Lecture 2: Crystals and symmetry. 2.1 Crystallization of Biological Molecules 2.1.1 Physical Principles of crystallization 2.1.2 Methods for growing crystals 2.1.3 Identifying crystallization conditions 2.2 Some Fundamentals of Symmetry 2.2.1 What is symmetry and why is it important in structural biology ? 2.2.2 Symmetry operations in 2D 2.2.3 Symmetry Groups 2.2.4 A critical concept : The asymmetric unit 2.2.5 Symmetry operations in 3D 2.2.6 Enantiomorphism and the symmetries of biological specimens 2.3 Three-dimensional Symmetry Groups. 2.3.1 Point groups 2.3.2 Lattices and Translational Periodicity 2.3.3 Line Groups 2.3.4 Layer Groups 2.3.5 Space Groups 2.4 “Racemic crystallography” 2.5 “Non-crystallographic” symmetry Thursday, 6 March 14 1 Physical principles of crystallization. •The ability of protein crystals to diffract X-rays provides the experimental data required to determine the three-dimensional structure of proteins at atomic resolution. •However a primary difficulty in the application of the technique is that it depends on on obtaining well-ordered crystals. This is not trivial. •We'll discuss crystallization of water soluble proteins. However, pretty much the same considerations apply to the crystallization of membrane proteins and nucleic acids, with some added complications. e.g. ★Detergents proteins. are required to shield the hydrophobic surfaces of membrane ★The crystallization of nucleic acids often requires addition of compounds to neutralize the charge on the phosphate backbone. Thursday, 6 March 14 2 Some examples of protein crystals From McPherson (1999) Very pretty, and very useful, but how do we make them ? Thursday, 6 March 14 3 Protein crystallization: The basic idea •Protein crystals form in supersaturated solutions in which the protein concentration exceeds it's equilibrium solubility. •Hence all the physical techniques for crystallizing proteins involve bringing a protein solution into the supersaturated state by alteration of some property of the system. •Typically this is accomplished by gradually increasing the concentration of substances which serve to reduce protein solubility (protein precipitants), via a diffusive process. Salts, simple organic compounds and long chain synthetic polymers are all used as protein precipitants. •From a supersaturated protein solution, equilibrium can be restored by phase separation. Thursday, 6 March 14 4 Standard crystallization techniques used for inorganic compounds are generally not applicable Crystallization by Temperature Manipulation Copper Sulfate crystals grown by cooling a saturated hot CuSO4 solution Crystallization by Uncontrolled Evaporation Salt crystals covering rocks, Great Salt Lake, Utah (Photo Credit: Eve Andersson) Proteins commonly denature (irreversibly unfold) at extreme temperatures, or under other harsh conditions Thursday, 6 March 14 5 How do protein precipitants work? Long chain synthetic polymers work by simple volume exclusion Finet et al. Controlling biomolecular crystallization by understanding the distinct effects of PEGs and salts on solubility. Meth. Enzymol. (2003) vol. 368 pp. 105-29 Around the protein there is a zone which water can readily penetrate, but is not accessible to the polymer (the light gray layer in the diagram). The associated loss in configurational entropy raises the Gibbs energy of the system. Protein-Protein interactions, which reduce the volume of the exclusion zone and minimize the Gibbs Energy, are favored. Thursday, 6 March 14 6 sion of Precipitation by long-chain synthetic polymers Proteins by Polyethylene Glycols reasing size of the 5 are given in Fig. n of the solubility emolecular weight ) to 20,000. There h appears to ap- I 1 1 1.6 Most commonly used polymer is Polyethylene Glycol (PEG) 1.2 HO-CH2-(CH2-O-CH2-)n-CH2-OH 0.8 3 5 0.4 v) - Ln 0.0 -0.4 -0.8 I 0 10 20 30 % PEG-4000 IwlvJ 40 FIG. 4. Solubility of various proteins in PEG-4000. Measurements were made in 0.05 M potassium phosphate, pH 7.0, containing 0.1 M KC1. Human fibrinogen (X), 20 mg/ml; human y-globulin (D), 20 mg/ml; lysozyme (A), 50 mg/ml; a-lactalbumin (+), 20 mg/ml; chymotrypsin (O),7.5 mg/ml; aldolase (A),13 mg/ml; thyroglobulin (0),14 mg/ml; human serum albumin (O),20 mg/ml. Downloaded from www.jbc.org at UNIVERSITY OF AUCKLAND on Ap obtained by least he corresponding able 11. The value eement with that min under similar intercepts are conh small decreases own in Fig. 3, a ained by assuming on point and that pe. This is reasonn the type of PEG. in, thyroglobulin, lubility curves are r these conditions, e term in Equation crease in the slope m 400 to 4000. At eater, the slope for 4.6. However, the ions (not shown) cess of 1/5indicatactions. of several proteins tions are shown in aracteristic linear tration. Lysozyme ologous and have r under these condiffer greatly (4.7 though the larger and thyroglobulin) ntrations than the I “PEG 400” Average Molecular Mass 400 Da Average n = 9 Atha and Ingham. Mechanism of precipitation of proteins by polyethylene