Internal Coordinate Simulation
q˙n 1/2 Tn 1/2Mn 11/2 p˜n 1/2 pn 1/2 Mn 1/2 q˙n 1/2
fn 1/2 |
h |
pn 1/2 p˜n 1/2 |
|
2 |
|||
|
|
qn 1 qn q˙n 1/2 h
127
(9d)
(9e)
(9f)
(9g)
where T is the matrix of the corresponding projection operator. It is computed for a halfstep conformation with correctly closed rings. These additional computations only slightly reduce the net performance because it is still largely determined by the cost of evaluation of atom–atom forces [54].
V.PRACTICAL EXAMPLES
A. Time Step Limitations
Testing time step limitations plays an important role in ICMD because, in many cases, several alternative models of the same system can be constructed, with different spectra of fast motions. In general, in MD the step-size maximum depends on the system being studied, and for a given algorithm it is determined by its ability to conserve the total energy in microcanonical ensemble conditions [26,30]. For the leapfrog discretization the most appropriate method consists in checking the variation of the average total energy computed with different time steps [55]. The same test trajectory is computed starting from the same constant-energy hypersurface, and the average total energy is compared with the ‘‘ideal’’ value, i.e., its low time step limit. With growing time steps, the average total energy normally deviates upward, and a deviation of 0.2D[U], where U is the potential energy and D[ ] denotes the operator of time variance, is taken as an upper acceptable level. The step-size maximum thus determined is denoted as hc and is said to be ‘‘characteristic.’’
Figure 4 shows the results of two such time step tests for a hydrated B-DNA dodecamer duplex [54]. All bases were rigid except for rotation of thymine methyls. Bond lengths were fixed as well as all bond angles except those centered at sugar ring atoms. It is always interesting to check whether the time step is limited by harmonic or anharmonic motions. To distinguish them, virtually harmonic conditions are simulated by reducing the temperature to very low values so that the time step becomes limited by the highest frequency normal mode. In Figure 4a, for instance, the two traces corresponding to low and normal temperatures leave the band of acceptable deviation with a similar time step of around 4 fs, which indicates the harmonic nature of the limitation. The expected fastest harmonic mode in this case is the scissors HECEH vibration with a frequency around 1600 cm 1, which according to theory [55] should limit hc to approximately 3.6 fs. In order to raise hc to the level of 10 fs, inertias of hydrogen-only rigid bodies, as well as rigid bodies in flexible rings, are increased by different empirically adjusted increments. In the case of a scissors hydrogen, for example, an inertia Iij µδij is added at the position
δ µ ˚ 2
of the carbon atom, where ij is the Kronecker delta and 9 amu A . This means that the hydrogen is no longer considered as a point mass but as a rigid body of the same, but redistributed, mass, which helps to scale down the scissors frequency by a factor of 3.
128 |
Mazur |
Figure 4 Time step dependence of the average total energy for two models of a partially hydrated dodecamer DNA duplex. Thinner traces show results for virtually harmonic conditions when temperature was lowered to 1 K. The DNA molecule has fixed bond lengths, rigid bases, and fixed valence angles except for the intraand extracyclic bond angles in sugars. (a) No modifications of inertia;
(b) inertia modified as explained in the text. (From Ref. 54.)
Testing of the resulting model system is shown in Figure 4b. We see that both the low and room temperature hc values have increased to the desired 10 fs level.
Modification of inertia of hydrogen-only rigid bodies is a simple and safe way to balance different frequencies in the system, and it usually allows one to raise hc to 10 fs. Unfortunately, the further increase appears problematic because of various anharmonic effects produced by collisions between non-hydrogen atoms [48].
