Generalized image charge solvation model for electrostatic interactions in molecular dynamics simulations of aqueous solutions

Ab initio force field for simulations of proteins and nucleic acids: Application to molecular dynamics simulations in aqueous solution

ICSM: An order [formula omitted] method for calculating electrostatic interactions added to TINKER

Dynamical properties of liquid water were studied using Car-Parrinello [Phys. Rev. Lett. 55, 2471 (1985)] ab initio molecular dynamics (AIMD) simulations within the Kohn-Sham (KS) density functional theory employing the Becke-Lee-Yang-Parr exchange-correlation functional for the electronic structure. The KS orbitals were expanded in a discrete variable representation basis set, wherein the complete basis set limit can be easily reached and which, therefore, provides complete convergence of ionic forces. In order to minimize possible nonergodic behavior of the simulated water system in a constant energy (NVE) ensemble, a long equilibration run (30 ps) preceded a 60 ps long production run. The temperature drift during the entire 60 ps trajectory was found to be minimal. The diffusion coefficient [0.055 A2/ps] obtained from the present work for 32 D2O molecules is a factor of 4 smaller than the most up to date experimental value, but significantly larger than those of other recent AIMD studies. Adjusting the experimental result so as to match the finite-sized system used in the present study brings the comparison between theory and experiment to within a factor of 3. More importantly, the system is not observed to become "glassy" as has been reported in previous AIMD studies. The computed infrared spectrum is in good agreement with experimental data, especially in the low frequency regime where the translational and librational motions of water are manifested. The long simulation length also made it possible to perform detailed studies of hydrogen bond dynamics. The relaxation dynamics of hydrogen bonds observed in the present AIMD simulation is slower than those of popular force fields, such as the TIP4P potential, but comparable to that of the TIP5P potential.

Accurate and efficient simulation of liquids, such as water and salt solutions, using high-level wave function theories is still a formidable task for computational chemists owing to the high computational costs. In this study, we develop a deep machine learning potential based on fragment-based second-order Møller-Plesset perturbation theory (DP-MP2) for water through neural networks. We show that the DP-MP2 potential predicts the structural, dynamical, and thermodynamic properties of liquid water in better agreement with the experimental data than previous studies based on density functional theory (DFT). The nuclear quantum effects (NQEs) on the properties of liquid water are also examined, which are noticeable in affecting the structural and dynamical properties of liquid water under ambient conditions. This work provides a general framework for quantitative predictions of the properties of condensed-phase systems with the accuracy of high-level wave function theory while achieving significant computational savings compared to ab initio simulations.

The Matsuoka-Clementi-Yoshimine (MCY) configuration interaction potential for rigid water-water interactions has been extended to include the intramolecular vibrations. The extended potential (MCYL), using no empirical parameters other than the atomic masses, electron charge, and Planck constant, is used in a molecular-dynamics simulation study of the static and dynamic properties of liquid water. Among the properties studied are internal energy, heat capacity, pressure, radial distribution functions, dielectric constant, static structure factor, velocity autocorrelation functions, self-diffusion coefficients, dipole autocorrelation function, and density and current fluctuations. Comparison with experiments is made whenever possible. Most of these properties are found to improve slightly relative to the MCY model. The simulated high-frequency sound mode seems to support the results and interpretation of a recent coherent inelastic neutron scattering experiment.

Molecular dynamics simulation of aqueous solution of acetone was made using Lennard–Jones 12-6-1 potentials to describe the solute–solvent interactions and TIP4P to describe the water–water interactions. The Morokuma decomposition scheme and the ESIE solute atomic charges were used to reproduce the molecular parameters of the solute–water interaction potential. The results showed that the functions that use the EX-DIS-ES model improve the values of the dielectric properties, in particular, the dielectric constant and the dielectric relaxation time.

