Publications

In many scientific and engineering problems, noise and nonlinearity are unavoidable, which could induce interesting mathematical problem such as transition phenomena. This paper focuses on efficiently discovering the most probable transition pathway for stochastic dynamical systems employing reinforcement learning. With the Onsager–Machlup action functional theory to quantify rare events in stochastic dynamical systems, finding the most probable pathway is equivalent to solving a variational problem on the action functional. When the action function cannot be explicitly expressed by paths near the reference orbit, the variational problem needs to be converted into an optimal control problem. First, by integrating terminal prediction into the reinforcement learning framework, we develop a Terminal Prediction Deep Deterministic Policy Gradient (TP-DDPG) algorithm to deal with the finite-horizon optimal control issue in a forward way. Next, we present the convergence analysis of our algorithm for the value function in terms of the neural network’s approximation error and estimation error. Finally, we conduct various experiments in different dimensions for the transition problems in applications to illustrate the effectiveness of our algorithm.

The Kidney Precision Medicine Project (KPMP) aims to create a kidney tissue atlas, define disease subgroups, and identify critical cells, pathways, and targets for novel therapies through molecular investigation of human kidney biopsies obtained from participants with AKI or CKD. We present the case of a 66-year-old woman with diabetic kidney disease who underwent a protocol KPMP kidney biopsy. Her clinical history included diabetes mellitus complicated by neuropathy and eye disease, increased insulin resistance, hypertension, albuminuria, and relatively preserved glomerular filtration rate (early CKD stage 3a). The patient's histopathology was consistent with diabetic nephropathy and arterial and arteriolar sclerosis. Three-dimensional, immunofluorescence imaging of the kidney biopsy specimen revealed extensive periglomerular neovascularization that was underestimated by standard histopathologic approaches. Spatial transcriptomics was performed to obtain gene expression signatures at discrete areas of the kidney biopsy. Gene expression in the areas of glomerular neovascularization revealed increased expression of genes involved in angiogenic signaling, proliferation, and survival of endothelial cells, as well as new vessel maturation and stability. This molecular correlation provides additional insights into the development of kidney disease in patients with diabetes and spotlights how novel molecular techniques used by the KPMP can supplement and enrich the histopathologic diagnosis obtained from a kidney biopsy.

Cytokinesis of animal and fungi cells depends crucially on the anillin scaffold proteins. Fission yeast anillin-related Mid1 anchors cytokinetic ring precursor nodes to the membrane. However, it is unclear if both of its Pleckstrin Homology (PH) and C2 C-terminal domains bind to the membrane as monomers or dimers, and if one domain plays a dominant role. We studied Mid1 membrane binding with all-atom molecular dynamics near a membrane with yeast-like lipid composition. In simulations with the full C terminal region started away from the membrane, Mid1 binds through the disordered L3 loop of C2 in a vertical orientation, with the PH away from the membrane. However, a configuration with both C2 and PH initially bound to the membrane remains associated with the membrane. Simulations of C2-PH dimers show extensive asymmetric membrane contacts. These multiple modes of binding may reflect Mid1’s multiple interactions with membranes, node proteins, and ability to sustain mechanical forces.

Transition metal dichalcogenide superlattices provide an exciting new platform for exploring and understanding a variety of phases of matter. The moiré continuum Hamiltonian, of two-dimensional jellium in a modulating potential, provides a fundamental model for such systems. Accurate computations with this model are essential for interpreting experimental observations and making predictions for future explorations. In this work, we combine two complementary quantum Monte Carlo (QMC) methods, phaseless auxiliary field quantum Monte Carlo and fixed-phase diffusion Monte Carlo, to study the ground state of this Hamiltonian. We observe a metal-insulator transition between a paramagnetic and a 120° Néel ordered state as the moiré potential depth and the interaction strength are varied. We find significant differences from existing results by Hartree-Fock and exact diagonalization studies. In addition, we benchmark density-functional theory, and suggest an optimal hybrid functional which best approximates our QMC results.

To generate actions in the face of physiological delays, the brain must predict the future. Here we explore how prediction may lie at the core of brain function by considering a neuron predicting the future of a scalar time series input. Assuming that the dynamics of the lag vector (a vector composed of several consecutive elements of the time series) are locally linear, normal mode decomposition decomposes the dynamics into independently evolving (eigen)modes allowing for straightforward prediction. We propose that a neuron learns the top mode and projects its input onto the associated subspace. Under this interpretation, the temporal filter of a neuron corresponds to the left eigenvector of a generalized eigenvalue problem. We mathematically analyze the operation of such an algorithm on noisy observations of synthetic data generated by a linear system. Interestingly, the shape of the temporal filter varies with the signal-to-noise ratio (SNR): a noisy input yields a monophasic filter and a growing SNR leads to multiphasic filters with progressively greater number of phases. Such variation in the temporal filter with input SNR resembles that observed experimentally in biological neurons.

