Power spectrum, growth velocities and cross-correlations of longitudinal and transverse oscillations of individual Nicotiana tabacum pollen tube

We report on our results concerning growth rate and oscillation modes of the individual pollen tube apex. The observed volumetric growth and growth rate periodicity in the longitudinal (axial) direction are accompanied by transverse oscillations with similar frequencies but higher energies than the axial modes. Examination of the time-domain coherence between oscillations in mutually perpendicular directions revealed minimal energy dissipation in the unperturbed (isotonic) case, opposite to the two remaining cases (hypertonic, hypotonic) with notable correlations. We conjecture that the minimal energy loss is therefore optimal in the natural growth conditions. The longitudinal growth velocity is also found to be the fastest in the unperturbed case. As a result, the isolated system (pollen tube tip) is conserving energy by transforming it from elastic potential energy of extending apical wall to the kinetic energy of periodical motion. The energy dissipation is found to be about 20 % smaller in axial direction than in lateral one, indicating that the main energy consumption is dedicated to the elongation. We further observe that the hypertonic spectrum is shifted towards lower and the hypotonic towards higher frequencies with respect to the isotonic spectrum. In consequence, the turgor pressure inside the growing cell influences monotonically the frequency of both modes of oscillations. The calculated power spectrum seen as a measure of the overall energy efficiency of tip growth under hypertonic, hypotonic and isotonic conditions implies that the biochemistry has been fine tuned to be optimal under normal growth conditions, which is the developmental implication of this work. A simple theoretical extension of the Ortega equation is derived and analysed with respect to its contribution to power spectrum. We show that the plastic term, related to the effective turgor pressure, with maximum contribution at frequency f = 0 is responsible for the steady growth. In turn, the elastic contribution dependent on the time derivative of pressure fluctuations tends to move the system into oscillatory mode (f > 0). None of those mechanisms is privileged over another. The coupling mechanism is naturally generated through a convolution of those two terms and will decide about the overall character of the growth for each particular case.