B. Standard Geometry Versus Unconstrained Simulations
In our last example we return to the issue of the possible damaging effects of the standard geometry constraints. Two long trajectories have been computed for a partially hydrated dodecamer DNA duplex of the previous example, first by using ICMD and second with Cartesian coordinate molecular dynamics without constraints [54]. Both trajectories started
˚
from the same initial conformation with RMSD of 2.6 A from the canonical B-DNA form. Figure 5 shows the time evolution of RMSD from the canonical A and B conformations. Each point in the figure corresponds to a 15 ps interval and shows an average RMSD value. We see that both trajectories approach the canonical B-DNA, while the RMSD
Internal Coordinate Simulation |
129 |
Figure 5 Time dependence of RMSD of atomic coordinates from canonical A- and B-DNA forms in two trajectories of a partially hydrated dodecamer duplex. The A and B (A and B correspond to A and B forms) trajectories started from the same state and were computed with internal and Cartesian coordinates as independent variables, respectively. (From Ref. 54.)
from A-DNA increases and reaches the level corresponding to the difference between the
˚
canonical A and B forms. The RMSD from B-DNA falls below the 2 A level, and in both
˚
cases the final RMSD from the crystallographic conformation is around 1.3 A. The RMSD
˚
between the two final computed states is around 1.1 A, which is within the range of short time scale fluctuations in dynamics, while the overall drift from the initial state goes
˚
beyond 2.5 A.
These two duplex models have 646 and 2264 internal degrees of freedom, respectively. In spite of this large difference they show very similar behavior in terms of atomic position fluctuations as well as in terms of fluctuations of torsions, sugar pseudorotation, and DNA helical parameters [54]. Apparently, the standard geometry model, which is allowed to move only along narrow paths in the full unconstrained configurational space, still keeps enough low energy tracks to sample from the main areas defined by a given temperature of 300 K. This example shows that the differences between the trajectories computed by ICMD and Cartesian MD at least are not readily seen, and, probably, for many applications they are not essential. It should be noted at the same time that the Cartesian coordinate trajectory was computed with a lower time step of 2 fs and took nearly five times as much computer time.
VI. CONCLUDING REMARKS
Internal coordinate molecular modeling is an efficient instrument with specific advantages that make it an indispensable complement to other existing approaches. It is best suited for simulation and analysis of large-scale structural transformations in biomacro-mole- cules, and at present ICMD is generally considered the most powerful tool in conforma-
130 |
Mazur |
tional searches, notably in NMR-based structural refinement [56]. Its application to physical problems involves certain controversial and unclear aspects that hold significant theoretical interest for future studies. The slow but steady progress in the development of these methods in recent years suggests that their performance and scope of application will continue to grow.
REFERENCES
1.HA Scheraga. Chem Rev 71:195–217, 1971.
2.KD Gibson, HA Scheraga. In: RH Sarma, MH Sarma, eds. Structure and Expression, Vol 1, From Proteins to Ribosomes. New York: Adenine Press, 1988, pp 67–94.
3.N Go, T Noguti, T Nishikawa. Proc Natl Acad Sci USA 80:3696–3700, 1983.
4.RA Abagyan. In: WF van Gunsteren, PK Weiner, AJ Wilkinson, eds. Computer Simulation of Biomolecular Systems. Dordrecht: Kluwer, 1998, pp 363–394.
5.R Lavery. Adv Comput Biol 1:69–145, 1994.
6.N Go, HA Scheraga. J Chem Phys 51:4751–4767, 1969.
7.R Abagyan, M Totrov, D Kuznetsov. J Comput Chem 15:488–506, 1994.
8.LM Rice, AT Bru¨nger. Proteins: Struct Funct Genet 19:277–290, 1994.
9.N Go, HA Scheraga. Macromolecules 9:535–542, 1976.
10.WF van Gunsteren. Mol Phys 40:1015–1019, 1980.
11.M Karplus, JN Kushick. Macromolecules 14:325–332, 1981.
12.WF van Gunsteren, M Karplus. Macromolecules 15:1528–1544, 1982.
13.IK Roterman, KD Gibson, HA Scheraga. J Biomol Struct Dyn 7:391–419, 1989.
14.IK Roterman, MH Lambert, KD Gibson, HA Scheraga. J Biomol Struct Dyn 7:421–453, 1989.
15.PA Kollman, KK Dill. J Biomol Struct Dyn 8:1103–1107, 1991.
