Publications

Accurately placing macromolecular assemblies in the cellular context is important in understanding their mechanistic role inside the cell. Previously, we developed a 2D template-matching (2DTM) approach (Rickgauer et al., 2017[Rickgauer, J. P., Grigorieff, N. & Denk, W. (2017). eLife, 6, e25648.]; Lucas et al., 2021[Lucas, B. A., Himes, B. A., Xue, L., Grant, T., Mahamid, J. & Grigorieff, N. (2021). eLife, 10, e68946.]) in cisTEM (Grant et al., 2018[Grant, T., Rohou, A. & Grigorieff, N. (2018). eLife, 7, e35383.]) to detect targets in cellular cryo-EM images with high positional and orientational accuracy. 2DTM not only detects targets such as ribosomes in cryo-EM images but also provides data that enable the in situ classification and high-resolution reconstruction of these targets (Lucas et al., 2022[Lucas, B. A., Zhang, K., Loerch, S. & Grigorieff, N. (2022). eLife, 11, e79272.], 2023[Lucas, B. A., Himes, B. A. & Grigorieff, N. (2023). eLife, 12, 12RP90486.]; Elferich et al., 2022[Elferich, J., Schiroli, G., Scadden, D. T. & Grigorieff, N. (2022). eLife, 11, e80980.]). Building on these successes, this work aims to improve the 2DTM framework to detect more challenging targets in various environments.

A 2DTM search yields a signal-to-noise ratio (SNR) for every location in the cryo-EM image that depends on the cross-correlation between the template and the image (Rickgauer et al., 2017[Rickgauer, J. P., Grigorieff, N. & Denk, W. (2017). eLife, 6, e25648.]). A target is detected when the SNR value exceeds a statistically defined threshold that limits the average false positives to one per image, based on the assumption that the cryo-EM image is dominated by noise and cellular background and that the cross-correlation values observed across the image after whitening the noise/background follow a Gaussian distribution. The 2DTM SNR can be further normalized by subtracting the mean and dividing by the standard deviation of cross-correlations calculated across all sampled orientations at each location in the image (Rickgauer et al., 2017[Rickgauer, J. P., Grigorieff, N. & Denk, W. (2017). eLife, 6, e25648.]). This step is often referred to as `z-score' normalization (Spiegel & Stephens, 1999[Spiegel, M. R. & Stephens, L. J. (1999). Schaum's Outline of Theory and Problems of Statistics, 3rd ed. New York: McGraw-Hill.]). Using the z-score instead of the SNR improves the detection of capsomers in rotavirus double-layered particles (DLPs; Rickgauer et al., 2017[Rickgauer, J. P., Grigorieff, N. & Denk, W. (2017). eLife, 6, e25648.]) and ribosomes in a crowded cellular environment (Lucas et al., 2022[Lucas, B. A., Zhang, K., Loerch, S. & Grigorieff, N. (2022). eLife, 11, e79272.]). In the following, we will refer to the outputs of 2DTM as the 2DTM SNR and 2DTM z-score, respectively.

Previous applications of 2DTM have shown that the 2DTM SNR and z-score function differently depending on the characteristics of the sample and target. For example, when low-resolution features were suppressed by using a near-focus image setting (70 nm), the 2DTM SNR map showed a flat background with sharp peaks indicating the locations of apoferritins, even in a dense protein (bovine serum albumin) background (Rickgauer et al., 2017[Rickgauer, J. P., Grigorieff, N. & Denk, W. (2017). eLife, 6, e25648.]). On the other hand, low-resolution features from the target itself when strongly defocused (>2000 nm), or from the background structural noise, can result in broader peaks or an uneven background in the SNR map, complicating target detection (Rickgauer et al., 2017[Rickgauer, J. P., Grigorieff, N. & Denk, W. (2017). eLife, 6, e25648.]; Lucas et al., 2022[Lucas, B. A., Zhang, K., Loerch, S. & Grigorieff, N. (2022). eLife, 11, e79272.]). The misleading low-resolution background can be suppressed by calculating the 2DTM z-score (Rickgauer et al., 2017[Rickgauer, J. P., Grigorieff, N. & Denk, W. (2017). eLife, 6, e25648.]), which removes spurious correlations between the template and the structural noise in the image, thereby flattening the background and improving the detectability of targets in cellular environments (Rickgauer et al., 2020[Rickgauer, J. P., Choi, H., Lippincott-Schwartz, J. & Denk, W. (2020). bioRxiv, 2020.04.22.053868.]; Lucas et al., 2022[Lucas, B. A., Zhang, K., Loerch, S. & Grigorieff, N. (2022). eLife, 11, e79272.]). In Fig. 1[link](a), a segment of a previously published micrograph of a yeast lamella near the nucleus is presented (Lucas et al., 2022[Lucas, B. A., Zhang, K., Loerch, S. & Grigorieff, N. (2022). eLife, 11, e79272.]). This image section contains various cellular compartments located from left to right, including the vacuole, cytoplasm and nucleus. Using the mature 60S as a search template, 2DTM outputs a 2DTM SNR map and a 2DTM z-score map [Figs. 1[link](b) and 1[link](c)]. The bright spots in the 2DTM SNR map are locations with high correlation values, indicating 60S ribosomes. However, the peaks are surrounded by halos of increased SNR values extending to other low-resolution features in the image, such as membranes. The z-score map removes these halos and spurious matches of high-contrast features, thereby reducing the number of false detections (membranes or partial overlap with ribosomes) while preserving locations with high-resolution matches from the riboso