glycols. Analysis in terms of excluded volume. Journal of Biological Chemistry (1981) vol. 256 (23) pp. 12108 "1 theory, protein solubility varies with polymer concentration in As predicted from volume-exclusion a fairly simple way Thursday, 6 March 14 7 Precipitation by salts Salts exhibit more complicated behavior, and no compete physical theory exists to explain their ability to precipitate proteins. Generally at low concentrations they serve to increase protein solubility (“salting-in” behavior) whereas at higher concentrations they force proteins from solution (“salting out” behavior). From Blundell and Johnson (1976) Thursday, 6 March 14 8 Precipitation by salts The ability of salts to precipitate proteins varies systemically, and is predicted by the Hofmeister series. The physical origins of the series are still being debated. The series predicts other properties too, such as the ability to stabilize and destabilize proteins. Precipitate proteins, Structure stabilizing Thursday, 6 March 14 Solubilize proteins, Structure destabilizing 9 Protein Conecntration Protein crystallization: A schematic phase diagram Supersaturation Protein crystals form in supersaturated solutions Undersaturation Solubility limit Precipitant Concentration Thursday, 6 March 14 10 Nucleation. A critical step in crystal growth. •Protein crystal growth is generally a nucleated process. The critical nucleus = smallest stable crystal. Once a critical nucleus has formed, crystal growth will follow spontaneously. In practice the critical nucleus may be 10 -100 molecules in size. •This occurs because in small aggregates, where a large fraction of the molecules are at the surface, too few favorable intermolecular contacts are created to compensate for the entropic cost of a molecule joining the crystal. •The idea is easy to illustrate with a helical assembly (which is in essence a 1D crystal). • A subunit adding to a small assembly experiences only lateral interactions with its neighbors. A subunit adding to a large assembly experiences both lateral and vertical interactions. We might guess that, in this system, a single turn of the helix might be a critical nucleus ... Thursday, 6 March 14 11 Nucleation. A critical step in crystal growth. Weber, P.C. (1991) Adv Prot. Chem. 41, 1-36 Thursday, 6 March 14 12 Protein crystallization: A more detailed (but still schematic) phase diagram Crystals will dissolve Crystals will grow but not nucleate Crystals will nucleate and grow Protein will precipitate Protein Conecntration PRECIPITATION ZONE NUCLEATION ZONE METASTABLE ZONE Precipitant Concentration Thursday, 6 March 14 13 The Batch Method for Crystallization In the batch method we simply mix protein and precipitant in fixed ratio (typically 1:1) A real example: Batch crystallization of lysozyme Feher and Kam (1985) Meth. Enzymol 114. NaCl concentration %(w/v) Thursday, 6 March 14 14 The Batch Method for Crystallization Returning to a schematic phase diagram, batch crystallization should proceed according to path labelled (i). Chayen & Saridakis (2008) Nature Methods 5, 147-153. Thursday, 6 March 14 15 The Vapor Diffusion method for crystallization A slightly gentler way of approaching supersaturation is the vapor diffusion method. This relies on water transport through the vapor phase as depicted below. From Crystallization of Biological Macromolecules, McPherson Thursday, 6 March 14 16 The Vapor Diffusion method for crystallization Returning to a schematic phase diagram, vapor diffusion should proceed according to path labelled (ii). Chayen & Saridakis (2008) Nature Methods 5, 147-153. Vapor diffusion is the most widely used technique for preparing protein crystals. There are other physical methods available (e.g. dialysis and free-interface diffusion). We will not discuss them here. Thursday, 6 March 14 17 Identifying crystallization conditions From a supersaturated protein solution a solid phase can result from: The formation of disordered protein aggregates, leading to an amorphous precipitate or flocculate (easy to do) ... Images courtesy of Andrea Kurtz, SLAC ... or the formation of ordered aggregates leading to nucleation and growth of crystals (hard to do) Image courtesy of Andrea Kurtz, SLAC For a water-soluble protein, a typical crystallization trial will involve at least 3 solution components; the protein; a buffer to control the pH, and a precipitant to exclude the protein from solution. What must be experimentally determined are an appropriate buffer and precipitant together with the ranges of temperature, pH and concentration of the solution components that will support crystal growth (vs amorphous precipitation). Thursday, 6 March 14 18 Identifying crystallization conditions Because many hundreds of trials may be required to identify protein crystallization conditions, the trials are usually carried out in multi-well plates, and often using robotics. Thursday, 6 March 14 19 Protein crystals are fragile Protein crystals are formed by a sparse network of weak molecular interactions. Hence protein crystals are generally fragile and easily damaged. We’ll discuss some of the practicalities of handling them in lecture 5. Rupp (2010) Thursday, 6 March 14 20 Some Fundamentals of Symmetry Notion of symmetry depends on the existence of equivalent parts in a pattern or object From Bernal, Hamilton & Ricci (1972) If there are motions which carry or map any part of an object to the original position of any other, while leaving the appearance of the object unchanged, then the object is symmetrical. Such motions are called symmetry operations for the object. Symmetry is a mathematical ideal and may be imperfectly realized in nature. Thursday, 6 March 14 21 Why is symmetry important in structural biology? 