= Independent permissive gates in an ion channel
{title2=$P_{\mathrm{open}}=p^q$}
A permissive gate allows conduction when it is open. Suppose an <ion channel> conducts only when all $q$ gates are open, where $q$ is a positive <integer>. If the gate-open <events> are mutually <independent events> and each has the same open <probability> $p$ at the time considered, then
$$
P_{\mathrm{open}}=\prod_{j=1}^q p=p^q.
$$
This follows from the multiplication rule for <independent events>. Identical transition rates alone do not guarantee independence: the initial gate states must also be independently prepared, and the model must not couple their transitions. Gates sharing a fluctuating <membrane potential> can be correlated; the simple product requires an appropriate independence assumption, for example independent gates at an externally fixed voltage.
At $p=0.5$, channels requiring $q=1,2,4$ permissive gates have open <probabilities> $0.5$, $0.25$ and $0.0625$. For four independent gates there are $2^4=16$ possible open/closed configurations, equally probable at $p=0.5$; only the all-open configuration conducts. Thus $0.5$ is the gate-open <probability>, while $0.0625$ is the whole-channel open <probability>. Neither is a fractional state of one observed channel. Among $160$ independently prepared identical four-gate channels, the <expected value> of the open count is $160/16=10$, although an actual count fluctuates.
\Image[/independent-ion-channel-gates.png]
{title=<Independent permissive gates in an ion channel>: an all-open requirement}
{description=Left: whole-channel open <probability> $p^q$ for one, two and four identical independent permissive gates. Right: all sixteen configurations of four gates at $p=0.5$, each with <probability> $1/16$; filled circles are open gates and empty circles closed gates. Only the highlighted all-open configuration permits conduction. This is an illustrative model, not a molecular structure or an ionic <electric current> measurement.}
For unequal independent gate-open <probabilities> $p_j$, use $P_{\mathrm{open}}=\prod_j p_j$ rather than $p^q$. This is the distinction behind the <Hodgkin-Huxley model>'s $m^3h$: $m$ and $h$ are different activation and inactivation-availability variables, rather than one common $p$. Its $n^4$ uses four identical activation factors. Independence and the all-open requirement are model assumptions; correlated gates or alternative conducting configurations require a different calculation. The simple product is useful because a modest gate-open <probability> can produce sharply suppressed whole-channel availability without any additional gate interaction.
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