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.     

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.     

Two-dimensional growth of a root system modelled as a diffusion process. I. Analytical solutions

For functioning of a root system, the temporal development of distribution of roots in the soil is important. For example, for computing uptake of water and nutrients the root length density distribution might be required. A way to describe root proliferation is to consider it as a diffusion process with a first-order sink term accounting for decay. In this paper, analytical solutions are derived for two-dimensional diffusion of roots both in a rectangular area, and in a cylindrical volume. The source of root dry matter is located at the surface. Root dry matter enters the soil domain through a part of the soil surface. It is shown that different distribution patterns are obtained, with different ratios of the diffusion coefficients in horizontal and vertical direction. From the solutions obtained it can be shown that for the situation where the dry matter enters through the complete surface eventually a steady-state occurs where root length density decreases exponentially with depth, as often is found in experiments.     

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.     

Theoretical study of mechano-chemical couplings in a compartmental enzyme system. I. Analytical treatment

This paper deals with theoretical aspects of the volume changes of a system in which diffusion, convection and reaction processes are coupled. This study involves a material able to swell in the presence of a chemical effector produced by an enzyme reaction. Three limiting factors of volume change rate were considered: fluid flow, diffusion or reaction limitations. Dimensionless diffusion-reaction and diffusion-convection parameters were introduced to allow quantitative predictions in limit cases. The steady states appear to be independent of convection processes; however, the transient states depend on diffusion, convection and reaction processes.     

Water and hermal diffusivity in a lipid-water smectic phase.

We report the first application of light scattering to measurement of the hydrodynamic relaxation of inhomogeneities in water concentration within a multilamellar, or smectic A, phospholipid water system (dipalmitoyl) phosphatidyl choline). Although the relaxation process in the multilamellar phase is different from the diffusion process in liquid phases, the relaxation rate can be described in terms of a diffusion coefficient. For diffusion parallel to the lamellae, diffusion coefficients ranging from 8 x 10(-7) to 2 x 10(-5) cm2/s were measured over a range of temperature and water concentrations. We describe a model that expresses the diffusion coefficient in terms of the chemical potential for water inside the multilamellar phase and the effective thickness of a "free water zone." The deduced thickness of this free water zone is in good agreement with estimates from X-ray diffraction results. The activation energy for the diffusion process is also deduced from the data, and is found to decrease monotonically with increasing water concentration. We also found the thermal diffusivity to be about 10(-3) cm2/s with only a weak temperature and water concentration dependence. The experimental technique is a new version of forced Rayleigh scattering. The method uses the phase information of the scattered light to improve the ability to detect weak signals. Experimental details are reported.     

Analysis of lateral diffusion from a spherical cell surface to a tubular projection.

Cell surfaces are often heterogeneous with respect to the lateral distribution and mobility of membrane components. Because lateral mobility is related to membrane structure, measurement of a particular component's local diffusion coefficient within a distinct surface region provides useful information about the formation and maintenance of that region. Many structurally interesting cell surface features can be described as narrow tubular projections from the body of the cell. In a companion paper, we consider the thin "tethers" that can be mechanically drawn from the red blood cell membrane, and we measure the transport of fluorescent integral proteins from the surface of the cell body onto the tether. In this paper we present an analysis to describe the surface diffusion of membrane particles from a spherical shell onto a thin cylindrical process. Provision is made for different rates of diffusion within the two morphologically distinct regions. The relative role of each region in controlling the diffusive flux between regions is determined primarily by a single dimensionless parameter. This parameter incorporates the ratio of the two diffusion coefficients as well as the dimensions of each region. The analysis can be applied to a fluorescence photobleaching experiment in which the extended process is bleached. If the dimensions of the spherical cell body and the cylindrical extension are known, then the diffusion coefficients of both regions can be determined from the experimental fluorescence recovery curve.     

Cell-specific constraints to the lateral diffusion of a membrane glycoprotein.