In this study, we trained a deep potential (DP) for H2O, an accurate machine learning (ML) potential. We performed molecular dynamics (MD) simulations of liquid water using the DP model (or DeePMD simulations). Our results showed that the DP model exhibits DFT-level accuracy, and the DeePMD simulation is a promising approach for modeling the structural properties of liquid water. Based on the DeePMD simulation trajectories, we calculated the isotropic Raman spectra of the O-H stretching mode using the surface-specific velocity-velocity correlation function (ssVVCF), showing that the DeePMD/ssVVCF approach can correctly capture the bimodal characteristics of the experimental Raman spectra, with one peak located near 3400 cm-1 and the other near 3250 cm-1. The success of the DeePMD/ssVVCF approach should be credited to (1) the DFT-level accuracy of the DP model for H2O, (2) the ssVVCF formulation considering the coupling between vibrational modes, and (3) non-Condon effects. Furthermore, the DeePMD simulations revealed that the anharmonic interactions between the coupled water molecules in the first and second hydration shells should play an essential role in the strong mixing of the H-O-H bending mode and the O-H stretching mode, leading to the delocalization of the O-H stretching band. In particular, increasing the strength of hydrogen bonds would enhance the bend-stretch coupling, leading to the red-shifting of the O-H vibrational spectra and the increase in the intensity of the shoulder around 3250 cm-1.

The term molecular mechanics (MM) refers to the use of simple potential-energy functions (e.g., harmonic oscillator or Coulombic potentials) to model molecular systems. Molecular mechanics approaches are widely applied in molecular structure refinement, molecular dynamics (MD) simulations, Monte Carlo (MC) simulations, and ligand-docking simulations. Typically, molecular mechanics models consist of spherical atoms connected by springs which represent bonds. Internal forces experienced in the model structure are described using simple mathematical functions. For example, Hooke’s law is commonly used to describe bonded interactions, and the nonbonded atoms might be treated as inelastic hard spheres or may interact according to a Lennard-Jones potential. Using these simple models, a molecular dynamics simulation numerically solves Newton’s equations of motion, thus allowing structural fluctuations to be observed with respect to time. Dynamic simulation methods are widely used to obtain information on the time evolution of conformations of proteins and other biological macromolecules1–4 and also kinetic and thermodynamic information. Simulations can provide fine detail concerning the motions of individual particles as a function of time. They can be utilized to quantify the properties of a system at a precision and on a time scale that is otherwise inaccessible, and simulation is, therefore, a valuable tool in extending our understanding of model systems. Theoretical consideration of a system additionally allows one to investigate the specific contributions to a property through “computational alchemy”,5 that is, modifying the simulation in a way that is nonphysical but nonetheless allows a model’s characteristics to be probed. One particular example is the artificial conversion of the energy function from that representing one system to that of another during a simulation. This is an important technique in free-energy calculations.6 Thus, molecular dynamics simulations, along with a range of complementary computational approaches, have become valuable tools for investigating the basis of protein structure and function. This review offers an outline of the origin of molecular dynamics simulation for protein systems and how it has developed into a robust and trusted tool. This review then covers more recent advances in theory and an illustrative selection of practical studies in which it played a central role. The range of studies in which MD has played a considerable or pivotal role is immense, and this review cannot do justice to them; MD simulations of biomedical importance were recently reviewed.4 Particular emphasis will be placed on the study of dynamic aspects of protein recognition, an area where molecular dynamics has scope to provide broad and far-ranging insights. This review concludes with a brief discussion of the future potential offered to advancement of the biological and biochemical sciences and the remaining issues that must be overcome to allow the full extent of this potential to be realized. 1.1. Historical Background MD methods were originally conceived within the theoretical physics community during the 1950s. In 1957, Alder and Wainwright7 performed the earliest MD simulation using the so-called hard-sphere model, in which the atoms interacted only through perfect collisions. Rahman8 subsequently applied a smooth, continuous potential to mimic real atomic interactions. During the 1970s, as computers became more widespread, MD simulations were developed for more complex systems, culminating in 1976 with the first simulation of a protein9,10 using an empirical energy function constructed using physics-based first-principles assumptions. MD simulations are now widely and routinely applied and especially popular in the fields of materials science11,12 and biophysics. As will be discussed later in this review, a variety of experimental conditions may be simulated with modern theories and algorithms. The initial simulations only considered single molecules in vacuo. Over time, more realistic or at least biologically relevant simulations could be performed. This trend is continuing today. The initial protein MD simulation, of the small bovine pancreatic trypsin inhibitor (BPTI), covered only 9.2 ps of simulation time. Modern simulations routinely have so-called equilibration periods much longer than that, and production simulations of tens of nanoseconds are routine, with the first microsecond MD simulation being reported in 1998.13 In addition, the original BPTI simulation included only about 500 atoms rather than the 104-106 atoms that are common today. While much of this advancement results from an immense increase in availability of computing power, major theoretical and methodological developments also contribute significantly. The number of publications regarding MD theory and application of MD to biological systems is growing at an extraordinary pace. A single review cannot do justice to the recent applications of MD. Using data from ISI Web of Science, the authors estimate that during 2005 at least 800 articles will be published that discuss molecular dynamics and proteins. The historical counts are shown in Figure 1. Open in a separate window Figure 1 Articles matching ISI Web of Science query “TS=(protein) AND TS=(molecular dynamics)”.

