arXiv subjects

Helmut Grubmüller

Publications and source records attributed to Helmut Grubmüller.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Intermediate Hamiltonians for Non-Equilibrium Free Energy Calculations: A Numerical Study of Markov Models

The Jarzynski relation enables the estimation of equilibrium free energy differences from non-equilibrium, finite-time switching simulations. These estimates usually converge poorly because rare trajectories dominate the exponential work average. Here, we numerically determined and explored the sequence of intermediate Hamiltonians connecting initial and final states that minimize the mean squared error (MSE) of the Jarzynski estimator and thereby enhance convergence. For discrete-time Markov models, an exact tilted-master-equation representation of the MSE in the large-sample limit, combined with automatic differentiation, enables efficient gradient-based minimization over all intermediate energies. We applied our approach to three model systems of increasing complexity: a two-state model, a double-well potential, and a shifted potential well. In all three systems, the optimal intermediate Hamiltonians jump at the initial and final times. Extensive Monte Carlo simulations show that optimal intermediates can reduce the MSE by more than an order of magnitude compared with linear and logarithmic interpolation, most strongly for large changes in the energy landscape. Remarkably, they need not dissipate less work than intermediates yielding larger errors. Our results suggest heuristics for more efficient non-equilibrium free energy calculations of realistic molecular systems: optimal intermediate Hamiltonians jump at the initial and final times; for barrier-crossing problems, the barrier should be lowered rapidly and raised again later; and minimizing dissipation does not guarantee faster convergence.

cond-mat.stat-mech↗

arXiv2025

Bayesian MINFLUX localization microscopy

MINFLUX microscopy allows for localization of fluorophores with nanometer precision using targeted scanning with an illumination profile with a minimum. However, current scanning patterns and the overall procedure are based on heuristics, and may therefore be suboptimal. Here we present a rigorous Bayesian that offers maximal resolutions from either minimal detected photons or minimal exposures. We estimate using simulated localization runs that this approach should reduce the number of photons required for 1 nm resolution by a factor of about four.

physics.comp-ph↗

arXiv2025

Structure determination from single-molecule X-ray scattering images using stochastic gradient ascent

Scattering experiments using ultrashort X-ray free electron laser (XFEL) pulses have opened a new path for structure determination of a wide variety of specimens, including nano-crystals and entire viruses, approaching atomistic spatial and femtoseconds time resolution. However, random and unknown sample orientations as well as low signal to noise ratios have so far prevented a successful application to smaller specimens like single biomolecules. We here present resolution-annealed stochastic gradient ascent (RASTA), a new approach for direct atomistic electron density determination, which utilizes our recently developed rigorous Bayesian treatment of single-particle X-ray scattering. We demonstrate electron density determination at 2\r{A} resolution of various small proteins from synthetic scattering images with as low as 15 photons per image.

physics.comp-ph↗

arXiv2025

The need to implement FAIR principles in biomolecular simulations

This letter illustrates the opinion of the molecular dynamics (MD) community on the need to adopt a new FAIR paradigm for the use of molecular simulations. It highlights the necessity of a collaborative effort to create, establish, and sustain a database that allows findability, accessibility, interoperability, and reusability of molecular dynamics simulation data. Such a development would democratize the field and significantly improve the impact of MD simulations on life science research. This will transform our working paradigm, pushing the field to a new frontier. We invite you to support our initiative at the MDDB community (https://mddbr.eu/community/) Now published as: Amaro, R.E., et al. The need to implement FAIR principles in biomolecular simulations. Nat Methods (2025) https://doi.org/10.1038/s41592-025-02635-0

q-bio.BM↗

arXiv2024

Constant pH Simulation with FMM Electrostatics in GROMACS. (A) Design and Applications

