Electrically triggered action potentials in the giant alga Chara corallina are associated with a transient rise in the concentration of free Ca2+ in the cytoplasm (Ca2+cyt). The present measurements of Ca2+cyt during membrane excitation show that stimulating pulses of low magnitude (subthreshold pulse) had no perceivable effect on Ca2+cyt. When the strength of a pulse exceeded a narrow threshold (suprathreshold pulse) it evoked the full extent of the Ca2+cyt elevation. This suggests an all-or-none mechanism for Ca2+ mobilization. A transient calcium rise could also be induced by one subthreshold pulse if it was after another subthreshold pulse of the same kind after a suitable interval, i.e., not closer than a few 100 ms and not longer than a few seconds. This dependency of Ca2+ mobilization on single and double pulses can be simulated by a model in which a second messenger is produced in a voltage-dependent manner. This second messenger liberates Ca2+ from internal stores in an all-or-none manner once a critical concentration (threshold) of the second messenger is exceeded in the cytoplasm. The positive effect of a single suprathreshold pulse and two optimally spaced subthreshold pulses on Ca2+ mobilization can be explained on the basis of relative velocity for second messenger production and decomposition as well as the availability of the precursor for the second messenger production. Assuming that inositol-1,4,5,-trisphosphate (IP3) is the second messenger in question, the present data provide the major rate constants for IP3 metabolism.

The membrane of the giant alga Chara corallina is electrically excitable. In the course of an action potential (AP), the concentration of free Ca2+ in the cytoplasm (Ca2+cyt) increases transiently from ∼100 nM to ∼1 μM (Williamson and Ashley 1982; Plieth et al. 1998). This transient rise in Ca2+cyt is considered central in the process of membrane excitation, because it is thought to activate Ca2+-sensitive Cl channels and, hence, initiate membrane depolarization (for review see Tazawa et al. 1987; Thiel et al. 1997).

The causal relationship between stimulation of APs by positive membrane voltage and elevation of Ca2+cyt is still unknown. The current view of this process is that membrane depolarization causes, by some unknown mechanism, a liberation of Ca2+ from internal stores (Plieth et al. 1998; Thiel and Dityatev 1998). The idea that Ca2+ is liberated from internal stores rather than entering through plasma membrane channels (Kikuyama and Tazawa 1998) is supported by experiments using Mn2+ as a quencher of fura-2 fluorescence. It was found that transient calcium rises were not associated with a quenching of the fura-2 fluorescence in the presence of extracellular Mn2+ (Plieth et al. 1998). Such a quenching would have been expected if the bulk change in Ca2+ was due to influx of Ca2+ via plasma membrane channels (Merrit et al. 1989). The consequent notion that liberation from internal stores is responsible for the transient calcium rises was further supported by experiments in which the cytoplasm of Chara cells was preloaded with Mn2+. With this preconditioning, APs were associated with a quenching of fura-2 even in the absence of any external Mn2+, suggesting that in this case quenching was due to liberation of Mn2+ together with Ca2+ from internal stores (Plieth et al. 1998).

Experiments with Chara cells have shown that inhibitors of PLC caused a delay and a suppression of the electrically stimulated elevation of the Cl conductance, i.e., the conductance that causes the depolarization in an AP (Biskup et al. 1999). Furthermore, elevation of the concentration of the second messenger inositol-1,4,5,-trisphosphate (IP3) in the cytoplasm of these cells was able to elicit APs (Thiel et al. 1990). Together these experiments lend support to the view that the mechanism linking electrical stimulation with mobilization of Ca2+ from internal stores includes IP3 as a second messenger.

In the present investigation, we examined the relationship between the electrical stimulation and the kinetics of Ca2+ mobilization in the course of an AP. We found that transient calcium rises were triggered by current pulses in an all-or-none fashion. Furthermore, we found that APs could be stimulated by a pair of two subthreshold pulses if the second pulse was neither too closely nor too far separated from the leading pulse. Together these data provide the basis for a kinetic model describing the voltage-dependent production of a second messenger and its transient elevation as a link between electrical stimulation and Ca2+ mobilization.

Plant Material and Ca2+cyt Measurements

Chara corallina Klein ex Wild was grown as reported previously (Thiel et al. 1993). The concentration of free Ca2+ in the cytoplasm (Ca2+cyt) was measured with a fluorescence ratio imaging method using the dual excitation dye fura-dextran as Ca2+ indicator (Grynkiewicz et al. 1985). Individual internodal cells of ∼40 mm in length were loaded with fluorescent dye via pressure injection using a custom-built injection device (Plieth and Hansen 1996). Cells loaded with dye were stored overnight in experimental solution (artificial pond water 0.5 mM KCl, 0.5 mM CaCl2, 1 mM NaCl, and 2 mM HEPES/NaOH, pH 7.5).

For Ca2+cyt measurements, the dye was excited with monochromatic light from a xenon lamp altering rapidly between 340 and 380 nm (T.I.L.L. Photonics). Emitted light from a square area of ∼10 μm2 to 25 μm2 was collected using a Zeiss objective (Fluar 40*/1.3 oil immersion) and transmitted through a 390-nm dichroic mirror before being detected by a photomultiplier (Seefelder Meßtechnik). A band pass filter (510 ± 30 nm; Schott) in the light path served to reduce auto fluorescence of chloroplasts. The EPC-9 unit with PULSE and X-chart software (Heka Elektronik) was used to control switching between excitation wavelength and recording of the photomultiplier output. Data were collected with a frequency of 5 or 10 Hz.

