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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.

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You have a few misconceptions about stem cells, I will try to explain where they are. First of all, cells are not independent. They influence each other with signals and secreted messenger substances. If you look at a human embryo the state of totipotency (where all cells can differentiate into each cell type of the body) ends after 3-4 days when the blastocyte has developed. This structure already has a differentiation into a inner cell mass (embryoblast) and a outer cell layer (tropoblast). If all cells would simply divide on their own, this would rather look like bacteria dividing in culture media, but not as a forming, multicellular organism.

From these few cells in the beginning all cell types of the body have to develop (around 200, if I remember the number correctly). To reach this, the pluripotent stem cells need to proliferate and differentiate (develop into more specialized cell types) quite a lot. So embryonic stem cells do proliferate. Adult stem cells are not proliferating at the same rate a embryonic, but they still do. A good example are the melanocyte stem cells (which are already pretty specialized as they can only further differentiate into melanocytes) which replenish the melanocyte population of the hair bulb at the beginning of each hair cycle. Differentiated cells can divide as well, when necessary, think only about wound healing.

Question 2: Stem cells (especially adult stem cells) need very defined living conditions. This includes specific signalling molecules (Oct or nanog are examples for that) to stay stem cells. If this signalling is missing (and it does in the normal human body) then these cells either differentiate or (more likely) die. The only stem cell transplant which takes place regulary is the transfer of hematopoietic stem cells to leukemia patients. There the problem is to keep the immune system from atacking these cells as foreign.

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