= Membrane charge separation
{title2=$Q_{\mathrm{inside}}=CV_m$}
A <membrane potential> can coexist with near <electroneutrality> in the bulk <cytosol> and external <chemical solution>. The small imbalance producing the voltage lies near the <cell membrane>; most positive and negative <electric charges> balance in the bulk.<Molecular Biology of the Cell, 4th edition by Bruce Alberts and others/Ion Channels and the Electrical Properties of Membranes>[Molecular Biology of the Cell, 4th edition by Bruce Alberts and others; The Resting Potential Decays Only Slowly When the Na+-K+ Pump Is Stopped; Figure 11-22: minute charge redistribution and interfacial excess]{ref}
For a separate original <capacitor> model, choose a total <membrane capacitance> $C=5\,\mathrm{pF}$ and $V_m=-70\,\mathrm{mV}$. These are stipulated parameters, not measured properties of a particular <biological cell>, and this voltage is not asserted to equal the preceding $E_{\mathrm K}$. Model the separated <electric charge> relative to an uncharged reference by
$$
Q_{\mathrm{inside}}=CV_m=(5\times10^{-12})(-70\times10^{-3})\,\mathrm C=-3.5\times10^{-13}\,\mathrm C,\qquad Q_{\mathrm{outside}}=-Q_{\mathrm{inside}}.
$$
Using the <elementary charge> $e=1.602176634\times10^{-19}\,\mathrm C$, this represents
$$
N_{\mathrm{excess}}=\frac{|Q_{\mathrm{inside}}|}{e}\approx2.18\times10^6
$$
excess monovalent negative-charge equivalents inside. These can represent a deficit of positive <ions>, rather than the arrival of that many negative <ions>.
Independently choose an internal reservoir <volume (thermodynamics)> $\mathcal V=1\,\mathrm{pL}=10^{-12}\,\mathrm L$ containing $140\,\mathrm{mM}=0.140\,\mathrm{mol\,L^{-1}}$ <potassium>, with balancing negative <electric charge>. With the <Avogadro constant> $N_A=6.02214076\times10^{23}\,\mathrm{mol^{-1}}$, its <potassium> inventory is
$$
N_{\mathrm K}=c\mathcal V N_A\approx8.43\times10^{10},\qquad
\frac{N_{\mathrm{excess}}}{N_{\mathrm K}}\approx2.59\times10^{-5}=0.00259\%.
$$
Thus, if charging were entirely due to outward monovalent <potassium> transport, about one in $38\,600$ of these <potassium> <ions> would suffice. This is a comparison with the chosen <potassium> inventory, not a fraction of all charged particles or a measurement of spatial <charge density>. No shape, surface area or thickness has been inferred from the chosen <capacitance> and <volume (thermodynamics)>. The calculation illustrates why substantial voltage does not require a strongly charged bulk <cytosol>.
\Image[/membrane-potassium-equilibrium-and-charge-budget.png]
{title=Chosen models: <potassium> equilibrium and the small charge budget of a membrane <capacitor>}
{description=Original computed <Nernst equilibrium potential> curve at $T=310\,\mathrm K$ with $c_{\mathrm{in}}=140\,\mathrm{mM}$, and an independent <capacitor> calculation using $C=5\,\mathrm{pF}$, $V_m=-70\,\mathrm{mV}$, and a $1\,\mathrm{pL}$ reservoir containing $140\,\mathrm{mM}$ <potassium>. All numerical inputs are chosen model stipulations. The two panels describe separate examples; neither is an experimental record.}
Back to article page
