^{1,a),b)}, Joel Ireta

^{2}and Jörg Neugebauer

^{1}

### Abstract

We have employed density functional theory to determine the temperature dependence of the intrinsic stability of an infinite poly-L-alanine helix. The most relevant helix types, i.e., the α- and the 3_{10} - helix, and several unfolded conformations, which serve as reference for the stability analysis, have been included. For the calculation of the free energies for the various chain conformations we have explicitly included both, harmonic and anharmonic contributions. The latter have been calculated by means of a thermodynamic integration approach employing stochastic Langevin molecular dynamics, which is shown to provide a dramatic increase in the computational efficiency as compared to commonly employed deterministic molecular dynamics schemes. Employing this approach we demonstrate that the anharmonic part of the free energy amounts to the order of 0.1–0.4 kcal/mol per peptide unit for all analysed conformations. Although small, the anharmonic contribution stabilizes the helical conformations with respect to the fully extended structure.

I. INTRODUCTION

II. SIMULATION APPROACH

A. Density functional theory supercell approach to model infinite helices

B. Locally stable conformations of the poly-L-alanine chain

C. Harmonic free energy—Phonons

D. Anharmonic corrections

1. Thermodynamic integration

2. Langevin dynamics

3. Friction parameter

4. Simpson's rule

III. RESULTS

IV. CONCLUDING REMARKS

### Key Topics

- Peptides
- 42.0
- Free energy
- 41.0
- Density functional theory
- 14.0
- Brownian dynamics
- 13.0
- Hydrogen bonding
- 13.0

## Figures

(a) Helix types and fully extended structure (FES). The red, dotted lines denote intra-chain N-H⋅⋅⋅O = C hydrogen bonds. ν denotes the number of peptide units to form a hydrogen bond. (b) Schematic representation of the L-alanine amino acid residue (peptide unit), with the side chain R = CH_{3}. (c) Schematic visualization of the parameters used to construct helices: pitch *L*, twist Θ, and radius *R*.

(a) Helix types and fully extended structure (FES). The red, dotted lines denote intra-chain N-H⋅⋅⋅O = C hydrogen bonds. ν denotes the number of peptide units to form a hydrogen bond. (b) Schematic representation of the L-alanine amino acid residue (peptide unit), with the side chain R = CH_{3}. (c) Schematic visualization of the parameters used to construct helices: pitch *L*, twist Θ, and radius *R*.

Phonon dispersion relations of the analyzed conformations of the poly-L-alanine chain.

Phonon dispersion relations of the analyzed conformations of the poly-L-alanine chain.

The statistical averages along the thermodynamic integration path determined with Langevin dynamics (black lines) and Nose Hoover dynamics (red lines). As test system a fully extended structure and the SCC-DFTB potential was used. The simulations were performed at *T* = 300 K. Statistical averages are shown after simulation times of 1 ps (dotted lines), 2.5 ps (dashed lines), and 50 ps (solid lines).

The statistical averages along the thermodynamic integration path determined with Langevin dynamics (black lines) and Nose Hoover dynamics (red lines). As test system a fully extended structure and the SCC-DFTB potential was used. The simulations were performed at *T* = 300 K. Statistical averages are shown after simulation times of 1 ps (dotted lines), 2.5 ps (dashed lines), and 50 ps (solid lines).

Anharmonic free energy contributions Δ*A* ^{ah} = *A* ^{ah} − *A* ^{harm} at room temperature, as evaluated by thermodynamic integration using different friction parameters γ in the Langevin equations of motion (Eq. (16)). Different simulation lengths are investigated: 10^{3} steps (dotted black line), 10^{4} steps (dashed black line), 10^{5} steps (solid black line), and the SCC-DFTB approach is used (see, Ref. 33). The reference potential is the harmonic potential for a FES in a supercell containing four peptide units. The target potential is the “full” anharmonic potential energy surface. The dashed gray horizontal line denotes the best guess for the free energy difference as determined after 10^{5} simulation steps with γ = 0.01.

Anharmonic free energy contributions Δ*A* ^{ah} = *A* ^{ah} − *A* ^{harm} at room temperature, as evaluated by thermodynamic integration using different friction parameters γ in the Langevin equations of motion (Eq. (16)). Different simulation lengths are investigated: 10^{3} steps (dotted black line), 10^{4} steps (dashed black line), 10^{5} steps (solid black line), and the SCC-DFTB approach is used (see, Ref. 33). The reference potential is the harmonic potential for a FES in a supercell containing four peptide units. The target potential is the “full” anharmonic potential energy surface. The dashed gray horizontal line denotes the best guess for the free energy difference as determined after 10^{5} simulation steps with γ = 0.01.

## Tables

Anharmonic corrections, Δ*A* ^{ah}, to the vibrational free energy as determined with DFT-GGA at room temperature denoted in kcal/mol per peptide unit. The anharmonic corrections are referred to the harmonic free energy determined with DFT-GGA at the equilibrium pitch *L* _{0} and twist Θ_{0}, i.e., Δ*A* ^{ah} = *A* ^{ah} − *A* ^{harm}(*L* _{0}, Θ_{0}). τ^{sim} denotes the number of simulation steps (Δ*t* = 1.5 × 10^{−15} s) and the remaining statistical error bar is indicated by δΔ*A* ^{ah}.

Anharmonic corrections, Δ*A* ^{ah}, to the vibrational free energy as determined with DFT-GGA at room temperature denoted in kcal/mol per peptide unit. The anharmonic corrections are referred to the harmonic free energy determined with DFT-GGA at the equilibrium pitch *L* _{0} and twist Θ_{0}, i.e., Δ*A* ^{ah} = *A* ^{ah} − *A* ^{harm}(*L* _{0}, Θ_{0}). τ^{sim} denotes the number of simulation steps (Δ*t* = 1.5 × 10^{−15} s) and the remaining statistical error bar is indicated by δΔ*A* ^{ah}.

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