Computational modeling of mitochondrial K⁺- and H⁺-driven ATP synthesis

Abstract

ATP synthase (F₁F_o) is a rotary molecular engine that harnesses energy from electrochemical-gradients across the inner mitochondrial membrane for ATP synthesis. Despite the accepted tenet that F₁F_o transports exclusively H⁺, our laboratory has demonstrated that, in addition to H⁺, F₁F_o ATP synthase transports a significant fraction of ΔΨ_m-driven charge as K⁺ to synthesize ATP. Herein, we utilize a computational modeling approach as a proof of principle of the feasibility of the core mechanism underlying the enhanced ATP synthesis, and to explore its bioenergetic consequences. A minimal model comprising the ‘core’ mechanism constituted by ATP synthase, driven by both proton (PMF) and potassium motive force (KMF), respiratory chain, adenine nucleotide translocator, Pi carrier, and K⁺/H⁺ exchanger (KHE_mito) was able to simulate enhanced ATP synthesis and respiratory fluxes determined experimentally with isolated heart mitochondria. This capacity of F₁F_o ATP synthase confers mitochondria with a significant energetic advantage compared to K⁺ transport through a channel not linked to oxidative phosphorylation (OxPhos). The K⁺-cycling mechanism requires a KHE_mito that exchanges matrix K⁺ for intermembrane space H⁺, leaving PMF as the overall driving energy of OxPhos, in full agreement with the standard chemiosmotic mechanism. Experimental data of state 4→3 energetic transitions, mimicking low to high energy demand, could be reproduced by an integrated computational model of mitochondrial function that incorporates the ‘core’ mechanism. Model simulations display similar behavior compared to the experimentally observed changes in ΔΨ_m, mitochondrial K⁺ uptake, matrix volume, respiration, and ATP synthesis during the energetic transitions at physiological pH and K⁺ concentration. The model also explores the role played by KHE_mito in modulating the energetic performance of mitochondria. The results obtained support the available experimental evidence on ATP synthesis driven by K⁺ and H⁺ transport through the F₁F_o ATP synthase.

1. Introduction

ATP synthesis in E. coli, mitochondria, and chloroplasts is catalyzed by F₁F_o ATP synthase that transduces phosphorylation and redox potentials into electrochemical (ion gradients) and chemical (ATP) energy [1, 2]. Foundationally, bioenergetics rests on the principle of F₁F_o ATP synthase harnessing the energy stored in H⁺ gradients to generate ATP.

Recently, our laboratory has described that a large part of the ATP synthesis flux is driven by K⁺ [3, 4]. In spite of the large F₁F_o selectivity for H⁺ over K⁺ (or Na⁺) (~10⁶-fold with regard to both alkali metal ions), the large abundance of cytoplasmic K⁺ over H⁺ (>10⁶-fold excess) in mammalian cells, reveals that, in principle, a non-trivial flux of K⁺ could actually be driving the F₁F_o ATP synthase in its normal operation. The portion of the H⁺ gradient not directly dissipated by the ATP synthase activity would be available to drive the efflux of K⁺ through the K⁺/H⁺ exchanger (KHE) restoring the osmotic balance. In this way, the K⁺- and H⁺- transporting function of ATP synthase remains fully compatible with Mitchell’s chemiosmotic mechanism [1, 5].

The K⁺ transport mechanism allows mitochondrial volume regulation of respiration since, unlike H⁺, K⁺ is, functionally, osmotically active owing to its 6 orders-of-magnitude greater concentration than H⁺. Purified F₁F_o ATP synthase reconstituted into proteoliposomes can harness energy from K⁺ flux to generate ATP, when subjected to a transmembrane K⁺ gradient[3, 4]. Pure K⁺ current-driven ATP synthase activity was accompanied by ATP generation recorded as an increase in the photon rate over background that, in turn, was significantly inhibited by specific F_o blockers venturicidin/oligomycin [3, 4].

Isolated rat heart mitochondria show that, compared to absence of K⁺, at constant osmolality (260 mOsm), physiological pH and K⁺ concentration drive significantly enhanced fluxes of ATP synthesis and respiration[3, 6]. Under the same conditions, and using radioactive tracers, we quantified the mitochondrial PMF, its individual components, ΔΨ_m and ΔpH, and volume, under both states 4 and 3 respiration, in the presence or absence of K⁺ [3, 6].

In the conventional view of cation flux cycles in mitochondria, a H⁺ gradient is driving solely H⁺ through F₁F_o to make ATP whereas K⁺ only enters the matrix through an “ordinary” channel, dissipating the energy change of this K⁺ movement as heat (see Fig 1A). In the new mechanism studied herein, in addition to that of H⁺, a significant fraction of the energy of ΔΨ_m-driven charge moves as K⁺ whose flux is directly harnessed by F₁F_o to make ATP (see Fig. 1B) rather than being wasted as heat in an ordinary channel, while the remainder of the H⁺ gradient energy is utilized to remove K⁺ through KHE_mito. The bioenergetic advantage in this mechanism over the conventional one is that more ATP is produced for the same input energy by not wasting some of that energy on maintaining what was originally thought to be a separate K⁺ cycle that would generate heat but not ATP.

Figure 1. Current and proposed views of cation flux cycles in mitochondria.

