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.     

Diffusion anisotropy in collagen gels and tumors: the effect of fiber network orientation

The interstitial matrix is comprised of cross-linked collagen fibers, generally arranged in nonisotropic orientations. Spatial alignment of matrix components within the tissue can affect diffusion patterns of drugs. In this study, we developed a methodology for the calculation of diffusion coefficients of macromolecules and nanoparticles in collagenous tissues. The tissues are modeled as three-dimensional, stochastic, fiber networks with varying degrees of alignment. We employed a random walk approach to simulate diffusion and a Stokesian dynamics method to account for hydrodynamic hindrance. We performed our analysis for four different structures ranging from nearly isotropic to perfectly aligned. We showed that the overall diffusion coefficient is not affected by the orientation of the network. However, structural anisotropy results in diffusion anisotropy, which becomes more significant with increase in the degree of alignment, the size of the diffusing particle, and the fiber volume fraction. To test our model predictions we performed diffusion measurements in reconstituted collagen gels and tumor xenografts. We measured fiber alignment and diffusion with second harmonic generation and multiphoton fluorescent recovery after photobleaching techniques, respectively. The results showed for the first time in tumors that the structure and orientation of collagen fibers in the extracellular space leads to diffusion anisotropy.     

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.     

Effector cells which mediate antibody-dependent cell-mediated cytotoxicity. II. Ontogeny of effector activity in murine spleen.

The specificity of T cells for syngeneic target cells is directed to both antigens and products of the major histocompatibility complex (MHC) on the target cell surface. This dual requirement is best accounted for by the altered-self hypothesis, which implies that the MHC products on a cell's surface are able to form complexes with many other proteins on the surface of the same cell. To account for the ability of MHC products to bind so many different cell surface antigens we propose that interactions in general among macromolecules on the surface of a membrane may be dramatically enhanced by a purely physical effect. This effect derives from the confinement of membrane macromolecules to an effective volume which is the product of membrane surface area times d, the distance over which the center of mass of the molecules can move in a vertical direction (perpendicular to the membrane surface). Because d is very small the effective concentrations of surface molecules are extremely high and their interactions are correspondingly enhanced.     

A physics-based approach of coarse-graining the cytoplasm of Escherichia coli (CGCYTO)

We have investigated protein stability in an environment of Escherichia coli cytoplasm using coarse-grained computer simulations. To coarse-grain a small slide of E. coli's cytoplasm consisting of over 16 million atoms, we have developed a self-assembled clustering algorithm (CGCYTO). CGCYTO uses the shape parameter and asphericity as well as a parameter λ (ranging from 0 to 1) that measures the covolume of a test protein and a macromolecule against the covolume of a test protein and a sphere of equal volume as that of a macromolecule for the criteria of coarse-graining a cytoplasmic model. A cutoff λ_c = 0.8 was chosen based on the size of a test protein and computational resources and it determined the resolution of a coarse-grained cytoplasm. We compared the results from a polydisperse cytoplasmic model (PD model) produced by CGCYTO with two other coarse-grained hard-sphere cytoplasmic models: 1), F70 model, macromolecules in the cytoplasm were modeled by homogeneous hard spheres with a radius of 55 Å, the size of Ficoll70 and 2), HS model, each macromolecule in the cytoplasm was modeled by a hard sphere of equal volume. It was found that the folding temperature T_f of a test protein (apoazurin) in a PD model is ～5° greater than that in a F70 model. In addition, the deviation of T_f in a PD model is twice as much as that in a HS model when an apoazurin is randomly placed at different voids formed by particle fluctuations in PD models.     

Bound-State Diffusion due to Binding to Flexible Polymers in a Selective Biofilter

Selective biofilters are used by cells to control the transport of proteins, nucleic acids, and other macromolecules. Biological filters demonstrate both high specificity and rapid motion or high flux of proteins. In contrast, high flux comes at the expense of selectivity in many synthetic filters. Binding can lead to selective transport in systems in which the bound particle can diffuse, but the mechanisms that lead to bound diffusion remain unclear. Previous theory has proposed a molecular mechanism of bound-state mobility based only on transient binding to flexible polymers. However, this mechanism has not been directly tested in experiments. We demonstrate that bound mobility via tethered diffusion can be engineered into a synthetic gel using protein fragments derived from the nuclear pore complex. The resulting bound-state diffusion is quantitatively consistent with theory. Our results suggest that synthetic biological filters can be designed to take advantage of tethered diffusion to give rapid, selective transport.     

