Energetics of Ion Transport in Dopaminergic Substantia nigra Neurons

Abstract

Cytosolic calcium ion levels are critical in sustaining neuronal activity. They have an intricate relationship with the neuronal energy systems, and Parkinson’s disease (PD) probably involves a dysfunctional energy system in the pacemaking neurons of the Substantia nigra. This chapter explores the association of repetitive firing pattern of these neurons and cytosolic calcium using a mathematical model. In particular, a theory is examined that proposes a role of low voltage activated L-type calcium channel in creating an energy stress within vulnerable neurons.

Appendix: Model Formulation

Membrane Dynamics

Membrane potential

$$ V = \displaystyle\frac{{F{v_{\rm{cyt}}}}}{{{C_{\rm{m}}}}}[{K_{\rm{i}}} - {K_{\rm{e}}} + N{a_{\rm{i}}} - N{a_{\rm{e}}} + 2(C{a_{\rm{i}}} - C{a_{\rm{e}}}) + a{n_{\rm{offset}}}]. $$

(5.15)

Intracellular cations

$$ \begin{array}{lll} {\displaystyle\frac{{{\rm{d}}C{a_i}}}{{{{d}}t}} = \displaystyle\displaystyle\frac{1}{{{z_{\rm{Ca}}}F{v_{\rm{cyt}}}}}\left[ {{I_{\rm{Ca}}} + 2{I_{\rm{pmca}}} - 2{I_{\rm{naca}}}} \right] - \left( {{J_{\rm{calb}}} + 4{J_{\rm{cam}}}} \right)} \hfill \\{\displaystyle\frac{{{\rm{d}}N{a_i}}}{{{\hbox{d}}t}} = \displaystyle\frac{1}{{{z_{\rm{Na}}}F{v_{\rm{cyt}}}}}\left[ {{I_{\rm{Na}}} + 3{I_{\rm{nak}}} + 3{I_{\rm{naca}}}} \right]} \hfill \\{\displaystyle\frac{{{\rm{d}}{K_i}}}{{{\hbox{d}}t}} = \displaystyle\frac{1}{{{z_{\rm{K}}}F{v_{\rm{cyt}}}}}\left[ {{I_{\rm{K}}} - 2{I_{\rm{nak}}}} \right]} \\\end{array} $$

(5.16)

Membrane currents from ion-channels

$$ \begin{array}{*{20}{c}} {{I_{\text{Ca}}} = \left[ {\sum\limits_{\text{c}} {{g_{{{\text{Ca,c}}}}}{O_{{{\text{Ca,c}}}}}} } \right]\sqrt {{C{a_{\text{e}}}C{a_{\text{i}}}}} \frac{{\sin {\rm h} \left( {{{{\left( {V - {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V}}}_{\text{Ca}}}} \right)}} \left/ {{{V_{\tau }}}} \right.}} \right)}}{{\sin {\rm h} c\left( {{{V} \left/ {{{V_{\tau }}}} \right.}} \right)}}} \hfill \\ {{I_{\text{Na}}} = \left[ {\sum\limits_{\text{c}} {{g_{{{\text{Na,c}}}}}{O_{{{\text{Na,c}}}}}} } \right]\sqrt {{N{a_{\text{e}}}N{a_{\text{i}}}}} \frac{{\sin {\rm h} \left( {{{{\left( {V - {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V}}}_{\text{Na}}}} \right)}} \left/ {{2{V_{\tau }}}} \right.}} \right)}}{{\sin {\rm h} c\left( {{{V} \left/ {{2{V_{\tau }}}} \right.}} \right)}}} \hfill \\ {{I_{\text{K}}} = \left[ {\sum\limits_{\text{c}} {{g_{{{\text{K,c}}}}}{O_{{{\text{K,c}}}}}} } \right]\sqrt {{{K_{\text{e}}}{K_{\text{i}}}}} \frac{{\sin {\rm h} \left( {{{{\left( {V - {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V}}}_{\text{K}}}} \right)}} \left/ {{2{V_{\tau }}}} \right.}} \right)}}{{\sin {\rm h} c\left( {{{V} \left/ {{2{V_{\tau }}}} \right.}} \right)}} + \left[ {\sum\limits_{\text{c}} {{\gamma_{{{\text{K,c}}}}}{O_{{{\text{K,c}}}}}} } \right]\left( {V - {{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{V}}}_{\text{K}}}} \right)} \hfill \\ \end{array} . $$

