Modeling control and transduction of electrochemical gradients in acid-stressed bacteria

Summary Transmembrane electrochemical gradients drive solute uptake and constitute a substantial fraction of the cellular energy pool in bacteria. These gradients act not only as “homeostatic contributors,” but also play a dynamic and keystone role in several bacterial functions, including sensing, stress response, and metabolism. At the system level, multiple gradients interact with ion transporters and bacterial behavior in a complex, rapid, and emergent manner; consequently, experiments alone cannot untangle their interdependencies. Electrochemical gradient modeling provides a general framework to understand these interactions and their underlying mechanisms. We quantify the generation, maintenance, and interactions of electrical, proton, and potassium potential gradients under lactic acid-stress and lactic acid fermentation. Further, we elucidate a gradient-mediated mechanism for intracellular pH sensing and stress response. We demonstrate that this gradient model can yield insights on the energetic limitations of membrane transport, and can predict bacterial behavior across changing environments.

Ion transporters interact emergently through shared electrochemical gradients pH-gated KcsA imparts pH sensing to F-ATPase, ultimately controlling cytosolic pH The pH sensing is mediated by Dc, which thermodynamically controls F-ATPase activity Sensitive to ion flux, Dc rapidly and efficiently responds to environmental changes

INTRODUCTION
Transmembrane electrochemical gradients act as rapid-response mediators between membrane transport, signaling, homeostasis, and metabolism in bacteria. [1][2][3] These gradients are maintained by a network of membrane-bound transporters, which together interconvert between ATP, the electrical gradient, and multiple ionic gradients. 4,5 This ''transportome'' consumes a large portion of total cellular maintenance cost (as much as 60%), but it enables more flexible and efficient transport than ATP-driven transport alone. 6 Membrane transporters frequently rely on the coupling of gradients in transport, where electrical and ionic gradients are consumed to drive otherwise unfavorable solute uptake or efflux. In addition to providing flexibility, such couplings enable survival in extreme environments. For example, acidophiles hold the electrical potential Dj to be inside-positive, enabling H + removal even at an external pH of 2 or below. 4 In this manner, electrochemical gradients provide cellular energy pools for nutrient uptake, maintenance of homeostasis, and stress response. 7,8 However, these gradients are challenging to study at the system level, as they interact via membrane transporters that both maintain the gradients and rely on them to drive transport. To understand membrane transport at a system level, the complex interplay between electrochemical gradients, transporter expression, extracellular conditions, and bacterial phenotype must be untangled. 1 The complex interactions and rapid dynamics of transport-gradient couplings have challenged experimental study at the system level. Complex interactions arise because a single transport system may influence multiple gradients, which in turn control the directionality and kinetics of many different transporters. [9][10][11] For example, the electrical gradient Dj arises from (and therefore influences) any net transport of charge across the cell membrane, including ion transport. These interactions are further confounded by rapid gradient dynamics, [12][13][14] to which the electrical gradient is particularly sensitive. A bacterial cell membrane is fully polarized to À200 mV by the movement of 10 5 ions across the cell membrane, 15,16 corresponding to only a $ 40 mM change in intracellular ion concentration. To untangle some of these interdependencies, ionophores are used to selectively extinguish ionic or electrical gradients in cells, [17][18][19] and fluorescent compounds are used to passively measure electrochemical potentials during cell growth 14,20 or in response to environmental changes. 13,21,22 These methods establish correlations between

Inherent limits of F-ATPase pH response under acid stress
The F 1 F 0 -ATPase (F-ATPase) consumes ATP to efflux H + and generate a proton-motive force (PMF) consisting of Dj and DpH. 51 Efflux through F-ATPase comprises a key part of the acid-stress response, 52-54 as evidenced by the necessity of F-ATPase for survival at low pH 55 and the increased expression of F-ATPase with decreasing pH. 56 However, PMF generation by F-ATPase becomes thermodynamically limited as the work required for H + efflux approaches the free energy of ATP hydrolysis. The amount of work required is determined by Dj, DpH, and the number of H + transported per ATP (n). The effects of these limitations have been demonstrated experimentally; for example, bacteria cannot simultaneously maintain a large Dj and a large DpH, particularly in the presence of a permeating weak acid. 57 Accordingly, we expect that Dj and the H + /ATP ratio n will limit the steady-state intracellular pH under acid-stress conditions.
