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Born in 1881 on a farm in Pennsylvania, Alice C. Evans dedicated her life to studying bacteria in dairy products. Early in her career, Alice became convinced that most bacteria display multicellular behavior as part of their life cycles. At the time, the morphological changes observed in bacterial life cycles created confusion among scientists. In 1928, as the first female president of the American Society for Microbiology, Alice wrote to the scientific community: “When one-celled organisms grow in masses, … individual cells influence and protect one another.” She continued, “Bacteriologists need not feel chagrinned … to admit that… forms they have considered as different genera are but stages in the life cycle of one species” (1). Nearly 100 years later, on page 71 of this issue, Qin et al. (2) make a substantial leap forward in deciphering cell dynamics in biofilms—groups of microorganisms that adhere to a surface, and each other, by excreting matrix components. 

The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length- and timescales. Often, it is computationally intractable to resolve the finest features in the solution. The only recourse is to use approximate coarse-grained representations, which aim to accurately represent long-wavelength dynamics while properly accounting for unresolved small-scale physics. Deriving such coarse-grained equations is notoriously difficult and often ad hoc. Here we introduce data-driven discretization, a method for learning optimized approximations to PDEs based on actual solutions to the known underlying equations. Our approach uses neural networks to estimate spatial derivatives, which are optimized end to end to best satisfy the equations on a low-resolution grid. The resulting numerical methods are remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in 1 spatial dimension at resolutions 4× to 8× coarser than is possible with standard finite-difference methods. 

Deep neural networks have achieved state-of-the-art accuracy at classifying molecules with respect to whether they bind to specific protein targets. A key breakthrough would occur if these models could reveal the fragment pharmacophores that are causally involved in binding. Extracting chemical details of binding from the networks could enable scientific discoveries about the mechanisms of drug actions. However, doing so requires shining light into the black box that is the trained neural network model, a task that has proved difficult across many domains. Here we show how the binding mechanism learned by deep neural network models can be interrogated, using a recently described attribution method. We first work with carefully constructed synthetic datasets, in which the molecular features responsible for “binding” are fully known. We find that networks that achieve perfect accuracy on held-out test datasets still learn spurious correlations, and we are able to exploit this nonrobustness to construct adversarial examples that fool the model. This makes these models unreliable for accurately revealing information about the mechanisms of protein–ligand binding. In light of our findings, we prescribe a test that checks whether a hypothesized mechanism can be learned. If the test fails, it indicates that the model must be simplified or regularized and/or that the training dataset requires augmentation.