# How to model multi-step cell differentiation

Can I better explain cell lineages using PDEs or stochastic?

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This is a major change to the question and renders the previous answers non-responsive. Please return to the prior question, then ask a new question if you want. I suspect this question needs much more detail to get a reasonable answer. –  Ross Millikan Feb 12 '13 at 18:51

In the limit of large numbers of cells, if $a(t)$, $b(t)$ and $c(t)$ are the numbers of cells of type $A$, $B$ and $C$ then the average dynamics are described by the ODEs

\begin{align} \dot{a} & = 0 \\ \dot{b} & = pa + 2pc \\ \dot{c} & = (1-p)a + (1-2p)c \end{align}

The first of these means that $a(t)=a_0$ always. The third can then be solved to give

$$c(t) = \frac{(1-2p)c_0+ (1-p)a_0}{1-2p} e^{(1-2p)t} - \frac{(1-p)a_0}{1-2p}$$

which leads to a solution for the second equation by integrating:

$$b(t) = \frac{(1-2p)c_0+ (1-p)a_0}{(1-2p)^2} e^{(1-2p)t} - \frac{(1-2p^2)a_0}{1-2p}t + \frac{(1-2p)^2b_0 - (1-2p)c_0 - (1-p)a_0}{(1-2p)^2}$$

This is assuming that $p\neq 1/2$. In that case the equations are somewhat simplified ($c(t)$ is linear in $t$ and $b(t)$ is quadratic).

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