Models of IP3 and Ca2+ oscillations: frequency encoding and identification of underlying feedbacks

Abstract

Hormones that act through the calcium-releasing messenger, inositol 1,4,5-trisphosphate (IP3), cause intracellular calcium oscillations, which have been ascribed to calcium feedbacks on the IP3 receptor. Recent studies have shown that IP3 levels oscillate together with the cytoplasmic calcium concentration. To investigate the functional significance of this phenomenon, we have developed mathematical models of the interaction of both second messengers. The models account for both positive and negative feedbacks of calcium on IP3 metabolism, mediated by calcium activation of phospholipase C and IP3 3-kinase, respectively. The coupled IP3 and calcium oscillations have a greatly expanded frequency range compared to calcium fluctuations obtained with clamped IP3. Therefore the feedbacks can be physiologically important in supporting the efficient frequency encoding of hormone concentration observed in many cell types. This action of the feedbacks depends on the turnover rate of IP3. To shape the oscillations, positive feedback requires fast IP3 turnover, whereas negative feedback requires slow IP3 turnover. The ectopic expression of an IP3 binding protein has been used to decrease the rate of IP3 turnover experimentally, resulting in a dose-dependent slowing and eventual quenching of the Ca2+ oscillations. These results are consistent with a model based on positive feedback of Ca2+ on IP3 production.

FIGURE 1

Interactions between Ca²⁺ transport processes and IP₃ metabolism included in the model. The solid and dashed arrows indicate transport/reaction steps and activations, respectively. The bold quantities indicate the model variables: IP₃, cytoplasmic IP₃; Ca(cyt), free cytoplasmic Ca²⁺; Ca(ER), free Ca²⁺ in the ER; IP₃R_a, active conformation of the IP₃R. The other abbreviations denote: IP₃R_i, inactive conformation of the IP₃R; rate of Ca²⁺ release through the IP₃R; active Ca²⁺ transport into the ER; and rate of Ca²⁺-induced IP₃R inactivation and recovery rate, respectively; production rate of IP₃; and rates of IP₃ dephosphorylation and phosphorylation, respectively; and rates of Ca²⁺ influx and extrusion across the plasma membrane, respectively.

FIGURE 2

Agonist-induced IP₃ and Ca²⁺ oscillations in the positive and negative feedback models. (A) Positive feedback model with Ca²⁺ activation of PLC. Changes in [Ca²⁺]_c, [IP₃], and in the fraction of active receptors (top, middle, and bottom panels) after stepwise increases in the agonist concentration (arrowheads), modeled by an increase in the maximal rate of PLC ( for t < 100 with successive increases to 0.787, 1, 1.5, and 2.5 μM/s). (B) Negative feedback model with Ca²⁺ activation of IP₃K. The response is shown for a step protocol with for t < 100, followed by increases to 0.45, 2.5, 5.8, and 10 nM/s. (C) Bifurcation diagram for positive feedback model; shown are the maxima and minima of the [Ca²⁺]_c oscillations (thick lines) and the [Ca²⁺]_c steady states (thin lines) as a function of the stimulus (). Solid and dashed lines indicate stable and unstable states, respectively. HB, Hopf bifurcation; HC, homoclinic bifurcation; SN, saddle-node bifurcation; FB, saddle node of periodics. (D) Bifurcation diagram for negative feedback model. PD, period doubling; TR, torus bifurcation. Between PD and HB₁ and TR and FB there exist complex oscillations (omitted for clarity). The parameter values used are listed in Table 1.

FIGURE 3

Frequency encoding of agonist stimulus. (A) Positive feedback: oscillation periods observed at different stimulation strengths (varying ). Increasing the half-saturation constant of PLC for Ca²⁺, K_PLC, from 0 (no functional positive feedback) to 0.2 μM (functional feedback) greatly enhances frequency encoding. (B) Negative feedback. Increasing the amount of IP₃K relative to IP₃P () enhances frequency encoding. (C, D) The feedback effects shown in panels A and B are preserved when plasma-membrane fluxes of Ca²⁺ are included in the models (). (E) Range of oscillation periods, in the presence (+) and absence (w/o) of positive feedback for two different strengths of the plasma-membrane Ca²⁺ fluxes (). (F) Range of oscillation periods in the presence (−) and absence (w/o) of negative feedback for two different strengths of the plasma-membrane Ca²⁺ fluxes (). We have found that the IP₃K has an impact on the oscillation period only when the Ca²⁺ fluxes between ER and cytoplasm are comparatively slow and the IP₃R is less sensitive to Ca²⁺ activation. To expose the period effect of the negative feedback, we have chosen different parameter values than in the positive feedback model (see Table 1).

FIGURE 4

IP₃ turnover time controls feedbacks. (A) Positive-feedback model. Dynamics of [Ca²⁺]_c, [IP₃], and the fraction of open IP₃Rs (solid, dashed, and dot-dashed lines, respectively) during an oscillation period; the fraction of open IP₃Rs is given by (see Eq. 6). Fast IP₃ turnover yields a pronounced spike (left panel, ), whereas slow IP₃ turnover supports only a small-amplitude response (right panel, ). (B) The negative-feedback model shows the opposite behavior, with a small-amplitude response for fast IP₃ turnover (left panel, ) and a sharp spike for slow IP₃ turnover (left panel, ). (C) Positive-feedback model. Bifurcation diagram showing the maxima and minima of the [Ca²⁺]_c oscillations as a function of the stimulus for different values of the IP₃ turnover. The bifurcation diagrams for different values of are compared by plotting them against the product ; in this way, the steady-state concentrations of Ca²⁺ and IP₃ are identical for a given (solid and dashed lines indicate stable and unstable states, respectively; the stability of the steady state is shown for ). Both amplitude and range of stimuli leading to oscillations increase with faster IP₃ turnover. (D) The corresponding bifurcation diagrams for the negative-feedback model show the opposite behavior. The amplitude and range of stimuli leading to oscillations increase with slower IP₃ turnover. (E, F) Dependence of frequency encoding on IP₃ turnover in the positive and negative feedback models, respectively. Shown are the differences between the largest (for low stimulation) and smallest (for high stimulation) oscillation period.

