CELLULAR AND AUTONOMIC PHYSIOLOGY 13 Case 3 Nernst Equation and Equilibrium Potentials This case will guide you through the principles underlying diffusion potentials and electrochemical equilibrium. veil QUESTIONS 1. A solution of 100 mmol/l KC1 is separated from a solution of 10 mmol/l KC1 by a membrane that is very permeable to K ions, but impermeable to Cl- ions. What are the magnitude and the direction (sign) of the potential difference that will be generated across this membrane? (Assume that 2.3 RT/F = 60 my.) Will the concentration of in either solution change as a result of the process that generates this potential difference? 2. If the same solutions of KC1 described in Question 1 are now separated by a membrane that is very permeable to Cl- ions, but impermeable to K' ions, what are the magnitude and the sign of the potential difference that is generated across the membrane? 3. A solution of 5 mmol/l CaC1 2 is separated from a solution of 1 umol/l CaC1 2 by a membrane that is selectively permeable to Ca t., but is impermeable to CV. What are the magnitude and the sign of the potential difference that is generated across the membrane? 4. A nerve fiber is placed in a bathing solution whose composition is similar to extracellular fluid. After the preparation equilibrates at 37 C, a microelectrode inserted into the nerve fiber records a potential difference across the nerve membrane as 70 mv, cell interior negative with respect to the bathing solution. The composition of the intracellular fluid and the extracellular fluid (bathing solution) is shown in Table 1-4. TABLE 1-4 Intracelluar and Extracellular Concentrations of Na, K, and Cl- in a Nerve Fiber Ion Intracellular Fluid Extracellular Fluid Na* 30 mmol/l 140 mmol/l K 100 mmol/l 4 mmol/l Cl- 5 mmol/l 100 mmol/l Assuming that 2.3 RT/F = 60 mv at 37'C, which ion is closest to electrochemical equilibrium? What can be concluded about the relative conductance of the nerve membrane to Na*, K*, and Cl- under these conditions?
14 PHYSIOLOGY CASES AND PROBLEMS pi ANSWERS AND EXPLANATIONS 1. Two solutions that have different concentrations of KC1 are separated by a membrane that is permeable to IQ-, but not to Cl-. Since in solution, KC1 dissociates into K* and Cl ions, there is also a concentration gradient for K+ and Cl- across the membrane. Each ion would "like" to diffuse down its concentration gradient. However, the membrane is permeable only to K. Thus, K F ions diffuse across the membrane from high concentration to low concentration, but Cl- ions do not follow. As a result of this diffusion, net positive charge is carried across the membrane, creating a potential difference (K + diffusion potential), as shown in Figure 1-3. The buildup of positive charge at the membrane retards further diffusion of K + (positive is repelled by positive). Eventually, sufficient positive charge builds up at the membrane to exactly counterbalance the tendency of to diffuse down its concentration gradient. This condition, called electrochemical equilibrium, occurs when the chemical and electrical driving forces on an ion (in this case, K-9 are equal and opposite and no further net diffusion of the ion occurs. Very few K` ions need to diffuse to establish electrochemical equilibrium. Because very few K4 ions are involved, the process does not change the concentration of K + in the bulk solutions. Stated differently, because of the prompt generation of the K. diffusion potential, I(' does not diffuse until the two solutions have equal concentrations of K* (as would occur with diffusion of an uncharged solute). K+- selective membrane 100 K+... 100 CI "" """" 10 K+ + 10 C1-1 2 Figure 1-3 IC + diffusion potential. The Nernst equation is used to calculate the magnitude of the potential difference generated by the diffusion of a,single permeant ion (in this case, K+). Thus, the Nernst equation is used to calculate the equilibrium potential of an ion for a given concentration difference across the membrane, assuming that the membrane is permeable only to that ion. E - 2.3 RT to [CI] g z F [C2] where E = equilibrium potential (mv) 2.3 RT/F = constants (60 mv at 37 C) z = charge on diffusing ion (including sign) C 1 = concentration of the diffusing ion in one solution (mmol/l) C2 = concentration of the diffusing ion in the other solution (mmol/l)