Model simulations performed with a minimal model comprising the respiratory chain (from constant levels of NADH, which is a model input), ATP synthase, ANT, Pi carrier and KHE_mito, comparatively tested two possible scenarios: (A) the current view of K⁺ cycling in mitochondria, involving a K_ATP channel unrelated to OxPhos (in orange) which dissipates the potential energy stored in the K⁺ gradient as heat (conventional model); (B) the proposed scheme of K⁺ cycling, in addition to H⁺, through ATP synthase, and KHE_mito mechanism, that harvests the ion motive force to drive ATP synthesis thus avoiding useless energy dissipation (updated minimal model). Extra-mitochondrial values of ions (K⁺ = 137 mM, pH= 7.0) and metabolites correspond to parameters (constant values in a simulation run, with the exception of ADP as detailed below). (C, D) ATP synthesis, respiratory flux obtained with the model in the corresponding column. (E,F) H⁺ and K⁺ uptake, the latter through either an independent K_ATP channel (E, conventional model) or the F₁F_o ATP synthase (F, updated minimal model). Both models were parameterized for K⁺ according to experimental data. The time course in panels C-F show the evolution of variables during the transition from state 4 (1 μM ADP) to state 3 (in the presence of constant ADP 50 μM) from the time indicated with an arrow.

The mechanism of K⁺- and H⁺-driven ATP synthesis remains fully compatible with Mitchell’s chemiosmotic mechanism (reviewed in [5]) and the present work computationally tests some of its major mechanistic and bioenergetic implications. We utilize a computational modeling approach that accounts for the mechanism of K⁺- and H⁺-driven ATP synthesis to address its feasibility to predict integrated biological outcomes. We explore the model’s ability to simulate experimental data obtained with isolated mitochondria that, in turn, had validated the proteoliposomal data [3]. We also seek to analyze the bioenergetic impact produced by K⁺-and H⁺-driven ATP synthesis as well as put forward testable predictions emerging from the model results. Herein, the mechanistic role(s) of K⁺ are highlighted. Overall, the results obtained support the experimental evidence available on ATP synthesis driven by K⁺ and H⁺ transport through the F₁F_o ATP synthase [3, 4, 6].

2. Model description

2.1. A minimal model that describes the core mechanism of K⁺- and H⁺-driven ATP synthesis

To investigate the potential bioenergetic advantage of a mechanism in which both K⁺ and H⁺ drive ADP phosphorylation through mitochondrial F₁F_o ATP synthase, we formulated an updated minimal model of OxPhos comprising the following inner membrane processes: H⁺ pumping by respiration coupled to NADH oxidation, adenine nucleotide translocator (ANT), phosphate carrier (Pi carrier), ATP synthase transport of H⁺ and K⁺, and the K⁺/H⁺ exchanger (KHE_mito) (Fig. 1B Updated Minimal Model). The behavior of the Updated Model was compared to a model in which mitochondrial K⁺ uptake occurs through a separated channel, unrelated to OxPhos acting as the K⁺ uniporter (Fig 1A Conventional Model).

The modules describing the dynamics of H⁺ pumping, ANT and Pi carrier were based on our model [7]. The KHE_mito rate expression was derived from previous work [8–10]. Additionally, the respiration rate expression includes a saturable factor that depends on the matrix volume, to account for matrix volume-associated-activation of respiration (see Appendix D: Eqs. D1–D2, D4–D5), as described in intact cardiomyocytes [3, 11]. The parameters representing the energy inputs of the model correspond to mitochondrial NADH and extra-mitochondrial ADP concentrations.

The following equations describe the rate of ATP synthesis driven by K⁺- and H⁺:

$$V_{ATPase}=-{\rho }^{F1}\frac{\left(p_a{f}_{MCa}+p_{c1}\exp \left(3F\mathrm{\Delta }{\mathrm{\Psi }}_B/RT\right)\right)\exp \left(A_{F1}F/RT\right)-\left(\begin{array}{l}{p}_a\exp \left(3F\mathrm{\Delta }{\mu }_{ion}/RT\right)\hfill \\ \cdots +p_{c2}\exp \left(A_{F1}F/RT\right)\exp \left(3F\mathrm{\Delta }{\mu }_{ion}/RT\right)\hfill \end{array}\right)}{\left(1+p_1\exp \left(A_{F1}F/RT\right)\right)\exp \left(3F\mathrm{\Delta }{\mathrm{\Psi }}_B/RT\right)+\left(p_2+p_3\exp \left(A_{F1}F/RT\right)\right)\exp \left(3F\mathrm{\Delta }{\mu }_{ion}/RT\right)}$$ Eq.(1)

$$V_{Ku}=-3{\rho }^{F1}\frac{p_a\left(f_K{10}^{-3\mathrm{\Delta }pK}\exp \left(A_{F1}F/RT\right)\right)-\left(p_a+p_b\right)f_K\exp \left(3F\mathrm{\Delta }{\mu }_K/RT\right)}{\left(1+p_1\exp \left(A_{F1}F/RT\right)\right)\exp \left(3F\mathrm{\Delta }{\mathrm{\Psi }}_B/RT\right)+\left(p_2+p_3\exp \left(A_{F1}F/RT\right)\right)\exp \left(3F\mathrm{\Delta }{\mu }_{ion}/RT\right)}$$ Eq.(2)

