Improved hidden Markov models for molecular motors, part 1: basic theory

DNA replication: In vitro single-molecule manipulation data analysis and models

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Data analysis allows to extract information from the noisy single-molecule data.

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Models provide insight in the underlying biochemical processes.

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Ligands can activate or inhibit DNA replication and DNA unwinding.

DNA replication is a key biochemical process of the cell cycle. In the last years, analysis of in vitro single-molecule DNA replication events has provided new information that cannot be obtained with ensembles studies. Here, we introduce crucial techniques for the proper analysis and modelling of DNA replication in vitro single-molecule manipulation data. Specifically, we review some of the main methods to analyze and model the real-time kinetics of the two main molecular motors of the replisome: DNA polymerase and DNA helicase. Our goal is to facilitate access to and understanding of these techniques to promotetheir use in the study of DNA replication at the single-molecule level. A proper analysis of single-molecule data is crucial to obtain a detailed picture of, among others, the kinetics rates, equilibrium contants and conformational changes of the system under study. The techniques presented here have been used or can be adapted to study the operation of other proteins involved in nucleic acids metabolism.

Direct Measurement of Single-Molecule Ligand-Receptor Interactions

Measuring the kinetics that govern ligand-receptor interactions is fundamental to our understanding of pharmacology. For ligand-gated ion channels, binding of an agonist triggers allosteric motions that open an integral ion-permeable pore. By mathematically modeling stochastic electrophysiological responses with high temporal resolution (ms), previous single channel studies have been able to infer the rate constants of ligands binding to these receptors. However, there are no reports of the direct measurement of the single-molecule binding events that are vital to how agonists exert their functional effects. For the first time, we report these direct measurements, the rate constants, and corresponding free energy changes, which describe the transitions between the different binding states. To achieve this, we use the super resolution technique: points accumulation for imaging in nanoscale topography (PAINT) to observe binding of ATP to orthosteric binding sites on the P2X1 receptor. Furthermore, an analysis of time-resolved single-molecule interactions is used to measure elementary rate constants and thermodynamic forces that drive the allosteric motions. These single-molecule measurements unequivocally establish the location of each binding states of the P2X1 receptor and the stochastic nature of the interaction with its native ligand. The analysis leads to the measurement of the forward and reverse rates from a weak ligand-binding state to a strong ligand binding state that is linked to allosteric motion and ion pore formation. These rates (k_α = 1.41 sec^-1 and k_β = 0.32 sec^-1) were then used to determine the free energy associated with this critical mechanistic step (3.7 kJ/mol). Importantly, the described methods can be readily applied to all ligand-gated ion channels, and more broadly to the molecular interactions of other classes of membrane proteins.

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.

Full molecular trajectories of RNA polymerase at single base-pair resolution

Optical tweezers enable scientists to follow the dynamics of molecular motors at high resolution. The ability to discern a motor’s discrete steps reveals important insights on its operation. Some motors operate at the scale of angstroms, rendering the observation of their steps extremely challenging. In some cases, such small steps have been observed sporadically; however, the full molecular trajectories of steps and intervals between steps remain elusive due to instrumental noise. Here, we eliminate the main source of noise of most high-resolution dual-trap optical tweezers and developed both a single-molecule assay and a self-learning algorithm to uncover the full trajectories of such a motor: RNA polymerase. Using this method, a whole new set of experiments becomes possible.

In recent years, highly stable optical tweezers systems have enabled the characterization of the dynamics of molecular motors at very high resolution. However, the motion of many motors with angstrom-scale dynamics cannot be consistently resolved due to poor signal-to-noise ratio. Using an acousto-optic deflector to generate a “time-shared” dual-optical trap, we decreased low-frequency noise by more than one order of magnitude compared with conventional dual-trap optical tweezers. Using this instrument, we implemented a protocol that synthesizes single base-pair trajectories, which are used to test a Large State Space Hidden Markov Model algorithm to recover their individual steps. We then used this algorithm on real transcription data obtained in the same instrument to fully uncover the molecular trajectories of Escherichia coli RNA polymerase. We applied this procedure to reveal the effect of pyrophosphate on the distribution of dwell times between consecutive polymerase steps.

