Effectors of the frequency of calcium oscillations in HEK-293 cells: wavelet analysis and a computer model

Abstract

Oscillations of the intracellular concentration of Ca²⁺ in cultured HEK-293 cells, which heterologously expressed the calcium-sensing receptor, were recorded with the fluorophore Fura-2 using fluorescence microscopy. HEK-293 cells are extremely sensitive to small perturbations in extracellular calcium concentrations. Resting cells were attached to cover slips and perifused with saline solution containing physiologically relevant extracellular Ca²⁺ concentrations in the range 0.5–5 mM. Acquired digitized images of the cells showed oscillatory fluctuations in the intracellular Ca²⁺ concentration over the time course, and were processed as a function of the change in Fura-2 excitation ratio and frequency at 12–37°C. Newly developed data processing techniques with wavelet analysis were used to estimate the frequency at which the rectified sinusoidal oscillations occurred; we estimated ~4 min⁻¹ under normal conditions. Temperature variations revealed an Arrhenius relationship in oscillation frequency. A critical Ca²⁺ concentration of ~2 mM was estimated, below which oscillations did not occur. These data were used to develop a kinetic model of the system that was simulated using Mathematica; kinetic parameter values were adjusted to match the experimentally observed oscillations of intracellular Ca²⁺ concentration as a function of extracellular Ca²⁺ concentration, and temperature; and from these, limit cycles were obtained and control coefficients were estimated for all parameters.

Appendix

Appendix 1

Mathematica program that solves the array of differential equations given in Eq. 12 to simulate the oscillations in Ca²⁺ concentration in HEK-293 cells.

• caConc = 5,000; (* Ca²⁺ concentration in μ mol L⁻¹ *)

• temp = 37; (* Temperature of the system in Celsius *)

• tempScalar = 0.173 + 0.00160882 × e ^{0.1684 temp}; (* Scalar value to account for frequency changes affected by temperature *)

• V _c = 0.33; V _ER = 0.33; V _e = 0.33; (* Cytosolic, ER, and extracellular volume ratios *)

• γ = 3.0; (* Cooperativity coefficient *)

• ϕ = 2.0; (* Hill coefficient for the PMCA *)

• β = 0.4; (* Stoichiometry coefficient *)

• ρ = 0.4; (* Fraction of extruded Ca²⁺ available to interact with the CaR *)

• $v_{1} = {\frac{3.854}{{{\text{caConc}}(1 + e^{{6 - 0.003{\text{caConc}}}} )}}}{\text{tempScalar}};$ (* Velocity of Ca²⁺ supply with logisitic-equation form to simulate critical Ca²⁺ concentration *)

• k ₁ = 3740.8 × tempScalar,

• k ₋₁ = 115.22 × tempScalar,

• k ₂ = 53.54 × tempScalar,

• k ₃ = 1007.4 × tempScalar,

• k ₋₃ = 18.1 × tempScalar,

• k ₄ = 58.35 × tempScalar,

• k ₋₄ = 633.01 × tempScalar,

• k ₅ = 500 × tempScalar,

• K _PMCA = 0.425;

• V _PMCA = 28 × tempScalar;

• K _SERCA = 0.18;

• ca[t] = caConc,

• v ₂ = 53.54 × tempScalar; (* Velocity of IP₃ degradation *)

• sol = NDSolve[{

• ca[t] = caConc,

• $s^{\prime}_{1} [t] = v_{1} {\text{ca}}[t] - {\frac{{V_{\text{c}} }}{{V_{\text{e}} }}}k_{1} s_{1} [t]x_{1} [t] + {\frac{{V_{\text{c}} }}{{V_{\text{e}} }}}k_{ - 1} x_{2} [t] + {\frac{{V_{\text{c}} }}{{V_{\text{e}} }}}\rho {\frac{{V_{\text{PMCA}} s_{4} [t]^{\phi } }}{{K_{\text{PMCA}}^{\phi } + s_{4} [t]^{\phi } }}},$ (* s ₁ is extracellular [Ca²⁺] sub-pool *)

• $s^{\prime}_{2} [t] = k_{2} x[t] - k_{3} s_{2} [t]^{\gamma } e_{1} [t] + \gamma k_{ - 3} x_{1} [t] - v_{2} s_{2} [t] + k_{ - 4} x_{3} [t] - {\frac{{V_{\text{ER}} }}{{V_{\text{c}} }}}k_{4} s_{2} [t]e_{2} [t],$ (* s ₂ is [IP₃] *)

• $s^{\prime}_{3} [t] = {\frac{{V_{\text{c}} }}{{V_{\text{ER}} }}}{\frac{{V_{\text{SERCA}} s_{4} [t]}}{{K_{\text{SERCA}} + s_{4} [t]}}}\;{\frac{1}{{s_{3} [t]}}} - \beta k_{5} s_{3} [t]^{\beta } x_{3} [t],$ (* s ₃ is ER [Ca²⁺] *)

• $s^{\prime}_{4} [t] = - {\frac{{V_{\text{c}} }}{{V_{\text{ER}} }}}{\frac{{V_{\text{SERCA}} s_{4} [t]}}{{K_{\text{SERCA}} + s_{4} [t]}}}\;{\frac{1}{{s_{3} [t]}}} - {\frac{{V_{\text{PMCA}} s_{4} [t]^{\phi } }}{{K_{\text{PMCA}}^{\phi } + s_{4} [t]^{\phi } }}} + {\frac{{V_{\text{ER}} }}{{V_{\text{c}} }}}\beta k_{5} s_{3} [t]^{\beta } x_{3} [t],$ (* s ₄ is cytosolic [Ca²⁺] *)

