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Molecular dynamics (MD) and Monte Carlo (MC) simulations are mainly used for the

Stochastic dynamics simulation is another solution for conformational search efficiency; it enables long time simulation because mild instabilities of certain long-time steps are masked [8, 9]. In Langevin dynamics simulation [8], one can treat the effects of solvent as a dissipative random force. Since the motion of polypeptide in aqueous solvent is slow, the solvent damping is large and the inertial memory is lost in a very short time. The relevant approximate equation is called the Brownian equation [9]. So far, each residue in a polypeptide has been mainly represented by one or two spheres connected by virtual bonds in Brownian dynamics of proteins [10, 11]. In order to obtain detailed information on the folding process, it is necessary to use more realistic models. So far, only a few attempts have been made using the atomistic model for 20-mer polyalanine [12] and 4-mer peptide [13]. In this report, we present an atomistic Brownian dynamics (BD) algorithm with implicit solvent model for protein folding simulation. We test the effectiveness of this algorithm using 28-mer bba peptide as a model. This is the first report of the stable simulation of an actual protein using an atomistic BD algorithm with implicit solvent model.

Here,

For the overdamped limit (the solvent damping is large and the inertial memory is lost in a very short time), we set the left side of Eq. 1 to zero,

The integrated equation of Eq. 2 is called Brownian dynamics [9];

where

In this simulation, the effective energy of the system

Intramolecular energy

The angle-dependent term,

where q

where

Figure 1. Ribbon representations of the pda8d NMR structure [15]. (a) A view from one side, with N and C termini labeled. (b) A view of the peptide rotated by 90o with respect to (a) around the vertical axis. The figures are generated with RasMol [43].

The BD simulation program was written in Fortran. All calculations were performed on an 800 MHz Duron processor.

Table 1. Computation Time for 1 nsec Dynamics of 28-mer bba Peptide

Solvent model | Cut-off radius(A) | Time(min) | Relative time |
---|---|---|---|

Vacuum MD^{a} | 9 | 85.1 | 1.0 |

BD without solvation energy^{b} | 9 | 52.5 | 0.6 |

DD/SA BD^{b} | 9 | 80.0 | 0.9 |

Explicit water MD^{c} | 9 | 4,230 | 49.7 |

Figure 2 shows Ca RMS deviations of the peptide from the NMR structure as a function of time. Snapshots of each trajectory at various periods are shown in Figure 3. The DD/SA model and explicit water model simulations appeared to give relatively stable trajectories, as judged by the smaller RMS deviations compared to that of

Figure 2. The Ca RMS deviations from NMR structure as a function of simulation time. From left to right: vacuum, DD/SA model, explicit water model.

Figure 3. Ribbon representations of the snapshots at 1 nsec (a, d, g), at 3 nsec (b, e, h), and at 5 nsec (c, f, i) in various solvent model simulations. First row is from vacuum simulation (a, b, c), second row is for DD/SA model (d, e, f) and third row is for explicit water model (g, e, f). The figures are generated with RasMol [43].

Figure 4 shows the radius of gyration (*R*_{g}) of the peptide as a function of time. The DD/SA solvent model gave stable *R*_{g}s at 9.5 A during 5 nsec simulation. The *R*_{g}s by DD/SA simulation were almost similar to those obtained by the explicit solvent simulation. *in vacuo* simulation gave much smaller *R*_{g}s.

Figure 4. The radii of gyration as a function of simulation time. From left to right: vacuum, DD/SA model, explicit water model.

We examined the Ca positional RMS fluctuations of the peptide during simulation using various solvent models (Figure 5). The RMS fluctuations observed in the simulation using DD/SA model were only a little smaller throughout the peptide than those observed in the simulation using the explicit solvent model, in which the fluctuations around the C terminus region of the peptide and the hairpin turn (residues 7 and 8) were specifically much larger. On the other hand, the fluctuations observed in the *in vacuo* simulation were very small throughout the peptide.

Figure 5. The RMS fluctuations of Ca atom positions obtained from simulations using different solvent models. The RMS fluctuation values were calculated from the average values over 2.5 nsec to 5 nsec period. From left to right: vacuum, DD/SA model, explicit water model.

