The movement of selected ions across biological membranes generates changes in the intracellular environment that, either directly or indirectly, result in the contraction of the muscle cell. This passage of ions can be studied from a variety of perspectives. A practical approach is to take advantage of the fact that ions carry an electrical charge. As such, the flow of ions across cell membranes can be studied using equipment designed to measure electrical flow, and the properties of excitable membranes can be modeled after the behavior of electric devices. In fact, the subject of electrophysiology is borne out, to a certain extent, by the similarities that can be established between the flow of ions across membranes and the behavior of electrical currents moving through cables. As an introduction to the subject of cardiac electrophysiology, we will first define some basic concepts of bioelectricity to establish the fundamental principles that govern electric currents across cell membranes.
On the Electricity of Biological Membranes
Charge
Most elements in nature tend to maintain an equal number of protons and electrons. However, occasionally electrons are transferred more or less permanently from one element to another, thus creating an imbalance. For example, sodium, potassium, and chloride ions have an unequal number of protons and electrons. This imbalance turns the element into a charged particle. Particles that are charged positively are called cations. Negatively charged particles are called anions. The unit of charge is the coulomb (C). Electricity is created by the attraction of charged particles of opposite sign.
Voltage Difference
The attracting (or repelling) force generated by a charged particle in space is called the electric field. If a negative charge is free to move within a given electric field, it will be strongly attracted to a positive charge, and the field will eventually become electroneutral. There is therefore a certain amount of work involved in keeping the negative particle from rejoining its positive counterpart. More formally, we say that the work involved in moving a charge from point "A" to point "B" in an electric field (Figure 1.1a) is called potential difference (or voltage difference). In more practical terms, from the point of view of the electrophysiologist, potential differences are created when charges accumulate unequally across an insulator. For example, a potential difference is created across the membrane of cardiac myocytes because more anions are present inside than outside the cell (Figure 1.1b).
Current
As illustrated in Figure 1.2a, when the two ends of a source of voltage are separated, the potential difference is maintained. If a conductive pathway is placed between them, charge will flow from the positive to the negative end. The negative plate will attract cations and is therefore referred to as the cathode (red). Conversely, the positively charged plate, which attracts anions, is called the anode (yellow). This movement of charges along a conductor is known as electric current. The unit of measure for electric current is the ampere (A). Current is more formally defined as the amount of charge passing through a conductor per unit time. By convention, positive current refers to the movement of cations toward the cathode.
In the cardiac cell (as in most living cells, for that matter) ions are constantly moving across the membrane, thus generating electric current. Ionic current is conceptualized as the flow of charge moving through selective hydrophilic pores or channels (Figure 1.2b). In the past, ion channels were studied only as functional entities, without any clear structural or biochemical correlate. Nowadays we know that channels are formed by integral membrane proteins that traverse the lipid bilayer and form a pathway for the transfer of selected ions between the intra- and extracellular spaces (see Chapter 2). Channels are conceptualized as electric resistors that connect the intra- and extracellular compartments. In the following section, we will describe the basic behavior of resistors in electric circuits. These concepts should be helpful in our subsequent review of the electrophysiological properties of the various ion channels in the membrane of the cardiac cell.
Resistance
Ohmic Resistors
All conductors offer a certain resistance (R) to the flow of current (if the flow of a fluid is used as an analogy, it can be said that a hose offers resistance to the flow of water). The unit of resistance is the ohm ([OMEGA]). Often, the properties of conductors are defined not by their resistance, but by their conductance. Conductance (G) is simply the inverse of resistance (i.e., G = 1/R) and it is expressed in siemens (S). The simplest resistors are those whose behavior is independent of time or voltage. These resistors are called "ohmic" because they follow Ohm's law:
I = V/R (1.1)
where I is current and V and R represent the magnitude of the voltage difference and the resistance, respectively. Ohm's law establishes that, given an increase in voltage across a constant resistance, there would be a linear increase in the amplitude of the current flowing through the circuit. Moreover, in an ohmic resistor, the time course of the change in current should be equal to the time course of the change in voltage. An example is illustrated in Figure 1.3. As shown by the simple circuit in panel (a), when a resistor (R) is placed between the anode and the cathode of a source of voltage (i.e., a battery), and the circuit is then closed by a switch, current flows toward the cathode. Moreover, a sudden increase in voltage induces an equivalent step in the amplitude of the current, as illustrated in panel (b). The bottom tracings represent three superimposed positive voltage steps of different magnitudes. The top tracings show recordings of the current flowing through the circuit in response to each voltage step. Thus, if current is plotted as a function of voltage (panel c), a linear function, of slope 1/R (or G) is obtained. This linear current voltage (I-V) relation would be the same for both positive and negative voltage steps.
