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>i</i>) the physical states are visited explicitly and over finite time periods; (<i>ii</i>) the extent of unphysical space required to ensure transitions is kept minimal and, in particular, one-dimensional; (<i>iii</i>) the setup protocol solely requires the topologies of the physical states; and (<i>iv</i>) 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

Connecting Macroscopic Observables and Microscopic Assembly Events in Amyloid Formation Using Coarse Grained Simulations

<div><p>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 -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.</p> </div

The time dependent distribution of oligomer mass, which captures the growth of individual fibers during one simulation with 0.51 mM peptide concentration.

<p>At the bottom, one can see the decreasing concentration of monomers (red/yellow), dimers (green/blue) and trimers (blue). Importantly, tetramers developed into a full fiber in every case. Only the beginning of the simulation is shown for clarity.</p

Free energy landscape for the formation of the oligomers with the marked free energy minimum path of the nucleus formation.

<p>The dashed line follows the free energy change to minimum within our largest cluster, where the cluster could also grow into a pentamer with an unknown free energy preference. The different oligomers are denoted as M: monomer, D: dimer, T: trimer, and Q: tetramer. The greek letters designate the composition of the clusters, where stand for random coil, while represents -sheet. The free energy path is schematically depicted at the bottom of the figure, where the state is represented by a blue circle with a red patch and state is displayed as a half orange/half grey circle, which represents the interaction parameters of the employed model. The displayed values are for 0.19 mM concentration.</p

Representative snapshot of the simulation box in the later stage of the fibril growth.

<p>The blue/red particles are in the random coil state, while the orange/grey particles are in the -sheet state. Note that the particles in the random coil state are mostly monomers in solution or at the end of the fibres, while the -sheet are forming chiral cross stacked fibrils.</p

The fibril growth from our simulation and corresponding fit by Oosawa's theory.

<p>The peptide concentrations are from left to right 7.97, 2.36, 1.00, 0.51, 0.30, and 0.19 mM. The fitted size of nucleus and growth rate are 3.8 particle and respectively. The inset shows the fit to obtain the nucleus size via the halftimes according to <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002692#pcbi.1002692.e067" target="_blank">Eq. 2</a>. The logarithmic time scale tends to visually over-emphasize the differences between the global fit (red curve) and data (black curve) at short time (highest concentrations).</p

The rigidity of the fibers can be reduced by removing the chirality of the employed model.

<p>In such systems, the fibers can bend to form a ring (depicted as a simulation snapshot in part B), which does not have a loose end available for further growth. This effects the whole growth process (shown in part A as a relative fiber mass concentration depending on time) and it results in a deviation from a global fit to Oosawa's theory, especially at later stages of the fibrillar growth, since there is no such effect included in theory. The fitted values are  = 4.0 and . The black curve represents simulation data and the red curve is the fit.</p

The fibrillar growth from our simulation fitted by Oosawa's theory with varying nucleation size.

<p>All the theoretical curves displayed with non-black colors were able to fit the simulation data very well by varying the growth rate from to and the nucleus size from 2.0 to 5.0; this overfitting can only be resolved by performing global fits as shown in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002692#pcbi-1002692-g005" target="_blank">Fig. 5</a>. The inset shows the time evolution of the mass concentrations of monomers (black), oligomers (dimers and trimers, green) and fibrils (aggregates with an aggregation number greater than four, red) for a simulation with a concentration 0.51 mM. The dots display the standard deviation calculated from the averaging of the five runs with different random initial configurations.</p

The configurations of small oligomers in the interaction minima with marked interactions and the total enthalpic contributions.

<p>The subscript and the size of the patch denotes the internal state of the PSC mode (: random coil and : -sheet).</p