Passive electrical properties of membranes

Objectives

1.

Define passive membrane electrical properties as those due to parameters that are constant near the resting potential of the cell.
2.

Explain why membranes behave, electrically, like a resistor in parallel with a capacitor.
3.

Explain why open ion channels are electrically equivalent to conductors (or resistors).
4.

Use Ohm’s Law to calculate current flow through ion channels.
5.

Explain why membranes have capacitive properties.
6.

Define membrane time constant and length constant, and describe the passive properties that influence them.

The time course and spread of membrane potential changes are predicted by the passive electrical properties of the membrane

The steady-state V _m of a membrane permeable to more than one ion can be estimated with the GHK equation ( Chapter 4 ). One of the main limitations of this approach, however, is that it cannot be used to predict how the V _m changes as a function of time or distance. In neurons, skeletal muscle cells, and other electrically excitable cells, certain changes in V _m have well-defined time courses. For example, the nerve AP ( Chapter 7 ) is a “spike” of depolarization that lasts 1 to 2 milliseconds. In addition, postsynaptic potentials ( V _m changes that result from neurotransmitter release at chemical synapses; Chapters 12 and 13 ) have relatively fast rising phases and exponential decays. The membrane properties that help determine the time course of these signals are described in this chapter. The passive spread of a change in V _m with distance along a membrane surface is also discussed.

Passive electrical properties refer to properties that are fixed, or constant, near the resting potential of the cell. Three such properties play important roles in determining the time course and spread of electrical activity: the membrane resistance, the membrane capacitance, and the internal or “axial” resistance of long thin processes or cells such as nerve axons and dendrites and skeletal muscle cells. By examining the membrane as an electrical circuit, we can deduce how these parameters can be used to describe changes in V _m . Equivalent circuit models of membranes are used to analyze potentials that vary with time and distance in a manner that depends only on the passive membrane properties. These potentials are called electrotonic potentials .

The membrane can be represented by an electrical equivalent circuit with a resistor and a capacitor in parallel

Membrane conductance is established by open ion channels

Many ion channels behave, in electrical terms, like conductors (or resistors, because conductance = 1/resistance). Each channel ( Fig. 6.1 A) can be modeled as a resistor, or conductor, with a single-channel conductance, γ, when the channel is open ( Fig. 6.1 B). Naturally, when the channel is closed, the conductance is zero.

Most permeant ions are distributed asymmetrically across the plasma membrane ( Chapter 4 ). This results in a chemical driving force that tends to push the ion through the open channel. This chemical force functions as a battery (with voltage equal to the equilibrium potential of the ion, E _K in the case of Fig. 6.1 ). The battery is in series with the resistor (γ _K ) representing the open channel, as shown for K ⁺ in Fig. 6.1 B. The current flow through the open channel obeys Ohm’s Law ( Appendix D ), which describes current flow through a resistor with a resistance of R in ohms (Ω), or with a conductance g = 1/ R in siemens (S):

?=?×?or?=??= g ×?V=l×Rorl=VR= g ×V
V=l×Rorl=VR= g ×V

where I is the current in amperes (A) for a potential difference of V in volts (V). For ion channels, Ohm’s Law must be modified, because the net ionic flux (and therefore the current) will be zero when V _m is equal to the equilibrium potential of the ion. Because the equilibrium potential is almost never 0 mV, Ohm’s Law for a single K ⁺ channel is

?K=γK(?m–?K)iK=γK(Vm-EK)
iK=γK(Vm-EK)

where i _K is the current through a single channel and V _m – E _K is called the driving force . Ohm’s Law predicts that the K ⁺ current is directly proportional to the driving force ( Fig. 6.1 C).

Membranes usually contain several different types of ion channels that are each present in large numbers. In electrical terms, single channels in the membrane represent conductors arranged in parallel; in this case the individual conductances are additive. In other words,

?Na=?o×γNagNa=No×γNa
gNa=No×γNa

where g _Na is the total conductance of the open sodium (Na ⁺) channels present in a unit area of membrane, γ _Na is the conductance of a single Na ⁺ channel, and N _o is the number of open Na ⁺ channels per unit area. In an equivalent circuit, we can then model a group of Na ⁺ channels as a resistor with conductance equal to g _Na , in series with a battery of voltage E _Na ( Fig. 6.2 ). A similar resistor-battery pair can be used to model a population of potassium (K ⁺ ) channels , or any other ion channels, in the membrane ( Fig. 6.2 ).

Capacitance reflects the ability of the membrane to separate charge

To complete the equivalent circuit, we need to account for the ability of the lipid bilayer to act as an electrical insulator that allows charges (ions such as K ⁺ , Na ⁺ , and Cl ⁻ ) to accumulate at the surface of the membrane. In electrical circuits a capacitor is an element that stores, or separates, charges across an insulator. Thus the equivalent circuit of the membrane has a capacitor ( C _m ) connected in parallel with the elements representing the ion channels ( Fig. 6.2 ).

The amount of charge, q , in coulombs (C), that can be separated across the membrane is directly proportional to V _m :

?=?m×?mq=Cm×Vm
q=Cm×Vm

where V _m is the potential difference in volts and C _m is the capacitance in farads (F). A 1 F capacitor can store 1 coulomb of charge per volt of potential difference. A farad is a very large quantity; all biological membranes have capacitances of approximately 1 × 10 ^–6 F (1 μF)/cm ² of membrane surface area ( Box 6.1 ).

