Energy landscapes and dynamics of ion translocation through membrane transporters: a meeting ground for physics, chemistry, and biology

Abstract

The dynamics of ion translocation through membrane transporters is visualized from a comprehensive point of view by a Gibbs energy landscape approach. The ΔG calculations have been performed with the Kirkwood–Tanford–Warshel (KTW) electrostatic theory that properly takes into account the self-energies of the ions. The Gibbs energy landscapes for translocation of a single charge and an ion pair are calculated, compared, and contrasted as a function of the order parameter, and the characteristics of the frustrated system with bistability for the ion pair are described and quantified in considerable detail. These calculations have been compared with experimental data on the ΔG of ion pairs in proteins. It is shown that, under suitable conditions, the adverse Gibbs energy barrier can be almost completely compensated by the sum of the electrostatic energy of the charge–charge interactions and the solvation energy of the ion pair. The maxima in ΔG_KTW with interionic distance in the bound H⁺ – A⁻ charge pair on the enzyme is interpreted in thermodynamic and molecular mechanistic terms, and biological implications for molecular mechanisms of ATP synthesis are discussed. The timescale at which the order parameter moves between two stable states has been estimated by solving the dynamical equations of motion, and a wealth of novel insights into energy transduction during ATP synthesis by the membrane-bound F_OF₁-ATP synthase transporter is offered. In summary, a unifying analytical framework that integrates physics, chemistry, and biology has been developed for ion translocation by membrane transporters for the first time by means of a Gibbs energy landscape approach.

Graphical abstract

1 Introduction

Genetic mechanisms that govern the perpetuation of living organisms and the dynamics of evolution have received considerable attention in the biological physics literature, especially from a biological perspective of “fitness” landscapes. This is primarily because biological entities, be they molecules, cell organelles, whole organisms, or ecological entities have complex interactions with their surroundings. Landscape theory is one way of describing, visualizing, characterizing, analyzing, and quantifying the dynamics of biological systems moving and evolving in space and time. The approach has been applied to evolution and natural selection [1], accelerated evolution dynamics [2], directed evolution [3], antibiotic resistance [4], and protein dynamics [5, 6].

However, the concept of energy landscapes is quite general and is not limited to being applicable only to evolutionary problems. It has been applied to diverse systems and multi-scale processes in many fields, such as spin glasses, biological macromolecules, and protein folding [7,8,9,10]. It has also been generalized for arbitrary finite-range potentials using a geometrical approach [11].

The dynamic regulation of the transport of ions and polar metabolites through lipid barriers represents a vital attribute of enormous significance to living systems. Such transport is carried out by specialized enzymes and transporters embedded in biological membranes. To the best of our knowledge, a fundamental analysis that develops an analytical framework using Gibbs energy landscape approaches for membrane transporters in biological membranes has not been done, although some computational studies using all-atom molecular dynamics simulations and recent experimental studies using fluorescence resonance energy transfer on a few systems are available, often only in preprint form [12,13,14,15,16]. An analytical study on membrane filtration and flow through porous media treating Darcy’s permeability law by the landscapes approach has recently been published [17]. However, we could not find any study that specifically treats the biomembrane-bound F_OF₁-ATP synthase transporter. The present article attempts to fill this gap.

The unfavorable energetics of placing a charge in a non-polar medium is a well-known effect [18,19,20,21,22,23,24,25], giving rise to an energy barrier that hinders or even prohibits charge transfer across the membrane. There exists a high desolvation Gibbs energy penalty for transferring ions from water into the membrane phase owing to this high self-energy of the charges.

Classically, the technique of voltammetry has been used to study ion-transfer reactions and transport processes at the water-organic interface and in membranes [20, 21]. Such interface reactions were found to be inconsistent with the assumption of simple, unassisted, single-step transfer processes and were proposed to involve the “formation of a transient ion pair at the interface” [21]. In a detailed analysis of previous works on the transport process, it was stated that “the role of the hydrophilic ions in aqueous phases which must be distributed into the bilayer membrane together with the hydrophobic ion as the counterion to hold the electroneutrality in the membrane and aqueous phases has not been taken into account. Hence it seems difficult to understand the different features of the ion transport based on the concepts so far proposed” [20].

Several models of proton translocation through the membrane-bound F_O portion of ATP synthase have been proposed reviewed in refs. [19, 26,27,28,29,30,31,32,33,34,35,36,37]. However, none of these models has considered the large desolvation Gibbs energy penalty incurred in transferring a protonic charge from the aqueous phase into the hydrophobic membrane phase in transporters [18,19,20,21,22,23,24,25]. There exists no obvious source of energy to translocate the proton over the high desolvation Gibbs energy barrier arising from the self-energy of the charge.

