Publications

When factorized approximations are used for variational inference (VI), they tend to underestimate the uncertainty -- as measured in various ways -- of the distributions they are meant to approximate. We consider two popular ways to measure the uncertainty deficit of VI: (i) the degree to which it underestimates the componentwise variance, and (ii) the degree to which it underestimates the entropy. To better understand these effects, and the relationship between them, we examine an informative setting where they can be explicitly (and elegantly) analyzed: the approximation of a Gaussian,~p, with a dense covariance matrix, by a Gaussian,~q, with a diagonal covariance matrix. We prove that q always underestimates both the componentwise variance and the entropy of p, \textit{though not necessarily to the same degree}. Moreover we demonstrate that the entropy of q is determined by the trade-off of two competing forces: it is decreased by the shrinkage of its componentwise variances (our first measure of uncertainty) but it is increased by the factorized approximation which delinks the nodes in the graphical model of p. We study various manifestations of this trade-off, notably one where, as the dimension of the problem grows, the per-component entropy gap between p and q becomes vanishingly small even though q underestimates every componentwise variance by a constant multiplicative factor. We also use the shrinkage-delinkage trade-off to bound the entropy gap in terms of the problem dimension and the condition number of the correlation matrix of p. Finally we present empirical results on both Gaussian and non-Gaussian targets, the former to validate our analysis and the latter to explore its limitations.

Multicellular organisms contain diverse tissues built from multiple cell types. It remains unclear how large numbers of interacting cells can precisely coordinate their gene expression during tissue self-organization. We develop a generalized model of multicellular gene expression that includes intracellular and intercellular gene interactions in tissue-like collectives. Motivated by modern transcriptomics, we represent multistable cellular phenotypes by mapping the binarized transcriptional patterns of individual cells onto Hopfield networks. We incorporate spatial cell-cell signaling by coupling transcriptional states of adjacent cells on a square lattice. We show that tuning the intercellular signaling strength results in a cascade of transitions toward different collective states with emergent single-cell phenotypes. Despite an enormous number of possible tissue states, we find that intercellular signaling tends to stabilize a small number of compositionally and spatially simple tissue types. These results establish a theoretical framework to investigate how cell collectives self-organize into distinct stable patterns.

We constrain the distribution of merging compact binaries across the celestial sphere using the GWTC-3 catalog from the LIGO-Virgo-KAGRA Collaborations' (LVK) third observing run. With 63 confident detections from O3, we constrain the relative variability (standard deviation) of the rate density across the sky to be ≲16% at 90\% confidence assuming the logarithm of the rate density is described by a Gaussian random field with correlation length ≥10∘. This tightens to ≲3.5% when the correlation length is ≥20∘. While the new O3 data provides the tightest constraints on anisotropies available to-date, we do not find overwhelming evidence in favor of isotropy, either. A simple counting experiment favors an isotropic distribution by a factor of isoani=3.7, which is nonetheless an improvement of more than a factor of two compared to analogous analyses based on only the first and second observing runs of the LVK.

The F-BAR protein Cdc15 is essential for cytokinesis in Schizosaccharomyces pombe and plays a key role in attaching the cytokinetic ring (CR) to the plasma membrane (PM). Cdc15’s abilities to bind to the membrane and oligomerize via its F-BAR domain are inhibited by phosphorylation of its intrinsically disordered region (IDR). Multiple cell polarity kinases regulate Cdc15 IDR phosphostate, and of these the DYRK kinase Pom1 phosphorylation sites on Cdc15 have been shown in vivo to prevent CR formation at cell tips. Here, we compared the ability of Pom1 to control Cdc15 phosphostate and cortical localization to that of other Cdc15 kinases: Kin1, Pck1, and Shk1. We identified distinct but overlapping cohorts of Cdc15 phosphorylation sites targeted by each kinase, and the number of sites correlated with each kinases’ abilities to influence Cdc15 PM localization. Coarse-grained simulations predicted that cumulative IDR phosphorylation moves the IDRs of a dimer apart and toward the F-BAR tips. Further, simulations indicated that the overall negative charge of phosphorylation masks positively charged amino acids necessary for F-BAR oligomerization and membrane interaction. Finally, simulations suggested that dephosphorylated Cdc15 undergoes phase separation driven by IDR interactions. Indeed, dephosphorylated but not phosphorylated Cdc15 undergoes liquid–liquid phase separation to form droplets in vitro that recruit Cdc15 binding partners. In cells, Cdc15 phosphomutants also formed PM-bound condensates that recruit other CR components. Together, we propose that a threshold of Cdc15 phosphorylation by assorted kinases prevents Cdc15 condensation on the PM and antagonizes CR assembly.

