Publications

Fast Ewald summation is the most widely used approach for computing long-range Coulomb interactions in molecular dynamics (MD) simulations. While the asymptotic scaling is nearly optimal, its performance on parallel architectures is dominated by the global communication required for the underlying fast Fourier transform (FFT). Here, we develop a novel method, ESP - Ewald summation with prolate spheroidal wave functions (PSWFs) - that, for a fixed precision, sharply reduces the size of this transform by performing the Ewald split via a PSWF. In addition, PSWFs minimize the cost of spreading and interpolation steps that move information between the particles and the underlying uniform grid. We have integrated the ESP method into two widely-used open-source MD packages: LAMMPS and GROMACS. Detailed benchmarks show that this reduces the cost of computing far-field electrostatic interactions by an order of magnitude, leading to better strong scaling with respect to number of cores. The total execution time is reduced by a factor of 2 to 3 when using more than one thousand cores, even after optimally tuning the existing internal parameters in the native codes. We validate the accelerated codes in realistic long-time biological simulations.

Designing RNA molecules that interact with specific proteins is a critical challenge in experimental and computational biology. Existing computational approaches require a substantial amount of experimentally determined RNA sequences for each specific protein or a detailed knowledge of RNA structure, restricting their utility in practice. To address this limitation, we develop RNA-BAnG, a deep learning-based model designed to generate RNA sequences for protein interactions without these requirements. Central to our approach is a novel generative method, Bidirectional Anchored Generation (BAnG), which leverages the observation that protein-binding RNA sequences often contain functional binding motifs embedded within broader sequence contexts. We first validate our method on generic synthetic tasks involving similar localized motifs to those appearing in RNAs, demonstrating its benefits over existing generative approaches. We then evaluate our model on biological sequences, showing its effectiveness for conditional RNA sequence design given a binding protein.

We introduce in-context denoising, a task that refines the connection between attention-based architectures and dense associative memory (DAM) networks, also known as modern Hopfield networks. Using a Bayesian framework, we show theoretically and empirically that certain restricted denoising problems can be solved optimally even by a single-layer transformer. We demonstrate that a trained attention layer processes each denoising prompt by performing a single gradient descent update on a context-aware DAM energy landscape, where context tokens serve as associative memories and the query token acts as an initial state. This one-step update yields better solutions than exact retrieval of either a context token or a spurious local minimum, providing a concrete example of DAM networks extending beyond the standard retrieval paradigm. Overall, this work solidifies the link between associative memory and attention mechanisms first identified by Ramsauer et al., and demonstrates the relevance of associative memory models in the study of in-context learning.

Large language models have been successful at tasks involving basic forms of in-context reasoning, such as generating coherent language, as well as storing vast amounts of knowledge. At the core of the Transformer architecture behind such models are feed-forward and attention layers, which are often associated to knowledge and reasoning, respectively. In this paper, we study this distinction empirically and theoretically in a controlled synthetic setting where certain next-token predictions involve both distributional and in-context information. We find that feed-forward layers tend to learn simple distributional associations such as bigrams, while attention layers focus on in-context reasoning. Our theoretical analysis identifies the noise in the gradients as a key factor behind this discrepancy. Finally, we illustrate how similar disparities emerge in pre-trained models through ablations on the Pythia model family on simple reasoning tasks.

A direct solver is introduced for solving overdetermined linear systems involving nonuniform discrete Fourier transform matrices. Such matrices can be transformed into a Cauchy-like form that has hierarchical low rank structure. The rank structure of this matrix is explained, and it is shown that the ranks of the relevant submatrices grow only logarithmically with the number of columns of the matrix. A fast rank-structured hierarchical approximation method based on this analysis is developed, along with a hierarchical least-squares solver for these and related systems. This result is a direct method for inverting nonuniform discrete transforms with a complexity that is usually nearly linear with respect to the degrees of freedom in the problem. This solver is benchmarked against various iterative and direct solvers in the setting of inverting the one-dimensional type-II (or forward) transform, for a range of condition numbers and problem sizes (up to (4 10

Amphipathic helices (AHs) are secondary structures that can facilitate binding of proteins to the membrane by folding into a helix with hydrophobic and hydrophilic faces that interact with the same surfaces in the lipid membrane. Septins are cytoskeletal proteins that preferentially bind to domains of micron-scale curvature on the cell membrane. Studies have shown that AH domains in septin are essential for curvature sensing. We present the first computational study of septin AH interactions with lipid bilayers. Using all-atom simulations and metadynamics-enhanced sampling, we study the effect of charge distribution at the flanking ends of septin AH on the energy for helical folding and its consequences on the binding configuration and affinity to the membrane. This is relevant to septins, since the net positive charge on the flanking C-terminal amino acids is a conserved property across several organisms. Simulations revealed that the energy barrier for folding in the neutral-capped AH is much larger than the charge-capped AH, leading to a small fraction of AH folding and integration to the membrane compared to a significantly folded configuration in the bound charge-capped AH. These observations are consistent with the binding measurements of synthetic AH constructs with variable helicity to lipid vesicles. Additionally, we examined an extended AH sequence including eight amino acids upstream and downstream of the AH to mimic the native protein. Again, simulations and experiments show that the extended peptide, with a net positive charge at C-terminus, adopts a strong helical configuration in solution, giving rise to a higher membrane affinity. Altogether, these results identify the energy cost for folding of AHs as a regulator of AH binding configuration and affinity and provide a basic template for parameterizing AH-membrane interactions as a starting point for the future multiscale simulations for septin-membrane interactions.

Despite the parallels between problems in computer vision and cryo-electron microscopy (cryo-EM), many state-of-the-art approaches from computer vision have yet to be adapted for cryo-EM. Within the computer-vision research community, implicits such as neural radiance fields (NeRFs) have enabled the detailed reconstruction of 3D objects from few images at different camera-viewing angles. While other neural implicits, specifically density fields, have been used to map conformational heterogeneity from noisy cryo-EM projection images, most approaches represent volume with an implicit function in Fourier space, which has disadvantages compared with solving the problem in real space, complicating, for instance, masking, constraining physics or geometry, and assessing local resolution. In this work, we build on a recent development in neural implicits, a multi-resolution hash-encoding framework called instant-NGP, that we use to represent the scalar volume directly in real space and apply it to the cryo-EM density-map reconstruction problem (InstaMap). We demonstrate that for both synthetic and real data, InstaMap for homogeneous reconstruction achieves higher resolution at shorter training stages than five other real-spaced representations. We propose a solution to noise overfitting, demonstrate that InstaMap is both lightweight and fast to train, implement masking from a user-provided input mask and extend it to molecular-shape heterogeneity via bending space using a per-image vector field.

Single-molecule experiments are a unique tool to characterize the structural dynamics of biomolecules. However, reconstructing molecular details from noisy single-molecule data is challenging. Simulation-based inference (SBI) integrates statistical inference, physics-based simulators, and machine learning and is emerging as a powerful framework for analysing complex experimental data. Recent advances in deep learning have accelerated the development of new SBI methods, enabling the application of Bayesian inference to an ever-increasing number of scientific problems. Here, we review the nascent application of SBI to the analysis of single-molecule experiments. We introduce parametric Bayesian inference and discuss its limitations. We then overview emerging deep-learning-based SBI methods to perform Bayesian inference for complex models encoded in computer simulators. We illustrate the first applications of SBI to single-molecule force-spectroscopy and cryo-electron microscopy experiments. SBI allows us to leverage powerful computer algorithms modeling complex biomolecular phenomena to connect scientific models and experiments in a principled way.

Simulating membrane proteins accurately combines two challenges into one: properly capturing the structure and dynamics of proteins as well as correctly representing the membrane environment in which they are usually embedded. Beginning with pioneering efforts in the 1980s and 1990s,1−7 both challenges have been met with increasing success over the years. Simulations of membrane proteins in realistic cellular contexts over many microseconds are now common.Concomitant advances in the determination of membrane protein structures, with over 50 unique structures determined 8 annually have further expanded the reach of simulations in this area. This Special Issue highlights a number of recent molecular dynamics (MD) simulations of membrane proteins and covers a wide range of applications and specialized techniques.

Hamiltonian Monte Carlo (HMC) is the mainstay of applied Bayesian inference for differentiable models. However, HMC still struggles to sample from hierarchical models that induce densities with multiscale geometry: a large step size is needed to efficiently explore low curvature regions while a small step size is needed to accurately explore high curvature regions. We introduce the delayed rejection generalized HMC (DR-G-HMC) sampler that overcomes this challenge by employing dynamic step size selection, inspired by differential equation solvers. In generalized HMC, each iteration does a single leapfrog step. DR-G-HMC sequentially makes proposals with geometrically decreasing step sizes upon rejection of earlier proposals. This simulates Hamiltonian dynamics that can adjust its step size along a (stochastic) Hamiltonian trajectory to deal with regions of high curvature. DR-G-HMC makes generalized HMC competitive by decreasing the number of rejections which otherwise cause inefficient backtracking and prevents directed movement. We present experiments to demonstrate that DR-G-HMC (1) correctly samples from multiscale densities, (2) makes generalized HMC methods competitive with the state of the art No-U-Turn sampler, and (3) is robust to tuning parameters.