Photon and fluorescence correlation spectroscopy and light scattering of eye-lens proteins at moderate concentrations

The bovine eye-lens protein, alpha L-crystallin, has been studied with photon correlation spectroscopy to obtain the mutual diffusion coefficient, Dm, with fluorescence correlation spectroscopy to determine the tracer diffusion coefficient, DT, and with light scattering to get the isothermal osmotic compressibility (delta pi/delta c) P,T. The concentration dependence of Dm, DT, and (delta pi/delta c) P,T up to a volume fraction phi of the protein of 2.5 x 10(-2) has been interpreted on the basis of four different interaction potentials: (a) an extended hard-sphere potential; (b) a shielded Coulomb potential; (c) a shielded Coulomb interaction where the effect of counterions is included; (d) a simple mixed potential. The three parameters Dm, DT, and (delta pi/delta c) P,T have also been combined in the generalized Stokes-Einstein equation, Dm = [(delta pi/delta c)P,T . (1--phi) . (DT)]/(kappa B . T). Our results indicate that, in the case that photon correlation spectroscopy gives the mutual diffusion coefficient Dm, the applicability of the Stokes-Einstein equation can be questioned; or that, when one assumes the Stokes-Einstein equation to be valid, there is significant discrepancy between the result of photon correlation spectroscopy and Dm.     

The concentration dependence of the hemoglobin mutual diffusion coefficient

Laser correlation Spectroscopy was used to measure the mutual diffusion coefficient, D, of human cyanomethemoglobin (Fe⁺⁺⁺:CN) at varying protein concentrations. These measurements were male at 20°C in a 0.1 M phosphate buffer solution at pH 7.0. For low protein concentrations we find D = (6.43 ± 0.26) × 10^?7 cm²/S and that there is a near linear decrease from this value at higher concentrations. The linear relation between the diffusion coefficient and protein concentration allows us to deduce the value of the linear frictional volume fraction coefficient, K_f= 7.75. and to extrapolate to hemoglobin concentrations equivalent to that in the red blood cell where we estimate D = 4.25 × 10^?7 cm²/s Various theoretical predictions of the dependence of the mutual diffusion coefficient on concentration are tested; we find that the generalized Stokes-Einstein relation can be made to fit our high concentration data if we assume a hard-sphere model and if we include a term involving a hydrodynamic interaction integral.     

Depletion theory of protein transport in semi-dilute polymer solutions

We consider the effect of polymer depletion on the transport (diffusion and electrophoresis) of small proteins through semi-dilute solutions of a flexible polymer. A self-consistent field theory may be set up in the important case of quasi-ideal interactions when the protein is small enough. Dynamic depletion, the reorganization of the depletion layer as the protein diffuses, is computed within a free-draining approximation. The transport of the dressed particle (protein + depletion layer) is tackled by extending Ogston's analysis of probe diffusion through fibrous networks to the case of a probe diffusing through a semi-dilute polymer inhomogeneous on the scale of the polymer correlation length. The resulting exponential retardation agrees almost quantitatively with that found in recent electrophoresis experiments of small proteins in polymer solutions that have been ascertained to be semi-dilute (S. P. Radko and A. Chrambach, Electrophoresis, 17:1094-1102, 1996; Biopolymers, 4:183-189, 1997).     

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.     

Diffusion of oxygen in water and hydrocarbons using an electron spin resonance spin-label technique.

The Smoluchowski equation for the bimolecular collision rate of dissolved oxygen molecules with spin labels yielded values for the diffusion constant of oxygen in water that are in agreement with the Stokes-Einstein equation (D infinity T/eta, where eta is the macroscopic viscosity) and with published values obtained by conventional methods. Heisenberg exchange at an interaction distance of 4.5 A occurs with a probability close to one for each encounter. In mixed hydrocarbons (olive oil, paraffin oils) and sec-butyl benzene, D infinity (T/eta)rho, where rho lies between 0.5 and 1. Oxygen diffuses in the hydrocarbons between 10 and 100 times more rapidly than predicted from the macroscopic viscosity. Similar results would be expected for diffusion of oxygen in model and biological membranes. Parallel measurements of rotational diffusion of the spin labels show little correlation with measurements of translational diffusion of oxygen. Dipolar interactions between spin labels and oxygen appear negligible except in the limit of highest viscosities.     