(5.17)

Gating Dynamics

L-type calcium channel [1]

$$ \begin{array}{lll} {{O_{\rm{cal}}} = {m_{\rm{cal}}}{h_{\rm{cal}}}} \hfill & {{\tau_{{{\rm{m,cal}}}}} = 0.943 + 10\exp \left( { - {{\left[ {\displaystyle\frac{{V + 86.4}}{{23.2}}} \right]}^2}} \right)} \hfill \\{{m_{{\infty, {\rm{cal}}}}} = {{\left[ {1 + \exp \left( { - \displaystyle\frac{{V + 15}}{7}} \right)} \right]}^{{ - 1}}}} \hfill & {{h_{\rm{cal}}} = \displaystyle\frac{{0.00045}}{{0.00045 + C{a_{\rm{i}}}}}} \hfill \\\end{array} {.} $$

High voltage activated calcium channel [1]

$$ \begin{array}{lll} {{O_{{{\rm{ca,hva}}}}} = {m_{{{\rm{ca,hva}}}}}{h_{{{\rm{ca,hva}}}}}} \hfill & {} \hfill \\{{m_{{\infty, {\rm{cahva}}}}} = {{\left[ {1 + \exp \left( { - \displaystyle\frac{{V + 10}}{{10}}} \right)} \right]}^{{ - 1}}}} \hfill & {{\tau_{{{\rm{m,cahva}}}}} = 0.05 + 0.1\exp \left( { - {{\left[ {\displaystyle\frac{{V + 62}}{{13}}} \right]}^2}} \right)} \hfill \\{{h_{{\infty, {\rm{cahva}}}}} = {{\left[ {1 + \exp \left( {\displaystyle\frac{{V + 48}}{5}} \right)} \right]}^{{ - 1}}}} \hfill & {{\tau_{{{\rm{h,cahva}}}}} = 0.5 + 0.5\exp \left( { - {{\left[ {\displaystyle\frac{{V + 55.6}}{{18}}} \right]}^2}} \right)} \hfill \\\end{array} $$

T-type calcium channel [6]

$$ \begin{array}{lll} {{O_{\rm{cat}}} = {m_{\rm{cat}}}{h_{\rm{cat}}}} \hfill & {} \hfill \\{{m_{{\infty, {\rm{cat}}}}} = {{\left[ {1 + \exp \left( { - \displaystyle\frac{{V + 63}}{{1.5}}} \right)} \right]}^{{ - 1}}}} \hfill & {{\tau_{{{\rm{m,cat}}}}} = 12 + 65\exp \left( { - {{\left[ {\displaystyle\frac{{V + 68}}{6}} \right]}^2}} \right)} \hfill \\{{h_{{\infty, {\rm{cat}}}}} = {{\left[ {1 + \exp \left( {\displaystyle\frac{{V + 76.2}}{3}} \right)} \right]}^{{ - 1}}}} \hfill & {{\tau_{{{\rm{h,cat}}}}} = 10 + 50\exp \left( { - {{\left[ {\displaystyle\frac{{V + 72}}{{10}}} \right]}^2}} \right)} \hfill \\\end{array} $$

Sodium channel

$$ \begin{array}{lll} {{O_{\rm{Na}}} = m_{\rm{na}}^3{h_{\rm{na}}}} \hfill & {{\alpha_{\rm{mna}}} = 1.965\exp \left( {1.713\displaystyle\frac{V}{{{V_{\tau }}}}} \right)} \hfill & {{\beta_{\rm{mna}}} = 0.0424\exp \left( {1.558\displaystyle\frac{V}{{{V_{\tau }}}}} \right)} \hfill \\\, \hfill & {{\alpha_{\rm{hna}}} = 9.566 \times {{10}^{{ - 5}}}\exp \left( { - 2.432\displaystyle\frac{V}{{{V_{\tau }}}}} \right)} \hfill & {{\beta_{\rm{mna}}} = 0.53\exp \left( { - 1.187\displaystyle\frac{V}{{{V_{\tau }}}}} \right)} \hfill \\\end{array} $$

