Single molecule force spectroscopy at high data acquisition: A Bayesian nonparametric analysis

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.

Biomembrane force probe (BFP): Design, advancements, and recent applications to live-cell mechanobiology

Mechanical forces play a vital role in biological processes at molecular and cellular levels, significantly impacting various diseases such as cancer, cardiovascular disease, and COVID-19. Recent advancements in dynamic force spectroscopy (DFS) techniques have enabled the application and measurement of forces and displacements with high resolutions, providing crucial insights into the mechanical pathways underlying these diseases. Among DFS techniques, the biomembrane force probe (BFP) stands out for its ability to measure bond kinetics and cellular mechanosensing with pico-newton and nano-meter resolutions. Here, a comprehensive overview of the classical BFP-DFS setup is presented and key advancements are emphasized, including the development of dual biomembrane force probe (dBFP) and fluorescence biomembrane force probe (fBFP). BFP-DFS allows us to investigate dynamic bond behaviors on living cells and significantly enhances the understanding of specific ligand-receptor axes mediated cell mechanosensing. The contributions of BFP-DFS to the fields of cancer biology, thrombosis, and inflammation are delved into, exploring its potential to elucidate novel therapeutic discoveries. Furthermore, future BFP upgrades aimed at improving output and feasibility are anticipated, emphasizing its growing importance in the field of cell mechanobiology. Although BFP-DFS remains a niche research modality, its impact on the expanding field of cell mechanobiology is immense.

Single-photon smFRET. I: Theory and conceptual basis

We present a unified conceptual framework and the associated software package for single-molecule Förster resonance energy transfer (smFRET) analysis from single-photon arrivals leveraging Bayesian nonparametrics, BNP-FRET. This unified framework addresses the following key physical complexities of a single-photon smFRET experiment, including: 1) fluorophore photophysics; 2) continuous time kinetics of the labeled system with large timescale separations between photophysical phenomena such as excited photophysical state lifetimes and events such as transition between system states; 3) unavoidable detector artefacts; 4) background emissions; 5) unknown number of system states; and 6) both continuous and pulsed illumination. These physical features necessarily demand a novel framework that extends beyond existing tools. In particular, the theory naturally brings us to a hidden Markov model with a second-order structure and Bayesian nonparametrics on account of items 1, 2, and 5 on the list. In the second and third companion articles, we discuss the direct effects of these key complexities on the inference of parameters for continuous and pulsed illumination, respectively.

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.

Single-photon smFRET. III. Application to pulsed illumination

Förster resonance energy transfer (FRET) using pulsed illumination has been pivotal in leveraging lifetime information in FRET analysis. However, there remain major challenges in quantitative single-photon, single-molecule FRET (smFRET) data analysis under pulsed illumination including 1) simultaneously deducing kinetics and number of system states; 2) providing uncertainties over estimates, particularly uncertainty over the number of system states; and 3) taking into account detector noise sources such as cross talk and the instrument response function contributing to uncertainty; in addition to 4) other experimental noise sources such as background. Here, we implement the Bayesian nonparametric framework described in the first companion article that addresses all aforementioned issues in smFRET data analysis specialized for the case of pulsed illumination. Furthermore, we apply our method to both synthetic as well as experimental data acquired using Holliday junctions.

Nonparametric Bayesian inference for meta-stable conformational dynamics

Analyses of structural dynamics of biomolecules hold great promise to deepen the understanding of and ability to construct complex molecular systems. To this end, both experimental and computational means are available, such as fluorescence quenching experiments or molecular dynamics simulations, respectively. We argue that while seemingly disparate, both fields of study have to deal with the same type of data about the same underlying phenomenon of conformational switching. Two central challenges typically arise in both contexts: (i) the amount of obtained data is large, and (ii) it is often unknown how many distinct molecular states underlie these data. In this study, we build on the established idea of Markov state modeling and propose a generative, Bayesian nonparametric hidden Markov state model that addresses these challenges. Utilizing hierarchical Dirichlet processes, we treat different meta-stable molecule conformations as distinct Markov states, the number of which we then do not have to set a priori. In contrast to existing approaches to both experimental as well as simulation data that are based on the same idea, we leverage a mean-field variational inference approach, enabling scalable inference on large amounts of data. Furthermore, we specify the model also for the important case of angular data, which however proves to be computationally intractable. Addressing this issue, we propose a computationally tractable approximation to the angular model. We demonstrate the method on synthetic ground truth data and apply it to known benchmark problems as well as electrophysiological experimental data from a conformation-switching ion channel to highlight its practical utility.

