# How to model multi-step cell differentiation

I have cell $A$ which will differentiate into either cells $A+B$ or cells $A+C$ with probability $p$ and $1-p$, respectively; and cell $C$, which will differentiate into cells $B+B$ or cells $C+C$ with probability $p$ and $1-p$; and the value of $p$ may change with time.

Can I more accurately explain these probabilistic cell lineages (with the aim of predicting an ultimate number of cells) using a deterministic (PDE/ODE) or stochastic model? What technique can be used to incorporate different probabilities at different steps of the lineage? Thank you for any advice.

• 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).

• This assumes that $p$ does not depend on $t$ although the question asks for varying $p$, and that $p$ and $1-p$ are transition rates instead of transition probabilities as in the question. – Did Mar 17 '17 at 9:40

First of all, I think you should find interesting this wiki.article on branching processes.

Next, the answer on your question depends on the precise description of the model you're dealing with - namely, would you like to work in a continuous time setting or in a discrete time. In both cases I would suggest to use Markov Chains: continuous or discrete since the distribution of number of cells for your model depends only on the current state, not on the whole history.

Dependence of probabilities on time may make analysis more difficult because time-dependent Markov Chains are not as popular in the literature as time-independent ones. Sometimes dependence on time may be changed to the more deep dependence on the state (say, when you have more cells than they consume more sources and hence they divide more rare).

With regards to ODE's or PDE's - I don't think that you need them unless you want to consider non-individual cells but rather the density of them spreading in some volume. Another useful wiki.article on Gillespie algorithm, you may also would like to take a look on papers cited there.