The filling of a finite region of a lattice by a random walk is of interest, fundamentally and for biophysical applications. Filling is described by the cover time, defined as the time required for the walker to visit all conducting sites at least once, and the blanket time, defined as the time required for the walker to visit the sites uniformly for a specified level of uniformity. For the unobstructed case, a theorem states that the blanket time is proportional to the cover time. That is, the cost of uniformity is just a multiplicative constant, not a more stringent time dependence.

In transient anomalous subdiffusion, diffusion is anomalous at short times, crossing over to normal at long times. This behavior is well-known for obstructed diffusion with obstacle concentrations below the percolation threshold, and is characterized by the anomalous exponent alpha, the limiting diffusion coefficient D(inf), and the crossover time t(cross), all functions of the obstacle concentration. In my work [Biophys J 66 (1994) 394], the anomalous and normal regions are chosen by eye in a log-log plot of D(t) = RSQ/t versus time t. Straight lines are found for each region by a least-squares fit, and t(cross) is defined as the intersection of these lines. Ellery et al. [J Chem Phys 144 (2016) 171104] criticized this approach and presented a method based on eigenvalues of the transition matrix. Their crossover times were significantly different from mine, and Ellery et al. claimed that their method provides a simpler and more objective evaluation of t(cross). I examine an alternative interpretation, that their method is an elegant measurement of a different characteristic time of the random walk. To test this, I compare results of the eigenvalue method with results of standard Monte Carlo calculations for various characteristic times, including distinct sites visited as a function of time, the number of visits per site (yielding the cover time and the blanket time), and the first passage time from the center of the lattice to the boundary of the region without periodic boundary conditions. The calculations are done for random walks on a square lattice with random obstacles and periodic boundary conditions. Importantly, I examine the dependence of the results on system size, and include percolation analysis so that higher obstacle concentrations can be included in some calculations. The so-called lazy random walk algorithm is also examined.

A key idea in SPT analysis is that the pure unhindered random walk is the null hypothesis. To claim that a diffusing tracer particle is doing something, one must evaluate, informally or formally, the probability that similar behavior would occur by chance in a pure random walk of the same duration. One biophysically important behavior of a tracer is immobilization. Immobilization is usually defined in terms of noise, specifically the observed scatter of positions of a tracer physically immobilized on the slide. But the minimal operational definition of immobilization is escape from the effective PSF, not the raw optical PSF but the optical PSF as narrowed by the SPT localization algorithm. A prominent feature of the PSF is the asymmetry between radial and axial resolution. Standard SPT algorithms narrow the effective PSF in the radial direction but not in the axial direction; increasing the axial resolution requires modifying the optics. The criterion for immobilization thus becomes, does the tracer remain in the effective PSF for a longer time than expected by chance? The PSF is modeled as a wall, corresponding to a prescribed radial-axial intensity contour. The dwell time is a well-defined first passage problem for a random walk. A tracer starts inside the wall, and carries out a random walk until it first reaches the wall. Wall geometries examined include circles, ellipses, generalized ellipses, and more complicated contours from the numerical solution for the PSF. Dwell times are obtained from analytical and Monte Carlo calculations. For tracers initially uniform, the probability density function is monotonically decreasing, power-law for short times and exponential for large times. For tracers initially at the origin, the distribution is unimodal. It resembles a log-normal distribution but is not.

Single-particle tracking (SPT) experiments have shown a truncated power-law distribution of dwell times for several DNA-binding species in the nucleus of living cells. Why this distribution? A standard explanation is Arrhenius escape from traps with exponentially distributed binding energies. But here the binding energies are roughly Gaussian, yielding roughly lognormal escape times. We therefore examine other mechanisms. The operational definition of SPT immobility is based on the dwell time within the volume of the effective SPT point spread function. The distribution of dwell times can be power-law at short times, but the exponent is constant and much less than observed. The dwell time problem has been treated in terms of revisits by a tracer to a trap, here recast as the number of revisits before the tracer leaves the volume of the effective SPT point spread function. The effect of revisits is limited because the sum of multiple escape times from a moderate trap is tame compared with single escapes from a deep trap, by the central limit theorem. Truncated power-law trapping yields transient anomalous subdiffusion, and obstructed diffusion includes two well-studied cases of this. In percolation the tracer concentration is low and the concentration of immobile obstacles is high. Anomalous subdiffusion is the result of static geometric trapping by dead ends, but how can these structures be made on the SPT length scale from DNA with a 50-nm persistence length? The other case is crowding, in which all species are mobile, the concentration is high, and the geometric traps are dynamic. We examine the effects of polydisperse crowders and of small concentrations of immobile species in a crowded system, to see whether crowding can bridge the gap between the SPT and DNA length scales.

