Anomalous diffusion of proteins due to molecular crowding

We have studied the diffusion of tracer proteins in highly concentrated random-coil polymer and globular protein solutions imitating the crowded conditions encountered in cellular environments. Using fluorescence correlation spectroscopy, we measured the anomalous diffusion exponent alpha characterizing the dependence of the mean-square displacement of the tracer proteins on time, r(2)(t) approximately t(alpha). We observed that the diffusion of proteins in dextran solutions with concentrations up to 400 g/l is subdiffusive (alpha < 1) even at low obstacle concentration. The anomalous diffusion exponent alpha decreases continuously with increasing obstacle concentration and molecular weight, but does not depend on buffer ionic strength, and neither does it depend strongly on solution temperature. At very high random-coil polymer concentrations, alpha reaches a limit value of alpha(l) approximately 3/4, which we take to be the signature of a coupling between the motions of the tracer proteins and the segments of the dextran chains. A similar, although less pronounced, subdiffusive behavior is observed for the diffusion of streptavidin in concentrated globular protein solutions. These observations indicate that protein diffusion in the cell cytoplasm and nucleus should be anomalous as well, with consequences for measurements of solute diffusion coefficients in cells and for the modeling of cellular processes relying on diffusion.     

Formation of 100 A filaments from purified glial fibrillary acidic protein in vitro.

Glial fibrillary acidic protein, which is specific to astroglia in the central nervous system, polymerizes in vitro into filaments similar to native ~ 100 Å filaments. Following purification from aqueous extracts of bovine brain by immunoaffinity chromatography, GFA ² protein is highly soluble in very low ionic strength solutions. Sedimentation equilibrium analysis of protein solutions in prefilament solvent conditions (2 mm-Tris · HCl, pH 7.8, 20 °C, containing 0.5 mm-dithiothreitol) indicates a paucidisperse mixture of species in solution with a typical range of apparent weight-average molecular weights from about 186,000 to 227,000. Between pH 6.0 and 8.0 the solubility is a function of pH and ionic strength as well as temperature, and precipitation is favored by lowering the pH or temperature and by raising the ionic strength. GFA protein associates in the form of filaments over a narrow range of pH and ionic strength; optimal conditions for polymerization of a 0.1 mg/ml protein solution are 100 mm-imidazole-HCl buffer (pH 6.8), at a temperature of 37 °C, and there is no requirement for co-factors. Filaments appear primarily as tangles of smooth curvilinear structures approximately 100 Å in diameter and of indefinite length, although some lateral association of filaments into thick bundles is also observed. While the formation of filaments is not affected by the presence or absence of reducing agent, under oxidizing conditions disulfide linkages form between protein subunits. Disassembly is achieved by dialysis against 2 mm-Tris · HCl buffer (pH 8.5), but this process is significantly enhanced by the addition of 0.5 mM-dithiothreitol during assembly and disassembly.These experiments clarify the role of GFA protein as the subunit of astroglialspecific intermediate filaments. In addition, they suggest that the 100 Å filament, as other components of the cytoskeleton, may assemble and disassemble in the glial cytoplasm.     

Electron microscopy of lanthanum in plasma

Summary Aqueous solutions of lanthanum nitrate may be used as electron microscopic tracers in vivo to study vascular permeability in the experimental animal. However, with this technique the size of the tracer particles is not known. To gain information about the tracer size, we injected lanthanum nitrate into the blood circulation of living rabbits. The plasma obtained from such animals 30 min later, was studied with the electron microscope. The plasma contained an electron-dense material, readily visible in the electron microscope. A precipitate obtained after centrifugation of the whole blood to separate the cells, also contained the tracer. Lanthanum was found in large amounts in the fibrin clot obtained after treating the plasma with thrombin. The tracer was not detected in the serum (i.e. thrombin-treated plasma). The study indicates that ionic lanthanum injected into the blood circulation of living rabbits, is to a great extent bound to fibrinogen, and that the smallest possible size of the tracer is that of the fibrinogen,molecule (m. w. 330,000). Larger particles are present as well.     

