Step-size Distribution Research Articles

Motor domain of condensin and step formation in extruding loop of DNA.

During the asymmetric loop extrusion of DNA by a condensin complex, one domain of the complex stably anchors to the DNA molecule, and another domain reels in the DNA strand into a loop. The DNA strand in the loop is fully relaxed, or there is no tension in the loop. Just outside of the loop, there is a tension that resists the extrusion of DNA. To maintain the extrusion of the DNA loop, the condensin complex must have a domain capable of generating a force to overcome the tension outside of the loop. This study proposes that the groove-shaped HEAT repeat domain Ycg1 plays the role of a molecular motor. A DNA molecule may bind to the groove electrostatically, and the weak binding force facilitates the random thermal motion of DNA molecules. A mechanical model that random collisions between DNA and the nonparallel inner surfaces of the groove may generate a directional force which is required for the loop extrusion to sustain. The hinge domain binds to the DNA molecule and acts as an anchor during asymmetric DNA loop extrusion. When the effects of ATP hydrolysis and the viscous drag of the fluid environment are considered, the motor-anchor model for the condensin complex and the mechanical model might explain the asymmetric loop extrusion, the formation of steps, the step size distribution in the loop extrusion, the tension-dependent extrusion speed, the interaction between coexisting loops on the DNA strand, and untying the knots during extrusion. This model can also explain the observed formation of the Z-loop.

Distinct Sub- to Superdiffuse Insulin Granule Transport Behaviors in β-Cells Are Strongly Affected by Granule Age.

Intracellular transport is a complex process that is difficult to describe by a single general model for motion. Here, we study the transport of insulin containing vesicles, termed granules, in live MIN6 cells. We characterize how the observed heterogeneity is affected by different intracellular factors by constructing a MIN6 cell line by CRISPR-CAS9 that constitutively expresses mCherry fused to insulin and is thus packaged in granules. Confocal microscopy imaging and single particle tracking of the granule transport provide long trajectories of thousands of single granule trajectories for statistical analysis. Mean squared displacement (MSD), angle correlation distribution, and step size distribution analysis allowed identifying five distinct granule transport subpopulations, from nearly immobile and subdiffusive to run-pause and superdiffusive. The subdiffusive subpopulation recapitulates the subordinated random walk we reported earlier (Tabei, 2013; ref 18). We show that the transport characteristics of the five subpopulations have a strong dependence on the age of insulin granules. The five subpopulations also reflect the effect of local microtubule and actin networks on transport in different cellular regions. Our results provide robust metrics to clarify the heterogeneity of granule transport and demonstrate the roles of microtubule versus actin networks with granule age since initial packaging in the Golgi.

Stability against fluctuations: a two-dimensional study of scaling, bifurcations and spontaneous symmetry breaking in stochastic models of synaptic plasticity

Stochastic models of synaptic plasticity must confront the corrosive influence of fluctuations in synaptic strength on patterns of synaptic connectivity. To solve this problem, we have proposed that synapses act as filters, integrating plasticity induction signals and expressing changes in synaptic strength only upon reaching filter threshold. Our earlier analytical study calculated the lifetimes of quasi-stable patterns of synaptic connectivity with synaptic filtering. We showed that the plasticity step size in a stochastic model of spike-timing-dependent plasticity (STDP) acts as a temperature-like parameter, exhibiting a critical value below which neuronal structure formation occurs. The filter threshold scales this temperature-like parameter downwards, cooling the dynamics and enhancing stability. A key step in this calculation was a resetting approximation, essentially reducing the dynamics to one-dimensional processes. Here, we revisit our earlier study to examine this resetting approximation, with the aim of understanding in detail why it works so well by comparing it, and a simpler approximation, to the system’s full dynamics consisting of various embedded two-dimensional processes without resetting. Comparing the full system to the simpler approximation, to our original resetting approximation, and to a one-afferent system, we show that their equilibrium distributions of synaptic strengths and critical plasticity step sizes are all qualitatively similar, and increasingly quantitatively similar as the filter threshold increases. This increasing similarity is due to the decorrelation in changes in synaptic strength between different afferents caused by our STDP model, and the amplification of this decorrelation with larger synaptic filters.

