Modeling Autoregulation of the Afferent Arteriole of the Rat Kidney

Abstract

One of the key autoregulatory mechanisms that control blood flow in the kidney is the myogenic response. Subject to increased pressure, the renal afferent arteriole responds with an increase in muscle tone and a decrease in diameter. To investigate the myogenic response of an afferent arteriole segment of the rat kidney, we extend a mathematical model of an afferent arteriole cell. For each cell, we include detailed Ca²⁺ signaling, transmembrane transport of major ions, the kinetics of myosin light chain phosphorylation, as well as cellular contraction and wall mechanics. To model an afferent arteriole segment, a number of cell models are connected in series by gap junctions, which link the cytoplasm of neighboring cells. Blood flow through the afferent arteriole is modeled using Poiseuille flow. Simulation of an inflow pressure up-step leads to a decrease in the diameter for the proximal part of the vessel (vasoconstriction) and to an increase in proximal vessel diameter (vasodilation) for an inflow pressure down-step. Through its myogenic response, the afferent arteriole segment model yields approximately stable outflow pressure for a physiological range of inflow pressures (100–160 mmHg), consistent with experimental observations. The present model can be incorporated as a key component into models of integrated renal hemodynamic regulation.

The original version of this chapter was revised. An erratum to this chapter can be found at https://doi.org/10.1007/978-3-319-60304-9_13

Appendix

This appendix contains the remaining equations besides the ones given in Sect. 2.1 for the afferent arteriole smooth muscle single-cell model of Edwards and Layton [5]. For further details and kinetic diagrams, refer to [5].

1.1 Transmembrane Ionic Transport

1.1.1 Ion and Charge Conservation Equations

The cytosolic concentrations of K⁺, Na⁺, Cl⁻, and Ca²⁺ are determined by considering the net sum of their respective fluxes into the cytosol (described in subsequent sections) and integrating

$$\displaystyle{ \frac{d[\mathrm{K}]_{\mathrm{cyt}}} {dt} = -\frac{\left (I_{\mathrm{K,b}} + I_{\mathrm{K,ir}} + I_{\mathrm{K,v}} + I_{\mathrm{K,Ca}} - 2I_{\mathrm{NaK}}\right )} {F \cdot \mathrm{ vol}_{\mathrm{cyt}}}, }$$

(12a)

$$\displaystyle{ \frac{d[\mathrm{Na}]_{\mathrm{cyt}}} {dt} = -\frac{\left (I_{\mathrm{Na,b}} + I_{\mathrm{Na,Pres}} + 3I_{\mathrm{NaK}} + 3I_{\mathrm{NCX}}\right )} {F \cdot \mathrm{ vol}_{\mathrm{cyt}}} + \frac{J_{\mathrm{NaCl}}} {\mathrm{vol}_{\mathrm{cyt}}}, }$$

(12b)

$$\displaystyle{ \frac{d[\mathrm{Cl}]_{\mathrm{cyt}}} {dt} = \frac{I_{\mathrm{Cl,b}} + I_{\mathrm{Cl,Ca}}} {F \cdot \mathrm{ vol}_{\mathrm{cyt}}} + \frac{J_{\mathrm{NaCl}}} {\mathrm{vol}_{\mathrm{cyt}}}, }$$

(12c)

$$\displaystyle\begin{array}{rcl} \frac{d[\mathrm{Ca}]_{\mathrm{cyt}}} {dt} & =& -\frac{\left (I_{\mathrm{Ca,b}} + I_{\mathrm{Ca,Pres}} + I_{\mathrm{PMCA}} + I_{\mathrm{Ca,L}} - 2I_{\mathrm{NCX}} + I_{\mathrm{SERCA}} - I_{\mathrm{RyR}} - I_{\mathrm{IP3R}}\right )} {2F \cdot \mathrm{ vol}_{\mathrm{cyt,Ca}}} \\ & +& R_{\mathrm{CaM}}^{\mathrm{cyt}} + R_{\mathrm{ Bf}}^{\mathrm{cyt}}. {}\end{array}$$

(12d)

Parameter values and definitions are given in Table 1.

In the sarcoplasmic reticulum (SR),

$$\displaystyle{ \frac{d[\mathrm{Ca}]_{\mathrm{SR}}} {dt} = \frac{I_{\mathrm{SERCA}} - I_{\mathrm{RyR}} - I_{\mathrm{IP3R}}} {2F \cdot \mathrm{ vol}_{\mathrm{SR}}} + R_{\mathrm{Calseq}}^{\mathrm{SR}}. }$$

(13)

The reaction terms R _CaM ^cyt, R _Bf ^cyt, and R _Calseq ^SR account for the buffering of Ca²⁺ by calmodulin, other cytosolic buffers, and calsequestrin, respectively, and are described in Eq. (35) below.

1.1.2 Background Currents

The background current of ion i, for i = K⁺, Na⁺, Cl⁻, Ca²⁺, is

$$\displaystyle{ I_{i,\mathrm{b}} = G_{i,\mathrm{b}}\left (V _{\mathrm{m}} - E_{i}\right ), }$$

(14)

where the Nernst potential of ion i with valence z _i is

$$\displaystyle{ E_{i} = \frac{RT} {z_{i}F}\ln \left (\frac{[i]_{\mathrm{out}}} {[i]_{\mathrm{cyt}}} \right ). }$$

(15)

Parameter values and definitions are given in Table 1.

