# 8.8 Exploring the biochemical and mechanical effects of intestinal  (Page 5/6)

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## Myosin activation

The activation of the ${M}_{K}$ complex includes 4 $C{a}^{{2}^{+}}$ ions bonded to calmodulin and myosin light chain kinase. Side reactions include the disassociation of the kinase from the complex and interactions with binding proteins. These interactions were modeled using mass action kinetics summarized in 9 reactions from [link] .

$\begin{array}{cc}\hfill \frac{d\left[C\right]}{dt}& ={r}_{1}\left[C{a}_{2}C\right]+{r}_{3}\left[{C}_{{M}_{K}}\right]+{r}_{8}\left[{C}_{{B}_{P}}\right]-{f}_{1}{\left[C{a}^{{2}^{+}}\right]}^{2}\left[C\right]\hfill \\ & -{f}_{3}\left[C\right]\left[{M}_{K}\right]-{f}_{8}\left[C\right]\left[{B}_{P}\right]\hfill \\ \hfill \frac{d\left[{M}_{K}\right]}{dt}& ={r}_{3}\left[{C}_{{M}_{K}}\right]+{r}_{4}\left[C{a}_{2}{C}_{{M}_{K}}\right]+{r}_{5}\left[C{a}_{4}{C}_{{M}_{K}}\right]-{f}_{3}\left[C\right]\left[{M}_{K}\right]\hfill \\ & -{f}_{4}\left[{M}_{K}\right]\left[C{a}_{2}C\right]-{f}_{5}\left[{M}_{K}\right]\left[C{a}_{4}C\right]\hfill \\ \hfill \frac{d\left[C{a}_{2}C\right]}{dt}& ={f}_{1}{\left[C{a}^{{2}^{+}}\right]}^{2}\left[C\right]+{r}_{2}\left[C{a}_{4}C\right]+{r}_{4}\left[C{a}_{2}{C}_{{M}_{K}}\right]\hfill \\ & +{f}_{9}{\left[C{a}^{{2}^{+}}\right]}^{2}\left[{C}_{{B}_{P}}\right]-{r}_{1}\left[C{a}_{2}C\right]-{f}_{2}{\left[C{a}^{{2}^{+}}\right]}^{2}\left[C{a}_{2}C\right]\hfill \\ & -{f}_{4}\left[{M}_{K}\right]\left[C{a}_{2}C\right]-{r}_{9}\left[C{a}_{2}C\right]\left[{B}_{P}\right]\hfill \\ \hfill \frac{d\left[C{a}_{4}C\right]}{dt}& ={f}_{2}{\left[C{a}^{{2}^{+}}\right]}^{2}\left[C{a}_{2}C\right]+{r}_{5}\left[C{a}_{4}{C}_{{M}_{K}}\right]-{r}_{2}\left[C{a}_{4}C\right]\hfill \\ & -{f}_{5}\left[{M}_{K}\right]\left[C{a}_{4}C\right]\hfill \\ \hfill \frac{d\left[{C}_{{M}_{K}}\right]}{dt}& ={f}_{3}\left[C\right]\left[{M}_{K}\right]+{r}_{6}\left[C{a}_{2}{C}_{{M}_{K}}\right]-{r}_{3}\left[{C}_{{M}_{K}}\right]\hfill \\ & -{f}_{6}{\left[C{a}^{{2}^{+}}\right]}^{2}\left[{C}_{{M}_{K}}\right]\hfill \\ \hfill \frac{d\left[C{a}_{2}{C}_{{M}_{K}}\right]}{dt}& ={f}_{4}\left[{M}_{K}\right]\left[C{a}_{2}C\right]+{f}_{6}{\left[C{a}^{{2}^{+}}\right]}^{2}\left[{C}_{{M}_{K}}\right]+{r}_{7}\left[C{a}_{4}{C}_{{M}_{K}}\right]\hfill \\ & -{r}_{4}\left[C{a}_{2}{C}_{{M}_{K}}\right]-{r}_{6}\left[C{a}_{2}{C}_{{M}_{K}}\right]-{f}_{7}{\left[C{a}^{{2}^{+}}\right]}^{2}\left[C{a}_{2}{C}_{{M}_{K}}\right]\hfill \\ \hfill \frac{d\left[C{a}_{4}{C}_{{M}_{K}}\right]}{dt}& ={f}_{5}\left[{M}_{K}\right]\left[C{a}_{4}C\right]+{f}_{7}{\left[C{a}^{{2}^{+}}\right]}^{2}\left[C{a}_{2}{C}_{{M}_{K}}\right]\hfill \\ & -{r}_{5}\left[C{a}_{4}{C}_{{M}_{K}}\right]-{r}_{7}\left[C{a}_{4}{C}_{{M}_{K}}\right]\hfill \\ \hfill \frac{d\left[{B}_{P}\right]}{dt}& ={r}_{8}\left[{C}_{{B}_{P}}\right]+{f}_{9}{\left[C{a}^{{2}^{+}}\right]}^{2}\left[{C}_{{B}_{P}}\right]-{f}_{8}\left[C\right]\left[{B}_{P}\right]\hfill \\ & -{r}_{9}\left[C{a}_{2}C\right]\left[{B}_{P}\right]\hfill \\ \hfill \frac{d\left[{C}_{{B}_{P}}\right]}{dt}& ={f}_{8}\left[C\right]\left[{B}_{P}\right]+{r}_{9}\left[C{a}_{2}C\right]\left[{B}_{P}\right]-{r}_{8}\left[{C}_{{B}_{P}}\right]\hfill \\ & -{f}_{9}{\left[C{a}^{{2}^{+}}\right]}^{2}\left[{C}_{{B}_{P}}\right]\hfill \end{array}$
 ${\left[C\right]}_{0}=$ $0.9285\phantom{\rule{0.277778em}{0ex}}\mu M$ Initial Calmodulin (C) concentration [link] ${\left[{M}_{K}\right]}_{0}=$ $9.6506\phantom{\rule{0.277778em}{0ex}}\mu M$ Initial Myosin LC Kinase ( ${M}_{K}$ ) concentration [link] ${\left[C{a}_{2}C\right]}_{0}=$ $0.0015\phantom{\rule{0.277778em}{0ex}}\mu M$ Initial $C{a}_{2}C$ complex concentration [link] ${\left[C{a}_{4}C\right]}_{0}=$ $0.00\phantom{\rule{0.277778em}{0ex}}\mu M$ Initial $C{a}_{4}C$ complex concentration [link] ${\left[{C}_{{M}_{K}}\right]}_{0}=$ $0.3332\phantom{\rule{0.277778em}{0ex}}\mu M$ Initial $C{M}_{K}$ complex concentration [link] ${\left[C{a}_{2}{C}_{{M}_{K}}\right]}_{0}=$ $0.2713\phantom{\rule{0.277778em}{0ex}}\mu M$ Initial $C{a}_{2}C{M}_{K}$ complex concentration [link] ${\left[C{a}_{4}{C}_{{M}_{K}}\right]}_{0}=$ $0.013\phantom{\rule{0.277778em}{0ex}}\mu M$ Initial $C{a}_{4}C{M}_{K}$ activated complex concentration [link] ${\left[{B}_{P}\right]}_{0}=$ $15.1793\phantom{\rule{0.277778em}{0ex}}\mu M$ Initial Binding Protein ( ${B}_{P}$ ) concentration [link] ${\left[{C}_{{B}_{P}}\right]}_{0}=$ $2.8207\phantom{\rule{0.277778em}{0ex}}\mu M$ Initial $C-{B}_{P}$ complex concentration [link] $\left[{f}_{1}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{r}_{1}\right]=$ $\left[12\phantom{\rule{0.277778em}{0ex}}\mu {M}^{-1}$ $12\right]{s}^{-1}$ Forward and reverse rates for reaction 1 [link] $\left[{f}_{2}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{r}_{2}\right]=$ $\left[480\phantom{\rule{0.277778em}{0ex}}\mu {M}^{-1}$ $1200\right]{s}^{-1}$ Forward and reverse rates for reaction 2 [link] $\left[{f}_{3}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{r}_{3}\right]=$ $\left[5\phantom{\rule{0.277778em}{0ex}}\mu {M}^{-1}$ $135\right]{s}^{-1}$ Forward and reverse rates for reaction 3 [link] $\left[{f}_{4}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{r}_{4}\right]=$ $\left[840\phantom{\rule{0.277778em}{0ex}}\mu {M}^{-1}$ $45.4\right]{s}^{-1}$ Forward and reverse rates for reaction 4 [link] $\left[{f}_{5}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{r}_{5}\right]=$ $\left[28\phantom{\rule{0.277778em}{0ex}}\mu {M}^{-1}$ $0.0308\right]{s}^{-1}$ Forward and reverse rates for reaction 5 [link] $\left[{f}_{6}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{r}_{6}\right]=$ $\left[120\phantom{\rule{0.277778em}{0ex}}\mu {M}^{-1}$ $4\right]{s}^{-1}$ Forward and reverse rates for reaction 6 [link] $\left[{f}_{7}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{r}_{7}\right]=$ $\left[7.5\phantom{\rule{0.277778em}{0ex}}\mu {M}^{-1}$ $3.75\right]{s}^{-1}$ Forward and reverse rates for reaction 7 [link] $\left[{f}_{8}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{r}_{8}\right]=$ $\left[5\phantom{\rule{0.277778em}{0ex}}\mu {M}^{-1}$ $25\right]{s}^{-1}$ Forward and reverse rates for reaction 8 [link] $\left[{f}_{9}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{r}_{9}\right]=$ $\left[7.6\phantom{\rule{0.277778em}{0ex}}\mu {M}^{-1}$ $22.8\right]{s}^{-1}$ Forward and reverse rates for reaction 9 [link]

## Force generation

The interactions between myosin, actin, and the activated ${M}_{K}$ complex were modeled using Henri-Michaelis-Menten Enzyme Kinetics from [link] .

