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Mao Y, Nielsen P, Ali J. Passive and Active Microrheology for Biomedical Systems. Front Bioeng Biotechnol 2022; 10:916354. [PMID: 35866030 PMCID: PMC9294381 DOI: 10.3389/fbioe.2022.916354] [Citation(s) in RCA: 14] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/09/2022] [Accepted: 06/08/2022] [Indexed: 12/12/2022] Open
Abstract
Microrheology encompasses a range of methods to measure the mechanical properties of soft materials. By characterizing the motion of embedded microscopic particles, microrheology extends the probing length scale and frequency range of conventional bulk rheology. Microrheology can be characterized into either passive or active methods based on the driving force exerted on probe particles. Tracer particles are driven by thermal energy in passive methods, applying minimal deformation to the assessed medium. In active techniques, particles are manipulated by an external force, most commonly produced through optical and magnetic fields. Small-scale rheology holds significant advantages over conventional bulk rheology, such as eliminating the need for large sample sizes, the ability to probe fragile materials non-destructively, and a wider probing frequency range. More importantly, some microrheological techniques can obtain spatiotemporal information of local microenvironments and accurately describe the heterogeneity of structurally complex fluids. Recently, there has been significant growth in using these minimally invasive techniques to investigate a wide range of biomedical systems both in vitro and in vivo. Here, we review the latest applications and advancements of microrheology in mammalian cells, tissues, and biofluids and discuss the current challenges and potential future advances on the horizon.
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Affiliation(s)
- Yating Mao
- Department of Chemical and Biomedical Engineering, FAMU-FSU College of Engineering, Tallahassee, FL, United States
- National High Magnetic Field Laboratory, Tallahassee, FL, United States
| | - Paige Nielsen
- Department of Chemical and Biomedical Engineering, FAMU-FSU College of Engineering, Tallahassee, FL, United States
- National High Magnetic Field Laboratory, Tallahassee, FL, United States
| | - Jamel Ali
- Department of Chemical and Biomedical Engineering, FAMU-FSU College of Engineering, Tallahassee, FL, United States
- National High Magnetic Field Laboratory, Tallahassee, FL, United States
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2
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Nguyen A, Brandt M, Muenker TM, Betz T. Multi-oscillation microrheology via acoustic force spectroscopy enables frequency-dependent measurements on endothelial cells at high-throughput. LAB ON A CHIP 2021; 21:1929-1947. [PMID: 34008613 PMCID: PMC8130676 DOI: 10.1039/d0lc01135e] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2020] [Accepted: 03/12/2021] [Indexed: 06/03/2023]
Abstract
Active microrheology is one of the main methods to determine the mechanical properties of cells and tissue, and the modelling of these viscoelastic properties is under heavy debate with many competing approaches. Most experimental methods of active microrheology such as optical tweezers or atomic force microscopy based approaches rely on single cell measurements, and thus suffer from a low throughput. Here, we present a novel method for frequency-dependent microrheology on cells using acoustic forces which allows multiplexed measurements of several cells in parallel. Acoustic force spectroscopy (AFS) is used to generate multi-oscillatory forces in the range of pN-nN on particles attached to primary human umbilical vein endothelial cells (HUVEC) cultivated inside a microfluidic chip. While the AFS was introduced as a single-molecule technique to measure mechanochemical properties of biomolecules, we exploit the AFS to measure the dynamic viscoelastic properties of cells exposed to different conditions, such as flow shear stresses or drug injections. By controlling the force and measuring the position of the particle, the complex shear modulus G*(ω) can be measured continuously over several hours. The resulting power-law shear moduli are consistent with fractional viscoelastic models. In our experiments we confirm a decrease in shear modulus after perturbing the actin cytoskeleton via cytochalasin B. This effect was reversible after washing out the drug. Additionally, we include critical information for the usage of the new method AFS as a measurement tool showing its capabilities and limitations and we find that for performing viscoelastic measurements with the AFS, a thorough calibration and careful data analysis is crucial, for which we provide protocols and guidelines.
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Affiliation(s)
- Alfred Nguyen
- Institute of Cell Biology, University of Münster, Münster, Germany.
| | - Matthias Brandt
- Institute of Cell Biology, University of Münster, Münster, Germany.
| | - Till M Muenker
- Institute of Cell Biology, University of Münster, Münster, Germany. and Third Institute of Physics-Biophysics, University of Göttingen, Göttingen, Germany.
| | - Timo Betz
- Institute of Cell Biology, University of Münster, Münster, Germany. and Third Institute of Physics-Biophysics, University of Göttingen, Göttingen, Germany.
