^{1,a)}and Yuko Okamoto

^{2,b)}

### Abstract

We give general formulations of the multidimensional multicanonical algorithm, simulated tempering, and replica-exchange method. We generalize the original potential energy function by adding any physical quantity of interest as a new energy term. These multidimensional generalized-ensemble algorithms then perform a random walk not only in space but also in space. Among the three algorithms, the replica-exchange method is the easiest to perform because the weight factor is just a product of regular Boltzmann-like factors, while the weight factors for the multicanonical algorithm and simulated tempering are not *a priori* known. We give a simple procedure for obtaining the weight factors for these two latter algorithms, which uses a short replica-exchange simulation and the multiple-histogram reweighting techniques. As an example of applications of these algorithms, we have performed a two-dimensional replica-exchange simulation and a two-dimensional simulated-tempering simulation using an -helical peptide system. From these simulations, we study the helix-coil transitions of the peptide in gas phase and in aqueous solution.

Some of the results were obtained by the computations on the super computers at the Institute for Molecular Science, Okazaki, Japan. This work was supported, in part, by Grants-in-Aid for Scientific Research in Priority Areas (“Water and Biomolecules” and “Molecular Theory for Real Systems”), for Scientific Research on Innovative Areas (“Fluctuations and Biological Functions”), and for the Next Generation Super Computing Project, Nanoscience Program from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.

I. INTRODUCTION

II. METHODS

A. Generalized energy function

B. Multidimensional multicanonical algorithm

C. Multidimensional simulated tempering

D. Multidimensional replica-exchange method

E. Weight factor determinations for multidimensional MUCA and ST

F. Averages of physical quantities

III. COMPUTATIONAL DETAILS

IV. RESULTS AND DISCUSSION

A. Two-dimensional REM simulation

B. Two-dimensional ST simulation

C. Detailed analysis of the solvent effects

V. CONCLUSIONS

### Key Topics

- Free energy
- 30.0
- Random walks
- 23.0
- Monte Carlo methods
- 20.0
- Molecular dynamics
- 10.0
- Electron densities of states
- 9.0

## Figures

Time series of the labels of , , (a) and , , (b) as functions of MC sweeps, and that of both and for the region from 400 000 MC sweeps to 700 000 MC sweeps (c). The results were from one of the replicas (replica 1). In (a) and (b), MC sweeps start at 100 000 and end at 1 100 000 because the first 100 000 sweeps have been removed from the consideration for thermalization purpose.

Time series of the labels of , , (a) and , , (b) as functions of MC sweeps, and that of both and for the region from 400 000 MC sweeps to 700 000 MC sweeps (c). The results were from one of the replicas (replica 1). In (a) and (b), MC sweeps start at 100 000 and end at 1 100 000 because the first 100 000 sweeps have been removed from the consideration for thermalization purpose.

Time series of the temperature (a), total energy (b), conformational energy (c), solvation free energy (d), and end-to-end distance (e) for the same replica as in Fig. 1. The temperature is in K, the energy is in kcal/mol, and the end-to-end distance is in Å.

Time series of the temperature (a), total energy (b), conformational energy (c), solvation free energy (d), and end-to-end distance (e) for the same replica as in Fig. 1. The temperature is in K, the energy is in kcal/mol, and the end-to-end distance is in Å.

Contour curves and histograms of distributions of the total energy and the solvation free energy [(a) and (b)] from the two-dimensional REM simulation.

Contour curves and histograms of distributions of the total energy and the solvation free energy [(a) and (b)] from the two-dimensional REM simulation.

The average total energy (a), average conformational energy (b), average of (c), and average end-to-end distance (d) with all the values as functions of temperature. The lines colored in red, green, blue, and purple are for , , , and , respectively.

The average total energy (a), average conformational energy (b), average of (c), and average end-to-end distance (d) with all the values as functions of temperature. The lines colored in red, green, blue, and purple are for , , , and , respectively.

The dimensionless free energy as a function of labels of temperature, , obtained by the two-dimensional REM simulation. The four curves correspond to , and 4 from top to bottom.

The dimensionless free energy as a function of labels of temperature, , obtained by the two-dimensional REM simulation. The four curves correspond to , and 4 from top to bottom.

Time series of the labels of , i.e., (a) and , i.e., (b) as functions of MC sweeps, and those of both and for the region from 350 000 MC sweeps to 550 000 MC sweeps (c). In (a) and (b), MC sweeps start at 100 000 and end at 1 100 000 because the first 100 000 sweeps have been removed from the consideration for thermalization.

Time series of the labels of , i.e., (a) and , i.e., (b) as functions of MC sweeps, and those of both and for the region from 350 000 MC sweeps to 550 000 MC sweeps (c). In (a) and (b), MC sweeps start at 100 000 and end at 1 100 000 because the first 100 000 sweeps have been removed from the consideration for thermalization.

Histogram of the distribution of the labels of , , and , , obtained by the two-dimensional ST simulation.

Histogram of the distribution of the labels of , , and , , obtained by the two-dimensional ST simulation.

Time series of the temperature (a), total energy (b), conformational energy (c), solvation free energy (d), and end-to-end distance (e) for the two-dimensional ST simulation.

Time series of the temperature (a), total energy (b), conformational energy (c), solvation free energy (d), and end-to-end distance (e) for the two-dimensional ST simulation.

