Accounting for diffusion in agent based models of reaction-diffusion systems with application to cytoskeletal diffusion

Diffusion plays a key role in many biochemical reaction systems seen in nature. Scenarios where diffusion behavior is critical can be seen in the cell and subcellular compartments where molecular crowding limits the interaction between particles. We investigate the application of a computational method for modeling the diffusion of molecules and macromolecules in three-dimensional solutions using agent based modeling. This method allows for realistic modeling of a system of particles with different properties such as size, diffusion coefficients, and affinity as well as the environment properties such as viscosity and geometry. Simulations using these movement probabilities yield behavior that mimics natural diffusion. Using this modeling framework, we simulate the effects of molecular crowding on effective diffusion and have validated the results of our model using Langevin dynamics simulations and note that they are in good agreement with previous experimental data. Furthermore, we investigate an extension of this framework where single discrete cells can contain multiple particles of varying size in an effort to highlight errors that can arise from discretization that lead to the unnatural behavior of particles undergoing diffusion. Subsequently, we explore various algorithms that differ in how they handle the movement of multiple particles per cell and suggest an algorithm that properly accommodates multiple particles of various sizes per cell that can replicate the natural behavior of these particles diffusing. Finally, we use the present modeling framework to investigate the effect of structural geometry on the directionality of diffusion in the cell cytoskeleton with the observation that parallel orientation in the structural geometry of actin filaments of filopodia and the branched structure of lamellipodia can give directionality to diffusion at the filopodia-lamellipodia interface.     

Size effects on diffusion processes within agarose gels

To investigate diffusion processes in agarose gel, nanoparticles with sizes in the range between 1 and 140 nm have been tested by means of fluorescence correlation spectroscopy. Understanding the diffusion properties in agarose gels is interesting, because such gels are good models for microbial biofilms and cells cytoplasm. The fluorescence correlation spectroscopy technique is very useful for such investigations due to its high sensitivity and selectivity, its excellent spatial resolution compared to the pore size of the gel, and its ability to probe a wide range of sizes of diffusing nanoparticles. The largest hydrodynamic radius (R(c)) of trapped particles that displayed local mobility was estimated to be 70 nm for a 1.5% agarose gel. The results showed that diffusion of particles in agarose gel is anomalous, with a diverging fractal dimension of diffusion when the large particles become entrapped in the pores of the gel. The latter situation occurs when the reduced size (R(A)/R(c)) of the diffusing particle, A, is >0.4. Variations of the fractal exponent of diffusion (d(w)) with the reduced particle size were in agreement with three-dimensional Monte Carlo simulations in porous media. Nonetheless, a systematic offset of d(w) was observed in real systems and was attributed to weak nonelastic interactions between the diffusing particles and polymer fibers, which was not considered in the Monte Carlo simulations.     

Active intranuclear movement of herpesvirus capsids

Although small molecules diffuse rapidly through the interphase nucleus, recent reports indicate that nuclear diffusion is limited for particles that are larger than 100 nm in diameter. Given the apparent size limits to nuclear diffusion, there is some debate as to whether the movement of large particles should be attributed to diffusion or to active transport. Here, we show that 125 nm-diameter herpes simplex virus 1 (HSV-1) capsids are actively transported within infected nuclei. Movement is directed, temperature- and energy-dependent, sensitive to the putative myosin inhibitor 2,3-butanedione monoxime (BDM) and to actin depolymerization with latrunculin-A, but insensitive to actin depolymerization with cytochalasin-D.     

Biological particles of given size,shape, and density for use in biological reactors

Novel biomass support particles containing growing cells have been developed for use in large-scale fermentation processes. The characteristic size of the entrapped biomass is identical to that of the physical structure of the support particle, and particles can be produced of any size, shape, and density with a wide variety of microorganisms. Use of the particles in fermentors leads to high biomass concentration independent of throughput, predetermined biomass concentrations, the use of novel types of fermentor with advantageous performance characteristics, possibilities for the optimization of advantageous diffusion effects, and new procedures for biomass recovery.     

Aqueous diffusion pathways as a part of the ventricular cell ultrastructure

The physical organization of the ventricular myocyte includes barriers for the movement of objects of varying dimensions ranging from ions to solid particles. There are two kinds of diffusion in the cell: lateral (in membranes) and aqueous. Here we examine the size constraints of aqueous diffusion pathways and discuss their impact on cellular physiology. Calibrated gold nanoparticles were used to probe the accessibility of the entire transverse-axial tubular system (TATS), the sarcoplasm, and intracellular structures. The TATS tubules, although up to 300 nm in diameter, permitted only particles 3 nm; 3), the mitochondrial voltage-dependent anion channel and the nuclear pore complex in ventricular cells could not be penetrated by particles >/=6 nm; and 4), there is a difference in size clearance between transversal and longitudinal sarcoplasmic diffusional pathways.     

