Cell membrane resistance and capacitance

Transcription

1 Cell membrane resistance and capacitance 1

2 Two properties of a cell membrane gives rise to two passive electrical properties: Resistance: Leakage pathways allow inorganic ions to cross the membrane. Capacitance: The ion-impermeant lipid bilayer can separate electrical charge. 2

3 Fig.1 Effect of Cell Size on its Input Resistance V small > V large V large I I Current Source V small Input Resistance of the Larger Cell is lower than that of the Smaller Cell 3

4 Membrane Resistance (R) If a step pulse of steady current is applied across the membrane, the membrane potential shifts by Vm from the resting value: V m R I It is useful when comparing membranes of different cells to correct for the effect of membrane area on the current density. The specific membrane resistance is calculated as: R m Vm RA A Ohms.m 2 I 4

5 Membrane Capacitance (C) Because they are very thin (~4 nm) and virtually impermeable to ions over most of their surface area, cell membranes can violate the principle of electroneutrality at the microscopic scale. Negative charges accumulated at or near one surface of a membrane will interact electrostatically, over the short distance of the membrane thickness, with posive charges on the other side of the membrane. The ability of the cell membrane to accumulate and separate electric charge is called its membrane capacitance. The relationship between potential V and time during the charging of the membrane capacitance is given by: V t V o e / ( ) RmCm 5

6 Fig.2 Equivalent Circuit for a Cell Membrane across which an abrupt pulse of constant current is passed Current source i c I m i r R m V m i r i c + C m - I m + - Time courses of resistive current i r, capacitive current i c, membrane potential V m (across membrane resistance and capacitance), and Membrane the total membrane current I m. 6

7 Membrane Resting Potential Extent of Ion Transfer Consider a cell of radius 10 m having a membrane capacitance of 10 mf/m 2. The total surface area of this cell is m 2, to give a total membrane capacitance of F. To set up a potential of 70 mv will require a charge Q given by Q = VC = 70x10-3 x = 8.8x10-13 C. Dividing by the Faraday constant (9.65x10 4 C/mol) leads to the result that we require the equivalent of 9.1x10-18 mol of monovalent ions to be transferred across the membrane to generate a membrane potential of -70 mv. 7

8 Membrane Resting Potential Extent of Ion Transfer The cell volume = ( )/3 m 3 = ( )/3 L. The potassium content of a cell of this volume, when the potassium is present at 150 mm, is roughly ( )/3 x 0.15 = 6.3 x moles (we will assume an activity coefficient of unity). Thus, to charge the membrane to -70 mv requires as little as 1.4x10-3 % of the cell s total potassium to be transferred across the membrane. The rule of electroneutrality - that positive charges must equal negative charges remains essentially unviolated at the macroscopic scale. The imbalance of charges exists only at the microscopic scale across the membrane thickness. 8

9 Membrane Equilibrium Potentials 9

10 Membrane Equilibrium Potentials The electrical energies of the transmembrane potentials of cells are responsible for nearly all the electrical phenomena that occur in the animal body. These potentials originate from two features of biological membranes: Assymetrical distribution of ions between the intracellular and extracellular compartments Selective permeability of the membrane. 10

11 Fig. 1 Electrochemical Equilibrium M 0.01 M 0.1 M 0.01 M [ ] [ ] (a) (Cl - cannot pass through membrane) 1 2 [ ] (b) 1 2 [ ] [ ] (c) emf [ ] 11

12 Membrane Equilibrium Potentials Each additional that diffuses from side 1 to 2 adds its positive charge to that side, and Cl - is left behind since it cannot cross this hypothetical membrane. This generates an electrical potential difference (a back emf). Thus, each ion now entering the membrane has two forces acting on it: a chemical p.d. favouring net flux from 1 to 2, and an electrical p.d. favouring net flux from 2 to 1. These two opposing forces come into equilibrium and remain balanced. The potassium ion is then said to be in electrochemical equilibrium. The p.d. that is established across a membrane in this way is termed the equilibrium potential for the ion in question. 12

13 Fig. 2 Electrochemical Equilibrium + E Cl - R Cl (infinite) Equivalent circuit for development of a potential across the membrane in Figure 1. - E K + Side 1 Side C m R K + + E K supplies the emf for to carry current through the membrane s potassium channels R K. This transport causes the buildup of positive charge on side 2 of the membrane capacitance C m. V m 13

14 Nernst Equation The Nernst equation is one of the most widely used mathematical relationships in studies of bioelectric phenomena. Its derivation is based on the concept of a thermodynamic equilibrium between the osmotic work that is required to move a given number of ions across a membrane in one direction, and the electrical work required to move the same number of charges back across the membrane in the opposite direction. The potential across the cell membrane that exactly opposes net diffusion of a particular ion through the membrane is called the Nernst potential for that ion. 14

15 Nernst Equation The magnitude of this potential is determined by the ratio of the concentrations of that specific ion on the two sides of the membrane. The greater this ratio, the greater the tendency for that ion to diffuse in one direction, and thus the greater the potential V required to prevent its diffusion. 15

16 Equilibrium Membrane Potential V m Determined by: - Different concentrations of ions across the membrane Na + and Cl - mainly outside the cell and organic anions mainly inside the cell - Different membrane permeabilities of the passive, selective, ion channels 16

17 Nernst Equation V RT zf ln [ ion conc outside ] [ ion conc inside] or V RT 2.3 log10 zf [ ion conc outside ] [ ion conc inside ] R : Universal gas constant: (R = 8.31 J.K 1 mol 1 ) T : Absolute temperature (Kelvin) F : Faraday constant (F = Cmol 1 ) z : Ionic charge (z = +1 for monovalent cation, z = -1 for monovalent anion) 17

18 Nernst Equation For a univalent cation, z = +1, and 2.3 = 59.1 mv at 25 o C. At 25 o C, we obtain a value for the Nernst Potential V of mv for C in /C out = 10. If a univalent cation diffuses down this concentration gradient to the outside of the cell, V will become less negative. For the situation C in /C out = 1, V = 0. RT zf 18

19 Ion Species Intracellular Conc. (mm) Extracellular Conc. (mm) Equilibrium Potential (mv) Ion channel Conductance G (S/m 2 ) Na x x 10-2 Cl x 10-2 Ionic Data for the Membrane of Frog Muscle (at 25 o C) (Ove Sten-Knudsen, Biological Membranes, Cambridge Univ. Press, 2002, pp.389 & 391) 19

20 For a membrane potential V m is -103 mv, the potassium ion current through the membrane would be zero. However, there would be a sodium ion current of (V m - V Na )G Na = -158 x (0.8 x 10-2 ) = -1.3 ma/m 2, as well as a chloride ion current = -17 ma/m 2. This gives a net membrane current of ma/m 2 (i.e., current flowing out of the cell across the membrane). How then, in the presence of these various ion concentrations, and ion channels with their different equilibrium potentials and conductance values, does the membrane potential ever attain an equilibrium value? Can we calculate such an equilibrium membrane potential? The answers form the subject matter of the next lecture. 20