Self-avoiding interaction

In the case of flexible polymer chains, the dominant problem for theory has been the understanding of the statistical properties of self-avoiding random walks in 2, 3 and 4 dimensions. The case of D>3 arises of course because the self-avoiding interaction only becomes perturbative in 4 dimensions and... 

Consider first the case of a random potential with a Gaussian distribution. For simplicity, the discussion in the rest of this section be limited to three spatial dimensions (d = 3). Recall that in the case when there is no self-avoiding interactions the optimal size of a chain is found by minimizing the free energy F in Eq. 23. This yields... 

We now add a self avoiding interaction and assume first that it is small, i.e. v g, or at least v < g. If the chain is still localized in the same well, which we will see momentarily not to hold when L is large, then... 

Here, besides assuming that v is small we assume for the moment that L is not too big so the last term in the free energy, resulting from the self-avoiding interaction, is small enough so one does not have to take into account the change in Rm due to the presence of u. If we plot F vs. L, we see that it is lowest when... 

For V > Vc the chain is delocalized. The above expression for the free energy may no longer be accurate, but the general picture is clear. There will be very few monomers in the low regions of the potential, and the chain will behave very much like an ordinary-chain with a self-avoiding interaction in the absence of a random potential. Any little perturbation can cause the chain to move to a different location in the medium (see Fig. 

Here x is a constant solution of other saddle point equations. These saddle-point equations have been analyzed in the repulsive force case (w > 0) [9,10]. We also recall that, in the phantom case, i.e. V (r ) = 0, the flat phase ( > 0) exists only for D > 2 and k > Kq, where Kc is a finite constant [9]. Here, phantom means that the self-avoiding interactions are neglected. By a formal expansion of Eq. (2.3) in powers of k, we obtain... 

From these analyses, it is clear that if the attractive interaction is finite ranged, then the membrane without self-avoiding interaction has the possibility to exhibit a continuous crumpling transition from the flat phase to the compact phase and there may exist a critical crumpled state at the transition point. However, in order to discuss the nature of the transition we must compare the corresponding free energies between the flat phase and the compact phase. It is a difficult task and we only make some reasoning based on the numerical studies of ref. [4] hereafter. 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

The basic features of folding can be understood in tenns of two fundamental equilibrium temperatures that detennine tire phases of tire system [7]. At sufficiently high temperatures (JcT greater tlian all tire attractive interactions) tire shape of tire polypeptide chain can be described as a random coil and hence its behaviour is tire same as a self-avoiding walk. As tire temperature is lowered one expects a transition at7 = Tq to a compact phase. This transition is very much in tire spirit of tire collapse transition familiar in tire theory of homopolymers [10]. The number of compact... 

Simplified models for proteins are being used to predict their stmcture and the folding process. One is the lattice model where proteins are represented as self-avoiding flexible chains on lattices, and the lattice sites are occupied by the different residues (29). When only hydrophobic interactions are considered and the residues are either hydrophobic or hydrophilic, simulations have shown that, as in proteins, the stmctures with optimum energy are compact and few in number. An additional component, hydrogen bonding, has to be invoked to obtain stmctures similar to the secondary stmctures observed in nature (30). 

Let us consider a simple self-avoiding walk (SAW) on a lattice. The net interaction of solvent-solvent, chain-solvent and chain-chain is summarized in the excluded volume between the monomers. The empty lattice sites then represent the solvent. In order to fulfill the excluded volume requirement each lattice site can be occupied only once. Since this is the only requirement, each available conformation of an A-step walk has the same probability. If we fix the first step, then each new step is taken with probability q— 1), where q is the coordination number of the lattice ( = 4 for a square lattice, = 6 for a simple cubic lattice, etc.). 

Self-avoiding random walks (SARW) statistics has been proposed [1] for single that is for non-interacting between themselves ideal polymeric chains (free-articulated Kuhn s chains [2]) into ideal solvents, in which the all-possible configurations of the polymeric chain are energetically equal. From this statistics follows, that under the absence of external forces the conformation of a polymeric chain takes the shape of the Flory ball, the most verisimilar radius Rf of which is described by known expression [3, 4]... 

Nidras, P.P., Brak, R. New Monte Carlo algorithms for interacting self-avoiding walks. J. Phys. A Math. Gen. 1997, 30,1457-69. 

In his paper Domb presents a detailed analysis of the statistical properties of self-avoiding walks on lattices.1 These walks serve as models for linear polymer chains with hard-core intramolecular interactions associated with the exclusion of multiple occupancies of the lattice sites by the chain so-called chains with excluded volume. 

Self-avoiding walks with nearest-neighbor interactions of attraction (corresponding to negative values of e), are of particular interest to us. As a consequence of the presence of the forces of attraction there is a possibility of configurational transition in the chain of the kind encountered... 

HI. STOCHASTIC MODEL FOR RESTRICTED SELF-AVOIDING CHAINS WITH NEAREST-NEIGHBOR INTERACTIONS... 

Consequently, this rather simple, Self-Avoiding MIDCO method incorporates some aspects of the non-bonded interactions resulting in a shape change of the MIDCO surfaces. 

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