Autocorrelation function and power spectrum of two-state random processes used in neurite guidance

During development neurons extend and retract cytoskeletal structures, chiefly microtubules and filopodia, to process informational cues from the extracellular environment and thereby guide growth cone migration toward an appropriate synaptic partner. This cytoskeleton-based exploration is achieved by stochastic switching, with microtubules and filopodia alternating between growing and shortening phases apparently at random. If stabilizing signals are not detected during the growth phase, then the structures switch to a shortening state, from which they can again return to a growth phase, and so forth. A useful means of characterizing these stochastic processes in a model-independent way is by autocorrelation and spectral analysis. Previously, we compared experiment to theory by performing Monte Carlo simulations and computing the autocorrelation function and power spectrum from the simulated dynamics, an approach that is computationally intensive and requires recalculation whenever model parameters are changed. Here we present analytical expressions for the autocorrelation function and power spectrum, which compactly characterize microtubule and filopodial dynamics based on the stochastic, two-state model. The model assumes that the phase times are of variable duration and gamma-distributed, consistent with experimental evidence for microtubules assembled in vitro from purified tubulin. The analytical expressions permit the precise quantitative characterization of changes in microtubule and filopodial searching behavior corresponding to changes in the shape of the gamma distribution.