# Electrical properties of cell membranes

Post-publication activity

Curator: Ernst Niebur

The fundamental unit of all biological life is the cell, a mass of biomolecules in watery solution surrounded by a cell membrane. One of the characteristic features of a living cell is that it controls the exchange of electrically charged ions across the cell membrane and therefore the electrical potential of its interior relative to the exterior. This Scholarpedia page is a basic introduction into the mechanisms underlying this process. Although most of the material applies to all biological cells, we will focus on those mechanisms which are of particular importance for neurons.

## Cells and cell membranes

All cells are surrounded by a cell membrane. We will neglect all the complexities of the metabolic and structural apparatus found in the interior of the cell and simply consider it as a little bag, formed by the cell membrane, and filled with saline (i.e., water with ions dissolved in it). Likewise, we will assume that the exterior of the cell is a bath of saline. This approximation is drastic but not unreasonable, particularly since we are here only interested in the basis of electrical information processing within the neuron and between neurons.

The crucial element (i.e., the only one that we did not abstract away!) is thus the cell membrane. In its simplest form, it is a phospholipid bilayer, i.e. a layer which is only two phospholipid molecules thick. Each of these molecules has two ends (one is a phosphate group, the other a hydrocarbon chain, i.e. a lipid) and these two ends have very different properties. The phosphate end is hydrophilic (it likes to be in a watery environment and to be surrounded by water molecules). In contrast, the lipid end is hydrophobic (it hates to be close to water; remember that oil is a hydrocarbon!). Love and hate for molecules means that they will achieve a lower energy if they attain the loved state and are able to avoid the hated states. Each molecule attempts to get into the lowest-energy state possible.

How can a phospholipid molecule be immersed in water at one of its ends and, at the same time, avoid to be in water at its other end? The answer is, as so often, team work! If enough phospholipid molecules get together, they can bundle up their oily (hydrocarbon) ends together, forming a double-layered sheet with the hydrocarbon ends in the center, and, at the same time, bath their phosphate ends in water on the outside of the sheet. This does not work at the borders of the sheet so it is best to have no ends, i.e., to close the sheet on itself, forming a closed sphere. The result is a certain volume of water (or saline) enclosed by a double layer of phospolipid molecules and -- Voilà! -- a cell! In fact, such artificial cells can be made from its constituent phospolipid molecules (for references see Scott 1975).

## Conductance

From the mentioned work on artificial membranes we know that pure phospholipid bilayers are quite good insulators (this is not surprising: there are no free ions in the membrane so there are no carriers to transport charges). Their specific conductance per unit area is only about $$g_{pure}=10^{-13}\Omega^{-1}m^{-2}$$ (Goldup et al, 1970).

The conductance of biological membranes is much higher, typically by several orders of magnitude even at rest (i.e., without synaptic influences etc). The reason is that there are all kinds of ion channels and other pores penetrating the membrane and allowing additional currents to flow. It is these currents that make cells behave in complex and interesting ways. We will discuss some of them below.

## Capacitance

According to our simplification, the inside and the outside of the cell are both solutions of various salts in water. As opposed to the cell membrane, salt water constitutes quite a good conductor because there are free ions that can transport electrical charges. What we have then is two conductors (the inside and the outside of the cell), separated by an insulator (the membrane). This makes it possible to have different amounts of electrical charges inside and outside the cell. If we can separate a charge $$Q$$ by applying an electrical potential $$V$$ across the membrane, the membrane has by definition a capacitance $$C=Q/V\ .$$ In fact, because the membrane is so thin (only two molecules thick, with a total thickness of about $$6\times 10^{-9}m$$), we don't need much voltage to separate the charges and therefore the membrane capacitance is quite high; per unit area, it is $\tag{1} c=\frac{C}{S} \approx 10^{-2}F m^{-2}$

where F is the unit of capacitance ("Farad").

The specific capacitance of biological membranes is very close to what is obtained simply from the dielectric constant of lipids and the thickness of the bilayer (for a simple derivation see Hobbie, 1997) and, unlike the conductance, the capacitance is very little influenced by all the complexities of biology.

## Electrical potentials across the membrane

Our interest is mainly on the function of neurons which is a class of cells that uses electrical signals for information processing. How can a cell generate such signals? The first thing we need is some way of generating different voltages at different parts of the system, in particular, inside and outside of each cell. Like all cells, neurons generate this difference by separating different ion species. More specifically, in the cell membrane of each neuron are ion pumps, which are protein molecules that span the membrane and use metabolic energy to transport some ions inside the cell and others outside. A typical one is the $$Na^+K^+$$ pump which moves two potassium ions into the cell and, at the same time, three sodium ions out of the cell. After this pump has been running for some time, the concentration of potassium inside the cell becomes larger than that outside, and the concentration of sodium becomes larger outside than inside. Running the pump requires energy, which is provided to the pump in the usual energy currency of the cell, the ATP$$\rightarrow$$ADP process.

How does this generate an electric potential? Let us assume we are in thermodynamic equilibrium which means in this context that the net flux of ions is vanishing. (Of course, this is a dynamic equilibrium, meaning that ions cross the membrane in both directions but on average, the number of ions flowing in the cell is the same as the number of ions flowing out of the cell. Therefore, the net flux, i.e. the difference between the numbers of ions going each direction, is zero but the numbers themselves are not). Then, the probability $$P_{in}$$ of finding a specific ion inside the cell, as compared to the probability $$P_{out}$$ of finding it outside, depends on the energy the ion has inside ($$E_{in}$$) vs. outside ($$E_{out}$$). From statistical mechanics, we know that the relation between these probabilities is given by the Boltzmann distribution:

$\tag{2} \frac{P_{in}}{P_{out}}= \frac{\exp(-\frac{E_{in}}{kT})}{\exp(-\frac{E_{out}}{kT})}$

where $$k$$ is a constant called the Boltzmann factor and $$T$$ is the temperature (in Kelvin). The energy of an ion is certainly a very complicated quantity, with all the interactions and biochemical complexities going on in a living system. Fortunately, the details are not important for our purposes: We can rewrite eq. (2) in the form

$\tag{3} \frac{P_{in}}{P_{out}}= \exp\{-\frac{E_{in}-E_{out}}{kT}\}$

and we see that only the difference of energies counts. The biochemical milieu is not very different inside the cell from outside the cell. Therefore, the chemical energies of ions inside and outside the cell are about the same, and the only real difference between the energy of the ions inside and outside is their electrical energy. This is computed directly as $$zeV\ ,$$ where $$z$$ is the valence of the ion, $$e$$ the electric unit charge (the charge of one electron), and $$V$$ the electrical potential. Therefore, all other energy terms cancel out and we are left with

$\frac{P_{in}}{P_{out}}= \exp\{-\frac{ze(V_{in}-V_{out})}{kT}\}$

Now we can turn things around: Instead of computing the probabilities from the voltages, we can obtain the voltages from the probabilities: The voltage difference between the inside and the outside of the cell is obtained as

$\tag{4} \Delta V = V_{in}-V_{out}=\frac{kT}{ze} \ln \frac{P_{out}}{P_{in}}$

This relation is know as the Nernst Equation. Usually, it is expressed in terms of the ion concentrations but since the probability of finding an ion at some location is proportional to its concentration at this location, the formulations are equivalent. It is also customary to set the voltage scale such that $$V_{out}=0$$ (the choice of the electric 'ground' is arbitrary; note that this convention changed over time and that in the classical papers by Hodgkin and Huxley which are cited below, the opposite sign is used).

We thus find that each ion species has its own voltage at which it is in statistical equilibrium. This voltage is commonly called the "reversal potential" of this ion because the current generated by these ions reverses its sign when this voltage is applied to the membrane.

## Membrane patch in equilibrium

In this section, we will consider a patch of cell membrane which may contain many ion channels but which is small enough such that the transmembrane voltage is approximately the same everywhere in the patch. Electrically, one single ion channel is equivalent to a resistance in series with a voltage source: The resistance $$r_{channel}$$ is simply the inverse of the conductance of the ion channel pore (let us assume for simplicity that the channel is permeable to a single ion species only), and the voltage $$V_i$$ is the reversal potential of the ion species $$i$$ which can pass through the channel. If we have many channels, say $$N\ ,$$ they are electrically all in parallel, so their total resistance is $$R=r_{channel}/N\ .$$ Likewise, if we have an ion channel that selectively lets different ions pass, it can be considered as several one-ion-only channels in parallel.