Single-particle tracking (SPT) experiments in other laboratories measured the dwell time distributions for several DNA-binding species in the nucleus of living cells, and found a truncated power law distribution. Why? The standard explanation of power-law dwell times is Arrhenius escape from traps with exponentially distributed binding energies. This explanation is often correct, but here the binding energy distribution is roughly Gaussian, which yields a lognormal distribution of dwell times. The lognormal distribution approximates a power law over a limited range but has the wrong exponent. Apparently the problem cannot be solved at the level of DNA-protein interactions, so we look further. One guiding principle is comparison of the length scales of the nucleus and the SPT resolution element (resel). The resel size is 25 nm and the persistence length of DNA is 50 nm, so DNA is approximately linear in a resel unless there are tightly bent structures requiring metabolic energy to assemble or disassemble. This length mismatch excludes percolation and constrains viscoelasticity. Standard fractal models are insufficent because they are based on coarse-grained DNA with a bead size around 1 resel. The length of a base pair is 0.3 nm, so there are enough traps on a single DNA segment in a single resel to yield binding dynamics below SPT resolution. Another guiding principle is that geometric explanations require geometric controls. Several laboratories have examined the effects of escape and recapture, and the geometric arrangement of traps, but in my opinion this topic is not yet settled. It is essential to identify a minimal model, and a necessary control for any specific arrangement of traps is a set of random point traps with the same dwell time distribution and concentrations.

Single-particle tracking experiments have measured escape times of DNA-binding species diffusing in living cells: CRISPR-Cas9, TetR, and LacI. The observed distribution is a truncated power law. Working backward from the experimental results, the observed distribution appears inconsistent with a Gaussian distribution of binding energies. Working forward, the observed distribution leads to transient anomalous subdiffusion, in which diffusion is anomalous at short times and normal at long times, here only mildly anomalous. Monte Carlo simulations are used to characterize the time-dependent diffusion coefficient D(t) in terms of the anomalous exponent α, the crossover time tcross, and the limits D(0) and D(∞) and to relate these quantities to the escape time distribution. The simplest interpretations identify the escape time as the actual binding time to DNA or the period of one-dimensional diffusion on DNA in the standard model combining one-dimensional and three-dimensional search, but a more complicated interpretation may be required. The model has several implications for cell biophysics. 1) The initial anomalous regime represents the search of the DNA-binding species for its target DNA sequence. 2) Non-target DNA sites have a significant effect on search kinetics. False positives in bioinformatic searches of the genome are potentially rate-determining in vivo. For simple binding, the search would be speeded if false-positive sequences were eliminated from the genome. 3) Both binding and obstruction affect diffusion. Obstruction ought to be measured directly, using as the primary probe the DNA-binding species with the binding site inactivated and eGFP as a calibration standard among laboratories and cell types. 4) Overexpression of the DNA-binding species reduces anomalous subdiffusion because the deepest binding sites are occupied and unavailable. 5) The model provides a coarse-grained phenomenological description of diffusion of a DNA-binding species, useful in larger-scale modeling of kinetics, FCS, and FRAP.

A key problem in the analysis of trajectories from two-dimensional single-particle tracking experiments is distinguishing actual structure due to anomalous subdiffusion, confinement, or directed motion from apparent structure due to fluctuations in random walks. To better analyze these trajectories we examine properties of descriptors based on the radius of gyration tensor of the trajectory. This work revisits and updates my work [Saxton, Biophys J 64 (1993) 1776] and work from the Amblard laboratory [Coscoy et al., Bull Math Biol 69 (2007) 2467]. The descriptors are chosen to be the time-scaled eigenvalues of the radius of gyration tensor. Both individual and joint probability density functions of the descriptors are considered, using analytical results from the literature and Monte Carlo simulations. Particular attention is paid to the time scaling of the histograms so one can easily obtain the histogram for a given number of experimental time steps.The current view in the physics literature on single-particle tracking is that the analysis should be done entirely in terms of the displacement over single time steps, Δt = 1. This approach is clearly appropriate for pure random walks. These are Markovian, and displacements for Δt > 1 by definition cannot add any information. But for non-Markovian motion, Δt > 1 is directly relevant, as the method of Thiel et al. [PRL 111 (2013) 010601] to identify the mechanism of anomalous diffusion by scaling analysis of displacements over pairs of Δt's. Biophysically interesting forms of non-Markovian motion include directed motion, which is positively correlated; anomalous subdiffusion, which is anticorrelated on all time scales; and confined motion, which is anticorrelated on the time scales of individual collisions with the corral walls and diffusive crossing of the corral. (Supported in part by NIH grant GM038133)

