EXEDOS

The simulation techniques presented above can be applied to all first order phase transitions provided that an appropriate order parameter is identified. For vapor-liquid equilibria, where the two coexisting phases of the fluid have the a similar structure, the density (a thermodynamic property) was an appropriate order parameter. More generally, the order parameter must clearly distinguish any coexisting phases from each other. Examples of suitable order parameters include the scalar order parameter for study of nematic-isotropic transitions in liquid crystals [87], a density-based order parameter for block copolymer systems [88], or a bond order parameter for study of crystallization [89]. Having specified a suitable order parameter, we now show how the EXEDOS technique introduced earlier can be used to obtain in a particularly effective manner for simulations of crystallization [33]. The Landau free energy of the system A( ) can then be related to P,g p( ((/"))... 

Depending on whether P( ) is obtained from EXEDOS in a constant n, V, T) or constant n,p, T) ensemble, A will correspond to the Helmholtz free energy or the Gibbs free energy of the system, respectively. 

For a system of 864 particles, the starting guess for coexistence temperature and pressure was T = 1.14 and p = 24.21. The order parameter range was set between 0.02 and 0.45. EXEDOS simulations were performed in multiple windows. The free energy curves obtained at this temperature and pressure (see Fig. 18) show that, as a result of finite-size effects, T = 1.14... 

Fig. 18. Free energy curve for crystallization of an 864-particle repulsive LJ system. EXEDOS simulations at T = 1.14 solid line, predicted free energy profile by reweighing T = 1.14 data to T = 1.03 dashed line, and EXEDOS simulations at T = 1.03 dot-dot-dashed line. From Chopra et al. [33]...

As discussed earlier, EXEDOS simulations can be used to determine free energy differences (or PMF s) with remarkable accuracy. If the reaction coordinate, is chosen to be the end-to-end distance between the N and C termini of the protein molecule being stretched, then the resulting PMF and its derivative should correspond to the actual force measured in the laboratory, provided the molecule is pulled slowly (i.e. reversibly) [32]. In this example, Monte Carlo simulations therefore provide an ideal bound against which results of molecular dynamics can be compared. We present results for a 15-segment polyalanine molecule, which adopts a stable a-helical conformation in an implicit solvent [26]. By applying an external stretching force. 

Fig. 26. Potentials of mean force as computed with SMD, SMD-NH and EXEDOS simulations. As the pulling velocity is reduced, the SMD estimates converges towards the EXEDOS results (from Rathore et al. [32])...

The corresponding force-extension profiles for these calculations are shown in Fig. 27. The data presented for SMD and SMD-NH correspond to an individual trajectory. In order to arrive at a meaningful basis for comparisons between different schemes, the PMF for SMD and SMD-NH simulations is time averaged over 0.1 A(the bin width in EXEDOS), thereby reducing some of the statistical noise. For higher pulling rates (rates comparable to those employed in the literature), the forces and the PMF obtained from steered MD without a cantilever (SMD-NH) exhibit less noise than those obtained from... 

The EXEDOS method that we use combines an expanded ensemble formalism with a density-of-states [9,10] scheme for the determination of the potential of mean force. In the expanded ensemble method, besides the canonical variables (number of mesogens N, volume V, and temperature T) the state of the system is also labelled by the value of C coordinate we consider M intervals of width S of the reaction coordinate, then the system is said to be in state m ii m - 1 < /5 < m. For suflSciently narrow intervals, we can associate the midpoint Cm = (m — 1/2)5 as the representative value. 

Once more, the EXEDOS method can be applied to find the potential of mean force AF(s) as a function of the separation of the two spheres. For convenience, we define the reaction coordinate s as the separation between the spheres surfaces... 

---