Improved motif-scaffolding with SE(3) flow matching

Protein design often begins with knowledge of a desired function from a motif which motif-scaffolding aims to construct a functional protein around. Recently, generative models have achieved breakthrough success in designing scaffolds for a diverse range of motifs. However, the generated scaffolds tend to lack structural diversity, which can hinder success in wet-lab validation. In this work, we extend FrameFlow, an SE(3) flow matching model for protein backbone generation, to perform motif-scaffolding with two complementary approaches. The first is motif amortization, in which FrameFlow is trained with the motif as input using a data augmentation strategy. The second is motif guidance, which performs scaffolding using an estimate of the conditional score from FrameFlow, and requires no additional training. Both approaches achieve an equivalent or higher success rate than previous state-of-the-art methods, with 2.5 times more structurally diverse scaffolds. Code: https://github.com/ microsoft/frame-flow.

Navigating protein landscapes with a machine-learned transferable coarse-grained model

The most popular and universally predictive protein simulation models employ all-atom molecular dynamics (MD), but they come at extreme computational cost. The development of a universal, computationally efficient coarse-grained (CG) model with similar prediction performance has been a long-standing challenge. By combining recent deep learning methods with a large and diverse training set of all-atom protein simulations, we here develop a bottom-up CG force field with chemical transferability, which can be used for extrapolative molecular dynamics on new sequences not used during model parametrization. We demonstrate that the model successfully predicts folded structures, intermediates, metastable folded and unfolded basins, and the fluctuations of intrinsically disordered proteins while it is several orders of magnitude faster than an all-atom model. This showcases the feasibility of a universal and computationally efficient machine-learned CG model for proteins.

Navigating the Design Space of Equivariant Diffusion-Based Generative Models for De Novo 3D Molecule Generation

Deep generative diffusion models are a promising avenue for de novo 3D molecular design in material science and drug discovery. However, their utility is still constrained by suboptimal performance with large molecular structures and limited training data. Addressing this gap, we explore the design space of E(3) equivariant diffusion models, focusing on previously blank spots. Our extensive comparative analysis evaluates the interplay between continuous and discrete state spaces. Out of this investigation, we introduce the EQGAT-diff model, which consistently surpasses the performance of established models on the QM9 and GEOM-Drugs datasets by a large margin. Distinctively, EQGAT-diff takes continuous atomic positions while chemical elements and bond types are categorical and employ a time-dependent loss weighting that significantly increases training convergence and the quality of generated samples. To further strengthen the applicability of diffusion models to limited training data, we examine the transferability of EQGAT-diff trained on the large PubChem3D dataset with implicit hydrogens to target distributions with explicit hydrogens. Fine-tuning EQGAT-diff for a couple of iterations further pushes state-of-the-art performance across datasets. We envision that our findings will find applications in structure-based drug design, where the accuracy of generative models for small datasets of complex molecules is critical.

Reaction coordinate flows for model reduction of molecular kinetics

In this work, we introduce a flow based machine learning approach, called reaction coordinate (RC) flow, for discovery of low-dimensional kinetic models of molecular systems. The RC flow utilizes a normalizing flow to design the coordinate transformation and a Brownian dynamics model to approximate the kinetics of RC, where all model parameters can be estimated in a data-driven manner. In contrast to existing model reduction methods for molecular kinetics, RC flow offers a trainable and tractable model of reduced kinetics in continuous time and space due to the invertibility of the normalizing flow. Furthermore, the Brownian dynamics-based reduced kinetic model investigated in this work yields a readily discernible representation of metastable states within the phase space of the molecular system. Numerical experiments demonstrate how effectively the proposed method discovers interpretable and accurate low-dimensional representations of given full-state kinetics from simulations.

Equivariant flow matching

Normalizing flows are a class of deep generative models that are especially interesting for modeling probability distributions in physics, where the exact likelihood of flows allows reweighting to known target energy functions and computing unbiased observables. For instance, Boltzmann generators tackle the long-standing sampling problem in statistical physics by training flows to produce equilibrium samples of many-body systems such as small molecules and proteins. To build effective models for such systems, it is crucial to incorporate the symmetries of the target energy into the model, which can be achieved by equivariant continuous normalizing flows (CNFs). However, CNFs can be computationally expensive to train and generate samples from, which has hampered their scalability and practical application. In this paper, we introduce equivariant flow matching, a new training objective for equivariant CNFs that is based on the recently proposed optimal transport flow matching. Equivariant flow matching exploits the physical symmetries of the target energy for efficient, simulation-free training of equivariant CNFs. We demonstrate the effectiveness of our approach on many-particle systems and a small molecule, alanine dipeptide, where for the first time we obtain a Boltzmann generator with significant sampling efficiency without relying on tailored internal coordinate featurization. Our results show that the equivariant flow matching objective yields flows with shorter integration paths, improved sampling efficiency, and higher scalability compared to existing methods.

Towards Predicting Equilibrium Distributions for Molecular Systems with Deep Learning

Advances in deep learning have greatly improved structure prediction of molecules. However, many macroscopic observations that are important for real-world applications are not functions of a single molecular structure, but rather determined from the equilibrium distribution of structures. Traditional methods for obtaining these distributions, such as molecular dynamics simulation, are computationally expensive and often intractable. In this paper, we introduce a novel deep learning framework, called Distributional Graphormer (DiG), in an attempt to predict the equilibrium distribution of molecular systems. Inspired by the annealing process in thermodynamics, DiG employs deep neural networks to transform a simple distribution towards the equilibrium distribution, conditioned on a descriptor of a molecular system, such as a chemical graph or a protein sequence. This framework enables efficient generation of diverse conformations and provides estimations of state densities. We demonstrate the performance of DiG on several molecular tasks, including protein conformation sampling, ligand structure sampling, catalyst-adsorbate sampling, and property-guided structure generation. DiG presents a significant advancement in methodology for statistically understanding molecular systems, opening up new research opportunities in molecular science.

Statistically Optimal Force Aggregation for Coarse-Graining Molecular Dynamics

Machine-learned coarse-grained (CG) models have the potential for simulating large molecular complexes beyond what is possible with atomistic molecular dynamics. However, training accurate CG models remains a challenge. A widely used methodology for learning CG force-fields maps forces from all-atom molecular dynamics to the CG representation and matches them with a CG force-field on average. We show that there is flexibility in how to map all-atom forces to the CG representation, and that the most commonly used mapping methods are statistically inefficient and potentially even incorrect in the presence of constraints in the all-atom simulation. We define an optimization statement for force mappings and demonstrate that substantially improved CG force-fields can be learned from the same simulation data when using optimized force maps. The method is demonstrated on the miniproteins Chignolin and Tryptophan Cage and published as open-source code.

Timewarp: Transferable Acceleration of Molecular Dynamics by Learning Time-Coarsened Dynamics

Molecular dynamics (MD) simulation is a widely used technique to simulate molecular systems, most commonly at the all-atom resolution where the equations of motion are integrated with timesteps on the order of femtoseconds ($1\textrm{fs}=10^{-15}\textrm{s}$). MD is often used to compute equilibrium properties, which requires sampling from an equilibrium distribution such as the Boltzmann distribution. However, many important processes, such as binding and folding, occur over timescales of milliseconds or beyond, and cannot be efficiently sampled with conventional MD. Furthermore, new MD simulations need to be performed from scratch for each molecular system studied. We present Timewarp, an enhanced sampling method which uses a normalising flow as a proposal distribution in a Markov chain Monte Carlo method targeting the Boltzmann distribution. The flow is trained offline on MD trajectories and learns to make large steps in time, simulating the molecular dynamics of $10^{5} - 10^{6}\:\textrm{fs}$. Crucially, Timewarp is transferable between molecular systems: once trained, we show that it generalises to unseen small peptides (2-4 amino acids), exploring their metastable states and providing wall-clock acceleration when sampling compared to standard MD. Our method constitutes an important step towards developing general, transferable algorithms for accelerating MD.

Two for One: Diffusion Models and Force Fields for Coarse-Grained Molecular Dynamics

Coarse-grained (CG) molecular dynamics enables the study of biological processes at temporal and spatial scales that would be intractable at an atomistic resolution. However, accurately learning a CG force field remains a challenge. In this work, we leverage connections between score-based generative models, force fields and molecular dynamics to learn a CG force field without requiring any force inputs during training. Specifically, we train a diffusion generative model on protein structures from molecular dynamics simulations, and we show that its score function approximates a force field that can directly be used to simulate CG molecular dynamics. While having a vastly simplified training setup compared to previous work, we demonstrate that our approach leads to improved performance across several small- to medium-sized protein simulations, reproducing the CG equilibrium distribution, and preserving dynamics of all-atom simulations such as protein folding events.

Rigid body flows for sampling molecular crystal structures

Normalizing flows (NF) are a class of powerful generative models that have gained popularity in recent years due to their ability to model complex distributions with high flexibility and expressiveness. In this work, we introduce a new type of normalizing flow that is tailored for modeling positions and orientations of multiple objects in three-dimensional space, such as molecules in a crystal. Our approach is based on two key ideas: first, we define smooth and expressive flows on the group of unit quaternions, which allows us to capture the continuous rotational motion of rigid bodies; second, we use the double cover property of unit quaternions to define a proper density on the rotation group. This ensures that our model can be trained using standard likelihood-based methods or variational inference with respect to a thermodynamic target density. We evaluate the method by training Boltzmann generators for two molecular examples, namely the multi-modal density of a tetrahedral system in an external field and the ice XI phase in the TIP4P-Ew water model. Our flows can be combined with flows operating on the internal degrees of freedom of molecules, and constitute an important step towards the modeling of distributions of many interacting molecules.