Module 30 X-ray Scattering of helix, analysis of structure of fibrous proteins, effect of inter molecular packing Learning Objectives Introduction Helix X-ray fibre diffraction Theory of X-ray fibre diffraction Humidity and Molecular orientation Introduction: 30.1. The Discovery of X-rays: Wilhelm Conrad Röntgen, a German Scientist, while conducting a number of experiments on cathode rays in 1895 (cathode ray is emanation of electrons from the cathode end of a vacuum-pumped glass tube) observed a new form of radiation other than cathode rays. This new form of radiation was named as X-rays by him. Fig 30.1: Wilhelm Conrad Röntgen 30.2. X-ray Diffraction: The properties of X-rays were studied by various scientists. In 1912, a group of physicists at the University of Munich were interested in both crystallography and the behaviour of X-rays. P. P. Ewald and A. Sommerfeld were studying the passage of light waves through crystals. At a colloquium, discussing some of this work, Max von Laue, pointed out that if the wavelength of the radiation became as small as the distance between the atoms in the crystal, a diffraction pattern should result. There was some evidence that, X-rays might have wavelength in this range, and W. Friedrich agreed to make an experimental test. An X-ray beam passed through a crystal of copper sulphate gave a definite diffraction pattern. 30.3. Helix: Helix is a smooth curve in a three dimensional space satisfying the condition that the tangent to the curve at any point makes a constant angle with the fixed line called axis. This axis is called helical axis. One can give numerous examples for helix. Common examples for helix are spiral staircase, and a spring (Fig 30.2). Fig 30.2: Spring - a basic model for Helix The helix can be classified into two categories: left handed helix and right handed helix. If the helix is advancing in the positive Z-direction and if the direction of the chain is in clockwise direction, then the helix is said to be right handed helix. If the direction of the chain is in the anti- clockwise direction, then the helix is said to be left handed helix. The above said definition and examples are for continuous helix. In the biological macromolecular system numerous biomolecules favour helical structures. In the molecular system, the atoms or molecules are arranged at equal distance from the helical axis. If the molecular chain forming helix is due to a single chain, the helix is called single helix. If two chains are involved in the helical structure, the helix is called double helix. If the helix is formed by three molecular chains, the helix is called triple helix. A chain molecule is the one, which has continuous covalent linkage. Alpha helix in protein structure is an example for single helix. The structure of DNA and Xanthan gum polysaccharide (a bacterial polysaccharide produced by xanthomonas campestris) are double helical structures. In the double helical structures, the chain can run in parallel direction or in antiparallel direction. In DNA, the chains are in the opposite directions and in Xanthan gum polysaccharide, the chains are in the parallel direction. Collagen and Xylem favour triple helical structures. In these triple helical structures, the chains are running in the same direction. X-ray fibre diffraction is the primary tool used to deduce the three dimensional structure of helical molecules. Fig 30.3: Types of Helices 30.4. X-ray Fiber Diffraction: X-ray fiber diffraction is the powerful tool which is used to study the three dimensional structure polymeric chain molecules which fail to form single crystals. However, the long chain molecule can allow orientation in a particular direction, preferably along the chain direction. The orientated molecules when allowed to diffract the x-rays in a direction perpendicular to the orientated direction, the recorded diffraction pattern can be used to deduce the helical structure. To enhance the chain orientation and chain packing, different techniques can be used. A stress can be applied to the fibrous structure which in turn will enhance the orientation. Fibre orientation can also be enhanced by applying mechanical stress in humidity environment. This technique is extensively used in X-ray fibre diffraction on polysaccharides and other fibrous proteins. 