In this problem, we consider a phenomenological model for steady state microtubule dynamics that was introduced by Dogterom and Leibler (1993). Note that there is a more interesting class of models that include GTP hydrolysis explicitly (see Flyvbjerg et al. 1996 ). The goal of such models is to respond to data such as those in Figure 15.34 This figure shows a record of the length of a single microtubule as a function of time and reveals the series of "catastrophes" and "rescues" as the polymer changes its length.
(a) Deduce the following equations for the probability distributions $p_{+}(n, t)$ and $p_{-}(n, t),$ which give the probability of finding a microtubule:
\[
\begin{array}{l}
\frac{\partial p_{+}}{\partial t}=-f_{+-} p_{+}+f_{-+} p_{-}-v_{+} \frac{\partial p_{+}}{\partial z} \\
\frac{\partial p_{-}}{\partial t}=+f_{+-} p_{+}-f_{-+} p_{-}+v_{-} \frac{\partial p_{-}}{\partial z}
\end{array}
\]
Write a master equation for $p_{+}(n, t)$ and $p_{-}(n, t)$ by noting that there are four processes that can change the probability at each instant. Consider the $+$ case: (i) the $n-1$ polymer can grow and become an $n$ polymer, characterized by a rate $v_{+} ;$ (ii) the $n$ polymer can grow and become an $n+1$ polymer, also characterized by a rate $v_{+} ;$ (iii) the $n+$ polymer can switch from growing to shrinking with a rate $f_{+-} ;$ and (iv) the $n-$ polymer can switch from shrinking to growing with a rate $f_{-+} .$ If you account for all four of these possibilities, you will have the correct master equation. Use a Taylor expansion on factors like $p_{+}(n-1, t)-p_{+}(n, t)$ to obtain Equations 15.124 and 15.125
(b) Solve these equations for $p_{\pm}(n)$ in the steady state (that is, $\partial p_{\pm}(n, t) / \partial t=0$ ). Show that in the steady state, the probabilities $p_{\pm}(n)$ decay exponentially with constant
\[
\sigma=\frac{v_{+} f_{-+}-v_{-} f_{+-}}{v_{+} v_{-}}
\]
(c) Use Figure 15.34 to estimate the parameters $v_{+}, v_{-}, f_{+-}, f_{-+},$ and then find the average length of the polymers that is predicted by this simple model. To find the average length, you will need to sum over all lengths with their appropriate probability. The slopes of the growth and decay regions tell you about the on and off rates, and the durations of the growth and decay periods tell you something about the parameters $f_{+-}$ and $f_{-+} .$ Note that by fitting the dynamical data, you are deducing/predicting something about the distribution of lengths.