CELLULAR AND AUTONOMIC PHYSIOLOGY 15 Now, to answer the question. What are the magnitude and the direction (sign) of the potential difference that is generated by the diffusion of K' ions down a concentration gradient of this magnitude? Stated differently, what is the K + equilibrium potential for this concentration difference? In practice, calculations involving the Nernst equation can be streamlined. Because these problems involve a logarithmic function, all signs in the calculation can be omitted, and the equation can be solved for the absolute value of the potential difference. For convenience, always put the higher concentration in the numerator and the lower concentration in the denominator. The correct sign of the potential difference is then determined intuitively, as illustrated in this question. The higher K+ concentration is 100 mmol/l, the lower K- concentration is 10 mmol/l, 2.3 RT/F is 60 mv at 37 C, and z for I(' is +1. Because we are determining the K + equilibrium potential in this problem, "E" is denoted as E k.. Remember that we agreed to omit all signs in the calculation and to determine the final sign intuitively later. E - x logio 10 mmol/l = 60 mv x log io 10 = 60 mv x 1 = 60 mv (absolute value of the equilibrium potential) To determine the direction (sign) of the equilibrium potential, see Figure 1-3. Ask: Which way does K' diffuse to create this potential difference? It diffuses from high concentration (Solution 1) to low concentration (Solution 2). Positive charge accumulates near the membrane in Solution 2; negative charge remains behind at the membrane in Solution 1. Thus, the potential difference (or the K equilibrium potential) is 60 mv, with Solution 1 negative with respect to Solution 2. (Or, stated differently, the potential difference is 60 mv, with Solution 2 positive with respect to Solution 1.) 2. All conditions are the same as for Question 1, except that the membrane is permeable to Cl- and impermeable to K+. Again, both K+ and Cl- ions have a large concentration gradient across the membrane, and both ions would "like" to diffuse down that concentration gradient. However, now only Cl- can diffuse. Cl- diffuses from the solution that has the higher concentration to the solution that has the lower concentration, carrying a net negative charge across the membrane and generating a Cl- diffusion potential, as shown in Figure 1-4. As negative charge builds up at the membrane, it prevents further net diffusion of Cl- (negative repels negative). At electrochemical equilibrium, the tendency for Cl- to diffuse down its concentration gradient is exactly counterbalanced by the potential difference that is generated. In other words, the chemical and electrical driving forces on Cl- are equal and opposite. Again, very few Cl- ions need to diffuse to create this potential difference; therefore, the process does not change the Cl- concentrations of the bulk solutions. CI - selective membrane z 100 K+ 100... 1 0 K+ 1 2 Figure 1-4 Cl- diffusion potential.
16 PHYSIOLOGY CASES AND PROBLEMS This time, we are using the Nernst equation to calculate the Cl- equilibrium potential (4 1-). The absolute value of the equilibrium potential is calculated by placing the higher Cl- concentration in the numerator, the lower Cl- concentration in the denominator, and ignoring all signs. = 1 x log, 10 mmol/l = 60 mv x logo 10 = 60 mv x 1 = 60 mv (absolute value of the equilibrium potential) The sign of the potential difference is determined intuitively from Figure 1-4. Cl- diffuses from high concentration in Solution 1 to low concentration in Solution 2. As a result, negative charge accumulates near the membrane in Solution 2, and positive charge remains behind at the membrane in Solution 1. Thus, the Cl- equilibrium potential (Ea-) is 60 mv, with Solution 2 negative with respect to Solution 1. 3. This problem is a variation on those you solved in Questions 1 and 2. There is a concentration gradient for CaC1 2 across a membrane that is selectively permeable to Ca 2+ ions. You are asked to calculate the Ca2+ equilibrium potential for the stated concentration gradient (i.e., the potential difference that would exactly counterbalance the tendency for Ca t' to diffuse down its concentration gradient). Ca2' ions diffuse from high concentration to low concentration, and each ion carries two positive charges. Again, the absolute value of the equilibrium potential is calculated by placing the higher Ca 21 concentration in the numerator, the lower Ca2' concentration in the denominator, and ignoring all signs. Remember that for Ca 2F, z is +2. 60 mv 5 mmol/l Ecz. = 2 x log umol/l 5 x 10-3 mol/l = 30 mv x login 1 x 10-6 mol/l = 30 mv x log /0 5 x 10 3 mol/l = 30 mv x 3.699 = 111 mv The sign of the equilibrium potential is determined intuitively from Figure 1-5. Ca 2' diffuses from high concentration in Solution 1 to low concentration in Solution 2, carrying positive charge across the membrane and leaving negative charge behind. Thus, the equilibrium potential for Ca2-' is 111 mv, with Solution 1 negative with respect to Solution 2.
