Fitting models to reconstruction
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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

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Fitting Models to Reconstruction

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Fitting models to reconstruction

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


Fitting models to reconstruction

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


Fitting models to reconstruction

Transverse sections of negatively stained

tarantula thick filaments


Fitting models to reconstruction1

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


Fitting models to reconstruction

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


Fitting models to reconstruction2

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


Calculating map from model

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)


Image processing software

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


Image processing programs

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


Blurring the model

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)


Aligning model reconstruction

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


Scoring model

Scoring model

  • Compare the aligned model & reconstruction volumes by cross correlation coefficient

    Lies between -1 and +1


Refining model

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


Downhill simplex method

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


Downhill simplex method1

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)


Downhill simplex method2

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)


Downhill simplex method3

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


Simulated annealing

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


Simulated annealing1

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


Surface views of reconstruction model of tarantula myosin filaments

Surface views of reconstruction & model of tarantula myosin filaments

reconstruction

model


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