1.1.3 Potassium Transport Pathways

The potassium current across inward-rectifier ( K _ir) channels is determined as

$$\displaystyle{ I_{\mathrm{K,ir}} = G_{\mathrm{Kir}}P_{\mathrm{Kir}}\left (\frac{[K]_{\mathrm{out}}} {[K]_{\mathrm{ref}}} \right )^{0.9}\left (V _{\mathrm{ m}} - E_{\mathrm{K}}\right ), }$$

(16a)

$$\displaystyle{ P_{\mathrm{Kir}} = \frac{1} {1 +\exp \left (\frac{V _{\mathrm{m}} - V _{\mathrm{Kir}}} {s_{\mathrm{Kir}}} \right )}, }$$

(16b)

where the exponential factor, 0.9, the potential of half-maximal activation, V _Kir, and the slope, s _Kir, were obtained by fitting K _ir currents in cerebral arterial smooth muscle cells (Wu et al. [18]). Parameter values are given in Tables 1 and 3.

Table 3 Parameters for potassium currents

The potassium current across delayed-rectifier ( K _v) channels is given by

$$\displaystyle{ I_{\mathrm{K,v}} = G_{\mathrm{Kv}}\left (P_{\mathrm{Kv}}\right )^{2}\left (V _{\mathrm{ m}} - E_{\mathrm{K}}\right ), }$$

(17a)

$$\displaystyle{ P_{\mathrm{Kv}} = 0.58P_{\mathrm{Kv1}} + 0.42P_{\mathrm{Kv2}}, }$$

(17b)

$$\displaystyle{ \frac{dP_{\mathrm{Kv1}}} {dt} = \frac{\bar{P}_{\mathrm{Kv}} - P_{\mathrm{Kv1}}} {\tau _{\mathrm{Kv1}}}, }$$

(17c)

$$\displaystyle{ \frac{dP_{\mathrm{Kv2}}} {dt} = \frac{\bar{P}_{\mathrm{Kv}} - P_{\mathrm{Kv2}}} {\tau _{\mathrm{Kv2}}}, }$$

(17d)

$$\displaystyle{ \bar{P}_{\mathrm{Kv}} = \frac{1} {1 +\exp \left (-\frac{V _{\mathrm{m}} + 1.77} {14.52} \right )}, }$$

(17e)

$$\displaystyle{ \tau _{\mathrm{kv1}} = 210.99\exp \left [-\left (\frac{V _{\mathrm{m}} + 214.34} {195.35} \right )^{2}\right ] - 20.59, }$$

(17f)

$$\displaystyle{ \tau _{\mathrm{kv2}} = 821.39\exp \left [-\left (\frac{V _{\mathrm{m}} + 31.59} {27.46} \right )^{2}\right ] + 0.189, }$$

(17g)

where P _Kv1 and P _Kv2 are the two components of the channel activation process and τ _Kv1 and τ _Kv2 are the respective time constants (in ms) (Yang et al. [19]). Variable$\bar{P}_{\mathrm{Kv}}$ is voltage-dependent and represents the steady-state value of P _Kv1 and P _Kv2.

The potassium current across Ca²⁺-activated K⁺ ( K _Ca) channels is computed as

$$\displaystyle{ I_{\mathrm{K,Ca}} = G_{\mathrm{KCa}}P_{\mathrm{KCa}}\left (V _{\mathrm{m}} - E_{\mathrm{K}}\right ), }$$

(18a)

$$\displaystyle{ P_{\mathrm{KCa}} = 0.65P_{\mathrm{f}} + 0.35P_{\mathrm{s}}, }$$

(18b)

$$\displaystyle{ \frac{dP_{\mathrm{f}}} {dt} = \frac{\bar{P}_{\mathrm{KCa}} - P_{\mathrm{f}}} {\tau _{\mathrm{Pf}}}, }$$

(18c)

$$\displaystyle{ \frac{dP_{\mathrm{s}}} {dt} = \frac{\bar{P}_{\mathrm{KCa}} - P_{\mathrm{s}}} {\tau _{\mathrm{Ps}}}, }$$

(18d)

$$\displaystyle{ \bar{P}_{\mathrm{KCa}} = \frac{1} {1 +\exp \left (-\frac{V _{\mathrm{m}} - V _{\mathrm{KCa}}} {21.70} \right )}, }$$

(18e)

$$\displaystyle{ V _{\mathrm{KCa}} = -45.0\,\,\,\log _{10}\left ([\mathrm{Ca}]_{\mathrm{cyt}}\right ) - 63.55, }$$

(18f)

where P _f and P _s are the fast and slow components of the channel activation process, respectively, and τ _Pf and τ _Ps are the corresponding time constants (Yang et al. [19]). The steady-state open probability of the channels is given by$\bar{P}_{\mathrm{KCa}}$.

The ATP-dependent K⁺ channels are not considered in the Edwards and Layton model [5] since it assumed that their conductance is negligible in well-perfused and oxygenated arterioles.

