Diffusion in macromolecular crowded media: Monte Carlo simulation of obstructed diffusion vs. FRAP experiments

The diffusion of tracer particles in 3D macromolecular crowded media has been studied using two methodologies, simulation and experimental, with the aim of comparing their results. First, the diffusion of a tracer in an obstructed 3D lattice with mobile and big size obstacles has been analyzed through a Monte Carlo (MC) simulation procedure. Secondly, fluorescence recovery after photobleaching (FRAP) experiments have been carried out to study the diffusion of a model protein (alpha-chymotrypsin) in in vitro crowded solution where two type of Dextran molecules are used as crowder agents. To facilitate the comparison, the relative size between the tracer and the crowder is the same in both studies. The results indicate a qualitative agreement between the diffusional behaviors observed in the two studies. The dependence of the anomalous diffusion exponent and the limiting diffusion coefficient on the obstacle size and excluded volume shows, in both cases, a similar tendency. The introduction of a reduced mobility parameter in the simulation model accounting for the short-range tracer?Cobstacle interactions allows obtaining a quantitative agreement between the limiting diffusion coefficient values yielded by both procedures. The simulation?Cexperiment quantitative agreement for the anomalous diffusion exponent requires further improvements. As far as we know, this is the first reported work where both techniques are used in parallel to study the diffusion in macromolecular crowded media.     

The effect of obstacle conductivity and electric field on effective mobility and dispersion in electrophoretic transport: a volume averaging approach

The method of volume averaging has been used to determine the effective electrophoretic mobility and dispersion coefficients for molecular transport of point-like solutes in a two-phase porous medium where the electrical conductivity and the diffusion and mobility coefficients may vary in both phases. The formal theory, derived in previous work, is numerically evaluated for cases where the obstacle phase has a large or small conductivity relative to the fluid phase and where the diffusion coefficient of the solute in the obstacle phase can be large or small relative to that in the fluid phase. In agreement with previous Monte Carlo methods, the effective electrophoretic mobility is not a function of media conductivity or electric field when the obstacles are impermeable to solute transport or have small diffusion solute diffusion coefficients. However, the dispersion coefficient is a strong function of electric field and varies with obstacle conductivity when diffusive transport is small in the obstacles relative to the fluid. In contrast, the effective electrophoretic mobility is a function of electric field when conductivity of the obstacles is much larger than the fluid and when the obstacles are very permeable to solute but have low electrical conductivity.     

Three-dimensional Brownian diffusion of rod-like macromolecules in the presence of randomly distributed spherical obstacles: molecular dynamics simulation

Brownian diffusion of rod-like polymers in the presence of randomly distributed spherical obstacles is studied using molecular dynamics simulations. It is observed that dependence of the reduced diffusion coefficient of these macromolecules on the available volume fraction can be described reasonably by a power law function. Despite the case of obstructed diffusion of flexible polymers in which reduced diffusion coefficient has a weak dependence on the polymer length, this dependence is noticeably strong in the case of rod-like polymers. Diffusion of these macromolecules in the presence of obstacles is observed that is anomalous at short time scales and normal at long times. Duration time of the anomalous diffusion regime is found that increases very rapidly with increasing both the polymer length and the obstructed volume fraction. Dynamics of diffusion of these polymers is observed that crosses over from Rouse to reptation type with increasing the density of obstacles.     

Diffusion amid random overlapping obstacles: similarities, invariants, approximations

Efficient and accurate numerical techniques are used to examine similarities of effective diffusion in a void between random overlapping obstacles: essential invariance of effective diffusion coefficients (D(eff)) with respect to obstacle shapes and applicability of a two-parameter power law over nearly entire range of excluded volume fractions (φ), except for a small vicinity of a percolation threshold. It is shown that while neither of the properties is exact, deviations from them are remarkably small. This allows for quick estimation of void percolation thresholds and approximate reconstruction of D(eff) (φ) for obstacles of any given shape. In 3D, the similarities of effective diffusion yield a simple multiplication "rule" that provides a fast means of estimating D(eff) for a mixture of overlapping obstacles of different shapes with comparable sizes.     

Monte Carlo simulation on polymer translocation in crowded environment

The effect of crowded environment with static obstacles on the translocation of a three-dimensional self-avoiding polymer through a small pore is studied using dynamic Monte Carlo simulation. The translocation time τ is dependent on polymer-obstacle interaction and obstacle concentration. The influence of obstacles on the polymer translocation is explained qualitatively by the free energy landscape. There exists a special polymer-obstacle interaction at which the translocation time is roughly independent of the obstacle concentration at low obstacle concentration, and the strength of the special interaction is roughly independent of chain length N. Scaling relation τ ~ N(1.25) is observed for strong driving translocations. The diffusion property of polymer chain is also influenced by obstacles. Normal diffusion is only observed in dilute solution without obstacles or in a crowded environment with weak polymer-obstacle attraction. Otherwise, subdiffusion behavior of polymer is observed.     

