Encoding prior knowledge in ensemble refinement.

The proper balancing of information from experiment and theory is a long-standing problem in the analysis of noisy and incomplete data. Viewed as a Pareto optimization problem, improved agreement with the experimental data comes at the expense of growing inconsistencies with the theoretical reference model. Here, we propose how to set the exchange rate a priori to properly balance this trade-off. We focus on gentle ensemble refinement, where the difference between the potential energy surfaces of the reference and refined models is small on a thermal scale. By relating the variance of this energy difference to the Kullback-Leibler divergence between the respective Boltzmann distributions, one can encode prior knowledge about energy uncertainties, i.e., force-field errors, in the exchange rate. The energy uncertainty is defined in the space of observables and depends on their type and number and on the thermodynamic state. We highlight the relation of gentle refinement to free energy perturbation theory. A balanced encoding of prior knowledge increases the quality and transparency of ensemble refinement. Our findings extend to non-Boltzmann distributions, where the uncertainty in energy becomes an uncertainty in information.

Complexes++: Efficient and versatile coarse-grained simulations of protein complexes and their dense solutions.

The interior of living cells is densely filled with proteins and their complexes, which perform multitudes of biological functions. We use coarse-grained simulations to reach the system sizes and time scales needed to study protein complexes and their dense solutions and to interpret experiments. To take full advantage of coarse-graining, the models have to be efficiently implemented in simulation engines that are easy to use, modify, and extend. Here, we introduce the Complexes++ simulation software to simulate a residue-level coarse-grained model for proteins and their complexes, applying a Markov chain Monte Carlo engine to sample configurations. We designed a parallelization scheme for the energy evaluation capable of simulating both dilute and dense systems efficiently. Additionally, we designed the software toolbox pycomplexes to easily set up complex topologies of multi-protein complexes and their solutions in different thermodynamic ensembles and in replica-exchange simulations, to grow flexible polypeptide structures connecting ordered protein domains, and to automatically visualize structural ensembles. Complexes++ simulations can easily be modified and they can be used for efficient explorations of different simulation systems and settings. Thus, the Complexes++ software is well suited for the integration of experimental data and for method development.

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.

Nanoporous Membranes of Densely Packed Carbon Nanotubes Formed by Lipid-Mediated Self-Assembly.

Nanofiltration technology faces the competing challenges of achieving high fluid flux through uniformly narrow pores of a mechanically and chemically stable filter. Supported dense-packed 2D-crystals of single-walled carbon nanotube (CNT) porins with ∼1 nm wide pores could, in principle, meet these challenges. However, such CNT membranes cannot currently be synthesized at high pore density. Here, we use computer simulations to explore lipid-mediated self-assembly as a route toward densely packed CNT membranes, motivated by the analogy to membrane-protein 2D crystallization. In large-scale coarse-grained molecular dynamics (MD) simulations, we find that CNTs in lipid membranes readily self-assemble into large clusters. Lipids trapped between the CNTs lubricate CNT repacking upon collisions of diffusing clusters, thereby facilitating the formation of large ordered structures. Cluster diffusion follows the Saffman-Delbrück law and its generalization by Hughes, Pailthorpe, and White. On longer time scales, we expect the formation of close-packed CNT structures by depletion of the intervening shared annular lipid shell, depending on the relative strength of CNT-CNT and CNT-lipid interactions. Our simulations identify CNT length, diameter, and end functionalization as major factors for the self-assembly of CNT membranes.

Global Structure of the Intrinsically Disordered Protein Tau Emerges from Its Local Structure.

