Avoiding matrix exponentials for large transition rate matrices.

Exact methods for the exponentiation of matrices of dimension N can be computationally expensive in terms of execution time (N3) and memory requirements (N2), not to mention numerical precision issues. A matrix often exponentiated in the natural sciences is the rate matrix. Here, we explore five methods to exponentiate rate matrices, some of which apply more broadly to other matrix types. Three of the methods leverage a mathematical analogy between computing matrix elements of a matrix exponential process and computing transition probabilities of a dynamical process (technically a Markov jump process, MJP, typically simulated using Gillespie). In doing so, we identify a novel MJP-based method relying on restricting the number of "trajectory" jumps that incurs improved computational scaling. We then discuss this method's downstream implications on mixing properties of Monte Carlo posterior samplers. We also benchmark two other methods of matrix exponentiation valid for any matrix (beyond rate matrices and, more generally, positive definite matrices) related to solving differential equations: Runge-Kutta integrators and Krylov subspace methods. Under conditions where both the largest matrix element and the number of non-vanishing elements scale linearly with N-reasonable conditions for rate matrices often exponentiated-computational time scaling with the most competitive methods (Krylov and one of the MJP-based methods) reduces to N2 with total memory requirements of N.

Learning Continuous 2D Diffusion Maps from Particle Trajectories without Data Binning.

Diffusion coefficients often vary across regions, such as cellular membranes, and quantifying their variation can provide valuable insight into local membrane properties such as composition and stiffness. Toward quantifying diffusion coefficient spatial maps and uncertainties from particle tracks, we use a Bayesian method and place Gaussian Process (GP) Priors on the maps. For the sake of computational efficiency, we leverage inducing point methods on GPs arising from the mathematical structure of the data giving rise to non-conjugate likelihood-prior pairs. We analyze both synthetic data, where ground truth is known, as well as data drawn from live-cell single-molecule imaging of membrane proteins. The resulting tool provides an unsupervised method to rigorously map diffusion coefficients continuously across membranes without data binning.

An accurate probabilistic step finder for time-series analysis.

Noisy time-series data-from various experiments, including Förster resonance energy transfer, patch clamp, and force spectroscopy, among others-are commonly analyzed with either hidden Markov models or step-finding algorithms, both of which detect discrete transitions. Hidden Markov models, including their extensions to infinite state spaces, inherently assume exponential-or technically geometric-holding time distributions, biasing step locations toward steps with geometric holding times, especially in sparse and/or noisy data. In contrast, existing step-finding algorithms, while free of this restraint, often rely on ad hoc metrics to penalize steps recovered in time traces (by using various information criteria) and otherwise rely on approximate greedy algorithms to identify putative global optima. Here, instead, we devise a robust and general probabilistic (Bayesian) step-finding tool that neither relies on ad hoc metrics to penalize step numbers nor assumes geometric holding times in each state. As the number of steps themselves in a time-series are a priori unknown, we treat these within a Bayesian nonparametric (BNP) paradigm. We find that the method developed, BNP Step (BNP-Step), accurately determines the number and location of transitions between discrete states without any assumed kinetic model and learns the emission distribution characteristic of each state. In doing so, we verify that BNP-Step can analyze sparser data sets containing higher noise and more closely spaced states than otherwise resolved by current state-of-the-art methods. What is more, BNP-Step rigorously propagates measurement uncertainty into uncertainty over state transition locations, numbers, and emission levels as characterized by the posterior. We demonstrate the performance of BNP-Step on both synthetic data as well as data drawn from force spectroscopy experiments.

Approaching Maximum Resolution in Structured Illumination Microscopy via Accurate Noise Modeling.

Biological images captured by microscopes are characterized by heterogeneous signal-to-noise ratios (SNRs) due to spatially varying photon emission across the field of view convoluted with camera noise. State-of-the-art unsupervised structured illumination microscopy (SIM) reconstruction algorithms, commonly implemented in the Fourier domain, do not accurately model this noise and suffer from high-frequency artifacts, user-dependent choices of smoothness constraints making assumptions on biological features, and unphysical negative values in the recovered fluorescence intensity map. On the other hand, supervised methods rely on large datasets for training, and often require retraining for new sample structures. Consequently, achieving high contrast near the maximum theoretical resolution in an unsupervised, physically principled, manner remains an open problem. Here, we propose Bayesian-SIM (B-SIM), an unsupervised Bayesian framework to quantitatively reconstruct SIM data, rectifying these shortcomings by accurately incorporating known noise sources in the spatial domain. To accelerate the reconstruction process, we use the finite extent of the point-spread-function to devise a parallelized Monte Carlo strategy involving chunking and restitching of the inferred fluorescence intensity. We benchmark our framework on both simulated and experimental images, and demonstrate improved contrast permitting feature recovery at up to 25% shorter length scales over state-of-the-art methods at both high- and low-SNR. B-SIM enables unsupervised, quantitative, physically accurate reconstruction without the need for labeled training data, democratizing high-quality SIM reconstruction and expands the capabilities of live-cell SIM to lower SNR, potentially revealing biological features in previously inaccessible regimes.

