Diffusion in colloidal media

When the molecules of a solute diffuse through a medium containing large colloidal particles, which absorb the diffusing molecules, the latter are transported in the diffusion flow not as free molecules, but as absorbtion compounds: solute+colloid. When the colloidal particle is much larger than the molecule of the solute, and has therefore a much smaller mobility, this results in a reduction of the apparent diffusion coefficient for the solute. The biological implications of this are discussed.     

Diffusional anisotropy in collagenous tissues: fluorescence imaging of continuous point photobleaching

Molecular transport in avascular collagenous tissues such as articular cartilage occurs primarily via diffusion. The presence of ordered structures in the extracellular matrix may influence the local transport of macromolecules, leading to anisotropic diffusion depending on the relative size of the molecule and that of extracellular matrix structures. Here we present what we believe is a novel photobleaching technique for measuring the anisotropic diffusivity of macromolecules in collagenous tissues. We hypothesized that macromolecular diffusion is anisotropic in collagenous tissues, depending on molecular size and the local organization of the collagen structure. A theoretical model and experimental protocol for fluorescence imaging of continuous point photobleaching was developed to measure diffusional anisotropy. Significant anisotropy was observed in highly ordered collagenous tissues such as ligament, with diffusivity ratios >2 along the fiber direction compared to the perpendicular direction. In less-ordered tissues such as articular cartilage, diffusional anisotropy was dependent on site in the tissue and size of the diffusing molecule. Anisotropic diffusion was also dependent on the size of the diffusing molecule, with greatest anisotropy observed for larger molecules. These findings suggest that diffusional transport of macromolecules is anisotropic in collagenous tissues, with higher rates of diffusion along primary orientation of collagen fibers.     

Fluorescence correlation spectroscopy. I. Conceptual basis and theory

We describe a method for determining chemical kinetic constants and diffusion coefficients by measuring the rates of decay of spontaneous concentration fluctuations. The equilibrium of the system is not disturbed during the measurement. We measure the number of molecules of a specified type in a defined open volume as a function of time and compute the time course of the deviations from the thermodynamic mean concentration. The method is based on the principle that the rates of decay of spontaneous microscopic fluctuations are determined by the same phenomenological rate coefficients as those of macroscopic departures from equilibrium which result from external perturbations. Hence, an analysis of fluctuations yields the same chemical rate constants and diffusion coefficients as are measured by conventional procedures. In practice the number of the specified molecules is measured by a property such as absorbance or fluorescence which is specific and sensitive to chemical change. The sample volume is defined by a light beam which traverses the cell. As the molecules appear in or disappear from the light beam, either due to diffusion or chemical reaction, their concentration fluctuations give rise to corresponding fluctuations of the intensity of absorbed or emitted light. This paper presents the theory needed to derive chemical rate constants and diffusion coefficients from these fluctuations in light intensity. The theory is applied to three examples of general interest: pure diffusion in the absence of chemical reaction; the binding of a small rapidly diffusing ligand to a larger slowly diffusing macromolecule; and a unimolecular isomerization. The method should be especially useful in studying highly cooperative systems, relatively noncooperative systems with intermediate states closely spaced in free energy, small systems, and systems not readily subject to perturbations of state.     

Revising Berg-Purcell for finite receptor kinetics

From nutrient uptake to chemoreception to synaptic transmission, many systems in cell biology depend on molecules diffusing and binding to membrane receptors. Mathematical analysis of such systems often neglects the fact that receptors process molecules at finite kinetic rates. A key example is the celebrated formula of Berg and Purcell for the rate that cell surface receptors capture extracellular molecules. Indeed, this influential result is only valid if receptors transport molecules through the cell wall at a rate much faster than molecules arrive at receptors. From a mathematical perspective, ignoring receptor kinetics is convenient because it makes the diffusing molecules independent. In contrast, including receptor kinetics introduces correlations between the diffusing molecules because, for example, bound receptors may be temporarily blocked from binding additional molecules. In this work, we present a modeling framework for coupling bulk diffusion to surface receptors with finite kinetic rates. The framework uses boundary homogenization to couple the diffusion equation to nonlinear ordinary differential equations on the boundary. We use this framework to derive an explicit formula for the cellular uptake rate and show that the analysis of Berg and Purcell significantly overestimates uptake in some typical biophysical scenarios. We confirm our analysis by numerical simulations of a many-particle stochastic system.     