$\begin{array}{cc}\hfill \frac{d\left[M\right]}{dt}& =-\frac{{k}_{1}\left[C{a}_{4}{C}_{{M}_{K}}\right]\left[M\right]}{{k}_{2}+\left[M\right]}+\frac{{k}_{5}\left[{M}_{L}\right]\left[{M}_{p}\right]}{{k}_{6}+\left[{M}_{p}\right]}+{k}_{7}\left[{A}_{M}\right]\hfill \\ \hfill \frac{d\left[{M}_{p}\right]}{dt}& =\frac{{k}_{1}\left[C{a}_{4}{C}_{{M}_{K}}\right]\left[M\right]}{{k}_{2}+\left[M\right]}-\frac{{k}_{5}\left[{M}_{L}\right]\left[{M}_{p}\right]}{{k}_{6}+\left[{M}_{p}\right]}-{k}_{3}\left[{M}_{p}\right]+{k}_{4}\left[{A}_{{M}_{p}}\right]\hfill \\ \hfill \frac{d\left[{A}_{{M}_{p}}\right]}{dt}& ={k}_{3}\left[{M}_{p}\right]-{k}_{4}\left[{A}_{{M}_{p}}\right]+\frac{{k}_{1}\left[C{a}_{4}{C}_{{M}_{K}}\right]\left[{A}_{M}\right]}{{k}_{2}+\left[{A}_{M}\right]}\hfill \\ & -\frac{{k}_{5}\left[{M}_{L}\right]\left[{A}_{{M}_{p}}\right]}{{k}_{6}+\left[{A}_{{M}_{p}}\right]}\hfill \\ \hfill \frac{d\left[{A}_{M}\right]}{dt}& =-\frac{{k}_{1}\left[C{a}_{4}{C}_{{M}_{K}}\right]\left[{A}_{M}\right]}{{k}_{2}+\left[{A}_{M}\right]}+\frac{{k}_{5}\left[{M}_{L}\right]\left[{A}_{{M}_{p}}\right]}{{k}_{6}+\left[{A}_{{M}_{p}}\right]}-{k}_{7}\left[{A}_{M}\right]\hfill \\ \hfill F\left(t\right)& ={F}_{max}\frac{\left[{A}_{M}\left(t\right)\right]+\left[{A}_{{M}_{p}}\left(t\right)\right]}{\left[{M}_{T}\right]}\hfill \end{array}$
 ${\left[M\right]}_{0}=$ $23.9558\phantom{\rule{0.277778em}{0ex}}\mu M$ Initial myosin concentration [link] ${\left[{M}_{p}\right]}_{0}=$ $0.0144\phantom{\rule{0.277778em}{0ex}}\mu M$ Initial phosphorylated myosin concentration [link] ${\left[{A}_{{M}_{p}}\right]}_{0}=$ $0.0166\phantom{\rule{0.277778em}{0ex}}\mu M$ Initial cross-bride concentration [link] ${\left[{A}_{M}\right]}_{0}=$ $0.0132\phantom{\rule{0.277778em}{0ex}}\mu M$ Initial latch-bridge concentration [link] $\left[{M}_{L}\right]=$ $7.5\phantom{\rule{0.277778em}{0ex}}\mu M$ Myosin light chain phosphatase concentration [link] $\left[{M}_{T}\right]=$ $24\phantom{\rule{0.277778em}{0ex}}\mu M$ Total myosin concentration [link] $F\left(t\right)=$ Force generated in $mN$ [link] ${F}_{max}=$ $70\phantom{\rule{0.277778em}{0ex}}mN$ Maximum force cell can generate [link] ${k}_{1}=$ $27{s}^{-1}$ Rate for reaction 10 [link] ${k}_{2}=$ $10\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mu M$ Rate for reaction 11 [link] ${k}_{3}=$ $15{s}^{-1}$ Forward rate for reaction 12 [link] ${k}_{4}=$ $5{s}^{-1}$ Reverse rate for reaction 12 [link] ${k}_{5}=$ $16{s}^{-1}$ Rate for reaction 13 [link] ${k}_{6}=$ $15\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mu M$ Rate for reaction 14 [link] ${k}_{7}=$ $10{s}^{-1}$ Rate for reaction 15 [link]

## Mechanical model

A variety of models which represent SMCs and other types of cells as mass-spring systems have been developed [link] , [link] , [link] , [link] , [link] , [link] . The most comparable of these models used a single contractile element and two passive elements: one to represent the elastic actin and myosin fibers and the other to represent the adjacent muscle cells and surroundings [link] . We present a novel mechanical model of the SMC which incorporates the previously described biochemical interactions to produce a comprehensive model of SMC contraction.

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