FIGURE 5

Control coefficients for the oscillation period. (A, B) Positive and negative feedback models, respectively, in the absence of Ca²⁺ fluxes across the plasma membrane (); control coefficients of Ca²⁺ exchange across the ER membrane (C_er, solid line), IP₃ metabolism (C_p, dashed line), and IP₃R dynamics (C_r, dotted line) as function of the period of the oscillations. A positive period control coefficient signifies that a slowing of the corresponding process increases the oscillation period. (C, D) Period control coefficients in the positive and negative feedback models, respectively, in the presence of plasma-membrane fluxes of Ca²⁺ (). The dash-dotted line indicates the control exerted by Ca²⁺ exchange across the plasma membrane (C_pm).

FIGURE 6

Slowing of the IP₃ turnover with an IP₃ buffer. (A) The maximal rate of [Ca²⁺]_c rise during a [Ca²⁺]_c spike decreases as a function of IP₃ buffer concentration in the positive-feedback model (solid line), whereas it is barely affected in the negative-feedback model (dashed line). The results are shown for (positive feedback) and (negative feedback); similar results are obtained for other values. (B) The intracellular wave speed decreases as a function of IP₃ buffer concentration in the positive-feedback model (solid line), whereas it is barely affected in the negative-feedback model (dashed line). A solitary Ca²⁺ wave is initiated by a local increase in [IP₃]; [Ca²⁺]_c and [IP₃] diffuse with diffusion constants of 20 and 280 μm²/s, respectively; (positive feedback) and 0.217 nM/s (negative feedback). (C) High IP₃ buffer concentration abolishes oscillations in the positive-feedback model ( μM, ). (D) Oscillations persist in the presence of IP₃ buffer in the negative feedback model ( μM, ). In all panels the IP₃ buffer dissociation constant is μM. For the positive-feedback calculations: s, s, /s. For the negative-feedback model Other parameters are as listed in Table 1.

FIGURE 7

Complex responses to an IP₃ buffer in the positive-feedback model. (A) Bifurcation diagram showing the region of oscillations as function of stimulus () and IP₃ buffer concentration (gray-shaded area; the solid lines indicate the locus where the steady state becomes unstable via a Hopf bifurcation). In region I, regular oscillations have a decreased rate of [Ca²⁺]_c rise with increased [IP₃ buffer] as shown in Fig. 6 A. In region II, the IP₃ buffer abolishes the Ca²⁺ oscillations completely, as shown in Fig. 6 C. In region III, bursting [Ca²⁺]_c oscillations are observed (the lower boundary of this region is determined by a period doubling bifurcation, dotted line). We have indicated an additional region IV, which is characterized by oscillations persisting even at high [IP₃ buffer]. The parameters are as in Fig. 6, with When the strength of the Ca²⁺ plasma-membrane fluxes is increased (), regions III and IV disappear (inset). (B) Example of bursting oscillations observed in region III (top panel, control without IP₃ buffer; bottom panel, [IP₃ buffer] = 10 μM; ). (C) Example of oscillations in region IV (; [IP₃ buffer] = 100 μM), which are characterized by high frequency and low amplitude.

FIGURE 8

The effects of a molecular IP₃ buffer on ATP-evoked [Ca²⁺]_i oscillations in CHO cells. CHO cells (n = 5 independent cultures) were transiently transfected with pEGFP-LBD (EGFP-LBD) or pEGFP-C1 (EGFP). The cells were loaded with fura-2/AM 16–48 h posttransfection and challenged with the indicated ATP concentrations. The traces (A) show typical ATP-evoked [Ca²⁺]_c spikes in CHO cells transiently expressing EGFP or different levels of EGFP-LBD. The intracellular EGFP-LBD concentration was estimated as described in Materials and Methods. (B) Data show the effects of increasing EGFP-LBD expression on ATP-evoked Ca²⁺signals. CHO cells expressing EGFP-LBD were arbitrarily divided into low (489 ± 66 units; n = 18 cells) or high (3170 ± 480 units; n = 27 cells) categories using a cutoff of 1000 fluorescence intensity units. The estimated mean EGFP-LBD concentration was 6 ± 0.8 or 38 ± 6 μM, respectively. The mean EGFP fluorescence intensity was 7400 ± 615 units (89 ± 7 μM; n = 198 cells). Truncated spikes are defined as low amplitude [Ca²⁺]_c oscillations similar to those shown in the bottom traces of panel A. The initial rates of [Ca²⁺]_c rise (C) and the widths of the [Ca²⁺]_i spike (D) were calculated in cells expressing EGFP (n = 52 cells) or EGFP-LBD (n = 20 cells) where ATP challenge (0.5 or 1 μM) evoked at least three sequential baseline-separated Ca²⁺ spikes. The width of the [Ca²⁺]_c spike were determined at half-peak height. (E) The positive feedback model, Ca²⁺ activation of PLC, with different IP₃ buffer concentrations (as indicated) shows a good agreement with the EGFP-LBD experimental data shown in panel A. An increase in ATP is simulated by an increase in the maximal activity of PLC (arrowheads). (F) In the positive feedback model we also observe a significant increase in spike width (calculated at half-peak height). To match the Ca²⁺ oscillations in CHO cells all variables have been slowed by a factor 10, reference parameter set as in Fig. 6 In panel E, 0.2, 0.4 μM/s. Initial condition at In panel F,