The structural dynamics of biological macromolecules, such as proteins, DNA/RNA, or complexes thereof, are strongly influenced by protonation changes of their typically many titratable groups, which explains their sensitivity to pH changes. Conversely, conformational and environmental changes of the biomolecule affect the protonation state of these groups. With few exceptions, conventional force field-based molecular dynamics (MD) simulations do not account for these effects, nor do they allow for coupling to a pH buffer. Here we present a GROMACS implementation of a rigorous Hamiltonian interpolation $λ$-dynamics constant pH method, which rests on GPU-accelerated Fast Multipole Method (FMM) electrostatics. Our implementation supports both CHARMM36m and Amber99sb*-ILDN force fields and is largely automated to enable seamless switching from regular MD to constant pH MD, involving minimal changes to the input files. Here, the first of two companion papers describes the underlying constant pH protocol and sample applications to several prototypical benchmark systems such as cardiotoxin V, lysozyme, and staphylococcal nuclease. Enhanced convergence is achieved through a new dynamic barrier height optimization method, and high p$K_a$ accuracy is demonstrated. We use Functional Mode Analysis and Mutual Information to explore the complex intra- and intermolecular couplings between the protonation states of titratable groups as well as those between protonation states and conformational dynamics. We identify striking conformation-dependent p$K_a$ variations and unexpected inter-residue couplings. Conformation-protonation coupling is identified as a primary cause of the slow protonation convergence notorious to constant pH simulations involving multiple titratable groups, suggesting enhanced sampling methods to accelerate convergence.

cs.DC↗

arXiv2024

Constant pH Simulation with FMM Electrostatics in GROMACS. (B) GPU Accelerated Hamiltonian Interpolation

The structural dynamics of biological macromolecules, such as proteins, DNA/RNA, or their complexes, are strongly influenced by protonation changes of their typically many titratable groups, which explains their pH sensitivity. In turn, conformational and environmental changes in the biomolecule affect the protonation state of these groups. With a few exceptions, conventional force field-based molecular dynamics (MD) simulations do not account for these effects, nor do they allow for coupling to a pH buffer. The $λ$-dynamics method implements this coupling and thus allows for MD simulations at constant pH. It uses separate Hamiltonians for the protonated and deprotonated states of each titratable group, with a $λ$ variable that continuously interpolates between them. However, rigorous implementations of Hamiltonian Interpolation (HI) $λ$-dynamics are prohibitively slow when used with Particle Mesh Ewald (PME). To circumvent this problem, it has been proposed to interpolate the charges instead of the Hamiltonians (QI). Here, we propose a rigorous yet efficient Multipole-Accelerated Hamiltonian Interpolation (MAHI) method to perform $λ$-dynamics in GROMACS. Starting from a charge-scaled Hamiltonian, precomputed with the Fast Multipole Method (FMM) or with PME, the correct HI forces are calculated with negligible computational overhead. We compare HI with QI and show that HI leads to more frequent transitions between protonation states, resulting in better sampling and accuracy. Our performance benchmarks show that introducing, e.g., 512 titratable sites to a one million atom MD system increases runtime by less than 20% compared to a regular FMM-based simulation. We have integrated the scheme into our GPU-FMM code for the simulation software GROMACS, allowing an easy and effortless transition from standard force field simulations to constant pH simulations.

cs.DC↗

arXiv2024

Bayesian electron density determination from sparse and noisy single-molecule X-ray scattering images

Single molecule X-ray scattering experiments using free electron lasers hold the potential to resolve both single structures and structural ensembles of biomolecules. However, molecular electron density determination has so far not been achieved due to low photon counts, high noise levels and low hit rates. Most analysis approaches therefore focus on large specimen like entire viruses, which scatter substantially more photons per image, such that it becomes possible to determine the molecular orientation for each image. In contrast, for small specimen like proteins, the molecular orientation cannot be determined for each image, and must be considered random and unknown. Here we developed and tested a rigorous Bayesian approach to overcome these limitations, and also taking into account intensity fluctuations, beam polarization, irregular detector shapes, incoherent scattering and background scattering. We demonstrate using synthetic scattering images that it is possible to determine electron densities of small proteins in this extreme high noise Poisson regime. Tests on published experimental data from the coliphage PR772 achieved the detector-limited resolution of $9\,\mathrm{nm}$, using only $0.01\,\%$ of the available photons per image.

