The Random Coil

According to H. Mark (5), the story of the development of the random coil began with the x-ray work of Katz on 

The Katz effect was particularly important because it was the first experiment to establish the relationship between mechanical deformation and concomitant molecular events in polymers (5). This led Mark and Valko (11) to carry out stress-strain studies over a wide temperature range together with x-ray studies in order to analyze the phenomenon of rubber reinforcement. This paper contained the first clear statement that the contraction of rubber was not caused by a decrease in energy but by the decrease in entropy on elongation. 

This finding can be explained by assuming that the rubber chains are in the form of flexible coils (12). These flexible coils have a high conformational entropy, but lose their conformational entropy on being straightened out. The fully extended chain, which is rod-shaped, can have only one conformation, and its entropy is zero. This 

The random coil model has remained essentially the same until today (17), although many mathematical treatments have refined its exact definition. Its main values are two-fold By all experiments, it appears to be the best model for amorphous polymers, and it is the only model that has been extensively treated mathematically. It is interesting to note that by its very randomness, the random coil model is easier to understand quantitatively and analytically than models introducing modest amounts of order (19). 

The random coil model has been supported by many experiments over the years. The most important of these ahs been light-scattering from diulte solutions, and more 

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The statistical model of the random coil discussed in Chap. 1 illustrates many of these items. 

We saw in Chap. 1 that the random coil is characterized by a spherical domain for which the radius of gyration is a convenient size measure. As a tentative approach to extending the excluded volume concept to random coils, therefore, we write for the volume of the coil domain (subscript d) = (4/3) n r, and combining this result with Eq. (8.90), we obtain... 

We begin our attempt to reconcile these two expressions for the excluded volume of the random coil by reviewing some ideas about random coils from Chap. 1 ... 

In addition to an array of experimental methods, we also consider a more diverse assortment of polymeric systems than has been true in other chapters. Besides synthetic polymer solutions, we also consider aqueous protein solutions. The former polymers are well represented by the random coil model the latter are approximated by rigid ellipsoids or spheres. For random coils changes in the goodness of the solvent affects coil dimensions. For aqueous proteins the solvent-solute interaction results in various degrees of hydration, which also changes the size of the molecules. Hence the methods we discuss are all potential sources of information about these interactions between polymers and their solvent environments. 

One thing that is apparent at the outset is that polymer molecules in solution are very different species from the rigid spheres upon which the Einstein theory is based. On the other hand, we saw in the last chapter that the random coil contributes an excluded volume to the second virial coefficient that is at least... 

Random coils. Equation (9.53) gives the Kirkwood-Riseman expression for the friction factor of a random coil. In the free-draining limit, the segmental friction factor can, in turn, be evaluated from f. In the nondraining limit the radius of gyration can be determined. We have already discussed f in Chap. 2 and (rg ) in this chapter and again in Chapter 10, so we shall not examine the information provided by D for the random coil any further. 

Small deformations of the polymers will not cause undue stretching of the randomly coiled chains between crosslinks. Therefore, the established theory of rubber elasticity [8, 23, 24, 25] is applicable if the strands are freely fluctuating. At temperatures well above their glass transition, the molecular strands are usually quite mobile. Under these premises the Young s modulus of the rubberlike polymer in thermal equilibrium is given by ... 

The typical shape of most polymer molecules in solution is the random coil. This is due to the relative ease of rotation around the bonds of the molecule and the resulting large number of possible conformations that the molecule can adopt. We should note in passing that where rotation is relatively hindered, the polymer may not adopt a random coil conformation until higher temperatures. 

This thermodynamic behaviour is consistent with stress-induced crystallisation of the rubber molecules on extension. Such crystallisation would account for the decrease in entropy, as the disorder of the randomly coiled molecules gave way to well-ordered crystalline regions within the specimen. X-Ray diffraction has confirmed that crystallisation does indeed take place, and that the crystallites formed have one axis in the direction of elongation of the rubber. Stressed natural rubbers do not crystallise completely, but instead consist of these crystallites embedded in a matrix of essentially amorphous rubber. Typical dimensions of crystallites in stressed rubber are of the order of 10 to 100 nm, and since the molecules of such materials are typically some 2000 nm in length, they must pass through several alternate crystalline and amorphous regions. 

Fig. 1 Vesicle construct formed from poly(L-lysine)-i)-poly(L-leucme) polypeptides where the poly(L-leucine) block corresponds to the a-helical hydrophobic segments and the poly (L-lysine) block corresponds to the random coil hydrophilic segments. Note that this is one specific example and not all vesicle constructs have a-helical and random coil blocks. Moreover, the amphiphilic copolymer can be comprised of either a pure block copolypeptide or a macromolecule consisting of a polypeptide and another type of polymer. Adapted from [20] with permission. Copyright 2010 American Chemical Society...

Apart from their utility in determining the correction factor 1/P( ), light-scattering dissymmetry measurements afford a measure of the dimensions of the randomly coiled polymer molecule in dilute solution. Thus the above analysis of measurements made at different angles yields the important ratio from which the root-mean-square... 

Thus we may retain the root-mean-square end-to-end distance as a measure of the size of the random-coiling polymer chain, and the parameter jS required to characterize the spatial distribution of polymer segments (not to be confused with the end-to-end distribution) can be calculated from It should be noted that the r used here... 

According to the interpretation given above, the intrinsic viscosity is considered to be proportional to the ratio of the effective volume of the molecule in solution divided by its molecular weight. In particular (see Eq. 23), this effective volume is represented as being proportional to the cube of a linear dimension of the randomly coiled polymer chain,... 

There are two broad kinds of polyion conformation the random coil and the ordered helix. In a helix there are regularly repeated structures along the coil there are none in the case of a random coil. In this book we are concerned with the latter where there are often several conformations with approximately equal free energies and, thus, conformational changes occur readily. 

For years, the reigning paradigm for the unfolded state has been the random coil, whose properties are given by statistical descriptors appropriate to a freely jointed chain. Is this the most useful description of the unfolded population for polypeptide length scales of biological interest The answer given by this volume is clear there is more to learn. But first a word about the occasion that prompted this volume. 

The random coil amide I VCD pattern is exacdy the same shape, but smaller in amplitude and shifted in frequency from the pattern characteristic of poly-L-proline II (PLP II) which is a left-handed 3ihelix of trans peptides (Kobrinskaya et al., 1988 Dukor and Keiderling, 1991 Dukor et al., 1991 Dukor and Keiderling, 1996 Keiderling et al., 1999b). This... 

Fig. 11. Amide F thermal denaturation spectra for ribonuclease A as followed by FTIR (left) and VCD (right), which show the IR peak shifting from the dominant /3-sheet frequency (skewed with a maximum at 1635 cm-1) to the random coil frequency ( 1645-1650 cm-1) and the VCD shape changing from the W-pattern characteristic of an a + p structure to a broadened negative couplet typical of a more disordered coil form. The process clearly indicates loss of one form and gain of another while encompassing recognition of an intermediate form. (This is seen here most easily as the decay and growth back of the 1630 cm-1 VCD feature, but is more obvious after factor analysis of the data set, Fig. 15).

IV. Reconciling the Random Coil with a Structured Denatured State. 257... 

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