Multistate λ-local-elevation umbrella-sampling (MS-λ-LEUS): method and application to the complexation of cations by crown ethers.

An extension of the λ-local-elevation umbrella-sampling (λ-LEUS) scheme [ Bieler et al. J. Chem. Theory Comput. 2014 , 10 , 3006 ] is proposed to handle the multistate (MS) situation, i.e. the calculation of the relative free energies of multiple physical states based on a single simulation. The key element of the MS-λ-LEUS approach is to use a single coupling variable Λ controlling successive pairwise mutations between the states of interest in a cyclic fashion. The Λ variable is propagated dynamically as an extended-system variable, using a coordinate transformation with plateaus and a memory-based biasing potential as in λ-LEUS. Compared to other available MS schemes (one-step perturbation, enveloping distribution sampling and conventional λ-dynamics) the proposed method presents a number of important advantages, namely: (i) the physical states are visited explicitly and over finite time periods; (ii) the extent of unphysical space required to ensure transitions is kept minimal and, in particular, one-dimensional; (iii) the setup protocol solely requires the topologies of the physical states; and (iv) the method only requires limited modifications in a simulation code capable of handling two-state mutations. As an initial application, the absolute binding free energies of five alkali cations to three crown ethers in three different solvents are calculated. The results are found to reproduce qualitatively the main experimental trends and, in particular, the experimental selectivity of 18C6 for K(+) in water and methanol, which is interpreted in terms of opposing trends along the cation series between the solvation free energy of the cation and the direct electrostatic interactions within the complex.

Flexible Boundaries for Multiresolution Solvation: An Algorithm for Spatial Multiscaling in Molecular Dynamics Simulations.

An algorithm is proposed for performing molecular dynamics (MD) simulations of a biomolecular solute represented at atomistic resolution surrounded by a surface layer of atomistic fine-grained (FG) solvent molecules within a bulk represented by coarse-grained (CG) solvent beads. The method, called flexible boundaries for multiresolution solvation (FBMS), is based on: (i) a three-region layering of the solvent around the solute, involving an FG layer surrounded by a mixed FG-CG buffer layer, itself surrounded by a bulk CG region; (ii) a definition of the layer boundary that relies on an effective distance to the solute surface and is thus adapted to the shape of the solute as well as adjusts to its conformational changes. The effective surface distance is defined by inverse-nth power averaging over the distances to all non-hydrogen solute atoms, and the layering is enforced by means of half-harmonic distance restraints, attractive for the FG molecules and repulsive for the CG beads. A restraint-free region at intermediate distances enables the formation of the buffer layer, where the FG and CG solvents can mix freely. The algorithm is tested and validated using the GROMOS force field and the associated FG (SPC) and CG (polarizable CGW) water models. The test systems include pure-water systems, where one FG molecule plays the role of a solute, and a deca-alanine peptide with two widely different solute shapes considered, α-helical and fully extended. In particular, as the peptide unfolds, the number of FG molecules required to fill its close-range solvation layer increases, with the additional molecules being provided by the buffer layer. Further validation involves simulations of four proteins in multiresolution FG/CG mixtures. The resulting structural, energetic, and solvation properties are found to be similar to those observed in corresponding pure FG simulations.

Orthogonal sampling in free-energy calculations of residue mutations in a tripeptide: TI versus λ-LEUS.