Estimating kinetic mechanisms with prior knowledge I: Linear parameter constraints

New mathematical tools are needed to incorporate existing knowledge into kinetic models of ion channels and other proteins. Salari et al. describe an algebraic transformation that can enforce linearly interdependent parameters into kinetic models in order to test new hypotheses.

To understand how ion channels and other proteins function at the molecular and cellular levels, one must decrypt their kinetic mechanisms. Sophisticated algorithms have been developed that can be used to extract kinetic parameters from a variety of experimental data types. However, formulating models that not only explain new data, but are also consistent with existing knowledge, remains a challenge. Here, we present a two-part study describing a mathematical and computational formalism that can be used to enforce prior knowledge into the model using constraints. In this first part, we focus on constraints that enforce explicit linear relationships involving rate constants or other model parameters. We develop a simple, linear algebra–based transformation that can be applied to enforce many types of model properties and assumptions, such as microscopic reversibility, allosteric gating, and equality and inequality parameter relationships. This transformation converts the set of linearly interdependent model parameters into a reduced set of independent parameters, which can be passed to an automated search engine for model optimization. In the companion article, we introduce a complementary method that can be used to enforce arbitrary parameter relationships and any constraints that quantify the behavior of the model under certain conditions. The procedures described in this study can, in principle, be coupled to any of the existing methods for solving molecular kinetics for ion channels or other proteins. These concepts can be used not only to enforce existing knowledge but also to formulate and test new hypotheses.

Enzyme-Mediated Directional Transport of a Small-Molecule Walker With Chemically Identical Feet

We describe a small-molecule "walker" that uses enzyme catalysis to discriminate between the relative positions of its "feet" on a track and thereby move with net directionality. The bipedal walker has identical carboxylic acid feet, and "steps" along an isotactic hydroxyl-group-derivatized polyether track by the formation/breakage of ester linkages. Lipase AS catalyzes the selective hydrolysis of the rear foot of macrocyclized walkers (an information ratchet mechanism), the rear foot producing an (R)-stereocenter at its point of attachment to the track. If the hydrolyzed foot reattaches to the track in front of the bound foot it forms an (S)-stereocenter, which is resistant to enzymatic hydrolysis. Only macrocyclic walker-track conjugates are efficiently hydrolyzed by the enzyme, leading to high processivity of the walker movement along the track. Conventional chemical reagents promote formation of the ester bonds between the walker and the track. Iterative macrocyclization and hydrolysis reactions lead to 68% of walkers taking two steps directionally along a three-foothold track.

Hidden Markov Modeling with Detailed Balance and Its Application to Single Protein Folding

Hidden Markov modeling (HMM) has revolutionized kinetic studies of macromolecules. However, results from HMM often violate detailed balance when applied to the transitions under thermodynamic equilibrium, and the consequence of such violation has not been well understood. Here, to our knowledge, we developed a new HMM method that satisfies detailed balance (HMM-DB) and optimizes model parameters by gradient search. We used free energy of stable and transition states as independent fitting parameters and considered both normal and skew normal distributions of the measurement noise. We validated our method by analyzing simulated extension trajectories that mimicked experimental data of single protein folding from optical tweezers. We then applied HMM-DB to elucidate kinetics of regulated SNARE zippering containing degenerate states. For both simulated and measured trajectories, we found that HMM-DB significantly reduced overfitting of short trajectories compared to the standard HMM based on an expectation-maximization algorithm, leading to more accurate and reliable model fitting by HMM-DB. We revealed how HMM-DB could be conveniently used to derive a simplified energy landscape of protein folding. Finally, we extended HMM-DB to correct the baseline drift in single-molecule trajectories. Together, we demonstrated an efficient, versatile, and reliable method of HMM for kinetics studies of macromolecules under thermodynamic equilibrium.