Role of diffusion potential on the flow of angiotensin II,bradykinin and related molecules through cellophane membranes

The diffusion of electrically charged peptides (angiotensin II, bradykinin and [Suc¹]angiotensin II) across tight cellophane membranes, obtained by different degrees of acetylation, shows a kinetic behaviour which was interpreted in the literature as indicative of the existence of different molecular conformations presenting slow interconversion velocities and different permeabilities across the membrane. A diffusion potential (Δψ) was found to be present across the membrane along diffusion experiments performed in low ionic strength. Upon annihilation of δψ by chemical voltage clamping (by equally increasing the ionic strength on both bathing solutions) the diffusion rate was decreased and the flow followed first order kinetics, indicating a major role of Δψ in the process. As the ionic strength increase could also affect molecular conformation, the role of Δψ on the diffusion of those molecules was tested by fitting flux and Δψ experimental results by an integrated form of Nernst-Planck flux equation. It is concluded that the deviation from first order diffusion kinetics, observed in low ionic strength, is solely due to the diffusion potential, and not to the existence of more than one molecular conformation in aqueous solution. This study was extended to amino acids and other related charged molecules.     

Lateral diffusion of membrane proteins in protein-rich membranes. A simple hard particle model for concentration dependence of the two-dimensional diffusion coefficient.

A model for the effect of protein concentration on the rate of lateral diffusion of integral membrane proteins is presented, in which the proteins are represented by equivalent hard circular particles on a surface. As the density of particles increases, the probability of finding a vacancy immediately adjacent to a tracer particle into which it may diffuse decreases, resulting in a concomitant reduction of the tracer diffusion coefficient. Using scaled particle theory to calculate the concentration-dependent probabilities, a simple approximate result is obtained in closed form, that is compared with the results of previously published Monte Carlo lattice simulations and experimental observations.     

Solution of linear one-dimensional diffusion equations

The paper introduces a basic mathematical form, which is characteristic of a number of linear one-dimensional diffusion equations with coefficients being represented as simple polynomials in the spatial coordinate. A number of particular diffusion equations are introduced and their corresponding exact mathematical solutions are obtained.     

Influence of double scattering in determination of rotational diffusion coefficients by depolarized dynamic light scattering: Application to the bacteriophages T7 and T4B

Dynamic light scattering experiments were performed on solutions of the bacteriophages T7 and T4B in order to obtain the rotational diffusion coefficients of these phages. Correlation functions were determined from the depolarized intensity scattered in the forward direction. The apparatus used in this study is described in detail. Particular attention is paid to the minimalization of the depolarized intensity due to double scattering. If double scattering cannot be neglected, the correlation function of the depolarized field is the sum of the correlation functions resulting from single and double scattering. It is shown that by correcting for double scattering, it is then possible to obtain the rotational diffusion coefficient of the macromolecules. Although the optical anisotropy of both T4B (retracted fibers) and T7 is very small, the experimental conditions could be chosen in such a way that no depolarized scattering due to double scattering was observed. The measured rotational diffusion coefficients for T4B and T7 are D = 258 ± 12 and 4528 ± 100 sec^?1, respectively. These values compare very well with those obtained by electric birefringence experiments.     

Mobility and diffusion in the plane of cell membrane

The mobility and diffusion of integral proteins in the plane of the membrane are studied in the light of recently known hydrodynamic equations of liquid crystals. We derive the formulas of mobility and diffusion constant in a two-dimensional liquid crystal, and give the solution of the diffusion equation in a spherical membrane. We then discuss the results of the Frye-Edidin (1970) experiments and compare the data with our theoretical values. We conclude that the (human and mouse cell) membranes of the Frye-Edidin experiments are in a phase transition or critical region at 37 °C. Further experiments are proposed to test our assertion.     

Intrinsic electric fields and proton diffusion in immobilized protein membranes. Effects of electrolytes and buffers.

We present a theory for proton diffusion through an immobilized protein membrane perfused with an electrolyte and a buffer. Using a Nernst-Planck equation for each species and assuming local charge neutrality, we obtain two coupled nonlinear diffusion equations with new diffusion coefficients dependent on the concentration of all species, the diffusion constants or mobilities of the buffers and salts, the pH-derivative of the titration curves of the mobile buffer and the immobilized protein, and the derivative with respect to ionic strength of the protein titration curve. Transient time scales are locally pH-dependent because of protonation-deprotonation reactions with the fixed protein and are ionic strength-dependent because salts provide charge carriers to shield internal electric fields. Intrinsic electric fields arise proportional to the gradient of an "effective" charge concentration. The field may reverse locally if buffer concentrations are large (greater to or equal to 0.1 M) and if the diffusivity of the electrolyte species is sufficiently small. The "ideal" electrolyte case (where each species has the same diffusivity) reduces to a simple form. We apply these theoretical considerations to membranes composed of papain and bovine serum albumin (BSA) and show that intrinsic electric fields greatly enhance the mobility of protons when the ionic strength of the salts is smaller than 0.1 M. These results are consistent with experiments where pH changes are observed to depend strongly on buffer, salt, and proton concentrations in baths adjacent to the membranes.     

