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BS22001 Nernst and GHK – Worked examples and self-assessment Q+As

WORKED EXAMPLES OF THE USE OF THE NERNST AND SIMPLIFIED GOLDMAN-HODGKIN-KATZ (GHK) EQUATIONS Nernst equation Example 1 Given extracellular (denoted subscript ‘o’) and intracellular (denoted subscript ‘i’) concentrations of K+ of 5 and 140 mM, respectively (i.e. [K +]o and [K+]i), employ the Nernst equation to calculate the equilibrium potential (E) for K + (EK) across a membrane with perfect selectivity for K+. The Nernst equation (at 37˚C, equivalent to 310 K) for potassium states: +

EK = 61 log10

[ K ]o +

[ K ]i

in this case….. [K+]o (extracellular concentration of K+ )= 5 mM [K+]i (extracellular concentration of K+ ) = 140 mM EK = 61 log10

[5] [140]

EK = 61 log10 0.0357 EK = 61 x -1.45 EK = -88.3 mV Note: entering concentrations in the unit milliMolar (mM) results in EK having the unit of milliVolts (mV). If concentrations where entered in Molar (M), EK would have the unit of Volts (V) - (which would be 0.0883 V, in this example). Example 2 Given extracellular and intracellular concentrations of Na + of 140 and 15 mM, respectively, employ the Nernst equation to calculate the equilibrium potential for Na + (ENa) across a membrane perfectly selective for Na+. The Nernst equation (at 37˚C) for sodium states: +

ENa = 61 log 10 in this case….. [Na+]o = 140 mM [Na+]i = 15 mM

[Na ]o [ Na+ ]i

BS22001 Nernst and GHK – Worked examples and self-assessment Q+As

[140] [15]

ENa = 61 log 10

ENa = 61 log 10 9.33

ENa = 61 x 0.97 ENa = 59.2 mV (by convention, the ‘+’ sign is omitted)

Example 3 Given extracellular and intracellular concentrations of Cl- of 140 and 10 mM, respectively, employ the Nernst equation to calculate the equilibrium potential for Cl - (ECl) across a membrane perfectly selective for Cl-. The Nernst equation (at 37˚C) for chloride states: -

ECl = -61 log10

[ Cl ]o -

[ Cl ]i

Note: the negative sign in the equation appears because z is negative (-1) for Cl-. in this case….. [Cl-]o = 140 mM [Cl-]i = 10 mM ECl = - 61 log10

[140] [10]

ECl = - 61 log10 14 ECl = -61 x 1.15 ECl = -69.9 mV

Example 4 Given extracellular and intracellular concentrations of Ca2+ of 1 and 0.0001 mM1, respectively, employ the Nernst equation to calculate the equilibrium potential for Ca 2+ (ECa) across a membrane perfectly selective for Ca2+. The Nernst equation (at 37˚C) for calcium states: ECa = 30.5 log10

[ Ca 2+ ]o [ Ca 2+ ]i

1 0.0001 mM is more conveniently expressed as 100 nanoMolar (i.e. 100 nM)

BS22001 Nernst and GHK – Worked examples and self-assessment Q+As

Note: the number 30.5, rather than 61, appears in the equation arises because z is 2+ for Ca2+. in this case….. [Ca2+]o = 1 mM [Ca2+]i = 0.0001 mM (or 100 nM), but remember to use consistent units in the calculation ECa = 30.5 log 10

[1] [0.0001]

ECa = 30.5 log 10 1000 0

ECa = 30.5 x 4 ECa = 122 mV

Simplified Goldman-Hodgkin-Katz (GHK) equation Strictly, this is the voltage equation as there is also a current equation that does not concern us here. Consider the above values for ENa and EK (i.e. 59 and -88 mV, respectively). The resting membrane potential in nerve cells is, to a good approximation, determined by the relative permeability of the membrane to Na+ and K+, and the concentration gradients of those permeant ions across it2. So, the resting membrane potential, V m, should be somewhere between these values. But where, exactly? A qualitative guess is nearer to EK than ENa, since the resting membrane is far more permeable to K + than to Na+ and the influence of K+ should thus dominate. If the permeability ratio ( PK/PNa is known), Vm can be predicted from the GHK (voltage) equation, given knowledge of the relevant ion concentration gradients. Note any contribution from PCl, which in axons for example is nil, is ignored in this simplified treatment. (Similarly, a very small contribution by the Na +/K+ ATPase is not included (remember, this is an electrogenic process in which the transporter pumps 3 Na + ions out of the cell in exchange for 2 K+ ions into the cell. Thus, the activity of the pump adds to the membrane potential, although very modestly, by the net movement of 1 positive charge out of the cell).

The simplified GHK equation states: V m = 61 log 10

PK / PNa x [ K + ] o + [ Na+ ] o PK / PNa x [ K + ] i + [ Na+ ] i

[K+]o = 5 mM [K+]i = 140 mM [Na+]o = 140 mM 2 Membranes are permeable, ions are permeant. As an aid the memory, an umbrella is impermeable to water.

BS22001 Nernst and GHK – Worked examples and self-assessment Q+As

[Na+]i = 15 mM Assume a permeability ratio ( PK/PNa) of 20 (or 1 : 0.05) for the nerve cell membrane at rest (a good estimate, but ignoring PCl, as stated above) V m = 61 log10

20 x [5] + [140] 20 x [140] + [15]

V m = 61 log 10

240 2815

V m = 61 log10 0.085 Vm = 61 x -1.07 Vm = - 65.2 mV Noting, as above, that entering concentrations in mM results in Vm having the unit of mV.

Self assessment 1. Calculate the equilibrium potential for potassium (EK) across a membrane perfectly selective for that ion using values for [K+]o and [K+]i of 7 and 140 mM, respectively3. Answer: -79.4 mV 2. Calculate the equilibrium potential for sodium (E Na) across a membrane perfectly selective for that ion using values for [Na +]o and [Na+]i of 125 and 15 mM, respectively4. Answer: 56.2 mV 3. Calculate the equilibrium potential for chloride (ECl) across a membrane perfectly selective for that ion using values for [Cl-]o and [Cl-]i of 135 and 8 mM, respectively Answer: -74.9 mV 4. Calculate the equilibrium potential for calcium (E Ca) across a membrane perfectly selective for that ion using values for [Ca 2+]o and [Ca2+]i of 1.2 and 0.0002 mM, respectively Answer: 115.2 mV

3 Clinical note: serum potassium concentration is normally within the range 3.5 to 5.0 mM. This is an example of a pathological situation in which raised serum potassium (hyperkalaemia) may have resulted from trauma such as severe burns, or damage to skeletal muscle. Consider the effect of hyperkalaemia upon the membrane potential (Vm) of cells, remembering that PK is the dominant influence. 4 Clinical note: serum sodium concentration is normally within the range 135 to 145 mM. This is an example of a pathological situation in which reduced serum sodium (hyponatraemia) may have resulted from excessive drinking of water, or extreme physical activity, such as running a marathon.

BS22001 Nernst and GHK – Worked examples and self-assessment Q+As

5. Using the above values for [Na+]o, [K+]o, [Na+]i and [K+]i,, employ the simplified GHK equation to estimate the potential across a membrane permeable only to those ions, neglecting any other influences. Assume PK/PNa = 20. Answer: -62.6 mV...