30.5. Theory of X-ray fibre diffraction There are fundamental differences between single crystal and fibre diffraction analysis. Much less order and regularity is present in the fiber than in the crystal, as a result of which less information will be obtained from the fibre diffraction pattern. Since the number of observed bragg reflections from single crystal X-ray diffraction experiment exceed far than the number of atoms in the molecular structure, one can find details of the structure with a high probability of correctness with such a surfeit of data. On the otherhand, in a poorly resolved fibre diagram, there may be only five to fifty distinguishable reflections in all, and these more often are diffused and ill-formed. As a result, if one already knows the nature of the subunits from which the fibre is constructed one can use X-ray data to determine how these subunits are arranged, but one cannot hope to solve fibrous protein structure in the complete absence of any assumptions as to compositions. Fibrous molecules are long chain polymers which need not show true structural repeats at finite intervals. For example, the short repeat distance of the collagen framework is about 30 Å, there is clear evidence that the exact aminoacid sequence in adjacent 30 Å segment differ considerably. The co-operative effect of that portion of the structure which does repeat contributes to the diffraction pattern and therefore represents an averaged structure. Hence, although one can discover the framework of the collagen structure by x-ray analysis, one cannot determine the exact aminoacid sequence. In fibrous proteins, the orderliness of aggregation of structural units is less than in the crystalline globular proteins. Stretching or rolling fibres or stroking films can align the polypeptide chain axis of the crystalline domains roughly in parallel, and to a lesser degree to make their orientation about these axis similar. In the most usual phase of a stretched fibre, however, the crystallites will be ordered along the fibre axis but we have random orientations about this axis. The effect on the diffraction pattern is therefore to smear out each spot into a ring around the fibre axis direction. The two- dimensional x-ray photograph obtained can be thought of as being roughly an axial slice through this rotated pattern. Because of this rotation effect, information about the entire pattern will be contained on this single photograph. Another difficulty is caused due to the alternation of crystalline and disordered regions in fibre. Only that fraction of the total material which is crystalline will contribute to the diffraction pattern, and the remaining amorphous material will produce an objectionable diffused background scattering. 30.6. X-ray diffraction of helical fibers: Fig 30.4: Geometrical representation of layer lines for fibre diffraction. For the interpretation of X-ray diagrams of fibres, an important unifying structural concept was first introduced by Michael Polanyi, namely that of layer-planes (LP) and layer-lines (LL) in reciprocal space. The concept is most easily explained with a simple model frequently used in the standard teaching of elementary diffraction theory. Assume a single linear, regular “polymer” consisting of identical “atoms” or point scatterers occupying equidistant positions 푧푗 = 푗푃 on the z- axis, where j is an integer and P is the polymer repeat period. For normal incidence, the scattering amplitude in 2 2 푖푞.푟푗 |퐴(푞)| = |∑푗 푓푗(푞)푒 | (30.1) reduces to 2휋 2휋 퐴(푞) = 푓(푞) ∑ 푒푖푞푧푗푃 = 푓(푞) ∑+∞ 훿(푞 − 1 ) (30.2) 푗 푃 1=−∞ 푍 푃 Where, we have made use of the identity (the fundamental relation for diffraction by ordered structures) +∞ 2휋푖푛푥 +∞ ∑푛=−∞ 푒 = ∑푚=−∞ 훿(푥 − 푚) (30.3) the exponential in the left-hand side is represented by a point on the unit circle in Gauss plane of complex numbers. If x is not an integer, the summation over n produces an infinite number of such points distributed evenly over the circle; by judiciously grouping the points, the sum is seen to vanish. On the other hand, if x is any integer m, all terms on the left are equal to 1 and the sum is then infinite. The vertical component 푞푧 of the transfer wave vector is therefore quantized to integer multiples of 2π/P. Eq. 30.2 expresses the familiar fact that the Fourier transform of a vertical set of points separated by P is a set of horizontal planes separated by 2π/P in reciprocal space. These planes of equation 푞푧 = 2πl/P are called layer planes and are labelled by the integer l. Now for elastic scattering, relevant for the diffraction of monochromatic X-rays, the allowed wave vector 푘푓 of X-ray which is scattered, in addition to lying on one of the layer planes, but also lies on Ewald sphere (ES), that is a sphere in reciprocal space of radius|푘푓| = |푘푖|.The intersections of the latter by the former are latitudes on the sphere and constitute the bases of cones with a common apex at the sphere centre (Fig.30.4). These conical surfaces intersect the observation screen along the loci of nonzero diffracted intensity. Such conical sections are hyperbolae called layer lines. Due to the finite radius of the Ewald sphere, the distance between the layer lines is chirped, that is it increases towards higher-l (Fig.30.4).