We have previously shown that the lateral diffusion, D, of the class I Major Histocompatibility Complex (MHC) glycoprotein H-2Ld is constrained by its glycosylation, when expressed in mouse L-cells. Removal of one or more of the 3 N-linked oligosaccharides of H-2Ld glycoproteins results in an increase in D. In order to further examine the influence of glycosylation on D, we compared lateral diffusion of H-2Ld expressed in wild-type CHO cells with lateral diffusion of the same molecule expressed in mutant CHO cells with aberrant surface glycosylation. In addition, we compared lateral diffusion of wild-type and unglycosylated H-2Ld antigens in these cells. In contrast to the large effect of glycosylation state on lateral diffusion of H-2Ld in mouse L-cells, there was little effect of glycosylation on lateral diffusion of H-2Ld in any of the CHO cells. This, together with similar results on hamster class I antigens, indicates that the constraints to D of H-2Ld and other class I MHC molecules are different in CHO cells than in L-cells. Measurements of lateral diffusion after treatment of cells with cytochalasin D make it clear that interactions between MHC class I molecules and a cytoskeleton are important in reducing the mobile fraction of diffusing molecules, R, though they cannot be shown to directly affect the diffusion coefficient, D.     

Replication of viruses in a growing plaque: a reaction-diffusion model.

An understanding of the viral replication process commonly referred to as "plaque growth" is developed in the context of a reaction-diffusion model. The interactions among three components: the virus, the healthy host, and the infected host are represented using rates of viral adsorption and desorption to the cell surface, replication and release by host lysis, and diffusion. The solution to the full model reveals a maximum in the dependence of the velocity of viral propagation on its equilibrium adsorption constant, suggesting that conditions can be chosen where viruses which adsorb poorly to their hosts will replicate faster in plaques than those which adsorb well. Analytic expressions for the propagation velocity as a function of the kinetic and diffusion parameters are presented for the limiting cases of equilibrated adsorption, slow adsorption, fast adsorption, and large virus yields. Hindered diffusion at high host concentrations must be included for quantitative agreement with experimental data.     

Use of a membrane-bound fluorophore to characterize diffusion boundary layers around human erythrocytes.

A novel method is used to demonstrate the presence of diffusion boundary layers around erythrocytes following rapid mixing in a stopped-flow spectrophotometer and to estimate the apparent dimensions of the diffusion boundary layers. Pink erythrocyte ghosts labeled on their external surfaces with tetramethyl rhodamine isothiocyanate (TRITC) were mixed in a stopped-flow apparatus with 50 mM NaI in Ringer's solutions. I- is an effective collisional quencher of TRITC fluorescence. TRITC fluorescence after flow stopped decreased monoexponentially with time. The concentration of I- at the cell surface as a function of time was estimated from the dependence of TRITC fluorescence on I- concentration in steady-state experiments. The kinetics of the increase in I- concentration at the cell surface was fit to two diffusional models: a planar erythrocyte ghost bounded by planar diffusion boundary layer and a spherical erythrocyte surrounded by a spherical shell diffusion boundary layer. The planar model best fits the experimental data with a diffusion boundary layer 4.68 microns thick. Using the spherical model the experimental data is best fit by a 6.9 microns diffusion boundary layer.     

Orientation constraints in diffusion-limited macromolecular association. The role of surface diffusion as a rate-enhancing mechanism.

Ligand association to a reactive site on a macromolecular surface could be very slow if the site is small. The effective capture radius of the reactive site can be significantly increased if the ligand can bind weakly to the nonspecific surface around the site and then slide in a two-dimensional diffusion along the surface. In this model, the diffusion along the surface has to be properly coupled with the free diffusion in solution and the effective bimolecular association rate constant to the reactive site can be calculated as a function of the nonspecific affinity. This is carried out both for a plane and spherical surface, modeling the association to a membrane receptor or to the catalytic site on an enzyme. The result of these calculations can be used to assign reasonable values to the parameters in the quasichemical approximation of K. Solc and W. H. Stockmayer (1973, Int. J. Chem. Kinet., 5:733-752). In this way a simple analytical expression can be derived for the diffusion-limited association rate constant of two asymmetrically reactive molecules, with or without surface diffusion contributing.     

