Macromolecules semiflexible

Some molecules, such as helical DNA, collagen, or PBLG, are stiffer than ordinary flexible polymers like polystyrene but are not rigid rods, and thus they are called semiflexible (Odijk 1986). Flexibility can be distributed either evenly or unevenly along the polymer backbone. 

A uniform distribution of flexibility describes some molecules such as DNA and PBG a simple model for such molecules is the flexible beam or worm-like chain (Saito 1967 Yamakawa 1971) which has a finite bending rigidity k that is uniform along the chain contour. When we define a contour variable s as the distance along the chain contour from one of the ends and define s) as the curve in space defining the conformation of the worm-like molecule, the bending free energy is 

The inextensibility of the worm-like chain implies that 9Rj/9 - Po flexible beam, 

For chain lengths L kp, the orientation of the chain as a whole becomes uncorrelated with that at one end of the chain, and the gross conformational properties of an isolated worm-like chain are the same as those of the freely jointed one. The persistence length Xp 

Nature produces the stiffest worm-like chains the long persistence length of DNA (Xp = 0.053 ixm) is short compared to that of F-actin filaments, Xp 17 /xm (Ott et al. 1993) the latter are components of the cellular cytoskeleton. Self-assembled cellular microtubules are even stiffer than this, with Xp on the order of millimeters  

---

Garcia Molina, J.J., Lopez Martinez, M.C. and Garcia de la Torre, J. (1990) Computer simulation of hydrodynamic properties of semiflexible macromolecules Randomly broken chains, wormlike chains, and analysis of properties... 

This article deals with some topics of the statistical physics of liquid-crystalline phase in the solutions of stiff chain macromolecules. These topics include the problem of the phase diagram for the liquid-crystalline transition in die solutions of completely stiff macromolecules (rigid rods) conditions of formation of the liquid-crystalline phase in the solutions ofsemiflexible macromolecules possibility of the intramolecular liquid-crystalline ordering in semiflexible macromolecules structure of intramolecular liquid crystals and dependence of die properties of the liquid-crystalline phase on the microstructure of the polymer chain. 

In Sect. 3, we will consider the orientational ordering in the solution of semiflexible macromolecules. In general, semiflexible macromolecules can have different flexibility distributions along the chain contour compare, for example, the freely-jointed chain of the long thin rods (Fig. 1 b) and the persistent chain, which is homogeneous along the contour (Fig. lc). We will see what properties of the liquid-crystalline transition do depend on the flexibility distribution along the drain contour and what properties are universal from this point of view. 

In this section we have presented the analysis of some features of the phase diagram for the solution of rigid rods with both repulsion and attraction. Besides self-dependent interest, this analysis will be important also for the consideration of the phase diagram of the solution of semiflexible macromolecules. This consideration will be the topic of the next section. 

Orientational Ordering in Solutions of Semiflexible Macromolecules 3.1 Model of a Semiflexible Macromolecule... 

Let us assume that the solution of semiflexible macromolecules occupies the volume V. Let i be the polymer volume fraction in the solution. Then, the average concentration of segments is c = 4 fibrpd3, the total number of segments N = Vc, and, finally, the average concentration of macromolecules is c//L, where L denotes the contour length of one macromolecule. 

Second, the effective virial coefficient B characterizing the interaction of segments differs from the usual virial coefficient B of the solution of disconnected rods -connectivity of segments into long chains is the reason. The corresponding renormalization of the virial coefficient has been studied in detail 32 33). In the application to the semiflexible macromolecule under consideration the result is (B - B)/B 1/p < 1, i.e. for the long rigid rods (p > 1) the renormalization is unessential. 

From the above it follows that the free energy of the solution of semiflexible macromolecules in the high temperature limit can be expressed as ... 

It is noteworthy that Eq. (3.1), as well as the other results of this section, can be applied not only to the model of freely jointed segments but also to any other model of semiflexible macromolecules it is necessary only to replace p in all equations by the ratio of the effective segment length to its width. In fact, the translational entropy... 

For the solution of semiflexible macromolecules, as well as for disconnected rods, when the isotropic and the anisotropic phases coexist at equilibrium at relatively low temperatures, the polymer volume fraction in the isotropic phase is very small while in the anisotropic phase it is dose to the maximum possible value, the chains in the anisotropic phase being practically parallel to one another. Thus, the lattice model may be applied to the description of the anisotropic phase. The free energy of the anisotropic phase can be written in the form analogous to Eq. (2.9) with the only difference that the role of rods is now played by the long chains of connected rods thus, in the first two terms of expression (2.9) it is necessary to perform the substitution p — pL/, N —> N /L. Hence, we obtain the low temperature expression for the free energy of the anisotropic phase in the form... 