Let us start by assuming that there is only one ion species, say sodium. The reversal potential $$V_{Na}$$ for sodium is computed from the Nernst equation, eq. (4). The conductance for sodium $$g_{Na}=R_{Na}^{-1}$$ is, as just discussed, the sum of the conductances of all channels that allow sodium ions to pass. According to Ohm's Law, the current across these channels is proportional to the difference in electrical potential (voltage) across the membrane, and the proportionality factor is just the conductance. Keeping in mind our convention that the outside voltage is zero (ground), the current flowing across the membrane is thus $\tag{5} I_{Na}=\frac{V_{Na}}{R_{Na}}=g_{Na}V_{Na}$

What if there is more than one type of ion, e.g. sodium and potassium? Currents will then flow across both types of channels until an equilibrium is established, with the voltage inside the cell somewhere between the reversal potentials of these two ion species. Common names for this equilibrium potential are resting potential or leakage potential, $$V_L\ .$$

In order to compute $$V_L\ ,$$ we assume that the system is already in equilibrium, with the inside of the cell at this potential (which is so far unknown). In order to compute the sodium current, we thus have to modify eq. (5) as follows,

$\tag{6} I_{Na}=\frac{V_{Na}-V_L}{R_{Na}}=g_{Na}(V_{Na}-V_L)$

since the difference between the outside (at zero voltage) and the inside (at voltage $$V_L$$) is $$V_{Na}-V_L\ .$$ By exactly the same argument, the current of $$K^+$$ ions is

$\tag{7} I_{K}=\frac{V_{K}-V_L}{R_{K}}=g_{K}(V_{K}-V_L)$

where, of course, $$G_K=R_K^{-1}$$ is the conductance across the potassium channels.

We can now compute $$V_L$$ from the requirement that the system is in equilibrium. If that is the case, then conservation of charge requires that all currents cancel each other out, or, in other words, that the sum of all currents is zero (this is known as Kirchhoff's Current Law), thus $$I_{Na}+I_{K}=0\ .$$ Together with eqs. (6) and (7), we therefore have $\tag{8} g_{K} V_{K}-g_{K}V_L + g_{Na}V_{Na} -g_{Na}V_L=0$

The only unknown in this equation is $$V_L$$ and we can solve for it to obtain

$\tag{9} V_L=\frac{ g_{Na}V_{Na}+ g_{K} V_{K}} { g_{Na}+g_{K}}$

Of course, voltages have to be entered in this equation with their correct signs, as they are obtained from eq. (4). In most physiological conditions, $$V_{Na}>0$$and $$V_K<0\ .$$

So far we have focused only on two ion species, sodium and potassium, just as as Hodgkin and Huxley did when they developed their famous model of the giant squid axon (Hodgkin et al, 1952; Hodgkin and Huxley, 1952a-d). However, it can be easily seen that equation (9) can be generalized for more than two ion species. The resting potential is then given by the quotient of two sums over all ion species,

$V_L=\frac{ \sum_i g_{i}V_{i}}{\sum_i g_{i}}$ where, of course, $$V_i$$ is the reversal potential of ion species $$i$$ and $$g_i$$ is its transmembrane conductance. It is found that for many neurons, the resting potential is close to $$-70mV\ .$$

## Membrane patch: temporal dynamics

We are now ready to go beyond the equilibrium state and to consider the behavior of a cell when its transmembrane voltage changes over time. In order to take into account the transients, we have to consider the membrane capacitance. As mentioned, the membrane itself is a quite good electrical isolator which means we can accumulate electrical charges on one side that can then not cross over to the other (of course, the same amount of charge with the opposite sign will be found on the other side). For instance, if we maintain an inward current $$I$$ for a time $$t\ ,$$ a charge of $$Q=I t$$ will accumulate on the inside. Conservation of charge requires that the rate of charge accumulation is equal to the current,

$$\frac{dQ}{dt}=I$$

The voltage in a capacitor is proportional to its charge, with the constant of proportionality being the capacity C. Since C is a constant for an ideal capacitor (and to an excellent approximation for a cell membrane as well), we have

$$\frac{dQ}{dt}=C \frac{dV}{dt}=I$$

The current $$I$$ in this equation is the sum of the ionic currents which we have computed in eqs. (6) and (7)

$$I= g_{Na}(V_{Na}-V)+g_{K}(V_{K}-V)$$

Combining the two previous equations, we obtain the following Ordinary Differential Equation for the membrane voltage:

$\tag{10} C\frac{dV}{dt} = g_{Na}(V_{Na}-V)+g_{K}(V_{K}-V)$

Note that, in equilibrium, the temporal derivative disappears and we get equation (8) again.

