Boltzmann Sampling Research Articles

Boltzmann sampling with quantum annealers via fast Stein correction

Despite the attempts to apply a quantum annealer to Boltzmann sampling, it is still impossible to perform accurate sampling at arbitrary temperatures. Conventional distribution correction methods such as importance sampling and resampling cannot be applied, because the analytical expression of sampling distribution is unknown for a quantum annealer. Stein correction [Q. Liu and J. Lee, in , Proceedings of Machine Learning Research (PMLR, 2017), Vol. 54, pp. 952–961] can correct the samples by weighting without the knowledge of the sampling distribution, but the naive implementation requires the solution of a large-scale quadratic program, hampering usage in practical problems. In this article, a fast and approximate method based on a random feature map and exponentiated gradient updates is developed to compute the sample weights and is used to correct the samples generated by D-Wave quantum annealers. In benchmarking problems, it is observed that the residual error of thermal average calculations is reduced significantly. If combined with our method, quantum annealers may emerge as a viable alternative to long-established Markov chain Monte Carlo methods. Published by the American Physical Society 2024

Irreversible Boltzmann samplers in dense liquids: Weak-coupling approximation and mode-coupling theory.

Exerting a nonequilibrium drive on an otherwise equilibrium Langevin process brings the dynamics out of equilibrium but can also speed up the approach to the Boltzmann steady state. Transverse forces are a minimal framework to achieve dynamical acceleration of the Boltzmann sampling. We consider a simple liquid in three space dimensions subjected to additional transverse pairwise forces, and quantify the extent to which transverse forces accelerate the dynamics. We first explore the dynamics of a tracer in a weak coupling regime describing high temperatures. The resulting acceleration is correlated with a monotonous increase of the magnitude of odd transport coefficients (mobility and diffusivity) with the amplitude of the transverse drive. We then develop a nonequilibrium version of the mode-coupling theory able to capture the effect of transverse forces, and more generally of forces created by additional degrees of freedom. Based on an analysis of transport coefficients, both odd and longitudinal, both for the collective modes and for a tracer particle, we find a systematic acceleration of the dynamics. Quantitatively, the gain, which is guaranteed throughout the ergodic phase, turns out to be a decreasing function of temperature beyond a temperature crossover, in particular as the glass transition is approached. Our theoretical results are in good agreement with available numerical results.

Infrared: a declarative tree decomposition-powered framework for bioinformatics.

Many bioinformatics problems can be approached as optimization or controlled sampling tasks, and solved exactly and efficiently using Dynamic Programming (DP). However, such exact methods are typically tailored towards specific settings, complex to develop, and hard to implement and adapt to problem variations. We introduce the Infrared framework to overcome such hindrances for a large class of problems. Its underlying paradigm is tailored toward problems that can be declaratively formalized as sparse feature networks, a generalization of constraint networks. Classic Boolean constraints specify a search space, consisting of putative solutions whose evaluation is performed through a combination of features. Problems are then solved using generic cluster tree elimination algorithms over a tree decomposition of the feature network. Their overall complexities are linear on the number of variables, and only exponential in the treewidth of the feature network. For sparse feature networks, associated with low to moderate treewidths, these algorithms allow to find optimal solutions, or generate controlled samples, with practical empirical efficiency. Implementing these methods, the Infrared software allows Python programmers to rapidly develop exact optimization and sampling applications based on a tree decomposition-based efficient processing. Instead of directly coding specialized algorithms, problems are declaratively modeled as sets of variables over finite domains, whose dependencies are captured by constraints and functions. Such models are then automatically solved by generic DP algorithms. To illustrate the applicability of Infrared in bioinformatics and guide new users, we model and discuss variants of bioinformatics applications. We provide reimplementations and extensions of methods for RNA design, RNA sequence-structure alignment, parsimony-driven inference of ancestral traits in phylogenetic trees/networks, and design of coding sequences. Moreover, we demonstrate multidimensional Boltzmann sampling. These applications of the framework-together with our novel results-underline the practical relevance of Infrared. Remarkably, the achieved complexities are typically equivalent to the ones of specialized algorithms and implementations. Infrared is available at https://amibio.gitlabpages.inria.fr/Infrared with extensive documentation, including various usage examples and API reference; it can be installed using Conda or from source.

