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Polyampholyte (PA) [5, 6, 15 - 18], a chain-like molecular model, is a typical case of the complex systems. We simulated the folding of the PA using the Replica Exchange Monte Carlo (REMC) method.

To keep the equilibrium of the overall system through the RE process, it is necessary that the system satisfies the detailed balance condition. For the exchanging configuration of the

Here, the symbols

Another virtue of the RE is easiness for parallel computing. The replicas are simulated independently except for the period of the exchange. Accordingly, the RE calculation can be fastened ideally in parallel with the number of replicas.

Here,

where

Figure 1. The illustration of the di-block polyampholyte.

The MC program was coded by Fortran 77, and the calculations performed with Alpha-Linux on non-parallelized system: because some variant calculations were performed simultaneously. We used the multiplicative method for random number generation.

Table 1. The parameters and conditions of the REMC and the SAMC calculations.

number of monomers: N | number of replicas | number of RE trials | MC steps^{a} / RE trial^{b} | MC steps for the SA^{c} |
---|---|---|---|---|

60 | 19 | 100 | 1.0 × 10^{6} ~ 3.0 × 10^{6} | 1.0 × 10^{8} ~ 3.0 × 10^{8} |

q / e | r / _{e}r_{0} | e / E_{0} | n | k / E_{0}r_{0}^{-4} |

±1 | 1 | 1 | 4 | 1.6 × 10^{4} |

Figure 2. The illustration of our procedure of the REMC.

**REMC.** The number of monomers of the di-block PA (*N*) is 60. The system contains one PA molecule in a vacuum. We consider the system to be a *replica*, and we assume 19 replicas to be an overall system. The initial configurations of all replicas are the same: a straight chain. The upper limit of the temperature is 100 *E*_{0}/*k _{B}*, and we set the temperatures in 5

In Figures 4, 5 we show the temperature dependences of the averaged intramolecular interaction, <

We performed 6 calculations for both REMC and SAMC respectively. The Stretched helix appeared not only in the RE but also in the SA, and the RE was hardly better than the SA for obtaining the probability of the Stretched helix: 3/6 in the RE and 2/6 in the SA. We conceive this result to be caused by the shape of the potential surface; if the potential barrier surrounding the local minimum is low, the local-minimum state easily surmounts the barrier without regard to application of the RE. It is considered that the potential function and its parameters determine the height of the potential barrier.

Shimizu, Hiwatari, and coworkers have performed multicanonical MD calculations for the folding transition of the di-block PA (

Figure 3. The final snapshots of the calculations at *T ^{r}* = 0.001. (a): The lowest energy configuration. (b): An example of the local-minima configuration. These two configurations were obtained by both RE and SA calculations.

Figure 4. The plots of the averaged intramolecular interaction <*U*> as a function of the temperature *T ^{r}*. This plot is the magnification of the low-temperature region.

Figure 5. The plots of the averaged distance between the center of mass of the PA and each monomer <r> as a function of the temperature *T ^{r}*.

[ 2] U. Wolff,

[ 3] B. A. Berg and T. Neuhaus,

[ 4] B. A. Berg and T. Neuhaus,

[ 5] H. Shimizu, K. Uehara, K. Yamamoto, and Y. Hiwatari,

[ 6] A. Baumketner, H. Shimizu, M. Isobe, and Y. Hiwatari,

[ 7] A. P. Lyubartsev, A. A. Martsinovski, S. V. Shevkunov, and P. N. Vorontsov-Velyaminov,

[ 8] E. Marinari and G. Parisi,

[ 9] W. Kerler and P. Rehberg,

[10] K. Hukushima and K. Nemoto,

[11] K. Hukushima, H. Takayama, and K. Nemoto,

[12] M. C. Tesi, E. J. J. van Rensburg, E. Orlandini, and S. G. Whittington,

[13] D. A.Kofke,

[14] A. Mitsutake, Y. Sugita, and Y. Okamoto,

[15] P. G. Higgs and J-F. Joanny,

[16] Y. Kantor, H. Li, and M. Kardar,

[17] Y. Kantor, M. Kardar, and H. Li,

[18] M. Tanaka, A. Y. Grosberg, V. S. Pande, and T. Tanaka,

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