Molecular crowding affects diffusion and binding of nuclear proteins in heterochromatin and reveals the fractal organization of chromatin

The nucleus of eukaryotes is organized into functional compartments, the two most prominent being heterochromatin and nucleoli. These structures are highly enriched in DNA, proteins or RNA, and thus thought to be crowded. In vitro, molecular crowding induces volume exclusion, hinders diffusion and enhances association, but whether these effects are relevant in vivo remains unclear. Here, we establish that volume exclusion and diffusive hindrance occur in dense nuclear compartments by probing the diffusive behaviour of inert fluorescent tracers in living cells. We also demonstrate that chromatin‐interacting proteins remain transiently trapped in heterochromatin due to crowding induced enhanced affinity. The kinetic signatures of these crowding consequences allow us to derive a fractal model of chromatin organization, which explains why the dynamics of soluble nuclear proteins are affected independently of their size. This model further shows that the fractal architecture differs between heterochromatin and euchromatin, and predicts that chromatin proteins use different target‐search strategies in the two compartments. We propose that fractal crowding is a fundamental principle of nuclear organization, particularly of heterochromatin maintenance.     

Better Cryo-EM Specimen Preparation: How to Deal with the Air–Water Interface?

Cryogenic electron microscopy (cryo-EM) is now one of the most powerful and widely used methods to determine high-resolution structures of macromolecules. A major bottleneck of cryo-EM is to prepare high-quality vitrified specimen, which still faces many practical challenges. During the conventional vitrification process, macromolecules tend to adsorb at the air–water interface (AWI), which is known unfriendly to biological samples. In this review, we outline the nature of AWI and the problems caused by it, such as unpredictable or uneven particle distribution, protein denaturation, dissociation of complex and preferential orientation. We review and discuss the approaches and underlying mechanisms to deal with AWI: 1) Additives, exemplified by detergents, forming a protective layer at AWI and thus preserving the native folds of target macromolecules. 2) Fast vitrification devices based on the idea to freeze in-solution macromolecules before their touching of AWI. 3) Thin layer of continuous supporting films to adsorb macromolecules, and when functionalized with affinity ligands, to specifically anchor the target particles away from the AWI. Among these supporting films, graphene, together with its derivatives, with negligible background noise and mechanical robustness, has emerged as a new generation of support. These strategies have been proven successful in various cases and enable us a better handling of the problems caused by the AWI in cryo-EM specimen preparation.     

Sizing particles with a Coulter counter

A theorem is presented which permits a determination of the amplitude of the signal generated by the passage of a particle of arbitrary shape through a Coulter counter. The theorem is applied to particles of two shapes, a sphere and a prolate spheroid. For the sphere the signal is directly proportional to the volume of the particle. For the spheroid the result is a complicated function of the shape. Two spheroids of the same volume but different shapes will give different signals.     

Treatment of mobile phase particulate matter in low-angle quasi-elastic light scattering

The method of quasi-elastic laser light scattering (QLS), particularly at low forward scattering angles, has been complicated by the transient presence of Mie or large Rayleigh scattering particles which contaminate the scattering volume. These large contaminating particles have substantial effects on photon correlation spectroscopy because the presence of these larger scatterers tends to decrease the value of the apparent diffusion coefficient of the particle of interest. A method is presented which yields more accurate diffusion constants by autocorrelation of selected photon count periods representative of minimal Mie or large Rayleigh particle contamination. This method was applied to the determination of the apparent diffusion constant for four proteins—ovalbumin, chymotrypsinogen-A, bovine serum albumin, and ribonuclease-A.     

Partitioning and diffusion of proteins and linear polymers in polyacrylamide gels.

The equilibrium partition coefficient (K) and diffusion coefficient (Dgel) of two proteins and two linear polymers were measured as a function of polymer content of a 2.7% cross-linked polyacrylamide (PA) gel. The gel concentration, expressed as a volume percentage of PA in the gel (phi), varied between 0 and 14%. The measurements were made by fluorescence spectroscopy; fluorescent dyes were covalently attached to the macromolecules. The dependence of K on phi for the proteins agrees with a model of the gel network as randomly placed, impenetrable rods. The diffusion data are interpreted in terms of an effective medium theory for the mobility of a sphere in a Brinkman fluid. Using values of the Brinkman parameter in the literature, the effective medium model with no adjustable parameters fits the diffusion data for the proteins very well but underpredicts Dgel for the linear polymers. The gel effect on partitioning is significantly greater than that on diffusion. The permeability (KDgel) of bovine serum albumin decreased by 10(3) over the range phi = 0 --> 8%, and the ratio of permeabilities for ribonuclease compared to BSA increased from 2 to 30.     

