Introduction

In natural conditions plants are exposed to various environmental factors and respond to their actions by the development of a number of functional changes. Among them action potential (AP) generation and propagation are considered to be the earliest responses. At present, many researchers focus their efforts on understanding the functional role of AP in vascular plants (Fromm and Lautner 2007; Pyatygin et al. 2008). For instance, the generation and propagation of APs are known to induce a number of changes in such physiological processes as photosynthesis, respiration, phloem transport, gene expression and others (Fromm and Lautner 2007). However, the way electrical signals are transformed into functional responses is still unclear, mainly due to a lack of understanding of the AP generation mechanism.

According to the current concept, the generation of AP is associated with passive fluxes of Ca2+, Cl− and K+ (Samejima and Sibaoka 1982; Opritov and Retivin 1982; Fromm and Spanswick 1993; Fromm and Bauer 1994; Felle and Zimmermann 2007). This hypothesis assumes the following sequence of events during the generation of AP in vascular plant cells (Davies 2006; Trebacz et al. 2006; Felle and Zimmermann 2007): depolarization to the threshold level of the membrane potential (E m) → activation of potential-dependent Ca2+ channels → influx of calcium ions into the cell → activation of potential-dependent Cl− channels → efflux of Cl− from the cell and development of the AP depolarization phase → activation of potential-dependent outward K+ channels → efflux of K+ and development of the repolarization phase. However, there are a number of facts which show participation of H+-ATPase in vascular plant AP generation: suppression of AP generation by inhibitors of metabolism and H+-ATPase (Hodick and Sievers 1988; Fromm and Spanswick 1993; Vodeneev et al. 2006), generation of AP under conditions with high concentration of external K+ and Cl− (Pyatygin et al. 1999a), a transitory change of extracellular pH during AP generation and a connection of this change with H+-ATPase regulation by calcium (Vodeneev et al. 2006), etc. There are several works (Gradmann 1976; Mummert and Gradmann 1991) which show that an electrogenic pump (Cl− pump) takes part in AP generation in the alga Acetabularia. On the basis of this idea and the experimental data on vascular plants, a modified hypothesis on the vascular plant AP mechanism has been proposed (Vodeneev et al. 2006). It assumes (Fig. 1) that the influx of calcium ions into the cell not only induces the activation of Cl− channels but reversibly inactivates H+-ATPase. As a result, this inactivation can take part in AP generation.

Fig. 1
figure 1

Proposed scheme of AP generation in excitable cells of vascular plants that takes into account passive ion fluxes and transient changes in the activity of electrogenic H+-ATPase (Vodeneev et al. 2006)

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The AP mathematical model, which takes into account Ca2+, Cl−, K+ and H+ transporters, can be used for a theoretical test of the hypothesis on the participation of H+-ATPase in AP generation. There are only few mathematical models of algal AP which include external stimuli. One of them takes into account passive fluxes and is based on the Hodgkin–Huxley model (Beilby 1982, 2007). It simulates AP in the alga Chara; however, possible participation of H+-ATPase and considerable changes of ion concentrations during AP generation in the vascular plants (Opritov and Retivin 1982; Fromm and Bauer 1994; Vodeneev et al. 2006; Felle and Zimmermann 2007) restrict the application of this model. Another model (Mummert and Gradmann 1991) takes into account the active flux (Cl− pump) as well as passive fluxes and simulates AP generation in Acetabularia acetabulum. This model more accurately simulates AP generation in vascular plants; however, application of the model requires a description of ion transport systems in the plasmalemma of vascular plant cells.

On the basis of the plasmalemma electrical process model by Gradmann et al. (Gradmann et al. 1993; Gradmann and Hoffstadt 1998; Gradmann 2001a, b), a preliminary model of AP generation taking into account active and passive ion transport systems has been worked out (Sukhov and Vodeneev 2005). The aim of this work was to improve the preliminary model and to check the hypothetical foundations (including the theoretical analysis of H+-ATPase participation in AP generation).