$$V_{ATPase}^{II}=-{\rho }^{F1}\frac{\left(p_a{10}^{3\mathrm{\Delta }pH}+p_{c1}\exp \left(3F\mathrm{\Delta }{\mathrm{\Psi }}_B/RT\right)\right)\exp \left(A_{F1}F/RT\right)-\left(\begin{array}{l}{p}_a\exp \left(3F\mathrm{\Delta }{\mu }_H/RT\right)\hfill \\ \cdots +p_{c2}\exp \left(A_{F1}F/RT\right)\exp \left(3F\mathrm{\Delta }{\mu }_H/RT\right)\hfill \end{array}\right)}{\left(1+p_1\exp \left(A_{F1}F/RT\right)\right)\exp \left(3F\mathrm{\Delta }{\mathrm{\Psi }}_B/RT\right)+\left(p_2+p_3\exp \left(A_{F1}F/RT\right)\right)\exp \left(3F\mathrm{\Delta }{\mu }_H/RT\right)}$$ Eq.(3)

$$V_{Ku}^{II}=P_K\frac{\Delta {\Psi }_{m}F}{RT}\left(\frac{{[K]}_m\exp \left(\frac{-\Delta {\Psi }_{m}F}{RT}\right)-{[K]}_i}{\exp \left(\frac{-\Delta {\Psi }_{m}F}{RT}\right)-1}\right)$$ Eq.(4)

Equations 1 and 2 are the rates of ATP synthesis and K⁺ uptake through F₁F_o ATP synthase in the updated model (Fig. 1B), whereas Eqs. 3 and 4 corresponds to the rates of ATP synthesis and K⁺ uptake in the conventional model in which K⁺ uniporter activity is independent of OxPhos (Fig 1A). The definitions and parameter values utilized in Eqns. 1–4 are presented in Appendix D. Equations 1–2 correspond to a modified rate expression from Magnus and Keizer [12] as described in Cortassa et al. [13]. In addition to purely H⁺-driven ATP synthesis through F₁F_o, these equations also account for the K⁺ motive force, KMF (analogous to PMF, linked via the ΔΨ_m-component), to describe K⁺-driven ion transport and ADP phosphorylation. The ATP synthase parameterization was based on the fluxes calculated according to Goldman-Hodgkin-Katz (GHK) for which the selective permeability has been calculated. More explicitly, fluxes of H⁺- and K⁺-driven ATP synthase rate expressions were adjusted according to our measurement of the relative permeabilities of H⁺:K⁺ at ~10⁶:1 for ATP synthase. Alternatively, a GHK equation (Eq. 4) was employed to represent K⁺ transport for simulation controls in which an ATP synthase-independent K⁺ transport mechanism was tested (conventional model).

The code was written in Matlab (The MathWorks, Natick, MA) and simulations were run with the built-in ODE15s integrator with 10⁻⁹ and 10⁻¹⁴ as relative and absolute tolerances, respectively.

2.2. Integrated bi-compartmental computational model of mitochondrial energetics, ROS generation/scavenging and ion-transport-driven ATP synthesis

After corroborating that K⁺, in addition to H⁺, transport through F₁F_o ATP synthase confers an energetic advantage to mitochondria, next we compared the behavior of our integrated model of energy metabolism to experimental data. Since the experimental system corresponded to isolated functional mitochondria, we utilized a comprehensive, bi-compartmental, computational model of mitochondria [7, 13, 14]. The model formulation includes OxPhos, matrix-, and extra-mitochondrial-based processes [14]. The energy inputs are provided by Acetyl CoA that will be consumed in the TCA cycle and the energy released will be stored as a redox potential, namely the redox couples NAD/NADH and NADP/NADPH, and extramitochondrial ADP providing the phosphorylation substrate, which is a parameter in the model that is adjusted to mimic different levels of energy demand.

In addition to energy metabolism and ion transport (H⁺, Ca²⁺, Na⁺, Pi) the model accounts for superoxide anion O₂^.− generation from both complex I- and complex II-derived electron transport. O₂^.− can be dismutated to H₂O₂ by matrix superoxide dismutase or be transported as such to the extra-matrix compartment through an inner membrane anion channel, where it will be scavenged by Cu,Zn superoxide dismutase generating H₂O₂. Mitochondrial matrix H₂O₂ can either diffuse to the extra-matrix compartment or be scavenged by the large capacity glutathione (GSH) and thioredoxin (TrxSH₂) systems [15, 16]. In the extra-matrix compartment H₂O₂ may be additionally scavenged by catalase besides the cytoplasmic GSH and TrxSH₂ systems. These scavenging systems (except for catalase) take the electrons from NADPH either in cytoplasm or in the mitochondrial matrix, where the model also accounts for the transhydrogenase that converts NADH to NADPH. Moreover, the model explicitly accounts for the stimulation of TCA cycle dehydrogenases by Ca²⁺ and modulation of their activities by other effectors such as NADH or ADP [13].

