Lateral diffusion in a mixture of mobile and immobile particles. A Monte Carlo study.

The lateral diffusion coefficient for mixtures of mobile and immobile particles is obtained from Monte Carlo calculations of random walks by mobile tracers in the presence of immobile obstacles on a triangular lattice. The diffusion coefficient of the mobile species is obtained as a function of the area fractions of mobile and immobile species. The results are applied to diffusion of band 3 in the erythrocyte membrane, and indicate that obstruction of diffusion of mobile band 3 by band 3 and glycophorin attached to the membrane skeleton is not sufficient to explain the observed diffusion coefficient.     

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.     

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.     

Cytoplasmic hydrogen ion diffusion coefficient.

The apparent cytoplasmic proton diffusion coefficient was measured using pH electrodes and samples of cytoplasm extracted from the giant neuron of a marine invertebrate. By suddenly changing the pH at one surface of the sample and recording the relaxation of pH within the sample, an apparent diffusion coefficient of 1.4 +/- 0.5 x 10(-6) cm2/s (N = 7) was measured in the acidic or neutral range of pH (6.0-7.2). This value is approximately 5x lower than the diffusion coefficient of the mobile pH buffers (approximately 8 x 10(-6) cm2/s) and approximately 68x lower than the diffusion coefficient of the hydronium ion (93 x 10(-6) cm2/s). A mobile pH buffer (approximately 15% of the buffering power) and an immobile buffer (approximately 85% of the buffering power) could quantitatively account for the results at acidic or neutral pH. At alkaline pH (8.2-8.6), the apparent proton diffusion coefficient increased to 4.1 +/- 0.8 x 10(-6) cm2/s (N = 7). This larger diffusion coefficient at alkaline pH could be explained quantitatively by the enhanced buffering power of the mobile amino acids. Under the conditions of these experiments, it is unlikely that hydroxide movement influences the apparent hydrogen ion diffusion coefficient.     

Lateral diffusion in inhomogeneous membranes. Model membranes containing cholesterol.

The problem of lateral diffusion in inhomogeneous membranes is illustrated by a theoretical calculation of the lateral diffusion of a fluorescent lipid probe in binary mixtures of phosphatidylcholine and cholesterol under conditions of temperature and composition such that this lipid mixture consists of alternating parallel domains of fluid and solid lipid, having separations that are small compared with the distance scale employed in photobleaching experiments. The theoretical calculations clearly illustrate how inhomogeneities in membrane composition affecting the lateral motion of membrane components on a small (10-100 nm) distance scale can give complex diffusive responses in experiments such as fluorescence photobleaching that employ comparatively macroscopic distances (10-100 micrometers) for the measurement of diffusive recovery. The theoretical calculations exhibit the unusual dependence of the apparent lateral diffusion coefficient of a fluorescent lipid probe on lipid composition in binary mixtures of cholesterol and phosphatidylcholines as reported by Rubenstein et al. (1979, Proc. Natl. Acad. Sci. U.S.A., 76:15-18).     

Intracellular diffusion in the presence of mobile buffers. Application to proton movement in muscle.

Junge and McLaughlin (1987) derived an expression for the apparent diffusion constant of protons in the presence of both mobile and immobile buffers. Their derivation applies only to cases in which the values of pH are considerably greater than the largest pK of the individual buffers, a condition that is not expected to hold in skeletal muscle or many other cell types. Here we show that, if the pH gradients are small, the same expression for the apparent diffusion constant of protons can be derived without such constraints on the values of the pK's. The derivation is general and can be used to estimate the apparent diffusion constant of any substance that diffuses in the presence of both mobile and immobile buffers. The apparent diffusion constant of protons is estimated to be 1-2 x 10(-6) cm2/s at 18 degrees C inside intact frog twitch muscle fibers. It may be smaller inside cut fibers, owing to a reduction in the concentration of mobile myoplasmic buffers, so that in this preparation a pH gradient, if established within a sarcomere following action potential stimulation, could last 10 ms or longer after stimulation ceased.     

