Electrical Propagation of Condensed and Diffuse Ions Along Actin Filaments

Abstract

In this article, we elucidate the roles of divalent ion condensation and highly polarized immobile water molecules on the propagation of ionic calcium waves along actin filaments. We introduced a novel electrical triple layer model and used a non-linear Debye-Huckel theory with a non-linear, dissipative, electrical transmission line model to characterize the physicochemical properties of each monomer in the filament. This characterization is carried out in terms of an electric circuit model containing monomeric flow resistances and ionic capacitances in both the condensed and diffuse layers. We considered resting and excited states of a neuron using representative mono and divalent electrolyte mixtures. Additionally, we used 0.05V and 0.15V voltage inputs to study ionic waves along actin filaments in voltage clamp experiments. Our results reveal that the physicochemical properties characterizing the condensed and diffuse layers lead to different electrical conductive mediums depending on the ionic species and the neuron state. This region specific propagation mechanism provides a more realistic avenue of delivery by way of cytoskeleton filaments for larger charged cationic species. A new direct path for transporting divalent ions might be crucial for many electrical processes found in localized neuron elements such as at mitochondria and dendritic spines.

Appendix

1.1 Validation on the mean electrostatic potential approximation

In Figures 13a, b we display in green color the approximate mean electrostatic potential solution used in the diffuse layer for $[Ca]=50nM$ and $[Ca]=1mM$ calcium concentrations, respectively. For comparison purposes, we also include the orange and blue lines, which correspond to the exact nonlinear Debye-Huckel and the linear Debye-Huckel solutions, respectively.

Fig. 13

Figure (a) is for $[Ca]=50nM$ and figure (b) is for calcium concentration $[Ca]=1mM$. The orange, green, and blue lines correspond to the exact nonlinear Debye-Huckel, the nonlinear Debye-Huckel approximation (this work), and the linear Debye-Huckel solutions

1.2 Non trivial contributions to the longitudinal conductivity coming the the diffuse layer

The explicit expression for $\Delta k_{l}^{DL}$ reads

$$\begin{aligned} \Delta k_{l}^{DL}=&-\frac{8HF^{2}B_{DL}\sum _{i}z_{i}u_{i}c_{i}^{DL}}{\varepsilon _{DL}((\frac{l_{B}^{DL}}{a}+x_{c})^{2}-xc^{2})K_{1}(\varepsilon _{DL})}(x_{c}K_{1}(x_{c}\varepsilon _{DL})\\& -(\frac{l_{B}^{DL}}{a}+x_{c})^{2}K_{1}((\frac{l_{B}^{DL}}{a}+x_{c})\varepsilon _{DL})\\& +(\frac{4B_{DL}\epsilon _{DL}\epsilon _{0}}{a\beta \,e}H)^{2}\frac{G}{\mu ((\frac{l_{B}^{DL}}{a}+x_{c})^{2}-x_{c}^{2})}\\& \end{aligned}$$

where

$$\begin{aligned} H(z_{p})=\frac{1}{z_{p}(1+q)(1+\sqrt{1+p^{2}\zeta ^{2}}} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{c} G(l_{B}^{DL},x_{c,}\varepsilon _{DL})K_{1}^{2}(\varepsilon _{DL})=\\ (\frac{l_{B}^{DL}}{a}+x_{c})^{2}K_{1}^{2}((\frac{l_{B}^{DL}}{a}+x_{c})\varepsilon _{DL})\\ -x_{c}^{2}K_{1}^{2}(x_{c}\varepsilon _{DL})+x_{c}^{2}K_{0}(x_{c}\varepsilon _{DL})K_{2}(x_{c}\varepsilon _{DL})\\ -(\frac{l_{B}^{DL}}{a}+x_{c})^{2}K_{0}((\frac{l_{B}^{DL}}{a}+x_{c})\varepsilon _{DL})K_{2}((\frac{l_{B}^{DL}}{a}+x_{c})\varepsilon _{DL}) \end{array} \end{aligned}$$

1.3 Condensed layer capacitance parameters

The expressions for the parameters $a_{0}$, $a_{1}$, $a_{2}$, and $a_{3}$ are $a_{0}$$=-\frac{L}{M+\frac{N}{2}}-\frac{L^{3}\,N\,\gamma ^{2}\,\zeta _{v}^{2}}{4(M+\frac{N}{2})^{3}\,(2M+N)}$, $a_{1}=\frac{1}{M+\frac{N}{2}}+\frac{3L^{2}\,N\,\gamma ^{2}\,\zeta _{v}^{2}}{4(M+\frac{N}{2})^{3}\,(2M+N)}$, $a_{2}=-\frac{3\ L\ N\,\gamma ^{2}\,\zeta _{v}^{2}}{4(M+\frac{N}{2})^{3}(2\,M+N)}$, and $a_{3}=\frac{N\,\gamma ^{2}\,\zeta _{v}^{2}}{4(M+\frac{N}{2})^{3}(2\,M+N)}$, where $M=\frac{4\,a\,\pi \,log(x_{c})}{\epsilon _{CL}}$, $N=\frac{8\,a\,\pi }{\epsilon _{DL}\,\varepsilon _{DL}}\frac{K_{0}(\varepsilon _{DL}\,x_{c})}{K_{1}(\varepsilon _{DL})}$ and $L=\frac{1}{8\,z_{p}\,\beta \,e}(x_{c}^{2}-1-2\,log(x_{c}))\varepsilon _{CL}^{2}$.

1.4 Impedance expression for the triple layer model

The expression for the impedance in the triple layer model reads

$$\begin{aligned} \begin{array}{c} Z=\frac{2}{15}\sqrt{2}\pi (5R_{l}^{eqv}+8b^{eqv}R_{t}^{eqv}V_{o})\\ \sqrt{4\pi ^{2}+\frac{\sqrt{-225(R_{l}^{eqv}+R_{t}^{eqv})^{2}+16(5R_{l}^{eqv}+8b^{eqv}R_{t}^{eqv}V_{o})^{2}\pi ^{4}}}{(5R_{l}^{eqv}+8b^{eqv}R_{t}^{eqv}V_{o})}} \end{array} \end{aligned}$$

For $R_{t}<<R_{l}$, this expression recovers the approximation $Z\simeq 25.1128R_{l}$ used in our previous work on the EDL in which the transversal resistance was neglectable.