Calcium-dependent lectin I from Pseudomonas aeruginosa (PA-IL) binds specifically to oligosaccharides presenting an alpha-galactose residue at their nonreducing end, such as the disaccharides alphaGal1-2betaGalOMe, alphaGal1-3betaGalOMe, and alphaGal1-4betaGalOMe. This provides a unique model for studying the effect of the glycosidic linkage of the ligands on structure and thermodynamics of the complexes by means of experimental and theoretical tools. The structural features of PA-IL in complex with the three disaccharides were established by docking and molecular dynamics simulations and compared with those observed in available crystal structures, including PA-IL.alphaGal1-2betaGalOMe complex, which was solved at 2.4 A resolution and reported herein. The role of a structural bridge water molecule in the binding site of PA-IL was also elucidated through molecular dynamics simulations and free energy calculations. This water molecule establishes three very stable hydrogen bonds with O6 of nonreducing galactose, oxygen from Pro-51 main chain, and nitrogen from Gln-53 main chain of the lectin binding site. Binding free energies for PA-IL in complex with the three disaccharides were investigated, and the results were compared with the experimental data determined by titration microcalorimetry. When the bridge water molecule was included in the free energy calculations, the simulations predicted the correct binding affinity trends with the 1-2-linked disaccharide presenting three times stronger affinity ligand than the other two. These results highlight the role of the water molecule in the binding site of PA-IL and indicate that it should be taken into account when designing glycoderivatives active against P. aeruginosa adhesion.

The properties of high‐pressure liquid water were investigated along the isobar of 25 MPa and in the temperature range 298.15‐623.15 K using classical molecular dynamics simulations. Particular attention has been given to the changes in the local structural and related dynamic properties of liquid water. The results obtained have revealed noticeable changes in the shape of the calculated radial distribution functions, as well as the existence of local extrema or crossovers in several structural descriptors and entropic quantities at temperatures around 423.15 K and 498.15 K, where also significant changes in the hydrogen bond network of liquid water have also been observed. The temperature dependence of translational, reorientational, and hydrogen bond dynamics of liquid water has also been investigated by calculating the translational self‐diffusion and the corresponding O‐H vector Legendre reorientational correlation times and hydrogen bond lifetimes. The corresponding activation energies for each investigated relaxation process have also been presented and discussed.