1. Many biological molecules assemble into symmetric or nearly symmetric structures … this is critical to their function. Thursday, 6 March 14 22 Why is symmetry important in structural biology? 1. We take advantage of symmetry in scattering-based techniques (e.g. electron microscopy and X-ray crystallography) … it massively increases the signal-to-noise ratio. Understanding symmetry is critical to proper data analysis in both methods. 3D Protein Crystals Electron micrograph of icosahedral virus particles (L) and a derived 3D reconstruction (R). Images courtesy Tuli Mukhopadhyay Thursday, 6 March 14 23 The purpose of this lecture … 1. To introduce the basic concepts and language of symmetry. 2.To provide some tables and notation … reference material for the subsequent lectures. Thursday, 6 March 14 24 Symmetry operations and symmetry elements A symmetry operation may keep some points fixed in place; the set of such points is called the symmetry element for the operation. In the case of reflection in the plane, the symmetry element is a line. Once upon a midnight dreary, while I pondered weak and weary, Over many a quaint and curious volume of forgotten lore... Symmetry operation: reflection in the plane Symmetry element Thursday, 6 March 14 25 Symmetry in 2D Rotations, Reflections and Translations Rotation and Reflection are “Point symmetry Operations”. They leave at least one point of the object fixed From Bernal, Hamilton & Ricci (1972) Translation is a “space symmetry operations”. It leaves no point of the object fixed. Symmetry operations are rigid motions, motions which do not alter the distances between points. Rigid motions are also called isometries Thursday, 6 March 14 26 Some group theory ... Consider this object: From Bernal, Hamilton & Ricci (1972) It has exactly 5 symmetries. We may rotate by 72, 144, 216 and 288° without changing its appearance. That makes 4. The fifth symmetry is to do nothing at all (or equivalently, to rotate through 360°). This is termed the identity Thursday, 6 March 14 27 Some group theory ... From Bernal, Hamilton & Ricci (1972) These symmetries can be combined. We may first rotate by 288° (4x72) and then by 144° (2x72). The net result would be the same as rotating by 432° (6x72) - or more simply - by 72° - which is one of our original 5 symmetries. Thursday, 6 March 14 28 Some group theory ... From Bernal, Hamilton & Ricci (1972) The identity is special: • When we combine the identity with any other symmetry we don’t change the result. •Also every symmetry has an inverse that is its undoing. Combining a symmetry with its inverse produces the identity e.g. combine a rotation of 288° (4x72) with a rotation of 72° and we get a rotation of 360° (The identity operation). A rotation of 72° is the inverse of a rotation of 288° Thursday, 6 March 14 29 Some group theory ... From Bernal, Hamilton & Ricci (1972) Combination of symmetries is associative. If we define “x” to mean “followed by” then: (72° x 144°) x 216° = 72° x (144° x 216°) = 72° Thursday, 6 March 14 30 This collection of movements forms a mathematical group. A group of movements is a set that satisfies the following conditions •The product of any two movements in the set, or of any movement with itself, is a member of the set. •The identity is included as a movement. •For every movement there is an inverse, a member of the set such that the product of the movement and its inverse is the identity. •For three successive movements the associative law applies (M1*M2)*M3 = M1*(M2*M3) Thursday, 6 March 14 31 Our object is described by a symmetry group The operations of the group From Bernal, Hamilton & Ricci (1972) Thursday, 6 March 14 In this context, “multiplication” means “followed by”. 