Calcium Oscillation Frequency-Sensitive Gene Regulation and Homeostatic Compensation in Pancreatic $\upbeta$-Cells

Abstract

Pancreatic islet $\upbeta$-cells are electrically excitable cells that secrete insulin in an oscillatory fashion when the blood glucose concentration is at a stimulatory level. Insulin oscillations are the result of cytosolic $\hbox {Ca}^{2+}$ oscillations that accompany bursting electrical activity of $\upbeta$-cells and are physiologically important. ATP-sensitive $\hbox {K}^{+}$ channels (K(ATP) channels) play the key role in setting the overall activity of the cell and in driving bursting, by coupling cell metabolism to the membrane potential. In humans, when there is a defect in K(ATP) channel function, $\upbeta$-cells fail to respond appropriately to changes in the blood glucose level, and electrical and $\hbox {Ca}^{2+}$ oscillations are lost. However, mice compensate for K(ATP) channel defects in islet $\upbeta$-cells by employing alternative mechanisms to maintain electrical and $\hbox {Ca}^{2+}$ oscillations. In a recent study, we showed that in mice islets in which K(ATP) channels are genetically knocked out another $\hbox {K}^{+}$ current, provided by inward-rectifying $\hbox {K}^{+}$ channels, is increased. With mathematical modeling, we demonstrated that a sufficient upregulation in these channels can account for the paradoxical electrical bursting and $\hbox {Ca}^{2+}$ oscillations observed in these $\upbeta$-cells. However, the question of determining the correct level of upregulation that is necessary for this compensation remained unanswered, and this question motivates the current study. $\hbox {Ca}^{2+}$ is a well-known regulator of gene expression, and several examples have been shown of genes that are sensitive to the frequency of the $\hbox {Ca}^{2+}$ signal. In this mathematical modeling study, we demonstrate that a $\hbox {Ca}^{2+}$ oscillation frequency-sensitive gene transcription network can adjust the gene expression level of a compensating $\hbox {K}^{+}$ channel so as to rescue electrical bursting and $\hbox {Ca}^{2+}$ oscillations in a model $\upbeta$-cell in which the key K(ATP) current is removed. This is done without the prescription of a target $\hbox {Ca}^{2+}$ level, but evolves naturally as a consequence of the feedback between the $\hbox {Ca}^{2+}$-dependent enzymes and the cell’s electrical activity. More generally, the study indicates how $\hbox {Ca}^{2+}$ can provide the link between gene expression and cellular electrical activity that promotes wild-type behavior in a cell following gene knockout.

Appendices

Appendix 1

The linear differential equation (Eq. 7) that governs the rate of change of the fraction of an activated enzyme has the following form:

$$\begin{aligned} \frac{\mathrm{d}E_a }{\mathrm{d}t}=p_E \frac{c^{n_E }}{c^{n_E }+K_E^{n_E } }\left( {1-E_a } \right) -d_E E_a . \end{aligned}$$

(29)

This can be solved in response to the following square-wave $\hbox {Ca}^{2+}$ stimulus:

$$\begin{aligned} c(t)=\left\{ {{\begin{array}{ll} c_0 =0.1,&{}\hbox { mod}({t,T})\le D \\ 0,&{}\hbox { mod}({t,T})>D \\ \end{array} }} \right. \end{aligned}$$

(30)

Derivation of the solution is similar to what was done in prior studies (Schuster et al. 2005; Salazar et al. 2008). The solution during the ith oscillation cycle is:

$$\begin{aligned} E_{a,i} \left( \theta \right) =\left\{ {{\begin{array}{ll} E_{ss} +\xi _i e^{-\left( {p_E^*+d_E } \right) \theta },&{} 0\le \theta <D \\ \psi _i e^{-d_E \theta }, &{} D\le \theta \le T \\ \end{array} }} \right. \end{aligned}$$

(31)

where $E_{a,i}$ is the solution of Eq. 29 for the ith stimulus cycle with the internal time $\theta \in \left[ {0,T} \right]$ and $E_{ss}$ and $p_E^*$ are given by:

$$\begin{aligned} p_E^*= & {} p_E \frac{c_0^{n_E } }{c_0^{n_E } +K_E^{n_E } }, \end{aligned}$$

