PX-CBMSO Course (2) The mathematical … · The mathematical descriptionThe mathematical...

PX-CBMSO Course (2)The mathematical descriptionThe mathematical descriptionof Symmetryy y

Cele Abad-ZapateroUniversity of Illinois at ChicagoC t f Ph ti lCenter for Pharmaceutical Biotechnology.Lecture no. 2This material copyrighted by py g yCele Abad-Zapatero.

PX-CBMSO-June 2011

Crystallography:i i fDescription of symmetry

Theory of Diffraction

Structure Determination

Refinement

Mathematical description of symmetry

r r’: thee-dimensional vectors

r’= A ·r + Tr, r : thee-dimensional vectorscomponents (x, y, z)

A: 3x3 matrix T: column matrix 3x1

r (x, y, z)r (x, y, z)

r’ (x’,y’,z’)Type of symmetry operationis encoded in the propertiesof the MATRICES:

Symmetryoperation

A: rotation, reflection(operator)

T: translation

Portions of this material have been t t d f L t 1 f thextracted from Lecture 1 of the

educational materials donated by yProf. JR Helliwell (Univ. Of Manchester UK) to the IUCr forManchester, UK) to the IUCr for educational purposes

Example of amino acid ienantiomers

The handedness of an object is changed by a mirror plane or anchanged by a mirror plane or an

inversion centre

xx

Mirror planey

z

Enantiomers descriptionEnantiomers description

• Optical isomers are Non SuperimposableOptical isomers are Non Superimposable Mirror Images of each other; a set of optical isomers are called enantiomersoptical isomers are called enantiomers.

Mathematical description of symmetry (from before):

r’= A ·r + Tr, r’: thee-dimensional vectorscomponents (x, y, z)

A: 3x3 matrix T: column matrix 3x1

Enantiomers: the determinant of matrix A will be negativeEnantiomers: 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 + 0y = 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 operationr’= A · r + T

x’ cos α sin α 0 x 0

y’ = i 0 + y 0x,y

x’,y’

αy = - sin α cos α 0 + y 0

z’ 0 0 1 z 0

α

The key equation to describe symmetry mathematicallyMatrix 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):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 axisperp to planeperp. to plane

Motif Motif+ syop unit Cell Motif+syop unit cell+ translationMotif Motif+ syop unit Cell Motif+syop unit cell+ translation symmetry

Unit cell and asymmetric unit (symmetry operators: 180º)(symmetry operators: α=180º)

5th motif6th motifasymmetric

unit cell

5 motif6th motifasymmetricunit a.u.

1st motif4th motif

2nd motif 3rd motif

What will be the form of the matrix A -1 0 0 0

2 motif 3 motif

Exercise:ops. for:

for 2nd motif symmetry operation above?and all the others?

0 -1 0 00 0 1 0

1-2

+ 1-3; 1-4;1-5; 1-6;

Understanding decorations in The Alhambra: planar crystal (and color) symmetry

Four identical pointswith identical environment define the unit cell.define the unit cell.

In crystals, in wallpaper, inf b i tt ifabric 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 toFor 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 the1) 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!), y )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) I d k d f i hi h i ll i i4) 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 LatticesLattices

P I F

CIP P

There are 7 crystal systems (unit cell shapes).

There are 14 Bra ais lattice t pesThere are 14 Bravais lattice types

There are 230 space groups, where the space group is the collection of symmetry

FIP C

collection of symmetry operations of the crystal. Proteins being enantiopure take up only 65 space groups (i.e. P

P P

missing mirror and inversion centre symmetry elements).

Pictorial and algebraic representation of a crystal

Th iti f i t ithi th ‘LOVE’ tif b d fi d b ’ A tThe position of any point within the ‘LOVE’ motif can be defined by r’ = A.r+ tRight? The vector algebra might be tedious but is conceptuallysimple. This is how you generate the ‘crystal’, by vector algebra.

One of the most importantconcept in crystallography:

THERECIPROCALLATTICE.

A LATTICE MADE UPOF INVERS DISTANCES:

1/d-(interplanar distance)

Analogy:Space of frequencies!Space of frequencies!

Not of wavelengths.

Sometimes calledFOURIER SPACE orRECIPROCAL SPACE

Matrices can also be usedTo go from REAL > RECIPROCALand RECIPROCAL>REAL

2-fold axis example and a.u.(very important)(very important)

5th motif6th motifasymmetric 5 motif6th motifasymmetricunit a.u.

1st motif4th motif

2nd motif 3rd motif

What will be the form of the matrix A -1 0 0 0

2 motif 3 motif

Exercise:ops. for:

for 2nd motif symmetry operation above?and all the others?

0 -1 0 00 0 1 0

1-2

+ 1-3; 1-4;1-5; 1-6;

The unit cell and Bravais LatticesLattices

P I F

CIP P

There are 7 crystal systems (unit cell shapes)

There are 14 Bra ais lattice t pesThere are 14 Bravais lattice types

There are 230 space groups, where the space group is the collection of symmetry

FIP C

collection of symmetry operations of the crystal. Proteins being enantiopure take up only 65 space groups (i.e. P

P P

missing mirror and inversion centre symmetry elements).