Tuning the voltage dependence of tetraethylammonium block with permeant ions in an inward-rectifier K+ channel.

To understand the role of permeating ions in determining blocking ion-induced rectification, we examined block of the ROMK1 inward-rectifier K+ channel by intracellular tetraethylammonium in the presence of various alkali metal ions in both the extra- and intracellular solutions. We found that the channel exhibits different degrees of rectification when different alkali metal ions (all at 100 mM) are present in the extra- and intracellular solution. A quantitative analysis shows that an external ion site in the ROMK1 pore binds various alkali metal ions (Na+, K+, Rb+, and Cs+) with different affinities, which can in turn be altered by the binding of different permeating ions at an internal site through a nonelectrostatic mechanism. Consequently, the external site is saturated to a different level under the various ionic conditions. Since rectification is determined by the movement of all energetically coupled ions in the transmembrane electrical field along the pore, different degrees of rectification are observed in various combinations of extra- and intracellular permeant ions. Furthermore, the external and internal ion-binding sites in the ROMK1 pore appear to have different ion selectivity: the external site selects strongly against the smaller Na+, but only modestly among the three larger ions, whereas the internal site interacts quite differently with the larger K+ and Rb+ ions.     

Effective diffusion rate through a polymer network: influence of nonspecific binding and intersegment transfer

The effective diffusion rate of a tracer molecule through a polymer network can be influenced by nonspecific binding. If such binding occurs, the local density fluctuations (segmental diffusion) of the network molecules will contribute to the net displacements of tracer molecules. If the network is strongly interconnected by entanglement or cross-linking, these local motions will only carry the tracer molecules over a small region, and effective transport would require dissociation and reassociation of the tracer molecule to another part of the network. Alternatively, tracer molecules could be transferred directly (intersegment transfer) between different parts of the network whenever they are brought sufficiently close by the density fluctuations. A wormlike-chain model for the segmental diffusion of a polymer is used to describe the network motions and to derive the effective diffusion rate for a tracer molecule as a function of network density and binding constant with or without intersegment transfer contributing. It is found that the density dependence for the effective diffusion of ethidium bromide through dense DNA solutions studied by photobleaching recovery [R. D. Icenogle and E. L. Elson (1983) Biopolymers 22 , 1949–1966] agrees with an intersegment-transfer mechanism limited by the segmental DNA motions. The calculations are also applied to a model for the intracellular diffusion of molecules loosely bound to the cytomatrix. If intersegment transfer dominates it can account for the observed size independence for the intracellular diffusion rates of various injected macromolecules.     

Mechanisms for the facilitated diffusion of substrates across cell membranes

Two classes of theoretical mechanisms for protein-mediated, passive, transmembrane substrate transport (facilitated diffusion) are compared. The simple carrier describes a carrier protein that exposes substrate influx and efflux sites alternately but never both sites simultaneously. Two-site models for substrate transport describe carrier proteins containing influx and efflux sites simultaneously. Velocity equations describing transport by these mechanisms are derived. These equations take the same general form, being characterized by five experimental constants. Simple carrier-mediated transport is restricted to hyperbolic kinetics under all conditions. Two-site carrier-mediated transport may deviate from hyperbolic kinetics only under equilibrium exchange conditions. When both simple- and two-site carriers display hyperbolic kinetics under equilibrium exchange conditions, these models are indistinguishable by using steady-state transport data alone. Seven sugar transport systems are analyzed. Five of these systems are consistent with both models for sugar transport. Uridine, leucine, and cAMP transport by human red cells are consistent with both simple- and two-site models for transport. Human erythrocyte sugar transport can be modeled by simple- and two-site carrier mechanisms, allowing for compartmentalization of intracellular sugars. In this instance, resolution of the intrinsic properties of the human red cell sugar carrier at 20 degrees C requires the use of submillisecond transport measurements.     

Quasi-elastic light scattering studies of hyaluronic acid solutions

Autocorrelation functions for the intensity of laser light scattered from solutions of hyaluronic acid have been measured under a variety of experimental conditions. The form of these functions is consistent with the large amount of long-range intra- and intermolecular interactions characteristic of polyelectrolyte solutions. Further experiments have been performed in order to study the effect of hyaluronic acid on the Brownian diffusion of polystyrene spheres. Using several different sphere sizes as solution probes, the importance of the ionic environment in determining hyaluronic confirmation has been demonstrated.     

Tracer diffusion through F-actin: effect of filament length and cross-linking.