CXCL-8 (Interleukin 8) is a CXC chemokine with a central role in the human immune response. We have undertaken extensive in silico analyses to elucidate the interactions of CXCL-8 with its various binding partners, which are crucial for its biological function. Sequence and structure analyses showed that residues in the thirdq β-sheet and basic residues in the heparin binding site are highly variable, while residues in the second β-sheet are highly conserved. Molecular dynamics simulations in aqueous solution of dimeric CXCL-8 have been performed with starting geometries from both X-ray and NMR structures showed shearing movements between the two antiparallel C-terminal helices. Dynamic conservation analyses of these simulations agreed with experimental data indicating that structural differences between the two structures at quaternary level arise from changes in the secondary structure of the N-terminal loop, the 3(10)-helix, the 30s, 40s, and 50s loops and the third β-sheet, resulting in a different interhelical separation. Nevertheless, the observation of these different states indicates that CXCL-8 has the potential to undergo conformational changes, and it seems likely that this feature is relevant to the mode of binding of glycosaminoglycan (GAG) mimetics such as cyclitols. Simulations of the receptor peptide fragment-CXCL-8 complex identified several specific interactions of the receptor peptide with CXCL-8 that could be exploited in the structure-based design of competitive peptides and nonpeptidic molecules targeting CXCL-8 for combating inflammatory diseases. Simulations of the CXCL-8 dimer complexed with a 24-mer heparin fragment and of the CXCL-8-receptor peptide complex revealed that Arg60, Lys64, and Arg68 in the dimer bind to cyclitols in a horseshoe pattern, defining a region which is spatially distinct from the receptor binding site. There appears to be an optimum number of sulfates and an optimum length of alkyl spacers required for the interaction of cyclitol inhibitors with the dimeric form of CXCL-8. Calculation of the binding affinities of cyclitol inhibitors reflected satisfactorily the ranking of experimentally determined inhibitory potencies. The findings of these molecular modeling studies will help in the search for inhibitors which can modulate various CXCL-8 biological activities and serve as an excellent model system to study CXC-inhibitor interactions.

Arginine-vasopressin was subjected to a long (11μs) molecular dynamics simulation in aqueous solution. Analysis of the results by DASH and principal components analyses revealed four main ring conformations that move essentially independently of the faster-moving tail region. Two of these conformations (labeled "saddle") feature well-defined β-turns in the ring and conserved transannular hydrogen bonds, whereas the other two ("open") feature neither. The conformations have been identified and defined and are all of sufficient stability to be considered candidates for biological conformations in their cognate receptors.

We investigate thoroughly the effect of confinement and solute topology on the orientational dynamics of water molecule in the interplate region between two nanoscopic hydrophobic paraffinlike plates. Results are obtained from molecular dynamics simulations of aqueous solutions of paraffinlike plates in the isothermal-isobaric ensemble. An analysis of survival time auto correlation function shows that the residence time of the water molecule in the confined region between two model nanoscopic hydrophobic plates depends on solute surface topology (intermolecular distance within the paraffinlike plate). As expected, the extent of confinement also changes the residence time of water molecules considerably. Orientational dynamics was analyzed along three different directions, viz., dipole moment, HH, and perpendicular to molecular plane vectors. It has been demonstrated that the rotational dynamics of the confined water does not follow the Debye rotational diffusion model, and surface topology of the solute plate and the extent of confinement have considerable effect on the rotational dynamics of the confined water molecules.

