Length fluctuations of long cell protrusions: statistics of passage times, random & extreme excursions

Long cell protrusions are effectively one-dimensional dynamic subcellular structures. The length of such a protrusion can keep fluctuating about the mean value even in the steady state. For its optimal performance as a sensor, the length of a protrusion must not cross a narrow band bounded by two thresholds. However, fluctuations may drive the length beyond these thresholds. Using the generic stochastic models of length control of long cell protrusions and exploiting the techniques of level crossings developed for random excursions of stochastic processes, we have derived analytical expressions of (a) passage times for hitting various thresholds, (b) sojourn times of random excursions beyond the threshold and (c) the extreme lengths attained during the lifetime of these protrusions. We apply our general formulae to flagella (also called cilia) of eukaryotic cells thereby getting estimates of the typical length scales and time scales associated with the various aspects of random length fluctuations. Since most of the earlier works investigated only the mean length of cell protrusions, our study of the fluctuations opens a new horizon in the field of subcellular size control.