In the last class we explored the properties of RC circuits that reflect the passive electrical properties of cells, including charging and discharging. We examined important intrinsic properties such as capacitance and conductance (resistance). Today we will build on that analysis by introducing new concepts that arise due to selective permeability of ion channels and the presence of concentration gradients. These factors will expand the repetoire of electrical behaviors that are possible in our basic cell model. We will use the Nernst-Planck equation (in several forms) to ultimately derive the very useful Nernst equation, which reflects the membrane potential at which the electrical and concentration gradients are equal and opposite and there is no net current flow for a given ion species that is permeable. The voltage given by the Nernst equation is called by various equivalent names, including: the Nernst potential, equilibrium potential, or reversal potential.

We will then discuss voltage-dependence in ion channels. Ohm's law specifies that conductance is constant, and that current is a linear function of voltage. We can accurately judge whether or not a particular conductance (resistor) obey's Ohm's law by plotting its current-voltage (IV) relationship, which should be a line with slope equal to the underlying conductance. However, most real channels violate Ohm's law (biologists should call it Ohm's dream). Conductance in most cases is a nonlinear voltage-dependent function. Currents, too, are nonlinear functions of the membrane potential, which is reflected in a nonlinear IV curve. The IV curve is not only, but it can also be nonmonotonic, i.e., N-shaped! Although most types of ion channels violate Ohm's law, their collective behavior in terms of current can be described mathematically using the chord-conductance equation. We will start the process of describing voltage-dependent properties today.

Four physical laws govern the movement of ions in membranes, which is the basis for cell voltage and ultimately signalling that takes advantage of concentration gradients and membrane potentials.

Diffusion
Fick's law describes the movement of ions based on concentration gradients: J_diff = -D(∂c/∂x), where J denotes flux of molecules in units of s^-1cm^-2, and D is the diffusion coefficient in units of cm²s^-1, c is concentration of molecules in units of cm^-3, and x is linear dimension in units of cm. The negative sign in front of D specifies that ions flow down their concentration gradient.

Drift
Also called electrophoresis, this force moves ions in electric fields. Even though an electric force constantly acts on the ion, it does not accelerate forever as Newton's laws predict because frictional forces in aqueous solution counteract the electric force and ions reach a terminal drift velocity where frictional and electric forces are equal and opposite. Flux due to drift is: J_drift = -μcz(∂v/∂x), where J is flux (as above), μ is mobility of ions in solution in units of cm²v^-1s^-1, z is valence in dimensionless units, v is voltage in units of V, and x is defined as above.

Einstein relation
Insights by Einstein in 1905 showed that diffusion and drift experience the same frictional resistance in solution, thus the diffusion coefficient (D) and mobility (μ) can be related by: D = μkT/q, where where k is Boltzmann's constant 1.38 E-23 J K^-1, T is temperature in kelvin, and q is the elementary charge 1.602 E-19 C. We use the Einstein relation to express diffusion as J_diff = -μkT/q (∂c/∂x).

Space-charge neutrality
The total concentration of positive and negative charges in a volume is approximately equal, Σ_iz_ic_iq = Σ_jz_jc_jq, where ion species i and j are cations and anions, respectively. The cell membrane is only exception to this rule, but since only a miniscule fraction of charges must be separated to result in significant membrane potential the overal principle of space-charge neutrality is not violated.

Nernst-Planck molar flux equation
Ions in solution in the context of biological cells experience both concentration gradients and voltages. the total flux is therefore the sum of both diffusion and drift: J_total = J_drift + J_diff. Below we rearrange and derive the molar flux equation:

J_total = -μkT/q (∂c/∂x) - μcz (∂v/∂x),

dividing all terms on the left and right by Avogadro's number (N_A = 6.02 E+23 mol^-1) gives,

J_total = -ukT/q (∂c/∂x) - ucz (∂v/∂x),

where J_total is molar flux in units of mol s^-1cm^-2 and u is molar mobility in units of mol cm² V^-1s^-1. (Note that Hille defines mobility u (which we named μ, above) with units expressed as (cm/s)/(V/cm) on p.317 Table 10.1.)

The first term on the right hand side can be further modified given that the gas constant (R = 8.314 J mol^-1K^-1) is equal to the product kN_A and Faraday's constant (F = 96,500 C mol^-1) is the product qN_A, thus we obtain the Nernst-Plank molar flux equation:

J_total = -uRT/F (∂c/∂x) - ucz (∂v/∂x).

