Statistical challenges in microrheology. (English) Zbl 1282.62232
Summary: Microrheology is the study of the properties of a complex fluid through the diffusion dynamics of small particles, typically latex beads, moving through that material. Currently, it is the dominant technique in the study of the physical properties of biological fluids, of the material properties of membranes or the cytoplasm of cells, or of the entire cell. The theoretical underpinning of microrheology was given in [T. G. Mason and D. A. Weitz, “Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids”, Phys. Rev. Lett. 74, 1250–1253 (1995; doi:10.1103/PhysRevLett.74.1250)], who introduced a framework for the use of path data of diffusing particles to infer viscoelastic properties of its fluid environment. The multi-particle tracking techniques that were subsequently developed have presented numerous challenges for experimentalists and theoreticians. This study describes some specific challenges that await the attention of statisticians and applied probabilists. We describe relevant aspects of the physical theory, current inferential efforts and simulation aspects of a central model for the dynamics of nano-scale particles in viscoelastic fluids, the generalized Langevin equation.
MSC:
| 62P10 | Applications of statistics to biology and medical sciences; meta analysis |
| 60G15 | Gaussian processes |
| 62M09 | Non-Markovian processes: estimation |
| 60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |
| 92C35 | Physiological flow |
Keywords:
Gaussian processes; microrheology; non-Newtonian fluids; generalized Langevin equation; local Whittle; fractional processes; simulation; waveletsReferences:
| [1] | Abry , P. Didier , G. 2012 On wavelet regression for operator fractional Brownian motions 1 50 · Zbl 1432.60045 |
| [2] | Adelman, Fokker-Planck equations for simple non-Markovian systems, The Journal of Chemical Physics 64 pp 124– (1976) · doi:10.1063/1.431961 |
| [3] | Asmussen, Stochastic Simulation: Algorithms and Analysis (2000) |
| [4] | Atzberger, A stochastic immersed boundary method for fluid-structure dynamics at microscopic length scales, Journal of Computational Physics 224 (2) pp 1255– (2007) · Zbl 1124.74052 · doi:10.1016/j.jcp.2006.11.015 |
| [5] | Bardet, Testing for the presence of self-similarity of Gaussian time series having stationary increments, Journal of Time Series Analysis 21 (5) pp 497– (2000) · Zbl 0972.62070 · doi:10.1111/1467-9892.00195 |
| [6] | Bardet, Statistical study of the wavelet analysis of fractional Brownian motion, IEEE Transactions on Information Theory 48 (4) pp 991– (2002) · Zbl 1061.60036 · doi:10.1109/18.992817 |
| [7] | Bardet, Theory and Applications of Long Range Dependence (2003) · Zbl 1005.00017 |
| [8] | Beran, Statistics for Long Memory Processes (1994) · Zbl 0869.60045 |
| [9] | Bondon, A class of antipersistent processes, Journal of Time Series Analysis 28 (2) pp 261– (2007) · Zbl 1150.62040 · doi:10.1111/j.1467-9892.2006.00509.x |
| [10] | Boysen, Consistencies and rates of convergence of jump-penalized least squares estimators, The Annals of Statistics 37 (1) pp 157– (2009) · Zbl 1155.62034 · doi:10.1214/07-AOS558 |
| [11] | Coppin, Detection of sub-8-nm movements of kinesin by high-resolution optical-trap microcopy, Proceedings of the National Academy of Sciences of the United States of America 93 (5) pp 1913– (1996) · doi:10.1073/pnas.93.5.1913 |
| [12] | Craigmile, Approximate wavelet-based simulation of long memory processes, Journal of Statistical Computation and Simulation 75 (5) pp 363– (2005) · Zbl 1061.62131 · doi:10.1080/0094965042000212916 |
| [13] | Craigmile, Simulating a class of stationary Gaussian processes using the Davies-Harte algorithm with application to long memory processes, Journal of Time Series Analysis 24 (5) pp 505– (2003) · Zbl 1035.68131 · doi:10.1111/1467-9892.00318 |
| [14] | Cramér, Stationary and Related Stochastic Processes (1967) |
| [15] | Davies, Tests for Hurst effect, Biometrika 74 (1) pp 95– (1987) · Zbl 0612.62123 · doi:10.1093/biomet/74.1.95 |