Molecular mechanisms that cause cell death in LHON

Mitochondrial impairments, caused by mtDNA mutations, lead to RGC death and optic atrophy causing blindness in LHON. It was once thought that this was a disease of impaired bioenergetics. More recent thinking, considering the tempo of damage, studies in cybrids and a faithful mouse genetic model of LHON, has led to the realization that the problem is more that of the pathological generation of Reactive Oxygen Species (ROS). For example, synaptosome analysis from the brains of LHON mice show normal levels of ATP, but the blockage of Complex I activity was associated with very high levels of ROS. We hypothesized that the mtDNA mutation, specifically for 3460 in ND1, changed the atomic structures around COQ10 such that ROS were generated. In order to test that hypothesis, we conducted a series of experiments with statistical ensembles, built with Monte Carlo and Boltzmann sampling for simulations in computational chemistry. We began with the single amino acid mutation in ND1 of Complex I that switches hydrophobic alanine to hydrophilic threonine, at the entry of the Coenzyme Q10 binding channel. We explicated the mechanistic consequences of this mutation, using the tools of Molecular Dynamics and Free Energy Perturbation simulations. We showed that the mutant remained capable of binding Coenzyme Q10. Indeed, the mutated channel was even more proficient than the wild type in reducing ubiquinone to ubiquinol due to the closer average proximity of the Coenzyme Q10 headgroup to the chain of iron–sulfur clusters, from which it received electrons via quantum mechanical tunnelling. However, the mutation kinetically hindered the exit of the reduced (ubiquinol) form of Q10 from the ND1 channel, slowing the passage of its headgroup due to unfavourable electrostatics. This effect lead to overproduction of ROS via re‐oxidation of Coenzyme Q10 as electrons were sent back up the Iron–sulfur cluster chain, where the ROS production occurs. Another line of studies has shown that ROS lead to the death of RGCs affecting the smallest axons first. In those areas of the optic nerve in which larger axons are preponderate, the ROS are absorbed by the larger axons and little more happens. However, there is a dearth of large axons inferior temporally where the papillomacular bundle enters the optic nerve. Then, the ROS begins a reiterative and vicious cycle of cell death and the production of more ROS. This explains both the very peculiar spatial and temporal features of the optic neuropathy in LHON.

Relative Entropy Regularized Sample-Efficient Reinforcement Learning With Continuous Actions.

In this article, a novel reinforcement learning (RL) approach, continuous dynamic policy programming (CDPP), is proposed to tackle the issues of both learning stability and sample efficiency in the current RL methods with continuous actions. The proposed method naturally extends the relative entropy regularization from the value function-based framework to the actor-critic (AC) framework of deep deterministic policy gradient (DDPG) to stabilize the learning process in continuous action space. It tackles the intractable softmax operation over continuous actions in the critic by Monte Carlo estimation and explores the practical advantages of the Mellowmax operator. A Boltzmann sampling policy is proposed to guide the exploration of actor following the relative entropy regularized critic for superior learning capability, exploration efficiency, and robustness. Evaluated by several benchmark and real-robot-based simulation tasks, the proposed method illustrates the positive impact of the relative entropy regularization including efficient exploration behavior and stable policy update in RL with continuous action space and successfully outperforms the related baseline approaches in both sample efficiency and learning stability.

Wavelength-division multiplexing optical Ising simulator enabling fully programmable spin couplings and external magnetic fields.

Recently various physical systems have been proposed for modeling Ising spin Hamiltonians appealing to solve combinatorial optimization problems with remarkable performance. However, how to implement arbitrary spin-spin interactions is a critical and challenging problem in unconventional Ising machines. Here, we propose a general gauge transformation scheme to enable arbitrary spin-spin interactions and external magnetic fields as well, by decomposing an Ising Hamiltonian into multiple Mattis-type interactions. With this scheme, a wavelength-division multiplexing spatial photonic Ising machine (SPIM) is developed to show the programmable capability of general spin coupling interactions. We exploit the wavelength-division multiplexing SPIM to simulate three spin systems: ±J models, Sherrington-Kirkpatrick models, and only locally connected J1-J2 models and observe the phase transitions. We also demonstrate the ground-state search for solving Max-Cut problem with the wavelength-division multiplexing SPIM. These results promise the realization of ultrafast-speed and high-power efficiency Boltzmann sampling to a generalized large-scale Ising model.