How well can the diffusion coefficient D of a globular biomolecule be predicted from its molecular mass MW? In "Wanted: Scalable Tracers for Diffusion Measurements" [J Phys Chem B, submitted], I propose that diffusion measurements in heterogeneous systems can be improved by the use of scalable tracers, in which the size is varied alone at constant shape, surface properties, diffusion mechanism, deformability, and other properties affecting diffusion. Before trying to design a de novo series of scalable globular proteins, it is appropriate to examine how scalable the commonly used de antiquo globular proteins are [ibid., supporting information]. The widely-used compilation of experimental diffusion coefficients by Tyn and Gusek [Biotech Bioeng 35 (1990) 327] was examined. This set -- ranging from ribonuclease, 12640 Da, to tobacco mosaic virus, 50 MDa -- was plotted as D versus log MW. The obviously linear species were removed, and values of D and MW for the outliers were examined. The plot yields a cloud of values of D versus log MW. In this plot, rigorously scalable tracers are expected to give a single smooth curve of D versus log MW, and the extent of the cloud represents scatter due to nonscalablity in the other properties, and to experimental error. Values of D from hydrodynamic calculations from various laboratories are remarkably consistent with the cloud. The cloud prediction is certainly good enough for semi-quantitative estimates or for designing single-particle tracking experiments. For a diffusion-controlled reaction in dilute solution, the prediction is close enough that the standard analysis of propagation of errors can be used. But arbitrary cloud proteins are not adequate for, say, measuring the percolation threshold of cytoplasm. The incomplete examination of the question here indicates what would be required for a complete examination.

first community experiment comparing the performance of analysis methods for single-particle tracking data declares no winner but reveals valuable information for users and developers.

Scalable tracers are potentially a useful tool to examine diffusion mechanisms and to predict diffusion coefficients, particularly for hindered diffusion in complex, heterogeneous, or crowded systems. Scalable tracers are defined as a series of tracers varying in size but with the same shape, structure, surface chemistry, deformability, and diffusion mechanism. Both chemical homology and constant dynamics are required. In particular, branching must not vary with size, and there must be no transition between ordinary diffusion and reptation. Measurements using scalable tracers yield the mean diffusion coefficient as a function of size alone; measurements using nonscalable tracers yield the variation due to differences in the other properties. Candidate scalable tracers are discussed for two-dimensional (2D) diffusion in membranes and three-dimensional diffusion in aqueous solutions. Correlations to predict the mean diffusion coefficient of globular biomolecules from molecular mass are reviewed briefly. Specific suggestions for the 3D case include the use of synthetic dendrimers or random hyperbranched polymers instead of dextran and the use of core-shell quantum dots. Another useful tool would be a series of scalable tracers varying in deformability alone, prepared by varying the density of crosslinking in a polymer to make say "reinforced Ficoll" or "reinforced hyperbranched polyglycerol."

One of the most common statements about random walks or Brownian motion is that they are statistically self-similar. Self-similarity is commonly depicted as an individual trajectory at different magnifications or plots of square displacement versus time at different magnifications. The first question is, what exactly does self-similarity mean here? That the trajectory itself and certain functionals of it are self-similar? That all functionals of it are self-similar? That all functionals of a random walk are self-similar in the limit as the random walk approaches Brownian motion? This problem is a form of the standard question of how a random walk (finite step length) approaches the limit of Brownian motion (infinitesimal step length). This problem is well understood mathematically but for applications it is useful to understand how the approach to the Brownian limit affects simulations. When does the histogram of a functional of a random walk approximate the corresponding distribution for Brownian motion? Second, what exactly does a random walk simulation yield? In the simulation the step size is chosen from a Rayleigh distribution with mean-square radius = 2d D₀ Δt, and the angle is random. This is exactly what the propagator for Brownian motion yields for a time interval Δt. Thus the usual random walk simulation is periodically sampled Brownian motion. Periodic sampling — in engineering terminology, downsampling — is a form of low-pass filtering. Likewise, an experimental single-particle tracking trajectory is to a first approximation periodically sampled Brownian motion, but there are deviations. At short distances, the dynamics is Newtonian, not Brownian, and at longer distances, complex behavior may occur, such as the Alder-Wainwright long-time tails from hydrodynamics, and the collective motions found by Vattulainen and collaborators. Supported in part by NIH grant GM038133.