Ratiometric measurements were calibrated in vitro as described in Plieth and Hansen 1996 using standard Ca2+ solutions (calibration kit, C-3722; Molecular Probes). APs were triggered by current pulses of variable amplitude and length via extracellular electrodes. These were placed close to the area for Ca2+cyt recording to assure that Ca2+cyt changes were picked up from the site of excitation.

The Model

The model to describe a transient calcium rise in response to short single and double pulses is based on the variation of the concentration of a second messenger (here termed Q2) in the cytoplasm. We assume that Q2 is generated from a pool Q1 and degraded to Q3. We further assume that a threshold concentration of Q2 is required for mobilization of Ca2+ from internal stores. Beyond this threshold, Q2 causes a transient calcium rise that is largely independent of pulse duration and strength. For the model no other parameters (e.g., diffusion, cell geometry) than production of Q2 and decay were considered. All calculated values are relative changes with reference to the resting concentrations of Q2 and Q1 that were set to 0 and 1, respectively.

The model is governed by the two following differential equations:

\begin{equation*}\frac{{\partial} \left \left[{\mathrm{Q}}_{1}\right] \right _{{\mathrm{t}}}}{{\partial}t}=k_{{\mathrm{Q1}}} \left \left( \left \left[{\mathrm{Q}}_{1}\right] \right _{0}- \left \left[{\mathrm{Q}}_{1}\right] \right _{{\mathrm{t}}}\right) \right -k_{{\mathrm{Q2}}} \left \left[{\mathrm{Q}}_{1}\right] \right _{{\mathrm{t}}}{\mathrm{,}}\end{equation*}
1
\begin{equation*}\frac{{\partial} \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{t}}}}{{\partial}t}=k_{{\mathrm{Q2}}} \left \left[{\mathrm{Q}}_{1}\right] \right _{{\mathrm{t}}}-k_{{\mathrm{Q3}}} \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{t}}} \left \left({\mathrm{Wang\;et\;al.,1995}}\right) \right {\mathrm{.}}\end{equation*}
2

where kQ1 and kQ3 are time- and voltage-independent rate constants for pool Q1 refilling and decay of Q2. kQ2 is the voltage-dependent rate constant for production of Q2 and associated depletion of Q1, respectively. In pulse intervals and after the second pulse, kQ2 is zero and the differential equations are then governed by kQ3 and kQ1, respectively.

The term kQ1([Q1]0 − [Q1]) in is equivalent to the assumption that a controller for Q1 homeostasis is used. Any deviation from the set point [Q1]0 (as caused by its conversion to Q2; ) leads to restoring of the set point value. The detailed mechanism used for this task may be complicated. A minimum scheme for such a homeostatic controller was suggested by Hansen 1990. The reaction there was furnished for homeostasis in a compartment by mean of transmembrane transport, however, the same equations hold for homeostasis by chemical reactions. The important feature is the requirement of ATP.

To simulate the effect of small increments in pulse strength we assumed:

\begin{equation*}k_{{\mathrm{Q2}}}=\frac{c_{2} \left \left(I-I_{0}\right) \right }{q}\end{equation*}
3

for I > I0 and kQ2 = 0 for II0, where I is the current of the stimulating pulse. The minimum current I0 and the charge q were determined by fitting the measured strength-duration relationship by . c2 is a fitting parameter without a dimension. The introduction of a threshold such as I0 is unusual for chemical reactions, as this does not fulfil the law of mass action. However, mechanisms can be assumed that can result in such a rate constant. For example, the presence of Ca2+-sensitive PLC in plants (Kopka et al. 1998) as well as the strong dependency of membrane excitation on extracellular Ca2+ (Williamson and Ashley 1982; Thiel et al. 1993) may suggest the following scenario: Ca2+ enters the cells in a voltage-dependent manner. This results in a local rise of Ca2+ in the vicinity of the plasma membrane, which remains undetected by our method. Significant PLC activity with a quasi-linear dependency on voltage would then only be stimulated for sufficient positive voltage excursions. The present linear approach is only an approximation but justified by the linear stimulus-quantity law that we found for triggering transient calcium rises in Chara (see Fig. 3). For larger I increments, we expect an exponential function for kQ2(I).

[Q1] and [Q2] was then calculated by:

\begin{equation*}\begin{matrix} \left \left[{\mathrm{Q}}_{1}\right] \right _{{\mathrm{t}}}=a_{{\mathrm{i0}}}-a_{{\mathrm{i}}2}{\cdot}{\mathrm{e}}^{- \left \left(k_{{\mathrm{Q1}}}+k_{{\mathrm{Q2}}}\right) \right {\cdot} \left \left(t-t_{{\mathrm{i0}}}\right) \right }\\ {\mathrm{and}}\end{matrix}\end{equation*}
4
\begin{equation*} \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{t}}}=b_{{\mathrm{i0}}}-b_{{\mathrm{i1}}}{\cdot}{\mathrm{e}}^{-k_{{\mathrm{Q\;3}}}{\cdot} \left \left(t-t_{{\mathrm{i0}}}\right) \right }-b_{{\mathrm{i0}}}{\cdot}{\mathrm{e}}^{- \left \left(k_{{\mathrm{Q1}}}+k_{{\mathrm{Q2}}}\right) \right {\cdot} \left \left(t-t_{{\mathrm{i0}}}\right) \right }{\mathrm{,}}\end{equation*}
5