To test whether our model captures these limitations, we modeled H + transport in response to weak acid stress, and calculated the dynamics of the cytosolic pH response (Figure 1) using a system of ordinary differential equations. Here, we considered the transport of H + into the cell by passive diffusion of lactic acid, and the removal of H + by F-ATPase. The rate of H + efflux by F-ATPase depends on substrate and product concentrations (ATP, ADP, and Pi) and the PMF. 58,59 To isolate DpH formation, we held constant the concentrations of cytosolic ATP, ADP, and Pi, and fixed Dj at a constant value of À100 mV (dashed lines) or À40 mV (solid lines). We chose these Dj values according to the polarized (large Dj) and depolarized (small Dj) membrane states of lactic acid bacteria under acid stress. 44 To model F-ATPase activity, we developed a kinetic expression that depends on Dj, DpH, and n, based on literature data and mechanism. 59, 60 We then considered the impact on steady-state pH of increased F-ATPase expression, which occurs during acid stress, and of changes in n, which differs between species and presents a potential target for genetic engineering. 61,62 We first modeled the dynamics of cytosolic pH recovery ( Figure 1A) and internal lactate concentration (Figure 1B) for varied F-ATPase expression level, with expression held at 1x (brown), 2x (red), or 5x (gold). In Figure 1A, the influx of lactic acid causes a rapid decrease in internal pH (initially 7.0). Dissociated lactate remained in the cytosol (1B), and the concentrations were consistent with the expected partition of lactate anion, given the pH difference. As F-ATPase pumped H + from the cytosol, the pH partially recovered in all cases. Although the steady-state pH depended strongly on Dj, it exhibited no dependence on F-ATPase expression level. Rather, the only benefit of increased F-ATPase expression level was an improvement in pH recovery dynamics, and only at small Dj. Although the expression level impacted dynamics, it did not shift equilibrium; for a given Dj, the steady-state internal pH and internal lactate concentration both remained the same.
In contrast to F-ATPase expression level, the H + /ATP ratio n controls the thermodynamics of H + efflux, as the work required for efflux is directly proportional to n. This ratio is determined by the stoichiometry of the F 0 c-ring, which can be modified by genetic engineering to produce a chimeric F 1 F 0 -ATPase with a different number of c-subunits. 62,63 To study how n influences the dynamics of cytosolic pH recovery, we repeated the simulation, but held the F-ATPase expression level constant, and instead varied n ( Figures 1C and 1D). As with the prior simulation, the cytosolic pH in 1C dropped sharply from 7.0, before partially recovering to a steady-state value ( Figure 1C). Likewise, the cytosolic lactate concentration in 1D was consistent with the expected partition ( Figure 1D). Unlike changing the F-ATPase expression level, increasing the H + /ATP ratio caused directionally opposite shifts in recovery dynamics and steady-state cytosolic pH. At small Dj, increased n produced a modest, transient improvement in pH recovery dynamics, because of an increased number of H + ions transported per ATP hydrolyzed and similar initial rates ll OPEN ACCESS iScience 26, 107140, July 21, 2023 3 iScience Article of ATP hydrolysis. However, the steady-state pH was lower (0.3 pH units), as the work required for proton efflux was directly proportional to both the ratio n and the PMF. 59 At large Dj, the increased ratio n causes a similar reduction in steady state pH (0.35 pH units), but with no apparent improvement in recovery dynamics ( Figure 1C). These results yielded insights on the energetic tradeoffs of the H + /ATP ratio in the F-ATPase acid-stress response system. At high H + /ATP ratio, F-ATPase produces greater initial H + efflux with more efficient use of ATP, but cytosolic pH recovery is reduced at steady-state. Furthermore, although changes in H + /ATP ratio can influence steady-state pH recovery, this ratio is a static quantity that is fixed for a particular species. [64][65][66] Therefore, the governing dynamic quantity for the F-ATPase response under acidstress is Dj, since the membrane depolarization (shift from large Dj to small Dj) enables cytosolic pH recovery. Figure 1. Cytosolic pH recovery driven by F-ATPase activity Dynamics of pH recovery and internal lactate accumulation under lactic acid shock at fixed membrane potential ðDjÞ.