1.1.4 Sodium Transport Pathways

The current across Na⁺/K⁺-ATPase pumps is determined as

$$\displaystyle{ I_{\mathrm{NaK}} = I_{\mathrm{NaK,max}}\left ( \frac{[K]_{\mathrm{out}}} {[K]_{\mathrm{out}} + K_{\mathrm{m,NaK}}^{\mathrm{K}}}\right )^{2}\left ( \frac{[\mathrm{Na}]_{\mathrm{cyt}}} {[\mathrm{Na}]_{\mathrm{cyt}} + K_{\mathrm{m,NaK}}^{\mathrm{Na}}}\right )^{3}. }$$

(19)

The current across Na⁺/Ca²⁺ (NCX) exchangers is given by

$$\displaystyle{ I_{\mathrm{NCX}} = I_{\mathrm{NCX,max}}A_{\mathrm{NCX}}\left (\frac{\varPhi _{\mathrm{F}}[\mathrm{Na}]_{\mathrm{cyt}}^{3}[\mathrm{Ca}]_{\mathrm{out}} -\varPhi _{\mathrm{R}}[\mathrm{Na}]_{\mathrm{out}}^{3}[\mathrm{Ca}]_{\mathrm{cyt}}} {G(1 + k_{\mathrm{sat}}\varPhi _{\mathrm{R}})} \right ), }$$

(20a)

$$\displaystyle{ A_{\mathrm{NCX}} = \frac{[\mathrm{Ca}]_{\mathrm{cyt}}^{2}} {\left (K_{\mathrm{m,NCX}}^{\mathrm{Ca}}\right )^{2} + [\mathrm{Ca}]_{\mathrm{cyt}}^{2}}, }$$

(20b)

$$\displaystyle{ \varPhi _{\mathrm{F}} =\exp \left (\frac{\gamma V _{\mathrm{m}}F} {RT} \right ), }$$

(20c)

$$\displaystyle{ \varPhi _{\mathrm{R}} =\exp \left (\frac{(\gamma -1)V _{\mathrm{m}}F} {RT} \right ), }$$

(20d)

$$\displaystyle\begin{array}{rcl} G& =& [\mathrm{Na}]_{\mathrm{out}}^{3}[\mathrm{Ca}]_{\mathrm{ cyt}} + [\mathrm{Na}]_{\mathrm{cyt}}^{3}[\mathrm{Ca}]_{\mathrm{ out}} + K_{\mathrm{mNao}}^{3}[\mathrm{Ca}]_{\mathrm{ cyt}} \\ & +& K_{\mathrm{mCao}}[\mathrm{Na}]_{\mathrm{cyt}}^{3} + K_{\mathrm{ mNai}}^{3}[\mathrm{Ca}]_{\mathrm{ out}}\left (1 + [\mathrm{Ca}]_{\mathrm{cyt}}/K_{\mathrm{mCai}}\right ) \\ & +& K_{\mathrm{mCai}}[\mathrm{Na}]_{\mathrm{out}}^{3}\left (1 + [\mathrm{Na}]_{\mathrm{ cyt}}^{3}/K_{\mathrm{ mNai}}^{3}\right ) {}\end{array}$$

(20e)

(Shannon et al. [17]). The flux across NaCl cotransporters is computed as

$$\displaystyle{ J_{\mathrm{NaCl}} = J_{\mathrm{NaCl,max}} \frac{\left (E_{\mathrm{Na}} - E_{\mathrm{Cl}}\right )^{4}} {\left (E_{\mathrm{Na}} - E_{\mathrm{Cl}}\right )^{4} + R_{\mathrm{NaCl}}^{4}} }$$

(21)

(Kneller et al. [11]). Parameter values are given in Tables 1 and 4.

1.1.5 Chloride Transport Pathways

The current across Ca²⁺-activated Cl⁻ (Cl_Ca) channels is

$$\displaystyle{ I_{\mathrm{Cl,Ca}} = G_{\mathrm{ClCa}}P_{\mathrm{ClCa}}(V _{\mathrm{m}} - E_{\mathrm{Cl}}), }$$

(22a)

$$\displaystyle{ \frac{dP_{\mathrm{ClCa}}} {dt} = \frac{\bar{P}_{\mathrm{ClCa}} - P_{\mathrm{ClCa}}} {\tau _{\mathrm{ClCa}}}, }$$

(22b)

$$\displaystyle{ \bar{P}_{\mathrm{ClCa}} = \frac{[\mathrm{Ca}]_{\mathrm{cyt}}^{3}} {[\mathrm{Ca}]_{\mathrm{cyt}}^{3} + K_{\mathrm{ClCa}}^{3}}, }$$

(22c)

where$\bar{P}_{\mathrm{ClCa}}$ is the steady-state open probability of the channel (Jacobson et al. [8]). Parameter values are given in Table 5.

Table 5 Parameters for chloride currents

1.1.6 Calcium Transport Pathways

Calcium is exchanged between the cytosol and the extracellular space, and between the cytosol and the SR, which acts as a storage compartment. Parameter values for calcium currents and buffer reactions are given in Tables 1 and 6.

Table 6 Parameters for calcium currents and buffers

The current through plasma membrane Ca²⁺ (PMCA) pumps is determined as

$$\displaystyle{ I_{\mathrm{PMCA}} = I_{\mathrm{PMCA,max}}\left ( \frac{[\mathrm{Ca}]_{\mathrm{cyt}}} {K_{\mathrm{m,PMCA}}^{\mathrm{Ca}} + [\mathrm{Ca}]_{\mathrm{cyt}}}\right ). }$$

(23)

The Ca_V1.2 model of Faber et al. [6] is used for the current across L-type Ca²⁺ channels. The voltage-dependent gating mode of the channel is considered, which includes four closed states ( c ₀, c ₁, c ₂, and c ₃), one open state ( p _o), and fast ( i _vf) and slow ( i _vs) inactivated states. The corresponding equations are

$$\displaystyle{ I_{\mathrm{Ca,L}} = G_{\mathrm{CaL}}p_{\mathrm{o}}\left (V _{\mathrm{m}} - E_{\mathrm{Ca}}\right ), }$$