Effect of Initial Conditions Sampling on Surface Hopping Simulations in the Ultrashort and Picosecond Time Range. Azomethane Photodissociation as a Case Study

We tested the effect of different ways of sampling the initial conditions in surface hopping simulations, with a focus on the initial energy distributions and on the treatment of the zero point energy (ZPE). As a test case, we chose the gas phase photodynamics of azomethane, which features different processes occurring in overlapping time scales: geometry relaxation in the excited state, internal conversion, photoisomerization, and fast and slow dissociation. The simulations, based on a semiempirical method, had a sufficiently long duration (10 ps) to encompass all of the above processes. We tested several variants of methods based on the quantum mechanical (QM) distributions of the nuclear coordinates q and momenta p, which yield, at least on the average over a large sampling set, the correct QM energy, namely the ZPE when starting from the ground vibrational state. We compared the QM samplings with the classical Boltzmann (CB) distribution obtained by a thermostated trajectory, whereby thermal effects are taken into account, but the ZPE is utterly ignored. We found that most QM and CB approaches yield similar results as to short time dynamics and decay lifetimes, whereas the rate of the ground state dissociation reaction CH3NNCH3 → CH3NN + CH3 is sharply affected by the sampling method. With QM samplings a large fraction of trajectories dissociate promply (<1 ps) after decay to the ground state and with rates of the order of 10-1 ps-1 after the first ps. Instead, the CB samplings yield a much smaller fraction of prompt dissociations and much lower rates at long times. We provided evidence that the ZPE "leaks" from high frequency modes to the reactive ones (N-C bond elongations), therefore unphysically increasing the dissociation rates with QM samplings. We show that an effective way to take into account the ZPE and to avoid the "leaking" problem is to add the ZPE to the potential energy surfaces as a function of the most relevant internal coordinates. Then, Boltzmann sampling can be done as usual, so this approach is suitable also for condensed state dynamics. In the tests we present here, the ZPE correction method yields dissociation rates intermediate between QM and uncorrected Boltzmann samplings.

Simulating actin networks in synaptic spine heads using dynamical graph grammars.

There is a morphodynamic component to synaptic learning by which dendritic (postsynaptic) spine head size changes can strengthen or weaken the synaptic connection between two neurons, in response to signals from the axon of a presynaptic neuron. These morphological factors are in turn sculpted by the graph-like dynamics of the actin cytoskeleton. In this project, we seek to use Dynamical Graph Grammars (DGGs) [1,2] implemented within a computer algebra system to model how networks of actin filaments can dynamically grow or shrink and reshape the spine head. We designed and implemented several DGG sub-grammar mathematical models including actin network growth, isotropic/anisotropic filament forces, filament-membrane mechanical interaction, and Hessian Boltzmann sampling of random molecular displacements, to regulate the generation and deletion of graph objects. From first principles expressed in about a dozen DGG rules we simulate emergent biomechanics of a simplified network of actin polymers and its interaction with membrane, using very rough parameter estimates, all in two dimensions. We refined this model by incorporating rate constants used in previous models (e.g. [3,4,5]) in each sector, and we are currently recapitulating the previously observed sub-grammar behaviors.

Quantitative views of protein dynamics through the computational microscope.

The complexity of the energy landscape of proteins and subjectivity in choosing reaction coordinates pose significant hurdles in the quantitative characterization of functional protein dynamics using computational techniques. We have proposed the cumulative variance of coordinate fluctuations (CVCF) as a parameter free and energy landscape sensitive metric to identify sections of Molecular dynamics (MD) and Monte-Carlo (MC) trajectories which achieve Boltzmann sampling conditions. Our CVCF-trace analysis can provide quantitative estimates of the effective curvature and roughness of the energy landscape of proteins. We have applied these conceptual advances to compare the overall and directional flexibilities of multiple protein systems in monomeric form as well as within complexes. The results not only correlate well with existing experimental single molecule force spectroscopy data but also provide clear predictions which can be tested by such experiments. Moving beyond CVCF, we propose a mode evolution metric (MEM) to track the evolution of protein dynamics along MD trajectories. MEM enables us to deduce the dimensionality of protein motions dictated by the underlying energy landscape, thereby providing an objective means to identify reaction coordinates to describe dynamics. We will discuss improved biomolecular energy landscape sampling formulations based on our set of dynamical metrics.