respectively for four different time intervals Δti = ti,endti0. ti0 and ti,end are the times for onset and end of each interval i (i = 1…4). During the first and the third interval, the cell is excited by a pulse, in the second and the forth interval the system is undisturbed. The constants aij and bij represents the boundary conditions for each interval. Their specific values are ai0 = kQ1/(kQ1 + kQ2)[Q1]0, ai1 = ai0 − [Q1]i0, wherein [Q1]i0 is the concentration of Q1 at the beginning of the i-th interval. Further are bi0 = kQ1kQ2/(kQ3(kQ1 + kQ2))[Q1]0], bi1 = bi0bi2 − [Q2]i0 and bi2 = kQ2ai2/(kQ3kQ1kQ2). With start of the first pulse at t10, [Q2]10 is zero and [Q1]10 is 1. In the pulse interval and after the second pulse, kQ2 and with it b0i and b2i are zero, and [Q2] is determined by the second right term of .

The time when [Q2] reaches its maximum value during the second pulse could be expressed in dependency of the time between both pulses Δt2 = t2,endt20 if pulse duration and pulse current were fixed. The time tmax2 is given by the null of the derivative of , which is resolved to t:

\begin{equation*}\begin{matrix}t_{{\mathrm{max\;2}}} \left \left({\mathrm{{\Delta}}}t_{2}\right) \right =t_{10}+{\mathrm{{\Delta}}}t_{1}+{\mathrm{{\Delta}}}t_{2}+\;\frac{1}{k_{{\mathrm{Q3}}}-k_{{\mathrm{Q1}}}-k_{{\mathrm{Q2}}}}{\cdot}\\ {\mathrm{ln}} \left \left[\frac{k_{{\mathrm{Q3}}}{\cdot} \left \left(b_{30}- \left \left[{\mathrm{Q}}_{2}\right] \right _{30}-b_{32} \left \left({\mathrm{{\Delta}}}t_{2}\right) \right \right) \right }{- \left \left(k_{{\mathrm{Q1}}}+k_{{\mathrm{Q2}}}\right) \right {\cdot}b_{32} \left \left({\mathrm{{\Delta}}}t_{2}\right) \right }\right] \right {\mathrm{,}}\end{matrix}\end{equation*}
6

where Δt1 = t1,endt10 is the duration of the stimulus. b30t2) and b32t2) are dependent of [Q1]30 and [Q2]30 at the beginning of the second pulse, which are functions of Δt2; their values are given by and , respectively. If the argument of the logarithm is equal to or smaller than zero, [Q2] does not reach a maximum even with infinite pulse duration. It rather approaches an asymptote. Then [Q2] reaches its maximum value at the end of the second pulse at tmax2 = t10 + Δt1 + Δt2 + Δt3.

[Q2]max2 is then:

\begin{equation*}\begin{matrix} \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max\;2}}} \left \left({\mathrm{{\Delta}}}t_{2}\right) \right =b_{30} \left \left({\mathrm{{\Delta}}}t_{2}\right) \right -b_{31} \left \left({\mathrm{{\Delta}}}t_{2}\right) \right {\cdot}{\mathrm{e}}^{-k_{{\mathrm{Q\;3}}}{\cdot} \left \left(t_{{\mathrm{max\;2}}}-t_{{\mathrm{i0}}}\right) \right }-\\ b_{32} \left \left({\mathrm{{\Delta}}}t_{2}\right) \right {\cdot}{\mathrm{e}}^{- \left \left(k_{{\mathrm{Q1}}}+k_{{\mathrm{Q2}}}\right) \right {\cdot} \left \left(t_{{\mathrm{max\;2}}}-t_{{\mathrm{i0}}}\right) \right }{\mathrm{.}}\end{matrix}\end{equation*}
7

With this equation, [Q2] can be plotted as a function of Δt2 for fixed pulse duration and voltages.

In the special case of a single pulse, as well as for the first pulse in a series ([Q2]s1 = 0), is simplified to:

\begin{equation*}t_{{\mathrm{max\;1}}}=t_{10}+\frac{1}{k_{{\mathrm{Q3}}}-k_{{\mathrm{Q1}}}-k_{{\mathrm{Q2}}}}{\mathrm{ln}} \left \left[\frac{k_{{\mathrm{Q3}}}-k_{{\mathrm{Q1}}}}{k_{{\mathrm{Q2}}}}\right] \right {\mathrm{.}}\end{equation*}
8

Its maximal value is t20. [Q2]max1 is then given by an analogous of :

\begin{equation*} \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max\;1}}}=b_{10}-b_{11}{\cdot}{\mathrm{e}}^{-k_{{\mathrm{Q\;3}}}{\cdot} \left \left(t_{{\mathrm{max\;1}}}-t_{{\mathrm{i0}}}\right) \right }-b_{12}{\cdot}{\mathrm{e}}^{- \left \left(k_{{\mathrm{Q1}}}+k_{{\mathrm{Q2}}}\right) \right {\cdot} \left \left(t_{{\mathrm{max\;1}}}-t_{{\mathrm{i0}}}\right) \right }\end{equation*}
9

Fig. 1 shows a recording of Ca2+cyt in a Chara internodal cell before and during electrical excitation. At rest, Ca2+cyt was typically ∼400 nM. Stimulation of the cell by a short current pulses (here 100 ms) triggered a transient calcium rise reaching within ∼3 s a maximum amplitude of 0.5–1 μM. Kinetics and amplitude of this electrically stimulated transient calcium rise is similar to those reported previously for excursions of Ca2+cyt during membrane excitation in Chara (Plieth et al. 1998).