(A) Cytosolic pH recovery for varied F-ATPase expression level. F-ATPase expression level is set to 1x (brown), 2x (red), or 5x (gold) expression, and membrane potential is fixed at À40 mV (solid lines) or À100 mV (dashed lines). Initial cytosolic pH is 7.0, and external acid-stress is 50 mM lactate at an extracellular pH of 5.0. (B) Internal lactate accumulation corresponding to (A). Initial internal lactate concentration is 10 mM. (C) Cytosolic pH recovery for varied H + /ATP ratio. H + /ATP ratio is set to 3.3 (black) or 4.0 (blue), and membrane potential is fixed at À40 mV (solid lines) or À100 mV (dashed lines). Initial cytosolic pH is 7.0, and external acid-stress is 50 mM lactate at an extracellular pH of 5.0. (D) Internal lactate accumulation corresponding to (C). Initial internal lactate concentration is 10 mM. iScience Article pH gating in KcsA imparts pH sensing and control to F-ATPase We modeled F-ATPase as the sole driver of cytosolic pH recovery under acid-stress, and demonstrated that this recovery is governed by shifts in the electrical potential Dj (Figure 1). We then modeled how shifts in Dj can occur through other ion transporters, and explore the additional properties that these shifts impart to the cytosolic pH response. Notably, our prior single-transporter model did not account for changes in Dj because of ion flux, including the H + efflux through F-ATPase. Without counter-ion transport, H + efflux through F-ATPase is Dj-limited and cannot generate a large DpH. For many bacteria, this Dj-limitation is overcome by K + influx, which depolarizes the cell membrane, thereby enabling the conversion of Dj to DpH. 17,40,44 However, this K + influx must be controlled to prevent complete depolarization, depletion of ATP, and over-alkalinization of the cytosol. 45,67 Control of K + channels can take the form of feedback regulation by ATP and ADP. 68 However, we hypothesize that this control can also be achieved by interactions between the DpH, D½K + , and Dj gradients. To investigate how this control can be effected, we extended our single-transporter model to include K + currents through KcsA. To isolate gradient interactions, we excluded consideration of K + movement through high-affinity K + transport systems such as Kdp; this exclusion is justified in environments where potassium is sufficient, as Kdp is expressed only at very low extracellular ½K + : 69 KcsA kinetics were calculated using a published kinetic expression, 12,70,71 modified here to include pH gating. 48 With this expression, we modeled the two-transporter response, where the acid stress was the same as in the prior single-transporter model ( Figure 2). In Figure 2A, we model the cytosolic pH (solid lines) and Dj (dashed lines) responses to acid stress, where KcsA is not expressed (brown) or KcsA is expressed (red).
As with the single transporter model, the cytosolic pH sharply decreased to near 5 because of the influx of lactic acid. As F-ATPase pumped H + from the cytosol, the membrane polarized; in the case where KcsA was not expressed, Dj decreased to near À100mV and prevented further H + efflux. In the case where KcsA was expressed, it activated at low cytosolic pH, enabling the influx of K + . The cationic influx caused a positive shift in Dj (Figure 2A, inset), reversing the polarization caused by F-ATPase. At a more positive Dj, H + efflux by F-ATPase resumed, and the cytosolic pH partially recovered. The response characteristics and steady-state values for Dj and cytosolic pH were largely insensitive to initial intracellular K + concentration (see Figure S3A), though experimental studies of similar transport interactions are typically conducted with K + -depleted cells. 13 Although steady-state required an hour or longer to reach, it is notable that the Dj-mediated response initiates within seconds of the acid stress event. The fast dynamics of this response are because of the cell membrane's low capacitance; as a result, small ionic currents can polarize or depolarize the cell membrane, and the membrane potential is almost completely insensitive to its initial value (see Figure S3B). The behavior of Dj and cytosolic pH are supported by the concentrations of intracellular lactate and K + , shown in Figure 2B. The lactate concentration matched the expected partition given the DpH, and the increase in K + concentration tracked closely with lactate when KcsA was expressed. At steady-state, the total PMF (sum of Dj and DpH) was constant with or without the expression of KcsA, though KcsA expression was required to transduce Dj to DpH.
Experimental measurements have demonstrated that the interchange of Dj and DpH requires extracellular potassium and occurs in a concentration-dependent manner. 44, 72 To investigate the dynamics of this behavior, we repeated the simulation where KcsA was expressed, but we varied external ½K + ( Figures 2C and 2D). Holding other conditions the same as previously, we set ½K + out to 2 mM (black), 20 mM (blue), or 200 mM (purple). We chose the lower bound for this concentration range (2 mM) such that we could exclude the consideration of additional, high-affinity K + -uptake systems. 69 In Figure 2C, the cytosolic pH sharply decreased because of lactic acid influx, accompanied by a simultaneous decrease in Dj. As the cytosol was acidified, K + influx through KcsA induced depolarization of the membrane (Figure 2C, inset), which maintained the cytosolic pH response through F-ATPase. As Dj reached a steady state, F-ATPase approached its thermodynamic limit, and the cytosolic pH recovery stalled for all K + out concentrations. Both the magnitude of the Dj depolarization response and the steady-state depolarization of the membrane increased with ½K + out . With a more depolarized membrane, F-ATPase generated a larger DpH; this is confirmed by the partition of lactate within the cytosol, which increases with ½K + out (1D). These results demonstrate that extracellular K + facilitates the recovery of cytosolic pH under acid stress, and they agree with known gradient interactions in acid-stressed bacteria: namely, that a large Dj prevents the generation of a DpH, 73 that K + -induced depolarization enables the formation of a DpH, 13,41 and that the distribution of PMF components favors DpH as ½K + out increases. 44 Membrane depolarization relaxed the PMF limitation on the F-ATPase and enabled pH recovery, such that DpH generation required KcsA expression and increased with ½K + out . When coupled, F-ATPase and KcsA ll OPEN ACCESS iScience 26, 107140, July 21, 2023 5 iScience Article generated two ionic gradients; F-ATPase generated a DpH and a Dj, whereas KcsA transduced Dj to a D½K + . KcsA mediated control of F-ATPase through Dj, because of the thermodynamic limitations imposed by the PMF. Conversely, F-ATPase controlled KcsA through the cytosolic pH, because of the pH-gated filter of KcsA (see Figure S3C for the Dj and cytosolic pH response where KcsA gating is removed). As such, these results demonstrate pH sensing by a two-way control between F-ATPase and KcsA.