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3
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Graybill PM, Bollineni RK, Sheng Z, Davalos RV, Mirzaeifar R. A constriction channel analysis of astrocytoma stiffness and disease progression. BIOMICROFLUIDICS 2021; 15:024103. [PMID: 33763160 PMCID: PMC7968935 DOI: 10.1063/5.0040283] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/11/2020] [Accepted: 02/23/2021] [Indexed: 05/12/2023]
Abstract
Studies have demonstrated that cancer cells tend to have reduced stiffness (Young's modulus) compared to their healthy counterparts. The mechanical properties of primary brain cancer cells, however, have remained largely unstudied. To investigate whether the stiffness of primary brain cancer cells decreases as malignancy increases, we used a microfluidic constriction channel device to deform healthy astrocytes and astrocytoma cells of grade II, III, and IV and measured the entry time, transit time, and elongation. Calculating cell stiffness directly from the experimental measurements is not possible. To overcome this challenge, finite element simulations of the cell entry into the constriction channel were used to train a neural network to calculate the stiffness of the analyzed cells based on their experimentally measured diameter, entry time, and elongation in the channel. Our study provides the first calculation of stiffness for grades II and III astrocytoma and is the first to apply a neural network analysis to determine cell mechanical properties from a constriction channel device. Our results suggest that the stiffness of astrocytoma cells is not well-correlated with the cell grade. Furthermore, while other non-central-nervous-system cell types typically show reduced stiffness of malignant cells, we found that most astrocytoma cell lines had increased stiffness compared to healthy astrocytes, with lower-grade astrocytoma having higher stiffness values than grade IV glioblastoma. Differences in nucleus-to-cytoplasm ratio only partly explain differences in stiffness values. Although our study does have limitations, our results do not show a strong correlation of stiffness with cell grade, suggesting that other factors may play important roles in determining the invasive capability of astrocytoma. Future studies are warranted to further elucidate the mechanical properties of astrocytoma across various pathological grades.
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Affiliation(s)
| | - R. K. Bollineni
- Department of Mechanical Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA
| | - Z. Sheng
- Department of Internal Medicine, Virginia Tech Carilion School of Medicine and Virginia Tech Fralin Biomedical Research Institute, Roanoke, Virginia 24016, USA
| | - R. V. Davalos
- Authors to whom correspondence should be addressed: and
| | - R. Mirzaeifar
- Department of Mechanical Engineering, Virginia Tech, Blacksburg, Virginia 24061, USA
- Authors to whom correspondence should be addressed: and
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Emile O, Emile J. Nanometer optical trap based on stimulated emission in evanescence of a totally reflected Arago spot : Nanometer optical trap for fluorescent nanoparticles. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2020; 43:68. [PMID: 33099687 DOI: 10.1140/epje/i2020-11991-6] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/01/2020] [Accepted: 09/28/2020] [Indexed: 06/11/2023]
Abstract
Optical tweezers have paved the way towards the manipulation of particles and living cells at the micrometer range. Its extension towards the nanometer world may create unprecedented potentialities in many areas of science. Following a letter (O. Emile, J. Emile, H. Tabuteau, EPL 129, 58001 (2020)) that reported the observation of the trapping of a single 200nm diameter fluorescent particle in a nanometric volume, we detail here our experimental findings. In particular, the trapping mechanism is shown to be based on the radiation pressure of light in one direction and on the stimulated emission of the particle in the evanescent wave of a nanometer Arago spot on a glass/liquid interface on the other directions. The trapping volume is a 200nm height cylinder whose radius varies with the spreading of the evanescent wave near the spot and can reach 50nm. The calculation of the force and the parameters limiting the lifetime are detailed. Applications to laser trapping of atoms and molecules are also discussed.
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Affiliation(s)
- Olivier Emile
- Université de Rennes 1, Campus de Beaulieu, F-35000, Rennes, France.
| | - Janine Emile
- Université de Rennes 1, CNRS IPR UMR 6251, F-35000, Rennes, France
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5
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Witzel P, Götz M, Lanoiselée Y, Franosch T, Grebenkov DS, Heinrich D. Heterogeneities Shape Passive Intracellular Transport. Biophys J 2019; 117:203-213. [PMID: 31278001 PMCID: PMC6700759 DOI: 10.1016/j.bpj.2019.06.009] [Citation(s) in RCA: 28] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2019] [Revised: 05/22/2019] [Accepted: 06/11/2019] [Indexed: 01/06/2023] Open
Abstract
A living cell's interior is one of the most complex and intrinsically dynamic systems, providing an elaborate interplay between cytosolic crowding and ATP-driven motion that controls cellular functionality. Here, we investigated two distinct fundamental features of the merely passive, non-biomotor-shuttled material transport within the cytoplasm of Dictyostelium discoideum cells: the anomalous non-linear scaling of the mean-squared displacement of a 150-nm-diameter particle and non-Gaussian distribution of increments. Relying on single-particle tracking data of 320,000 data points, we performed a systematic analysis of four possible origins for non-Gaussian transport: 1) sample-based variability, 2) rarely occurring strong motion events, 3) ergodicity breaking/aging, and 4) spatiotemporal heterogeneities of the intracellular medium. After excluding the first three reasons, we investigated the remaining hypothesis of a heterogeneous cytoplasm as cause for non-Gaussian transport. A, to our knowledge, novel fit model with randomly distributed diffusivities implementing medium heterogeneities suits the experimental data. Strikingly, the non-Gaussian feature is independent of the cytoskeleton condition and lag time. This reveals that efficiency and consistency of passive intracellular transport and the related anomalous scaling of the mean-squared displacement are regulated by cytoskeleton components, whereas cytoplasmic heterogeneities are responsible for the generic, non-Gaussian distribution of increments.