Macrocycles are a promising therapeutic class. The incorporation of heterochiral and non-natural chemical building-blocks presents challenges for rational design, however. With no existing machine learning methods tailored for heterochiral macrocycle design, we developed a novel convolutional autoencoder model to rapidly generate energetically favorable macrocycle backbones for heterochiral design and structure prediction. Our approach surpasses the current state-of-the-art method, Generalized Kinematic loop closure (GenKIC) in the Rosetta software suite. Given the absence of large, available macrocycle datasets, we created a custom dataset in-house and in silico. Our model, CyclicCAE, produces energetically stable backbones and designable structures more rapidly than GenKIC. It enables users to perform energy minimization, generate structurally similar or diverse inputs via MCMC, and conduct inpainting with fixed anchors or motifs. We propose that this novel method will accelerate the development of stable macrocycles, speeding up macrocycle drug design pipelines.

Granular flows in small-outlet hoppers exhibit several characteristic but poorly understood behaviors: temporary clogs (pauses) where flow stops before later spontaneously restarting, permanent clogs that last indefinitely, and non-Gaussian, nonmonotonic flow-rate statistics. These aspects have been studied independently, but a model of hopper flow that explains all three has not been formulated. Here, we introduce a phenomenological model that provides a unifying dynamical mechanism for all three behaviors: coupling between the flow rate and a hidden mode that controls the stability of clogs. In the theory, flow rate evolves according to Langevin dynamics with multiplicative noise and an absorbing state at zero flow, conditional on the hidden mode. The model fully reproduces the statistics of pause and clog events of a large (>40000 flows) experimental dataset, including nonexponentially distributed clogging times and non-Gaussian flow rate distribution, and explains the stretched-exponential growth of the average clogging time with outlet size. Further, we identify the physical nature of the hidden mode in microscopic configurational features, including size and smoothness of the static arch structure formed during pauses and clogs. Our work provides a unifying framework for several poorly understood clogging phenomena, and suggests numerous new paths toward further understanding of this complex system.

Cell populations must adjust their phenotypic composition to adapt to changing environments. One adaptation strategy is to maintain distinct phenotypic subsets within the population and to modulate their relative abundances via gene regulation. Another strategy involves genetic mutations, which can be augmented by stress-response pathways. Here, we studied how a migrating bacterial population regulates its phenotypic distribution to traverse diverse environments. We generated isogenic Escherichia coli populations with varying distributions of swimming behaviors and observed their phenotype distributions during migration in liquid and porous environments. We found that the migrating populations became enriched with high-performing swimming phenotypes in each environment, allowing the populations to adapt without requiring mutations or gene regulation. This adaptation is dynamic and rapid, reversing in a few doubling times when migration ceases. By measuring the chemoreceptor abundance distributions during migration toward different attractants, we demonstrated that adaptation acts on multiple chemotaxis-related traits simultaneously. These measurements are consistent with a general mechanism in which adaptation results from a balance between cell growth generating diversity and collective migration eliminating underperforming phenotypes. Thus, collective migration enables cell populations with continuous, multidimensional phenotypes to flexibly and rapidly adapt their phenotypic composition to diverse environmental conditions.

Resolving continuous conformational heterogeneity in single-particle cryo-electron microscopy (cryo-EM) is a field in which new methods are now emerging regularly. Methods range from traditional statistical techniques to state-of-the-art neural network approaches. Such ongoing efforts continue to enhance the ability to explore and understand the continuous conformational variations in cryo-EM data. One of the first methods was the manifold embedding approach or ManifoldEM. However, comparing it with more recent methods has been challenging due to software availability and usability issues. In this work, we introduce a modern Python implementation that is user-friendly, orders of magnitude faster than its previous versions and designed with a developer-ready environment. This implementation allows a more thorough evaluation of the strengths and limitations of methods addressing continuous conformational heterogeneity in cryo-EM, paving the way for further community-driven improvements.

Resolving continuous conformational heterogeneity in single-particle cryo-electron microscopy (cryo-EM) is a field in which new methods are now emerging regularly. Methods range from traditional statistical techniques to state-of-the-art neural network approaches. Such ongoing efforts continue to enhance the ability to explore and understand the continuous conformational variations in cryo-EM data. One of the first methods was the manifold embedding approach or ManifoldEM. However, comparing it with more recent methods has been challenging due to software availability and usability issues. In this work, we introduce a modern Python implementation that is user-friendly, orders of magnitude faster than its previous versions and designed with a developer-ready environment. This implementation allows a more thorough evaluation of the strengths and limitations of methods addressing continuous conformational heterogeneity in cryo-EM, paving the way for further community-driven improvements.

