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The conformational dynamics of single-stranded nucleic acids are fundamental for nucleic acid folding and function. However, their elementary chain dynamics have been difficult to resolve experimentally. Here we employ a combination of single-molecule F\"orster resonance energy transfer, nanosecond fluorescence correlation spectroscopy, fluorescence lifetime analysis, and nanophotonic enhancement to determine the conformational ensembles and rapid chain dynamics of short single-stranded nucleic acids in solution. To interpret the experimental results in terms of end-to-end distance dynamics, we utilize the hierarchical chain growth approach, simple polymer models, and refinement with Bayesian inference of ensembles to generate structural ensembles that closely align with the experimental data. The resulting chain reconfiguration times are exceedingly rapid, in the 10-ns range. Solvent viscosity-dependent measurements indicate that these dynamics of single-stranded nucleic acids exhibit negligible internal friction and are thus dominated by solvent friction. Our results provide a detailed view of the conformational distributions and rapid dynamics of single-stranded nucleic acids.

Disordered proteins and nucleic acids play key roles in cellular function and disease. Here we review recent advances in the computational exploration of the conformational dynamics of flexible biomolecules. We focus on hierarchical chain growth (HCG) from fragment libraries built with atomistic molecular dynamics simulations. HCG combines chain fragments in a statistically reproducible manner into ensembles of full-length atomically detailed biomolecular structures. The input fragment structures are typically collected from molecular dynamics simulations, but could also come from structural databases. Experimental data can be integrated during and after chain assembly. Applications to the neurodegeneration-linked proteins $\alpha$-synuclein, tau, and TDP-43, including as condensate, illustrate the use of HCG. We conclude by highlighting the emerging connections to AI-based structural modeling., Comment: 21 pages, 4 figures

Molecular self-organization driven by concerted many-body interactions produces the ordered structures that define both inanimate and living matter. Understanding the physical mechanisms that govern the formation of molecular complexes and crystals is key to controlling the assembly of nanomachines and new materials. We present an artificial intelligence (AI) agent that uses deep reinforcement learning and transition path theory to discover the mechanism of molecular self-organization phenomena from computer simulations. The agent adaptively learns how to sample complex molecular events and, on the fly, constructs quantitative mechanistic models. By using the mechanistic understanding for AI-driven sampling, the agent closes the learning cycle and overcomes time-scale gaps of many orders of magnitude. Symbolic regression condenses the mechanism into a human-interpretable form. Applied to ion association in solution, gas-hydrate crystal formation, and membrane-protein assembly, the AI agent identifies the many-body solvent motions governing the assembly process, discovers the variables of classical nucleation theory, and reveals competing assembly pathways. The mechanistic descriptions produced by the agent are predictive and transferable to close thermodynamic states and similar systems. Autonomous AI sampling has the power to discover assembly and reaction mechanisms from materials science to biology.

In single-molecule force spectroscopy experiments, a biomolecule is attached to a force probe via polymer linkers, and the total extension -- of molecule plus apparatus -- is monitored as a function of time. In a typical unfolding experiment at constant force, the total extension jumps between two values that correspond to the folded and unfolded states of the molecule. For several biomolecular systems the committor, which is the probability to fold starting from a given extension, has been used to extract the molecular activation barrier (a technique known as "committor inversion"). In this work, we study the influence of the force probe, which is much larger than the molecule being measured, on the activation barrier obtained by committor inversion. We use a two-dimensional framework in which the diffusion coefficient of the molecule and of the pulling device can differ. We systematically study the free energy profile along the total extension obtained from the committor, by numerically solving the Onsager equation and using Brownian dynamics simulations. We analyze the dependence of the extracted barrier on the linker stiffness, molecular barrier height, and diffusion anisotropy, and thus, establish the range of validity of committor inversion. Along the way, we showcase the committor of 2-dimensional diffusive models and illustrate how it is affected by barrier asymmetry and diffusion anisotropy., Comment: 22 pages, 7 figures

