Efficient, nonparametric removal of noise and recovery of probability distributions from time series using nonlinear-correlation functions: Photon and photon-counting noise

A preceding paper [M. Dhar, J. A. Dickinson, and M. A. Berg, J. Chem. Phys. 159, 054110 (2023)] shows how to remove additive noise from an experimental time series, allowing both the equilibrium distribution of the system and its Green's function to be recovered. The approach is based on nonlinear-correlation functions and is fully nonparametric: no initial model of the system or of the noise is needed. However, single-molecule spectroscopy often produces time series with either photon or photon-counting noise. Unlike additive noise, photon noise is signal-size correlated and quantized. Photon counting adds the potential for bias. This paper extends noise-corrected-correlation methods to these cases and tests them on synthetic datasets. Neither signal-size correlation nor quantization is a significant complication. Analysis of the sampling error yields guidelines for the data quality needed to recover the properties of a system with a given complexity. We show that bias in photon-counting data can be corrected, even at the high count rates needed to optimize the time resolution. Using all these results, we discuss the factors that limit the time resolution of single-molecule spectroscopy and the conditions that would be needed to push measurements into the submicrosecond region.

Revealing in real-time a multistep assembly mechanism for SV40 virus-like particles

Many viruses use their genome as template for self-assembly into an infectious particle. However, this reaction remains elusive because of the transient nature of intermediate structures. To elucidate this process, optical tweezers and acoustic force spectroscopy are used to follow viral assembly in real time. Using Simian virus 40 (SV40) virus-like particles as model system, we reveal a multistep assembly mechanism. Initially, binding of VP1 pentamers to DNA leads to a significantly decreased persistence length. Moreover, the pentamers seem able to stabilize DNA loops. Next, formation of interpentamer interactions results in intermediate structures with reduced contour length. These structures stabilize into objects that permanently decrease the contour length to a degree consistent with DNA compaction in wild-type SV40. These data indicate that a multistep mechanism leads to fully assembled cross-linked SV40 particles. SV40 is studied as drug delivery system. Our insights can help optimize packaging of therapeutic agents in these particles.

Real-Time Assembly of Viruslike Nucleocapsids Elucidated at the Single-Particle Level

While the structure of a multitude of viral particles has been resolved to atomistic detail, their assembly pathways remain largely elusive. Key unresolved issues are particle nucleation, particle growth, and the mode of genome compaction. These issues are difficult to address in bulk approaches and are effectively only accessible by the real-time tracking of assembly dynamics of individual particles. This we do here by studying the assembly into rod-shaped viruslike particles (VLPs) of artificial capsid polypeptides. Using fluorescence optical tweezers, we establish that small oligomers perform one-dimensional diffusion along the DNA. Larger oligomers are immobile and nucleate VLP growth. A multiplexed acoustic force spectroscopy approach reveals that DNA is compacted in regular steps, suggesting packaging via helical wrapping into a nucleocapsid. By reporting how real-time assembly tracking elucidates viral nucleation and growth principles, our work opens the door to a fundamental understanding of the complex assembly pathways of both VLPs and naturally evolved viruses.

Statistical Learning of Discrete States in Time Series

Time series obtained from time-dependent experiments contain rich information on kinetics and dynamics of the system under investigation. This work describes an unsupervised learning framework, along with the derivation of the necessary analytical expressions, for the analysis of Gaussian-distributed time series that exhibit discrete states. After the time series has been partitioned into segments in a model-free manner using the previously developed change-point (CP) method, this protocol starts with an agglomerative hierarchical clustering algorithm to classify the detected segments into possible states. The initial state clustering is further refined using an expectation-maximization (EM) procedure, and the number of states is determined by a Bayesian information criterion (BIC). Also introduced here is an achievement scalarization function, usually seen in artificial intelligence literature, for quantitatively assessing the performance of state determination. The statistical learning framework, which is comprised of three stages, detection of signal change, clustering, and number-of-state determination, was thoroughly characterized using simulated trajectories with random intensity segments that have no underlying kinetics, and its performance was critically evaluated. The application to experimental data is also demonstrated. The results suggested that this general framework, the implementation of which is based on firm theoretical foundations and does not require the imposition of any kinetics model, is powerful in determining the number of states, the parameters contained in each state, as well as the associated statistical significance.

Detection of kinetic change points in piece-wise linear single molecule motion

Single-molecule approaches present a powerful way to obtain detailed kinetic information at the molecular level. However, the identification of small rate changes is often hindered by the considerable noise present in such single-molecule kinetic data. We present a general method to detect such kinetic change points in trajectories of motion of processive single molecules having Gaussian noise, with a minimum number of parameters and without the need of an assumed kinetic model beyond piece-wise linearity of motion. Kinetic change points are detected using a likelihood ratio test in which the probability of no change is compared to the probability of a change occurring, given the experimental noise. A predetermined confidence interval minimizes the occurrence of false detections. Applying the method recursively to all sub-regions of a single molecule trajectory ensures that all kinetic change points are located. The algorithm presented allows rigorous and quantitative determination of kinetic change points in noisy single molecule observations without the need for filtering or binning, which reduce temporal resolution and obscure dynamics. The statistical framework for the approach and implementation details are discussed. The detection power of the algorithm is assessed using simulations with both single kinetic changes and multiple kinetic changes that typically arise in observations of single-molecule DNA-replication reactions. Implementations of the algorithm are provided in ImageJ plugin format written in Java and in the Julia language for numeric computing, with accompanying Jupyter Notebooks to allow reproduction of the analysis presented here.

Parallelization of Change Point Detection

The change point detection method ( Watkins , L. P. ; Yang , H. J. Phys. Chem. B 2005 , 109 , 617 ) allows the objective identification and isolation of abrupt changes along a data series. Because this method is grounded in statistical tests, it is particularly powerful for probing complex and noisy signals without artificially imposing a kinetics model. The original algorithm, however, has a time complexity of [Formula: see text], where N is the size of the data and is, therefore, limited in its scalability. This paper puts forth a parallelization of change point detection to address these time and memory constraints. This parallelization method was evaluated by applying it to changes in the mean of Gaussian-distributed data and found that time decreases superlinearly with respect to the number of processes (i.e., parallelization with two processes takes less than half of the time of one process). Moreover, there was minimal reduction in detection power. These results suggest that our parallelization algorithm is a viable scheme that can be implemented for other change point detection methods.

Competition between Bending and Internal Pressure Governs the Mechanics of Fluid Nanovesicles

Nanovesicles (∼100 nm) are ubiquitous in cell biology and an important vector for drug delivery. Mechanical properties of vesicles are known to influence cellular uptake, but the mechanism by which deformation dynamics affect internalization is poorly understood. This is partly due to the fact that experimental studies of the mechanics of such vesicles remain challenging, particularly at the nanometer scale where appropriate theoretical models have also been lacking. Here, we probe the mechanical properties of nanoscale liposomes using atomic force microscopy (AFM) indentation. The mechanical response of the nanovesicles shows initial linear behavior and subsequent flattening corresponding to inward tether formation. We derive a quantitative model, including the competing effects of internal pressure and membrane bending, that corresponds well to these experimental observations. Our results are consistent with a bending modulus of the lipid bilayer of ∼14kbT. Surprisingly, we find that vesicle stiffness is pressure dominated for adherent vesicles under physiological conditions. Our experimental method and quantitative theory represents a robust approach to study the mechanics of nanoscale vesicles, which are abundant in biology, as well as being of interest for the rational design of liposomal vectors for drug delivery.

Extraction of Protein Conformational Modes from Distance Distributions Using Structurally Imputed Bayesian Data Augmentation

Protein conformational changes are known to play important roles in assorted biochemical and biological processes. Driven by thermal motions of surrounding solvent molecules, such a structural remodeling often occurs stochastically. Yet, regardless of how random the conformational reconfiguration may appear, it could in principle be described by a linear combination of a set of orthogonal modes which, in turn, are contained in the intramolecular distance distributions. The central challenge is how to obtain the distribution. This contribution proposes a Bayesian data-augmentation scheme to extract the predominant modes from only few distance distributions, be they from computational sampling or directly from experiments such as single-molecule Förster-type resonance energy transfer (smFRET). The inference of the complete protein structure from insufficient data was recognized as isomorphic to the missing-data problem in Bayesian statistical learning. Using smFRET data as an example, the missing coordinates were deduced, given protein structural constraints and multiple but limited number of smFRET distances; the Boltzmann weighing of each inferred protein structure was then evaluated using computational modeling to numerically construct the posterior density for the global protein conformation. The conformational modes were then determined from the iteratively converged overall conformational distribution using principal component analysis. Two examples were presented to illustrate these basic ideas as well as their practical implementation. The scheme described herein was based on the theory behind the powerful Tanner-Wang algorithm that guarantees convergence to the true posterior density. However, instead of assuming a mathematical model to calculate the likelihood as in conventional statistical inference, here the protein structure was treated as a statistical parameter and was imputed from the numerical likelihood function based on structural information, a probability model-free method. The framework put forth here is anticipated to be generally applicable, offering a new way to articulate protein conformational changes in a quantifiable manner.