5 x 5 = A rotation of 72°, followed by a rotation of 72° 32 A more complicated point group in the plane. From Bernal, Hamilton & Ricci (1972) Thursday, 6 March 14 33 The asymmetric unit / fundamental region of a symmetric object. From Vainshtein (1994) The asymmetric unit of an object, described by a symmetry group, is the smallest compact region from which the whole object can be reconstructed through application of the symmetry. Thursday, 6 March 14 34 Symmetry operations in 3D One useful way to classify symmetry operations is by the number of points they keep fixed Point symmetry operations The identity Fixes 4 non-coplanar points Reflection Fixes 3 non-colinear points Rotation Fixes 2 points Rotary inversion/ Rotary reflection* Fixes 1 point Translation Fixes no points Glide Reflection Fixes no points Screw Rotation Fixes no points Space symmetry operations * N.B Rotary inversion and Rotary reflection are mathematically equivalent. Straight inversion is a special case Thursday, 6 March 14 35 Symmetry operations in 3D: Reflection From Vainshtein (1994) Thursday, 6 March 14 36 Symmetry operations in 3D: Rotation From Glusker & TrueBlood (1985) Thursday, 6 March 14 37 Symmetry operations in 3D: Inversion and Rotary Inversion (Rotary Reflection) From Vainshtein (1994) From Glusker & TrueBlood (1985) From Blow (2002) Thursday, 6 March 14 Anything that can be achieved by rotary reflection can be done by rotary inversion. For example rotation by 0° coupled with reflection is the same as rotation by 180° coupled with inversion 38 Symmetry operations in 3D: Translation From Glusker & TrueBlood (1985) Thursday, 6 March 14 39 Symmetry operations in 3D: Glide reflection From Vainshtein (1994) From Glusker & TrueBlood (1985) Thursday, 6 March 14 40 Symmetry operations in 3D: Screw Rotation From Glusker & TrueBlood (1985) From Glusker & TrueBlood (1985) An nm screw axis involves a rotation of 360/n° accompanied by a translation of m/n of the fundamental repeat distance. Thursday, 6 March 14 41 Symmetry operations in 3D: Screw Rotation From Glusker & TrueBlood (1985) An nm screw axis involves a rotation of 360/n° accompanied by a translation of m/n of the fundamental repeat distance. An 21 screw axis involves a rotation of 360 / 2 = 180° accompanied by a translation of 1/2 of the fundamental repeat distance. Thursday, 6 March 14 42 Symmetry operations in 3D: Screw Rotation An nm screw axis involves a rotation of 360/n° accompanied by a translation of m/n of the fundamental repeat distance. An 41 screw axis involves a rotation of 360/4 = 90° accompanied by a translation of 1/4 of the fundamental repeat distance. An 43 screw axis involves a rotation of 360/4 = 90° accompanied by a translation of 3/4 of the fundamental repeat distance. Thursday, 6 March 14 43 Graphical representations of symmetry operations From Giacovazzo (1992) Thursday, 6 March 14 44 ble pointgroup symmetries for biological macromolecules, the reader is refe complete list of all possible pointgroup symmetries including those with m Enantiomorphism and the etry, to the International Tables of Crystallography, and to Wikipedia: “Li symmetries of biological specimens cal symmetry groups”. Short story: There is no mirror symmetry in proteins Image courtesy of Marin van Heel. There is no mirror symmetry in Proteins! Thursday, 6 March 14 45 Slightly longer story: From Rupp (2010) Many biological molecules are chiral. This includes the amino acids - the building blocks of proteins. Natural proteins are made of L-amino acids. The enantiomers - D-amino acids - are used very rarely in nature. This means that for proteins and protein assemblies reflections and inversions are not “permitted” since they switch the handedness of an object. They would convert a protein made up of L-amino acids into a protein made up of D-amino acids. Enantiomorphism therefore limits the symmetries we observe with biological specimens Thursday, 6 March 14 46 While a biological assembly cannot accommodate mirror planes and centers of inversion, its diffraction pattern may … Tetragonal crystals of lysozyme From Fraser & Macrae (1969) Thursday, 6 March 14 47 Symmetry groups in 3D • We’ll deal first with the symmetry groups that don’t include translational symmetry (Point groups) • Then we’ll discuss translational symmetry and the concept of the lattice. • Finally we’ll deal with the symmetry groups that include translational symmetry (Line groups, Layer Groups, Space groups) Thursday, 6 March 14 48 Point groups The symmetries of non-periodic objects •A group of symmetry operations that leaves at least one point invariant (in the same place) is known as a point group •There are three families of point groups that can accommodate chiral molecules of fixed hand … the cyclic, dihedral and cubic point groups. They involve only rotations (reflections and inversions being disallowed). •Unluckily for you, point groups containing reflections and inversions are important for describing the symmetry of diffraction patterns, so we can’t ignore them entirely. Thursday, 6 March 14 49 A note on notation ... To describe the point groups we are going to use the International (Hermann-Mauguin) notation. This is the standard in crystallography ... Carl Hermann (1898-1961) A German crystallographer http://www.staff.uni-marburg.de/ Charles-Victor Mauguin (1878-1958) A French Mineralogist http://www-int.impmc.upmc.fr/public/Associations/afc/ A alternative notation - The Schönflies system - is commonly used for isolated molecules, particularly in chemistry and spectroscopy Thursday, 