TriMem: A parallelized hybrid Monte Carlo software for efficient simulations of lipid membranes.

Lipid membranes are integral building blocks of living cells and perform a multitude of biological functions. Currently, molecular simulations of cellular-scale membrane remodeling processes at atomic resolution are extremely difficult, due to their size, complexity, and the large times-scales on which these processes occur. Instead, elastic membrane models are used to simulate membrane shapes and transitions between them and to infer their properties and functions. Unfortunately, an efficiently parallelized open-source simulation code to do so has been lacking. Here, we present TriMem, a parallel hybrid Monte Carlo simulation engine for triangulated lipid membranes. The kernels are efficiently coded in C++ and wrapped with Python for ease-of-use. The parallel implementation of the energy and gradient calculations and of Monte Carlo flip moves of edges in the triangulated membrane enable us to simulate large and highly curved membrane structures. For validation, we reproduce phase diagrams of vesicles with varying surface-to-volume ratios and area difference. We also compute the density of states to verify correct Boltzmann sampling. The software can be used to tackle a range of large-scale membrane remodeling processes as a step toward cell-scale simulations. Additionally, extensive documentation make the software accessible to the broad biophysics and computational cell biology communities.

Ring polymer molecular dynamics in gas-surface reactions: tests on initial sampling and potential energy landscape

Given the exponential scaling of the standard quantum wavepacket method, accurate quantum dynamics calculations for gas-surface scattering remain a daunting challenge. Recent development of the ring polymer molecular dynamics (RPMD) approach has shed some light on calculating microcanonical dynamical quantities in non-thermal conditions, although most studied systems feature a notable barrier. Here we investigate the performance of RPMD reaction probability calculations in two gas-surface systems, one is an example for weakly-activated reactions (H2+Co(0001)) and the other is a prototypical non-activated reaction (H2+Pd(111)). Initial conditions of the molecule have been obtained with the Boltzmann sampling on the realistic potential energy surface or the approximate harmonic potential. The two sampling schemes lead to little difference in the weakly-activated H2+Co(0001) system, both giving results in good agreement with the quantum ones. However, they yield distinguishable dynamics in the non-activated system, where harmonic Boltzmann sampling-based RPMD behaves similarly as the quasi-classical trajectory method and underestimates the reactivity at low energies. In comparison, the realistic Boltzmann sampling-based RPMD overestimates the reactivity but reasonably captures the trapping-mediated reaction mechanism. Possible reasons of this deficiency have been discussed.

RNAxplorer: harnessing the power of guiding potentials to sample RNA landscapes.

MotivationPredicting the folding dynamics of RNAs is a computationally difficult problem, first and foremost due to the combinatorial explosion of alternative structures in the folding space. ions are therefore needed to simplify downstream analyses, and thus make them computationally tractable. This can be achieved by various structure sampling algorithms. However, current sampling methods are still time consuming and frequently fail to represent key elements of the folding space.MethodWe introduce RNAxplorer, a novel adaptive sampling method to efficiently explore the structure space of RNAs. RNAxplorer uses dynamic programming to perform an efficient Boltzmann sampling in the presence of guiding potentials, which are accumulated into pseudo-energy terms and reflect similarity to already well-sampled structures. This way, we effectively steer sampling toward underrepresented or unexplored regions of the structure space.ResultsWe developed and applied different measures to benchmark our sampling methods against its competitors. Most of the measures show that RNAxplorer produces more diverse structure samples, yields rare conformations that may be inaccessible to other sampling methods and is better at finding the most relevant kinetic traps in the landscape. Thus, it produces a more representative coarse graining of the landscape, which is well suited to subsequently compute better approximations of RNA folding kinetics.Availabilityand implementationhttps://github.com/ViennaRNA/RNAxplorer/.Supplementary informationSupplementary data are available at Bioinformatics online.

Physics-Based Computational Protein Design: An Update.

We describe methods for physics-based protein design and some recent applications from our work. We present the physical interpretation of a MC simulation in sequence space and show that sequences and conformations form a well-defined statistical ensemble, explored with Monte Carlo and Boltzmann sampling. The folded state energy combines molecular mechanics for solutes with continuum electrostatics for solvent. We usually assume one or a few fixed protein backbone structures and discrete side chain rotamers. Methods based on molecular dynamics, which introduce additional backbone and side chain flexibility, are under development. The redesign of a PDZ domain and an aminoacyl-tRNA synthetase enzyme were successful. We describe a versatile, adaptive, Wang-Landau MC method that can be used to design for substrate affinity, catalytic rate, catalytic efficiency, or the specificity of these properties. The methods are transferable to all biomolecules, can be systematically improved, and give physical insights.

The challenge of stochastic Størmer-Verlet thermostats generating correct statistics.

In light of the recently developed complete GJ set of single random variable stochastic, discrete-time Størmer-Verlet algorithms for statistically accurate simulations of Langevin equations [N. Grønbech-Jensen, Mol. Phys. 118, e1662506 (2020)], we investigate two outstanding questions: (1) Are there any algorithmic or statistical benefits from including multiple random variables per time step and (2) are there objective reasons for using one or more methods from the available set of statistically correct algorithms? To address the first question, we assume a general form for the discrete-time equations with two random variables and then follow the systematic, brute-force GJ methodology by enforcing correct thermodynamics in linear systems. It is concluded that correct configurational Boltzmann sampling of a particle in a harmonic potential implies correct configurational free-particle diffusion and that these requirements only can be accomplished if the two random variables per time step are identical. We consequently submit that the GJ set represents all possible stochastic Størmer-Verlet methods that can reproduce time step-independent statistics of linear systems. The second question is thus addressed within the GJ set. Based on numerical simulations of complex molecular systems, as well as on analytic considerations, we analyze apparent friction-induced differences in the stability of the methods. We attribute these differences to an inherent, friction-dependent discrete-time scaling, which depends on the specific method. We suggest that the method with the simplest interpretation of temporal scaling, the GJ-I/GJF-2GJ method, be preferred for statistical applications.

Development of New Methods for Enhanced Conformational Sampling of GPCRs

G protein coupled receptors (GPCRs) are integral membrane proteins that enable a cell to respond to extracellular stimuli like light, small molecules, peptides, and proteins. These receptors have evolved as highly flexible proteins that possess multiple functionally important conformational states, which enable these receptors to activate pleiotropic signaling events inside the cells. Recent progress in membrane protein structural biology techniques and cryo‐electron microscopy is providing structural evidence of two or more distinct receptor conformations for these receptors. Computational structural modeling approaches can complement and supplement these studies by predicting all functionally important states of a receptor, however, they run into the classical protein conformational sampling problem. We are developing a computational biophysical method called Enhanced Conformational Markov‐state Sampling in Membrane BiLayer Environment (EnCoMSeMBLE) to solve this sampling problem for GPCRs. We are utilizing Markov‐State‐Models to combine the Boltzmann sampling of receptor conformations from molecular dynamics (MD) with the brute‐force sampling of receptor conformations from our previously developed ActiveGEnSeMBLE method. The method was applied to the A2A receptor starting from the active state by only using traditional MD simulations. This is not surprising because the pre‐activce states is a downhill direction on the energy landscape. However, applying both MD and ActivateGEnSeMBLE methods was able to predict pre‐active states from the inactive states which, classical MD cannot achieve. Our results indicated that our methods provide an unbiased method to sample the conformational landscape. This can protentional sample unexplored regions of the conformational landscape and find novel conformations that can elucidate the mechanistic properties that GPCRs possess.Support or Funding InformationThis project was funded through a grant from the National Institutes of Health (NIH) Building Infrastructure Leading to Diversity (BUILD) #5TL4GM118977 or # 5RL5GM118975.

Complete set of stochastic Verlet-type thermostats for correct Langevin simulations

ABSTRACTWe present the complete set of stochastic Verlet-type algorithms that can provide correct statistical measures for both configurational and kinetic sampling in discrete-time Langevin systems. The approach is a brute-force general representation of the Verlet-algorithm with free parameter coefficients that are determined by requiring correct Boltzmann sampling for linear systems, regardless of time step. The result is a set of statistically correct methods given by one free functional parameter, which can be interpreted as the one-time-step velocity attenuation factor. We define the statistical characteristics of both true on-site and true half-step velocities, and use these definitions for each statistically correct Størmer-Verlet method to find a unique associated half-step velocity expression, which yields correct kinetic Maxwell-Boltzmann statistics for linear systems. It is shown that no other similar, statistically correct on-site velocity exists. We further discuss the use and features of finite-difference velocity definitions that are neither true on-site, nor true half-step. The set of methods is written in convenient and conventional stochastic Verlet forms that lend themselves to direct implementation for, e.g. Molecular Dynamics applications. We highlight a few specific examples, and validate the algorithms through comprehensive Langevin simulations of both simple nonlinear oscillators and complex Molecular Dynamics.