Regulation of Mutagenic DNA Polymerase V Activation in Space and Time

Spatial regulation is often encountered as a component of multi-tiered regulatory systems in eukaryotes, where processes are readily segregated by organelle boundaries. Well-characterized examples of spatial regulation are less common in bacteria. Low-fidelity DNA polymerase V (UmuD′₂C) is produced in Escherichia coli as part of the bacterial SOS response to DNA damage. Due to the mutagenic potential of this enzyme, pol V activity is controlled by means of an elaborate regulatory system at transcriptional and posttranslational levels. Using single-molecule fluorescence microscopy to visualize UmuC inside living cells in space and time, we now show that pol V is also subject to a novel form of spatial regulation. After an initial delay (~ 45 min) post UV irradiation, UmuC is synthesized, but is not immediately activated. Instead, it is sequestered at the inner cell membrane. The release of UmuC into the cytosol requires the RecA* nucleoprotein filament-mediated cleavage of UmuD→UmuD′. Classic SOS damage response mutants either block [umuD(K97A)] or constitutively stimulate [recA(E38K)] UmuC release from the membrane. Foci of mutagenically active pol V Mut (UmuD′₂C-RecA-ATP) formed in the cytosol after UV irradiation do not co-localize with pol III replisomes, suggesting a capacity to promote translesion DNA synthesis at lesions skipped over by DNA polymerase III. In effect, at least three molecular mechanisms limit the amount of time that pol V has to access DNA: (1) transcriptional and posttranslational regulation that initially keep the intracellular levels of pol V to a minimum; (2) spatial regulation via transient sequestration of UmuC at the membrane, which further delays pol V activation; and (3) the hydrolytic activity of a recently discovered pol V Mut ATPase function that limits active polymerase time on the chromosomal template.

Escherichia coli, and many other bacteria, respond to high levels of DNA damage with an inducible system called the SOS response. In this response, bacteria first try to restart replication using non-mutagenic DNA repair strategies. If that fails, replication can be restored using DNA polymerases that simply replicate over DNA lesions, a desperation strategy that results in mutations. DNA polymerase V (pol V) is responsible for most mutagenesis that accompanies the SOS response. Because of the risk inherent to elevated mutation levels, pol V activation is tightly constrained. This report introduces a new layer of regulation on pol V activation, with a novel spatial component. After synthesis, the UmuC subunit of pol V is sequestered transiently at the membrane. Release into the cytosol and final activation depends on the activity of RecA protein and the autocatalytic cleavage of UmuD to generate the UmuD' subunit of pol V. The resulting delay in activation represents an additional molecular mechanism that limits the amount of time that this sometimes necessary but potentially detrimental enzyme spends on the DNA.

smFRET studies of the 'encounter' complexes and subsequent intermediate states that regulate the selectivity of ligand binding

The selectivity with which a biomolecule can bind its cognate ligand when confronted by the vast array of structurally similar, competing ligands that are present in the cell underlies the fidelity of some of the most fundamental processes in biology. Because they collectively comprise one of only a few methods that can sensitively detect the 'encounter' complexes and subsequent intermediate states that regulate the selectivity of ligand binding, single-molecule fluorescence, and particularly single-molecule fluorescence resonance energy transfer (smFRET), approaches have revolutionized studies of ligand-binding reactions. Here, we describe a widely used smFRET strategy that enables investigations of a large variety of ligand-binding reactions, and discuss two such reactions, aminoacyl-tRNA selection during translation elongation and splice site selection during spliceosome assembly, that highlight both the successes and challenges of smFRET studies of ligand-binding reactions. We conclude by reviewing a number of emerging experimental and computational approaches that are expanding the capabilities of smFRET approaches for studies of ligand-binding reactions and that promise to reveal the mechanisms that control the selectivity of ligand binding with unprecedented resolution.