(32)

$$\begin{aligned} E_{ss}= & {} \frac{1}{1+\frac{d_E }{p_E^*}}. \end{aligned}$$

(33)

For consecutive oscillation cycles $i-1$ and i,

$$\begin{aligned} E_{a,i-1} (T)=E_{a,i} (0) \end{aligned}$$

(34)

and $E_{a,i}$ is continuous at D. Therefore, these relations yield the following difference equations for coefficients $\xi _i$ and $\psi _i$:

$$\begin{aligned} \xi _{i+1}= & {} E_{ss} \left( {e^{-d_E \left( {T-D} \right) }-1} \right) +e^{-\left( {p_E^*D+d_E T} \right) }\xi _i \end{aligned}$$

(35)

$$\begin{aligned} \psi _i= & {} e^{-p_E^*D}\xi _i +E_{ss} e^{-d_E D}. \end{aligned}$$

(36)

Assuming that the enzyme is completely in its inactive form at the beginning, $E_{a,0} \left( 0 \right) =0,$ we get $\xi _0 =-E_{ss}$. The difference equation in Eq. 35 has the form,

$$\begin{aligned} x_{i+1} =ax_i +b \end{aligned}$$

(37)

and with initial condition $x_0$:

$$\begin{aligned} x_i= & {} ax_{i-1} +b\\= & {} a(ax_{i-2} +b)+b\\= & {} a^{2}x_{i-2} +ab+b\\= & {} a^{2}\left( {ax_{i-3} +b} \right) +ab+b\\= & {} a^{3}x_{i-3} +a^{2}b+ab+b\\&{\ldots }&\\ x_i= & {} a^{i}x_0 +b {\underbrace{\left( {a^{i-1}+\ldots a^{2}+a+1} \right) }_{\frac{\left( {a^{i}-1} \right) }{a-1}}} \end{aligned}$$

Hence,

$$\begin{aligned} x_i =a^{i}x_0 +\frac{b\left( {a^{i}-1} \right) }{a-1}. \end{aligned}$$

(38)

Therefore, the solution to Eq. 35 is:

$$\begin{aligned} \xi _i =-e^{-\left( {p_E^*D+d_E T} \right) i}E_{ss} +\frac{E_{ss} \left( {e^{-d_E \left( {T-D} \right) }-1} \right) \left( {e^{-\left( {p_E^*D+d_E T} \right) i}-1} \right) }{e^{-\left( {p_E^*D+d_E T} \right) }-1} \end{aligned}$$

(39)

For $i\rightarrow \infty$,

$$\begin{aligned} \xi _i \rightarrow \xi _\infty =-E_{ss} \frac{e^{-d_E \left( {T-D} \right) }-1}{e^{-\left( {p_E^*D+d_E T} \right) }-1} \end{aligned}$$

(40)

and consequently,

$$\begin{aligned} \psi _i \rightarrow \psi _\infty =E_{ss} \frac{e^{d_E D}-e^{-p_E^*D}}{1-e^{-\left( {p_E^*D+d_E T} \right) }}. \end{aligned}$$

(41)

Thus, over many stimulus cycles the solution to Eq. 31 approaches:

$$\begin{aligned} E_{a,\infty } (\theta )=\left\{ {\begin{array}{ll} E_{ss} +\xi _\infty e^{-\left( {p_E^*+d_E } \right) \theta }, &{} 0\le \theta <D^{-} \\ \psi _\infty e^{-d_E \theta },&{} D^{+}\le \theta \le T \\ \end{array} }\right. . \end{aligned}$$

(42)

The mean fraction of activated enzyme concentration during this stimulus cycle is then given by:

$$\begin{aligned} \bar{E}_a =\frac{1}{T}\mathop {\int }\nolimits _0^T E_{a,\infty } (\theta )\mathrm{d}\theta , \end{aligned}$$

(43)

or upon integration:

$$\begin{aligned} \bar{E}_a =E_{ss} \left( {\frac{D}{T}+\frac{1}{d_E T}E_{ss} \frac{\left( {1-e^{-D\left( {d_E +p_E^*} \right) }} \right) \left( {1-e^{-d_E \left( {T-D} \right) }} \right) }{1-e^{-\left( {p_E^*D+d_E T} \right) }}} \right) . \end{aligned}$$

(44)