To investigate the variability of electrically triggered transient calcium rises, one Chara cell was repetitively stimulated and Ca2+cyt recorded. Fig. 1 illustrates an overlay of transient calcium rises after eight successive stimulations with pulses of the same strength-duration. The plot shows that identical stimulations triggered within one cell transient calcium rises of very similar amplitude and kinetics. This result was confirmed in experiments with five other cells.

To quantify changes in Ca2+cyt during excitation, transient calcium rises were fitted by the sum of two exponentials with the form:

\begin{equation*}{\mathrm{Ca}}_{{\mathrm{cyt}}}^{2+}\;=A_{1}e^{ \left \left(-{\mathrm{{\lambda}}}_{1} \left \left(t-t_{0}\right) \right \right) \right }+A_{2}e^{ \left \left(-{\mathrm{{\lambda}}}_{2} \left \left(t-t_{0}\right) \right \right) \right }{\mathrm{,}}\end{equation*}
10

with a positive amplitude A1 and a negative amplitude A2, and time relaxation coefficients λ1 and λ2. The start of the fitted interval is given by t0.

As shown in the example in Fig. 2, fitting with two exponentials was adequate for an ad hoc description of transient calcium rises. Therefore, it was used throughout for quantitative description of a transient calcium rise. Notably, in the late phase of transient calcium rises (i.e., at which Ca2+cyt had already decayed back one third of the maximum), the fit often deviated from the data (not shown) indicating that a more complex model is required for description of the real events. However, this did not affect the present analysis.

Transient Calcium Rise Is an All-or-None Response

To determine the relationship between the strength of the stimulation and the evoked elevation of Ca2+cyt, we measured the amplitude of transient calcium rises as a function of stimulus strength. Fig. 2 shows three exemplary transient calcium rises after application of an electrical stimulus (100 ms) with a low (A), medium (B), and (C) high amplitude (Fig. 2 B). The data reveal that a small pulse (now termed subthreshold pulse) caused no detectable change in Ca2+cyt (further details see below). Increasing the pulse amplitude (here by a factor of 1.2) caused a typical transient calcium rise. Further increase of the pulse amplitude (here by a factor of 1.25) triggered a transient calcium rise of about the same magnitude as that after the medium sized pulse. To illustrate the relationship between pulse strength and transient calcium rise, the amplitudes of transient Ca2+cyt elevations were plotted versus the stimulus strength. Fig. 2 E shows that a narrow threshold exists for the stimulus strength. Below this threshold, Ca2+cyt remains unaffected (further details see below). After passing the threshold, the Ca2+cyt response already approaches its maximum. The same narrow threshold in pulse strength was found in all cells tested. This finding stresses that an all-or-none mechanism is the underlining Ca2+ mobilization.

Strength-duration Relationship

To investigate the relationship between pulse strength-duration and transient calcium rises, we monitored Ca2+cyt in response to pulses with different current amplitudes and/or duration. Fig. 3 illustrates a strength-duration curve for the effective stimulation of Ca2+cyt transients as a function of pulse duration and pulse strength. Shown are strength-duration values, which did (filled symbols) and which did not (open symbols) cause transient elevation of Ca2+cyt. The threshold for stimulation follows a hyperbolic function:

\begin{equation*}I=I_{0}+{q}/{{\mathrm{{\Delta}}}t}{\mathrm{,}}\end{equation*}
11

that is plotted in Fig. 3. Fitting the data yields a minimum current I0 of 2.5 μA and a minimal charge qmin of 115 nC.

Double Pulse Experiments

In the following experiments, we examined more closely the effect of subthreshold stimulation on Ca2+cyt. At the beginning of an experiment, the strength-duration of a stimulus was adjusted such that it was close to, but still lower than, the stimulation threshold. Fig. 4 shows representative recordings from an experiment in one Chara cell with the response of Ca2+cyt to a single or a series of subthreshold pulses. The data in Fig. 4 (A and B) show that Ca2+cyt is not affected appreciably by subthreshold pulses. To examine the possibility that such changes might be small and therefore unresolved, we analyzed the noise in Ca2+cyt recording in response to subthreshold pulses. Fig. 5 reports the variance of Ca2+cyt from the mean Ca2+cyt data obtained before, during and after subthreshold pulses. The absence of any appreciable change in variance in correlation with the pulses further shows that Ca2+cyt is not affected by subthreshold pulses.

Fig. 4 B shows that the effect of subthreshold pulses can be additive. In the present case, a subthreshold pulse was followed after 3 s by a second pulse of the same strength and duration. In this case, the second pulse triggered a transient calcium rise (Fig. 4 B), and this was similar in magnitude and shape to the transient calcium rise obtained by a large pulse in the same cell (Fig. 4 A). This shows that the stimulation encoded in any pulse is additive.

