Local Molecular Field Theory for Nonequilibrium Systems.

We provide a framework for extending equilibrium local molecular field (LMF) theory to a statistical ensemble evolving under a time-dependent applied field. In this context, the self-consistency of the original LMF equation is achieved dynamically, which provides an efficient method for computing the equilibrium LMF potential, in addition to providing the nonequilibrium generalization. As a concrete example, we investigate water confined between hydrophobic or charged walls, systems that are very sensitive to the treatment of long-ranged electrostatics. We then analyze confined water in the presence of a time-dependent applied electric field, generated by a sinusoidal or abrupt variation of the magnitudes of uniform charge densities on each wall. Very accurate results are found from the time-dependent LMF formalism even for strong static fields and for time-dependent systems that are driven far from equilibrium where linear response methods fail.

Adaptations for Pressure and Temperature Effects on Loop Motion in Escherichia coli and Moritella profunda Dihydrofolate Reductase.

Determining how enzymes in piezophilic microbes function at high pressure can give insights into how life adapts to living at high pressure. Here, the effects of pressure and temperature on loop motions are compared Escherichia coli (Ec) and Moritella profunda (Mp) dihydrofolate reductase (DHFR) via molecular dynamics simulations at combinations of the growth temperature and pressure of the two organisms. Analysis indicates that a flexible CD loop in MpDHFR is an adaptation for cold because it makes the adenosine binding subdomain more flexible. Also, analysis indicates that the Thr113-Glu27 hydrogen bond in MpDHFR is an adaptation for high pressure because it provides flexibility within the loop subdomain compared to the very strong Thr113-Asp27 hydrogen bond in EcDHFR, and affects the correlation of the Met20 and GH loops. In addition, the results suggest that temperature might affect external loops more strongly while pressure might affect motion between elements within the protein.

Effects of Pressure and Temperature on the Atomic Fluctuations of Dihydrofolate Reductase from a Psychropiezophile and a Mesophile.

Determining the effects of extreme conditions on proteins from "extremophilic" and mesophilic microbes is important for understanding how life adapts to living at extremes as well as how extreme conditions can be used for sterilization and food preservation. Previous molecular dynamics simulations of dihydrofolate reductase (DHFR) from a psychropiezophile (cold- and pressure-loving), Moritella profunda (Mp), and a mesophile, Escherichia coli (Ec), at various pressures and temperatures indicate that atomic fluctuations, which are important for enzyme function, increase with both temperature and pressure. Here, the factors that cause increases in atomic fluctuations in the simulations are examined. The fluctuations increase with temperature not only because of greater thermal energy and thermal expansion of the protein but also because hydrogen bonds between protein atoms are weakened. However, the increase in fluctuations with pressure cannot be due to thermal energy, which remains constant, nor the compressive effects of pressure, but instead, the hydrogen bonds are also weakened. In addition, increased temperature causes larger increases in fluctuations of the loop regions of MpDHFR than EcDHFR, and increased pressure causes both increases and decreases in fluctuations of the loops, which differ between the two.

Quasiharmonic Analysis of the Energy Landscapes of Dihydrofolate Reductase from Piezophiles and Mesophiles.

A quasiharmonic analysis (QHA) method is used to compare the potential energy landscapes of dihydrofolate reductase (DHFR) from a piezophile (pressure-loving organism), Moritella profunda (Mp), and a mesophile, Escherichia coli (Ec). The QHA method considers atomic fluctuations of the protein as motions of an atom in a local effective potential created by neighboring atoms and quantitates it in terms of effective force constants, isothermal compressibilities, and thermal expansivities. The analysis indicates that the underlying potential energy surface of MpDHFR is inherently softer than that of EcDHFR. In addition, on picosecond time scales, the energy surfaces become more similar under the growth conditions of Mp and Ec. On these time scales, DHFR behaves as expected; namely, increasing temperature makes the effective energy minimum less steep because thermal fluctuations increase the available volume, whereas increasing pressure steepens it because compression reduces the available volume. Our longer simulations show that, on nanosecond time scales, increasing temperature has a similar effect as on picosecond time scales because thermal fluctuations increase the volume even more on a longer time scale. However, these simulations also indicate that, on nanosecond time scales, pressure makes the local potential less steep, contrary to picosecond time scales. Further examination of the QHA indicates the nanosecond pressure response may originate at picosecond time scales at the exterior of the protein, which suggests that protein-water interactions may be involved. The results may lead to understanding adaptations in enzymes made by piezophiles that enable them to function at higher pressures.