The average total energy (a), average conformational energy (b), average of (c), and average end-to-end distance (d) with all the values as functions of temperature. The lines colored in red, green, blue, and purple are for , , , and , respectively.

Time series of the total energy [(a) and (b)], conformational energy [(c) and (d)], and solvation free energy [(e) and (f)] with (in gas phase) and with (in aqueous solution), respectively. The red, green, and blue curves are for the fixed temperatures (300 K), (549 K), and (700 K), respectively. In gas phase, the total energy of (a) is the same as the conformational energy of (c). The scales of the ordinate in (a) and (c) are different from each other.

Time series of the total energy [(a) and (b)], conformational energy [(c) and (d)], and solvation free energy [(e) and (f)] with (in gas phase) and with (in aqueous solution), respectively. The red, green, and blue curves are for the fixed temperatures (300 K), (549 K), and (700 K), respectively. In gas phase, the total energy of (a) is the same as the conformational energy of (c). The scales of the ordinate in (a) and (c) are different from each other.

The lowest-total-energy conformations obtained at the lowest temperature (300 K) with (in gas phase) (a) and (in aqueous solution) (b), and the two super imposed conformations (c). VMD software (Ref. 86) and RASTER 3D software (Ref. 87) were used to create the figures. The solid spheres are the oxygen atoms (in red) and the nitrogen atoms (in blue) in Table V.

The lowest-total-energy conformations obtained at the lowest temperature (300 K) with (in gas phase) (a) and (in aqueous solution) (b), and the two super imposed conformations (c). VMD software (Ref. 86) and RASTER 3D software (Ref. 87) were used to create the figures. The solid spheres are the oxygen atoms (in red) and the nitrogen atoms (in blue) in Table V.

The atomistic solvation free energy as a function of labels of heavy atoms for the lowest-total-energy conformations obtained at the lowest temperature (300 K) with (in gas phase) and (in aqueous solution).

The atomistic solvation free energy as a function of labels of heavy atoms for the lowest-total-energy conformations obtained at the lowest temperature (300 K) with (in gas phase) and (in aqueous solution).

Snapshots at 200 000 MC sweeps [(a) and (d)], at 600 000 MC sweeps [(b) and (e)], and at 1 000 000 MC sweeps [(c) and (f)] at the highest temperature . (a)–(c) correspond to (in gas phase) and (d)–(f) to (in aqueous solution). VMD software (Ref. 86) and RASTER 3D software (Ref. 87) were used to create the figures.

Snapshots at 200 000 MC sweeps [(a) and (d)], at 600 000 MC sweeps [(b) and (e)], and at 1 000 000 MC sweeps [(c) and (f)] at the highest temperature . (a)–(c) correspond to (in gas phase) and (d)–(f) to (in aqueous solution). VMD software (Ref. 86) and RASTER 3D software (Ref. 87) were used to create the figures.

Time series of the end-to-end distance with (in gas phase) (a) and with (in aqueous solution) (b) at the highest temperature (700 K).

Time series of the end-to-end distance with (in gas phase) (a) and with (in aqueous solution) (b) at the highest temperature (700 K).

## Tables

Acceptance ratios of replica exchanges between pairs of temperatures, , , with fixed from the two-dimensional REM simulation. In the case of , is set to be .

Acceptance ratios of replica exchanges between pairs of temperatures, , , with fixed from the two-dimensional REM simulation. In the case of , is set to be .

Acceptance ratios of replica exchanges between pairs of parameters, , , with fixed temperatures from the two-dimensional REM simulation. In the case of , is set to be .

Acceptance ratios of replica exchanges between pairs of parameters, , , with fixed temperatures from the two-dimensional REM simulation. In the case of , is set to be .

Acceptance ratios of -updates, , with fixed in the two-dimensional ST simulation. In the cases of and 8, and are set to be and , respectively.

Acceptance ratios of -updates, , with fixed in the two-dimensional ST simulation. In the cases of and 8, and are set to be and , respectively.

Acceptance ratios of -updates, , with fixed in the two-dimensional ST simulation. In the cases of and 4, and are set to be and , respectively.

Acceptance ratios of -updates, , with fixed in the two-dimensional ST simulation. In the cases of and 4, and are set to be and , respectively.

Differences in the atomistic solvent-accessible surface area and the atomistic solvation free energy of heavy atoms between the lowest-total-energy conformations obtained in aqueous solution and in gas phase [i.e., and ]. We list heavy atoms of N in Ser1, OG in Ser1, OG in Ser2, OD in Asp3, NE in Gln8, OH in Tyr12, NZ in Lys13, OE in Glu16, and OD in Asp17, of which the absolute values of are larger than 1.2 kcal/mol. The solvent-accessible surface area is in and the solvation free energy is in kcal/mol.

Differences in the atomistic solvent-accessible surface area and the atomistic solvation free energy of heavy atoms between the lowest-total-energy conformations obtained in aqueous solution and in gas phase [i.e., and ]. We list heavy atoms of N in Ser1, OG in Ser1, OG in Ser2, OD in Asp3, NE in Gln8, OH in Tyr12, NZ in Lys13, OE in Glu16, and OD in Asp17, of which the absolute values of are larger than 1.2 kcal/mol. The solvent-accessible surface area is in and the solvation free energy is in kcal/mol.

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