Multiscale master equation for dividing cells I have a conceptual problem trying to build a master equation for dividing cells which have a certain surface receptor. Each cell has its own receptor dynamics, and they divide with receptor-dependent rates. 
For each cell, receptors $n$ are produced and degraded. This is reflected in the following reactions
$$A \xrightarrow{\alpha} n$$
$$n\xrightarrow{\beta} \emptyset$$
where $A$ is assumed to be a large reservoir from which receptors can be constantly produced. The master equation for this system is
$$\frac{dP(n,t)}{dt}=\alpha P(n-1,t)-\alpha P(n,t) + \beta(n+1)P(n+1,t) - \beta n P(n,t)$$
whose stationary solution, if I'm not mistaken, is the Poisson distribution:
$$P(n)=\frac{(\alpha/\beta)^n}{n!}e^{\alpha/\beta}$$
Thus, for each cell, the distribution of receptors will be Poisson-distributed. On the other hand, if we consider that cells divide, die and there is immigration, we can write the following set of reactions:
$$B \xrightarrow{\lambda_0} m$$
$$m\xrightarrow{\lambda_1} 2m$$
$$m\xrightarrow{\mu} \emptyset$$
where $B$ is a large reservoir of cells. Then we can construct the master equation for this system and find the probability of the system to have $m$ cells, $P(m,t)$. But at this point I'm stuck trying to find a way to join the fact that cells divide, but at the same time each cell produces and degrades receptor.
So the obvious question is: how can I join these two levels of description?. Is there any way to incorporate the fact that cells divide with receptor-dependent rates? Thanks!
 A: One solution would be to keep track of the number of cells with a certain number of receptors. In other words, instead of $P(m,t)$ you would consider $P(\{m_k\}_{k=0}^\infty,t)$, where $k$ is an index for the number of receptors on each cell (and $\sum_{k=0}^{k_{max}} m_k=m$).  Depending on the details of cell division, you would have some $k$ dependent transition rate where a single cell in state $k$ divides into two cells in states $k'$ and $k''$. For example, a simple division rule would be for $(m_0,m_1,\dots,m_k,\dots)\underset{\alpha_k}{\mapsto} (m_0+1,m_1,\dots,m_k,\dots)$. That is, one cell from the set of cells with $k$ receptors produces a new cell with no receptors at rate $\alpha_k$. You could also consider more general transition rules $\alpha_k^{(j)}$ or $\alpha_k^{(j,j')}$ depending on how receptors are distributed between daughter cells, or whether receptors are conserved during cell division.
In this formalism the production/degradation of receptors corresponds to a separate transition rule, whereby $(\dots,m_k,m_{k+1},\dots)\underset{\Gamma_{\mp}}{\mapsto} (\dots,m_k\pm 1,m_{k+1}\mp 1,\dots)$. In other words, modulo intricacies of actual biological processes the model can be described in the framework of birth-death processes on a 1D lattice.