CELLULAR AND AUTONOMIC PHYSIOLOGY 17 Ca2+- selective membrane Ca2+ CI - ''''...... ca2+ 1 2 Figure 1-5 Ca" diffusion potential. 4. The problem gives the intracellular and extracellular concentrations of Na t, and Cl and the measured membrane potential of a nerve fiber. The question asks which ion is closest to electrochemical equilibrium under these conditions. Indirectly, you are being asked which ion has the highest permeability or conductance in the membrane. The approach is to first calculate the equilibrium potential for each ion at the stated concentration gradient. (As before, use the Nernst equation to calculate the absolute value of the equilibrium potential, and determine the sign intuitively). Then, compare the calculated equilibrium potentials with the actual measured membrane potential. If the calculated equilibrium potential for an ion is close or equal to the measured membrane potential, then that ion is close to (or at) electrochemical equilibrium; that ion must have a high permeability or conductance. If the equilibrium potential for an ion is far from the measured membrane potential, then that ion is far from electrochemical equilibrium and must have a low permeability or conductance. Figure 1-6 shows the nerve fiber and the concentrations of the three ions in the intracellular fluid and extracellular fluid. The sign of the equilibrium potential for each ion (determined intuitively) is superimposed on the nerve membrane in its correct orientation. It is important to know that membrane potentials and equilibrium potentials are always expressed as intracellular potential with respect to extracellular potential. For example, in this question, the membrane potential is 70 my, cell interior negative; by convention, that is called -70 mv. 140 Na+ ECF 4 K+ A +++ 100.C1- + + + + + + V 30 Na+ 100 K+ 5 CI- ICF Figure 1-6 Orientation of equilibrium potentials for Na', K-, and CI- in a nerve fiber.
18 PHYSIOLOGY CASES AND PROBLEMS Now the equilibrium potential for each ion can be calculated with the Nernst equation. Figure 1-6 can be referenced for the signs. 60 mv 140 mmol/l ENa+ X logio 1 30 trunol/l = 60 mv x log10 4.67 = 60 mv x 0.669 = 40 mv (or +40 mv, cell interior positive) = x logio 1 4 mmol/l = 60 mv x log 10 25 = 60 mv x 1.40 = 84 mv (or 84 mv, cell interior negative) Ea_ x log 1 5 mmol/l = 60 mv x log 10 20 = 60 mv x 1.3 = 78 mv (or 78 mv, cell interior negative) These calculations are interpreted as follows. The equilibrium potential for Na+ at the stated concentration gradient is +40 mv. In other words, for Na + to be at electrochemical equilibrium, the membrane potential must be +40 mv. However, the actual membrane potential of 70 mv is far from that value. Thus, we can conclude that Na+, because it is far from electrochemical equilibrium, must have a low conductance or permeability. For K + to be at electrochemical equilibrium, the membrane potential must be 84 mv. The actual membrane potential is reasonably close, at 70 my. Thus, we can conclude that K + is close to electrochemical equilibrium. The ion closest to electrochemical equilibrium is C1-; its calculated equilibrium potential of 78 mv is closest to the measured membrane potential of 70 mv. Thus, the conductance of the nerve cell membrane to Cl- is highest, the conductance to K* is next highest, and the conductance to Na+ is the lowest. Key topics Conductance Diffusion potential Electrochemical equilibrium Equilibrium potential Membrane potential Nernst equation Permeability