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

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

Bending Rigidities and Interdomain Forces in Membranes with Coexisting Lipid Domains

To precisely quantify the fundamental interactions between heterogeneous lipid membranes with coexisting liquid-ordered (Lo) and liquid-disordered (Ld) domains, we performed detailed osmotic stress small-angle x-ray scattering experiments by exploiting the domain alignment in raft-mimicking lipid multibilayers. Performing a Monte Carlo-based analysis allowed us to determine with high reliability the magnitude and functional dependence of interdomain forces concurrently with the bending elasticity moduli. In contrast to previous methodologies, this approach enabled us to consider the entropic undulation repulsions on a fundamental level, without having to take recourse to crudely justified mean-field-like additivity assumptions. Our detailed Hamaker-coefficient calculations indicated only small differences in the van der Waals attractions of coexisting Lo and Ld phases. In contrast, the repulsive hydration and undulation interactions differed significantly, with the latter dominating the overall repulsions in the Ld phase. Thus, alignment of like domains in multibilayers appears to originate from both, hydration and undulation repulsions.     

Phospholipid interactions in model membrane systems. II. Theory.

We describe statistical thermodynamic theory for the lateral interactions among phospholipid head groups in monolayers and bilayers. Extensive monolayer experiments show that at low surface densities, PC head groups have strong lateral repulsions which increase considerably with temperature, whereas PE interactions are much weaker and have no significant temperature dependence (see the preceding paper). In previous work, we showed that the second virial coefficients for these interactions can be explained by: (a) steric repulsions among the head groups, and (b) a tilting of the P-N+ dipole of PC so that the N+ end enters the oil phase, to an extent that increases with temperature. It was also predicted that PE interactions should be weaker and less temperature dependent because the N+ terminal of the PE head-group is hydrophilic, hence, it is tilted into the water phase, so dipolar contributions among PE's are negligible due to the high dielectric constant of water. In the present work, we broaden the theory to treat phospholipid interactions up to higher lateral surface densities. We generalize the Hill interfacial virial expansion to account for dipoles and to include the third virial term. We show that to account for the large third virial coefficients for both PC and PE requires that the short range lateral attractions among the head groups also be taken into account. In addition, the third virial coefficient includes fluctuating head group dipoles, computed by Monte Carlo integration assuming pairwise additivity of the instantaneous pair potentials. We find that because the dipole fluctuations are correlated, the average triplet interactions do not equal the sum of the average dipole pair potentials. This is important for predicting, the magnitude and the independence of temperature of the third virial coefficients for PC. The consistency of the theory with data of both the second and the third virial coefficients extends the applicability of the head-group model to semiconcentrated monolayers, in agreement with the surface potential data in the foregoing paper.     

Effect of hydrodynamic interactions on the diffusion of integral membrane proteins: tracer diffusion in organelle and reconstituted membranes.

A persistent discrepancy exists between theoretical predictions and experimental observations for the diffusion coefficients of integral membrane proteins in lipid bilayers free of immobilized proteins. Current thermodynamic theories overestimate tracer diffusion coefficients at high area fractions. We explore the hypothesis that the combined effect of hydrodynamic and thermodynamic interactions reconciles theory with experiment. We have determined previously the short- and long-time tracer diffusivities, Ds and Dl, respectively, of integral membrane proteins in lipid bilayers as a function of their area fraction, phi. The results are based on two-particle hydrodynamic and thermodynamic interactions and are precise to O(phi). Here we extend the results for Dl to high phi by combining the hydrodynamic results for Ds into theories for Dl based on many-particle thermodynamic interactions. The results compare favorably with the experimental measurements of Dl as a function of protein area fraction for bacteriorhodopsin in reconstituted membranes and for complex III of the mitochondrial inner membrane. The agreement suggests that both hydrodynamic and thermodynamic interactions are important determinants of diffusion coefficients of proteins in lipid bilayers. Additional experiments are required to verify the role of hydrodynamic interactions in protein diffusion in reconstituted systems.     

Protein denaturation with guanidine hydrochloride or urea provides a different estimate of stability depending on the contributions of electrostatic interactions.

