^{1,a)}and Gregory A. Voth

^{1,b)}

### Abstract

A new reductionist coarse-grained model is presented for double-helix molecules in solution. As with such models for lipid bilayers and micelles, the level of description is both particulate and mesoscopic. The particulate (bead-and-spring) nature of the model makes for a simple implementation in standard molecular dynamics simulation codes and allows for investigation of thermomechanic properties without preimposing any (form of) response function. The mesoscopic level of description—where groups of atoms are condensed into coarse-grained beads—causes long-range interactions to be effectively screened, which greatly enhances the efficiency and scalability of simulations. Without imposing local or global order parameters, a linear initial configuration of the model molecule spontaneously assembles into a double helix due to the interplay between three contributions: hydrophobic/hydrophilic interactions between base pairs, backbone, and solvent; phosphate-phosphate repulsion along the backbone; and favorable base-pair stacking energy. We present results for the process of helix formation as well as for the equilibrium properties of the final state, and investigate how both depend on the input parameters. The current model holds promise for two routes of investigation: First, within a limited set of generic parameters, the effect of local (atomic-scale) perturbations on overall helical properties can be systematically studied. Second, since the efficiency allows for a direct simulation of both small and large ( base pairs) systems, the model presents a testground for systematic coarse-graining methods.

The authors thank Dr. Gary Ayton and Professor Tom Cheatham for stimulating discussions. This research was supported by a grant from the National Science Foundation (Grant No. CHE-0218739). The computational resources for this project have been provided by the National Institutes of Health (Grant No. NCRR 1 S10 RR17214-01) on the Arches Metacluster, administered by the University of Utah Center for High Performance Computing.

I. INTRODUCTION

II. THE REDUCTIONIST COARSE-GRAINED MODEL

A. The base pair

B. The backbone

C. The solvent and nonbonded interactions

III. SIMULATION METHOD

IV. RESULTS

A. Spontaneous helix formation

B. Effect of the solute mass distribution and the nature of the solvent

C. Inter-base-pair interactions

D. Equilibrium properties of the end state

V. DISCUSSION

### Key Topics

- Solvents
- 36.0
- DNA
- 19.0
- Hydrophobic interactions
- 13.0
- Numerical modeling
- 11.0
- Chiral symmetries
- 10.0

## Figures

Schematic picture of the base pair (light-gray beads), with its connection to the backbone (dark gray). The numbers along the springs refer to the base-pair bond potentials of Table I, and the symbols above and below the base pair denote the angular potentials. The straight angle corresponds to in Table I and the oblique one to .

Schematic picture of the base pair (light-gray beads), with its connection to the backbone (dark gray). The numbers along the springs refer to the base-pair bond potentials of Table I, and the symbols above and below the base pair denote the angular potentials. The straight angle corresponds to in Table I and the oblique one to .

Schematic picture of the backbone connectivity. The springs and the (straight) angle correspond to the parameter definitions in Table I.

Schematic picture of the backbone connectivity. The springs and the (straight) angle correspond to the parameter definitions in Table I.

Representative snapshots of a spontaneous helix formation, starting from a straight ladder conformation (first frame). Only the solute molecule is shown, the surrounding solvent is removed. In each frame, the molecule is displayed with its long axis vertically aligned, although in the actual simulation the molecule is free to rotate. Note the appearance of a major (bottom half) and a minor groove (top half) in the final state.

Representative snapshots of a spontaneous helix formation, starting from a straight ladder conformation (first frame). Only the solute molecule is shown, the surrounding solvent is removed. In each frame, the molecule is displayed with its long axis vertically aligned, although in the actual simulation the molecule is free to rotate. Note the appearance of a major (bottom half) and a minor groove (top half) in the final state.

Coarse description of the helix with only the four sugar beads (grey spheres) at the ends of the chain. The white spheres denote the midpoints of two sugar beads. Shown are the twist angle and the end-to-end distance .

Coarse description of the helix with only the four sugar beads (grey spheres) at the ends of the chain. The white spheres denote the midpoints of two sugar beads. Shown are the twist angle and the end-to-end distance .