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Affiliation(s)
- Patrick Witzel
- Faculty for Chemistry and Pharmacy, Julius-Maximilians-Universität Würzburg, Würzburg, Germany; Fraunhofer Institute for Silicate Research ISC, Würzburg, Germany
| | - Maria Götz
- Faculty for Chemistry and Pharmacy, Julius-Maximilians-Universität Würzburg, Würzburg, Germany; Fraunhofer Institute for Silicate Research ISC, Würzburg, Germany
| | - Yann Lanoiselée
- Laboratoire de Physique de la Matière Condensée, CNRS-Ecole Polytechnique, Palaiseau, France
| | - Thomas Franosch
- Institut für Theoretische Physik, Universität Innsbruck, Innsbruck, Austria
| | - Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée, CNRS-Ecole Polytechnique, Palaiseau, France
| | - Doris Heinrich
- Fraunhofer Institute for Silicate Research ISC, Würzburg, Germany; Leiden Institute of Physics, Huygens-Kamerlingh Onnes Laboratory, Leiden University, Leiden, the Netherlands.
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Zhang K, Crizer KPR, Schoenfisch MH, Hill DB, Didier G. Fluid heterogeneity detection based on the asymptotic distribution of the time-averaged mean squared displacement in single particle tracking experiments. JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL 2018; 51:445601. [PMID: 31037119 PMCID: PMC6486181 DOI: 10.1088/1751-8121/aae0af] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
A tracer particle is called anomalously diffusive if its mean squared displacement grows approximately as σ 2 t α as a function of time t for some constant σ 2, where the diffusion exponent satisfies α ≠ 1. In this article, we use recent results on the asymptotic distribution of the time-averaged mean squared displacement [20] to construct statistical tests for detecting physical heterogeneity in viscoelastic fluid samples starting from one or multiple observed anomalously diffusive paths. The methods are asymptotically valid for the range 0 < α < 3/2 and involve a mathematical characterization of time-averaged mean squared displacement bias and the effect of correlated disturbance errors. The assumptions on particle motion cover a broad family of fractional Gaussian processes, including fractional Brownian motion and many fractional instances of the generalized Langevin equation framework. We apply the proposed methods in experimental data from treated P. aeruginosa biofilms generated by the collaboration of the Hill and Schoenfisch Labs at UNC-Chapel Hill.
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Affiliation(s)
- Kui Zhang
- Department of Mathematics, Tulane University
| | | | | | - David B Hill
- The Marsico Lung Institute and Department of Physics and Astronomy, University of North Carolina at Chapel Hill
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7
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Lanoiselée Y, Moutal N, Grebenkov DS. Diffusion-limited reactions in dynamic heterogeneous media. Nat Commun 2018; 9:4398. [PMID: 30353010 PMCID: PMC6199324 DOI: 10.1038/s41467-018-06610-6] [Citation(s) in RCA: 62] [Impact Index Per Article: 10.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2018] [Accepted: 09/11/2018] [Indexed: 12/02/2022] Open
Abstract
Most biochemical reactions in living cells rely on diffusive search for target molecules or regions in a heterogeneous overcrowded cytoplasmic medium. Rapid rearrangements of the medium constantly change the effective diffusivity felt locally by a diffusing particle and thus impact the distribution of the first-passage time to a reaction event. Here, we investigate the effect of these dynamic spatiotemporal heterogeneities onto diffusion-limited reactions. We describe a general mathematical framework to translate many results for ordinary homogeneous Brownian motion to heterogeneous diffusion. In particular, we derive the probability density of the first-passage time to a reaction event and show how the dynamic disorder broadens the distribution and increases the likelihood of both short and long trajectories to reactive targets. While the disorder slows down reaction kinetics on average, its dynamic character is beneficial for a faster search and realization of an individual reaction event triggered by a single molecule. “Diffusing diffusivity” concept has been recently put forward to account for rapid structural rearrangements in soft matter and biological systems. Here the authors propose a general mathematical framework to compute the distribution of first-passage times in a dynamically heterogeneous medium.
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Affiliation(s)
- Yann Lanoiselée
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, University Paris-Saclay, 91128, Palaiseau, France
| | - Nicolas Moutal
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, University Paris-Saclay, 91128, Palaiseau, France
| | - Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, University Paris-Saclay, 91128, Palaiseau, France.