The objective of this study was to address the question of whether or not urea and guanidine hydrochloride (GdnHCl) give the same estimates of the stability of a particular protein. We previously suspected that the estimates of protein stability from GdnHCl and urea denaturation data might differ depending on the electrostatic interactions stabilizing the proteins. Therefore, 4 coiled-coil analogs were designed, where the number of intrachain and interchain electrostatic attractions (A) were systematically changed to repulsions (R): 20A, 15A5R, 10A10R, and 20R. The GdnHCl denaturation data showed that the 4 coiled-coil analogs, which had electrostatic interactions ranging from 20 attractions to 20 repulsions, had very similar [GdnHCl]1/2 values (average of congruent to 3.5 M) and, as well, their delta delta Gu values were very close to 0 (0.2 kcal/mol). In contrast, urea denaturation showed that the [urea]1/2 values proportionately decreased with the stepwise change from 20 electrostatic attractions to 20 repulsions (20A, 7.4 M; 15A5R, 5.4 M; 10A10R, 3.2 M; and 20R, 1.4 M), and the delta delta Gu values correspondingly increased with the increasing differences in electrostatic interactions (20A-15A5R, 1.5 kcal/mol; 20A-10A10R, 3.7 kcal/mol; and 20A-20R, 5.8 kcal/mol). These results indicate that the ionic nature of GdnHCl masks electrostatic interactions in these model proteins, a phenomenon that was absent when the unchanged urea was used. Thus, GdnHCl and urea denaturations may give vastly different estimates of protein stability, depending on how important electrostatic interactions are to the protein.     

Diffusion of Particles in the Extracellular Matrix: The Effect of Repulsive Electrostatic Interactions

Diffusive transport of macromolecules and nanoparticles in charged fibrous media is of interest in many biological applications, including drug delivery and separation processes. Experimental findings have shown that diffusion can be significantly hindered by electrostatic interactions between the diffusing particle and charged components of the extracellular matrix. The implications, however, have not been analyzed rigorously. Here, we present a mathematical framework to study the effect of charge on the diffusive transport of macromolecules and nanoparticles in the extracellular matrix of biological tissues. The model takes into account steric, hydrodynamic, and electrostatic interactions. We show that when the fiber size is comparable to the Debye length, electrostatic forces between the fibers and the particles result in slowed diffusion. However, as the fiber diameter increases the repulsive forces become less important. Our results explain the experimental observations that neutral particles diffuse faster than charged particles. Taken together, we conclude that optimal particles for delivery to tumors should be initially cationic to target the tumor vessels and then change to neutral charge after exiting the blood vessels.     

Concentration dependence of proteoglycan diffusion

The mutual diffusion coefficient of the bovine nasal cartilage proteoglycan subunit is found to increase rapidly with increasing concentration and decreasing ionic strength. These results have been obtained by analysis of the boundary relaxation of concentration gradients in the analytical ultracentrifuge by schlieren optics. The diffusion behavior can be understood in terms of the nonideality of the proteoglycan. The magnitude of the nonideality is dominated by charge interactions, whereas the influence of molecular size and associated excluded-volume interactions is small. The concentration dependence of the apparent diffusion coefficient of the proteoglycan subunit from dynamic light scattering was found, in contrast, to decrease with increasing concentration. Computer simulation of the dynamic light scattering suggests that the presence of a small population of aggregates may account for the difference in the two types of diffusion measurement due to their marked influence on the scattering.     

Dielectric behavior of polyelectrolytes II. The cylinder

The dipolar correlation function for a system of coupterions diffusing on the surface of a polyelectrolyte cylinder is computed. The influence of screened coulombic repulsions on the dielectric increment is determined. Dissociation and reassociation of the counterions to the cylinder is treated microscopically and the coupled bulk diffusion is solved in the presence of the Poisson-Boltzmann potential. It is found that the correlation function contains a small, fast decaying, molecular weight independent part arising from diffusion around the cylinder and a large, slowly decaying, molecular weight dependent part arising from diffusion along the cylinder axis. The dissociation-reassociation kinetics can play a large, possible dominant, role in determining the relaxation rates.     