Fig. 4 C further shows that the additive effect of multiple subthreshold pulses is only effective for triggering a transient calcium rise if the interval between two pulses is not too long. In the case reported in Fig. 4 C, the two subthreshold pulses were separated by 4 s. In this case, Ca2+cyt remained entirely unaffected. Fig. 4 E summarizes the effect of dual pulses on transient calcium rises in one cell. Plotted are the amplitudes of Ca2+cyt changes as a function of the pulse interval. It is apparent that intervals must be shorter than ∼4 s to assure an additive effect of subthreshold pulses. The same pattern for stimulation of a transient calcium rise by subthreshold pulses was observed in five other cells tested. This result was independent on whether the experiment was started with a short or a long interval. This renders an endogenous decrease in excitability of the cell unlikely as explanation for the results.

To further test the hypothesis that two subthreshold pulses could be additive in their ability to stimulate Ca2+ mobilization, we compared the minimum charge (qmin) required for stimulation with single and double pulses. Therefore, one cell was stimulated (as in Fig. 4) with a double pulse protocol. However, in this case, strength and duration of the second pulse were varied, whereas the parameters of the first pulse as well as Δt2 were kept constant. For comparison, the same cell was also stimulated with single pulses of variable strength and duration. The plot in Fig. 6 illustrates the strength-duration relationship in one cell for the two different modes of stimulation, i.e., stimulation with a single pulse (closed symbols) and stimulation with a variable second pulse after a leading constant pulse (open symbols). Fitting of both data sets with yielded very similar values for I0. However, the qmin value from the double pulse stimulation was 1.19 times lower than that for single pulse stimulation. The same result was confirmed in three similar experiments showing that the qmin required for effective stimulation was on average 1.2 ± 0.06 times smaller, when the stimulating pulse was preceded by a subthreshold pulse. These data and the finding that the strength-duration plot for the second pulse shows the same hyperbolic relationship as that obtained for single pulse stimulation is best explained by the fact that individual pulses are indeed additive.

Fig. 7 shows another surprising observation with respect to a minimum interval between two effective subthreshold stimuli. In this case, pulses with low strength and long duration were chosen. As a single pulse, these were not able to stimulate a transient calcium rise (not shown). When two pulses of the same kind were applied in series with an interval of 300 ms, a transient calcium rise was stimulated. Subsequently, the interval between the two stimuli was shortened and the effect on Ca2+cyt was monitored. Fig. 7 (B and C) shows that also a reduction in the interval between two subthreshold pulses resulted in a loss of the additive effect of subthreshold pulses as trigger for Ca2+cyt mobilization. Fig. 7 D summarizes the effects of dual pulses on transient calcium rises tested in the same cell. The plot shows the amplitudes of Ca2+cyt changes as a function of the pulse interval. It is apparent that intervals must be longer than ∼200 ms to assure an additive effect of subthreshold pulses. The same pattern for stimulation of transient calcium rises by subthreshold was observed in four other cells tested. The result was independent on whether the experiment was started with a short or a long interval. This renders an endogenous decrease in excitability of the cell unlikely as explanation for the results.

In conclusion, the present data show that two subthreshold pulses have an additive effect on the stimulation of a transient calcium rise. Summation of the effect of single subthreshold pulses is only possible if the intervals are neither too long nor to short.

It has long been known that electrical excitation in Chara is associated with a transient calcium rise (Williamson and Ashley 1982; Kikuyama and Tazawa 1983). The present data now provide information on the mechanisms linking electrical stimulation and cytoplasmic Ca2+ mobilization.

One key finding is that the mechanism of Ca2+ elevation has a very steep dependency on the strength of the stimulation pulse. Over a very narrow range of pulse strength ΔCa2+cyt varies between zero and the maximal amplitude. This threshold-like dependency of Ca2+cyt on pulse strength fosters the view that the electrical stimulation causes elevation of Ca2+cyt by an all-or-none type mechanism.

Previously it had been suggested that the transient calcium rise during excitation is due to an influx of Ca2+ via voltage-sensitive Ca2+ channels in the plasma membrane (Kikuyama and Tazawa 1998). However, the steep dependency of Ca2+cyt on the stimulating pulse as well as the all-or-none type behavior of transient calcium rises is not in accordance with the operation mode of any known voltage-dependent channels (Hille 1992). This excludes Ca2+ influx via voltage- sensitive Ca2+ channels in the plasma membrane as the source of the bulk rise in calcium during excitation. The present data are better explained by a second messenger–operated release of Ca2+ from internal stores. This is in accordance with previous reports stressing that Ca2+ is mobilized from internal stores in the course of an AP (Beilby 1984; Thiel et al. 1993; Plieth et al. 1998; Thiel and Dityatev 1998).

The double pulse experiments show that the stimulation by a pulse is longer lived than the duration of the pulse itself. Furthermore the information encoded by individual subthreshold pulses is additive. The best explanation for these data is that membrane depolarization causes production of an intermediate second messenger with a lifetime in the order of seconds. This cannot be Ca2+ since we did not even detect minor changes in global Ca2+cyt upon subthreshold pulses. Furthermore the life time of Ca2+ in the cytoplasm is at least one order of magnitude shorter (Lipp and Niggli 1996) stressing that Ca2+ is not the intermittent messenger.

