Calcium signaling modulates the dynamics of cilia and flagella

Abstract

To adapt to changing environments cells must signal and signaling requires messengers whose concentration varies with time in space. We here consider the messenger role of calcium ions implicated in regulation of the wave-like bending dynamics of cilia and flagella. The emphasis is on microtubules as polyelectrolytes serving as transmission lines for the flow of Ca²⁺ signals in the axoneme. This signaling is superimposed with a geometric clutch mechanism for the regulation of flagella bending dynamics and our modeling produces results in agreement with experimental data.

Appendices

Appendix 1

Finding the second derivative in Eq. (3.32).

$$\frac{{d}^{2}}{d{s}^{2}}\left[tanh\left(1\right)-tanh\left(\frac{s}{L}-\tau \right)\right]\frac{{\text{d}}\phi }{ds}=\frac{d}{ds}\left[\frac{-1}{L}\frac{1}{{cosh}^{2}\left(\frac{s}{L}-\tau \right)}\left(\frac{{\text{d}}\phi }{ds}\right)+\left[tanh\left(1\right)-tanh\left(\frac{s}{L}-\tau \right)\right]\frac{{d}^{2}\phi }{d{s}^{2}}\right]= \frac{-1}{L}\frac{1}{{cosh}^{2}\left(\frac{s}{L}-\tau \right)}\left(\frac{{d}^{2}\phi }{d{s}^{2}}\right)+\frac{2}{{L}^{2}}\frac{tanh\left(\frac{s}{L}-\tau \right)}{{cosh}^{2}\left(\frac{s}{L}-\tau \right)}\left(\frac{{\text{d}}\phi }{ds}\right)-\frac{1}{L}\frac{1}{{cosh}^{2}\left(\frac{s}{L}-\tau \right)}\left(\frac{{d}^{2}\phi }{d{s}^{2}}\right)+\left[tanh\left(1\right)-tanh\left(\frac{s}{L}-\tau \right)\right]\frac{{d}^{3}\phi }{d{s}^{3}}=\frac{2}{L}\frac{1}{{cosh}^{2}\left(\frac{s}{L}-\tau \right)}\left(\frac{{d}^{2}\phi }{d{s}^{2}}\right)+\frac{2}{{L}^{2}}\frac{tanh\left(\frac{s}{L}-\tau \right)}{{cosh}^{2}\left(\frac{s}{L}-\tau \right)}\left(\frac{{\text{d}}\phi }{ds}\right)+\left[tanh\left(1\right)-tanh\left(\frac{s}{L}-\tau \right)\right]\frac{{d}^{3}\phi }{d{s}^{3}}$$

(A.1.1)

Appendix 2

We start with the dimensionless variable $\xi$

$$\xi =\frac{s}{L};\frac{{\text{d}}\phi }{ds}=\frac{1}{L}\frac{{\text{d}}\phi }{{\text{d}}\xi }; \frac{{d}^{2}\phi }{d{s}^{2}}=\frac{1}{{L}^{2}}\frac{{d}^{2}\phi }{d{\xi }^{2}}; \frac{{d}^{3}\phi }{d{s}^{3}}=\frac{1}{{L}^{3}}\frac{{d}^{3}\phi }{d{\xi }^{3}}.$$

(A.2.1)

so that Eq. (3.30) now reads:

$$\frac{{d}^{4}\phi }{d{\xi }^{4}}-\frac{{C}_{1}\left(\sigma \right)L}{\kappa }\left[tanh\left(1\right)-tanh\left(\xi -\tau \right)\right]\left(\frac{{d}^{3}\phi }{d{\xi }^{3}}\right)+\left[\frac{2L{C}_{1}\left(\sigma \right)}{\kappa }\frac{1}{{cosh}^{2}\left(\xi -\tau \right)}-\frac{{C}_{2}\left(\sigma \right){L}^{2}}{\kappa }\right]\left(\frac{{d}^{2}\phi }{d{\xi }^{2}}\right)-\frac{2L{C}_{1}\left(\sigma \right)}{\kappa }\left[\frac{tanh\left(\xi -\tau \right)}{{cosh}^{2}\left(\xi -\tau \right)}\right]\left(\frac{{\text{d}}\phi }{{\text{d}}\xi }\right)+\frac{\sigma {C}_{N}{L}^{4}}{\kappa }\phi =0.$$

(A.2.2)

Using the abbreviations for ${C}_{1}\left(\sigma \right)$, ${C}_{2}\left(\sigma \right)$, Eqs. (3.34, 3.35) we go over to new ones ${\lambda }_{1}$, ${\lambda }_{2}$ and $\gamma$ as expressed in the set of Eqs. (3.38, 3.39, 3.40). We just illustrate the specified numerical calculation for parameter:

$$\gamma =\frac{i\omega {C}_{N}{L}^{4}}{\kappa }=\frac{310\frac{1}{s}\cdot 2.5\cdot {10}^{-3}\frac{Ns}{{m}^{2}}\cdot {\left(12\cdot {10}^{-6}\right)}^{4}{m}^{4}}{5.22\cdot {10}^{-22}N{m}^{2}}\left(i\right)=32i.$$

(A.2.3)

Similarly, we completed the explicit form of Eq. (3.41).