Modeling Non-additive Effects in Neighboring Chemically Identical Fluorophores

Quantitative fluorescence analysis is often used to derive chemical properties, including stoichiometries, of biomolecular complexes. One fundamental underlying assumption in the analysis of fluorescence data─whether it be the determination of protein complex stoichiometry by super-resolution, or step-counting by photobleaching, or the determination of RNA counts in diffraction-limited spots in RNA fluorescence in situ hybridization (RNA-FISH) experiments─is that fluorophores behave identically and do not interact. However, recent experiments on fluorophore-labeled DNA origami structures such as fluorocubes have shed light on the nature of the interactions between identical fluorophores as these are brought closer together, thereby raising questions on the validity of the modeling assumption that fluorophores do not interact. Here, we analyze photon arrival data under pulsed illumination from fluorocubes where distances between dyes range from 2 to 10 nm. We discuss the implications of non-additivity of brightness on quantitative fluorescence analysis.

Extraction of rapid kinetics from smFRET measurements using integrative detectors

Hidden Markov models (HMMs) are used to learn single-molecule kinetics across a range of experimental techniques. By their construction, HMMs assume that single-molecule events occur on slower timescales than those of data acquisition. To move beyond that HMM limitation and allow for single-molecule events to occur on any timescale, we must treat single-molecule events in continuous time as they occur in nature. We propose a method to learn kinetic rates from single-molecule Förster resonance energy transfer (smFRET) data collected by integrative detectors, even if those rates exceed data acquisition rates. To achieve that, we exploit our recently proposed "hidden Markov jump process" (HMJP), with which we learn transition kinetics from parallel measurements in donor and acceptor channels. HMJPs generalize the HMM paradigm in two critical ways: (1) they deal with physical smFRET systems as they switch between conformational states in continuous time, and (2) they estimate transition rates between conformational states directly without having recourse to transition probabilities or assuming slow dynamics. Our continuous-time treatment learns the transition kinetics and photon emission rates for dynamic regimes that are inaccessible to HMMs, which treat system kinetics in discrete time. We validate our framework's robustness on simulated data and demonstrate its performance on experimental data from FRET-labeled Holliday junctions.

Residence time analysis of RNA polymerase transcription dynamics: A Bayesian sticky HMM approach

The time spent by a single RNA polymerase (RNAP) at specific locations along the DNA, termed "residence time," reports on the initiation, elongation, and termination stages of transcription. At the single-molecule level, this information can be obtained from dual ultrastable optical trapping experiments, revealing a transcriptional elongation of RNAP interspersed with residence times of variable duration. Successfully discriminating between long and short residence times was used by previous approaches to learn about RNAP's transcription elongation dynamics. Here, we propose an approach based on the Bayesian sticky hidden Markov model that treats all residence times for an Escherichia coli RNAP on an equal footing without a priori discriminating between long and short residence times. Furthermore, our method has two additional advantages: we provide full distributions around key point statistics and directly treat the sequence dependence of RNAP's elongation rate. By applying our approach to experimental data, we find assigned relative probabilities on long versus short residence times, force-dependent average residence time transcription elongation dynamics, ∼10% drop in the average backtracking durations in the presence of GreB, and ∼20% drop in the average residence time as a function of applied force in the presence of RNaseA.

