Some TMNT stuff really isn't for little kids.
In the IDW continuity, Natsu is the granddaughter of Yakuza boss Hayashi Toru. She incurred the wrath of Oroku Karai when she shot Karai in the shoulder, merely because she was standing in front of a yakuza target. She was outraged when her grandfather apologized for her behavior, charging at Karai and nearly being decapitated for her insolence.
Toru intervened, pleading for his granddaughter's life to be spared, and told Karai that he wanted a legendary sword that he believed she could find.
Despite her brutishness, Natsu was not without honor. Upon hearing that her grandfather planned to have Karai ambushed and killed, she was horrified at the dishonor of it, and the idea that he had simply been eliminating a threat. She was also horrified when she realized the contempt her grandfather had for the idea of honor itself. She went to Karai's home to warn her teacher Toshiro of what was about to happen, and was brought along by him to get rid of her grandfather's assassins.
When Karai emerged from the den where the sword had been kept she found all of Toru's assassins murdered by Toshiro and Natsu. Natsu declaring that by going against her grandfather's wishes, she repaid a dept and that she and Karai were even now. "Not quite." was Karai's answer before grabbing one of the assassin's guns and shooting Natsu in the shoulder as the "price" for having done the same to her earlier, and declares "NOW we're even" and that they would part as neither friend nor enemy, since Natsu was unwilling to stop Karai from killing her grandfather.
Despite this declaration, a few weeks later, Natsu stood at Karai's side when she disposed of the yakuza bosses. Sometime later, she accompanied Karai to the United States.
- The original plan was to name her Tatsu, but according to Erik Burnham, they were unable to use the name. Her finalized name is short for "Not Tatsu" as a nod to this. However, "natsu" is a real Japanese word meaning "summer" as well as a legitimate feminine Japanese name.