This upgraded version of our mitochondrial model (Fig. 3) accounts for ATP synthase transporting K⁺ (and Na⁺) besides H⁺ to drive ATP synthesis, and a KHE_mito, to cycle K⁺ across the inner mitochondrial membrane, with the same kinetic parameters as a Na⁺/H⁺ exchanger (NHE) [7] as described in the minimal updated OxPhos model (see Fig. 1B). The formulation of the KHE_mito kinetics is the same as the one used to describe the NHE, in agreement with reports about a general univalent cation/H⁺ exchanger [17, 18].

Figure 3. Scheme of the integrated mitochondrial model accounting for the “core” mechanism of K⁺ and H⁺ driven ATP synthesis.

The scheme presented in Figure 1B was integrated into a bi-compartmental mitochondrial model encompassing energetics, redox, ion exchange, ROS production and scavenging (see Model description, for details). TCA, tricarboxylic acid; MCU, mitochondrial Ca²⁺ uniporter; ETC, electron transport chain.

Additionally, the model incorporates volume regulation, according to [19], that involves the dynamics of Cl⁻ anion through a charge conservation relation and matrix volume (Appendix: Eqns. D31–D32) [11]. We have also modified our original rate expression of the Na⁺ Ca²⁺ exchanger, NCX_mito [14], to include the sensitivity to matrix Na⁺ in addition to cytoplasmic Na⁺ [20].

Several simulation protocols were employed, mimicking the addition to isolated mitochondria of exogenous pulses of: (i) increasing concentrations of K⁺ to state 4 mitochondria in the presence of baseline 2 mM K⁺, or (ii) 5μM to 500μM ADP simultaneously with 0.1 μM to 0.35 μM Ca²⁺ to simulate state 4 to state 3 transitions in respiration during low-to-high energy demand. Model simulations also mimicked the assay medium of mitochondria with respect to 7.2 pH, 2mM Pi, 137mM KCl or very low (2mM) K⁺ levels (the latter chosen to avoid numerical issues related to zero K⁺).

3. Results

3.1. Simulation of the core mechanism of K⁺ and H⁺-driven ATP synthesis

First, we tested the essential mechanistic components able to sustain K⁺ and H⁺-driven ATP synthesis through F₁F_o ATP synthase in an updated minimal model of OxPhos (Fig. 1B) compared to a similar, but conventional model in which K⁺ is transported via a channel with a Goldman-Hodgkin-Katz (GHK) mechanism not linked to OxPhos (conventional GHK model) (Fig. 1A).

Compared to the conventional K⁺-transport described by the GHK model, the updated minimal model shows that, upon transition from state 4 (i.e., absence of ADP) to state 3 respiration the ATP synthesis and respiration fluxes are 2-fold and 1.6-fold larger when ATP synthase is driven by both K⁺ and H⁺ vs. by H⁺ alone (Fig. 1, compare panels C and D). This result is expected from a mechanism in which the H⁺ gradient is used either entirely through F₁F_o or partly through the KHE_mito avoiding the dissipation as heat produced by K⁺ transport not linked to ATP synthesis. Consistently, the ATP synthase H⁺ flux was larger in the GHK model both in states 3 and 4 respiration although the parameters for the ATP synthase were the same in both models. By harnessing KMF to make ATP, essentially driven by ΔΨ_m, and continuously restored by the respiratory chain-generated Δμ_H through the KHE_mito activity (in the minimal updated model, Fig. 1B), F₁F_o generates additional ATP proportional to the amount of energy that would have been dissipated as heat (in the conventional model, Fig. 1A) by the same K⁺ current in passing through a separate molecular entity functioning exclusively as a K⁺ uniporter.

The updated minimal model simulations reproduce the experimental ΔΨ_m and K⁺ uptake behavior, both under control conditions and in the presence of inhibitors (Juhaszova et al [3]). Figure 2 depicts control conditions where ADP addition triggers the transition from state 4 → 3 that elicits ΔΨ_m depolarization (~25mV), due to ATP synthase consumption of PMF and KMF to synthesize ATP (from both H⁺ and K⁺), and the mitochondrial K⁺ influx via F₁F_o resulting in matrix K⁺ accumulation (Fig. 2B). When both changes in ΔΨ_m and K⁺ influx are inhibited by preincubation of mitochondria with 10μM atractyloside, an inhibitor of the adenine nucleotide translocator (ANT), or 10μM oligomycin, an inhibitor of F₁F_o ATP synthase, the model simulations show qualitatively and semi-quantitatively the expected changes in ΔΨ_m and K⁺ influx that would happen under conditions in which mitochondrial ADP was negligible (due to ANT inhibition) or accumulated to higher levels (with ATP synthesis inhibited) because of lack of consumption by ATP synthase (Fig. 2C). The ATP synthesis flux increases under control conditions from 0.25 mM s⁻¹ to 0.939 mM s⁻¹ before and after ADP addition respectively. When the ATP synthase was inhibited the level of activity was negligible < 10⁻⁴ mM s⁻¹ whereas upon inhibition of ANT the ATP synthase reverted its activity, hydrolyzing ATP at a constant flux of 0.094 mM s⁻¹.

Figure 2. Simulations performed with the updated minimal model in the absence or presence of inhibition of ATP synthase or ANT.

Mitochondrial inner membrane potential (ΔΨ_m panel A), ADP concentration (Panel C), and K⁺ accumulation (Panel B) according to the minimal model during the state 4→3 transition, triggered by addition of 50 μM ADP. Simulations of the minimal model run without (Control, black line) or with ATP synthase (magenta line) or ANT (green line) inhibition, mimicked by decreasing the concentration of ATP synthase units (RhoF1) from 0.5 to 1^.10⁻⁵ or the Vmax of ANT from 3.15 to 0.125, respectively (see also Appendix table A1). Model simulation conditions are detailed in the caption of Figure 1. Note that when ANT was inhibited, mitochondrial ADP levels are negligible and do not change upon ADP addition. Panel D shows the rates of K⁺ uptake via ATP synthase and the KHE_mito that drive the matrix K⁺ accumulation during the transition in the absence of inhibition.

Regarding the dynamics of K⁺ transport inwards through F₁F_o and outwards through KHE_mito, the experimentally observed net K⁺ accumulation suggests a transiently faster rate of K⁺-influx via F₁F_o compared to K⁺-efflux via KHE_mito during the transition state 4→3, which was simulated by the updated minimal model (Fig. 2D). The transiently higher speed exhibited by F₁F_o over KHE_mito during the energetic transition enables matrix K⁺ accumulation until both rates converge allowing K⁺ to reach a steady state concentration.

The results presented show that an updated minimal model, capturing the essential, core, mechanistic elements, is able to display an enhanced energy (ATP and respiratory fluxes) output when, in addition to H⁺, a K⁺-transporting mechanism coupled to ATP synthesis is present. Unlike in the conventional model, the minimal model simulations show that this happens by the energy harnessed when K⁺ is transported through the ATP synthase rather than a non-ATP generating process. Importantly, the KHE_mito role, a key component of the core mechanism, is observed to operate through the kinetic lag of the exchanger with respect to ATP synthase, during the transient accumulation of K⁺ in the matrix when the synthase’s rate outpaces the exchanger’s. The effect produced by inhibitors of ATP synthase and ANT on the dynamic behavior of ΔΨ_m and K⁺ uptake during energetic transitions lend further support to the core mechanism.

3.2. Integrating the core mechanism of K⁺- and H⁺-driven ATP synthesis to a comprehensive computational model of mitochondrial function

Next, we asked whether an integrated computational model of mitochondria (Fig. 3), including the novel core mechanism above described, was able to simulate experimental data obtained with isolated rat heart mitochondria [3, 6].

First, we assessed the model’s ability to simulate experimental data in response to the addition of serial K⁺ pulses (increasing from 5 to 20 mM concentrations) to energized (St4) heart mitochondria, initially in the presence of 2 mM K⁺. Figure 4 displays the experimental time course of the signal of the K⁺ sensor PBFI (Fig. 4B), volume (90° light scattering) (Fig. 4D), and ΔΨ_m (Fig. 4F) after pulses of KH₂PO₄ [21]. Compared to the experimental data, model simulations show that, qualitatively, they reproduce well the matrix K⁺ accumulation and accompanying mitochondrial volume change (Fig. 4A, C), but the dynamic response of ΔΨ_m is only partially recapitulated (i.e., the magnitude-change is generally accurately predicted, but the kinetics are different) (Fig. 4E). Measuring the initial K⁺ uptake rate from the slope of the curves simulating matrix K⁺ accumulation (Fig. 4A) and plotting vs. K⁺ pulse concentration (Fig. 4G) the average and the maximum uptake rate following the K⁺ pulse (Fig. 4E, inset), we found a linear relationship without saturation (Fig. 4G), similar to the experimental data (Fig. 4H) with the exception of the last point that corresponds to the saturation part of the curve.

Figure 4. Simulations performed with the integrated mitochondrial model of K⁺, ΔΨ_m, and volume dynamics in response to increasing K⁺ concentrations.

Simulation conditions represent state 4 mitochondria in the presence of 2 mM K⁺ to which pulses of increasing K⁺ concentration, enough to reach final concentrations of: 5, 10, 15 or 20 mM, were added. The key to the line colors of the model results are depicted in Panel C. Panels A, C, E and G show simulation results whereas panels B, D, F and H show the corresponding experimental data of the same variable from reference [21]. The slope of K⁺ accumulation (A) was represented as an inset next to panel E together with the evolution of ΔΨ_m. Panel E and its inset show that K⁺ accumulation precedes ΔΨ_m depolarization. Plotting the maximal value of the rate of K⁺ accumulation (solid square) or its average value across the mid-range (empty square) vs. K⁺ concentration (Panel G), renders a linear relationship similar to the experimental data (H).

The disparity in the behavior exhibited by ΔΨ_m with respect to experimental and model simulations data is likely due to distinct determinants that control the net K⁺ uptake rate by mitochondria. While in the experiment ΔΨ_m showed a slow and steady depolarization (Fig. 4F), in the simulations it behaved like a mirror image of the rate of K⁺ uptake which, after a transient, returned to the value before the perturbation introduced by the K⁺ pulse (Fig. 4E). A plausibly rate-controlling step by chloride (Cl⁻) flux (e.g., large iontophoretic forces could be involved in small mismatched charge balance) and/or water flux that could cause a kinetic limitation in the transport of K⁺ that is not accounted for by the model, could underlie the differential behavior observed (see further discussion under: Model limitations).

Next, the updated integrated computational model was utilized to simulate results obtained with freshly isolated rat heart mitochondria subjected to energetic transitions in state 4 → 3 respiration, in the presence or absence of 137mM K⁺ under isosmotic conditions. Experimentally, the driving forces, ΔΨ_m and PMF, were analyzed, and simulations with the model were run to test its ability to reproduce them. Figure 5A shows that during the transition, in the presence of K⁺, ΔΨ_m decreased ~38 mV, from 206 mV in St4 to 168 mV in St3, whereas in, very low, 2mM K⁺ (simulating K⁺ “absence”, i.e., isosmotic sucrose medium) ΔΨ_m decreased ~24 mV, from 208mV to 184mV. This result was very similar to the experimental values (Fig. 5, table A1), except for ΔΨ_m which was 10 mV more polarized in the model than in the experiment (Fig. 5A). The proton motive force was similar with respect to the magnitude of change in the St4→St3 transition (9 mV difference in the presence of K⁺ vs. 5 mV in its “absence”).

Figure 5. Driving forces and fluxes of ATP synthesis and ions (H⁺ and K⁺) transported by ATP synthase in the presence of 137mM or 2mM K⁺.

The initial conditions employed for the simulations of the integrated mitochondrial model correspond to state 4 (1.10⁻³ mM cytoplasmic ADP) to which 0.5 μM ADP (State 3) was added at the time indicated by an arrow. Panel A displays ΔΨ_m following the addition of ADP. Panel B depicts the ATP synthesis calculated in the same simulations shown in A-D. In panel C the proton motive force (PMF) was calculated from ΔΨ_m and the temporal evolution of ΔpH with Eqs. C5 and D24 in Appendices C and D. The right y-axis in panel D corresponds to the K⁺ flux (solid line) and H⁺ flux (dashed line). The numbers next to the color lines in the legend represent the extramitochondrial K⁺ concentration time course during the simulation of state 4 to state 3 transition.

In comparison, model simulations showed that the ATP synthesis and V_O2 fluxes in the presence vs. absence of K⁺ were 1.8-fold and ~1.2-fold higher, respectively (Fig 5B). Experimentally, the ATP synthase flux was 3.5-fold larger in the presence vs. absence of K⁺, whereas the respiratory flux was enhanced 2.6-fold at a P:O ratio of 2.0 [3, 6]. Regarding ion fluxes through the ATP synthase, the H⁺ uptake flux was similar under both conditions whereas the K⁺ uptake rate was negligible in 2mM K⁺ and comparable to the H⁺ flux at 137mM K⁺.

Overall, in the integrated bi-compartmental mitochondrial model, the core mechanism of K⁺- and H⁺-driven ATP synthesis was able to recapitulate key experimental findings such as the increase in ATP synthesis rate, and the behavior of the ΔΨ_m and ΔμH during the state 4→3 transition. Additionally, the dynamics of key energetic variables such as volume, and rate of K⁺ uptake by energized state 4 mitochondria following K⁺ pulses, were qualitatively well simulated.

3.3. The role of the KHE_mito: From simulation to prediction

Based on this level of validation of the computational model, we proceeded to further test its predictive capabilities. Specifically, we aimed at predictions that could reveal new, hitherto unforeseen, but experimentally-testable physiological phenomena such as the potential impact of up- or down-modulation of KHE_mito, a key mechanistic element in the core mechanism of K⁺- and H⁺-driven ATP synthesis.

We assessed the bioenergetic impact produced by 2- and 4-fold down modulation of KHE_mito, simulated by decreasing the maximal rate of the exchanger in the presence of 137 mM K⁺. Figure 6 displays the time course of mitochondrial ΔΨ_m, volume, ATP synthesis, respiratory flux, and respiratory coupling ratio (RCR) during the St4-to-St3 transition at 137 mM K⁺, which, shows that all these outcomes are significantly modulated by changes in KHE_mito activity (Fig. 6A–D). As a function of KHE_mito down-modulation, ATP synthase activity decreased (by up to ~30%) (Fig. 6B) concomitantly with the rate of respiration from complex I substrates, namely NADH derived from the degradation of AcCoA and glutamate, revealing a decrease in the OxPhos coupling (Fig. 6D, Inset). With the decrease in KHE_mito activity, both ΔΨ_m and mitochondrial volume remained higher, as expected from less consumption of PMF (Fig. 6A, B). Unlike at 137mM K⁺, in 2mM K⁺ the KHE_mito down-modulation did not exert any bioenergetic effect as revealed by ΔΨ_m and the ATP synthesis rate, as would be expected (results not shown).

Figure 6. Effect of the KHE_mito activity on ΔΨ_m, matrix volume, and OxPhos fluxes during the state 4 → 3 transition.

Under similar parametric conditions to those utilized in Figure 5 in the presence of 137 mM extra-mitochondrial K⁺, model simulations were run at different values of KHE_mito activity (given as the parameter of KHE_mito protein concentration): 1.95 (black lines, used in the simulations represented in Figure 4 and 5), 1 (red lines) and 0.5 (blue lines). The Respiratory Control Ratio (inset next to panel D) represents the ratio of state 3 over state 4 respiratory flux obtained for each KHE_mito protein concentration as indicated in the x-axis.

Overall, the model simulations reveal the crucial role played by KHE_mito in modulating the energetic output of mitochondria, as indicated by the OxPhos rates (ATP synthase and respiration) and coupling efficiency. This in silico experiment predicts a significant energetic impact of KHE_mito activity down-modulation, in mitochondria subjected to energetic transitions in the presence of physiologic (137mM) K⁺, but not in sucrose (isosmotic) medium or even in the virtual absence (2mM) of K⁺ (not shown), while providing suitable readouts that should facilitate the design of a future biological experiment to address this potentially important question.

4. Discussion

Using a computational modeling approach to assess the mechanism of K⁺- and H⁺-driven ATP synthesis through F₁F_o ATP synthase, the comparison of simulations with experimental data obtained in isolated mitochondria, and its feasibility to predict integrated biological outcomes, the present work reports the following main findings: (i) the experimentally-demonstrated natural capacity of F₁F_o ATP synthase to transport K⁺ in addition to H⁺ to generate ATP bestows mitochondria with a significant energetic advantage (Fig. 1D) compared to H⁺-only utilization by F₁F_o together with K⁺ transport occurring through a putative K⁺ channel not linked to OxPhos (Fig. 1C); (ii) a minimal updated model, including the respiratory chain, ATP synthase, KHE_mito, ANT and Pi carrier, captures the essential (‘core’) mechanistic components able to reproduce the enhanced ATP synthesis and respiratory fluxes (Fig. 1B); (iii) an integrated computational model of mitochondrial function accounting for the ‘core’ mechanism (Fig. 3) was able to simulate in isolated mitochondria: 1) the mitochondrial K⁺ uptake rate and volume expansion in response to K⁺ additions (Fig. 4), 2) the increase in fluxes of ATP synthesis and respiration, and the behavior of K⁺ uptake, ΔΨ_m, PMF, and volume, during state 4→3 energetic transitions mimicking low to high energy demand at physiological K⁺ concentration (Figs. 5,6); (iv) model and confirm the fundamental role of KHE_mito as a modulator of OxPhos rate and the coupling of energy transduction, and make testable predictions about the pathophysiological relevance of down-modulating its activity (Fig. 6).

The herein newly described function of the ATP synthase is fully compatible with Peter Mitchell’s chemiosmotic theory and, by linking OxPhos to K⁺ transport coupled with the role of KHE_mito, procures a significant energetic advantage while offering a mechanistic insight into mitochondrial volume regulation and energy supply and demand matching.

Mitchell postulated that the mitochondrial membrane should have low permeability to H⁺ and other ions to allow the build-up of ΔpH and ΔΨ_m [1, 17]. However, K⁺ transport and H⁺ leak do occur which could, potentially, collapse the ion motive force driving ATP synthesis, if such transport occurs through unregulated transporters [17]. Given the large cytoplasmic K⁺ concentration, a path to avoiding this drawback would be K⁺ transport through the ATP synthase as a mechanism capable of harnessing the ion motive force, including ΔΨ_m, by taking advantage of the naturally occurring vast K⁺ abundance (vs H⁺) which overcomes the high H⁺-selectivity property of F₁F_o to drive ATP generation using K⁺.

Another potential disadvantage of uncontrolled mitochondrial swelling associated with the movement of K⁺ inward could be given by the accompanying water transport; such swelling could lead to mitochondrial lysis in a few minutes. This risk is precluded by the KHE_mito antiporter activity that restores the K⁺ gradient in exchange for H⁺ and, in turn, mitochondrial volume by restoring the osmotic balance [17].

The mitochondrial KHE has been functionally characterized since 1969 [22]. Nakashima and Garlid characterized two monovalent cation/H⁺ antiporter electroneutral activities: Na⁺/H⁺ exchanger highly selective for Na⁺ and Li⁺ and a more promiscuous one accepting K⁺, Na⁺ and Li⁺ [23]. In this century LETM1, a transmembrane mitochondrial protein, has been found to be part of the KHE_mito [24]. LetM1 is an essential gene in animals (knockout animals were not viable) [25]. In several organisms reduced expression of LETM1 brings about matrix swelling, mitochondrial fragmentation, loss of cristae, and defects of metabolism and ATP synthesis, while overexpression is characterized by matrix shrinkage and cristae swelling [24]. Note that the present predictions regarding reductions in mitochondrial ATP synthesis capacity accompanying downmodulation of KHE_mito are compatible with reduced ATP levels seen in experimental models of Letm1 deficiency and the Wolf-Hirshhorn syndrome (reviewed in [24, 26]). Although LETM1 was isolated as a component of the K⁺/H⁺ exchanger activity, it was characterized also to participate in Ca²⁺/H⁺ exchange [27]. Another possible interpretation of the altered Ca²⁺ concentrations observed in mitochondria with reduced LETM1 expression [27], is that the effect of Na⁺ that would accumulate in the mitochondrial matrix could, in turn, affect the activity of the Na⁺/Ca²⁺ exchanger as previously shown [20].

It has been claimed that an Oxphos-independent mitochondrial ATP-dependent K⁺ channel (mitoK_ATP) was responsible for K⁺ uptake across the inner mitochondrial membrane ([28]; reviewed recently [29]). Previous work about the molecular nature of the mitoK_ATP channel identified ROMK, a K⁺ channel from kidney [30]. However, very recent work from the same lab has ruled out that possibility for the heart [31]. A cardiac specific ROMK knockout did not show exacerbated damage in the hearts after ischemia reperfusion, challenging the ROMK identity as the pore-forming unit of the mitoK_ATP channel, since it is thought that cardioprotection is its main function [31]. Moreover, mitochondria isolated from ROMK KO mice exhibit similar volume regulation characteristics compared to the wild type in response to K⁺ channel openers and during oxidative phosphorylation [31].

Another line of enquiry about an OxPhos-independent molecular identity of a “mitoK_ATP” channel has recently been proposed, identified as a protein complex comprised of CCDC51, a protein of unknown function (called by the authors, “MITOK”), and the ABCB8 protein (named, “MITOSUR”) that binds ATP [32]. When reconstituted in liposomes, this protein complex displays sensitivity to diaxozide and glibenclamide. These authors provided evidence of morphological and functional alterations in K⁺ transport following deletion of the subunits of the complex in mitochondria from Hela cells.

Importantly, it has been demonstrated that F₁F_o ATP synthase (via its natural K⁺ flux) responds to the all the mitoK ATP channel pharmacology and meets all the characteristics thereof, including serving the described functions including cardioprotection [3], and in light of the energetic advantages of K⁺ flux though ATP synthase vs ordinary K⁺ channels, supports the notion that it is fully suited to serve all the known functions of a “mitoK_ATP channel.” Additionally, we have verified that purified F₁F_o is devoid of contamination with ATP-dependent ROMK potassium channel [3] as well as CCDC51 and ABCB8 (unpublished observation).

Overall, it remains unclear whether the function of putative K_ATP channels, not linked directly to the OxPhos machinery, would compromise the energetics and volume regulation of mitochondria as above mentioned. The evidence available suggests that the energetically wasteful nature of persistent inner membrane K⁺ leaks of even small amounts of electrical charge, such as through inner membrane ordinary K⁺ channels (should they exist), are unlikely to be in any abundance, and thus only serve as fine-tuning mechanisms.

4.1. Model limitations

Differences in the time course of K⁺ uptake rate and ΔΨ_m kinetics between experiments and simulations suggests that the model is not accounting for limitations introduced by another ion such as Cl⁻, modeled as instantaneously adapting its intramitochondrial concentration (calculated from the charge balance expression, see Eq. D31) rather than being a state variable. This aspect of the model needs improvement which, for now, is limited by lack of quantitative kinetic information about the dynamics of Cl⁻ channels [33]. Data from our laboratory [11] blocking Cl⁻ transport in cardiomyocytes suggested chloride as a plausible rate-controlling step of K⁺ uptake. This is of potential importance due to the large iontophoretic forces involved in small mismatches in discrete charge balance.

In the present model, the regulation of mitochondrial matrix volume mainly depends on the dynamics of K⁺ which is assumed to be the main driver of water flux across the inner mitochondrial membrane. Although the model can correctly simulate changes in matrix volume during state 4→3 energetic transitions at physiological K⁺ concentrations, there is still disagreement between experiments and simulations at very low K⁺. The model does not explicitly account for water flux across the inner mitochondrial matrix which potentially could limit ion transport. Thus, this model’s inability to reproduce the experimental data suggests that other unaccounted mechanism(s) could be involved. Possible candidates could be associated with water movement, not necessarily related to major metal cations, and the counter movement of anions.

In summary the model presented here represents a first endeavor to analyze the consequences of an updated view of ion cycles in OxPhos. According to this view, F₁F_o ATP synthase utilizes energy derived from Δμ_H to transport monovalent cations, namely K⁺ in addition to H⁺, and in turn harvests this energy to drive ATP synthesis rather than wasting it as heat. The quantity of these alkali metal ion fluxes is directly proportional to ATP synthesis and is utilized to serve various mitochondrial regulatory mechanisms that perform important metabolic and ATP supply-demand matching functions. This tool enables us to test the role of the various players in OxPhos regulation and volume control as well as interactions between several transport systems to facilitate a better understanding of the dynamics of mitochondrial bioenergetics.

Highlights.

A computational model of H⁺ and K⁺ driven F₁F_o ATP synthase was formulated.

Model simulations corroborate energetic advantage conferred by K⁺ transport through F₁F_o.

Integrated model of mitochondrial function simulates experimental data during energetic transitions.

Simulations predict key role for the K⁺/H⁺ exchanger in mitochondrial energetic performance.

Abbreviations:

PMF

proton motive force

KMF

potassium motive force

NCX_mito

mitochondrial Na⁺/Ca²⁺ exchanger

ANT

adenine nucleotide translocator

KHE_mito

mitochondrial K⁺/H⁺ exchanger

NHE_mito

mitochondrial Na⁺/H⁺ exchanger

ATP synth

F₁F_o ATP synthase

OxPhos

oxidative phosphorylation

KCO

K⁺ channel openers

GHK

Goldman-Hodgkin-Katz