Structure refinement

In the final stages of the determination of a crystal structure, the parameters describing the structure (atom positions, for instance) are adjusted systematically to give the best fit between the observed intensities and those calculated from the model of the structure deduced from the diffraction pattern. This process is called structure refinement. Not only does the procedure give accurate positions for all the atoms in the unit cell, but it also gives an estimate of the errors in those positions and in the bond lengths and angles derived from them. The procedure also provides information on the vibrational amplitudes of the atoms.

X-ray crystallography is the deployment of X-ray diffraction techniques for the determination of the location of all the atoms in molecules as complicated as biopolymers. Bragg’s law helps us understand the features of one of the most seminal X-ray images of all time, the characteristic X-shaped pattern obtained by Rosalind Franklin and Maurice Wilkins from strands of DNA and used by James Watson and Francis Crick in their construction of the double-helix model of DNA (Fig. 20.26). To interpret this image by using the Bragg law we have to be aware that it was obtained by using a fibre consisting of many DNA molecules oriented with their axes parallel to the axis of the fibre, with X-rays incident from a perpendicular direction. All the molecules in the fibre are parallel (or nearly so), but are randomly distributed in the perpendicular directions; as a result, the diffraction pattern exhibits the periodic structure parallel to the fibre axis superimposed on a general background of scattering from the distribution of molecules in the perpendicular directions. There are two principal features in Fig. 20.26: the strong ‘meridional’ scattering upward and downward by the fibre and the X-shaped distribution at smaller scatter ing angles. Because scattering through large angles occurs for closely spaced features (from λ =2dsinθ, if d is small, then θ must be large to preserve the equality), we can infer that the meridional scattering arises from closely spaced components and that the inner X-shaped pattern arises from features with a longer periodicity. Because the meridional pattern occurs at a distance of about 10 times that of the innermost spots of the X-pattern, the large-scale structure is about 10 times bigger than the small-scale structure. From the geometry of the instrument, the wavelength of the radiation, and Bragg’s law, we can infer that the periodicity of the small-scale feature is 340 pm whereas that of the large-scale feature is 3400 pm (that is, 3.4 nm). To see that the cross is characteristic of a helix, look at Fig. 20.27. Each turn of the helix defines two planes, one orientated at an angle α to the horizontal and the other at −α. As a result, to a first approximation, a helix can be thought of as consisting of an array of planes at an angle α together with an array of planes at an angle −α with a separation within each set determined by the pitch of the helix. Thus, a DNA molecule is like two arrays of planes, each set corresponding to those treated in the derivation of Bragg’s law, with a perpendicular separation d = p cos α, where p is the pitch of the helix, each canted at the angles ±α to the horizontal. The diffraction spots from one set of planes therefore occur at an angle α to the vertical, giving one leg of the X, and those of the other set occur at an angle −α, giving rise to the other leg of the X. The experimental arrangement has up–down symmetry, so the diffraction pattern repeats to produce the lower half of the X. The sequence of spots outward along a legcor responds to first-, second-,...order diffraction (n = 1, 2,...in eqn 20.4). Therefore from the X-ray pattern, we see at once that the molecule is helical and we can measure the angle α directly, and find α = 40°. Finally, with the angle α and the pitch p determined, we can determine the radius r of the helix from tan α = p/4r, from which it follows that r = (3.4 nm)/(4 tan 40°) = 1.0 nm. To derive the relation between the helix and the cross-like pattern we have ignored the detailed structure of the helix, the fact that it is a periodic array of nucleotide bases, not a smooth wire. In Fig. 20.28 we represent the bases by points, and see that there is an additional periodicity of separation h, forming planes that are perpendicular to the axis to the molecule (and the fibre). These planes give rise to the strong meridional diffraction with an angle that allows us to determine the layer spacing from Bragg’s law in the form λ = 2h sin θ as h = 340 pm. The success of modern biochemistry in explaining such processes as DNA replication, protein biosynthesis, and enzyme catalysis is a direct result of developments in preparatory, instrumental, and computational procedures that have led to the determination of large numbers of structures of biological macromolecules by techniques based on X-ray diffraction. Most work is now done not on fibres but on crystals, in which the large molecules lie in orderly ranks. A technique that works well for charged proteins consists of adding large amounts of a salt, such as (NH4)2SO4, to a buffer solution containing the biopolymer. The increase in the ionic strength of the solution decreases the solubility of the protein to such an extent that the protein precipitates, sometimes as crystals that are amenable to analysis by X-ray diffraction. Other common strategies for inducing crystallization involve the gradual removal of solvent from a biopolymer solution, either by dialysis (Impact I5.2) or vapour diffusion. In one implementation of the vapour diffusion method, a single drop of biopolymer solution hangs above an aqueous solution (the reservoir), as shown in Fig. 20.29. If the reservoir solution is more concentrated in a non-volatile solute (for example, a salt) than is the biopolymer solution, then solvent will evaporate slowly from the drop until the vapour pressure of water in the closed container reaches a constant, equilibrium value. At the same time, the concentration of biopolymer in the drop increases gradually until crystals begin to form. Special techniques are used to crystallize hydrophobic proteins, such as those spanning the bilayer of a cell membrane. In such cases, surfactant molecules, which like phospholipids contain polar head groups and hydrophobic tails, are used to encase the protein molecules and make them soluble in aqueous buffer solutions. Dialysis or vapour diffusion may then be used to induce crystallization. After suitable crystals are obtained, X-ray diffraction data are collected and ana lysed as described in the previous sections. The three-dimensional structures of a very large number of biological polymers have been determined in this way. However, the techniques discussed so far give only static pictures and are not useful in studies of dynamics and reactivity. This limitation stems from the fact that the Bragg rotation method requires stable crystals that do not change structure during the lengthy data acquisition times required. However, special time-resolved X-ray diffraction techniques have become available in recent years and it is now possible to make exquisitely detailed measurements of atomic motions during chemical and biochemical reactions. Time-resolved X-ray diffraction techniques make use of synchrotron sources, which can emit intense polychromatic pulses of X-ray radiation with pulse widths varying from 100 ps to 200 ps (1 ps = 10−12 s). Instead of the Bragg method, the Laue method is used because many reflections can be collected simultaneously, rotation of the sample is not required, and data acquisition times are short. However, good diffraction data cannot be obtained from a single X-ray pulse and reflections from several pulses must be averaged together. In practice, this averaging dictates the time resolution of the experiment, which is commonly tens of microseconds or less. An example of the power of time-resolved X-ray crystallography is the elucidation of structural changes that accompany the activation by light of the photoactive yellow protein of the bacterium Ectothiorhodospira halophila. Within 1 ns after absorption of a photon of 446 nm light, a protein-bound phenolate ion undergoes trans–cis isomerization to form the intermediate shown in Fig. 20.30. A series of rearrangements then follows, which includes the ejection of the ion from its binding site deep in the protein, its return to the site, and re-formation of the cis conformation. The physio logical outcome of this cycle is a negative phototactic response, or movement of the organism away from light. Time-resolved X-ray diffraction studies in the nanosecond to millisecond ranges identified a number of structural changes that follow electronic excitation of the phenolate ion with a laser pulse: isomerization, ejection, protonation of the exposed ion, and a number of amino acid motions.