Particle Transport through Hydrogels Is Charge Asymmetric

Transport processes within biological polymer networks, including mucus and the extracellular matrix, play an important role in the human body, where they serve as a filter for the exchange of molecules and nanoparticles. Such polymer networks are complex and heterogeneous hydrogel environments that regulate diffusive processes through finely tuned particle-network interactions. In this work, we present experimental and theoretical studies to examine the role of electrostatics on the basic mechanisms governing the diffusion of charged probe molecules inside model polymer networks. Translational diffusion coefficients are determined by fluorescence correlation spectroscopy measurements for probe molecules in uncharged as well as cationic and anionic polymer solutions. We show that particle transport in the charged hydrogels is highly asymmetric, with diffusion slowed down much more by electrostatic attraction than by repulsion, and that the filtering capability of the gel is sensitive to the solution ionic strength. Brownian dynamics simulations of a simple model are used to examine key parameters, including interaction strength and interaction range within the model networks. Simulations, which are in quantitative agreement with our experiments, reveal the charge asymmetry to be due to the sticking of particles at the vertices of the oppositely charged polymer networks.     

Temperature dependence of macromolecular interactions in dilute and concentrated hemoglobin solutions

We studied the temperature-dependent effects of intramolecular interactions on the mutual diffusion coefficient of normal human oxygenated hemoglobin in salt solution. We used photon correlation spectroscopy to observe this temperature dependence of the mutual diffusion coefficient of two protein concentrations (1.25 and 17.0 g %) between 13 and 37°C. This coefficient was our probe for monitoring temperature-dependent structural changes of hydrated hemoglobin in solution. Comparison of our measured diffusion coefficient with that predicted by the Stokes-Einstein relationship in terms of solvent or solution viscosity showed a clear transition in the conformation of hemoglobin at approximately 22°C, independent of the hemoglobin concentration. We postulated that at this physiological temperature, a considerable quaternary rearrangement of the hemoglobin chains takes place. We believe this rearrangement changes the effective volume and the hydration sphere of the hemoglobin macromolecule.     

Monomolecular assembly of siRNA and poly(ethylene glycol)-peptide copolymers

In this work, we design and investigate the complex formation of highly uniform monomolecular siRNA complexes utilizing block copolymers consisting of a cationic peptide moiety covalently bound to a poly(ethylene glycol) (PEG) moiety. The aim of the study was to design a shielded siRNA construct containing a single siRNA molecule to achieve a sterically stabilized complex with enhanced diffusive properties in macromolecular networks. Using a 14 lysine-PEG (K14-PEG) linear diblock copolymer, formation of monomolecular siRNA complexes with a stoichiometric 1:3 grafting density of siRNA to PEG is realized. Alternatively, similar PEGylated monomolecular siRNA particles are achieved through complexation with a graft copolymer consisting of six cationic peptide side chains bound to a PEG backbone. The hydrodynamic radii of the resulting complexes as measured by fluorescence correlation spectroscopy (FCS) were found to be in good agreement with theoretical predictions using polymer brush scaling theory of a PEG decorated rodlike molecule. It is furthermore demonstrated that the PEG coating of the siRNA-PEG complexes can be rendered biodegradable through the use of a pH-sensitive hydrazone or a reducible disulfide bond linker between the K14 and the PEG blocks. To model transport under in vivo conditions, diffusion of these PEGylated siRNA complexes is studied in various charged and uncharged matrix materials. In PEG solutions, the diffusion coefficient of the siRNA complex is observed to decrease with increasing polymer concentration, in agreement with theory of probe diffusion in semidilute solutions. In charged networks, the behavior is considerably more complex. FCS measurements in fibrin gels indicate complete dissociation of the diblock copolymer from the complex, while transport in collagen solutions results in particle aggregation.     

Diffusion of macromolecules in agarose gels: comparison of linear and globular configurations.