Appendix 2

The $\upbeta$-cell model is from (Bertram and Sherman 2004) with the following ionic currents:

$$\begin{aligned} I_\mathrm{Ca}= & {} g_\mathrm{Ca} m_\infty \left( {V-V_\mathrm{Ca} } \right) , \end{aligned}$$

(45)

$$\begin{aligned} I_\mathrm{K}= & {} g_K n\left( {V-V_K } \right) ,\end{aligned}$$

(46)

$$\begin{aligned} I_{K_\mathrm{ATP} }= & {} g_{K_\mathrm{ATP} } a\left( {V-V_K } \right) , \end{aligned}$$

(47)

$$\begin{aligned} I_{K_\mathrm{Ca} }= & {} g_{K_\mathrm{Ca} } \omega \left( {V-V_K } \right) ,\end{aligned}$$

(48)

$$\begin{aligned} I_l= & {} g_l \left( {V-V_l } \right) , \end{aligned}$$

(49)

$$\begin{aligned} I_{\mathrm{cmp}}= & {} g_{\mathrm{cmp}} k_\infty \left( {V-V_K } \right) . \end{aligned}$$

(50)

For each ionic current $I_i , g_i$ is the maximal conductance, $V_i$ is the reversal potential and $({V-V_i })$ is the driving force. The rates of changes of the delayed rectifier $\hbox {K}^{+}$ current activation, n, and the K(ATP) current activation, a, are:

$$\begin{aligned} \frac{\mathrm{d}n}{\mathrm{d}t}=\left( {n_\infty \left( V \right) -n} \right) /\tau _n ,\end{aligned}$$

(51)

$$\begin{aligned} \frac{\mathrm{d}a}{\mathrm{d}t}=\left( {a_\infty \left( c \right) -a} \right) /\tau _a , \end{aligned}$$

(52)

where $\tau _n$ and $\tau _a$ are the time constants. Steady-state activation functions, $m_\infty , n_\infty$, $a_\infty$ and $k_\infty$, are:

$$\begin{aligned} m_\infty (V)= & {} \frac{1}{1+e^{\left( {-20-V} \right) /12}}, \end{aligned}$$

(53)

$$\begin{aligned} n_\infty (V)= & {} \frac{1}{1+e^{\left( {-16-V} \right) /5}}, \end{aligned}$$

(54)

$$\begin{aligned} k_\infty (V)= & {} \frac{1}{1+e^{\left( {-49-V} \right) /15}}, \end{aligned}$$

(55)

$$\begin{aligned} a_\infty (c)= & {} \frac{1}{1+e^{\left( {0.14-c} \right) /0.1}}, \end{aligned}$$

(56)

where $m_\infty , n_\infty , a_\infty$ and $k_\infty$ are sigmoidal functions of V and c. $\omega$ is the $\hbox {Ca}^{2+}$-dependent activation variable of $I_{K_\mathrm{Ca} }$ and given with the following Hill equation:

$$\begin{aligned} \omega =\frac{c^{5}}{c^{5}+K_\omega ^5 }, \end{aligned}$$

(57)

where $K_{\upomega }$ is the dissociation constant. $\hbox {Ca}^{2+}$ fluxes across the plasma and endoplasmic reticulum (ER) membranes are:

$$\begin{aligned} J_\mathrm{mem}= & {} -\left( {\alpha I_\mathrm{Ca} +k_\mathrm{pmca} c} \right) , \end{aligned}$$

(58)

$$\begin{aligned} J_\mathrm{er}= & {} p_\mathrm{leak} \left( {c_\mathrm{er} -c} \right) -k_\mathrm{serca} c,\end{aligned}$$

(59)

where parameter $\alpha$ converts ionic current to flux and provides $\hbox {Ca}^{2+}$ influx through voltage-gated $\hbox {Ca}^{2+}$ channels and $k_\mathrm{pmca}$ is the plasma membrane $\hbox {Ca}^{2+}$-ATPase pumping rate and mediates $\hbox {Ca}^{2+}$ efflux from the cytosol. $\hbox {Ca}^{2+}$ leaks from the ER with a rate proportional to $p_\mathrm{leak}$. $k_\mathrm{serca}$ is the $\hbox {Ca}^{2+}$ pumping rate into the ER by SERCA pumps.