Potassium uptake maintains PMF during fermentation
With the single-transporter and two-transporter models, we demonstrated how the cytosolic pH response is mediated by Dj and how the pH-gating characteristics of KcsA impart cytosolic pH control to F-ATPase. The model allows tuning of transporter expression levels, thermodynamic/gating characteristics of transporters, and extracellular stress conditions; consequently, our results generalize across systems where the same electrochemical gradients predominate and where these transporters are conserved as the means of iScience Article gradient transduction. To demonstrate a specific case application of this model, we simulated the generation of electrochemical gradients and transduction of energy during homolactic fermentation. In tandem with our two-protein stress response model, we simulated biomass, ATP, and lactic acid formation from glucose in phosphate-buffered medium containing ammonium and amino acids. To calculate growth and lactic acid production rates, we modified Monod-based expressions to include inhibition at low pH in , such that biomass formation ceases at a pH in of 6 and metabolism ceases at a pH in of 5. These values were chosen according to evidence that biomass formation ceases at a higher pH than metabolism, 19 and that lactic acid bacteria can maintain activity near an internal pH of 5. 74,75 In addition, we calculated the total (per cell) energy stored as ATP, D½K + , DpH, and Dj. From literature measurements and our prior models, we expected that K + influx would facilitate the generation of a DpH, and maintain the PMF at a near-constant value during fermentation. 13,42, 76 We also expected that the Dj and DpH would contribute a small fraction of cellular energy, because of the low electrical capacitance of the cell and the low cytosolic concentrations of H + . iScience Article Simulation results are presented in Figure 3. During fermentation, glucose was consumed to generate biomass or to generate ATP and internal lactic acid ( Figure 3A), which resulted in acidification of the cytosol ( Figure 3B). Initially, H + efflux through F-ATPase generated a large and negative Dj, which arrested further efflux ( Figure 3C). As the cytosolic pH decreased, the pH-gated KcsA channel opened, causing an influx of K + ( Figure 3B) and a corresponding positive shift in Dj. At a more positive Dj, F-ATPase generated a larger DpH, maintaining the cytosolic pH as the extracellular medium was acidified. Even though the Dj and DpH components interchange, the total PMF remained constant near 120 mV over the course of the fermentation (about 14 h, Figure 3C). As a result of the combined activities of F-ATPase and KcsA, the cytosolic pH was maintained at near neutral values for most of the fermentation.
As the DpH increased above 1.0 (about 12 h), the combined activities of F-ATPase and KcsA could no longer maintain the cytosolic pH at optimal values. The potential required to influx K + against the concentration gradient approached the value of Dj, such that no further K + influx occurred and Dj stabilized. Consequently, H + efflux through F-ATPase was slowed and eventually arrested by PMF limitations, because of simultaneous contributions of Dj and DpH. As the internal pH decreased to 6.0, biomass formation ceased. Metabolism continued to consume glucose and generate ATP and lactate until the internal pH reached 5.0, at which point ATP production ceased, DpH collapsed, and the membrane was depolarized. The cessation of metabolism and collapse of energetic gradients, or ''cell de-energization'' (Figure 3), is the direct result of the form of the Monod-based expression for metabolism, where the glycolysis rate approaches zero as the internal pH approaches 5 (see STAR Methods).
