An important finding has been the discovery that the rate of pollen tube growth oscillates (Pierson et al. 1995). In vitro, lily pollen tubes longer than 600-700 ^m exhibit a change in growth rate from 100 to 500 nm/s over a period of 15-50 s (Pierson et al. 1996). Additionally, the intracellular activities of both Ca2+ and H+, as well as the extracellular fluxes of Ca2+, H+, K+, and Cl-, all exhibited oscillatory profiles which possessed the same period as that of growth, but usually not the same phase (Holdaway-Clarke and Hep-ler 2003). These observations allow us to decipher the phase relationship between an ion activity or its extracellular flux and the underlying rate of growth. Further, by determining whether an ion expression precedes or follows growth, information can be gained about those events or processes that anticipate and possibly regulate growth, as opposed to those that follow and appear to be governed by the preceding growth event.
When two processes oscillate, but not with the same phase, it is not immediately obvious which one precedes the other. To solve this problem, cross-correlation analysis has been used. Cross-correlation analysis of simultaneously collected time series processes, e.g., tip growth and process "x", allows the strength of correlation and the lag between two processes to be established (Brillinger 1981).
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