Hydraulic conductivity of polymer matrices

We demonstrate that the chondroitin sulfate proteoglycan exhibits enhanced sensitivity to the flow of water compared to other macromolecules which is in accord with their functional role in conferring compressive resistance to cartilage. In order to understand factors that may contribute to its low hydraulic conductivity, a comparative study of hydraulic conductivity, as measured by the sedimentation velocity technique is made of various macromolecules representing variations in charge density, chemical composition, thermodynamic nonideality, size and flexibility. The polymers examined were dextran, poly(ethylene glycol), poly(vinyl alcohol), albumin, and dextran sulfate. The differences in hydraulic conductivity between the various macromolecules could not be explained by conventional theories which included prediction of hydraulic conductivity related to the radius of the molecule regarded as a uniform cylinder, nor the absolute charge density of the molecule and nor to the steric hindrance offered by the macromolecule to the diffusion of tritiated water. A qualitative relationship is established, however, between the noncounterion polymer contribution to osmotic activity and the resistance to water flow for polymers with high osmotic activity.     

Diffusion rate limitations in actin-based propulsion of hard and deformable particles

The mechanism by which actin polymerization propels intracellular vesicles and invasive microorganisms remains an open question. Several recent quantitative studies have examined propulsion of biomimetic particles such as polystyrene microspheres, phospholipid vesicles, and oil droplets. In addition to allowing quantitative measurement of parameters such as the dependence of particle speed on its size, these systems have also revealed characteristic behaviors such a saltatory motion of hard particles and oscillatory deformation of soft particles. Such measurements and observations provide tests for proposed mechanisms of actin-based motility. In the actoclampin filament end-tracking motor model, particle-surface-bound filament end-tracking proteins are involved in load-insensitive processive insertion of actin subunits onto elongating filament plus-ends that are persistently tethered to the surface. In contrast, the tethered-ratchet model assumes working filaments are untethered and the free-ended filaments grow as thermal ratchets in a load-sensitive manner. This article presents a model for the diffusion and consumption of actin monomers during actin-based particle propulsion to predict the monomer concentration field around motile particles. The results suggest that the various behaviors of biomimetic particles, including dynamic saltatory motion of hard particles and oscillatory vesicle deformations, can be quantitatively and self-consistently explained by load-insensitive, diffusion-limited elongation of (+)-end-tethered actin filaments, consistent with predictions of the actoclampin filament-end tracking mechanism.     

A model of effective diffusion and tortuosity in the extracellular space of the brain

Tortuosity of the extracellular space describes hindrance posed to the diffusion process by a geometrically complex medium in comparison to an environment free of any obstacles. Calculating tortuosity in biologically relevant geometries is difficult. Yet this parameter has proved very important for many processes in the brain, ranging from ischemia and osmotic stress to delivery of nutrients and drugs. It is also significant for interpretation of the diffusion-weighted magnetic resonance data. We use a volume-averaging procedure to obtain a general expression for tortuosity in a complex environment. A simple approximation then leads to tortuosity estimates in a number of two-dimensional (2D) and three-dimensional (3D) geometries characterized by narrow pathways between the cellular elements. It also explains the counterintuitive fact of lower diffusion hindrance in a 3D environment. Comparison with Monte Carlo numerical simulations shows that the model gives reasonable tortuosity estimates for a number of regular and randomized 2D and 3D geometries. Importantly, it is shown that addition of dead-end pores increases tortuosity in proportion to the square root of enlarged total extracellular volume fraction. This conclusion is further supported by the previously described tortuosity decrease in ischemic brain slices where dead-end pores were partially occluded by large macromolecules introduced into the extracellular space.     