During lactic acid fermentation, the cell continuously exchanges between the energetic ''pools'' of ATP, DpH, Dj, and D½K + . 76 These pools represent the total amount of energy available to the cell through ATP hydrolysis or through dissipation of an electrical or ionic gradient. To understand the sizes and dynamics of these different energy pools, we calculate the total and specific energies stored as ATP, DpH, Dj, and D½K + gradients. Total energy pools, on a per-cell basis, are presented in Figure 3D, whereas specific energies are shown in Figure S4. The specific energy of each gradient, defined as the free energy per mole of ions transported, was about 10-20% of the energy of ATP hydrolysis. In comparing total energy capacity (pools), the energy stored in Dj was more than 1000-fold lower than the ATP pool, as expected. However, the ionic gradients held a substantial energy capacity: the D½K + pool held 10-fold greater energy than the ATP pool, and the DpH pool held as much energy as the ATP pool near the end of the fermentation. Although the capacity of the D½K + pool can be attributed to the large internal concentration of K + , the capacity of the DpH pool was surprising because of the small number of free H + per cell. 1 These results demonstrate that maintenance of homeostasis during lactic acid fermentation requires dynamic shifts in the Dj, DpH, and D½K + gradients, and that these gradients can hold a total energy comparable to, or greater than, the energy contained as intracellular ATP.

DISCUSSION
Computational models of bacteria have grown in sophistication and complexity, fueled by vast troves of available genomic, transcriptomic, and metabolomic data. [77][78][79] Yet, relatively few models have focused on the role of electrical and ionic gradients, even though these gradients constitute a substantial fraction of the cellular energy pool, 7,30 and drive transport-based uptake and stress response. 11,44,80 This is partially because of the challenges of gradient measurement, and partially because of assumptions that these gradients are mere ''homeostatic contributors.'' 1 With the advent of improved gradient indicators and imaging techniques, 14,81 studies have since highlighted the dynamic and keystone role of electrochemical gradients in environmental sensing, 2,82 cell-cell communication, 22,39 metabolism, 43 and sporulation. 83 Outside of such well-studied systems, however, much remains unclear concerning the interplay between transporters, electrochemical gradients, and bacterial behavior. 1 Due to the complex and rapid interactions of gradients and transporters, experiment alone cannot bridge this gap. 14 Electrochemical gradient modeling provides a general framework to understand these interactions and their underlying mechanisms, complementing the existing suite of bacterial models. Here, we demonstrated that a simple gradient model yields insights into the energetic limitations of transport, elucidates mechanisms for bacterial survival under stress, and predicts bacterial behavior across different environments.
In this model, we quantified the generation, maintenance, and interactions of electrical, proton, and potassium potential gradients under lactic acid-stress and lactic acid fermentation. Using kinetic models of KcsA and F-ATPase, we demonstrated that electrogenic K + transport imparts pH-sensing to the acid-stress iScience Article response. Under acid-stress, K + transport is necessary for H + efflux 40 and pH homeostasis. 11,44,79,84 This has long-been understood to be because of membrane polarization (a large and negative Dj), as PMF limitations on F-ATPase prevent the simultaneous maintenance of large Dj and DpH. 4,42 Addition of extracellular K + to potassium-starved cells depolarizes the cell membrane by dissipating Dj, enabling generation of a DpH. 17,40,44 This agrees with our two-protein model, where K + influx through KcsA depolarizes the cell membrane, enabling an increase in cytosolic pH because of F-ATPase activity. Membrane depolarization can also be accomplished by addition of a K + ionophore; at low external pH, this causes the PMF components to interchange in a similar manner. 40 At neutral pH, however, K + ionophores cause over-alkalinization of the cytosol, which does not occur with channel-mediated K + transport. 45 Our model indicates that this discrepancy is caused by pH-gating in K + transport. At high cytosolic pH, the K + channel KcsA exists in a closed state, which blocks K + influx even for large Dj. 48,85 As relatively few ions are required to polarize the cell membrane, F-ATPase generates a PMF with Dj as the major component. 40 As the cytosolic pH decreases, KcsA shifts to the open state, 85 permitting the influx of K + and subsequent depolarization of the membrane. By this mechanism, the pH-gating of KcsA controls H + efflux through F-ATPase, and therefore functions as a cytosolic pH sensor for the acid-stress response. This sensing occurs without direct interaction between F-ATPase and KcsA, and without indirect metabolite-based gating, as has been found in other systems such as TrkA/TrkH. 68 Both metabolite-based and charge transport-based emergent interactions are known to occur at the cell-cell level; such interactions drive much of the robustness and metabolic flexibility found within mixed microbial communities. 86-89 Notably, however, gradient-mediated pHsensing occurs within a single cell, arising from emergent interactions because of a shared gradient (Dj) between different transporters.
