Coordinate Systems Coordinate systems We have to deal with many coordinate systems in crystallographic software. The book-keeping is not exciting, but it is vital. Simplifications (e.g. Coordinate systems, operators, and assuming orthogonal grids) must usually be paid for transformations. later. Examples: Kevin Cowtan [email protected] Orthogonal Ångstrom coordinates. Fractional coordinates. Reciprocal orthogonal coordinates. Reflection (Miller) indices, i.e. HKLs. Grid coordinates. Kevin Cowtan, [email protected] Sienna/Coordinate Systems Kevin Cowtan, [email protected] Sienna/Coordinate Systems Coordinate Systems Coordinate Systems In addition, we need to handle: Conventions for this lecture: Transformations between these coordinate systems. scalars are italic, lower case, e.g. s, r Transformations within a coordinate system vectors are italic, lower case and underlined, e.g. v, x, h (i.e. rotation and translation operators). matrices are italic, uppercase and bold, e.g. O, F, M Rotation representations. In addition to matrix notation, most equations are Derivatives of functions with respect to different also given as explicit sums of terms. coordinate systems. The elements of a vector or matrix are given by the same This lecture will give a basic overview of the issues. symbol in italic lower case with an appropriate number of Implementations of all these data types and subscripts. e.g. xi , Oi,j . transformations are a part of both the CCTBX and All vectors and matrices are of rank 3. Clipper crystallographic libraries. Kevin Cowtan, [email protected] Sienna/Coordinate Systems Kevin Cowtan, [email protected] Sienna/Coordinate Systems Coordinate Systems Coordinate Systems: Real space Coordinate systems: Orthogonal Ångstrom Coordinates: Coordinate are used to describe the positions of 3 orthogonal distances in Ångstroms along elements within the crystal system. e.g. the position directions x, y, z. (Formally: basis vectors) of a particular atom within a unit cell, or a particular grid point within a grid. y z 3 dimensions -> rank 3. y Coordinates represented by a vector of 3 numbers. x x Note: there is no reference to the unit cell at this point. z Kevin Cowtan, [email protected] Sienna/Coordinate Systems Kevin Cowtan, [email protected] Sienna/Coordinate Systems Coordinate Systems: Real space Coordinate Systems: Real space Fractional coordinates: Relating orthogonal and fractional coordinates: Position in the unit cell described as a fractional We can orient and position one coordinate system position along each cell edge: however we want with respect to the other, but... It is convenient to adopt some convenient b 1 convention. The most common convention in the PDB (also the w CCP4 default) is: v Note: since the cell repeats, Align the a axis along x 0 1 u,v,w repeat on the range 0...1. Align the b axis in the x-y plane u a We often standardize on the Or equivalently align z axis perpendicular to a and b 1 range 0...1 (or -1/2...1/2), but this may split a molecule. c Kevin Cowtan, [email protected] Sienna/Coordinate Systems Kevin Cowtan, [email protected] Sienna/Coordinate Systems Coordinate Systems: Real space Coordinate Systems: Real space Relating orthogonal and fractional coordinates: Relating orthogonal and fractional coordinates: An orthogonal coordinate may be determined from b a fractional coordinate by: y x = O u i.e. x = O u z w Σ xi = j Oij uj y v Where O is the orthogonalization matrix. For the common convention, x u x a c z Kevin Cowtan, [email protected] Sienna/Coordinate Systems Kevin Cowtan, [email protected] Sienna/Coordinate Systems Coordinate Systems: Real space Coordinate Systems: Real space Relating orthogonal and fractional coordinates: Measuring distances: A fractional coordinate may be determined from an We do this all the time, e.g. inter-atomic distances. orthogonal coordinate by: In orthogonal coordinates, the squared distance u = F x between two points is given by r2 = Δx2+Δy2+Δz2 i.e. u = F x i.e: Δx =x -x Σ 2 1 ui = j Fij xj r2 = ΔxTΔx 2 T Where F is the fractionalization matrix. r = x x or: -1 Clearly F = O 2 Σ Δ 2 r = i xi Note: a,b,c,α,β,γ are the cell constants. a*,b*,c*,α*,β*,γ* are the reciprocal cell constants. Kevin Cowtan, [email protected] Sienna/Coordinate Systems Kevin Cowtan, [email protected] Sienna/Coordinate Systems Coordinate Systems: Real space Coordinate Systems: Real space Measuring distances: Measuring distances: For the distance between two fractional Simplified form using the metric tensor: coordinates, convert to orthogonal first: 2 Δ 2 Δ 2 Δ 2 r =M11 u1 +M22 u2 +M33 u3 2 Δ T T Δ r = u O O u 2 T T +2M Δu Δu +2M Δu Δu +2M Δu Δu r = u O O u 12 1 2 13 1 3 23 2 3 or: (since the matrix is symmetric, we just use the r2 = Σ Σ Σ O O Δu Δu i j k ij ik j k upper triangle and double the off-diagonal terms). Simplify by pre-calculating the central product: (This is often a performance critical task). M = OTO Σ Mjk = i Oij Oik M is a symmetric matrix, called the “real-space metric tensor”. Kevin Cowtan, [email protected] Sienna/Coordinate Systems Kevin Cowtan, [email protected] Sienna/Coordinate Systems Coordinate Systems: Real space Coordinate Systems: Real space Other coordinate types: Other coordinate types: Grid coordinates: Electron density maps are usually Grid coordinates: g calculated on a grid which samples the unit cell. Each grid point is indexed using 3 integer indices, gi, b usually starting from 0. 