Equation for the Membrane Potential

As most of the J and K ions are ion-paired, then cj + c a = where Cq is the overall concentration of ion-exchanger ion A". The same procedure as used above yields the equations for the membrane potential and the ISE potential. 

As similar relationships hold for the second membrane/solution boundary, the equation for the membrane potential can be obtained from (3.4.6) ... 

Cationic response. The equation for the membrane potential for higher concentrations of cation J is obtained in a similar manner ... 

It is then necessary to assign a numerical value to Cx(0), which can then be substituted into the equation for the membrane potential (3.4.9)... 

While historically things were described a little differently, all in all, Hodgkin and Huxley had one equation for the membrane potential and three more for describing conductance. They had a total of four differential equations. Four differential equations is a manageable number and one that might benefit from reduction because two is very close to four. Indeed, two prevents chaos and allows two dimensional qualitative analysis of the differential equation. The reduction to two dimensions is precisely what Nagumo et al. (1962) and Fitzhugh (1961) have done. They have done a... 

Mullin and Noda have derived an equation for the membrane potential in the presence of electrogenic pumping under a steady state condition/... 

The ISEs are often use a special thin membrane for signal generation. The membrane separates the tubular lumen of the electrode from the sample solution. There is an internal filling solution inside containing the Oi activity of the detected ion. If the activity of the detected ion in the sample is 02, then the Nernst equation for the membrane potential E is... 

When the expressions (6.2.5) are substituted into the Henderson equation (2.5.34) A0l is obtained. Both contributions A0D are calculated from the Donnan equation. From Eq. (6.2.3) we obtain, for the membrane potential,... 

Because a similar equation holds for the membrane/solution 2 phase boundary and no diffusion potential is formed within the membrane, (3.2.3) is valid for the membrane potential. 

If the diffusion potential in the membrane is neglected, this equation yields relationships for the membrane potential, for the ISE potential and for the selectivity coefficient. It is apparent that formation of complexes with various numbers of ions in the membrane does not affect the dependence of the ISE potential on the activities of the determinand and interferent according to the Nikolsky equation. 

Activity aK (m) is eliminated using (3.2.23). It can be seen that the solution is similar to that ven in (3.2.22) to (3.2.27). The resultant equation for the membrane/analyte potential yields an expression for the selectivity coefficient,... 

Exchange reaction (3.2.13) takes place between the ions in solution and those in the membrane. The procedure is the same as for (3.2.30) to (3.2.33), so that (assuming that the activity coefficients of the ions are equal to one in the membrane) an equation is obtained for the membrane potential ... 

These equations offer an adequate basis for the development of the negative membrane potential of 70 to 90 mV. Excitation as a process characterizing nerve and muscle cells is associated with a transient reduction or abolition of this membrane potential, and in some cases with a temporary "overshoot" or reversal of its polarity. Just as for the membrane potential, these major but transient perturbations in the production of action potentials have been adequately modeled in dynamics of ionic equilibria by Hodgkin and Huxley (2). 

This procedure results in the following equation for a membrane potential (Koryta, 1991) ... 

Equation 3.17 describes the net ionic flux densities leading to the electrical potential differences across a membrane (Fig. 3-7). After substituting the expressions for the variousiy s into Equation 3.17, we will solve the resulting equation for the diffusion potential across a membrane, EM. 

Here we have assumed that the relative permittivity is assumed to take the same value inside and outside the membrane. We also assume that the distribution of counterions n(x) obeys Eq. (18.4) and thus the charge density PeiW is given by Eq. (18.5).Thus, we obtain the following Poisson-Boltzmann equations for the scaled potential y x) = ze l/(x)/kT ... 

When two electrolyte solutions at different concentrations are separated by an ion--permeable membrane, a potential difference is generally established between the two solutions. This potential difference, known as membrane potential, plays an important role in electrochemical phenomena observed in various biomembrane systems. In the stationary state, the membrane potential arises from both the diffusion potential [1,2] and the membrane boundary potential [3-6]. To calculate the membrane potential, one must simultaneously solve the Nernst-Planck equation and the Poisson equation. Analytic formulas for the membrane potential can be derived only if the electric held within the membrane is assumed to be constant [1,2]. In this chapter, we remove this constant held assumption and numerically solve the above-mentioned nonlinear equations to calculate the membrane potential [7]. 

Szymczyk et al. [55] published an equation for the streaming potential of the n-th layer of multilayer membrane when the streaming potential of the entire membrane and of a membrane composed of the remaining n-1 layers is known. A number of IEP values was obtained by means of this equation, but the results are not used in Tables 3.1 and 3.3. 

The derivation presented here gives a greatly oversimplified picture of the actual situation in a membrane with fixed ion exchange sites. Use of the Henderson equation implies that the diffusion process in the membrane has reached a steady state with a linear concentration distribution. This is certainly not the case for solid membranes such as glass which are thick with respect to the diffusion length of the ion. Moreover, it is probably not valid to assume that the ionic mobility is independent of position in the membrane. More complex models for the membrane potential have been developed but they lead to essentially the same result. More details can be found in monographs devoted to this subject [9]. 

The equilibrium determining the Donnan potential is given by equation (9.8.4). Assuming that there is no diffusion potential in the membrane, the Donnan potentials on either side of the membrane are given by the same equations obtained for the ion-exchanging system, namely, equations (9.8.36) and (9.8.38). In addition, the expression for the membrane potential is the simple result given by equation (9.8.40). 

This is the general formula of the Nemst equation for the membrane concentration potential. 

In solving for the membrane potential it is necessary to determine from the transcendental equation [Eq. 136]. This can be done by trial and error. 