The paradigmatic disordered protein tau plays an important role in neuronal function and neurodegenerative diseases. To disentangle the factors controlling the balance between functional and disease-associated conformational states, we build a structural ensemble of the tau K18 fragment containing the four pseudorepeat domains involved in both microtubule binding and amyloid fibril formation. We assemble 129-residue-long tau K18 chains with atomic detail from an extensive fragment library constructed with molecular dynamics simulations. We introduce a reweighted hierarchical chain growth (RHCG) algorithm that integrates experimental data reporting on the local structure into the assembly process in a systematic manner. By combining Bayesian ensemble refinement with importance sampling, we obtain well-defined ensembles and overcome the problem of exponentially varying weights in the integrative modeling of long-chain polymeric molecules. The resulting tau K18 ensembles capture nuclear magnetic resonance (NMR) chemical shift and J-coupling measurements. Without further fitting, we achieve very good agreement with measurements of NMR residual dipolar couplings. The good agreement with experimental measures of global structure such as single-molecule Förster resonance energy transfer (FRET) efficiencies is improved further by ensemble refinement. By comparing wild-type and mutant ensembles, we show that pathogenic single-point P301L, P301S, and P301T mutations shift the population from the turn-like conformations of the functional microtubule-bound state to the extended conformations of disease-associated tau fibrils. RHCG thus provides us with an atomically detailed view of the population equilibrium between functional and aggregation-prone states of tau K18, and demonstrates that global structural characteristics of this intrinsically disordered protein emerge from its local structure.

Membrane fusion and drug delivery with carbon nanotube porins.

Drug delivery mitigates toxic side effects and poor pharmacokinetics of life-saving therapeutics and enhances treatment efficacy. However, direct cytoplasmic delivery of drugs and vaccines into cells has remained out of reach. We find that liposomes studded with 0.8-nm-wide carbon nanotube porins (CNTPs) function as efficient vehicles for direct cytoplasmic drug delivery by facilitating fusion of lipid membranes and complete mixing of the membrane material and vesicle interior content. Fusion kinetics data and coarse-grained molecular dynamics simulations reveal an unusual mechanism where CNTP dimers tether the vesicles, pull the membranes into proximity, and then fuse their outer and inner leaflets. Liposomes containing CNTPs in their membranes and loaded with an anticancer drug, doxorubicin, were effective in delivering the drug to cancer cells, killing up to 90% of them. Our results open an avenue for designing efficient drug delivery carriers compatible with a wide range of therapeutics.

Solution Structure and Conformational Flexibility of a Polyketide Synthase Module.

Polyketide synthases (PKSs) are versatile C-C bond-forming enzymes that are broadly distributed in bacteria and fungi. The polyketide compound family includes many clinically useful drugs such as the antibiotic erythromycin, the antineoplastic epothilone, and the cholesterol-lowering lovastatin. Harnessing PKSs for custom compound synthesis remains an open challenge, largely because of the lack of knowledge about key structural properties. Particularly, the domains-well characterized on their own-are poorly understood in their arrangement, conformational dynamics, and interplay in the intricate quaternary structure of modular PKSs. Here, we characterize module 2 from the 6-deoxyerythronolide B synthase by small-angle X-ray scattering and cross-linking mass spectrometry with coarse-grained structural modeling. The results of this hybrid approach shed light on the solution structure of a cis-AT type PKS module as well as its inherent conformational dynamics. Supported by a directed evolution approach, we also find that acyl carrier protein (ACP)-mediated substrate shuttling appears to be steered by a nonspecific electrostatic interaction network.

MDBenchmark: A toolkit to optimize the performance of molecular dynamics simulations.

Despite the impending flattening of Moore's law, the system size, complexity, and length of molecular dynamics (MD) simulations keep on increasing, thanks to effective code parallelization and optimization combined with algorithmic developments. Going forward, exascale computing poses new challenges to the efficient execution and management of MD simulations. The diversity and rapid developments of hardware architectures, software environments, and MD engines make it necessary that users can easily run benchmarks to optimally set up simulations, both with respect to time-to-solution and overall efficiency. To this end, we have developed the software MDBenchmark to streamline the setup, submission, and analysis of simulation benchmarks and scaling studies. The software design is open and as such not restricted to any specific MD engine or job queuing system. To illustrate the necessity and benefits of running benchmarks and the capabilities of MDBenchmark, we measure the performance of a diverse set of 23 MD simulation systems using GROMACS 2018. We compare the scaling of simulations with the number of nodes for central processing unit (CPU)-only and mixed CPU-graphics processing unit (GPU) nodes and study the performance that can be achieved when running multiple simulations on a single node. In all these cases, we optimize the numbers of message passing interface (MPI) ranks and open multi-processing (OpenMP) threads, which is crucial to maximizing performance. Our results demonstrate the importance of benchmarking for finding the optimal system and hardware specific simulation parameters. Running MD simulations with optimized settings leads to a significant performance increase that reduces the monetary, energetic, and environmental costs of MD simulations.

