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My biology book says, that the equilibrium potential for an ion with a charge of +1 is: $$E_{ion}= 62mV \biggl(\log\frac{[ion]_{outside}}{[ion]_{inside}}\biggr)$$
Where does the 62 mV come from? How was this value derived? I understand that it's the equilibrium voltage of $\ce{Na+}$.
The System intracellular/membrane/extracellular space is well described by the model of a Concentration cell (see more on Wikipedia). The equation you mentioned is also called the Nernst equation.
$$ E_{ion}= 62mV \biggl(\log\frac{[ion]_{outside}}{[ion]_{inside}}\biggr)= \frac{k_\mathrm{B} T}{z \mathrm{e}} \biggl(\ln\frac{[ion]_{outside}}{[ion]_{inside}}\biggr) $$
where $k_\mathrm{B}$ is the Boltzmann constant, $T$ temperature (310 K), $\mathrm{e}$ elementary charge and $z$ the number of elementary charges per ion (1 in the case of Na+)
Thus the $62 mV$ come from $\frac{k_\mathrm{B} T}{\mathrm{e}} ln(10)$
As you can see, as the temperature increases, entropic effects gain strength and thus more ions move against the electric field, raising the membrane potential.
So your 62 mV are not only empirical but also based on a well defined physical model.
And the equilibrium voltage of Na+ is the same because at physiological conditons the intracellular concentration is ten times smaller than the extracellular. Thus $\log\frac{[ion]_{outside}}{[ion]_{inside}}$ equals to $log(10)$ which is $1$. The total membrane reversal potential is well described by the Goldman Equation.
That quasi-travesty is the Nernst equation in $\log_{10}$ for a positive monovalent ion at physiological temperatures (37 degrees celsius), but they've hidden all that from you. Shame on them.
The canonical form of the Nernst equation, for an ion $S$ is
$$ E_{S} = \frac{RT}{z_{S}F}\ln{\frac{[S]_{out}}{[S]_{in}}} $$
where $R$ is the gas constant, $T$ is temperature expressed in Kelvin, $F$ is Faraday's constant, and $z_S$ is the charge of ion $S$. This is the actually useful form of the equation that can be used for any ion for any temperature.
The Nernst equation is a limiting case of the Goldman-Hodgkin-Katz equation for a single ion. The Nernst equation is useful for directly determining the equilibrium potential of a single ionic species. The GHK equation is used for determining the reversal potential of a membrane or channel in multi-ion cases.
Note that the terms "reversal potential" and "equilibrium potential" are not synonymous, except in the case of a single ion system. The reversal potential is where the direction of current switches. An equilibrium is when net ion flux is zero (although, in a living cell, it is more appropriate to call even this situation a steady state). If membrane were only permeable to one ion, the reversal potential will be at the equilibrium potential (given by Nernst) for that single ion. However, when there are several permeant ions, usually none of the ions will be in equilibrium at the reversal potential--that is, all the ions will have a measurable flux across the membrane even though the sum of those fluxes (weighted by permeabilities) is zero at the reversal potential.
Converting natural log to log base 10, you can use the identity $\log_b{a} = \frac{\log_{10}{a}}{\log_{10}{b}}$, which gives
$$ E_{S} = \frac{1}{\log_{10}{e}} \cdot \frac{RT}{z_{S}F}\log_{10}{\frac{[S]_{out}}{[S]_{in}}} $$
In the case of sodium ions (Na+) at 37 degrees (physiological temperature), $T$ is 310 and $z_S$ is +1. Substituting in these values, we have
$$ E_{S} = 2.303 \cdot \frac{(8.3145) (310)}{(+1) (96485) }\log_{10}{\frac{[S]_{out}}{[S]_{in}}} \\ E_{S} = 61.5 \text{ mV} \cdot\log_{10}{\frac{[S]_{out}}{[S]_{in}}} $$
YAK uses a less common form of the Nernst equation in membrane physiology which uses the Boltzmann constant directly and deals with elementary charges instead of moles ($R = N_A k_B$). It is a wonderful exercise to derive the Nernst equation from the Boltzmann equation,
$$ \frac{p_2}{p_1} = \exp{(-\frac{u_2 - u_1}{k_BT})} $$
where $p_i$ is the probability of a particle being in state $i$, and $u_i$ is the energy of state $i$. All it takes is some rearranging and keeping careful track of the units.
