Unbiased Sampling of Globular Lattice Proteins in Three Dimensions

We present a Monte Carlo method that allows efficient and unbiased sampling of Hamiltonian walks on a cubic lattice. Such walks are self-avoiding and visit each lattice site exactly once. They are often used as simple models of globular proteins, upon adding suitable local interactions. Our algorithm can easily be equipped with such interactions, but we study here mainly the flexible homopolymer case where each conformation is generated with uniform probability. We argue that the algorithm is ergodic and has dynamical exponent $z=0$. We then use it to study polymers of size up to ${64}^3=262144$ monomers. Results are presented for the effective interaction between end points, and the interaction with the boundaries of the system.

Figure 1

Move between two of the 2 480 304 possible Hamiltonian walks on a $3\times 3\times 3$ cubic lattice.

Figure 2

Normalized probability distribution of $x_{\mathrm{ee}}\equiv x_{12}/L$, where $x_{12}$ is the end-to-end distance.

Figure 3

Normalized probability distribution of $x_S\equiv d_S/L$, where $d_S$ is the distance to the nearest side of the cube.

Figure 4

Normalized probability distribution of $x_E\equiv d_E/L$, where $d_E$ is the distance to the nearest edge of the cube.

Figure 5

Normalized probability distribution of $x_C\equiv d_C/L$, where $d_C$ is the distance to the nearest corner of the cube.

Figure 6

Normalized probabilities for an end point being a corner site ($p_C$), an edge site ($p_E$), a side site ($p_S$), or a bulk site ($p_B$), as functions of $1/L$.