PX-CBMSO Course (2) The mathematical description of Symmetry y PX-CBMSO-June 2011 Cele Abad-Zapatero University of Illinois at Chicago Center for Pharmaceutical Biotechnology. Lecture no. 2 This material copyrighted by Cele Abad-Zapatero.
Crystallography: Description of symmetry Theory of Diffraction Structure Determination Refinement
Mathematical description of symmetry r = A r + T rr : r, r thee-dimensional vectors components (x, y, z) A: 3x3 matrix T: column matrix 3x1 Symmetry operation (operator) r (x, y, z) r (x,y,z ) Type of symmetry operation is encoded in the properties of the MATRICES: A: rotation, reflection T: translation
Portions of this material have been extracted t from Lecture 1 of the educational materials donated by Prof. JR Helliwell (Univ. Of Manchester, UK) to the IUCr for educational purposes
Example of amino acid enantiomers
The handedness of an object is changed by a mirror plane or an inversion centre x y Mirror plane z
Enantiomers description Optical isomers are Non Superimposable Mirror Images of each other; a set of optical isomers are called enantiomers. Mathematical description of symmetry (from before): r = A r + T r, r : thee-dimensional vectors components (x, y, z) A: 3x3 matrix T: column matrix 3x1 Enantiomers: the determinant of matrix A will be negative Enantiomers: the determinant of matrix A will be negative (change in hand: reflection or inversion).
A simple example (mirror in x-y plane) r = A r + T x 1 0 0 x 0 y = 0 1 0 y + 0 z 0 0-1 z 0 The Key equation to describe symmetry mathematically Matrix A carries the information of the symmetry operation: (rotation, inversion, or distortion ) Vector T: carries the information of the translational component
A generalized pure rotation operation r = A r + T x cos α sin α 0 x 0 y = - sin α cos α 0+ y 0 z 0 0 1 z 0 x,y α x,y The key equation to describe symmetry mathematically Matrix A carries the information of the symmetry operation: (rotation, inversion, or distortion ) in the coefficients of the matrix. Vector T: carries the information of the translational component Pure rotation matrices are: ORTHOGONAL (preserve angles, distances): Exercise: demonstrate that the matrix A, above, is orthogonal (for any α). Rotation matrices can be combined (multiplied) to produce new sym ops.
The concepts of Unit Cell and Asymmetric Unit (a.u) 2-fold axis perp. to plane Motif Motif+ syop unit Cell Motif+syop unit cell+ translation Motif Motif+ syop unit Cell Motif+syop unit cell+ translation symmetry
Unit cell and asymmetric unit (symmetry operators: α=180º) unit cell asymmetric unit a.u. 6 th motif 5 th motif 1 st motif 4 th motif 2 nd motif 3 rd motif Exercise: What will be the form of the matrix A -1 0 0 0 ops. for: for 2 nd motif symmetry operation above? 0-1 0 + 0 1-3; 1-4; and all the others? 0 0 1 0 1-5; 1-6; 1-2
Understanding decorations in The Alhambra: planar crystal (and color) symmetry Four identical points with identical environment define the unit cell. In crystals, in wallpaper, in fabric patterns, ceramic, tiles, in periodic drawings..etc.
Crystallography in Islamic Patterns of the Alhambra in Granada, Spain* real cell recip. cell
Hard facts: Space Symmetry (or Space Group Symmetry) For crystals the concepts used with molecular (point) symmetry have to be extended to include translational symmetry. 1) An unit-cell is repeated in the a, b and c directions to produce the macroscopic crystal. Atom locations are described in terms of fractional coordinates x, y, z relative to origin of a, b, c. (x, y,z value ranges 0-1.0, thus they are called fractional coordinates!) 2) The repeat of the unit cell (basic box) is portrayed by using a crystal lattice. 3) Each unit-cell must belong to one of the 7 crystal classes: triclinic, monoclinic, orthorhombic, tetragonal, (from less to high symmetry)etc., 4) In order to take advantage of symmetry within the unit-cell it is sometimes better to use centered cells e.g. F, I. When unit-cell centering is added to the 7 crystal classes 14 Bravais lattices are generated. TYPES OF CELLS: P: PRIMITIVE, F: FACE CENTERED; I; BODY CENTERED; C: C CENTERED
The unit cell and Bravais Lattices P I F P I P C There are 7 crystal systems (unit cell shapes). There are 14 Bravais ais lattice types There are 230 space groups, where the space group is the collection of symmetry operations of the crystal. Proteins being enantiopure take up only 65 space groups (i.e. missing mirror and inversion centre symmetry elements). P P I P F C P
Pictorial and algebraic representation of a crystal Th iti f i t ithi th LOVE tif b d fi d b A t The position of any point within the LOVE motif can be defined by r = A.r+ t Right? The vector algebra might be tedious but is conceptually simple. This is how you generate the crystal, by vector algebra.