A pore‐hindered diffusion and reaction model can help explain the importance of pore size distribution in enzymatic hydrolysis of biomass

Until now, most efforts to improve monosaccharide production from biomass through pretreatment and enzymatic hydrolysis have used empirical optimization rather than employing a rational design process guided by a theory‐based modeling framework. For such an approach to be successful a modeling framework that captures the key mechanisms governing the relationship between pretreatment and enzymatic hydrolysis must be developed. In this study, we propose a pore‐hindered diffusion and kinetic model for enzymatic hydrolysis of biomass. When compared to data available in the literature, this model accurately predicts the well‐known dependence of initial cellulose hydrolysis rates on surface area available to a cellulase‐size molecule. Modeling results suggest that, for particles smaller than 5 × 10^?3 cm, a key rate‐limiting step is the exposure of previously unexposed cellulose occurring after cellulose on the surface has hydrolyzed, rather than binding or diffusion. However, for larger particles, according to the model, diffusion plays a more significant role. Therefore, the proposed model can be used to design experiments that produce results that are either affected or unaffected by diffusion. Finally, by using pore size distribution data to predict the biomass fraction that is accessible to degradation, this model can be used to predict cellulose hydrolysis with time using only pore size distribution and initial composition data. Biotechnol. Bioeng. 2013; 110: 127–136. © 2012 Wiley Periodicals, Inc.     

Probe diffusion in aqueous poly(vinyl alcohol) solutions studied by fluorescence correlation spectroscopy

We report fluorescence correlation spectroscopy measurements of the translational diffusion coefficient of various probe particles in dilute and semidilute aqueous poly(vinyl alcohol) solutions. The range of sizes of the particles (fluorescent molecules, proteins, and polymers) was chosen to explore various length scales of the polymer solutions as defined by the polymer-polymer correlation length. For particles larger than the correlation length, we find that the diffusion coefficient, D, decreases exponentially with the polymer concentration. This can be explained by an exponential increase in the solution viscosity, consistent with the Stokes-Einstein equation. For probes on the order of the correlation length, the decrease of the diffusion coefficient cannot be accounted for by the Stokes-Einstein equation, but can be fit by a stretched exponential, D approximately exp(-alphacn), where we find n = 0.73-0.84 and alpha is related to the probe size. These results are in accord with a diffusion model of Langevin and Rondelez (Polymer 1978, 19, 1875), where these values of n indicate a good solvent quality.     

DETACHMENT OF BACTERIOPHAGE FROM ITS CARRIER PARTICLES

The active substance (phage) present in the lytic broth filtrate is distributed through the medium in the form of particles. These particles vary in size within broad limits. The average size of these particles as calculated on the basis of the rate of diffusion approximates 4.4 mµ in radius. Fractionation by means of ultrafiltration permits partial separation of particles of different sizes. Under conditions of experiments here reported the particles varied in the radius size from 0.6 mµ to 11.4 mµ. The active agent apparently is not intimately identified with these particles. It is merely carried by them by adsorption, and under suitable experimental conditions it can be detached from the larger particles and redistributed on smaller particles of the medium.     

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.     

The Effect of Antisite Disorder and Particle Size on Li Intercalation Kinetics in Monoclinic LiMnBO3

In materials containing 1D lithium diffusion channels, cation disorder can strongly affect lithium intercalation processes. This work presents a model to explain the unusual transport properties of monoclinic LiMnBO₃, a material determined by scanning electron microscopy and synchrotron X‐ray diffraction to contain a wide particle size distribution and Mn/Li antisite disorder. First‐principles calculations indicate that Mn occupying Li sites obstruct the 1D lithium diffusion channel along the [001] direction. While channel blockage by the antisites significantly lowers Li mobility in large particles, Li kinetics in small particles and particle surfaces are found to be less sensitive to the presence of antisite disorder. Thus, in an electrode containing a large particle size distribution, smaller particles have higher Li mobility, and the measured Li diffusivity as determined by potentiostatic intermittent titration test varies as a function of particle size. The Li capacity in monoclinic LiMnBO₃ is kinetically controlled by the fraction of large particles with antisite disorder, but is not intrinsically limited. These results strongly suggest that particle nanosizing will significantly enhance the electrochemical performance of LiMnBO₃.     

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.     