Distribution and diffusivity of a hydrophobic probe molecule in the interior of a membrane: theory and simulation.

We propose a simple model for the distribution of position and orientation and the diffusion of a hydrophobic probe molecule embedded in a membrane. The molecule experiences both a Maier-Saupe orienting potential as well as an enclosing potential of repulsion from the membrane walls. A statistical thermodynamics treatment of the model provides predictions of the location and orientation of the molecule within the membrane. In particular, we evaluate the order parameter of the molecule in terms of the model constants. The diffusivity of the probe is studied by Brownian dynamics simulation. For rotational diffusion, we check an available analytical approximate treatment that allows for the prediction of the dynamics in terms of equilibrium quantities. We also pay attention to quantities related to the initial and mean reorientational rate of the probe. For translational diffusion, we use the simulation results to analyze some general aspects of lateral and transversal diffusion.     

Lateral diffusion in biological membranes. A normal-mode analysis of diffusion on a spherical surface.

A new approach is described for the analysis of lateral diffusion in biological membranes. It is shown that a suitably defined first moment of the concentration distribution on a spherical surface decays as a single exponential with a relaxation rate proportional to the diffusion coefficient and inversely proportional to the square of the radius of the sphere. The approach is illustrated with an example of fluorescence redistribution after photobleaching of membrane proteins in a spectrin-deficient spherocytic mouse erythrocyte membrane.     

Resonance energy transfer study using a rhenium metal-ligand lipid conjugate as the donor in a model membrane.

We measured steady state and time-resolved resonance energy transfer between donors and acceptors in model membranes. The donor was a long lifetime rhenium-lipid complex, which displayed a mean lifetime of 1 microsecond and lifetime components as long as 3 microseconds in the labeled DOPC membranes. The transfer efficiencies were found to be substantially larger than those predicted without consideration of lateral diffusion. The larger transfer efficiencies are consistent with a mutual lateral diffusion coefficient in the membrane near 2 x 10(-8) cm2/s. These results demonstrate that lateral diffusion in membranes can be detected with microsecond lipid probes.     

Transport properties of particles with segmental flexibility. I. Hydrodynamic resistance and diffusion coefficients of a freely hinged particle

Expressions are derived for the hydrodynamic resistance tensor and the diffusion tensor of a particle consisting of two rigid subunits connected by a free hinge. No restrictions are placed on the shapes of the subunits. The resistance tensor is obtained by using two independent approaches: first, from the Rayleigh dissipation function and, second, from an examination of the generalized forces for the appropriate seven-dimensional coordinate system. For the derivation of the generalized Einstein equation connecting the diffusion and resistance tensors, the Brownian motion is treated as a stochastic process. That derivation is based on the assumption that the restoring force for bending is negligible, and the Einstein relation holds instantaneously only if that assumption is true. The relationship between these tensors and the macroscopically observable parameters is discussed, and it is shown that the separate measurement of resistance and diffusion coefficients can be used to detect macromolecular flexibility. One example is treated, the diffusion of a particle composed of two long rods joined at a free hinge. Those calculations are carried out with the first-order assumption of negligible hydrodynamic interactions between the subunits. For the hinged rod, the bending degree of freedom produces a 34% increase in the translational diffusion coefficient over that of a stiff rod of the same total length, while the rotational diffusion coefficient about the axis perpendicular to the plane of bending is increased by 125%.     

Normal-mode analysis of lateral diffusion on a bounded membrane surface.

