^{1}, Taner Z. Sen

^{2}, Robert L. Jernigan

^{2}and Andrzej Kloczkowski

^{2}

### Abstract

We enumerated all compact conformations within simple geometries on the two-dimensional (2D) triangular and three-dimensional (3D) face centered cubic (fcc) lattice. These compact conformations correspond mathematically to Hamiltonian paths and Hamiltonian circuits and are frequently used as simple models of proteins. The shapes that were studied for the 2D triangular lattice included parallelograms, regular equilateral triangles, and various hexagons. On the 3D fcc lattice we generated conformations for a limited class of skewed parallelepipeds. Symmetries of the shape were exploited to reduce the number of conformations. We compared surface to volume ratios against protein length for compact conformations on the 3D cubic lattice and for a selected set of real proteins. We also show preliminary work in extending the transfer matrix method, previously developed by us for the 2D square and the 3D cubic lattices, to the 2D triangular lattice. The transfer matrix method offers a superior way of generating all conformations within a given geometry on a lattice by completely avoiding attrition and reducing this highly complicated geometrical problem to a simple algebraic problem of matrix multiplication.

The authors acknowledge the financial support provided by NIH Grant No. R01GM072014 to two of the authors (A.K. and R.L.J.).

INTRODUCTION

Symmetries

Extension of the transfer matrix method to the triangular lattice

The extension of the transfer matrix method to the triangular lattice

RESULTS

DISCUSSION

## Figures

The 2D triangular lattice (a) and a unit cell in the face centered cubic lattice (b).

The 2D triangular lattice (a) and a unit cell in the face centered cubic lattice (b).

Examples of a Hamiltonian circuit (a) and a Hamiltonian path (b) within a parallelogram of size on the 2D triangular lattice.

Examples of a Hamiltonian circuit (a) and a Hamiltonian path (b) within a parallelogram of size on the 2D triangular lattice.

Examples of protein shapes on the triangular and the fcc lattices studied in the present work. (a) a parallelogram, (b) a regular (equilateral) hexagon having side lengths of 1 (in lattice units), (c) a regular (equilateral) triangle with sides of length 3, and (d) a skewed parallelepiped on the fcc lattice.

Examples of protein shapes on the triangular and the fcc lattices studied in the present work. (a) a parallelogram, (b) a regular (equilateral) hexagon having side lengths of 1 (in lattice units), (c) a regular (equilateral) triangle with sides of length 3, and (d) a skewed parallelepiped on the fcc lattice.

Shapes embedded in the 3D fcc lattice. (a) , (b) , (c) , (d) , (e) , and (f) . The symmetries of each are referred to in the preceding table. Cross sections are shown on the right side.

Shapes embedded in the 3D fcc lattice. (a) , (b) , (c) , (d) , (e) , and (f) . The symmetries of each are referred to in the preceding table. Cross sections are shown on the right side.

Dealing with symmetric conformations. (a) An example of two conformations exhibiting head-tail symmetry. The two structures are equivalent upon rotation by 180° in the plane. Shown in (b) is the method we use to eliminate symmetries. If we start our path from the central node then only one of the six equivalent nodes is chosen as the first step and only one of the two equivalent nodes as the second step (the first two steps are shown as dark lines). The other step, shown as a broken line, would produce conformations symmetrical to the first one.

Dealing with symmetric conformations. (a) An example of two conformations exhibiting head-tail symmetry. The two structures are equivalent upon rotation by 180° in the plane. Shown in (b) is the method we use to eliminate symmetries. If we start our path from the central node then only one of the six equivalent nodes is chosen as the first step and only one of the two equivalent nodes as the second step (the first two steps are shown as dark lines). The other step, shown as a broken line, would produce conformations symmetrical to the first one.

All possible connectivity states (only physically acceptable states are labeled from 1 to 6) (a) and bond distributions (b) for generation of Hamiltonian circuits within rectangles of size .

All possible connectivity states (only physically acceptable states are labeled from 1 to 6) (a) and bond distributions (b) for generation of Hamiltonian circuits within rectangles of size .

The extension of the transfer matrix method to the triangular lattice must take into account nodes (such as the central one in the figure) that are already occupied during the process of piecewise building of the chain. The consideration of such nodes on the triangular lattice leads to an extension of the definition of connectivity states compared to the square lattice.

The extension of the transfer matrix method to the triangular lattice must take into account nodes (such as the central one in the figure) that are already occupied during the process of piecewise building of the chain. The consideration of such nodes on the triangular lattice leads to an extension of the definition of connectivity states compared to the square lattice.

All possible connectivity states for Hamiltonian circuits on parallelograms on the triangular lattice. The cross symbol denotes connectivity states containing excluded nodes, such as the central node in Fig. 7.

All possible connectivity states for Hamiltonian circuits on parallelograms on the triangular lattice. The cross symbol denotes connectivity states containing excluded nodes, such as the central node in Fig. 7.

Bond distributions (a) for each of the five connectivity states. All other distributions will not lead to valid conformations. (b) shows an example of a valid transfer from one state to another while (c) shows an invalid transfer from state 5 to state 4 because of a triple-bonded node.

Bond distributions (a) for each of the five connectivity states. All other distributions will not lead to valid conformations. (b) shows an example of a valid transfer from one state to another while (c) shows an invalid transfer from state 5 to state 4 because of a triple-bonded node.

All possible transitions between various connectivity states for Hamiltonian circuits on parallelograms on the triangular lattice. The notation means the transition from connectivity state 1 (in Fig. 8) to connectivity state 3.

All possible transitions between various connectivity states for Hamiltonian circuits on parallelograms on the triangular lattice. The notation means the transition from connectivity state 1 (in Fig. 8) to connectivity state 3.

Connectivity states that are the starting states (a) or the ending states (b) for Hamiltonian circuits on the parallelograms on the triangular lattice.

Connectivity states that are the starting states (a) or the ending states (b) for Hamiltonian circuits on the parallelograms on the triangular lattice.

All possible connectivity states for Hamiltonian circuits on parallelograms on the triangular lattice.

All possible connectivity states for Hamiltonian circuits on parallelograms on the triangular lattice.

Connectivity states that are starting states for Hamiltonian circuits on the parallelograms on the triangular lattice.

Connectivity states that are starting states for Hamiltonian circuits on the parallelograms on the triangular lattice.

Connectivity states that are ending states for Hamiltonian circuits on the parallelograms on the triangular lattice.

Connectivity states that are ending states for Hamiltonian circuits on the parallelograms on the triangular lattice.

The plot of the number of possible Hamiltonian chains vs the length for varying widths of the parallelogram.

The plot of the number of possible Hamiltonian chains vs the length for varying widths of the parallelogram.

The volume/area ratio comparison of 3D cubic lattice conformations and real proteins as a function of protein length (length meaning number of residues).

The volume/area ratio comparison of 3D cubic lattice conformations and real proteins as a function of protein length (length meaning number of residues).

## Tables

Symmetries for several classes of shapes on the 2D triangular (top) and 3D face centered cubic (fcc) (bottom) lattices.

Symmetries for several classes of shapes on the 2D triangular (top) and 3D face centered cubic (fcc) (bottom) lattices.

Numbers of Hamiltonian circuits within parallelograms of size on the triangular lattice and the ratio .

Numbers of Hamiltonian circuits within parallelograms of size on the triangular lattice and the ratio .

Enumerations of all paths and circuits for various geometries for the 2D triangular lattice and the 3D fcc lattice.

Enumerations of all paths and circuits for various geometries for the 2D triangular lattice and the 3D fcc lattice.

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