Generalizing HMMs to Continuous Time for Fast Kinetics: Hidden Markov Jump Processes

The hidden Markov model (HMM) is a framework for time series analysis widely applied to single-molecule experiments. Although initially developed for applications outside the natural sciences, the HMM has traditionally been used to interpret signals generated by physical systems, such as single molecules, evolving in a discrete state space observed at discrete time levels dictated by the data acquisition rate. Within the HMM framework, transitions between states are modeled as occurring at the end of each data acquisition period and are described using transition probabilities. Yet, whereas measurements are often performed at discrete time levels in the natural sciences, physical systems evolve in continuous time according to transition rates. It then follows that the modeling assumptions underlying the HMM are justified if the transition rates of a physical process from state to state are small as compared to the data acquisition rate. In other words, HMMs apply to slow kinetics. The problem is, because the transition rates are unknown in principle, it is unclear, a priori, whether the HMM applies to a particular system. For this reason, we must generalize HMMs for physical systems, such as single molecules, because these switch between discrete states in "continuous time". We do so by exploiting recent mathematical tools developed in the context of inferring Markov jump processes and propose the hidden Markov jump process. We explicitly show in what limit the hidden Markov jump process reduces to the HMM. Resolving the discrete time discrepancy of the HMM has clear implications: we no longer need to assume that processes, such as molecular events, must occur on timescales slower than data acquisition and can learn transition rates even if these are on the same timescale or otherwise exceed data acquisition rates.

Direct Photon-by-Photon Analysis of Time-Resolved Pulsed Excitation Data using Bayesian Nonparametrics

Lifetimes of chemical species are typically estimated by either fitting time-correlated single-photon counting (TCSPC) histograms or phasor analysis from time-resolved photon arrivals. While both methods yield lifetimes in a computationally efficient manner, their performance is limited by choices made on the number of distinct chemical species contributing photons. However, the number of species is encoded in the photon arrival times collected for each illuminated spot and need not be set by hand a priori. Here, we propose a direct photon-by-photon analysis of data drawn from pulsed excitation experiments to infer, simultaneously and self-consistently, the number of species and their associated lifetimes from a few thousand photons. We do so by leveraging new mathematical tools within the Bayesian nonparametric. We benchmark our method for both simulated and experimental data for 1-4 species.

Single-Molecule Studies of Protein Folding with Optical Tweezers

Manipulation of individual molecules with optical tweezers provides a powerful means of interrogating the structure and folding of proteins. Mechanical force is not only a relevant quantity in cellular protein folding and function, but also a convenient parameter for biophysical folding studies. Optical tweezers offer precise control in the force range relevant for protein folding and unfolding, from which single-molecule kinetic and thermodynamic information about these processes can be extracted. In this review, we describe both physical principles and practical aspects of optical tweezers measurements and discuss recent advances in the use of this technique for the study of protein folding. In particular, we describe the characterization of folding energy landscapes at high resolution, studies of structurally complex multidomain proteins, folding in the presence of chaperones, and the ability to investigate real-time cotranslational folding of a polypeptide.

Top-down machine learning approach for high-throughput single-molecule analysis

Single-molecule approaches provide enormous insight into the dynamics of biomolecules, but adequately sampling distributions of states and events often requires extensive sampling. Although emerging experimental techniques can generate such large datasets, existing analysis tools are not suitable to process the large volume of data obtained in high-throughput paradigms. Here, we present a new analysis platform (DISC) that accelerates unsupervised analysis of single-molecule trajectories. By merging model-free statistical learning with the Viterbi algorithm, DISC idealizes single-molecule trajectories up to three orders of magnitude faster with improved accuracy compared to other commonly used algorithms. Further, we demonstrate the utility of DISC algorithm to probe cooperativity between multiple binding events in the cyclic nucleotide binding domains of HCN pacemaker channel. Given the flexible and efficient nature of DISC, we anticipate it will be a powerful tool for unsupervised processing of high-throughput data across a range of single-molecule experiments.

Inferring effective forces for Langevin dynamics using Gaussian processes

Effective forces derived from experimental or in silico molecular dynamics time traces are critical in developing reduced and computationally efficient descriptions of otherwise complex dynamical problems. This helps motivate why it is important to develop methods to efficiently learn effective forces from time series data. A number of methods already exist to do this when data are plentiful but otherwise fail for sparse datasets or datasets where some regions of phase space are undersampled. In addition, any method developed to learn effective forces from time series data should be minimally a priori committal as to the shape of the effective force profile, exploit every data point without reducing data quality through any form of binning or pre-processing, and provide full credible intervals (error bars) about the prediction for the entirety of the effective force curve. Here, we propose a generalization of the Gaussian process, a key tool in Bayesian nonparametric inference and machine learning, which meets all of the above criteria in learning effective forces for the first time.