Digital backpropagation based on binary logarithmic step size distribution for fibre non‐linearity compensation

AbstractCapacity crunch has become critical in recent years as commercial communication systems approach their theoretical data rate limits. This work presents a low‐complexity digital backpropagation (DBP) implementation approach based on step size distribution that uses a binary logarithmic step size method to achieve high data rate optical transmission. The proposed scheme shows performance improvements (∆Q) of 2.36, 1.19, and 0.71 dB over linear compensation, constant step size (CSS) DBP, and logarithmic step size DBP techniques in a 2400 km 112 Gbit/s DP‐16 quadrature amplitude modulation (QAM) system, respectively. At 13 dBm, a high performance (Q) of 10.9 dB (BER = 2.25 × 10−4) is achieved, above the 3.80 × 10−3 hard‐decision forward error correction (HD‐FEC) limit, using the proposed scheme. Also, the allowable transmission distance is extended by 960 km at the HD‐FEC limit over the linear compensation technique. The optimization achieves a 38% saving in the number of DBP calculation steps compared to the CSS DBP, which considerably reduces the computational cost since a few steps are required for effective non‐linearity compensation.

Multi-order analytical solving computation of rainstorm causal decomposition during typhoons using a designed key–lock quasi-Newton optimizing derivation

To precisely identify multi-dimensional spatiotemporal rain-making parameters, generate an approximate Hessian matrix, and solve the nonlinear ill-posed problem, this study uses composite logical tangent hyperbolic functions to construct the rain-generating simulation model as nonlinear algebraic equations with designed key–lock quasi-Newton optimization for deriving multi-order objective functional derivatives for rainstorm causal decomposition into advanced functional, analytical solution (lock) and Newton's conditional constraints. Specifically, the rank-two approximate structure of the Levenberg–Marquardt and Broyden–Fletcher–Goldfarb–Shanno quasi-Newton algorithms are modified as the symmetric rank-four structure to efficiently calculate a positive definite stable Hessian and solve the constrained nonlinear rain-making threshold. The model projects various rain-making factors into multi-rank loading scores, characterizing rain-generating mechanisms and causal components as associated DNAs. To accelerate/modify directional convergence, avoid local minimum, and detect global optimum, the devised vectorized limited switchable step sizes are optimized using advanced double-bracketing approaches combined with candidate parameters' correction vectors (key) and referenced step-size distributions solved by Newton's constrained analytical solution to reduce heterogeneous differences and eliminate the conventional overestimated Hessian. The identified rain-making DNAs reveal that typhoons with similar DNAs move in similar directions. Specifically, rain-making DNAs in Taipei Category 1 were correlated with wind force/direction and cloud height along PCs 1, 3, 4, and 7, and those in Category 2 were correlated with cloud-cover distribution along PCs 1, 2, and 5. The identified rain-making thresholds of typhoons with constant direction/structure showed a weaker steady state, whereas the unsteady rest produced multi-peak rainfall hydrographs. Rain evolution analysis reveals that cloudy rainbands, carried by the wind field, move along the Tamsui River valley when traveling between northeast and south-southeast of Taipei; converge with gradient and geostrophic winds when traveling between east-northeast and southwest; merge with southwest monsoon when traveling between west-southwest and northeast of Kaohsiung.

Asymptotic Properties of Extremal Markov Processes Driven by Kendall Convolution

This paper is devoted to the analysis of the finite-dimensional distributions and asymptotic behavior of extremal Markov processes connected with the Kendall convolution. In particular, we provide general formulas for the finite dimensional distributions of the random walk driven by the Kendall convolution for a large class of step size distributions. Moreover, we prove limit theorems for random walks and associated continuous-time stochastic processes.

Micromechanical Study of Hyperacetylated Nucleosomes Using Single Molecule Transverse Magnetic Tweezers.

Nucleosomes are stable complexes of DNA and histone proteins that are essential for the proper functioning of the genome. These structures must be unwrapped and disassembled for processes such as gene expression, replication, and repair. Histone post-translational modifications (PTMs) are known to play a significant role in regulating the structural changes of nucleosomes. However, the underlying mechanisms by which these modifications function remain unclear. In this study, we report the results of single molecule micromanipulation experiments on DNA-protein complexes composed of hyperacetylated histone proteins using transverse magnetic tweezers. The experiments were conducted by pre-extending λ-DNA with a force less than 4 pN before introducing hyperacetylated histones into the sample chamber. The DNA shortened as the histones formed complexes with it and the nucleosome arrays were then exposed to increasing tension, resulting in quantized changes in the DNA's extension with step sizes of (integral multiples of) ~50 nm. We also compared results of experiments using PTM histones and native histones with data collected for both types of histones for the same force ranges (2-80 pN) and loading rates. Our data show that hyperacetylated nucleosomes require an unbinding force of around ~2.5 pN, which is similar to that required for native histones. Moreover, we identified clear differences between the step-size distributions of native and hyperacetylated histones and found that in contrast to tethers reconstituted with native histones, the majority of nucleosomes in tethers compacted with hyperacetylated histones underwent disassembly at forces significantly lower than 6 pN.