The normal-mode analysis of fluorescence redistribution after photobleaching, introduced for the characterization of lateral diffusion on spherical membrane surfaces, has been generalized and extended to other surface geometries. Theoretical expressions are derived for the characteristic values and orthogonal characteristic functions of the diffusion equations for cylindrical surfaces, ellipsoids of revolution and dimpled discoidal surfaces. On the basis of these results, a simple analytical function is proposed as an empirical solution for the analysis of photobleaching data on a variety of discoidal surfaces. Special experimental and computational methods for determining the surface-diffusion coefficient are described, and demonstrated with data for lipid diffusion in erythrocyte membranes.     

Private alleles in a partially isolated population. II. Distribution of persistence time and probability of emigration

Two diffusion limits were derived from a discrete Wright-Fisher model of migration, mutation, and selection with an arbitrary degree of dominance. Instantaneous killing of the process due to emigration of a mutant leads to one of two diffusion processes with a killing term. One (weak gene flow) is the boundary case of the other (strong gene flow), which can cover a wide range of gene flow. The diffusion process subject to strong gene flow is similar to that studied by S. Karlin and S. Tavaré (1983, SIAM J. Appl. Math. 43, 31-41). The spectral decomposition of the transition probability density of "private" allele frequencies is presented in the case of strong gene flow. The fate of mutant in a deme is discussed in terms of the probabilities of survival and emigration.     

Competitive processes in a monoculture of Pinus radiata D. Don

Summary The changes between 9 and 40 years of age of the frequency distribution of tree diameters were studied in an experiment in a plantation of Pinus radiata D. Don in South Australia in which the spacing between trees at planting varied. Empirical functions were developed to relate tree diameter to growth rate in diameter, variance of growth rate in diameter and mortality rate. These functions were used in solving a forward diffusion equation to predict future diameter distributions. The contributions from each of the terms in the diffusion equation were examined and it was found that the drift (growth rate in diameter) term made the greatest contribution to the solution, whilst the diffusion (variance in growth rate) term had neglible effect. This implied that competition between individual trees for light was the dominant competitive process operating in this plantation, rather than below ground competition for soil resources. The ramifications of this finding to forest growth modelling are discussed.     

The 5' and 3' splice sites come together via a three dimensional diffusion mechanism.

We present evidence that the splice sites in mammalian pre-mRNAs are brought together via a three dimensional diffusion mechanism. We tested two mechanisms for splice site pairing: a lateral diffusion ('scanning') model and the currently favored three dimensional diffusion ('jumping') model. Two lines of evidence that distinguish between these two models are presented. The first utilized bipartite splicing substrates tethered by double-stranded RNA stems predicted to provide either a moderate or severe block to splice site pairing via a scanning mechanism. Splice site pairing via a jumping mechanism was expected to be unaffected or affected minimally. The second approach utilized a flexible poly(ethylene glycol) moiety within the intron. This insertion was predicted to reduce scanning efficiency but not the efficiency of a three dimensional diffusion mechanism. The best explanation for the data with the bipartite RNAs is that splice site pairing occurs through three dimensional diffusion. Kinetic analysis of the poly(ethylene glycol) containing substrate showed that neither the lag phase nor the initial rates of mRNA production and spliceosome assembly were affected by this insertion. Therefore, both experimental approaches supported the three dimensional diffusion model of splice site pairing.     

Models for chromatin remodeling: a critical comparison.

Nucleosome remodeling has been shown, in many cases, to involve cis displacement of nucleosomes on the DNA. This process seems similar to the long-recognized random diffusion of nucleosomes along DNA, but the remodeling process is unidirectional and ATP dependent. Several years ago, we developed a model for nucleosome migration, based on the diffusion of "twist-defects" within the nucleosomal DNA. This has been modified into a model that incorporates ATP-dependent defect generation, and can account for many observations concerning remodeling. However, certain experimental studies in recent years have cast doubt on the applicability of the twist-diffusion model for remodeling, and seem to favor instead a "reptation" model. We discuss herein these problems and propose a resolution.