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The patch-clamp technique is a powerful tool that allows for a long observation of transport protein activity in real time. Experimental traces of single-channel currents can be considered as a record of the channel's conformational switching related to its activation and gating. In this work, we present a mathematically simple method of patch-clamp data analysis that assesses the connectivity and occupancy of distinct conformational substates of the channel. The proposed approach appears to be a big step forward due to its possible applications in the determination of channel substates related to disease and in the analysis of drug-channel interactions on the level of repetitive sequences of channel conformations. This is especially important in cases when molecular dynamics docking is impossible and Markovian modeling requires ambiguous optimization tasks.

Using the results from previous modeling of the ball and chain inactivation of the potassium channels we try to model the inactivation of the NaV 1.5 sodium channels in adult and neonatal form. The (fast) inactivation of sodium channels differs from the inactivation of the potassium channels by the use of a inactivating hinge rather than a ball on a chain. The adult and neonatal variants of the channel differ mostly in a charged amino acid residue located on the extracellular side. We show that a drift caused by this residue is sufficient to describe the differences in inactivation between the two forms of the NaV 1.5. We use the survival probability, the patch-clamp measurable parameter, to discriminate between the cells of different metastaticity.

The large-conductance voltage- and Ca2+-activated K+ channels (BK) are encoded in humans by the Kcnma1 gene. Nevertheless, BK channel isoforms in different locations can exhibit functional heterogeneity mainly due to the alternative splicing during the Kcnma1 gene transcription. Here, we would like to examine the existence of dynamic diversity of BK channels from the inner mitochondrial and cellular membrane from human glioblastoma (U-87 MG). Not only the standard characteristics of the spontaneous switching between the functional states of the channel is discussed, but we put a special emphasis on the presence and strength of correlations within the signal describing the single-channel activity. The considered short- and long-range memory effects are here analyzed as they can be interpreted in terms of the complexity of the switching mechanism between stable conformational states of the channel. We calculate the dependencies of mean dwell-times of (conducting/non-conducting) states on the duration of the previous state, Hurst exponents by the rescaled range R/S method and detrended fluctuation analysis (DFA), and use the multifractal extension of the DFA (MFDFA) for the series describing single-channel activity. The obtained results unraveled statistically significant diversity in gating machinery between the mitochondrial and cellular BK channels.

To improve the process of ethanol dehydration, asymmetric chitosan membranes were prepared and applied in the pervaporation process. A porous structure was obtained by adding absolute ethanol into chitosan solution. Fabricated membranes were comprehensively characterized by means of contact angle measurements, degree of swelling analysis, scanning electron microscopy, positron annihilation lifetime spectroscopy and wide angle X-rays scattering, to provide an extensive analysis of their physicochemical properties. The results showed a strong correlation between structure, amount and size of pores, and separation effectiveness of investigated membranes. Additionally, an interesting phenomenon of changing in membrane structure during pervaporation process was observed and was found to cause a nonlinear time dependence of transport and separation parameters. Consequently, the best performance of PV separation of 96 vol% ethanol (feed) was obtained for a membrane prepared from casting solution containing 0.1 ml of absolute ethanol, for which the values of separation factor and permeate flux reached 599.7 and 3.534 kg∙m−2∙h−1, respectively.

Kv 1.2 are voltage-dependent potassium channels of great biological importance. Despite the existence of many reports considering structure - function relations of the Kv 1.2 channel's quantitative domains, some details of the voltage gating remain ambiguous, or even unknown. One of the examples is the range of the S4-S6 domains motions involved in channel activation and gating. Another important question is to what extent the channel geometry influences the observable channel conductance at different voltages, and what mechanism stands behind. Does the narrowing of the pore reduce the conductance by ohmic resistance growth? The answer is surprisingly negative. But it can be explained in an alternative way by considering the fluctuations. To address these problems, we formulate geometric models that mimic the generic features of voltage sensor movement and trigger the movement of the other domains involved in gating. We carry out a complete simulation of S4-S6 domains translations and tilts. The obtained pore profiles allow to estimate the (ohmic) conductance dependency on the voltage. From a family of analysed models, we choose the one most concurring with the experimental data. The results allow to suggest the most probable scenario of S4-S6 domains movement during channel activation by membrane depolarization.

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.

A procedure to estimate the diffusion coefficient in solution&ndash, diffusion models of hydrophilic membranes used in pervaporation-based purification experiments is presented. The model is based on a series solution of the general permeation problem. It considers a membrane that can be filled with water or with the feed solution before the measurement. Furthermore, the length of the tubing between the permeation cell and the place of cold traps is also addressed. To illustrate the parameter estimation procedure, we have chosen the data for the separation of water and ethanol by chitosan membranes. It is shown that the diffusion coefficient can be estimated effectively from the time course of the transported mass and by the analysis of certain well defined time lags of the permeation curve.