(24a)

$$\displaystyle{ \frac{dc_{0}} {dt} =\beta c_{1} - (4\alpha )c_{0}, }$$

(24b)

$$\displaystyle{ \frac{dc_{1}} {dt} = (4\alpha )c_{0} + (2\beta )c_{2} - (3\alpha +\beta )c_{1}, }$$

(24c)

$$\displaystyle{ \frac{dc_{2}} {dt} = (3\alpha )c_{1} + (3\beta )c_{3} - (2\alpha + 2\beta )c_{2}, }$$

(24d)

$$\displaystyle{ \frac{dc_{3}} {dt} = (2\alpha )c_{2} + (4\beta )p_{\mathrm{o}} +\omega _{\mathrm{f}}i_{\mathrm{vf}} +\omega _{\mathrm{s}}i_{\mathrm{vs}} - (\alpha +3\beta +\gamma _{\mathrm{f}} +\gamma _{\mathrm{s}})c_{3}, }$$

(24e)

$$\displaystyle{ \frac{dp_{\mathrm{o}}} {dt} =\alpha c_{3} +\lambda _{\mathrm{f}}i_{\mathrm{vf}} +\lambda _{s}i_{\mathrm{vs}} - (4\beta +\phi _{\mathrm{f}} +\phi _{\mathrm{s}})p_{\mathrm{o}}, }$$

(24f)

$$\displaystyle{ \frac{di_{\mathrm{vf}}} {dt} =\gamma _{\mathrm{f}}c_{3} +\phi _{\mathrm{f}}p_{\mathrm{o}} +\omega _{\mathrm{sf}}i_{\mathrm{vs}} - (\omega _{\mathrm{f}} +\lambda _{\mathrm{f}} +\omega _{\mathrm{fs}})i_{\mathrm{vf}}, }$$

(24g)

$$\displaystyle{ \frac{di_{\mathrm{vs}}} {dt} =\gamma _{\mathrm{s}}c_{3} +\phi _{\mathrm{s}}p_{\mathrm{o}} +\omega _{\mathrm{fs}}i_{\mathrm{vf}} - (\omega _{\mathrm{s}} +\lambda _{\mathrm{s}} +\omega _{\mathrm{sf}})i_{\mathrm{vs}}, }$$

(24h)

where

$$\displaystyle\begin{array}{rcl} \alpha & =& 0.925\exp (V _{\mathrm{m}}/30),\qquad \qquad \,\,\,\,\,\,\beta = 0.390\exp (V _{\mathrm{m}}/40), {}\\ \gamma _{\mathrm{f}}& =& 0.245\exp (V _{\mathrm{m}}/10),\qquad \qquad \,\,\,\,\,\,\gamma _{\mathrm{s}} = 0.005\exp (-V _{\mathrm{m}}/40), {}\\ \phi _{\mathrm{f}}& =& 0.020\exp (V _{\mathrm{m}}/500),\qquad \qquad \,\,\,\,\phi _{\mathrm{s}} = 0.030\exp (-V _{\mathrm{m}}/280), {}\\ \lambda _{\mathrm{f}}& =& 0.035\exp (-V _{\mathrm{m}}/300),\qquad \qquad \lambda _{\mathrm{s}} = 0.0011\exp (V _{\mathrm{m}}/500), {}\\ \omega _{\mathrm{f}}& =& (4\beta \lambda _{\mathrm{f}}\gamma _{\mathrm{f}})/(\alpha \phi _{\mathrm{f}}),\qquad \qquad \qquad \,\,\,\omega _{\mathrm{s}} = (4\beta \lambda _{\mathrm{s}}\gamma _{\mathrm{s}})/(\alpha \phi _{\mathrm{s}}), {}\\ \omega _{\mathrm{sf}}& =& (\lambda _{\mathrm{s}}\phi _{\mathrm{f}})/\lambda _{\mathrm{f}},\qquad \qquad \qquad \qquad \quad \omega _{\mathrm{fs}} =\phi _{\mathrm{s}}. {}\\ \end{array}$$

T-type Ca²⁺ channels are not considered in the Edwards and Layton [5] model.

The current across sarco/endoplasmic reticulum Ca²⁺ (SERCA) pumps is given by

$$\displaystyle{ I_{\mathrm{SERCA}} = I_{\mathrm{SERCA,max}}\left ( \frac{[\mathrm{Ca}]_{\mathrm{cyt}}^{2}} {\left (K_{\mathrm{m,SERCA}}^{\mathrm{Ca}}\right )^{2} + [\mathrm{Ca}]_{\mathrm{cyt}}^{2}}\right ). }$$

(25)

The RyR model of Keizer and Levine [10] is used to determine the RyR-mediated release current into the cytosol,

$$\displaystyle{ I_{\mathrm{RyR}} = v_{\mathrm{RyR}}P_{\mathrm{RyR}}\left ([\mathrm{Ca}]_{\mathrm{SR}} - [\mathrm{Ca}]_{\mathrm{cyt}}\right )\left (2F \cdot \mathrm{ vol}_{\mathrm{SR}}\right ), }$$

