Next: Chapter 3 Up: Dynamic and Stochastic Protein Previous: Conclusions

Leapfrog Verlet for Internal Coordinates

The Verlet algorithm is probably the most popular algorithm for integrating the equations of motions in standard Cartesian-space molecular dynamics. Currently, NEIMO dynamics uses a modification of one formulation of the Verlet algorithm, the ``leapfrog'' formulation. Regardless of the formulation, the Verlet algorithm requires that accelerations be calculated at a time , before the velocities at time are known. However, NEIMO accelerations at time depend upon the velocities at time , so these velocities must first be estimated. Iteration allows for a more accurate determination of the velocities.

A ``leapfrog'' Verlet calculation with a timestep of determines the accelerations at integer timesteps (, simulation time = ) and uses these to determine the velocities at timestep (). These velocities are then used to determine the coordinates at timestep (). The next dynamics timestep then proceeds with . However, in order to calculate , NEIMO requires the unknown as well as known . It is, therefore, first necessary to estimate the velocities from the previously determined velocities:

It is then possible to calculate the NEIMO accelerations . Accelerations are calculated by solving the spatial operator (SO) equations described in detail in References [33] and [32]. The accelerations are a function of the coordinates , the velocities , the torques , and/or the forces . The forces and torques are

calculated from the derivatives of the potential energy functions with respect to Cartesian and dihedral coordinates, respectively. The accelerations are used to update the velocities as they are in Cartesian dynamics (cf. Equation ()).

Because is estimated in Equation (), is inaccurate and such errors could build up as the simulation progresses. In order to eliminate such errors, is re-estimated from the new and the known .

Repeating Equations through until converges produces a very accurate value for . Sufficient convergence is generally reached after a single iteration, so the effect on overall computational speed is minimal. The converged values of in Equation () give values for which are then used to update the coordinates.

The dynamics step is completed by updating the Cartesian coordinates from the new internal coordinates .


I.T. Weber et al., Science, 243, 928-931 (1989).
R.S. Struthers, D.H. Kitson, and A.T. Hagler, Proteins: Structure, Function, Genet., 9, 1-11 (1991).
J.F. Leszczynski and G.D. Rose, Science, 234, 849-855 (1986).
C. Chothia and A.M. Lesk, J. Mol. Biol., 196, 901-917 (1987).
T.A. Jones and S. Thirup, EMBO J., 5, 819-822 (1986).
R.E. Bruccoleri and M. Karplus, Biopolymers, 26, 137-168 (1987).
P.S. Shenkin et al., Biopolymers, 26, 2053-2085 (1987).
A.C.R. Martin, J.C. Cheetham, and A.R. Rees, Proc. Natl. Acad. Sci., USA, 86, 9268-9272 (1989).
S.L. Mayo, B.D. Olafson, and W.A. Goddard III, J. Phys. Chem., 94, 8897-8909 (1990).
BIOGRAF/POLYGRAF. Copyright by Molecular Simulations, Inc. (1992).
N. Go and H.A. Scheraga, Macromolecules, 3, 178-187 (1970).
N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, and A.H. Teller, J. Chem. Phys., 21, 1087-1092 (1953).
D. Eisenberg and A.D. McLachlan, Nature, 319, 199-203 (1986).
Y. Satow et al., J. Mol. Biol., 190, 593-604 (1986).
S. Sheriff et al., Proc. Natl. Acad. Sci., USA, 84, 8075-8079 (1987).
C. Chothia et al., Nature, 342, 887-883 (1989).
R.M. Fine et al., Proteins: Structure, Function, Genet., 1, 342-362 (1986).
R.E. Bruccoleri, E. Haber, and J. Novotný, Nature, 335, 564-568 (1988).
D.R. Davies, E.A. Padlan, and S. Sheriff, Annu. Rev. Biochem., 59, 439-473, (1990).
E.A. Padlan, G.H. Cohen, and D.R. Davies, Ann. Inst. Pasteur/Immunol., 136C, 259-294 (1985).

Next: Chapter 3 Up: Dynamic and Stochastic Protein Previous: Conclusions
Sat Jun 18 14:06:11 PDT 1994