4 The sampling usually involves a grid in which each cell edge is divided into a set number of equal divisions, with 7 the number of divisions roughly proportional to the cell edge. 6 Symmetry and FFT requirements may constrain these a values(e.g.multipleof2,3,4,nolargeprimefactors). c Kevin Cowtan, [email protected] Sienna/Coordinate Systems Kevin Cowtan, [email protected] Sienna/Coordinate Systems Coordinate Systems: Real space Coordinate Systems: Real space Other coordinate types: Other coordinate types: Grid coordinates: Grid coordinates: Additional complications: Convert to grid coordinates by scaling the fractional M. Rowicka, A. Kudlicki and Z. Otwinowski, [Acta Cryst. coordinates by the samplings, and taking nearest integer: (2002). A58, 574-579] use grids which do not intersect theorigintoimprovesymmetryhandlingintheFFT Hexagonal close packed grids give a more efficient sampling of real space. How are they best indexed? For orthogonal coordinates , convert to/from fractional first. As an optimization, the two steps can be combined. Grid coordinates repeat every ni along the i'th axis. Kevin Cowtan, [email protected] Sienna/Coordinate Systems Kevin Cowtan, [email protected] Sienna/Coordinate Systems Coordinate Systems: Real space Coordinate Systems: Real space Other coordinate types: Implementation issues: Map coordinates: (Cowtan) It is very easy when programming to make Non-integer grid coordinates. mistakes over coordinate types. For crystallographic maps, fractional coordinates do the job just When using strongly typed languages (e.g. C++), fine implement each coordinate type as a different class For non-crystallographic maps, fractional coordinates are to prevent such errors. undefined. Used for interpolation in non-crystallographic maps. (Use inheritance for common behaviors) Coordinate types or cell and sampling classes may then implement all the required conversions. See CCTBX or Clipper. Kevin Cowtan, [email protected] Sienna/Coordinate Systems Kevin Cowtan, [email protected] Sienna/Coordinate Systems Coordinate Systems: Reciprocal space Coordinate Systems: Reciprocal space In reciprocal space we mainly deal with reflections, In reciprocal space we mainly deal with reflections, indexed by integer h,k,l: indexed by integer h,k,l: h,k,l are coordinates on a non-orthogonal grid, like grid coordinates. h,k,l do not repeat. They are centered (+/-) about the origin h=k=l=0. May be referred to as Miller indices (by correspondence with the indexing of crystal cell faces). Kevin Cowtan, [email protected] Sienna/Coordinate Systems Kevin Cowtan, [email protected] Sienna/Coordinate Systems Coordinate Systems: Reciprocal space Coordinate Systems: Reciprocal space We frequently need to determine the (reciprocal) Relating reciprocal orthogonal and fractional distance from a reflection to the origin (resolution). coordinates: We may also need to An reciprocal orthogonal coordinate may be determine a reciprocal determined from an HKL by: orthogonal coordinate s = O* h e.g. to make this picture. As in real space, O* is determined by the orthogonalization convention. A convenient choice is to use the transpose of the real space fractionalizing matrix: O* = FT i.e. z parallel to c*, y in the b*-c* plane Kevin Cowtan, [email protected] Sienna/Coordinate Systems Kevin Cowtan, [email protected] Sienna/Coordinate Systems Coordinate Systems: Reciprocal space Coordinate Systems: Reciprocal space Relating reciprocal orthogonal and fractional Measuring distances in reciprocal space: coordinates: As before, to calculate squared distances (in For display purposes, the “Cambridge Convention” inverse squared Ångstroms), we first need is more common: reciprocal orthogonal coordinates, or a reciprocal x parallel to a* metric tensor: s2 =hTO*TO*h 2 T T y in the a*-b* plane s = h O* O* h or: In this case O* is calculated using the equivalent 2 Σ Σ Σ s = i j k O*ij O*ik hj hk formula to O. Simplify by pre-calculating the central product: Don't mix conventions between real and M* = O*TO* reciprocal space – it will only end in tears.