Previously it has been observed that elevation of IP3 is effective in triggering membrane excitation in Chara (Thiel et al. 1990). Also, inhibition of IP3 production by inhibitors of PLC was reported to suppress membrane excitation (Biskup et al. 1999). This fostered the hypothesis that membrane depolarization causes production of the second messenger IP3 and consequent mobilization of Ca2+ from internal stores. Thus, the best candidate for the second messenger Q2 linking electrical stimulation and Ca2+ mobilization is IP3. In this context, it can now be assumed that membrane depolarization causes, by a yet unknown mechanism, a rapid production of IP3 drawing from the PIP2 pool. The latter would be equivalent to the pool Q1 in our model. If the IP3 level remains below a threshold, no Ca2+ is mobilized from the stores. Above this critical value, IP3 causes complete mobilization from the internal stores. This view of IP3 action is consistent with the finding, that IP3 is indeed known to cause an all-or-none type calcium liberation from internal stores of animal cells (Parker and Ivorra 1990).

Upon elevation in the cytoplasm, IP3 is known to be subjected to degradation to the inactive IP2 (Berridge 1987), equivalent to pool Q3 in our model. The lifetime of IP3 was determined in animal cells and was found to be of the order of ∼1 s (Wang et al. 1995; Fink et al. 1999). From this long lifetime of IP3 it can be assumed that any further mobilization of IP3 during this decay time will add to the IP3 remaining from the first stimulation. By summation, the cytoplasmic concentration of IP3 could then exceed the threshold. In the present experiments, we found that double pulses were only effective if the pulse intervals were not longer than ∼3 s. This time is within the lifetime of IP3 and, thus, supports our notion that IP3 can act as the intermittent second messenger in question.

The view of IP3 production as second messenger is also consistent with the observation that two low amplitude subthreshold pulses must have a minimum interval to be effective as trigger. This experimental result can be explained by the fact that IP3 production during a subthreshold pulse draws on the pool of PIP2. If the refilling of the PIP2 pool from phosphatidylinositolphosphate is not too fast relative to the decay of IP3, a second pulse can meet the system in a situation in which the PIP2 pool is so far depleted that the second pulse is unable to generate enough IP3 to exceed the threshold required for Ca2+ mobilization.

Simulation

Here, we described the present experimental data in the context of a second messenger, Q2, linking electrical stimulation and Ca2+ mobilization. On the basis of the aforementioned evidence, we assume that IP3 is the second messenger in question. But in principle the model is valid for any other chemical second messenger with a metabolism similar to IP3. We assume that electrical stimulation causes a graded production of Q2 and that the concentration of Q2 needs to exceed a threshold for complete mobilization of Ca2+ from internal stores. The concentration of Q2 upon stimulation is given by the rate of production from Q1 and by the rate of decay to Q3 (see materials and methods). For estimation of the relative magnitude of the rate constants, it is important to note that transient calcium rises can be elicited by single pulses as short as 10 ms. The interval between two stimulating subthreshold pulses, on the other hand, can be in the range of seconds. This means that the rate of Q2 production is much larger than the rate of decay (kQ2kQ3). The existence of a minimum interval between subthreshold pulses can be explained with the fact that pool Q1 needs refilling to allow sufficient production of Q2 during the second pulse. Under the condition that kQ1is larger than kQ3, two pulses can be additive.

Fig. 8 (A–D) shows that this model is able to explain the body of the present data. A pair of subthreshold pulses is unable to stimulate a transient calcium rise if the interval is too long. The reason for the failure is that the concentration of Q1 has decreased so far that production of Q2 during the second pulse is not able to add sufficient new Q2 required for exceeding the threshold. A pair of subthreshold pulses is also not able to stimulate a transient calcium rise when the pulses are too close together (Fig. 8 C). In this case, stimulation fails because pool Q1 is so far empty that the second pulse is not able to produce sufficient new Q2 for exceeding the threshold. Only an intermediate spacing of the two pulses guarantees that the second pulse can produce enough Q2 to propel it over the threshold.

To examine the dependency of the concentration of Q2 on pulse intervals, we calculated with the appropriate rate constants the maximal concentration of Q2 achieved at the second pulse as a function of the pulse intervals. The plot in Fig. 9 shows that only pulse intervals between 0.3 and 3 s cause elevation of Q2 over the threshold and, thus, are able to trigger a transient calcium rise. Shorter and longer intervals are predicted to not stimulate a transient calcium rise. This features are in good agreement with the experimental data.

In double pulse experiments, we found that summation of stimulation by long single pulses with low strength (pl with strength il, duration Δt1,l) was within a single cell only possible if the pulse intervals were neither too long (interval Δt2,max) nor too short (interval Δt2,min) (Fig. 4 and Fig. 6, respectively). Moreover, we found for each cell also short single suprathreshold pulse (ph with ih, Δt1,h).

To quantify kQ1, kQ3, and the factor determining kQ2, c2 from such a set of experimental data (pl, ph, I0, q, Δt2,min, and Δt2,max), we derived some conditions that have to be complied with by and , respectively, for different pulse intervals assuming that gives a good description of the kinetics of Q2. The exclamation marks above the (equal) signs in the following equations shows the condition/character of the equations. They are not fulfilled a priori.