We present a new framework for the fast solution of inhomogeneous elliptic boundary value problems in domains with smooth boundaries. High-order solvers based on adaptive box codes or the fast Fourier transform can efficiently treat the volumetric inhomogeneity, but require care to be taken near the boundary to ensure that the volume data is globally smooth. We avoid function extension or cut-cell quadratures near the boundary by dividing the domain into two regions: a bulk region away from the boundary that is efficiently treated with a truncated free-space box code, and a variable-width boundary-conforming strip region that is treated with a spectral collocation method and accompanying fast direct solver. Particular solutions in each region are then combined with Laplace layer potentials to yield the global solution. The resulting solver has an optimal computational complexity of O(N) for an adaptive discretization with N degrees of freedom. With an efficient two-dimensional (2D) implementation we demonstrate adaptive resolution of volumetric data, boundary data, and geometric features across a wide range of length scales, to typically 10-digit accuracy. The cost of all boundary corrections remains small relative to that of the bulk box code. The extension to 3D is expected to be straightforward in many cases because the strip

A lack of tools for detecting receptor activity in vivo has limited our ability to fully explore receptor-level control of developmental patterning. Here, we extend a new class of biosensors for receptor tyrosine kinase (RTK) activity, the pYtag system, to visualize endogenous RTK activity in Drosophila. We build biosensors for three Drosophila RTKs that function across developmental stages and tissues. By characterizing Torso::pYtag during terminal patterning in the early embryo, we find that Torso activity differs from downstream ERK activity in two surprising ways: Torso activity is narrowly restricted to the poles but produces a broader gradient of ERK, and Torso activity decreases over developmental time while ERK activity is sustained. This decrease in Torso activity is driven by ERK pathway-dependent negative feedback. Our results suggest an updated model of terminal patterning where a narrow domain of Torso activity, tuned in amplitude by negative feedback, locally activates signaling effectors which diffuse through the syncytial embryo to form the ERK gradient. Altogether, this work highlights the usefulness of pYtags for investigating receptor-level regulation of developmental patterning.

Although optimal control (OC) has been studied in stochastic thermodynamics for systems with continuous state variables, less is known in systems with discrete state variables, such as chemical reaction networks (CRNs). Here, we develop a general theoretical framework to study OC of CRNs for changing the system from an initial distribution of states to a final distribution with minimum dissipation. We derive a “Kirchhoff’s law” for the probability current in the adiabatic limit, from which the optimal kinetic rates are determined analytically for any given probability trajectory allowed by local rate constraints. By using the optimal rates, we show that the total dissipation is determined by a 𝐿2-distance measure in the probability space and derive an analytical expression for the metric tensor that depends on the probability distribution, network topology, and capacity of each link. Minimizing the total dissipation leads to the geodesic trajectory in the probability space and the corresponding OC protocol is determined by the Kirchhoff’s law. To demonstrate our general approach, we use it to find a lower bound for the minimum dissipation that is tighter than existing bounds obtained with only global constraints in the adiabatic limit. We also apply it to simple networks, e.g., fully connected three-state CRNs with different local constraints and show that indirect pathway and nonfunctional transient state can play a crucial role in switching between different probability distributions efficiently. Future directions in studying OC in CRNs by using our general framework are discussed.

The centrosomal aster is a mobile and adaptable cellular organelle that exerts and transmits forces necessary for tasks such as nuclear migration and spindle positioning. Recent experimental and theoretical studies of nematode and human cells demonstrate that pulling forces on asters by cortically anchored force generators are dominant during such processes. Here, we present a comprehensive investigation of the S-model (S for stoichiometry) of aster dynamics based solely on such forces. The model evolves the astral centrosome position, a probability field of cell-surface motor occupancy by centrosomal microtubules (under an assumption of stoichiometric binding), and free boundaries of unattached, growing microtubules. We show how cell shape affects the stability of centering of the aster, and its transition to oscillations with increasing motor number. Seeking to understand observations in single-cell nematode embryos, we use highly accurate simulations to examine the nonlinear structures of the bifurcations, and demonstrate the importance of binding domain overlap to interpreting genetic perturbation experiments. We find a generally rich dynamical landscape, dependent upon cell shape, such as internal constant-velocity equatorial orbits of asters that can be seen as traveling wave solutions. Finally, we study the interactions of multiple asters which we demonstrate an effective mutual repulsion due to their competition for surface force generators. We find, amazingly, that centrosomes can relax onto the vertices of platonic and nonplatonic solids, very closely mirroring the results of the classical Thomson problem for energy-minimizing configurations of electrons constrained to a sphere and interacting via repulsive Coulomb potentials. Our findings both explain experimental observations, providing insights into the mechanisms governing spindle positioning and cell division dynamics, and show the possibility of new nonlinear phenomena in cell biology.