Toward building comprehensive particle tracking tools with u-track 3D.

In this issue of Cell Reports Methods, Roudot et al. present u-track 3D, a package geared toward improving the workflow of offline widefield multi-molecule tracking. The package is tailored for visualization of tracks, tracking, and assessment of trackability in tracking particles in biological systems.

Gene expression model inference from snapshot RNA data using Bayesian non-parametrics.

Gene expression models, which are key towards understanding cellular regulatory response, underlie observations of single-cell transcriptional dynamics. Although RNA expression data encode information on gene expression models, existing computational frameworks do not perform simultaneous Bayesian inference of gene expression models and parameters from such data. Rather, gene expression models-composed of gene states, their connectivities and associated parameters-are currently deduced by pre-specifying gene state numbers and connectivity before learning associated rate parameters. Here we propose a method to learn full distributions over gene states, state connectivities and associated rate parameters, simultaneously and self-consistently from single-molecule RNA counts. We propagate noise from fluctuating RNA counts over models by treating models themselves as random variables. We achieve this within a Bayesian non-parametric paradigm. We demonstrate our method on the Escherichia coli lacZ pathway and the Saccharomyces cerevisiae STL1 pathway, and verify its robustness on synthetic data.

An accurate probabilistic step finder for time-series analysis.

Noisy time-series data is commonly collected from sources including Förster Resonance Energy Transfer experiments, patch clamp and force spectroscopy setups, among many others. Two of the most common paradigms for the detection of discrete transitions in such time-series data include: hidden Markov models (HMMs) and step-finding algorithms. HMMs, including their extensions to infinite state-spaces, inherently assume in analysis that holding times in discrete states visited are geometrically-or, loosely speaking in common language, exponentially-distributed. Thus the determination of step locations, especially in sparse and noisy data, is biased by HMMs toward identifying steps resulting in geometric holding times. In contrast, existing step-finding algorithms, while free of this restraint, often rely on ad hoc metrics to penalize steps recovered in time traces (by using various information criteria) and otherwise rely on approximate greedy algorithms to identify putative global optima. Here, instead, we devise a robust and general probabilistic (Bayesian) step-finding tool that neither relies on ad hoc metrics to penalize step numbers nor assumes geometric holding times in each state. As the number of steps themselves in a time-series are, a priori unknown, we treat these within a Bayesian nonparametric (BNP) paradigm. We find that the method developed, Bayesian Nonparametric Step (BNP-Step), accurately determines the number and location of transitions between discrete states without any assumed kinetic model and learns the emission distribution characteristic of each state. In doing so, we verify that BNP-Step can analyze sparser data sets containing higher noise and more closely-spaced states than otherwise resolved by current state-of-the-art methods. What is more, BNP-Step rigorously propagates measurement uncertainty into uncertainty over state transition locations, numbers, and emission levels as characterized by the posterior. We demonstrate the performance of BNP-Step on both synthetic data as well as data drawn from force spectroscopy experiments.

Single-Molecule Reaction-Diffusion.

We propose to capture reaction-diffusion on a molecule-by-molecule basis from the fastest acquirable timescale, namely individual photon arrivals. We illustrate our method on intrinsically disordered human proteins, the linker histone H1.0 as well as its chaperone prothymosin α, as these diffuse through an illuminated confocal spot and interact forming larger ternary complexes on millisecond timescales. Most importantly, single-molecule reaction-diffusion, smRD, reveals single molecule properties without trapping or otherwise confining molecules to surfaces. We achieve smRD within a Bayesian paradigm and term our method Bayes-smRD. Bayes-smRD is further free of the average, bulk, results inherent to the analysis of long photon arrival traces by fluorescence correlation spectroscopy. In learning from thousands of photon arrivals continuous spatial positions and discrete conformational and photophysical state changes, Bayes-smRD estimates kinetic parameters on a molecule-by-molecule basis with two to three orders of magnitude less data than tools such as fluorescence correlation spectroscopy thereby also dramatically reducing sample photodamage.