Nernst-Planck current density equation
A more useful version of this equation from the perspective of neurophysiology can be obtained by multiplying molar flux by valence (z) and Faraday's constant (F), which changes flux of material to flux of charge, which is (by definition) electric current:

I = zF(J_total) = -uzRT (∂c/∂x) - ucFz² (∂v/∂x), where I is defined as current density in units of A cm^-2. Note that amperes (A) are defined as C s^-1.

Probably the most widely used form of the Nernst-Planck equation is obtained by asking what is the membrane potential at which concentration and voltage gradients are equal and opposite such that there is no net flux of ions across the membrane? This form is called the plain-old Nernst equation, and the voltage where there is no net current is called the Nernst potential (or equilibrium potential, or reversal potential).

To obtain a useful expression, assume that a permeable cation is permeable across a membrane. The concentration on side 1 equals c₁ and the concentration on side 2 equals c₂. We start by setting I=0 in the Nernst-Planck current density equation (above), which reflects the steady-state condition where diffusion and drift are equal and opposite and there is no net current:

0 = -uzRT (∂c/∂x) - ucFz² (∂v/∂x)

czF (∂v/∂x) = -RT (∂c/∂x)

∫(∂v/∂x)∂x = -RT/zF ∫(∂c/c)

We can integrate the left from x₁ to x₂ and the right from c₁ to c₂

But first we will change variables on the left and integrate from v₁ to v₂

∫∂v = -RT/zF ∫∂c/c

v₂ - v₁ = -(RT/zF) LN(c₂/c₁)

v₂ - v₁ = (RT/zF) LN(c₁/c₂)

If we define side 1 = OUT and side 2 = IN, then v₂ - v₁ = V_IN - V_OUT = V_M by definition.

And finally, V_M = (RT/zF) LN(C_OUT/C_IN).

In real cells, typically we are concerned with K⁺, Na⁺, Ca²⁺, and Cl^-, which are the major permeable ions with relatively fixed concentration gradients. We can determine their reversal potentials using the Nernst equation (Note how z affects the final expression for Ca²⁺ and Cl^-:

E_K = (RT/F) LN([K⁺]_O/[K⁺]_I)

E_Na = (RT/F) LN([Na⁺]_O/[Na⁺]_I)

E_Ca = (RT/2F) LN([Ca²⁺]_O/[Ca²⁺]_I)

E_Cl = (RT/F) LN([Cl^-]_I/[Cl^-]_O)

Driving Force
So far we have used Ohm's law to describe current as the product of conductance and voltage (I = gV). Now that we know about reversal potentials, we must modify this relationship empirically. First, each permeable ion is subject to the electric field of the membrane (V_M) and also possesses its own reversal potential. Therefore, we will define the driving force for the current as the difference between these two electromotive forces, (V_M - E_rev). When V_M = E_rev there is no driving force, and thus no current. The chord conductance equation is the product of conductance and driving force: I_X = g_X(V_M - E_X), where X is denotes current of ion X, g_X is conductance of ion X, and E_X is the equilibrium potential of ion X determined using the Nernst equation.

Nonlinear properties and voltage-dependence
Biological conductances do not generally have a fixed conductance, denoted by the constant g. Instead, ion channels are frequently voltage-dependent, the conductance changes as membrane potential changes. Typically there is a characteristic range of voltages for which the conductance equals zero, and then a range of voltages within which the conductance activates, i.e., the ion channels open, so we will write g(V) to reflect this voltage dependence. Obviously we will need to consider both voltage dependence and driving force to determine the ionic current across the membrane. Examples in Hille Figure 1.6 illustrate this point.

(1) In the Nernst-Planck molar flux equation, analyze the quantities in the terms of the equation and verify that the units of molar mobility (u) must be mol cm² V^-1s^-1.

(2) Describe in words how it is possible that IV curves 2a and 2b in Hille Fig. 1.6(D) show increasing inward (i.e., negative) current on the left-hand side of the origin, but show increasingly more outward-going (i.e., positive) current on the right-hand side of the origin.

(3) How do you know that the reversal potential for the current in 2a is E₂, and not E₁? (It is not sufficient to say that the current is zero in both cases.)

(4) Use the chord-conductance equation to draw IV curves for the conductances g_A and g_B, shown in A and B (below). For both g_A and g_B, draw the IV curve if the reversal potential were -70 mV and +40 mV, respectively. Therefore, in total you will show four IV curves:

(5 and 6) Do Fall et al. 2.8 Exercises 1(a-b) and 2(a-e), on pp. 50-1.

(7) Answer the following questions for the system CV′ = I_app - g_K(V_M - E_K) L6_gKpassive.ode, where I have set C=1, g_K=1, E_K=-90 mV, and I_app=50 pA.

(8) The relative conductances for K⁺, Na⁺, and Cl^- of a photoreceptor cell are given below: