Protein simulations using techniques suitable for very large systems: The cell multipole method for nonbond interactions and the Newton‐Euler inverse mass operator method for internal coordinate dynamics

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Abstract
Two new methods developed for molecular dynamics simulations of very large proteins are applied to a series of proteins ranging up to the protein capsid of tomato bushy stunt virus (TBSV). For molecular dynamics of very large proteins and polymers, it is useful to carry out the dynamics using internal coordinates (say, torsions only) rather than Cartesian coordinates. This allows larger time steps, eliminates problems with the classical description of high energy modes, and focuses on the important degrees of freedom. The resulting equation of motion has the form where for T is the vector of generalized forces, ℳ︁(θ) is the moments of inertia tensor, is the vector of torsions, and 𝒞 is a vector containing Coriolis forces and nonbond forces. The problem is that to calculate the acceleration vector from ℳ︁, 𝒞, and Trequires inverting. ℳ︁(θ), an order 𝒩3calculation. Since the number of degrees of freedom might be 300,000 for a million atom system, solving these equations every time step is impractical, restricting internal coordinate methods to small systems. The new method, Newton-Euler Inverse Mass Operator (NEIMO) dynamics, constructs the torsional accelerations vector directly by an order 𝒩 process, allowing internal-coordinate dynamics to be solved for super larger (million atom) systems, The first use of the NEIMO method for molecular dynamics of proteins is presented here. A second serious difficulty for large proteins is calculation of the nonbond forces. We report here the first application to proteins of the new Cell Multipole Method (CMM) to evaluate the Coulomb and van der Waals interactions. The cost of CMM scales linearly with the number of particles while retaining an accuracy significantly better than standard non bond methods (involving cutoffs). Results for NEIMO and CMM are given for simulations of a wide range of peptide and protein systems, including the protein capsid of TBSV with 488,000 atoms. The computational times for NEIMO and CMM are demonstrated to scale linearly with size. With NEIMO the dynamics time steps can be as large as 20 fs (for small peptides), much larger than possible with standard Cartesian coordinate dynamics. For TBSV we considered both the normal form and the high pH form, in which the Ca2+ ions are removed. These calculations lead to a contraction of the protein for both forms (probably because of ignoring the RNA core not observed in the X-ray).