This paper proposes the use of carriers with hierarchical porous structures as novel monolithic tablets for modified drug release. The influence of pore structure on the tamsulosin release profile is presented. The hierarchical arrangement of porous structure in monolithic tablets and the deposition of tamsulosin inside the silica carrier enable to control the kinetic of release and the amount of tamsulosin released. We developed a mathematical model of tamsulosin release from two carriers with different hierarchy of meso- and macropores. A model of this nature will allow to predict the release of tamsulosin from other carriers with a similar pore structure. We hope this research will improve the design process of novel carriers, and thus, will allow to tailor the porous structure of a carrier to achieve the desired release profile.

The physicochemical basic laws and their suitable mathematical formulations form a frame of the batch pervaporation process, i.e. the cycled process of removing small traces of volatile impurities from large amounts of a volatile solvent, is described and analyzed in detail. The generic nature of the process, i.e. its exclusively transient behaviour is described as an IBV (initial boundary value) problem with a feed side boundary condition dependent on time. The long time asymptotics is investigated to determine the partition and diffusion coefficients of the trace.

An idea of using magnetic field to enhance oxygen transport, as an effective factor for the air enrichment, has already been successfully demonstrated. Direct consequence of this concept was to produce polymeric magnetic membranes by casting technique. An analysis of experimental data, obtained from transient and stationary states of mass transport, allowed to formulate a hypothesis of a “dead thickness” of magnetic membranes in question, i.e. the bulk polymer phase as a part of membrane with no magnetic channels but limiting the overall transport through it. It will be shown that removing this part increases the rate of transport and enhances the air separation. To complete the overall approach, the theory is provided both on phenomenological and molecular levels. The simulation by Monte Carlo and Brownian dynamics, to show some molecular properties of magnetic field and its activity during separation, was performed.

A previously studied drug delivery systems based on 4-arm star shaped polyacids with pentaerythritol core have been proposed as excellent micellar nanocarriers with adjustable characteristics for drug loading and release, including anti-inflammatory drugs. A series of experiments to measure the release of drugs by micelles of various composition were conducted. Here we model this release process by mathematical methods and conclude about the structure and loading capability of the obtained micelles in relation to their composition. Our method is a random walk simulation in a potential field. The results predict a decreasing dependency of the micelle core size on hydrophobic content which relates to the loading capability and release kinetics. We link this phenomenon to the packing quality of the micelle’s core, which is reflected by the values of the drug’s binding energy to the core and the probability to incorporate the drug into the core.

The effect of magnetic field on the solubility of magnetic membranes is investigated. As a result, a conclusion is drawn that the solubility is not altered on the gas–polymer interface but may be altered to a significant degree in the vicinity of the, so-called, magnetic channels that form close to the magnetic particles suspended in the polymer matrix.

Passive calcium influx is one of the theories to explain the cathodal galvanotaxis of cells that utilize the electric field to guide their motion. When exposed to an electric field, the intracellular fluid becomes polarized, leading to positive charge accumulation on the cathodal side and negative charge accumulation on the anodal side. The negative charge on the anodal side attracts extracellular calcium ions, increasing the anodal calcium concentration, which is supposed to decrease the mobile properties of this side. Unfortunately, this model does not capture the Ca2+ dynamics after its presentation to the intracellular fluid. The ions cannot permanently accumulate on the anodal side because that would build a potential drop across the cytoplasm leading to an ionic current, which would carry positive ions (not only Ca2+) from the anodal to the cathodal part through the cytoplasm. If the cytoplasmic conductance for Ca2+ is low enough compared to the membrane conductance, the theory could correctly predict the actual behavior. If the ions move through the cytoplasm at a faster rate, compensating for the passive influx, this theory may fail. This paper contains a discussion of the regimes of validity for this theory.

A set of methods (D1–D8 transient sorption and permeation analysis system) previously used for the regular inhomogeneous membranes, that is, the membranes where the diffusive properties vary regula...

We investigate the relationship between the Freundlich adsorption exponent and the fractal dimension of the adsorption sites for quasi-monolayer adsorption, and of the adsorbed aggregate for a simple case of multilayer adsorption. We further check whether the Freundlich adsorption mechanism may contribute to anomalous diffusion in the transport through porous materials.

Rat prostate cancer cells have been previously investigated using two cell lines: a highly metastatic one (Mat-Ly-Lu) and a nonmetastatic one (AT-2). It turns out that the highly metastatic Mat-Ly-Lu cells exhibit a phenomenon of cathodal galvanotaxis in an electric field which can be blocked by interrupting the voltage-gated sodium channel (VGSC) activity. The VGSC activity is postulated to be characteristic for metastatic cells and seems to be a reasonable driving force for motile behavior. However, the classical theory of cellular motion depends on calcium ions rather than sodium ions. The current research provides a theoretical connection between cellular sodium inflow and cathodal galvanotaxis of Mat-Ly-Lu cells. Electrical repulsion of intracellular calcium ions by entering sodium ions is proposed after depolarization starting from the cathodal side. The disturbance in the calcium distribution may then drive actin polymerization and myosin contraction. The presented modeling is done within a continuous one-dimensional Poisson-Nernst-Planck equation framework.