Exascale computing holds great opportunities for molecular dynamics (MD) simulations. However, to take full advantage of the new possibilities, we must learn how to focus computational power on the discovery of complex molecular mechanisms, and how to extract them from enormous amounts of data. Both aspects still rely heavily on human experts, which becomes a serious bottleneck when a large number of parallel simulations have to be orchestrated to take full advantage of the available computing power. Here, we use artificial intelligence (AI) both to guide the sampling and to extract the relevant mechanistic information. We combine advanced sampling schemes with statistical inference, artificial neural networks, and deep learning to discover molecular mechanisms from MD simulations. Our framework adaptively and autonomously initializes simulations and learns the sampled mechanism, and is thus suitable for massively parallel computing architectures. We propose practical solutions to make the neural networks interpretable, as illustrated in applications to molecular systems., Comment: 11 pages, 5 figures, supporting information

We describe and implement iMapD, a computer-assisted approach for accelerating the exploration of uncharted effective Free Energy Surfaces (FES), and more generally for the extraction of coarse-grained, macroscopic information from atomistic or stochastic (here Molecular Dynamics, MD) simulations. The approach functionally links the MD simulator with nonlinear manifold learning techniques. The added value comes from biasing the simulator towards new, unexplored phase space regions by exploiting the smoothness of the (gradually, as the exploration progresses) revealed intrinsic low-dimensional geometry of the FES.

We present a new method that enables the identification and analysis of both transition and metastable conformational states from atomistic or coarse-grained molecular dynamics (MD) trajectories. Our algorithm is presented and studied by using both analytical and actual examples from MD simulations of the helix-forming peptide Ala5, and of a larger system, the epidermal growth factor receptor (EGFR) protein. In all cases, our method identifies automatically the corresponding transition states and metastable conformations in an optimal way, with the input of a set of relevant coordinates, by capturing accurately the intrinsic slowest relaxation rate. Our approach provides a general and easy to implement analysis method that provides unique insight into the molecular mechanism and the rare but crucial rate limiting conformational pathways occurring in complex dynamical systems such as molecular trajectories.

We describe different Bayesian ensemble refinement methods, examine their interrelation, and discuss their practical application. With ensemble refinement, the properties of dynamic and partially disordered (bio)molecular structures can be characterized by integrating a wide range of experimental data, including measurements of ensemble-averaged observables. We start from a Bayesian formulation in which the posterior is a functional that ranks different configuration space distributions. By maximizing this posterior, we derive an optimal Bayesian ensemble distribution. For discrete configurations, this optimal distribution is identical to that obtained by the maximum entropy "ensemble refinement of SAXS" (EROS) formulation. Bayesian replica ensemble refinement enhances the sampling of relevant configurations by imposing restraints on averages of observables in coupled replica molecular dynamics simulations. We show that the strength of the restraint should scale linearly with the number of replicas to ensure convergence to the optimal Bayesian result in the limit of infinitely many replicas. In the "Bayesian inference of ensembles" (BioEn) method, we combine the replica and EROS approaches to accelerate the convergence. An adaptive algorithm can be used to sample directly from the optimal ensemble, without replicas. We discuss the incorporation of single-molecule measurements and dynamic observables such as relaxation parameters. The theoretical analysis of different Bayesian ensemble refinement approaches provides a basis for practical applications and a starting point for further investigations., Comment: Paper submitted to The Journal of Chemical Physics (15 pages, 4 figures); updated references; expanded discussions of related formalisms, error treatment, and ensemble refinement with and without replicas; appendix

We present a recently developed one-dimensional dipole lattice model that accurately captures the key properties of water in narrow nanopores. For this model, we derive three equivalent representations of the Hamiltonian that together yield a transparent physical picture of the energetics of the water chain and permit efficient computer simulations. In the charge representation, the Hamiltonian consists of nearest-neighbor interactions and Coulomb-like interactions of effective charges at the ends of dipole ordered segments. Approximations based on the charge picture shed light on the influence of the Coulomb-like interactions on the structure of nanopore water. We use Monte Carlo simulations to study the system behavior of the full Hamiltonian and its approximations as a function of chemical potential and system size and investigate the bimodal character of the density distribution occurring at small system sizes.

