Mini-κ calibration studies Kristopher I. White & Sandor Brockhauser Kappa Workgroup Meeting...

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mini-κ calibration studies Kristopher I. White & Sandor Brockhauser Kappa Workgroup Meeting • 22.4.2010

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

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

TC: number of points req’d for optimal scale factor calculation

points in subset

szsy

TC: application of Sy (70% anisotropy)

x-axis (µm)

y-a

xis

m)

TC: error calculation

TC: examplesty

pic

al

valu

es

70

%

anis

otr

opy

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: maths

The magnitude of rotation in each axis can then be calculated from the correct R:

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?

RC: total images required

Images collected

An

gu

lar

devia

tion (

º)

RC: Angle between images

Rotation

Rota

tion

err

or

RC: axis accuracy/precision

RC: stability

RC: no. images/wedge