Statistical mechanics of the two-dimensional hydrogen-bonding self-avoiding walk including solvent effects

A two-dimensional square-lattice model for the formation of secondary structures in proteins, the hydrogen-bonding model, is extended to include the effects of solvent quality. This is achieved by allowing configuration-dependent nearest-neighbor interactions. The phase diagram is presented and found to have a much richer variety of phases than either the pure hydrogen-bonding self-avoiding walk model or the standard $\Theta$-point model.

Figure 1

The phase diagram for the standard $\Theta$-point model, showing a low-$K$ zero density (finite walk length) phase and a high-$K$ (critical) collapsed phase, where the walk density is finite. At high temperatures (low $\beta$) the transition is second order (solid line), while at low temperatures (high $\beta$) the transition becomes first order (dashed line). These two behaviors are separated by a tricritical point, the $\Theta$ point.

Figure 2

The nearest-neighbor interactions are split into two classes, those of type (a) where four bonds forming two parallel lines model the hydrogen bonds, while the others [(b), (c) and (d)] model the solvent interactions. Configuration (a) induces a preferred orientation, while the other configurations do not.

Figure 3

A schematic version of the phase diagram for the bond-interacting self-avoiding walk model, proposed by Buzano and Pretti [11] and confirmed by Foster [12]. Phase I is the low-$K$ finite walk phase, II is the critical collapsed (liquid) phase, and III is the crystalline oriented phase.

Figure 4

Phase diagram for $\alpha =0.2$ calculated using eigenvalue crossings (lines), and the phenomenological RG method (points) for the high-$\rho$ transition.

Figure 5

Density calculated for $\alpha =0.2$, $K=2$ using CTMRG with $L=1000$

Figure 6

(Color online) Data collapse for the density close to the transition using data from the CTMRG method with $\alpha =0.2$, $K=2$. The finite-size scaling form of the density is taken with ${\rho }_{\infty }=0.989$, ${\beta }_c=0.6222$, and $\nu =0.87$

Figure 7

Plot of $\delta \rho =\left|{\rho }_h-{\rho }_v\right|$ for $\alpha =0.2$ and $K=2$.

Figure 8

Estimates for ${\beta }_c$ using various methods for $\alpha =0.2$ and $K=2$. Arrow shows the value found using the data collapse method ${\beta }_c=0.6222$. $●$ represent the peaks of the fluctuations of $\delta \rho$, while the triangles correspond to the position of the peaks of the fluctuations in the corner density, calculated with CTMRG. $*$ gives the position of the solutions to the Nightingale RG method, and the squares the position estimated using the condition that ${\lambda }_e={\lambda }_o$.

Figure 9

Phase diagram for $\alpha =0.8$ calculated using Nightingale’s phenomenological RG method. The special transitions along the low-$K$ line are shown. The first is the $\theta$ point for $\alpha =0.8$ while the second is the collapsed-crystalline transition. The solution of the condition ${\lambda }_e={\lambda }_o$ is shown for various sizes using points, and can be seen to give a distinct line, which does not converge to the collapsed-crystalline transition line.

Figure 10

(Color online) Fluctuations of the number of corners in the walk for $\alpha =0.8$ calculated along the lower critical line showing clearly the existence of two transitions. Both transitions appear to be critical.

Figure 11

Bond density calculated for $K=2$, $\alpha =0.8$ using CTMRG with $L=1000$. The circle indicates the location of the collapsed-crystalline phase transition calculated using data collapse (see Fig. 12). The transition can be seen to occur at a density $\rho =1$.

Figure 12

(Color online) Data collapse of the corner density ${\rho }_c$ for $\alpha =0.8$, $K=2$ fitted with ${\rho }_{c,\infty }=0.2744$, ${\beta }_c=2.349$, and $\nu =0.96$.

Figure 13

Estimates for ${\beta }_c$ using various methods for $\alpha =0.8$, $K=2$. The arrow shows the value ${\beta }_c=2.349$ found using the data collapse method.

Figure 14

Phase diagram in the $\alpha \text{−}\beta$ plane calculated using phenomenological RG and eigenvalue crossings. The solid lines show the solutions to the condition ${\lambda }_e={\lambda }_o$, expected to coincide with the self-avoiding walk-crystalline transition.

Figure 15

Phase diagram in the $\alpha \text{−}\beta$ plane calculated using the peaks of ${\chi }_{\rho }$ and ${\chi }_{\delta \rho }$ calculated using transfer matrices. The peaks of $\rho$ fluctuations pick out the line of $\Theta$ points, while the peak of $\delta \rho$ fluctuations pick out the transition between the isotropic collapsed phase and the anisotropic crystalline phase. These two lines merge to form the first order transition line separating the SAW phase from the crystalline phase.