It ought to be stressed that most Gibbs energy simulations in physics have focused on ion channels [38,39,40,41,42]. However, all membrane transporters including the F_OF₁-ATP synthase have very different characteristics from ion channels. Thus, transporters are alternating access, unlike channels, and translocate ions at a rate that is ~10⁴ times smaller than for channels. Every known membrane transporter family is either a cotransporter or an exchanger that is engaged in coupled co-transport/exchange of at least two species of ions or charged metabolites, unlike channels that selectively transport a single ion species. Hence, the analysis made on ion channels cannot be extrapolated to transporters, which show unique properties and conformational changes.

A novel way to solve the conundrum is offered by Nath’s two-ion theory of energy coupling and ATP synthesis [30,31,32,33,34,35,36,37, 43,44,45,46,47,48,49,50,51], formulated during a 30-year long research program on the mechanism and thermodynamics of biological molecular machines. The theory postulates the involvement of a proton and a membrane-permeable anion that approach and bind to their respective binding sites with a separation distance of ~0.5-1 nm in the access half-channels of membrane transporters and subsequently unbind to form a neutral species and exit [37, 49, 50]. This catalytic act leads to the generation of H⁺-A^- charge pair on the enzyme as the “high-energy intermediate” and results in coupling with addition and joint utilization of energy of both ionic species, leading to rotation by ion-protein interactions [30, 34, 37, 48,49,50]. Nath’s two-ion theory of energy coupling and Nath’s torsional mechanism of energy transduction and ATP synthesis [30, 34, 37, 49, 52,53,54,55, 66] thus used the transient, local electric field arising from an ordered, sequential nature of the coupled ion transport steps for mechanical rotation, yet also satisfied the constraint of overall electroneutrality of the membrane and aqueous phases [for a review, see ref. 34, 54].

We have therefore calculated the Gibbs energy landscape/barrier for ion translocation through membrane transporters and examined to what extent the stabilizing interactions due to the coupled movement of two species of ions, e.g., anions and protons, facilitate transport and contribute to lowering of the adverse desolvation Gibbs energy barrier due to the individual ions. The ΔG calculations have been performed by using the Kirkwood–Tanford–Warshel (KTW) electrostatic theory that properly takes into account the self-energies of the ions. These calculations have been compared with experimental data on the ΔG of ion pairs in several proteins. It is shown that, under suitable conditions, the adverse ΔG_desolvation can be compensated by the sum of the electrostatic Gibbs energy of the charge–charge interactions and the solvation energy of the ion pair. Hence, the system is shown to become frustrated at this point via a first order phase transition involving bistability. The maxima occurring in ΔG_KTW as a function of the interionic distance R in the bound H⁺ - A^- charge pair on the enzyme is interpreted in general thermodynamic and molecular terms, and several implications for molecular mechanisms of ATP synthesis have been discussed. Finally, the timescale at which the order parameter moves between the two stable states has been estimated by solving the dynamical equations of motion, and a wealth of novel insights into energy transduction during ATP synthesis by F_OF₁-ATP synthase have been provided.

The article is organized as follows. In the Methods, Sect. 2.1 derives the leading n = 0 and n = 1 order terms of the original Kirkwood electrostatic theory that was developed for a spherical geometry. In Sect. 2.2, the Kirkwood theory is extended to the case of a membrane with planar geometry.

In the Results and Discussion, Sect. 3.1, Poisson’s equation is integrated twice to obtain the local field and local potential profiles in the double lattice of a membrane containing localized positive and negative space charge regions. The profile of the Gibbs energy desolvation barrier for transport of a single charged species through a membrane is quantified in Sect. 3.2. The Gibbs energy landscape for translocation of a single charge and an ion pair is calculated as a function of the order parameter in Sects. 3.3, and 3.4 respectively, and the characteristics of the frustrated system with bistability for the ion pair are described in detail in Sect. 3.4. The theoretically calculated Gibbs energy landscape is compared with experimental data from NMR spectroscopy on a number of globular proteins in Sect. 3.5.

A general thermodynamic interpretation of analysis using continuum electrostatic Gibbs energy theories is carried out in Sect. 3.6. A molecular-level interpretation of the Gibbs energy landscape along with biological implications for ATP mechanism are spelled out in Sect. 3.7. The dynamics of movement of the system between two states is quantified by solving the equations of motion for torque production in ATP synthase, and a large number of novel insights into the energy transduction mechanism in ATP synthesis are given in Sect. 3.8. Finally, Sect. 3.9 briefly discusses how work on Gibbs energy landscapes in membrane transporters provides a meeting ground for physics, chemistry, and biology. The main conclusions of the study are summarized in Sect. 4.