Anomalous subdiffusion in cells and model systems is an active area of research. The main questions are whether diffusion is anomalous or normal, and if it is anomalous, its mechanism. The subject is controversial, especially the hypothesis that crowding causes anomalous subdiffusion. Anomalous subdiffusion measurements would be strengthened by an experimental standard, particularly one able to cross-calibrate the different types of measurements. Criteria for a calibration standard are proposed. First, diffusion must be anomalous over the length and timescales of the different measurements. The length-scale is fundamental; the time scale can be adjusted through the viscosity of the medium. Second, the standard must be theoretically well understood, with a known anomalous subdiffusion exponent, ideally readily tunable. Third, the standard must be simple, reproducible, and independently characterizable (by, for example, electron microscopy for nanostructures). Candidate experimental standards are evaluated, including obstructed lipid bilayers; aqueous systems obstructed by nanopillars; a continuum percolation system in which a prescribed fraction of randomly chosen obstacles in a regular array is ablated; single-file diffusion in pores; transient anomalous subdiffusion due to binding of particles in arrays such as transcription factors in randomized DNA arrays; and computer-generated physical trajectories.

The fundamental principle in interpreting single-particle trajectories is that a pure random walk is the control and null hypothesis. In order to make any claim about a putative physical or biological event in an observed single-particle trajectory, one must evaluate the probability that the event could have occurred by chance in the corresponding pure random walk. By eye, a random walk often shows alternating periods of “moving” and “dithering.” We quantify this apparent structure by Monte Carlo calculations of pure random walks in which various measures of apparent directed motion ("moving") and apparent confinement ("dithering") are compared. A key point is that long dithering or moving events are rare in themselves. But in examining a trajectory for these events, we do not specify the starting point of the event, and all that we specify about the duration is that it be above some minimum threshold of detectability. The large number of potential starting points and durations of these events compensates for their rarity. An essential feature of the approach is to separate three distinct aspects of the problem. First, we separate characterization from trajectory segmentation, and treat only characterization here. Second, we separate characterization from the effect of noise, and assume that the particle positions are exactly known. In practical applications to experimental single-particle trajectories, the random noise in the position measurement must be taken into account. The move-dither analysis will ultimately form the basis for new tests to identify directed and confined motion in single-particle trajectories, and to distinguish anomalous from normal diffusion. Supported by NIH grant GM038133.

Lateral diffusion in the plasma membrane is obstructed by proteins bound to the cytoskeleton. The most important parameter describing obstructed diffusion is the percolation threshold. The thresholds are well known for point tracers, but for tracers of nonzero radius, the threshold depends on the excluded area, not just the obstacle concentration. Here thresholds are obtained for circular obstacles on the continuum. Random obstacle configurations are generated by Brownian dynamics or Monte Carlo methods, the obstacles are immobilized, and the percolation threshold is obtained by solving a bond percolation problem on the Voronoi diagram of the obstacles. The percolation threshold is expressed as the diameter of the largest tracer that can cross a set of immobile obstacles at a prescribed number density. For random overlapping obstacles, the results agree with the known analytical solution quantitatively. When the obstacles are soft disks with a 1/r12 repulsion, the percolating diameter is ∼20% lower than for overlapping obstacles. A percolation model predicts that the threshold is highly sensitive to the tracer radius. To our knowledge, such a strong dependence has so far not been reported for the plasma membrane, suggesting that percolation is not the factor controlling lateral diffusion. A definitive experiment is proposed.

Single-particle techniques are a powerful approach to study systems with spatial or temporal inhomogeneity. The living cell is a prime example of both. Single-particle measurements give much more detailed information than ensemble-averaged measurements. One can find the distribution of properties or behavior, not just the average. Is the distribution Gaussian or does it have more extreme wings? Are there rare events or rare intermediates or interesting subpopulations? Here the term particle is defined very broadly, to include, for example, a lipid, a protein, a subnuclear body, a vesicle, an organelle, a virus, or a colloidal particle. In the discussion of dynamics a particle will be taken to be any object small enough to undergo Brownian motion.

