Intercellular calcium signalling in cultured renal epithelia: a theoretical study of synchronization mode and pacemaker activity

Abstract

We investigate a two-dimensional lattice model representation of intercellular Ca²⁺ signalling in a population of epithelial cells coupled by gap junctions. The model is based on and compared with Ca²⁺ imaging data from globally bradykinin-stimulated MDCK-I (Madin-Darby canine kidney)-I cell layers. We study large-scale synchronization of relevance to our laboratory experiments. The system is found to express a wealth of dynamics, including quasiperiodic, chaotic and multiply-periodic behaviour for intermediate couplings. We take a particular interest in understanding the role of “pacemaker cells” in the synchronization process. It has been hypothesized that a few highly hormone-sensitive cells control the collective frequency of oscillation, which is close to the natural frequencies (without coupling) of these cells. The model behaviour is consistent with the conjectures of the pacemaker cell hypothesis near the critical coupling where the cells lock onto a single frequency. However, the simulations predict that the frequency in globally connected systems decreases with increasing coupling. It is found that a pacemaker is not defined by its natural frequency alone, but that other intrinsic or local factors must be considered. Inclusion of partly sensitized cells that do not oscillate autonomously in the cell layer increases the coupling necessary for global synchronization. For not excessively high coupling, these cells oscillate irregularly and with distinctive lower frequencies. In summary, the present study shows that the frequency of synchronized oscillations is not dictated by one or few fast-responding cells. The collective frequency is the result of a two-way communication between the phase-advanced pacemaker and its environment.

Appendix

Model expressions

The expression for the Ca²⁺ release from the ER (J _{ER_out}) is taken from Li and Rinzel (1994):

$$ \begin{aligned} & J_{{{\text{ER\_out}}}} = {\left( {c_{1} P_{{IP3R}} + c_{2} } \right)}{\left( {y - x} \right)}{\text{ with}} \\ & P_{{IP3R}} = \frac{{{\left( {d_{2} \frac{{d_{1} + IP_{{3ij}} }} {{d_{3} + IP_{{3ij}} }}IP_{{3ij}} x_{{ij}} } \right)}^{3} }} {{{\left( {d_{p} + IP_{{3ij}} } \right)}^{3} {\left( {d_{a} + x_{{ij}} } \right)}^{3} {\left( {d_{2} \frac{{d_{1} + IP_{{3ij}} }} {{d_{3} + IP_{{3ij}} }} + x_{{ij}} } \right)}}} \\ \end{aligned} $$

(12)

where the driving force is the concentration difference (y=x). The channel opening probability, $P_{{IP_{3} R}}$, is described by considering three binding sites on each IP₃R subunit: one for IP₃ and one activating and one de-activating Ca²⁺ binding site. The constants c ₁ and c ₂ are rate constants of the maximal Ca²⁺ channel release and Ca²⁺ leakage from the ER.

The Ca²⁺-ATPase pumps on the ER membrane (J _{ER_in}) and plasma membrane (J _out) are both modelled by simple Hill expressions in cytosolic Ca²⁺ x according to

$$ J_{{{\text{ER\_in}}}} = \nu _{1} \frac{{x^{2} }} {{K^{2}_{1} + x^{2} }}\,\,\,\,\,J_{{{\text{out}}}} = \nu _{2} \frac{{x^{2} }} {{K^{2}_{2} + x^{2} }} $$

(13)

where ν ₁ and ν ₂ are maximal rate constants and $K^{2}_{1} {\text{ and }}K^{2}_{2}$ represent the Ca²⁺ pumping for which the activity is half-maximum.

The Ca²⁺ entry rate is given as a sum of a small Ca²⁺ leakage and a store-operated Ca²⁺ influx. The store-operated inflow is controlled by the level of IP₃ in the cell as suggested by Dupont and Goldbeter (1993):

$$ J_{{{\text{in}}}} = \nu _{1} + \nu _{0} \frac{{IP_{3} }} {{K_{0} + IP_{3} }} $$

(14)

where ν _l is a small-leak term, v ₀ is the maximal flux of Ca²⁺ through the store-operated channel, and K ₀ is the Ca²⁺ level for half-maximal pumping activity.