Lon protease degrades stable substrates slowly but with enhanced processivity, redefining the model of a strong and robust AAA+ protease.

AAA+ proteases function in all domains of life using an ATP-dependent unfoldase/polypeptide-translocase motor. Lon is a highly conserved AAA+ protease especially important for degrading oxidatively-damaged proteins. To address if Lon is a weak vs. robust unfoldase, as its substrates are often poorly folded, and elucidate whether its mechanism of translocation is similar to the previously-characterized AAA+ proteases, ClpXP and ClpAP, which have a distinct subunit assembly/architecture, we used a modified single-bead optical trapping assay. We find that Lon: (i) unfolds stable proteins using substantially longer pre-unfolding dwell times compared to ClpXP/ClpAP but (ii) has a higher probability of fully degrading a multi-domain substrate than the other enzymes. (iii) Although a slower translocase, Lon's fundamental step size is shared by ClpXP/ClpAP and its step-size distribution mirrors that of ClpAP. (iv) Finally, both ClpXP and Lon hydrolyze ∼1 ATP per translocation step. Thus, the mechanism of the ATP hydrolysis-driven ‘power stroke’ appears largely conserved between three distinct AAA+ proteases, regardless of their speed, processivity, or whether the ATPase and peptidase modules are physically linked (Lon) or assembled from distinct polypeptides (ClpXP and ClpAP). Lon therefore likely uses a common set of operating principles but is ‘geared differently’ than other AAA+ proteases.

Levy skimmer in an asymmetric ratchet channel

We numerically study the dynamics of a model Levy walker, moving in a 2dimensional medium confined by the walls of an asymmetric channel. We show that, as a resultof both asymmetric potential due to the channel and also the power law step size distribution,the Levy walker will achieve a net directed velocity in the direction preferred by the channel.Other statistical properties of the walker such as mean first passage time and the mean squaredisplacements are also examined.

Distributed medium viscosity yields quasi-exponential step-size probability distributions in heterogeneous media.

The analysis of the statistics of random walks undertaken by passive particles in complex media has important implications in a number of areas including pathogen transport and drug delivery. In several systems in which heterogeneity is important, the distribution of particle step-sizes has been found to be exponential in nature, as opposed to the Gaussian distribution associated with Brownian motion. Here, we first develop a theoretical framework to study a simplified version of this problem: the motion of passive tracers in a range of sub-environments with different viscosity. We show that in the limit of a large number of equi-distributed sub-environments spanning a broad viscosity range, an exact analytical expression for the underlying particle step-size distribution can be derived, which approaches an exponential distribution when step sizes are small. We then validate this using a simple experimental system of glycerol-water mixtures, in which the volume fraction of glycerol is systematically varied. Overall, the assumption of exponentially distributed step sizes may substantially over-estimate the incidence of large steps in heterogeneous systems, with important implications in the analysis of various biophysical processes.

Persistence of heavy-tailed sample averages: principle of infinitely many big jumps

We consider the sample average of a centered random walk in Rd with regularly varying step size distribution. For the first exit time from a compact convex set A not containing the origin, we show that its tail is of lognormal type. Moreover, we show that the typical way for a large exit time to occur is by having a number of jumps growing logarithmically in the scaling parameter.