Key to this pH-sensing is the limitation of F-ATPase activity by Dj. H + efflux through F-ATPase is electrogenic, and therefore controlled by Dj. 40,59 Consistent with this notion, we found that Dj limits steady-state pH recovery under acid stress, and that over-expression of F-ATPase cannot overcome this limitation. However, F-ATPase expression does increase under acid stress, 56,80,90 though this increase is part of a multifaceted response that includes membrane depolarization. 8,19,91 With a polarized membrane (large Dj), F-ATPase is thermodynamically limited, and the kinetic differences between F-ATPase expression levels become imperceptible. With a depolarized membrane (small Dj), we found that increased F-ATPase expression caused a transient increase in cytosolic pH, which may benefit cells by limiting acid stressinduced damage. Therefore, the limitation imposed by F-ATPase expression level is purely kinetic, as the equilibrium pH is not a function of expression level. Our findings give context to a recent review, 52 which notes that increased F-ATPase expression maintains pH in in acid-stressed bacteria. To maintain H + efflux, increased F-ATPase expression must be accompanied by a depolarization response to avoid PMF-limitation. Notably, PMF-limitation of F-ATPase is multiplicative in effect with the H + /ATP ratio, which is determined by the number of c-subunits in the F 0 domain. 32,59,66 Although the c-subunit number is fixed for a given species, it varies greatly between organisms, and promotes survival specific to an extracellular environment. 61 The H + /ATP ratio determines the work required for H + efflux, which equals the free energy of ATP hydrolysis at equilibrium (no net transport). As the ratio increases, there is an increase in the efficiency: that is, the number of H + ions removed per molecule of ATP. However, there is a corresponding decrease in the power, or the maximal PMF at which H + can be removed. We found that this efficiency/power tradeoff favors a high H + /ATP ratio at low Dj. However, acidic environments do not exclude the existence of high H + /ATP ratios, which are found in some acidophilic bacteria. 63 Rather, a high H + /ATP ratio provides more efficient use of ATP, at the cost of requiring a small or positive Dj, or tolerance of a more acidic cytosol. The bacterial acid-stress and acid-shock responses can potentially be improved by manipulating the H + /ATP ratio, such as by engineering chimeric ATPases with variable c-subunit stoichiometries. 62 By quantifying the tradeoffs between cytosolic pH maintenance, Dj maintenance, and consumption of ATP, this model predicts the conditions where such engineering strategies are likely to succeed.
As previously discussed, H + efflux is controlled by the interactions of the DpH, D½K + , and Dj gradients. However, these gradients are not static quantities; rather, they function as energy pools that vary with cellular metabolism and the extracellular environment. 1,7 Accordingly, we modeled these gradients in concert with acid-fermentation metabolism and calculated the size of each energy pool. We found that the Dj and DpH gradients interchanged over the course of the fermentation, maintaining a near-constant PMF. These results show good agreement with literature data over the same external pH range, 76 94 In addition, we found that the Dj pool contains far less energy than available from ATP hydrolysis or ionic gradient dissipation; in contrast, the specific energy of Dj (or energy per monovalent ion transported) is more than 20% of the specific energy of ATP hydrolysis 95 (Figure S4). Together, these findings demonstrate the energetic advantages of Dj-mediated stress response.
Owing to its small size, the Dj pool exhibits rapid turnover and high sensitivity to changes in ion flow. 16 However, since transport favorability depends on specific energy, Dj controls transport rates while undergoing rapid changes at little energetic cost to the cell. Consequently, Dj offers a sensitive and efficient means of sensing changes to the extracellular environment and maintaining homeostasis.
The model presented here elucidates how electrochemical gradients can control the acid-stress response and how Dj can function as a highly sensitive and dynamic pH sensor and regulator. By capturing and predicting the dynamics and interactions of electrochemical gradients, this model bridges the experimental gap caused by the large number of potential interactions and by measurement techniques that fail to capture rapid dynamics. 14 The identified gradient interactions offer targets for future experimental studies, supported by the development of specialized measurement methods with improved dynamics. 96,97 By including additional transport systems and ionic gradients, this model can be adapted to other organisms and extracellular environments. This model offers insights on the control of electrochemical gradient interactions and their influence on bacterial behavior, which will inform strategies for organism engineering and growth optimization.