We describe a reverse integration approach for the exploration of low-dimensional effective potential landscapes. Coarse reverse integration initialized on a ring of coarse states enables efficient "navigation" on the landscape terrain: escape from local effective potential wells, detection of saddle points, and identification of significant transition paths between wells. We consider several distinct ring evolution modes: backward stepping in time, solution arc--length, and effective potential. The performance of these approaches is illustrated for a deterministic problem where the energy landscape is known explicitly. Reverse ring integration is then applied to "noisy" problems where the ring integration routine serves as an outer "wrapper" around a forward-in-time inner simulator. Three versions of such inner simulators are considered: a system of stochastic differential equations, a Gillespie--type stochastic simulator, and a molecular dynamics simulator. In these "equation-free" computational illustrations, estimation techniques are applied to the results of short bursts of "inner" simulation to obtain the unavailable (in closed form) quantities (local drift and diffusion coefficient estimates) required for reverse ring integration; this naturally leads to approximations of the effective landscape., Comment: article submitted to J Chem Phys

We show how accurate kinetic information, such as the rates of protein folding and unfolding, can be extracted from replica-exchange molecular dynamics (REMD) simulations. From the brief and continuous trajectory segments between replica exchanges, we estimate short-time propagators in conformation space and use them to construct a master equation. For a helical peptide in explicit water, we determine the rates of transitions both locally between microscopic conformational states and globally for folding and unfolding. We show that accurate rates in the ~1/(100 ns) to ~1/(1 ns) range can be obtained from REMD with exchange times of 5 ps, in excellent agreement with results from long equilibrium molecular dynamics., Comment: 4 pages, 4 figures; v2: updated abstract and Fig. 3

We present a ``coarse molecular dynamics'' approach and apply it to studying the kinetics and thermodynamics of a peptide fragment dissolved in water. Short bursts of appropriately initialized simulations are used to infer the deterministic and stochastic components of the peptide motion parametrized by an appropriate set of coarse variables. Techniques from traditional numerical analysis (Newton-Raphson, coarse projective integration) are thus enabled; these techniques help analyze important features of the free-energy landscape (coarse transition states, eigenvalues and eigenvectors, transition rates, etc.). Reverse integration of (irreversible) expected coarse variables backward in time can assist escape from free energy minima and trace low-dimensional free energy surfaces. To illustrate the ``coarse molecular dynamics'' approach, we combine multiple short (0.5-ps) replica simulations to map the free energy surface of the ``alanine dipeptide'' in water, and to determine the ~ 1/(1000 ps) rate of interconversion between the two stable configurational basins corresponding to the alpha-helical and extended minima., Comment: The article has been submitted to "The Journal of Chemical Physics."

A theoretical treatment based upon the quasi-chemical theory of solutions predicts the most probable number of water neighbors in the inner shell of a Li+ ion in liquid water to be four. The instability of a six water molecule inner sphere complex relative to four-coordinated structures is confirmed by an `ab initio' molecular dynamics calculation. A classical Monte Carlo simulation equilibrated 26 water molecules with a rigid six-coordinated Li(H2O)6+ complex with periodic boundary conditions in aqueous solution. With that initial configuration for the molecular dynamics, the six-coordinated structure relaxed into four-coordinated arrangements within 112 fs and stabilized. This conclusion differs from prior interpretations of neutron and X-ray scattering results on aqueous solutions., Comment: 9 pages, 2 figures

Ewald summation and physically equivalent methods such as particle-mesh Ewald, kubic-harmonic expansions, or Lekner sums are commonly used to calculate long-range electrostatic interactions in computer simulations of polar and charged substances. The calculation of pressures in such systems is investigated. We find that the virial and thermodynamic pressures differ because of the explicit volume dependence of the effective, resummed Ewald potential. The thermodynamic pressure, obtained from the volume derivative of the Helmholtz free energy, can be expressed easily for both ionic and rigid molecular systems. For a system of rigid molecules, the electrostatic energy and the forces at the atom positions are required, both of which are readily available in molecular dynamics codes. We then calculate the virial and thermodynamic pressures for the extended simple point charge (SPC/E) water model at standard conditions. We find that the thermodynamic pressure exhibits considerably less system size dependence than the virial pressure. From an analysis of the cross correlation between the virial and thermodynamic pressure, we conclude that the thermodynamic pressure should be used to drive volume fluctuations in constant-pressure simulations., Comment: RevTeX, 19 pages, 2 EPS figures; in press: Journal of Chemical Physics, 15-August-1998