Morphodynamics facilitate cancer cells to navigate 3D extracellular matrix

Cell shape is linked to cell function. The significance of cell morphodynamics, namely the temporal fluctuation of cell shape, is much less understood. Here we study the morphodynamics of MDA-MB-231 cells in type I collagen extracellular matrix (ECM). We systematically vary ECM physical properties by tuning collagen concentrations, alignment, and gelation temperatures. We find that morphodynamics of 3D migrating cells are externally controlled by ECM mechanics and internally modulated by Rho/ROCK-signaling. We employ machine learning to classify cell shape into four different morphological phenotypes, each corresponding to a distinct migration mode. As a result, we map cell morphodynamics at mesoscale into the temporal evolution of morphological phenotypes. We characterize the mesoscale dynamics including occurrence probability, dwell time and transition matrix at varying ECM conditions, which demonstrate the complex phenotype landscape and optimal pathways for phenotype transitions. In light of the mesoscale dynamics, we show that 3D cancer cell motility is a hidden Markov process whereby the step size distributions of cell migration are coupled with simultaneous cell morphodynamics. Morphological phenotype transitions also facilitate cancer cells to navigate non-uniform ECM such as traversing the interface between matrices of two distinct microstructures. In conclusion, we demonstrate that 3D migrating cancer cells exhibit rich morphodynamics that is controlled by ECM mechanics, Rho/ROCK-signaling, and regulate cell motility. Our results pave the way to the functional understanding and mechanical programming of cell morphodynamics as a route to predict and control 3D cell motility.

Theory and simulations of condensin mediated loop extrusion in DNA

Condensation of hundreds of mega-base-pair-long human chromosomes in a small nuclear volume is a spectacular biological phenomenon. This process is driven by the formation of chromosome loops. The ATP consuming motor, condensin, interacts with chromatin segments to actively extrude loops. Motivated by real-time imaging of loop extrusion (LE), we created an analytically solvable model, predicting the LE velocity and step size distribution as a function of external load. The theory fits the available experimental data quantitatively, and suggests that condensin must undergo a large conformational change, induced by ATP binding, bringing distant parts of the motor to proximity. Simulations using a simple model confirm that the motor transitions between an open and a closed state in order to extrude loops by a scrunching mechanism, similar to that proposed in DNA bubble formation during bacterial transcription. Changes in the orientation of the motor domains are transmitted over ~50 nm, connecting the motor head and the hinge, thus providing an allosteric basis for LE.

КОМПАКТНАЯ ГОЛОВКА СКАНИРУЮЩЕГО ЗОНДОВОГО МИКРОСКОПА НА ОСНОВЕ ИНЕРЦИАЛЬНЫХ ДВИЖИТЕЛЕЙ, ИСПОЛЬЗУЮЩИХ ПЬЕЗОПАКЕТЫ

The instrumental principles of constructing a compact measuring head of a scanning probe microscope are discussed. To minimize the overall dimensions, a piezo package is used, which makes it possible to implement in one unit both the functions of the actuating element of the SPM tracking system and the functions of the module for approaching the probe to the sample. A control scheme for a piezo-inertial linear step mover is presented, and histograms of the step size distribution are measured. Diagrams of compact SPM heads operating in tunneling and semi-contact power modes using self-sensing probe sensors are presented. To increase the size of the survey SPM images, scanning over several overlapping areas was implemented with further stitching of the SPM images in a single frame. The algorithm for stitching SPM images is discussed. The deviations in the directions of the stage and probe movements, arising during the new scanning frame selection, are being corrected within the framework of the SPM image stitching algorithm. The results of capturing SPM images using a compact SPM head and stitching SPM images by means of elaborated algorithms are presented.

Levy Swimmer In An Asymmetric Channel

In this article, we numerically study the dynamics of a model Levy walker, moving in a 2 dimensional medium confined by the walls of an asymmetric channel. We show that, as a result of both asymmetric potential due to the channel and also the power law step size distribution, the Levy walker will achieve a net directed velocity in the direction preferred by the channel. Other statistical properties of the walker such as mean first passage time and the mean square displacements are also examined

Direct Measurement of Stepping Dynamics of E. coli UvrD Helicase

UvrD is a prototypical Superfamily 1 (SF1) helicase from E. coli involved in DNA repair. We report herein the directmeasurement of the stepping behavior of UvrD at thesingle-molecule level as it unwinds and re-zips a DNA hairpin usinghigh-resolution optical tweezers. We measure an averageunwinding and re-zipping step size of ∼3 base pairs, but also observe large variation in step size.Analysis of the step size distribution reveals the presence of multiple sub-steps smaller than 3 bp,suggesting that 3 bp is not the elemental unit of step sizefor UvrD.

