Worm-like chain model

The worm-like chain model (sometimes called the Kratky-Porod model) is a special case of the freely rotating chain model for very small values of the bond angle. This is a good model for very stiff polymers, such as double-stranded DNA for which the flexibility is due to fluctuations of the contour of the chain from a straight line rather than to trans-gauche bond rotations. For small values of the bond angle ( 1), the cos 9 in Eq. (2.23) can be expanded about its value of unity at = 0  

Since 9 is small, the persistence segment of the chain [Eq. (2.23)] contains a large number of main-chain bonds. 

The persistence length is the length of this persistence segment  

The Flory characteristic ratio of the worm-like chain is very large  

The corresponding Kuhn length (see Eq. (2.16)] is twice the persistence length  

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The molar mass dependence of the intrinsic viscosity of rigid chain polymers cannot be described by a simple scaling relation in the form of Equation (36) with molar mass independent of K and a. over a broad molar mass range. Starting from the worm-like chain model, Bohdanecky proposed [29] the linearizing equation... 

The analysis described above is useful for modelling colligative properties but does not address polyelectrolyte conformations. Polyelectrolyte conformations in dilute solution have been calculated using the worm-like chain model [103,104], Here, the polymer conformation is characterized by a persistence length (a measure of the local chain stiffness) [96]. One consequence of the... 

From their light-scattering measurements Holtzer, Benoit, and Doty (126) concluded that the short-range interactions control the dimensions of cellulose nitrate chains, and they discussed their results in terms of the worm-like chain model of Kratky and Porod (142), obtaining a persistence length of about 34.7 A. In Fig. 21 these data are shown as a plot of (S yjMw against Mw. The open circles are the experimental points and the broken curve is that calculated from the equations for the worm-like chain model. The theoretical curve is claimed to reproduce the data to within the probable experimental error in all but two cases. 

Table 8). This permits the interpretation of experimental data by using the electro-optical properties of flexible-chain polymers in terms of a worm-like chain model However, EB in solutions of polyelectrolytes is of a complex nature. The high value of the observed effect is caused by the polarization of the ionic atmosphere surrounding the ionized macromolecule rather than by the dipolar and dielectric structure of the polymer chain. This polarization induced by the electric field depends on the ionic state of the solution and the ionogenic properties of the polymer chain whereas its dependence on the chain structure and conformation is slight. Hence, the information on the optical, dipolar and conformational properties of macromoiecules obtained by using EB data in solutions of flexible-chain polyelectrolytes is usually only qualitative. Studies of the kinetics of the Kerr effect in polyelectrolytes (arried out by pulsed technique) are more useful since in these... 

The freely jointed chain model is most appropriate for synthetic polymers, such as polyethylene and polystyrene. For other molecules, such as DNA and polypeptides, the molecular flexibility is better described by the worm-like chain model (described in Section 2.2.4), whose force law can be approximated by a simple expression due to Marko and Siggia (1995), namely. 

Figure 3-21 Distribution of bead mass as a function of position downstream of the tether point of a DNA molecule of length L -67.2 tm for various velocities measured in experiments similar to those described in the caption to Fig. 3-1. The lines are the predictions of Monte Carlo molecular simulations using the elastic force from the worm-like chain model, Eq. (3-57), and conformation-dependent drag, as described in the text. The value of the parameter fcoii/ ksT — 4.8 sec(/im) is obtained from the diffusivity measurements of Smith et al. (1995) Crod/ a = 9.1 sec(/Ltm) 2 is obtained from Eq. (3-62) for a fully stretched filament (From Larson et al. 1997, reprinted with permission from the American Physical Society.)...

A qualitatively different mechanism of flexibility of many polymers, such as double-helix DNA is uniform flexibility over the whole polymer length. These chains are well described by the worm-like chain model (see Section 2.3.2). 

Table 2.2 summarizes the assumptions of the ideal chain models. The worm-like chain model is a special case of the freely rotating chain with a small value of the bond angle 6. Moving from left to right in Table 2.2, the models become progressively more specific (and more realistic). As more constraints are adopted, the chain becomes stiffer, reflected in larger Coo-... 

In the case of the worm-like chain model (Section 2.3.2), the extensional force diverges reciprocally proportional to the square of R ax — (R) ... 

There is no simple analytical solution for the worm-like chain model at all extensions, but there is an approximate expression valid both for small and for large relative extensions --------------------------------------------... 

Comparison of experimental force for 97 kilobase >.-DNA dimers with the worm-like chain model [solid curve is Eq. (2,119) with = 33 pm and b — 100 nm]. The dotted curve corresponds to the Langevin function of the freely jointed chain model [Eq. (2.112)]. Data are from R. H. Austin et al., Phys. Today, Feb.—... 

Fig. 3.9. This divergence is discussed in Section 2.6.2 for freely jointed and worm-like chain models.

Fig. 2.15. Owing to the finite extensibility of chains, the force diverges at the maximum end-to-end distance Rinax = bN [Eq. (2.116)]. A different relation with an even stronger divergence [Eq. (2.117)] has been used for the worm-like chain model.

Flgure 5.5. qg q) plot of p-3BCMU in a good solvent, The full line represents the fit by the theoretical structure factor g q) = go(q) cxp(-q Rc /2). The worm-like chain model is assumed for goiq)- The dashed line corresponds to the height of the q plateau which yields the monomer unit length. (Reprinted with permission from ref, 34)... 

Since the extension of the neck-linker made of 15-amino acids can be obtained from the simulation (contour length L = 5.7 nm, extension x = 3.1 0.8 nm), with a proper range of persistence length for the polypeptide chain (/p = 0.4-1. Onm) [40,41,56] one can estimate the internal tension on the neck-linker as/= 7-15pN by using the force-extension (/- x) relation of the worm-like chain model [44]) ... 

Since the values of (s2)/m for C(3.00)A are large, it is convenient to discuss them in terms of the worm-like chain model for which (5)... 

Single Chain Stretching Revisited Worm-Like Chain Model and dsDNA... 

Interestingly, there was a sort of over-reaction among researchers now many people consider worm like chain model either more realistic or just real — while in fact many polymers (e.g., proteins) should be described by other models, with rotational isomers etc. 

Because of great importance of this subject, we will sketch below the derivation of f R) for both freely-jointed and worm like chain models and explain the physical source of difference between them. In both cases our approach will be based on the following idea. 

Worm-like chain model is good not only for dsDNA, but for a number of other polymers. The notable example is so-called F-actin. Strictly speaking, calling F-actin a polymer is a little bit of a stretch it does not have a covalent backbone it is actually a chain-like assembly of protein globules (called G-actin). The diameter of F-actin chain is about 5 nm, and the chain is rather stiff, its effective segment is close to 30 pm, about three... 

The success of worm-like chain model for dsDNA made this model and formula (7.42) fashionable among the scientists (somewhat surprising fact of life is that there is such thing as fashion in science ) nowadays formula (7.42) is often used to fit the data for which it is not at all the most appropriate. We sincerely hope that Langevin formula (7.41) and its underlying freely-jointed model, as well as rotational isomers and other models will soon regain their rights and will be used where appropriate. 

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