Correlating anomalous diffusion with lipid bilayer membrane structure using single molecule tracking and atomic force microscopy

Anomalous diffusion has been observed abundantly in the plasma membrane of biological cells, but the underlying mechanisms are still unclear. In general, it has not been possible to directly image the obstacles to diffusion in membranes, which are thought to be skeleton bound proteins, protein aggregates, and lipid domains, so the dynamics of diffusing particles is used to deduce the obstacle characteristics. We present a supported lipid bilayer system in which we characterized the anomalous diffusion of lipid molecules using single molecule tracking, while at the same time imaging the obstacles to diffusion with atomic force microscopy. To explain our experimental results, we performed lattice Monte Carlo simulations of tracer diffusion in the presence of the experimentally determined obstacle configurations. We correlate the observed anomalous diffusion with obstacle area fraction, fractal dimension, and correlation length. To accurately measure an anomalous diffusion exponent, we derived an expression to account for the time-averaging inherent to all single molecule tracking experiments. We show that the length of the single molecule trajectories is critical to the determination of the anomalous diffusion exponent. We further discuss our results in the context of confinement models and the generating stochastic process.     

Anomalous and Normal Diffusion of Tracers in Crowded Environments: E ect of Size Disparity between Tracer and Crowders

The dynamics of tracers in crowded matrix is of interest in various areas of physics, such as the diffusion of proteins in living cells. By using two-dimensional (2D) Langevin dynamics simulations, we investigate the diffusive properties of a tracer of a diameter in crowded environments caused by randomly distributed crowders of a diameter. Results show that the emergence of subdiffusion of a tracer at intermediate time scales depends on the size ratio of the tracer to crowders δ. If δ falls between a lower critical size ratio and a upper one, the anomalous diffusion occurs purely due to the molecular crowding. Further analysis indicates that the physical origin of subdiffusion is the "cage effect". Moreover, the subdiffusion exponent α decreases with the increasing medium viscosity and the degree of crowding, and gets a minimum αmin=0.75 at δ=1. At long time scales, normal diffusion of a tracer is recovered. For δ≤1, the relative mobility of tracers is independent of the degree of crowding. Meanwhile, it is sensitive to the degree of crowding for δ>1. Our results are helpful in deepening the understanding of the diffusive properties of biomacromolecules that lie within crowded intracellular environments, such as proteins, DNA and ribosomes.     

Time dependent diffusion in a disordered medium with partially absorbing walls: a perturbative approach

We present an analytical study of the time dependent diffusion coefficient in a dilute suspension of spheres with partially absorbing boundary condition. Following Kirkpatrick [J. Chem. Phys. 76, 4255 (1982)] we obtain a perturbative expansion for the time dependent particle density using volume fraction f of spheres as an expansion parameter. The exact single particle t operator for partially absorbing boundary condition is used to obtain a closed form time dependent diffusion coefficient D(t) accurate to first order in the volume fraction f. Short and long time limits of D(t) are checked against the known short time results for partially or fully absorbing boundary conditions and long time results for reflecting boundary conditions. For fully absorbing boundary condition the long time diffusion coefficient is found to be D(t)=5a(2)/(12fD(0)t)+O((D(0)t/a(2))(-2)) to the first order of perturbation theory. Here f is small but nonzero, D(0) the diffusion coefficient in the absence of spheres, and a the radius of the spheres. The validity of this perturbative result is discussed.     

Modeling diffusional transport in the interphase cell nucleus

In this paper a lattice model for the diffusional transport of particles in the interphase cell nucleus is proposed. Dense networks of chromatin fibers are created by three different methods: Randomly distributed, noninterconnected obstacles, a random walk chain model, and a self-avoiding random walk chain model with persistence length. By comparing a discrete and a continuous version of the random walk chain model, we demonstrate that lattice discretization does not alter particle diffusion. The influence of the three dimensional geometry of the fiber network on the particle diffusion is investigated in detail while varying the occupation volume, chain length, persistence length, and walker size. It is shown that adjacency of the monomers, the excluded volume effect incorporated in the self-avoiding random walk model, and, to a lesser extent, the persistence length affect particle diffusion. It is demonstrated how the introduction of the effective chain occupancy, which is a convolution of the geometric chain volume with the walker size, eliminates the conformational effects of the network on the diffusion, i.e., when plotting the diffusion coefficient as a function of the effective chain volume, the data fall onto a master curve.     

