Principles of Helical Reconstruction

David Stokes2DX Workshop

University of Washington8/15-8/19/2011

2D Lattice

Helical Lattice

equator

meridian

6-start

7-start

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

equator

meridian

6-start

7-start

13-start

diffraction from 2D lattice

equator

d

normal to crystal planes

1/d

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

diffraction from helices

equator

d

c/l

2r/n

lcnr

//2tan

nrd /2)(cos

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

cylindrical vs. flattened

planar cylindrical

d=rd=2r

Bessel functions

-4

-2

0

2

4

6

8

10

12

14

-20 -15 -10 -5 0 5 10 15 20 25

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)

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

Overlapping lattices (near and far sides) mirror symmetry

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

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

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

n

ln inRGclRF )2/(exp[)()/,,( ,

R

Z

),( ZRGn

n

ln inRGclRF )2/(exp[)()/,,( ,

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)

repe

at d

ista

nce

=c

(uni

t ce

ll)

pitc

h=p=

c/8

subunits/turn=3.x

n>0 => right-handed helix

frozen-hydrated Ca-ATPase tubes

15Å

10Å

Chen Xu : 2002: 70/58 tubes, 6.5 Å

TM domain