One of the most important concept in crystallography: THE RECIPROCAL LATTICE. A LATTICE MADE UP OF INVERS DISTANCES: 1/d-(interplanar distance) Analogy: Space of frequencies! Not of wavelengths. Sometimes called FOURIER SPACE or RECIPROCAL SPACE Matrices can also be used To go from REAL > RECIPROCAL and RECIPROCAL>REAL
2-fold axis example and a.u. (very important) asymmetric 6 th motif 5 th motif unit a.u. 1 st motif 4 th motif 2 nd motif 3 rd motif Exercise: What will be the form of the matrix A -1 0 0 0 ops. for: for 2 nd motif symmetry operation above? 0-1 0 + 0 1-3; 1-4; and all the others? 0 0 1 0 1-5; 1-6; 1-2
The unit cell and Bravais Lattices P I F P I P C There are 7 crystal systems (unit cell shapes) There are 14 Bravais ais lattice types There are 230 space groups, where the space group is the collection of symmetry operations of the crystal. Proteins being enantiopure take up only 65 space groups (i.e. missing mirror and inversion centre symmetry elements). P P I P F C P
KEY points to remember about crystallographic symmetry. All the possible Space Groups are presented in their entirety in International Tables using various diagrams including projections to aid 3D visualisation. They are also coded in the various crystallographic packages by symbol (i. e. P21) or number (i.e. No. 4). There is a listing of possible atomic symmetrically y related locations x,y,x ; -x, y+1/2,-z. for that space group (i. e. P21). The actual atomic positions within one molecule form the core unit of the unit cell known as the asymmetric unit. Motif The rest of the crystal can be generated by applying the symmetry y operator matrices and translations to the basic motif.
The simplest space group: P1 For a triclinic cell it is possible to have only the identity operator (x,y,z); A point of inversion i (-x,-y,-z), not allowed for biological macromolecules. (i)p1 is the simplest Space Group: only translation symmetry. P1 is pure translational symmetry
OTHER SYMMETRY OPERATIONS POSSIBLE IN THE MONOCLINIC SYSTEM For monoclinic systems the symmetry possibilities increase with C- NOT IN PROTEINS centered as well as Primitive cells, 2-fold axes and mirror 2-fold axis planes in addition to points of inversion can occur. mirror ALSO operations, which combine point operations with a translation, are possible :- (i) c-glide plane (denoted c), a reflection in the xz plane and translation half way down c (1/2+x,-y,1/2+z). c/2 2-fold screw (ii) 2-fold screw axis (denoted 2 1 ), a 2-fold rotation about b and translation half way down b c-glide (-x,1/2+y,-z). b/2
Screw axes operator (VERY COMMON IN PROTEIN XTALS!) c b a A projection along the 2 1 screw axis gives rise to a zigzag pattern of rotation-related related molecules. Operation has two parts: 1. 180 degree rotation 2. Translation by ½ of the axis.
A screw-axis operation along b r = A r + t x cos α 0 sin α x 0 x,z y = 0 1 0 y 1/2 z -sin α 0 cos α z 0 positions: x,y,z initial; after operation: -x, y+1/2, -z b/2 x,z The Key equation to describe symmetry mathematically Matrix A carries the information of the symmetry operation: (rotation, inversion, or distortion ) in the coefficients of the matrix. Vector t: carries the information of the translational component
A screw-axis operation along b r = A (α=180º) r + t x -1 0 0 x 0 x,z y = 0 1 0 y 1/2 z 0 0-1 z 0 positions: x,y,z initial; after operation: -x, y+1/2, -z b/2 x,z The Key equation to describe symmetry mathematically Matrix A carries the information of the symmetry operation: (rotation, inversion, or distortion ) in the coefficients of the matrix. Vector t: carries the information of the translational component Exercise: deduce equivalent positions for the P21 space group.