Ionoids: Macrocyclic, organic and small inorganic building blocks electrostatically self-assemble in solution. The formed nano-objects feature a mobile interior. The molecules inside the aggregates seem to rotate freely on certain spatial positions. The hydrodynamic radius of the nano-assemblies, however, is precisely defined. Many societal challenges at the beginning of the 21st century lead to an apparent and growing need for functional materials and novel ways of materials synthesis and assembly. Rising to the challenge, the utilization of small, self-assembling building blocks for the bottom-up construction of new types of polymers and nanostructures has enjoyed increasing popularity among materials researchers in the recent past. Supramolecular materials like foldamers, surface films, nanoparticles, etc. are created by exploiting noncovalent forces1 leading to an ordered arrangement of nanoscale building blocks.2 In the search for new polymers based on noncovalent molecular forces, we are motivated by the idea of supramolecular or even polymer-like structures by self-assembly of small ionic monomers, merely formed from electrostatic and solvent interactions. We, in particular, focus on applying intrinsically nondirected electrostatic interactions to create nonetheless well- organized supramolecular structures.3 To achieve this aim, we here make use of a recently highlighted, multicationic molecular box (14+; Figure 1 a), developed by Sessler and co-workers, which has already been applied in self-assembly applications of metal–organic rotaxane frameworks.4 The molecular box 14+ seems to facilitate structure generation through self- assembly, indicated by, for example, the surprising ability to build supramolecular necklaces through incorporation of electron-rich aromatic guest molecules.5 However, not focused on short-ranged interactions but on new applications of long-ranged electrostatic forces at the nanoscale,6 we combine 14+ with small di-anionic salts, (KSO3)2CH2 (22−) and (KSO3)2NO (32−), to generate a defined assembly of cations and anions in solution (Figure 1 a). However, self-assembly due to electrostatic attraction is complex to describe qualitatively as well as quantitatively, because not only Coulomb forces but also entropy changes due to counterion release, solvation of ions, depletion forces, etc., contribute to the total free energy of a system. A central question in any application of solution-based ionic self-assembly therefore is: does any kind of ordered arrangement of the ionic building blocks take place?7 Intuitively one would assume the distribution of ions in solution to be of random, rather homogeneous nature. a) Molecular structure of 14+ (left) and 22−/32− (top right). b) A typical monoexponential fit of an autocorrelation function gained from DLS on 22−/14+ ionoids. c) Hydrodynamic radii of the 22−/14+ ionoids plotted against time, as derived from DLS measurements. t=0 days refers to the date of initial preparation. The upper and lower lines indicate error estimates derived from the fits of the autocorrelation functions. In the following, we show dynamic light scattering (DLS) data, which clearly illustrate that low-concentrated mixtures of anionic 22− and cationic 14+ self-assemble in solution to highly defined, monodisperse ionic clusters. In this special case, the distribution of ions in solution is therefore not at all random but well-ordered due to long-range electrostatic correlations. Performing molecular dynamics (MD) and Monte Carlo (MC) simulations, we further elucidate the internal constitution of the ionic clusters and show that a long-ranged order may be induced by electrostatic correlations of the cationic macrocycles. The simulations indicate that the monodisperse objects are constituted of relatively loosely bound ions that, however, occupy certain fixed coordinates in the clusters for at least orders of nanoseconds. The reliability of the computational data is additionally supported by double electron–electron resonance (DEER)8 spectroscopy data. Taking long-ranged (DLS, MC) and molecular (MD, DEER) insights together, we demonstrate, to the best of our knowledge for the first time, that size-controlled self-assembly through electrostatic interaction between small ions in solution is possible. The so formed objects can be regarded as loosely bound ion-based colloids, thus we propose the name ionoids. These ionoids are peculiar as they result from ion–ion correlations of moderately charged constituents at low concentrations, while long-range electrostatic correlations have so far only been discussed in the context of macroions.9 DLS experiments were performed on a system containing a mixture of 22−/14+ at an approximate ratio of 3 mM:1 mM in DMSO/88 % aqueous glycerol (1:1). Note that the exact concentration of 22− within the light-scattering sample is difficult to determine, as the solubility of the pure component 22−, in contrast to 32−, is only 3 mM in our solvent mixture. The measurements revealed monodisperse small objects in the solution, after an initial period of several days where no correlation could be detected. The fit of one correlation function measured at a scattering angle of 90 ° is depicted in Figure 1 b. Note the monoexponential nature of this fit, which indicates monodispersity of the aggregates in the solution. The intercept of the correlation function is unusually low, due to the small signal-to-noise ratio of <3, that is, the scattered intensity from the pure solvent mixture is nearly 40 % of the overall scattered intensity. As, in the absence of 22−, we found no correlation at all but only the DLS signature of the pure solvent mixture, it is at hand to deduce that these small objects stem from self- assembly of 22−–14+ clusters. The hydrodynamic radius of the self-assembled 22−–14+ cluster is quite small, that is, only 6.2 nm±0.2 nm can be observed 10 days after sample preparation, thus, growing very slow (+2 nm after 140 days; see Figure 1 c) and fostering an analogy with charge-stabilized colloids. It is important to note that all correlation functions measured over time can be fitted by a single exponential decay, and the resulting diffusion coefficient is independent of the scattering angle, which underlines the monodispersity in aggregate size. Thus, one can conclude that molecules 22− are necessary to overcome like-charge repulsion and to bring together 3–4 molecules 14+ (estimated from the size of approx. 