MEMBRANE POTENTIAL - PAPER

Introduction

The manner in which physiology textbooks, including some for medical physiology courses, deal with the origin of membrane potentials is unnecessarily simplistic. They usually start with a discussion of the potassium equilibrium potential and, to justify the lesser negativity of the resting membrane potential, follow with statements about how a sodium inward flux would reduce this negativity. More or less detail may be added concerning the sodium equilibrium potential and the difference between the sodium and potassium conductances (or permeabilities). The advantage of this strategy is that a significant amount of effort on the part of both students and instructors is saved while developing a partial qualitative sense of the phenomenon. Still, the opportunity to acquire a good command of the common mechanisms that underlie phenomena such as synaptic potentials, action potentials, pacemaker potentials, etc., as well as the relative stability of the resting membrane potential, is being missed. This more general and deeper knowledge requires the better understanding of the dynamics of the system that can be reached only after examining moderately rigorous models for the generation of the resting membrane potential.

Recently Moran et al. (1) described a laboratory exercise based on the selective diffusion of ions through a dialysis membrane with a small molecular weight cutoff value instead of the ion-selective membranes used by Manalis and Hastings (2). These experimental approaches should help clarify notions about how ionic diffusion leads to the generation of membrane potentials and the relevance of the membrane capacitance. Associated quantitative manipulations would introduce some of the theoretical framework.

 (1) Moran, W. M., J. Denton, K. Wilson, M. Williams, and S. W. Runge. A simple, inexpensive method for teaching how membrane potentials are generated. Advances in Physiology Education 22: S51-S59, 1999.

(2) Manalis, R. S. and L. Hastings. Electrical gradients across an ion-exchange membrane in student’s artificial cell. J. Appl. Physiol. 36:769-770, 1974.

I have developed an approach for discussing these matters in an undergraduate course entitled Physiological Mechanisms, which consists of a discussion of the physics of a variety of physiological processes. The models discussed in it are, obviously, not elaborate, but sufficiently realistic to lead to an acceptable predictive ability. Still, although quantitative results may be derived from them, students are alerted to their limitations. The approach I follow here is to first derive the equations that describe the behavior of the model and then look at their consequences. It is based on the premise that we do not benefit students by protecting them from the efforts involved in finding out why they were required to take physics and calculus during their freshman and sophomore years. Contrary to that, they should be expected to use that knowledge as soon as possible in contexts that make it relevant to biology. This should not only significantly improve the students’ understanding of physical principles, but also illustrate how they can be profitably used to generate reasonable models of physiological processes. At the same time, students should begin to appreciate the generality of these models, which contributes to the unification of knowledge by allowing their application to apparently dissimilar phenomena.

Initial concepts

The first section in this course, Material Transport, deals first with the energetics of passive and active transports, and then develops a general treatment of the kinetics of passive transport that is first applied to diffusion and later on to blood flow. The concepts developed during its application to diffusion lead directly to a model of passive ionic fluxes. Those concepts can be briefly summarized as follows:

1) The concept of gradient is introduced and applied to the definition of field as the negative gradient of a potential. The concepts of potential and field, as those of energy and force are general and applicable to any system in which potential energy is stored, be it electrical, hydraulic, gravitational or chemical. Their relationships may be more clearly shown by the following diagram.

2) The chemical potential for an ion is defined as the portion of the free energy of the solution due to the presence of that ion at a given concentration. When possible, the rigorous definition ( :_i = ( MG / Mn_i ) _{T, P, n j} ) is given. It is then shown that the chemical potential, or free energy per mole, is the potential whose corresponding field may be considered to generate a force responsible for the directed movement of the particles.

3) The analysis of the flow of particles driven by a concentration gradient in a viscous fluid shows that, in the steady state, the mean velocity of particles in a direction opposite to that of the gradient acquires a constant value when the frictional force, which is a dissipative force, and the driving force become equal and opposite.

4) The flux j is shown to be the product of the mean velocity of the particles times their concentration, and the mean velocity is expressed as the product of the field (force per mole) times a mobility u. The mobility reflects the influence of the frictional force and is equal to         u = n_i / k 0, where n_i is the number of moles of i, k is a constant that depends on the shape of the particle, and 0 is the viscosity of the medium. The units of mobility are moles @ m @ N^-1 @ s^-1. Then, the equation for the flux becomes 

                                j = - u c [d :/dx)] = - u c RT [d (ln c) /dx] = - u RT [d c /dx]

in which : is the chemical potential and c is the concentration. This equation is then compared to Fick’s first law of diffusion.

5) The equation of continuity is introduced and applied to the steady state to show that, when the concentration at any point does not vary in time, the flux will remain constant throughout the entire path of diffusion. Or, using a formal expression, when  Mc/Mt = 0,  Mj /Mx = 0.  For diffusion across membranes this means that when the membrane is not considered homogeneous the mobility of successive layers of the membrane may not be the same. Yet, when the steady state is reached, different concentration gradients will have developed such that the above equation and Fick’s first law would still apply for each thin layer of the membrane. This is not that useful in discussing diffusion, but, when applied to blood flow in the circulatory system, it helps explain the different blood pressures along the vascular system.

Electrodiffusion

The basis established by the earlier development of these concepts leads naturally to its expansion to deal with the movement of ions when exposed to both a concentration and an electrical potential gradient. The electrochemical potential is defined as the sum of the two terms:

           :_EC = : + zFR

where :_EC is the electrochemical potential of an ion in joules per mole, F is Faraday’s constant in coulombs per mole, and R is the electrical potential in joules per coulomb. Then, when a membrane separates two solutions with different concentrations of an ion and different electrical potentials, the field to which these ions will be exposed within the membrane is given by the following differential equation:

- d :_EC /dx = - [d : /dx + zF (d R /dx)] = - [RT d (ln c) /dx + zF d R /dx)]

and the flux is

                j = - u c (d :_EC /dx) = - u c [RT d (ln c) /dx + zF (d R /dx)]           Eq 1

which is known as the Nernst-Planck equation. In it, the electrochemical, chemical and electrical potentials as well as the concentration of the ion are functions of time and of distance along a chosen direction x normal to the membrane, but the dependence on time can be eliminated by imposing steady state conditions, which also renders j  constant, since at steady state the conservation of mass requires that, when  Mc /Mt = 0, then   Mj /Mx = 0, so j is a constant).

The presence of the concentration as a factor in the right side of the equation renders it nonlinear and impossible to integrate directly and, therefore, to obtain an equation useful to calculate the flux as a function of time as was done for diffusion. Two integrations of equation 1 have been made by incorporating certain assumptions: 0ne by Henderson to calculate the junction potential at the interface between different electrolytic solutions (P. Henderson, Z. Physk. Chem. 59: 118 (1907); 40: 325 (1908) and another by Goldman for electrodiffusion across a membrane (D. E. Goldman, J. Gen. Physiol, 27: 37 (1943). Goldman assumed a constant electrical field within the membrane (d R /dx) = constant = )R / )x) so that integration became possible. The resulting flux equation was then applied by Hodgkin and Katz in the steady state and assuming zero net current to derive an equation to calculate the resting membrane potential (A. L. Hodgkin and B. Katz, J. Physiol. (London) 170: 541 (1949)). This equation, frequently referred to as the Goldman equation, yields values that are a good approximation to experimental values obtained with squid axons for a wide range of external potassium ion concentrations.