Effects of Multisolute Steric Interactions on Membrane Partition Coefficients

A key parameter in membrane and chromatographic separations is the partition coefficient, the equilibrium ratio of the solute concentration in a porous or fibrous material to that in bulk solution. The theoretical effects of solute size on partition coefficients in straight pores or randomly oriented fiber matrices have been investigated previously for very dilute solutions, where solute-solute interactions are negligible, and also for more concentrated solutions consisting of spherical solutes of uniform size. For concentrated solutions it has been found that steric and other repulsive interactions among solutes increase the partition coefficient above the dilute limit. To extend the results for porous or fibrous media to include concentrated mixtures of solutes with different sizes or shapes, we used an excluded volume approach. In this formulation, which describes steric interactions only, partition coefficients were computed by summing all volumes excluded to a solute molecule by virtue of its finite size, the finite size of other solutes, and the presence of fixed obstacles (pore walls or fibers). For a mixture of two spherical solutes, the addition of any second solute at finite concentration increased the partition coefficient of the first solute. That increase was sensitive to the size of the second solute; for a given volume fraction of the second solute, the smaller its radius, the larger the effect. When the total volume fraction of solutes was fixed, an increase in the amount of a second, smaller solute increased the partition coefficient of the first solute, whereas an increase in the amount of a second, larger solute had the opposite effect. Results were obtained also for oblate or prolate spheroidal solutes and for fibrous media containing fibers of different radii. For constant total fiber volume fraction, an increase in the amount of a second, smaller fiber decreased the partition coefficient of a spherical solute, whereas an increase in the amount of a second, larger fiber had the opposite effect. Overall, the theory suggests that the introduction of heterogeneities, whether as mixtures of solute sizes or mixtures of fiber sizes, may cause partition coefficients to differ markedly from those of uniform systems. Copyright 2000 Academic Press.     

Diffusion in the presence of cylindrical obstacles arranged in a square lattice analyzed with generalized Fick-Jacobs equation

The generalized Fick-Jacobs equation is widely used to study diffusion of Brownian particles in three-dimensional tubes and quasi-two-dimensional channels of varying constraint geometry. We show how this equation can be applied to study the slowdown of unconstrained diffusion in the presence of obstacles. Specifically, we study diffusion of a point Brownian particle in the presence of identical cylindrical obstacles arranged in a square lattice. The focus is on the effective diffusion coefficient of the particle in the plane perpendicular to the cylinder axes, as a function of the cylinder radii. As radii vary from zero to one half of the lattice period, the effective diffusion coefficient decreases from its value in the obstacle free space to zero. Using different versions of the generalized Fick-Jacobs equation, we derive simple approximate formulas, which give the effective diffusion coefficient as a function of the cylinder radii, and compare their predictions with the values of the effective diffusion coefficient obtained from Brownian dynamics simulations. We find that both Reguera-Rubi and Kalinay-Percus versions of the generalized Fick-Jacobs equation lead to quite accurate predictions of the effective diffusion coefficient (with maximum relative errors below 4% and 7%, respectively) over the entire range of the cylinder radii from zero to one half of the lattice period.     

Lateral diffusion of proteins in the plasma membrane: spatial tessellation and percolation theory

The obstructed diffusion of proteins in the plasma membrane is studied using computer simulation and an analysis using spatial tessellation and percolation theory. The membrane is modeled as a two-dimensional space with fixed hard disc obstacles, and the proteins are modeled as hard discs. The simulations show a transition from normal to anomalous diffusion as the area fraction, phim, of obstacles is increased and to confined diffusion for area fractions above the pecolation threshold, which occurs for phim=0.22. A Voronoi tessellation procedure is used to map the continuous space system onto an effective lattice model, with the connectivity of bonds determined from a geometric criterion. The estimate of the percolation threshold obtained from this lattice model is in excellent agreement with the simulation results, although the nature of the dynamics in the continuous space model is different from lattice models. At high obstacle area fractions (but below the percolation threshold), the primary mode of transport is a hopping motion between voids, consistent with experiment. The simulations and analysis emphasize the importance of structural correlations between obstacles.     

