Why does the Doctor always hold his companions' hands?
In all his time the Doctor had learned one sure thing about human beings: They liked physical contact. Not necessarily the sort categorised as promiscuous, lecherous fumblings that can either degrade or make relationships, but more the necessity for comfort and safety that so often stemmed from a subtle shoulder squeeze or lingering embrace.
He mainly held his own human friends' hands because he was a darn fast runner and needed to drag them along when they were too caught up staring in disbelief at the danger in question to move fast enough. Martha had been like that – always caught up in her fascination; he'd had to tear her away on numerous occasions with a tug on her arm. But sometimes it was because he felt like they needed that little bit of reassurance when something so unorthodox was looking them straight in the eye.
He'd never held Jack's hand. That wasn't because he didn't want him to evade those dangers, but because it never really seemed appropriate. The liberal soul he was, the Doctor had no qualms with platonically touching a man, but he always felt like he should only hold those who needed him to look after them. And whilst he was sure he was perhaps a little wiser than the Captain, it never really felt right.
Rose's hand had always magnetically attached itself to his - not that he was complaining, mind. Her small, warm palm encased in his always filled him with a strange mixture of adrenaline and soothing. Just feeling it there let him know she was unfailingly on the end of that hand, smiling that lop-sided grin with her tongue poking through her teeth and it seemed like the most natural thing in the universe. But then that hand slipped out of his reach and he was left reaching into a bleak emptiness.
He'd found Martha's cool, elegant hands later. He felt strange grasping it when he first met her, so he didn't, but soon he found himself having to pull her along as her afore mentioned enthralment would get the better of her, of both of them. As for Donna, physical contact had seemed strange to start with, like he was reading her diary or personal letters. But when they settled into normality – or perhaps became accustomed to such abnormality – clutching her strong fists made him feel like some of her inner strength transferred to him.
Above all he held their hands so they knew he was there to protect them. But most of the time he had to do it so he knew they were there. To realise he had company. To realise that he wasn't alone, no matter how 'human' it sounded.