Hydration and Hydrodynamic Interactions of Lysozyme: Effects of Chaotropic versus Kosmotropic Ions

Using static and dynamic light scattering we have investigated the effects of either strongly chaotropic, nearly neutral or strongly kosmotropic salt ions on the hydration shell and the mutual hydrodynamic interactions of the protein lysozyme under conditions supportive of protein crystallization. After accounting for the effects of protein interaction and for changes in solution viscosity on protein diffusivity, protein hydrodynamic radii were determined with ±0.25 Å resolution. No changes to the extent of lysozyme hydration were discernible for all salt-types, at any salt concentration and for temperatures between 15-40°C. Combining static with dynamic light scattering, we also investigated salt-induced changes to the hydrodynamic protein interactions. With increased salt concentration, hydrodynamic interactions changed from attractive to repulsive, i.e., in exact opposition to salt-induced changes in direct protein interactions. This anti-correlation was independent of solution temperature or salt identity. Although salt-specific effects on direct protein interactions were prominent, neither protein hydration nor solvent-mediated hydrodynamic interactions displayed any obvious salt-specific effects. We infer that the protein hydration shell is more resistant than bulk water to changes in its local structure by either chaotropic or kosmotropic ions.     

Local and translational dynamics in DNA-lipid assemblies monitored by solid-state and diffusion NMR

The influence of electrostatic interactions on the dynamic properties of complexes containing DNA and mixtures of cationic- (DDA) and zwitterionic (DLPC) lipids are studied by means of NMR. The systems are arranged in lamellar membrane stacks intercalated by DNA molecules. This is confirmed by 31P-NMR, where a superposition of an axially symmetric powder pattern arising from the phospholipid membrane and an asymmetric tensor due to DNA can be fitted to the experimentally observed lineshape. The local mobility and order is assessed using two solid-state NMR techniques applicable to samples with natural isotopic abundance: WIdeline SEparation (WISE) and Separated Local Field (SLF) spectroscopy. Both experiments yield highly resolved 13C spectra in the direct dimension. The indirect dimension contains information about molecular dynamics through the 1H dipolar linewidth (WISE) or the 1H(-13)C dipolar coupling constant (SLF). The experiments suggest that DNA is static while it induces an increased disorder in the hydrocarbon chains as compared to the parent lipid case. DDA chain order is more affected than DLPC due to the attractive electrostatic interaction between DNA and the cationic lipid. Translational dynamics of the lipids and the water was measured with the Pulsed Field Gradient STimulated Echo (PFG STE) technique. The influence of lamellar domain size and the angular dependence of the diffusion coefficients and nuclear relaxation times on the results of the PFG STE experiments are discussed. The local water diffusion coefficient is reduced by a factor four from the value of bulk water, and increases as the DLPC content is increased. We observe two lipid components with an order of magnitude difference in diffusion coefficients in the DNA:DDA:DLPC precipitate and these are assigned to DLPC (fast) and DDA (slow). Cationic lipid (DDA) diffusion is decreasing a factor of 2 when DLPC is added to the pure DNA:DDA system, indicating DNA-induced lipid segregation within the bilayer and the transition from locally 2D to 1D diffusion of the DDA. The results show that DNA-lipid electrostatic interactions reduce the long-range lipid mobility but locally enhance the hydrocarbon chain dynamics by perturbing the preferred lipid packing.     

Protein effective rotational correlation times from translational self-diffusion coefficients measured by PFG-NMR

Molecular rotational correlation times are of interest for many studies carried out in solution, including characterization of biomolecular structure and interactions. Here we have evaluated the estimates of protein effective rotational correlation times from their translational self-diffusion coefficients measured by pulsed-field gradient NMR against correlation times determined from both collective and residue-specific (15)N relaxation analyses and those derived from 3D structure-based hydrodynamic calculations. The results show that, provided the protein diffusive behavior is coherent with the Debye-Stokes-Einstein model, translational diffusion coefficients provide rapid estimates with reasonable accuracy of their effective rotational correlation times. Effective rotational correlation times estimated from translational diffusion coefficients may be particularly beneficial in cases where i) isotopically labelled material is not available, ii) collective backbone (15)N relaxation rates are difficult to interpret because of the presence of flexible termini or loops, or iii) a full relaxation analysis is practically difficult because of limited sensitivity owing to low protein concentration, high molecular mass or low temperatures.     

A simple analytical model of water

Water is an unusual liquid. It expands upon freezing, has minima in its volume, heat capacity, and isothermal compressibility with temperature, and shows signs of a first-order phase transition when supercooled. These anomalies disappear at high pressures. We review a recent analytical theory that predicts water's thermal properties and the main features of its phase diagram, including multiple crystalline phases and a fluid-fluid transition in the supercooled liquid. It also predicts a fragile-to-strong crossover in supercooled water's temperature-dependent relaxation processes. The theory is based on a simplified model for how triplets of waters interact via hydrogen bonds, steric repulsions, and dispersion attractions. It is designed to give simple insights into the microscopic origins of water's properties.     