For a given subthreshold pulse pl, the maximal concentration of Q2 ([Q2]max2) induced by the minimal pulse interval Δt2,min has to be equal to [Q2]max2 evoked by the maximal stimulating pulse interval Δt2,max. This must be the case because shorter (for Δt2,min) and longer (for Δt2,max) pulse intervals evoked no transient calcium rise. Hence,

\begin{equation*} \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max2}}} \left \left({\mathrm{{\Delta}}}t_{2,{\mathrm{max}}},p_{1}\right) \right - \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max2}}} \left \left({\mathrm{{\Delta}}}t_{2,{\mathrm{min}}},p_{1}\right) \right =0{\mathrm{.!}}\end{equation*}
12a

The concentration [Q2]max2t2,min,pl) is then assumed as the threshold concentration [Q2]thres.

The function described by is steady for Δt2 ≥ 0. For Δt2 longer than Δt2,min but shorter than Δt2,max, [Q2]max2(Δt2) must consequently be higher than [Q2]thres because all Δt2 ε [Δt2,min, Δt2,max] induce a transient calcium rise.

\begin{equation*}\begin{matrix} \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max2}}} \left \left({\mathrm{{\Delta}}}t_{2},p_{1}\right) \right - \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{thres}}}>0{\mathrm{,!}}\\ {\mathrm{for\;{\Delta}}}t_{2}{\in} \left \left[{\mathrm{{\Delta}}}t_{2,{\mathrm{min}}},{\mathrm{{\Delta}}}t_{2,{\mathrm{max}}}\right] \right {\mathrm{.}}\end{matrix}\end{equation*}
12b

Moreover, for pulse intervals Δt2 < Δt2,min and for pulse intervals longer than Δt2,max, [Q2]max2 has to be lower than [Q2]thres. This is because pulse intervals with these lengths do not induce a transient calcium rise.

\begin{equation*}\begin{matrix}{\mathrm{Q}}_{2{\mathrm{max2}}} \left \left({\mathrm{{\Delta}}}t_{2},p_{1}\right) \right -{\mathrm{Q}}_{2{\mathrm{thres}}}<0{\mathrm{,!}}\\ {\mathrm{for\;{\Delta}}}t_{2}<{\mathrm{{\Delta}}}t_{2,{\mathrm{min}}}{\mathrm{or\;{\Delta}}}t_{2}>{\mathrm{{\Delta}}}t_{2,{\mathrm{max}}}{\mathrm{.}}\end{matrix}\end{equation*}
12c

Finally, [Q2]max induced by a single suprathreshold pulse ph has to reach at least [Q2]thres.

\begin{equation*} \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max1}}} \left \left(p_{{\mathrm{h}}}\right) \right - \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{thres}}}{\geq}0{\mathrm{.!}}\end{equation*}
12d

We use this system of (in)equations to construct an error function (see  ) in which only the three parameters kQ1, kQ3, and c2 are variables. By minimization of the error function through variation of these three parameters, we found for a given set of experimental data appropriate values for which a–d are fulfilled. The results are listed in Table.

Conclusion

The bulk of the experimental data on electrically stimulated elevation of Ca2+cyt can be simulated with a model, which is based on the voltage-dependent production of a second messenger. The distinct relationship between strength/duration of electrical stimuli and an all-or none mobilization of Ca2+ from internal stores can be explained in context of the velocity for second messenger production and decomposition as well as the availability of the precursor for the second messenger production. The data further allow approximation of the major rate constants kQ1, kQ2, and kQ3, which are relevant for production and decay of the second messenger. Assuming that IP3 is this second messenger in question (Thiel et al. 1990; Biskup et al. 1999), the present data provide some quantitative information on the metabolism of this second messenger.

We are grateful to U.-P. Hansen (University of Kiel), D. Gradmann (University of Göttingen), J. Dainty (Norwich) for comments on the manuscript.

We are grateful to the Deutsche Forschungsgemeinschaft for financial support.

Error Function

This section describes the algorithms that were used to obtain values for kQ1, kQ3, and [Q2]thres as well as the constant c2 (). The rate constant kQ2 was calculated from c2, q, and I. I0 was determined from , and the parameters q and I0 were obtained from a fit of the measured strength duration plots (3). I was set in the experiment. Under these circumstances, c2 is the only free parameter for determining kQ2 and, hence, is used as a fitting parameter.

As limiting conditions, the following experimentally determined values were used. (a) A single superthreshold pulse ph (with duration Δt1h and amplitude ih), which is only just sufficient to stimulate an AP, provides the value [Q2]max1(ph). This unknown value is considered as the threshold. (b) A pair of subthreshold pulses pl with identical duration Δt1l and Δt3l and intensity il, for which the dynamics of [Q2]max2t2) were calculated in relation to Δt2. (c) The rheobase I0 from the strength duration plot of an individual cell. (d) The integral q of the pulse strength over time of stimulation (Δt) used in the strength duration experiments. (e) The minimal distance Δt2min required for a second pulse to stimulate a transient rise in Ca2+. (f) The maximal distance Δt2max allowed between two subthreshold pulses without the second pulse losing its ability to stimulate a transient rise in Ca2+.