Entry in the International Tables for Space Group P21 (no. 4) (b-is the unique axis)
A screw-axis operation along b r = A r + t x cos α 0 sin α x 0 x,z y = 0 1 0 y 1/2 z -sin α 0 cos α z 0 positions: x,y,z initial; after operation: -x, y+1/2, -z b/2 x,z The Key equation to describe symmetry mathematically Matrix A carries the information of the symmetry operation: (rotation, inversion, or distortion ) in the coefficients of the matrix. Vector t: carries the information of the translational component Exercise: deduce equivalent positions for the P21 space group.
Entry in the International Tables for Space Group P21 (no. 4) (b-is the unique axis)
The most common protein space group is P22 2 P212121 Four equivalent positions:- x,y,z -x+1/2,-y,z+1/2 yz+1/2 -x,y+1/2,-z+1/2 x+1/2,-y+1/2,-z Reflection absence conditions: for h00 h=2n+1 absent; for 0k0 k=2n+1 absent; for 00l l=2n+1 absent. [absent means an exactly zero intensity.]
Entry in the International Tables for P212121
Diffraction pattern: Precession photo of the reciprocal lattice How do crystallographers know about details of the lattice and the space group? b* axis Diffraction pattern is a photograph p of the reciprocal lattice. It does show the internal symmetry of the particular crystal lattice d d in mm (or pixels)/24 = the a* axis
Systematic absences Notice that all odd reflections along both the h and k axes are absent. This shows there must be 2 1 screw axes along these directions. From Prof S Hovmoller ppt via the www.
A non-orthogonal crystal with all reflections present! k 0 0 reflections Notice: 4,4 Non-orthogonal Lattice No-systematic absences h 0 0 reflections This diffraction pattern contains a portion of the hk0 plane of the reciprocal lattice of a monoclinic crystal This precession diffraction image shows a 2D projection of spots; the axial reflections are marked and all are present : no systematic absences. The centre of the pattern is this larger overexposed centre spot (usually captured by a piece of lead!).
Screw axes A projection perpendicular to a 2 1 screw axis gives rise to a zigzag pattern of mirror-related molecules. Operation has two parts: 1. 180 degree rotation 2. Translation by ½ of the axis.
Systematic absences Notice that all odd reflections along both the h and k axes are absent. This shows there must be 2 1 screw axes along these directions. From Prof S Hovmoller ppt via the www.
Centering leads to some reflections being systematically absent:- (i) Primitive (P) no absence conditions apply in this the simplest case (ii) Body-centered (I) h+k+l 2n (iii) Face-centered (F) h+k and h+l and k+l 2n Other space operations also lead to absences e.g. Other absence conditions occur, even in P cases (i) due to c-glide, h 0 l, l 2n (ii) due to 2 1, 0 k 0, k 2n The systematic absences are listed in International Tables as an aid to Space Group identification. Using the above it is usually possible to assign a crystal to its Space Group and hence take advantage of the symmetry information during structure solution and refinement. Crystal structures are reported in terms of unit-cell dimensions, space-group and fractional atomic coordinates.
Crystal systems and space groups for proteins; triclinic through to orthorhombic (excluding centering absence conditions for brevity). Bravais Lattice Candidates Axial Reflection Conditions Primitive Orthorhombic 16 P222 17 P2221 18 P21212 19 P212121 (0,0,2n) (2n,0,0),(0,2n,0) (2n,0,0),(0,2n,0),(0,0,2n) C222 Centered orthorhombic C Centered Orthorhombic 20 C2221 21 C222 (0,0,2n) I212121 I Centered Orthorhombic 23 I222 * Body centered 24 I212121 * orthorhombic F Centered Orthorhombic 22 F222 Primitive Monoclinic C Centered Monoclinic Primitive Triclinic 3 P2 4 P21 (0,2n,0) 5 C2 1 P1 Space Group: S.G. number or SG. symbol.
