Teorell

The most popular theoretical description of the potentiometric behavior of ion-selective membranes makes use of the three-segmented membrane model introduced by Sollner53), Teorell 30,54), and Meyer and Sievers 31-5S). In this model the two phase boundaries and the interior of the membrane are treated separately. Here, the... 

Teorell T. Kinetics of distribution of substances administered to the body. Arch Intern Pharmacodyn 1937 57 205-40. 

T. Teorell, Kinetics of distribution of substances administered to the body. 1. The extravascular modes of administration. Arch. Int. Pharmacodyn., 57 (1937) 205-225. II. The intravascular modes of administration. Arch. Int. Pharmacodyn., 57 (1937) 226-240. 

The distribution of electric potential across the membrane and the dependence of the membrane potential on the concentration of fixed ions in the membrane and of the electrolyte in the solutions in contact with the membrane is described in the model of an ion-exchanger membrane worked out by T. Teorell, and K. H. Meyer and J. F. Sievers. 

The search for models of biological membranes among porous membranes continued in the twenties and thirties. Here, Michaelis [67] and Sollner (for a summary of his work, see [90] for development in the field, [89]) should be mentioned. The existence and characteristics of Donnan membrane equilibria could be confirmed using this type of membrane [20]. The theory of porous membranes with fixed charges of a certain sign was developed by Teorell [93], and Meyer and Sievers [65]. 

Finally, in 3.4 we present a calculation of membrane potential in terms of the classical Teorell-Meyer-Sievers (TMS) [2], [3] model of a charged permselective membrane. In spite of its extreme simplicity, this calculation yields a practically useful result and is typical for numerous membrane computations, some more of which will be touched upon subsequently in Chapter 4. 

Below we present a well-known calculation of membrane potential based on the classical Teorell-Meyer-Sievers (TMS) membrane model [2], [3]. The essence of this model is in treating the ion-selective membrane as a homogeneous layer of electrolyte solution with constant fixed charge density and with local ionic equilibrium at the membrane/solution interfaces. In spite of the obvious idealization involved in the first assumption the TMS model often yields useful results and represents in fact the main tool for practical membrane calculations. We shall return to TMS once again in 4.4 when discussing the electric current effects upon membrane selectivity. In the case of our present interest, the simplest TMS model of membrane potential for a 1,2 valent electrolyte reads... 

Different versions of the above calculations, carried out for particular ionic contexts, form the basis of numerous studies of the ion-selective membrane transport, starting with classical papers by Teorell [7], and Meyer and Sievers [8]. Without attempting to give a full or merely fair account of all these studies, we shall mention here just a typical few Schlogl [5] (arbitrary number of ions in a monopolar membrane), Spiegler [9] (am-bipolar ionic transport in a unipolar membrane and solution layers adjacent to it — concentration polarization), Oren and Litan [10], Brady and Turner [11], Rubinstein [12] (multipolar transport in a unipolar membrane and the adjacent solution layers—effects of concentration polarization upon... 

T. Teorell, An attempt to formulate a quantitative theory of membrane permeability, Proc. Soc. Expt. Biol. Med., 33 (1935), p. 282. 

For comparison, we present in Fig. 5.3.2 some numerical results for the following non-locally-electro-neutral generalization of the classical Teorell-Meyer-Sievers (TMS) model of membrane transport (see [11], [12] and 3.4 of this text). 

Preliminaries. This entire chapter is devoted to one physical phenomenon—electro-osmotic (Teorell) oscillations. As opposed to phenomena discussed in previous chapters, electro-convection will be of importance here in its interaction with electro-diffusion. 

Electro-osmotic oscillation (first observed by Teorell [1]—[4] in a laboratory set-up devised to mimic nerve excitation) may likely represent a common source of oscillations in various natural or synthetic electrokinetic systems such as solid microporous filters, synthetic ion-exchange membranes or their biological counterparts. The original experimental set-up, which contained all essential elements to look for when the electro-osmotic oscillations are suspected in a natural system, is schematically as follows. 

Here m is the liquid density in vessel II and g is the gravitational constant. For each new current value sufficient time is allowed for the establishment of the steady state if possible. Phenomenology observed by Teorell was roughly as follows. 

To rationalize his observations Teorell suggested the following model in terms of ordinary differential and algebraic equations for the average dynamic characteristics of the system concerned. 