physics.comp-ph↗

arXiv2024

Stochastic Algorithms for Large-Scale Composite Optimization: the Case of Single-Shot X-FEL Imaging

We apply a recently developed framework for analyzing the convergence of stochastic algorithms to the general problem of large-scale nonconvex composite optimization more generally, and nonconvex likelihood maximization in particular. Our theory is demonstrated on a stochastic gradient descent algorithm for determining the electron density of a molecule from random samples of its scattering amplitude. Numerical results on an idealized synthetic example provide a proof of concept. This opens the door to a broad range of algorithmic possibilities and provides a basis for evaluating and comparing different strategies. While this case study is very specific, it shares a structure that transfers easily to many problems of current interest, particularly in machine learning.

math.OC↗

arXiv2023

Why solvent response contributions to solvation free energies are compatible with Ben-Naim's theorem

We resolve a seeming paradox arising from a common misinterpretation of Ben-Naim's theorem, which rests on a pairwise decomposition of the Hamiltonian of a molecular solute/solvent system into pairwise solute-solvent and solvent-solvent interactions. According to this theorem, also the solvation entropy can be decomposed into a solute-solute term and a remaining term that is -- perhaps misleadingly -- referred to as "solvent reorganization entropy". Crucially, the latter equals the average solvent-solvent interaction energy, such that these two solvent-solvent terms do not change the total solvation free energy. This analytical result has often been used to argue that the reorganization of the solvent cannot play a role in the solvation process, and thus to rule out "iceberg"-type solvent shell ordering as a solvation driving force. However, recent calculations based on atomistic simulations of a solvated globular protein and spatially resolved mutual information expansions revealed substantial contributions of many-body solvent correlations to the solvation free energy. Here we resolved this seeming contradiction and illustrate by two examples -- a simple Ising model and a solvated Lennard-Jones particle -- that the solvent reorganization entropy and the actual entropy contribution arising from many-body solvent correlations differ both conceptually and numerically. Whereas the solvent reorganization entropy in fact arises from both solvent-solvent as well as solute-solvent interactions and thus fails to describe what the name suggests, the mutual information expansion permits a straightforward interpretation in terms of the entropy contribution of solvent-solvent correlations to the solvation free energy.

physics.comp-ph↗

arXiv2023

De novo structural ensemble determination from single-molecule X-ray scattering: A Bayesian approach

Single molecule X-ray scattering experiments with free electron lasers have opened a new route to the structure determination of biomolecules. Because typically only very few photons per scattering image are recorded and thus the signal-to-noise ratio is very low in this extreme Poisson regime, structure refinement is quite challenging. In addition, in each scattering event the orientation of the biomolecule is random and unknown. As a further layer of complexity, many biomolecules show structural heterogeneity and conformational transitions between different distinct structures; these structural dynamics are averaged out by existing refinement methods. To overcome these limitations, here we developed and tested a rigorous Bayesian approach and demonstrate that it should be possible to determine not only a single structure, but an entire structural ensemble from these experiments. Using $10^6$ synthetic scattering images generated from molecular dynamics trajectories, our approach was able to resolve ensembles of eight alanine dipeptide conformers at $2\,$Å resolution; similarly, we determined the unfolded ensemble of the protein chignolin at $4-7\,$Å resolution using $1.2\cdot 10^7$ images. Unexpectedly, much fewer images are required to determine an ensemble of $n$ structures of $m$ atoms each than a single structure of $n\times m$ atoms, i.e., of the same total number of degrees of freedom. These findings show that X-ray scattering experiments using state-of-the-art free electron lasers should allow one to determine not only biomolecular structures, but whole structure ensembles and, ultimately, 'molecular movies'.

