^{1}, Stephen Quirk

^{2}and Rigoberto Hernandez

^{1,a)}

### Abstract

The potential of mean force (PMF) for stretching decaalanine in vacuum was determined earlier by Park and Schulten [J. Chem. Phys.120, 5946 (2004)] in a landmark article demonstrating the efficacy of combining steered molecular dynamics and Jarzynski's nonequilibrium relation. In this study, the recently developed adaptive steered molecular dynamics (ASMD) algorithm [G. Ozer, E. Valeev, S. Quirk, and R. Hernandez, J. Chem. Theory Comput.6, 3026 (2010)] is used to reproduce the PMF of the unraveling of decaalanine in vacuum by averaging over fewer nonequilibrium trajectories. The efficiency and accuracy of the method are demonstrated through the agreement with the earlier work by Park and Schulten, a series of convergence checks compared to alternate SMD pulling strategies, and an analytical proof. The nonequilibrium trajectories obtained through ASMD have also been used to analyze the intrapeptide hydrogen bonds along the stretching coordinate. As the decaalanine helix is stretched, the initially stabilized *i* → *i* + 4 contacts (α-helix) is replaced by *i* → *i* + 3 contacts (3_{10}-helix). No significant formation of *i* → *i* + 5 hydrogen bonds (π-helix) is observed.

This work has been partially supported by the National Science Foundation (NSF) through Grant No. CHE 1112067. The computing resources necessary for this research were provided in part by the National Science Foundation through TeraGrid resources provided by the Purdue Dell PowerEdge Linux Cluster (Steele) under grant number TG-CTS090079, and by the Center for Computational Molecular Science & Technology through Grant No. CHE 0946869.

I. INTRODUCTION

II. MODELS AND METHODS

A. Steered molecular dynamics (SMD)

B. Adaptive steered molecular dynamics

C. Simulation parameters

III. RESULTS AND DISCUSSION

A. The thermodynamics of decaalanine stretching in vacuum

B. Decaalanine structure and hydrogen bonding along the stretching coordinate

IV. CONCLUSIONS

### Key Topics

- Hydrogen bonding
- 34.0
- Free energy
- 22.0
- Peptides
- 15.0
- Molecular dynamics
- 9.0
- Solvents
- 5.0

## Figures

Representative ribbon and atomically detailed snapshots of decaalanine in vacuum are displayed along the steered path. From top to bottom, the structures correspond to: (a) a compact structure at the NN–NC distance of 13 Å, (b) the minimum energy conformation—an α-helix with an end-to-end distance of 15.2 Å, (c) a structure at the kind of the PMF shown in Fig. 2—at circa 26 Å, and (d) a coil structure at the end of one of the pulled trajectories at the end-to-end distance of 33 Å.

Representative ribbon and atomically detailed snapshots of decaalanine in vacuum are displayed along the steered path. From top to bottom, the structures correspond to: (a) a compact structure at the NN–NC distance of 13 Å, (b) the minimum energy conformation—an α-helix with an end-to-end distance of 15.2 Å, (c) a structure at the kind of the PMF shown in Fig. 2—at circa 26 Å, and (d) a coil structure at the end of one of the pulled trajectories at the end-to-end distance of 33 Å.

The comparison of the PMFs obtained from the adaptive SMD method pulling at 100 Å/ns (left column of panels) and 10 Å/ns (right column of panels) when a different selection criterion is used to choose the configuration from the structures at the end of each segment. The configuration is chosen according to the JA, MW, and RC criterion and displayed in the bottom, middle and top panels, respectively. Dashed curves in red, yellow, green, brown and blue represents 50, 100, 200, 400 and 800 trajectories per segment, respectively. The solid black curve is the PMF obtained from averaging 10,000 standard SMD simulations. Note that the standard PMF for the 10 Å/ns pulling simulations (solid black curves in the right column) largely overlaps onto the reversible PMF (although not shown).