(26)

where v _RyR is the RyR rate constant. The open probability of RyR ( P _RyR) is calculated as

$$\displaystyle{ P_{\mathrm{RyR}} = \frac{\omega \left (1 + \left ([\mathrm{Ca}]_{\mathrm{cyt}}/K_{\mathrm{b}}\right )^{3}\right )} {1 + \left (K_{\mathrm{a}}/[\mathrm{Ca}]_{\mathrm{cyt}}\right )^{4} + \left ([\mathrm{Ca}]_{\mathrm{cyt}}/K_{\mathrm{b}}\right )^{3}}, }$$

(27a)

$$\displaystyle{ \frac{d\omega } {dt} = \frac{k_{\mathrm{c}}^{-}\left (\omega ^{\infty }-\omega \right )} {\omega ^{\infty }}, }$$

(27b)

$$\displaystyle{ \omega ^{\infty } = \frac{1 + \left (K_{\mathrm{a}}/[\mathrm{Ca}]_{\mathrm{cyt}}\right )^{4} + \left ([\mathrm{Ca}]_{\mathrm{ cyt}}/K_{\mathrm{b}}\right )^{3}} {1 + 1/K_{\mathrm{c}} + \left (K_{\mathrm{a}}/[\mathrm{Ca}]_{\mathrm{cyt}}\right )^{4} + \left ([\mathrm{Ca}]_{\mathrm{cyt}}/K_{\mathrm{b}}\right )^{3}}. }$$

(27c)

The IP₃R model of De Young and Keizer [2] is used to determine the IP₃R-mediated release current in the cytosol,

$$\displaystyle{ I_{\mathrm{IP3R}} = v_{\mathrm{IP3R}}(x_{110})^{3}\left ([\mathrm{Ca}]_{\mathrm{ SR}} - [\mathrm{Ca}]_{\mathrm{cyt}}\right )\left (2F \cdot \mathrm{ vol}_{\mathrm{SR}}\right ), }$$

(28)

where v _IP3R is the IP₃R rate constant and x ₁₁₀ is the fraction of receptors bound by one activated Ca²⁺ and one IP₃, calculated as described below. The cytosolic concentration of IP₃ is calculated as

$$\displaystyle{ \frac{d[\mathrm{IP}_{3}]_{\mathrm{cyt}}} {dt} = k_{+}^{\mathrm{IP3}}[\mathrm{IP}_{ 3}]_{\mathrm{ref}}\left (\frac{[\mathrm{Ca}]_{\mathrm{cyt}} + (1 -\alpha _{4})k_{4}} {[\mathrm{Ca}]_{\mathrm{cyt}} + k_{4}} \right ) - k_{-}^{\mathrm{IP3}}[\mathrm{IP}_{ 3}]_{\mathrm{cyt}}, }$$

(29)

where k ₊ ^IP3 and k ₋ ^IP3 are the rate constants for IP₃ formation and consumption, respectively, [IP₃]_ref is a reference IP₃ concentration, and α ₄ determines the strength of the Ca²⁺ feedback on IP₃ production.

Three equivalent and independent IP₃R subunits are assumed to be involved in conduction, and each subunit has one IP₃ binding site (denoted as site 1) and two Ca²⁺ binding sites, one for activation ( site 2) and one for inhibition ( site 3). The fraction of receptors in state S$_{i_{1}i_{2}i_{3}}$ is denoted by$x_{i_{1}i_{2}i_{3}}$, where i _j equals 0 if the j-th binding site is unoccupied or 1 if it is occupied. All three subunits must be in the state S₁₁₀ (corresponding to the binding of one IP₃ and one activating Ca²⁺) for the IP₃R channel to be open. Assuming rapid equilibrium for IP₃ binding,

$$\displaystyle{ a_{1}[\mathrm{Ca}]_{\mathrm{cyt}}x_{0k0} = b_{1}x_{1k0},\qquad k = 0,1, }$$

(30a)

$$\displaystyle{ a_{3}[\mathrm{Ca}]_{\mathrm{cyt}}x_{0k1} = b_{3}x_{1k1},\qquad k = 0,1. }$$

(30b)

Defining d _k = b _k ∕ a _k , the conservation equations for$x_{0i_{2}i_{3}}$ are

$$\displaystyle{ \frac{dx_{000}} {dt} = -a_{4}\left ([\mathrm{Ca}]_{\mathrm{cyt}}x_{000} - d_{4}x_{001}\right ) - a_{5}\left ([\mathrm{Ca}]_{\mathrm{cyt}}x_{000} - d_{5}x_{010}\right ), }$$

(31a)

$$\displaystyle{ \frac{dx_{001}} {dt} = +a_{4}\left ([\mathrm{Ca}]_{\mathrm{cyt}}x_{000} - d_{4}x_{001}\right ) - a_{5}\left ([\mathrm{Ca}]_{\mathrm{cyt}}x_{001} - d_{5}x_{011}\right ), }$$

(31b)

$$\displaystyle{ \frac{dx_{010}} {dt} = +a_{5}\left ([\mathrm{Ca}]_{\mathrm{cyt}}x_{000} - d_{5}x_{010}\right ) - a_{4}\left ([\mathrm{Ca}]_{\mathrm{cyt}}x_{010} - d_{4}x_{011}\right ), }$$

(31c)

$$\displaystyle{ x_{011} = 1 -\left (x_{000} + x_{001} + x_{010} + x_{100} + x_{101} + x_{110} + x_{111}\right ). }$$

(31d)