Equation (6.1.4) asserts that the volumetric flow rate is a superposition of two components. They are the electro-osmotic component proportional to the electric field intensity (voltage) with the proportionality factor u> and the filtrational Darcy s component proportional to —P with the hydraulic permeability factor i>. Teorell assumed both w and t> constant. Finally another equation, crucial for Teorell s model, was postulated for the dynamics of instantaneous electric resistance of the filter R(t). Teorell assumed a relaxation law of the type... 

Teorell studied the system (6.1-1)—(6.1.5) graphically, by the isocline method, and also numerically. He recovered most of the features observed experimentally. This study was further elaborated by several investigators. Thus, Kobatake and Fujita [5], [6] criticized the original model for invoking the ad hoc equation (6.1.5). These authors assumed instead instantaneous relaxation of the resistance to its stationary value while preserving the overall order of the relevant ordinary differential equation (ODE) system by including consideration of the mechanical inertia of the liquid column in the manometer tube. 

Here i = ujR0I/V0 is the dimensionless current —the control parameter of this model. Furthermore, f(pt) is the dimensionless stationary resistance function of a general shape sketched in Fig. 6.1.3 and k = k/sv 1 is the dimensionless relaxation parameter, assumed large in accordance with Teorell s intuitive ideas. 

To elaborate somewhat on the above issues and mainly to pave the ground for the treatment of the PDEs. version of the Teorell model in the next section we conclude here with the standard weakly nonlinear analysis of the vicinity of bifurcation i — ic in the model (6.2.7). 

Generalized local Darcy s model of Teorell s oscillations (PDEs) [12]. In this section we formulate and study a local analogue of Teorell s model discussed previously. The main difference between the model to be discussed and the original one is the replacement of the ad hoc resistance relaxation equation (6.1.5) or (6.2.5) by a set of one-dimensional Nernst-Planck equations for locally electro-neutral convective electro-diffusion of ions across the filter (membrane). This filter is viewed as a homogenized aqueous porous medium, lacking any fixed charge and characterized... 

We thus observe that M is positive for all w, A > 0 and, therefore, the bifurcation is supercritical. This stands in contrast with the result for the classical Teorell model discussed in the previous section. Recall that there the type of bifurcation depended on the properties of the postulated stationary resistance function /( ), specifically on the sign of / "(0) whenever... 

To stress this difference between the local and the classical Teorell s model let us work out the overall stationary resistance of the filter R(v, A, D) for a given time-independent flow rate v. We have analogously to (6.3.30)... 

Bifurcation analysis in the local Teorell model accounting for negative osmosis and concentration dependence of the electro-osmotic factor as prescribed by (6.4.44). The analysis should be essentially identical to that of 6.3 with the generalized Darcy s law (6.3.11) replaced by the expression... 

K. R. Page and P. Meares, Factors controlling the frequency and amplitude of the Teorell oscillator, Faraday Symp. Chem. Soc., 9 (1974), pp. 166-173. 

T. Erneux and I. Rubinstein, Hopf bifurcation in a local model of Teorell oscillations, to appear. 

Gelperin NI, Nosov GA, Makotkin LV (1971) Teorel. Osnovy Khimich. Technologii 5 429... 

In 1935 T. Teorell (168) and in 1936 K. H. Meyer and J. F. Sie-vers (97) independently pictured a model for an ion-selective membrane which can account for its properties and the main features of which are therefore generally accepted now. 

Taking for the diffusion potential the Henderson diffusion potential and adding the two Donnan potentials one gets the well-known formula of K. H. Meyer (97, 98), J. F. Sievers (97, 98) and T. Teorell (168). 

Concentration- or Dialysis Potential. The refined formula of Meyer, Sievers and Teorell for the concentration potential has been experimentally tested by several investigators. G. Manecke and K. F. Bonhoeffer (89) and K. F. Bonhoeffer and U. Schindewolf (21) found that for external concentrations smaller than appr. x/3 of the concentration at fixed charged groups the general trend of the potential with the external concentration was as indicated by the formula of K. H. Meyer and J. F. Sievers (97) [equation (44)]. Above this value there are deviations. 

M.S.T.-model = Meyer-Sievers-Teorell-model (see Introduction, sec. 1). 

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