Young policeman Greg Lestrade arrests an underage teen for prostitution (Mycroft is about 15) and learns that Mycroft is from a very rich family, but something bad has happened, his parents died and someone (an evil family member perhaps) took possession of everything that the Holmes' family owned. Failing to find any help, Mycroft and his little brother (Sherlock can be 4-5 y.o.) ended up in the streets. And, eventually, Mycroft ended up selling his body, too young and not qualified for any real job, despite his cleverness. It is the only way for him to feed himself and his little brother, who's all alone in their tiny flat/bedsit while Mycroft is under arrest.

Greg ends up helping them somehow. I prefer platonic mystrade in this, since I don't suppose Mycroft is interested in sexual stuff, especially after being used and abused by his clients. But it's up to you.