Biopolyelectrolyte Surface Diffusion Within a Planar Slit Geometry

The effective surface diffusion of biopolyelectrolytes within nanoscale confinement is integral to biosensing, biocatalysis, bioseparations (such as chromatography columns utilizing nanoporous media), as well as biophysical processes. Confinement influences the surface-mediated diffusion of biopolyelectrolyte chains due to the presence of electric double layers and the increased frequency of transient polymer/surface interactions. Here, the diffusion of poly-L-lysine (PLL) in a nanoslit was studied using Convex Lens-induced Confinement (CLiC) single-molecule tracking microscopy. Three surface chemistries were employed to understand and compare the effects of electrostatic and short-range interactions, respectively. In all cases, the effective surface diffusion coefficient increased rapidly with slit height until it saturated at its value for a semi-infinite interface for slit heights <30 nm. The evolution of the diffusion coefficient with slit height was faster for amine-functionalized surfaces with which PLL exhibited stronger short-range interactions. These hydrogen-bonding interactions influenced the intermittent random walk behavior by increasing the waiting times between flights and the re-adsorption probability (a.k.a. sticking coefficient) during flights. Intermittent random walks were simulated within a planar slit geometry, using the appropriate adsorption probabilities and waiting time distributions (obtained from diffusion at a single interface) to account for the characteristic short-range interactions between PLL and each type of surface chemistry. The simulated step-size distributions were in good agreement with experimental measurements, indicating that the fine details of diffusion in nanoscale confinement can be quantitatively described by a model that combines confined Brownian motion with transient surface adsorption and desorption, using parameters obtained only from the behavior at a single interface in a semi-infinite geometry. These results highlight the importance of short-range biopolyelectrolyte/surface interactions on diffusion within confined environments and moreover, demonstrate the complex relationship between electrostatics, confinement, and local polymer and surface chemistry on biopolyelectrolyte surface diffusion within a nanoslit.

Myosin V executes steps of variable length via structurally constrained diffusion.

The molecular motor myosin V transports cargo by stepping on actin filaments, executing a random diffusive search for actin binding sites at each step. A recent experiment suggests that the joint between the myosin lever arms may not rotate freely, as assumed in earlier studies, but instead has a preferred angle giving rise to structurally constrained diffusion. We address this controversy through comprehensive analytical and numerical modeling of myosin V diffusion and stepping. When the joint is constrained, our model reproduces the experimentally observed diffusion, allowing us to estimate bounds on the constraint energy. We also test the consistency between the constrained diffusion model and previous measurements of step size distributions and the load dependence of various observable quantities. The theory lets us address the biological significance of the constrained joint and provides testable predictions of new myosin behaviors, including the stomp distribution and the run length under off-axis force.

Crystalline chitin hydrolase is a burnt-bridge Brownian motor

Motor proteins are essential units of life and are well-designed nanomachines working under thermal fluctuations. These proteins control moving direction by consuming chemical energy or by dissipating electrochemical potentials. Chitinase A from bacterium Serratia marcescens (SmChiA) processively moves along crystalline chitin by hydrolysis of a single polymer chain to soluble chitobiose. Recently, we directly observed the stepping motions of SmChiA labeled with a gold nanoparticle by dark-field scattering imaging to investigate the moving mechanism. Time constants analysis revealed that SmChiA moves back and forth along the chain freely, because forward and backward states have a similar free energy level. The similar probabilities of forward-step events (83.5%=69.3%+14.2%) from distributions of step sizes and chain-hydrolysis (86.3%=(1/2.9)/(1/2.9+1/18.3)×100) calculated from the ratios of time constants of hydrolysis and the backward step indicated that SmChiA moves forward as a result of shortening of the chain by a chitobiose unit, which stabilizes the backward state. Furthermore, X-ray crystal structures of sliding intermediate and molecular dynamics simulations showed that SmChiA slides forward and backward under thermal fluctuation without large conformational changes of the protein. Our results demonstrate that SmChiA is a burnt-bridge Brownian ratchet motor.

Large deviations for the maximum of a branching random walk with stretched exponential tails

We prove large deviation results for the position of the rightmost particle, denoted by $M_{n}$, in a one-dimensional branching random walk in a case when Cramér’s condition is not satisfied. More precisely we consider step size distributions with stretched exponential upper and lower tails, i.e. both tails decay as $e^{-\Theta (|t|^{r})}$ for some $r\in ( 0,1)$. It is known that in this case, $M_{n}$ grows as $n^{1/r}$ and in particular faster than linearly in $n$. Our main result is a large deviation principle for the laws of $n^{-1/r}M_{n}$ . In the proof we use a comparison with the maximum of (a random number of) independent random walks, denoted by $\tilde{M} _{n}$, and we show a large deviation principle for the laws of $n^{-1/r}\tilde{M} _{n}$ as well.