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

Hidden Markov models (HMMs) provide an excellent analysis of recordings with very poor signal/noise ratio made from systems such as ion channels which switch among a few states. This method has also recently been used for modeling the kinetic rate constants of molecular motors, where the observable variable—the position—steadily accumulates as a result of the motor's reaction cycle. We present a new HMM implementation for obtaining the chemical-kinetic model of a molecular motor's reaction cycle called the variable-stepsize HMM in which the quantized position variable is represented by a large number of states of the Markov model. Unlike previous methods, the model allows for arbitrary distributions of step sizes, and allows these distributions to be estimated. The result is a robust algorithm that requires little or no user input for characterizing the stepping kinetics of molecular motors as recorded by optical techniques.     

Random Walk of Processive, Quantum Dot-Labeled Myosin Va Molecules within the Actin Cortex of COS-7 Cells

Myosin Va (myoVa) is an actin-based intracellular cargo transporter. In vitro experiments have established that a single myoVa moves processively along actin tracks, but less is known about how this motor operates within cells. Here we track the movement of a quantum dot (Qdot)-labeled myoVa HMM in COS-7 cells using total internal reflectance fluorescence microscopy. This labeling approach is unique in that it allows myoVa, instead of its cargo, to be tracked. Single-particle analysis showed short periods (≤0.5 s) of ATP-sensitive linear motion. The mean velocity of these trajectories was 604 nm/s and independent of the number of myoVa molecules attached to the Qdot. With high time (16.6 ms) and spatial (15 nm) resolution imaging, Qdot-labeled myoVa moved with sequential 75 nm steps per head, at a rate of 16 s⁻¹, similarly to myoVa in vitro. Monte Carlo modeling suggests that the random nature of the trajectories represents processive myoVa motors undergoing a random walk through the dense and randomly oriented cortical actin network.     

Tracking single particles and elongated filaments with nanometer precision

Recent developments in image processing have greatly advanced our understanding of biomolecular processes in vitro and in vivo. In particular, using Gaussian models to fit the intensity profiles of nanometer-sized objects have enabled their two-dimensional localization with a precision in the one-nanometer range. Here, we present an algorithm to precisely localize curved filaments whose structures are characterized by subresolution diameters and micrometer lengths. Using surface-immobilized microtubules, fluorescently labeled with rhodamine, we demonstrate positional precisions of ∼2 nm when determining the filament centerline and ∼9 nm when localizing the filament tips. Combined with state-of-the-art single particle tracking we apply the algorithm 1), to motor-proteins stepping on immobilized microtubules, 2), to depolymerizing microtubules, and 3), to microtubules gliding over motor-coated surfaces.     

Extracting dwell time sequences from processive molecular motor data

Processive molecular motors, such as kinesin, myosin, or dynein, convert chemical energy into mechanical energy by hydrolyzing ATP. The mechanical energy is used for moving in discrete steps along the cytoskeleton and carrying a molecular load. Single-molecule recordings of motor position along a substrate polymer appear as a stochastic staircase. Recordings of other single molecules, such as F1-ATPase, RNA polymerase, or topoisomerase, have the same appearance. We present a maximum likelihood algorithm that extracts the dwell time sequence from noisy data, and estimates state transition probabilities and the distribution of the motor step size. The algorithm can handle models with uniform or alternating step sizes, and reversible or irreversible kinetics. A periodic Markov model describes the repetitive chemistry of the motor, and a Kalman filter allows one to include models with variable step size and to correct for baseline drift. The data are optimized recursively and globally over single or multiple data sets, making the results objective over the full scale of the data. Local binary algorithms, such as the t-test, do not represent the behavior of the whole data set. Our method is model-based, and allows rapid testing of different models by comparing the likelihood scores. From data obtained with current technology, steps as small as 8 nm can be resolved and analyzed with our method. The kinetic consequences of the extracted dwell sequence can be further analyzed in detail. We show results from analyzing simulated and experimental kinesin and myosin motor data. The algorithm is implemented in the free QuB software.     