Non-ohmic Resistors: Rectification
It is common to find that the resistance of a conductor varies with the polarity of the current that flows through it. An example is illustrated in Figure 1.4. Panel (a) shows three superimposed negative tracings of current (top) obtained from our electric circuit in response to voltage steps of negative polarity (bottom). A linear relation similar to the one obtained from a purely ohmic resistor (Figure 1.3) is obtained. However, as shown in Figure 1.4b, a different behavior is observed for pulses of positive polarity. In that case, voltage steps induce only a small current step whose amplitude is essentially constant for any voltages being applied. This property of some conductors to allow the passage of current only (or largely) in one direction is called rectification. Rectification is one example of voltage dependence.
Slope Resistance and Chord Resistance
Some cardiac membrane channels rectify. In most cases, the channel allows the passage of current more effectively in the inward (i.e., from the extra- to the intracellular space) than in the outward direction. For this reason, this property is called inward-going rectification. Figure 1.5 shows the example of a current-voltage relation of an inward-rectifier cardiac membrane channel (in this case, the potassium current [I.sub.K1]; see Chapter 2). It is clear that in this case the resistance of the channel is not constant. Indeed, the slope of the I-V relation changes with the voltage. There are basically two approaches to evaluate the conductive properties of these channels. One is to determine the slope resistance. This is done by calculating the slope of a line that is tangential to the I-V relation at a certain point. In the case of Figure 1.5, the slope of the dashed line that touches the I-V function at a voltage of -60 mV is the slope conductance of that channel at that particular voltage. The inverse of the slope conductance is the slope resistance. Clearly, in a nonlinear I-V relation, the slope resistance varies appreciably, depending on the voltage at which it is measured. The other approach is to measure the chord resistance. In that case, two specific points (A and B) are chosen, and resistance is measured from the slope of the line (or "chord") that joins those two points. In a linear I-V relation, slope resistance and chord resistance are the same; however, in a nonlinear I-V relation, the two parameters may be different from each other, and their individual values should depend on the points chosen for measurement.
Time Dependence
Thus far, we described the properties of resistors that respond instantaneously to the changes in voltage. However, in some cases, the amplitude of the current in response to a voltage change may vary also as a function of time. An example is illustrated in Figure 1.6. In this case, a sudden change in voltage causes a progressive increase in the amplitude of the current. Because the voltage is constant, the increase in current is not due to voltage changes but rather to the intrinsic ability of the conductor to allow the passage of varying amounts of current as a function of time. Many cardiac membrane currents are time-dependent. In some cases, the current progressively increases during a voltage step, whereas, in other cases, the current decreases, and yet in other instances, completely disappears even if the voltage step is held constant for an extended period of time.
Capacitance
Capacitance is the property of an electric nonconductor that permits the storage of energy as a result of electric displacement when opposite surfaces of the nonconductor are maintained at a different potential. The measure of capacitance is the ratio of the change in the charge on either surface to the potential difference between the surfaces. Thus, a capacitor is formed when two conducting materials are separated by a thin layer of nonconducting material, an insulator (or dielectric). Cell membranes are capacitors in that the thin lipid bilayer (which is a very poor electric conductor) behaves like a dielectric interposed between the intracellular and the extracellular spaces, both of which are capable of conducting electricity. As opposed to resistors (ion channels in the case of cells), a voltage step imposed through a capacitor causes only a temporary current. This is illustrated in Figure 1.7. In panel (a), the top diagram shows an electric circuit consisting of a variable voltage generator (i.e., a battery of variable voltage), a capacitor, and a switch. Initially (step 1), the switch is off and no voltage difference is set between the anode and the cathode. When the switch is turned on, a voltage difference is established across the circuit (step 2), charge travels toward the cathode until it encounters the capacitor. Because the conductive pathway is interrupted by the dielectric that separates the two plates of the capacitor, positive charge accumulates at the plate that is closer to the anode. A steady-state condition is rapidly reached (step 3), and the flow of current stops. The tracings in panel (b) show the time course of positive capacitive current in response to voltage in this circuit. The voltage step elicits a rapid surge of current; however, the current rapidly returns to zero. When the voltage difference is switched back to zero (step 4), the capacitor is gradually discharged (i.e., charges now flow in the opposite direction) and a negative capacitive current is observed.
