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WIREs Comput Mol Sci
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Exploiting the potential energy landscape to sample free energy

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We review a number of recently developed strategies for enhanced sampling of complex systems based on knowledge of the potential energy landscape. We describe four approaches, replica exchange, Kirkwood sampling, superposition‐enhanced nested sampling, and basin sampling, and show how each of them can exploit information for low‐lying potential energy minima obtained using basin‐hopping global optimization. Characterizing these minima is generally much faster than equilibrium thermodynamic sampling, because large steps in configuration space between local minima can be used without concern for maintaining detailed balance. WIREs Comput Mol Sci 2015, 5:273–289. doi: 10.1002/wcms.1217 This article is categorized under: Structure and Mechanism > Molecular Structures Structure and Mechanism > Computational Biochemistry and Biophysics Structure and Mechanism > Computational Materials Science
Acceptance profile for REX simulation of LJ31: The average REX acceptance probability for pair (Ti, Ti + 1) is plotted versus temperature index i, for temperature spacings chosen using standard geometric progression (empty black circles), and chosen by HSA optimization (filled blue circles). The dip in the geometric case coincides with a heat‐capacity peak. At higher temperatures, both profiles deviate from uniform behavior, as the HSA becomes less accurate. Results replotted from Ref .
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Calculated values for the ln Mst(V) and ln MPI(V)/2N ! as a function of potential energy for LJ75. Mst(V) and MPI(V) are the potential energy densities of distinct local minima structures and permutation–inversion isomers, respectively.
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LJ75 probability distribution P(Q6, T) and the corresponding free energy surface F(Q6, T) = − kBT ln P(Q6, T) for a database containing 8,391,630 structures. Four structures are illustrated, namely the global minimum (decahedron, Q6 = 0.31), the lowest minima based upon icosahedral packing with anti‐Mackay and Mackay overlayers (Q6 = 0.02 and 0.15, respectively), and a minimum associated with the liquid‐like phase (Q6 = 0.10).
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Heat capacity as a function of reduced temperature for LJ75. The lowest and second‐lowest minima based on a Marks decahedron and an incomplete Mackay icosahedron are illustrated on the left and right of the low‐temperature heat‐capacity peak, respectively. The atoms are coloured according to their contribution to the total energy: the most tightly bound atoms are blue, the least tightly bound are red, with intermediate binding energies in green. The curve‐marked PT is for the parallel tempering data only from the BSPT run. The inset shows a magnification of the low‐temperature peak corresponding to the solid–solid transition and a comparison with the harmonic superposition result. The curves marked BSPT+ result when the original statistics are combined with longer runs aimed at converging the potential energy densities of local minima, and provide a consistency check.
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Comparison of heat‐capacity curves for LJ31 obtained by exact SENS using different numbers of replicas. The PT and HSA curves were obtained by parallel tempering and the harmonic superposition approximation, respectively. Figure reproduced from Ref .
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Heat capacity curves for LJ31. HSA corresponds to the harmonic superposition approximation. All NS and SENS calculations were performed using K = 20, 000 replicas.
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The nested sampling procedure is illustrated for a two‐dimensional energy landscape where darker colors indicate lower energies. The image is drawn after 26 iterations of the nested sampling procedure. The circles represent the replica positions (K = 15), which are distributed uniformly in the two‐dimensional space. The lines are constant energy contours at for each iteration i. The cross gives the location of the highest energy replica, which defines Emax for the next iteration. The dotted line is the path of the MC walk of the newly generated replica, which is shown as a solid circle. The lower panel shows the one‐dimensional representation of phase space, which is used to derive the compression factor of Eqs and . In this configuration, there are no replicas in the basin with the global minimum, which will cause the density of states to be overestimated. This major drawback of the NS algorithm can be overcome by superposition enhanced nested sampling if the basin‐hopping procedure correctly identifies the relevant low‐lying minima.
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Potential energy distributions (lines marked by circles) from the 5 × 106 steps temperature replica exchange simulations (a) without and (b) with a doublet‐level Kirkwood reservoir. The replica temperatures are 20 (black), 30 (red), 50 (blue), and 100 K (green). The distributions from the Kirkwood reservoir simulation are much closer to the reference distributions (unmarked lines), indicating improved convergence upon coupling to a reservoir.
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Biased MC simulations for a nine‐atom chain molecule using doublet‐level Kirkwood sampling distribution as the biasing distribution. Distribution of energy from one million step biased MC simulations (black unmarked line) performed at T = 500, 400, 300, and 200 K, is overlaid on the corresponding reference Boltzmann distributions (in blue unmarked lines). The distribution of energy of the Kirkwood samples is marked by boxes. The acceptance ratio for the different simulations was 0.29, 0.25, 0.07, and 0.009 for T = 500, 400, 300, and 200 K, respectively. These results show that this approach can generate Boltzmann distributions even at temperatures lower than that of the original MD simulation used to populate the input pdfs.
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Doublet‐level Kirkwood sampling results for 52‐atom tetra alanine peptide. The input singlet and doublet pdfs were populated using 5 million conformations from a 500‐ns vacuum MD simulation performed at 1000 K. (a) The backbone (blue tube) of 100 Kirkwood sampled conformations aligned on the backbone atoms; one conformation is also shown in licorice. Potential energy for a million Kirkwood samples was computed using the MD energy function. (b) The energy distribution of the Kirkwood samples (crosses) overlaid on the Boltzmann energy distribution obtained from the MD simulation (unmarked). The two distributions have substantial overlap indicating overlap of the Kirkwood sampled and MD‐sampled conformational space.
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