Interferometric Scattering Microscopy for the Study of Molecular Motors

Our understanding of molecular motor function has been greatly improved by the development of imaging modalities, which enable real-time observation of their motion at the single-molecule level. Here, we describe the use of a new method, interferometric scattering microscopy, for the investigation of motor protein dynamics by attaching and tracking the motion of metallic nanoparticle labels as small as 20nm diameter. Using myosin-5, kinesin-1, and dynein as examples, we describe the basic assays, labeling strategies, and principles of data analysis. Our approach is relevant not only for motor protein dynamics but also provides a general tool for single-particle tracking with high spatiotemporal precision, which overcomes the limitations of single-molecule fluorescence methods.

Energy-based scheme for reconstruction of piecewise constant signals observed in the movement of molecular machines

Analyzing the physical and chemical properties of single DNA-based molecular machines such as polymerases and helicases requires to track stepping motion on the length scale of base pairs. Although high-resolution instruments have been developed that are capable of reaching that limit, individual steps are oftentimes hidden by experimental noise which complicates data processing. Here we present an effective two-step algorithm which detects steps in a high-bandwidth signal by minimizing an energy-based model (energy-based step finder, EBS). First, an efficient convex denoising scheme is applied which allows compression to tuples of amplitudes and plateau lengths. Second, a combinatorial clustering algorithm formulated on a graph is used to assign steps to the tuple data while accounting for prior information. Performance of the algorithm was tested on Poissonian stepping data simulated based on published kinetics data of RNA polymerase II (pol II). Comparison to existing step-finding methods shows that EBS is superior in speed while providing competitive step-detection results, especially in challenging situations. Moreover, the capability to detect backtracked intervals in experimental data of pol II as well as to detect stepping behavior of the Phi29 DNA packaging motor is demonstrated.

Kinetics of nucleotide-dependent structural transitions in the kinesin-1 hydrolysis cycle

To dissect the kinetics of structural transitions underlying the stepping cycle of kinesin-1 at physiological ATP, we used interferometric scattering microscopy to track the position of gold nanoparticles attached to individual motor domains in processively stepping dimers. Labeled heads resided stably at positions 16.4 nm apart, corresponding to a microtubule-bound state, and at a previously unseen intermediate position, corresponding to a tethered state. The chemical transitions underlying these structural transitions were identified by varying nucleotide conditions and carrying out parallel stopped-flow kinetics assays. At saturating ATP, kinesin-1 spends half of each stepping cycle with one head bound, specifying a structural state for each of two rate-limiting transitions. Analysis of stepping kinetics in varying nucleotides shows that ATP binding is required to properly enter the one-head-bound state, and hydrolysis is necessary to exit it at a physiological rate. These transitions differ from the standard model in which ATP binding drives full docking of the flexible neck linker domain of the motor. Thus, this work defines a consensus sequence of mechanochemical transitions that can be used to understand functional diversity across the kinesin superfamily.

Effective rates from thermodynamically consistent coarse-graining of models for molecular motors with probe particles

Many single-molecule experiments for molecular motors comprise not only the motor but also large probe particles coupled to it. The theoretical analysis of these assays, however, often takes into account only the degrees of freedom representing the motor. We present a coarse-graining method that maps a model comprising two coupled degrees of freedom which represent motor and probe particle to such an effective one-particle model by eliminating the dynamics of the probe particle in a thermodynamically and dynamically consistent way. The coarse-grained rates obey a local detailed balance condition and reproduce the net currents. Moreover, the average entropy production as well as the thermodynamic efficiency is invariant under this coarse-graining procedure. Our analysis reveals that only by assuming unrealistically fast probe particles, the coarse-grained transition rates coincide with the transition rates of the traditionally used one-particle motor models. Additionally, we find that for multicyclic motors the stall force can depend on the probe size. We apply this coarse-graining method to specific case studies of the F(1)-ATPase and the kinesin motor.