Now let us discuss the applicability of the results obtained for other models of semiflexible macromolecules. It is clear that the qualitative form of the phase diagram does not depend on the model adopted. The low-temperature behavior of the phase diagram is independent of the flexibility distribution along the chain contour as well, since at low temperatures the two coexisting phases are very dilute, nearly ideal solution and the dense phase composed of practically completely stretched chains. The high temperature behavior is also universal (see Sect. 3.2). So, some unessential dependence of the parameters of the phase diagram on the chosen polymer chain model (with the same p) can be expected only in the intermediate temperature range, i.e. in the vicinity of the triple point. 

Thus, at T > T3, the long semiflexible macromolecule is in the coil state while at T < T3 it is in the globular state. Consequently, the temperature T3, which is determined by Eq. (3.7), is the temperature of the coil-globule transition for the long freely jointed macromolecule (see Fig. 6). It is clear that this transition is the first order phase transition with a considerable bound of the coil dimensions, it leads simultaneously to the transformation of the coil into the globule and to the formation of the liquid-crystalline ordering in the globule. 

In particular, it is well known that, if the macromolecule is supercooled below the 0 temperature, the phase transition isotropic coil-isotropic globule occurs. We emphasize that for the semiflexible macromolecule this is the peculiar phase transition between two metastable states. It should be recalled that the theory of the transition isotropic coil-isotropic globule for the model of beads is formulated in terms of the second and third virial coefficients of the interactions of beads , B and C24). This transition takes place slightly below the 0 point and its type depends on the value of the ratio C1/2/a3 if Cw/a3 I, the coil-globule transition is the first order phase transition with the bound of the macromolecular dimensions, and if C1/2/a3 1, it is a smooth second order phase transition (see24, 25)). 

The orientational ordering in the solutions of semiflexible macromolecules which was considered in Sect. 3 has been insufficiently studied from the experimental point of view. Direct measurements of the properties of the corresponding liquid-crystalline transition have practically not been reported in spite of the general interest in this problem. 

The 6 temperature of the solution of semiflexible macromolecules in the limit p 1 practically coincides with the 6 temperature of the solution of disconnected rods with the same p value - the reason is the small degree of renormalization of the virial coefficients (for a more detailed discussion see ). Thus, the result of Eq. (2.18) for el6 also remains valid for this case. From the comparison of Eqs. (2.18) and (3.6) it can be concluded that the 0 point is always (independently of L/ ) situated in the low temperature region of the phase diagram, well below the triple point. 

It should be recalled that the qualitative form of the phase diagram for the solution of disconnected rods was obtained in Sect. 2 using the Onsager method, which was generalized to take into account the attraction of the rods. The same approach can be applied to the solution of semiflexible macromolecules. 

For the model of semiflexible macromolecule accepted in Sect. 3.1, the attractive part of the second virial coefficient of the interactions of segments is given directly by Eqs. (2.16) and (2.21). Thus, the procedure for the determination of the most stable homogeneous phase of the solution of freely jointed semiflexible macromolecules is absolutely identical with the corresponding procedure for the solution of disconnected rods (compare Eqs. (2.3) and (3.1)) consequently, this procedure leads to Eqs. (2.25) (see also Fig. 5). 

The general qualitative form of the phase diagram can now be determined from the known high temperatiu-e and low temperature behaviours and from the fact that the curves of Fig. 5 must lie entirely within the phase separation region (cf. Sect. 2). The phase diagram for the solution of semiflexible macromolecules obtained in this way is shown in Fig. 6. 

The dotted curves in Fig. 6 show the phase diagram for the corresponding solution of disconnected rods which was found in Sect. 2. We see that the phase separation region for the solution of semiflexible macromolecules is generally broader - this is due to the obvious fact that the connection of rods into long chain favours their... 

Let us begin with the estimation of the polymer volume fraction inside the coil formed by one long semiflexible macromolecule. It is well known that this estimation depends essentially on the strength of the excluded volume effect, i.e. on the value of the parameter z = vN /a, where v is the excluded volume of a monomer and a is the spatial distance between two neighbouring monomers. To be definite let us adopt for a moment the model shown in Fig. 1 b. Then, if we choose one segment as an elementary monomer, v 6. (see Eq. (2.4)) and a i.e. z p Consequently, the excluded volume effect is pronounced at N p and negligible at N [Pg.77]

---