# An over-simplified model of cell division and its relation to stem cells

I am a mathematician, and I decided to try to model cell division with math and see what conclusions came out of it. I had some questions concerning stem cells that came up in the model.

In an extremely simplified model of cell division, each cell decides what cells it will divide into with no output from the outside workd, including from neighboring cells. I also assume the time frame is small enough that cells do not die, and that there are only finitely many 'programs' telling different cells how to divide.

Given these assumptions, I came to the conclusion that one of two things could happen: 1. The number of cells grows exponentially, or 2. Most cells will never divide into other cells.

In fact, if growth is going to remain constant (e.g. 300,000,000 cells per month, every month in a sample), then eventually there will be two kinds of cells: 1. Normal cells, which never divide, and 2. 'Starter cells', which divide into normal cells but NEVER create other starter cells. The number of starter cells is eventually constant.

I hypothesized that these starter cells would eventually be far outnumbered and become a very small part if the body proportionally. I hypothesized that hair follicles, nail roots, growth plates, and bone marrow contained starter cells. I further hypothesized that these are adult stem cells.

Question 1: Is this what stem cells are? Are they the cells that divide to create other, non-dividing cells?

After this, I realized that if the starter cells die, they would not be replaced under my model, because no cell is allowed to divide into new starter cells. I hypothesized that stem cell transplants fail because the host body kills off stem cells one by one and they can it be replaced. I also hypothesized that in the host, outside signals like hormones (which weren't allowed in my model) tell the body to create new stem cells as needed.

Question 2: Do stem cell transplants fail because isolated stem cells do not replicate themselves?

Finally, I noted above that the alternative to a bounded amount of stem cells is exponential growth, where stem cells divide into stem cells.

Question 3: Do embryo stem cells divide into other stem cells exponentially? Is this why embryo stem cells are easier to work with?

Thanks for your time. Even answers like "no, this is obviously wrong because ..." would be appreciated.