3.1. Voltage changes

At first, we considered in a concrete scenario an electrical signal that significantly changed the membrane voltage (by $\Delta V(0)=V(0)-{V}_{ss}$ ) but not the K⁺ gradient ( $\Delta {E}_K=0$ ). Such voltage changes have their physiological equivalent in action potentials (Brownlee, Reference Brownlee2022; Cuin et al., Reference Cuin, Dreyer and Michard2018; Hedrich et al., Reference Hedrich, Salvador-Recatalà and Dreyer2016; Scherzer et al., Reference Scherzer, Böhm, Huang, Iosip, Kreuzer, Becker, Heckmann, Al-Rasheid, Dreyer and Hedrich2022) but are also induced in optogenetic experiments (Huang et al., Reference Huang, Ding, Roelfsema, Dreyer, Scherzer, Al-Rasheid, Gao, Nagel, Hedrich and Konrad2021; Papanatsiou et al., Reference Papanatsiou, Petersen, Henderson, Wang, Christie and Blatt2019; Reyer et al., Reference Reyer, Häßler, Scherzer, Huang, Pedersen, KAS, Bamberg, Palmgren, Dreyer and Nagel2020). For illustration, we chose a physiological model scenario, in which a steady state has been established at ${V}_{ss}=-150\; mV$ , ${E}_K^{ss}=-115\; mV$ ([K⁺]_in = 150 mM, [K⁺]_out = 1 mM), and ${E}_H^{ss}=92.1\; mV$ (pH_in = 7.2, pH_out = 5.6). This could be achieved with the parameter set g_KC = 8 V^-1, g_HKs = 0.02 V^-1, and g_HKa = 1.378 V^-1. Together with an I_Hmax = 50 pA and a membrane surface of 1000 μm² (C = 10 pF), the respective H⁺ and K⁺ current densities were similar to those observed in guard cells (Dietrich et al., Reference Dietrich, Dreyer, Wiesner and Hedrich1998; Lohse & Hedrich, Reference Lohse and Hedrich1992). This system was allowed to equilibrate in its steady state and then challenged with a very short (1 ms) voltage stimulus in the range of $\Delta V=\pm 50\; mV$ (Figure 1a,e). Such a change in the membrane voltage altered the driving forces for K⁺ and H⁺, which caused a net current (Figure 1b,f), a net K⁺ flux (Figure 1c,g) and a net H⁺ flux (Figure 1d,h; Supplementary Text S2) via the different transporters. A hyperpolarization (ΔV < 0; Figure 1a–d) provoked a negative net current and a K⁺ and H⁺ influx, while a depolarization (ΔV > 0; Figure 1e–h) provoked a positive current and effluxes. In this scenario, we have excluded any dependence of the transporter activities on the membrane voltage to clearly work out the effect of the voltage change on the driving forces. The effects of changes in transporter activity that would superimpose the consequences presented here are discussed below.

Figure 1. Response of the K homeostat to a very rapid voltage pulse (ΔV). The K homeostat in steady state is challenged at t = 0 s by a rapid (1 ms) hyperpolarization (a–d) or depolarization (e–h) and is then left to its own devices again. The induced net currents (b, f) cause the membrane voltage to reset over time (a, e). The net currents are carried by net fluxes of K⁺ (J_K ; c, g) and H⁺ (J_H ; d, h) across the membrane.

The net charge transport had a rapid effect on the membrane voltage, with a tendency to restore the original voltage within tens of milliseconds, that is, $\Delta V(t)$ and the ion fluxes varied with time. The time until the steady state (dV/dt = 0) was reached, varied with the K-channel activity. With higher activity, the reset time was shorter than with low activity. For comparison, we chose a second scenario, in which the K-channel activity was 10-fold lower than in the first scenario but with the same steady state condition ( ${V}_{ss}=-150\; mV$ , ${E}_K^{ss}=-115\; mV$ ). This was achieved with g_KC = 0.8 V^-1, g_HKs = 0.474 V^-1, and g_HKa = 0.77 V^-1 (Supplementary Figure S2). The slightly longer relaxation time in the second scenario was mainly due to the lower net current, which restored the membrane voltage. As expected for a system with fewer K channels, the K⁺ flux (J_K ; Supplementary Figure S2c and g) in particular was reduced and was only partially compensated by a larger H⁺ flux (J_H ; Supplementary Figure S2d and h).