Identifying potential sources of artifactual anomalous diffusion is an important contribution, particularly in the case of transient anomalous subdiffusion. Martin et al. (1.Martin D.S. Forstner M.B. Käs J.A. Apparent subdiffusion inherent to single particle tracking.Biophys. J. 2002; 83: 2109-2117Abstract Full Text Full Text PDF PubMed Scopus (160) Google Scholar), for example, showed that noise in single-particle tracking (SPT) measurements can lead to a period of spurious anomalous subdiffusion. This work originated from experimental evidence of anomalous subdiffusion in a system for which diffusion ought to have been purely normal. In their Comment, Destainville et al. (2.Destainville N. Saulière A. Salomé L. Comment to the article by Michael J. Saxton: a biological interpretation of transient anomalous subdiffusion. I. Qualitative model.Biophys. J. 2008; : 3117-3119Abstract Full Text Full Text PDF PubMed Scopus (21) Google Scholar) point out that a period of spurious anomalous diffusion can result from the transition between two limiting cases, normal diffusion within a corral (or a cage in three dimensions) at short times, and normal hop diffusion among corrals at long times. (For a review of anomalous diffusion see Metzler and Klafter (3.Metzler R. Klafter J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics.J. Phys. A. 2004; 37: R161-R208Crossref Scopus (1709) Google Scholar) and for a discussion in a biological context see Condamin et al. (4.Condamin S. Tejedor V. Voituriez R. Bénichou O. Klafter J. Probing microscopic origins of confined subdiffusion by first-passage observables.Proc. Natl. Acad. Sci. USA. 2008; 105: 5675-5680Crossref PubMed Scopus (154) Google Scholar).) How can true transient anomalous subdiffusion be identified in modeling? In some cases one can conclude that transient anomalous subdiffusion is real from the behavior of the model as a parameter is tuned. For example, for obstructed diffusion on a lattice, there is an initial period of anomalous subdiffusion and a crossover to normal diffusion at long times (5.Saxton M.J. Anomalous diffusion due to obstacles: a Monte Carlo study.Biophys. J. 1994; 66: 394-401Abstract Full Text PDF PubMed Scopus (475) Google Scholar). As shown in Fig. 1, as the obstacle concentration is increased, diffusion becomes more anomalous over longer times. At the percolation threshold, diffusion becomes anomalous at all times, a well-known result, and the anomalous diffusion exponent becomes equal to its known value for diffusion on the percolation cluster. In the case of a finite hierarchy of traps, the parameter to be tuned is the number of layers in the hierarchy (6.Saxton M.J. A biological interpretation of transient anomalous subdiffusion. I. Qualitative model.Biophys. J. 2007; 92: 1178-1191Abstract Full Text Full Text PDF PubMed Scopus (248) Google Scholar). For even a single trap, there is necessarily an inflection point in the plot of log〈r2〉/t versus log t, and the linear region around the inflection point is best interpreted as an artifactual period of anomalous subdiffusion (see Fig. 5 of Saxton (6.Saxton M.J. A biological interpretation of transient anomalous subdiffusion. I. Qualitative model.Biophys. J. 2007; 92: 1178-1191Abstract Full Text Full Text PDF PubMed Scopus (248) Google Scholar)). But Fig. 2 shows that as the hierarchy is built up, diffusion becomes more anomalous over longer times. Here the limit of an infinite trap hierarchy is similar to the well-known continuous-time random walk (CTRW) model, which gives anomalous subdiffusion at all times. In both the CTRW and the trap hierarchy models, the escape times are given by a power-law distribution. The difference is that in a CTRW, the trap at the occupied site is newly generated from a random distribution at each move (dynamic or annealed disorder), but in the trap hierarchy model, the traps are permanent and immobile (static or quenched disorder). In the CTRW the distribution is continuous; in the trap hierarchy model it is discrete, although this is not essential to the model. Parameter tuning can be done experimentally as well. In measurements of diffusion of a colloidal probe in an actin gel, Wong et al. (7.Wong I.Y. Gardel M.L. Reichman D.R. Weeks E.R. Valentine M.T. Bausch A.R. Weitz D.A. Anomalous diffusion probes microstructure dynamics of entangled F-actin networks.Phys. Rev. Lett. 2004; 92: 178101Crossref PubMed Scopus (38) Google