Magnetic and electric fields across sodium and potassium channels

Transcription

1 Magnetic and electric fields across sodium and potassium channels Marília A. G. Soares, Frederico A. O. Cruz, and Dilson Silva Citation: AIP Conference Proceedings 1702, (2015); doi: / View online: View Table of Contents: Published by the AIP Publishing Articles you may be interested in Electric-field-induced phase switching in the lead free piezoelectric potassium sodium bismuth titanate Appl. Phys. Lett. 97, (2010); / Augmentation of macromolecular adsorption rates through transverse electric fields generated across patterned walls of a microfluidic channel J. Appl. Phys. 100, (2006); / Nitrogen 14 and sodium 23 nuclear magnetic resonance of sodium and potassium cyanide J. Chem. Phys. 58, 3018 (1973); / Electric Fields Caused by the Diffusion of Charged Particles across a Magnetic Field Phys. Fluids 9, 2244 (1966); / Measurement of Magnetically Induced Sound Absorption in Liquid Sodium and Potassium Phys. Fluids 7, 375 (1964); /

2 Magnetic and Electric Fields across Sodium and Potassium Channels Marília A. G. Soares a, Frederico A. O. Cruz b and Dilson Silva a a Postgraduation in Computational Sciences, Rio de Janeiro State University, dilsons@uerj.br Rua São Francisco Xavier, , RJ, Brazil b Department of Physics, Rural Federal University of Rio de Janeiro Rodovia BR 465, Km 7, Zona Rural , RJ, BR Abstract. We determined the magnetic field around sodium and potassium ionic channels based on a physicomathematical model that took into account charges in the surface bilayer. For the numerical simulation, we applied the finite element method. Results show that each channel produces its specific and individual response to the ion transport, according to its individual intrinsic properties. The existence of a number of active Na + -channels in a given membrane region seems not to interfere directly in the functioning of K + -channel located among them, and vice-versa. Keywords: ion channels, biological membrane, finite element method. PACS: D INTRODUCTION The ion movement throughout the membrane has been considered mainly driven by diffusion and electrostatic interactions between moving ions, protein charges, and an externally applied electric field. Here, we discuss the influence of surface charges over the lipid bilayer on the behavior of the magnetic field generated by the ion flux across the ionic channel, based on a physico-mathematical model. Ionic channels regulate the flow of ions, such as Na +, K + e Ca +2, adjust cellular volume, keep the resting membrane potential, and regulate the electric excitability of excitable cells. It is known that ion channels control many vital physiological functions, and therefore many medicines act by blocking the action of endogenous ligands that regulate the ionic flow across these channels [1]. Thus, ion channels form an area of intensive research in the fields of Biology and Medicine. The biological membrane is formed by a lipid matrix basically constituted by a phospholipidic bilayer, inserted by cholesterol, proteins, and complex macromolecules. The distribution of phospholipids in the two lipid monolayers of the bilayer is asymmetric and the cholesterol arrangement in bilayer stabilizes the membrane structure [2,3]. The lipid bilayer confers to the membrane a high impermeability, and its semi-permeability is due to the presence of proteins in its structure. Associated to the outer surface of the bilayer there is the glycocalyx, an additional layer rich in polysaccharides, which presents a net negative charge in a solution of NaCl at ph 7.0 (10) [4-7]. Investigating the electrical activity of cells is a fascinating journey, and the part concerning the ion channels began in 1949 with Hodgkin, Huxley and Katz. Hodgkin and Huxley showed the relationship between resting potential, potassium and sodium permeability ratio and conductance for a squid giant axon [8-10]. More recently, analyses based on several computational methods have represented a powerful tool to model ionic channels, in especial to devoted to the understanding of some cellular dysfunctions. In this work we present a physico-mathematical model representing the ion transport across ionic channels, in special sodium and potassium channels. Our study deals in special with the magnetic field behavior related to the ionic flux, analyzing the influence of the surface bilayer charges on this behavior. For the numerical simulation, we applied the finite element method (FEM). International Conference of Computational Methods in Sciences and Engineering 2015 (ICCMSE 2015) AIP Conf. Proc. 1702, ; doi: / AIP Publishing LLC /$30.00 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: Downloaded to IP:

3 ADOPTED MODEL FOR IONIC CHANNEL Considering the membrane as a physical system formed by three regions (Figure 1): extracellular region (region 1), lipidic bilayer of thickness h (region 2), and cytoplasmatic region (region 3), the ionic channel was modeled as a cylinder of radius r crossing the lipidic bilayer. Potential equation of Poisson-Boltzmann. The total charge in each region can be given by the sum of electrolytic charges dispersed in the solution and fixed charges (fixed on lipid bilayer surfaces and glycocalyx or cytoplasmic proteins). Electrolytic charge density can be represented as a Boltzmann distribution. Thus, the equation to describe the variation of electric potential (φ i ) in any point of the region i (i = 1, 2, 3) is (1) where φ i is the electric potential at any point within region i and φ li is the limiting electric potential; q j is the electric charge, and j is the molar concentration of ion j, k and T are, respectively, the Boltzmann constant and the absolute temperature. FIGURE 1. Adopted model for the ionic channel (r is the channel radius). The adopted boundary conditions were: (a) when z tends to infinity, z - (phase 1) or z (phase 3), the electric potential and ionic concentration tend to limiting values, and, in these regions, the electroneutrality condition is valid; and (b) the conditions of continuity of electric potential and of discontinuity of the electric displacement vector are assumed for all surfaces. Using the boundary conditions and values given in table 1, the second order approximation of equation 1 for region 1 is, (2) and for region 3: (3) Permeability equation. Permeability is a membrane property that depends on the relative ionic mobility and solubility [9]. According to Armstrong [11], the membrane permeability to potassium (P k ) and sodium (P Na ) can be given by (4) (5) where V m is the membrane potential, F is Faraday constant and R is the gas universal constant. The factors 0.2 and refer to the resting potential at -80 mv. Considering that depends on the difference between the inner and the outer surface charge densities (Δσ) of lipid bilayer, in addition the dependence of electrolytic charges, equations 4 and 5 can write as (6) (7)

4 Ionic current density. For a system conservative system, the diffusion force ( ) is related with potential energy ( ) as Thus, for ions moving in a channel along axis x, which is under action of osmotic and electric forces, we can write that, where is electrochemical potential, defined as the sum of the chemical (u c ) and electrical potentials (u e ), and is unitary vector. The electrochemical potential (), which is the electrochemical potential for each ionic solute in a diluted solution is given by, where u 0 is pattern chemical potential. Substituting the last in the precedent equation and deriving, we have (8) The velocity v imposed to the molecules by osmotic and electric fields across channel is given by, where E is the osmotic or electric field related with its respective force actuating on the ion (E=F e /q) and μ is the ionic mobility (μ=u c /q). Considering the relation between ionic flux and ionic current is, the current density is, and that, for E constant, we can write that, being V m=δv and h the bilayer thickness. Taking into account only monovalent ions, z=±1, after the integration, we can write, that (10) where η ou and η in are the outer and inner molar concentrations of bilayer, respectively, and P is the permeability, Introducing Eqs. (6) and (7) in Eq (10), we will have that density equation of current for sodium and potassium in function of electrochemical potential and surface potential. In this way, (11) (12) Magnetic field equation for sodium and potassium ionic channels. Finally, substituting Eqs (11) and (12) in the classical Ampère equation for the magnetic field,, we have finally the expressions: (13) (14) RESULTS AND DISCUSSION The model was tested by applying numerical values in its equations using the FEM. In this, a complex system of partial differential equations is solved by discretization methods, and a continuous domain can be discretized into many small subdomains, or finite elements, conserving the original properties, resulting in a complex system of points or nodes that make a mesh. The computation time depend on the node density. This numerical analysis method has been considered a computational power tool in several science areas [12,13]

5 Using Eqs. (13) and (14) it was possible to determine the Na + and K + -current densities taking into account the classic Ampère, Armstrong equations, and our considerations on surface bilayer charge, making our model closer to the biological profile. Figures 2 (a) and (b) illustrate the representation of magnetic flux density around the Na + current carrying channel and the K + current carrying channel, respectively. As shown these figures, the magnetic field profiles surrounding both Na + and K + -current carrying channels are similar to that in the current carrying conductor: the maximal intensity at channel border and a quick fall with increase of radial distance [14]. It is important to note that the magnetic field intensity for the Na + -channel is higher than that for the K + -channel, being the radial magnetic flux fall in the first channel slower than in the second. Different microscopic modelling approaches have been presented to simulate ionic channel dynamics, and all model types have advantages and disadvantages. Traditionally, the standard model based on the Poisson-Nernst- Planck equations have been used to describe electrodiffusion in various systems [14-16], including the transport of ions across an open channel [17]. These equations extend the first Fick s law of diffusion for the case in which the diffusing particles are also moved in a fluid by electrostatic forces [18]. (a) (b) FIGURE 2. Magnetic flux density around the (a) Na + channel and (b) K + channel. CONCLUSIONS The results allow us to conclude that each channel produces its specific and individual response to the ion transport, according to its individual intrinsic properties. The existence of a number of active Na + -channels a given membrane region does not appear to interfere directly in the functioning of K + -channel located among them, and vice-versa, generating only a short-range perturbation. Finally, the simulation using FEM attended our expectative and seems to be a adequate method to simulate and analyze cases more complex cases of the problem, depending only on the adopted physico-mathematical model

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