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Long cell protrusions, which are effectively one-dimensional, are highlydynamic subcellular structures. Length of many such protrusions keepfluctuating about the mean value even in the the steady state. We develop herea stochastic model motivated by length fluctuations of a type of appendage ofan eukaryotic cell called flagellum (also called cilium). Exploiting thetechniques developed for the calculation of level-crossing statistics of randomexcursions of stochastic process, we have derived analytical expressions ofpassage times for hitting various thresholds, sojourn times of randomexcursions beyond the threshold and the extreme lengths attained during thelifetime of these model flagella. We identify different parameter regimes ofthis model flagellum that mimic those of the wildtype and mutants of a wellknown flagellated cell. By analysing our model in these different parameterregimes, we demonstrate how mutation can alter the level-crossing statisticseven when the steady state length remains unaffected by the same mutation.Comparison of the theoretically predicted level crossing statistics, inaddition to mean and variance of the length, in the steady state with thecorresponding experimental data can be used in near future as stringent testsfor the validity of the models of flagellar length control. The experimentaldata required for this purpose, though never reported till now, can becollected, in principle, using a method developed very recently for flagellarlength fluctuations.

Level crossing statistics in a biologically motivated model of a long dynamic protrusion: passage times, random and extreme excursions