This manual describes the use of the Portable, Extensible Toolkit for Scientific Computation (PETSc) and the Toolkit for Advanced Optimization (TAO) for the numerical solution of partial differential equations and related problems on high-performance computers. PETSc/TAO is a suite of data structures and routines that provide the building blocks for the implementation of large-scale application codes on parallel (and serial) computers. PETSc uses the MPI standard for all distributed memory communication.

Segmentation of the Drosophila embryo is initiated by localized maternal signals. In this context, anteriorly localized Bicoid activates the gap genes in the anterior half of the embryo while posteriorly localized Nanos represses the translation of maternal hunchback mRNA to pattern the posterior half. The non-segmented termini are patterned by the localized activation of mitogen-activated protein kinase. Yet, the spatial extent of the terminal patterning system in regulating gap genes beyond poles remains unknown. We investigated the patterning potential of the terminal system using mutagenized embryos that lack both the anterior and the posterior maternal signaling systems. Using a combination of quantitative imaging and mathematical modeling, we analyzed the spatial patterns of gap genes in the early Drosophila embryo. We found that this mutant embryo develops symmetric cuticle patterns along the anteroposterior axis with two segments on each side. Notably, the terminal system can affect the expression of Krüppel in the torso region. Our mathematical model recapitulates the experimental data and reveals the potential bistability in the terminal patterning system. Collectively, our study suggests that the terminal system can act as a long-range inductive signal and establish multiple gene expression boundaries along the anteroposterior axis of the developing embryo.

The previously reported Q is a thermoresponsive coiled-coil protein capable of higher-order supramolecular assembly into fibers and hydrogels with upper critical solution temperature (UCST) behavior. Here, we introduce a new coiled-coil protein that is redesigned to disfavor lateral growth of its fibers and thus achieve a higher crosslinking density within the formed hydrogel. We also introduce a favorable hydrophobic mutation to the pore of the coiled-coil domain for increased thermostability of the protein. We note that an increase in storage modulus of the hydrogel and crosslinking density is coupled with a decrease in fiber diameter. We further fully characterize our α-helical coiled-coil (Q2) hydrogel for its structure, nano-assembly, and rheology relative to our previous single domain protein, Q, over the time of its gelation demonstrating the nature of our hydrogel self-assembly system. In this vein, we also characterize the ability of Q2 to encapsulate the small hydrophobic small molecule, curcumin, and its impact on the mechanical properties of Q2. The design parameters here not only show the importance of electrostatic potential in self-assembly but also provide a step towards predictable design of electrostatic protein interactions.

We survey the phase diagram of high-pressure molecular hydrogen with path integral molecular dynamics using a machine-learned interatomic potential trained with Quantum Monte Carlo forces and energies. Besides the HCP and C2/c-24 phases, we find two new stable phases both with molecular centers in the Fmmm-4 structure, separated by a molecular orientation transition with temperature. The high temperature isotropic Fmmm-4 phase has a reentrant melting line with a maximum at higher temperature (1450K at 150GPa) than previously estimated and crosses the liquid-liquid transition line around 1200K and 200GPa.

The acoustic inverse obstacle scattering problem consists of determining the shape of a domain from measurements of the scattered far field due to some set of incident fields (probes). For a penetrable object with known sound speed, this can be accomplished by treating the boundary alone as an unknown curve. Alternatively, one can treat the entire object as unknown and use a more general volumetric representation, without making use of the known sound speed. Both lead to strongly nonlinear and nonconvex optimization problems for which recursive linearization provides a useful framework for numerical analysis. After extending our shape optimization approach developed earlier for impenetrable bodies, we carry out a systematic study of both methods and compare their performance on a variety of examples. Our findings indicate that the volumetric approach is more robust, even though the number of degrees of freedom is significantly larger. We conclude with a discussion of this phenomenon and potential directions for further research.

A new scheme is presented for imposing periodic boundary conditions on unit cells with arbitrary source distributions. We restrict our attention here to the Poisson, modified Helmholtz, Stokes and modified Stokes equations. The approach extends to the oscillatory equations of mathematical physics, including the Helmholtz and Maxwell equations, but we will address these in a companion paper, since the nature of the problem is somewhat different and includes the consideration of quasiperiodic boundary conditions and resonances. Unlike lattice sum-based methods, the scheme is insensitive to the unit cell's aspect ratio and is easily coupled to adaptive fast multipole methods (FMMs). Our analysis relies on classical “plane-wave” representations of the fundamental solution, and yields an explicit low-rank representation of the field due to all image sources beyond the first layer of neighboring unit cells. When the aspect ratio of the unit cell is large, our scheme can be coupled with the nonuniform fast Fourier transform (NUFFT) to accelerate the evaluation of the induced field. Its performance is illustrated with several numerical examples.