Extracting physics of life at the molecular level: A review of single-molecule data analyses

Studying individual biomolecules at the single-molecule level has proved very insightful recently. Single-molecule experiments allow us to probe both the equilibrium and nonequilibrium properties as well as make quantitative connections with ensemble experiments and equilibrium thermodynamics. However, it is important to be careful about the analysis of single-molecule data because of the noise present and the lack of theoretical framework for processes far away from equilibrium. Biomolecular motion, whether it is free in solution, on a substrate, or under force, involves thermal fluctuations in varying degrees, which makes the motion noisy. In addition, the noise from the experimental setup makes it even more complex. The details of biologically relevant interactions, conformational dynamics, and activities are hidden in the noisy single-molecule data. As such, extracting biological insights from noisy data is still an active area of research. In this review, we will focus on analyzing both fluorescence-based and force-based single-molecule experiments and gaining biological insights at the single-molecule level. Inherently nonequilibrium nature of biological processes will be highlighted. Simulated trajectories of biomolecular diffusion will be used to compare and validate various analysis techniques.

Fluorescence tracking of motor proteins in vitro

Motor proteins convert the chemical energy of adenosine triphosphate (ATP) hydrolysis into directed movement along filamentous tracks, such as DNA, microtubule, and actin. The motile properties of motors are essential to their wide variety of cellular functions, including cargo transport, mitosis, cell motility, nuclear positioning, and ciliogenesis. Detailed understanding of the biophysical mechanisms of motor motility is therefore essential to understanding the physical basis of these processes. In which direction is the motor going? How fast and how far can a single motor walk down its track? How is ATP hydrolysis coupled to directed motion? How do multiple subunits of a motor coordinate with each other during motility? These questions can be addressed directly by tracking motors at a single-molecule level. This chapter will focus on high-resolution fluorescence tracking techniques of the processive cytoskeletal motors: myosins, kinesins, and cytoplasmic dynein. We outline the theoretical and practical considerations for studying these motors in vitro using fluorescence tracking at nanometer precision.

Determination of parameter identifiability in nonlinear biophysical models: A Bayesian approach

A major goal of biophysics is to understand the physical mechanisms of biological molecules and systems. Mechanistic models are evaluated based on their ability to explain carefully controlled experiments. By fitting models to data, biophysical parameters that cannot be measured directly can be estimated from experimentation. However, it might be the case that many different combinations of model parameters can explain the observations equally well. In these cases, the model parameters are not identifiable: the experimentation has not provided sufficient constraining power to enable unique estimation of their true values. We demonstrate that this pitfall is present even in simple biophysical models. We investigate the underlying causes of parameter non-identifiability and discuss straightforward methods for determining when parameters of simple models can be inferred accurately. However, for models of even modest complexity, more general tools are required to diagnose parameter non-identifiability. We present a method based in Bayesian inference that can be used to establish the reliability of parameter estimates, as well as yield accurate quantification of parameter confidence.

SOLVING ION CHANNEL KINETICS WITH THE QuB SOFTWARE

The ability to record the currents from single ion channels led to the need to extract the underlying kinetic model from such data. This inverse hidden Markov problem is difficult but led to the creation of a software suite called QuB utilizing likelihood optimization. This review presents the software. The software is open source and, in addition to solving kinetic models, has many generic database operations including report generation with publishable graphics, function fitting and scripting for new and repeated processing and AD/DA I/O. The core algorithms allow for constraints such as fixed rates or maintaining detailed balance in the model. All rate constants can be driven by a stimulus and the system can analyze nonstationary data. QuB also can analyze the kinetics of multichannel data where individual events cannot be discriminated, but the fitting algorithms utilize the signal variance as well as the mean to fit models. QuB can be applied to any data appropriately modeled with Markov kinetics and has been utilized to solve ion channels but also the movement of motor proteins, the sleep cycles in mice, and physics processes.

[Formula: see text]Special Issue Comment: This is a review about the software QuB that can extract a model from the trajectory. It is connected with the review about treatments when solving single molecules,60 and the reviews about enzymes.61,62

High-resolution optical tweezers for single-molecule manipulation

Forces hold everything together and determine its structure and dynamics. In particular, tiny forces of 1-100 piconewtons govern the structures and dynamics of biomacromolecules. These forces enable folding, assembly, conformational fluctuations, or directional movements of biomacromolecules over sub-nanometer to micron distances. Optical tweezers have become a revolutionary tool to probe the forces, structures, and dynamics associated with biomacromolecules at a single-molecule level with unprecedented resolution. In this review, we introduce the basic principles of optical tweezers and their latest applications in studies of protein folding and molecular motors. We describe the folding dynamics of two strong coiled coil proteins, the GCN4-derived protein pIL and the SNARE complex. Both complexes show multiple folding intermediates and pathways. ATP-dependent chromatin remodeling complexes translocate DNA to remodel chromatin structures. The detailed DNA translocation properties of such molecular motors have recently been characterized by optical tweezers, which are reviewed here. Finally, several future developments and applications of optical tweezers are discussed. These past and future applications demonstrate the unique advantages of high-resolution optical tweezers in quantitatively characterizing complex multi-scale dynamics of biomacromolecules.

Step detection in single-molecule real time trajectories embedded in correlated noise

Single-molecule real time trajectories are embedded in high noise. To extract kinetic or dynamic information of the molecules from these trajectories often requires idealization of the data in steps and dwells. One major premise behind the existing single-molecule data analysis algorithms is the gaussian 'white' noise, which displays no correlation in time and whose amplitude is independent on data sampling frequency. This so-called 'white' noise is widely assumed but its validity has not been critically evaluated. We show that correlated noise exists in single-molecule real time trajectories collected from optical tweezers. The assumption of white noise during analysis of these data can lead to serious over- or underestimation of the number of steps depending on the algorithms employed. We present a statistical method that quantitatively evaluates the structure of the underlying noise, takes the noise structure into account, and identifies steps and dwells in a single-molecule trajectory. Unlike existing data analysis algorithms, this method uses Generalized Least Squares (GLS) to detect steps and dwells. Under the GLS framework, the optimal number of steps is chosen using model selection criteria such as Bayesian Information Criterion (BIC). Comparison with existing step detection algorithms showed that this GLS method can detect step locations with highest accuracy in the presence of correlated noise. Because this method is automated, and directly works with high bandwidth data without pre-filtering or assumption of gaussian noise, it may be broadly useful for analysis of single-molecule real time trajectories.

Signal processing for molecular and cellular biological physics: an emerging field

Recent advances in our ability to watch the molecular and cellular processes of life in action--such as atomic force microscopy, optical tweezers and Forster fluorescence resonance energy transfer--raise challenges for digital signal processing (DSP) of the resulting experimental data. This article explores the unique properties of such biophysical time series that set them apart from other signals, such as the prevalence of abrupt jumps and steps, multi-modal distributions and autocorrelated noise. It exposes the problems with classical linear DSP algorithms applied to this kind of data, and describes new nonlinear and non-Gaussian algorithms that are able to extract information that is of direct relevance to biological physicists. It is argued that these new methods applied in this context typify the nascent field of biophysical DSP. Practical experimental examples are supplied.

Single reconstituted neuronal SNARE complexes zipper in three distinct stages

Soluble N-ethylmaleimide-sensitive factor attachment protein receptor (SNARE) proteins drive membrane fusion by assembling into a four-helix bundle in a zippering process. Here, we used optical tweezers to observe in a cell-free reconstitution experiment in real time a long-sought SNARE assembly intermediate in which only the membrane-distal amino-terminal half of the bundle is assembled. Our findings support the zippering hypothesis, but suggest that zippering proceeds through three sequential binary switches, not continuously, in the amino- and carboxyl-terminal halves of the bundle and the linker domain. The half-zippered intermediate was stabilized by externally applied force that mimicked the repulsion between apposed membranes being forced to fuse. This intermediate then rapidly and forcefully zippered, delivering free energy of 36 k(B)T (where k(B) is Boltzmann's constant and T is temperature) to mediate fusion.