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8
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Goychuk I. Viscoelastic subdiffusion in a random Gaussian environment. Phys Chem Chem Phys 2018; 20:24140-24155. [PMID: 30206605 DOI: 10.1039/c8cp05238g] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022]
Abstract
Viscoelastic subdiffusion governed by a fractional Langevin equation is studied numerically in a random Gaussian environment modeled by stationary Gaussian potentials with decaying spatial correlations. This anomalous diffusion is archetypal for living cells, where cytoplasm is known to be viscoelastic and a spatial disorder also naturally emerges. We obtain some first important insights into it within a model one-dimensional study. Two basic types of potential correlations are studied: short-range exponentially decaying and algebraically slow decaying with an infinite correlation length, both for a moderate (several kBT, in the units of thermal energy), and strong (5-10kBT) disorder. For a moderate disorder, it is shown that on the ensemble level viscoelastic subdiffusion can easily overcome the medium's disorder. Asymptotically, it is not distinguishable from the disorder-free subdiffusion. However, a strong scatter in single-trajectory averages is nevertheless seen even for a moderate disorder. It features a weak ergodicity breaking, which occurs on a very long yet transient time scale. Furthermore, for a strong disorder, a very long transient regime of logarithmic, Sinai-type diffusion emerges. It can last longer and be faster in the absolute terms for weakly decaying correlations as compared with the short-range correlations. Residence time distributions in a finite spatial domain are of a generalized log-normal type and are reminiscent also of a stretched exponential distribution. They can be easily confused for power-law distributions in view of the observed weak ergodicity breaking. This suggests a revision of some experimental data and their interpretation.
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Affiliation(s)
- Igor Goychuk
- Institute for Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany.
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9
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Cherstvy AG, Thapa S, Mardoukhi Y, Chechkin AV, Metzler R. Time averages and their statistical variation for the Ornstein-Uhlenbeck process: Role of initial particle distributions and relaxation to stationarity. Phys Rev E 2018; 98:022134. [PMID: 30253569 DOI: 10.1103/physreve.98.022134] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2018] [Indexed: 06/08/2023]
Abstract
How ergodic is diffusion under harmonic confinements? How strongly do ensemble- and time-averaged displacements differ for a thermally-agitated particle performing confined motion for different initial conditions? We here study these questions for the generic Ornstein-Uhlenbeck (OU) process and derive the analytical expressions for the second and fourth moment. These quantifiers are particularly relevant for the increasing number of single-particle tracking experiments using optical traps. For a fixed starting position, we discuss the definitions underlying the ensemble averages. We also quantify effects of equilibrium and nonequilibrium initial particle distributions onto the relaxation properties and emerging nonequivalence of the ensemble- and time-averaged displacements (even in the limit of long trajectories). We derive analytical expressions for the ergodicity breaking parameter quantifying the amplitude scatter of individual time-averaged trajectories, both for equilibrium and out-of-equilibrium initial particle positions, in the entire range of lag times. Our analytical predictions are in excellent agreement with results of computer simulations of the Langevin equation in a parabolic potential. We also examine the validity of the Einstein relation for the ensemble- and time-averaged moments of the OU-particle. Some physical systems, in which the relaxation and nonergodic features we unveiled may be observable, are discussed.
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Affiliation(s)
- Andrey G Cherstvy
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Samudrajit Thapa
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Yousof Mardoukhi
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
| | - Aleksei V Chechkin
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
- Institute for Theoretical Physics, Kharkov Institute of Physics and Technology, 61108 Kharkov, Ukraine
| | - Ralf Metzler
- Institute for Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany
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10
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Goychuk I. Sensing Magnetic Fields with Magnetosensitive Ion Channels. SENSORS (BASEL, SWITZERLAND) 2018; 18:E728. [PMID: 29495645 PMCID: PMC5877195 DOI: 10.3390/s18030728] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/25/2018] [Revised: 02/13/2018] [Accepted: 02/13/2018] [Indexed: 12/23/2022]
Abstract
[-15]Magnetic nanoparticles are met across many biological species ranging from magnetosensitive bacteria, fishes, bees, bats, rats, birds, to humans. They can be both of biogenetic origin and due to environmental contamination, being either in paramagnetic or ferromagnetic state. The energy of such naturally occurring single-domain magnetic nanoparticles can reach up to 10-20 room k B T in the magnetic field of the Earth, which naturally led to supposition that they can serve as sensory elements in various animals. This work explores within a stochastic modeling framework a fascinating hypothesis of magnetosensitive ion channels with magnetic nanoparticles serving as sensory elements, especially, how realistic it is given a highly dissipative viscoelastic interior of living cells and typical sizes of nanoparticles possibly involved.
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Affiliation(s)
- Igor Goychuk
- Institute of Physics and Astronomy, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam-Golm, Germany.