We have determined diffusion coefficients for small (50- to 70-nm diameter) fluorescein-thiocarbamoyl-labeled Ficoll tracers through F-actin as a function of filament length and cross-linking. fx45 was used to regulate filament length and avidin/biotinylated actin or ABP-280 was used to prepare cross-linked actin gels. We found that tracer diffusion was generally independent of filament length in agreement with theoretical predictions for diffusion through solutions of rods. However, in some experiments diffusion was slower through short (< or = 1.0 micron) filaments, although this result was not consistently reproducible. Measured diffusion coefficients through unregulated F-actin and filaments of lengths > 1.0 micron were more rapid than predicted by theory for tracer diffusion through rigid, random networks, which was consistent with some degree of actin bundling. Avidin-induced cross-linking of biotinylated F-actin did not affect diffusion through unregulated F-actin, but in cases where diffusion was slower through short filaments this cross-linking method resulted in enhanced tracer diffusion rates indistinguishable from unregulated F-actin. This finding, in conjunction with increased turbidity of 1.0-micron filaments upon avidin cross-linking, indicated that this cross-linking method induces F-actin bundling. By contrast, ABP-280 cross-linking retarded diffusion through unregulated F-actin and decreased turbidity. Tracer diffusion under these conditions was well approximated by the diffusion theory. Both cross-linking procedures resulted in gel formation as determined by falling ball viscometry. These results demonstrate that network microscopic geometry is dependent on the cross-linking method, although both methods markedly increase F-actin macroscopic viscosity.     

Design and analysis of mixture systems: Applications in hydroponic,plant nutrition research

This study demonstrates that nutrient solutions can be defined as mixture systems. A general methodology for design and analysis of mixture optimization experiments is developed. The emphasis is centered on multivariate investigation of the zone of optimal solution properties as a function of the ion composition and the total ionic strength of the solution. The study of the effects of ion interaction on well-defined solution properties is also possible by this multivariate approach. This work is a valuable tool in mineral nutritional research, because for the first time the chemical feasibility conditions of such solution, combined with additional chemical, physiological or economical constraints, form the foundation of the statistical experimental design theory, which makes the optimization of complex mixtures of ions in relation to well-defined response variables possible.     

Transport of macromolecules in arterial wallin vivo: A mathematical model and analytical solutions

A mathematical model has been developed to simulatein vivo transmural accumulation of an intravenously injected tracer in the aortic wall of experimental animals. Parameters have been included to represent the following processes that affect tracer distribution: permeation of the blood-tissue interface, diffusion through the layers of the artery wall,convective solute drag through the same, and degradation. Of particular interest for thein vivo situation situation is the inclusion of boundary conditions that account for the variation in the plasma concentration of injected tracer as a function of time. Two analytical solutions are presented. The first describes a system in which two boundaries must be delineated; it pertains if the tracer is allowed to circulate until it enters the avascular media of the artery wall both across its luminal boundary and from the capillaries in its outer layer. The second applies to shorter duration experiments in which entry across only the luminal boundary is considered. A limiting case of the solution for short circulation times is presented, compared with a previously published solution, and examined for its potential utility in parameter estimation. Because of its treatment of time-dependent boundary conditions, the model has unique application toin vivo experiments related to macromolecular transport in atherosclerosis that may otherwise elude proper interpretation. This work was supported by National Institutes of Health Grants HL-29582 and HL-07242.     

A theory for the effects of neutral carriers such as the macrotetralide actin antibiotics on the electric properties of bilayer membranes

Summary To develop a quantitiative theoretical treatment for the effects of neutral macrocyclic antibiotics on the electrical properties of phospholipid bilayer membranes, this paper proceeds from the known ability of such molecules to form stoichiometric, lipid-soluble complexes with cations and deduces the electrical properties that a simple organic solvent phase would have if it were made into a membrane of the thinness of the phospholipid bilayer. In effect, we postulate that the essential barrier to ion movement across a bilayer membrane is its liquid-like hydrocarbon interior and that the neutral macrocyclic antibiotics bind monovalent cations and solubilize them in the membrane as mobile positively charged complexes. Using the Poisson-Boltzmann equation to describe the equilibrium profile of the electrical potential, it is shown that an excess of the positive complexes over all the other ions is expected in the membrane as a net space charge for appropriate conditions of membrane thickness and values of the partition coefficients of the various ionic species and without requiring the presence of fixed charges. Describing the fluxes of these complexes by the Nernst-Planck equation and neglecting the contribution to the electric current of uncomplexed ions, theoretical expressions are derived for the membrane potential in ionic mixtures, as well as for the limiting value of the membrane conductance at zero current when the membrane is interposed between identical solutions. The expressions are given in terms of the ionic activities and antibiotic concentrations in the aqueous solutions so as to be accessible to direct experimental test. Under suitable experimental conditions, the membrane potential is described by an equation recognizible as the Goldman-Hodgkin-Katz equation, in which the permeability ratios are combinations of parameters predicted from the present theory to be independently determinable from the ratio of membrane conductances in single salt solutions. Since this identity between permeability and conductance ratios is expected also for systems obeying the Independence Principle of Hodgkin and Huxley, the applicability of this principle to membranes exposed to antibiotics is discussed, and it is shown that this principle is compatible with the permeation mechanism proposed here.     

