1. Principles of Helical Reconstruction David Stokes 2 DX Workshop University of Washington 8/15-8/19/2011

2. 2D Lattice

4. Helical Lattice

5. equator meridian 6-start 7-start 13-start

6. Helical start l=2 l=1 n: 3 2 1 0 -1 -2 -3

9. n=7 with 2-fold symmetry normal to the helical axis

10. n=8 with 4-fold rotational symmetry down the axis and 2-fold symmetry normal to the axis

11. equator meridian 6-start 7-start 13-start

12. diffraction from 2D lattice equator d  normal to crystal planes  1/d

13. n,l plot = FFT of 2D lattice n=num crosses of equator l=num crosses of meridian

14. diffraction from helices equator d   c/l 2  r/n

15. scaling of n,l plot  1/d x y  n/2  r l/c diffraction pattern = n,l plot in units of 1/c and 1/2  r

16. cylindrical vs. flattened planar cylindrical d=  r d=2r

17. Bessel functions

18. Bessel Functions are solution to partial differential equation solve for functions “y” that satisfy this equation another example of a differential equation: Laplace’s equation: or solutions (u(x,y,z)) are “harmonic equations” relevant in many fields of physics (e.g. pendulum)

19. 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). general solution to differential equation:for integer values of alpha:

21. Overlapping lattices (near and far sides)  mirror symmetry

22. mirror symmetry in diffraction pattern: near and far sides of helix

23. Bessel Functions J n (2  Rr) 1)wrapping into cylinder  mirror symmetry 2) cylindrical shape  smearing of spots n/2  r J n (2  Rr), 1 st max at 2  rR  n+2; R=(n+2)/2  r

24. 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 20 Each layer line: G n (R,Z) Diaz et al, 2010, Methods Enzym. 482:131

25. R  Z

27. 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

30. Data from (0,1) Layer Line (after averaging ~15 tubes)

31. repeat distance =c (unit cell) pitch=p=c/8 subunits/turn=3.x n>0 => right-handed helix

36. frozen-hydrated Ca-ATPase tubes 15Å 10Å Chen Xu : 2002: 70/58 tubes, 6.5 Å TM domain