Solute exclusion from cellulose in packed columns: process modeling and analysis

In this investigation, process modeling and analysis were used to explore the behavior of solute exclusion from cellulose in packed columns. The study focused on modeling the effects of dispersion, mass transport, and pore diffusion. Three mathematical models were used to predict the behavior of the columns: an equilibrium model, a mass transfer model, and a combined mass transfer and pore diffusion model. Computer implementations of these models were tested against experimental conditions where cellulose particle size and solution velocity were used to either amplify or minimize dispersion or skewness in the elution curves. For small cellulose particles (200-300 mesh), all three models accurately predicted the shape of the elution curve and the particle porosity. For larger particles (45-60 mesh), the mass transfer model and the combined mass and pore diffusion model best represented the behavior of the column. At high solution velocities (0.63 cm(3) min(-1)) and large particles, only the combined mass transfer and pore diffusion model accurately represent the column behavior. Sensitivity analysis revealed that the mass transfer coefficient had little effect on the elution curves for the range of values (10(-6)-10(-3) cm s(-1)) calculated from the experimental data. The combined mass transfer and pore diffusion model presented in this article can be used to design solute exclusion measurement experiments for the larger cellulose particles found in a commercial cellulose-to-ethanol plant.     

Sizing biological samples by photosedimentation techniques.

The performance of the Joyce-Loebl disk centrifuge in the sizing of Escherichia coli cells, protein inclusion bodies, and cell debris is evaluated. The need for a density gradient that extends throughout the entire spin fluid is highlighted, and a set of standard conditions that fulfill this requirement is defined. E. coli cells experience a reduction in their Stokes diameter when exposed to ethanol, indicating that a spin-buffer fluid combination such as glycerol-water is to be preferred for the sizing of bacteria. The instrument baseline is influenced by the presence of particles, and a method of estimating the baseline is described. The sizing of small particles is further complicated by baseline drift due to temperature sensitivity of the optical yoke. An analysis of diffusion in the spin fluid is conducted, and an expression for the sedimentation:diffusive flux ratio is derived. For the current samples, it is shown that diffusion within the spin fluid does not lead to significant errors for 0.15-microns particles, whereas the phenomenon may be significant at the manufacturer's size limit of 0.01 micron.     

Diffusion Constant and Dimension of Bacteriophage φX174 as Determined by Self-Beat Laser Light Spectroscopy and Electron Microscopy

The diffusion constant of bacteriophage phiX174 was determined by laser light self-beat spectroscopy. The method allows one to establish the diffusion constant in comparatively short time and with high accuracy. From the diffusion constant D(37) = 1.96 +/- 0.08 x 10(-7) cm(2)/s, the size of the virus particles within their aqueous milieu can be calculated. For phiX174, a diffusional diameter of 31.4 +/- 1.0 nm was found, in good agreement with measurements of diameters of freezeetched (32.3 +/- 1.8 nm) and negatively stained particles (33.8 +/- 2.1 nm), provided that the entire spikes of the virion are included. Other isometric viruses may show a complex interaction of the virion with the surrounding water and its ions.     

Effect of cytoskeletal geometry on intracellular diffusion.

A method is presented for determining the retardation of diffusion of particles inside cells owing to cytoskeletal barriers. The cytoskeletal meshwork is treated as a repeating periodic two-dimensional or three-dimensional lattice composed of elements of given size, shape, and spacing. We derive an analytic expression for the diffusion coefficient relative to that of the cytosol. This expression is evaluated by solving numerically an appropriate boundary-value problem for the Laplace equation. For the two-dimensional case, e.g., diffusion in a membrane, the results are quantitatively similar to those obtained by Saxton (1987. Biophys. J. 52:989-997) using Monte Carlo methods. The three-dimensional results are quantitatively similar to experimental results reported by Luby-Phelps et al. (1987. Proc. Natl. Acad. Sci. USA. 84:4910-4913) for the diffusion of dextran and Ficoll particles in Swiss 3T3 cells. By accounting for geometrical factors, these results allow one to assess the relative contributions of geometrical hindrance and of binding to the cytoskeletal lattice from measurements of intracellular diffusion coefficients of proteins.     

Applications of dielectrophoretic/electrohydrodynamic "zipper" electrodes for detection of biological nanoparticles

A major problem for surface-based detection techniques such as surface plasmon resonance and quartz crystal microbalances is that at low concentrations, diffusion is an insufficient driving force to bring colloidal submicron-scale particles to the detection surface. In order to overcome this, it has previously been demonstrated that a combination of dielectrophoresis and AC-electro-hydrodynamic flow can be used to focus cell-sized particles from suspension onto a large metal surface, in order to improve the detection capabilities of such systems. In this paper we describe how the combination of these two phenomena, using the so-called "zipper" electrode array, can be used to concentrate a wide range of nanoparticles of biological interest, such as influenza virus, dissolved albumin, and DNA molecules as well as latex beads of various sizes. We also demonstrate that the speed at which particles are transported towards the centre of the electrode pads by dielectrophoresis and electro-hydrodynamic flow is not related to the particle size for colloidal particles.     

