On the significance of sodium ionic channels in analysis of the breast cancer metastaticity.

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.

Mechanosensitivity of the BK Channels in Human Glioblastoma Cells: Kinetics and Dynamical Complexity.

BK channels are potassium selective and exhibit large single-channel conductance. They play an important physiological role in glioma cells: they are involved in cell growth and extensive migrating behavior. Due to the fact that these processes are accompanied by changes in membrane stress, here, we examine mechanosensitive properties of BK channels from human glioblastoma cells (gBK channels). Experiments were performed by the use of patch-clamp method on excised patches under membrane suction (0-40 mmHg) at membrane hyper- and depolarization. We have also checked whether channel's activity is affected by possible changes of membrane morphology after a series of long impulses of suction. Unconventionally, we also analyzed internal structure of the experimental signal to make inferences about conformational dynamics of the channel in stressed membranes. We examined the fractal long-range memory effect (by R/S Hurst analysis), the rate of changes in information by sample entropy, or correlation dimension, and characterize its complexity over a range of scales by the use of Multiscale Entropy method. The obtained results indicate that gBK channels are mechanosensitive at membrane depolarization and hyperpolarization. Prolonged suction of membrane also influences open-closed fluctuations-it decreases channel's activity at membrane hyperpolarization and, in contrary, increases channel's activity at high voltages. Both membrane strain and its "fatigue" reduce dynamical complexity of channel gating, which suggest decrease in the number of available open conformations of channel protein in stressed membranes.

The role of entropic potential in voltage activation and K+ transport through Kv 1.2 channels.

We analyze the entropic effects of inner pore geometry changes of Kv 1.2 channel during membrane depolarization and their implications for the rate of transmembrane transport of potassium ions. We base this on the idea that spatial confinements within the channel pore give rise to entropic barriers which can both effectively affect the stability of open macroconformation and influence channel's ability to conduct the potassium ions through the membrane. First, we calculate the differences in entropy between voltage-activated and resting states of the channel. As a template, we take a set of structures of channel pore in an open state at different membrane potentials generated in our previous research. The obtained results indicate that tendency to occupy open states at membrane depolarization is entropy facilitated. Second, we describe the differences in rates of K+ transport through the channel pore at different voltages based on the results of appropriate random walk simulations in entropic and electric potentials. The simulated single channel currents (I) suggest that the geometry changes during membrane depolarization are an important factor contributing to the observed flow of potassium ions through the channel. Nevertheless, the charge distribution within the channel pore (especially at the extracellular entrance) seems most prominent for the observed I/Imax relation at a qualitative level at analyzed voltages.

The temperature dependence of the BK channel activity - kinetics, thermodynamics, and long-range correlations.

Large-conductance, voltage dependent, Ca2+-activated potassium channels (BK) are transmembrane proteins that regulate many biological processes by controlling potassium flow across cell membranes. Here, we investigate to what extent temperature (in the range of 17-37°C with ΔT=5°C step) is a regulating parameter of kinetic properties of the channel gating and memory effect in the series of dwell-time series of subsequent channel's states, at membrane depolarization and hyperpolarization. The obtained results indicate that temperature affects strongly the BK channels' gating, but, counterintuitively, it exerts no effect on the long-range correlations, as measured by the Hurst coefficient. Quantitative differences between dependencies of appropriate channel's characteristics on temperature are evident for different regimes of voltage. Examining the characteristics of BK channel activity as a function of temperature allows to estimate the net activation energy (Eact) and changes of thermodynamic parameters (ΔH, ΔS, ΔG) by channel opening. Larger Eact corresponds to the channel activity at membrane hyperpolarization. The analysis of entropy and enthalpy changes of closed to open channel's transition suggest the entropy-driven nature of the increase of open state probability during voltage activation and supports the hypothesis about the voltage-dependent geometry of the channel vestibule.

Impact of geometry changes in the channel pore by the gating movements on the channel's conductance.

