Structural basis of protein stability at poly extreme crystal structure of amya at 1 6 a resolution

1.2.2 Lattice, point group and space group A lattice is classically defined as a group of points organized in space in such a way that each point has the same environment.. They can be

CHAPTER 1

INTRODUCTION TO MACROMOLECULAR X-RAY

CRYSTALLOGRAPHY

The 1901 Nobel Prize for physics was awarded to Roentgen for his discovery

of X-rays X-rays are electromagnetic waves whose wavelengths are in the range of 0.1-100 Å They are produced when rapidly moving electrons strike a solid metal target and their kinetic energy is converted into radiation The wavelength of the emitted radiation depends on the energy of the electrons In 1912, von Laue’s group discovered X-ray diffraction by crystals and this discovery gave rise to the development of a very rich scientific period and created a new academic branch – X-ray crystallography One year later, W L Bragg determined the first crystal structure From then on, crystal structure determination has been broadly undertaken on inorganic and organic molecules

X-ray crystallography is now a commonly used technique for determination of the three-dimensional structure of biomolecules The methodology is fairly robust in that the experimental and computational methods for these studies are now well developed The use of advanced protein expression and purification procedures, crystallization robots and powerful synchrotron radiation sources has enabled high-throughput structure determination This chapter briefly discusses the concepts and methodologies used in macromolecular X-ray crystallography

1.1 MACROMOLECULAR CRYSTALLIZATION

To perform X-ray crystallography, it is necessary to grow crystals with edge lengths around 0.1-0.3 mm Crystals are formed as the conditions in a supersaturated solution slowly change For small molecules, growing large crystals is relatively simple Proteins are difficult to crystallize because of their complexity, molecular weight and flexibility Also purification of a protein to homogeneity is a very tedious process The strategy to crystallize a protein is to guide a protein/solvent system very slowly toward a state of reduced solubility by modifying the properties of the solvent

or the character of the macromolecule This is most frequently accomplished by increasing the concentration of precipitating agents or by altering some physical properties (e.g., pH, temperature) to achieve supersaturation Efforts then have to be put into the refinement and optimization of the crystallization conditions that will encourage and promote specific bonding interactions between molecules, bigger single crystal formation and to stabilize the crystals once they are formed

The ‘salting in’ and ‘salting out’ properties of proteins have been used to push them into supersaturation Although the ‘salting in’ effect can be used as a method for crystallization, however, most proteins are not stable at a low salt environment Therefore, exploration of the protein ‘salting out’ property is more commonly used A number of methods have been attempted to bring proteins in an unsaturated state gradually into a supersaturated state The most commonly used method to crystallize proteins is the vapor diffusion method A drop of protein solution is suspended over a reservoir containing buffer and precipitant Water diffuses from the drop to the reservoir solution leaving the drop with optimal crystal growth conditions The other methods include batch crystallization, micro-batch crystallization and dialysis

1.2 BASIC CONCEPTS OF CRYSTALLOGRAPHY

1.2.1 Crystal, unit-cell and asymmetric unit

Protein crystals are usually about 40-60% solvent by weight and are thus fragile and sensitive to drying out In a crystal, molecules are arranged with regular repeats of symmetry A unit-cell is defined as the smallest possible volume that when repeated, represents the entire crystal The dimensions of a unit-cell can be described with 3 edge lengths (a, b, c) and 3 interaxial angles (α, β, γ) The location of atoms within a unit-cell can be listed in the Cartesian coordinate system

The smallest volume within the unit-cell that can be rotated and translated to generate one unit-cell is called the asymmetric unit Only the symmetry operators that are allowed by the crystallographic symmetry must be used for the construction of the entire unit-cell Even though the asymmetric unit may commonly contain only one molecule or one subunit of a multimeric protein, it can also be more than one

1.2.2 Lattice, point group and space group

A lattice is classically defined as a group of points organized in space in such

a way that each point has the same environment There are 14 types of unit-cells in crystallography that lead to 14 Bravais lattices The Bravais lattices are the distinct lattice types which, when repeated can fill the whole space They can be classified as primitive (simple unit-cell), face centered (equals the simple lattice with the addition

of a lattice point in the center of each of the six faces of each unit-cell), body centered (point at the center of the cell) and end centered (point at the center of one face) The cubic crystal system (which warrants a cubic unit-cell) can have a primitive, body centered and face centered lattice; the tetragonal system can have a primitive and

body centered lattice; the orthorhombic system can have a primitive, face centered, body centered and end centered lattice; the hexagonal crystal system can only have a primitive lattice while the trigonal system can have a rhombohedral lattice; the monoclinic system needs a primitive or an end centered lattice while the triclinic system can only have a primitive lattice

Molecules follow certain symmetry operations when they are packed into a crystal Beside unit translations along the three unit-cell axes, called three-dimensional translation symmetry, other symmetry elements are rotation, reflection, and inversion The combination of these symmetry elements that acts on a unit-cell is commonly called a crystallographic point group The simplest point groups are composed of proper rotation around the symmetry axis These are the point groups 1,

2, 3, 4, and 6 The total number of crystallographic point groups involving proper rotation is 11 Point groups also contain improper rotations, which are conformed to

one of the six general types: n , n n, PII, IPI, IIP, and P/I P/I P/I There are 21 improper rotations Thus there are totally 32 crystallographic point groups (Buerger, 1956)

Rotation or reflection combined with translation will generate screw or glide symmetry, respectively The combination of lattices and points groups (including their allowed screw axes and glide planes) leads to 230 different ways to combine the allowed symmetry operations in a crystal, known as space groups Because only L-amino acids are present in proteins and application of the mirror plane and inversion center to an L-amino acid would demand a D-amino acid not all the 230 space groups are allowed in protein crystals and only 65 space groups are applicable (McRee, 1999)

1.2.3 hkl plane

A convenient way to study the crystalline lattice is through the use of hkl planes The index h gives the number of planes in the set per unit-cell in the X direction, or equivalently, the number of parts into which the set of planes cuts the X edge of each cell Similarly, the indices k and l specify how many such planes exist per unit-cell in the Y and Z directions The family of planes having indices hkl is the (hkl) family of planes This concept is very useful in explaining the diffraction of X-rays by crystals

Diffraction occurs as waves interact with a regular structure whose repeat distance is about the same as the wavelength It happens that X-rays have wavelengths

in the order of Angstroms, same as typical interatomic distances in crystalline solids That means X-rays can be diffracted by minerals, which, by definition, are crystalline and have regularly repeating atomic structures When certain geometric requirements are satisfied, X-rays that are scattered from a crystalline solid can constructively interfere, thereby producing a diffracted beam These geometric requirements were first explained by Bragg

Diffraction depends on spacing between scattering bodies and wavelengths of incident radiation In Bragg’s model of diffraction as reflection from parallel sets of planes, (Fig 1.1) any of these sets of planes can be the source of one diffracted X-ray beam Bragg showed that a set of parallel planes with indices hkl and interplanar

spacing dhkl produces a diffracted beam when X-rays of wavelength λ impinge on the planes at an angle θ and are reflected at the same angle, only if θ meets the condition

Figure 1.1 The Bragg’s law The condition that produces diffracted

rays sin θ = BC/AB, BC = AB sinθ = dhkl sinθ If the additional

distance (2BC) travelled by the more deeply penetrating ray R2 is an

integral multiple of λ, then rays R1 and R2 interfere constructively

Notice that the angle of diffraction θ is inversely related to the interplanar spacing dhkl (sinθ is proportional to 1/dhkl) This implies that large unit-cells, with large spacing, give small angles of diffraction and hence produce many reflections that fall within a convenient angle from the incident beam On the other hand, small unit-cells give a large angle of diffraction, producing fewer measurable reflections In a sense, the number of measurable reflections depends on how many reflections are possible from

a unit-cell under a given experimental condition

Each set of parallel planes in a crystal produces one reflection The intensity of

a reflection depends on the summation of the electron distribution in the unit-cell along the direction of the planes that produce that reflection

θ θ

A

B C

R

R

hkl 1

C

d 2

1.3.2 Reciprocal lattice

Although Bragg’s law gives a simple and convenient method for calculating the separation of crystallographic planes, further analysis is necessary to calculate the intensity of scattering from a spatial distribution of electrons within each unit-cell A reciprocal lattice is defined as a discrete set of diffracted rays (reflections) The reciprocal lattice vectors are perpendicular to the real lattice planes from which they are derived The dimensions of the reciprocal lattice are inversely related to those of the real lattice Thus large unit-cells result in a very closely spaced reciprocal lattice and small unit-cells result in a reciprocal lattice with large intervals

Fig 1.2 explains how a reciprocal lattice is generated Take O as the origin Through a neighboring crystal lattice point N, draw one plane each of the set (110),

(120) and so forth, whose interplanar distances will be d110, d120 and so on From the origin, draw a line normal to the (110) plane The point at a distance, 1/d110, on this line will define the reciprocal lattice point 110 Do the same for (120) and so on Note that the points defined by this operation form a lattice, with the chosen origin This new lattice is the reciprocal lattice If the real unit-cell angles α, β and γ are 90°, the

reciprocal unit-cell has axes a* lying along the real unit-cell edge with the corresponding length of 1/a Similarly, the other parameters, b* and c* are defined If

the axial lengths are expressed in Angstroms, then the reciprocal lattice spacing is in the unit 1/Å or Å-1 (reciprocal Angstroms)

1.3.3 Ewald sphere

Reciprocal lattice points give the crystallographer a convenient way to compute the direction of diffracted beams from all sets of parallel planes in the

b*

θ

θθθ

B X

(010)110

120130

140

b*

x y

crystalline lattice (real space) The following geometrical interpretation of diffraction was formulated by Ewald

Figure 1.2 The reciprocal lattice

Assume that an X-ray beam (arrow XO in Fig 1.3) impinges on the crystal on

a plane Point O is arbitrarily chosen as the origin of the reciprocal lattice O is also

the real lattice origin in the crystal Draw a circle of radius 1/λ with its center C on XO

and passes through O This circle represents the wavelength of X-rays in the

Figure 1.3 The Ewald sphere

reciprocal space Rotating the crystal about O will also rotate the reciprocal lattice about O, successively bringing the reciprocal lattice points P and P' into contact with

the circle Because the triangle PBO is inscribed in a semicircle, it is a right angled

triangle and sinθ = OP/ BO = OP/ (2/λ) Because P is a reciprocal lattice point, the

length of line OP is 1/d hkl , where h, k and l are the indices of the set of planes represented by P So, 1/OP = d hkl and 2d hkl sinθ = λ, which is Bragg's law with n = 1

The line defining a reciprocal lattice point is normal to the set of planes having

the same indices as the point BP, which is perpendicular to OP, is parallel to the planes that are producing the reflection P in Fig 1.3 If we draw a line parallel to BP and passing through C, the center of the circle, this line represents a plane in the set

that reflects the X-ray beam under these conditions The beam impinges on this plane

at an angleθ, reflected at the same angle and diverges from the plane at C by an angle

2θ, which takes it precisely through the point P CP gives the direction of the reflected ray R In conclusion, reflection occurs in the direction CP when the reciprocal lattice point P comes in contact with this circle As the crystal is rotated in

the X-ray beam, all reciprocal lattice points come into contact with this sphere Each reciprocal lattice point produces a beam in the direction of a line from the center of the sphere of reflection through the reciprocal lattice point that is in contact with the sphere

This model of diffraction also implies that the directions of reflections, as well

as the number of reflections, depend only on the unit-cell dimensions, and not on the contents of the unit-cell

1.4 FOURIER TRANSFORM

1.4.1 The Fourier series

A Fourier series, named after Joseph Fourier, is an expansion of a periodic

function f(x) in terms of an infinite sum of sines and cosines and makes use of the

orthogonality relationships of the sine and cosine functions The computation and study of the Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain a solution to the original problem or an approximation to it to whatever accuracy is desired in practice

Each reflection is the result of diffraction from atoms in the unit-cell As a wave is periodic, Fourier analysis is the approximation of periodic functions by sine and cosine The basic idea of Fourier analysis is that any function f(x) of period 1 can

be approximated by sums of the type

f

0

)]

(2sin)(2[cos

|

|)

Here f (x)specifies the resulting diffracting wave and it is the sum of n Fourier terms

or diffraction from n atoms Each term is a simple wave with its own amplitude |Fh|, its own frequency h, and implicitly, its own phase αh Since

h e F x

When the above Fourier series is derived as a three dimensional Fourier series, the equation will be

∑∑∑ + +

=

h k l

lz ky hx i hkl e F z

y x

Here each term in the series is a simple three-dimensional wave whose frequency is h

in the X direction, k in the Y direction and l in the Z direction For each possible set

of value h, k and l, the associated wave has an amplitude |Fhkl|

1.4.2 The Fourier transform

The Fourier transform defines a relationship between a signal in the time domain and its representation in the frequency domain Being a transform, no information is created or lost in the process, so the original signal can be recovered

from knowing the Fourier transform, and vice versa Fourier demonstrated that for

any function f(x), there exists another Function F(h) such that

The Fourier transform operation is reversible That is, the same mathematical

operation that gives F(h) from f(x) can be carried out in the opposite direction to give

f(x) from F(h), if x and h are reciprocal to each other

The above functions f(x) and F(h) are one-dimensional If stated in three dimensions,

the Fourier transform would be:

=

x y z

lz ky hx i

dxdydz e

z y x f l

k h

and in turn the reverse Fourier transform is

∫∫∫ − + +

=

h k l

lz ky hx i

dhdkdl e

l k h F z

y x

1.4.3 Electron density and structure factor

The Fourier series is directly applicable in the study of crystals because the electron density function in a crystal is periodic Although the information about a protein structure is presented in the Cartesian coordinates of each atom, in reality what the crystallographer sees is the electron density, the cloud of electrons surrounding the nucleus of an atom with which X-rays interact

The unit-cell can be represented as an assembly of electron density in several

defined volume elements The electron density of each volume element centered at (x,

y, z) is roughly the average value of ρ (x, y, z) in that region Smaller the volume

elements, the more precisely these averages approach the correct value of ρ (x, y, z) at

all points The electron density is written as

) ( 21

) , ,

h k l

hkle F v

z y

x = ∑∑∑ −π + +

where F hkl is called structure factor, whose Fourier transform is the electron density

and vice versa In turn, the structure factor is written as

=

h k l

lz ky hx i

In other words, the structure factor is the resultant of N waves scattered in the

direction of the reflection hkl by the N atoms in the unit-cell Each of these waves has

an amplitude, which is proportional to the sum of f j , the scattering factor of atom j,

and a phase angle αj with respect to the origin of the unit-cell

Crystallographers represent each structure factor as a complex vector The

length of this vector represents the amplitude of the structure factor F , which is