For time-independent conductances (and reversal potentials), it is customary to lump together the ionic conductances and voltages. Indeed, we can write eq. (10) as follows:

$\tag{11} C\frac{dV}{dt} = g_{Na}V_{Na} + g_{K}V_K - (g_{Na} + g_K)V = ({g_{Na} + g_K}) \times (\frac{g_{Na}V_{Na}+g_{K}V_K}{g_{Na} + g_K}-V)$

Making use of the definition of the leakage potential $$V_L$$ in eq. (9) and defining the leakage conductance $$g_L$$ as

$\tag{12} g_L= {g_{Na} + g_K}$

we can rewrite eq. (11) as

$\tag{13} C\frac{dV}{dt} = g_{L}(V_{L}-V)$

Frequently, eq. (13) is written as $\tag{14} \tau\frac{dV}{dt} = V_{L}-V$

where $$\tau=C/g_L$$ is the time constant of the cell. From this equation it is obvious that the membrane will approach the resting potential $$V_L$$ exponentially, with a characteristic time $$\tau\ .$$

So far, we have been considering only conductances that have no voltage or time dependence. There are many other types of conductances which play a role in neurons. One important class of conductances results from different types of synapses which are responsible for most of the communication between neurons. In the case of chemical synapses, the channel is closed (conductance g = 0), until a chemical emitted from another neuron causes the channel to open (conductance > 0). In the case of electrical synapses, there is a fixed conductance between the two coupled neurons (conductance g > 0 always) and the current flowing between them is proportional to the difference of the voltages in these two cells (see eq.(15) below). Another important class of conductances are due to channels that open dependent on the voltage of the cell itself (see below).

All of these currents have the same form as the leakage current in eq. (13), i.e., they can all be written as

$\tag{15} g(V_c-V)$

In this equation, $$V$$ is the voltage of the neuron under study and $$g$$ is the conductance for the ion species (one or several) which carry the current. $$V_c$$ is a voltage which is specific for this current; it can be the reversal voltage of the ions carried by the conductance $$g$$ or, in the case of an electrical synapse (gap junction), it is the voltage inside the partner neuron. Note that both $$g$$ and $$V_c$$ can depend on time and other variables, including the transmembrane voltage itself (see Excitability). For instance, Hodgkin-Huxley type conductances depend on the transmembrane voltage, and they also have their own specific kinetics for opening and closing. There can be a large number of terms of the form (15), one for each current, and all are added to the right hand side of eq. (13)

## Interactions in space

We have so far considered the electrical activity in a patch of membrane that is small enough (or homogeneous enough) to behave everywhere the same. Many neurons are large (or inhomogeneous) enough that their membranes cannot be described by a single membrane patch. Instead, the interactions between different parts of the cell membrane need to be taken into account. This is the topic of Cable Theory.

## References

• Goldup, A., Ohki, S. and Danielli, J. F. 1970. Recent progress in surface science 3:193
• Hobbie, R. K. 1997. Intermediate physics for medicine and biology. 3rd edition. American Institute of Physics, New York.
• Hodgkin, A. L., Huxley, A. F. and Katz, B. 1952. Measurement of current-voltage relations in the membrane of the giant axon of Loligo. J. Physiol. 1952. 116: 424.
• Hodgkin, A. L. and Huxley, A. F. 1952a. Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo. J. Physiol. 116: 449.
• Hodgkin, A. L. and Huxley, A. F. 1952b. The components of membrane conductance in the giant axon of Loligo. J. Physiol. 116: 473.
• Hodgkin, A. L. and Huxley, A. F. 1952c. The dual effect of membrane potential on sodium conductance in the giant axon of Loligo. J. Physiol. 116: 497.
• Hodgkin, A. L. and Huxley, A. F. 1952d. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117: 500
• Scott, A.C., 1975. The electrophysics of a nerve fiber. Reviews of Modern Physics 47: 487.

Internal references

• Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.