2 Methods

The pioneering theory of Kirkwood [56] is employed in this work for evaluation of the Gibbs energy of charged particles. The theory was further explicated and developed by Tanford and Kirkwood [57], and modified later by Warshel [19, 38], as briefly described in Sect. 2.1. The Kirkwood theory was originally developed for a spherical membrane phase [56]. In Sect. 2.2, we derive an extension of the Kirkwood electrostatic theory to a planar membrane geometry, which provides a more accurate model of ion translocation through membrane transporters.

2.1 Kirkwood theory for calculation of electrostatic Gibbs energy

As mentioned above, in the original work [56], Kirkwood obtained the Gibbs energy of an arbitrary charge distribution placed inside a spherical cavity. The spherical cavity had a dielectric constant ${\varepsilon }_{\mathrm{m}}$ and radius L and was surrounded by an electrolyte solution of dielectric constant ${\varepsilon }_{\mathrm{w}}$ (Fig. 1). He solved the Laplace equation ${\nabla }^{2}\Phi =0$ in polar coordinates $\left(r,\theta ,\varphi \right)$ for the potential due to n charges ${q}_{1},{q}_{2},\dots ,{q}_{n}$ at radial positions $r,r,\dots ,{r}_{n}$ and for the reaction potential due to interaction of the charges with the surrounding solution. The work employed a combination of Dirichlet and Neumann boundary conditions which ensured that the potential and the field were continuous across the boundary of the sphere at r = L. The coefficients of the distance terms containing rⁿ in his general solution for $\Phi$ were chosen to ensure that the potential is 0 at infinity [56]. Kirkwood’s final equation for the total energy of the system, including the energy of the charge–charge interactions and the interaction between the charges and the reaction potential, can be written as

Fig. 1

Ion pair in a membrane phase in the original Kirkwood–Tanford–Warshel (KTW) electrostatic theory [19, 38, 56, 57, 61]. The membrane has a characteristic length $L$ and dielectric ${\varepsilon }_{\mathrm{m}}$. The inter-ionic distance between the charges measures R. The dimension $a$ represents the finite radius of the charges. The apolar phase is surrounded by a continuum aqueous phase approximated by the dielectric constant of water ${\varepsilon }_{w}$

$${\Delta G}_{\mathrm{K}}=\frac{1}{2}\sum_{k}^{n}\sum_{l\ne k}^{n}\frac{{q}_{k}{q}_{l}}{{\varepsilon }_{\mathrm{m}}{r}_{kl}}+\frac{1}{2{\varepsilon }_{\mathrm{m}}}\sum_{n=0}^{\infty }\frac{\left(n+1\right)(1-\gamma )}{\left(n+1\right)\gamma +n}\left(\frac{1}{{L}^{2n+1}}\right)\sum_{k}^{n}\sum_{l}^{n}{q}_{k}{q}_{l}{r}_{k}^{n}{r}_{l}^{n}{P}_{n}(\mathrm{cos}{\theta }_{kl})$$

(1)

where $\gamma =\frac{{\varepsilon }_{\mathrm{w}}}{{\varepsilon }_{\mathrm{m}}}$ and ${P}_{n}(\mathrm{cos}{\theta }_{kl})$ represents the Legendre polynomial.

The leading 1/L term (n = 0 in Eq. (1)) gives the difference in Born energy [58] when the dielectric around a charged sphere of radius L is altered from ${\varepsilon }_{\mathrm{m}}$ to ${\varepsilon }_{\mathrm{w}}$. The r²/L³ term (n = 1 in Kirkwood’s expression) represents the energy of a dipole of length r in a sphere of radius L, where r is the distance of the charge from the center of the protein. Writing the leading n = 0 and n = 1 terms in Kirkwood’s formula (Eq. (1)) for ${\varepsilon }_{\mathrm{m}}\ll {\varepsilon }_{\mathrm{w}}$ leads to the expression (in kJ mol⁻¹)

$${\Delta G}_{\mathrm{K}}=-694\left[\frac{1}{{\varepsilon }_{\mathrm{m}}}-\frac{1}{{\varepsilon }_{\mathrm{w}}}\right]\left(\frac{1}{L}+\frac{{r}^{2}}{{L}^{3}}\right)$$

(2)

Warshel later noticed that Kirkwood’s expression in Eqs. (1) and (2) did not incorporate the self-energy of the charges or the finite size of the charges [19, 38]. He therefore added the electrostatic Gibbs energy of transferring a charge of radius $a$ from the reference state (water) with $\varepsilon ={\varepsilon }_{\mathrm{w}}$ into a membranous medium with $\varepsilon ={\varepsilon }_{\mathrm{m}}$ to Eq. (2) so as to properly calculate the reference energy. Thus, the final equation of the Kirkwood–Tanford–Warshel (KTW) theory in kilojoules per mole for a single charge after including the self-energy of the charge works out to be

$${\Delta G}_{\mathrm{KTW}}={\Delta G}_{\mathrm{K}}+\frac{694}{a}\left(\frac{1}{{\varepsilon }_{\mathrm{m}}}-\frac{1}{{\varepsilon }_{\mathrm{w}}}\right)$$

(3)

Strictly speaking, Kirkwood’s expression is not applicable for contact of charges with the surrounding aqueous solution. However, the electrostatic Gibbs energy profile is continuous, and the Gibbs energy of charges in water beyond the boundary approaches 0.

Now consider the configuration shown in Fig. 1 of a positive–negative charge pair of two ions of size $a$ separated by a finite distance R embedded in a sphere of radius L of low dielectric constant, ${\varepsilon }_{\mathrm{m}}$, and surrounded by a solvent of high dielectric constant, ${\varepsilon }_{\mathrm{w}}$. For ${\varepsilon }_{\mathrm{m}}\ll {\varepsilon }_{\mathrm{w}}$, and R and L in Å, we have (in kJ mol⁻¹)

$${\Delta G(R)}_{\mathrm{K}}=-\frac{1388}{{\varepsilon }_{\mathrm{m}}R}-694\left[\frac{1}{{\varepsilon }_{\mathrm{m}}}-\frac{1}{{\varepsilon }_{\mathrm{w}}}\right]\left(\frac{{R}^{2}}{{L}^{3}}\right)$$

(4)

Adding to Eq. (4) the Warshel self-energies of taking the two ions from water to the membrane protein, we obtain in kilojoules per mole

$${\Delta G(R)}_{\mathrm{KTW}}={\Delta G(R)}_{\mathrm{K}}+\frac{1388}{a}\left(\frac{1}{{\varepsilon }_{\mathrm{m}}}-\frac{1}{{\varepsilon }_{\mathrm{w}}}\right)$$

(5)

2.2 Extension of Kirkwood theory to a planar membrane

Figure 2 shows a double lattice containing localized regions of positive and negative charges due to an ion pair that is separated by an abrupt junction. It is shown in this section that ΔG(R)_KTW can also be derived analytically for the case of a planar membrane.

Fig. 2

Extension of the KTW theory to a planar membrane. The extremities of the model lattice are taken to lie at -L and +L. Distance r is taken to be 0 at the center of the double lattice, and the electrostatic potential due to the ions is fixed as 0 at r = 0. The distance between the positive and negative charges is taken as R. The profiles of the parabolic electrostatic potential ($\psi$) and the linear electrical field (E) about an abrupt junction/interface at r = 0 are sketched. The exact equations for $\psi$ and E are obtained by solving the Poisson equation directly with the appropriate boundary conditions; for the equations, please consult the text (Sect. 3.1). The physically interesting case in which electrical double layers of the positive and negative charges interact in highly localized space charge regions, e.g., for $R<{\lambda }_{\mathrm{D}}$, or if the transporter half-aqueous access channels are characterized by the inequality $2L<{\lambda }_{\mathrm{D}}$, where ${\lambda }_{\mathrm{D}}$ is the Debye length is also treated in detail in the text

In order to evaluate the net ΔG modeled by ΔG(R)_KTW, we first consider the Gibbs energy ΔG_m of two opposite charges interacting with each other and separated by a distance R inside an infinite membrane of dielectric ${\varepsilon }_{\mathrm{m}}$ (Fig. 2). To ΔG_m, we need to add the stabilizing Gibbs energy ΔG_w of replacing the dielectric ${\varepsilon }_{\mathrm{m}}$ by the dielectric ${\varepsilon }_{\mathrm{w}}$ at distances $r>\left|L\right|$ (Fig. 2) and subtract ΔG_c, the Gibbs energy of the charges in water at infinite separation.