Quasiharmonic analysis of protein energy landscapes from pressure-temperature molecular dynamics simulations.

Positional fluctuations of an atom in a protein can be described as motion in an effective local energy minimum created by the surrounding protein atoms. The dependence of atomic fluctuations on both temperature (T) and pressure (P) has been used to probe the nature of these minima, which are generally described as harmonic in experiments such as x-ray crystallography and neutron scattering. Here, a quasiharmonic analysis method is presented in which the P-T dependence of atomic fluctuations is in terms of an intrinsic isobaric thermal expansivity αP and an intrinsic isothermal compressibility κT. The method is tested on previously reported mean-square displacements from P-T molecular dynamics simulations of lysozyme, which were interpreted to have a pressure-independent dynamical transition Tg near 200 K and a change in the pressure dependence near 480 MPa. Our quasiharmonic analysis of the same data shows that the P-T dependence can be described in terms of αP and κT where below Tg, the temperature dependence is frozen at the Tg value. In addition, the purported transition at 480 MPa is reinterpreted as a consequence of the pressure dependence of Tg and the quasiharmonic frequencies. The former also indicates that barrier heights between substates are pressure dependent in these data. Furthermore, the insights gained from this quasiharmonic analysis, which was of the energy landscape near the native state of a protein, suggest that similar analyses of other simulations may be useful in understanding such phenomena as pressure-induced protein unfolding.

Extreme biophysics: Enzymes under pressure.

A critical question about piezophilic (pressure-loving) microbes is how their constituent molecules maintain function under high pressure. Here, factors are examined that may lead to the increased activity under pressure in dihydrofolate reductase from the piezophilic Moritella profunda compared to the homologous enzyme from the mesophilic Escherichia coli. Molecular dynamics simulations are performed at various temperatures and pressures to examine how pressure affects the flexibility of the enzymes from these two microbes, since both stability and flexibility are necessary for enzyme activity. The results suggest that collective motions on the 10-ns timescale are responsible for the flexibility necessary for "corresponding states" activity at the growth conditions of the parent organism. In addition, the results suggest that while the lower stability of many enzymes from deep-sea microbes may be an adaptation for greater flexibility at low temperatures, high pressure may enhance their adaptation to low temperatures. © 2017 Wiley Periodicals, Inc.

Understanding the phase behavior of coarse-grained model lipid bilayers through computational calorimetry.

We study the phase behavior of saturated lipids as a function of temperature and tail length for two coarse-grained models: the soft-repulsive model typically employed with dissipative particle dynamics (DPD) and the MARTINI model. We characterize the simulated transitions through changes in structural properties, and we introduce a computational method to monitor changes in enthalpy, as is done experimentally with differential scanning calorimetry. The lipid system experimentally presents four different bilayer phases - subgel, gel, ripple, and fluid - and the DPD model describes all of these phases structurally while MARTINI describes a single order-disorder transition between the gel and the fluid phases. Given both models' varying degrees of success in displaying accurate structural and thermodynamic signatures, there is an overall satisfying extent of agreement for the coarse-grained models. We also study the lipid dynamics displayed by these models for the various phases, discussing this dynamics with relation to fidelity to experiment and computational efficiency.

On the Equivalence of Schemes for Simulating Bilayers at Constant Surface Tension.

