Stochastic Dynamics of Eukaryotic Flagellar Growth

Abstract

We study the dynamics of flagellar growth in eukaryotes where intraflagellar transporters (IFT) play a crucial role. First we investigate a stochastic version of the original balance point model where a constant number of IFT particles move up and down the flagellum. The detailed model is a discrete event vector-valued Markov process occurring in continuous time. First the detailed stochastic model is compared and contrasted with a simple scalar ordinary differential equation (ODE) model of flagellar growth. Numerical simulations reveal that the steady-state mean value of the stochastic model is well approximated by the ODE model. Then we derive a scalar stochastic differential equation (SDE) as a first approximation and obtain a “small noise” approximation showing flagellar length to be Gaussian with mean and variance governed by simple ODEs. The accuracy of the small noise model is compared favorably with the numerical simulation results of the detailed model. Secondly, we derive a revised SDE for flagellar length following the revised balance point model proposed in 2009 in which IFT particles move in trains instead of in isolation. Small noise approximation of the revised SDE yields the same approximate Gaussian distribution for the flagellar length as the SDE corresponding to the original balance point model.

Appendix: A Renewal Processes and Central Limit Theorem

In this section, we briefly summarize some results from the theory of renewal processes. For details, we refer to Ross (2002). We define a renewal process N(t) to be a process that takes integer values, is increasing and the increments are always 1. Also we take $N(0)=0$. Thus N(t) is completely characterized by (random) waiting times $T_n$, $n=1,2,\dots$ between increments. The $T_n$ are further assumed to be iid, and say distributed like T. When T is an exponential, N(t) becomes the familiar Poisson process. The Central limit theorem for renewal processes states that as $t \rightarrow \infty$, $\frac{N(t)}{t}$ converges weakly to a Gaussian with mean $\frac{1}{E(T)}$ and variance $\frac{\text {Var}(T)}{E^3(T)}$. Thus for large t, we may approximate N(t) as

$$\begin{aligned} N(t) \sim \mathcal {N}\left( \frac{1}{E(T)} t,\frac{\text {Var}(T)}{E^3(T)} t\right) . \end{aligned}$$