1. Introduction

Hodgkin and Huxley ^[1] won the Nobel Prize for their description of action potential (AP) in animal kingdom. In the nerve the stimulus depolarization of the membrane potential difference (PD) directly activates Na⁺ selective channels, Na⁺ inflow further depolarizes the membrane and spontaneous inactivation of the same channels and the outflow of K⁺ returns the cell to status quo in several ms. Beilby ^[2] and Beilby and Coster ^[3,4] fitted the Hodgkin Huxley (HH) equations to AP in giant cells of salt sensitive Characeae Chara australis, with the outflow of Cl^- replacing the inflow of Na⁺ as the depolarizing ionic flux. In this“green axon”the AP transient takes about 2 s in normal artificial pond water (APW) and at room temperature. Later research found that the Cl^- channels are activated by rise of Ca²⁺ in the cytoplasm ^[5,6], which in turn may be mediated by inositol 1, 4, 5-triphosphate (IP₃)-activated Ca²⁺ influx from internal stores ^[7]. Biskup et al. ^[8] investigated the effects of inhibitors of IP₃ formation on Cl^- excitation currents in voltage-clamped Chara cells. Wacke and Thiel ^[9] and Wacke et al. ^[10] measured Ca²⁺ rise in the cytoplasm following a PD stimulus and formulated a new Chara AP model, where the threshold, “all-or-none”AP form and refractory period all originate from formation of second messenger (IP₃), opening of the Ca²⁺ channels on internal stores and re-sequestration of the Ca²⁺ by calcium pumps. Interestingly, parts of this model are also“borrowed”from animal cells ^[11]. Tazawa and Kikuyama ^[12] were not able to replicate the Thiel experiments: IP₃ did not stimulate excitation, nor did Neomycin and U73122 inhibit it. Future experiments, suggested by the present modelling, will provide more details of this signalling pathway.

The present work was initially inspired by observation of prolonged APs in salt-stressed Chara. The modeling departs from the Thiel model of transient Ca²⁺ concentration rise and activation of Cl^- channels to calculate the form of the action potential and compare it to the experimental data under normal conditions and at the time of saline stress. The restoration of the resting PD depends on the drop in cytoplasmic Ca²⁺ concentration and inactivation of the Cl^- current, the outward rectifier current (activated by depolarization), the background current and the proton pump current (Figure 1). The PD dependence of the last three transporters was presumed unchanged by the brief transient Ca²⁺ concentration rise. The effect of salinity stress on the shape of the AP is explored in the terms of the combined Thiel-Beilby model.

---

2. Materials and Methods

---

2.1. Materials

The experimental AP data were obtained from Chara australis. The“standard”AP, designated“AP_av”, in APW (NaCl 1 mM, KCl 0.1 mM, CaCl₂ 0.1 mM, neutral pH) and at room temperature was adopted from previous AP modeling (see ^[2] and Results section). To elicit excitation under such conditions a short depolarizing current pulse was applied to the cell. The AP, which displayed slight separation from the excitation pulse, was selected and the pulse was excluded from the data to obtain what was then considered the“pure”AP form.

C. australis is salt sensitive and cells die in 5 days in comparatively mild salinity of 50-100 mM NaCl added to APW ^[13]. Electrophysiology, particularly current-voltage (I/V) scans, add to the cell stress and we often observed loss of cell PD and streaming after a few hours in Saline APW ^[13,14,15]. The survival of the saline-stressed cells was prolonged by applying the osmotic stress separately and exposing the cell to Sorbitol APW for 1 hr, with sorbitol concentration adjusted to the same osmolarity as the Saline APW with 50 mM NaCl and 0.1 mM Ca²⁺. The spontaneous APs, designated“AP_sal”, with prolonged durations were only observed in the Saline APW. Repetitive periodic APs were recorded when the membrane PD depolarized to the excitation threshold of-100 mV. In the investigations of the salinity effect on the cell electrophysiology the membrane PD was continuously data-logged at 0.1 s intervals throughout each experiment and stored as text files. In the present modeling studies the APs were selected from the whole PD record and entered into Mathematica 10 programs.

