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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):
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
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,
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 :
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
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 .
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