Random chain model

It is important to realize that the random-chain model need not imply an absence of residual structure in the unfolded population. Formative articles—many of them appearing on the pages of Advances in Protein Chemistry—recognized this fact. Kauzmann s famous review raised the central question about structure in the unfolded state (Kauzmann, 1959) ... 

Fig. 12. The s weighted reduced intensity functions si (s) calculated for a complete random chain model. The intrachain structure is that used in Fig. lOe. The interchain interactions are modeled as if they arise from randomly packed spheres. The curves are calculated using a distribution of sphere sizes. The numbers stated on the curves give the standard deviation of that distribution. The sphere diameter distribution is centered on 5.3 A. The packing density of the spheres is 0.6.

The structural analysis described above and in more detail elsewhere, shows the x-ray scattering functions to be sensitive to intrachain correlations. In fact, a more "random chain model (with a delocalized rotation state for one bond) than the "random coil" chain model is required to give a satisfactory match between the experimental and model si (s) functions. A model in which the interchain correlations are minimal with no orientational correlations provides a scattering function which is in good agreement with the observed scattering. Thus there seems to be no evidence to require more local order than inherent in a dense molecular system. This is perhaps not suprising. The polyisoprene molecule has a compact cross section, almost cylindrical in nature and corresponds to the "typical molecule drawn in schematic views of the noncrystalline state. 

Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

Another simplified model is the freely jointed or random flight chain model. It assumes all bond and conformation angles can have any value with no energy penalty, and gives a simplified statistical description of elasticity and average end-to-end distance. 

The molecules used in the study described in Fig. 2.15 were model compounds characterized by a high degree of uniformity. When branching is encountered, it is generally in a far less uniform way. As a matter of fact, traces of impurities or random chain transfer during polymer preparation may result in a small amount of unsuspected branching in samples of ostensibly linear molecules. Such adventitious branched molecules can have an effect on viscosity which far exceeds their numerical abundance. It is quite possible that anomalous experimental results may be due to such effects. 

We close these introductory remarks with a few comments on the methods which are actually used to study these models. They will for the most part be mentioned only very briefly. In the rest of this chapter, we shall focus mainly on computer simulations. Even those will not be explained in detail, for the simple reason that the models are too different and the simulation methods too many. Rather, we refer the reader to the available textbooks on simulation methods, e.g.. Ref. 32-35, and discuss only a few technical aspects here. In the case of atomistically realistic models, simulations are indeed the only possible way to approach these systems. Idealized microscopic models have usually been explored extensively by mean field methods. Even those can become quite involved for complex models, especially for chain models. One particularly popular and successful method to deal with chain molecules has been the self-consistent field theory. In a nutshell, it treats chains as random walks in a position-dependent chemical potential, which depends in turn on the conformational distributions of the chains in... 

Before discussing details of their model and others, it is useful to review the two main techniques used to infer the characteristics of chain conformation in unordered polypeptides. One line of evidence came from hydrodynamic experiments—viscosity and sedimentation—from which a statistical end-to-end distance could be estimated and compared with values derived from calculations on polymer chain models (Flory, 1969). The second is based on spectroscopic experiments, in particular CD spectroscopy, from which information is obtained about the local chain conformation rather than global properties such as those derived from hydrodynamics. It is entirely possible for a polypeptide chain to adopt some particular local structure while retaining characteristics of random coils derived from hydrodynamic measurements this was pointed out by Krimm and Tiffany (1974). In support of their proposal, Tiffany and Krimm noted the following points ... 

To explain the difference between the experimental results and theory, Doherty et al. (4J have given an empirical and a theoretical hypothesis. The theoretical hypothesis concerns the question of the meaning to be attached to the concept of the "equivalent random link" in the statistical theory of the randomly-jointed chain. According to Doherty et al., the assumption that the optical properties of the chain are describable by a randomly jointed model, using the same value of n, as for the description of stress has no strictly logical foundation. 

A second problem with the random walk model concerns the interaction between segments far apart along the contour of the chain but which are close together in space. This is the so-called "excluded volume" effect. The inclusion of this effect gives rise to an expansion of the chain, and in three-dimensions, 2 a, r3/5 (9), rather than the r dependence given in equation (I). 

The favoured dihedral angles for protein main chains were derived from energy considerations of steric clashes in peptides giving the well known Ramachandran plot (Ramachandran and Sasisekharan, 1968). These phi/psi combinations characterize the elements of secondary structure. Accurate main chain models can be constructed from spare parts, that is short pieces of helices, sheets, turns, and random coils taken from highly refined structures, provided a series of C-alpha positions can be established from the electron density map... 

More quantitative chemical evidence for random coil configuration comes from cyclization equilibria in chain molecules (49). According to the random coil model there must be a very definite relationship among the concentrations of x-mer rings in an equilibrated system, since the cyclization equilibrium constant Kx should depend on configurational entropy and therefore on equilibrium chain and ring dimensions. Values of /Af deduced from experimental values on Kx for polydimethylsiloxane, both in bulk and in concentrated solution, agree very well with unperturbed dimensions deduced from dilute solution measurements(49). 

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