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
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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
Maps
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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
Transverse sections of negatively stained
tarantula thick filaments
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
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Example Fitting of S1 & actin molecules to contour plot of
actoS1
Fitting of atomic models of actin & S1 to 3d
reconstruction of actoS1
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
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Example for myosin filament use tilt, slew, radius, rotation,
flex1, flex2 of molecules
Calculating map from model
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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)
Image processing software
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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
Image processing programs
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Can choose from large number of routines
Example survey html list of SPIDER routines
show window routine
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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)
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)
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– with either top-hat function (abrupt truncation) or
– Gaussian function (avoids ripples)
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Low-pass filter a length > one repeat then window
to one repeat (avoid end effects)
Aligning model & reconstruction
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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
Scoring model
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Compare the aligned model & reconstruction
volumes by cross correlation coefficient
Lies between -1 and +1
Refining model
Want to find model with better score
 Only two methods can be used to find the
minimum (maximum) of a function with >1
variable if gradients not available
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– (1) Powells method
– (2) Downhill simplex method
Downhill Simplex method
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A simplex is a polyhedron in n-dimensional space, one
dimension for each parameter defining the model.
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For two dimensions the simplex is a triangle, for three
dimensions a tetrahedron.
In general, the simplex has n+1 vertices, each vertex
corresponding to one of the models currently under
consideration
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Downhill Simplex method
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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)
Downhill Simplex method
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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)
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
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Simulated annealing
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The downhill simplex method finds only the local
minimum
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Hence the best model may never be tried
The downhill simplex method can be modified so
sometimes uphill moves are tried
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Simulated annealing
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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
Surface views of reconstruction & model of
tarantula myosin filaments
reconstruction
model