Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington...
-
Upload
patience-ball -
Category
Documents
-
view
214 -
download
0
Transcript of Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington...
![Page 1: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/1.jpg)
Principles of Helical Reconstruction
David Stokes2DX Workshop
University of Washington8/15-8/19/2011
![Page 2: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/2.jpg)
2D Lattice
![Page 3: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/3.jpg)
![Page 4: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/4.jpg)
Helical Lattice
![Page 5: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/5.jpg)
equator
meridian
6-start
7-start
13-start
![Page 6: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/6.jpg)
Helical start
l=2
l=1
n: 3 2 1 0 -1 -2 -3
![Page 7: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/7.jpg)
![Page 8: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/8.jpg)
![Page 9: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/9.jpg)
n=7
with 2-fold symmetry normal to the helical
axis
![Page 10: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/10.jpg)
n=8
with 4-fold rotational symmetry down the axis and 2-fold symmetry normal to the axis
![Page 11: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/11.jpg)
equator
meridian
6-start
7-start
13-start
![Page 12: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/12.jpg)
diffraction from 2D lattice
equator
d
normal to crystal planes
1/d
![Page 13: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/13.jpg)
n,l plot = FFT of 2D latticen=num crosses of equatorl=num crosses of meridian
-4
-2
0
2
4
6
8
10
12
14
-20 -15 -10 -5 0 5 10 15 20 25
![Page 14: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/14.jpg)
diffraction from helices
equator
d
c/l
2r/n
lcnr
//2tan
nrd /2)(cos
![Page 15: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/15.jpg)
scaling of n,l plot
lcnr
//2tan
nrd /2)(cos
1/d
x
y
rn
dx 2)(cos
1
cl
nr
nrd
y
2tan
cos2sin
)(sin1
1/d
n/2r
l/c
diffraction pattern = n,l plotin units of 1/c and 1/2r
![Page 16: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/16.jpg)
cylindrical vs. flattened
planar cylindrical
d=rd=2r
![Page 17: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/17.jpg)
Bessel functions
-4
-2
0
2
4
6
8
10
12
14
-20 -15 -10 -5 0 5 10 15 20 25
![Page 18: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/18.jpg)
Bessel Functions are solution to partial differential equation
0)( 222
22 nx
dx
dyx
dx
ydx solve for functions “y”
that satisfy this equation
another example of a differential equation: Laplace’s equation:
02 u 02
2
2
2
2
2
dz
ud
dy
ud
dx
udor
solutions (u(x,y,z)) are “harmonic equations” relevant in many fields of physics (e.g. pendulum)
![Page 19: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/19.jpg)
Applications of Bessel Functions
Bessel functions are especially important for many problems of wave propagation and static potentials. In solving problems in cylindrical coordinate systems, one obtains Bessel functions of integer order (α = n); in spherical problems, one obtains half-integer orders (α = n + ½). For example:
•Electromagnetic waves in a cylindrical waveguide•Heat conduction in a cylindrical object•Modes of vibration of a thin circular (or annular) artificial membrane (such as a drum)•Diffusion problems on a lattice•Solutions to the radial Schrödinger equation (in spherical and cylindrical coordinates) for a free particle•Solving for patterns of acoustical radiation•Bessel functions also have useful properties for other problems, such as signal processing (e.g., see FM synthesis, Kaiser window, or Bessel filter).
0
2
21
)1(!
)1()(
m
mm
xmm
xJ
general solution to differential equation: for integer values of alpha:
0
)sincos(1
)( dxnxJ n
![Page 20: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/20.jpg)
![Page 21: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/21.jpg)
Overlapping lattices (near and far sides) mirror symmetry
![Page 22: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/22.jpg)
mirror symmetry in diffraction pattern:near and far sides of helix
-4
-2
0
2
4
6
8
10
12
14
-25 -20 -15 -10 -5 0 5 10 15 20 25
![Page 23: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/23.jpg)
Bessel Functions Jn(2Rr)
1) wrapping into cylindermirror symmetry
2) cylindrical shape smearing of spots
n/2r
Jn(2Rr), 1st max at 2rRn+2; R=(n+2)/2r
-4
-2
0
2
4
6
8
10
12
14
-25 -20 -15 -10 -5 0 5 10 15 20 25
![Page 24: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/24.jpg)
Jn(2Rr), 1st max at 2rRn+2; R=(n+2)/2r
Use radial position to determine Bessel order (approximation) - radius hard to measure with defocus fringes - different radii of contrast for different helical families - particle may be flattened
0 5 10 15 200 5 10 15 20
Each layer line: Gn(R,Z)
Diaz et al, 2010, Methods Enzym. 482:131
![Page 25: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/25.jpg)
n
ln inRGclRF )2/(exp[)()/,,( ,
R
Z
),( ZRGn
![Page 26: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/26.jpg)
n
ln inRGclRF )2/(exp[)()/,,( ,
![Page 27: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/27.jpg)
Out of plane tilt gives rise to systematic changes in phases along the layer lines, which can be corrected if tilt angle and indexing of layer lines are known
![Page 28: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/28.jpg)
![Page 29: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/29.jpg)
![Page 30: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/30.jpg)
Data from (0,1) Layer Line(after averaging ~15 tubes)
![Page 31: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/31.jpg)
repe
at d
ista
nce
=c
(uni
t ce
ll)
pitc
h=p=
c/8
subunits/turn=3.x
n>0 => right-handed helix
![Page 32: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/32.jpg)
![Page 33: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/33.jpg)
![Page 34: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/34.jpg)
![Page 35: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/35.jpg)
![Page 36: Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011.](https://reader036.fdocuments.us/reader036/viewer/2022062717/56649e5c5503460f94b543cf/html5/thumbnails/36.jpg)
frozen-hydrated Ca-ATPase tubes
15Å
10Å
Chen Xu : 2002: 70/58 tubes, 6.5 Å
TM domain