Role of phosphocreatine in energy transport in skeletal muscle of bullfrog studied by 31P-NMR

To evaluate the energy-shuttle hypothesis of the phosphocreatine/creatine kinase system, diffusion rates for ATP, phosphocreatine and flux through the creatine kinase reaction were determined by 31P-NMR in resting bullfrog biceps muscle. The diffusion coefficient of phosphocreatine measured by 31P-pulsed gradient NMR was 1.4-times larger than ATP in the muscle, indicating the advantage of phosphocreatine molecules for the intracellular energy transport. The flux of the creatine kinase reaction measured by 31P-saturation transfer NMR was 3.6 mmol/kg wet wt. per s in the resting muscle. The flux is equal to the turnover rate of ATP, ADP, phosphocreatine and creatine molecules, therefore, the life-times of these substrates and the average distance traversed after the life-times by the diffusing molecules were calculated using the diffusion coefficients obtained by 31P-NMR. The mean square length of one-dimensional diffusion was 22 microns in ATP molecules and the minimum diffusion length was 1.8 microns in ADP molecules. The latter was calculated using free ADP concentration, 30 mumol/kg wet wt., obtained from the equilibrium constant of the creatine kinase reaction and the diffusion coefficient assumed to be the same of ATP in muscle. Similar diffusion lengths of ADP were calculated using the reported values for the flux of the creatine kinase reaction in heart and smooth-muscle. The diffusion lengths of all substrates involved in the creatine kinase reaction were larger than the radii of myofibrils. Therefore, in the muscles with an alternating arrangement of mitochondria and myofibrils, such as heart and certain skeletal muscles, ATP and ADP molecules can move freely between myofibrils and mitochondria without the aid of the creatine kinase reaction; thus, we conclude that the energy-shuttle hypothesis is not obligatory for energy transport between the mitochondria and the myofibrils.     

The effect of hydration of lecithin-water lamellar phases: a comparative study of solute diffusion and ESR spectroscopic measurements

An analysis of the paramagnetic resonance spectra of spin labels in the lipidic region of lecithin-water lamellar phases as a function of phase water content has been carried out. The observed variation of the local organization and mobility of the lipids is consonant with previous results obtained from solute diffusion measurements. The previously observed sudden changes of solute diffusion for hydration of 9 and 18 molecules water per lecithin molecule are compared with the concomitant sudden changes as seen by ESR spectroscopy. The results also indicate that there is a gradient of fluidity across the lipid leaflets which are therefore not homogeneous to diffusing molecules.     

Diffusion of molecules on biological membranes of nonplanar form. II. Diffusion anisotropy.

Molecules diffusing on nonplanar membranes, which have different amounts of corrugation in different directions, may experience dissimilar diffusion coefficients in each direction. Smith et al. (1979, Proc. Natl. Acad. Sci. USA, 76:5641-5644) measured diffusion anisotropy on fibroblast cell membranes in which the ratio of the diffusion coefficients, in different directions, was 0.27. In the present work we calculate the effect of anisotropic corrugation on the rate of diffusion of molecules on membranes. We find that part of the anisotropy reported by Smith et al. (1979, Proc. Natl. Acad. Sci. USA, 76:5641-5644) can be explained by the membrane nonplanarity, and we present the way of calculating this geometric factor.     

FCS cell surface measurements--photophysical limitations and consequences on molecular ensembles with heterogenic mobilities.

BACKGROUND: Fluorescence Correlation Spectroscopy is a powerful method to analyze densities and diffusive behavior of molecules in membranes, but effects of photodegradation can easily be overlooked. METHOD: Based on experimental photophysical parameters, calculations were performed to analyze the consequences of photobleaching in fluorescence correlation spectroscopy (FCS) cell surface experiments, covering a range of standard measurement conditions. RESULTS: Cumulative effects of photobleaching can be prominent, although an absolute majority of the fluorescent molecules would pass the laser excitation beam without being photo-bleached. Given a distribution of molecules on a cell surface with different diffusive properties, the fraction of molecules that is actually analyzed depends strongly on the excitation intensities and measurement times, as well as on the size of the reservoir of freely diffusing molecules. Both the slower and the faster diffusing molecules can be disfavored. CONCLUSIONS: Apart from quantifying photobleaching effects, the calculations suggest that the effects can be used to extract additional information, for instance about the size of the reservoirs of free diffusion. By certain choices of measurement conditions, it may be possible to more specifically analyze certain species within a population, based on their different diffusive properties, different areas of free diffusion, or different kinetics of possible transient binding.     

Development of time-integrated multipoint moment analysis for spatially resolved fluctuation spectroscopy with high time resolution

Spatial gradients in the behaviors of soluble proteins are thought to underlie many phenomena in cell and developmental biology, but the nature and even the existence of these gradients are often unclear because few techniques can adequately characterize them. Methods with sufficient temporal resolution to study the dynamics of diffusing molecules can only sample relatively small regions, whereas methods that are capable of imaging larger areas cannot probe fast timescales. To overcome these limitations, we developed and implemented time-integrated multipoint moment analysis (TIMMA), a form of fluorescence fluctuation spectroscopy that is capable of probing timescales down to 20 μs at hundreds of different locations simultaneously in a sample. We show that TIMMA can be used to measure the diffusion of small-molecule dyes and fluorescent colloids, and that it can create spatial maps of the behavior of soluble fluorescent proteins throughout mammalian tissue culture cells. We also demonstrate that TIMMA can characterize internal gradients in the diffusion of freely moving proteins in single cells.     

Transition regime diffusion and the structure of the insect tracheolar system

A model is developed for the diffusion of oxygen along insect tracheoles. It is shown that reduction of the effective tracheolar diameter is advantageous until this quantity approaches the mean free path of diffusing oxygen molecules, and that, once this transition regime has been entered, further reduction of tracheolar diameter is disadvantageous. This finding offers a plausible explanation of Weis-Fogh's observation that the respiratory pathway in insects continues to branch until the smallest ‘twigs’ approach a mean free path in diameter.     

Simulation of diffusion time of small molecules in protein crystals

A simple model for evaluation of diffusion times of small molecule into protein crystals has been developed, which takes into account the physical and chemical properties both of protein crystal and the diffusing molecules. The model also includes consideration of binding and the binding affinity of a ligand to the protein. The model has been validated by simulation of experimental set-ups of several examples found in the literature. These experiments cover a wide range of situations: from small to relatively large diffusing molecules, crystals having low, medium, or high protein density, and different size. The reproduced experiments include ligand exchange in protein crystals by soaking techniques. Despite the simplifying assumptions of the model, theoretical and experimental data are in agreement with available data, with experimental diffusion times ranging from a few seconds to several hours. The method has been used successfully for planning intermediate cryotrapping experiments in maltodextrin phosphorylase crystals.     

Multi-scale stochastic simulation of diffusion-coupled agents and its application to cell culture simulation

Many biological systems consist of multiple cells that interact by secretion and binding of diffusing molecules, thus coordinating responses across cells. Techniques for simulating systems coupling extracellular and intracellular processes are very limited. Here we present an efficient method to stochastically simulate diffusion processes, which at the same time allows synchronization between internal and external cellular conditions through a modification of Gillespie's chemical reaction algorithm. Individual cells are simulated as independent agents, and each cell accurately reacts to changes in its local environment affected by diffusing molecules. Such a simulation provides time-scale separation between the intra-cellular and extra-cellular processes. We use our methodology to study how human monocyte-derived dendritic cells alert neighboring cells about viral infection using diffusing interferon molecules. A subpopulation of the infected cells reacts early to the infection and secretes interferon into the extra-cellular medium, which helps activate other cells. Findings predicted by our simulation and confirmed by experimental results suggest that the early activation is largely independent of the fraction of infected cells and is thus both sensitive and robust. The concordance with the experimental results supports the value of our method for overcoming the challenges of accurately simulating multiscale biological signaling systems.     

Exchange efflux of [3H]palmitate from human red cell ghosts to bovine serum albumin in buffer. Effects of medium volume and concentration of bovine serum albumin.

[3H]Palmitate, PA, exchange efflux kinetics is recorded from human erythrocyte ghosts to buffer with bovine serum albumin, BSA, at 0 degrees C. The effects have been investigated of three medium/ghost volume ratios: 36, 80 and 500, of six BSA concentrations, [BSA]: 0.01, 0.02, 0.05, 0.2, 1 and 2% (1.5, 3.0, 7.5, 30, 150 and 300 microM) and of various v, molar ratios of palmitate to BSA, between 0.15 and 0.94. Data are analyzed in terms of a virtually closed three-compartment model. In theory, the tracer efflux is biexponential and the rate coefficients differ at least 20 fold [1]. The efflux rate at 2% BSA is monoexponential beyond our resolution time of about 1 s, but nearly biexponential at or below 0.2% BSA with a well-defined smallest-rate coefficient beta. beta depends strongly on [BSA] but is remarkably v independent. The medium/ghost volume ratio has no effect on beta when [BSA] > or = 0.2%, although beta measured at 2% BSA is almost 2-fold higher than at 0.2%. This suggests the presence of an unstirred layer, USL. According to our model, the observations are understood quantitatively on basis of our previously published dissociation rate constants of the PA-BSA complex, as well as PA equilibrium bindings to ghost membranes (Bojesen, I.N. and Bojesen, E. (1991) Biochim. Biophys. Acta 1069, 297-307). Essentially, beta is theoretically a function of two terms, one comprising the membrane transport parameters and the other the medium-dependent variables. Most important is the clearance with respect to monomer concentration adjacent to the membrane. The clearance is calculated on basis of quasi-stationary diffusion in USL. The data are compatible with a planar USL of 6 microns depth and with the same area as a ghost but not with a spherical USL.     

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.     