6–7 nm) in an extraordinarily uniform manner. To support the reliability of the DLS data and to elucidate the internal constitution of the ionic clusters, we conducted all-atom MD simulations corresponding to a 32−/14+ system at a molar ratio of 6:2. In order to check the reliability of the MD simulation, we supplement it with DEER data and therefore replaced 22− by paramagnetic 32−. (This change may have an influence on special interactions in the ionoids, yet long-range electrostatic interactions remain unchanged. Note that the concentration in the simulations was three times higher than in the DEER measurements, to reduce the computation time). Starting from a random distribution of all molecules, we let the system equilibrate for 0.2 ns, and subsequently we extracted the trajectories for 8 ns with the aim of calculating the radial distribution function, g(r), of 32−. Figure 2 a shows a snapshot of the simulation after 2 ns, illustrating the molecular arrangement of the ions in such an ionoid-type cluster. Figure 2 b depicts the radial distance distribution of 32− summed up over 8 ns of simulation and over different time intervals of 1 ns. Prominent features in the overall g(r) at 1.9 nm, 2.4 nm, 3.5 nm are observable. Thus, the distribution of 32− in the simulated system is far from being random. Also, the distributions summed up in each case over 1 ns of simulation indicate favored population of distinct distances in the 32−/14+ system. At first sight, it is astonishing that the maxima in g(r) occur with a spacing of approximately 0.5 nm, indicating a periodic arrangement of the spin probes (Figure 2 b) . a) Snapshot after 2 ns of MD of the system 32−/14+ at 6:2 molar ratio. b) Pair correlation function of 32− molecules gained from MD simulation of the system 32−/14+ (6:2) summed up over different time intervals. The overall distance distribution is shown in orange (—). Note that over time, certain maxima in the distributions appear frequently, as indicated by the dashed lines. The black distribution depicts the distances gained from DEER on the same system (—). c) The Monte Carlo-derived radial distribution function for 32− calculated for two 14+ and six 32− molecules (—). The box size was 263 nm3. As can be seen, there are no distinct maxima in the distance distribution. Hence, the aggregation of 32− and 14+ is not based on Coulomb interaction and excluded volume effects alone. Solvent entropy may possibly contribute significantly to the generation of the ionoids, as the MD-derived distributions lead to reasonable agreements with the experimental data. The 14+–14+ distribution (—) reveals a long-range correlation at approximately 15 nm. For more details, see the Supporting Information. Although the inner constitution of the 32−–14+ aggregates could be assessed with MD, the global size of the so formed aggregates cannot be deduced from the simulation and compared with RH because of a box size of only 83 nm3. Consequently, we performed Monte Carlo (MC) simulations with a box size of 263 nm3 to elucidate the individual cluster size. These simulations included electrostatic potentials as well as excluded volume effects. However, the MC simulation showed that, due to mere electrostatic interactions and excluded volume, no distinct short-range association takes place (Figure 2 c). Yet, the MC simulations do reveal a surprising long-range correlation of 14+ at about 15 nm, suggesting a defined size of the observed ionic clusters, electrostatic and excluded volume effects to be essential for the formation of the observed ionic clusters (Figure 2 c and the Supporting Information), because the experimentally determined radius (6.2 nm–8.2 nm) and the size suggested from MC simulations (R≈7.5 nm) are in good agreement. Yet the distinct 32−–32− distances, as gained from MD, seem to be a consequence of additional entropic or solvent effects next to mere electrostatic effects. To further support the reliability of the computational data, we performed electron paramagnetic resonance (EPR)-based nanoscale distance measurement (DEER) experiments on a freeze-quenched 32−/14+ system to extract the experimental distance distribution of 32−. The resulting distance distribution is depicted in Figure 2 b (black curve). It shows the population of all 32−–32− distances of 6 mM molecules 32− around 2 mM molecules 14+ and can be compared directly with the g(r) gained from MD. Taking into account that the simulation features significantly higher concentrations than the experimentally measured system (the 14+ concentration was three times higher than in the experiment to increase the speed of calculation) and the restricted number of simulated 14+ and 32− molecules, the simulation-derived and the experimental distribution overall are in reasonable agreement. However, there are no distinct dipolar modulations observable in the DEER time traces (see the Supporting Information). Hence, the extracted distances (by Tikhonov regularization) have to be treated with