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. 2015 Dec 8;11(12):5624-37.
doi: 10.1021/acs.jctc.5b00648. Epub 2015 Nov 25.

A Stochastic Algorithm for the Isobaric-Isothermal Ensemble with Ewald Summations for All Long Range Forces

Michele Di Pierro  1 , Ron Elber  2   3 , Benedict Leimkuhler  4
Affiliations

A Stochastic Algorithm for the Isobaric-Isothermal Ensemble with Ewald Summations for All Long Range Forces

Michele Di Pierro et al. J Chem Theory Comput. .

Abstract

We present an algorithm termed COMPEL (COnstant Molecular Pressure with Ewald sum for Long range forces) to conduct simulations in the NPT ensemble. The algorithm combines novel features recently proposed in the literature to obtain a highly efficient and accurate numerical integrator. COMPEL exploits the concepts of molecular pressure, rapid stochastic relaxation to equilibrium, exact calculation of the contribution to the pressure of long-range nonbonded forces with Ewald summation, and the use of Trotter expansion to generate a robust, highly stable, symmetric, and accurate algorithm. Explicit implementation in the MOIL program and illustrative numerical examples are discussed.

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Figures

Figure 1
Figure 1
When a molecule extends across the periodic boundary atoms that belong to the same molecule interact through different periodic copies. When computing the molecular virial, atoms i1, j1, k1 are associated with the molecule centered in r→1, while the atoms l2, m2, n2 are associated with a copy that is positioned at r→2 = r→1 + δ→. Therefore, in implementing the molecular virial, it is necessary to allocate multiple molecular coordinates to a single molecule.
Figure 2
Figure 2
The peptide system is compressed to a target pressure of 1200 Atm (see main text for details). In Figure 2A the extended energy is shown as a function of time. In Figure 2B the pressure is shown in red and the black horizontal line marks the target pressure of 1200 Atm. Figures 2C and 2D show volume and temperature also as functions of time.
Figure 3
Figure 3
The membrane system is expanded to a target pressure of 1 Atm (see main text for details). In Figure 3A the extended energy is shown as a function of time. In Figure 3B the pressure is shown in red and the black horizontal line marks the target pressure. Figures 3C and 3D show volume and temperature also as function of time. After the initial expansion periodic ringing is evident in both figures 3C and 3D.
Figure 4
Figure 4
The volume as a function of time in constant pressure simulations. The results of the deterministic simulation are shown in green while the simulations with friction γ equal to 0.01 and 0.1 are shown in blue and red, respectively. In panel A we plot a nanosecond simulation while in panel B only the first 100 ps are shown. Ringing phenomenon is clearly present in the deterministic simulation and is reduced in the stochastic algorithm depending on the size of the friction coefficient. The relaxation time to the preset pressure also depends on the value of the friction coefficient.
Figure 5
Figure 5
Simulation of a water box under constant pressure conditions. Pressure and temperature are shown for 3 values of the friction constant 0, 0.01 and 0.1. Panels A, B and C show the instantaneous value of the pressure in red. The blue line indicates the value of the pressure averaged over 10 ps windows while the green line indicates the running average. The black horizontal line is the target pressure. Convergence to the target pressure is evident in all the runs. Panel D shows the temperature as a function of time for the three simulations: green correspond to friction coefficient γ equal to 0 and blue and red to 0.01 and 0.1 respectively. Once again, the horizontal black line indicates the target temperature.
Figure 6
Figure 6
Volume distribution for the water box (see text for details). Histograms are reported histogram for the three different simulations: green bins are the results of the the deterministic simulations, blue and red are results of simulations with friction coefficients of 0.01 and 0.1 respectively. The three simulations with different level of stochasticity converge to the same average volume and share the similar distributions for the volume variable.
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References

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