BOX 6.1

Cell Size and Total Cell Capacitance

The capacitance of all biological membranes is typically C _m = 1 μF/cm ² . It is important to note, however, that for a single cell, even 1 μF is a large quantity, as illustrated by a calculation of the capacitance of a cell 10 μm in diameter (slightly larger than a red blood cell). The surface area of the cell is A _cell = π d ² , where d is the diameter. For d = 10 μm (10 ⁻³ cm), A _cell = 3.1 × 10 ⁻⁶ cm ² .

The total cell capacitance is

?cell=?m×?cellCcell=Cm×Acell
Ccell=Cm×Acell

=10–6F⋅cm–2(3.1 ×10–6cm2)=3.1×10–12F=3.1 pF=10-6F⋅cm-2(3.1 ×10-6cm2)=3.1×10-12F=3.1 pF
=10-6F⋅cm-2(3.1 ×10-6cm2)=3.1×10-12F=3.1 pF

Because cell surface area is on the order of hundreds of μm ² , it may be more relevant to use units of pF/μm ² for C _m where 1 pF (picofarad) = 10 ⁻¹² F. Thus C _m = 0.01 pF/μm ² .

Passive membrane properties produce linear current-voltage relationships

The passive properties of cell membranes can be studied by passing current across the cell membrane using microelectrodes ( Box 6.2 and Fig. 6.3 A). When an inward pulse of current is passed across the membrane ( Fig. 6.3 B), V _m approaches a more negative, or hyperpolarized , value following an exponential time course ( Fig. 6.3 B) and eventually reaches a constant, steady-state level. The larger the applied current, the greater the hyperpolarization. A graph of the current versus the steady-state V _m is a straight line ( Fig. 6.3 C). Therefore in the steady state, the resting membrane behaves electrically like a resistor. The slope of the linear current-voltage relationship (i.e., ΔI/Δ V _m ) is a measure of the resting conductance of the membrane. The V _m at which the current-voltage line crosses the voltage axis is called the reversal potential (E _rev) . At potentials negative to E _rev the current is negative (inward) and at potentials positive to E _rev the current is positive (outward); thus the current reverses direction at E _rev .

BOX 6.2

Measuring and Manipulating the Membrane Potential

The most common technique for recording V _m involves the use of microelectrodes. The microelectrode is a glass capillary tube tapered to a fine, sharp tip with a diameter smaller than 1 μm. The electrode is filled with a concentrated salt solution, and a wire is placed in the back of the electrode to allow connection to electronic devices. The sharp tip of the microelectrode allows it to be pushed through the cell membrane without damaging the cell, thereby allowing the measurement of the intracellular potential. A pair of these electrodes (one intracellular and one extracellular) is connected to an amplifier to record V _m , as shown in Fig. 6.3 A.

A second pair of electrodes can be used to pass current across the cell membrane. Fig. 6.3 A shows passage of an inward current: positive charges flow out of the extracellular electrode and are deposited on the outside surface of the cell, whereas positive charges move away from the inside surface of the membrane and enter the intracellular electrode. Thus this inward current makes V _m become more negative.

Membrane capacitance affects the time course of voltage changes

Ionic and capacitive currents flow when a channel opens

When current starts to flow across the membrane, V _m does not instantaneously reach a new steady-state level. Instead, the presence of membrane capacitance causes V _m to approach the steady-state level gradually ( Fig. 6.3 B). To understand the effect of the membrane capacitance, consider a cell that contains only a single closed K ⁺ channel ( Fig. 6.4 A). If V _m is 0 mV when the K ⁺ channel opens, and E _K is −90 mV, K ⁺ ions will flow out of the cell down their electrochemical gradient. The removal of positive charge from the cell makes V _m move in the negative direction. Because the cell is permeable only to K ⁺ , V _m will eventually reach E _K and stop changing. In terms of current flow in the equivalent circuit ( Fig. 6.4 B), an outward ionic K ⁺ current, I _K (K ⁺ ions moving out of the cell), produces an inward capacitive current , I _c . This capacitive current consists of positive charges moving away from the inside surface of the membrane and an equal number of positive charges moving up to the outside surface of the membrane. Because it takes time for ions to move through the channel and accumulate at the membrane surface, V _m can change only gradually. The general term ionic current ( I _i) refers to current generated by ions crossing the membrane, such as through channels.

The exponential time course of the membrane potential can be understood in terms of the passive properties of the membrane

The equivalent circuit of a resting (passive) membrane ( Fig. 6.5 A) can help explain the role of the membrane capacitance in the exponential time course of the V _m change. All open ion channels are combined into a single conducting pathway, R _m , which is in series with battery E _RP , representing the resting potential of the cell. The membrane is connected to a constant current generator through a switch. With no current flowing from the external source, V _m will be at E _RP with an excess of negative charges at the inside surface of the capacitor. At the instant the switch is closed and a constant inward current is turned on, all of the current flows to the capacitor ( Fig. 6.5 B). Perhaps the simplest way to understand this is to recognize that at the instant the current is turned on, there is no driving force for current flow through the resistor (i.e., V _m – E _RP = 0). The inward capacitive current further increases the charge separation across the capacitor because positive charges build up at the external surface and move away from the internal surface. As a result, V _m moves in the negative direction, which in turn produces a driving force for current flow through the resistor. Because the total current being controlled by the constant current generator must remain constant, the capacitive current ( I _c ) decreases in magnitude as the resistive (ionic) current ( I _i ) increases ( Fig. 6.5 C). Finally, a new steady-state V _m is reached, and all the applied current flows through the resistor. Analysis of this circuit ( Appendix D ) produces the following relationship between V _m and time: