Publications

We introduce an efficient numerical method for second-order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory phase function that obeys the nonlinear Riccati equation. We propose a defect correction iteration that gives an asymptotic series for such a phase function; this is numerically approximated on a Chebyshev grid with a small number of nodes. For analytic coefficients we prove that each iteration, up to a certain maximum number, reduces the residual by a factor of order of the local frequency. The algorithm adapts both the stepsize and the choice of method, switching to a conventional spectral collocation method away from oscillatory regions. In numerical experiments we find that our proposal outperforms other state-of-the-art oscillatory solvers, most significantly at low to intermediate frequencies and at low tolerances, where it may use up to \(10^6\) times fewer function evaluations. Even in high-frequency regimes, our implementation is on average 10 times faster than other specialized solvers.

Huntington’s disease (HD) is a fatal neurodegenerative disorder resulting from an abnormal expansion of polyglutamine (polyQ) repeats in the N-terminus of the huntingtin protein. When the polyQ tract surpasses 35 repeats, the mutated protein undergoes misfolding, culminating in the formation of intracellular aggregates. Research in mouse models suggests that HD pathogenesis involves the aggregation of N-terminal fragments of the huntingtin protein (htt). These early oligomeric assemblies of htt, exhibiting diverse characteristics during aggregation, are implicated as potential toxic entities in HD. However, a consensus on their specific structures remains elusive. Understanding the heterogeneous nature of htt oligomers provides crucial insights into disease mechanisms, emphasizing the need to identify various oligomeric conformations as potential therapeutic targets. Employing coarse-grained molecular dynamics, our study aims to elucidate the mechanisms governing the aggregation process and resultant aggregate architectures of htt. The polyQ tract within htt is flanked by two regions: an N-terminal domain (N17) and a short C-terminal proline-rich segment. We conducted self-assembly simulations involving five distinct N17 + polyQ systems with polyQ lengths ranging from 7 to 45, utilizing the ProMPT force field. Prolongation of the polyQ domain correlates with an increase in β-sheet-rich structures. Longer polyQ lengths favor intramolecular β-sheets over intermolecular interactions due to the folding of the elongated polyQ domain into hairpin-rich conformations. Importantly, variations in polyQ length significantly influence resulting oligomeric structures. Shorter polyQ domains lead to N17 domain aggregation, forming a hydrophobic core, while longer polyQ lengths introduce a competition between N17 hydrophobic interactions and polyQ polar interactions, resulting in densely packed polyQ cores with outwardly distributed N17 domains. Additionally, at extended polyQ lengths, we observe distinct oligomeric conformations with varying degrees of N17 bundling. These findings can help explain the toxic gain-of-function that htt with expanded polyQ acquires.

Mathematical models are indispensable for disentangling the interactions through which biological components work together to generate the forces and flows that position, mix, and distribute proteins, nutrients, and organelles within the cell. To illuminate the ever more specific questions studied at the edge of biological inquiry, such models inevitably become more complex. Solving, simulating, and learning from these more realistic models requires the development of new analytic techniques, numerical methods, and scalable software. In this review, we discuss some recent developments in tools for understanding how large numbers of cytoskeletal filaments, driven by molecular motors and interacting with the cytoplasm and other structures in their environment, generate fluid flows, instabilities, and material deformations which help drive crucial cellular processes.

Recent developments in parallel Markov chain Monte Carlo (MCMC) algorithms allow us to run thousands of chains almost as quickly as a single chain, using hardware accelerators such as GPUs. While each chain still needs to forget its initial point during a warmup phase, the subsequent sampling phase can be shorter than in classical settings, where we run only a few chains. To determine if the resulting short chains are reliable, we need to assess how close the Markov chains are to their stationary distribution after warmup. The potential scale reduction factor Rˆ is a popular convergence diagnostic but unfortunately can require a long sampling phase to work well. We present a nested design to overcome this challenge and a generalization called nested Rˆ. This new diagnostic works under conditions similar to Rˆ and completes the workflow for GPU-friendly samplers. In addition, the proposed nesting provides theoretical insights into the utility of Rˆ, in both classical and short-chains regimes.

Habituation – a phenomenon in which a dynamical system exhibits a diminishing response to repeated stimulations that eventually recovers when the stimulus is withheld – is universally observed in living systems from animals to unicellular organisms. Despite its prevalence, generic mechanisms for this fundamental form of learning remain poorly defined. Drawing inspiration from prior work on systems that respond adaptively to step inputs, we study habituation from a nonlinear dynamics perspective. This approach enables us to formalize classical hallmarks of habituation that have been experimentally identified in diverse organisms and stimulus scenarios. We use this framework to investigate distinct dynamical circuits capable of habituation. In particular, we show that driven linear dynamics of a memory variable with static nonlinearities acting at the input and output can implement numerous hallmarks in a mathematically interpretable manner. This work establishes a foundation for understanding the dynamical substrates of this primitive learning behavior and offers a blueprint for the identification of habituating circuits in biological systems.

In a probabilistic latent variable model, factorized (or mean-field) variational inference (F-VI) fits a separate parametric distribution for each latent variable. Amortized variational inference (A-VI) instead learns a common inference function, which maps each observation to its corresponding latent variable’s approximate posterior. Typically, A-VI is used as a cog in the training of variational autoencoders, however it stands to reason that A-VI could also be used as a general alternative to F-VI. In this paper we study when and why A-VI can be used for approximate Bayesian inference. We derive conditions on a latent variable model which are necessary, sufficient, and verifiable under which A-VI can attain F-VI’s optimal solution, thereby closing the amortization gap. We prove these conditions are uniquely verified by simple hierarchical models, a broad class that encompasses many models in machine learning. We then show, on a broader class of models, how to expand the domain of AVI’s inference function to improve its solution, and we provide examples, e.g. hidden Markov models, where the amortization gap cannot be closed.

Organisms must perform sensory-motor behaviors to survive. What bounds or constraints limit behavioral performance? Previously, we found that the gradient-climbing speed of a chemotaxing Escherichia coli is near a bound set by the limited information they acquire from their chemical environments (1). Here we ask what limits their sensory accuracy. Past theoretical analyses have shown that the stochasticity of single molecule arrivals sets a fundamental limit on the precision of chemical sensing (2). Although it has been argued that bacteria approach this limit, direct evidence is lacking. Here, using information theory and quantitative experiments, we find that E. coli’s chemosensing is not limited by the physics of particle counting. First, we derive the physical limit on the behaviorally-relevant information that any sensor can get about a changing chemical concentration, assuming that every molecule arriving at the sensor is recorded. Then, we derive and measure how much information E. coli’s signaling pathway encodes during chemotaxis. We find that E. coli encode two orders of magnitude less information than an ideal sensor limited only by shot noise in particle arrivals. These results strongly suggest that constraints other than particle arrival noise limit E. coli’s sensory fidelity.

The recognizable shapes of landforms arise from processes such as erosion by wind or water currents. However, explaining the physical origin of natural structures is challenging due to the coupled evolution of complex flow fields and three-dimensional (3D) topographies. We investigate these issues in a laboratory setting inspired by yardangs, which are raised, elongate formations whose characteristic shape suggests erosion of heterogeneous material by directional flows. We combine experiments and simulations to test an origin hypothesis involving a harder or less erodible inclusion embedded in an outcropping of softer material. Optical scans of clay objects fixed within flowing water reveal a transformation from a featureless mound to a yardang-like form resembling a lion in repose. Phase-field simulations reproduce similar shape dynamics and show their dependence on the erodibility contrast and flow strength. Through visualizations of the flow fields and analysis of the local erosion rate, we identify effects associated with flow funneling and the turbulent wake that are responsible for carving the unique geometrical features. This highly 3D scouring process produces complex shapes from simple and commonplace starting conditions and is thus a candidate explanation for natural yardangs. The methods introduced here should be generally useful for geomorphological problems and especially those for which material heterogeneity is a primary factor.

With dedicated exoplanet surveys underway for multiple extreme-precision radial velocity (EPRV) instruments, the near-future prospects of RV exoplanet science are promising. These surveys' generous time allocations are expected to facilitate the discovery of Earth analogs around bright, nearby Sun-like stars. But survey success will depend critically on the choice of observing strategy, which will determine the survey's ability to mitigate known sources of noise and extract low-amplitude exoplanet signals. Here we present an analysis of the Fisher information content of simulated EPRV surveys, accounting for the most recent advances in our understanding of stellar variability on both short and long timescales (i.e., oscillations and granulation within individual nights, and activity-induced variations across multiple nights). In this analysis, we capture the correlated nature of stellar variability by parameterizing these signals with Gaussian process kernels. We describe the underlying simulation framework and the physical interpretation of the Fisher information content, and we evaluate the efficacy of EPRV survey strategies that have been presented in the literature. We explore and compare strategies for scheduling observations over various timescales, and we make recommendations to optimize survey performance for the detection of Earth-like exoplanets.

Diffuse midline glioma (DMG), including Diffuse intrinsic pontine glioma (DIPG), is an aggressive brain tumor in children with limited treatment options. Recent developments of phase 1 clinical trials have shown early promise for chimeric antigen receptor (CAR) T cells in patients with DMG/DIPG. However, several barriers such as the absence of tumor-specific antigens, restricted trafficking to the tumor site, and poor persistence hinder the full therapeutic potential of CAR T cell therapy in DMG/DIPG.