In a recent article (Bieler et al., J. Chem. Theory Comput. 2014, 10, 3006), we introduced a combination of λ-dynamics and local-elevation umbrella-sampling termed λ-LEUS to calculate free-energy changes associated with alchemical processes using molecular dynamics simulations. This method was suggested to be more efficient than thermodynamic integration (TI), because the dynamical variation of the alchemical variable λ opens up pathways to circumvent barriers in the orthogonal space (defined by the N - 1 degrees of freedom that are not subjected to the sampling enhancement), a feature λ-LEUS shares with Hamiltonian replica-exchange (HR) approaches. However, the mutation considered, hydroquinone to benzene in water, was no real challenge in terms of orthogonal-space properties, which were restricted to solvent-relaxation processes. In the present article, we revisit the comparison between TI and λ-LEUS considering non-trivial mutations of the central residue X of a KXK tripeptide in water (with X = G, E, K, S, F, or Y). Side-chain interactions that may include salt bridges, hydrogen bonds or steric clashes lead to slow relaxation in the orthogonal space, mainly in the two-dimensional subspace spanned by the central φ and ψ dihedral angles of the peptide. The efficiency enhancement afforded by λ-LEUS is confirmed in this more complex test system and can be attributed explicitly to the improved sampling of the orthogonal space. The sensitivity of the results to the nontrivial choices of a mass parameter and of a thermostat coupling time for the alchemical variable is also investigated, resulting in recommended ranges of 50 to 100 u nm(2) and 0.2 to 0.5 ps, respectively.

On the ambiguity of conformational states: A B&S-LEUS simulation study of the helical conformations of decaalanine in water.

Estimating the relative stabilities of different conformational states of a (bio-)molecule using molecular dynamics simulations involves two challenging problems: the conceptual problem of how to define the states of interest and the technical problem of how to properly sample these states, along with achieving a sufficient number of interconversion transitions. In this study, the two issues are addressed in the context of a decaalanine peptide in water, by considering the 310-, α-, and π-helical states. The simulations rely on the ball-and-stick local-elevation umbrella-sampling (B&S-LEUS) method. In this scheme, the states are defined as hyperspheres (balls) in a (possibly high dimensional) collective-coordinate space and connected by hypercylinders (sticks) to ensure transitions. A new object, the pipe, is also introduced here to handle curvilinear pathways. Optimal sampling within the so-defined space is ensured by confinement and (one-dimensional) memory-based biasing potentials associated with the three different kinds of objects. The simulation results are then analysed in terms of free energies using reweighting, possibly relying on two distinct sets of collective coordinates for the state definition and analysis. The four possible choices considered for these sets are Cartesian coordinates, hydrogen-bond distances, backbone dihedral angles, or pairwise sums of successive backbone dihedral angles. The results concerning decaalanine underline that the concept of conformational state may be extremely ambiguous, and that its tentative absolute definition as a free-energy basin remains subordinated to the choice of a specific analysis space. For example, within the force-field employed and depending on the analysis coordinates selected, the 310-helical state may refer to weakly overlapping collections of conformations, differing by as much as 25 kJ mol(-1) in terms of free energy. As another example, the π-helical state appears to correspond to a free-energy basin for three choices of analysis coordinates, but to be unstable with the fourth one. The problem of conformational-state definition may become even more intricate when comparison with experiment is involved, where the state definition relies on spectroscopic or functional observables.

Communication: estimating the initial biasing potential for λ-local-elevation umbrella-sampling (λ-LEUS) simulations via slow growth.

In a recent article [Bieler et al., J. Chem. Theory Comput. 10, 3006-3022 (2014)], we introduced a combination of the λ-dynamics (λD) approach for calculating alchemical free-energy differences and of the local-elevation umbrella-sampling (LEUS) memory-based biasing method to enhance the sampling along the alchemical coordinate. The combined scheme, referred to as λ-LEUS, was applied to the perturbation of hydroquinone to benzene in water as a test system, and found to represent an improvement over thermodynamic integration (TI) in terms of sampling efficiency at equivalent accuracy. However, the preoptimization of the biasing potential required in the λ-LEUS method requires "filling up" all the basins in the potential of mean force. This introduces a non-productive pre-sampling time that is system-dependent, and generally exceeds the corresponding equilibration time in a TI calculation. In this letter, a remedy is proposed to this problem, termed the slow growth memory guessing (SGMG) approach. Instead of initializing the biasing potential to zero at the start of the preoptimization, an approximate potential of mean force is estimated from a short slow growth calculation, and its negative used to construct the initial memory. Considering the same test system as in the preceding article, it is shown that of the application of SGMG in λ-LEUS permits to reduce the preoptimization time by about a factor of four.

