Lévy Flights Diffusion with Drift in Heterogeneous Membranes.

The modelling of diffusion in membranes is essential to understanding transport processes through membranes, especially when it comes to improving process efficiency. The purpose of this study is to understand the relationship between membrane structures, external forces, and the characteristic features of diffusive transport. We investigate Cauchy flight diffusion with drift in heterogeneous membrane-like structures. The study focuses on numerical simulation of particle movement across different membrane structures with differently spaced obstacles. Four studied structures are similar to real polymeric membranes filled with inorganic powder, while the next three structures are designed to show which distribution of obstacles can cause changes in transport. The movement of particles driven by Cauchy flights is compared to a Gaussian random walk both with and without additional drift action. We show that effective diffusion in membranes with an external drift depends on the type of the internal mechanism that causes the movement of particles as well as on the properties of the environment. In general, when movement steps are provided by the long-tailed Cauchy distribution and the drift is sufficiently strong, superdiffusion is observed. On the other hand, strong drift can effectively stop Gaussian diffusion.

Numerical Study of Drift Influence on Diffusion Transport through the Hybrid Membrane.

Sodium alginate membranes filled with iron oxide nanoparticles consist of a mixture of organic and inorganic phases. This design offers the possibility to combine the polymer's easy processability and superior separation performance. For a better understanding of the mechanisms of mixture separation, we analyze the diffusion motion of a particle in the hybrid membrane environment. We model structures of two-dimensional heterogenic membranes, which resemble real membrane structures, and then we simulate a random walk on them. We investigate how the additional action of drift changes the motion properties of the diffusing particles through the polymeric membrane filled with inorganic powder. We test the effect of two parameters: the distribution of obstacles (filling) in the membrane and the value of drift on the nature of diffusion. It appears that the synergy between drift, the diffusion, and the membrane structure affect the occurrence of the superdiffusive and subdiffusive character of particle motion as measured by the time-averaged mean square displacement. An important point is the observation that the strong drift supports subdiffusive motion as it increases the chances of particle trapping. Moreover, there exists the optimal value of drift, for which the transport through a membrane speeds up and does not cause trapping.

Characterization of the Structure and Transport Properties of Alginate/Chitosan Microparticle Membranes Utilized in the Pervaporative Dehydration of Ethanol.

The structure and transport properties of alginate/chitosan microparticle membranes used in ethanol dehydration processes were investigated. The membranes were characterized based on images obtained from high-resolution microscopy. The following parameters were determined: the observed total amount of void space, the average size of the void domains, their length and diameter, the fractal dimension, and the generalized stochastic fractal parameters. The total amount of void space was determined to be between 54% and 64%. The average size of the void domains is smaller for alginate membranes containing neat (CS) and phosphorylated (CS-P) chitosan particles when compared to those membranes filled with glycidol-modified (CS-G) and glutaraldehyde crosslinked (CS-GA) chitosan particles. Furthermore, the transport of ethanol and water particles through the studied membranes was modelled using a random walk framework. It was observed that the results from the theoretical and experimental studies are directly correlated. The smallest values of water to ethanol diffusion coefficient ratios (i.e., 14) were obtained for Alg (sodium alginate) membranes loaded with the CS and CS-P particles, respectively. Significantly larger values (27 and 19) were noted for membranes filled with CS-G and CS-GA particles, respectively. The simulation results show that the size of channels which develop in the alginate matrix is less suited for ethanol molecules compared to water molecules because of their larger size. Such a situation facilitates the separation of water from ethanol. The comparison of the structural analysis of the membranes and random walk simulations allows one to understand the factors that influence the transport phenomena, in the studied membranes, and comment on the effect of the length, diameter, number of channels, and variations in the pore diameters on these transport parameters.

Structure-diffusion relationship of polymer membranes with different texture.

Two-dimensional diffusion in heterogenic composite membranes, i.e., materials comprising polymer with dispersed inorganic fillers, composed of ethylcellulose and magnetic powder is studied. In the experimental part, the morphology of membranes is described by the following characteristics: the amount of polymer matrix, the fractal dimension of polymer matrix, the average size of polymer matrix domains, the average number of obstacles in the proximity of each polymer matrix pixel. The simulation work concentrates on the motion of a particle in the membrane environment. The focus is set on the relationship between membranes morphology characterized by polymer matrix density, its fractal dimension, the average size of domains, and the average number of near obstacles and the characteristics of diffusive transport in them. The comparison of diffusion driven by Gaussian random walk and Lévy flights shows that the effective diffusion exponent at long time limits is subdiffusive and it does not depend on the details of the underlying random process causing diffusion. The analysis of the parameters describing the membrane structure shows that the most important factor for the diffusion character is the average size of a domain penetrated by diffusing particles. The presented results may be used in the design and preparation of membrane structures with specific diffusion properties.

