What is an example of the connection between sex and politics in the Roman Empire?

There is no more notorious example than that of Valeria Messalina, typically just called "Messalina," the third wife of Emperor Claudius. She, too, was of royal blood. She was the great grand-niece of the lauded Emperor Augustus and cousin to the notorious Emperors Nero and Caligula.

Messalina was rumored to have been so promiscuous that she held a competition with a prostitute at a Roman bordello. She wagered that she could manage to have sex...

There is no more notorious example than that of Valeria Messalina, typically just called "Messalina," the third wife of Emperor Claudius. She, too, was of royal blood. She was the great grand-niece of the lauded Emperor Augustus and cousin to the notorious Emperors Nero and Caligula.

Messalina was rumored to have been so promiscuous that she held a competition with a prostitute at a Roman bordello. She wagered that she could manage to have sex all night, outlasting the prostitute. It is said that Messalina won the competition. 

Historians, now understanding the sexism that existed in Ancient Rome, suspect that Messalina, though probably adulterous, was probably not as promiscuous as historical records claim. She was a powerful woman who was directly and ruthlessly involved in politics, even causing her husband Claudius to condemn numerous senators to death. The reason for this is said to have been their refusal of her sexual advances, but one cannot be certain.

As is true today, when women assert their political power in ways that the patriarchy does not like, their characters are smeared or simplified by one transgression.