| [16] | Dawson, Enhanced viscoelasticity of human cystic fibrotic sputum correlates with increasing microheterogeneity in particle transport, Journal of Biological Chemistry 278 (50) pp 50393– (2003) · doi:10.1074/jbc.M309026200 |
| [17] | Didier , G. Fricks , J. 2012 On the wavelet-based simulation of subdiffusions |
| [18] | Didier, Gaussian stationary processes: discrete approximations adaptive wavelet decompositions and their convergence, Journal of Fourier Analysis and Applications 14 (2) pp 203– (2008) · Zbl 1157.60028 · doi:10.1007/s00041-008-9012-6 |
| [19] | Didier, Integral representations and properties of operator fractional Brownian motions, Bernoulli 17 (1) pp 1– (2011) · Zbl 1284.60079 · doi:10.3150/10-BEJ259 |
| [20] | Dietrich, Fast and exact simulation of stationary Gaussian processes through circulant matrix embedding of the covariance matrix, SIAM Journal of Scientific Computation 18 (4) pp 1088– (1997) · Zbl 0890.65149 · doi:10.1137/S1064827592240555 |
| [21] | Dixit, Differential regulation of dynein and kinesin motor proteins by tau, Science’s STKE 319 (5866) pp 1086– (2008) |
| [22] | Durbin, Time Series Analysis by State Space Methods (2001) · Zbl 0995.62504 |
| [23] | Embrechts, Selfsimilar Processes (2002) |
| [24] | Ferry, Viscoelastic Properties of Polymers (1980) |
| [25] | Fricks, Time domain methods for diffusive transport in soft matter, SIAM Journal of Applied Mathematics 69 pp 1277– (2009) · Zbl 1186.62113 · doi:10.1137/070695186 |
| [26] | Gardel, Stress-dependent elasticity of composite actin networks as a model for cell behavior, Physical Review Letters 96 (8) pp 88102– (2006) · doi:10.1103/PhysRevLett.96.088102 |
| [27] | Grigolini, A generalized Langevin equation for dealing with nonadditive fluctuations, Journal of Statistical Physics 27 pp 283– (1982) · Zbl 0512.60057 · doi:10.1007/BF01008940 |
| [28] | Gripenberg, Volterra Integral and Functional Equations 34 (1990) · doi:10.1017/CBO9780511662805 |
| [29] | Gubner, Theorems and fallacies in the theory of long-range-dependent processes, IEEE Transactions on Information Theory 51 (3) pp 1234– (2005) · Zbl 1301.60037 · doi:10.1109/TIT.2004.842768 |
| [30] | Guigas, Probing the nanoscale viscoelasticity of intracellular fluids in living cells, Biophysical journal 93 (1) pp 316– (2007) · doi:10.1529/biophysj.106.099267 |
| [31] | Hohenegger, The XVth International Congress on Rheology Part Two(AIP Conference Proceedings Volume 1027 Part 2) 1027 pp 1093– (2008a) |
| [32] | Hohenegger, Two-bead microrheology: modeling protocols, Physical Review E 78 (3) pp 031501– (2008b) · doi:10.1103/PhysRevE.78.031501 |
| [33] | Johnson, Constructions of particular random processes, Proceedings of the IEEE 82 (2) pp 270– (1994) · doi:10.1109/5.265353 |
| [34] | Kim, State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications (1999) |
| [35] | Kloeden, Numerical Solutions of Stochastic Differential Equations (1995) |
| [36] | Kou, Stochastic modeling in nanoscale biophysics: subdiffusion within proteins, Annals of Applied Statistics 2 (2) pp 501– (2008) · Zbl 1400.62272 · doi:10.1214/07-AOAS149 |
| [37] | Kou, Generalized Langevin equation with fractional Gaussian noise: subdiffusion within a single protein molecule, Physical review letters 93 (18) pp 180603-1– (2004) · doi:10.1103/PhysRevLett.93.180603 |
| [38] | Kremer, Dynamics of entangled linear polymer melts: a molecular-dynamics simulation, The Journal of Chemical Physics 92 pp 5057– (1990) · doi:10.1063/1.458541 |
| [39] | Kubo, The fluctuation-dissipation theorem, Reports on Progress in Physics 29 pp 255– (1966) · Zbl 0163.23102 · doi:10.1088/0034-4885/29/1/306 |
| [40] | Künsch, Probability Theory and Applications pp 67– (1987) |
| [41] | Lai, Altering mucus rheology to solidify human mucus at the nanoscale, PLoS One 4 (1) pp 4294– (2009a) · doi:10.1371/journal.pone.0004294 |
| [42] | Lai, Mucus-penetrating nanoparticles for drug and gene delivery to mucosal tissues, Advanced Drug Delivery Reviews 61 (2) pp 158– (2009b) · doi:10.1016/j.addr.2008.11.002 |
| [43] | Larson, The structure and rheology of complex fluids (2001) |
| [44] | Levine, One-and two-particle microrheology, Physical Review Letters 85 (8) pp 1774– (2000) · doi:10.1103/PhysRevLett.85.1774 |