Theory

Figure 2 shows the scheme of the vascular plant cell electrophysiological model (Sukhov and Vodeneev 2005) which includes plasmalemma basic transport systems (H+, Ca2+, Cl− and K+). Five of these carriers (outward and inward K+ channels, Cl− channels, H+-ATPase, 2H+/Cl− symporter) as well as cytoplasmic and apoplastic buffers have been taken into account on the basis of the model by Gradmann et al. (Gradmann et al. 1993; Gradmann and Hoffstadt 1998; Gradmann 2001a, b). The H+/K+ antiporter (Rodriguez-Navarro 2000) has been included in the model since it maintains a stationary concentration of K+ at rest, when E m is more negative than the potassium equilibrium potential (Opritov et al. 1991). Other additional elements in our model are systems of Ca2+ transport: Ca2+ channels (Zimmermann and Sentenac 1999) and Ca2+-ATPase (Bush 1993) since the influx of calcium triggers the initiation of AP in vascular plant cells (Davies 2006; Trebacz et al. 2006; Vodeneev et al. 2006; Felle and Zimmermann 2007), inducing the activation of Cl− channels and, possibly, the inactivation of H+-ATPase.

Fig. 2
figure 2

Scheme of a plant cell electrophysiological model. E m is the electrical potential of the plasma membrane; [K+], [Cl−] and [Ca2+] are the concentrations of potassium, chlorine and calcium ions, respectively (M); B −in and BHin are free and H+-bound proton-buffer molecules in the cell; B −out , BHout and BKout are free, H+- and K+-bound buffer molecules in the apoplast. Above is the schema of Ca2+ channel states; below is the schema of Cl−- and K+ channel states; on the right is the schema of the “two-state” model of primary active transport (see text)

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The changes of the membrane potential are described by the Hodgkin–Huxley equation, which has been modified for ion fluxes in plants (Gradmann et al. 1993; Gradmann and Hoffstadt 1998; Gradmann 2001a, b; Sukhov and Vodeneev 2005):

$$ {\frac{{dE_{m} }}{dt}} = {\frac{1}{C}} \cdot Fj_{r} \,r \in \left( {{\text{K}}^{ + } ,{\text{Cl}}^{ - } ,{\text{H}}^{ + } ,{\text{Ca}}^{2 + } } \right) $$
(1)

where E m is the membrane potential, j r is the transmembrane ion flux r, C is the specific capacitance and F is the Faraday constant.

Passive ion transport is described on the basis of the Goldman–Hodgkin–Katz equation (Gradmann and Hoffstadt 1998):

$$ j = p_{o}^{r} P_{ \max }^{r} zu{\frac{{[r]_{in} - [r]_{\text{out}} \exp \left( { - zu} \right)}}{{1 - \exp \left( { - zu} \right)}}} $$
(2)

where \( p_{o}^{r} \) is the probability of the r-ion channel open state; \( P_{ \max }^{r} \) is the r-ion maximum permeability; [r]in and [r]out are free r concentrations in the cytoplasm and apoplast, respectively; u = FE m/(RT) is the normalized membrane potential; z is the ion charge; and R and T have their standard thermodynamic values.

There are two states of K+ and Cl− channels in our model: the closed state and the open state (Barbier-Brygoo et al. 2000; Gradmann 2001a, b). Ca2+ channels have three states: closed, open and inactivated (Huang et al. 1994). The dynamics of open state probabilities (\( p_{o}^{r} \)) are described (Gradmann and Hoffstadt 1998) by Eq. 3 for K+ and Cl− channels and by Eq. 4 for Ca2+ channels:

$$ {\frac{{dp_{o}^{r} }}{dt}} = k_{ + o}^{r} \left( {1 - p_{o}^{r} } \right) - k_{ - o}^{r} p_{o}^{r} $$
(3)
$$ {\frac{{dp_{o}^{r} }}{dt}} = k_{ + o}^{r} \left( {1 - p_{o}^{r} - p_{i}^{r} } \right) - k_{ - o}^{r} p_{o}^{r} + k_{ - i}^{r} p_{i}^{r} - k_{ + i}^{r} p_{o}^{r} $$
(4)
$$ {\frac{{dp_{i}^{o} }}{dt}} = k_{ + i}^{r} p_{o}^{r} - k_{ - i}^{r} p_{i}^{r} $$

