Macromolecules crystallization problems with

In the crystallographic case, the limited radius of convergence of refinement arises not only from the high dimensionality of the parameter space, but also from what is known as the crystallographic phase problem . With monochromatic diffraction experiments on single crystals, one measures the amplitudes of the reflections, not the phases. The phases, however, are required to compute electron density maps, which are obtained by Fourier transformation of the structure factor (described by a complex number for each reflection). Phases for new crystal structures are usually obtained from experimental methods such as multiple isomorphous replacement. However, electron density maps computed from a combination of native crystal amplitudes and multiple isomorphous replacement phases are sometimes insufficiently accurate to allow a complete and unambiguous tracing of the macromolecule. Furthermore, electron density maps for macromolecules are... 

The most demanding element of macromolecular crystallography (except, perhaps, for dealing with macromolecules that resist crystallization) is the so-called phase problem, that of determining the phase angle ahkl for each reflection. In the remainder of this chapter, I will discuss some of the common methods for overcoming this obstacle. These include the heavy-atom method (also called isomorphous replacement), anomalous scattering (also called anomalous dispersion), and molecular replacement. Each of these techniques yield only estimates of phases, which must be improved before an interpretable electron-density map can be obtained. In addition, these techniques usually yield estimates for a limited number of the phases, so phase determination must be extended to include as many reflections as possible. In Chapter 7,1 will discuss methods of phase improvement and phase extension, which ultimately result in accurate phases and an interpretable electron-density map. 

This article deals with some topics of the statistical physics of liquid-crystalline phase in the solutions of stiff chain macromolecules. These topics include the problem of the phase diagram for the liquid-crystalline transition in die solutions of completely stiff macromolecules (rigid rods) conditions of formation of the liquid-crystalline phase in the solutions ofsemiflexible macromolecules possibility of the intramolecular liquid-crystalline ordering in semiflexible macromolecules structure of intramolecular liquid crystals and dependence of die properties of the liquid-crystalline phase on the microstructure of the polymer chain. 

If the intensity and phase of each individual reflection could be determined, the electron density at any position in the unit cell could be calculated, hence revealing the molecular structure. Unfortunately, the phase cannot be measured using X-rays. This is not a problem for simple crystals with only a few atoms, since there are only a very limited number of ways that the phase can be assigned. However, this is not possible for macromolecules like proteins or nucleic acids, and various approaches are used to get around this phase problem . 

The use of force fields, as described in the preceding section, demands that their potentials be summed over all internal coordinates of the system of interest. Such a summation is straightforward for small molecules. For calculations on large systems (e.g., crystal structures, macromolecules), however, the summation of the long-range nonbonded interactions becomes a problem because their number increases rapidly (as for pair interaction potentials between N particles) with the size of the system. Therefore, one needs methods to minimize the range over which the summation of the nonbonded interactions is performed. 

The measurement of diffraction data from crystalline macromolecules presents additional problems. In the first place, the intensity of the diffraction is related to the size of the unit cell. Crystals with large unit cells diffract less strongly than do crystals with small unit cells. This is because there are fewer unit cells per unit volume for a macromolecular crystal. As a result, there is a need for very sensitive detection devices that can measure the intensities of weak Bragg reflections with high precision. A second related complication that arises with large unit cells is that the number of Bragg reflections is increased, and therefore the... 

It is often the case in the X-ray crystallographic studies of biological macromolecules that only noisy or insufficient experimental data is available. If an approximation of the expected macromolecular structure is available beforehand, the situation can be remedied without recourse to further more complete or more accurate data collection. However the remedy requires that the independently available rough model be correctly oriented with respect to the crystal axes. In principle, the formulation of this orientation problem involves exhaustive search calculations in vast multi-dimensional spaces. In practice, such enormous calculations cannot be done with present-day computers. However, simulated annealing strategies can overcome such limitations. This article will focus on such strategies. 

The Patterson synthesis (Patterson, 1935), or Patterson map as it is more commonly known, will be discussed in detail in the next chapter. It is important in conjunction with all of the methods above, except perhaps direct methods, but in theory it also offers a means of deducing a molecular structure directly from the intensity data alone. In practice, however, Patterson techniques can be used to solve an entire structure only if the structure contains very few atoms, three or four at most, though sometimes more, up to a dozen or so if the atoms are arranged in a unique motif such as a planar ring structure. Direct deconvolution of the Patterson map to solve even a very small macromolecule is impossible, and it provides no useful approach. Substructures within macromolecular crystals, such as heavy atom constellations (in isomorphous replacement) or constellations of anomalous scattered, however, are amenable to direct Patterson interpretation. These substructures may then be used to solve the phase problem by one of the other techniques described below. 