Time evolution of the twist angle in five sample runs of Case 1.

Time evolution of the twist angle in five sample runs of Case 1.

Time evolution of the twist angle between base pair 6 and base pair for 1 representative run of Case 1. Solid lines correspond to and dotted lines to .

Time evolution of the twist angle between base pair 6 and base pair for 1 representative run of Case 1. Solid lines correspond to and dotted lines to .

Time evolution of the distance between base pair 1 and base pair with increasing from top to bottom.

Time evolution of the distance between base pair 1 and base pair with increasing from top to bottom.

Kinetically trapped conformation of the molecule with a chiral defect in the center. The bottom half of the molecule has a left-handed twist while the top half is right handed.

Kinetically trapped conformation of the molecule with a chiral defect in the center. The bottom half of the molecule has a left-handed twist while the top half is right handed.

Average decay of twist angle as a function of time for Case 1 (solid lines), Case 2 (dotted lines), and Case 3 (dashed lines). In each case, the upper line denotes the average over the whole ensemble of initial conditions, the lower line only over the subset of molecules that completed a full helical turn during the length of the run.

Average decay of twist angle as a function of time for Case 1 (solid lines), Case 2 (dotted lines), and Case 3 (dashed lines). In each case, the upper line denotes the average over the whole ensemble of initial conditions, the lower line only over the subset of molecules that completed a full helical turn during the length of the run.

Average decay of twist angle as a function of time for Case 1b (solid line), Case 1c (dotted line), and Case 1b with diminished strength of the backbone angular potential (dashed line).

Average decay of twist angle as a function of time for Case 1b (solid line), Case 1c (dotted line), and Case 1b with diminished strength of the backbone angular potential (dashed line).

Average twist angle (circles, left scale) and average end-to-end distance (diamonds, right scale) as a function of the inter-base-pair Lennard-Jones energy parameter .

Average twist angle (circles, left scale) and average end-to-end distance (diamonds, right scale) as a function of the inter-base-pair Lennard-Jones energy parameter .

Average twist angle (circles, left scale) and average end-to-end distance (diamonds, right scale) as a function of the strength of the angular potential along the backbone.

Average twist angle (circles, left scale) and average end-to-end distance (diamonds, right scale) as a function of the strength of the angular potential along the backbone.

Wall clock time for simulations of the generic model (circles, dashed line, left scale) and an atomistic model (diamonds, dotted line, right scale) as a function of box length in the direction. Reported clock times are for simulation time.

Wall clock time for simulations of the generic model (circles, dashed line, left scale) and an atomistic model (diamonds, dotted line, right scale) as a function of box length in the direction. Reported clock times are for simulation time.

## Tables

Bond and angle parameters for the coarse-grained model [Eqs. (1) and (2)]. The bond harmonic force constants are given in and equilibrium lengths in . The angular harmonic force constants are given in and their equilibrium angles in degrees.

Bond and angle parameters for the coarse-grained model [Eqs. (1) and (2)]. The bond harmonic force constants are given in and equilibrium lengths in . The angular harmonic force constants are given in and their equilibrium angles in degrees.

Nonbonded interaction parameters for the different bead types of the model: (backbone beads), (base-pair beads), and (solvent beads).

Nonbonded interaction parameters for the different bead types of the model: (backbone beads), (base-pair beads), and (solvent beads).

Different variations of the models of Table II to study the effect of the ratio of attractive and repulsive interactions. ShLJ refers to the shifted Lennard-Jones potential [Eqs. (3) and (4)], -12 to the first (repulsive) term of the Lennard-Jones potential, and WCA to the Weeks–Chandler–Andersen potential [Eq. (5)].

Different variations of the models of Table II to study the effect of the ratio of attractive and repulsive interactions. ShLJ refers to the shifted Lennard-Jones potential [Eqs. (3) and (4)], -12 to the first (repulsive) term of the Lennard-Jones potential, and WCA to the Weeks–Chandler–Andersen potential [Eq. (5)].

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