Recent progress on molecular theories of hydration of nonpolar solutes in aqueous solution has led to new ways of thinking about the old issue of free, or "available," volume in liquids. This article surveys the principal new results with particular attention to general issues of packing in liquids., Comment: 14 pages included 3 figures; submitted to the proceedings of the NATO Advanced Study Institute: New Approaches to Old and New Problems in Liquid State Theory, Patti Marina (Messina) Italy, July 1998 replacement: minor typographical corrections

Recent developments in molecular theories and simulation of ions and polar molecules in water are reviewed. The hydration of imidazole and imidazolium solutes is used to exemplify the theoretical issues. The treatment of long-ranged electrostatic interactions in simulations is discussed extensively. It is argued that the Ewald approach is an easy way to get correct hydration free energies in the thermodynamic limit from molecular calculations; and that molecular simulations with Ewald interactions and periodic boundary conditions can also be more efficient than many common alternatives. The Ewald treatment permits a conclusive extrapolation to infinite system size. The picture that emerges from testing of simple models is that the most prominent failings of the simplest theories are associated with solvent proton conformations that lead to non-gaussian fluctuations of electrostatic potentials. Thus, the most favorable cases for the second-order perturbation theories are monoatomic positive ions. For polar and anionic solutes, continuum or gaussian theories are less accurate. The appreciation of the specific deficiencies of those simple models have led to new concepts, multistate gaussian and quasi-chemical theories, that address the cases for which the simpler theories fail. It is argued that, relative to direct dielectric continuum treatments, the quasi-chemical theories provide a better theoretical organization for the computational study of the electronic structure of solution species., Comment: invited J. Phys. Chem. feature article, 30 pages including 10 figures, minor stylistic revisions

Free energies of ionic solvation calculated from computer simulations exhibit a strong system size dependence. We perform a finite-size analysis based on a dielectric-continuum model with periodic boundary conditions. That analysis results in an estimate of the Born ion size. Remarkably, the finite-size correction applies to systems with only eight water molecules hydrating a sodium ion and results in an estimate of the Born radius of sodium that agrees with the experimental value., Comment: 2 EPS figures

Theories of solvation free energies often involve electrostatic potentials at the position of a solute charge. Simulation calculations that apply cutoffs and periodic boundary conditions based on molecular centers result in center-dependent contributions to electrostatic energies due to a systematic sorting of charges in radial shells. This sorting of charges induces a surface-charge density at the cutoff sphere or simulation-box boundary that depends on the choice of molecular centers. We identify a simple solution that gives correct, center-independent results, namely the radial integration of charge densities. Our conclusions are illustrated for a Lennard-Jones solute in water. The present results can affect the parameterization of force fields., Comment: 10 pages, 3 Encapsulated Postscript figures, LaTeX, uses RevTeX macro

An information theory model is used to construct a molecular explanation why hydrophobic solvation entropies measured in calorimetry of protein unfolding converge at a common temperature. The entropy convergence follows from the weak temperature dependence of occupancy fluctuations for molecular-scale volumes in water. The macroscopic expression of the contrasting entropic behavior between water and common organic solvents is the relative temperature insensitivity of the water isothermal compressibility. The information theory model provides a quantitative description of small molecule hydration and predicts a negative entropy at convergence. Interpretations of entropic contributions to protein folding should account for this result., Comment: Phys. Rev. Letts. (in press 1996), 3 pages, 3 figures

A closed form of the electrostatic potential of a homogeneously charged cube is derived by integration. The exact result is compared with multipole expansions for the exterior and interior of the cube. The electrostatic potential of a homogeneously charged square in two-dimensional electrostatics is also determined., Comment: 5 pages, 1 Postscript figure, uses revtex.sty