Lipid bilayers are simulated using flexible simulation cells in order to allow for relaxations in area per lipid as bilayer content and temperature are varied. We develop a suite of Monte Carlo (MC) moves designed to generate constant surface tension γ and constant pressure P and find that the NPT partition function proposed by Attard [J. Chem. Phys.1995, 103, 9884-9885] leads to an NPγT partition function with a form invariant to choice of independent shape variables. We then compare this suite of MC moves to NPγT MC moves previously employed in our group as well as a pair of MC moves designed to replicate the NP∥P⊥T "ensemble" often used in molecular dynamics simulations to yield zero surface tension and constant pressure. A detailed analysis of shape fluctuations in a small bilayer system reveals that the two latter MC move sets are different from one another as well as from our new suite of MC moves, as justified by careful analysis of the partition functions. However, the study of a larger bilayer system reveals that, for practical purposes for this system, all six MC move sets are comparable to one another.

Waste-recycling Monte Carlo with optimal estimates: application to free energy calculations in alloys.

The estimator proposed recently by Delmas and Jourdain for waste-recycling Monte Carlo achieves variance reduction optimally with respect to a control variate that is evaluated directly using the simulation data. Here, the performance of this estimator is assessed numerically for free energy calculations in generic binary alloys and is compared to those of other estimators taken from the literature. A systematic investigation with varying simulation parameters of a simplified system, the anti-ferromagnetic Ising model, is first carried out in the transmutation ensemble using path-sampling. We observe numerically that (i) the variance of the Delmas-Jourdain estimator is indeed reduced compared to that of other estimators; and that (ii) the resulting reduction is close to the maximal possible one, despite the inaccuracy in the estimated control variate. More extensive path-sampling simulations involving an FeCr alloy system described by a many-body potential additionally show that (iii) gradual transmutations accommodate the atomic frustrations; thus, alleviating the numerical ergodicity issue present in numerous alloy systems and eventually enabling the determination of phase coexistence conditions.

Molecular Monte Carlo Simulations Using Graphics Processing Units: To Waste Recycle or Not?

In the waste recycling Monte Carlo (WRMC) algorithm, (1) multiple trial states may be simultaneously generated and utilized during Monte Carlo moves to improve the statistical accuracy of the simulations, suggesting that such an algorithm may be well posed for implementation in parallel on graphics processing units (GPUs). In this paper, we implement two waste recycling Monte Carlo algorithms in CUDA (Compute Unified Device Architecture) using uniformly distributed random trial states and trial states based on displacement random-walk steps, and we test the methods on a methane-zeolite MFI framework system to evaluate their utility. We discuss the specific implementation details of the waste recycling GPU algorithm and compare the methods to other parallel algorithms optimized for the framework system. We analyze the relationship between the statistical accuracy of our simulations and the CUDA block size to determine the efficient allocation of the GPU hardware resources. We make comparisons between the GPU and the serial CPU Monte Carlo implementations to assess speedup over conventional microprocessors. Finally, we apply our optimized GPU algorithms to the important problem of determining free energy landscapes, in this case for molecular motion through the zeolite LTA.

Molecular simulation of the effect of cholesterol on lipid-mediated protein-protein interactions.

Experiments and molecular simulations have shown that the hydrophobic mismatch between proteins and membranes contributes significantly to lipid-mediated protein-protein interactions. In this article, we discuss the effect of cholesterol on lipid-mediated protein-protein interactions as function of hydrophobic mismatch, protein diameter and protein cluster size, lipid tail length, and temperature. To do so, we study a mesoscopic model of a hydrated bilayer containing lipids and cholesterol in which proteins are embedded, with a hybrid dissipative particle dynamics-Monte Carlo method. We propose a mechanism by which cholesterol affects protein interactions: protein-induced, cholesterol-enriched, or cholesterol-depleted lipid shells surrounding the proteins affect the lipid-mediated protein-protein interactions. Our calculations of the potential of mean force between proteins and protein clusters show that the addition of cholesterol dramatically reduces repulsive lipid-mediated interactions between proteins (protein clusters) with positive mismatch, but does not affect attractive interactions between proteins with negative mismatch. Cholesterol has only a modest effect on the repulsive interactions between proteins with different mismatch.

Molecular simulation of the DMPC-cholesterol phase diagram.