Enhanced viscoelasticity of human cystic fibrotic sputum correlates with increasing microheterogeneity in particle transport

Current biochemical characterizations of cystic fibrosis (CF) sputum do not address the high degree of microheterogeneity in the rheological properties of the mucosal matrix and only provide bulk-average particle diffusion coefficients. The viscoelasticity of CF sputum greatly reduces the diffusion rates of colloidal particles, limiting the effectiveness of gene delivery to underlying lung cells. We determine diffusion coefficients of hundreds of individual amine-modified and carboxylated polystyrene particles (diameter 100-500 nm) embedded in human CF sputum with 5 nm and 33 ms of spatiotemporal resolution. High resolution multiple particle tracking is used to calculate the effective viscoelastic properties of CF sputum at the micron scale, which we relate to its macroscopic viscoelasticity. CF sputum microviscosity, as probed by 100- and 200-nm particles, is an order of magnitude lower than its macroviscosity, suggesting that nanoparticles dispersed in CF sputum are transported primarily through lower viscosity pores within a highly elastic matrix. Multiple particle tracking provides a non-destructive, highly sensitive method to quantify the high heterogeneity of the mucus pore network. The mean diffusion coefficient becomes dominated by relatively few but fast-moving particles as particle size is reduced from 500 to 100 nm. Neutrally charged particles with a diameter <200 nm undergo more rapid transport in CF sputum than charged particles. Treatment with recombinant human DNase (Pulmozyme) reduces macroviscoelastic properties of CF sputum by up to 50% and dramatically narrows the distribution of individual particle diffusion rates but surprisingly does not significantly alter the ensemble-average particle diffusion rate.     

Estimation of the shape and size of fribrinogen in solution from its hydrodynamic properties using theories for bead models and cylinders

The translational and rotational diffusion coefficients and the intrinsic viscosity of fibrinogen in solution are used to estimate its size, shape and hydration. Experimental data of the three hydrodynamic properties taken from the literature are compared with theoretical predictions for several molecular geometries that have been observed by electron microscopy. Modern theories for the hydrodynamics of bead models and cylindrical particles are employed in the calculations. The discrepancy between experimental results and theoretical predictions for spherical particles rules out the dodecahedral model and indicates that fibrinogen is elongated. The Hall-Slayter nodular model and its refinements perform better but still underestimate the size of the hydrated molecule. The best agreement between theoretical and experimental values is found for a cylindrical particle with length and diameter of about 48 and 6.8 nm, respectively. The hydration is calculated to be 3 g water/g protein. We speculate that, to accommodate such a large amount of water, fibrinogen in solution should be appreciably hydrated.     

Subcellular compartmentalization by local differentiation of cytoplasmic structure

The compartmentalization of eukaryotic cells by internal membranes and the subcellular localization of endogenous macromolecules by specific binding mechanisms are familiar concepts. In this report we present evidence that the cytoplasmic ground substance, which surrounds and contains the membrane-bound compartments, may also be compartmentalized by local differentiations of its submicroscopic structure that sort subcellular particles on the basis of size. The subcellular distribution of size-fractionated, fluorescent tracer particles was studied in living cells by ratio imaging and fluorescence recovery after photobleaching (FRAP). Large and small particles showed different distributions within the cytoplasmic volume, suggesting that the large particles were relatively excluded from some domains. While the structural basis for this phenomenon is not yet understood in detail, ratio imaging of large and small particles can be used as an empirical tool to identify cytoplasmic compartments for further study. The cytoplasmic diffusion coefficient (Dcyto) and % mobile fraction of the large particles showed considerable spatial variation over the projected area of the cell, while Dcyto and % mobile fraction of the small particles did not. A model is presented to account for this difference. Based on this model, a method is proposed by which FRAP can be used to detect sol-gel transitions in the cytoplasmic ground substance of living cells.     

Components of the plasma membrane of growing axons. I. Size and distribution of intramembrane particles

The plasmalemma of mature and growing olfactory axons of the bullfrog has been studied by freeze-fracture. Intramembrane particles (IMPs) of mature olfactory axons are found to be uniformly distributed along the shaft. However, during growth, a decreasing gradient of IMP density is evident along the somatofugal axis. The size histograms of axolemmal IMPs from different segments of growing nerve reveal regional differences in the particle composition. The distribution of each individual size class of particles along the growing nerve forms a decreasing gradient in the somatofugal direction; the slope of these gradients varies directly with particle diameter. These size-dependent density gradients are consistent with a process of lateral diffusion of membrane components that are inserted proximally into the plasma membrane. The membrane composition of the growth cone, however, appears to be independent of these diffusion gradients; it displays a mosaic pattern of discrete domains of high and low particle densities. The relative IMP profiles of these growth cone regions are similar to one another but contain higher densities of large IMPs than the neighboring axonal shaft. The shifting distributions of intramembrane particles that characterize the sprouting neuron give new insights into cellular processes that may underlie the establishment of the functional polarity of the neuron and into the dynamics of axolemmal maturation.