caution. Yet, as they are clearly indicative of nonhomogeneous/nonrandom distributions of 32− and in addition are even in reasonably good agreement with the simulated distributions, one can assume that the distances extracted from the MD simulation resemble the situation inside the ionoids to a certain degree. Thus, judging from the simulation, the situation as shown in Figure 2 a can be taken as a representative snapshot of the distribution of ions in the solution. As it does not change significantly over time, it seems like the molecules 14+ are forced into a short-distance arrangement in order to minimize the potential electrostatic energy of the whole system. Divalent molecules 32−, thereby, could be acting as 'ionic glue' between macrocycles 14+. In doing so, the positions of local maxima in the distance distribution as derived from the MD simulation do not change significantly over time. This means that the spatial conformational ensemble covered seems to be relatively small. Once 32− is 'trapped' in an ensemble, it remains at a certain position for at least several nanoseconds. Continuous wave (CW)-EPR data at room temperature heavily support this hypothesis: the 32− probes still rotate relatively freely, although their rotational mobility is slightly restricted when small amounts of 14+ are added to a 32− solution (see the Supporting Information for the CW-EPR spectra and corresponding spectral simulations). At the same time, they still favor certain coordinates in the ionoids because of electrostatic attraction to 14+ and repulsion among themselves. It is likely that the combination of larger organic and small, yet doubly charged, inorganic building blocks is necessary for the emergence of ordered ionic assemblies, which persist in solution over time. Also, solvent effects could play a role, as DMSO is known to feature anomalous properties because of self-association.10 Taking all the experimental and computational data into account, one can conclude that the self-assembled objects feature a defined radius, although, on a molecular scale, their interior seems not to be strongly fixed: 'gluing' 32− molecules can still rotate relatively undisturbed. Yet, on the supramolecular scale, certain defined coordinates inside the aggregates are preferentially populated for at least some nanoseconds. Hence, the position of the 'glue′ is not arbitrary. From a physico-chemical point of view it is highly intriguing that an intrinsically nondirected force like electrostatic attraction/repulsion leads to the formation of highly defined nanoscopic objects. One can even control the initial size of the self-assembled ionoids precisely and keep them stable in solution for months. Thus, by choosing the right ions and counterions, their distribution in solution is not more or less homogeneous as one would expect intuitively for electrolyte solutions.10–11 Contrarily, they assemble into discrete arrangements as has been observed for macroions such as in DNA-condensation years ago .12 As the observed ionoids are based on noncovalent interaction among constituents, they probably are sensitive towards even minor changes in the solvent properties, ionic strength of the solution, and concentration of the constituents. These factors are likely to influence their composition and size. Furthermore, we expect that the observed growth of the ionoids will be fostered by a decrease in solvent viscosity, and that their occurrence is limited to certain solvents. Altogether, we have presented experimental and computational evidence for the formation of electrostatically self-assembled supramolecular structures in solution. The nano-assemblies have a well-defined size, and molecules in their interior populate, at least transiently, certain fixed positions in the cluster, while keeping a certain rotational freedom. The so formed ionoids can possibly open new routes towards the design and self-assembly of novel materials by exploiting electrostatic interactions between small molecules which have not attracted much attention in this field of research so far. In addition, these nano-objects are intriguing from a theoretical point of view, as long-range electrostatic correlations in solution have been postulated but, to the best of our knowledge, have not yet been observed directly for low-molecular electrolytes (unlike macroions).13 From the slow but steady size increase over the course of two months (Figure 1 c) one could even speculate that ionoids could be some sort of solvated precursor for precipitation or even crystallization nuclei. More experimental evidence would be needed to further substantiate such a picture, which currently is a mere speculation. Considering the small size of the ionoids, it is likely that X-ray and neutron scattering experiments will yield substantial insight into the specific properties of these solution structures. Such measurements are currently on our agenda. The responsiveness of ionoids towards stimuli such as concentration, solvent-mixture or charge ratio could also become interesting in the course of the search for new responsive materials. We thank Christian Bauer for technical support and Prof. Hans W. Spiess for continuing support. D.K. acknowledges support by the Gutenberg Academy of the University of Mainz. This work was financially supported by the Deutsche Forschungsgemeinschaft (DFG) under grant number HI 1094/2-1 and the Max Planck Graduate Center with the University of Mainz (MPGC). Detailed facts of importance to specialist readers are published as "Supporting Information". Such documents are peer-reviewed, but not copy-edited or typeset. They are made available as submitted by the authors. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.