Quantifying Protein-Protein Interactions in Molecular Simulations.

Interactions among proteins, nucleic acids, and other macromolecules are essential for their biological functions and shape the physicochemcial properties of the crowded environments inside living cells. Binding interactions are commonly quantified by dissociation constants Kd, and both binding and nonbinding interactions are quantified by second osmotic virial coefficients B2. As a measure of nonspecific binding and stickiness, B2 is receiving renewed attention in the context of so-called liquid-liquid phase separation in protein and nucleic acid solutions. We show that Kd is fully determined by B2 and the fraction of the dimer observed in molecular simulations of two proteins in a box. We derive two methods to calculate B2. From molecular dynamics or Monte Carlo simulations using implicit solvents, we can determine B2 from insertion and removal energies by applying Bennett's acceptance ratio (BAR) method or the (binless) weighted histogram analysis method (WHAM). From simulations using implicit or explicit solvents, one can estimate B2 from the probability that the two molecules are within a volume large enough to cover their range of interactions. We validate these methods for coarse-grained Monte Carlo simulations of three weakly binding proteins. Our estimates for Kd and B2 allow us to separate out the contributions of nonbinding interactions to B2. Comparison of calculated and measured values of Kd and B2 can be used to (re-)parameterize and improve molecular force fields by calibrating specific affinities, overall stickiness, and nonbinding interactions. The accuracy and efficiency of Kd and B2 calculations make them well suited for high-throughput studies of large interactomes.

Fragment Binding Pose Predictions Using Unbiased Simulations and Markov-State Models.

Predicting the costructure of small-molecule ligands and their respective target proteins has been a long-standing problem in drug discovery. For weak binding compounds typically identified in fragment-based screening (FBS) campaigns, determination of the correct binding site and correct binding mode is usually done experimentally via X-ray crystallography. For many targets of pharmaceutical interest, however, establishing an X-ray system which allows for sufficient throughput to support a drug discovery project is not possible. In this case, exploration of fragment hits becomes a very laborious and consequently slow process with the generation of protein/ligand cocrystal structures as the bottleneck of the entire process. In this work, we introduce a computational method which is able to reliably predict binding sites and binding modes of fragment-like small molecules using solely the structure of the apoprotein and the ligand's chemical structure as input information. The method is based on molecular dynamics simulations and Markov-state models and can be run as a fully automated protocol requiring minimal human intervention. We describe the application of the method to a representative subset of different target classes and fragments from historical FBS efforts at Boehringer Ingelheim and discuss its potential integration into the overall fragment-based drug discovery workflow.

Inferring Structural Ensembles of Flexible and Dynamic Macromolecules Using Bayesian, Maximum Entropy, and Minimal-Ensemble Refinement Methods.

The flexible and dynamic nature of biomolecules and biomolecular complexes is essential for many cellular functions in living organisms but poses a challenge for experimental methods to determine high-resolution structural models. To meet this challenge, experiments are combined with molecular simulations. The latter propose models for structural ensembles, and the experimental data can be used to steer these simulations and to select ensembles that most likely underlie the experimental data. Here, we explain in detail how the "Bayesian Inference Of ENsembles" (BioEn) method can be used to refine such ensembles using a wide range of experimental data. The "Ensemble Refinement of SAXS" (EROS) method is a special case of BioEn, inspired by the Gull-Daniell formulation of maximum entropy image processing and focused originally on X-ray solution scattering experiments (SAXS) and then extended to integrative structural modeling. We also briefly sketch the "minimum ensemble method," a maximum-parsimony refinement method that seeks to represent an ensemble with a minimal number of representative structures.

Finite-Size-Corrected Rotational Diffusion Coefficients of Membrane Proteins and Carbon Nanotubes from Molecular Dynamics Simulations.

We investigate system-size effects on the rotational diffusion of membrane proteins and other membrane-embedded molecules in molecular dynamics simulations. We find that the rotational diffusion coefficient slows down relative to the infinite-system value by a factor of one minus the ratio of protein and box areas. This correction factor follows from the hydrodynamics of rotational flows under periodic boundary conditions and is rationalized in terms of Taylor-Couette flow. For membrane proteins like transporters, channels, or receptors in typical simulation setups, the protein-covered area tends to be relatively large, requiring a significant finite-size correction. Molecular dynamics simulations of the protein adenine nucleotide translocase (ANT1) and of a carbon nanotube porin in lipid membranes show that the hydrodynamic finite-size correction for rotational diffusion is accurate in standard-use cases. The dependence of the rotational diffusion on box size can be used to determine the membrane viscosity.

Efficient Ensemble Refinement by Reweighting.

Ensemble refinement produces structural ensembles of flexible and dynamic biomolecules by integrating experimental data and molecular simulations. Here we present two efficient numerical methods to solve the computationally challenging maximum-entropy problem arising from a Bayesian formulation of ensemble refinement. Recasting the resulting constrained weight optimization problem into an unconstrained form enables the use of gradient-based algorithms. In two complementary formulations that differ in their dimensionality, we optimize either the log-weights directly or the generalized forces appearing in the explicit analytical form of the solution. We first demonstrate the robustness, accuracy, and efficiency of the two methods using synthetic data. We then use NMR J-couplings to reweight an all-atom molecular dynamics simulation ensemble of the disordered peptide Ala-5 simulated with the AMBER99SB*-ildn-q force field. After reweighting, we find a consistent increase in the population of the polyproline-II conformations and a decrease of α-helical-like conformations. Ensemble refinement makes it possible to infer detailed structural models for biomolecules exhibiting significant dynamics, such as intrinsically disordered proteins, by combining input from experiment and simulation in a balanced manner.

Precision DEER Distances from Spin-Label Ensemble Refinement.

Double electron-electron resonance (DEER) experiments probe nanometer-scale distances in spin-labeled proteins and nucleic acids. Rotamer libraries of the covalently attached spin-labels help reduce position uncertainties. Here we show that rotamer reweighting is essential for precision distance measurements, making it possible to resolve Ångstrom-scale domain motions. We analyze extensive DEER measurements on the three N-terminal polypeptide transport-associated (POTRA) domains of the outer membrane protein Omp85. Using the "Bayesian inference of ensembles" maximum-entropy method, we extract rotamer weights from the DEER measurements. Small weight changes suffice to eliminate otherwise significant discrepancies between experiments and model and unmask 1-3 Å domain motions relative to the crystal structure. Rotamer-weight refinement is a simple yet powerful tool for precision distance measurements that should be broadly applicable to label-based measurements including DEER, paramagnetic relaxation enhancement, and fluorescence resonance energy transfer (FRET).

Hydrodynamics of Diffusion in Lipid Membrane Simulations.

By performing molecular dynamics simulations with up to 132 million coarse-grained particles in half-micron sized boxes, we show that hydrodynamics quantitatively explains the finite-size effects on diffusion of lipids, proteins, and carbon nanotubes in membranes. The resulting Oseen correction allows us to extract infinite-system diffusion coefficients and membrane surface viscosities from membrane simulations despite the logarithmic divergence of apparent diffusivities with increasing box width. The hydrodynamic theory of diffusion applies also to membranes with asymmetric leaflets and embedded proteins, and to a complex plasma-membrane mimetic.

Molecular dynamics simulations of carbon nanotube porins in lipid bilayers.

Artificial channels made of carbon nanotube (CNT) porins are promising candidates for applications in filtration and molecular delivery devices. Their symmetric shape and high mechanical, chemical, and thermal stability ensure well-defined transport properties, and at the same time make them ideal model systems for more complicated membrane protein pores. As the technology to produce and tune CNTs advances, simulations can aid in the design of customized membrane porins. Here we concentrate on CNTs embedded in lipid membranes. To derive design guidelines, we systematically studied the interaction of CNT porins with their surrounding lipids. For our simulations, we developed an AMBER- and Lipid14-compatible parameterization scheme for CNTs with different chirality and with functional groups attached to their rim, and a flexible coarse-grained description for open-ended CNTs fitting to the MARTINI lipid model. We found that the interaction with lipid acyl chains is independent of the CNT chirality and the chemical details of functional groups at the CNT rims. The latter, however, are important for the interactions with lipid head groups, and for water permeability. The orientation and permeability of the pore are mainly determined by how well its hydrophobicity pattern matches the membrane. By identifying the factors that determine the structure both of isolated CNTs in lipid membranes and of CNT clusters, we set the foundation for a targeted design of CNT-membrane systems.

Rotational Diffusion Depends on Box Size in Molecular Dynamics Simulations.

We show that the rotational dynamics of proteins and nucleic acids determined from molecular dynamics simulations under periodic boundary conditions suffer from significant finite-size effects. We remove the box-size dependence of the rotational diffusion coefficients by adding a hydrodynamic correction kB T/6 ηV with kB Boltzmann's constant, T the absolute temperature, η the solvent shear viscosity, and V the box volume. We show that this correction accounts for the finite-size dependence of the rotational diffusion coefficients of horse-heart myoglobin and a B-DNA dodecamer in aqueous solution. The resulting hydrodynamic radii are in excellent agreement with experiment.

Fully Anisotropic Rotational Diffusion Tensor from Molecular Dynamics Simulations.

We present a method to calculate the fully anisotropic rotational diffusion tensor from molecular dynamics simulations. Our approach is based on fitting the time-dependent covariance matrix of the quaternions that describe the rigid-body rotational dynamics. Explicit analytical expressions have been derived for the covariances by Favro, which are valid irrespective of the degree of anisotropy. We use these expressions to determine an optimal rotational diffusion tensor from trajectory data. The molecular structures are aligned against a reference by optimal rigid-body superposition. The quaternion covariances can then be obtained directly from the rotation matrices used in the alignment. The rotational diffusion tensor is determined by a fit to the time-dependent quaternion covariances, or directly by Laplace transformation and matrix diagonalization. To quantify uncertainties in the fit, we derive analytical expressions and compare them with the results of Brownian dynamics simulations of anisotropic rotational diffusion. We apply the method to microsecond long trajectories of the Dickerson-Drew B-DNA dodecamer and of horse heart myoglobin. The anisotropic rotational diffusion tensors calculated from simulations agree well with predictions from hydrodynamics.

Carbon Nanotubes Mediate Fusion of Lipid Vesicles.

The fusion of lipid membranes is opposed by high energetic barriers. In living organisms, complex protein machineries carry out this biologically essential process. Here we show that membrane-spanning carbon nanotubes (CNTs) can trigger spontaneous fusion of small lipid vesicles. In coarse-grained molecular dynamics simulations, we find that a CNT bridging between two vesicles locally perturbs their lipid structure. Their outer leaflets merge as the CNT pulls lipids out of the membranes, creating an hourglass-shaped fusion intermediate with still intact inner leaflets. As the CNT moves away from the symmetry axis connecting the vesicle centers, the inner leaflets merge, forming a pore that completes fusion. The distinct mechanism of CNT-mediated membrane fusion may be transferable, providing guidance in the development of fusion agents, e.g., for the targeted delivery of drugs or nucleic acids.

Direct observation of ultrafast collective motions in CO myoglobin upon ligand dissociation.

The hemoprotein myoglobin is a model system for the study of protein dynamics. We used time-resolved serial femtosecond crystallography at an x-ray free-electron laser to resolve the ultrafast structural changes in the carbonmonoxy myoglobin complex upon photolysis of the Fe-CO bond. Structural changes appear throughout the protein within 500 femtoseconds, with the C, F, and H helices moving away from the heme cofactor and the E and A helices moving toward it. These collective movements are predicted by hybrid quantum mechanics/molecular mechanics simulations. Together with the observed oscillations of residues contacting the heme, our calculations support the prediction that an immediate collective response of the protein occurs upon ligand dissociation, as a result of heme vibrational modes coupling to global modes of the protein.