| [45] | Lieleg, Characterization of particle translocation through mucin hydrogels, Biophysical Journal 98 (9) pp 1782– (2010) · doi:10.1016/j.bpj.2010.01.012 |
| [46] | Mallat, A Wavelet Tour of Signal Processing (1999) |
| [47] | Mason, Optical measurements of the linear viscoelastic moduli of complex fluids, Physical Review Letters 74 pp 1250– (1995) · doi:10.1103/PhysRevLett.74.1250 |
| [48] | Matsui, A physical linkage between cystic fibrosis airway surface dehydration and pseudomonas aeruginosa biofilms, Proceedings of the National Academy of Sciences 103 (48) pp 18131– (2006) · doi:10.1073/pnas.0606428103 |
| [49] | McKinley, Transient anomalous diffusion of tracer particles in soft matter, Journal of Rheology 53 (6) pp 1487– (2009) · doi:10.1122/1.3238546 |
| [50] | Min, Observation of a power-law memory kernel for fluctuations within a single protein molecule, Physical review letters 94 (19) pp 198302– (2005) · doi:10.1103/PhysRevLett.94.198302 |
| [51] | Morgado, Relation between anomalous diffusion and normal diffusion in systems with memory, Physical Review Letters 89 (10) pp 100601– (2002) · doi:10.1103/PhysRevLett.89.100601 |
| [52] | Mori, A continued-fraction representation of the time-correlation functions, Progress of Theoretical Physics 34 (3) pp 399– (1965) · doi:10.1143/PTP.34.399 |
| [53] | Moulines, On the spectral density of the wavelet coefficients of long-memory time series with application to the log-regression estimation of the memory parameter, Journal of Time Series Analysis 28 (2) pp 155– (2007) · Zbl 1150.62058 · doi:10.1111/j.1467-9892.2006.00502.x |
| [54] | Neuenkirch, Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion, Stochastic Processes and their Applications 118 (12) pp 2294– (2008) · Zbl 1154.60338 · doi:10.1016/j.spa.2008.01.002 |
| [55] | Palma, Long-memory Time Series: Theory and Methods (2007) · Zbl 1183.62153 · doi:10.1002/9780470131466 |
| [56] | Percival, University of Washington Technical Report pp 1– (2002) |
| [57] | Pipiras, Wavelet-based simulation of fractional Brownian motion revisited, Applied and Computational Harmonic Analysis 19 (1) pp 49– (2005) · Zbl 1074.60048 · doi:10.1016/j.acha.2005.01.002 |
| [58] | Robinson, Gaussian semiparametric estimation of long range dependence, Annals of Statistics 23 (5) pp 1630– (1995a) · Zbl 0843.62092 · doi:10.1214/aos/1176324317 |
| [59] | Robinson, Log-periodogram regression of time series with long range dependence, Annals of Statistics 23 (3) pp 1048– (1995b) · Zbl 0838.62085 · doi:10.1214/aos/1176324636 |
| [60] | Rubinstein, Polymer Physics 105 (2003) |
| [61] | Saxton, Anomalous diffusion due to obstacles: a Monte Carlo study, Biophysical Journal 66 (2) pp 394– (1994) · doi:10.1016/S0006-3495(94)80789-1 |
| [62] | Saxton, Anomalous diffusion due to binding: a Monte Carlo study, Biophysical Journal 70 (3) pp 1250– (1996) · doi:10.1016/S0006-3495(96)79682-0 |
| [63] | Shumway, Time Series Analysis and its Applications (2006) · Zbl 1096.62088 |
| [64] | Squires, Fluid mechanics of microrheology, Annual Review of Fluid Mechanics 42 pp 413– (2010) · doi:10.1146/annurev-fluid-121108-145608 |
| [65] | Suh, Real-time multiple-particle tracking: applications to drug and gene delivery, Advanced Drug Delivery Reviews 57 pp 63– (2005) · doi:10.1016/j.addr.2004.06.001 |
| [66] | Suk, The penetration of fresh undiluted sputum expectorated by cystic fibrosis patients by non-adhesive polymer nanoparticles, Biomaterials 30 (13) pp 2591– (2009) · doi:10.1016/j.biomaterials.2008.12.076 |
| [67] | Valentine, Investigating the microenvironments of inhomogeneous soft materials with multiple particle tracking, Physical Review E 64 (6) pp 061506– (2001) · doi:10.1103/PhysRevE.64.061506 |
| [68] | Veitch, A wavelet-based joint estimator of the parameters of long-range dependence, IEEE Transactions on Information Theory 45 (3) pp 878– (1999) · Zbl 0945.94006 · doi:10.1109/18.761330 |
| [69] | Wood, Simulation of stationary Gaussian processes in [0,1]d, Journal of Computational and Graphical Statistics 3 (4) pp 409– (1994) |
| [70] | Zwanzig, Nonequilibrium Statistical Mechanics (2001) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