Limitations of the study
This model captures aspects of transport-gradient-behavior interactions in acid-stressed lactic acid bacteria, but extending these results to other extracellular conditions and organisms would require further theoretical development and experimental validation. For example, modeling bacterial behavior in a potassium-deficient medium would necessitate the inclusion of high affinity transporters, such as the potassium pump Kdp. Kdp can generate a 50,000-fold ½K + gradient 98 through the combined driving force of Dj and ATP hydrolysis, corresponding to an outward potassium ion-motive force (IMF) of greater than 250 mV. But, Kdp is repressed at extracellular potassium concentrations above 2 mM, 69 such that its effect on potassium IMF and ATP consumption can be excluded in this model. Lactic acid bacteria express other high-affinity K + -transport systems such as KupA/KupB, as well as constitutively expressed transporters such as Trk/Ktr, which predominate at near-neutral pH. 72 However, both KupA/KupB and Trk/Ktr are inhibited by cyclic-di-AMP, 99,100 which is expected to increase under acid stress. 101 Inclusion of these K + transport systems would be necessary to extend the model to potassium-deficient or higher-pH environments, or to organisms with reduced or non-existent cyclic-di-AMP. 102,103 Similarly, extending this model to respiring bacteria would require inclusion of the electron-transport chain and its effect on Dj. Additional challenges are posed by other acidophiles, such as archaea, which possess K + uptake systems that are poorly characterized. 104 In this case, both a theoretical model of the transporter and experimental validation of the transport rate expression are required for the model to be valid. Experimental validation is also particularly important for conditions where other ionic gradients predominate (e.g. saline conditions), as models for both additional transporters and additional ionic gradients (D½Na + ) would need to be developed. Lastly, multi-component model extensions are required for eukaryotes such as yeast, which encounter acid-stress in many renewable and industrial bioprocesses. [105][106][107][108][109] In this case, multiple cellular components (e.g., vacuole and plasma membrane) must be modeled in tandem, because of the combined effect of vacuolar and plasma membrane ATPases in maintaining cytosolic pH under acid-stress. 109,110 For the wide range of extracellular conditions and cellular physiologies, the increasing availability of kinetic and expression data for transport systems will aid in extending the theoretical framework developed herein. 11,30,78

STAR+METHODS
Detailed methods are provided in the online version of this paper and include the following:

Materials availability
This study did not generate new unique reagents.

Data and code availability
This paper analyzes existing, publicly available data. The references for the data are listed in Table S1. All original code is available in the supplemental information. Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

METHOD DETAILS
We model the dynamics of D pH, Dj, and D½K + during lactic acid stress, and then extend this model to include biomass and lactic acid production during fermentation. To do so, we first review the calculation of Dj and lactic acid influx rates, and derive rate equations for the transport proteins F-ATPase and KcsA. Using these rate equations, we build a system of ordinary differential equations (ODEs) to predict intracellular/extracellular solute concentrations and growth rate. We then solve these ODEs using the MATLAB solver ode15s, adapting the code framework and pHtools solver from Liao et al. 35 and Dougherty et al., 111 respectively. Model parameters are given in Table S1.

Membrane potential
Changes in Dj occur by any net movement of charge across the cell membrane; proteins that facilitate such charge transport (such as ion transporters) are considered electrogenic. We can express Dj as a function of ions accumulated across the cell membrane, 16 as given by Equation 1. For n different ionic solutes, Z is the ionic charge (Coulombs per mol), m is the amount of ions (mol) accumulated in transport across the cell membrane, and C is the cell membrane capacitance (Farads).
In electrogenic transport, ions remain close to the cell membrane or exchange with ions in the bulk solution. 16 As a result, m in Equation 1 corresponds to transport-accumulated ions, rather than the bulk ionic concentration. To relate Dj to the rate of ionic transport, we take the derivative of 1, and find, for a constant cell membrane capacitance:

Lactic acid diffusion
Under lactic acid-stress, extracellular lactate is present at an acidic external pH. Following Liao et al., 35 we assume that only undissociated lactic acid permeates the cell membrane. Accordingly, undissociated acid concentrations for external and internal lactate are calculated using Equations 3 and 4, respectively.
½HLac out = ½Lac total out 1+10 pHout À pKa Lac (Equation 3) The rate of lactic acid influx is directly proportional to the concentration difference and the permeability constant P (s À 1 ), as given in Equation 5.
For both the cytosol and extracellular medium, phosphate acts as the buffering agent, and acidification is caused by the presence or production of lactic acid. The buffering capacity of other medium constituents such as amino acids are expected to be small 112 and are therefore omitted from pH calculations.
is a membrane-bound enzyme complex that couples the efflux of protons across the cell membrane to the hydrolysis of ATP. The complex consists of the catalytic F 1 domain and the membrane-integral rotary F 0 domain, which rotates in distinct 120 + steps each catalytic cycle. 58 H + efflux generates a PMF composed of D pH and Dj; consequently, we expect F-ATPase kinetics to exhibit interdependence with both substrate/product concentrations (ATP, ADP, and Pi) and driving forces (PMF).
