Response to Comment by Destainville et al.

Abstract

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.