Few-body hydrodynamic interactions probed by optical trap pulling experiment.

We study the hydrodynamic coupling of neighboring micro-beads placed in a multiple optical trap setup allowing us to precisely control the degree of coupling and directly measure time-dependent trajectories of entrained beads. We performed measurements on configurations with increasing complexity starting with a pair of entrained beads moving in one dimension, then in two dimensions, and finally a triplet of beads moving in two dimensions. The average experimental trajectories of a probe bead compare well with the theoretical computation, illustrating the role of viscous coupling and setting timescales for probe bead relaxation. The findings also provide direct experimental corroborations of hydrodynamic coupling at large, micrometer spatial scales and long, millisecond timescales, of relevance to, e.g., microfluidic device design and hydrodynamic-assisted colloidal assembly, improving the capability of optical tweezers, and understanding the coupling between micrometer-scale objects within a living cell.

Monte Carlo samplers for efficient network inference.

Accessing information on an underlying network driving a biological process often involves interrupting the process and collecting snapshot data. When snapshot data are stochastic, the data's structure necessitates a probabilistic description to infer underlying reaction networks. As an example, we may imagine wanting to learn gene state networks from the type of data collected in single molecule RNA fluorescence in situ hybridization (RNA-FISH). In the networks we consider, nodes represent network states, and edges represent biochemical reaction rates linking states. Simultaneously estimating the number of nodes and constituent parameters from snapshot data remains a challenging task in part on account of data uncertainty and timescale separations between kinetic parameters mediating the network. While parametric Bayesian methods learn parameters given a network structure (with known node numbers) with rigorously propagated measurement uncertainty, learning the number of nodes and parameters with potentially large timescale separations remain open questions. Here, we propose a Bayesian nonparametric framework and describe a hybrid Bayesian Markov Chain Monte Carlo (MCMC) sampler directly addressing these challenges. In particular, in our hybrid method, Hamiltonian Monte Carlo (HMC) leverages local posterior geometries in inference to explore the parameter space; Adaptive Metropolis Hastings (AMH) learns correlations between plausible parameter sets to efficiently propose probable models; and Parallel Tempering takes into account multiple models simultaneously with tempered information content to augment sampling efficiency. We apply our method to synthetic data mimicking single molecule RNA-FISH, a popular snapshot method in probing transcriptional networks to illustrate the identified challenges inherent to learning dynamical models from these snapshots and how our method addresses them.

Two-state swimming: Strategy and survival of a model bacterial predator in response to environmental cues.

Bdellovibrio bacteriovorus is a predatory bacterium preying upon Gram-negative bacteria. As such, B. bacteriovorus has the potential to control antibiotic-resistant pathogens and biofilm populations. To survive and reproduce, B. bacteriovorus must locate and infect a host cell. However, in the temporary absence of prey, it is largely unknown how B. bacteriovorus modulate their motility patterns in response to physical or chemical environmental cues to optimize their energy expenditure. To investigate B. bacteriovorus' predation strategy, we track and quantify their motion by measuring speed distributions as a function of starvation time. While an initial unimodal speed distribution relaxing to one for pure diffusion at long times may be expected, instead we observe a bimodal speed distribution with one mode centered around that expected from diffusion and the other centered at higher speeds. What is more, for an increasing amount of time over which B. bacteriovorus is starved, we observe a progressive reweighting from the active swimming state to an apparent diffusive state in the speed distribution. Distributions of trajectory-averaged speeds for B. bacteriovorus are largely unimodal, indicating switching between a faster swim speed and an apparent diffusive state within individual observed trajectories rather than there being distinct active swimming and apparent diffusive populations. We also find that B. bacteriovorus' apparent diffusive state is not merely caused by the diffusion of inviable bacteria as subsequent spiking experiments show that bacteria can be resuscitated and bimodality restored. Indeed, starved B. bacteriovorus may modulate the frequency and duration of active swimming as a means of balancing energy consumption and procurement. Our results thus point to a reweighting of the swimming frequency on a trajectory basis rather than a population level basis.

Fluorescence Microscopy: a statistics-optics perspective.

Fundamental properties of light unavoidably impose features on images collected using fluorescence microscopes. Modeling these features is ever more important in quantitatively interpreting microscopy images collected at scales on par or smaller than light's wavelength. Here we review the optics responsible for generating fluorescent images, fluorophore properties, microscopy modalities leveraging properties of both light and fluorophores, in addition to the necessarily probabilistic modeling tools imposed by the stochastic nature of light and measurement.

BNP-Track: A framework for multi-particle superresolved tracking.

When tracking fluorescently labeled molecules (termed "emitters") under widefield microscopes, point spread function overlap of neighboring molecules is inevitable in both dilute and especially crowded environments. In such cases, superresolution methods leveraging rare photophysical events to distinguish static targets nearby in space introduce temporal delays that compromise tracking. As we have shown in a companion manuscript, for dynamic targets, information on neighboring fluorescent molecules is encoded as spatial intensity correlations across pixels and temporal correlations in intensity patterns across time frames. We then demonstrated how we used all spatiotemporal correlations encoded in the data to achieve superresolved tracking. That is, we showed the results of full posterior inference over both the number of emitters and their associated tracks simultaneously and self-consistently through Bayesian nonparametrics. In this companion manuscript we focus on testing the robustness of our tracking tool, BNP-Track, across sets of parameter regimes and compare BNP-Track to competing tracking methods in the spirit of a prior Nature Methods tracking competition. We explore additional features of BNP-Track including how a stochastic treatment of background yields greater accuracy in emitter number determination and how BNP-Track corrects for point spread function blur (or "aliasing") introduced by intraframe motion in addition to propagating error originating from myriad sources (such as criss-crossing tracks, out-of-focus particles, pixelation, shot and camera artefact, stochastic background) in posterior inference over emitter numbers and their associated tracks. While head-to-head comparison with other tracking methods is not possible (as competitors cannot simultaneously learn molecule numbers and associated tracks), we can give competing methods some advantages in order to perform approximate head-to-head comparison. We show that even under such optimistic scenarios, BNP-Track is capable of tracking multiple diffraction-limited point emitters conventional tracking methods cannot resolve thereby extending the superresolution paradigm to dynamical targets.

BNP-Track: A framework for superresolved tracking.

Assessing dynamic processes at single molecule scales is key toward capturing life at the level of its molecular actors. Widefield superresolution methods, such as STORM, PALM, and PAINT, provide nanoscale localization accuracy, even when distances between fluorescently labeled single molecules ("emitters") fall below light's diffraction limit. However, as these superresolution methods rely on rare photophysical events to distinguish emitters from both each other and background, they are largely limited to static samples. In contrast, here we leverage spatiotemporal correlations of dynamic widefield imaging data to extend superresolution to simultaneous multiple emitter tracking without relying on photodynamics even as emitter distances from one another fall below the diffraction limit. We simultaneously determine emitter numbers and their tracks (localization and linking) with the same localization accuracy per frame as widefield superresolution does for immobilized emitters under similar imaging conditions (≈50nm). We demonstrate our results for both in cellulo data and, for benchmarking purposes, on synthetic data. To this end, we avoid the existing tracking paradigm relying on completely or partially separating the tasks of emitter number determination, localization of each emitter, and linking emitter positions across frames. Instead, we develop a fully joint posterior distribution over the quantities of interest, including emitter tracks and their total, otherwise unknown, number within the Bayesian nonparametric paradigm. Our posterior quantifies the full uncertainty over emitter numbers and their associated tracks propagated from origins including shot noise and camera artefacts, pixelation, stochastic background, and out-of-focus motion. Finally, it remains accurate in more crowded regimes where alternative tracking tools cannot be applied.

How many submissions are needed to discover friendly suggested reviewers?

It is common in scientific publishing to request from authors reviewer suggestions for their own manuscripts. The question then arises: How many submissions are needed to discover friendly suggested reviewers? To answer this question, as the data we would need is anonymized, we present an agent-based simulation of (single-blinded) peer review to generate synthetic data. We then use a Bayesian framework to classify suggested reviewers. To set a lower bound on the number of submissions possible, we create an optimistically simple model that should allow us to more readily deduce the degree of friendliness of the reviewer. Despite this model's optimistic conditions, we find that one would need hundreds of submissions to classify even a small reviewer subset. Thus, it is virtually unfeasible under realistic conditions. This ensures that the peer review system is sufficiently robust to allow authors to suggest their own reviewers.