Especially the smaller radii of gyration often observed by

We used LINCS algorithm [35] to constrain bond lengths instead of SHAKE algorithms [38] in our BD simulations. Both algorithm saved simulation time similarly, thus were useful. However, when we used SHAKE algorithm in BD simulation, the simulation was sometimes suspended since SHAKE did not converge for large atomic displacements, especially at high temperatures. Therefore, we implemented the more stable constraint algorithm, LINCS, in this study.

Although a simple surface area model was used for solvation energy calculation in this initial development of the atomistic Brownian dynamics method, the Generalized Born [30] or Poisson-Boltzmann solvation model can be utilized to improve accuracy of solvation energy calculation. Furthermore, the replica exchange method [7] can be combined to increase sampling efficiency in future development. The results shown in this paper indicate that BD simulation with the implicit solvent model is a promising approach that replaces MD simulation with explicit water molecules.

This work was supported (I. Y.) by a Grant-in-Aid for Scientific Research on Priority Area (C) Genome Information Science from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

[ 2] U. H. Hansmann and Y. Okamoto,

[ 3] T. Lazaridis and M. Karplus,

[ 4] D. J. Osguthorpe,

[ 5] C. Hardin, T. V. Pogorelov and Z. Luthey-Schulten,

[ 6] B. Nolting,

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

[ 8] W. F. van Gunsteren,

[ 9] D. L. Ermak and J. A. McCammon,

[10] J. A. MaCammon, S. H. Northrup, M. Karplus and R. M. Levy,

[11] S. Takada, Z. Luthey-Schulten and P. G. Wolynes,

[12] N. Gronbech-Jensen and S. Doniach,

[13] T. Shen, C. F. Wong and J. A. McCammon,

[14] S. J. Weiner, P. A. Kollman, D. A. Case, U. C. Singh, C. Ghio, G. Alagona, S. J. Profeta and P. Weiner,

[15] B. I. Dahiyat, C. A. Sarisky and S. L. Mayo,

[16] B. I. Dahiyat, D. B. Gordon and S. L. Mayo,

[17] B. I. Dahiyat and S. L. Mayo,

[18] B. I. Dahiyat and S. L. Mayo,

[19] S. M. Malakauskas and S. L. Mayo,

[20] S. W. Rick and B. J. Berne,

[21] R. M. Levy and E. Gallicchio,

[22] B. R. Gelin and M. Karplus,

[23] J. Vila, R. L. Williams, M. Vasquez and H. A. Scheraga,

[24] L. Wesson and D. Eisenberg,

[25] B. von Freyberg, T. J. Richmond and W. Braun,

[26] F. Fraternali and W. F. van Gunsteren,

[27] T. Lazaridis and M. Karplus,

[28] W. C. Still, A. Tempczyk, R. C. Hawley and T. Hendrickson,

[29] B. N. Dominy and C. L. Brooks III,

[30] Y. Liu and D. L. Beveridge,

[31] J. Zhu, Y. Shi and H. Liu,

[32] M. Schaefer and M. Karplus,

[33] M. Kinoshita, Y. Okamoto and F. Hirata,

[34] W. Hasel, T. F. Hendrickson and W. C. Still,

[35] B. Hess, H. Beker, H. J. C. Berendsen and J. G. E. M. Fraaije,

[36] W. D. Cornell, P. Cieplak, C. I. Bayly, I. R. Gould, K. M. Merz Jr., D.M. Ferguson, D. C. Spellmeyer, T. Fox, J. W. Caldwell and P. A. Kollman,

[37] H. J. C. Berendsen, J. P. M. Postma, W. F. van Gunsteren, A. DiNola and J. R. Haak,

[38] J. P. Ryckaert, G. Ciccotti and H. J. C. Berendsen,

[39] C. L. Brooks III, M. Karplus and B.M. Pettitt,

[40] P. Ferrara and A. Caflisch,

[41] P. Ferrara, J. Apostolakis and A. Caflisch,

[42] P. Ferrara, J. Apostolakis and A. Caflisch,

[43] R. A. Sayle and E. J. Milner-White,

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