1.2 Intracellular Ca²⁺ Dynamics

1.2.1 Calcium Buffers

Calcium buffering by cytosolic Ca²⁺-binding proteins other than calmodulin is described as a first-order dynamic process,

$$\displaystyle{ \frac{d[\mathrm{Bf} \cdot \mathrm{ Ca}]_{\mathrm{cyt}}} {dt} = k_{\mathrm{on}}^{\mathrm{Bf}}[\mathrm{Ca}]_{\mathrm{ cyt}}\left ([\mathrm{Bf}]_{\mathrm{cyt}}^{\mathrm{tot}} - [\mathrm{Bf} \cdot \mathrm{ Ca}]_{\mathrm{ cyt}}\right ) - k_{\mathrm{off}}^{\mathrm{Bf}}[\mathrm{Bf} \cdot \mathrm{ Ca}]_{\mathrm{ cyt}}, }$$

(32)

where [Bf]_cyt ^tot is the total concentration of Ca²⁺-binding proteins other than calmodulin in the cytosol and [Bf⋅ Ca]_cyt is the concentration of the calcium-bound sites of these other buffering elements.

Calcium buffering by calsequestrin in the SR is described as

$$\displaystyle\begin{array}{rcl} \frac{d[\mathrm{Calseq} \cdot \mathrm{ Ca}]_{\mathrm{SR}}} {dt} & =& k_{\mathrm{on}}^{\mathrm{Calseq}}[\mathrm{Ca}]_{\mathrm{ SR}}\left ([\mathrm{Calseq}]_{\mathrm{SR}}^{\mathrm{tot}} - [\mathrm{Calseq} \cdot \mathrm{ Ca}]_{\mathrm{ SR}}\right ) \\ & & \qquad \qquad \qquad \qquad \qquad \quad \,\,\,\,\, - k_{\mathrm{off}}^{\mathrm{Calseq}}[\mathrm{Calseq} \cdot \mathrm{ Ca}]_{\mathrm{ SR}}, {}\end{array}$$

(33)

where [Calseq]_SR ^tot is the total concentration of calsequestrin sites available for Ca²⁺ binding in the SR and [Calseq⋅ Ca]_SR is the concentration of Ca²⁺-bound calsequestrin sites in that compartment. Parameter values are given in Table 6.

1.3 Kinetics of Myosin Light Chain Phosphorylation

1.3.1 CaM Activation of MLCK

Calmodulin (CaM) has four Ca²⁺ binding sites, with two at the NH₂ terminus (low affinity) and two at the COOH terminus (high affinity). Binding of Ca²⁺ to those sites yields the CaM.Ca₄ complex, and CaM.Ca₄ binds to myosin light chain kinase (MLCK) to form MLCK.CaM.Ca₄, which is the active form of MLCK that phosphorylates MLCs.

The scheme proposed by Fajmut et al. [7] is used to determine the kinetics of formation of MLCK.CaM.Ca₄. Subscripts N and C represent two binding sites each for Ca²⁺ at the NH₂ and COOH terminus of CaM, respectively, and the subscript M represents the CaM binding site occupied by MLCK. An underscore (_) denotes an unoccupied site for each of these binding sites. For example, CaM_NCM designates MLCK.CaM.Ca₄. The kinetic equations for the formation of MLCK are

$$\displaystyle\begin{array}{rcl} \frac{d[\mathrm{CaM}_{\mathrm{-C-}}]} {dt} & =& \left (-k_{-1}^{\mathrm{CaM}} - k_{ 4}^{\mathrm{CaM}}[\mathrm{Ca}]^{2} - k_{ 5}^{\mathrm{CaM}}[\mathrm{MLCK}]_{\mathrm{ free}}\right )[\mathrm{CaM}_{\mathrm{-C-}}] \\ & +& k_{1}^{\mathrm{CaM}}[\mathrm{Ca}]^{2}[\mathrm{CaM}_{\mathrm{ ---}}] + k_{-4}^{\mathrm{CaM}}[\mathrm{CaM}_{\mathrm{ NC-}}] + k_{-5}^{\mathrm{CaM}}[\mathrm{CaM}_{\mathrm{ -CM}}],{}\end{array}$$

(34a)

$$\displaystyle\begin{array}{rcl} \frac{d[\mathrm{CaM}_{\mathrm{N--}}]} {dt} & =& \left (-k_{-2}^{\mathrm{CaM}} - k_{ 3}^{\mathrm{CaM}}[\mathrm{Ca}]^{2}\right )[\mathrm{CaM}_{\mathrm{ N--}}] + k_{2}^{\mathrm{CaM}}[\mathrm{Ca}]^{2}[\mathrm{CaM}_{\mathrm{ ---}}] \\ & +& k_{-3}^{\mathrm{CaM}}[\mathrm{CaM}_{\mathrm{ NC-}}], {}\end{array}$$

(34b)

$$\displaystyle\begin{array}{rcl} \frac{d[\mathrm{CaM}_{\mathrm{NC-}}]} {dt} & =& \left (-k_{-3}^{\mathrm{CaM}} - k_{ -4}^{\mathrm{CaM}} - k_{ 7}^{\mathrm{CaM}}[\mathrm{MLCK}]_{\mathrm{ free}}\right )[\mathrm{CaM}_{\mathrm{NC-}}] \\ & +& k_{3}^{\mathrm{CaM}}[\mathrm{Ca}]^{2}[\mathrm{CaM}_{\mathrm{ N--}}] + k_{4}^{\mathrm{CaM}}[\mathrm{Ca}]^{2}[\mathrm{CaM}_{\mathrm{ -C-}}] \\ & +& k_{-7}^{\mathrm{CaM}}[\mathrm{Ca}]^{2}[\mathrm{CaM}_{\mathrm{ NCM}}], {}\end{array}$$