Probe diffusion in cross-linked actin gels

The diffusion of monodisperse polystyrene latex spheres (PLS) in column-purified 0.65 mg/mL actin solutions, polymerized with 100 mM KCl in the absence and presence of a cross-linker, actin binding protein (ABP), has been studied using dynamic light scattering. Measurements over a wide range of scattering angles from 90 degrees to 8 degrees, corresponding to inverse scattering vector probing distances of about 40-400 nm, respectively, give a measure of both the fraction of PLS mobile over the probing distance (from the normalized time autocorrelation function amplitude) and the average diffusion coefficient of the mobile PLS. Both 100- and 500-nm diameter PLS are fully mobile in polymerized actin solutions over distances of less than 100 nm, as reported previously (Newman, J., Schick, K. S. & Gukelberger, G. Biophys. J. 53, 573a, and Newman, J., Mroczka, N. & Schick, K. L. Biopolymers, 28, 655-666). At increasing probing distances, or when ABP is added at molar ratios of 1:750 or 1:150, greater fractions of the PLS are immobilized, up to almost 99% at the conditions of a 400-nm probing distance with 500-nm probes and at a ratio of 1:150 added ABP to actin. The degree of immobilization correlated well with the amount of added ABP, the size of the PLS, and the probing distance. At increasing probe distances, as the degree of immobilization increases, the remaining mobile fraction of PLS has an increasing average diffusion coefficient. These results suggest a range of pore sizes in the actin gels with a mean size of a few hundred nanometers.     

Approximation method for electrolyte diffusion through a fixed charge medium

The diffusion of a binary salt through media containing uniformly dispersed immobile ions is formally equivalent to the diffusion of a nonelectrolyte with a concentration-dependent diffusion coefficient. This permits the application of a quasi-steady-state approximation, developed in a previous paper (Bull. Math. Biophysics,21, 19–32, 1959), to problems of transient diffusion. Free diffusion across an interface involving a discontinuity in immobile ion concentration is considered and expressions for the salt concentration and electrical potential at any point are developed. The dependence of electrical potential on the fixed charge density, ionic mobilities, as well as on the boundary concentrations is illustrated by a numerical example. This particular example is suggested by experiments on connective tissue.     

Effect of hydrodynamic interactions on the diffusion of integral membrane proteins: diffusion in plasma membranes.

Tracer diffusion coefficients of integral membrane proteins (IMPs) in intact plasma membranes are often much lower than those found in blebbed, organelle, and reconstituted membranes. We calculate the contribution of hydrodynamic interactions to the tracer, gradient, and rotational diffusion of IMPs in plasma membranes. Because of the presence of immobile IMPs, Brinkman's equation governs the hydrodynamics in plasma membranes. Solutions of Brinkman's equation enable the calculation of short-time diffusion coefficients of IMPs. There is a large reduction in particle mobilities when a fraction of them is immobile, and as the fraction increases, the mobilities of the mobile particles continue to decrease. Combination of the hydrodynamic mobilities with Monte Carlo simulation results, which incorporate excluded area effects, enable the calculation of long-time diffusion coefficients. We use our calculations to analyze results for tracer diffusivities in several different systems. In erythrocytes, we find that the hydrodynamic theory, when combined with excluded area effects, closes the gap between existing theory and experiment for the mobility of band 3, with the remaining discrepancy likely due to direct obstruction of band 3 lateral mobility by the spectrin network. In lymphocytes, the combined hydrodynamic-excluded area theory provides a plausible explanation for the reduced mobility of sIg molecules induced by binding concanavalin A-coated platelets. However, the theory does not explain all reported cases of "anchorage modulation" in all cell types in which receptor mobilities are reduced after binding by concanavalin A-coated platelets. The hydrodynamic theory provides an explanation of why protein lateral mobilities are restricted in plasma membranes and why, in many systems, deletion of the cytoplasmic tail of a receptor has little effect on diffusion rates. However, much more data are needed to test the theory definitively. We also predict that gradient and tracer diffusivities are the same to leading order. Finally, we have calculated rotational diffusion coefficients in plasma membranes. They decrease less rapidly than translational diffusion coefficients with increasing protein immobilization, and the results agree qualitatively with the limited experimental data available.     

The spectrin network as a barrier to lateral diffusion in erythrocytes. A percolation analysis.