In principle and allow us to derive the temporal variation of the pool sizes [Q1](t) and [Q2](t) in the model. However, the problem is that the dynamics of the pool sizes cannot be determined, because the experiments only provide data in a situation, in which Ca2+ is released. Thus, the fitting algorithm has no reference to a continuous function. The only guides for the improvement of the fit are the above stated conditions.

To write , which should fulfil numerically conditions a–c, in such a way that it depends explicitly on Δt2, the coefficients b31 and b32 have to be calculated, because their values depend on [Q2]30 and [Q1]30 respectively. Both these values could be written (with and for i = 2) as functions of Δt2.

In the double pulse experiments, in which only the parameter Δt2 was varied, [Q2]max2t2) () could then be written (with the assumption tmax2 = t0 + Δt1 + Δt2 + Δt3) as a sum of two exponentials with constant coefficients. This function depends only on Δt2. With these constraints, the trajectory of the function is more or less determined because the sum of two exponents can have only one extreme. In the present case, this extreme must be, because of conditions a and d, between the pulse intervals Δt2min and Δt2max.

In this way the four conditions for fitting [Q2]max2t2,pl) and [Q2]max1(ph) can be reduced to the following condition:

\begin{equation*}\begin{matrix} \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max2}}} \left \left({\mathrm{{\Delta}}}t_{2,{\mathrm{max}}},p_{1}\right) \right = \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max2}}} \left \left({\mathrm{{\Delta}}}t_{2,{\mathrm{min}}},p_{1}\right) \right =\;{\mathrm{!!}}\\ \\ \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max1}}} \left \left(p_{{\mathrm{h}}}\right) \right = \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{thres.}}}{\mathrm{!}}\end{matrix}\end{equation*}

This equation has more than one solution because it can be solved for different values of [Q2]thres,kQ1, kQ3, and c2.

The possible solutions for kQ1, kQ3 and c2 depend strongly on the threshold [Q2]thres. To obtain a criterion for a unique solution, the following extra criterion was considered in the fitting. The relative deviation between the threshold [Q2]thres and the calculated concentration [Q2]max2(0) in response to an experimentally measured subthreshold double pulse with the distance 0,

\begin{equation*}\frac{ \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max2}}} \left \left(0\right) \right - \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{thres}}}}{ \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{thres}}}}{\mathrm{,}}\end{equation*}

should be maximal. The rational behind this is that a signal transduction system should produce a signal which is large enough to be recognized by the next downstream step in the cascade to avoid a false alarm.

On the background of these considerations, the error function that should be minimized here for obtaining the values in 1 can be written as :

\begin{equation*}\begin{matrix}{\mathrm{f}}_{{\mathrm{err}}} \left \left({\mathrm{k1,k3,c2}}\right) \right =\\ {\mathrm{{\Delta}t}}_{2{\mathrm{max}}} \left \left(\begin{matrix} \left \left( \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max2}}} \left \left({\mathrm{{\Delta}}}t_{2{\mathrm{max}}}\right) \right - \left \left( \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max2}}} \left \left({\mathrm{{\Delta}}}t_{2{\mathrm{min}}}\right) \right \right) \right \right) \right ^{2}\\ + \left \left( \left \left[{\mathrm{Q}}_{2}\right] \right \right _{{\mathrm{max2}}} \left \left({\mathrm{{\Delta}}}t_{2{\mathrm{max}}}\right) \right - \left \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max2}}} \left \left(p_{{\mathrm{h}}}\right) \right \right) \right ^{2}\end{matrix}\right) \right \\ +\; \left \left( \left \left[{\mathrm{Q}}_{2}\right] \right \right _{{\mathrm{max2}}} \left \left({\mathrm{{\Delta}}}t_{2{\mathrm{min}}}\right) \right - \left \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max1}}} \left \left(p_{{\mathrm{h}}}\right) \right \right) \right ^{2}\\ +{1}/{ \left \left( \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max2}}} \left \left({\mathrm{{\Delta}}}t_{2{\mathrm{min}}}\right) \right - \left \left[{\mathrm{Q}}_{2}\right] \right _{{\mathrm{max2}}} \left \left(0\right) \right \right) \right ^{2}}{\mathrm{.}}\end{matrix}\end{equation*}
13

For evaluation of the variable parameters in question, ferr was minimized by a downhill-simplex algorithm (Press et al. 1989).

Because the number of APs that can be induced in an experiment on a single cell is limited, it is in practice not possible to determine the exact values of Δt2,min and Δt2,max. To nonetheless approximate the kinetic parameters from our experiments, we used the following boundary values: for Δt2,min the longest subthreshold interval <Δt2,min and the shortest suprathreshold interval >Δt2,min and for Δt2,max the shortest subthreshold interval >Δt2,max and the longest suprathreshold interval <Δt2,max.

For calculation of the parameters in question, we assumed these boundary values to be Δt2,min and Δt2,max and fitted the parameters for each of the four possible combinations of the boundary-values. The averaged solutions of the four data sets were used as the approximation for the kinetic parameters.

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The present address for G. Thiel is Department of Botany, TU-Darmstadt, Schnittspahnstrasse 3, 64287 Darmstadt, Germany. Fax: 49-6151-164630; E-mail: thiel@bio.tu-darmstadt.de

Abbreviations used in this paper: AP, action potential; IP3, inositol-1,4,5,-trisphosphate.