Small conductance calcium gated potassium channel

$$ {O_{{{\rm{k,sk}}}}} = \displaystyle\frac{{Ca_{\rm{i}}^4}}{{{{\left( {0.00035} \right)}^4} + Ca_{\rm{i}}^4}}. $$

Delayed rectifier potassium channel [1]

$$ \begin{array}{lll} {{O_{\rm{kdr}}} = m_{\rm{kdr}}^3} \hfill & {} \hfill \\{{m_{{\infty, {\rm{kdr}}}}} = {{\left[ {1 + \exp \left( { - \displaystyle\frac{{V + 25}}{{12}}} \right)} \right]}^{{ - 1}}}} \hfill & {{\tau_{{{\rm{m,kdr}}}}} = 18{{\left[ {1 + \exp \left( {\displaystyle\frac{{V + 39}}{8}} \right)} \right]}^{{ - 1}}} + 1} \hfill \\\end{array} $$

Transient A-type potassium channel [1]

$$ \begin{array}{lll} {{O_{\rm{ka}}} = m_{\rm{ka}}^3{h_{\rm{ka}}}} \hfill & {} \hfill \\{{m_{{\infty, {\rm{ka}}}}} = {{\left[ {1 + \exp \left( { - \displaystyle\frac{{V + 43}}{{24}}} \right)} \right]}^{{ - 1}}}} \hfill & {{\tau_{{{\rm{m,ka}}}}} = 1.1 + 2\exp \left( { - {{\left[ {\displaystyle\frac{{V + 50}}{{23.45}}} \right]}^2}} \right)} \hfill \\{{h_{{\infty, {\rm{ka}}}}} = {{\left[ {1 + \exp \left( {\displaystyle\frac{{V + 56}}{8}} \right)} \right]}^{{ - 1}}}} \hfill & {{\tau_{{{\rm{h,ka}}}}} = 20} \hfill \\\end{array} $$

Internal rectifying potassium channel [6]

$$ {O_{\rm{kdr}}} = {\left[ {1 + \exp \left( {\displaystyle\frac{{V + 90}}{{12.1}}} \right)} \right]^{{ - 1}}} $$

HCN channels (based on [6]; see Fig. 5.1f)

$$ \begin{array} {lll}{{O_{\rm{hcn}}} = y} \hfill & {\displaystyle\frac{{{\text{d}}y}}{{{\hbox{d}}t}} = {k_{{{\rm{f,hcn}}}}}y - {k_{{{\rm{r,hcn}}}}}\left( {1 - y} \right)} \hfill \\{{k_{{{\rm{f,hcn}}}}} = {k_{{{\rm{f,free}}}}}{\rm P}(C) + {k_{{{\rm{f,bnd}}}}}\left[ {1 - {\rm P}(C)} \right]} \hfill & {{k_{{{\rm{r,hcn}}}}} = {k_{{{\rm{r,free}}}}}{\rm P}(O) + {k_{{{\rm{r,bnd}}}}}\left[ {1 - {\rm P}(O)} \right]} \hfill \\{{\rm P}(C) = {{\left[ {1 + \displaystyle\frac{\text{cAMP}}{{1.163 \times {{10}^{{ - 3}}}({\hbox{mM}})}}} \right]}^{{ - 1}}}} \hfill & {{\rm P}(O) = {{\left[ {1 + \displaystyle\frac{\text{cAMP}}{{1.45 \times {{10}^{{ - 3}}}({\hbox{mM}})}}} \right]}^{{ - 1}}}} \hfill \\{{k_{{{\rm{f,free}}}}} = \displaystyle\frac{{0.006}}{{1 + \exp \left( {\displaystyle\frac{{V + 87.7}}{{6.45}}} \right)}}} \hfill & {{k_{{{\rm{r,free}}}}} = \displaystyle\frac{{0.08}}{{1 + \exp \left( { - \displaystyle\frac{{V + 51.7}}{7}} \right)}}} \hfill \\{{k_{{{\rm{f,bnd}}}}} = \displaystyle\frac{{0.0268}}{{1 + \exp \left( {\displaystyle\frac{{V + 94.2}}{{13.3}}} \right)}}} \hfill & {{k_{{{\rm{r,bnd}}}}} = \displaystyle\frac{{0.08}}{{1 + \exp \left( { - \displaystyle\frac{{V + 35.5}}{7}} \right)}}} \hfill \\\end{array} $$