---

2.2. The Thiel Model

Zherelova ^[16] and Thiel et al. ^[5] elicited APs in Chara and Nitella by artificial elevation of IP₃ in the cytoplasm. Biskup et al. ^[8] found that the inhibitors of phospholipase C (PLC), Neomycin and U73122, inhibited excitation in Chara. In animal cells IP₃ is cleaved from the membrane bound phosphatidyl inositol 4, 5-biphosphate (PIP₂) by PLC. Wacke and Thiel ^[9] measured cytoplasmic Ca²⁺ concentration as a function of depolarising current pulses. To liberate Ca²⁺ from internal stores, IP₃ had to reach a critical concentration. The spacing of a double pulse provided data about the IP₃ formation and decay into IP₂ and replenishing of the PIP₂ pool.

Following Othmer ^[11], Wacke et al. ^[10] modelled the Ca²⁺ channels on the internal stores as transiting between 4 states: unbound (R), bound to IP₃ (IR), bound to IP₃ and activating Ca²⁺ molecule (IRC₊), bound to IP₃ and second inactivating Ca²⁺ molecule (IRC₊C_-). The channel is only conductive in IRC₊ state. The long lifetime of the last state is thought to be responsible for the refractory period.

We can set up rate equations for change in Ca²⁺ cytoplasmic concentration, C, and fractions of channels in each state: x₂ (R), x₃ (IR), x₄ (IRC₊) and x₅ (IRC₊C_-):

$\begin{gathered} {\text{R}} + {\text{I}}\overset {{{\text{k}}_{\text{1}}}{\text{, }}{{\text{k}}_{{\text{-1}}}}} \longleftrightarrow {\text{RI}} \hfill \\ {\text{RI}} + {\text{C}}\overset {{{\text{k}}_{\text{2}}}{\text{, }}{{\text{k}}_{{\text{-2}}}}} \longleftrightarrow {\text{RI}}{{\text{C}}_ + } \hfill \\ {\text{RI}}{{\text{C}}_ + } + {\text{C}}\overset {{{\text{k}}_{\text{3}}}{\text{, }}{{\text{k}}_{{\text{-3}}}}} \longleftrightarrow {\text{RI}}{{\text{C}}_ + }{{\text{C}}_-} \hfill \\ \end{gathered}$

(1)

The initial chemical stimulus is the appearance of IP₃ in the system, initiated by depolarising current pulse. The concentration of IP₃, I, decays from the initial amount, I₀, with time t ^[10]:

${\text{I=}}{{\text{I}}_0}{{\text{e}}^{{\text{-0}}{\text{.2 t}}}}$

(2)

Scaled Ca²⁺ concentration:

$\begin{gathered} {x_1}=\frac{{\text{C}}}{{{{\text{C}}_{\text{0}}}}} \hfill \\ {{\text{C}}_{\text{0}}}=\frac{{{\text{C + }}{{\text{ν }}_{\text{r}}}{{\text{C}}_{\text{s}}}}}{{1 + {{\text{ν }}_{\text{r}}}}} \hfill \\ \end{gathered}$

(3a, b)

where C₀=average calcium concentration, C_s=Ca²⁺ concentration in the store, ν_r=ratio of endoplasmic reticulum (ER) volume to cytoplasmic volume.

$\frac{{{\text{d}}{{\text{x}}_{\text{1}}}}}{{{\text{dt}}}}=\lambda ({\gamma _0} + {\gamma _1}{x_4})(1-{x_1})-\frac{{{{\text{p}}_{\text{1}}}{\text{x}}_{\text{1}}^{\text{4}}}}{{{\text{p}}_{\text{2}}^{\text{4}} + {\text{x}}_{\text{1}}^{\text{4}}}}$

(4)

where γ₀=permeability of Ca²⁺ store in absence of IP₃, γ₁=density of IP₃ activated channels, λ=1 + ν_r. The last ratio in the equation (4) is the Hill function, which describes the calcium conductance of the Ca²⁺ pump with scaled Hill coefficients p₁=p’₁/C₀ and p₂=p’₂/C₀ ^[11].

