Hydrostatic pressure and interfacial tension induce mode instability in wave propagation along a liquid-filled microtubule

Abstract

Wave propagation in microtubules plays an important role in cell function and engineering applications. Interfacial tension and hydrostatic pressure significantly affect such wave propagation in liquid-filled microtubules, but it remains elusive how they influence the dispersion relation. To address this, we develop a theoretical model based on Flügge’s theory, with interfacial tension and hydrostatic pressure duly accounted for. We then employ the model to analyze the dispersion relation of axisymmetric and non-axisymmetric waves. The difference between interfacial tension and hydrostatic pressure is found to affect the dispersion relation. With the increase in interfacial tension, wave velocity increases for all modes of axisymmetric waves under different hydrostatic pressures. With the increase in interfacial tension or decrease in hydrostatic pressure, wave velocity increases for the first mode of the non-axisymmetric wave but non-monotonously changes for the second and third modes of the non-axisymmetric wave. Notably, increasing the difference between dimensionless hydrostatic pressure (μ) and dimensionless interfacial tension (λ) can lead to mode instability. For the axisymmetric wave, the second mode becomes unstable when |μ-λ| is sufficiently large. For the non-axisymmetric wave, the first mode becomes unstable when |μ-λ| is large enough and the second mode becomes unstable only when μ-λ is positive and large enough. The developed theory enables a better understanding of the effect of the environment on signal transmission in cells and provides guidelines in nondestructive testing with microtubules.