Differential effects of severe water stress on linear and cyclic electron fluxes through Photosystem I in spinach leaf discs in CO<Subscript>2</Subscript>-enriched air

Linear and cyclic electron fluxes through Photosystem I in 1% CO(2) were quantified in spinach leaf tissue under severe water stress. Using actinic light with a peak at 697 nm for preferential light absorption by Photosystem I while also stimulating Photosystem II to improve redox poising, the cyclic electron flux after 60 s of illumination was a substantial proportion (33-44%) of the total electron flux through PSI at irradiances up to ~1,070 mumol photons m(-2) s(-1). At the maximum irradiance, the cyclic electron flux changed little with the progressive water loss from leaf tissue up to ~60%; by contrast, the linear electron flux was approximately halved. A reason for this differential effect of water stress on the capacity for cyclic and linear electron flow could be the increased crowding of soluble proteins in the stroma due to chloroplast shrinkage. Indeed the confinement of soluble proteins to a smaller chloroplast volume was indicated by cryo-scanning electron microscopy. It is known that the diffusion coefficient of large proteins is decreased when the background concentration of small proteins is raised; by contrast, the diffusion coefficient of small proteins is not affected by increasing the concentration of a large protein (Muramatsu and Minton in Proc Natl Acad Sci USA 85:2984-2988, 1988). Therefore, we suggest that linear electron flow, being coupled to the Calvin-Benson cycle, is limited by the diffusion of large macromolecules, especially the ribulose 1, 5-bisphosphate carboxylase/oxygenase complex. By contrast, cyclic electron flow, involving relatively small macromolecules such as ferredoxin, is less susceptible to inhibition by crowding in the stroma.     

Diffusion behavior of lipid vesicles in entangled polymer solutions.

Dynamic light scattering was used to follow the tracer diffusion of phospholipid/cholesterol vesicles in aqueous polyacrylamide solutions and compared with the diffusive behavior of polystyrene (PS) latex spheres of comparable diameters. Over the range of the matrix concentration examined (Cp = 0.1-10 mg/ml), the diffusivities of the PS spheres and the large multilamellar vesicles exhibited the Stokes-Einstein (SE) relation, while the diffusivity of the unilamellar vesicles did not follow the increase of the solution's viscosity caused by the presence of the matrix molecules. The difference between the diffusion behaviors of unilamellar vesicles and hard PS spheres of similar size is possibly due to the flexibility of the lipid bilayer of the vesicles. The unilamellar vesicles are capable of changing their shape to move through the entangled polymer solution so that the hindrance to their diffusion due to the presence of the polymer chains is reduced, while the rigid PS spheres have little flexibility and they encounter greater resistance. The multilamellar vesicles are less flexible, thus their diffusion is similar to the hard PS spheres of similar diameter.     

Calculation of the solution properties of flexible macromolecules: methods and applications

While the prediction of hydrodynamic properties of rigid particles is nowadays feasible using simple and efficient computer programs, the calculation of such properties and, in general, the dynamic behavior of flexible macromolecules has not reached a similar situation. Although the theories are available, usually the computational work is done using solutions specific for each problem. We intend to develop computer programs that would greatly facilitate the task of predicting solution behavior of flexible macromolecules. In this paper, we first present an overview of the two approaches that are most practical: the Monte Carlo rigid-body treatment, and the Brownian dynamics simulation technique. The Monte Carlo procedure is based on the calculation of properties for instantaneous conformations of the macromolecule that are regarded as if they were instantaneously rigid. We describe how a Monte Carlo program can be interfaced to the programs in the HYDRO suite for rigid particles, and provide an example of such calculation, for a hypothetical particle: a protein with two domains connected by a flexible linker. We also describe briefly the essentials of Brownian dynamics, and propose a general mechanical model that includes several kinds of intramolecular interactions, such as bending, internal rotation, excluded volume effects, etc. We provide an example of the application of this methodology to the dynamics of a semiflexible, wormlike DNA.     

Rotational diffusion coefficients of a small,spherical subunit flexibly tethered to a larger sphere

The rotational diffusion coefficients of a small spherical particle, which is flexibly anchored to the surface of a much larger sphere, are calculated using the hydrodynamic theory of segmentally flexible particles. The model is intended for representing the rotational mobility of a small residue or chromophore in the surface of a globular macromolecule. The coefficients are found to be essentially independent, or to vary slowly with the relative dispositions of the spheres. They are also insensitive to the size ratio when this ratio is high enough. These findings support the use of an approximative treatment proposed by Wegener in which the small conformation dependence is averaged out. The resulting averages are tentatively used in the Lipari-Szabo model for restricted rotational diffusion in a cone. It is concluded that the rotational relaxation of the small sphere has three components: (i) a torsional rotation with the same diffusion coefficient as the free sphere; (ii) a perpendicular wobbling with a diffusion coefficient several (five in a typical case) times smaller; and (iii) an overall rotation of the whole macromolecule, that will appear in a much longer time scale if the two spheres have quite distinct sizes.