$\frac{{{\text{d}}{{\text{x}}_{\text{2}}}}}{{{\text{dt}}}}=-{{\text{k}}_{\text{1}}}{\text{I}}{x_2} + {{\text{k}}_{{\text{-1}}}}{x_3}$

(5)

$\frac{{{\text{d}}{{\text{x}}_{\text{3}}}}}{{{\text{dt}}}}=-({{\text{k}}_{{\text{-1}}}} + {{\text{k}}_{\text{2}}}{x_1}){x_3} + {{\text{k}}_{\text{1}}}{\text{I}}{x_2} + {{\text{k}}_{{\text{-2}}}}{x_4}$

(6)

$\frac{{{\text{d}}{{\text{x}}_{\text{4}}}}}{{{\text{dt}}}}={{\text{k}}_{\text{2}}}{x_1}{x_3} + {{\text{k}}_{{\text{-3}}}}{x_5}-({{\text{k}}_{{\text{-2}}}} + {{\text{k}}_{\text{3}}}{x_1}){x_4}$

(7)

$\frac{{{\text{d}}{{\text{x}}_{\text{5}}}}}{{{\text{dt}}}}={{\text{k}}_{\text{3}}}{x_1}{x_4}-{{\text{k}}_{{\text{-3}}}}{x_5}$

(8)

where k_i (i=±1, ±2, ±3) are the rate constants for forward and backward transitions between the channel states (see equation 1). Some of the rate constants are also scaled by C₀: k₂=k’₂C₀ and k₃=k’₃C₀. With the scaling, all x_i range between 0 and 1. As the channels have to be in one of the four states, conservation condition applies: $\sum\nolimits_{{\text{k}}=2}^5 {{x_{\text{k}}}}=1$. Wacke et al. ^[10] started with parameter values used by Othmer ^[11] and gradually modified some of them to suit Chara Ca²⁺ concentration data (see Table 1a).

To re-construct the transient increase in cytoplasmic Ca²⁺ the Thiel model was programmed into Mathematica 10. The numerical integration of equations (4)-(8) was performed by three concentric DO-loops, one to apply 4^th order Runge-Kutta method, the second to advance time and the third to test different time intervals and parameter values. In this study, we are not interested in the stimulus properties, so the excitation was initiated by introducing the initial IP₃ concentration, I₀, into the model (equation 2). We tested different IP₃ concentrations, I₀, and selected 2.5 μM for saturating Ca²⁺ concentration increase (see Figure 4 of Wacke et al. ^[10]) and a shorter delay before the steep Ca²⁺ concentration rise (Figure 2). The changes to Thiel model parameters are summarized in Table 1a. The programs can be found on MJ Beilby website link“Action Potential models”: http://newt.phys.unsw.edu.au/~mjb/APproj.html.

---

2.3. The Thiel-Beilby Model

To calculate the PD transient we start from the rate of change of the membrane PD, V:

$\frac{{{\text{dV}}}}{{{\text{dt}}}}=-\frac{1}{{{{\text{C}}_{\text{m}}}}}\left[{\frac{{{{\text{G}}_{{\text{Cl, max}}}}(V-{{\text{E}}_{{\text{Cl}}}})}}{{1 + \frac{{{{\text{k}}_{\text{i}}}}}{{{{\text{k}}_{\text{a}}}{x_1}}}}} + {{\text{I}}_{\text{p}}} + {{\text{I}}_{{\text{orc}}}} + {{\text{I}}_{{\text{bkg}}}} + {{\text{I}}_{{\text{TRP, Ca}}}}} \right]$

(9)

where C_m is the membrane capacitance (Table 1b) and the terms inside the square bracket are all the currents flowing at the time of the AP. The first ratio in the square brackets is the Ca²⁺-activated chloride current, I_Cl. This model was formulated by Biskup et al. ^[8] and fitted to excitation voltage clamp currents. The rate constants k_a and k_i describe the activation and inactivation of I_Cl by increased Ca²⁺ concentration in the cytoplasm (Table 1a). G_{Cl, max} was set in range suggested by previous modeling (^[2,4] and Table 1b).

