^{1}, Yanzhao Huang

^{1}, Xiaofeng Ji

^{2}and Yi Xiao

^{1,a)}

### Abstract

Minimum Free Energy Path (MFEP) is very important in computational biology and chemistry. The barrier in the path is related to the reaction rate, and the start-to-end difference gives the relative stability between reactant and product. All these information is significant to experiment and practical application. But finding MFEP is not an easy job. Lots of degrees of freedom make the computation very complicated and time consuming. In this paper, we use the Steepest Descent Path (SDP) to accelerate the sampling of MFEP. The SHAKE algorithm and the * Lagrangian * multipliers are used to control the optimization of both SDP and MFEP. These strategies are simple and effective. For the former, it is more interesting. Because as we known, SHAKE algorithm was designed to handle the constraints in molecular dynamics in the past, has never been used in geometry optimization. Final applications on ALA dipeptide and 10-ALA peptide show that this combined optimization method works well. Use the information in SDP, the initial path could reach the more optimal MFEP. So more accurate free energies could be obtained and the amount of computation time could be saved.

This work is supported partially by the National Science Foundation of China (NSFC) under Grant Nos. 30800166, 11074084, and 11174093.

INTRODUCTION

MATERIALS AND METHODS

Models

Build the initial transition path

Optimization of reaction path

Optimization of MFEP

Simulation details

RESULTS AND DISCUSSIONS

CONCLUSION

### Key Topics

- Free energy
- 51.0
- Peptides
- 23.0
- Lagrangian mechanics
- 14.0
- Optimization
- 12.0
- Determinants
- 6.0

##### C08

## Figures

Two model peptides in our work. (a) ALA dipeptide, from C7_{eq} to C7_{ax}. (b) 10-ALA peptide, from *α*-helix to *β*-hairpin. The figure is produced by VMD. ^{ 74 }

Two model peptides in our work. (a) ALA dipeptide, from C7_{eq} to C7_{ax}. (b) 10-ALA peptide, from *α*-helix to *β*-hairpin. The figure is produced by VMD. ^{ 74 }

Free energy landscape for ALA dipeptide in vacuum (obtained by Adaptively Biased Molecular Dynamics (ABMD), ^{ 13,14 } in kcal/mol). The backbone angles, Φ and Ψ, are chosen as the degrees of freedom. Two minima on the surface, C7_{eq} and C7_{ax}, are marked by labels. The solid lines on the surface represent the optimized path after 0, 100, 200 and 300 iterations, respectively, from straight line to the most bending curve. The path is optimized by Eq. (2) .

Free energy landscape for ALA dipeptide in vacuum (obtained by Adaptively Biased Molecular Dynamics (ABMD), ^{ 13,14 } in kcal/mol). The backbone angles, Φ and Ψ, are chosen as the degrees of freedom. Two minima on the surface, C7_{eq} and C7_{ax}, are marked by labels. The solid lines on the surface represent the optimized path after 0, 100, 200 and 300 iterations, respectively, from straight line to the most bending curve. The path is optimized by Eq. (2) .

(a) Potential energy profiles for ALA dipeptide in vacuum from state C7_{eq} to C7_{ax}. The lines from top to bottom are the paths after 0, 100, 200 and 300 iterations (unit: kcal/mol). (b) The gradient of the path during the reaction path optimization (unit: kcal/mol). (c) The determinant of the generalized inverse of transformation matrix from Cartesian coordinate to collective variables.

(a) Potential energy profiles for ALA dipeptide in vacuum from state C7_{eq} to C7_{ax}. The lines from top to bottom are the paths after 0, 100, 200 and 300 iterations (unit: kcal/mol). (b) The gradient of the path during the reaction path optimization (unit: kcal/mol). (c) The determinant of the generalized inverse of transformation matrix from Cartesian coordinate to collective variables.

The free energy gradients in the constrained dynamics, obtained from Eq. (6) . (a) The gradient of the first term. (b) The gradient of the second term. See the details of the computation in the main text.

The free energy gradients in the constrained dynamics, obtained from Eq. (6) . (a) The gradient of the first term. (b) The gradient of the second term. See the details of the computation in the main text.

The movement of the path on the free energy landscape during the MFEP optimization. The solid lines represent the path after 0, 100, 200 and 300 iterations from SDP (see SDP in Fig. 2 ), and the dashed lines represent the path after 0, 100, 200 and 300 iterations from the initial straight path.

The movement of the path on the free energy landscape during the MFEP optimization. The solid lines represent the path after 0, 100, 200 and 300 iterations from SDP (see SDP in Fig. 2 ), and the dashed lines represent the path after 0, 100, 200 and 300 iterations from the initial straight path.

(a) Free energy profiles for ALA dipeptide in vacuum from state C7_{eq} to C7_{ax}. The lines from top to bottom are the data of the initial straight path, SDP, MFEP from initial path and MFEP from SDP, respectively. (b) The gradient of the path during the MFEP optimization from the initial straight path (dotted line) or SDP (solid line).

(a) Free energy profiles for ALA dipeptide in vacuum from state C7_{eq} to C7_{ax}. The lines from top to bottom are the data of the initial straight path, SDP, MFEP from initial path and MFEP from SDP, respectively. (b) The gradient of the path during the MFEP optimization from the initial straight path (dotted line) or SDP (solid line).

Optimization of 10-ALA peptide from *α*-helix to *β*-hairpin by three typical methods: Conjugate Gradient (CG, ^{ 51,52 } dotted line), Limited Broyden-Fletcher-Goldfarb-Shanno (LBFGS, ^{ 50 } dashed line), and Truncated Newton (TN, ^{ 53 } solid line). (a) Energy variance during the optimization (unit: kcal/mol). (b) Gradient variance during the optimization (unit: kcal/mol). (c) Structural RMSD during the optimization (unit: Å).

Optimization of 10-ALA peptide from *α*-helix to *β*-hairpin by three typical methods: Conjugate Gradient (CG, ^{ 51,52 } dotted line), Limited Broyden-Fletcher-Goldfarb-Shanno (LBFGS, ^{ 50 } dashed line), and Truncated Newton (TN, ^{ 53 } solid line). (a) Energy variance during the optimization (unit: kcal/mol). (b) Gradient variance during the optimization (unit: kcal/mol). (c) Structural RMSD during the optimization (unit: Å).

(a) Potential energy profiles for 10-ALA peptide in solvent from *α*-helix to *β*-hairpin. The lines from top to bottom indicate the paths after 0, 50, 100, 200, 300, and 500 iterations (unit: kcal/mol). (b) The gradient of the path during the reaction path optimization (unit: kcal/mol). (c) The determinant of the generalized inverse of transformation matrix from Cartesian coordinate to collective variables.

(a) Potential energy profiles for 10-ALA peptide in solvent from *α*-helix to *β*-hairpin. The lines from top to bottom indicate the paths after 0, 50, 100, 200, 300, and 500 iterations (unit: kcal/mol). (b) The gradient of the path during the reaction path optimization (unit: kcal/mol). (c) The determinant of the generalized inverse of transformation matrix from Cartesian coordinate to collective variables.

(a) Free energy profiles for 10-ALA peptide in solvent from *α*-helix to *β*-hairpin. The lines from top to bottom are the data of the initial straight path, SDP, MFEP from initial path and MFEP from SDP, respectively. (b) The gradient of the path during the MFEP optimization from the initial path (dotted line) or SDP (solid line).

(a) Free energy profiles for 10-ALA peptide in solvent from *α*-helix to *β*-hairpin. The lines from top to bottom are the data of the initial straight path, SDP, MFEP from initial path and MFEP from SDP, respectively. (b) The gradient of the path during the MFEP optimization from the initial path (dotted line) or SDP (solid line).

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