6 March 14 50 A note on notation ... The basics of Hermann-Mauguin notation. Operation: Rotation of 360°/n Symmetry element: A Rotation Axis Denoted by n (1,2,3,4,5,6,7,...) Operation: Reflection Symmetry Element: A Mirror plane Denoted by m Operation: Rotary inversion (Rotation of 360°/n coupled with inversion) Symmetry Elements: Combination of a Rotation axis and a point. - - -- -- - Denoted by n (1,2,3,4,5,6,7,...) Thursday, 6 March 14 51 “Biological” Point Groups part 1. Cyclic point groups From Vainshtein (1994) From Blow (2002) Thursday, 6 March 14 52 seen in negatively stained EM preparations. However, it is actu distinguish C4 from some of the other pointgroups (like 22 proper symmetry analysis. Head-to-tail types of inter-subu helical polymers rather than to closed Cn pointgroup assembl “n” values. To form a Cn structure requires “n” identical units A particularly common symmetry for pore-forming proteins polypeptide chains. Cyclic point groups: examples Top view Side view α-hemolysin; A heptameric pore-forming protein with 7-fold rotational symmetry Thursday, 6 March 14 The 3D reconstruction of the LTX oligomers. a, The top and Image courtesy of Marin van Heel C4 tetramer. c, The side and cut-open views of the tetramer. calculated from 128 class averages that included 1,900 ori Spider α-latrotoxin: The tetrameric (E.V. Orlova, M.A. Rahman, B. Gowen, K.E. Volynski, F. Meunie neurotoxin from the widow spider Ushkaryov: Structure of αblack -Latrotoxin: Dimers Assemble into T with 4-fold rotational Large, Gated Membrane Pores. symmetry Nature Struct. Biol. 7 (2000) 4853 “Biological” Point Groups part 2. Dihedral point groups From Vainshtein (1994) From Blow (2002) Thursday, 6 March 14 54 Dihedral point groups: examples Top view Side view Glutamine synthetase: Dodecamer with 622 point group symmetry Thursday, 6 March 14 Three orthogonal views, looking down the 2fold rotation axes of the tetramer Glucose fructose oxidoreductase: Tetramer with 222 point group symmetry 55 density maps of aromatic residues are clearly shown in the regions with higher correlation coefficients (~0.95). An electron density map of each 80.0% were in the most favorable region of the Ramachandran plot (25), 19.7% in the allowed region, and 0.3% in the disallowed region. Out of Dihedral point groups: examples Fig. 1. Overall structure of the vault shell. One molecule of MVP is colored in tan, and the others are colored in purple. (Left) Side view the mysterious ribbon representation. Thecomplex whole vault shell comprises a 78Theofvery rat liver vault oligomer polymer of MVP molecules. The size is ~670 Å from the top to the bottom 78-mer withof39the2 whole point particle group symmetry and ~400 Å in maximum diameter. The particle has two protruding caps, two shoulders, and a body with Tanaka et The structure liver vault at 3.5 angstrom resolution. Science vol. 323 (5912) pp. 384-8 an invaginated waist. Two half-vaults are associated atal.the waistof rat with N-terminal domains of(2009) MVP. (Right) Thursday, March 14 Top 6view of the ribbon representation. The maximum diameter of the cap is ~200 Å. The outer and the 56 23 … tetrahedral Order of the group: 24 432 … octahedral Order of the group: 48 “Biological” Point Groups part 3. Cubic point groups Images courtesy Dr Stefan Immel http://csi.chemie.tu-darmstadt.de/ak/immell 532 … icosahedral Order of the group: 60 Thursday, 6 March 14 57 Cubic point groups: examples Ferritin particle Cut-away view Ferritin - an iron storage protein 432 (octahedral) symmetry Image courtesy David Goodsell http://www.pdb.org/ Thursday, 6 March 14 58 Cubic point groups: examples http://www.virology.wisc.edu/virusworld/ Human echovirus 532 icosahedral symmetry Thursday, 6 March 14 Norovirus (“Norwalk virus”) 532 icosahedral symmetry 59 Summary: The Biological point groups CYCLIC 138 Schönflies notation J. Janin, R. P. Bahadur and P. Chakrabarti ... 1 2 3 4 5 Hermann-Mauguin notation Adapated from Janin et al. Protein-protein interaction and quaternary structure. Q Rev Biophys (2008) vol. 41 (2) pp. 133-180 Thursday, 6 March 14 60 Summary: The Biological point groups 138 J. Janin, R. P. Bahadur and P. Chakrabarti DIHEDRAL Schönflies notation ... 222 32 422 52 Hermann-Mauguin notation Fig. 1. Symmetry of oligomeric proteins. An oligomeric protein with n identical subunits may have the symmetries of the cyclic Cn point group (top row), one with 2n subunits, the symmetries of the dihedral Dn point group (middle row) cubic row) requirestructure. the protein to have 12,vol.2441or identical Adapated from; Janin et al.symmetries Protein-protein(bottom interaction and quaternary Q Rev Biophys (2008) (2)60 pp. 133-180 subunits. Symmetry axes of different types are marked as dotted lines. Courtesy of E. Lévy (Cambridge, Thursday, 6 March 14 UK). 