Approximation methods, nonbonded interactions

In LN, the bonded interactions are treated by the approximate linearization, and the local nonbonded interactions, as well as the nonlocal interactions, are treated by constant extrapolation over longer intervals Atm and At, respectively). We define the integers fci,fc2 > 1 by their relation to the different timesteps as Atm — At and At = 2 Atm- This extrapolation as used in LN contrasts the modern impulse MTS methods which only add the contribution of the slow forces at the time of their evaluation. The impulse treatment makes the methods symplectic, but limits the outermost timestep due to resonance (see figures comparing LN to impulse-MTS behavior as the outer timestep is increased in [88]). In fact, the early versions of MTS methods for MD relied on extrapolation and were abandoned because of a notable energy drift. This drift is avoided by the phenomenological, stochastic terms in LN. 

Schulten238 outlined the development of a multiple-time-scale approximation (distance class algorithm) for the evaluation of nonbonded interactions, as well as the fast multipole expansion (FME). efficiency of the FME was demonstrated when the method outperformed the direct evaluation of Coulomb forces for 5000 atoms by a large margin and showed, for systems of up to 24,000 atoms, a linear dependence on atom number. 

The motions of proteins are usually simulated in aqueous solvent. The water molecules can be represented either explicitly or implicitly. To include water molecules explicitly implies more time-consuming calculations, because the interactions of each protein atom with the water atoms and the water molecules with each other are computed at each integration time step. The most expensive part of the energy and force calculations is the nonbonded interactions because these scale as 77 where N is the number of atoms in the system. Therefore, it is common to neglect nonbonded interactions between atoms separated by more than a defined cut-off ( 10 A). This cut-off is questionable for electrostatic interactions because of their 1/r dependence. Therefore, in molecular dynamics simulations, a Particle Mesh Ewald method is usually used to approximate the long-range electrostatic interactions (71, 72). 

The simplest approach to obtaining optimized bond angles close to the reference value 0() (Fig. 1) is to introduce a quadratic energy penalty, the harmonic approximation, similar to the representation of bond energies (Eq. (4)), although some methods use nonbonded interactions to model angle forces [3]. 

The foregoing method enjoys great versatility. The chain may be of any specified length and structure. If it comprises a variety of skeletal bonds and repeat units, the factors entering into the serial products have merely to be fashioned to introduce the characteristics of the bond represented by each of the successive factors. The mathematical methods are exact the procedure is free of approximations beyond that involved in adoption of the rotational isometric state scheme. With judicious choice of rotational states, the error here Involved is generally within the limits of accuracy of basic information on bond rotations, nonbonded interact ions, etc. 

Parallel molecular dynamics codes are distinguished by their methods of dividing the force evaluation workload among the processors (or nodes). The force evaluation is naturally divided into bonded terms, approximating the effects of covalent bonds and involving up to four nearby atoms, and pairwise nonbonded terms, which account for the electrostatic, dispersive, and electronic repulsion interactions between atoms that are not covalently bonded. The nonbonded forces involve interactions between all pairs of particles in the system and hence require time proportional to the square of the number of atoms. Even when neglected outside of a cutoff, nonbonded force evaluations represent the vast majority of work involved in a molecular dynamics simulation. 

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