Mini-κ calibration studies Kristopher I. White & Sandor Brockhauser Kappa Workgroup Meeting...
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Transcript of Mini-κ calibration studies Kristopher I. White & Sandor Brockhauser Kappa Workgroup Meeting...
mini-κ calibration studiesKristopher I. White & Sandor Brockhauser
Kappa Workgroup Meeting • 22.4.2010
TRANSLATION CALIBRATION(TC)
Calibration of motors responsible for sample re-centering following rotation.
ROTATION CALIBRATION(RC)
Calibration of motors responsible for rotation about (ω,κ,ϕ).
TC: overview
We need a fast, reliable method for calibrating motors responsible for performing translational re-centering.
Criteria:
• Use current hardware/software(microscopy-based; use “three-click-centering”)
• Provide rapid means for troubleshooting alignment issues(Anisotropy, non-orthogonal motor axes)
• Minimize time required to perform(limit to 10–15 minutes max, ensure that accuracy/precision are “good enough”)
• Easy to perform
TC: maths
D describes rotation axis direction.
T describes rotation axis location.
R is a rotation matrix describing rotation about a given axis over the angular range α1 → α2.
tα is a translation vector describing the motor current motor positions for a given α.
tα2
=T −Rα1 −α2
D T −tα1( )
for rotation about the κ- or φ-axis,
TC: basic method
I. Rotate motors such that (ω, κ, φ) = (0º, 0º, 0º).
II. Perform projection-based centering on a well-defined point that is clearly recognizable at a variety of angles. After centering, the translation motor positions are registered such that .
III. For the κ- and φ-axes, separately perform the following:
i) Rotate about the given axis by angle α.
I. Re-center the reference point such that the translation position corresponding with the rotation is registered as tα.
IV. Repeat (III) at least four more times, recording a total of at least six unique points per axis; points should be evenly distributed and paired with another point 180º degrees away.
α=0º
−κ
t =α =0º
−ϕ
t
TC: processing
ellipseellipsefitfit
scalingscaling
planeplanefitfit
planeplaneprojectiprojecti
onon
3D MOTOR 3D MOTOR POSITIONSPOSITIONS3D MOTOR 3D MOTOR POSITIONSPOSITIONS
2D POSITIONS 2D POSITIONS IN PLANEIN PLANE
2D POSITIONS 2D POSITIONS IN PLANEIN PLANE
IDEAL 3D IDEAL 3D POSITIONSPOSITIONSIDEAL 3D IDEAL 3D
POSITIONSPOSITIONS
ellipse fit statsellipse fit statsellipse fit statsellipse fit stats
plane fit statsplane fit statsplane fit statsplane fit stats
scale factorsscale factorsscale factorsscale factors
linear error, linear error, angular errorangular errorlinear error, linear error,
angular errorangular error
aside: improving calibration precision/accuracy
Would be ideal with implementation of circular centering reticule for manual centering, and easy to detect for auto-centering.
aside: making calibration pins
Materials:•Glass capillaries (borosilicate, 0.78mm ID,
9mm OD)
•Pins (Hampton HR4-923)
•Microbeads Calibre® calibration standard polystyrene beads (10 µm, other sizes avail.)
Instructions:Cut pins ~5 mm from base. Pull glass
capillaries at ~60ºC with fast separation to ensure a short needle (maximizes stability). Break needle base to appropriate length and glue to pin on base. Under microscope, fill needle tip with glue, then use sticky tip to grab a bead.
RC: overview
We need to ensure the accuracy and precision of rotational motion.
Let’s use the orientation matrix for a common protein to measure rotation.
Same generic criteria as before—easy to implement and perform, and, most importantly, fast.
RC: maths
Hlab describes some scattering vector in laboratory space.
Φ describes rotation as a function of rotation axis angles.
The product UB represents the orientation of the sample, where U is a rotation matrix and B is a square orthogonalization matrix.
[h, k, l]’ represents the Miller indices for an observed scattering vector in reciprocal space.
RC: maths
The transformation matrix T between any two UB matrices Oref and Oi that differ by a rotation about the ω, κ, or ϕ axes can thus be determined:
These two orientation matrices were calculated from different indexing solutions; consider non-rotational instabilities.
As such, reorthogonalize T to a pure rotation matrix R using an SVD-based method such that
RC: maths
The angle θi between Oref and Oi in laboratory space is given by
Multiple equivalent lattice indexing transformations can be generated from equivalent solutions for OMs, so use the Oi that minimizes θi.
RC: method
Orientation matrix calculation optimization
• Wedge angle?
• Number of images in wedge?
• Optimal angular distance between wedges?
• Protein choice?
Rotation calibration studies
• Expected precision/accuracy for different axes?
• Stability?