61 Summary: The Biological point groups CUBIC Schönflies notation T (Tetrahedral) O (Octahedral) I (Icosahedral) Fig. 1. Symmetry of oligomeric proteins. An oligomeric protein with n identical subunits may have the (top row), one with 2n subunits, the 532 symmetries of the dihedral Dn symmetries 23 of the cyclic Cn point group 432 point group (middle row) ; cubic symmetries (bottom row) require the protein to have 12, 24 or 60 identical subunits. Symmetry axes of different types are marked as dotted lines. Courtesy of E. Lévy (Cambridge, UK). Hermann-Mauguin notation Thursday, 6 March 14 (Fig. 1). Oligomers that display the symmetries of a cyclic Cn group have an n-fold axis : their subunits are related by 360x/n rotations. The dihedral Dm groups require an even number of subunits, n=2m ; they possess an m2-fold axis and m2-fold axes orthogonal to it. The T (tetrahedral) cubic point group has non-orthogonal twofold and threefold axes ; in addition, the O (octahedral) point has fourfold and the I (icosahedral) point group, fivefold axes. Adapated from Janin etgroup al. Protein-protein interactionaxes, and quaternary structure. Q Rev Biophys (2008) vol. 41 (2) pp. 133-180 Symmetry is a general property of oligomeric proteins (Goodsell & Olson, 2000). The most 62 Other Families of Point Groups, incorporating the “forbidden” inversions and mirror planes Images from Vainshtein (1994) 32 Thursday, 6 March 14 63 Important sets of point groups I. The crystallographic point groups, which describe the rotational symmetries of crystals Crystals (which we’ll come too shortly) can only accommodate rotation axes of a certain order. 1,2,3,4 and 6 fold symmetries are okay. 5 fold symmetry is out. So is 7-fold, 8-fold and all higher rotational symmetries. There are consequently only 11 “crystallographic” point groups involving pure rotations. Here they are … From Blow (2002) Thursday, 6 March 14 64 Important sets of point groups II. The Laue groups which describe the point symmetry of X-ray diffraction patterns Notice there are 11 crystallographic point groups and 11 Laue groups - and there is an obvious correspondence between them. If you add a center of inversion to a crystallographic point group you generate one of the Laue groups. Images from Vainshtein (1994) Thursday, 6 March 14 65 Lattices and Translational periodicity • A lattice is a convenient representation of translational symmetry. • Lattices are the framework on which much crystallographic theory is built. It’s all best illustrated by example … Thursday, 6 March 14 66 A crystal is the convolution of an object and a lattice From Holmes and Blow (1965) Thursday, 6 March 14 67 A crystal is the convolution of an object and a lattice From Chiu, Schmid and Prasad (1993) Thursday, 6 March 14 68 Lattices in 1 and 2 dimensions There is 1 unique 1-dimensional lattice … a There are 5 unique 2-dimensional lattices A 1D lattice is generated by a single translation A 2D lattice is generated by 2 linearly independent translations From Vainshtein (1994) Thursday, 6 March 14 69 The unit cell The generators of a 2D lattice form two edges of a box … if we complete the box we have a simple unit cell. This is a parallelogram with a lattice point at each vertex and no lattice point anywhere else inside it or on its surface. The orthorhombic primitive lattice a b Conventional unit cell choice The entire lattice can be constructed from a single unit cell by stacking these boxes together. Since each lattice point is shared by four boxes and each box has four vertices, this works out to be one lattice point per unit cell. Thursday, 6 March 14 70 The unit cell The lattice generators are not unique, and the unit cell can be chosen in infinitely many different ways. This is easy to see in two dimensions. However these choices do not reflect the symmetry of the lattice. By convention we choose a unit cell which does reflect this symmetry The orthorhombic primitive lattice Some alternate cell choices Thursday, 6 March 14 71 The unit cell Following this line of reasoning crystallographers select a “non-primitive” or “centered” unit cell for one of the 2D lattices This cell contains 2 lattice points per cell. This choice drives mathematicians crazy*, but is actually quite sensible and convenient for crystallography. The orthorhombic centered lattice a a b A primitive unit cell b The Centered unit cell standard in crystallography. * The cell edges no longer generate the lattice, which destroys the mathematical theory Thursday, 6 March 14 72 Lattices in 3 dimensions. The 14 Bravais Lattices A 3D lattice is generated by 3 linearly independent translations. There are exactly 14 of them, no more, no less. Figures from Woolfson (1970) Thursday, 6 March 14 73 The Unit Cell in 3 Dimensions The generators of a 3D lattice form three edges of a box … if we complete the box we have a simple unit cell. This is a parallelepiped with a lattice point at each vertex and no lattice point anywhere else inside it or on it’s surface. From Drenth (2002) The entire lattice can be constructed from a single unit cell by stacking these boxes together. Since each lattice point is shared by eight boxes and each box has eight vertices, this works out to be one