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.

On the fractality of the Freundlich adsorption isotherm in equilibrium and non-equilibrium cases.

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.

On Application Of Langevin Dynamics In Logarithmic Potential To Model Ion Channel Gate Activity.

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.

On the simple random-walk models of ion-channel gate dynamics reflecting long-term memory.

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.

On the magnetic channels in polymer membranes.

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.

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.

Electrostatic interactions during Kv1.2 N-type inactivation: random-walk simulation.

N-type inactivation of the Kv1.2 voltage-gated potassium channel is a process in which the N-terminal of the protein (its first 20 amino acids) binds to the open-channel surface, extends and occludes its pore. This process has been experimentally studied in both intact and ShBDelta6-46 channels in which the inactivating peptides are supplied in the bath solution. In this work we provide a qualitative description of N-type inactivation by simulating the random walk of charged inactivating peptides in the electrostatic field that originates from the charges present in the channel and in the cellular membrane. Our results give a deeper insight into the previously reported influence of electrostatics on the rate of N-type inactivation of ShBDelta6-46. We also show how the enchaining of the peptides, i.e., considering the intact form of the channel, influences the N-type inactivation with different charges of those peptides.

Classification of antituberculosis herbs for remedial purposes by using fuzzy sets.

Using fuzzy set theory, we created a system, that assesses a herb's usefulness for the treatment of tuberculosis, based on ethnobotanical data. We analysed two systems which contain different amount of inputs. The first system contains four inputs, the second one contains six inputs. We used the Takagi-Sugeno-Kanga model. Mamdani model is poor at representation as it needs more fuzzy rules than that of TSK to model a real world system where accuracy is demanded. It has been employed a fuzzy controller, and a fuzzy model, in successfully solving difficult control and modelling problems in practice. It is implemented in the Fuzzy Logic Toolbox in Matlab. The data for inputs are gathered in the database named SOPAT (selection of plants against tuberculosis), which is part of a project coordinated by the Oxford International Biomedical Centre. In this database there could be up to one million plant species. It would be cumbersome to select a remedy from one (or some) of these species looking at the data base one-by-one. By means of the fuzzy set theory this remedy can be chosen very quickly.

A diffusive model of the ball and chain inactivation.

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.

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.

On the role of ball and chain interactions in recovery from the inactivation of the shaker potassium channel.

We describe a new factor in the recovery from inactivation in the ball and chain model. We propose a model in which the tension from the chain may help pull the ball away from its binding site, reducing the duration of the inactivation period. A corresponding model was built and analysed.

On the possible methods for the mathematical description of the ball and chain model of ion channel inactivation.

Ion channels are large transmembrane proteins that are able to conduct small inorganic ions. They are characterized by high selectivity and the ability to gate, i.e. to modify their conductance in response to different stimuli. One of the types of gating follows the ball and chain model, according to which a part of the channel's protein forms a ball connected with the intracellular side of the channel by a polypeptide chain. The ball is able to modify the conductance of the channel by properly binding to and plugging the channel pore. In this study, the polypeptide ball is treated as a Brownian particle, the movements of which are limited by the length of the chain. The probability density of the ball's position is resolved by different diffusional operators--parabolic (including the case with drift), hyperbolic, and fractional. We show how those different approaches shed light on different aspects of the movement. We also comment on some features of the survival probabilities (which are ready to be compared with electrophysiological measurements) for issues based on the above operators.

On the mathematical reconstruction of two dimensional plants.

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(alpha)-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.

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.

Tenth Oxford Conference organized by the Oxford International Biomedical Centre (OIBC).

A series of events organized at Mansfield College and Magdalen College School by Oxford International Biomedical Centre from March 29th to April 1st 2004, is reported. There were more than 60 active participants (speakers, moderators and discussants) plus the general audience. A special programme for the group of five youngsters from developing and restructuring countries, for the first time in the 10 years history of OIBC conferences, was also run.