There is also a psionic component, which is the thing that determines who the bloodline's "current representative" is (that is, the person who is the line's current "power cable" for charging the psionic energy of the bloodline; in the case of the bloodline on the throne, this would normally be the political Hegemon). There is some kind of psionic connection among all living members of a bloodline. When the representative dies, their "psionic hegemon token" automatically passes to some other living bloodline member--Katya's guess is their closest genetic relative, active-gene preffered over inactive-gene (see below). Political passing-the-torch or heir-declaration ceremonies have ritual elements relating to this, and can be used for things like activating a new hegemon before the old one dies or picking a specific person to get the token, but without the politics, the psionics still happens.
(Jayla: The declaration of an heir is not a psionic ritual per se, but it has great resonance in the hearts of the people who follow the Hegemon.)
The gene is active in the current representative. Any children concieved by the current representative inherit two active hegemon-genes. However, any children concieved by someone who is not the current representative get their hegemon-genes in inactive state. (Thus, e.g.: Gabriel Nomarche is Hegemon when he conceives Vissarion, Ambros, and Arista, so he's got two active genes and they all get two active genes. If Vissarion has a child before Gabriel dies and passes on the "psionic token," the child will get two copies of the gene, but they'll be inactive. If that child has a child while Gabriel is still active hegemon, this new child will get only one, inactive, copy of the gene.)
The "psionic token" will jump to people with inactive genes if there's no one with an active gene to receive it; if this happens, the new representative's gene(s) are activated. But what the "token" actually is/does is still a little fuzzy.