Movement of proteins in an environment crowded by surfactant micelles: anomalous versus normal diffusion

Small proteins move in crowded cell compartments by anomalous diffusion. In many of them, e.g., the endoplasmic reticulum, the proteins move between lipid membranes in the aqueous lumen. Molecular crowding in vitro offers a systematic way to study anomalous and normal diffusion in a well controlled environment not accessible in vivo. We prepared a crowded environment in vitro consisting of hexaethylene glycol monododecyl ether (C(12)E(6)) nonionic surfactant and water and observed lysozyme diffusion between elongated micelles. We have fitted the data obtained in fluorescence correlation spectroscopy using an anomalous diffusion model and a two-component normal diffusion model. For a small concentration of surfactant (below 4 wt %) the data can be fitted by single-component normal diffusion. For larger concentrations the normal diffusion fit gave two components: one very slow and one fast. The amplitude of the slow component grows with C(12)E(6) concentration. The ratio of diffusion coefficients (slow to fast) is on the order of 0.1 for all concentrations of surfactant in the solution. The fast diffusion is due to free proteins while the slow one is due to the protein-micelle complexes. The protein-micelle interaction is weak since even in a highly concentrated solution (35% of C(12)E(6)) the amplitude of the slow mode is only 10%, despite the fact that the average distance between the micelles is the same as the size of the protein. The anomalous diffusion model gave the anomaly index (r(2)(t) approximately t(alpha)), alpha monotonically decreasing from alpha = 1 (at 4% surfactant) to alpha = 0.88 (at 37% surfactant). The fits for two-component normal diffusion and anomalous diffusion were of equally good quality, but the physical interpretation was only straightforward for the former.     

Experimental evidence of distance-dependent diffusion coefficients of a globular protein observed in polymer aqueous solution forming a network structure on nanometer scale

The distance dependence of the diffusion coefficient (DDDC) of a globular protein (cytochrome c) in aqueous hyaluronan (HA) solution, which is a model system for extracellular matrices (ECMs), was measured by a combination of three kinds of spectroscopic measurements of diffusion coefficients, the time and space samplings of which are different. The results of the three methods are plotted against the diffusion distance derived from the consideration of each experimental condition. Due to the characteristic morphology of HA with an effective mesh structure, the proteins showed two extreme diffusion modes: (1) short (<10 nm) diffusion with rare contact with polymer chains; (2) long (>100 nm) diffusion significantly disrupted by polymer chains showing an approximately 30% reduction in diffusion coefficient. The transition from the short diffusion to the long one occurs in a very narrow range (10-100 nm) of diffusion distance and this unique character of HA realizing anomalous diffusion should provide suitable environments for various bioactivities when involved in ECM.     

Self-diffusion coefficients of ions in the presence of charged obstacles

The self-diffusion coefficient of ions of the charge- and size-symmetric +1:-1 (or +2:-2) electrolyte was studied in the presence of ionic obstacles (matrix) representing disordered media. For this purpose the Brownian dynamics method was used, complemented with the replica Ornstein-Zernike theory for the partly-quenched systems. The matrix was prepared by a rapid quench of the size-symmetric +1:-1 (in few cases also of +2:-2) electrolyte solution being in equilibrium at (temperature, relative permittivity) T0, epsilon. Within the matrix the charge- and size-symmetric (+1:-1 or +2:-2) electrolyte at T1, epsilon1 was distributed. This component was fully mobile (annealed) and in thermodynamic equilibrium with the matrix. In this study a special attention was paid to the self-diffusion of the annealed ions. The ratio D/D degrees, where D degrees is the self-diffusion coefficient of ions at infinite dilution, has been studied for various model parameters varying the concentration of all species in the system. The presence of charged obstacles decreases the self-diffusion of the annealed electrolyte; the D/D degrees values are lower in the partly-quenched mixtures than in the fully annealed electrolyte of the same concentration. In the investigated range of concentrations and solvent dielectric constants, the D/D degrees values first increased with the increased concentration of annealed electrolyte present and then decreased. An increase of the strength of the Coulomb interaction between annealed ions, or between annealed and quenched charges, yielded a decrease of the self-diffusion. In the range of concentrations investigated in this work, the decrease is mainly due to the Coulomb interaction with the matrix, since the presence of neutral obstacles did not modify the diffusion properties with respect to the situation without obstacles.     

Formation of columns of anomalous water by condensation in quartz capillaries

It is shown that these columns are formed in the kinetic mode from unsaturated water vapor. The condensation coefficient for an anomalous column is much less than for ordinary water. The growth rate is dependent on the equilibration time for the concentration of the anomalous component in the column. The diffusion coefficient for the anomalous component in water is less than the self-diffusion coefficient of water by nearly an order of magnitude; this and the low volatility are to be ascribed to large molecular size. It is supposed that the molecules of the anomalous component are strong groups of H₂O molecules that are only slowly converted to monomers.     

Rotational and translational self-diffusion in concentrated suspensions of permeable particles

In our recent work on concentrated suspensions of uniformly porous colloidal spheres with excluded volume interactions, a variety of short-time dynamic properties were calculated, except for the rotational self-diffusion coefficient. This missing quantity is included in the present paper. Using a precise hydrodynamic force multipole simulation method, the rotational self-diffusion coefficient is evaluated for concentrated suspensions of permeable particles. Results are presented for particle volume fractions up to 45% and for a wide range of permeability values. From the simulation results and earlier results for the first-order virial coefficient, we find that the rotational self-diffusion coefficient of permeable spheres can be scaled to the corresponding coefficient of impermeable particles of the same size. We also show that a similar scaling applies to the translational self-diffusion coefficient considered earlier. From the scaling relations, accurate analytic approximations for the rotational and translational self-diffusion coefficients in concentrated systems are obtained, useful to the experimental analysis of permeable-particle diffusion. The simulation results for rotational diffusion of permeable particles are used to show that a generalized Stokes-Einstein-Debye relation between rotational self-diffusion coefficient and high-frequency viscosity is not satisfied.     

The role of chromatin conformations in diffusional transport of chromatin-binding proteins: Cartesian lattice simulations

In this paper, a lattice model for the diffusional transport of chromatin-binding particles in the interphase cell nucleus is proposed. Sliding effects are studied in dense networks of chromatin fibers created by three different methods: Randomly distributed, noninterconnected obstacles, a random walk chain model with an attractive step potential, and a self-avoiding random walk chain model with a hard repulsive core and attractive surroundings. By comparing a discrete and continuous version of the random walk chain model, we demonstrate that lattice discretization does not alter the diffusion of chromatin-binding particles. The influence of conformational properties of the fiber network on the particle sliding is investigated in detail while varying occupation volume, sliding probability, chain length, and persistence length. It is observed that adjacency of the monomers, the excluded volume effect incorporated in the self-avoiding random walk model, and the persistence length affect the chromatin-binding particle diffusion. It is demonstrated that sliding particles sense local chain structures. When plotting the diffusion coefficient as a function of the accessible volume for diffusing particles, the data fall onto master curves depending on the persistence length. However, once intersegment transfer is involved, chromatin-binding proteins no longer perceive local chain structures.     

Motion of star‐branched chains in polymer networks: a Monte Carlo study

A simple model of branched polymers in confined space is developed. Star‐branched polymer molecules are built on a simple cubic lattice with excluded volume and no attractive interactions (good solvent conditions). A single star molecule is trapped in a network of linear polymer chains of restricted mobility. The simulations are carried out using the classical Metropolis algorithm. Static and dynamic properties of the star‐branched polymer are determined using various networks. The dependence of the longest relaxation time and the self‐diffusion coefficient on chain length and network properties are discussed and the proper scaling laws formulated. The possible mechanism of motion is discussed. The differences between the motion of star‐branched polymers in such a network are compared with the cases of a dense matrix of linear chains and regular rod‐like obstacles.     

Enhancement of the recollision rate in diffusion-influenced reactions in an inhomogeneous medium

Brownian dynamics simulations were performed to determine the first collision and recollision rates of spherical reagent particles in a reaction medium made heterogeneous by the presence of randomly located inert spherical obstacles in a continuum solvent. The recollision rate vp (and hence the overall reactive collision rate when activation energy is high) was always enhanced by the presence of obstacles, the degree of enhancement increasing with the volume fraction occupied by obstacles (phi) and with decreasing reagent concentration phiR. Enhancement increased with obstacle size at high phiR, and fell with increasing obstacle size at low phiR. The vp-phiR data follow a power law, where the scaling factor betap decreased with decreasing obstacle size and increasing phi, and the prefactor kp initially increased with phi and then fell (except for large obstacles). The behavior of betap appears to be largely due to the obstacles reducing the probability that reagent particles escape from each other after collision, while the dominant factors responsible for the behavior of kp appear to be initially the effect of obstacles in enhancing effective local reagent concentration, and then (for small obstacles), their reduction of the reagent-particle coordination number. As the energy of activation falls, the reactive collision rate becomes less influenced by the reagent recollision rate and more influenced by the rate of first collision. For low-activation-energy reactions, the presence of obstacles depresses the reactive collision rate if reagent concentration is low or if the obstacles are small and their concentration high. The fall in the reactive collision rate with decreasing activation energy is steeper, the lower the reagent concentration and the smaller the obstacles.