A three-compartment chemically-driven molecular information ratchet

We describe a three-compartment rotaxane information ratchet in which the macrocycle can be directionally transported in either direction along an achiral (disregarding isotopic labeling) track. Chiral DMAP-based catalysts promote a benzoylation reaction that ratchets the displacement of the macrocycle, transporting it predominantly to a particular end compartment determined by the handedness of the catalyst.

Fluorescence imaging with one nanometer accuracy: in vitro and in vivo studies of molecular motors

Traditional microscopy techniques are limited by the wave-like characteristics of light, which dictate that about 250 nm (or roughly half the wavelength of the light) is the smallest distance by which two identical objects can be separated while still being able to distinguish between them. Since most biological molecules are much smaller than this limit, traditional light microscopes are generally not sufficient for single-molecule biological studies. Fluorescence Imaging with One Nanometer Accuracy (FIONA) is a technique that makes possible localization of an object to approximately one nanometer. The FIONA technique is simple in concept; it is built upon the idea that, if enough photons are collected, one can find the exact center of a fluorophore's emission to within a single nanometer and track its motion with a very high level of precision. The center can be localized to approximately (λ/2)/Ö-N, where λ is the wavelength of the light and N is the number of photons collected. When N  =  10,000, FIONA achieves an accuracy of 1-2 nm, assuming the background is sufficiently low. FIONA, thus, works best with the use of high-quality dyes and fluorescence stabilization buffers, sensitive detection methods, and special microscopy techniques to reduce background fluorescence. FIONA is particularly well suited to the study of molecular motors, which are enzymes that couple ATP hydrolysis to conformational change and motion. In this chapter, we discuss the practical application of FIONA to molecular motors or other enzymes in biological systems.

The RSC chromatin remodelling ATPase translocates DNA with high force and small step size

ATP-dependent chromatin remodelling complexes use the energy of ATP hydrolysis to reposition and reconfigure nucleosomes. Despite their diverse functions, all remodellers share highly conserved ATPase domains, many shown to translocate DNA. Understanding remodelling requires biophysical knowledge of the DNA translocation process: how the ATPase moves DNA and generates force, and how translocation and force generation are coupled on nucleosomes. Here, we characterize the real-time activity of a minimal RSC translocase 'motor' on bare DNA, using high-resolution optical tweezers and a 'tethered' translocase system. We observe on dsDNA a processivity of ∼35 bp, a speed of ∼25 bp/s, and a step size of 2.0 (±0.4, s.e.m.) bp. Surprisingly, the motor is capable of moving against high force, up to 30 pN, making it one of the most force-resistant motors known. We also provide evidence for DNA 'buckling' at initiation. These observations reveal the ATPase as a powerful DNA translocating motor capable of disrupting DNA-histone interactions by mechanical force.

Improved hidden Markov models for molecular motors, part 2: extensions and application to experimental data

Unbiased interpretation of noisy single molecular motor recordings remains a challenging task. To address this issue, we have developed robust algorithms based on hidden Markov models (HMMs) of motor proteins. The basic algorithm, called variable-stepsize HMM (VS-HMM), was introduced in the previous article. It improves on currently available Markov-model based techniques by allowing for arbitrary distributions of step sizes, and shows excellent convergence properties for the characterization of staircase motor timecourses in the presence of large measurement noise. In this article, we extend the VS-HMM framework for better performance with experimental data. The extended algorithm, variable-stepsize integrating-detector HMM (VSI-HMM) better models the data-acquisition process, and accounts for random baseline drifts. Further, as an extension, maximum a posteriori estimation is provided. When used as a blind step detector, the VSI-HMM outperforms conventional step detectors. The fidelity of the VSI-HMM is tested with simulations and is applied to in vitro myosin V data where a small 10 nm population of steps is identified. It is also applied to an in vivo recording of melanosome motion, where strong evidence is found for repeated, bidirectional steps smaller than 8 nm in size, implying that multiple motors simultaneously carry the cargo.