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

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

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) Examplesurvey 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 > 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 >1 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 n+1 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