The diffusion coefficients (D) of different types of macromolecules (proteins, dextrans, polymer beads, and DNA) were measured by fluorescence recovery after photobleaching (FRAP) both in solution and in 2% agarose gels to compare transport properties of these macromolecules. Diffusion measurements were conducted with concentrations low enough to avoid macromolecular interactions. For gel measurements, diffusion data were fitted according to different theories: polymer chains and spherical macromolecules were analyzed separately. As chain length increases, diffusion coefficients of DNA show a clear shift from a Rouse-like behavior (DG congruent with N0-0.5) to a reptational behavior (DG congruent with N0-2.0). The pore size, a, of a 2% agarose gel cast in a 0.1 M PBS solution was estimated. Diffusion coefficients of the proteins and the polymer beads were analyzed with the Ogston model and the effective medium model permitting the estimation of an agarose gel fiber radius and hydraulic permeability of the gels. Not only did flexible macromolecules exhibit greater mobility in the gel than did comparable-size rigid spherical particles, they also proved to be a more useful probe of available space between fibers.     

Viscosity of concentrated solutions and of human erythrocyte cytoplasm determined from NMR measurement of molecular correlation times. The dependence of viscosity on cell volume

Metabolically active human erythrocytes were incubated with [alpha-13C]glycine which led to the specific enrichment of intracellular glutathione. The cells were then studied using 13C-NMR in which the longitudinal relaxation times (T1) and nuclear Overhauser enhancements of the free glycine and glutathione were measured. The T1 values of labelled glycine were also determined in various-concentration solutions of bovine serum albumin and glycerol and also of the natural abundance 13C of glycerol in glycerol solutions. From the T1 estimates the rotational correlation time (tau r) was calculated using a formula based on a model of an isotropic spherical rotor or that of a symmetrical ellipsoidal rotor; for glycine the differences in estimates of tau r obtained using the two models were not significant. From the correlation times and by use of the Stokes-Einstein equations viscosity and translational diffusion coefficients were calculated; thus comment can be made on the likelihood of diffusion control of certain enzyme-catalysed reactions in the erythrocyte. Bulk viscosities of the erythrocyte cytoplasm and the above-mentioned solutions were measured using Ostwald capillary viscometry. Large differences existed between the latter viscosity estimates and those based upon NMR-T1 measurements. We derived an equation from the theory of the viscosity of concentrated solutions which contains two phenomenological interaction parameters, a 'shape' factor and a 'volume' factor; it was fitted to data relating to the concentration dependence of viscosity measured by both methods. We showed, by using the equation and interaction-parameter estimates for a particular probe molecule in a particular solution, that it was possible to correlate NMR viscosity and bulk viscosity; in other words, given an estimate of the bulk viscosity, it was possible to calculate the NMR 'micro' viscosity or vice versa. However, the values of the interaction parameters depend upon the relative sizes of the probe and solute molecules and must be separately determined for each probe-solute-solvent system. Under various conditions of extracellular osmotic pressure, erythrocytes change volume and thus the viscosity of the intracellular milieu is altered. The volume changes resulted in changes in the T1 of [alpha-13C]glycine. Conversely, we showed that alterations in T1, when appropriately calibrated, could be used for monitoring changes in volume of metabolically active cells.     

Dynamic light scattering by kappa- and lambda-carrageenan solutions

The intensity correlation functions of kappa- and lambda-carrageenan in various salt solutions and at different concentrations have been determined with the help of dynamic light scattering. From the first cumulant of these correlation functions the values of the translational diffusion coefficients D have been derived. They increase with macromolecular concentration. The extrapolated values to infinite dilution of the diffusion coefficients increase with increasing salt concentration as expected from the salt concentration dependence of the r.m.s. radii of gyration determined previously by static light scattering. The translational diffusion coefficient of lambda-carrageenan in 0.1 M NaCl is smaller than the corresponding value for the kappa species. This is consistent with the difference in contour length and linear charge density of the two samples used. No satisfactory interpretation for the concentration dependence of the diffusion coefficient seems to be possible at present. Although current theories for the macromolecular and salt concentration dependence of D, taking into account charge effects, seem to be applicable, they do not allow for a consistent interpretation of the data. No specific difference between the solution behaviour of kappa- and lambda-carrageenan has been detected.     

