Modelling of Macromolecules

The complexity of the polymer structure is reflected in the large number of dimensions needed to describe it. Alexander and Orbach [28] proposed the use of spectral or fracton dimension for the description of the density of states on a fractal. The necessity of introducing is due to the fact that the fractal dimension defined by Equation (11.1) does not reflect this parameter. The investigators made use of the fact that anomalous diffusion of particles is expected on a fractal and, hence  

The spectral dimension d is a true property of a fractal and is determined only by its connectivity. It differs from the mass scaling index [see Equation (11.1)] or fractal dimension df and from the scaling index of the diffusion constant 8 by the fact that it does not depend on the way in which a fractal has been inserted into the Euclidean space with dimension d. The dependence of d and 8 on df is described by Equation (11.3). 

Alexander and Orbach [28] found that for a separate linear polymer chain, d =. n the case of a critical percolation cluster, the ds value does not depend on d and amounts to -4/3 (Table 11.1). 

The dimensions of a linear macromolecule in different states have been considered [51]. The following states are known [51, 52] to he the most typical 1 - compact globule, 2 - coil at the 0-point, 3 - impermeable coil in a good solvent, 4 - permeable coil (the state typical of rigid-chain macromolecules), 5 - completely uncoiled rod-like macromolecule. 

The radius of gyration (Rg) of a macromolecule depends on the molecular mass (M) and on the geometry of distribution of the molecule in space and is characterised by the 

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The book can be used in a one semester course for senior undergraduate and graduate students who are interested in understanding physical aspects of biochemistry and computer modeling of macromolecules. It can also be... 

The same geometric and mathematical principles lie at the root of all types of diffraction experiments, whether the samples are powders, solutions, fibers, or crystals, and whether the experiments involve electromagnetic radiation (X rays, visible light) or subatomic particles (electrons, neutrons). My aim in this chapter was to show the common ground shared by all of these probes of molecular structure. Note in particular how the methods complement each other and can be used in conjunction with each other to produce more inclusive models of macromolecules. For example, phases from X-ray work can serve as starting phase estimates for neutron work, and the resulting accurate... 

Whilst these difficulties do not invalidate application of molecular mechanics methods to such systems, they do mean that the interpretation of the results must be different from what is appropiate for small-molecule systems. For these reasons, the real value of molecular modeling of macromolecule systems emerges when the models are used to make predictions that can be tested experimentally or when the modeling is used as an adjunct to the interpretation of experiments. Alternatively, the relatively crude molecular mechanics models, while not of quantitative value, are an excellent aid to the visualization of problems not readily accessible in any other way. Molecular dynamics is needed, especially for large molecules, to scan the energy surface and find low-energy minima. The combination of computational studies with experimental data can help to assign the structure. 

Kirkwood and Riseman (1948) did not encounter this problem, because they used the bead-rod or, in other words, pearl-necklace model of macromolecule (Kramers 1946), in which A is a number of Kuhn s stiff segments, so that N present the length of the macromolecule. 

Now, we have to return to the subchain model of macromolecule, which was used to calculate the stresses in the polymeric system, and express the tensor of the mean orientation of the segments of the macromolecule in terms of the subchain model. 

Of course, both microporous structure and the rest nonempty solid phase can be considered as fractals. For processes of pyrolytic treatment of organic materials, in most cases it is more useful to apply fractal methods to pores. For polymerization, it is more effective to apply fractal modeling of macromolecules. 

In both equations, d is the separation between the atoms. The Lennard-Jones potential is simpler and computationally less demanding and is therefore favored for models of macromolecules such as proteins and DNA. The Buckingham function more closely resembles the energy relationship and is preferred when higher accuracy is required. The latter function is available in MOMEC and we will concentrate on this. 

This model of macromolecule HPLC has not been developed in any detail. It simply recognizes that solute molecules are attached in some fashion to the stationary phase, presumably through the various constituent groups of the molecule. Since a large molecule will have more groups, there will be several points of attachment. 

Calculation of entropies of both types is straightforward and efficient because simulations are not required. This allows one to study the conformational stability of many localized microstates of the same molecule by comparison of their harmonic free energies. However, the method is limited only to models of macromolecules in vacuum. 

For the determination of individual bond mpture, the qnan-titative assessment of spacer elasticity may prove to be useful. Hence, it is worthwhile to discuss two models of polymer elasticity for the mechanical modeling of macromolecules that are frequently employed the freely jointed chain (FJC) and wormlike chain (WLC) models. Both these models are employed to describe the polymer chain deformation under an external force. 

Importance and General Overview.—The usefulness of thermodynamic information gleaned from studying polymer solutions has long been recognized. For the theoretician it is essential to his construction and assessment of statistical mechanical models of macromolecules. For the technologist, as is well illustrated... 

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