(34c)

$$\displaystyle\begin{array}{rcl} \frac{d[\mathrm{CaM}_{\mathrm{-CM}}]} {dt} & =& \left (-k_{-5}^{\mathrm{CaM}} - k_{ 6}^{\mathrm{CaM}}[\mathrm{Ca}]^{2}\right )[\mathrm{CaM}_{\mathrm{ -CM}}] + k_{5}^{\mathrm{CaM}}[\mathrm{MLCK}]_{\mathrm{ free}}[\mathrm{CaM}_{\mathrm{-C-}}] \\ & +& k_{-6}^{\mathrm{CaM}}[\mathrm{Ca}]^{2}[\mathrm{CaM}_{\mathrm{ NCM}}], {}\end{array}$$

(34d)

$$\displaystyle\begin{array}{rcl} \frac{d[\mathrm{CaM}_{\mathrm{NCM}}]} {dt} & =& \left (-k_{-6}^{\mathrm{CaM}} - k_{ -7}^{\mathrm{CaM}}[\mathrm{Ca}]^{2}\right )[\mathrm{CaM}_{\mathrm{ NCM}}] + k_{6}^{\mathrm{CaM}}[\mathrm{Ca}]^{2}[\mathrm{CaM}_{\mathrm{ -CM}}] \\ & +& k_{7}^{\mathrm{CaM}}[\mathrm{MLCK}]_{\mathrm{ free}}[\mathrm{CaM}_{\mathrm{NC-}}], {}\end{array}$$

(34e)

$$\displaystyle\begin{array}{rcl} [\mathrm{CaM}]^{\mathrm{tot}}& =& [\mathrm{CaM}_{\mathrm{ ---}}] + [\mathrm{CaM}_{\mathrm{-C-}}] + [\mathrm{CaM}_{\mathrm{N--}}] + [\mathrm{CaM}_{\mathrm{NC-}}], \\ & +& [\mathrm{CaM}_{\mathrm{-CM}}] + [\mathrm{CaM}_{\mathrm{NCM}}], {}\end{array}$$

(34f)

$$\displaystyle{ [\mathrm{MLCK}]^{\mathrm{tot}} = [\mathrm{CaM}_{\mathrm{ -CM}}] + [\mathrm{CaM}_{\mathrm{NCM}}] + [\mathrm{MLCK}]_{\mathrm{free}}, }$$

(34g)

where the on- and off-rate constants are denoted by k _i ^CaM and k _{− i} ^CaM, respectively, [MLCK]_free is the concentration of free (unbound) MLCK, [CaM]^tot is the total concentration of calmodulin, [MLCK]^tot is the total concentration of MLCK, and the subscript “cyt” that denotes the cytosolic compartment is omitted for simplicity. Parameter values are given in Table 7.

Table 7 Parameters for MLCK and MLCP kinetics

The buffering terms in the Ca²⁺ conservation equations (12d) and (13) are given by

$$\displaystyle\begin{array}{rcl} R_{\mathrm{CaM}}^{\mathrm{cyt}}& =& -2\frac{d[\mathrm{CaM}_{\mathrm{N--}}]} {dt} - 2\frac{d[\mathrm{CaM}_{\mathrm{-C-}}]} {dt} - 2\frac{d[\mathrm{CaM}_{\mathrm{-CM}}]} {dt} \\ & -& 4\frac{d[\mathrm{CaM}_{\mathrm{NC-}}]} {dt} - 4\frac{d[\mathrm{CaM}_{\mathrm{NCM}}]} {dt}, {}\end{array}$$

(35a)

$$\displaystyle{ R_{\mathrm{Bf}}^{\mathrm{cyt}} = -\frac{d[\mathrm{Bf} \cdot \mathrm{ Ca}]_{\mathrm{cyt}}} {dt}, }$$

(35b)

$$\displaystyle{ R_{\mathrm{Calseq}}^{\mathrm{SR}} = -\frac{d[\mathrm{Calseq} \cdot \mathrm{ Ca}]_{\mathrm{SR}}} {dt}. }$$

(35c)

1.3.2 Rho-Kinase Inhibition of MLCP

Myosin light chain phosphatase (MLCP) consists of three subunits, one of which, MYPT1, can be phosphorylated by Rho kinase (RhoK). Rho-K-induced phosphorylation of MYPT1 inactivates MLCP, which promotes contraction. The cytosolic concentration of active MLCP (denoted MLCP^∗) is given by

$$\displaystyle{ \frac{d[\mathrm{MLCP}^{{\ast}}]} {dt} = k_{+}^{\mathrm{MLCP}}\left ([\mathrm{MLCP}]^{\mathrm{tot}} - [\mathrm{MLCP}^{{\ast}}]\right ) - k_{ -}^{\mathrm{MLCP}}[\mathrm{MLCP}^{{\ast}}], }$$

(36)

where [MLCP]^tot is the total concentration of MLCP in the cytosol and the inactivation of MLCP by RHoK is given by

$$\displaystyle{ k_{-}^{\mathrm{MLCP}} = k_{\mathrm{ cat}}[\mathrm{RhoK}] }$$

(37)

(Mbikou et al. [13]). The concentration of RhoK, [RhoK], is assumed to be fixed at 30 nM (Kaneko-Kawano et al. [9]) except in the presence of specific inhibitors. Parameter values are given in Table 7.