Fixed-parameter tractable sampling for RNA design with multiple target structures

BackgroundThe design of multi-stable RNA molecules has important applications in biology, medicine, and biotechnology. Synthetic design approaches profit strongly from effective in-silico methods, which substantially reduce the need for costly wet-lab experiments.ResultsWe devise a novel approach to a central ingredient of most in-silico design methods: the generation of sequences that fold well into multiple target structures. Based on constraint networks, our approach supports generic Boltzmann-weighted sampling, which enables the positive design of RNA sequences with specific free energies (for each of multiple, possibly pseudoknotted, target structures) and GC-content. Moreover, we study general properties of our approach empirically and generate biologically relevant multi-target Boltzmann-weighted designs for an established design benchmark. Our results demonstrate the efficacy and feasibility of the method in practice as well as the benefits of Boltzmann sampling over the previously best multi-target sampling strategy—even for the case of negative design of multi-stable RNAs. Besides empirically studies, we finally justify the algorithmic details due to a fundamental theoretic result about multi-stable RNA design, namely the #P-hardness of the counting of designs.Conclusion introduces a novel, flexible, and effective approach to multi-target RNA design, which promises broad applicability and extensibility.Our free software is available at: https://github.com/yannponty/RNARedPrintSupplementary data are available online.

On Obtaining Boltzmann-Distributed Configurational Ensembles from Expanded Ensemble Simulations with Fast State Mixing.

In Expanded Ensemble (EXE) or Simulated Tempering simulations, the system's (effective) temperature is frequently updated to enhance configurational sampling. We investigated how short the EXE state update interval τ can become before too frequent updates impede Boltzmann sampling. Simulating alanine dipeptide in explicit water, we show that a hybrid MC/MD integrator reliably yields Boltzmann-distributed configurations regardless of τ. However, in MD-driven EXE simulations with short τ, configurational ensembles depend on the thermostat settings.

Boltzmann sampling from the Ising model using quantum heating of coupled nonlinear oscillators

A network of Kerr-nonlinear parametric oscillators without dissipation has recently been proposed for solving combinatorial optimization problems via quantum adiabatic evolution through its bifurcation point. Here we investigate the behavior of the quantum bifurcation machine (QbM) in the presence of dissipation. Our numerical study suggests that the output probability distribution of the dissipative QbM is Boltzmann-like, where the energy in the Boltzmann distribution corresponds to the cost function of the optimization problem. We explain the Boltzmann distribution by generalizing the concept of quantum heating in a single nonlinear oscillator to the case of multiple coupled nonlinear oscillators. The present result also suggests that such driven dissipative nonlinear oscillator networks can be applied to Boltzmann sampling, which is used, e.g., for Boltzmann machine learning in the field of artificial intelligence.

Path-integral isomorphic Hamiltonian for including nuclear quantum effects in non-adiabatic dynamics.

We describe a path-integral approach for including nuclear quantum effects in non-adiabatic chemical dynamics simulations. For a general physical system with multiple electronic energy levels, a corresponding isomorphic Hamiltonian is introduced such that Boltzmann sampling of the isomorphic Hamiltonian with classical nuclear degrees of freedom yields the exact quantum Boltzmann distribution for the original physical system. In the limit of a single electronic energy level, the isomorphic Hamiltonian reduces to the familiar cases of either ring polymer molecular dynamics (RPMD) or centroid molecular dynamics Hamiltonians, depending on the implementation. An advantage of the isomorphic Hamiltonian is that it can easily be combined with existing mixed quantum-classical dynamics methods, such as surface hopping or Ehrenfest dynamics, to enable the simulation of electronically non-adiabatic processes with nuclear quantum effects. We present numerical applications of the isomorphic Hamiltonian to model two- and three-level systems, with encouraging results that include improvement upon a previously reported combination of RPMD with surface hopping in the deep-tunneling regime.