lattice point per unit cell. Thursday, 6 March 14 74 “Centered unit cells” Just as in 2D, crystallographers choose the unit cell so that it reflects the symmetry of the lattice. Sometimes this puts lattice points at the center of a unit cell, or on its faces. Once again this generates “non-primitive”, “centered” unit cells containing more than one lattice point per cell. From Drenth (2002) Thursday, 6 March 14 75 Symmetry of the lattice imposes constraints on the unit cell dimensions From Blow (2002) Lattices are assigned to crystal systems according to their symmetry though mathematicians and crystallographers do this slightly differently. This time we’re going to side with the mathematicians (The argument centers over where to place the Rhombohedral Lattice). Thursday, 6 March 14 76 The seven crystal systems System Essential rotational symmetry Conventional Unit Cell choice of unit cell restrictions axes Possible Lattices Triclinic None No constraints None P Monoclinic Two-fold axis b parallel to 2-fold α = γ = 90° P,C Orthorhombic Three perpendicular 2-fold axes a, b, c parallel to 2fold axes α = β = γ = 90° P,C,I,F Trigonal/Hexagonal 3-fold or 6-fold axis c parallel to 3-fold or 6-fold a=b α = β = 90°, γ = 120° P Rhombohedral 3-fold axis a, b, c related by three fold axis a=b=c α=β=γ R Tetragonal 4-fold axis c parallel to 4-fold a=b α = β = γ = 90° P,I Cubic 4 3-fold axes a, b, c related by three fold axis a=b=c α = β = γ = 90° P,I,F Thursday, 6 March 14 77 Symmetry groups involving translational periodicity. Skipping over the mathematical complexities, these groups are derived by combining point group symmetry with a lattice. The lattice might be 1, 2 or 3 dimensional leading to … The Line (or Rod) groups … the symmetries of helices The Layer (or Two-sided Plane) groups … the symmetries of 2D crystals The Space groups … the symmetries of 3D crystals. Thursday, 6 March 14 78 Line Groups: The symmetries of 3D objects periodic in 1 Dimension These groups describe the symmetries of helices and rods. They are derived by combination of a 3D point group and a 1D lattice. Very important in X-ray Fiber diffraction and Helical image reconstruction. They are constructed by combination of a point group and a 1D lattice. We do not consider them in detail in this course. Tobacco Mosaic Virus … image from Don Caspar Thursday, 6 March 14 79 Layer Groups (Two-sided plane groups): The symmetries of 3D objects periodic in 2 Dimensions These groups describe the symmetries of “2D” crystals. They are constructed through combination of a 3D point group and a 2D lattice. From Mosser (2001) Thursday, 6 March 14 80 Transmission electron microscopy … The relationship between the Layer groups and the Plane groups Electrons Layer Group Crystal Projection Plane Group Thursday, 6 March 14 47 81 Layer Groups (Two-sided plane groups) The symmetries of 3D objects periodic in 2 Dimensions These groups describe the symmetries of “2D” protein crystals. There are 80 Layer groups in total, but only 17 can accommodate biological specimens. From Engelhardt (1988) Thursday, 6 March 14 82 Plane Groups The symmetries of 2D objects periodic in 2 Dimensions These groups describe the symmetries of crystals in projection. They are therefore of critical importance in electron crystallography. Happily, the 17 “Biological” Layer groups each have a unique corresponding Plane group. From Engelhardt (1988) Thursday, 6 March 14 83 The 17 “Biological” Layer Groups 2D Bravais Lattice Oblique 3D point group Layer group 1 p1 2 p21 Primitive orthorhombic 2 p12, p121 222 p222, p2212, p22121 Centered orthorhombic 2 c12 222 c222 Square 4 p4 422 p422, p4212 3 p3 32 p312, p321 6 p6 622 p622 Hexagonal Thursday, 6 March 14 84 A closer look at the Layer groups based on point group 222 From Engelhardt (1988) Tshudy, D. K. & Litvin, D. B. VRML general position/symmetry diagrams of the 80 layer groups. J Appl Cryst 31, 973–973 (1998). There are three different ways to combine the point group 222 with the primitive orthorhombic lattice Thursday, 6 March 14 85 Space Groups: The symmetries of 3D objects periodic in 3 Dimensions These groups describe the symmetries of 3D crystals. They underpin X-ray crystallography. They are constructed by combining a 3D point group with a 3D lattice. There are 230 Space groups in total. 