Cell membranes are modeled with electrical circuits
The plasma membrane can be modeled as an electrical circuit. Fig. AppD.1 A is a schematic view of the structure of a biological membrane with a single open K + –selective ion channel. This physical entity is electrically equivalent to the circuit shown in Fig. AppD.1 B. The circuit consists of a resistor in series with a battery, and this combination is in parallel with a capacitor. In Fig. AppD.1 , we follow the convention that the positive pole of a battery is represented by a longer bar. The electrical behavior of most resting biological membranes is, in fact, indistinguishable from a circuit similar to that shown in Fig. AppD.1 B. Therefore we can use such equivalent circuits to obtain quantitative descriptors of membrane electrical behavior, such as the length constant and the membrane time constant ( Chapter 6 ). The equivalent circuit is also used to describe current flow across membranes and the effect of current flow on membrane potential ( V m ), such as how the action potential is generated and propagated along an elongated structure such as an axon or a skeletal muscle cell ( Chapter 7 ).
Definitions of electrical parameters
Electrical potential and potential difference
The potential difference between two points is the amount of work done per unit charge to move charge from one point to the other. The unit of measure of potential difference is the volt. Potential difference (often called voltage ) reflects the electrostatic force exerted on a charge. We use the symbol E to represent theoretical potentials, such as the equilibrium potential of an ion ( Chapter 4 ), and the symbol V for actual voltages, such as V m .
Current
When a potential difference exists across space, positive charges will move toward the region with the more negative potential and negative charges will move toward the region with the more positive potential. This movement of charge is a current ( I ). The current at a point in a circuit is defined as the net movement of positive charge past that point per unit time. The movement of 1 coulomb of charge per second is a current of 1 ampere (A). In wires and electronic devices, the current is carried by electrons. In biological systems current is carried by ions (e.g., Na + , K + , Cl − , Ca 2+ ) moving in an aqueous environment.
By convention, when an arrow is used to indicate current flow, it shows the direction of the net movement of positive charge. The physical reality corresponding to such a representation could be either positive charges moving in the direction of the arrow or negative charges moving in the opposite direction.
Resistance and conductance
The current that flows through a conductive material is proportional to the potential difference ( V ) across the material. We refer to such a material as either a conductor or a resistor. In electronic circuits resistors are made of carbon or some other low-conductivity material and are characterized by their resistance ( R ). Intuition tells us that the current through the resistor should increase as the driving force ( V ) across the resistor increases. This relationship is quantitatively given by Ohm’s Law, which states that the current through a resistor is directly proportional to the voltage across it:
V=I⋅R
Resistance is measured in ohms (Ω): A current of 1 ampere will flow through a 1-ohm resistor with a driving force of 1 volt. Open ion channels behave electrically like conductors (or resistors): the membrane potential drives ion movement through open channels. The ability of a channel to pass current is characterized as its conductance ( g or γ), which is the reciprocal of resistance: g = 1/ R. The unit of conductance is the siemens (S): 1 S = 1/Ω = Ω −1 .
Capacitance
A capacitor is a device that can store, or separate, charges of opposite sign. A parallel plate capacitor has two parallel conducting plates (e.g., made of metal foil) separated by an insulator (e.g., mica, Mylar, glass, air). Biological membranes have capacitive properties because the lipid bilayer is an effective electrical insulator that allows ions to be separated across the membrane. The amount of charge stored on a capacitor is directly proportional to the voltage across the capacitor:
q=C⋅V
where q is the charge in coulombs, V is the voltage in volts, and C is the capacitance of the device in farads. Note that + q coulombs are stored on one plate and − q coulombs on the other plate. The capacitance of a given device, or a given area of membrane, is a constant, so Equation AD.2 shows that if V increases, q must increase.
By taking the time derivative of Equation AD.2 , we obtain
dqdt=CdVdtorIc=CdVdt
The quantity dq/dt (charge per unit time, or coulombs per second) is a current ( I c , for capacitive current). Thus in contrast to a resistor, where the current is proportional to the voltage, in a capacitor the current is proportional to the rate of change of the voltage. This important relationship shows that if the voltage is changing (i.e., dV/dt ≠ 0), capacitive current is flowing. Conversely, I c = 0 when the voltage is constant (i.e., when dV/dt = 0).
Current flow in simple circuits
The circuits in the following sections are analyzed by using the rules described in Box AppD.1 .
Two relationships, known as Kirchhoff’s Laws, are useful in circuit analysis.
Kirchhoff’s current law
When an arrow is used to indicate the direction of current flow, it shows the direction of net positive charge movement. Thus the end of the resistor that the current enters is at a positive voltage relative to the end the current leaves.
Charge can never accumulate at any point in a circuit. Thus Kirchhoff’s First Law states that the sum of the currents flowing into a point in a circuit must equal the sum of the currents flowing out of that point (i.e., there is conservation of charge). Another way to state this law is that the sum of all currents entering and leaving a point in a circuit is zero: we adopt the convention that currents entering are positive and currents leaving are negative. Using Kirchhoff’s Current Law for the point labeled 1 in Fig. AppD.2 gives the result
IA-IB-IC=0
We will come back to this example later in Box AppD.2 . From Kirchhoff’s Current Law we can also deduce that in a series circuit (where elements are connected end to end), the current must be the same everywhere.
Kirchhoff’s voltage law
Kirchhoff’s Voltage Law states that the algebraic sum of all voltage drops around a closed loop is zero. As a result, elements connected in parallel have the same voltage across them. The following sign conventions should be used: (1) the voltage drop across a resistor is positive in the direction of current flow and (2) the voltage drop across a battery is taken as positive if we meet the + pole of the battery going around the loop and negative if we meet the – pole. In the circuit in Fig. AppD.2 the voltage drops across resistors R A and R B are equal to I A R A and I B R B , respectively. If we apply Kirchhoff’s Voltage Law to the loop with battery V b and the resistors R A and R B , we would write
IA⋅RA+IB⋅RB-Vb=0
For the loop with R B and R C the voltage law gives
IC⋅RC-IB⋅RB=0orIC⋅RC=IB⋅RB
Using Equations AD.4 , AD.5 , and AD.6 , we can solve for I A , I B , and I C as follows. Rearranging Equation AD.5 gives
IA=Vb-IBRBRA
Substitution of Equations AD.6 and AD.7 into Equation AD.4 gives the following, which can be solved for I B :
Vb-IBRBRA-IB-IBRBRA=0
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