Condensin extrudes DNA loops in steps up to hundreds of base pairs that are generated by ATP binding events

The condensin SMC protein complex organizes chromosomal structure by extruding loops of DNA. Its ATP-dependent motor mechanism remains unclear but likely involves steps associated with large conformational changes within the ∼50 nm protein complex. Here, using high-resolution magnetic tweezers, we resolve single steps in the loop extrusion process by individual yeast condensins. The measured median step sizes range between 20–40 nm at forces of 1.0–0.2 pN, respectively, comparable with the holocomplex size. These large steps show that, strikingly, condensin typically reels in DNA in very sizeable amounts with ∼200 bp on average per single extrusion step at low force, and occasionally even much larger, exceeding 500 bp per step. Using Molecular Dynamics simulations, we demonstrate that this is due to the structural flexibility of the DNA polymer at these low forces. Using ATP-binding-impaired and ATP-hydrolysis-deficient mutants, we find that ATP binding is the primary step-generating stage underlying DNA loop extrusion. We discuss our findings in terms of a scrunching model where a stepwise DNA loop extrusion is generated by an ATP-binding-induced engagement of the hinge and the globular domain of the SMC complex.     

Improved profile HMM performance by assessment of critical algorithmic features in SAM and HMMER

Background  

Profile hidden Markov model (HMM) techniques are among the most powerful methods for protein homology detection. Yet, the critical features for successful modelling are not fully known. In the present work we approached this by using two of the most popular HMM packages: SAM and HMMER. The programs' abilities to build models and score sequences were compared on a SCOP/Pfam based test set. The comparison was done separately for local and global HMM scoring.     

Force–Velocity Curves of Motor Proteins Cooperating In Vivo

Motor proteins convert chemical energy into work, thereby generating persistent motion of cellular and subcellular objects. The velocities of motor proteins as a function of opposing loads have been previously determined in vitro for single motors. These single molecule “force–velocity curves” have been useful for elucidating motor kinetics and for estimating motor performance under physiological loads due to, for example, the cytoplasmic drag force on transported organelles. Here we report force–velocity curves for single and multiple motors measured in vivo. Using motion enhanced differential interference contrast (MEDIC) movies of living NT2 (neuron-committed teratocarcinoma) cells at 37°C, three parameters were measured—velocity (v), radius (a), and effective cytoplasmic viscosity (η′)—as they applied to moving vesicles. These parameters were combined in Stokes’ equation, F = 6πaη′v, to determine the force, F, required to transport a single intracellular particle at velocity, v. In addition, the number of active motors was inferred from the multimodal pattern seen in a normalized velocity histogram. Using this inference, the resulting in vivo force–velocity curve for a single motor agrees with previously reported in vitro single motor force–velocity curves. Interestingly, however, the curves for two and three motors lie significantly higher in both measured velocity and computed force, which suggests that motors can work cooperatively to attain higher transport forces and velocities. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Maximum likelihood estimation of molecular motor kinetics from staircase dwell-time sequences

Molecular motors, such as kinesin, myosin, or dynein, convert chemical energy into mechanical energy by hydrolyzing ATP. The mechanical energy is used for moving in discrete steps along the cytoskeleton and carrying a molecular load. High resolution single molecule recordings of motor steps appear as a stochastic sequence of dwells, resembling a staircase. Staircase data can also be obtained from other molecular machines such as F1 -ATPase, RNA polymerase, or topoisomerase. We developed a maximum likelihood algorithm that estimates the rate constants between different conformational states of the protein, including motor steps. We model the motor with a periodic Markov model that reflects the repetitive chemistry of the motor step. We estimated the kinetics from the idealized dwell-sequence by numerical maximization of the likelihood function for discrete-time Markov models. This approach eliminates the need for missed event correction. The algorithm can fit kinetic models of arbitrary complexity, such as uniform or alternating step chemistry, reversible or irreversible kinetics, ATP concentration and mechanical force-dependent rates, etc. The method allows global fitting across stationary and nonstationary experimental conditions, and user-defined a priori constraints on rate constants. The algorithm was tested with simulated data, and implemented in the free QuB software.     

Alternating-Site Mechanism of Kinesin-1 Characterized by Single-Molecule FRET Using Fluorescent ATP Analogues

Kinesin-1 motor proteins move along microtubules in repetitive steps of 8 nm at the expense of ATP. To determine nucleotide dwell times during these processive runs, we used a Förster resonance energy transfer method at the single-molecule level that detects nucleotide binding to kinesin motor heads. We show that the fluorescent ATP analog used produces processive motility with kinetic parameters altered <2.5-fold compared with normal ATP. Using our confocal fluorescence kinesin motility assay, we obtained fluorescence intensity time traces that we then analyzed using autocorrelation techniques, yielding a time resolution of ∼1 ms for the intensity fluctuations due to fluorescent nucleotide binding and release. To compare these experimental autocorrelation curves with kinetic models, we used Monte-Carlo simulations. We find that the experimental data can only be described satisfactorily on the basis of models assuming an alternating-site mechanism, thus supporting the view that kinesin's two motor domains hydrolyze ATP and step in a sequential way.     

