The subjects the three teachers in the show deals with is very relavent to the progress of the show.
Takahashi majors in biology, and the whole show is based on his curiosity at researching the biology of the demis.
Satou majors in math, not only it represents her formulaic life, but also explains his good work relationship with Takahashi as they can (and do) have discussions based on bouncing logic off each other.
As of Souma the physics professor, it does take some heavy duty quantum physics/grand unified theory to explain Machi’s neck.
Machi's birth, or more accurately, her mother's likely reaction to it. No matter how prepared you are, giving birth to a headless body and disembodied head has to be profoundly disturbing at best. Crosses with Fridge Heartwarming, when you realize Kyouko's parents must be extremely supportive of their daughter for her to be so well-adjusted.
It is possible that all demi powers manifest during puberty. If this is the case Machi would be born normally but find herself waking up with her head and body suddenly separated one day.
Alternatively, this may be one of (very few) powers that manifest before birth, in response to the fetus having the umbilical cord wrapped around its neck.
Fortunately, Volume 8 of the manga addresses this and explains that Machi's neck essentially disappeared when she was a toddler, having been born normally; they also show the instance where it occurred, with Machi playing in the park with her father.
