistical Physics (Pergamon, Oxford, 1958). CR. Cantor and P.R. Schimmel, Biophysical Chemistry III (Freeman, San Francisco, 1980). F. Oosawa, Biophys. Chem. 11 (1980) 443. J. Trinick, J. Cooper, J. Seymour and E.H. Egelman, J. Microscopy 141 (1986) 349. E.H. Egelman and A. Stasiak, J. Mol. Biol., in press. A. Stasiak and E.H. Egelman, Biophys. J. 49 (1986) 5. L.D. Landau and E.M. Lifshitz, Mechanics (Pergamon. Oxford, 1959). J.H. Ahlberg, E.N. Nilson and J.L. Walsh, The Theory of Splines and Their Applications (Academic Press, New York, 1967). R.A. Crowther, CM. Wischik and M. Stewart, in: Proc. 43rd Annual EMSA Meeting, Louisville, KY, 1985, Ed. G.W. Bailey (San Francisco Press, San Francisco, 1985). R.A. Crowther and CM. Wischik, EMBO J. 4 (1985) 3661. A.C. Steven, R. Stall, P.M. Steinert and B.L. Trus, in: Proc. 43rd Annual EMSA Meeting, Louisville, KY, 1985, Ed. G.W. Bailey (San Francisco Press, San Francisco, 1985). D.L.D. Caspar and K.C. Holmes, J. Mol. Biol. 46 (1969) 99.