physics.comp-ph↗

arXiv2022

GROMACS in the cloud: A global supercomputer to speed up alchemical drug design

We assess costs and efficiency of state-of-the-art high performance cloud computing compared to a traditional on-premises compute cluster. Our use case are atomistic simulations carried out with the GROMACS molecular dynamics (MD) toolkit with a focus on alchemical protein-ligand binding free energy calculations. We set up a compute cluster in the Amazon Web Services (AWS) cloud that incorporates various different instances with Intel, AMD, and ARM CPUs, some with GPU acceleration. Using representative biomolecular simulation systems we benchmark how GROMACS performs on individual instances and across multiple instances. Thereby we assess which instances deliver the highest performance and which are the most cost-efficient ones for our use case. We find that, in terms of total costs including hardware, personnel, room, energy and cooling, producing MD trajectories in the cloud can be as cost-efficient as an on-premises cluster given that optimal cloud instances are chosen. Further, we find that high-throughput ligand-screening for protein-ligand binding affinity estimation can be accelerated dramatically by using global cloud resources. For a ligand screening study consisting of 19,872 independent simulations, we used all hardware that was available in the cloud at the time of the study. The computations scaled-up to reach peak performances using more than 4,000 instances, 140,000 cores, and 3,000 GPUs simultaneously around the globe. Our simulation ensemble finished in about two days in the cloud, while weeks would be required to complete the task on a typical on-premises cluster consisting of several hundred nodes. We demonstrate that the costs of such and similar studies can be drastically reduced with a checkpoint-restart protocol that allows to use cheap Spot pricing and by using instance types with optimal cost-efficiency.

cs.DC↗

arXiv2021

Small Sample Limit of the Bennett Acceptance Ratio Method and the Variationally Derived Intermediates

Free energy calculations based on atomistic Hamiltonians provide microscopic insight into the thermodynamic driving forces of biophysical or condensed matter systems. Many approaches use intermediate Hamiltonians interpolating between the two states for which the free energy difference is calculated. The Bennett Acceptance Ratio (BAR) and Variationally Derived Intermediates (VI) methods are optimal estimator and intermediate states in that the mean-squared error of free energy calculations based on independent sampling is minimized. However, BAR and VI have been derived based on several approximations that do not hold for very few sample points. Analyzing one-dimensional test systems we show that in such cases BAR and VI are suboptimal and that established uncertainty estimates are inaccurate. Whereas for VI to become optimal less than seven samples per state suffice in all cases, for BAR the required number increases unboundedly with decreasing configuration space densities overlap of the end states. We show that for BAR the required number of samples is related to the overlap through an inverse power law. Because this relation seems to hold universally and almost independent of other system properties, these findings can guide the proper choice of estimator for free energy calculations.

physics.comp-ph↗

arXiv2020

Atomistic simulations of the human proteasome inhibited by a covalent ligand

The proteasome is a large biomolecular complex responsible for protein degradation. It is under intense research due to its fundamental role in cellular homeostasis, and tremendous potential for medicinal applications. Recent data from X-ray crystallography and cryo-electron microscopy have suggested that there is a large-scale structural change upon binding of an inhibitor. We carried out atomistic molecular dynamics simulations of the native and inhibited proteasomes to understand the molecular details of the inhibition. Here we describe the technical details of the simulations and assess the quality of the trajectories obtained. The biochemical aspects of the proteasome are under further investigation and will be published elsewhere. This work was a part of the GCS-Prot project at the HLRS, run on the Cray XC40 supercomputing system.

physics.bio-ph↗

arXiv2020

GROMACS Implementation of Free Energy Calculations with Non-Pairwise Variationally Derived Intermediates