65 of these do not involve reflection or inversion, and can accommodate biological molecules of a fixed hand. Thursday, 6 March 14 86 Space Groups: The symmetries of 3D objects periodic in 3 Dimensions The simplest space groups to understand are those that result from “stacking” the layer groups (if you stack an object periodic in two dimensions, you get an object periodic in three dimensions). To illustrate here’s a schematic of a protein molecule with P4 layer group symmetry. If we stacked these structures, one on top of another, to generate a 3D crystal - this crystal would have P4 space group symmetry. Not too difficult too visualize. Rupp (2010) Thursday, 6 March 14 87 Space Groups: The symmetries of 3D objects periodic in 3 Dimensions While we are here, let’s consider what the asymmetric unit of this crystal would be. Recall that the asymmetric unit of a crystal is the region from which the entire crystal can be generated through application of the symmetry operations. Rupp (2010) The asymmetric unit of layer group/space group P4 covers 1/4 of the unit cell. Above are two possible asymmetric units. The choice on the left is ideal for visualizing the tiling, but the choice on the right is much better suited for visualizing the molecule. The asymmetric unit needs to cover 1/4 of the cell - it doesn’t matter mathematically which 1/4 you choose Thursday, 6 March 14 88 Another very simple space group … P2 From Blow (2002) Thursday, 6 March 14 89 The asymmetric unit of P2 From Outline of Crystallography for Biologists, Blow The asymmetric unit of a crystal with P2 symmetry, covers 1/2 of the unit cell. In this case it contains in totality, 1 duck. Note that again - the duck doesn’t fit neatly in the bounds of the crystallographic cell. Thursday, 6 March 14 90 A unit cell’s worth of some other simple space groups Alas, not all the 3D space groups are so simply related to the 2D plane groups. In general, the space groups can be quite hard to visualize. The great compendium of information on the space groups is the International Tables for Crystallography. From Cantor and Schimmel (1980) Thursday, 6 March 14 91 International Tables:Space Group P2 space group point group crystal system Symmetry of the diffraction pattern Projection diagrams showing the position of the symmetry elements in the unit cell Bounds of the asymmetric unit Mathematical operations which will generate the cell contents from the asymmetric unit. Specifies diffraction data which will be “systematically absent” ... this is critical for space group determination For Further information see Dauter, Z. & Jaskolski, M. How to read (and understand) Volume A of International Tables for Crystallography: An introduction for nonspecialists. J Appl Cryst 43, 1150-1171 (2010). Thursday, 6 March 14 92 The 65 “Biological” Space Groups Bravais Lattice Possible space groups Associated point group Primitive Cubic P23 (195), P213 (198) P432 (207), P4132 (213), P4232 (208), P4332 (212) 23 432 I centered Cubic I23 (197), I213 (199) I432 (211), I4132 (214) 23 432 F centered Cubic F23 (196) F432 (209),F4132 (210) 23 432 Rhombohedral R3 (146) R32 (155) 3 32 Primitive Hexagonal P3 (143), P31 (144), P32 (145) P312 (149), P3112 (151), P3212 (153), P321 (150), P3121 (152), P3221 (154) P6 (168), P61 (169), P62 (171), P63 (173), P64 (172), P65 (170) P622 (177), P6122 (178), P6222 (180), P6322 (182), P6422 (181), P6522 (179) 3 32 6 622 Primitive Tetragonal P4 (75), P41 (76), P42 (77), P43 (78) P422 (89), P4212 (90), P4122 (91), P41212 (92), P4222 (93), P42212 (94), P4322, (95), P43212 (96) 4 422 I centred Tetragonal I4 (79), I41 (80) I422 (97), I4122 (98) 4 422 Primitive Orthorhombic P222 (16), P2221 (17), P21212 (18), P212121 (19) 222 C Centered Orthorhombic C222 (21), C2221 (20) 222 I Centered Orthorhombic I222 (23), I212121 (24) 222 F Centered Orthorhombic F222 (22) 222 Primitive Monoclinic P2 (3), P21 (4) 2 C Centered Monoclinic C2 (5) 2 Triclinic P1 (1) 1 Thursday, 6 March 14 93 “Racemic” crystallography Having emphasized that space groups containing mirror planes and centers of inversion are not relevant to the crystallography of biological molecules, we note some circumstances in which this is not true. Using total chemical synthesis one can produce small proteins entirely from D- amino acids. This may fold into the mirror image of the biologically occurring L-protein If you create a racemic mixture of L- and Dproteins, the 165 “disallowed” space groups can be accessed. L- and D- plectasin crystallized in space group P1, which contains a center of inversion. It appears that racemic mixtures of proteins may be easier to crystalize than the single biological L-enantiomers. There are also other crystallographic advantages which we will touch on later. The principal problem is that total chemical synthesis of proteins, is time consuming, difficult, and not applicable to large polypeptides. Mandal, K. et al. Racemic crystallography of synthetic protein enantiomers used to determine the X-ray structure of plectasin by direct methods. Protein Sci 18, 1146-1154 (2009). Thursday, 6 March 14 94 Non-crystallographic symmetry If there’s more than one equivalent copy of a molecule in the asymmetric unit of a crystal, then there will be some symmetry operation(s) which relate the molecules. These symmetry operations are called non-crystallographic or “local” because they extend only a finite distance through the crystal. The diagram on the right illustrates the basic idea, in 2 dimensions. Non-crystallographic symmetry occurs frequently with protein cr ystals. Depending on it’s nature, it can either help or hinder structure determination. We’ll return to this in the later lectures. Thursday, 6 March 14 95