Capacitive current ([I.sub.C]) is thus defined as
[I.sub.C] = CdV/dt (1.2)
where ITLITL is the capacitance and dV/dt represents the first derivative of voltage with respect to time. The latter can be roughly thought of as the rate at which voltage changes. When voltage is constant, dV/dt is zero (because voltage is not changing), and the amplitude of the capacitive current is also zero.
It is important to note that the capacitive properties of the cell are essential for the maintenance of a voltage difference across the membrane. Indeed, the lipid bilayer allows for the separation of charge. The voltage difference across the membrane is established by the fact that charge is unequally distributed. Therefore, the magnitude of the membrane potential reflects the extent of the disparity in charge distribution across the capacitor. Changes in membrane potential occur when ions, normally moving through the membrane channels, charge or discharge the membrane capacitance, thus changing the number of charges in the intra- and the extracellular spaces.
Parallel RC Circuits
In the previous sections, we described the behavior of the lipid bilayer of the membrane as a capacitor. We also equated membrane channels with resistors, because they allow the movement of ionic currents. Because the channels are formed by proteins that span the membrane, they are usually modeled in equivalent circuits as resistors in parallel with the membrane capacitor.
Therefore, the basic membrane circuit is that of a resistor and a capacitor in parallel (Figure 1.8a) and is usually referred to as an RC circuit. In an RC circuit, the total current flow ([I.sub.t]) is equal to the sum of the current that moves through the capacitor ([I.sub.C]) and the current that flows through the resistor ([I.sub.R]).
[I.sub.t] = [I.sub.C] + [I.sub.R] (1.3)
Consequently, if one combines Equations 1.1, 1.2, and 1.3, then
[I.sub.t] = CdV/dt + V/R (1.4)
Figure 1.8b depicts the change in current in response to a voltage step in a parallel RC circuit (assuming that the resistor shows no time-dependent properties). The current flowing through the capacitor ([I.sub.C]) has the properties depicted in Figure 1.7, whereas the current moving through the resistor ([I.sub.R]) is directly proportional to the voltage step itself (as in Figure 1.3). Because both currents add, the total current ([I.sub.t]) in Figure 1.8b shows an initial transient change, which is due to the flow of capacitive current, but rapidly reaches a steady state. The steady state corresponds to the magnitude of the current flowing through the resistor, and it is maintained for as long as the voltage step is maintained. Termination of the voltage step elicits the discharge of the capacitor, and then the current trace returns to the baseline value.
As noted earlier, cell membranes are modeled as parallel RC circuits. Accordingly, when a voltage change is imposed across the cell membrane, there is an initial transient surge of capacitive current, also called the capacitive transient. In the case of a square voltage pulse, the capacitive current rapidly drops to zero. Hence, all currents recorded after the end of the capacitive transient are currents that move through ion channels.
Origin of the Membrane Potential
Electrical current is driven by the voltage difference across a conductor. In the case of cells, this driving force is generated by the unequal distribution of electric charges and ion concentrations across the membrane. In other words, the membrane potential is electrochemical in origin. The physical basis for the establishment of electrochemical potentials is defined by the Nernst equation.
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Excerpted from Basic Cardiac Electrophysiology for the Clinician by Jose Jalife Mario Delmar Justus Anumonwo Omer Berenfeld Jerome Kalifa Copyright © 2009 by Jose Jalife. Excerpted by permission.
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