Observation of processive telomerase catalysis using high-resolution optical tweezers

Telomere maintenance by telomerase is essential for continuous proliferation of human cells and is vital for the survival of stem cells and 90% of cancer cells. To compensate for telomeric DNA lost during DNA replication, telomerase processively adds GGTTAG repeats to chromosome ends by copying the template region within its RNA subunit. Between repeat additions, the RNA template must be recycled. How telomerase remains associated with substrate DNA during this critical translocation step remains unknown. Using a newly developed single-molecule telomerase activity assay utilizing high-resolution optical tweezers, we demonstrate that stable substrate DNA binding at an anchor site within telomerase facilitates the processive synthesis of telomeric repeats. The product DNA synthesized by telomerase can be recaptured by the anchor site or fold into G-quadruplex structures. Our results provide detailed mechanistic insights into telomerase catalysis, a process of critical importance in aging and cancer.

Pitching single-focus confocal data analysis one photon at a time with Bayesian nonparametrics

Fluorescence time traces are used to report on dynamical properties of molecules. The basic unit of information in these traces is the arrival time of individual photons, which carry instantaneous information from the molecule, from which they are emitted, to the detector on timescales as fast as microseconds. Thus, it is theoretically possible to monitor molecular dynamics at such timescales from traces containing only a sufficient number of photon arrivals. In practice, however, traces are stochastic and in order to deduce dynamical information through traditional means-such as fluorescence correlation spectroscopy (FCS) and related techniques-they are collected and temporally autocorrelated over several minutes. So far, it has been impossible to analyze dynamical properties of molecules on timescales approaching data acquisition without collecting long traces under the strong assumption of stationarity of the process under observation or assumptions required for the analytic derivation of a correlation function. To avoid these assumptions, we would otherwise need to estimate the instantaneous number of molecules emitting photons and their positions within the confocal volume. As the number of molecules in a typical experiment is unknown, this problem demands that we abandon the conventional analysis paradigm. Here, we exploit Bayesian nonparametrics that allow us to obtain, in a principled fashion, estimates of the same quantities as FCS but from the direct analysis of traces of photon arrivals that are significantly smaller in size, or total duration, than those required by FCS.

Energetic dependencies dictate folding mechanism in a complex protein

Large proteins composed of multiple domains are abundant in all proteomes, but their folding and structural dynamics remain poorly understood. Using single-molecule force spectroscopy, we have defined how stabilizing interfaces among the domains of elongation factor G (EF-G) shape its folding pathway. Contrary to the expectation that multidomain proteins fold sequentially as they emerge from the ribosome, we find that folding cannot be completed until the full protein has been synthesized. This posttranslational folding mechanism results in a propensity for misfolding. It is dictated by an energetic coupling among domains that enables conformational flexibility crucial for EF-G function. EF-G thus provides an example of how distinct biological ends—robust folding and functionally important flexibility—come into conflict during protein biogenesis.

Large proteins with multiple domains are thought to fold cotranslationally to minimize interdomain misfolding. Once folded, domains interact with each other through the formation of extensive interfaces that are important for protein stability and function. However, multidomain protein folding and the energetics of domain interactions remain poorly understood. In elongation factor G (EF-G), a highly conserved protein composed of 5 domains, the 2 N-terminal domains form a stably structured unit cotranslationally. Using single-molecule optical tweezers, we have defined the steps leading to fully folded EF-G. We find that the central domain III of EF-G is highly dynamic and does not fold upon emerging from the ribosome. Surprisingly, a large interface with the N-terminal domains does not contribute to the stability of domain III. Instead, it requires interactions with its folded C-terminal neighbors to be stably structured. Because of the directionality of protein synthesis, this energetic dependency of domain III on its C-terminal neighbors disrupts cotranslational folding and imposes a posttranslational mechanism on the folding of the C-terminal part of EF-G. As a consequence, unfolded domains accumulate during synthesis, leading to the extensive population of misfolded species that interfere with productive folding. Domain III flexibility enables large-scale conformational transitions that are part of the EF-G functional cycle during ribosome translocation. Our results suggest that energetic tuning of domain stabilities, which is likely crucial for EF-G function, complicates the folding of this large multidomain protein.

An alternative framework for fluorescence correlation spectroscopy

Fluorescence correlation spectroscopy (FCS), is a widely used tool routinely exploited for in vivo and in vitro applications. While FCS provides estimates of dynamical quantities, such as diffusion coefficients, it demands high signal to noise ratios and long time traces, typically in the minute range. In principle, the same information can be extracted from microseconds to seconds long time traces; however, an appropriate analysis method is missing. To overcome these limitations, we adapt novel tools inspired by Bayesian non-parametrics, which starts from the direct analysis of the observed photon counts. With this approach, we are able to analyze time traces, which are too short to be analyzed by existing methods, including FCS. Our new analysis extends the capability of single molecule fluorescence confocal microscopy approaches to probe processes several orders of magnitude faster and permits a reduction of photo-toxic effects on living samples induced by long periods of light exposure.

Fluorescence correlation spectroscopy is widely used for in vivo and in vitro applications, yet extracting information from experiments still requires long acquisition times. Here, the authors exploit Bayesian non-parametrics to directly analyze the output of confocal fluorescence experiments thereby probing physical processes on much faster timescales.

A Bayesian Nonparametric Approach to Single Molecule Förster Resonance Energy Transfer

We develop a Bayesian nonparametric framework to analyze single molecule FRET (smFRET) data. This framework, a variation on infinite hidden Markov models, goes beyond traditional hidden Markov analysis, which already treats photon shot noise, in three critical ways: (1) it learns the number of molecular states present in a smFRET time trace (a hallmark of nonparametric approaches), (2) it accounts, simultaneously and self-consistently, for photophysical features of donor and acceptor fluorophores (blinking kinetics, spectral cross-talk, detector quantum efficiency), and (3) it treats background photons. Point 2 is essential in reducing the tendency of nonparametric approaches to overinterpret noisy single molecule time traces and so to estimate states and transition kinetics robust to photophysical artifacts. As a result, with the proposed framework, we obtain accurate estimates of single molecule properties even when the supplied traces are excessively noisy, subject to photoartifacts, and of short duration. We validate our method using synthetic data sets and demonstrate its applicability to real data sets from single molecule experiments on Holliday junctions labeled with conventional fluorescent dyes.

Preface: Special Topic on Single-Molecule Biophysics

Single-molecule measurements are now almost routinely used to study biological systems and processes. The scope of this special topic emphasizes the physics side of single-molecule observations, with the goal of highlighting new developments in physical techniques as well as conceptual insights that single-molecule measurements bring to biophysics. This issue also comprises recent advances in theoretical physical models of single-molecule phenomena, interpretation of single-molecule signals, and fundamental areas of statistical mechanics that are related to single-molecule observations. A particular goal is to illustrate the increasing synergy between theory, simulation, and experiment in single-molecule biophysics.

Accurate Protein-Folding Transition-Path Statistics from a Simple Free-Energy Landscape

A central goal of protein-folding theory is to predict the stochastic dynamics of transition paths-the rare trajectories that transit between the folded and unfolded ensembles-using only thermodynamic information, such as a low-dimensional equilibrium free-energy landscape. However, commonly used one-dimensional landscapes typically fall short of this aim, because an empirical coordinate-dependent diffusion coefficient has to be fit to transition-path trajectory data in order to reproduce the transition-path dynamics. We show that an alternative, first-principles free-energy landscape predicts transition-path statistics that agree well with simulations and single-molecule experiments without requiring dynamical data as an input. This "topological configuration" model assumes that distinct, native-like substructures assemble on a time scale that is slower than native-contact formation but faster than the folding of the entire protein. Using only equilibrium simulation data to determine the free energies of these coarse-grained intermediate states, we predict a broad distribution of transition-path transit times that agrees well with the transition-path durations observed in simulations. We further show that both the distribution of finite-time displacements on a one-dimensional order parameter and the ensemble of transition-path trajectories generated by the model are consistent with the simulated transition paths. These results indicate that a landscape based on transient folding intermediates, which are often hidden by one-dimensional projections, can form the basis of a predictive model of protein-folding transition-path dynamics.