KEY points to remember about crystallographic symmetrysymmetry.

All the possible Space Groups are presented in their entirety in “International Tables” using various diagrams including projections toInternational Tables using various diagrams including projections to aid 3D visualisation.

They are also coded in the various crystallographic packagesThey are also coded in the various crystallographic packagesby symbol (i. e. P21) or number (i.e. No. 4).

There is a listing of possible atomic symmetrically related locations g p y yx,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’. MotifMotif

The rest of the crystal can be generated by applyingthe symmetry operator matrices and translations to the y y pbasic motif.

The simplest space group: P1p p g p

For a triclinic cell it is possible to have only the identity operator (x,y,z); A point of i i ( ) ll d finversion (-x,-y,-z), not allowed for biological macromolecules.

(i)P1 is the simplest Space Group: only translation symmetry.

P1 is pure translational symmetry

For monoclinic systems the symmetry

OTHER SYMMETRY OPERATIONS POSSIBLE IN THE MONOCLINIC SYSTEM

For monoclinic systems the symmetry possibilities increase with C-centered as well as Primitive cells, 2-fold axes and mirror

2-fold axis

NOT IN PROTEINS

planes in addition to points of inversion can occur.

ALSO operations, which combine point mirror

operations with a translation, are possible :-

(i) c-glide plane (denoted c), a r fl ti i th l dreflection in the xz plane and translation half way down c (1/2+x,-y,1/2+z).

(ii) 2-fold screw axis (denoted 2 ) a

c/22-fold screw

(ii) 2 fold screw axis (denoted 21) , a 2-fold rotation about b and translation half way down b (-x,1/2+y,-z).

c-glide

b/2( , y, ) b/2

Screw axes operator(VERY COMMON IN PROTEIN XTALS!)(VERY COMMON IN PROTEIN XTALS!)

A projection along the 2 screw axis gives21 screw axis gives rise to a zigzag pattern of rotation-relatedof rotation related molecules.Operation has two pparts:

1. 180 degree rotationg2. Translation by ½ of

the axis.a

b

ac

A screw-axis operation along b r’= A · r + t

’ ’x’ cos α 0 sin α x 0

y’ = 0 1 0 y 1/2

x’,z’

y = 0 1 0 y 1/2

z’ - sin α 0 cos α z 0x,z

b/2


positions: x,y,z initial; after operation: -x, y+1/2, -z

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 componentVector 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

y’ = 0 1 0 y 1/2

x’,z’

y = 0 1 0 y 1/2

z’ 0 0 -1 z 0x,z

b/2





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

y’ = 0 1 0 y 1/2

x’,z’

y = 0 1 0 y 1/2

z’ - sin α 0 cos α z 0x,z

b/2





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 i P2 2 2group is P212121

• Four equivalent positions:-Four equivalent positions:• x,y,z• x+1/2 y z+1/2• -x+1/2,-y,z+1/2• -x,y+1/2,-z+1/2

+1/2 +1/2• x+1/2,-y+1/2,-z• Reflection absence conditions: for h00

h 2 +1 b t f 0k0 k 2 +1 b t fh=2n+1 absent; for 0k0 k=2n+1 absent; for 00l l=2n+1 absent. [absent means an exactly zero intensity ]exactly zero intensity.]

Entry in the International Tables for P212121

Diffraction pattern: Precession photo of the reciprocal lattice

How do crystallographersknow about details of the lattice and

Diffractionpattern is a photograph

the space group?

b* axisp p g pof the reciprocallattice.

It does show the internalIt does show the internalsymmetry of theparticular crystal lattice

d in mm (or pixels)/24 = the a* axis

d

Systematic absencesSystematic absencesNotice that all odd

reflections along both the hreflections along both the h and k axes are absent. This shows there must be 21screw axes along thesescrew 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:

Non-orthogonalLattice

4,4

No-systematic absences

h 0 0 reflections

This diffraction pattern containsa portion of the hk0 plane of the

This ‘precession diffraction image’ shows a 2D projection of spots; the axial

reciprocal lattice of a monoclinic crystal

p g p j preflections 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 axesScrew axesA projection

di l t 2perpendicular to a 21screw axis gives rise to a zigzag pattern ofa zigzag pattern of mirror-related molecules.Operation has two parts: p

1. 180 degree rotation2. Translation by ½ of y

the axis.

Systematic absencesSystematic absences

Notice that all oddNotice that all odd reflections along both the h and k axes are absent. This shows there must be 21shows there must be 21screw 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 casecase(ii) Body-centered (I) h+k+l ≠2n(iii) Face-centered (F) h+k and h+l and k+l ≠2n Oth ti l l d t bOther 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( ) g , ,(ii) due to 21, 0 k 0, k≠2nThe systematic absences are listed in “International Tables” as an aid to Space Group identificationaid to Space Group identification.