The comparison of the PMFs obtained from the adaptive SMD method pulling at 100 Å/ns (left column of panels) and 10 Å/ns (right column of panels) when a different selection criterion is used to choose the configuration from the structures at the end of each segment. The configuration is chosen according to the JA, MW, and RC criterion and displayed in the bottom, middle and top panels, respectively. Dashed curves in red, yellow, green, brown and blue represents 50, 100, 200, 400 and 800 trajectories per segment, respectively. The solid black curve is the PMF obtained from averaging 10,000 standard SMD simulations. Note that the standard PMF for the 10 Å/ns pulling simulations (solid black curves in the right column) largely overlaps onto the reversible PMF (although not shown).

The convergence of the PMF as a function of the number of sampled trajectories in ASMD (JA, MW, and RC simulations) is shown through the relative root-mean-square (RMS) error in the total free energy difference between the initial and final points. In the top panel, the reference is the reversible PMF for which the total energy difference between the end points is 21.271 kcal/mol. In the bottom panel, the reference is the SMD for which the total energy difference between the end points is 30.134 kcal/mol at 100 Å/ns pulling speed, and 21.516 kcal/mol at Å/ns pulling speed. The abscissa displays the number of trajectories along a range from 50 (=50 × 2^{0}) to 3 200 (=50 × 2^{6}). Circle data points connected by the same colored solid lines were obtained from simulation with 100 Å/ns pulling rate; whereas, square data points connected by the same colored dashed lines were obtained from simulation with 10 Å/ns pulling rate. Black, red, green represent JA, MW, RC selection criteria, respectively.

The convergence of the PMF as a function of the number of sampled trajectories in ASMD (JA, MW, and RC simulations) is shown through the relative root-mean-square (RMS) error in the total free energy difference between the initial and final points. In the top panel, the reference is the reversible PMF for which the total energy difference between the end points is 21.271 kcal/mol. In the bottom panel, the reference is the SMD for which the total energy difference between the end points is 30.134 kcal/mol at 100 Å/ns pulling speed, and 21.516 kcal/mol at Å/ns pulling speed. The abscissa displays the number of trajectories along a range from 50 (=50 × 2^{0}) to 3 200 (=50 × 2^{6}). Circle data points connected by the same colored solid lines were obtained from simulation with 100 Å/ns pulling rate; whereas, square data points connected by the same colored dashed lines were obtained from simulation with 10 Å/ns pulling rate. Black, red, green represent JA, MW, RC selection criteria, respectively.

The convergence of the PMF as a function of the number of sampled trajectories in ASMD (JA selection criterion) and SMD is shown through the relative root-mean-square (RMS) error in the total free energy difference between the initial and final points. As in the top panel of Fig. 3, the reference is the reversible PMF and the number of trajectories are displayed with the same scales in the abscissa. Circle data points connected by the same colored solid lines were obtained from simulation with 100 Å/ns pulling rate; whereas, square data points connected by the same colored dashed lines were obtained from simulation with 10 Å/ns pulling rate. Black and red represent ASMD (JA) and SMD, respectively.

The convergence of the PMF as a function of the number of sampled trajectories in ASMD (JA selection criterion) and SMD is shown through the relative root-mean-square (RMS) error in the total free energy difference between the initial and final points. As in the top panel of Fig. 3, the reference is the reversible PMF and the number of trajectories are displayed with the same scales in the abscissa. Circle data points connected by the same colored solid lines were obtained from simulation with 100 Å/ns pulling rate; whereas, square data points connected by the same colored dashed lines were obtained from simulation with 10 Å/ns pulling rate. Black and red represent ASMD (JA) and SMD, respectively.

The comparison of the PMFs obtained from the adaptive SMD method pulling at 100 Å/ns (top) and 10 Å/ns (bottom) when the overall simulation window is divided into 10, 20, 40, and 80 segments.

