Our study is divided into two parts: The first part studies effects, which are observable within simulation times covering a few ps, and addresses questions like the accuracy of force approximation and the size of the energy drift caused by algorithmic noise. As our sample system we have chosen here the small protein bovine pancreatic trypsin inhibitor (BPTI), which is frequently used as a test system for evaluation of simulation methods [53, 7, 54, 3]. We embedded BPTI in a water shell of 27Å radius, comprising a total of 7,147 atoms.
The second part serves to estimate algorithmic accuracy with respect to effects on longer time scales. For that purpose the sample system introduced above is prohibitively large as far as the intended reference simulations are concerned, since these require the computationally very expensive evaluation of the Coulomb sum, eq. (1). Therefore we had to choose here a much smaller sample system, namely BPTI in vacuo comprising only 568 atoms. The test simulations from which we have extracted regular statistical observables for our quality estimates covered 1.2ns each.
Since it is well known that for realistic protein simulations the natural environment has to be included [6, 7, 55, 4] a note on our choice of an in vacuo test system is necessary: Concerning our long time simulations we do not aim to evaluate the quality of the physical model of the protein, which, here, may be poor. Rather, we want to check to what extent our algorithmic approximations affect the molecular dynamics of our given model. For that purpose, the physical model used here does not necessarily have to be very accurate as long as possible artifacts in realistic applications are likely to show up in our test simulations, too.
All test simulations were carried out with the MD program
EGO_VIII [51], which employs the CHARMM force
field [14].
Non-polar hydrogen atoms were represented by compound atoms [14],
and the lengths of chemical bonds involving polar hydrogen atoms were fixed
using the SHAKE algorithm [3].
An integration step size of 1fs was used.
Water molecules were described by the TIP3 model [56].
The interaction list was updated every 64-th integration step.
For the cutoff simulations a cutoff distance
of 9Å and a switching function [57] were chosen.
Translations and rotations of the protein were eliminated as
described in refs. [58] and [59].
Prior to the test simulations, BPTI in vacuo
and the BPTI-water system were equilibrated for 50ps and
70ps, respectively, using the reference method.
For the long-time simulations of BPTI in vacuo
the temperature was kept fixed at 300K by
weakly coupling the system to a heat bath
using a coupling constant of 
s as defined in
refs. [60] and [61].
In contrast, the short-time simulations of the large BPTI-water system
were carried out in the microcanonical ensemble as to enable an estimate, to
what extent algorithmic artifacts entail violations of energy conservation.
For the first part of our quality estimates we will compare results of simulations obtained by the supposedly exact reference method with results obtained by SAMM, FAMUSAMM/linear and FAMUSAMM/DC-1d. Here, comparisons with the cutoff method are superfluous, since related accuracy tests comparing the various versions of SAMM with that method have been published previously [19, 18]. By these tests the superiority of the SAMM algorithms as compared to the conventional cutoff procedure has been convincingly demonstrated. Thus, it solely remains to be checked, to what extent our FAMUSAMM approach preserves the advantages of its parent method.
For the second part we have performed 1.2ns test simulations utilizing all methods under consideration. Here we have included the cutoff approach, since related investigations have not yet been published. Due to the statistical nature of our observables a second 1.2ns test simulation was necessary for the reference method in order to enable an estimate of the sizes of statistical fluctuations. Only upon such estimates algorithmic artifacts can be identified and can be distinguished from inevitable fluctuations of the necessarily statistical observables compared. The initial conditions of the first reference simulation and of the test simulations for the approximate methods were chosen as the atomic positions and velocities obtained after equilibration. For the second reference simulation these initial positions were modified by extremely small random amounts. Note that the chaotic character of protein dynamics ensures complete decorrelation of the two reference simulations within a few picoseconds.