Life in complex systems, such as cities and organisms, comes to a standstill when global coordination of mass, energy and information flows is disrupted. Global coordination is no less important in single cells, especially in large oocytes and newly formed embryos, which commonly use fast fluid flows for dynamic reorganization of their cytoplasm. These cytoplasmic streaming flows have been proposed to spontaneously arise from hydrodynamic interactions among cortically anchored microtubules loaded with cargo-carrying molecular motors. Here, we combine modelling and simulation with live imaging to investigate such flows in the Drosophila oocyte. Using a fast, accurate and scalable numerical approach to investigate fluid–structure interactions of thousands of flexible fibres, we demonstrate the robust emergence and evolution of cell-spanning vortices—or twisters—in three-dimensional cellular geometries. These twister flows, dominated by a near-rigid-body rotation with secondary toroidal components, reproduce the variety of experimental observations. In cells, these flows are probably involved in rapid mixing and transport of ooplasmic components.

Single same cell RNAseq/ATACseq multiome data provide unparalleled potential to develop high resolution maps of the cell-type specific transcriptional regulatory circuitry underlying gene expression. We present CREMA, a framework that recovers the full cis-regulatory circuitry by modeling gene expression and chromatin activity in individual cells without peak-calling or cell type labeling constraints. We demonstrate that CREMA overcomes the limitations of existing methods that fail to identify about half of functional regulatory elements which are outside the called chromatin ‘peaks’. These circuit sites outside called peaks are shown to be important cell type specific functional regulatory loci, sufficient to distinguish individual cell types. Analysis of mouse pituitary data identifies a Gata2-circuit for the gonadotrope-enriched disease-associated Pcsk1 gene, which is experimentally validated by reduced gonadotrope expression in a gonadotrope conditional Gata2-knockout model. We present a web accessible human immune cell regulatory circuit resource, and provide CREMA as an R package.

We report an accurate and efficient classical simulation of a kicked Ising quantum system on the heavy hexagon lattice. A simulation of this system was recently performed on a 127-qubit quantum processor using noise-mitigation techniques to enhance accuracy [Y. Kim et al., Nature, 618, 500–5 (2023)]. Here we show that, by adopting a tensor network approach that reflects the geometry of the lattice and is approximately contracted using belief propagation, we can perform a classical simulation that is significantly more accurate and precise than the results obtained from the quantum processor and many other classical methods. We quantify the treelike correlations of the wave function in order to explain the accuracy of our belief propagation-based approach. We also show how our method allows us to perform simulations of the system to long times in the thermodynamic limit, corresponding to a quantum computer with an infinite number of qubits. Our tensor network approach has broader applications for simulating the dynamics of quantum systems with treelike correlations.

We derive basic differential geometric formulae for surfaces in hyperbolic space represented as envelopes of horospheres. The dual notion of parallel hypersurfaces is also studied. The representation is applied to prove existence and regularity theorems for Weingarten surfaces in H^3, which satisfy (1-a)K = a(2-H), for an a < 0, and have a specified boundary curve at infinity. These surfaces are shown to be closely connected to conformal mappings of domains in S^2 into the unit disk and provide Riemannian interpretations for some conformal invariants associated to such mappings.
This paper was originally written in 1984, before I learned to use TeX, and was typed by one of the secretaries in the Princeton Math Department. It was more or less, my first original work after my dissertation. For some reason, I was not able to get this paper published in a timely manner. The results and perspective in this paper have proved to be useful to a variety of people, some of whom asked me to render the article into TeX and post it to the arXiv. I had been seriously thinking about doing this, when Martin Bridgemen sent me a transcription of my original article into TeX. I am extremely grateful to him for the effort he has put into this project.
The paper is now formatted in a more or less modern AMS-article style, but for lots of additional punctuation, a few corrections and some minor stylistic changes, the content has been largely reproduced as it originally was. Remarks about the 'state-of-the-art' in hyperbolic geometry are obviously way out of date, as there has been enormous progress in many aspects of this still rich subject.

Fluid-structure interactions between active and passive components are important for many biological systems to function. A particular example is chromatin in the cell nucleus, where ATP-powered processes drive coherent motions of the chromatin fiber over micron lengths. Motivated by this system, we develop a multiscale model of a long flexible polymer immersed in a suspension of active force dipoles as an analog to a chromatin fiber in an active fluid—the nucleoplasm. Linear analysis identifies an orientational instability driven by hydrodynamic and alignment interactions between the fiber and the suspension, and numerical simulations show activity can drive coherent motions and structured conformations. These results demonstrate how active and passive components, connected through fluid-structure interactions, can generate coherent structures and self-organize on large scales.