ΔG_m in kilojoules per mole works out to be

$${\Delta G}_{\mathrm{m}}=-\frac{694}{a}\left(1-\frac{1}{{\varepsilon }_{\mathrm{m}}}\right)-\frac{694}{a}\left(1-\frac{1}{{\varepsilon }_{\mathrm{m}}}\right)-\frac{1388}{{\varepsilon }_{\mathrm{m}}R}$$

(6)

where the first term on the right-hand side is for the positive charge, the second term is for the negative charge, and the third term on the right is a measure of the charge–charge interaction. Thus,

$${\Delta G}_{\mathrm{m}}=-\frac{1388}{a}\left(1-\frac{1}{{\varepsilon }_{\mathrm{m}}}\right)-\frac{1388}{{\varepsilon }_{\mathrm{m}}R}$$

(7)

Equation (7) represents the electrostatic Gibbs energy of the charges at infinite separation and the work of bringing them to approach each other to a separation distance, R.

ΔG_w is the stabilization of a dipole inside the membrane of dielectric ${\varepsilon }_{\mathrm{m}}$ by the aqueous solution (dielectric ${\varepsilon }_{\mathrm{w}}$) with respect to its stabilization if the medium surrounding the membrane phase also possessed a dielectric ${\varepsilon }_{\mathrm{m}}$. This is the Onsager energy of dipole solvation [59] shown to measure, for ${\varepsilon }_{\mathrm{m}}\ll {\varepsilon }_{\mathrm{w}}$ as

$${\Delta G}_{w}=-\frac{694}{{\varepsilon }_{m}}\frac{{R}^{2}}{{L}^{3}}$$

(8)

Hence,

$$\Delta G={\Delta G}_{\mathrm{m}}+{\Delta G}_{\mathrm{w}}-{\Delta G}_{c}=-\frac{1388}{a}+\frac{1388}{{\varepsilon }_{\mathrm{m}}a}-\frac{1388}{{\varepsilon }_{\mathrm{m}}R}-\frac{694}{{\varepsilon }_{\mathrm{m}}}\frac{{R}^{2}}{{L}^{3}}-\left(-\frac{694}{a}-\frac{694}{a}\right)$$

(9)

i.e.,

$$\Delta G=\frac{1388}{{\varepsilon }_{\mathrm{m}}a}-\frac{1388}{{\varepsilon }_{\mathrm{m}}R}-\frac{694}{{\varepsilon }_{\mathrm{m}}}\frac{{R}^{2}}{{L}^{3}}$$

(10)

which is the same as Eq. (5) for ${\varepsilon }_{\mathrm{m}}\ll {\varepsilon }_{\mathrm{w}}$. Hence, we have shown by analytical derivation that the KTW theory is applicable to both spherical and planar membrane phases.

Variation of the electrostatic Gibbs energy of the system is given by Eqs. (2) and (3) and Eqs. (4) and (5) (or by Eq. (10) for ${\varepsilon }_{m}\ll {\varepsilon }_{w}$) for a single ion or ion pair respectively as a function of R or by R/L in dimensionless terms (Figs. 1 and 2). This is important because, generally, change of Gibbs energy is reported by biochemical experimentalists as a function of R or R/L. In other words, the latter function is a kind of “order parameter” that is well known in the phase transition literature to physicists.

In Sect. 3.1, the order parameter used in chemistry and biology is related to the variation as a function of distance in the double lattice of the local field, E (Fig. 2), a variable used in physics.

3 Results and discussion

Figure 2 shows a lattice containing localized positive charge separated by a definite distance from a lattice with negative charge. Since the aspect ratio, i.e., the length/diameter ratio of membrane transporter access channels is large, and typically measures ~5–10, a one-dimensional configuration space is a good approximation and presents a useful simplification. The electrical potential and electrical field profiles can therefore be obtained by direct solution of the one-dimensional Poisson equation. In essence, our approach represents a generalization of the Shockley analysis in semiconductor physics that uses “electrons and holes” [60] to classical particles in ionic systems containing localized fixed negative and positive charges and mobile ions.

3.1 Profile of local electrical field and potential in a biological membrane containing localized space-charge regions of positive and negative charge

For both geometries shown in Figs. 1 and 2, we can write Poisson’s equation as

$$\frac{{d}^{2}\psi }{d{r}^{2}}=-\frac{4\pi \rho }{\varepsilon }$$

(11)

On one side of the midpoint/center at r = 0, since the intrinsic field is the negative gradient of the local electrical potential, i.e., $E=-\frac{d\psi }{dr}$, we can write, for a single molecule with unit charge density

$$\frac{dE}{dr}=\frac{4\pi q}{\varepsilon }$$

(12)

which upon integration gives

$$E=\frac{4\pi q}{\varepsilon }r+C$$

(13)

Noting the boundary condition that

$$E\ =\ 0\ \mathrm{at}\ r=-L$$

(14)

$$E=\frac{4\pi q}{\varepsilon }\left(r+L\right)$$

(15)