Kinesin steps do not alternate in size

Kinesin is a two-headed motor protein that transports cargo inside cells by moving stepwise on microtubules. Its exact trajectory along the microtubule is unknown: alternative pathway models predict either uniform 8-nm steps or alternating 7- and 9-nm steps. By analyzing single-molecule stepping traces from “limping” kinesin molecules, we were able to distinguish alternate fast- and slow-phase steps and thereby to calculate the step sizes associated with the motions of each of the two heads. We also compiled step distances from nonlimping kinesin molecules and compared these distributions against models predicting uniform or alternating step sizes. In both cases, we find that kinesin takes uniform 8-nm steps, a result that strongly constrains the allowed models.     

Lateral Motion and Bending of Microtubules Studied with a New Single-Filament Tracking Routine in Living Cells

The cytoskeleton is involved in numerous cellular processes such as migration, division, and contraction and provides the tracks for transport driven by molecular motors. Therefore, it is very important to quantify the mechanical behavior of the cytoskeletal filaments to get a better insight into cell mechanics and organization. It has been demonstrated that relevant mechanical properties of microtubules can be extracted from the analysis of their motion and shape fluctuations. However, tracking individual filaments in living cells is extremely complex due, for example, to the high and heterogeneous background. We introduce a believed new tracking algorithm that allows recovering the coordinates of fluorescent microtubules with ∼9 nm precision in in vitro conditions. To illustrate potential applications of this algorithm, we studied the curvature distributions of fluorescent microtubules in living cells. By performing a Fourier analysis of the microtubule shapes, we found that the curvatures followed a thermal-like distribution as previously reported with an effective persistence length of ∼20 μm, a value significantly smaller than that measured in vitro. We also verified that the microtubule-associated protein XTP or the depolymerization of the actin network do not affect this value; however, the disruption of intermediate filaments decreased the persistence length. Also, we recovered trajectories of microtubule segments in actin or intermediate filament-depleted cells, and observed a significant increase of their motion with respect to untreated cells showing that these filaments contribute to the overall organization of the microtubule network. Moreover, the analysis of trajectories of microtubule segments in untreated cells showed that these filaments presented a slower but more directional motion in the cortex with respect to the perinuclear region, and suggests that the tracking routine would allow mapping the microtubule dynamical organization in cells.     

Temperature Dependence of the DNA Double Helix at the Nanoscale: Structure,Elasticity, and Fluctuations

Biological organisms exist over a broad temperature range of −15°C to +120°C, where many molecular processes involving DNA depend on the nanoscale properties of the double helix. Here, we present results of extensive molecular dynamics simulations of DNA oligomers at different temperatures. We show that internal basepair conformations are strongly temperature-dependent, particularly in the stretch and opening degrees of freedom whose harmonic fluctuations can be considered the initial steps of the DNA melting pathway. The basepair step elasticity contains a weaker, but detectable, entropic contribution in the roll, tilt, and rise degrees of freedom. To extend the validity of our results to the temperature interval beyond the standard melting transition relevant to extremophiles, we estimate the effects of superhelical stress on the stability of the basepair steps, as computed from the Benham model. We predict that although the average twist decreases with temperature in vitro, the stabilizing external torque in vivo results in an increase of ∼1°/bp (or a superhelical density of Δσ?+0.03$\text{Δ}\sigma ?+0.03$) in the interval 0–100°C. In the final step, we show that the experimentally observed apparent bending persistence length of torsionally unconstrained DNA can be calculated from a hybrid model that accounts for the softening of the double helix and the presence of transient denaturation bubbles. Although the latter dominate the behavior close to the melting transition, the inclusion of helix softening is important around standard physiological temperatures.     