Dynamics of Facilitated Transport

Sodium–calcium exchanger (Model modified from [24])

$$ {I_{\rm{naca}}} = {k_{\rm{naca}}}\left[ {{\beta_{ - }}{\rm P}\left( {{{E \prime}_{\!\!\!1}}} \right)y - {\beta_{ + }}{\rm P}\left( {E_2^{\prime}} \right)\left( {1 - y} \right)} \right]. $$

(5.18)

$$ \begin{array}{llll} {\displaystyle\frac{{{\text{d}}y}}{{{\hbox{d}}t}} = {k_{ - }}\left( {1 - y} \right) - {k_{ + }}y} \hfill & \, \hfill \\{{k_{ + }} = {\alpha_{ + }}{\rm P}\left( {E_1^{*}} \right) + {\beta_{ - }}{\rm P}\left( {{{E \prime}_{\!\!\!\!1}}} \right)} \hfill & {{k_{ - }} = {\alpha_{ - }}{\rm P}\left( {E_2^{*}} \right) + {\beta_{ + }}{\rm P}\left( {{{E^{\prime}}_{\!\!\!2}}} \right)} \hfill \\{{\rm P}\left( {E_1^{*}} \right) = {{\left[ {1 + \displaystyle\frac{{0.00138}}{{C{a_{\rm{i}}}}}\left( {1 + {{\left( {\displaystyle\frac{{N{a_{\rm{i}}}}}{{8.75}}} \right)}^3}} \right)} \right]}^{{ - 1}}}} \hfill & {{\rm P}\left( {E_2^{*}} \right) = {{\left[ {1 + \displaystyle\frac{{1.38}}{{C{a_{\rm{e}}}}}\left( {1 + {{\left( {\displaystyle\frac{{N{a_{\rm{e}}}}}{{87.5}}} \right)}^3}} \right)} \right]}^{{ - 1}}}} \hfill \\{{\rm P}\left( {{{E\prime}_{\!\!\!\!1}}} \right) = {{\left[ {1 + {{\left( {\displaystyle\frac{{8.75}}{{N{a_{\rm{i}}}}}} \right)}^3}\left( {1 + \displaystyle\frac{{C{a_{\rm{i}}}}}{{0.00138}}} \right)} \right]}^{{ - 1}}}} \hfill & {{\rm P}\left( {{{E^{\prime}}_{\!\!\!2}}} \right) = {{\left[ {1 + {{\left( {\displaystyle\frac{{87.5}}{{N{a_{\rm{e}}}}}} \right)}^3}\left( {1 + \displaystyle\frac{{C{a_{\rm{e}}}}}{{1.38}}} \right)} \right]}^{{ - 1}}}} \hfill \\{{\alpha_{ + }} = \exp \left( {{{{\left( {1 - 0.32} \right)V}} \left/ {{{V_{\tau }}}} \right.}} \right)} \hfill & {{\beta_{ + }} = \exp \left( {{{{ - 0.32V}} \left/ {{{V_{\tau }}}} \right.}} \right)({\hbox{m}}{{\hbox{s}}^{{ - 1}}})} \hfill \\{{\alpha_{ - }} = \exp \left( {{{{ - 0.32V}} \left/ {{{V_{\tau }}}} \right.}} \right)} \hfill & {{\beta_{ - }} = \exp \left( {{{{\left( {1 - 0.32} \right)V}} \left/ {{{V_{\tau }}}} \right.}} \right)({\hbox{m}}{{\hbox{s}}^{{ - 1}}})} \hfill \\\end{array}. $$

