Globule toroidal

Vasilevskaya, V.V., Khokhlov, A.R., Kidoaki, S. and Yoshikawa, K. (1997) Structure of collapsed persistent macromolecule toroid vs. spherical globule. Biopolymers, 41, 51-60. 

Although toroids are the most common morphology in DNA condensation, different manipulations can produce various morphologies, including rods, spherical globules and fibrous networks (Bloomfield, 1998 ... 

Since the toroidal globule can only be formed as a result of a strong attraction of the parts of the macromolecule (in any case, it must be (e/T) > (e/T)3), the volume of the torus is approximately equal to the selfvolume of the macromolecule, i.e. 

As to the surface free energy, obviously it is of order AFsurf cRr/d2. So we can write down the expression for the total free energy of the toroidal globule as follows ... 

By substituting the equilibrium torus radii (5.41) to Eq. (5.40), we obtain the expression for the equilibrium value of the globular free energy, from which it can be seen that upon heating the toroidal globule transforms into the coil state, the corresponding phase transition being of the first order. For the temperature and the heat of the transition we obtain... 

It should be noted that the toroidal globule can be stable only if the formation of the structural defect of the loop type (compare with the structures A and B in the previous section) is improbable. (In the presence of this defect two parts of the chain can twine round the torus cavity in the opposite directions). Indeed, if such defects were possible, the following equalities would be valid y>(x. u) = ip+(x. — u) = t/)(x, —0) and in this case the first term of the conformational entropy (5.38) would become zero. From the physical point of view, this means that, due to these defects, the cavity in the middle of the globule is destroyed. A detailed analysis shows (see Ref.16 ) that the formation of the defects of this type is practically forbidden until... 

Some experimental data show that the double-helix DNA macromolecules exist in poly(ethylene oxide) solution in the so-called compact form, which is very similar to the toroidal globule considered in Sect. 5. Electron micrographs definitely reveal the existence of cavities in the center of the globules. Of course, the development of an adequate theory of the compact form of DNA requires the consideration of some more refined peculiarities of the system and, primarily, of the role of poly-(ethylene oxide) (see recent works ). However, we emphasize that the existence of the cavity in the middle of the globule is apparently the fundamental consequence of the absence of the points of easy bending in the polymer chain. Hence, more refined theories must use as.a starting point the simple consideration of Sect. 5.6 of this paper. 

FIGURE 1.22 Typical conformations (I, random coil II, molten globule III, toroid IV, rod V, defect coil VI, defect cylinder) of a 100-segment homopolymer generated by Monte Carlo simulations. (Source Ref. [32].)... 

Macromolecules capable of internal local orientational ordering, usually of nematic type, in the globular state, like in toroid DNA case. Macromolecules with other types of symmetry or organization in the globule, up to a crystal. 

A typical configuration of a toroidal globule looks like depicted in Fig. 5. To identify these structures in the course of the simulation one has to partition the simulated equilibrium structures into sub-ensembles, for instance according to their value of the winding measure a. The distribution of a-values is bimodal, allowing for the definition of two sub-ensembles with a < 150 and a > 150, respectively, for chain length N = 40. We expect the toroidal structures to be contained in the subensemble with a > 150. All the structures in this sub-ensemble are characterized by a shape parameter Ki around 0.5. Looking now at the density profiles obtain for the structures in this sub-ensemble in Fig. 7 we can identify one set of structures with a hole in the middle (toroids) and another one which is densely packed in the middle. Since these structures also have Ki 0.5 they have to be disks, which can be verified... 

FigiiTe 5. Snapshot figure of a toroidal globule for a chain of length JV = 240 at stiffness 6 = 15 and inverse temperature P = 1. 

Ou ZY, Muthukumar M (2005) Langevin dynamics of semifiexible polyelectrolytes rod-toroid-globule-coil stmctuies and counterion distribution. J Chem Phys 123(7) 074905. doi 10.1063/1.1940054... 

Fig. 3 Morphological change after the sudden change of the chain stilhiess with Monte Carlo simulation (IV = 100, e = 2.5). (a) Transformation from toroid to spherical globule after the change of the stiffness parameter from k = 2 into k = 0, (b) transformation from spherical the toroid globule with the change from k = 0 into...

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