The integration of Equation 1 is necessary to transform the derivatives of the potentials, which are point functions with possibly different values at different points in x, into potential differences between the two surfaces of the membrane, which are macroscopic functions with specified values over an extended region such as the entire thickness of the membrane. The values of macroscopic quantities can be measured directly. That integration would change the units of both derivatives from newtons per mole to joules per mole, and those of the coefficient from                            moles @s^-1 @ m^-2 @ mole @ N^-1   to mole @s ^-1 @ m^-2 @ mole @ N^-1 @ m^-1, while those of j remain moles @ s ^-1 @ m^-2. The approach that I follow in presenting these ideas is suggested by rewriting Equation 1 as follows:

                j =  - u c [d : /dx + zF (d R /dx)]                         Eq.  2

where the quantities being added in the square brackets are the derivatives of two potentials and, if x is taken in the direction of the flux, these derivatives represent the intensity of the corresponding fields in newtons per coulomb. So, they represent two independent forces acting on the ions, one due to a chemical potential gradient and another due to an electrical potential gradient both with units of newtons per mole. Equation 2 may be rearranged as follows (Weiss, T. F., Cellular Biophysics, Vol 2, Transport, The MIT Press, Cambridge, Massachusetts, London. 1996, p474):

                j = - u c [RT d/dx ln c + zF dR/dx]

                j @ 1/ u c =  - d/dx [RT ln c + zF R]

                j @ (1 /u) @ dx/c = - d [RT ln c + zF R]

Integrating both sides of the equation across the thickness of the membrane which is equal to the distance from x = 0 to x = b and keeping in mind that in the steady state j is constant

                j @ (1/u) @I^b₀ dx /c  = - [RT ln (c_b /c₀) + zF (R_b - R₀)]  

and

j  =  - [u (1/I^b₀ dx /c)] @ [RT ln (c_b /c₀₎ +  zF (R_b - R₀)]            Eq.  2'

The coefficient in the above equation  g = u (1/I^b₀ dx /c) is a proportionality constant between the potential difference in joules per mole and flux, and its units are mole @ s ^-1 @ m^-2 @ mole @ joule^-1.  Then, Eq.  2'   may be written as follows 

                j =  -  g ( P_C + P_E )               Eq.     3

where P_C and P_E are the chemical and electrical potential differences P(x=d) - P(x=0). An equivalent expression can be obtained by noticing that I^b₀ dx /c) is dividing the potentials and, therefore, so is the linear dimension of dx. Then, we could write

                 j =  - L (FC + F_E)                   Eq.   3'

in which the intensities of the two fields are assumed to be constant in x and that F_C and F_E are fields equal to P_C / b and P_E / b, respectively. Then, the units of L are mole @ s ^-1 @ m^-2 @ mole @ N^-1 and those of F_C and F_E  are N @ mole ^-1 instead of potential differences in joules @ mole ^-1.

Equations 3 and 3’ provide students with an instrument useful in facilitating quantitative reasoning when analyzing problems that involve ionic fluxes. Since the intention of this paper is to describe an approach to teaching, the use of Equations 3 and 3’ for much of what follows instead of the more elaborate Eq. 2' and others can be justified because of their simplicity and intuitive appeal. Obviously, students are expected to be able to calculate P_C,  P_E,  F_C, and F_E  from the appropriate information.

Application of the Flux Equation: The ionic Equilibrium Potential   

Equations 3 and 3' can be applied to situations in which ions may cross a membrane under the influence of electrochemical potential gradients. For example, consider the one depicted in Fig 1. Two potassium chloride solutions of different concentrations are poured separated by a membrane permeable only to potassium ions. At time = 0, Equation 3' should be written  j _K = g (FC), since there is no electrical potential difference across the membrane to generate the corresponding force. Since chloride ions do not permeate the membrane, the diffusion of K+ driven by its concentration gradient generates a charge separation across the membrane and the corresponding electrical

potential gradient. Choosing the positive direction of x from right to left determines that the signs of both F_C for potassium and j _K are negative: Equation 2 shows that the force due to the chemical potential gradient is given by - RT d / dx (ln c) and the slope of the logarithm of c is a positive quantity since the concentration in side 2 is larger than in side 1 and j _K must have the same direction as the predominating force. On the other hand,  F_E, given by  -zF d / dx (R) in Equation 2, has a positive value since the slope of R is negative.

Figure 2 shows these different directions and the effect of adding an opposing electrical force on the intensity of the flux. Clearly, as this electrical force increases due to a continuous charge transfer by the potassium flux, the net force F_C + F_E  acting on the ions will decrease. The rate of charge transfer, on the other hand, is decreasing because of the decreasing flux. Eventually, the two forces cancel each other, the flux is reduced to zero and the system reaches a stable potential difference at thermodynamic equilibrium.  The electrical potential difference established across the membrane is the potassium equilibrium potential which depends on the ratio of the concentrations of the ion on both sides of the membrane. In equation 2', once   j = 0,

                                    Eq.    4

which defines the equilibrium potential for K⁺. This same equation has been derived during an earlier part of the course by imposing the condition of equilibrium to the equation for the change in free energy during ionic transport. It is customary to define the membrane potential difference of a cell membrane as V_M = R(in) - R(out) since this is the way in which it is measured, and the same convention is used for the equilibrium potential defined by Equation 4. Then, in Figs.2 and 3, side 1 becomes the outside of the cell and side 2 is the inside. This convention is equivalent to defining the positive direction of x as going from the outside of the cell to the inside so that it also applies to chemical potential differences. It is a lot easier on students to maintain this convention unchanged even though they should be made aware of its arbitrariness.

The Equivalent Electrical Circuit

The transformation of the concentration force from a difference in chemical potential into a difference in electrical potential as shown in Equation 4 suggests that Equation 3 may be expressed in purely electrical terms, which is frequently convenient. In Equation 2' we could factor out zF from both potential differences which gives us

j_K  =  - [ u z F (1/I^b₀ dx /c )] @ [ RT/zF @ ln {c(x = b) / c(x = 0)} +  {R(x = b) - R(x = 0)}]

If now we multiply the entire equation by zF, instead of a flux of moles of the ion we get a flux of coulombs. For the example above, Equation 2’ becomes

J_K = -  [ u z² F² (1/I^b₀ dx /c )] @ [RT/z_K F @ ln {c(x = b) / c(x = 0)} + {R(x = b) - R(x = 0)}]

in which J_K is current density in C @s^-2 @m^-2 and the coefficient has the units

C @s^-1 @ m^-2 @ N ^-1  @ m^-1 @ C  or  S @ m^-2 @ V ^-1.

Then, an equation similar to Equation 3 but for current density (charge flux) instead of ionic flux may be written as follows.

                J_K =  - G_K ( V_M  - V_K )            Eq.  5

where the coefficient, which is an electrical conductivity, is represented by G_K. This equation is just another form of Ohm’s law.

The above discussion should suggest the possibility of representing ionic flows across the cell membrane as electrical currents that flow through simple circuits that reflect the electrical properties of the membrane. Cell membranes show resistance to current flow from one surface to the other and capacitance. The resistance is inversely related to its permeability to ions and the capacitance is due to its structure with two hydrophilic conducting surfaces separated by a dielectric layer made out of lipids.

Fig. 3 shows a circuit diagram that corresponds to the situation presented in Fig. 1 and 2. V_K is the ionic (potassium in this case) equilibrium potential which is the electromotive force that drives the current density or current flux J_K through the conductance G_K (here represented with the symbol used for resistance) and the transient capacitaive current through the capacitor C_M. This current will decay rapidly as the capacitor is charged and develops a voltage difference that opposes V_K. When both EMFs become equal, the system reaches equilibrium and the current is zero. When the membrane is fully charged and the potential difference across the capacitor is equal to V_K, the charge accumulated at the capacitor’s plates is Q = C @ V. In many cells, the membrane capacitance is 1 x 10 ^-2 F @ m^-2 and V_K = 0.086 V. Therefore, the charge required is 0.86 x 10^-3 coul @ m^-2 . The number of moles of ions that need to cross the membrane to reach this charge is that number divided by Faraday’s constant: 8.91 x 10^-8  @ m^-2, which converted to a more reasonable unit for membrane area becomes 8.91 x 10^-12  @ cm^-2. The time it takes for this charge to be transferred and V_K to be reached depend on the membrane capacitance and its conductance to the ion. The equation is  V_M = V_K (1 - e ^{-(G/C) t})  and is derived in Appendix I. It shows that when  t $ 3 (G_K / C)   V_M = 0.95 V_K and continues approaching this value asymptotically as time passes. The value of G_K for the squid giant axon membrane is