Scholar) tuned from normal to anomalous to elastic regimes by increasing the ratio of the probe size to the average mesh size in the gel. Consider the experimental curves (Fig. 1 of Destainville et al. (2.Destainville N. Saulière A. Salomé L. Comment to the article by Michael J. Saxton: a biological interpretation of transient anomalous subdiffusion. I. Qualitative model.Biophys. J. 2008; : 3117-3119Abstract Full Text Full Text PDF PubMed Scopus (21) Google Scholar)) showing apparent transient anomalous subdiffusion. On physical grounds a corral model is plausible in both cases, so the analysis proposed by Destainville et al. (2.Destainville N. Saulière A. Salomé L. Comment to the article by Michael J. Saxton: a biological interpretation of transient anomalous subdiffusion. I. Qualitative model.Biophys. J. 2008; : 3117-3119Abstract Full Text Full Text PDF PubMed Scopus (21) Google Scholar) may be applicable. Two-dimensional diffusion in the plasma membrane is likely to be obstructed by cytoskeletal elements, as proposed in the corral models of Sheetz (8.Sheetz M.P. Membrane skeletal dynamics: role in modulation of red cell deformability, mobility of transmembrane proteins, and shape.Semin. Hematol. 1983; 20: 175-188PubMed Google Scholar) and Kusumi et al. (9.Kusumi A. Nakada C. Ritchie K. Murase K. Suzuki K. Murakoshi H. Kasai R.S. Kondo J. Fujiwara T. Paradigm shift of the plasma membrane concept from the two-dimensional continuum fluid to the partitioned fluid: high-speed single-molecule tracking of membrane molecules.Annu. Rev. Biophys. Biomol. Struct. 2005; 34: 351-378Crossref PubMed Scopus (863) Google Scholar). Likewise three-dimensional diffusion in the nucleus may be obstructed by chromatin. In both cases, however, binding is also plausible. Proteins permanently or transiently bound to the cytoskeleton form the pickets in the Kusumi picket fence model (9.Kusumi A. Nakada C. Ritchie K. Murase K. Suzuki K. Murakoshi H. Kasai R.S. Kondo J. Fujiwara T. Paradigm shift of the plasma membrane concept from the two-dimensional continuum fluid to the partitioned fluid: high-speed single-molecule tracking of membrane molecules.Annu. Rev. Biophys. Biomol. Struct. 2005; 34: 351-378Crossref PubMed Scopus (863) Google Scholar), and binding of certain proteins to sites on chromatin is essential to the function of the nucleus. How can one distinguish true transient anomalous subdiffusion from artifactual subdiffusion? Several approaches are possible.1.Modeling. One approach would be to construct a model of corrals including the dynamics of corral walls and diffusion. The model would express the escape time in terms of the probabilities of gate-opening events of various widths and durations, the probability that the diffusing particle will reach an open gate, and the probability that the particle will exit through the gate. The key questions would be, does the distribution of escape times imply anomalous, transient anomalous, or normal diffusion, and does diffusion become more anomalous as one tunes a parameter such as the stiffness of the corral wall or the density of crosslinks?2.SPT measurements of 〈r2(t)〉. The equation proposed by Destainville et al. (2.Destainville N. Saulière A. Salomé L. Comment to the article by Michael J. Saxton: a biological interpretation of transient anomalous subdiffusion. I. Qualitative model.Biophys. J. 2008; : 3117-3119Abstract Full Text Full Text PDF PubMed Scopus (21) Google Scholar) might be able to distinguish the mechanisms. The curves of Figure 1, Figure 2 are not well fit by (segments of) that equation, but a conclusive test would require Monte Carlo results for the continuum, not a lattice, because a lattice model integrates out behavior over distances less than the lattice constant. The curves in Figure 1, Figure 2 start in the anomalous region because some diffusing particles are initially in contact with obstacles or at a binding site.3.Refined SPT measurements. The most direct experimental approach would be SPT measurements at high enough resolution to detect motion within the corrals in order to distinguish binding from corralling. Measuring histograms of escape times is essential. Simultaneous measurements of the position of the corral walls is highly useful. The data analysis must distinguish trapping or confinement from the apparent localization that occurs by chance in a pure random walk (10.Saxton M.J. Lateral diffusion in an archipelago: single-particle diffusion.Biophys. J. 1993; 64: 1766-1780Abstract Full Text PDF PubMed Scopus (225) Google Scholar, 11.Meilhac N. Le Guyader L. Salomé L. Destainville N. Detection of confinement and jumps in single-molecule membrane trajectories.Phys. Rev. E. 2006; 73: 011915Crossref Scopus (56) Google Scholar, 12.Simson R. Sheets E.D. Jacobson K. Detection of temporary lateral confinement of membrane proteins using single-particle tracking analysis.Biophys. J. 1995; 69: 989-993Abstract Full Text PDF PubMed Scopus (206) Google Scholar).Motion within corrals and jumps between them have been observed by SPT in the plasma membrane (9.Kusumi A. Nakada C. Ritchie K. Murase K. Suzuki K. Murakoshi H. Kasai R.S. Kondo J. Fujiwara T. Paradigm shift of the plasma membrane concept from the two-dimensional continuum fluid to the partitioned fluid: high-speed single-molecule tracking of membrane molecules.Annu. Rev. Biophys. Biomol. Struct. 2005; 34: 351-378Crossref PubMed Scopus (863) Google Scholar). Similar observations were made for colloidal particles in actin gels (7.Wong I.Y. Gardel M.L. Reichman D.R. Weeks E.R. Valentine M.T. Bausch A.R. Weitz D.A. Anomalous diffusion probes microstructure dynamics of entangled F-actin networks.Phys. Rev. Lett. 2004; 92: 178101Crossref PubMed Scopus (38) Google Scholar). The observed anomalous subdiffusion in actin gels was attributed to large rare jumps between cages; the escape time from a cage had a power-law distribution over ∼2 1/2 orders of magnitude. Andrews et al. (13.Andrews N.L. Lidke K.A. Pfeiffer J.R. Burns A.R. Wilson B.S. Oliver J.M. Lidke D.S. Actin restricts FcϵRI diffusion and facilitates antigen-induced receptor immobilization.Nat. Cell Biol. 2008; 10: 955-963Crossref PubMed Scopus (227) Google Scholar) reported caging in their careful SPT measurements on the high-affinity IgE receptor with simultaneous imaging of the actin cortex. SPT measurements of Cajal bodies and chromatin in the nucleus were interpreted in terms of transient binding by Platani et al. (14.Platani M. Goldberg I. Lamond A.I. Swedlow J.R. Cajal body dynamics and association with chromatin are ATP-dependent.Nat. Cell Biol. 2002; 4: 502-508Crossref PubMed Scopus (217) Google Scholar) though related measurements by Görisch et al. (15.Görisch S.M. Wachsmuth M. Ittrich C. Bacher C.P. Rippe K. Lichter P. Nuclear body movement is determined by chromatin accessibility and dynamics.Proc. Natl. Acad. Sci. USA. 2004; 101: 13221-13226Crossref PubMed Scopus (87) Google Scholar) were taken to indicate caging.4.Inhibitors. In the finite trap hierarchy model, anomalous subdiffusion occurs only for a nonequilibrium initial state (6.Saxton M.J. A biological interpretation of transient anomalous subdiffusion. I. Qualitative model.Biophys. J. 2007; 92: 1178-1191Abstract Full Text Full Text PDF PubMed Scopus (248) Google Scholar). In principle, one could use metabolic energy inhibitors to test for this mechanism. However, in cells this test will not distinguish binding from corralling if the actin or chromatin corral walls are constantly remodeled by processes requiring metabolic energy. Inhibitors affecting the stiffness of the corral walls would still be useful. According to one formulation of Occam's razor, “Entities are not to be multiplied without necessity”. But given the known structural components of cells and their known or plausible interactions, diffusion in a cell involves obstruction, binding, and hydrodynamic interactions with obstacles, all in a crowded system. One must be cautious in invoking Occam's razor to constrain cellular mechanisms when nature has already multiplied the entities, presumably for various biological necessities. This work was supported by National Institutes of Health grant GM038133.

lgorithms for analyzing single-particle tracking images to obtain the paths of individual particles are challenged by high-density data. Improvements in algorithms help to overcome these limitations.

Modeling obstructed diffusion is essential to the understanding of diffusion-mediated processes in the crowded cellular environment. Simple Monte Carlo techniques for modeling obstructed random walks are explained and related to Brownian dynamics and more complicated Monte Carlo methods. Random number generation is reviewed in the context of random walk simulations. Programming techniques and event-driven algorithms are discussed as ways to speed simulations.