On the theory of diffusion of electrolytes

In the theory of diffusion of electrolytes the following assumptions are frequently made: (i) the electrolytic solution is electrically neutral everywhere, (ii) the ionic concentrations and the electric potential all depend on a single Cartesian coordinate as the only space variable. Often the electric potential of the solution is determined on the basis of the Poisson equation alone, disregarding any other relation between this potential and the ionic concentrations. Since the Poisson equation only represents a condition which the potential fulfills, the use of this equation alone may lead to error unless the explicit relation for the potential involving a space integration of ionic concentrations is also taken into account. But if this relation is used the Poisson equation becomes redundant and, more important, assumptions (i) and (ii) appear unacceptable, the former because it leads to a zero electric potential everywhere, the latter because it is mathematically incorrect. The present paper is based on general equations of diffusion of ions, excluding the Poisson equation. These equations form a system of nonlinear integrodifferential quations whose number equals the number of ionic species present in the solution. It appears that when all ions are distributed symmetrically around a point all functions related to the above system of equations can be made dependent on a single space coordinate: the distance from the center of symmetry. Two methods of successive approximations are given for the solution of the equations in the case of spherical symmetry with limitation to the steady state. These methods are then applied to the study of the distribution of ionic concentrations and electrical potentials inside a cell of spherical shape in equilibrium with its surroundings. These methods are rapidly convergent; exact theoretical values of the electric potential are calculable on the boundary of the cell. It appears that the potential at the center of the cell is not more than ∼50% higher than at its boundary and that variation of concentration inside the cell is not very large. For instance, with 100 mV on the boundary the ionic concentration there is about four times higher than at the center. Calculations show that extremely small amounts of electricity are sufficient to account for the electric potentials currently observed. In a cell of 100 micra diameter an average concentration of only 10⁻¹⁴ mole/cm³ of a monovalent ion would be sufficient to give 1 millivolt on the boundary. This concentration is directly proportional to the voltage and inversely proportional to the square of the cell diameter. Most of the numerical results given above are obtained by considering only those ions whose electrical charge is not compensated for by ions of an opposite sign. The total concentrations may be much higher than those quoted. The theory does not take into account possible effects of structural heterogeneities which may exist in the cell, particularly of various phase boundaries. An incidental result shows that the Boltzmann distribution function in the form employed in modern theory of electrolytes is fundamentally a consequence of the mathematical theory of diffusion alone. It is pointed out, however, that Boltzmann distribution is not always compatible with the definition of the electric potential.     

Single-file diffusion multi-ion mechanism of permeation in paracellular epithelial channels

Summary The transepithelial fluxes, conductances and permeabilities of Li⁺, Na⁺, K⁺, Cs⁺, NH ₄ ⁺ and H₃CNH ₃ ⁺ were studied under ionic concentrations ranging from 12 to 250mm inBufo arenarum gallbladders. When these measurements are carefully corrected in order to get only the component due to the paracellular cation channels, the following results are obtained: (1) The permeability ratios (cationic/anionic) are a decreasing function of salt concentration. (2) The partial conductances through paracellular cationic channels show nonlinear saturable concentration kinetics. (3) Moreover, partial conductance kinetics of K⁺, Cs⁺ and NH ₄ ⁺ present a maximum followed, at higher concentratons, by a negative-slope region. (4) The selectivity sequences obtained from biionic potentials do not agree with those obtained from partial conductance measurements. (5) The unidirectional²²Na tracer flux (serosal to mucosal) is inhibited by 63% when the K⁺ symmetrical concentration in the bathing solutions is raised from 25 to 200mm. (6) When the unidirectional⁴²K fluxes (serosal to mucosal) at 200mm KCl Na-free solutions are compared with K⁺ partial conductance by means of the Hodgkin and Keynes (Hodgkin, A.L., Keynes, R.D. 1955.J. Physiol London 128:61–88) expression, then factor is 2.0. These results indicate that cations do not follow the independence principle and behave as in single-file diffusion multi-ion pores when crossing the paracellular cation channels ofBufo arenarum gallbladder epithelium.