3.7 x 10 ^-4 S @ cm^-2 or                       3.7  C @ s^-1 @ m^-2  @ C @ N^-1 @ m^-1.

Two different yet intimately related models for the description of passive ionic transport across the plasma membrane have been illustrated. Whenever time permits, I use both, as has been done here for the equilibrium potential, moving from one to the other to emphasize their equivalence. The rationale is that they reinforce each other in leading to a better understanding of these phenomena. For the sake of brevity, I will here continue using only the electrical model.

The Resting Membrane Potential

The ionic composition of the fluid that surrounds cells is quite different than that of the cytoplasm.  Large differences in the concentrations of many ions coupled to non-zero membrane conductances to these ions result in passive ionic fluxes. These differences in concentration are of such importance that, for many cells, around 50% of their energy expenditure is devoted to maintaining them through active transport systems that run against the passive ones. The passive ionic fluxes run in directions such that they tend to independently establish equilibrium for each ion. In other words, each ionic flux tends to charge the capacitor to a potential difference equal to the corresponding ion’s equilibrium potential. Clearly, not all ions can simultaneously be at equilibrium since their equilibrium potentials differ. Still, cells not subjected to certain external influences will maintain a stable electrical potential difference across their plasma membrane called the resting membrane potential (V_R) which is determined by the various ion concentrations and their respective conductances. Changes in these conductances result in deviations of V_M from the resting value. These deviations constitute the electrical signals used characteristically, but not exclusively, by nerve and muscle cells. In this context, the control of the conductances of membranes to Na⁺, K⁺, Cl^-, and Ca⁺⁺, is critical.

The condition for having a stable V_M is that the total ionic current be zero. When a single ion permeates the membrane, this is possible only when the system is at equilibrium, i.e., when the sum of the electrical and chemical potential differences of the permeating ion is zero. In other words, when  V_M is equal to the ion’s equilibrium potential. If various ions permeate the membrane, the stable membrane potential is achieved when the sum of all the ionic currents is zero, since then, the capacitative current        J_C  = J_K + J_Na + J_Cl   is zero and the charge across the capacitor remains constant.

Figure 4 illustrates a circuit diagram for a membrane permeable to potassium, sodium and chloride ions. The three batteries stand for the ions’ equilibrium potentials and the resistors for their respective conductances. When J_C becomes zero, none of the individual ionic currents needs to have vanished. In fact, it is rare to find ions that are at equilibrium when V_M = V_R. The various ionic currents running in opposite directions cancel each other and no current runs to the capacitor.

We may write the capacitative current as

J_C  =  - G_K ( V_M  - V_K ) -  G_Na ( V_M  - V_Na ) - G_Cl ( V_M  - V_Cl )

and, when it becomes zero V_M = V_R, so

V_R ( G_K + G_Na + G_Cl ) = G_K  V_K + G_Na V_Na  +  G_Cl  V_Cl

which shows that the membrane resting potential is a weighted sum of the equilibrium potentials of the participating ions.

Electrical signals in excitable cells are generated by inducing changes in the ionic conductances. This is accomplished by opening (and sometimes closing) the gates of specific ionic channels. Three main types of these channels have been found: Voltage-gated, ligand-gated and mechanically gated channels.

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