A new method, named ‘‘spherical mapping approximation’’ (SMA), is developed for the evaluation of displacement fields of body waves and surface waves from explosions in nonspherical cavities embedded in elastic media. Under SMA, the explosion-generated stress distribution on the surface of an arbitrary cavity is mapped onto the surface of an equivalent virtual spherical cavity having the volume of the true cavity. The analytical results express the displacement field in terms of a multipole double-sum expansion of spherical eigenvectors with coefficients in the form of a finite Legendre transform of the components of the normal vector of the cavity boundary. These ‘‘cavity integrals’’ can be evaluated exactly for spheroidal and cylindrical inclusions. In the long-wave far-field approximation, symmetric finite cavities are shown to be equivalent to a linear combination of point dipoles directed along the principal cavity axes. The ensuing radiation patterns yield, in general, four-lobe patterns for S waves, two-lobe patterns for P waves, and single to two-lobe Rayleigh-wave pattern, independent of the details of the cavities’ shape. However, all radiation patterns are modulated by a frequency-dependent ‘‘cavity factor’’ that embodies the boundary conditions on the cavity surface. Moreover, it is shown that the radiation pattern for P waves from a nonspherical symmetrical cavity in the long-wave far-field approximation is always dipolar. Since the radiation pattern of radiated P waves from a standard earthquake is always quadrupolar, the cavity explosion behaves like a non-double-couple earthquake. Thus the examination of the deviatoric moment tensor of a given seismic event enables one, in principle, to state whether it is a standard earthquake or perhaps (if the S-wave pattern is quadrupolar) an evasion of the test-ban treaty. It is shown that SMA can be easily fitted to a multilayered elastic half-space if the equivalent source is placed in one of the layers. In that case, Love waves will be generated in addition to P, S, and Rayleigh waves. Displacement patterns for body and surface waves are calculated for spheroidal and cylindrical cavities for a wide range of aspects ratios and corresponding aperture angles, exhibiting the whole gamut of cavity shapes from a line source to a disk. The moments of the equivalent dipoles are shown to depend on the corresponding cavity integrals, the elastic constants of the medium in the neighborhood of the source, and the initial energy injection. All nonspherical cavities generate strong shear waves except for special aperture angles at which a spherical P wave is generated, unaccompanied by S waves. The wave-spectra of body waves (surface waves) exhibit a corner frequency (peak frequency) at a wavelength equal to the radius of the equivalent sphere. This enables one to deduce the size of the cavity from the spectrum of its far-field displacement signals, provided that the explosion is fully decoupled and that the interaction of the shock wave with the medium occurred in the elastic regime. The results of the present research are applicable to the detection and identification of seismic signals from clandestine underground nuclear explosions.