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Fitting Models to Reconstruction

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- 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

- 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

- 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

Fitting of atomic models of actin & S1 to 3d reconstruction of actoS1

- 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

- 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)

- 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

- 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)

Examplesurvey html list of SPIDER routines

show window routine

- 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 > one repeat then window to one repeat (avoid end effects)

- 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

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

- 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
- (1) Powells method
- (2) 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 n+1 vertices, each vertex corresponding to one of the models currently under consideration

- 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)

- 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)

- reflection (through opposite

- 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

- 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

- 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

reconstruction

model