Conservation of water and hydronium lead to Nerst-Plank equations for the membrane water content, M, and the electric potential . 

Since the Poisson equation for the electric potential has to be solved at every time step, an efficient and robust Poisson solver is crucial for the total performance of the P M scheme. Beckers et al. [5] adopted the conventional SOR method to solve the Poisson equation in their scheme, while more recently a multi-grid Poisson solver has been implemented into the P M scheme to simulate ionic liquids. Using this scheme, the authors have successfully calculated the thermodynamic properties of electrolyte solutions and examined the charge distribution profile for systems containing a lipid membrane. In this chapter, we introduce the multi-grid Poisson solver due to its superior convergence properties relative to the SOR method. 

The Donnan equilibrium allows the evaluation of the distribution of M and X" over both sides of the semipermeable membrane. If electrodes responding to either M+ or X" were inserted at either side of the membrane there would be no potential difference between them. This is a consequence of the system being in equilibrium which implies that no work can be performed. Nevertheless, because of the different ion concentrations, the potentials at the respective electrodes i/o,i and /o n are not equal. Consequently, there must be a compensating potential difference across the membrane, Axi/. If the electrodes respond reversibly to the ion concentrations so that Nernst s law (Equation 9.14) applies, it follows for the membrane potential... 

In order to accurately interpret the resirlts of EMF measurements at supercritical temperatures, it is necessary to estimate the activity of water in the fluid, as noted above, because the activity of water appears in the equation for the electrode potential of the YSZ membrane as °"... 

Here, i , represents the chemical potential of water, and a , is the transport coefficient of water. The equation for the membrane is ignored, because it is dependent on the other two of Gibbs-Duhem of equations. From many models in the literatures, these equations were used in a Stefan-Maxwell framework. 

The equation for the proton potential is derived from Ohm s law. It represents the proton flux divided by the membrane conductivity. The electroneutrality assumption allows the total molar proton flux to be related directly to the current density and velocity that represents the convective flux of protons. This results in the following equation ... 

The transmembrane potential derived from a concentration gradient is calculable by means of the Nemst equation. If K+ were the only permeable ion then the membrane potential would be given by Eq. 1. With an ion activity (concentration) gradient for K+ of 10 1 from one side to the other of the membrane at 20 °C, the membrane potential that develops on addition of Valinomycin approaches a limiting value of 58 mV87). This is what is calculated from Eq. 1 and indicates that cation over anion selectivity is essentially total. As the conformation of Valinomycin in nonpolar solvents in the absence of cation is similar to that of the cation complex 105), it is quite understandable that anions have no location for interaction. One could with the Valinomycin structure construct a conformation in which a polar core were formed with six peptide N—H moieties directed inward in place of the C—O moieties but... 

We shall write (p) and (q) for the membrane surface layers adjacent to solutions (a) and (p), respectively. Using the equations reported in Section 5.3, we can calculate the ionic concentrations in these layers as well as the potential differences and between the phases. According to Eq. (5.1), the expression for the total membrane potential additionally contains the diffusion potential within the membrane itself, where equilibrium is lacking. Its value can be found with the equations of Section 5.2 when the values of and have first been calculated. 

The conclusion above is valid for ideally selective membranes. Real membranes in most cases have limited selectivity. A quantitative criterion of membrane selectivity for an ion to be measured, relative to another ion M +, is the selectivity coefficient The lower this coefficient, the higher the sefectivity wifi be for ions relative to ions An electrolyte system with an imperfectly selective membrane can be described by the scheme (5.16). We assume, for the sake of simplicity, that ions and have the same charge. Then the membrane potential is determined by Eq. (5.17), and the equation for the full cell s OCV becomes... 

If there is a net transport of charge across the membrane, the membrane potential will influence the solute transfer and also be affected by it, complicating the data treatment. The starting point for most descriptions of the internalisation flux of permeant ions, i, is the one-dimensional Nernst-Planck equation (cf. equation (10)) that combines a concentration gradient with the corresponding electric potential gradient [270] ... 

The action of so-called active transport, also known as ion pumps, facilitates larger Na" /K" gradients than those possible considering calculations of Donnan equilibria. For instance, the concentration of K+ in red blood cells equals 92 mM versus 10 mM in blood plasma. Calculation of the membrane potential using equation 5.11 would lead to a large negative potential ... 

Until the 1950s, the rare periodic phenomena known in chemistry, such as the reaction of Bray [1], represented laboratory curiosities. Some oscillatory reactions were also known in electrochemistry. The link was made between the cardiac rhythm and electrical oscillators [2]. New examples of oscillatory chemical reactions were later discovered [3, 4]. From a theoretical point of view, the first kinetic model for oscillatory reactions was analyzed by Lotka [5], while similar equations were proposed soon after by Volterra [6] to account for oscillations in predator-prey systems in ecology. The next important advance on biological oscillations came from the experimental and theoretical studies of Hodgkin and Huxley [7], which clarified the physicochemical bases of the action potential in electrically excitable cells. The theory that they developed was later applied [8] to account for sustained oscillations of the membrane potential in these cells. Remarkably, the classic study by Hodgkin and Huxley appeared in the same year as Turing s pioneering analysis of spatial patterns in chemical systems [9]. 

The usefulness of the ion-selective electrode for determining ion J is Umited by the distribution equilibrium for salt JA between the membrane and solution 1. If the activity of ion J which is present in solution 1 only as a result of this distribution equihbrium, is much smaller than the overall activity aj+(l), then the dependence of the membrane potential on the activity of determinand J has Nernstian slope. If this condition is not fulfilled, then the ISE behaviour can be obtained from the equation for the distribution equihbrium... 

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