F-actin, a model polymer for semiflexible chains in dilute, semidilute, and liquid crystalline solutions.

Single actin filaments were analyzed in solutions ranging from dilute (0.2 microgram/ml), where filaments interact only with solvent, to concentrations (4.0 mg/ml) at which F-actin forms a nematic phase. A persistence length of approximately 1.8 microns and an average length of approximately 22 microns (Kaufmann et al., 1992) identify actin as a model for studying the dynamics of semiflexible polymers. In dilute solutions the filaments exhibit thermal bending undulations in addition to diffusive motion. At higher semidilute concentrations (1.4 mg/ml) three-dimensional reconstructions of confocal images of fluorescently labeled filaments in a matrix of unlabeled F-actin reveal steric interactions between filaments, which account for the viscoelastic behavior of these solutions. The restricted undulations of these labeled chains reveal the virtual tube formed around a filament by the surrounding actin. The average tube diameter <a> scales with monomer concentration c as <a> varies; is directly proportional to c-(0.5 +/- 0.15). The diffusion of filaments in semidilute solutions (c = (0.1-2.0) mg/ml) is dominated by diffusion along the filament contour (reptation), and constraint release by remodeling of the surrounding filaments is rare. The self-diffusion coefficient D parallel along the tube decreases linearly with the chain length for semidilute solutions. For concentrations > 2.5 mg/ml a transition occurs from an isotropic entangled phase to a coexistence between isotropic and nematic domains. Analysis of the molecular motions of filaments suggests that the filaments in the aligned domains are in thermal equilibrium and that the diffusion coefficient parallel to the director D parallel is nearly independent of filament length. We also report the novel direct observation of u-shaped defects, called hairpins, in the nematic domains.     

Prediction of diffusion coefficients of proteins

A correlation for predicting the diffusion coefficients of proteins at standard conditions is proposed by adapting the Stokes-Einstein equation to a model for the equivalent hydrodynamic sphere. The radius of gyration, which accounts for the size and shape of protein molecules in solution, was used in deriving the new correlation. The correlation successfully predicted the diffusion coefficients of proteins, for which experimental data were available, with a higher degree of accuracy than achieved by previous correlation methods.     

Measurement of the translational mobility of concanavalin A in glycerol-saline solutions and on the cell surface by fluorescence recovery after photobleaching.

The fluorescence recovery kinetics of succinyl-fluorescein Concanavalin A (S-F-ConA) in glycerol-physiological saline solutions of high viscosity and when bound to the surface of mouse fibroblasts were measured following brief photobleaching using a laser excited fluorescence microscope. In the high viscosity solutions, the recovery kinetics, interpreted on the basis of a simple diffusion model, yielded a diffusion coefficient in close agreement with the values predicted by the Stokes-Einstein equation. Recovery kinetics for S-F-ConA bound to the surface of mouse 3T3 and SV3T3 cells cultured in vitro yielded diffusion coefficients in the range of 5-10-10(-11) cm2/s, values considerably lower than those reported previously for membrane proteins. These measurements indicated that a considerable fraction of the S-F-ConA molecules bound to the cell surface are immobilized. These results are discussed in relation to current concepts of lateral motion of protein components within natural membranes.     

Type I collagen fibrillogenesis: initiation via reversible linear and lateral growth steps

Type I collagen fibrillogenesis in vitro has been studied by laser light scattering, and the results indicate that initiation of aggregation involves at least two steps. Step I of aggregation involves no change in the intensity of scattered light at an angle of 90° and is accompanied by a decrease in the diffusion coefficient. Step II is characterized by an increased intensity of scattered light and decreased diffusion coefficients. Theoretical calculations using the Stokes-Einstein equation for the translational diffusion coefficient and the Perrin equation for the frictional coefficient of a prolate ellipsoid indicate that the step I aggregates are 4D staggered linear dimers and trimers 570 and 845 nm long, whereas step II aggregates are greater than 950 nm in length. These dimensions are similar to those previously reported based on physicochemical measurements and electron microscopy. It is proposed that the rate and extent of fibrillogenesis in vitro is controlled by the concentration of the linear aggregates and that the effects of temperature and collagen concentration on fibrillogenesis previously observed are qualitatively explained in terms of their effects on the concentration of these aggregates.     