In this paper, we present a coarse-grained model of a hydrated saturated phospholipid bilayer (dimyristoylphosphatidylcholine, DMPC) containing cholesterol that we study using a hybrid dissipative particle dynamics-Monte Carlo method. This approach allows us to reach the time and length scales necessary to study structural and mechanical properties of the bilayer at various temperatures and cholesterol concentrations. The properties studied are the area per lipid, condensation, bilayer thickness, tail order parameters, bending modulus, and area compressibility. Our model quantitatively reproduces most of the experimental effects of cholesterol on these properties and reproduces the main features of the experimental phase and structure diagrams. We also present all-atom simulation results of the system and use these results to further validate the structure of our coarse-grained bilayer. On the basis of the changes in structural properties, we propose a temperature-composition structure diagram, which we compare with the experimental phase and structure diagrams. Attention is paid to the reliability and interpretation of the model and simulation method and of the different experimental techniques. The lateral organization of cholesterol in the bilayer is discussed.

Alcohol solubility in a lipid bilayer: Efficient grand-canonical simulation of an interfacially active molecule.

We derive a new density-biased Monte Carlo technique which preserves detailed balance and improves the convergence of grand-canonical simulations of a species with a strong preference for an interfacial region as compared to the bulk. This density-biasing technique is applied to the solubility of "alcohol" molecules in a mesoscopic model of the lipid bilayer, a system which has anesthetic implications but is poorly understood.

Accurate thermodynamics for short-ranged truncations of Coulomb interactions in site-site molecular models.

Coulomb interactions are present in a wide variety of all-atom force fields. Spherical truncations of these interactions permit fast simulations but are problematic due to their incorrect thermodynamics. Herein we demonstrate that simple analytical corrections for the thermodynamics of uniform truncated systems are possible. In particular, results for the simple point charge/extended (SPC/E) water model treated with spherically truncated Coulomb interactions suggested by local molecular field theory [J. M. Rodgers and J. D. Weeks, Proc. Natl. Acad. Sci. U.S.A. 105, 19136 (2008)] are presented. We extend the results developed by Chandler [J. Chem. Phys. 65, 2925 (1976)] so that we may treat the thermodynamics of mixtures of flexible charged and uncharged molecules simulated with spherical truncations. We show that the energy and pressure of spherically truncated bulk SPC/E water are easily corrected using exact second-moment-like conditions on long-ranged structure. Furthermore, applying the pressure correction as an external pressure removes the density errors observed by other research groups in NPT simulations of spherically truncated bulk species.

Interplay of local hydrogen-bonding and long-ranged dipolar forces in simulations of confined water.

Spherical truncations of Coulomb interactions in standard models for water permit efficient molecular simulations and can give remarkably accurate results for the structure of the uniform liquid. However, truncations are known to produce significant errors in nonuniform systems, particularly for electrostatic properties. Local molecular field (LMF) theory corrects such truncations by use of an effective or restructured electrostatic potential that accounts for effects of the remaining long-ranged interactions through a density-weighted mean field average and satisfies a modified Poisson's equation defined with a Gaussian-smoothed charge density. We apply LMF theory to 3 simple molecular systems that exhibit different aspects of the failure of a naïive application of spherical truncations-water confined between hydrophobic walls, water confined between atomically corrugated hydrophilic walls, and water confined between hydrophobic walls with an applied electric field. Spherical truncations of 1/r fail spectacularly for the final system, in particular, and LMF theory corrects the failings for all three. Further, LMF theory provides a more intuitive way to understand the balance between local hydrogen bonding and longer-ranged electrostatics in molecular simulations involving water.

Attraction between like-charged walls: Short-ranged simulations using local molecular field theory.

Effective attraction between like-charged walls mediated by counterions is studied using local molecular field (LMF) theory. Monte Carlo simulations of the "mimic system" given by LMF theory, with short-ranged "Coulomb core" interactions in an effective single particle potential incorporating a mean-field average of the long-ranged Coulomb interactions, provide a direct test of the theory, and are in excellent agreement with more complex simulations of the full Coulomb system by Moreira and Netz [Eur. Phys. J. E 8, 33 (2002)]. A simple, generally applicable criterion to determine the consistency parameter sigma(min) needed for accurate use of the LMF theory is presented.