Similarly in the region on the other side of r = 0, we obtain

$$E=-\frac{4\pi q}{\varepsilon }\left(r-L\right)$$

(16)

Integrating again to obtain the potential, with the boundary condition that

$$\psi =0\ \mathrm{at}\ r=0$$

(17)

we finally have, on one side of the center,

$$\psi =-\frac{2\pi q}{\varepsilon }\left[{\left(r+L\right)}^{2}-{L}^{2}\right]$$

(18)

and on the other side of the center/midpoint,

$$\psi =\frac{2\pi q}{\varepsilon }\left[{\left(r-L\right)}^{2}-{L}^{2}\right]$$

(19)

The linear local field profile and the parabolic profile of the local electrical potential about r = 0 is sketched in Fig. 2. Hence, the order parameter given by R or by R/L can also be related to the local field, E, given its linear profile. Experimental chemists and biologists who work with NMR spectroscopy or X-ray diffraction conventionally show their results as a function of R or R/L, probably because of the difficulty of measurement of the local field within the coupling membrane, so we shall continue to follow this convention. This is all the more reasonable given the fact that we shall compare Gibbs energy landscape predictions for membrane transport with experimental data on ion pairs in various proteins subsequently (Sect. 3.5; Fig. 7).

3.2 Calculation of the desolvation Gibbs energy barrier for ion transport through membranes

The desolvation Gibbs energy (kJ mol⁻¹) arising from the self-energy of a single charge can be calculated by the Born-type formula [19, 24, 38, 61]

$${\Delta G}_{\mathrm{desolvation}}=\frac{694{q}^{2}}{a}\left(\frac{1}{{\varepsilon }_{\mathrm{m}}}-\frac{1}{{\varepsilon }_{\mathrm{w}}}\right)$$

(20)

where q = +1 for H⁺, $a$ is the hydrated radius of the ion in angstroms (on the order of 1 Å), and ${\varepsilon }_{m}$ and ${\varepsilon }_{w}$ are the membrane and water dielectrics, respectively. For ${\varepsilon }_{\mathrm{m}}$ of 20, this approximately yields a maximum value of ΔG_{desolvation energy} of ~34 kJ mol⁻¹ (Fig. 3). The Gibbs energy profile at an arbitrary point z within a membrane of thickness l (in Å) is given in kilojoules per mole by [61]

Fig. 3

Desolvation Gibbs energy barrier for unassisted translocation of a single ion species (e.g., H⁺) through a biological membrane. The profile is calculated as a function of distance, z, for transfer of a unit H⁺ charge from water into a membrane of thickness l = 50 Å with an effective dielectric constant ${\varepsilon }_{\mathrm{m}}=20$ (bold blue curve). A “protonmotive force” (Δp) of 200 mV (~ 20 kJ mol⁻¹) is shown by the dotted black line. The Gibbs energy barrier after superposition of the Δp is also illustrated (dashed orange curve), still revealing a substantial barrier to non-facilitated ion transport. Estimations of the effect of surface charges on the membrane at the edges z = 0, l are also shown

$${\Delta G(z)}_{\mathrm{desolvation}}=\frac{694{q}^{2}}{{\varepsilon }_{\mathrm{m}}}\left[\frac{1}{a}-\left(\frac{1}{4z}+\frac{1}{4\left(l-z\right)}\right)\right]$$

(21)

which is plotted in Fig. 3 for l = 50 Å with the constraints that ΔG tends to 0 beyond the membrane edges, i.e., at $z=-a$ and $z=l+a$. Several workers use ${\varepsilon }_{\mathrm{m}}=4-8$ and estimate considerably higher values for the desolvation Gibbs energy barrier in the range of 70–135 kJ mol⁻¹ [24, 25]. Here we have taken a higher value of the dielectric in order to estimate the minimum value of the Gibbs energy barrier/landscape for unassisted proton translocation through biological membranes (Fig. 3). An estimate of the effect of surface charges at the membrane boundaries at z = 0 and z = l is also shown in Fig. 3.

Current models of ion transport through membranes do not explain how this large desolvation Gibbs energy penalty is paid for by proton-translocating enzyme systems, such as the F_OF₁-ATP synthase. There exists no obvious source of energy to translocate the proton over the high desolvation Gibbs energy barrier. It should be noted that the delocalized potential, φ or the delocalized “protonmotive force” of the chemiosmotic theory, Δp of ~200 meV (~20 kJ mol⁻¹), is far smaller than the estimated desolvation Gibbs energy and cannot flatten the barrier (Fig. 3). Moreover, if Δp is used for reducing the height of the barrier, then no energy is available for ATP synthesis as per the chemiosmotic theory, because the driving force for ATP synthesis, Δp, has already been consumed to reduce the desolvation Gibbs energy penalty arising from the ion self-energy.