Irrespective of the scenario, the K homeostat showed a very fundamental short-term behavior of homeostats in general: a voltage change ΔV induced a net current that was proportional to the voltage change (Figure 2) and which counteracted the deflection from a steady state. The positive current induced by a depolarization (ΔV > 0) drove the membrane voltage in the negative direction while the negative current induced by a hyperpolarization (ΔV < 0) caused a positive shift of the voltage. Thus, the dynamics of the homeostat stabilized the membrane voltage at the steady state value. The more K channels were active in the K homeostat, the stronger was this autoregulative feedback. Current–voltage (I/V) curves of the type shown in Figure 2 are not limited to K homeostats, but a characteristic feature of all types of homeostats. Actually, they are the fundamental basis for setting the membrane voltage in the steady state.

Figure 2. Fundamental response of a homeostat to a voltage stimulus. A voltage stimulus (ΔV) induces a net current, which in turn causes a negative restoring force for the membrane voltage. The system is stable at ΔV = 0.

The K⁺ and H⁺ fluxes induced by the voltage stimulus were in the range of hundreds of attomols (10⁻¹⁸ mols) per second. Over the time course of the relaxation process, they summed up to a transmembrane net transfer of up to 1–4 attomols (Supplementary Figure S3). To estimate the potential consequences of these at first sight extremely tiny fluxes, it needed to be put in relation to the cellular volume. Considering in our model scenarios a spherical cell with C = 10 pF (cell surface = 1000 μm²), the internal volume would be ~3 pL (3000 μm³). Considering further that the vacuole could occupy about 90% of the internal cellular space, we calculated with a cytosolic volume of 0.3 – 3 pL. A mass transfer of 3 attomols into or out of such a volume would change the concentration by 1 μM (3 pL) or 10 μM (0.3 pL). For K⁺ with a concentration of [K⁺]_in = 100 mM, such a change was insignificant. In contrast, for H⁺ with a resting concentration of [H⁺]_in = 0.063 μM (pH 7.2) such a change would have implied an alteration of 1500%–15,000% in the absence of any pH buffer system. So far, the cellular simulations were carried out with a potent pH buffer ([Buf^-] = [HBuf] ≥ 10 mM). Under this condition, the H⁺ fluxes did not change pH_in in the considered short time interval. However, when we repeated the simulations with 100-fold smaller buffer concentrations ([Buf^-] = [HBuf] ~ 0.1 mM), the cytosolic pH was strongly affected (Figure 3). Although this low buffer capacity might not be physiological (Felle, Reference Felle2001), this example is intended to illustrate the consequences of small buffering capacities. A depolarization alkalinized the cytosol, while a hyperpolarization caused an acidification. The magnitude of the pH changes was different in the two homeostat scenarios. It was more pronounced in scenario 2 with fewer K-channel activity than in scenario 1 because the smaller K⁺ fluxes were partially compensated by larger H⁺ fluxes in the second scenario. Thus, a homeostat has a resetting/stabilizing effect on a disturbance of the membrane voltage. In addition, it could in principle convert a voltage signal into a pH_cyt signal. How sensitive this conversion is, depends on the buffer capacity and on the composition of the homeostat. Interestingly, also the feature of converting an electrical into a chemical signal could be generalized to other homeostats. For example, a homeostat with Ca²⁺-permeable transporters could convert a voltage signal into a cytosolic Ca²⁺ signal. Resting [Ca²⁺]_cyt – like [H⁺]_cyt – is very low and can therefore be quickly changed by fluxes in the range of tens to hundreds of attomoles per second. The time course of the conversion of the electrical signal into a chemical signal depends strongly on the respective chemical buffer capacity of the cell. The lower the buffer capacity, the more sensitive is the conversion process.

Figure 3. Buffer-dependent changes of the internal proton concentration in response to a voltage stimulus. Hyperpolarization (a) or depolarization (b) provoke H⁺ fluxes that may significantly modify the internal pH (pH_in). The magnitude of this pH change depends on the buffer capacity of the cell and the composition of the homeostat. Compared are cells with a strong (≥10 mM; black curves) and a very weak (0.1 mM) buffer capacity, as well as a K homeostat with high () and one with low () K-channel activity.