The spectrin network on the cytoplasmic surface of an erythrocyte can be modeled as a triangular lattice of spectrin tetramers (Tsuji, A., and S. Ohnishi, 1986. Biochemistry. 25:6133-6139). The tetramers act as barriers to protein diffusion, while dissociated dimer pairs, single dimers, and missing tetramers do not. Diffusion in the presence of these barriers is shown to be equivalent to bond percolation on the honeycomb lattice. Monte Carlo calculations for this system then yield the relative diffusion constant of a mobile integral protein as a function of the fraction of spectrin tetramers. At high concentrations of spectrin tetramer, long-range diffusion is blocked, but short-range diffusion is still possible. Monte Carlo calculations yield the average distance over which short-range diffusion can occur, as a function of the fraction of spectrin tetramers. Applications to erythrocyte development and hereditary hemolytic anemia are discussed.     

Aldolase exists in both the fluid and solid phases of cytoplasm [published erratum appears in J Cell Biol 1988 Dec;107(6 Pt 1):following 2463]

We have prepared a functional fluorescent analogue of the glycolytic enzyme aldolase (rhodamine [Rh]-aldolase), using the succinimidyl ester of carboxytetramethyl-rhodamine. Fluorescence redistribution after photobleaching measurements of the diffusion coefficient of Rh-aldolase in aqueous solutions gave a value of 4.7 x 10(-7) cm2/S, and no immobile fraction. In the presence of filamentous actin, there was a 4.5-fold reduction in diffusion coefficient, as well as a 36% immobile fraction, demonstrating binding of Rh-aldolase to actin. However, in the presence of a 100-fold molar excess of its substrate, fructose 1,6-diphosphate, both the mobile fraction and diffusion coefficient of Rh-aldolase returned to control levels, indicating competition between substrate binding and actin cross-linking. When Rh-aldolase was microinjected into Swiss 3T3 cells, a relatively uniform intracellular distribution of fluorescence was observed. However, there were significant spatial differences in the in vivo diffusion coefficient and mobile fraction of Rh-aldolase measured with fluorescence redistribution after photobleaching. In the perinuclear region, we measured an apparent cytoplasmic diffusion coefficient of 1.1 x 10(-7) cm2/s with a 23% immobile fraction; while measurements in the cell periphery gave a value of 5.7 x 10(-8) cm2/s, with no immobile fraction. Ratio imaging of Rh-aldolase and FITC-dextran indicated that FITC-dextran was relatively excluded excluded from stress fiber domains. We interpret these data as evidence for the partitioning of aldolase between a soluble fraction in the fluid phase and a fraction associated with the solid phase of cytoplasm. The partitioning of aldolase and other glycolytic enzymes between the fluid and solid phases of cytoplasm could play a fundamental role in the control of glycolysis, the organization of cytoplasm, and cell motility. The concepts and experimental approaches described in this study can be applied to other cellular biochemical processes.     

Calcium diffusion coefficient in rod photoreceptor outer segments.

Calcium (Ca(2+)) modulates several of the enzymatic pathways that mediate phototransduction in the outer segments of vertebrate rod photoreceptors. Ca(2+) enters the rod outer segment through cationic channels kept open by cyclic GMP (cGMP) and is pumped out by a Na(+)/Ca(2+),K(+) exchanger. Light initiates a biochemical cascade, which leads to closure of the cGMP-gated channels, and a concomitant decline in the concentration of Ca(2+). This decline mediates the recovery from stimulation by light and underlies the adaptation of the cell to background light. The speed with which the decline in the Ca(2+) concentration propagates through the rod outer segment depends on the Ca(2+) diffusion coefficient. We have used the fluorescent Ca(2+) indicator fluo-3 and confocal microscopy to measure the profile of the Ca(2+) concentration after stimulation of the rod photoreceptor by light. From these measurements, we have obtained a value of 15 +/- 1 microm(2)s(-1) for the radial Ca(2+) diffusion coefficient. This value is consistent with the effect of a low-affinity, immobile buffer reported to be present in the rod outer segment (L.Lagnado, L. Cervetto, and P.A. McNaughton, 1992, J. Physiol. 455:111-142) and with a buffering capacity of approximately 20 for rods in darkness(S. Nikonov, N. Engheta, and E.N. Pugh, Jr., 1998, J. Gen. Physiol. 111:7-37). This value suggests that diffusion provides a significant delay for the radial propagation of the decline in the concentration of Ca(2+). Also, because of baffling by the disks, the longitudinal Ca(2+) diffusion coefficient will be in the order of 2 microm(2)s(-1), which is much smaller than the longitudinal cGMP diffusion coefficient (30-60 microm(2)s(-1); ). Therefore, the longitudinal decline of Ca(2+) lags behind the longitudinal spread of excitation by cGMP.     