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11
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Lanoiselée Y, Grebenkov DS. Unraveling intermittent features in single-particle trajectories by a local convex hull method. Phys Rev E 2017; 96:022144. [PMID: 28950648 DOI: 10.1103/physreve.96.022144] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2017] [Indexed: 01/01/2023]
Abstract
We propose a model-free method to detect change points between distinct phases in a single random trajectory of an intermittent stochastic process. The local convex hull (LCH) is constructed for each trajectory point, while its geometric properties (e.g., the diameter or the volume) are used as discriminators between phases. The efficiency of the LCH method is validated for six models of intermittent motion, including Brownian motion with different diffusivities or drifts, fractional Brownian motion with different Hurst exponents, and surface-mediated diffusion. We discuss potential applications of the method for detection of active and passive phases in the intracellular transport, temporal trapping or binding of diffusing molecules, alternating bulk and surface diffusion, run and tumble (or search) phases in the motion of bacteria and foraging animals, and instantaneous firing rates in neurons.
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Affiliation(s)
- Yann Lanoiselée
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, University Paris-Saclay, 91128 Palaiseau, France
| | - Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, University Paris-Saclay, 91128 Palaiseau, France and Interdisciplinary Scientific Center Poncelet (ISCP), Bolshoy Vlasyevskiy Pereulok 11, 119002 Moscow, Russia
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12
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Sikora G, Teuerle M, Wyłomańska A, Grebenkov D. Statistical properties of the anomalous scaling exponent estimator based on time-averaged mean-square displacement. Phys Rev E 2017; 96:022132. [PMID: 28950534 DOI: 10.1103/physreve.96.022132] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2017] [Indexed: 06/07/2023]
Abstract
The most common way of estimating the anomalous scaling exponent from single-particle trajectories consists of a linear fit of the dependence of the time-averaged mean-square displacement on the lag time at the log-log scale. We investigate the statistical properties of this estimator in the case of fractional Brownian motion (FBM). We determine the mean value, the variance, and the distribution of the estimator. Our theoretical results are confirmed by Monte Carlo simulations. In the limit of long trajectories, the estimator is shown to be asymptotically unbiased, consistent, and with vanishing variance. These properties ensure an accurate estimation of the scaling exponent even from a single (long enough) trajectory. As a consequence, we prove that the usual way to estimate the diffusion exponent of FBM is correct from the statistical point of view. Moreover, the knowledge of the estimator distribution is the first step toward new statistical tests of FBM and toward a more reliable interpretation of the experimental histograms of scaling exponents in microbiology.
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Affiliation(s)
- Grzegorz Sikora
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wybrzeże Wyspianskiego 27, 50-370 Wroclaw, Poland
| | - Marek Teuerle
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wybrzeże Wyspianskiego 27, 50-370 Wroclaw, Poland
| | - Agnieszka Wyłomańska
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wroclaw University of Science and Technology, Wybrzeże Wyspianskiego 27, 50-370 Wroclaw, Poland
| | - Denis Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-École Polytechnique, Université Paris-Saclay, 91128 Palaiseau, France
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Rigato A, Miyagi A, Scheuring S, Rico F. High-frequency microrheology reveals cytoskeleton dynamics in living cells. NATURE PHYSICS 2017; 13:771-775. [PMID: 28781604 PMCID: PMC5540170 DOI: 10.1038/nphys4104] [Citation(s) in RCA: 135] [Impact Index Per Article: 19.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
Living cells are viscoelastic materials, with the elastic response dominating at long timescales (≳1 ms)1. At shorter timescales, the dynamics of individual cytoskeleton filaments are expected to emerge, but active microrheology measurements on cells accessing this regime are scarce2. Here, we develop high-frequency microrheology (HF-MR) to probe the viscoelastic response of living cells from 1Hz to 100 kHz. We report the viscoelasticity of different cell types and upon cytoskeletal drug treatments. At previously inaccessible short timescales, cells exhibit rich viscoelastic responses that depend on the state of the cytoskeleton. Benign and malignant cancer cells revealed remarkably different scaling laws at high frequency, providing a univocal mechanical fingerprint. Microrheology over a wide dynamic range up to the frequency of action of the molecular components provides a mechanistic understanding of cell mechanics.
