Michael J. Saxton

Anomalous diffusion due to obstacles: a Monte Carlo study

In normal lateral diffusion, the mean-square displacement of the diffusing species is proportional to time. But in disordered systems anomalous diffusion may occur, in which the mean-square displacement is proportional to some other power of time. In the presence of moderate concentrations of obstacles, diffusion is anomalous over short distances and normal over long distances. Monte Carlo calculations are used to characterize anomalous diffusion for obstacle concentrations between zero and the percolation threshold. As the obstacle concentration approaches the percolation threshold, diffusion becomes more anomalous over longer distances; the anomalous diffusion exponent and the crossover length both increase. The crossover length and time show whether anomalous diffusion can be observed in a given experiment.

Single-particle tracking: the distribution of diffusion coefficients

In single-particle tracking experiments, the diffusion coefficient D may be measured from the trajectory of an individual particle in the cell membrane. The statistical distribution of single-trajectory diffusion coefficients is examined by Monte Carlo calculations. The width of this distribution may be useful as a measure of the heterogeneity of the membrane and as a test of models of hindered diffusion in the membrane. For some models, the distribution of the short-range diffusion coefficient is much narrower than the observed distribution for proteins diffusing in cell membranes. To aid in the analysis of single-particle tracking measurements, the distribution of D is examined for various definitions of D and for various trajectory lengths.

Anomalous diffusion due to binding: a Monte Carlo study.

In classical diffusion, the mean-square displacement increases linearly with time. But in the presence of obstacles or binding sites, anomalous diffusion may occur, in which the mean-square displacement is proportional to a nonintegral power of time for some or all times. Anomalous diffusion is discussed for various models of binding, including an obstruction/binding model in which immobile membrane proteins are represented by obstacles that bind diffusing particles in nearest-neighbor sites. The classification of binding models is considered, including the distinction between valley and mountain models and the distinction between singular and nonsingular distributions of binding energies. Anomalous diffusion is sensitive to the initial conditions of the measurement. In valley models, diffusion is anomalous if the diffusing particles start at random positions but normal if the particles start at thermal equilibrium positions. Thermal equilibration leads to normal diffusion, or to diffusion as normal as the obstacles allow.

Lateral Diffusion in an Archipelago: Effects of Impermeable Patches on Diffusion in a Cell Membrane

Lateral diffusion of molecules in lipid bilayer membranes can be hindered by the presence of impermeable domains of gel-phase lipid or of proteins. Effective-medium theory and percolation theory are used to evaluate the effective lateral diffusion constant as a function of the area fraction of fluid-phase lipid and the permeability of the obstructions to the diffusing species. Applications include the estimation of the minimum fraction of fluid lipid needed for bacterial growth, and the enhancement of diffusion-controlled reactions by the channeling effect of solid patches of lipid.

Lateral diffusion in an archipelago. Single-particle diffusion

Several laboratories have measured lateral diffusion of single particles on the cell surface, and these measurements may reveal an otherwise inaccessible level of submicroscopic organization of cell membranes. Pitfalls in the interpretation of these experiments are analyzed. Random walks in unobstructed systems show structure that could be interpreted as free diffusion, obstructed diffusion, directed motion, or trapping in finite domains. To interpret observed trajectories correctly, one must consider not only the trajectories themselves but also the probabilities of occurrence of various trajectories. Measures of the asymmetry of obstructed and unobstructed random walks are calculated, and probabilities are evaluated for random trajectories that resemble either directed motion or diffusion in a bounded region.

Anomalous Subdiffusion in Fluorescence Photobleaching Recovery: A Monte Carlo Study

Anomalous subdiffusion is hindered diffusion in which the mean-square displacement of a diffusing particle is proportional to some power of time less than one. Anomalous subdiffusion has been observed for a variety of lipids and proteins in the plasma membranes of a variety of cells. Fluorescence photobleaching recovery experiments with anomalous subdiffusion are simulated to see how to analyze the data. It is useful to fit the recovery curve with both the usual recovery equation and the anomalous one, and to judge the goodness of fit on log-log plots. The simulations show that the simplest approximate treatment of anomalous subdiffusion usually gives good results. Three models of anomalous subdiffusion are considered: obstruction, fractional Brownian motion, and the continuous-time random walk. The models differ significantly in their behavior at short times and in their noise level. For obstructed diffusion the approach to the percolation threshold is marked by a large increase in noise, a broadening of the distribution of diffusion coefficients and anomalous subdiffusion exponents, and the expected abrupt decrease in the mobile fraction. The extreme fluctuations in the recovery curves at and near the percolation threshold result from extreme fluctuations in the geometry of the percolation cluster.

Lateral diffusion in an archipelago. The effect of mobile obstacles.

Lateral diffusion of mobile proteins and lipids (tracers) in a membrane is hindered by the presence of proteins (obstacles) in the membrane. If the obstacles are immobile, their effect may be described by percolation theory, which states that the long-range diffusion constant of the tracers goes to zero when the area fraction of obstacles is greater than the percolation threshold. If the obstacles are themselves mobile, the diffusion constant of the tracers depends on the area fraction of obstacles and the relative jump rate of tracers and obstacles. This paper presents Monte Carlo calculations of diffusion constants on square and triangular lattices as a function of the concentration of obstacles and the relative jump rate. The diffusion constant for particles of various sizes is also obtained. Calculated values of the concentration-dependent diffusion constant are compared with observed values for gramicidin and bacteriorhodopsin. The effect of the proteins as inert obstacles is significant, but other factors, such as protein-protein interactions and perturbation of lipid viscosity by proteins, are of comparable importance. Potential applications include the diffusion of proteins at high concentrations (such as rhodopsin in rod outer segments), the modulation of diffusion by release of membrane proteins from cytoskeletal attachment, and the diffusion of mobile redox carriers in mitochondria, chloroplasts, and endoplasmic reticulum.

SINGLE-PARTICLE TRACKING

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. Why are these techniques so useful? One reason is that single-particle techniques give the ultimate resolution. The discovery of Brownian motion in 1828 was based on single-particle observations of particles in pollen grains and particles of similar size. The work of Perrin in the early 1900s on Brownian motion established the molecular nature of matter, evaluated Avogadros number, and identified one of the first fractals known by describing the selfsimilarity of the path of a Brownian particle [1,2]. Another reason is that single-particle techniques provide an alternative approach to a basic requirement in kinetics measurements. Bulk measurements require synchronization, for example, by rapid mixing of reactants, a temperature shift, or photoactivation, to define experimentally the initial time t = 0. Living cells are harder to synchronize, though in some cases it may be done by these methods or by drug or nutrient treatment. Furthermore, even if the system can be synchronized initially, the initial synchronization is lost quickly in a multistep reaction [3]. A single-molecule approach resolves the steps in the sequence and makes it possible to define t = 0 for each reaction step after the fact. The proliferation of single-particle techniques was made possible by a combination of technical advances: digital imaging, computer image enhancement, high-sensitivity video cameras (as well as avalanche photodetectors and better photomultipliers for nonimaging instruments), improved microscope optics and lasers, and improved fluorescent labels. Also important was a

Lateral diffusion in an archipelago. Distance dependence of the diffusion coefficient

An understanding of the distance dependence of the lateral diffusion coefficient is useful in comparing the results of diffusion measurements made over different length scales, and in analyzing the kinetics of mobile redox carriers in organelles. A distance-dependent, concentration-dependent diffusion coefficient is defined, and it is evaluated by Monte Carlo calculations of a random walk by mobile point tracers in the presence of immobile obstacles on a triangular lattice, representing the diffusion of a lipid or a small protein in the presence of immobile membrane proteins. This work confirms and extends the milling crowd model of Eisinger, J., J. Flores, and W. P. Petersen (1986. Biophys J. 49:987-1001). Similar calculations for diffusion of mobile particles interacting by a hard-core repulsion yield the distance dependence of the self-diffusion coefficient. An expression for the range of short-range diffusion is obtained, and the distance scales for various diffusion measurements are summarized.

Lateral diffusion in an archipelago. Dependence on tracer size

In a pure fluid-phase lipid, the dependence of the lateral diffusion coefficient on the size of the diffusing particle may be obtained from the Saffman-Delbrück equation or the free-volume model. When diffusion is obstructed by immobile proteins or domains of gel-phase lipids, the obstacles yield an additional contribution to the size dependence. Here this contribution is examined using Monte Carlo calculations. For random point and hexagonal obstacles, the diffusion coefficient depends strongly on the size of the diffusing particle, but for fractal obstacles--cluster-cluster aggregates and multicenter diffusion-limited aggregates--the diffusion coefficient is independent of the size of the diffusing particle. The reason is that fractals have no characteristic length scale, so a tracer sees on average the same obstructions, regardless of its size. The fractal geometry of the excluded area for tracers of various sizes is examined. Percolation thresholds are evaluated for a variety of obstacles to determine how the threshold depends on tracer size and to compare the thresholds for compact and extended obstacles.