A novel machine learning unsupervised algorithm for sleep/wake identification using actigraphy

ABSTRACT

Actigraphy is widely used in sleep studies but lacks a universal unsupervised algorithm for sleep/wake identification. An unsupervised algorithm is useful in large-scale population studies and in cases where polysomnography (PSG) is unavailable, as it does not require sleep outcome labels to train the model but utilizes information solely contained in actigraphy to learn sleep and wake characteristics and separate the two states. In this study, we proposed a machine learning unsupervised algorithm based on the Hidden Markov Model (HMM) for sleep/wake identification. The proposed algorithm is also an individualized approach that takes into account individual variabilities and analyzes each individual actigraphy profile separately to infer sleep and wake states. We used Actiwatch and PSG data from 43 individuals in the Multi-Ethnic Study of Atherosclerosis study to evaluate the method performance. Epoch-by-epoch comparisons and sleep variable comparisons were made between our algorithm, the unsupervised algorithm embedded in the Actiwatch software (AS), and the pre-trained supervised UCSD algorithm. Using PSG as the reference, the accuracy was 85.7% for HMM, 84.7% for AS, and 85.0% for UCSD. The sensitivity was 99.3%, 99.7%, and 98.9% for HMM, AS, and UCSD, respectively, and the specificity was 36.4%, 30.0%, and 31.7%, respectively. The Kappa statistic was 0.446 for HMM, 0.399 for AS, and 0.311 for UCSD, suggesting fair to moderate agreement between PSG and actigraphy. The Bland–Altman plots further show that the total sleep time, sleep latency, and sleep efficiency estimates by HMM were closer to PSG with narrower 95% limits of agreement than AS and UCSD. All three methods tend to overestimate sleep and underestimate wake compared to PSG. Our HMM approach is also able to differentiate relatively active and sedentary individuals by quantifying variabilities in activity counts: individuals with higher estimated activity variabilities tend to show more frequent sedentary behaviors. Our unsupervised data-driven HMM algorithm achieved better performance than the commonly used Actiwatch software algorithm and the pre-trained UCSD algorithm. HMM can help expand the application of actigraphy in cases where PSG is hard to acquire and supervised methods cannot be trained. In addition, the estimated HMM parameters can characterize individual activity patterns and sedentary tendencies that can be further utilized in downstream analysis.     

Power line interference rejection from surface electromyography signal using an adaptive algorithm

This work focuses on the power line interference (PLI) rejection from surface EMG signal. It contains three parts: the algorithm, the experimental setting and the results. This study begins with describing the new technique, which consists in filtering respiratory surface electromyogram signals (EMG + PLI), then, becoming familiar with it. The proposed algorithm requires only one channel to both estimating the adaptive filter input reference noise and the EMG signal. The algorithm of PLI rejection has been organized into two steps. The first step insists to apply adaptive filter, especially the LMS one, in which the reference input is mathematically constructed using two different cosine functions; 50 Hz (the fundamental) function and 150 Hz (the first harmonic) function. Whereas, the second step applies the matching pursuit algorithm that uses the cosine packet dictionary to improve the result of PLI obtained at the first step. After trying statistical, as well as mathematical analysis, the complete investigation ensures that all details and steps make proof that our rigorous method is appropriate, we have also compared our method with the previous known techniques.     

Stochastic Approximation Boosting for Incomplete Data Problems

Summary Boosting is a powerful approach to fitting regression models. This article describes a boosting algorithm for likelihood‐based estimation with incomplete data. The algorithm combines boosting with a variant of stochastic approximation that uses Markov chain Monte Carlo to deal with the missing data. Applications to fitting generalized linear and additive models with missing covariates are given. The method is applied to the Pima Indians Diabetes Data where over half of the cases contain missing values.     

Virus safety of plasma products using 20 nm instead of 15 nm filtration as virus removing step

During the manufacture of human plasma derivatives, a series of complementary measures are undertaken to prevent transmission of blood-borne viruses. Virus filtration using 15 nm (Planova15N) filters has successfully been implemented in manufacturing processes for various plasma derivatives primarily because virus filtration is a technique, mild for proteins, that can effectively remove even small non-lipid-enveloped viruses, such as HAV and parvovirus B19. However, the use of 15 nm filters has limitations with regard to protein capacity of the filters and the process flow, resulting in an expensive manufacturing step. Therefore, studies were performed to test whether the use of 20 nm (Planova20N) filters, having different characteristics compared to 15 nm filters, can be an alternative for the use of 15 nm filters.It is shown that 20 nm filtration can be an alternative for 15 nm filtration. However, the virus removal capacity of the 20 nm filters depends on the plasma product that is filtered. Therefore, an optimisation study must be performed with regard to process parameters such as pressure, pH and protein concentration for each plasma product. In this study, using optimised conditions, the virus removal capacity of 20 nm filters appears to be comparable or even better when compared to that of 15 nm filters.     

