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

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 α 1 → α 2. t α is a translation vector describing the motor current motor positions for a given α. 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. ii) Re-center the reference point such that the translation position corresponding with the rotation is registered as t α. 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.

TC: processing

ellipsefit scaling plane fit plane projectio n 3D MOTOR POSITIONS 2D POSITIONS IN PLANE IDEAL 3D POSITIONS ellipse fit stats plane fit stats scale factors linear error, angular error

TC: number of points req’d for optimal scale factor calculation points in subset szsz sysy

TC: application of S y (70% anisotropy) x-axis (µm) y-axis (µm)

TC: error calculation

TC: examples typical values 70% anisotropy

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 H lab 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. \mathbf{H}_{\text{lab}} = \mathbf{\Phi}\left(\omega,\kappa,\phi\ri ght) \mathbf{U} \mathbf{B} \left[\begin{array}{c} h \\ k \\ l \\ \end{array}\right] \mathbf{H}_{\text{lab}} = \mathbf{\Phi}\left(\omega,\kappa,\phi\ri ght) \mathbf{U} \mathbf{B} \left[\begin{array}{c} h \\ k \\ l \\ \end{array}\right]

RC: maths The transformation matrix T between any two UB matrices O ref and O i that differ by a rotation about the ω, κ, or ϕ axes can thus be determined: \mathbf{T}=\mathbf{O}^{- 1}_{ref}\mathbf{O}_i 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 \mathbf{RR}'=\mathbf{R}^{-1} \text{, } \mathbf{R}'\mathbf{R}=\mathbf{I} \text{, and } \mathbf{R}'=\mathbf{R}^{-1} \mathbf{RR}'=\mathbf{R}^{-1} \text{, } \mathbf{R}'\mathbf{R}=\mathbf{I} \text{, and } \mathbf{R}'=\mathbf{R}^{-1}

RC: maths The angle θ i between O ref and O i in laboratory space is given by Multiple equivalent lattice indexing transformations can be generated from equivalent solutions for OMs, so use the O i that minimizes θ i. \theta_i = \arccos \left( \frac{ \text{trace} \left( \mathbf{R}_{i} \right) ^{-1} } {2} \right) \theta_i = \arccos \left( \frac{ \text{trace} \left( \mathbf{R}_{i} \right) ^{-1} } {2} \right)

RC: maths The magnitude of rotation in each axis can then be calculated from the correct R: \omega_i= \frac{1}{2\sin\left(\theta_i\right)} \left[ \begin{array}{c} \mathbf{R}\left(3,2\right)- \mathbf{R}\left(2,3\right) \\ \mathbf{R}\left(1,3\right)- \mathbf{R}\left(3,1\right) \\ \mathbf{R}\left(2,1\right)- \mathbf{R}\left(1,2\right) \\ \end{array} \right] \omega_i= \frac{1}{2\sin\left(\theta_i\right)} \left[ \begin{array}{c} \mathbf{R}\left(3,2\right)- \mathbf{R}\left(2,3\right) \\ \mathbf{R}\left(1,3\right)- \mathbf{R}\left(3,1\right) \\ \mathbf{R}\left(2,1\right)- \mathbf{R}\left(1,2\right) \\ \end{array} \right]

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 Angular deviation (º)

RC: Angle between images Rotation Rotation error

RC: axis accuracy/precision

RC: stability

RC: no. images/wedge