SynopsisThis paper is motivated by experiments in which time series of tracer particles in viscoelastic liquids are recorded using advanced microscopy. The experiments seek to infer either viscoelastic properties of the sample ͓Mason and Weitz, Phys. Rev. Lett. 74, 1250-1253 ͑1995͔͒ or diffusive properties of the specific tracer particle in the host medium ͓Suh et al., Adv. Drug Delivery Rev. 57, 63-78 ͑2005͒; Matsui et al., Proc. Natl. Acad. Sci. U.S.A. 103, 18131-18136 ͑2006͒; Lai et al., Proc. Natl. Acad. Sci. U.S.A. 104, 1482-1487 ͑2007͒; Fricks et al., SIAM J. Appl. Math. 69, 1277-1308 ͑2009͔͒. Our focus is the latter. Experimentalists often fit data to transient anomalous diffusion: a sub-diffusive power law scaling ͑t , with 0 Ͻ Ͻ 1͒ of mean-squared displacement ͑MSD͒ over a finite time interval, with longtime viscous scaling ͑t 1 ͒. The time scales of sub-diffusion and exponents are observed to vary with particle size, particle surface chemistry, and viscoelastic properties of the host material. Until now, explicit models for transient subdiffusive MSD scaling behavior ͓Doi and Edwards, The Theory of Polymer Physics ͑Oxford University Press, New York, 1986͒; Kremer and Grest, J. Chem. Phys. 92, 5057-5086 ͑1990͒; Rubinstein and Colby, Polymer Physics ͑Oxford University Press, New York, 2003͔͒ are limited to precisely three exponents: monomer diffusion in Rouse chain melts ͑t 1/2 ͒, in Zimm chain solutions ͑t 2/3 ͒, and in reptating chains ͑t 1/4 ͒. In this paper, we present an explicit parametrized family of stochastic processes ͑generalized Langevin equations with prescribed memory kernels͒ and derive their closed-form solutions which ͑1͒ span the complete range of transient sub-diffusive behavior and ͑2͒ possess the flexibility to tune both the time window of sub-diffusive scaling and the power law exponent . These results establish a robust family of sub-diffusive models, for which the inverse problem of parameter inference from experimental data ͓Fricks et al., SIAM J. Appl. Math. I. DIFFUSION IN A VISCOELASTIC MEDIUM A. The generalized Langevin equationOver the past half-century, the molecular theory of polymer dynamics ͓Rouse ͑1953͒; Zimm ͑1956͒; Kubo ͑1985͒; Kremer and Grest ͑1990͒; Levine and Lubensky ͑2001͔͒ has developed the correspondence between diffusive dynamics and linear viscoelastic relaxation of polymer melts and solutions. Excellent accounts of the sources and scaling properties of diffusive time scales and associated normal modes of stress relaxation are given in the monographs of Doi and Edwards ͑1986͒ and Rubinstein and Colby ͑2003͒. Stochastic models were likewise developed to understand atomic and molecular fluctuation spectra in media where viscous and elastic collisions are comparable ͓Mori ͑1965͒; Zwanzig and Bixon ͑1970͔͒.A standard model for diffusion in a viscoelastic medium is a generalized Langevin equation ͑GLE͒. Following Mori ͑1965͒, Zwanzig and Bixon ͑1970͒, and Mason and Weitz ͑1995͒, the Stokes drag law is generalized by convolution of the velocity with a memory func...

Passive microrheology [12] utilizes measurements of noisy, entropic fluctuations (i.e., diffusive properties) of micron-scale spheres in soft matter to infer bulk frequency-dependent loss and storage moduli. Here, we are concerned exclusively with diffusion of Brownian particles in viscoelastic media, for which the Mason-Weitz theoretical-experimental protocol is ideal, and the more challenging inference of bulk viscoelastic moduli is decoupled. The diffusive theory begins with a generalized Langevin equation (GLE) with a memory drag law specified by a kernel [7, 16, 22, 23]. We start with a discrete formulation of the GLE as an autoregressive stochastic process governing microbead paths measured by particle tracking. For the inverse problem (recovery of the memory kernel from experimental data) we apply time series analysis (maximum likelihood estimators via the Kalman filter) directly to bead position data, an alternative to formulas based on mean-squared displacement statistics in frequency space. For direct modeling, we present statistically exact GLE algorithms for individual particle paths as well as statistical correlations for displacement and velocity. Our time-domain methods rest upon a generalization of well-known results for a single-mode exponential kernel [1, 7, 22, 23] to an arbitrary M-mode exponential series, for which the GLE is transformed to a vector Ornstein-Uhlenbeck process.

In this work, we explore fundamental energy requirements during mammalian cell movement. Starting with the conservation of mass and momentum for the cell cytosol and the actin-network phase, we develop useful identities that compute dissipated energies during extensions of the cell boundary. We analyze 2 complementary mechanisms of cell movement: actin-driven and water-driven. The former mechanism occurs on 2-dimensional cell-culture substrate without appreciable external hydraulic resistance, while the latter mechanism is prominent in confined channels where external hydraulic resistance is high. By considering various forms of energy input and dissipation, we find that the water-driven cell-migration mechanism is inefficient and requires more energy. However, in environments with sufficiently high hydraulic resistance, the efficiency of actin-polymerization-driven cell migration decreases considerably, and the water-based mechanism becomes more efficient. Hence, the most efficient way for cells to move depends on the physical environment. This work can be extended to higher dimensions and has implication for understanding energetics of morphogenesis in early embryonic development and cancer-cell metastasis and provides a physical basis for understanding changing metabolic requirements for cell movement in different conditions.

SUMMARY A novel augmented forcing point method is presented to solve the problem of chemical transport in the fluid outside of a collection of suspended cells coupled with chemical reactions on the surfaces of the cells. In this method, the chemical concentrations and the forcing function values are determined simultaneously from an augmented system of equations. The method is more stable and accurate than predictor‐corrector‐type forcing point methods, yields the same solution as a corresponding ghost cell method with much less computational cost, and provides solutions that are pointwise second‐order accurate in space and time, even for closely‐spaced cells. Copyright © 2011 John Wiley & Sons, Ltd.

The classical oscillatory shear wave model of Ferry et al. [J. Polym. Sci. 2:593-611, (1947)] is extended for active linear and nonlinear microrheology. In the Ferry protocol, oscillation and attenuation lengths of the shear wave measured from strobe photographs determine storage and loss moduli at each frequency of plate oscillation. The microliter volumes typical in biology require modifications of experimental method and theory. Microbead tracking replaces strobe photographs. Reflection from the top boundary yields counterpropagating modes which are modeled here for linear and nonlinear viscoelastic constitutive laws. Furthermore, bulk imposed strain is easily controlled, and we explore the onset of normal stress generation and shear thinning using nonlinear viscoelastic models. For this paper, we present the theory, exact linear and nonlinear solutions where possible, and simulation tools more generally. We then illustrate errors in inverse characterization by application of the Ferry formulas, due to both suppression of wave reflection and nonlinearity, even if there were no experimental error. This shear wave method presents an active and nonlinear analog of the two-point microrheology of Crocker et al. [Phys. Rev. Lett. 85: 888 -891 (2000)]. Nonlocal (spatially extended) deformations and stresses are propagated through a small volume sample, on wavelengths long relative to bead size. The setup is ideal for exploration of nonlinear threshold behavior.