# Why are cell populations maintained via multiple intermediate compartments?

## Abstract

We consider the maintenance of ‘product’ cell populations from progenitor cells via one or more intermediate compartments. Every cell in each compartment undergoes one of three fates: the cell may divide, die or make a transition to the next compartment. As well as identifying an individual cell by the compartment it belongs to, \$c=1,\ldots,C\$, we label it by generation, \$n=0,1,\ldots\$.

The progenitor cell is said to be in generation \$0\$. Whenever a cell in generation \$n\$ divides, the result is two cells in generation \$n+1\$. From this point of view, the population of product cells is heterogeneous because it is made up of cells of different generations. The number of cells from one family that become product cells is the random variable \$R\$ with mean value \$N\$. We find the probability distribution of \$R\$ as the ultimate state of a multitype branching process. The variance of \$R\$ is proportional to \$N^{2+1/C}\$ when \$N\$ is large. If there is only one compartment, a large ratio of product cells to progenitors comes with the cost of the product cell population being dominated by large families of cells descended from individual progenitors, and large average number of generations separating product cells from progenitors.

These undesirable features can be avoided if there are multiple intermediate compartments. A sequence of compartments is, in fact, an efficient way to maintain a product cell population from a progenitor population, avoiding excessive clonality and minimising the number of rounds of division en route. Given \$N\$, the prevalence of large families of cells in the product cell population decreases as the number of compartments increases, and the mean generation number, \$D\$, decreases. With \$C\$ compartments, \$D_{min} = 2C(N^{1/C}-1)\$.

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