A Cross-Bridge Cycle with Two Tension-Generating Steps Simulates Skeletal Muscle Mechanics

We examined whether cross-bridge cycle models with one or two tension-generating steps can account for the force-velocity relation of and tension response to length steps of frog skeletal muscle. Transition-state theory defined the strain dependence of the rate constants. The filament stiffness was non-Hookean. Models were refined against experimental data by simulated annealing and downhill simplex runs. Models with one tension-generating step were rejected, as they had a low efficiency and fitted the experimental data relatively poorly. The best model with two tension-generating steps (stroke distances 5.6 and 4.6 nm) and a cross-bridge stiffness of 1.7 pN/nm gave a good account of the experimental data. The two tensing steps allow an efficiency of up to 38% during shortening. In an isometric contraction, 54.7% of the attached heads were in a pre-tension-generating state, 44.5% of the attached heads had undergone the first tension-generating step, and only 0.8% had undergone both tension-generating steps; they bore 34%, 64%, and 2%, respectively, of the isometric tension. During slow shortening, the second tensing step made a greater contribution. During lengthening, up to 93% of the attached heads were in a pre-tension-generating state yet bore elevated tension by being dragged to high strains before detaching.     

Dwell time symmetry in random walks and molecular motors

The statistics of steps and dwell times in reversible molecular motors differ from those of cycle completion in enzyme kinetics. The reason is that a step is only one of several transitions in the mechanochemical cycle. As a result, theoretical results for cycle completion in enzyme kinetics do not apply to stepping data. To allow correct parameter estimation, and to guide data analysis and experiment design, a theoretical treatment is needed that takes this observation into account. In this article, we model the distribution of dwell times and number of forward and backward steps using first passage processes, based on the assumption that forward and backward steps correspond to different directions of the same transition. We extend recent results for systems with a single cycle and consider the full dwell time distributions as well as models with multiple pathways, detectable substeps, and detachments. Our main results are a symmetry relation for the dwell time distributions in reversible motors, and a relation between certain relative step frequencies and the free energy per cycle. We demonstrate our results by analyzing recent stepping data for a bacterial flagellar motor, and discuss the implications for the efficiency and reversibility of the force-generating subunits.     

A flexible domain is essential for the large step size and processivity of myosin VI

Myosin VI moves processively along actin with a larger step size than expected from the size of the motor. Here, we show that the proximal tail (the approximately 80-residue segment following the IQ domain) is not a rigid structure but, rather, a flexible domain that permits the heads to separate. With a GCN4 coiled coil inserted in the proximal tail, the heads are closer together in electron microscopy (EM) images, and the motor takes shorter processive steps. Single-headed myosin VI S1 constructs take nonprocessive 12 nm steps, suggesting that most of the processive step is covered by a diffusive search for an actin binding site. Based on these results, we present a mechanical model that describes stepping under an applied load.     

A Cross-Bridge Cycle with Two Tension-Generating Steps Simulates Skeletal Muscle Mechanics

We examined whether cross-bridge cycle models with one or two tension-generating steps can account for the force-velocity relation of and tension response to length steps of frog skeletal muscle. Transition-state theory defined the strain dependence of the rate constants. The filament stiffness was non-Hookean. Models were refined against experimental data by simulated annealing and downhill simplex runs. Models with one tension-generating step were rejected, as they had a low efficiency and fitted the experimental data relatively poorly. The best model with two tension-generating steps (stroke distances 5.6 and 4.6 nm) and a cross-bridge stiffness of 1.7 pN/nm gave a good account of the experimental data. The two tensing steps allow an efficiency of up to 38% during shortening. In an isometric contraction, 54.7% of the attached heads were in a pre-tension-generating state, 44.5% of the attached heads had undergone the first tension-generating step, and only 0.8% had undergone both tension-generating steps; they bore 34%, 64%, and 2%, respectively, of the isometric tension. During slow shortening, the second tensing step made a greater contribution. During lengthening, up to 93% of the attached heads were in a pre-tension-generating state yet bore elevated tension by being dragged to high strains before detaching.