Simulation of intracellular $\hbox {Ca}^{2+}$ transients in osteoblasts induced by fluid shear stress and its application

Abstract

Intracellular $\hbox {Ca}^{2+}$ transient induced by fluid shear stress (FSS) plays an important role in mechanical regulation of osteoblasts, but the cellular mechanism remains incompletely understood. Here, we constructed a mathematical model combined with experiments to elucidate it. Our simulated and experimental results showed that it was the delay of membrane potential repolarization to produce the refractory period of FSS-induced intracellular calcium transients in osteoblasts. Moreover, the results also demonstrated that the amplitude of FSS-induced intracellular calcium transient is crucial to the proliferation, while its duration is critical to the differentiation, of osteoblasts. Overall, the present study provides a way to understand the cellular mechanism of intracellular calcium transients in osteoblast induced by FSS and explains some of related physiological events.

Appendix: Model Equations

FSS-induced $\hbox {Ca}^{2+}$ transients in osteoblasts were described as the following equations:

$$\begin{aligned}&\frac{d[Na^{+}]_i }{\hbox {d}t}=-(I{ }_{\mathrm{SAC}}+I_{\mathrm{Nab}} +3I_{\mathrm{NaK}} ), \end{aligned}$$

(1)

$$\begin{aligned}&\frac{d[\hbox {K}^{+}]_i }{\hbox {d}t}=-(I{ }_{\mathrm{SAC}}+I_{\mathrm{Kb}} -2I_{\mathrm{NaK}} ), \end{aligned}$$

(2)

$$\begin{aligned}&\frac{dV}{\hbox {d}t}=-(I_{\mathrm{SAC,Na}} +I_{\mathrm{SAC,K}} +I_{\mathrm{NaK}} +I_{\mathrm{Nab}} \nonumber \\\quad&+I_{\mathrm{Kb}} +I_{\mathrm{CaL}} +I_{\mathrm{Cab}} ), \end{aligned}$$

(3)

$$\begin{aligned}&\frac{dATP}{\hbox {d}t}=k_{\mathrm{ATP}} \phi \left( {Ca} \right) -k_{-\mathrm{ATP}} ATP, \end{aligned}$$

(4)

$$\begin{aligned}&\frac{dP}{\hbox {d}t}=\frac{k_p B_{\max } ATP}{k_d +ATP}\frac{Ca_{\mathrm{cyt}} }{1+kK}-k_{-\mathrm{p}} P, \end{aligned}$$

(5)

$$\begin{aligned}&\frac{dK}{\hbox {d}t}=k_K Ca_{\mathrm{cyt}} \left( {K_T -K} \right) -k_{-\mathrm{K}} K, \end{aligned}$$

(6)

$$\begin{aligned}&\frac{dCa_{\mathrm{nano}} }{\hbox {d}t}=\frac{1}{\gamma _s }\left[ {J_{\hbox {CaL}} -k_{\mathrm{ATP}} \left( {Ca_{\mathrm{nano}} -Ca_{\mathrm{cyt}} } \right) } \right] , \end{aligned}$$

(7)

$$\begin{aligned}&\frac{dCa_{\mathrm{cyt}} }{\hbox {d}t}=k_{\mathrm{dif}} \left( {Ca_s -Ca_{\mathrm{cyt}} } \right) +\left( {k_{\mathrm{IP}_3 \mathrm{R}} +k_{\mathrm{er}} } \right) \left( Ca_{\mathrm{er}}\right. \nonumber \\&\left. \quad -Ca_{\mathrm{cyt}} \right) -J_{\mathrm{erca}} , \end{aligned}$$

(8)

$$\begin{aligned}&\frac{1}{\gamma _{\mathrm{er}} }\frac{dCa_{\mathrm{er}} }{\hbox {d}t}=-\left( {k_{\mathrm{IP}_3 R} +k_{\mathrm{er}} } \right) \left( {Ca_{\mathrm{er}} -Ca_{\mathrm{cyt}} } \right) +J_{\mathrm{erca}},\nonumber \\ \end{aligned}$$

(9)

where

$$\begin{aligned} \phi \left( {Ca} \right)= & {} \frac{Ca_s ^{4}}{Ca_s ^{4}+k_{\mathrm{Ca}} ^{4}}\\ k_{\mathrm{IP}_3 \mathrm{R}}= & {} k_f P_{\mathrm{IP}_3 \mathrm{R}} ; \end{aligned}$$

$k_{\mathrm{ATP} }$=100 $\upmu$M$^{-1}$s$^{-1}$, if Ca $_{\mathrm{s}}$ was higher than 1 $\upmu$M, else $k_{\mathrm{ATP} }$= 0;

$I_{\mathrm{NaK}}$ was computed with the model of NaK ATPase taken from the model of Luo and Rudy (1994);

$\hbox {Ca}^{2+ }$ current of L-VSCCs, $I_{\mathrm{CaL}}$ was obtained with the model of Tusscher et al. (2004) and $\hbox {Ca}^{2+}$ influx through L-VSCCs was described by

$$\begin{aligned} J_\mathrm{{CaL}}= & {} -I_\mathrm{{CaL}} \frac{A_\mathrm{{cap}} C_m}{2F\times 10^{-6}{\upmu } \mathrm{l/pl}} \end{aligned}$$

IP$_{3}$ production, P, and $\hbox {Ca}^{2+}$ activating PKC, K, were from the model of Kang and Othmer (2009).

$\gamma _{s}$ was the ratio of cytoplasmic volume to the volume of small space;

Na$^{+}$ and K$^{+}$ current of SACs was described by the equations:

$$\begin{aligned} I_{\mathrm{SAC,Na}}= & {} G_{\mathrm{SAC,Na}} P_{\mathrm{SAC}} (V-E_{\mathrm{Na}} ), \end{aligned}$$

(10)

$$\begin{aligned} I_{\mathrm{SAC,K}}= & {} G_{\mathrm{SAC,K}} P_{\mathrm{SAC}} (V-E_K ), \end{aligned}$$

(11)

$G_{\mathrm{SAC,Na}}$ and $G_{\mathrm{SAC,K}}$ were the max conductance of Na$^{+}$ and K$^{+}$ current through SACs. $P_{\mathrm{SAC}}$ was the open probability of SACs, which was set to one or zero based on our experimental results (Fig. 3). $E_{\mathrm{Na}}$ and $E_{\mathrm{K}}$ are obtained with Nernst equation.