Increasing ${\varepsilon }_{\mathrm{m}}$ to 15 or 20 decreases the equilibrium desolvation Gibbs energy barrier; however, it also increases the proton transfer reorganization energy, and hence, the activation barrier for proton translocation remains almost the same. Hence, the basic contradiction of the problem of too high a Gibbs energy barrier for charge transfer across the membrane is not resolved.

Nath’s two-ion theory of energy coupling and ATP synthesis [30, 34, 37, 47,48,49,50] offers a way out of the above impasse. Since, according to the theory, the H⁺ approaches the A⁻ very closely (r ~0.5 – 1 nm) in access half-channels in the membrane, and the H⁺ binds within the electrostatic field of the bound A⁻, this anionic electrical field should help lower the desolvation Gibbs energy barrier considerably. The anionic field will decrease the equilibrium value of the H⁺ energy but will not increase the medium reorganization energy. It may even be possible to decrease the desolvation Gibbs energy penalty and barrier to 0 under special conditions.

Hence, in subsequent sections, we shall explore the electrostatic properties of ion pairs in a medium of low dielectric (membrane) surrounded by a medium of high dielectric (e.g., water).

3.3 Calculation of Gibbs energy landscapes for a single-charged species in a membrane transporter

The electrostatic Gibbs energy landscape for unassisted translocation of a single ion (e.g., H⁺) in a membrane transporter is calculated based on Eq. (3) of the KTW continuum electrostatic theory. The dependence of the Gibbs energy as a function of distance of the charge from the center is shown in Fig. 4 for ${\varepsilon }_{\mathrm{m}}=20$ and various parameter values of $L/a$. An important feature that emerges from Fig. 4 is that the Gibbs energy asymptotes at $\frac{r}{L}=0$ to the destabilizing value tabulated in Table 1; therefore, the energy barrier cannot be lowered to 0 for such a charge configuration involving only a single type of ion. Furthermore, no maxima in the Gibbs energy landscape as a function of the dimensionless distance or order parameter is found. These results are robust with respect to the choice of parameters. This arises primarily from an absence of coupling, a lack of synergistic action, and the inability of formation, in the absence of the anion, of the low Gibbs energy HA species, and the diffusion of this neutral species thereafter (i.e., after the elementary event of molecular energy transduction is completed) from the membrane phase to the outside aqueous medium. Further interpretation from the point of view of molecular mechanism is given in Sect. 3.7. The results (Fig. 4; Table 1) are quite in contrast to those obtained for an ion pair (Sect. 3.4).

Fig. 4

Electrostatic Gibbs energy for a single type of ion/charged group of diameter $a$ placed inside a membrane phase of characteristic length L and dielectric ${\varepsilon }_{\mathrm{m}}=20$ surrounded by water of dielectric ${\varepsilon }_{\mathrm{w}}$ $\left({\varepsilon }_{\mathrm{m}}\ll {\varepsilon }_{\mathrm{w}}\right)$. Calculations of the Gibbs energy profiles are shown as a function of the order parameter r/L for various parameter values of $L/a$ (from top to bottom) of 5.0 (gray), 3.5 (orange), and 2.5 (blue). The ΔG value is very sensitive to the position of placement of the charge. Note also that the electrostatic Gibbs energy asymptotes at $\frac{r}{L}=0$ to the value tabulated in Table 1; therefore, the energy barrier cannot be lowered to 0 in such a charge configuration

Table 1 Expressions used and typical values of the asymptote for the electrostatic Gibbs energy change, ΔG, and the order parameter, r/L, at which the asymptotic value of ΔG is reached for a single ion $({\varepsilon }_{\mathrm{m}}\ll {\varepsilon }_{\mathrm{w}})$. Similarly, the maxima in ΔG and the typical value of the order parameter, R/L, for which the maxima is obtained, are tabulated for the case of a monovalent cation–anion ion pair. Both calculations are performed based on the KTW electrostatic theory