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Affiliation(s)
- Annafrancesca Rigato
- U1006 INSERM, Aix-Marseille Université, Parc Scientifique et Technologique de Luminy, 163 avenue de Luminy, 13009 Marseille, France
| | - Atsushi Miyagi
- U1006 INSERM, Aix-Marseille Université, Parc Scientifique et Technologique de Luminy, 163 avenue de Luminy, 13009 Marseille, France
| | - Simon Scheuring
- U1006 INSERM, Aix-Marseille Université, Parc Scientifique et Technologique de Luminy, 163 avenue de Luminy, 13009 Marseille, France
| | - Felix Rico
- U1006 INSERM, Aix-Marseille Université, Parc Scientifique et Technologique de Luminy, 163 avenue de Luminy, 13009 Marseille, France
- Correspondence to:
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14
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Alibert C, Goud B, Manneville JB. Are cancer cells really softer than normal cells? Biol Cell 2017; 109:167-189. [DOI: 10.1111/boc.201600078] [Citation(s) in RCA: 168] [Impact Index Per Article: 24.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2016] [Accepted: 02/23/2017] [Indexed: 12/21/2022]
Affiliation(s)
- Charlotte Alibert
- Institut Curie; PSL Research University, CNRS; UMR 144 Paris France
- Sorbonne Universités, UPMC University Paris 06, CNRS; UMR 144 Paris France
| | - Bruno Goud
- Institut Curie; PSL Research University, CNRS; UMR 144 Paris France
- Sorbonne Universités, UPMC University Paris 06, CNRS; UMR 144 Paris France
| | - Jean-Baptiste Manneville
- Institut Curie; PSL Research University, CNRS; UMR 144 Paris France
- Sorbonne Universités, UPMC University Paris 06, CNRS; UMR 144 Paris France
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15
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Lanoiselée Y, Grebenkov DS. Revealing nonergodic dynamics in living cells from a single particle trajectory. Phys Rev E 2016; 93:052146. [PMID: 27300868 DOI: 10.1103/physreve.93.052146] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/18/2015] [Indexed: 06/06/2023]
Abstract
We propose the improved ergodicity and mixing estimators to identify nonergodic dynamics from a single particle trajectory. The estimators are based on the time-averaged characteristic function of the increments and can thus capture additional information on the process as compared to the conventional time-averaged mean-square displacement. The estimators are first investigated and validated for several models of anomalous diffusion, such as ergodic fractional Brownian motion and diffusion on percolating clusters, and nonergodic continuous-time random walks and scaled Brownian motion. The estimators are then applied to two sets of earlier published trajectories of mRNA molecules inside live Escherichia coli cells and of Kv2.1 potassium channels in the plasma membrane. These statistical tests did not reveal nonergodic features in the former set, while some trajectories of the latter set could be classified as nonergodic. Time averages along such trajectories are thus not representative and may be strongly misleading. Since the estimators do not rely on ensemble averages, the nonergodic features can be revealed separately for each trajectory, providing a more flexible and reliable analysis of single-particle tracking experiments in microbiology.
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Affiliation(s)
- Yann Lanoiselée
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
| | - Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
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Bouzat S. Models for microtubule cargo transport coupling the Langevin equation to stochastic stepping motor dynamics: Caring about fluctuations. Phys Rev E 2016; 93:012401. [PMID: 26871095 DOI: 10.1103/physreve.93.012401] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2015] [Indexed: 01/03/2023]
Abstract
One-dimensional models coupling a Langevin equation for the cargo position to stochastic stepping dynamics for the motors constitute a relevant framework for analyzing multiple-motor microtubule transport. In this work we explore the consistence of these models focusing on the effects of the thermal noise. We study how to define consistent stepping and detachment rates for the motors as functions of the local forces acting on them in such a way that the cargo velocity and run-time match previously specified functions of the external load, which are set on the base of experimental results. We show that due to the influence of the thermal fluctuations this is not a trivial problem, even for the single-motor case. As a solution, we propose a motor stepping dynamics which considers memory on the motor force. This model leads to better results for single-motor transport than the approaches previously considered in the literature. Moreover, it gives a much better prediction for the stall force of the two-motor case, highly compatible with the experimental findings. We also analyze the fast fluctuations of the cargo position and the influence of the viscosity, comparing the proposed model to the standard one, and we show how the differences on the single-motor dynamics propagate to the multiple motor situations. Finally, we find that the one-dimensional character of the models impede an appropriate description of the fast fluctuations of the cargo position at small loads. We show how this problem can be solved by considering two-dimensional models.
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Affiliation(s)
- Sebastián Bouzat
- Consejo Nacional de Investigaciones Científicas y Técnicas, Centro Atómico Bariloche (CNEA), (8400) Bariloche, Río Negro, Argentina
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Aekbote BL, Fekete T, Jacak J, Vizsnyiczai G, Ormos P, Kelemen L. Surface-modified complex SU-8 microstructures for indirect optical manipulation of single cells. BIOMEDICAL OPTICS EXPRESS 2016; 7:45-56. [PMID: 26819816 PMCID: PMC4722909 DOI: 10.1364/boe.7.000045] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/20/2015] [Revised: 11/19/2015] [Accepted: 11/20/2015] [Indexed: 05/24/2023]
Abstract
We introduce a method that combines two-photon polymerization (TPP) and surface functionalization to enable the indirect optical manipulation of live cells. TPP-made 3D microstructures were coated specifically with a multilayer of the protein streptavidin and non-specifically with IgG antibody using polyethylene glycol diamine as a linker molecule. Protein density on their surfaces was quantified for various coating methods. The streptavidin-coated structures were shown to attach to biotinated cells reproducibly. We performed basic indirect optical micromanipulation tasks with attached structure-cell couples using complex structures and a multi-focus optical trap. The use of such extended manipulators for indirect optical trapping ensures to keep a safe distance between the trapping beams and the sensitive cell and enables their 6 degrees of freedom actuation.
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Affiliation(s)
- Badri L. Aekbote
- Biological Research Centre of the Hungarian Academy of Sciences, Temesvári krt. 62, Szeged 6726, Hungary
| | - Tamás Fekete
- Biological Research Centre of the Hungarian Academy of Sciences, Temesvári krt. 62, Szeged 6726, Hungary
| | - Jaroslaw Jacak
- University of Applied Sciences Upper Austria, Garnisonstraße 21, 4020 Linz, Austria
| | - Gaszton Vizsnyiczai
- Biological Research Centre of the Hungarian Academy of Sciences, Temesvári krt. 62, Szeged 6726, Hungary
| | - Pál Ormos
- Biological Research Centre of the Hungarian Academy of Sciences, Temesvári krt. 62, Szeged 6726, Hungary
| | - Lóránd Kelemen
- Biological Research Centre of the Hungarian Academy of Sciences, Temesvári krt. 62, Szeged 6726, Hungary
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Smelser AM, Macosko JC, O'Dell AP, Smyre S, Bonin K, Holzwarth G. Mechanical properties of normal versus cancerous breast cells. Biomech Model Mechanobiol 2015; 14:1335-47. [PMID: 25929519 DOI: 10.1007/s10237-015-0677-x] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/22/2014] [Accepted: 04/04/2015] [Indexed: 12/21/2022]
Abstract
A cell's mechanical properties are important in determining its adhesion, migration, and response to the mechanical properties of its microenvironment and may help explain behavioral differences between normal and cancerous cells. Using fluorescently labeled peroxisomes as microrheological probes, the interior mechanical properties of normal breast cells were compared to a metastatic breast cell line, MDA-MB-231. To estimate the mechanical properties of cell cytoplasms from the motions of their peroxisomes, it was necessary to reduce the contribution of active cytoskeletal motions to peroxisome motion. This was done by treating the cells with blebbistatin, to inhibit myosin II, or with sodium azide and 2-deoxy-D-glucose, to reduce intracellular ATP. Using either treatment, the peroxisomes exhibited normal diffusion or subdiffusion, and their mean squared displacements (MSDs) showed that the MDA-MB-231 cells were significantly softer than normal cells. For these two cell types, peroxisome MSDs in treated and untreated cells converged at high frequencies, indicating that cytoskeletal structure was not altered by the drug treatment. The MSDs from ATP-depleted cells were analyzed by the generalized Stokes-Einstein relation to estimate the interior viscoelastic modulus G* and its components, the elastic shear modulus G' and viscous shear modulus G", at angular frequencies between 0.126 and 628 rad/s. These moduli are the material coefficients that enter into stress-strain relations and relaxation times in quantitative mechanical models such as the poroelastic model of the interior regions of cancerous and non-cancerous cells.
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Affiliation(s)
- Amanda M Smelser
- Department of Biochemistry and Molecular Biology, Wake Forest University School of Medicine, Winston-Salem, NC, 27157, USA
| | - Jed C Macosko
- Department of Biochemistry and Molecular Biology, Wake Forest University School of Medicine, Winston-Salem, NC, 27157, USA
- Department of Physics, Wake Forest University, Winston-Salem, NC, 27109, USA
| | - Adam P O'Dell
- Department of Physics, Wake Forest University, Winston-Salem, NC, 27109, USA
| | - Scott Smyre
- Department of Physics, Wake Forest University, Winston-Salem, NC, 27109, USA
| | - Keith Bonin
- Department of Physics, Wake Forest University, Winston-Salem, NC, 27109, USA
| | - George Holzwarth
- Department of Physics, Wake Forest University, Winston-Salem, NC, 27109, USA.
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Bouzat S. Influence of molecular motors on the motion of particles in viscoelastic media. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:062707. [PMID: 25019814 DOI: 10.1103/physreve.89.062707] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2013] [Indexed: 06/03/2023]
Abstract
We study theoretically and by numerical simulations the motion of particles driven by molecular motors in a viscoelastic medium representing the cell cytoplasm. For this, we consider a generalized Langevin equation coupled to a stochastic stepping dynamics for the motors that takes into account the action of each motor separately. In the absence of motors, the model produces subdiffusive motion of particles characterized by a power-law scaling of the mean square displacement versus the lag time as t^{α}, with 0<α<1, similar to that observed in cells. Our results show how the action of the motors can induce a transition to a superdiffusive regime at large lag times with the characteristics of those found in experiments reported in the literature. We also show that at small lag times, the motors can act as static crosslinkers that slow down the natural subdiffusive transport. An analysis of previously reported experimental data in the relevant time scales provides evidence of this phenomenon. Finally, we study the effect of a harmonic potential representing an optical trap, and we show a way to approach to a macroscopic description of the active transport in cells. This last point stresses the relevance of the molecular motors for generating not only directed motion to specific targets, but also fast diffusivelike random motion.
