Helical wormlike chain model

Yamakawa and co-workers have formulated a discrete helical wormlike chain model that is mechanically equivalent to that described above for twisting and bending/79111 117) However, their approach to determining the dynamics is very different. They do not utilize the mean local cylindrical symmetry to factorize the terms in r(t) into products of correlation functions for twisting, bending, and internal motions, as in Eq. (4.24). Instead, they... 

Finally, some rather recent devdopments must be noted. Several years ago, Yamakawa and co-workers [25-27] developed the wormlike continuous cylinder model. This approach models the polymer as a continuous cylinder of hydrodynamic diameter d, contour length L, and persistence length q (or Kuhn length / ). The axis of the cylinder conforms to wormlike chain statistics. More recently, Yamakawa and co-workers [28] have developed the helical wormlike chain model. This is a more complicated and detailed model, which requires a total of five chain parameters to be evaluated as compared to only two, q and L, for the wormlike chain model and three for a wormlike cylinder. Conversely, the helical wormlike chain model allows a more rigorous description of properties, and especially of local dynamics of semi-flexible chains. In large part due to the complexity of this model, it has not yet gained widespread use among experimentalists. Yamakawa and co-workers [29-31] have interpreted experimental data for several polymers in terms of this model. 

Moha et al. (54) considered that polypeptides in helicogenic solvents can assume intact helices but that the resulting rods can be weakly bent, and proposed to represent such a flexible rod by a wormlike chain. It was shown that the nonlinear dependence of 1/2 on N as depicted in Fig. 21 can be fitted by a suitable choice of the two parameters characterizing the wormlike-chain model. However, the necessary value for the length per monomeric unit forces us to accept the conclusion that the helical conformation is not of the a-type but of the 310-type. This is at variance with the ample experimental evidence now available for many synthetic polypeptides. 

A consideration of the molecular conformation using the wormlike chain model suggests that the curdlan molecule may contain helical portions but, as a whole, takes a random-coil conformation ( ) ... 

Yamakawa and co-workers developed the quasi-two-parameter (QTP) theory from the earlier two-parameter (TP) theory, to incorporate chain stiffness into the model. Specifically, the QTP theory computes C o via the helical wormlike coil model (HW),... 

We may constrain a freely rotating chain in such a way that the internal rotation of each segment is allowed to occur only over a limited range of rotational angle. If the limiting process leading to the wormlike chain is carried out with this additional constraint taken into account, we obtain a continuous chain model first considered by Miyake and Hoshino [2] and also by Burchard [3]. Yet another continuous chain model called the helical wormlike chain was proposed by Yamakawa and Fuji [4]. This chain is derived from a discrete chain somewhat different from the random flight chain [5]. 

The present chapter aims to describe some typical contributions from recent studies on stiff polymers in dilute solution. We will be mainly interested in (1) applicability of the wormlike chain model to actual polymers, (ii) validity of the hydrodynamic theories [2-4] recently developed for this model, and (iii) the onset of the excluded-volume effect on the dimensions of semi-flexible polymers. Yamakawa [5, 6] has generalized the wormlike chain model to one that he named the helical wormlike chain. In a series of papers he and his collaborators have made a great many efforts to formulate its static and dynamic properties in dilute solution. In fact, the theoretical information obtained is now comparable in both breadth and depth to that of the wotmlike chain (see Ref. [6] for an overview). Unfortunately, however, most of the derived expressions are too complex to be of use for quantitative anal) sis and interpretation of experimental data. Thus, we only have a few to be considered with reference to the practical aspects of the helical wormlike chain, and have to be content with mentioning the definition and some basic features of this novel model. 

M. Fujii, K. Nagasaka, J. Shimada, and H. Yamakawa. More on the model parameters of helical wormlike chains. Macromolecules, 16 (1983), 1613-1623. 

Figure 1. F(ii) = NiJP(ii) vi. ii = (4t/ K) sin (e/2) for helical amylosic chain nwdels A, B, and C, wormlike amylosic chain model W, jointed helical model J, and realistic random coil model R. Details of the models are described in the text.

A perfect helical main chain conformation always leads to a rodlike or cylindrical external shape. But each monomeric unit in such a rod contributes a certain flexibility. So, the flexibility of the rod, as a whole, must increase with increasing degree of polymerization, even when the flexibility per monomeric unit remains constant. A macroscopic example of this would be the flexibility of steel wires of equal diameter but different lengths. Thus, even a perfect helix will adopt coil shape if the molecular mass is very high. Because of this, helically occurring macromolecules, and other stiff macromolecules, can often be well represented by what is known as the wormlike screw model for macromolecular chains at low molecular masses, the chains behave like a stiff rod, but for high molecular masses, the behavior is more coil-like. Examples are nucleic acids, many poly(a-amino acids), and highly tactic poly(a-olefins). 

The continuous wormlike (persistence) chain model is more suitable for the description of large-scale conformational properties of macromolecules with large intrinsic stiffness, for example, a-helical polypeptides or DNA. In this polymer, rotation around the backbone C-C bonds is reduced to small oscillations around one (ground state) conformation because... 

This book offers a detailed explanation on conformation and dynamics of wormlike chains and helical wormhke chains by an expert of the model. 

Rods and Helices.—Because of the considerable advances being made in biopolymer characterization, theories and experimental methods particularly applied to stiffened chains and wormlike models, as well as to polyelectrolytes, have continued to proliferate. At the same time work has continued to progress in the synthesis and description of novel synthetic rod-like polymers, many of which have interesting liquid crystalline properties. The publication of the proceedings of a recent conference devoted to the latter materials provides a state of the art description of work in this field. 

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