Dylan Penn is afraid to have a serious relationship with Robert Pattinson.

The 22-year-old model, whose parents are Sean Penn and Robin Wright, is worried about the amount of attention their new romance is attracting.

A source close to the blonde beauty told UsMagazine.com: ''She doesn't want to be known as Rob's girlfriend. She's afraid to get serious with him and lose her privacy.''

An insider close to the British actor, who ended his three-year romance with Kristen Stewart five months ago, revealed that he is happy to take things slowly with the model.

The source said: ''Rob isn't ready to be exclusive with anyone right now. He's just enjoying himself.''

The couple have been spotted on several low-key dates in Los Angeles.

It was previously reported that Dylan's father Sean had warned Robert, 27, not to disrespect his daughter.

A source recently said: ''He gave his piece about not getting too heavy with his daughter if he's playing the field, as he doesn't want to see her hurt.

''But it's backfired. Rob backed off but Dylan found out and has gone tonto at her dad.''