Diagnostic of precipitant for biomacromolecule crystallization by quasi-elastic light-scattering

The translational diffusion coefficient D25,w of hen egg-white lysozyme and concanavalin A from the jack bean is measured in various precipitating agent solutions as a function of salt and protein concentration using quasi-elastic light-scattering. With some precipitants, in undersaturated protein solutions, a protein or salt concentration dependence of the diffusion coefficient of the scatters is observed. It can be correlated with the inability of the protein to crystallize in this precipitant once the solution is supersaturated. These variations of D25,w are interpreted in terms of non-specific interactions and/or aggregation that prevent the protein from making appropriate contacts to form a crystal. With other precipitants known to lead to crystallization, no significant variation of the diffusion coefficient with increasing concentration was observed, indicating that under such conditions up to saturation the proteins remain essentially monodisperse. Application of this technique to find crystallization conditions of other proteins is discussed.     

Quasi-elastic laser light scattering from solutions and gels of hemoglobin S.

Quasi-elastic light scattering has been used to examine solutions and gels of deoxyhemoglobin S. The autocorrelation function is found to decay with a characteristic exponential relaxation which can be ascribed to the diffusion of monomer (64,000 molecular weight) hemoglobin S molecules. In the absence of polymers, the relaxation time is in good agreement with previous measurements of the diffusion coefficient for solutions of normal human hemoglobin. In the presence of the polymer phase, a large (greater than 200-fold) increase in the scattered intensity is observed but no contribution to the decay of the autocorrelation function from the motion of the aligned polymer phase can be detected. Heterodyning between the time-independent scattering amplitude from the polymers and the time-dependent scattering of the diffusing monomers results in a twofold increase in the relaxation time arising from monomer diffusion.     

Obtaining the concentration-dependent diffusion coefficient from ultrafiltration experiments: Application to hyaluronate

A method for determining the concentration-dependent mutual diffusion coefficient D(C) of a macromolecule-solvent combination over a wide macromolecular concentration range is presented. All necessary data are gathered from a single experiment, in which polymer concentration profiles are measured during one-dimensional dead-end ultrafiltration. Based on these profiles, the convection-diffusion equation is used to deduce the dependence of D on C. To demonstrate the utility of the approach, studies using sodium hyaluronate dissolved in either 10 mM NaCl or a phosphate buffer were carried out. For hyaluronate (HA) in 10 mM NaCl, the mutual diffusion coefficient varies approximately linearly with concentration according to D(C) = 4.1 × 10⁻⁶ C^0.96 cm²/s in the range 0 ≤ C ≤ 0.6 mass %, for C expressed in mass %. However, transition points (slope changes) in the D(C) curve are present at C ≅ 0.7 mass % and C ≅ 1.4 mass %. For HA in the phosphate buffer, the mutual diffusion coefficient is well described by D(C) = 1.9 × 10⁻⁶ C^0.825, for 0 ≤ C ≤ 1.8 mass %. These values agree well with previously published data. The technique is robust, and permits reasonably high polymer concentrations to be easily studied. © 1996 John Wiley & Sons, Inc.     

An asymptotic solution for traveling waves of a nonlinear-diffusion Fisher's equation

We examine traveling-wave solutions for a generalized nonlinear-diffusion Fisher equation studied by Hayes [J. Math. Biol. 29, 531–537 (1991)]. The density-dependent diffusion coefficient used is motivated by certain polymer diffusion and population dispersal problems. Approximate solutions are constructed using asymptotic expansions. We find that the solution will have a corner layer (a shock in the derivative) as the diffusion coefficient approaches a step function. The corner layer at z = 0 is matched to an outer solution for z < 0 and a boundary layer for z > 0 to produce a complete solution. We show that this model also admits a new class of nonphysical solutions and obtain conditions that restrict the set of valid traveling-wave solutions. Supported by a National Science Foundation graduate fellowship. This work was performed under National Science Foundation grant DMS-9024963 and Air Force Office of Scientific Research grant AFOSR-F49620-94-1-0044.