Transcript Stokes-2dx-workshop_day2

Principles of Helical Reconstruction
David Stokes
2DX Workshop
University of Washington
8/15-8/19/2011
2D Lattice
Helical Lattice
7-start
meridian
6-start
equator
13-start
Helical start
l=2
l=1
n:
3
2
1
0
-1
-2
-3
n=7
with 2-fold
symmetry
normal to
the helical
axis
n=8
with 4-fold
rotational
symmetry
down the axis
and 2-fold
symmetry
normal to the
axis
7-start
meridian
6-start
equator
13-start
diffraction from 2D lattice
1/d
normal to crystal planes
d 
equator

n,l plot = FFT of 2D lattice
n=num crosses of equator
l=num crosses of meridian
14
12
10
8
6
4
2
0
-20
-15
-10
-5
0
-2
-4
5
10
15
20
25
diffraction from helices
tan   2r / n
c/l
d  (cos  )  2r / n

c/l

d
2r/n
equator
scaling of n,l plot
tan   2r / n
c/l
d  (cos  )  2r / n
1/d
l/c

n/2r
1/d
y

x
x
1
(cos  )  n
2r
d
1
sin 
tan  l
y  (sin  ) 


c
2

r
2

r
d
cos 
n
n
diffraction pattern = n,l plot
in units of 1/c and 1/2r
cylindrical vs. flattened
planar
d=r
cylindrical
d=2r
Bessel functions
14
12
10
8
6
4
2
0
-20
-15
-10
-5
0
-2
-4
5
10
15
20
25
Bessel Functions are solution to partial
differential equation
2
d
y
dy
2
2
2
x

x

(
x

n
)0
2
dx
dx
solve for functions “y”
that satisfy this equation
another example of a differential equation: Laplace’s equation:
 2u  0
or
d 2u d 2u d 2u
 2  2 0
2
dx
dy
dz
solutions (u(x,y,z)) are “harmonic equations”
relevant in many fields of physics (e.g. pendulum)
Applications of Bessel Functions
general solution to differential equation:
(1) m
 12 x2m
J  ( x)  
m 0 m!(m    1)
for integer values of alpha:

J n ( x) 
1

cos( n  x sin  )d


0
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).
Overlapping lattices (near and far sides)
 mirror symmetry
mirror symmetry in diffraction pattern:
near and far sides of helix
14
12
10
8
6
4
2
0
-25
-20
-15
-10
-5
0
-2
-4
5
10
15
20
25
Bessel Functions Jn(2Rr)
14
12
10
8
6
4
2
0
-25
-20
-15
-10
-5
0
5
10
15
20
25
-2
-4
1) wrapping into cylinder
mirror symmetry
2) cylindrical shape
 smearing of spots
n/2r
Jn(2Rr), 1st max at 2rRn+2; R=(n+2)/2r
Each layer line: Gn(R,Z)
0
5
10
15
20
0
5
10
15
20
Jn(2Rr), 1st max at 2rRn+2; R=(n+2)/2r
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
Diaz et al, 2010, Methods Enzym. 482:131
Z
R

Gn ( R, Z )
F ( R, , l / c)   Gn,l ( R) exp[ in (   / 2)
n
F ( R, , l / c)   Gn,l ( R) exp[ in (   / 2)
n
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
Data from (0,1) Layer Line
(after averaging ~15 tubes)
pitch=p=c/8
repeat distance =c (unit cell)
n>0 => right-handed helix
subunits/turn=3.x
frozen-hydrated Ca-ATPase tubes
10Å
15Å
TM domain
Chen Xu : 2002: 70/58 tubes, 6.5 Å