Using the above it is usually possible to assign a crystal to it S G d h t k d t f th tits Space Group and hence take advantage of the symmetry information during structure solution and refinement. Crystal structures are reported in terms of unit-cell y pdimensions, 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 P22217 P222118 P2121219 P212121

(0,0,2n)(2n,0,0),(0,2n,0)(2n,0,0),(0,2n,0),(0,0,2n)

C222C Centered Orthorhombic 20 C2221

21 C222(0,0,2n)

I Centered Orthorhombic 23 I222 *

C222Centeredorthorhombic

I212121

B d t d 24 I212121 *

F Centered Orthorhombic 22 F222

Primitive Monoclinic 3 P2

Body centeredorthorhombic

4 P21 (0,2n,0)

C Centered Monoclinic 5 C2

Primitive Triclinic 1 P1

Space Group: S.G. number or SG. symbol.

Entry in the International Tables for P212121

Tetragonal systems

Primitive Tetragonal 75 P4

Tetragonal systems…. 4-foldor4 i

z

76 P4177 P4278 P43

89 P42290 P4212

(0,0,4n)*(0,0,2n)(0,0,4n)*

(0 2n 0)

41-screw axis

90 P421291 P412292 P4121293 P422294 P4221295 P4322

(0,2n,0)(0,0,4n)*(0,0,4n),(0,2n,0)**(0,0,2n)(0,0,2n),(0,2n,0)(0,0,4n)*2

y

96 P43212( , , )(0,0,4n),(0,2n,0)**

I Centered Tetragonal 79 I480 I41

97 I422

(0,0,4n)21

2x

97 I42298 I4122 (0,0,4n)

Space groups of Tetragonal crystals:

symbolnumber

Space groups of Tetragonal crystals:Example P43212

Hexagonal systems….

Primitive Hexagonal 143 P3144 P31145 P32

149 P312

(0,0,3n)*(0,0,3n)*3,6-fold

or3 6 i

z149 P312151 P3112153 P3212

150 P321152 P3121

(0,0,3n)*(0,0,3n)*

(0,0,3n)*

3,61-screw axis

152 P3121154 P3221

168 P6169 P61170 P65

(0,0,3n)(0,0,3n)*

(0,0,6n)*(0,0,6n)*2

y

171 P62172 P64173 P63

177 P622178 P6122

(0,0,3n)**(0,0,3n)**(0,0,2n)

(0 0 6n)*

2

2x

178 P6122179 P6522180 P6222181 P6422182 P6322

(0,0,6n)*(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 P23198 P213 (2n,0,0)

207 P432208 P4232212 P4332213 P4132

(2n,0,0)(4n,0,0)*(4n,0,0)*

I Centered Cubic 197 I23199 I213

211 I432214 I4132 (4n,0,0)

F Centered Cubic 196 F23

209 F432210 F4132

(4n,0,0)

Primitive Rhombohedral 146 R3Primitive Rhombohedral 146 R3

155 R32

Assignment of space groups: Systematic absences are very important

• No. 16 P222No. 17 P2221

No. 18 P21212No. 19 P212121

• 16 has no absence conditions•

17 (0 0 2n+1) ) (ONLY EVEN REFLECTIONS ARE PRESENT ALONG THE c-axis17 (0,0,2n+1) ) (ONLY EVEN REFLECTIONS ARE PRESENT ALONG THE c-axis18 (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 degreesangles 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 P212121p p g p

Crystal packing; monoclinic insulin-h ( i f di )hexamers (trimer of dimers)

U it ll

Notice largesolvent channels

Unit cell

solvent channelsallowing flow of water, ligands, ions,etc.

Point symmetry ofhexamer: 32

INSULIN HEXAMERS IN THE CRYSTAL LATTICE

SummaryCrystallographic symmetry can be described in a concise d ff i b i i / l b hand 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

h theach other.

All subsequent work will be based on thisso ‘be careful’. In qthe early times of protein crystallography, determining the space group of your crystals was a major milestone.

Go to: http://img.chem.ucl.ac.uk/sgp/mainmenu.htmS l t ith i t

ASSIGNMENT (Exercise):

Select a space group with main symmetryfourfold (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). p p p g p ( , q p )Turn it in after next class (June 13, 2011) with your name on it. Advanced: demonstrate that the Sym. Ops. form a ‘Group’.

High- Medium-HighResolution Space GroupDiagrams and T bl

MediumResolution Space GroupDiagrams and T blTables

(1280 × 1024) pixel screens)

Tables(1024 × 768) pixel screens)

This outlines and illustrates the mathematicaldescription of symmetry as applied to protein crystalsdescription of symmetry as applied to protein crystals.

Next Lecture will the outline of diffraction theoryNext Lecture will the outline of diffraction theory

and next

structure determination

Questions?

This leads to a total of 13 unique monoclinic Space Groups:-Groups:P2, P21, Pm, Pc, P2/m, P21/m, P2/c, P21/c, C2, Cm, Cc, C2/m, C2/c.

P21/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 (21)(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 G t d I t ti l T bl f XGroups are generated - see International 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 covertiling a surface with 50 (UK) pence coins; you can t cover the surface completely).

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