Surface diffusion of interacting proteins. Effect of concentration on the lateral mobility of adsorbed bovine serum albumin.

Surface diffusion of bovine serum albumin absorbed from aqueous solution to poly(methylmethacrylate) surfaces is significantly hindered by protein-protein lateral interactions. The long-time self diffusion coefficient measured by fluorescence recovery after pattern photobleaching decreases by approximately one order of magnitude as the surface area fraction occupied by protein increases from 0.10 to 0.69. Qualitative features of the surface concentration dependence of the self diffusion coefficient can be described by several recent models for lateral diffusion of interacting species. The mobile fraction is independent of the surface concentration, and both the self diffusion coefficient and the mobile fraction are constant between 15 min and 7 h of adsorption.     

Cyclic AMP diffusion coefficient in frog olfactory cilia

Cyclic AMP (cAMP) is one of the intracellular messengers that mediate odorant signal transduction in vertebrate olfactory cilia. Therefore, the diffusion coefficient of cAMP in olfactory cilia is an important factor in the transduction of the odorous signal. We have employed the excised cilium preparation from the grass frog (Rana pipiens) to measure the cAMP diffusion coefficient. In this preparation an olfactory cilium is drawn into a patch pipette and a gigaseal is formed at the base of the cilium. Subsequently the cilium is excised, allowing bath cAMP to diffuse into the cilium and activate the cyclic nucleotide-gated channels on the plasma membrane. In order to estimate the cAMP diffusion coefficient, we analyzed the kinetics of the currents elicited by step changes in the bath cAMP concentration in the absence of cAMP hydrolysis. Under such conditions, the kinetics of the cAMP-activated currents has a simple dependence on the diffusion coefficient. From the analysis we have obtained a cAMP diffusion coefficient of 2.7 +/- 0.2. 10(-6) cm2 s-1 for frog olfactory cilia. This value is similar to the expected value in aqueous solution, suggesting that there are no significant diffusional barriers inside olfactory cilia. At cAMP concentrations higher than 5 microM, diffusion slowed considerably, suggesting the presence of buffering by immobile cAMP binding sites. A plausible physiological function of such buffering sites would be to prolong the response of the cell to strong stimuli.     

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.     

Protein diffusion and macromolecular crowding in thylakoid membranes

The photosynthetic light reactions of green plants are mediated by chlorophyll-binding protein complexes located in the thylakoid membranes within the chloroplasts. Thylakoid membranes have a complex structure, with lateral segregation of protein complexes into distinct membrane regions known as the grana and the stroma lamellae. It has long been clear that some protein complexes can diffuse between the grana and the stroma lamellae, and that this movement is important for processes including membrane biogenesis, regulation of light harvesting, and turnover and repair of the photosynthetic complexes. In the grana membranes, diffusion may be problematic because the protein complexes are very densely packed (approximately 75% area occupation) and semicrystalline protein arrays are often observed. To date, direct measurements of protein diffusion in green plant thylakoids have been lacking. We have developed a form of fluorescence recovery after photobleaching that allows direct measurement of the diffusion of chlorophyll-protein complexes in isolated grana membranes from Spinacia oleracea. We show that about 75% of fluorophores are immobile within our measuring period of a few minutes. We suggest that this immobility is due to a protein network covering a whole grana disc. However, the remaining fraction is surprisingly mobile (diffusion coefficient 4.6 +/- 0.4 x 10(-11) cm(2) s(-1)), which suggests that it is associated with mobile proteins that exchange between the grana and stroma lamellae within a few seconds. Manipulation of the protein-lipid ratio and the ionic strength of the buffer reveals the roles of macromolecular crowding and protein-protein interactions in restricting the mobility of grana proteins.     