3.2. Change in K ⁺ concentration

Next, we considered in our model scenarios a sudden change in the concentrations of the permeating ions, that is, in the case of the K homeostat K⁺ (this section) and H⁺ (next section). Also these changes affected the driving forces of transport. Although a slower change is more in line with physiological reality, it is by analyzing abrupt changes that one can learn the most about the dynamics of the system. First of all, we changed [K⁺]_out, which affected the K⁺ gradient ([K⁺]_out increase → $\Delta {E}_K>0$ ; [K⁺]_out decrease → $\Delta {E}_K<0$ ). When we started with the K homeostat in steady state conditions and then doubled or halved [K⁺]_out we induced a $\Delta {E}_K=\pm 17.3 mV$ . The cellular system responded to such a disturbance in two temporally separated phases (Supplementary Text S3). In a first phase of a few tens of milliseconds, the membrane voltage quickly reached a new, transient peak (i.e. dV/dt ≈ 0; Figure 4a), while in the second phase, a stable, long-lasting H⁺/K⁺ antiport occurred (Figure 4c,f), which could theoretically last for several tens of minutes (up to hours) with a correspondingly strong pH buffer. To illustrate this process free from secondary pH-effects, we clamped in our simulations [H⁺]_in and [H⁺]_out. The effects of a limited buffer capacity are illustrated further below. The condition dV/dt = 0 implied J_K = −J_H , i.e. an electroneutral H⁺/K⁺ antiport [equation (24)]. Thus, in the second phase, after equalization of the membrane voltage, the K homeostat mediated an exchange of K⁺ for H⁺, which slowly changed the potassium and proton concentrations (in our simulations clamped) until also here a new steady was established. Exactly, such a H⁺/K⁺ antiport has been observed experimentally in guard cells and mesophyll cells (Li et al., Reference Li, Grauschopf, Hedrich, Dreyer and Konrad2024), which underpins the correctness of the theoretical considerations. It should be noted that the H⁺/K⁺ antiport occurred regardless of whether the K homeostat contained antiporters or not. Further inspection revealed that the H⁺/K⁺ antiport was not exactly electroneutral, which is illustrated for a positive ΔE_K in the following. After the first depolarization caused by a decaying negative net current with an amplitude in the pA-range (Figure 4b), in the following, the net current was not exactly zero but slightly positive with an amplitude in the sub-fA-range (Figure 4e, inset), that is, the H⁺-efflux was slightly larger in amplitude than the K⁺-influx. This positive net current led to a very slow return of the membrane voltage to the stable steady state at ${V}_{ss}=-150\; mV$ (Figure 4d). The negative K⁺ flux (J_K , Figure 4c,f) caused an increase of [K⁺]_in with the tendency of $\Delta {E}_K\to 0$ and restauration of the original ${E}_K^{ss}$ , that is, the larger [K⁺]_out was compensated by a larger [K⁺]_in. When we repeated the simulations with the K homeostat of the second scenario (fewer K channels), we observed essentially a similar behavior (Supplementary Figure S4a–f). Just the amplitudes of the transient voltage change, the net currents and the H⁺ and K⁺-fluxes were smaller indicating that in this case the K homeostat of scenario 1 was more sensitive than that of scenario 2. Qualitatively identical results, but reversed in sign, were obtained when a $\Delta {E}_K=-17.3\; mV$ (reduction of [K⁺]_out) was applied (Supplementary Figure S4g–l).

Figure 4. Response of the K homeostat to a rapid change in the K⁺ gradient (ΔE_K ). The K homeostat in steady state is challenged at t = 0 s by a doubling of [K⁺]_out (ΔE_K > 0). The induced net currents (b, e) cause the membrane voltage to reset over time (a, d). The net currents are carried by net fluxes of K⁺ (J_K , green) and H⁺ (J_H , blue; c, f) across the membrane. Please note the different answers in the millisecond (short-term response, a–c) and minutes time scale (long-term response, d–f).

To illustrate the effect of the pH buffer, we repeated the experiment shown in Figure 4 (clamped pH_in, i.e., infinite buffer capacity), where [K⁺]_out has been doubled at t = 0, with finite buffer capacities (100 mM, 50 mM, and 10 mM; Kurkdjian & Guern, Reference Kurkdjian and Guern1989) (Supplementary Figure S5). When the buffer capacity had no limit (Figure 4), the H⁺/K⁺ exchange proceeded until also [K⁺]_in had doubled, the K⁺ gradient was as before (ΔE_K = 0), and the original membrane voltage was restored (ΔV = 0, i.e., V = −150 mV; Supplementary Figure S5, black curves). However, when the buffer capacity reached its limits, the efflux of protons could only be partially compensated and the internal pH value increased (Supplementary Figure S5e; gray curves). This in turn changed the transmembrane proton gradient (E_H ) and caused the H⁺/K⁺ exchange to terminate earlier (Supplementary Figure S5b and d), so that neither the K⁺ gradient (Supplementary Figure S5c) nor the membrane voltage (Supplementary Figure S5a) was fully restored. Instead, a new pH_in-dependent steady-state condition was established. Thus, a homeostat can convert a concentration gradient signal into a voltage and a pH_in signal. The sensitivity of this conversion depends on the pH buffer capacity and the composition of the homeostat.

3.3. Change in H ⁺ concentrations

Our considerations have shown so far that a K homeostat could process (i) a voltage signal into a pH_in signal and (ii) a change in the K⁺ gradient into a voltage and a pH_in signal. As next we considered a change in the proton gradient (ΔE_H ). We could have simulated similar scenarios as shown before for ΔV (Figures 1– 3 and Supplementary Figures S2 and S3) or ΔE_K (Figure 4 and Supplementary Figures S4 and S5). However, closer mathematical inspection revealed that the scenario of ΔE_H could be explained by a combination of the previously considered perturbations (ΔV and ΔE_K ). Because the steady state of the K homeostat ( ${V}_{ss},{E}_K^{ss}$ ) depends on E_H [equations (8, 9)], a change in this value sets a new steady-state condition of the homeostat ( ${V}_{ss, new}$ , ${E}_K^{ss, new}$ ). Consequently, the homeostat, which was originally in dynamic equilibrium ( ${V}_{ss, old}$ , ${E}_K^{ss, old}$ ), was set by ΔE_H at t = 0 to a state corresponding to a deflection from its new dynamic equilibrium. After the change, the situation of the homeostat could therefore also be explained by $\Delta V={V}_{ss, old}-{V}_{ss, new}$ and ${\Delta E}_K={E}_K^{ss, old}-{E}_K^{ss, new}$ , causing responses such as those described in the previous two sections. In a rapid reaction, the membrane voltage and, under very weak buffer conditions, also pH_in were adjusted within tens of milliseconds to a temporary steady state. In the following, the net K⁺ and H⁺ fluxes as well as the tiny, but non-zero net current caused by the concentration disequilibrium changed then slowly [K⁺]_in, [H⁺]_in (depending on the buffer capacity) and V until a new steady state was reached (compare Figures 1– 4 and Supplementary Figures S2–S5). Thus, from a mechanistic point of view, the cases of ΔE_H and ΔE_K did not differ.

3.4. Changes in transporter activities (parameters g _X )

Finally, we considered what happened if internal or external stimuli modified the activities of the membrane transporters of the homeostat. Physiologically, this happens very frequently, for example, when expression levels change or when posttranslational modifications occur. However, physical parameters such as voltage or hydraulic pressure can also be regulators of the transporter proteins.

As in the previous case, we were able to gain important insights by taking a closer look at the mathematical representation of the homeostat. When being in a steady state, the values ( ${V}_{ss},$ ${E}_K^{ss}$ ) were determined according to equations (8, 9) by the activity of the proton ATPase, the K⁺ channel, the H⁺/K⁺ symporter and the H⁺/K⁺ antiporter, represented by (g_KC , g_HKs , g_HKa ). A change in any of the g_X implied a new steady state condition, which was determined by the altered parameter set. The current condition ( ${V}_{ss, old}$ , ${E}_K^{ss, old}$ ) could thus be interpreted, as in the case of ΔE_H , as a deviation by ΔV and ΔE_K from this new steady state ( ${V}_{ss, new}$ , ${E}_K^{ss, new}$ , Supplementary Text S4).

Changes in g_X might have been caused by an altered expression level, protein turnover, or posttranslational modifications, for instance. These processes are usually rather slow (minutes to hours) suggesting that in these cases also ( ${V}_{ss, new}$ , ${E}_K^{ss, new}$ ) are slowly adjusted. But they could also be faster as in the case of wounding when a hydraulic wave is assumed to propagate quickly within the tissues and modifies the activity of tension-regulated channels/transporters (Moe-Lange et al., Reference Moe-Lange, Gappel, Machado, Wudick, C. S. A., Schott-Verdugo, Bonus, Mishra, Hartwig and Bezrutczyk2021). To classify the consequences of such quicker processes, we exemplarily altered the parameter of the K channel, g_KC , in scenarios 1 and 2 considered before. In scenario 1, we decreased g_KC by a factor of 10 to the value of g_KC of scenario 2, and in scenario 2, we increased g_KC by a factor of 10 to the g_KC value of scenario 1. These alterations changed the steady-state values from ( ${V}_{ss, old}=-150\; mV$ , ${E}_K^{ss, old}=-115\; mV$ ) to ( ${V}_{ss, new1}=-168.3\; mV$ , ${E}_K^{ss, new1}=-7.3\; mV$ ) and ( ${V}_{ss, new2}=-149.2\; mV$ , ${E}_K^{ss, new2}=-141.5\; mV$ ), respectively. Instead of an abrupt change of g_KC , we chose an exponential decay kinetics characterized by a time constant τ. We did this because ensembles of ion channels or transporters, which individually are stochastically acting objects, adapt their activity to external stimuli with exponential kinetics (Dreyer, Reference Dreyer2017; Dreyer et al., Reference Dreyer, Müller-Röber, Köhler and Blatt2004).

To understand the short-term effects of the g_KC changes on the membrane voltage, we first considered the consequences of the equivalent changes in ΔV and ΔE_K (Supplementary Text S3). A decrease of g_KC caused a ${V}_{ss, new1}=-168.3\; mV$ more negative than the original ${V}_{ss, old}=-150\; mV$ , that is, $\Delta V={V}_{ss, old}-{V}_{ss, new1}>0$ , and a ${E}_K^{ss, new1}=-7.3\; mV$ more positive than the original ${E}_K^{ss, old}=-115\; mV$ , that is, $\Delta {E}_K={E}_K^{ss, old}-{E}_K^{ss, new1}<0$ . Both, $\Delta V>0$ and $\Delta {E}_K<0$ provoked a positive net current (Figures 1f, 2 and Supplementary Figure S4h and k) that hyperpolarized the membrane toward a temporary minimum. The time course of this hyperpolarization depended on τ, the reaction time constant of the channel ensemble (Figure 5a). With τ = 100 ms, the first rapid adjustment of the voltage took less than a second, while with a nonunrealistic τ = 1s (Brüggemann et al., Reference Brüggemann, Dietrich, Dreyer and Hedrich1999; Dietrich et al., Reference Dietrich, Dreyer, Wiesner and Hedrich1998; Maksaev & Haswell, Reference Maksaev and Haswell2012) it could take around 10 s. Please note that the change in g_KC provoked a ${\Delta E}_K<0$ , which caused a temporary membrane voltage peak (−182 mV, Figure 5a) that was negative of the new final steady state ( ${V}_{ss, new1}=-168.3\; mV$ ), a phenomenon observed already before (Supplementary Figure S4g and j). This imbalance would equilibrate in the slow phase of the adjustment (tens of minutes) that is dominated by a change of the concentrations due to the net fluxes of K⁺ and H⁺ (compare Supplementary Figure S4g–l).

Figure 5. Changes in the membrane voltage as a response to changes in g_KC of the K homeostat. Shown is the first phase of the biphasic adjustment of the voltage. (a) In scenario 1, g_KC started at t = 0 to decay exponentially with time constant τ toward the g_KC value of scenario 2 (decay by factor 10). (b) In scenario 2, g_KC started at t = 0 to increase with exponential saturation with time constant τ toward the g_KC value of scenario 1 (increase by factor 10).

Similar conclusions as those illustrated for a decrease of g_KC could also be obtained when increasing g_KC , albeit with inverse signs. Compared to the starting conditions ( ${V}_{ss, old}=-150\; mV$ , ${E}_K^{ss, old}=-115\; mV$ ) the new steady state ${V}_{ss, new2}$ was more positive i.e. $\Delta V={V}_{ss, old}-{V}_{ss, new2}<0$ , and ${E}_K^{ss, new2}$ more negative than the starting value ${E}_K^{ss, old}$ , i.e. $\Delta {E}_K={E}_K^{ss, old}-{E}_K^{ss, new2}>0$ . Both deviations from the steady state caused a negative current (Figures 1b and 2, S4b and e) that triggered a temporary depolarization of the membrane voltage (Figure 5b). In the long term (tens of minutes), the membrane voltage would repolarize to the new steady state ( ${V}_{ss, new2}=-149.2\; mV$ ) alongside with an increase of [K⁺]_in ( ${E}_K\to {E}_K^{ss, new2}$ ) due to the net K⁺ influx (Supplementary Figure S4c and f). The long-term fluxes were strongly influenced by the buffer capacity of the cell (compare Supplementary Figure S5), as a K⁺ influx was accompanied by an H⁺ efflux and a K⁺ efflux by an H⁺ influx. In summary, these considerations indicated that a change in the activity of an ion channel type involved in a homeostat is a strong signal that affects the membrane voltage, the pH conditions and in the long-term the ionic conditions of the cell.