Local Elevation Umbrella Sampling Applied to the Calculation of Alchemical Free-Energy Changes via λ-Dynamics: The λ-LEUS Scheme.

When using molecular dynamics (MD) simulations for the calculation of alchemical free-energy changes, the extended-dynamics method called λ-dynamics (λD) in its currently available implementations does not compare very favorably with thermodynamic integration (TI) in terms of robustness, efficiency, and accuracy, although it is in principle easier to set up, postprocess, and automatize. In the present article, the main shortcomings of the λD approach are carefully analyzed, and possible remedies are proposed. The resulting scheme, called λ-LEUS, involves: (i) the use of a simple noninvertible coordinate transformation ensuring a finite sampling of the two physical end states; (ii) the application of the local elevation umbrella sampling (LEUS) memory-based biasing scheme to enforce homogeneous sampling and overcome barriers along the alchemical coordinate; (iii) recommendations concerning the choice of the mass parameter and of the temperature-coupling scheme for this coordinate; (iv) the use of a second-order splines basis set for the memory-based biasing functions. The λ-LEUS scheme is described and tested considering the perturbation of hydroquinone to benzene in water. The results are compared to those of TI calculations and exhibit a superior accuracy-to-efficiency ratio, presumably because dynamic variations in the alchemical coordinate open up pathways to circumvent orthogonal barriers, these pathways being inaccessible when the coordinate is constrained. Therefore, λ-LEUS combines the practical advantages of λD with the robustness of TI, simultaneously affording a slightly enhanced computational efficiency.

Connecting macroscopic observables and microscopic assembly events in amyloid formation using coarse grained simulations.

The pre-fibrillar stages of amyloid formation have been implicated in cellular toxicity, but have proved to be challenging to study directly in experiments and simulations. Rational strategies to suppress the formation of toxic amyloid oligomers require a better understanding of the mechanisms by which they are generated. We report Dynamical Monte Carlo simulations that allow us to study the early stages of amyloid formation. We use a generic, coarse-grained model of an amyloidogenic peptide that has two internal states: the first one representing the soluble random coil structure and the second one the [Formula: see text]-sheet conformation. We find that this system exhibits a propensity towards fibrillar self-assembly following the formation of a critical nucleus. Our calculations establish connections between the early nucleation events and the kinetic information available in the later stages of the aggregation process that are commonly probed in experiments. We analyze the kinetic behaviour in our simulations within the framework of the theory of classical nucleated polymerisation, and are able to connect the structural events at the early stages in amyloid growth with the resulting macroscopic observables such as the effective nucleus size. Furthermore, the free-energy landscapes that emerge from these simulations allow us to identify pertinent properties of the monomeric state that could be targeted to suppress oligomer formation.

Analysis of the Cartesian Tensor Transfer Method for Calculating Vibrational Spectra of Polypeptides.

The Cartesian Tensor Transfer Method (CTTM) was proposed as an efficient way to calculate infrared, Raman, and Raman Optical Activity (ROA) spectra for large molecules from the Hessian matrix and property tensor derivatives calculated for smaller molecular fragments. Although this approach has been widely used, its reliability has not been analyzed in depth yet. Especially for ROA spectra, such an analysis became only recently possible because of methodological advances that allow for the calculation of full ROA spectra of fairly large molecules with large basis sets. In this work, we investigate an α-helical polypeptide of 20 alanine amino acids, for which we reported the full ROA spectra earlier, in order to study the CTTM for protein subunits. By comparing the full first-principles calculation of the vibrational spectra with spectra reconstructed with the CTTM from different fragment sizes, we find that infrared and Raman spectra are mostly well reproduced. However, this is not the case for the ROA spectrum. This might have implications for peptide and protein CTTM ROA spectra that have already been published in the literature.