16.KD Gibson, HA Scheraga. J Biomol Struct Dyn 8:1108–1111, 1991.
17.M Fixman. J Chem Phys 69:1527–1537, 1978.
18.E Helfand. J Chem Phys 71:5000–5007, 1979.
19.MR Pear, JH Weiner. J Chem Phys 71:212–224, 1979.
20.NG van Kampen. Phys Rep 124:69–160, 1985.
21.S Toxvaerd. J Chem Phys 87:6140–6143, 1987.
22.A Jain. J Comput Phys 136:289–297, 1997.
23.RA Laskowski, DS Moss, JM Thornton. J Mol Biol 231:1049–1067, 1993.
24.RA Marcus. Ber Bunsenges Phys Chem 136:190–197, 1977.
25.M Ben-Nun, RD Levine. J Chem Phys 105:8136–8141, 1996.
26.MP Allen, DJ Tildesley. Computer Simulation of Liquids. Oxford, UK: Clarendon Press, 1987.
27.H Goldstein. Classical Mechanics. Reading, MA: Addison-Wesley, 1980.
28.H Abe, W Braun, T Noguti, N Go. Comput Chem 8:239–247, 1984.
29.RW Hockney, JW Eastwood. Computer Simulation Using Particles. New York: McGraw-Hill, 1981.
30.JM Haile. Molecular Dynamics Simulations: Elementary Methods. New York: Wiley-Inter- science, 1992.
31.JP Ryckaert, A Bellemans. Chem Phys Lett 30:123–125, 1975.
32.JP Ryckaert, G Ciccotti, HJC Berendsen. J Comput Phys 23:327–341, 1977.
33.E Barth, K Kuczera, B Leimkuhler, RD Skeel. J Comput Chem 16:1192–1209, 1995.
34.JW Perram, HG Petersen. Mol Phys 65:861–874, 1988.
35.A Jain, N Vaidehi, G Rodriguez. J Comput Phys 106:258–268, 1993.
36.AM Mathiowetz, A Jain, N Karasawa, WA Goddard III. Proteins: Struct Funct Genet 20:227– 247, 1994.
37.GR Kneller, K Hinsen. Phys Rev E 50:1559–1564, 1994.
Internal Coordinate Simulation |
131 |
38.AK Mazur, RA Abagyan. J Biomol Struct Dyn 6:815–832, 1989.
39.AK Mazur, VE Dorofeyev, RA Abagyan. J Comput Phys 92:261–272, 1991.
40.VE Dorofeyev, AK Mazur. J Comput Phys 107:359–366, 1993.
41.KD Gibson, HA Scheraga. J Comput Chem 11:468–486, 1990.
42.AK Mazur. J Comput Chem 18:1354–1364, 1997.
43.AF Vereshchagin. Eng Cybernet 6:1343–1346, 1974.
44.R Featherstone. Robot Dynamics Algorithms. Boston: Kluwer, 1987.
45.G Rodriguez. IEEE J Robot Automat RA-3:624–639, 1987.
46.G Rodriguez, K Kreutz-Delgado. IEEE Trans Robot Automat 8:65–75, 1992.
47.WF van Gunsteren, HJC Berendsen. Angew Chem 29:992–1023, 1990.
48.AK Mazur. J Phys Chem B 102:473–479, 1998.
49.JM Sanz-Serna, MP Calvo. Numerical Hamiltonian Problems. London: Chapman and Hall, 1994.
50.VI Arnold. Mathematical Methods of Classical Mechanics. New York: Springer-Verlag, 1978.
51.A Jain. J Guidance, Control Dyn 14:531–542, 1991.
52.RA Abagyan, AK Mazur. J Biomol Struct Dyn 6:833–845, 1989.
53.AK Mazur. J Chem Phys 111:1407–1414, 1999.
54.AK Mazur. J Am Chem Soc 120:10928–10937, 1998.
55.AK Mazur. J Comput Phys 136:354–365, 1997.
56.P Gu¨ntert. Quart Rev Biophys 31:145–237, 1998.