We study the chemical potential of water as a function of charge based on perturbation theory. By calculating the electrostatic-energy fluctuations of two states (fully charged and uncharged) we are able to determine accurate values for the dependence of the chemical potential on charge. We find identical results for the chemical-potential difference of fully charged and uncharged water from overlapping-histogram and acceptance-ratio methods and by smoothly connecting the curves of direct exponential averages. Our results agree with those of Rick and Berne (J. Am. Chem. Soc., 1994, 116, 3949) with respect to both the chemical-potential difference and its dependence on the charge coupling parameter. We observe significant deviations from simple Gaussian-fluctuation statistics. The dependence on the coupling parameter is not quadratic, as would be inferred from linear continuum models of electrostatics., Comment: 10 pages, 4 Postscript figures, uses revtex.sty

The hydration free energies of ions exhibit an approximately quadratic dependence on the ionic charge, as predicted by the Born model. We analyze this behavior using second-order perturbation theory. This provides effective methods to calculating free energies from equilibrium computer simulations. The average and the fluctuation of the electrostatic potential at charge sites appear as the first coefficients in a Taylor expansion of the free energy of charging. Combining the data from different charge states allows calculation of free-energy profiles as a function of the ionic charge. The first two Taylor coefficients of the free-energy profiles can be computed accurately from equi- librium simulations; but they are affected by a strong system-size dependence. We apply corrections for these finite-size effects by using Ewald lattice sum- mation and adding the self-interactions consistently. Results are presented for a model ion with methane-like Lennard-Jones parameters in SPC water. We find two very closely quadratic regimes with different parameters for positive and negative ions. We also studied the hydration free energy of potassium, calcium, fluoride, chloride, and bromide ions. We find negative ions to be solvated more strongly compared to positive ions of equal size, in agreement with experimen- tal data. We ascribe this preference of negative ions to their strong interac- tions with water hydrogens, which can penetrate the ionic van der Waals shell without direct energetic penalty in the models used. We also find a positive electrostatic potential at the center of uncharged Lennard-Jones particles in water, which favors negative ions. Regarding finite-system-size effects, we show that even using only 16 water molecules it is possible to calculate accu- rately the hydration free energy of sodium if self-interactions are considered., Comment: 17 pages, 6 Postscript figures, uses revtex.sty

Ewald summation is widely used to calculate electrostatic interactions in computer simulations of condensed-matter systems. We present an analysis of the errors arising from truncating the infinite real- and Fourier-space lattice sums in the Ewald formulation. We derive an optimal choice for the Fourier-space cutoff given a screening parameter $\eta$. We find that the number of vectors in Fourier space required to achieve a given accuracy scales with $\eta^3$. The proposed method can be used to determine computationally efficient parameters for Ewald sums, to assess the quality of Ewald-sum implementations, and to compare different implementations., Comment: 6 pages, 3 figures (Encapsulated PostScript), LaTex

Recent molecular simulation and integral equation results alkali-halide ion pair potentials-of-mean-force in water are discussed. Dielectric model calculations are implemented to check that these models produce that characteristic structure of contact and solvent-separated minima for oppositely charged ions in water under physiological thermodynamic conditions. Comparison of the dielectric model results with the most current molecular level information indicates that the dielectric model does not, however, provide an accurate description of these potentials-of-mean-force. We note that linear dielectric models correspond to modelistic implementations of second-order thermodynamic perturbation theory for the excess chemical potential of a distinguished solute molecule. Therefore, the molecular theory corresponding to the dielectric models is second-order thermodynamic perturbation theory for that excess chemical potential. The second-order, or fluctuation, term raises a technical computational issue of treatment of long-ranged interactions similar to the one which arises in calculation of the dielectric constant of the solvent. It is contended that the most important step for further development of dielectric models would be a separate assessment of the first-order perturbative term (equivalently the {\it potential at zero charge} ) which vanishes in the dielectric models but is generally nonzero. Parameterization of radii and molecular volumes should then be based of the second-order perturbative term alone. Illustrative initial calculations are presented and discussed., Comment: 37 pages and 8 figures. LA-UR-93-4205