where \( p_{i}^{r} \) is the probability of the r-ion channel inactivated state; \( k_{ + o( + i)}^{r} = k_{o(i)}^{r} \exp \left( {{{uc_{o(i)}^{r} } \mathord{\left/ {\vphantom {{uc_{o(i)}^{r} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right) \) and \( k_{ - o( - i)}^{r} = k_{o(i)}^{r} \exp \left( {c_{o(i)}^{r} u_{o(i)}^{r} - u{{c_{o(i)}^{r} } \mathord{\left/ {\vphantom {{c_{o(i)}^{r} } 2}} \right. \kern-\nulldelimiterspace} 2}} \right) \) are the velocity constants of transitions from the closed state to the open one and vice versa (subscript o) or from the open state to the inactivated one and vice versa (subscript i); \( c_{o}^{r} \) and \( c_{i}^{r} \) are the constants which represent a portion of the membrane potential acting on the gating mechanisms and their charge; \( u_{o}^{r} \) and \( u_{i}^{r} \) are the normalized potential barriers for the transition of the channel from the closed state to the open one and from the open state to the inactivated one, respectively; \( k_{o}^{r} \) and \( k_{i}^{r} \) are the constants representing transition velocities between the closed and open states and between the open and inactivated states at u, \( u_{o}^{r} \) and \( u_{i}^{r} \) equal to 0.

Equations 3 and 4 have numerically been solved for outward K+ channels and Ca2+ channels. Open probabilities of Cl− channels and inward K+ channels are described by equilibrium distributions since activation and deactivation velocities of these channels are faster than those of membrane potential changes during AP in vascular plants (Blatt 1992; Thomine et al. 1995). The model takes into account the dependence of Cl− channel activity on the Ca2+ concentration in the cytoplasm (Lewis et al. 1997; Berestovsky and Kataev 2005). The Ca2+ influence on Cl− channel activity has been described by addition of coefficient A into Eq. 2:

$$ A = {\frac{{[{\text{Ca}}^{2 + } ]_{\text{in}}^{2} }}{{K^{2} + [{\text{Ca}}^{2 + } ]_{\text{in}}^{2} }}} $$
(5)

where A is the portion of Cl− channels activated by Ca2+, [Ca2+]in is the Ca2+ concentration in the cytoplasm, K is the constant equal to [Ca2+]in at A = 0.5 and the power 2 reflects activation of the Cl− channel by two Ca2+ (Berestovsky and Kataev 2005). The open probability of Cl− channels has been assumed to be very small (\( P_{\text{Ca}}^{\text{o}} \approx 0 \)) for low Ca2+ concentrations (<10−7 M) (Barbier-Brygoo et al. 2000).

Primary active transport of ions (H+-ATPase and Ca2+-ATPase) is described by the “two-state model” (Hansen et al. 1981; Beilby and Shepherd 2001; Sukhov and Vodeneev 2005), which takes into account two states of pump: with an ion and without it (Fig. 2). Unlike some other models (the ohmic description, the description on the basis of the Goldman–Hodgkin–Katz equation [Gradmann et al. 1993; Gradmann and Hoffstadt 1998; Gradmann 2001a, b]), the model not only simulates the dependence of the transport on the ion concentration and E m but also describes the enzyme saturation by high concentrations of H+ and Ca2+ as well as by the membrane potential. This can be important for the description of AP in vascular plants since substantial changes of ion concentrations during AP have been observed (Opritov and Retivin 1982; Fromm and Bauer 1994; Felle and Zimmermann 2007).