Resolving Molecular Heterogeneity with Single-Molecule Centrifugation.

For many classes of biomolecules, population-level heterogeneity is an essential aspect of biological function─from antibodies produced by the immune system to post-translationally modified proteins that regulate cellular processes. However, heterogeneity is difficult to fully characterize for multiple reasons: (i) single-molecule approaches are needed to avoid information lost by ensemble-level averaging, (ii) sufficient statistics must be gathered on both a per-molecule and per-population level, and (iii) a suitable analysis framework is required to make sense of a potentially limited number of intrinsically noisy measurements. Here, we introduce an approach that overcomes these difficulties by combining three techniques: a DNA nanoswitch construct to repeatedly interrogate the same molecule, a benchtop centrifuge force microscope (CFM) to obtain thousands of statistics in a highly parallel manner, and a Bayesian nonparametric (BNP) inference method to resolve separate subpopulations with distinct kinetics. We apply this approach to characterize commercially available antibodies and find that polyclonal antibody from rabbit serum is well-modeled by a mixture of three subpopulations. Our results show how combining a spatially and temporally multiplexed nanoswitch-CFM assay with BNP analysis can help resolve complex biomolecular interactions in heterogeneous samples.

Fluorescence lifetime: Beating the IRF and interpulse window.

Fluorescence lifetime imaging captures the spatial distribution of chemical species across cellular environments employing pulsed illumination confocal setups. However, quantitative interpretation of lifetime data continues to face critical challenges. For instance, fluorescent species with known in vitro excited-state lifetimes may split into multiple species with unique lifetimes when introduced into complex living environments. What is more, mixtures of species, which may be both endogenous and introduced into the sample, may exhibit 1) very similar lifetimes as well as 2) wide ranges of lifetimes including lifetimes shorter than the instrumental response function or whose duration may be long enough to be comparable to the interpulse window. By contrast, existing methods of analysis are optimized for well-separated and intermediate lifetimes. Here, we broaden the applicability of fluorescence lifetime analysis by simultaneously treating unknown mixtures of arbitrary lifetimes-outside the intermediate, Goldilocks, zone-for data drawn from a single confocal spot leveraging the tools of Bayesian nonparametrics (BNP). We benchmark our algorithm, termed BNP lifetime analysis, using a range of synthetic and experimental data. Moreover, we show that the BNP lifetime analysis method can distinguish and deduce lifetimes using photon counts as small as 500.

Learning continuous potentials from smFRET.

Potential energy landscapes are useful models in describing events such as protein folding and binding. While single-molecule fluorescence resonance energy transfer (smFRET) experiments encode information on continuous potentials for the system probed, including rarely visited barriers between putative potential minima, this information is rarely decoded from the data. This is because existing analysis methods often model smFRET output assuming, from the onset, that the system probed evolves in a discretized state space to be analyzed within a hidden Markov model (HMM) paradigm. By contrast, here, we infer continuous potentials from smFRET data without discretely approximating the state space. We do so by operating within a Bayesian nonparametric paradigm by placing priors on the family of all possible potential curves. As our inference accounts for a number of required experimental features raising computational cost (such as incorporating discrete photon shot noise), the framework leverages a structured-kernel-interpolation Gaussian process prior to help curtail computational cost. We show that our structured-kernel-interpolation priors for potential energy reconstruction from smFRET analysis accurately infers the potential energy landscape from a smFRET binding experiment. We then illustrate advantages of structured-kernel-interpolation priors for potential energy reconstruction from smFRET over standard HMM approaches by providing information, such as barrier heights and friction coefficients, that is otherwise inaccessible to HMMs.

Single-photon smFRET: II. Application to continuous illumination.

Here we adapt the Bayesian nonparametrics (BNP) framework presented in the first companion article to analyze kinetics from single-photon, single-molecule Förster resonance energy transfer (smFRET) traces generated under continuous illumination. Using our sampler, BNP-FRET, we learn the escape rates and the number of system states given a photon trace. We benchmark our method by analyzing a range of synthetic and experimental data. Particularly, we apply our method to simultaneously learn the number of system states and the corresponding kinetics for intrinsically disordered proteins using two-color FRET under varying chemical conditions. Moreover, using synthetic data, we show that our method can deduce the number of system states even when kinetics occur at timescales of interphoton intervals.