Lateral diffusion of saturated phosphatidylcholines in cholesterol-containing bilayers

Lateral diffusion in oriented bilayers of saturated cholesterol-containing phosphatidylcholines, dipalmitoylphosphatidylcholine and dimyrilstoylphosphatidylcholine upon their limiting hydration has been studied by NMR with impulse gradient of magnetic field. For both systems, similar dependences of the coefficient of lateral diffusion on temperature and cholesterol concentration were observed, which agree with the phase diagram showing the presence of regions of ordered and unordered liquid-crystalline phases and a two-phase region. Under similar conditions, the coefficient of lateral diffusion for dipalmytoylphosphatidylcholine has lower values, which is in qualitative agreement with its greater molecular mass. A comparison of data for dipalmytoylphosphatidylcholine with the results obtained earlier for dipalmytoylsphyngomyelin/cholesterol under the same conditions shows, despite a similarity in phase diagrams, greater (two- to threefold) differences in the values of the coefficient of lateral diffusion and a different mode of dependence of the coefficient on cholesterol concentration. A comparison of data for dimyrilstoylphosphatidylcholine with the results obtained previously shows that the values of the coefficient of lateral diffusion and the mode of its dependence on cholesterol concentration coincide in the region of higher concentrations (more than 15 mole %) and differ in the region of lower concentrations (below 15 mole %). The discrepancies may be explained by different contents of water in the systems during the measurements. At a limiting hydration (more than 35%) of water, the coefficient of lateral diffusion decreases with increasing cholesterol concentration. If the content of water is about 25% (as a result of equilibrium hydration from vapors), the coefficient of lateral diffusion of phosphatidylcholine is probably independent of cholesterol concentration. This results from a denser packing of molecules in the bilayer at a lower water concentration, an effect that competes with the ordering effect of cholesterol.     

Lateral diffusion of saturated phosphatidylcholines in cholesterol-containing bilayers

A pulsed field gradient NMR was used to study lateral diffusion in the cholesterol-containing oriented bilayers of saturated (dipalmitoyl- and dimyristoyl-) phosphatidylcholines, upon their limiting hydration. Similar dependences of lateral diffusion coefficients on temperature and cholesterol concentration were observed, which agree with phase diagram showing the presence of the regions of disordered and ordered liquid-crystalline phases and a two-phase region. Under the same conditions, the lateral diffusion coefficient of dipalmitoylphosphatidylcholine is lower, which agrees qualitatively with its larger molecular weight. The comparison of data for dipalmitoylphosphatidylcholine with previous results for dipalmitoylsphingomyelin-cholesterol bilayers under the same conditions, in spite of similarity of phase diagrams, shows large (two–three times) differences in the lateral diffusion coefficient and a different profile of its dependence on cholesterol concentration. The comparison of data for dimyristoylphosphatidylcholine with previous results shows that the values of lateral diffusion coefficient and the shape of its dependence on cholesterol concentration coincide at high concentrations (>15 mol%) but differ at lower concentrations The revealed disagreement may be caused by the fact that the measurements were carried out at different water content in the system. At limiting hydration (more than 35% of water), the lateral diffusion coefficient for lipids decreases when cholesterol concentration rises, while at water content about 25% (as a result of equilibrium hydration from vapors) the lateral diffusion coefficient of phosphatidylcholine may be independent of cholesterol concentration. This is the consequence of the denser packing of molecules in the bilayer at reduced water content, an effect that competes with the ordering effect of cholesterol.     

Multiple diffusion mechanisms due to nanostructuring in crowded environments

One of the key questions regarding intracellular diffusion is how the environment affects molecular mobility. Mostly, intracellular diffusion has been described as hindered, and the physical reasons for this behavior are: immobile barriers, molecular crowding, and binding interactions with immobile or mobile molecules. Using results from multi-photon fluorescence correlation spectroscopy, we describe how immobile barriers and crowding agents affect translational mobility. To study the hindrance produced by immobile barriers, we used sol-gels (silica nanostructures) that consist of a continuous solid phase and aqueous phase in which fluorescently tagged molecules diffuse. In the case of molecular crowding, translational mobility was assessed in increasing concentrations of 500 kDa dextran solutions. Diffusion of fluorescent tracers in both sol-gels and dextran solutions shows clear evidence of anomalous subdiffusion. In addition, data from the autocorrelation function were analyzed using the maximum entropy method as adapted to fluorescence correlation spectroscopy data and compared with the standard model that incorporates anomalous diffusion. The maximum entropy method revealed evidence of different diffusion mechanisms that had not been revealed using the anomalous diffusion model. These mechanisms likely correspond to nanostructuring in crowded environments and to the relative dimensions of the crowding agent with respect to the tracer molecule. Analysis with the maximum entropy method also revealed information about the degree of heterogeneity in the environment as reported by the behavior of diffusive molecules.