Equation 6 describes stationary active fluxes of H+ and Ca2+:

$$ j_{P} = E_{o} {\frac{{k_{ + 1} k_{ + 2} - k_{ - 1} k_{ - 2} }}{{k_{ + 1} + k_{ - 1} + k_{ + 2} + k_{ - 2} }}} $$
(6)

where E 0 is the total enzyme concentration and k +1, k −1, k +2 and k −2 are velocity constants of forward (+) and reverse (−) transitions between two states of the pump (indices 1 and 2 are used for potential independent and dependent constants, respectively). The Goldman–Hodgkin–Katz equation is used to describe the dependence of constants k +2 and k −2 on the membrane potential.

Constants (k) for H+-ATPase have been calculated using Eq. 7:

$$ k_{ + 1} = k_{1} [{\text{H}}^{ + } ]_{\text{in}}, \,k_{ - 1} = k_{1} \exp \left( { - {{\Updelta G_{\text{ATP}} } \mathord{\left/ {\vphantom {{\Updelta G_{\text{ATP}} } {RT}}} \right. \kern-\nulldelimiterspace} {RT}}} \right),\,k_{ + 2} = k_{2} u{\frac{1}{{1 - \exp \left( { - u} \right)}}}, \, k_{ - 2} = k_{2} [H^{ + } ]_{\text{out}} u{\frac{{\exp \left( { - u} \right)}}{{1 - \exp \left( { - u} \right)}}} $$
(7)

where k 1 and k 2 are the velocity constants at u = 0 and [H+]in = [H+]out = 1 M, [H+]in and [H+]out are the H+ concentrations in the cytoplasm and apoplast, respectively; ΔG ATP is the energy of ATP hydrolysis.

The dependence of pump activity on the Ca2+ concentration in the cytoplasm (Kinoshita et al. 1995; Vodeneev et al. 2006) has been taken into account in the description of H+-ATPase using Eq. 8:

$$ A = {\frac{{K^{n} }}{{K^{n} + [{\text{Ca}}^{2 + } ]_{in}^{n} }}} $$
(8)

where A is the portion of H+-ATPase inactivated by Ca2+, K is the constant equal to [Ca2+]in at A = 0.5 and n is the number of Ca2+ needed to inactivate one molecule of the enzyme. It is assumed that n = 2.

Ca2+-ATPase is described as a system which exchanges Ca2+ and H+ (Bush 1993). Constants (k) are given by Eq. 9:

$$ k_{ + 1} = k_{1} [{\text{Ca}}^{2 + } ]_{\text{in}} [{\text{H}}^{ + } ]_{\text{out}}, \,k_{ - 1} = k_{1} \exp ( - {{\Updelta G_{\text{ATP}} } \mathord{\left/ {\vphantom {{\Updelta G_{\text{ATP}} } {RT}}} \right. \kern-\nulldelimiterspace} {RT}}),\,k_{ + 2} = k_{2} u{\frac{1}{{1 - \exp \left( { - u} \right)}}},\,k_{ - 2} = k_{2} [{\text{Ca}}^{2 + } ]_{\text{out}} [{\text{H}}^{ + } ]_{\text{in}} u{\frac{{\exp \left( { - u} \right)}}{{1 - \exp \left( { - u} \right)}}} $$
(9)

where [Ca2+]in and [Ca2+]out are the Ca2+ concentrations in the cytoplasm and apoplast, respectively.

Ion fluxes associated with secondary active transport are described as a difference between fluxes directed to the apoplast (j in) and to the cytoplasm (j out): \( j_{P} = j_{\text{in}} - j_{\text{out}} \), flux velocities being proportional to ion concentrations (Eqs. 10 and 11). The electrogenic active flux of Cl− also depends on the membrane potential (Eq. 10) (Gradmann 2001a).

$$ j_{P}^{\text{Cl}} =\; V^{\text{Cl}} \left( {[{\text{Cl}}^{ - } ]_{\text{in}} [{\text{H}}^{ + } ]_{\text{in}}^{2} u{\frac{1}{{1 - \exp \left( { - u} \right)}}} - [{\text{Cl}}^{ - } ]_{\text{out}} [{\text{H}}^{ + } ]_{\text{out}}^{2} u{\frac{{\exp \left( { - u} \right)}}{{1 - \exp \left( { - u} \right)}}}} \right) $$
(10)
$$ j_{P}^{\text{K}} = V^{\text{K}} \left( {[{\text{K}}^{ + } ]_{\text{in}} [{\text{H}}^{ + } ]_{\text{out}} - [{\text{K}}^{ + } ]_{\text{out}} [{\text{H}}^{ + } ]_{\text{in}} } \right) $$
(11)

where \( j_{P}^{\text{Cl}} \)and \( j_{P}^{\text{K}} \) are the fluxes of Cl− and K+ and V Cl and V K are the total velocity constants of 2H+/Cl− symporter and H+/K+ antiporter, respectively. For ion r, V r = k r E r, where E r is the concentration of the enzyme which transports ion r, k r is the velocity constant at u = 0 and [r]in = [r]out = 1 M.

This description does not take into account different states of transporters, so it cannot describe a number of transport system properties, such as ion transport saturation; however, it is simpler than that of the two-state model and requires only one parameter.

The temperature influence on primary and secondary systems of active transport is described by temperature coefficient Q10, which has been introduced into Eqs. 6, 10 and 11 (Sukhov and Vodeneev 2005). It has been assumed that Q10 = 3, which is typical for enzymatic processes. The dependence of passive transport systems on temperature is weak; therefore, the temperature coefficient has not been used for their description.

Ion concentration changes and the buffer capacity of cytoplasm and apoplast. Changes of ion concentrations in the cytoplasm and apoplast are described on the basis of Eq. 2 by adding coefficients S cell /V cell (a ratio between the cell surface and the cell volume) and V cell /V ap (a ratio between the apoplast and cell volume) (Sukhov and Vodeneev 2005).

Gradmann’s model (2001a, b) (without the apoplast Ca2+ buffer since the calcium extracellular concentration has been assumed to be constant) has been used to simulate the buffer capacity of the cytoplasm and apoplast (the scheme of buffer states is shown in Fig. 2). The values of concentrations of H+ and K+ buffers ([B]in and [B]out for the cytoplasm and apoplast, respectively) and dissociation constants (\( K_{\text{H}}^{\text{in}} , \, K_{\text{H}}^{\text{in}} ,K_{\text{K}}^{\text{out}} \) for H+ in the cytoplasm and apoplast and for K+ in the apoplast) have been taken from Gradmann (2001a).

The model parameters have been taken from the literature or assumed (Table 1; Fig. 2). In the latter case, we have used the parameter values which are typical for the transport processes in the plant cell and describe the basic properties of these processes. A quantitative agreement between model and experimental results was the second criterion for the correctness of the accepted values.

Table 1 Values of parameters and initiation concentrations of ions in the mathematical model of AP
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The total system of equations (on the basis of Eqs. 1–11) has been solved by Euler’s method. Δt has been taken to equal 5 ms since Δt < 10 ms does not influence the simulated results. An initial value of the membrane potential has been accepted as −180 mV (Opritov et al. 1991); initial distributions of closed, open and inactivated states have been calculated as the equilibrium ones for this value of the membrane potential; and the ion initial concentrations have been taken from the literature (Table 1).

Results and Discussion

The numerical solution of the model equations has shown that E m and ion concentrations in the cell and the apoplast are stationary without stimulation and correspond to the real values in vascular plants (values shown in Fig. 2).

Simulated changes of E m, induced by electrical stimulation (temporary increase of the electrical potential on the plasma membrane to various constant values), are shown in Fig. 3. The imitation of the electrical stimulation more than a threshold potential induces AP generation, the amplitude of which does not depend on the stimulation magnitude (Fig. 3). This result is in good agreement with the “all-or-none” law, which is an important characteristics of AP (Davies 2006; Trebacz et al. 2006). It is necessary to note that the magnitude of the threshold electrical stimulus depends on its duration: the long-term stimulus magnitude has been shown to be less than that of the short-term one. This result fits well with the classic relationship strength–duration, which is typical for excitable structures including vascular plant cells (Zawadzki et al. 1991; Fromm and Spanswick 1993).

Fig. 3
figure 3

APs induced by electrical stimulation with Ca2+ in the apoplast (5 × 10−4 m) (solid line) and without it (dashed line). Electrical stimulation was simulated by an increase of E m to the constant level (the value of E m changes is shown under the curve; the duration of stimuli was 4 s)

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At the next stage of the research, we investigated the influence of gradual cooling on E m. It should be noted that gradual cooling is one of the effective methods of AP induction, which allows one to distinguish the subthreshold changes of E m and AP generation (Pyatygin et al. 1992). Application of this method to simulate AP generation makes it possible to compare theoretical results with our experimental data (Opritov et al. 1991, 2002; Pyatygin et al. 1999a, b; Vodeneev et al. 2006).

Gradual cooling is considered a linear decrease of temperature (4°C min−1) from an initial value (25°C). The imitation of cooling induces typical changes of E m: a weak depolarization up to threshold level of E m, followed by AP generation (Fig. 4a). Simulated changes of the membrane potential are very similar to those obtained in the gradual cooling experiments (Fig. 4c).

Fig. 4
figure 4

APs induced by gradual cooling. a Simulated AP; b simulated E m change without Ca2+ in the apoplast; c experimental AP (Pyatygin et al. 1999b). Solid line, E m changes; dashed line, changes of temperature

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As seen from Figs. 3 and 4a, the simulated AP has two stages of repolarization, which is typical for vascular plants (Samejima and Sibaoka 1982; Hodick and Sievers 1988; Fromm and Spanswick 1993; Opritov et al. 2002).

Unlike other descriptions of electrical reactions in plants (Beilby 1982; Mummert and Gradmann 1991; Gradmann et al. 1993), our model takes into account the participation of Ca2+ in AP generation noted in a number of works (Davies 2006; Trebacz et al. 2006). Figures 3 and 4b show that the absence of extracellular Ca2+ suppresses AP generation by the model, which is in good agreement with the experimental data (Iijima and Sibaoka 1985; Hodick and Sievers 1988; Vodeneev et al. 2006; Felle and Zimmermann 2007) and supports the correctness of our model.

Thus, our results show that the elaborated model simulates the basic properties of AP generation in vascular plants; therefore, it can be used for subsequent analyses. Simulated kinetics of electrogenic ion fluxes before and during AP generation are shown in Fig. 5. K+ and H+ fluxes are basic at rest; Cl− and Ca2+ fluxes increase at excitation. However, the increase of the Ca2+ flux is of short duration, and its magnitude remains two orders lower than that of K+, H+ or Cl−. This correlates well with an idea on about a regulatory role of Ca2+ during AP (Davies 2006; Trebacz et al. 2006; Felle and Zimmermann 2007). Changes of K+ and Cl− fluxes are of long duration and develop simultaneously. The latter possibly provides a significant efflux of these ions during AP: an increase of the Cl− concentration in the apoplast is 2 mM and that of K+ is 1.5 mM, which are close to the experimental data (Felle and Zimmermann 2007).

Fig. 5
figure 5

Simulated kinetics of ion fluxes at AP generation. Gray line schematically shows the changes of E m (without coordinate axis). +, influx; −, efflux

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H+ flux changes during AP generation are also shown in Fig. 5. The H+ flux magnitude decreases at the AP depolarization phase. The development of the repolarization phase is connected with H+ and K+ fluxes at the first stage of repolarization (outward currents of K+ and H+) and only with the H+ flux at the second stage, when E m is more negative than the equilibrium potential for K+ (the inward current of K+ and the outward current of H+). This result is in compliance with the hypothesis concerning H+-ATPase participation in AP generation.

The next stage of our investigation shows that a decrease of electrogenic pump activity induces depolarization of E m and reduction of the simulated AP amplitude (Fig. 6a). AP does not develop if the portion of inactivated enzyme molecules is >50% (data not shown). This result is compatible with the influence of the H+-ATPase inhibitor DCCD on the electrogenesis of plant cells shown in Fig. 6b (Vodeneev et al. 2006). From that it follows that AP development requires some level of H+-ATPase activity.

Fig. 6
figure 6

Influence of H+-ATPase inactivation on the generation of cooling-induced APs. a Theoretical AP simulated by the model when H+-ATPase activity is equal to 100, 65 and 50% (inactivation magnitude shown in the figure). b Experimental AP (Vodeneev et al. 2006) measured when H+-ATPase activity is decreased by the inhibitor DCCD (concentrations shown in the figure)

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To check the role of Ca2+-dependent inactivation of H+-ATPase in AP generation, we studied the changes of pH in the apoplast since they are very important to support the participation of H+-ATPase in AP development (Vodeneev et al. 2006). Extracellular pH–simulated changes shown in Fig. 7a are in good qualitative agreement with the experimental curve (Fig. 7c). The quantitative differences are possibly related to the measurement of the surface potential and the registration of pH changes in the bathing solution (Vodeneev et al. 2006) since these methods cause a decrease of response magnitude. A connection of these changes with Ca2+-dependent inactivation of H+-ATPase has been tested by a decrease of the enzyme activity dependence on calcium ions. The reduction of the Ca2+ influence on H+-ATPase is described as a decrease of the enzyme affinity for calcium ions (K for H+-ATPase is changed from 4 × 10−7 M to 10 × 10−7 M). The apoplast alkalization is suppressed in this case (Fig. 7b). This supports the necessity to describe the Ca2+-dependent inactivation of H+-ATPase for the simulation of ion concentration changes (in particular, H+ concentration) during AP. However, the development of the electrical response is not suppressed by the reduction of the Ca2+ influence on H+-ATPase. This result agrees with the experimental data, which show that a disruption of the Ca2+ influence on H+-ATPase suppresses the changes of H+ concentration but weakly modifies the electrical reaction (Vodeneev et al. 2006). It demonstrates that the Ca2+-dependent inactivation of the electrogenic pump is not strictly required for AP generation. The reduction of the Ca2+-dependent inactivation of H+-ATPase can possibly stimulate activation of Cl− channels, but this idea needs checking.

Fig. 7
figure 7

Changes in extracellular pH at cooling-induced AP generation. a, b Simulated changes in E m and apoplastic pH at the high (a, K = 4 × 10−7 M) and low (b, K = 10 × 10−7 m) affinity of H+-ATPase for Ca2+. c Experimental changes in the surface potential and pH in the extracellular solution (Vodeneev et al. 2006)

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Thus, the elaborated model describes well the AP generation induced by external stimuli (electrical stimulation and gradual cooling) in vascular plants. The model analysis has shown the activity of H+-ATPase to be a necessary condition for AP generation. Ca2+ influx induces the changes of this activity (inactivation and reactivation); however, these changes are not strictly required for AP generation. The compliance of the simulated and experimental data confirms the correctness of the hypothesis on H+-ATPase participation in AP generation. The model, therefore, can be used for further investigation of AP mechanisms.

In conclusion, some directions of the model development should be noted. (1) Additional cell compartments could be taken into account: the vacuole, which is the main store of ions in mature plant cells and takes part in AP generation in Chara alga (Shimmen et al. 1994; Beilby 2007); chloroplasts, which are also associated with AP generation in different plants (Bulychev and Turovetsky 1983; Shimmen et al. 1994; Bulychev et al. 2004; Fromm and Lautner 2007), etc. These modifications of the model could be a tool for theoretical analysis of a functional role of AP; however, elaboration of these models is restricted by the lack of experimental data on a relation between AP generation and the processes in the organelles of vascular plant cells. (2) Ensembles of a number of interrelated individual AP models can be elaborated. This system can be used to analyze AP propagation.