The proton pump current, I_p, the background current, I_bkg, and the outward rectifier current, I_orc, are part of the electrophysiological make up of the Chara plasma membrane (Figure 1 and Chapter 2 of Beilby and Casanova ^[17] for the full description of the modeling of I/V characteristics of these transporters). It is assumed that the brief increase in the cytoplasmic Ca²⁺ concentration does not influence the PD dependencies of these transporters.

Briefly: The PD dependence of I_p is calculated from HGSS (Hansen, Gradmann, Sanders and Slayman) single cycle two state model, based on cyclic enzyme-mediated transport ^[18]. The voltage-independent steps, such as ATP-, ADP-, inorganic phosphate-and H⁺ binding and de-binding steps and carrier recycling, are represented by rate constants κ_ioand κ_oi, while the charge transit is characterised by the rate constants k_io and k_oi, across a symmetrical Eyring barrier.

The dependence of the pump current, I_p, on membrane PD, V, is given by:

${{\text{I}}_{\text{p}}}={{\text{z}}_{\text{p}}}{\text{FN}}\frac{{{{\text{k}}_{{\text{io}}}}{{\text{κ }}_{{\text{oi}}}}-{{\text{k}}_{{\text{oi}}}}{{\text{κ }}_{{\text{io}}}}}}{{{{\text{k}}_{{\text{io}}}} + {{\text{k}}_{{\text{oi}}}} + {{\text{κ }}_{{\text{io}}}} + {{\text{κ }}_{{\text{oi}}}}}}$

(10)

$\begin{gathered} {{\text{k}}_{{\text{io}}}}={\text{k}}_{{\text{io}}}^0{{\text{e}}^{\frac{{{{\text{z}}_{\text{p}}}{\text{FN}}}}{{{\text{2RT}}}}}} \hfill \\ {{\text{k}}_{{\text{oi}}}}={\text{k}}_{{\text{oi}}}^0{{\text{e}}^{-\frac{{{{\text{z}}_{\text{p}}}{\text{FN}}}}{{{\text{2RT}}}}}} \hfill \\ \end{gathered}$

(11a, b)

F, R, T symbols have their usual meaning, z_p is the pump stoichiometry of 1, N is a scaling factor set to 2×10^-8 and ${\text{k}}_{{\text{io}}}^0{\text{ and k}}_{{\text{oi}}}^0$ are defined at 0 PD. The parameter values were adjusted to simulate the resting PD of each cell (Figure 1 and Table 1b).

The background current, I_bkg, is modelled by an empirical equation:

${{\text{I}}_{{\text{bkg}}}}={{\text{G}}_{{\text{bkg}}}}(V-{{\text{E}}_{{\text{bkg}}}})$

(12)

where the background conductance G_bkg is independent of membrane PD. The reversal PD, E_bkg, is near-100 mV (±20 mV), a puzzling PD level, as it does not correspond to Nernst PD of the most abundant ions in the cytoplasm, vacuole or medium. Further, G_bkg increases markedly in Saline (but not Sorbitol) APW without changes to the E_bkg ^[13]. For parameter values see Table 1b.

The outward rectifier current, I_orc, is thought to be carried mainly by K⁺ and can be fitted by the Goldmann-Hodgkin-Katz (GHK) equation, multiplied by the Boltzmann distribution of open probabilities, P_o- and P_o+, to make the PD-dependence stronger ^[19,20]:

${{\text{I}}_{\text{X}}}=\frac{{{{\text{P}}_{{\text{o + }}}}{{\text{P}}_{{\text{o-}}}}{{\text{N}}_{\text{X}}}{{\text{P}}_{\text{X}}}{{(zF)}^2}V({{[X]}_{\text{i}}}-{{[X]}_{\text{o}}}{e^{-\frac{{{\text{zFV}}}}{{{\text{RT}}}}}})}}{{{\text{RT}}(1-{e^{-\frac{{{\text{zFV}}}}{{{\text{RT}}}}}})}}$

(13)

$\begin{gathered} {{\text{P}}_{{\text{o-}}}}=1-\frac{1}{{1 + {e^{-\frac{{{{\text{z}}_{\text{g}}}{\text{F(V-}}{{\text{V}}_{{{50}^-}}})}}{{{\text{RT}}}}}}}} \hfill \\ {{\text{P}}_{{\text{o + }}}}=1-\frac{1}{{1 + {e^{-\frac{{{{\text{z}}_{\text{g}}}{\text{F(V-}}{{\text{V}}_{{{50}^ + }}})}}{{{\text{RT}}}}}}}} \hfill \\ \end{gathered}$

(14a, b)

where z=valence of the transported ion, [X]_o and [X]_i are the ion concentrations in the medium and cytoplasm, respectively, N_XP_X=number of X ion channels and their permeability as a single parameter; z_g=number of gating charges, V_50- and V₅₀₊=the half activation potentials, V₅₀, at the negative and positive PDs of channel closure (see Figure 1). The parameter values can be found in Table 1b. The total current, I_tot, was fitted to the I/V characteristics of many characean cells under range of conditions ^[17].

I_{TRP, Ca} was included, as the depolarizing phase of the AP was too gradual compared to data (see the Results section). The modeling was supported by the sharp spike to positive PD levels in the experimental spontaneous saline APs (see the Results and the Discussion sections). The I_{TRP, Ca} was simulated by I_Ca=G_Ca(V-E_Ca) (see Table 1b for parameter values), turned on as a square pulse of Ca²⁺ for 15 sec in the early stages of the simulated AP (see Table 1b for time intervals). It is not known how much of this Ca²⁺ inflow reaches the channels on the internal stores: a small fraction was added to x₁ in the same time interval to increase the rate of depolarization to match the data (see the Results section).

The numerical integration of equation (9) was simply added to the innermost DO loop of the“Ca²⁺ concentration”program. Starting from resting PD at time zero, the potential was calculated for the rising Ca²⁺ concentration, which increased the I_Cl, depolarizing the membrane PD. As Ca²⁺ concentration decreased to steady state level, I_CL diminished, I_orc also decreased, as PD became more negative and I_p eventually restored the RPD.

---

3. Results

---

3.1. Modeling AP_av

In the Thiel model the transient increase in cytoplasmic Ca²⁺ starts gradually after stimulation (see Figure 1 of Wacke et al. ^[10]). This is the nature of the model: initially there is not enough activating Ca²⁺, but as more channels open, more Ca²⁺ flows into cytoplasm and positive feedback leads to a steep rise. Then the second inactivating Ca²⁺ starts to close the channels and the event becomes self-limiting.

Using Thiel model parameters (Table 1a) and adjusting the pump current, I_p, to provide the RPD at-170 mV of the AP_av (Figure 1), equation (9) was integrated and resulting AP compared to AP_av (Figure 3a). Clearly, the model AP started with a delay. To compare the AP forms, artificial delay of 0.6 s was added to AP_av, revealing a surprisingly good correspondence for a first attempt! The fit was improved by recasting the Hill function with exponents of 2 and decreasing parameters p₂’from 0.065 to 0.035 μM and p₁’from 8.5 to 8 μM.s ^-1 (Table 1a). These changes, however, resulted in a more gradual depolarizing phase in the model and necessitated in artificial shift of AP_av by 1.39 s to compare with the model form (Figure 3c).

After observing APs in many experiments and under different conditions, the depolarizing phase is usually very prompt. It was tempting to hypothesize that Chara plasma membrane contains transient receptor potential (TRP)-like channels, which may be activated by diacylglycerol (DAG) formed at the same time as IP₃ by hydrolysis of PIP₂. DAG mediates rapid inflow of either Ca²⁺ or Na⁺ from the outside through TRP channels in many animal systems, such as cardiomyocytes ^[21]. I_Ca was allowed to flow for a brief interval 0.04-0.16 s and small fraction of Ca²⁺, Ca²⁺, was added to C during that time. The Ca²⁺ values were manipulated to fit the data. Without changing any other model parameters, we obtained very good correspondence to AP_av form (Figure 4a,Table 1b). Slight changes in Ca²⁺ resulted in strong effects on the AP form (Figure 4b, c).

---

3.2. Modeling AP_sal

The duration of the AP started to change after only a few s in 50 mM NaCl Saline APW (Figure 5a). Prior to salinity stress, the RPD of this cell was very negative, so the pump parameters were adjusted for greater amplitude I_p in generating the model for AP_av, designated“AP_{av, hyper}”(Figure 1). For the AP_{av, hyper} peak to remain at the same level, G_{Cl, max} had to be increased (Figure 5 and Table 1b). To approximate the prolonged AP_sal, the Ca²⁺ pump parameters in the Hill function were decreased: p₁’=1.59 μM.s ^-1and p₂’=0.009 μM (Figure 5b, dashed gray line). The fit could be manipulated by decreasing the initial IP₃ concentration to 0.5 mM (Figure 5b, continuous gray line, values of the Hill coefficients in brackets in Table 1b).

The medium Cl^- concentration changed from 1.3 mM to 50 mM in Saline APW, decreasing the driving force on I_Cl. However, the AP peak remained at the similar level of-30 mV in APW and Saline APW. To achieve this effect in the model, the Cl^- concentration in the cytoplasm was increased to 50 mM and G_{Cl, max} to 25 S.m^-2. The smaller values of Hill coefficients prolonged the AP duration (Figure 5b,Table 1b).

Figure 6 shows the diversity of the AP_sal form. Figure 6a depicts one of the longest AP_sal observed in Saline APW (black points). The approximate model AP (4) was obtained by making the Hill coefficients very small (see Table 1a). Despite the extreme duration of the AP, the cell was only exposed to Saline APW for 2 hr and RPD was still more negative than -100 mV. In the model the decrease in the magnitude of Hill coefficients prolongs the AP duration (APs 1-3, Table 1a). Figure 6b shows AP durations more commonly encountered in a Saline APW stressed cell. It is clear that the duration is not directly dependent on the time of saline exposure and not sensitive to pH changes from neutral to higher values. All the APs show very fast depolarizing phase and a sharp initial spike, reaching up to +50 mV. E_Ca was estimated as +120 mV, providing sufficient driving force for membrane PD to become temporarily positive. The data-logging speed was too slow to record more detailed shape of this initial feature, which was completed within 0.2-0.3 s. Consequently, the I_{TRP, Ca} was approximated by a short square pulse across plasma membrane (see Methods section). As Ca²⁺ is not very mobile in the cytoplasm ^[22], the Ca²⁺ fraction reaching the Ca²⁺ channels on ER was adjusted to maximise the fit of the model AP (Figure. 4a).

The modeled AP_sal shapes are approximate at this point: the peak is not as flat as the data and the repolarizing phase is slightly different. We have also tried alternative approach: to introduce a delay before starting the Ca²⁺ pumps. This calculation did provide a flat peak, but the repolarizing phase was also difficult fit (results not shown). In the next paper we will aim to provide more detailed fit of the AP_sal, exploring different initial concentrations of IP₃, as well as gathering statistics on the saline AP form as a function of exposure to Saline APW of different pH.

---

4. Discussion

---

4.1. Second messengers

Second messengers are intracellular components of signaling sequences initiated by extracellular signals, the first messengers. In animal systems, where these sequences are better known, the first messengers are often hormones or neurotransmitters. The first messengers may not be able to cross the lipid membrane in absence of specific channels and activation of second messengers acts as trans-membrane transducer and sometimes as amplification. Typical animal second messengers are cyclic AMP (adenosine monophosphate), IP₃, DAG and, similar to plants, Ca²⁺. In animal systems the binding of a ligand to a G-protein activates PLC, which in turn cleaves PIP₂ into IP₃ and DAG. The signaling sequence is interesting in the Chara AP, as it is initiated by an electrical signal, membrane depolarization, which, by yet unknown process, mobilizes PLC, IP₃ and DAG. The second messengers then open Ca²⁺ channels on the ER and plasma membrane. The increased Ca²⁺ concentration in the cytoplasm activates Cl^- channels on the plasma membrane, mediating another electrical signal, the AP. In the nerve the AP is 1000 times faster with the second messenger system bypassed and the Na⁺ channel activated by depolarization and inactivated spontaneously. It appears that plants just do not have a suitable channel to produce an AP in this way. While the higher plant genomes lack IP₃ and IP₆ receptors and the DAG target, protein kinase C (see ^[23] for review), there is increasing experimental evidence that IP₃ is involved in Ca²⁺ release in the plant cytoplasm (^[24,25] and references within). DAG cell content was also observed to change in response to a variety of developmental and environmental stimuli ^[26]. Mikami ^[27] suggests that the receptor genes were eliminated on multiple occasions in plant evolution: after divergence of Chlorophyta and Rhodophyta and of Chlorophyta and Charophyta. So, plants might employ IP₃, IP₆ ^[28] and DAG as messenger molecules, but the receptors might be different to those of animals. This is clearly important area of plant signaling needing further analysis. In this study we show that second messenger model, based on animal paradigm, yields good correspondence to Chara AP under normal conditions and at the time of salinity stress.

---

4.2. Insights from the model

The model indicates that there are two sources of Ca²⁺ in Chara AP: short transient inflow from outside through TRP-like channels at the same time as IP₃-activated channel opening on the internal stores (Figures 3, 4a). Such early prompt Ca²⁺ transient is supported by experiments of Berestovsky and Kataev ^[29] and by positive going spikes in the spontaneous AP_sal (Figure 6b). These spikes were not noticed in previous AP studies: in the APW the spontaneous excitation is rare and short depolarizing current pulse was often used ^[2,4]. In such experiments, the depolarizing threshold pulse and the I_Ca transient may be difficult to distinguish. Baudenbacher et al. ^[30] also suggested that higher Ca²⁺ concentration in the cytoplasm speeds up the rate of depolarization of the AP. In their experiments the concentration increase resulted from exposing the cell to dark/light transition. In terms of the AP form, the increase in background Ca²⁺ concentration indeed improves the fit of the depolarizing phase, but the repolarizing phase does not return the PD to resting level (this can be expected from the long tail of the Ca²⁺ concentration transient in Figure 8 of their paper).

TRP channels seem to be absent from genomes of land plants, but were found in some chlorophyte algae ^[31]. Preliminary search through recently sequenced Chara braunii genome did not yield any homologies to chlorophyte TRP channels (Rensing pers. com). However, TRP channels descriptions are relatively recent and new types are still being discovered.

The conservative AP peak level (-30 mV) in both APW and Saline was highlighted by the modeling. In APW, the more negative the RPD (due to activity of the proton pump—see Figure 1), the greater Cl^- conductance is necessary (compare G_{Cl, max} for AP_av and for AP_{av, hyper} before saline exposure, Figure 5a and Table 1b). It is possible that more negative PDs“prime”more Cl^- channels to be available upon activation by increased Ca²⁺ concentration. The AP peak constancy is even more interesting, as the cells are exposed to Saline APW (Figures 5 and 6). The change in the medium Cl^- concentration from 1.3 mM to 50 mM in Saline APW diminishes the driving force on I_Cl. To obtain the same AP peak level, the cytoplasmic concentration had to be increased. We used equilibration with the outside concentration of 50 mM in the model. Teakle and Tyerman ^[32] suggested a possibility of passive Cl^- entry upon exposure to saline media.

The spontaneous occurrence of the AP_sal can be mostly explained by the depolarization of the membrane PD to and above excitation threshold. This depolarization is a result of increased background conductance and progressive inhibition of the proton pump (^[13,17] and Figure 1). However, even with negative RPD, the APs may occur spontaneously (e.g. the AP in Figure 5, just after the cell was exposed to Saline APW) or with very slight stimuli, such as refreshing the outside medium—which would not elicit APs in APW. This greater excitability may be explained by some cells becoming more sensitive to mechanical stimuli in Saline APW with 0.1 Ca^{2+ [13]}. The presence of higher Na⁺ on the outside and in the cytoplasm might optimize production of IP₃. As the mechanism for cleavage of PIP₂ upon depolarization is not yet known, this information might aid the future investigation.

Why does salinity prolong the duration of the AP? Several mechanisms return the cytoplasm to steady state low Ca²⁺. Plant P-type 2B Ca²⁺ ATPases have been found in Arabidopsis, which share high homology to animal plasma membrane Ca²⁺ ATPase with auto-inhibitory N-terminus ^[33]. The pumps become active at high Ca²⁺ concentration and deactivate at low Ca²⁺. 2A P-type ATPases with 2 Ca²⁺/ATP stoichiometry were described on endoplasmic reticulum, similar to animal pumps on sarcoplasmic reticulum. Arabidopsis genome also contains Ca²⁺/H⁺ antiporters, CAX, probably on the vacuolar membrane. It is not clear which of these mechanisms are active in Chara in maintenance of the low cytoplasmic Ca²⁺ and the“mop up”of sudden Ca²⁺ spills. Not all the Ca²⁺ returns to the stores that supply the AP. Mitochondria, microsomes and chloroplasts all sequester Ca²⁺ under various circumstances (for review see ^[34]). Saline exposure increases the Na⁺ concentration in the cytoplasm and, after longer exposures, erodes the electrochemical gradient for H⁺, which would affect the Ca²⁺/H⁺ antiporter. However, the effect of saline exposure is apparent almost immediately (Figure 5), where the RPD, pump and cytoplasmic pH are not yet affected. The mechanism still works (although slowly) after the proton pump was inhibited and RPD dropped to above-100 mV. Thus we suggest that Ca²⁺ pump/s are involved and preliminary modelling with the Hill equation produced long APs by decreasing the values of the coefficients (Figures 5b, 6a and Table 1b). The initial IP₃ concentration also affects the shape of the AP (Figure 5b) and will be further investigated in future detailed modelling of AP_sal.

---

5. Conclusion

This work is a theoretical exploration of the Chara AP form, based on transient Ca²⁺ concentration rise in the cytoplasm, modeled by second messenger system partially derived from animal kingdom paradigms ^[8,9,10,11]. The Thiel-Beilby model produces a good fit to“normal”AP at room temperature and simulates the long AP trend under saline stress. A range of future experiments can test the model:

(1) Two sources of Ca²⁺ in Chara AP: the removal of Ca²⁺ from the external medium, block of the TRP-like channel or inhibition of DAG signaling (separately from IP₃ pathway) should produce slow depolarizing phase APs (such as in Figure 3b or c).

(2) The AP will be data-logged at higher speed to resolve the time characteristics of the TRP-like current.

(3) The light/dark transition experiments of Baudenbacher et al. ^[30] need to be revisited, measuring APs under different light conditions.

(4) The increase in cytoplasmic Cl^- concentration upon exposure to Saline APW with a range of Ca²⁺ concentrations can be measured as a function of duration of salinity exposure.

(5) Ca²⁺ pumps inhibitors can be applied and should produce long APs in normal APW.

---

Conflicts of Interest

All authors declare no conflicts of interest in this paper.

---