• $x^{\prime}_{1} [t] = - {\frac{{V_{\text{e}} }}{{V_{\text{c}} }}}k_{1} s_{1} [t]x_{1} [t] + (k_{ - 1} + k_{2} )x_{2} [t] + k_{3} e_{1} [t]s_{2} [t]^{\gamma } - k_{ - 3} x_{1} [t],$ (* x ₁ is [$E_{1} S_{2}^{\gamma }$] *)

• $x^{\prime}_{2} [t] = {\frac{{V_{\text{e}} }}{{V_{\text{c}} }}}k_{1} s_{1} [t]x_{1} [t] - (k_{ - 1} + k_{2} )x_{2} [t],$ (* x ₂ is [$S_{1} E_{1} S_{2}^{\gamma }$] *)

• $x^{\prime}_{3} [t] = - {\frac{{V_{\text{c}} }}{{V_{\text{ER}} }}}k_{ - 4} x_{3} [t] + {\frac{{V_{\text{c}} }}{{V_{\text{ER}} }}}k_{4} s_{2} [t]e_{2} [t],$ (* x ₃ is [$E_{2} S_{2}$] *)

• $e^{\prime}_{1} [t] = - k_{3} e_{1} [t]s_{2} [t]^{\gamma } + k_{ - 3} x_{1} [t],$ (* e ₁ is [CaR] *)

• $e^{\prime}_{2} [t] = k_{ - 4} x_{3} [t] - {\frac{{V_{\text{c}} }}{{V_{\text{ER}} }}}k_{4} s_{2} [t]e_{2} [t],$ (* e ₂ is [IP₃R] *)

• s ₁[0] = 0.5, s ₂[0] = 0.44, s ₃[0] = 500, s ₄[0] = 0.0, x ₁[0] = 0.0, x ₂[0] = 0.0,

• x ₃[0] = 0.0, ca[0] = caConc, e ₁[0] = 0.16, e ₂[0] = 0.2,

• {s ₁, s ₂, s ₃, s ₄, x ₁, x ₂, x ₃, e ₁, e ₂, ca},

• {t, 0, 400}, MaxSteps → 20,000, MaxStepSize → 0.01,

• MaxStepFraction → 0.01, StartingStepSize → 0.1]; (* “sol” contains the solution to the kinetic equations with the given starting conditions for t = 0–100 s. *)

• Plot[s ₄[t]/.sol,{t, 0, 400},PlotRange → {{0, 400},{0, 0.4}}, PlotPoints → 2,000] (* Plot the intracellular [Ca²⁺], S ₄ *)

Appendix 2

Program written in Mathematica for the wavelet analysis used in the present work:

• data = Import[″data.dat″]; (* Import the data set as a list of values *)

• Hw := If [ω > 0, 1, 0]; (* Definition of the Heaviside step function *)

• ω ₀ = 6; (* Nondimensional frequency of the wavelet *)

• $\psi f[\omega_{ - } ,s_{ - } ] = \sqrt {{\frac{2\pi s}{{{\delta}t}}}} \pi^{{ - {\frac{1}{4}}}} {\text{Hw}}\,e^{{{\frac{{ - (s\omega - \omega_{0} )^{2} }}{2}}}} ;$ (* Definition of the Morlet wavelet as a function of ω, time and s, the scale *)

• ${\delta}t = {\frac{1}{1.087454}};$ (* Difference in time between successive data points. For our Fura-2 experiments, this was slightly over 1 s *)

• s ₀ = 2δt; (* Value of the first wavelet scale *)

• δj = 0.125; (* Difference in power between successive wavelet scales *)

• nN = Length[data]; (* Number of data points in the data set *)

• fdata = Fourier[data]; (* Calculate the Fourier transform of the data set *)

• J = 60; (* Number of scales to include in the wavelet transform *)

• wt = Table[Table[0,{x, 0, nN}],{y, 1, J};] (* Initialization of the table receiving the wavelet transform result *)

• For[j = 0, j < J, j++], {pdata = 

• $$ {\text{Table}}\left[ {\psi f\left[ {\omega ,2{\delta}t \times 2^{{j{\delta}j}} } \right],\left\{ {\omega ,0,{\frac{2\pi }{{{\delta}t}}},{\frac{{\left( {{\frac{2\pi }{{{\delta}t}}}} \right)}}{{{\text{nN}} - 1}}}} \right\}} \right]//N; $$

• $wt[[j + 1]] = {\frac{{({\text{Abs}}[{\text{InverseFourier}}[{\text{fdata}} \times {\text{pdata}}]])^{2} }}{{{\text{Variance}}[{\text{pdata}}]}}};$ (* Calculation and insertion of wavelet transform result *)

• ${\text{sca}} = {\text{Round}}\left[ {\sqrt 2 \left( {{\text{Table}}\left[ {2{\delta}t \times 2^{{j{\delta}j}} ,\,\{ j,0,J\} } \right]} \right) \times {\frac{1}{{{\delta}t}}}} \right.;$ (* Table of wavelet scales *)

• coi = Table[Table[0,{x, 1, nN}],{y, 0, J − 1}];

• Table[coi[[q, Table[x, {x, 1, sca[[q]]}]]] = Table[1,{x, 1, sca[[q]]}], {q, 1, J}; (* Initialization of the table receiving the cone of influence result *)

• Table[coi[[q,Table[x,{x, Length[coi[[q]]] − sca[[q]] + 1, Length[coi[[q]]]}]]] = Table[1,{x, 1, sca[[q]]}], {q, 1, J}]; (* Calculation of cone of influence *)

• coi = Mod[coi + 1,2]; (* Inversion of cone of influence *)

• wtcoi = wt coi; (* Wavelet transform multiplied by cone of influence *)