1.3.3 MLCK- and MLCP-Dependent Phosphorylation of Myosin

The contractile force of the vessels is determined by the fraction of myosin cross-bridges that are phosphorylated. The four types of cross-bridges considered are free cross-bridges (Myo), phosphorylated cross-bridges (MyoP), attached phosphorylated cycling cross-bridges (AMyoP), and attached dephosphorylated, non-cycling cross-bridges (AMyo), and the corresponding equations for their concentrations are

$$\displaystyle{ \frac{d[\mathrm{Myo}]} {dt} = -k_{1}^{\mathrm{Myo}}[\mathrm{Myo}] + k_{ 2}^{\mathrm{Myo}}[\mathrm{MyoP}] + k_{ 7}^{\mathrm{Myo}}[\mathrm{AMyo}], }$$

(38a)

$$\displaystyle{ \frac{d[\mathrm{MyoP}]} {dt} = +k_{1}^{\mathrm{Myo}}[\mathrm{Myo}] - (k_{ 2}^{\mathrm{Myo}} + k_{ 3}^{\mathrm{Myo}})[\mathrm{MyoP}] + k_{ 4}^{\mathrm{Myo}}[\mathrm{AMyoP}], }$$

(38b)

$$\displaystyle{ \frac{d[\mathrm{AMyoP}]} {dt} = +k_{3}^{\mathrm{Myo}}[\mathrm{MyoP}] - (k_{ 4}^{\mathrm{Myo}} + k_{ 5}^{\mathrm{Myo}})[\mathrm{AMyoP}] + k_{ 6}^{\mathrm{Myo}}[\mathrm{AMyo}], }$$

(38c)

$$\displaystyle{ [\mathrm{Myo}]^{\mathrm{tot}} = [\mathrm{Myo}] + [\mathrm{MyoP}] + [\mathrm{AMyoP}] + [\mathrm{AMyo}]. }$$

(38d)

The rate constants k ₃ ^Myo, k ₄ ^Myo, and k ₇ ^Myo are fixed (Yang et al. [19]). Parameter values are given in Table 7.

The rate constants k ₁ ^Myo and k ₆ ^Myo represent the activity of MLCK and are assumed to be proportional to the fraction of the fully activated form of the enzyme, while the rate constants k ₂ ^Myo and k ₅ ^Myo represent the activity of MLCP. The corresponding equations are

$$\displaystyle{ k_{1}^{\mathrm{Myo}} = k_{ 6}^{\mathrm{Myo}} = k_{\mathrm{ MLCK}}^{\mathrm{Myo}}\frac{[\mathrm{CaM}_{\mathrm{NCM}}]} {[\mathrm{CaM}]^{\mathrm{tot}}}, }$$

(39a)

$$\displaystyle{ k_{2}^{\mathrm{Myo}} = k_{ 5}^{\mathrm{Myo}} = k_{\mathrm{ MLCP}}^{\mathrm{Myo}} \frac{[\mathrm{MLCP}^{{\ast}}]} {[\mathrm{MLCP}]^{\mathrm{tot}}}, }$$

(39b)

where k _MLCK ^Myo and k _MLCP ^Myo are fixed.

1.4 Mechanical Behavior of Cell

The vasomotion of the afferent arteriole is affected by the variations in the number of cross-bridges, which induce variations in the contractile force and thus alter the diameter of the vessel. Edwards and Layton [5] implemented the model of Carlson et al. [1] that represents vessel wall tension as the sum of a passive component and an active myogenic component. The passive component, T _pass, is a function of the vessel diameter, D,

$$\displaystyle{ T_{\mathrm{pass}} = C_{\mathrm{pass}}\exp \left [C_{\mathrm{pass}}^{{\prime}}\left ( \frac{D} {D_{0}} - 1\right )\right ], }$$

(40)

where D ₀ is the reference vessel diameter.

The active myogenic component is the product of the maximal active tension generated at a given vessel circumference, T _act ^max, given by

$$\displaystyle{ T_{\mathrm{act}}^{\mathrm{max}} = C_{\mathrm{ act}}\exp \left [-\left (\frac{D/D_{0} - C_{\mathrm{act}}^{{\prime}}} {C_{\mathrm{act}}^{{\prime\prime}}} \right )^{2}\right ], }$$

(41)

and the fraction of myosin light chains that are phosphorylated, Ψ. Therefore, the total tension in the wall, T _wall, is

$$\displaystyle{ T_{\mathrm{wall}} = T_{\mathrm{pass}} +\varPsi T_{\mathrm{act}}^{\mathrm{max}}. }$$

(42)

The change in vessel diameter depends on the difference between the tension resulting from intravascular pressure p, T _pres = pD∕2, and the tension generated in the wall, T _wall, so that

$$\displaystyle{ \frac{dD} {dt} = \frac{1} {\tau _{\mathrm{d}}} \frac{D_{\mathrm{c}}} {T_{\mathrm{c}}} (T_{\mathrm{pres}} - T_{\mathrm{wall}}), }$$

(43)

where D _c is a reference diameter, T _c is the tension at a pressure of 100 mmHg and diameter D _c, and τ _d is a time constant. Parameter values are given in Table 8.

Table 8 Parameters for smooth muscle cell mechanics