Cross-Strain Quorum Sensing Inhibition by Staphylococcus Aureus. Part 2: A Spatially Inhomogeneous Model

Staphylococcus aureus uses quorum sensing (QS) to enhance its pathogenicity. An intriguing aspect of this is that different strains are capable of inactivating the QS systems of opposing strains. In Part 1 of this study, we presented a model of this phenomenon in a well-mixed environment; here, we incorporate spatial structure. Two competitive strains occupying adjacent habitats with freely diffusing QS signal molecules (QSSMs) are considered. We investigate the effect of the QSSM diffusion coefficient and the relative size of the two populations on the ability of one population to dominate the other. Regarding population size, a larger population is generally at an advantage (initial conditions permitting), while the implications of different diffusivities are more complex and depend upon the sizes of the populations.     

The fence and picket structure of the plasma membrane of live cells as revealed by single molecule techniques (Review)

Models of the organization of the plasma membrane of live cells as discovered through diffusion measurements of integral membrane molecules (transmembrane and GPI-anchored proteins, and lipid) at the single molecule level are discussed. Diffusion of transmembrane protein and, indeed, even lipid is anomalous in that the molecules tend to diffuse freely in limited size compartments, with infrequent intercompartment transitions. This average residency time in a compartment is dependent on the diffusing species and on its state of oligomerization, becoming completely confined to a single compartment upon sufficient oligomerization. This will be of great importance in determining cellular mechanisms for controlling the random diffusive motion of membrane molecules and in understanding signalling processes.     

The position of regenerating cambia: auxin/sucrose ratio and the gradient induction hypothesis.

To account for the positions in which vascular cambia regenerate in wound callus, a gradient induction hypothesis was proposed in 1961 in terms of gradients in 'some factor as yet unknown'. It now seems likely that the gradient is based on morphogen diffusion between source and sink on opposite sides of existing cambia, with morphogen diffusing into the adjoining wound callus. It is specifically proposed that there are two morphogens, auxin diffusing centrifugally and sucrose diffusing centripetally. The cambium then regenerates along a path where the ratio of auxin to sucrose concentration is similar to that at the original cambium, and its orientation (as regards xylem and phloem formation) is determined by the direction of the gradient in this ratio. These proposals are supported by published evidence on auxin and sucrose concentration gradients across the cambium, and on their sources, movements, and known effects on vascular differentiation. Simulations of the proposed positional control system predict patterns of cambial regeneration and orientation corresponding to those observed in four different types of wound and graft.     

Solute size effects on the diffusion in biofilms of Streptococcus mutans

The diffusion of poly(ethylene glycol) (PEG) (MW varying between 200 and 10,000), and of three different types of micelles was examined in Streptococcus mutans biofilms using infrared spectroscopy. PEGs were used because they show limited interactions with biological materials and their weight can be selected in order to cover a wide range of size. The study showed that a considerable fraction at the base of the biofilm was not accessible to the diffusing solute molecules and this inaccessible fraction was very dependent on the size of the diffusing molecules. In parallel, it was found that the diffusion coefficients of these solutes in the biofilms were less than those in water and this reduction was less pronounced for large macromolecules, an effect proposed to be related to their limited penetration. Triton X-100, a neutral detergent, forms micelles that behave like PEG, suggesting that the behaviour observed for neutral macromolecules can be extrapolated to neutral macroassemblies. However, the diffusion, as well as the penetration of sodium dodecylsulphate micelles (a negatively charged surfactant) and cetylpyridinium chloride micelles (positively charged), in the biofilms appeared to be significantly influenced by electrostatic interactions with biofilm components. The present findings provide useful insights associated with the molecular parameters required to efficiently penetrate bacterial biofilms. The study suggests a rationale for the limited bactericidal power of some antibiotics (the large ones). The restricted accessibility of macromolecules and macroassemblies to biofilms must be examined carefully in order to offer guidelines in the development of novel antibacterial treatments.     

Calculation of diffusion-limited kinetics for the reactions in collision coupling and receptor cross-linking

Both enzyme (e.g., G-protein) activation via a collision coupling model and the formation of cross-linked receptors by a multivalent ligand involve reactions between two molecules diffusing in the plasma membrane. The diffusion of these molecules is thought to play a critical role in these two early signal transduction events. In reduced dimensions, however, diffusion is not an effective mixing mechanism; consequently, zones in which the concentration of particular molecules (e.g., enzymes, receptors) becomes depleted or enriched may form. To examine the formation of these depletion/ accumulation zones and their effect on reaction rates and ultimately the cellular response, Monte Carlo techniques are used to simulate the reaction and diffusion of molecules in the plasma membrane. The effective reaction rate at steady state is determined in terms of the physical properties of the tissue and ligand for both enzyme activation via collision coupling and the generation of cross-linked receptors. The diffusion-limited reaction rate constant is shown to scale with the mean square displacement of a receptor-ligand complex. The rate constants determined in the simulation are compared with other theoretical predictions as well as experimental data.