Brownian dynamics simulation of probe diffusion in DNA: effects of probe size, charge and DNA concentration

We have used Brownian dynamics simulation to study probe diffusion in solutions of short chain DNA using our previously developed simulation algorithm. We have examined the effect of probe size, charge, and DNA concentration on the probe diffusion coefficient, with the aim of gaining insight into the diffusion of proteins in a concentrated DNA environment. In these simulations, DNA was modeled as a worm-like chain of hydrodynamically equivalent spherical frictional elements while probe particles were modeled as spheres of given charge and hydrodynamic radius. The simulations allowed for both short range Lennard-Jones interactions and long ranged electrostatic interactions between charged particles. For uncharged systems, we find that the effects of probe size and DNA concentration on the probe diffusion coefficient are consistent with excluded volume models and we interpret our results in terms of both empirical scaling laws and the predictions of scaled particle theory. For charged systems, we observe that the effects of probe size and charge are most pronounced for the smallest probes and interpret the results in terms of the probe charge density. For an ionic strength of 0.1 M we find that, below a critical probe surface charge density, the probe diffusion coefficient is largely independent of probe charge and only weakly dependent on the DNA charge. These effects are discussed in terms of the interactions between the probe and the DNA matrix and are interpreted in terms of both the underlying physics of transport in concentrated solutions and the assumptions of the simulation model.     

Restricted diffusion in biophysical systems: Theory

Theoretical results are presented on measurements of restricted diffusion in biophysical systems by the pulsed gradient spin echo nuclear magnetic resonance (PGSENMR) technique. A Fokker-Planck equation is developed to describe restricted diffusion, and it is shown that only two basic types of penetrable diffusion barriers exist, those in which the diffusing particles are partially excluded from the barrier region because of an increased free energy, and those in which the diffusing particles are not excluded but experience increased viscosity in the region. The Fokker-Planck equation is used to obtain expressions for the spin echo amplitude in PGSENMR experiments, and it is shown that for restricted diffusion the average diffusion coefficient measured in these experiments over short intervals is larger than that measured over long intervals. The possibility of distinguishing between the two types of barriers is considered. The experimental parameters required for intracellular restricted diffusion measurements are discussed, and it is shown that the interpretation of PGSENMR results in animal tissues should include the possibility of penetrable barriers rather than just the impenetrable barriers of previous PGSENMR calculations.     

Microscopic diffusion and hydrodynamic interactions of hemoglobin in red blood cells

The cytoplasm of red blood cells is congested with the oxygen storage protein hemoglobin occupying a quarter of the cell volume. The high protein concentration leads to a reduced mobility; the self-diffusion coefficient of hemoglobin in blood cells is six times lower than in dilute solution. This effect is generally assigned to excluded volume effects in crowded media. However, the collective or gradient diffusion coefficient of hemoglobin is only weakly dependent on concentration, suggesting the compensation of osmotic and friction forces. This would exclude hydrodynamic interactions, which are of dynamic origin and do not contribute to the osmotic pressure. Hydrodynamic coupling between protein molecules is dominant at short time- and length scales before direct interactions are fully established. Employing neutron spin-echo-spectroscopy, we study hemoglobin diffusion on a nanosecond timescale and protein displacements on the scale of a few nanometers. A time- and wave-vector dependent diffusion coefficient is found, suggesting the crossover of self- and collective diffusion. Moreover, a wave-vector dependent friction function is derived, which is a characteristic feature of hydrodynamic interactions. The wave-vector and concentration dependence of the long-time self-diffusion coefficient of hemoglobin agree qualitatively with theoretical results on hydrodynamics in hard spheres suspensions. Quantitative agreement requires us to adjust the volume fraction by including part of the hydration shell: Proteins exhibit a larger surface/volume ratio compared to standard colloids of much larger size. It is concluded that hydrodynamic and not direct interactions dominate long-range molecular transport at high concentration.     