Plasma membrane calcium ATP-ase

$$ {I_{\rm{pmca}}} = {k_{\rm{pmca}}}{A_{\rm{pmca}}}\left[ {{\alpha_{ + }}{\rm P}\left( {E_1^{*}} \right)y - {\alpha_{ - }}{\rm P}\left( {E_2^{*}} \right)\left( {1 - y} \right)} \right]. $$

(5.19)

$$ \begin{array} {lll}{\displaystyle\frac{{{\text{d}}y}}{{{\hbox{d}}t}} = {k_{ - }}\left( {1 - y} \right) - {k_{ + }}y} \hfill & \, \hfill \\{{k_{ + }} = {\alpha_{ + }}{\rm P}\left( {E_1^{*}} \right) + {\beta_{ - }}{\rm P}\left( {{E_1}} \right)} \hfill & {{k_{ - }} = {\alpha_{ - }}{\rm P}\left( {E_2^{*}} \right) + {\beta_{ + }}{\rm P}\left( {{E_2}} \right)} \hfill \\{{\rm P}\left( {E_1^{*}} \right) = {{\left[ {1 + \displaystyle\frac{{{k_{{{\rm{pmca,cai}}}}}}}{{C{a_i}}}} \right]}^{{ - 1}}}} \hfill & {{\rm P}\left( {{E_1}} \right) = 1 - {\rm P}\left( {E_1^{*}} \right)} \hfill \\{{\rm P}\left( {E_2^{*}} \right) = {{\left[ {1 + \displaystyle\frac{2}{{C{a_e}}}} \right]}^{{ - 1}}}} \hfill & {{\rm P}\left( {{E_2}} \right) = 1 - {\rm P}\left( {E_2^{*}} \right)} \hfill \\{{A_{\rm{pmca}}} = \displaystyle\frac{{10.56 \times [cacam]}}{{[cacam] + 0.00005}} + 1.2(pA)} \hfill & \hfill \\\end{array} $$

$$ \begin{array}{lll} {{{k_{{{\rm{pmca,cai}}}}} = \frac{{\left( {180 - 6.4} \right) \times {{10}^{{ - 5}}}}}{{1 + {{{[cacam]}} \left/ {{0.00005}} \right.}}} + 6.4 \times {{10}^{{ - 5}}}({{mM}})}} \hfill \\ {{\alpha_{ + }} = {{\left( {1 + \displaystyle\frac{{0.1}}{{[ATP]}}} \right)}^{{\!\! - 1}}}\quad {\beta_{ + }} = 0.001({{m}}{{{s}}^{{ - 1}}})} \\ {{\alpha_{ - }} = 0.001\quad {\beta_{ - }} = 1({{m}}{{{s}}^{{ - 1}}})} \end{array}$$

Sodium–potassium ATP-ase

$$ {I_{\rm{nak}}} = {k_{\rm{nak}}}\left[ {{\alpha_{ + }}{\rm P}\left( {E_1^{*}} \right)y - {\alpha_{ - }}{\rm P}\left( {E_2^{*}} \right)\left( {1 - y} \right)} \right]. $$

(5.20)