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Affiliation(s)
- Sebastián Bouzat
- Consejo Nacional de Investigaciones Científicas y Técnicas, Centro Atómico Bariloche (CNEA), (8400) Bariloche, Río Negro, Argentina
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Grebenkov DS, Vahabi M. Analytical solution of the generalized Langevin equation with hydrodynamic interactions: subdiffusion of heavy tracers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012130. [PMID: 24580195 DOI: 10.1103/physreve.89.012130] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/07/2013] [Indexed: 06/03/2023]
Abstract
We consider a generalized Langevin equation that can be used to describe thermal motion of a tracer in a viscoelastic medium by accounting for inertial and hydrodynamic effects at short times, subdiffusive scaling at intermediate times, and eventual optical trapping at long times. We derive a Laplace-type integral representation for the linear response function that governs the diffusive dynamics. This representation is particularly well suited for rapid numerical computation and theoretical analysis. In particular, we deduce explicit formulas for the mean and variance of the time averaged (TA) mean square displacement (MSD) and velocity autocorrelation function (VACF). The asymptotic behavior of the TA MSD and TA VACF is investigated at different time scales. Some biophysical and microrheological applications are discussed, with an emphasis on the statistical analysis of optical tweezers' single-particle tracking experiments in polymer networks and living cells.
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Affiliation(s)
- Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
| | - Mahsa Vahabi
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
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Gupta S, Rosso A, Texier C. Dynamics of a tagged monomer: effects of elastic pinning and harmonic absorption. PHYSICAL REVIEW LETTERS 2013; 111:210601. [PMID: 24313470 DOI: 10.1103/physrevlett.111.210601] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2013] [Indexed: 06/02/2023]
Abstract
We study the dynamics of a tagged monomer of a Rouse polymer for different initial configurations. In the case of free evolution, the monomer displays subdiffusive behavior with strong memory of the initial state. In the presence of either elastic pinning or harmonic absorption, we show that the steady state is independent of the initial condition that, however, strongly affects the transient regime, resulting in nonmonotonic behavior and power-law relaxation with varying exponents.
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Affiliation(s)
- Shamik Gupta
- Laboratoire de Physique Théorique et Modèles Statistiques (CNRS, UMR 8626), Université Paris-Sud, Orsay, France
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Grebenkov DS, Vahabi M, Bertseva E, Forró L, Jeney S. Hydrodynamic and subdiffusive motion of tracers in a viscoelastic medium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:040701. [PMID: 24229100 DOI: 10.1103/physreve.88.040701] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/15/2013] [Revised: 07/25/2013] [Indexed: 06/02/2023]
Abstract
We investigate the diffusive motion of micron-sized spherical tracers in a viscoelastic actin filament network over the time span of 8 orders of magnitude using optical-tweezers single-particle tracking. The hydrodynamic interactions of a tracer with the surrounding fluid are shown to dominate at microsecond time scales, while subdiffusive scaling due to viscoelastic properties of the medium emerges at millisecond time scales. The transition between these two regimes is analyzed in the frame of a minimal phenomenological model which combines the Basset force and the generalized Stokes force. The resulting Langevin equation accounts for various dynamical features of the thermal motion of endogenous or exogenous tracers in viscoelastic media such as inertial and hydrodynamic effects at short times, subdiffusive scaling at intermediate times, and eventual optical trapping at long times. Simple analytical formulas for the mean-square displacement and velocity autocorrelation function are derived.
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Affiliation(s)
- Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS - Ecole Polytechnique, 91128 Palaiseau, France
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Grebenkov DS. Optimal and suboptimal quadratic forms for noncentered Gaussian processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:032140. [PMID: 24125246 DOI: 10.1103/physreve.88.032140] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2013] [Revised: 09/06/2013] [Indexed: 06/02/2023]
Abstract
Individual random trajectories of stochastic processes are often analyzed by using quadratic forms such as time averaged (TA) mean square displacement (MSD) or velocity auto-correlation function (VACF). The appropriate quadratic form is expected to have a narrow probability distribution in order to reduce statistical uncertainty of a single measurement. We consider the problem of finding the optimal quadratic form that minimizes a chosen cumulant moment (e.g., the variance) of the probability distribution, under the constraint of fixed mean value. For discrete noncentered Gaussian processes, we construct the optimal quadratic form by using the spectral representation of cumulant moments. Moreover, we obtain a simple explicit formula for the smallest achievable cumulant moment that may serve as a quality benchmark for other quadratic forms. We illustrate the optimality issues by comparing the optimal variance with the variances of the TA MSD and TA VACF of fractional Brownian motion superimposed with a constant drift and independent Gaussian noise.
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Affiliation(s)
- Denis S Grebenkov
- Laboratoire de Physique de la Matière Condensée (UMR 7643), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
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