Fractal analysis of heterogeneous polymer networks formed by photopolymerization of dental dimethacrylates.

OBJECTIVES: In this work the influence of the dimethacrylate monomer chemical structure on structural heterogeneity and physico-mechanical properties of the resulting polymer networks was investigated. Rigid aromatic dimethacrylate (Bis-GMA), triethylene glycol dimethacrylate (TEGDMA) and flexible aliphatic urethane-dimethacrylate (UDMA) were chosen for room-temperature homopolymerizations and copolymerizations induced by camphorquinone/N,N-dimethylaminoethyl methacrylate photoinitiating system. METHODS: Atomic force microscopy (AFM) was used for visualizing the morphology of poly(dimethacrylate)s, which was described by: the fractal dimension (D(F)), the generalized fractal dimensions (D(q) and ΔD) as well as the modified fractal dimension (D(ß)). Estimated fractal characteristics were correlated with polymer density, hardness and impact strength. RESULTS: AFM images of fractured surfaces revealed the highly complex morphology of dimethacrylate polymer networks. They were found to possess the fractal character. The fractal parameters were observed to be proportional to the density, hardness and impact resistance of investigated polymers. ΔD appeared to be a good indicator of the structural heterogeneity of dimethacrylate networks. The results suggest that the fracture behavior of poly(dimethacrylate) matrix of dental materials can be controlled by the fractal morphology. SIGNIFICANCE: Correlating the morphological studies with the mechanical tests would be beneficial in defining the role of morphology in the mechanical behavior of dimethacrylate networks and consequently, lead to the development of a reliable method for identifying the cause of dental material failures under stress. Thus, fractal analysis could become one of the key elements in designing and developing dental materials.

Fractal analysis and ionic dependence of endocytotic membrane activity of human breast cancer cells.

The endocytic membrane activities of two human breast cancer cell lines (MDA-MB-231 and MCF-7) of strong and weak metastatic potential, respectively, were studied in a comparative approach. Uptake of horseradish peroxidase was used to follow endocytosis. Dependence on ionic conditions and voltage-gated sodium channel (VGSC) activity were characterized. Fractal methods were used to analyze quantitative differences in vesicular patterning. Digital quantification showed that MDA-MB-231 cells took up more tracer (i.e., were more endocytic) than MCF-7 cells. For the former, uptake was totally dependent on extracellular Na(+) and partially dependent on extracellular and intracellular Ca(2+) and protein kinase activity. Analyzing the generalized fractal dimension (D(q )) and its Legendre transform f(alpha) revealed that under control conditions, all multifractal parameters determined had values greater for MDA-MB-231 compared with MCF-7 cells, consistent with endocytic/vesicular activity being more developed in the strongly metastatic cells. All fractal parameters studied were sensitive to the VGSC blocker tetrodotoxin (TTX). Some of the parameters had a "simple" dependence on VGSC activity, if present, whereby pretreatment with TTX reduced the values for the MDA-MB-231 cells and eliminated the differences between the two cell lines. For other parameters, however, there was a "complex" dependence on VGSC activity. The possible physical/physiological meaning of the mathematical parameters studied and the nature of involvement of VGSC activity in control of endocytosis/secretion are discussed.

Lacunarity as a novel measure of cancer cells behavior.

An important goal in many branches of science, especially in molecular biology and medicine is the quantitative analysis of the structures and their morphology. The morphology can be analyzed in many ways, in particular by the fractal analysis. Apart from the fractal dimension, an important part of the fractal analysis is the lacunarity measurement which, roughly speaking, characterizes the distribution of gaps in the fractal: a fractal with high lacunarity has large gaps. In this paper, we present an extension of the lacunarity measure to objects with nonregular shapes that enables us to provide a successful discrimination of cancer cell lines. The cell lines differ in the shape of vacuole (the gaps in their body) which is perfectly suited for the lacunarity analysis.