Here, we describe the stoichiometry and driving forces for F-ATPase, and derive a rate equation for F-ATPase kinetics using the equilibrium approximation. 113 We first develop a simplified reaction schematic from the reported mechanism, identifying fast steps and a rate limiting (slow) step. We then identify electrogenic steps, where ions move through the transmembrane potential, and incorporate PMF-dependence into those steps. Finally, we determine equilibrium K-values for fast steps from published free-energy profiles, and check the resulting rate expression with published kinetic data. This derivation should generalize to building rate expressions for other electrogenic ATPase pumps.
The stoichiometry of H + transport by F-ATPase is given in Equation 6, where n is the H + /ATP ratio: Based on the published F-ATPase mechanism, 114 we list simplified reaction steps (S1)-(S5) for the F 1 catalytic trimer complex. Each step involves one of three processes: binding/unbinding (steps (S1) and (S4), hydrolysis reaction (step (S3), and conformational change (steps (S2) and (S5). Our notation omits binding/unbinding to different catalytic subunits; rather, we denote the F 1 complex to exist in either the E1 or E2 conformation. The shift from E1 to E2 occurs by an 80 + rotation, and the return to E1 occurs upon a further 40 + rotation. As it rotates, the complex is bound to ATP, ADP, and Pi as denoted in the subscript. In accordance with published kinetic studies, Pi dissociation at step (S4) is assumed to be the rate-determining step (r.d.s.). 59,115 At step (S5), the F 1 subunit returns to the E1 state after a total of 120 + of rotation, requiring two additional catalytic cycles to complete a full rotation. Since H + transport is associated with conformational changes that occure in steps (S2) and (S5), we assume that these steps are electrogenic. 59 Under the equilibrium approximation, we assume steps S1, S2, S3, and S5 to be fast and at equilibrium. For each fast step, an equilibrium K-value relates the ratio of reactant and product concentrations, as given in Equations 7,8,9,and 11. For rate-determining step, the rate is expressed in terms of reaction intermediates, and is shown in Equation 10. In Equations 7-11, solute concentration terms (bracketed) represent unbound ATP, ADP, and Pi, while enzyme concentration terms (unbracketed) represent the fraction of total enzyme that exists in each state. ) ) ) ) Since enzyme concentrations in 7-11 are fractions of the total, we assume the total enzyme concentration is unity and write Equation 12. ) )

(Equation 19)
With Equation 19, the reaction rate is expressed in terms of solute concentrations and equilibrium K-values. For the rate-determining step, the forward rate constant k f is averaged from literature values, 116 while the backward rate constant is calculated to satisfy the equilibrium condition in (20), where the forward and backward rates must be equal: The external load of (23) is distributed between steps (S2) and (S5). We parameterize this distribution as x, where x is the fraction of the load assigned to (S2). Since the torque profile of F-ATPase is constant, 60 we assign 2/3 of the load to the 80 + rotation of (S2) and the remaining 1/3 to the 40 + rotation of (S5). Since work is equivalent to DG + i for these steps, K 2 and K 5 are recalculated using (23)  r ATPase = k f K 5;load K 3 K 2;load K 1 $½ATP À k b $½Pi½ADP ½ADPðK 5;load +1Þ+½ATPK 1 K 5;load ð1+K 2;load +K 3 K 2;load Þ (Equation 27) With Equation 27, we have developed a rate expression for F-ATPase that depends on substrate concentrations and PMF. The forward (ATP hydrolysis) reaction in expression (27) resembles Michaelis-Menten kinetics with respect to ½ATP, and the backwards (ATP synthesis) reaction resembles Michaelis-Menten kinetics with respect to ½ADP and first-order kinetics with respect to ½Pi. While the rate decreases without bound as ½Pi increases, this is a consequence of assuming that phosphate dissociation is the ratedetermining step. As we hold cytosolic phosphate concentration to be constant, the assumption is reasonable in this model.
We plot the ATP hydrolysis rate as a function of PMF in Figure S1 for different PMF values and ADP concentrations. At non-limiting ATP and ADP concentrations, the maximum F-ATPase hydrolysis/synthesis rates at high and low PMF show good agreement with other studies. 59 We can calculate the specific energy of ion transport of both H + and K + using (44). For K + , the total energy for ion transport is given by Equation 45, assuming constant extracellular ½K + : In the case of H + , equillibrium is reached when the cytosolic pH and extracellular pH are equal. However, only a small fraction of H + is available as free protons, 1 with the rest bound to phosphate or other intracellular buffers. If we consider the buffer capacity of the cytosol to provide a source of H + , we can calculate the total energy available through H + transport. The buffer capacity b is calculated using the pHtools toolkit and is given by Equation 46, where M is the concentration of acid or base: Therefore, integrating to calculate the total energy, we find: