^{1,a)}

### Abstract

Free-energyperturbation calculation is frequently used to calculate free-energy differences because it is easy to implement and the computation is fast. However, the calculation is subject to large inaccuracies in some circumstances due to the insufficient sampling of the relevant tails of the energy-difference distributions. Here we expand this knowledge of insufficient sampling into a two-dimensional (2D) energy space using a model of harmonic oscillators. We show analytically the relation between the energies of the sampling system and those of the desired target energy spaces, which provide the basis to understand the difficulties in free-energyperturbation calculations. We clarify the reasons of the inaccurate calculation in the different harmonic cases that stem from the spatial separations of the reference and the target energy pairs located in the two-dimensional energy space. The potential-energy space introduced into this 2D energy-space model provides additional clues to improve the sampling efficiency. Based on this understanding, we propose two ways to calculate the free-energy differences using the two schemes of the distribution method. We show that the distribution method implemented in the appropriate energy space—the energy-difference space and the potential-energy space, respectively—can improve the calculation of free energies in different circumstances. This analysis implies that the sampling can be improved if it is directed toward the appropriate region in the potential-energy space, which is easily implemented in various types of free-energy calculations. To test this, we calculate the free-energy surface of alanine dipeptide in gas phase and in aqueous phase, respectively. We demonstrate that the free-energy surface calculation is improved when the biased sampling of the potential energy is integrated into the sampling scheme.

The author is grateful to Professor David Kofke for the helpful discussions and suggestions. This work was supported by Fudan University, Tan Jiazhen Young Investigator Grant, CAS-MPG, Natural Science Foundation of Shanghai (Grant No. 09ZR1403800) and Shanghai Leading Academic Discipline Project, Project Number: B111.

I. INTRODUCTION

II. THEORY

A. Free-energyperturbation calculation

B. Analytical model system

C. Four harmonic cases

III. METHODS

A. Distribution sampling schemes

B. Summary of the methods

IV. RESULTS AND DISCUSSION

A. FEP calculation of the harmonic-oscillator system

B. Free-energy surface of alanine dipeptide

V. CONCLUSION

### Key Topics

- Free energy
- 62.0
- Peptides
- 20.0
- Oscillators
- 5.0
- Phase space methods
- 5.0
- Spatial dimensions
- 5.0

## Figures

Analytical solutions of the reference and target energy-space distributions of the four harmonic cases. The red and blue contours in each case, obtained from the distributions *p* _{ A }(*U* _{ A },*W*) and *p* _{ B }(*U* _{ B },*W*), represent the reference energy spaces of system *A* and system *B*, respectively. The cyan and magenta contours (e.g., in cases B and D, they are located to the right of the blue and the red contours), obtained from *p* _{ A }(*U* _{ A },*W*) × *e* ^{−} ^{ βW } and *p* _{ B }(*U* _{ B },*W*) × *e* ^{+} ^{ βW }, represent the target energy spaces of system *A* and system *B*, respectively. The two insets show the rescaled plots of cases A and D. *U* and *W* have units of *kT*.

Analytical solutions of the reference and target energy-space distributions of the four harmonic cases. The red and blue contours in each case, obtained from the distributions *p* _{ A }(*U* _{ A },*W*) and *p* _{ B }(*U* _{ B },*W*), represent the reference energy spaces of system *A* and system *B*, respectively. The cyan and magenta contours (e.g., in cases B and D, they are located to the right of the blue and the red contours), obtained from *p* _{ A }(*U* _{ A },*W*) × *e* ^{−} ^{ βW } and *p* _{ B }(*U* _{ B },*W*) × *e* ^{+} ^{ βW }, represent the target energy spaces of system *A* and system *B*, respectively. The two insets show the rescaled plots of cases A and D. *U* and *W* have units of *kT*.

Comparison of *p* _{ A }(*U* _{ A },*W*) and *p* _{ B }(*U* _{ B },*W*) distributions obtained by the different methods for the harmonic case A. All the yellow distributions represent the analytical solutions. The red and blue distributions are obtained by conducting the sampling in the reference systems *A* and *B*, respectively. The method used for obtaining the distribution is labeled above each distribution. The contours in the figure have been described in Fig. 1, here the red and cyan contours are plotted together and the blue and magenta contours are plotted together. *U* and *W* have units of *kT*.

Comparison of *p* _{ A }(*U* _{ A },*W*) and *p* _{ B }(*U* _{ B },*W*) distributions obtained by the different methods for the harmonic case A. All the yellow distributions represent the analytical solutions. The red and blue distributions are obtained by conducting the sampling in the reference systems *A* and *B*, respectively. The method used for obtaining the distribution is labeled above each distribution. The contours in the figure have been described in Fig. 1, here the red and cyan contours are plotted together and the blue and magenta contours are plotted together. *U* and *W* have units of *kT*.

Comparison of the free energies calculated by the different methods for the harmonic case A. All the symbols represent the average results of ten independent calculations with the error bars attached to them. The symbols colored red or blue represent the results calculated by the corresponding red or blue distributions plotted in Fig. 2.

Comparison of the free energies calculated by the different methods for the harmonic case A. All the symbols represent the average results of ten independent calculations with the error bars attached to them. The symbols colored red or blue represent the results calculated by the corresponding red or blue distributions plotted in Fig. 2.

Comparison of *p* _{ A }(*U* _{ A },*W*) and *p* _{ B }(*U* _{ B },*W*) distributions obtained by the different methods for the harmonic case B. The symbol descriptions are the same as in Fig. 2.

Comparison of *p* _{ A }(*U* _{ A },*W*) and *p* _{ B }(*U* _{ B },*W*) distributions obtained by the different methods for the harmonic case B. The symbol descriptions are the same as in Fig. 2.

Comparison of the free energies calculated by the different methods for the harmonic case B. The symbol descriptions are the same as in Fig. 3 except that the symbols colored red or blue represent the results calculated by the corresponding red or blue distributions plotted in Fig. 4.

Comparison of the free energies calculated by the different methods for the harmonic case B. The symbol descriptions are the same as in Fig. 3 except that the symbols colored red or blue represent the results calculated by the corresponding red or blue distributions plotted in Fig. 4.

Comparison of *p* _{ A }(*U* _{ A },*W*) and *p* _{ B }(*U* _{ B },*W*) distributions obtained by the different methods for the harmonic case C. The symbol descriptions are the same as in Fig. 2.

Comparison of *p* _{ A }(*U* _{ A },*W*) and *p* _{ B }(*U* _{ B },*W*) distributions obtained by the different methods for the harmonic case C. The symbol descriptions are the same as in Fig. 2.

Comparison of the free energies calculated by the different methods for the harmonic case C. The symbol descriptions are the same as in Fig. 3 except that the symbols colored red or blue represent the results calculated by the corresponding red or blue distributions plotted in Fig. 6.

Comparison of the free energies calculated by the different methods for the harmonic case C. The symbol descriptions are the same as in Fig. 3 except that the symbols colored red or blue represent the results calculated by the corresponding red or blue distributions plotted in Fig. 6.

Comparison of *p* _{ A }(*U* _{ A },*W*) and *p* _{ B }(*U* _{ B },*W*) distributions obtained by the different methods for the harmonic case D. The symbol descriptions are the same as in Fig. 2.

Comparison of *p* _{ A }(*U* _{ A },*W*) and *p* _{ B }(*U* _{ B },*W*) distributions obtained by the different methods for the harmonic case D. The symbol descriptions are the same as in Fig. 2.

Comparison of the free energies calculated by the different methods for the harmonic case D. The symbol descriptions are the same as in Fig. 3 except that the symbols colored red or blue represent the results calculated by the corresponding red or blue distributions plotted in Fig. 8.

Comparison of the free energies calculated by the different methods for the harmonic case D. The symbol descriptions are the same as in Fig. 3 except that the symbols colored red or blue represent the results calculated by the corresponding red or blue distributions plotted in Fig. 8.

Schematic plot of alanine dipeptide and the *ϕ*, *ψ* dihedral angles.

Schematic plot of alanine dipeptide and the *ϕ*, *ψ* dihedral angles.

The *ϕ*–*ψ* free-energy surface of alanine dipeptide in gas phase calculated by the Metropolis-sampling method *M*1. Energy levels are in kcal/mol.

The *ϕ*–*ψ* free-energy surface of alanine dipeptide in gas phase calculated by the Metropolis-sampling method *M*1. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in gas phase calculated by the distribution method *M*4. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in gas phase calculated by the distribution method *M*4. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in gas phase calculated by the distribution method *M*3. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in gas phase calculated by the distribution method *M*3. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in gas phase calculated by the distribution method *M*5. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in gas phase calculated by the distribution method *M*5. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in aqueous phase calculated by the distribution method *M*4. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in aqueous phase calculated by the distribution method *M*4. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in aqueous phase calculated by the distribution method *M*3. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in aqueous phase calculated by the distribution method *M*3. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in aqueous phase calculated by the distribution method *M*5-a. The method is described in the text. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in aqueous phase calculated by the distribution method *M*5-a. The method is described in the text. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in aqueous phase calculated by the distribution method *M*5-b. The method is described in the text. Energy levels are in kcal/mol.

The free-energy surface of alanine dipeptide in aqueous phase calculated by the distribution method *M*5-b. The method is described in the text. Energy levels are in kcal/mol.

## Tables

Parameter sets for the four harmonic-model cases. In all cases, *N* = 10, *ω* _{ A } = 1, and *β* = 1. The *W* _{ A }, *W* _{ B }, *U _{A},* and

*U*

_{ B }energy ranges are utilized in the distribution methods

*M*2

_{ A },

*M*2

_{ B },

*M*3

_{ A }, and

*M*3

_{ B }, respectively. The subscript denotes the reference system

*A*or

*B*.

*U*and

*W*have units of

*kT*.

Parameter sets for the four harmonic-model cases. In all cases, *N* = 10, *ω* _{ A } = 1, and *β* = 1. The *W* _{ A }, *W* _{ B }, *U _{A},* and

*U*

_{ B }energy ranges are utilized in the distribution methods

*M*2

_{ A },

*M*2

_{ B },

*M*3

_{ A }, and

*M*3

_{ B }, respectively. The subscript denotes the reference system

*A*or

*B*.

*U*and

*W*have units of

*kT*.

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