This paper investigates a known problem of gas permeation into a finite volume. The literature treatment of this problem provides a solution, but misses the well-posedness analysis, which is of crucial importance, especially when a nonstandard problem is formulated. We have demonstrated that the Hadamard sense of well-posedness for this problem is satisfied by making use of the Duhamel's theorem. The solution we have obtained using the Duhamel's theorem and the solution available in the literature have different analytical representations, yet they are mathematically equivalent. In parallel, this paper provides some practical comments on the well-posedness analysis.

We consider a deterministic process described by a discrete one-dimensional chaotic map and study its diffusive-like properties. Starting with the corresponding Frobenius-Perron equation we derive an approximate evolution equation for the probability distribution which is a partial differential equation of a hyperbolic type. Consequently, the process is correlated, non-Markovian, non-Gaussian and the information propagates with a finite velocity. This is in clear contrast to conventional diffusion processes described by a standard parabolic diffusion equation with an infinite velocity of information propagation. Our approach allows for a more complete characterisation of diffusion dynamics of deterministic systems.

Some important physical processes represented by the shape of voltammograms of polyaniline in the phase of oxidation are identified and modelled. The main process, i.e. an angular diffusion by which the motion of conducting polymer subunits take place is described and modelled. To fit a molecular picture of the oxidation process a nontrivial random walk of polymer subunits in configurational space of angles has been performed. A very good agreement between theoretical predictions and experimental data has been obtained.

In this paper, we report on the new method of image compression. The method is based on LZ77 dictionary algorithm. We introduce two modifications such as quantization and noise levels. Experimental results presented in this paper prove that the new method of image compression gives promising results as compared with original LZ77 dictionary algorithm and JPEG2000.

We model the activity of an ion channel gate by Langevin dynamics in a logarithmic potential. This approach enables one to describe the power-law dwell-time distributions of the considered system, and the long-term correlations between the durations of the subsequent channel states, or fractal scaling of statistical characteristics of the gate's movement with time. Activity of an ion channel gate is described as an overdamped motion of the reaction coordinate in a confining logarithmic potential, which ensures great flexibility of the model. Depending on the chosen parameters, it allows one to reproduce many types of gate dynamics within the family of non-Markovian, anomalous conformational diffusion processes. In this study we apply the constructed model to largeconductance voltage and Ca2+-activated potassium channels (BKCa). The interpretation of model assumptions and parameters is provided in terms of this biological system. Our results show good agreement with the experimental data.

A transition from nondiffusive growth of variance of a particles position to diffusive one of various type, in nonlinear deterministic maps, is discussed. The details of generating regular, subdiffusive and superdiffusive cases are shown.

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.

The ball and chain mechanism is a widely accepted theory for the inactivation of the Shaker K + channel. In this paper we propose a diffusive model that predicts a rate of inactivation that is comparable to the experimental measurements.

A set of 10, chosen medicinal plants (some of them with a reputation as remedies for tuberculosis) has been investigated through Partitioned Iterated Function Systems-Semi Fractals with Angle (PIFS-SFA) coding, Lempel, Ziv, Welch with quantization and noise (LZW-QN) compression, and surface density statistics (f(α)-SDS) discrimination techniques. The final outcomes of this quantitative analysis were, firstly: the linear ordering of the plants in question accompanied by the hope that it reflects their medical significance, secondly: the mathematical representation of each of the plants, and thirdly: the impressive compression achieved, leading to remarkable computer memory saving, and still permitting successful pattern recognition i.e., proper identification of the plant from the compressed image.

In recent years we have proposed magnetic membranes (membranes with dispersed granules of a permanent magnet) for the separation of oxygen and nitrogen in the air mixture. The idea was to utilize oxygen's paramagnetic properties which are absent in the nitrogen. The experimental results were promising but they demanded a theoretical description which is given in the present paper. By a detailed analysis of the available data we arrive at four important conclusions: (1) magnetic channels are formed around the magnetic granules, (2) the channels provide high permeability "highways" for the diffusion of permeating molecules, (3) the oxygen molecules, due to their paramagnetic properties, stick to these "highways" for a longer time than the nitrogen, which is probably based on the interaction with the Weiss molecular field of the permanent magnet, (4) the magnetic field induces aggregation between oxygen and nitrogen which enhances the transport of both nitrogen and oxygen by prolonging their residence in the channel. In addition to these findings, the most suggestive theory of magnetic membrane operation is ruled out: the forcing of the oxygen molecules due to the gradients of magnetic field generated by the granules is too weak to account for the observed effects.

Several approaches to ion-channel gating modelling have been proposed. Although many models describe the dwell-time distributions correctly, they are incapable of predicting and explaining the long-term correlations between the lengths of adjacent openings and closings of a channel. In this paper we propose two simple random-walk models of the gating dynamics of voltage and Ca(2+)-activated potassium channels which qualitatively reproduce the dwell-time distributions, and describe the experimentally observed long-term memory quite well. Biological interpretation of both models is presented. In particular, the origin of the correlations is associated with fluctuations of channel mass density. The long-term memory effect, as measured by Hurst R/S analysis of experimental single-channel patch-clamp recordings, is close to the behaviour predicted by our models. The flexibility of the models enables their use as templates for other types of ion channel.