Spatial proteomics technologies have transformed our understanding of complex tissue architectures by enabling simultaneous analysis of multiple molecular markers and their spatial organization. The high dimensionality of these data, varying marker combinations across experiments and heterogeneous study designs pose unique challenges for computational analysis. Here, we present Virtual Tissues (VirTues), a foundation model framework for biological tissues that operates across the molecular, cellular and tissue scale. VirTues introduces innovations in transformer architecture design, including a novel tokenization scheme that captures both spatial and marker dimensions, and attention mechanisms that scale to high-dimensional multiplex data while maintaining interpretability. Trained on diverse cancer and non-cancer tissue datasets, VirTues demonstrates strong generalization capabilities without task-specific fine-tuning, enabling cross-study analysis and novel marker integration. As a generalist model, VirTues outperforms existing approaches across clinical diagnostics, biological discovery and patient case retrieval tasks, while providing insights into tissue function and disease mechanisms.

Equivariant neural networks have considerably improved the accuracy and data-efficiency of predictions of molecular properties. Building on this success, we introduce EquiReact, an equivariant neural network to infer properties of chemical reactions, built from three-dimensional structures of reactants and products. We illustrate its competitive performance on the prediction of activation barriers on the GDB7-22-TS, Cyclo-23-TS and Proparg-21-TS datasets with different regimes according to the inclusion of atom-mapping information. We show that, compared to state-of-the-art models for reaction property prediction, EquiReact offers: (i) a flexible model with reduced sensitivity between atom-mapping regimes, (ii) better extrapolation capabilities to unseen chemistries, (iii) impressive prediction errors for datasets exhibiting subtle variations in three-dimensional geometries of reactants/products, (iv) reduced sensitivity to geometry quality and (iv) excellent data efficiency.

Sampling all possible transition paths between two 3D states of a molecular system has various applications ranging from catalyst design to drug discovery. Current approaches to sample transition paths use Markov chain Monte Carlo and rely on time-intensive molecular dynamics simulations to find new paths. Our approach operates in the latent space of a normalizing flow that maps from the molecule's Boltzmann distribution to a Gaussian, where we propose new paths without requiring molecular simulations. Using alanine dipeptide, we explore Metropolis-Hastings acceptance criteria in the latent space for exact sampling and investigate different latent proposal mechanisms.

Diffusion Schr\"odinger bridges (DSB) have recently emerged as a powerful framework for recovering stochastic dynamics via their marginal observations at different time points. Despite numerous successful applications, existing algorithms for solving DSBs have so far failed to utilize the structure of aligned data, which naturally arises in many biological phenomena. In this paper, we propose a novel algorithmic framework that, for the first time, solves DSBs while respecting the data alignment. Our approach hinges on a combination of two decades-old ideas: The classical Schr\"odinger bridge theory and Doob's $h$-transform. Compared to prior methods, our approach leads to a simpler training procedure with lower variance, which we further augment with principled regularization schemes. This ultimately leads to sizeable improvements across experiments on synthetic and real data, including the tasks of rigid protein docking and temporal evolution of cellular differentiation processes.

Schr\"odinger bridges (SBs) provide an elegant framework for modeling the temporal evolution of populations in physical, chemical, or biological systems. Such natural processes are commonly subject to changes in population size over time due to the emergence of new species or birth and death events. However, existing neural parameterizations of SBs such as diffusion Schr\"odinger bridges (DSBs) are restricted to settings in which the endpoints of the stochastic process are both probability measures and assume conservation of mass constraints. To address this limitation, we introduce unbalanced DSBs which model the temporal evolution of marginals with arbitrary finite mass. This is achieved by deriving the time reversal of stochastic differential equations with killing and birth terms. We present two novel algorithmic schemes that comprise a scalable objective function for training unbalanced DSBs and provide a theoretical analysis alongside challenging applications on predicting heterogeneous molecular single-cell responses to various cancer drugs and simulating the emergence and spread of new viral variants.

The static optimal transport $(\mathrm{OT})$ problem between Gaussians seeks to recover an optimal map, or more generally a coupling, to morph a Gaussian into another. It has been well studied and applied to a wide variety of tasks. Here we focus on the dynamic formulation of OT, also known as the Schr\"odinger bridge (SB) problem, which has recently seen a surge of interest in machine learning due to its connections with diffusion-based generative models. In contrast to the static setting, much less is known about the dynamic setting, even for Gaussian distributions. In this paper, we provide closed-form expressions for SBs between Gaussian measures. In contrast to the static Gaussian OT problem, which can be simply reduced to studying convex programs, our framework for solving SBs requires significantly more involved tools such as Riemannian geometry and generator theory. Notably, we establish that the solutions of SBs between Gaussian measures are themselves Gaussian processes with explicit mean and covariance kernels, and thus are readily amenable for many downstream applications such as generative modeling or interpolation. To demonstrate the utility, we devise a new method for modeling the evolution of single-cell genomics data and report significantly improved numerical stability compared to existing SB-based approaches.

Optimal transport (OT) theory describes general principles to define and select, among many possible choices, the most efficient way to map a probability measure onto another. That theory has been mostly used to estimate, given a pair of source and target probability measures $(\mu, \nu)$, a parameterized map $T_\theta$ that can efficiently map $\mu$ onto $\nu$. In many applications, such as predicting cell responses to treatments, pairs of input/output data measures $(\mu, \nu)$ that define optimal transport problems do not arise in isolation but are associated with a context $c$, as for instance a treatment when comparing populations of untreated and treated cells. To account for that context in OT estimation, we introduce CondOT, a multi-task approach to estimate a family of OT maps conditioned on a context variable, using several pairs of measures $\left(\mu_i, \nu_i\right)$ tagged with a context label $c_i$. CondOT learns a global map $\mathcal{T}_\theta$ conditioned on context that is not only expected to fit all labeled pairs in the dataset $\left\{\left(c_i,\left(\mu_i, \nu_i\right)\right)\right\}$, i.e., $\mathcal{T}_\theta\left(c_i\right) \sharp \mu_i \approx \nu_i$, but should also generalize to produce meaningful maps $\mathcal{T}_\theta\left(c_{\text {new }}\right)$ when conditioned on unseen contexts $c_{\text {new }}$. Our approach harnesses and provides a novel usage for partially input convex neural networks, for which we introduce a robust and efficient initialization strategy inspired by Gaussian approximations. We demonstrate the ability of CondOT to infer the effect of an arbitrary combination of genetic or therapeutic perturbations on single cells, using only observations of the effects of said perturbations separately.

Learning representations that capture the underlying data generating process is a key problem for data efficient and robust use of neural networks. One key property for robustness which the learned representation should capture and which recently received a lot of attention is described by the notion of invariance. In this work we provide a causal perspective and new algorithm for learning invariant representations. Empirically we show that this algorithm works well on a diverse set of tasks and in particular we observe state-of-the-art performance on domain generalization, where we are able to significantly boost the score of existing models.

Optimal transport tools (OTT-JAX) is a python toolbox that can solve optimal transport problems between point clouds and histograms. The toolbox builds on various JAX features, such as automatic and custom reverse mode differentiation, vectorization, just-in-time compilation and accelerators support. The toolbox covers elementary computations, such as the resolution of the regularized OT problem, and more advanced extensions, such as barycenters, Gromov-Wasserstein, low-rank solvers, estimation of convex maps, differentiable generalizations of quantiles and ranks, and approximate OT between Gaussian mixtures. The toolbox code is available at https://github.com/ott-jax/ott

Protein complex formation is a central problem in biology, being involved in most of the cell's processes, and essential for applications, e.g. drug design or protein engineering. We tackle rigid body protein-protein docking, i.e., computationally predicting the 3D structure of a protein-protein complex from the individual unbound structures, assuming no conformational change within the proteins happens during binding. We design a novel pairwise-independent SE(3)-equivariant graph matching network to predict the rotation and translation to place one of the proteins at the right docked position relative to the second protein. We mathematically guarantee a basic principle: the predicted complex is always identical regardless of the initial locations and orientations of the two structures. Our model, named EquiDock, approximates the binding pockets and predicts the docking poses using keypoint matching and alignment, achieved through optimal transport and a differentiable Kabsch algorithm. Empirically, we achieve significant running time improvements and often outperform existing docking software despite not relying on heavy candidate sampling, structure refinement, or templates.