Energetics of cell-cell and cell-biopolymer interactions

The energy vs distance balance of cell suspensions (in the presence and in the absence of extracellular biopolymer solutions) is studied, not only in the light of the classical Derjaguin-Landau-Verwey-Overbeek (DLVO) theory (which considered just the electrostatic (EL) and Lifshitz-van der Waals (LW) interactions), but also by taking electron-acceptor/electron-donor, or Lewis acid-base (AB) and osmotic (OS) interactions into account. Since cell surfaces, as well as many biopolymers tend to have strong monopolar electron-donor properties, they are able to engage in a strong mutual AB repulsion when immersed in a polar liquid such as water. The effects of that repulsion have been observed earlier in the guise of hydration pressure. The AB repulsion is, at close range, typically one or two orders of magnitude stronger than the EL repulsion, but its rate of decay is much steeper. In most cases, AB interactions are quantitatively the dominant factor in cell stability (when repulsive) and in "hydrophobic interactions" (when attractive). OS interactions exerted by extracellularly dissolved biopolymers are weak, but their rate of decay is very gradual, so OS repulsions engendered by biopolymer solutions may be of importance in certain long-range interactions. OS interactions exerted by biopolymers attached to cells or particles (e.g., by glycocalix glycoproteins), are very short-ranged and usually are negligibly small in comparison with the other interaction forces, in aqueous media.     

Potential influence of tubicolous worms on the bottom roughness length z0 in the south-western Baltic Sea

The influence of three different species of tubicolous worms (Pygospio elegans, Polydora ciliata and Lagis koreni) on the hydrodynamic bottom roughness length (z₀) was analysed in this study. Flume experiments and geospatial methods were combined to determine the potential interactions between worm tubes and the near-bed flow regime and the resulting effects on sediment transport in the south-western Baltic Sea. The three selected species are common in the area of interest (3539 km²). Their species-specific population densities were taken from existing macrozoobenthos datasets and transferred into a Geographic Information System (GIS). In analogy to the sediment roughness length, the hydrodynamic roughness lengths generated by the tubicolous worms were calculated and corresponding sediment transport values, derived from flume experiments with artificial tube lawns, were geospatially analysed using GIS. In order to show the direct influence of worm tubes on the surrounding sediment surface flume experiments were conducted at two given current velocities of 20 cm s^− 1 for sediment displacement effects and 5 cm s^− 1 for deposition effects. The roughness length was shown to increase by a factor of 2 to 30 in the presence of biogenic structures such as the worm tubes. The near-bed hydrodynamic conditions are significantly influenced at low roughness densities through independent or isolated flow conditions at 0.7 to 1.9% and at high roughness densities between 4.2 and 7.5%, resulting in unaffected sediment surfaces through “skimming flow”, as well. The GIS analysis revealed that this effect occurs over 4% (137 km²) of the area of investigation, whereas sediment displacement at roughness densities between 0.7 and 1.9% due to increased turbulence is the predominant effect over 33% (present on 1172 km²) of the area of investigation. These findings reveal the important influence of species-generated microtopography on sediment transport processes.     

Energetics of cell-cell and cell-biopolymer interactions

The energy vs distance balance of cell suspensions (in the presence and in the absence of extracellular biopolymer solutions) is studied, not only in the light of the classical Derjaguin-Landau-Verwey-Over-beek (DLVO) theory (which considered just the electrostatic (EL) and Lifshitz-van der Waals (LW) interactions), but also by taking electron-acceptor/electron-donor, or Lewis acid-base (AB) and osmotic (OS) interactions into account. Since cell surfaces, as well as many biopolymers tend to have strong monopolar electron-donor properties, they are able to engage in a strong mutual AB repulsion when immersed in a polar liquid such as water. The effects of that repulsion have been observed earlier in the guise of hydration pressure. The AB repulsion is, at close range, typically one or two orders of magnitude stronger than the EL repulsion, but its rate of decay is much steeper. In most cases, AB interactions are quantitatively the dominant factor in cell stability (when repulsive) and in “hydrophobic interactions” (when attractive). OS interactions exerted by extracellularly dissolved biopolymers are weak, but their rate of decay is very gradual, so OS repulsions engendered by biopolymer solutions may be of importance in certain long-range interactions. OS interactions exerted by biopolymers attached to cells or particles (e.g., by glycocalix glycoproteins), are very short-ranged and usually are negligibly small in comparison with the other interaction forces, in aqueous media.