Fitting Models to Reconstruction

1
Fitting Models to Reconstruction

• Suppose we have a 3d reconstruction
• We want to explain the reconstruction in terms of
  the atomic structure of the molecule
• May want to fit with rigid molecule or allow
  domains of molecule to flex

Examples Fitting a 3d reconstruction of spherical
virus by atomic model of coat protein Fitting a
3d reconstruction of F-actin by atomic model of
actin
2
(No Transcript)
3
Maps

• A density map is a description of the protein
  density in 3 dimensions (a 3d image) pixel by
  pixel
• Might be set of files each representing a slice
  or a
• single file representing a volume
• It can be displayed as
• a) a set of slices
• b) contour plots
• c) surface views

Example Map of helical reconstruction of
negatively stained tarantula myosin filaments
4
Transverse sections of negatively stained
tarantula thick filaments
5
Fitting Models to Reconstruction

• Can fit interactively by eye
• Choose a contour level which encloses a volume
  equal to that of the molecule
• Position the molecule to lie within this contour
• Advantages - quick simple get feeling of
  problem
• Disadvantage - not use highest density features

Example Fitting of S1 actin molecules to
contour plot of actoS1
6
Fitting of atomic models of actin S1 to 3d
reconstruction of actoS1
7
Fitting Models to Reconstruction

• For objective method of fitting first need to
  parameterise model ie describe model in numerical
  terms - what are the variables?
• Usually variables define orientation, radius
  any internal flexing not translation rotation
  between molecules

Example for myosin filament use tilt, slew,
radius, rotation, flex1, flex2 of molecules
8
(No Transcript)
9
Calculating map from model

• A density map must be calculated for each model
• Choose pixel size to match reconstruction
  (resolution)
• Overall size one repeating unit
• Represent each (non-hydrogen) atom in model by
  sphere eg radius 3 angstroms
• Calculate volume contribution of each sphere to
  each pixel of the map. Hence calculate density
  of each pixel. Convenient if scale 0-255 (1
  pixel 1 byte)

10
Image processing software

• IMAGIC SUPRIM SPIDER etc
• Can view maps eg as slices or volumes
• Stack slices
• Window
• Interpolate
• Fourier transform
• Low-pass filter
• Translate rotate
• Align etc

11
Image processing programs

• Can choose from large number of routines
• Write own programs using these routines
• eg to extend a volume
• break down volume into slices (ps)
• make copies of the set (copy)
• stack all the slices (sk)

Example survey html list of SPIDER routines show
window routine
12
Blurring the model

• To compare with reconstruction need to blur model
  to similar resolution
• Use low-pass filter (truncation of Fourier
  transform to chosen radius)
• with either top-hat function (abrupt truncation)
  or
• Gaussian function (avoids ripples)
• Low-pass filter a length gt one repeat then window
  to one repeat (avoid end effects)

13
Aligning model reconstruction

• Make end projection of model reconstruction
  volumes one repeat long
• determine rotation required for alignment
  apply this to model volume
• Now make longitudinal projection of volumes
• determine translation required for alignment
  apply this to model volume

14
Scoring model

• Compare the aligned model reconstruction
  volumes by cross correlation coefficient
• Lies between -1 and 1

15
Refining model

• Want to find model with better score
• Only two methods can be used to find the minimum
  (maximum) of a function with gt1 variable if
  gradients not available
• (1) Powells method
• (2) Downhill simplex method

16
Downhill Simplex method

• A simplex is a polyhedron in n-dimensional space,
  one dimension for each parameter defining the
  model.
• For two dimensions the simplex is a triangle, for
  three dimensions a tetrahedron.
• In general, the simplex has n1 vertices, each
  vertex corresponding to one of the models
  currently under consideration

17
Downhill Simplex method

• Start by making a reasonable model by manual
  fitting
• Make another n models by allowing each parameter
  to change by a small increment eg tilt by 5,
  radius by 5 Å
• Score each of these models rank them (worst,
  next worst best)

18
Downhill Simplex method

• For each iteration up to 4 new models tried with
  the following moves
• reflection (through opposite
• face from high point)
• extension (further in same
• direction as reflection)
• contraction
• (away from high point)
• shrinkage
• (towards low point)

19
Downhill Simplex method

• Downhill simplex program considers these new
  models scores them.
• If reflection point better than previous best
  model try out extension.
• Replace poorest scoring model by reflection or
  extension or contraction points if these are
  improvements
• Otherwise replace all models by shrinkage points

20
Simulated annealing

• The downhill simplex method finds only the local
  minimum
• Hence the best model may never be tried
• The downhill simplex method can be modified so
  sometimes uphill moves are tried

21
Simulated annealing

• This is equivalent to giving the system thermal
  energy so it can overcome energy barriers
• This is done at the stage of deciding on a new
  move by adding a random number (proportional to
  temperature) to existing scores and subtracting
  a random number to new score. So moves are made
  which may be unfavourable.
• Gradually the temperature is reduced to zero so
  final stage is a simple downhill simplex
  refinement.
• Can repeat with different sets of random numbers
  to get new trajectories

Example Result of refining model of tarantula
myosin filaments by simulated annealing
22
Surface views of reconstruction model of
tarantula myosin filaments
reconstruction
model