Excellent and detailed treatments of the use of anomalous dispersion data in the deduction of phase information can be found elsewhere (Smith et al., 2001), and no attempt will be made to duplicate them here. The methodology and underlying principles are not unlike those for conventional isomorphous replacement based on heavy atom substitution. Here, however, the anomalous scatterers may be an integral part of the macromolecule sulfurs (or selenium atoms incorporated in place of sulfurs), the iron in heme groups, Ca++, Zn++, and so on. Anomalous scatterers can also be incorporated by diffusion into the crystals or by chemical means. With anomalous dispersion techniques, however, all data necessary for phase determination are collected from a single crystal (but at different wavelengths) hence non-isomorphism is less of a problem. 

To identify the macroconformation of an equihbrium crystal, one must first find the helix of lowest energy and then pack such hehces densely, side-by-side as described in Sect. 5.1.8. Such crystals are of an extended-chain macroconformation, as illustrated schematically at comer C of Fig. 5.42. Two problems prevent the common occurrence of extended-chain crystals. First, unlike in short, small molecules, macromolecules are not all of the same length. They could, thus, not produce a smooth, low-energy surface with their chain ends. Second, flexible molecules in the melt or in solution are not sufficiendy extended to immediately go to the equilibrium crystal. To produce the extended-chain macroconformation, a substantial reduction of its entropy is necessary, producing a barrier of positive free enthalpy of extension (AG = -TAS, see the discussion of entropy elasticity in Sect. 5.6.5). The ultimate compensation of the entropy of extension can occur only after the packing of the molecule into the crystal is achieved and all heat of crystallization has been absorbed (AH t = TmAS yst)- If the major disentanglement and close packing do not occur in close succession, i.e., are decoupled processes, the nonequilibrium path to the crystal leads to arrested, metastable states. 

MD simulation (2) is a computer experiment in which the atoms of a stipulated system execute Newtonian dynamics on an assumed potential energy surface. The model system chosen for study, the assumed energy surface, and the simulation protocol are all operational variables. The simulation per se begins with the choice of an initial configuration, typically the crystal structure of the macromolecule, and an arbitrary arrangement of solvent. In DNA problems, the canonical forms obtained from fiber diffraction studies are sometimes chosen to serve as unbiased... 

Herman Mark is famous for his tremendous contributions to the field of polymer science. He is not so well known for his early work in the field of crystal structure, nevertheless, I think that it was his experience in the crystal structure field that gave him the badkground that permitted him to make his important contributions to the understanding of polymers. He was, in fact, one of the leading investigators in the field of Uie use of x-ray diffraction for the determination of the structure of crystals in the years 1923 to 1928, and it was through this work that he developed the feeling for atoms and their interaction with one another that permitted him, later on, to make an effective attack on the problem of the structure and properties of macromolecules. 

The differences between standard thermotropic LCs and macromolecular condis crystals are summarized in Fig. 8. The first three and the last two points make it easy to experimentally identify low molecular mass LCs. For macromolecules, however, the viscosity may be suflBciently large to lose the obvious liquid character the birefringence does not always show the well-known LC texture (55) the small ASj of LCs may be confused with partial crystallinity of the condis crystals and in polymers, some larger main-chain rigid groups are not always easily identifiable as mesogens. This leaves points four and eight for differentiation between the two mesophases. Points five and six are more difficult to establish, and solid state NMR and detailed X-ray structure-determinations may be necessary for full characterization. Furthermore, borderline structures may be possible between thermotropic LCs, amphiphilic LCs, and condis crystals. A few examples and the resolution of their structures are discussed next, to illustrate the resolution of some of these problems. 

The present volume was suggested and stimulated by the aforementioned thoughts. We shall be concerned here with the phenomena and problems associated with the participation of macromolecules in phase transitions. The term crystallization arises from the fact that ordered structures are involved in at least one of the phases. The book is composed of three major portions which, however, are of unequal length. After a deliberately brief introduction into the nature of high polymers, the equilibrium aspects of the subject are treated from the point of view of thermodynamics and statistical mechanics, with recourse to a large amount of experimental observation. The second major topic discussed is the kinetics of crystallization. The treatment is intentionally very formal and allows for the deduction... 

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