Entry in the International Tables for P212121
Tetragonal systems. 4-fold z or 41-screw axis Primitive Tetragonal 75 P4 76 P41 77 P42 78 P43 (0,0,4n)* (0,0,2n) (0,0,4n)* 2 y x 89 P422 90 P4212 91 P4122 92 P41212 93 P4222 94 P42212 95 P4322 96 P43212 (0,2n,0) 0) (0,0,4n)* (0,0,4n),(0,2n,0)** (0,0,2n) (0,0,2n),(0,2n,0) (0,0,4n)*, (0,0,4n),(0,2n,0)** 21 I Centered Tetragonal 79 I4 80 I41 (0,0,4n) 97 I422 98 I4122 (0,0,4n) number symbol Space groups of Tetragonal crystals: Example P43212
Hexagonal systems. 3,6-fold z or 36 3,61-screw axis Primitive Hexagonal 143 P3 144 P31 145 P32 149 P312 151 P3112 153 P3212 (0,0,3n)* (0,0,3n)* (0,0,3n)* (0,0,3n)* 150 P321 152 P3121 154 P3221 (0,0,3n)* (0,0,3n)* 2 2 y x 168 P6 169 P61 170 P65 171 P62 172 P64 173 P63 (0,0,6n)* (0,0,6n)* (0,0,3n)** (0,0,3n)** (0,0,2n) 177 P622 178 P6122 179 P6522 180 P6222 181 P6422 182 P6322 (0,0,6n)* 0 (0,0,6n)* (0,0,3n)** (0,0,3n)** (0,0,2n) Space groups of Hexagonal-trigonal crystals: Example P6322, number 182
Rhombohedral through cubic. Primitive Cubic 195 P23 198 P213 (2n,0,0) I Centered Cubic 207 P432 208 P4232 212 P4332 213 P4132 197 I23 199 I213 (2n,0,0) (4n,0,0)* (4n,0,0)* F Centered Cubic 211 I432 214 I4132 (4n,0,0) 196 F23 Primitive Rhombohedral 209 F432 210 F4132 146 R3 155 R32 (4n,0,0)
Assignment of space groups: Systematic absences are very important No. 16 P222 No. 17 P2221 No. 18 P21212 No. 19 P212121 16 has no absence conditions 17 (0,0,2n+1) 0 ) (ONLY EVEN REFLECTIONS ARE PRESENT ALONG THE c-axis 18 (2n+1,0,0), (0,2n+1,0) (ONLY EVEN REFLECTIONS ALONG a-axis) 19 (2n+1,0,0), (0,2n+1,0), (0,0,2n+1) (ONLY EVEN ALONG a,b,c, axis) Orthorhomic is assigned first of all via the fact that a, b, c are different and the unit cell angles are each equal to 90 degrees. P means a primitive unit cell ie no face or body centering The most common protein space group is P212121
Crystal packing; monoclinic insulin- hexamers (trimer of dimers) Unit cell Notice large solvent channels allowing flow of water, ligands, ions, etc. Point symmetry of hexamer: 32 INSULIN HEXAMERS IN THE CRYSTAL LATTICE
Summary Crystallographic symmetry can be described in a concise and effective way by matrix operations/algebra among the vectors describing the atomic positions. Determining the crystal lattice/geometry is critical because it imposes the constraints on how your protein is packed. In a way, establishes what size/type of box your protein is and how the different copies of the protein target relate to each other. All subsequent work will be based on thisso be careful. In q the early times of protein crystallography, determining the space group of your crystals was a major milestone.
ASSIGNMENT (Exercise): Go to: http://img.chem.ucl.ac.uk/sgp/mainmenu.htm Select a space group with main symmetry fourfold (4) or higher (3-6). Print the page. Find the matrices corresponding to the crystal Symmetry Operators present in that space group (i.e., equivalent positions). Turn it in after next class (June 13, 2011) with your name on it. Advanced: demonstrate that the Sym. Ops. form a Group. High- Resolution Space Group Diagrams and Tbl Tables (1280 1024) pixel screens) Medium- Resolution Space Group Diagrams and Tbl Tables (1024 768) pixel screens)
This outlines and illustrates the mathematical description of symmetry as applied to protein crystals. Next Lecture will the outline of diffraction theory and next structure determination Questions?
This leads to a total of 13 unique monoclinic Space Groups:- P2, P2 1, Pm, Pc, P2/m, P2 1 /m, P2/c, P2 1 /c, C2, Cm, Cc, C2/m, C2/c. P2 1 /c is by far the most common Space Group in the world of crystals and includes four operations:- (i) x, y, z (identity) (ii) -x, 0.5+y, 0.5-z (2 1 ) (iii) x, 0.5-y, 0.5+z (c) (iv) -x, -y, -z (inversion) When all the Bravais lattices are combined with all the possible symmetry operations a grand total of 230 Space Groups are generated - see International ti Tables for X- ray crystallography. (Note no 5-fold or 7-fold axes; try tiling a surface with 50 (UK) pence coins; you can t cover the surface completely).