Fig. 20.26 The X-ray diffraction pattern obtained from a fibre of B-DNA. The black dots are the reflections, the points of maximum constructive interference, that are used to determine the structure of the molecule. (Adapted from an illustration that appears in J.P. Glusker and K.N. Trueblood, Crystal structure analysis: A primer. Oxford University Press (1972).)

Fig. 20.27 The origin of the X pattern characteristic of diffraction by a helix. (a) A helix can be thought of as consisting of an array of planes at an angle α together with an array of planes at an angle −α. (b) The diffraction spots from one set of planes appear at an angle α to the vertical, giving one leg of the X, and those of the other set appear at an angle −α, giving rise to the other leg of the X. The lower half of the X appears because the helix has up–down symmetry in this arrangement. (c) The sequence of spots outward along a leg of the X corresponds to first-, second-,... order diffraction (n = 1, 2, . . .).

Fig. 20.28 The effect of the internal structure of the helix on the X-ray diffraction pattern. (a) The residues of the macromolecule are represented by points. (b) Parallel planes passing through the residues are perpendicular to the axis of the molecule. (c) The planes give rise to strong diffraction with an angle that allows us to determine the layer spacing h from λ = 2h sin θ

Fig. 20.29 In a common implementation of the vapour diffusion method of biopolymer crystallization, a single drop of biopolymer solution hangs above a reservoir solution that is very concentrated in a non-volatile solute. Solvent evaporates from the more dilute drop until the vapour pressure of water in the closed container reaches a constant equilibrium value. In the course of evaporation (denoted by the downward arrows), the biopolymer solution becomes more concentrated and, at some point, crystals may form.

Fig. 20.30 Light-induced isomerization of a protein-bound phenolate ion in the photoactive yellow protein of the bacterium Ectothiorhodospira halophila.

اخر الاخبار

اخبار العتبة العباسية المقدسة

المجمع العلمي يواصل إقامة مسابقة الثقلين القرآنية لطلبة مدارس كربلاء

عضو مجلس إدارة العتبة العباسية المقدسة يطلع على آليات تطوير البرامج التعليمية في مجموعة العميد

مجموعة مناحل الكفيل تختتم مشاركتها في معرض السليمانية الدولي للزراعة والصناعات الغذائية

لتوثيق التراث العلمي.. وفد قسم شؤون المعارف يلتقي عددًا من أعلام مدينة النجف

المزيد

الآخبار الصحية

ثورة طبية.. الذكاء الاصطناعي يكشف سرطان الكلى في وقت قياسي

لتعزيز الطاقة والتغلب على التعب.. عليك بـ7 طرق بسيطة

لتحسين الهضم والتحكم في الشهية.. ما أفضل وقت لتناول التمر؟

خبراء يكشفون تأثير الاستحمام في الظلام على جودة النوم !

المزيد

الآخبار التكنلوجية

إزالة ثاني أكسيد الكربون من الغلاف الجوي.. ما دور المحيطات والبحار؟

نظائر الهيليوم مقياس لاكتشاف رواسب الذهب.. كيف ذلك؟

الصين تطلق أول ناقلة نفط خام ذكية في العالم تعمل بالميثانول

الزمن على المريخ أسرع من الأرض.. دراسة علمية تكشف الفوارق

المزيد

مواضيع ذات صلة

Energetics

Structure

Close packing

The phase problem

Bragg’s law

X-ray diffraction

Lattices and unit cells

Surface pressure

Membrane formation

Micelle formation

The electrical double layer

Classification and preparation