Gradients in free energies are the driving forces of physical and biochemical systems. To predict free energy differences with high accuracy, Molecular Dynamics (MD) and other methods based on atomistic Hamiltonians conduct sampling simulations in intermediate thermodynamic states that bridge the configuration space densities between two states of interest ('alchemical transformations'). For uncorrelated sampling, the recent Variationally derived Intermediates (VI) method yields optimal accuracy. The form of the VI intermediates differs fundamentally from conventional ones in that they are non-pairwise, i.e., the total force on a particle in an intermediate states cannot be split into additive contributions from the surrounding particles. In this work, we describe the implementation of VI into the widely used GROMACS MD software package (2020, version 1). Furthermore, a variant of VI is developed that avoids numerical instabilities for vanishing particles. The implementation allows the use of previous non-pairwise potential forms in the literature, which have so far not been available in GROMACS. Example cases on the calculation of solvation free energies, and accuracy assessments thereof, are provided.

physics.comp-ph↗

arXiv2020

Variationally Derived Intermediates for Correlated Free Energy Estimates between Intermediate States

Free energy difference calculations based on atomistic simulations generally improve in accuracy when sampling from a sequence of intermediate equilibrium thermodynamic states that bridge the configuration space between two states of interest. For reasons of efficiency, usually the same samples are used to calculate the step-wise difference of such an intermediate to both adjacent intermediates. However, this procedure violates the assumption of uncorrelated estimates that is necessary to derive both the optimal sequence of intermediate states and the widely used Bennett acceptance ratio (BAR) estimator. In this work, via a variational approach, we derive the sequence of intermediate states and the corresponding estimator with minimal mean squared error that account for these correlations and assess its accuracy.

physics.comp-ph↗

arXiv2019

Computing spatially resolved rotational hydration entropies from atomistic simulations

For a first principles understanding of macromolecular processes, a quantitative understanding of the underlying free energy landscape and in particular its entropy contribution is crucial. The stability of biomolecules, such as proteins, is governed by the hydrophobic effect, which arises from competing enthalpic and entropic contributions to the free energy of the solvent shell. While the statistical mechanics of liquids, as well as molecular dynamics simulations have provided much insight, solvation shell entropies remain notoriously difficult to calculate, especially when spatial resolution is required. Here, we present a method that allows for the computation of spatially resolved rotational solvent entropies via a non-parametric k-nearest-neighbor density estimator. We validated our method using analytic test distributions and applied it to atomistic simulations of a water box. With an accuracy of better than 9.6%, the obtained spatial resolution should shed new light on the hydrophobic effect and the thermodynamics of solvation in general.

physics.bio-ph↗

arXiv2019

Determining Free Energy Differences Through Variational Morphing

Free energy calculations based on atomistic Hamiltonians and sampling are key to a first principles understanding of biomolecular processes, material properties, and macromolecular chemistry. Here, we generalize the Free Energy Perturbation method and derive non-linear Hamiltonian transformation sequences for optimal sampling accuracy that differ markedly from established linear transformations. We show that our sequences are also optimal for the Bennett Acceptance Ratio (BAR) method, and our unifying framework generalizes BAR to small sampling sizes and non-Gaussian error distributions. Simulations on a Lennard-Jones gas show that an order of magnitude less sampling is required compared to established methods.

physics.comp-ph↗

arXiv2019

More Bang for Your Buck: Improved use of GPU Nodes for GROMACS 2018

We identify hardware that is optimal to produce molecular dynamics trajectories on Linux compute clusters with the GROMACS 2018 simulation package. Therefore, we benchmark the GROMACS performance on a diverse set of compute nodes and relate it to the costs of the nodes, which may include their lifetime costs for energy and cooling. In agreement with our earlier investigation using GROMACS 4.6 on hardware of 2014, the performance to price ratio of consumer GPU nodes is considerably higher than that of CPU nodes. However, with GROMACS 2018, the optimal CPU to GPU processing power balance has shifted even more towards the GPU. Hence, nodes optimized for GROMACS 2018 and later versions enable a significantly higher performance to price ratio than nodes optimized for older GROMACS versions. Moreover, the shift towards GPU processing allows to cheaply upgrade old nodes with recent GPUs, yielding essentially the same performance as comparable brand-new hardware.

cs.DC↗