Voltage-gated sodium channel expression and potentiation of human breast cancer metastasis.

PURPOSE: Ion channel activity is involved in several basic cellular behaviors that are integral to metastasis (e.g., proliferation, motility, secretion, and invasion), although their contribution to cancer progression has largely been ignored. The purpose of this study was to investigate voltage-gated Na(+) channel (VGSC) expression and its possible role in human breast cancer. EXPERIMENTAL DESIGN: Functional VGSC expression was investigated in human breast cancer cell lines by patch clamp recording. The contribution of VGSC activity to directional motility, endocytosis, and invasion was evaluated by in vitro assays. Subsequent identification of the VGSC alpha-subunit(s) expressed in vitro was achieved using reverse transcription-PCR, immunocytochemistry, and Western blot techniques and used to investigate VGSCalpha expression and its association with metastasis in vivo. RESULTS: VGSC expression was significantly up-regulated in metastatic human breast cancer cells and tissues, and VGSC activity potentiated cellular directional motility, endocytosis, and invasion. Reverse transcription-PCR revealed that Na(v)1.5, in its newly identified "neonatal" splice form, was specifically associated with strong metastatic potential in vitro and breast cancer progression in vivo. An antibody specific for this form confirmed up-regulation of neonatal Na(v)1.5 protein in breast cancer cells and tissues. Furthermore, a strong correlation was found between neonatal Na(v)1.5 expression and clinically assessed lymph node metastasis. CONCLUSIONS: Up-regulation of neonatal Na(v)1.5 occurs as an integral part of the metastatic process in human breast cancer and could serve both as a novel marker of the metastatic phenotype and a therapeutic target.

Endogenous and exogenous citrate transport and release in prostatic preparations: semi-polarized two-dimensional cultures of human PNT2-C2 cells and isolated tubules and segments of rat prostate.

BACKGROUND: Electrophysiological characterization of normal human prostate epithelial cells showed exogenous trivalent citrate transport (release) to be K(+)-dependent. METHODS: (1) Ussing chamber recordings of short circuit current (SCC) were used to study citrate transport in the same (PNT2-C2) cell line grown on micro-pore filters as a monolayer. (2) Release of endogenous citrate from confluent cultures and tubules and segments of rat prostate was measured using a fluorescence technique. (3) Enzyme-spectrophotometry was employed to detect citrate release from segments of rat prostate. RESULTS: Citrate transport across the PNT2-C2 monolayer was asymmetrical, consistent with release into the lumen-side. Fluorescence and/or enzyme-spectrophotometric measurements showed that time-dependent citrate release (endogeneous and preabsorbed) occurred from rat prostate (tubules and segments), but not kidney or lung. The release was dependent on extracellular K(+) but not Na(+). CONCLUSIONS: Citrate release from prostatic cells and tissues (rat and human) was K(+)-dependent, consistent with the previous electrophysiological data.

Patterning of endocytic vesicles and its control by voltage-gated Na+ channel activity in rat prostate cancer cells: fractal analyses.

Fractal methods were used to analyze quantitative differences in secretory membrane activities of two rat prostate cancer cell lines (Mat-LyLu and AT-2) of strong and weak metastatic potential, respectively. Each cell's endocytic activity was determined by horseradish peroxidase uptake. Digital images of the patterns of vesicular staining were evaluated by multifractal analyses: generalized fractal dimension (Dq) and its Legendre transform f(alpha), as well as partitioned iterated function system -- semifractal (PIFS-SF) analysis. These approaches revealed consistently that, under control conditions, all multifractal parameters and PIFS-SF codes determined had values greater for Mat-LyLu compared with AT-2 cells. This would agree generally with the endocytic/vesicular activity of the strongly metastatic Mat-LyLu cells being more developed than the corresponding weakly metastatic AT-2 cells. All the parameters studied were sensitive to tetrodotoxin (TTX) pre-treatment of the cells, which blocked voltage-gated Na+ channels (VGSCs). Some of the parameters had a "simple" dependence on VGSC activity, whereby pre-treatment with TTX reduced the values for the MAT-LyLu cells and eliminated the differences between the two cell lines. For other parameters, however, there was a "complex" dependence on VGSC activity. The possible physical/physiological meaning of the mathematical parameters studied and the nature of involvement of VGSC activity in control of endocytosis/secretion are discussed.