The comparison of the PMFs obtained from the adaptive SMD method pulling at 100 Å/ns (top) and 10 Å/ns (bottom) when the overall simulation window is divided into 10, 20, 40, and 80 segments.

The PMFs obtained from the adaptive SMD method pulling at 100 Å/ns (top) and 10 Å/ns (bottom) are shown as a function of decaalanine end-to-end distance. The ensemble of trajectories is relaxed at constant temperature and end-to-end distance for 2 ps (red), 100 ps (green), and 200 ps (blue) between pulling segments. The solid black curve is the PMF obtained using ASMD with the JA selection criterion (and no relaxation between segments). The solid grey curve is the reversible PMF.

The PMFs obtained from the adaptive SMD method pulling at 100 Å/ns (top) and 10 Å/ns (bottom) are shown as a function of decaalanine end-to-end distance. The ensemble of trajectories is relaxed at constant temperature and end-to-end distance for 2 ps (red), 100 ps (green), and 200 ps (blue) between pulling segments. The solid black curve is the PMF obtained using ASMD with the JA selection criterion (and no relaxation between segments). The solid grey curve is the reversible PMF.

The average number of internal hydrogen bonds in decaalanine in vacuum is shown as a function of decaalanine end-to-end distance from 100 Å/ns (top) and 10 Å/ns (bottom) pulling simulations. All curves are labeled as in Fig. 6.

The average number of internal hydrogen bonds in decaalanine in vacuum is shown as a function of decaalanine end-to-end distance from 100 Å/ns (top) and 10 Å/ns (bottom) pulling simulations. All curves are labeled as in Fig. 6.

The average number of internal hydrogen bonds in decaalanine as a function of decaalanine end-to-end distance is shown for fast pulling (top panel) and slow pulling (bottom panel) simulations. Black represents *i* → *i* + 4 (α-helix), red represents *i* → *i* + 3 (3_{10}-helix), and green represents *i* → *i* + 5 (π-helix). The semi-transparent curves in the top panel correspond to five additional independent simulations and indicate the spread in the error. Each of these gave rise to a PMF which is the same as that shown earlier within the resolution of the plots.

The average number of internal hydrogen bonds in decaalanine as a function of decaalanine end-to-end distance is shown for fast pulling (top panel) and slow pulling (bottom panel) simulations. Black represents *i* → *i* + 4 (α-helix), red represents *i* → *i* + 3 (3_{10}-helix), and green represents *i* → *i* + 5 (π-helix). The semi-transparent curves in the top panel correspond to five additional independent simulations and indicate the spread in the error. Each of these gave rise to a PMF which is the same as that shown earlier within the resolution of the plots.

Ramachandran plots of the middle eight residues (excluding the termini residues because they do not have a pair of ϕ and ψ angles) is displayed for the 100 Å/ns stretching simulations (top) for the 10 Å/ns stretching simulations (bottom) (the selection criterion is JA, the number of trajectories sampled per step is 400 for each of them). Each diagram has a total of 3200 data points: 8 ϕ-ψ angle pairs for each of the 400 trajectories. The reaction coordinate begins from top left and goes towards bottom right by walking along each row. Coloring is as follows: ALA2, black; ALA3, red; ALA4, green; ALA5, blue; ALA6, yellow; ALA7, brown; ALA8, gray; ALA9, purple.

Ramachandran plots of the middle eight residues (excluding the termini residues because they do not have a pair of ϕ and ψ angles) is displayed for the 100 Å/ns stretching simulations (top) for the 10 Å/ns stretching simulations (bottom) (the selection criterion is JA, the number of trajectories sampled per step is 400 for each of them). Each diagram has a total of 3200 data points: 8 ϕ-ψ angle pairs for each of the 400 trajectories. The reaction coordinate begins from top left and goes towards bottom right by walking along each row. Coloring is as follows: ALA2, black; ALA3, red; ALA4, green; ALA5, blue; ALA6, yellow; ALA7, brown; ALA8, gray; ALA9, purple.

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