$$ \begin{array}{lll} {\displaystyle\frac{{{\text{d}}y}}{{{\hbox{d}}t}} = {k_{ - }}\left( {1 - y} \right) - {k_{ + }}y} \hfill & \, \hfill \\{{k_{ + }} = {\alpha_{ + }}{\rm P}\left( {E_1^{*}} \right) + {\beta_{ - }}{\rm P}\left( {{{E^{\prime}}_{\!\!\!1}}} \right)} \hfill & {{k_{ - }} = {\alpha_{ - }}{\rm P}\left( {E_2^{*}} \right) + {\beta_{ + }}{\rm P}\left( {{{E^{\prime}}_{\!\!2}}} \right)} \hfill \\{{\rm P}\left( {E_1^{*}} \right) = {{\left[ {1 + \displaystyle\frac{{4.05}}{{N{a_{\rm{i}}}}}\left( {1 + \displaystyle\frac{{{K_{\rm{i}}}}}{{32.88}}} \right)} \right]}^{{ - 1}}}} \hfill & {{\rm P}\left( {{{E^{\prime}}_{\!\!\!1}}} \right) = {{\left[ {1 + \displaystyle\frac{{32.88}}{{{K_{\rm{i}}}}}\left( {1 + \displaystyle\frac{{N{a_{\rm{i}}}}}{{4.05}}} \right)} \right]}^{{ - 1}}}} \hfill \\{{\rm P}\left( {E_2^{*}} \right) = {{\left[ {1 + \displaystyle\frac{{69.8}}{{N{a_{\rm{eff}}}}}\left( {1 + \displaystyle\frac{{{K_{\rm{e}}}}}{{0.258}}} \right)} \right]}^{{ - 1}}}} \hfill & {{\rm P}\left( {{{E^{\prime}}_{\!\!2}}} \right) = {{\left[ {1 + \displaystyle\frac{{0.258}}{{{K_{\rm{e}}}}}\left( {1 + \displaystyle\frac{{N{a_{\rm{eff}}}}}{{69.8}}} \right)} \right]}^{{ - 1}}}} \hfill \\\end{array} $$

$$ \begin{array}{lll} {N{a_{\rm{eff}}} = N{a_{\rm{e}}}\exp \left( {{{{ - 0.82V}} \left/ {{{V_{\tau }}}} \right.}} \right)} \hfill & \, \hfill \\{{\alpha_{ + }} = 0.37{{\left( {1 + \displaystyle\frac{{0.094}}{{[ATP]}}} \right)}^{{ - 1}}}} \hfill & {{\beta_{ + }} = 0.165({\hbox{m}}{{\hbox{s}}^{{ - 1}}})} \hfill \\{{\alpha_{ - }} = 0.04} \hfill & {{\beta_{ - }} = 0.01({\hbox{m}}{{\hbox{s}}^{{ - 1}}})} \hfill \\\end{array} $$

Buffer Dynamics

Calbindin

The dynamics of binding of calcium to the fast buffer, calbindin is modelled using mass action kinetics (Fig. 5.1d) and the kinetic parameters for the high affinity binding adopted from Nagerl et al. [27],

$$ {J_{\rm{calb}}} = {k_{{{\rm{calb,b}}}}}{[Ca]_{\rm{i}}}[Calb] - {k_{{{\rm{calb,d}}}}}[cacalb]. $$

Calmodulin

Calcium has four binding sites on calmodulin. Two of these are located on the C-terminal lobe and two on the N-terminal. However, the binding and dissociation rates to each of these lobes are different. We model the simultaneous binding of calcium [42] as a four state Markov process, and further reduce the model to two states, assuming quasi-steady state for the intermediary states (Fig. 5.1e).

$$ \begin{array}{lll} {{J_{\rm{cam}}} = {\alpha_{\rm{cam}}}[Cam] - {\beta_{\rm{cam}}}[cacam]} \hfill & \, \hfill \\{{\alpha_{\rm{cam}}} = {k_{\rm{cb}}}{k_{\rm{nb}}}\left[ {\displaystyle\frac{1}{{{k_{\rm{cb}}} + {k_{\rm{nd}}}}} + \displaystyle\frac{1}{{{k_{\rm{cd}}} + {k_{\rm{nb}}}}}} \right]} \hfill & {{\beta_{\rm{cam}}} = {k_{\rm{cd}}}{k_{\rm{nd}}}\left[ {\displaystyle\frac{1}{{{k_{\rm{cb}}} + {k_{\rm{nd}}}}} + \displaystyle\frac{1}{{{k_{\rm{cd}}} + {k_{\rm{nb}}}}}} \right]} \hfill \\{{k_{\rm{cb}}} = {k_{{{\rm{cam,cb}}}}}[Ca]_{\rm{i}}^2} \hfill & {{k_{\rm{nb}}} = {k_{{{\rm{cam,nb}}}}}[Ca]_{\rm{i}}^2} \hfill \\\end{array} $$