3.4 Calculation of Gibbs energy landscapes for an ion pair in a membrane transporter

In this and subsequent sections, we have attempted to determine the extent to which stabilizing interactions due to the coupled movement of two species of ions, e.g., anions and protons, facilitate transport, and contribute to lowering of the adverse desolvation Gibbs energy barrier due to the individual ions. The ΔG calculations have been performed by using the Kirkwood–Tanford–Warshel (KTW) electrostatic theory that properly takes into account the self-energies of the ions. These calculations have been compared with experimental data on the ΔG of ion pairs in a subsequent Section (Sect. 3.5). It is shown that, under suitable conditions, the adverse ΔG_desolvation can be almost completely compensated by the sum of the electrostatic Gibbs energy of the charge–charge interactions and the solvation energy of the ion pair. The maxima occurring in ΔG_KTW as a function of the interionic distance R in the H⁺ – A^– charge pair (Table 1) is interpreted in thermodynamic and molecular terms (Sects. 3.6 and 3.7, respectively) and various biological implications for molecular mechanisms of energy transduction and ATP synthesis that arise have been highlighted (Sect. 3.7). The results provide an illustration of frustrated systems with interesting bistable behavior in the physics of first-order phase transitions. The timescale at which the system moves between the two states has also been quantified for membrane transporters for the first time (Sect. 3.8).

As derived in Sect. 2 for both spherical and planar membranes (see Eqs. (5) and (10)), the basic equation of the KTW theory for the Gibbs energy of an ion pair separated by a finite distance R, which includes the Gibbs energy of taking the ions from water into the membrane, ΔG_KTW(R), can be recast as follows:

$${\Delta G}_{\mathrm{KTW}}\left(R\right)=-\frac{1388}{{\varepsilon }_{\mathrm{m}}R}-694\frac{{R}^{2}}{{L}^{3}}\left[\frac{1}{{\varepsilon }_{\mathrm{m}}}-\frac{1}{{\varepsilon }_{\mathrm{w}}}\right]+\frac{1388}{a}\left[\frac{1}{{\varepsilon }_{\mathrm{m}}}-\frac{1}{{\varepsilon }_{\mathrm{w}}}\right]$$

(22)

The Gibbs energy balance associated with charge separation in membrane transporters is depicted schematically in Fig. 5. The dependence of the various components of ΔG_KTW (Eq. (22)) on the ion pair interionic distance, R, with respect to the reference Gibbs energy of the ions at infinite separation in water is quantified. The specific calculations shown are for ${\varepsilon }_{\mathrm{m}}=20$, $a$ = 2 Å , and a spherical size (or separation 2L in the planar geometry) of 1 nm, but the same trends are obtained for any reasonable set of parameters. ΔG_{desolvation energy} due to the self-energy of the charges shows up as a large positive destabilizing value that is independent of R. On the other hand, ΔG_{charge-charge} is negative (stabilizing), scales as –(1/R), and goes from a negative infinity value as the bonding distance tends to 0, as would occur, for example, in the neutral substrate HA, to a value close to 0 for long-range ion pairs (Fig. 5). ΔG_{dipole solvation}, the stabilizing Onsager energy, follows a parabolic –(R²) relationship that goes from 0 at a bonding distance between the ion pair to a large negative value for long-range ion pairs (Fig. 5). From the various contributions shown in Fig. 5, it can be readily inferred that an apparently small value of ΔG_KTW(R) can result at small (R < 4 Å), for a neutral substrate HA, and for long-range ion pairs (R ∼ 8 – 10 Å)  due to compensation of the destabilizing self-energy term and the sum of stabilizing Gibbs energy contributions of ΔG_KTW(R) (Fig. 5). Furthermore, a maxima in ΔG_KTW(R) is expected to be found at an intermediate value of the interionic distance, R, corresponding to a H⁺ – A^– “high-energy charge-pair intermediate”.

Fig. 5

Energy balance associated with Gibbs energy landscapes for ion pair translocation through membrane transporters. The variation of the components of ΔG on the ion pair interionic distance, R, is plotted. The large destabilizing desolvation Gibbs energy penalty due to the self-energy of the charges is shown (blue). The stabilizing Gibbs energy of the charge–charge interaction scales as –(1/R) (black), while the stabilizing Onsager energy for dipole solvation (red) follows a –(R²) relationship. Calculations are based on Eq. (22) for ${\varepsilon }_{\mathrm{m}}=20$, $a$ = 2 Å  and a protein characteristic size of 1 nm. Similar trends are obtained for other sets of parameter values. The reference Gibbs energy of the ions at infinite separation in water is taken as 0

Figure 6 calculates the Gibbs energy of an ion pair with inter-ionic separation R placed inside a membrane phase ${(\varepsilon }_{\mathrm{m}}=20)$ surrounded by water of dielectric ${\varepsilon }_{\mathrm{w}}$ $\left({\varepsilon }_{\mathrm{m}}\ll {\varepsilon }_{\mathrm{w}}\right)$. Calculations of the Gibbs energy landscapes are shown as a function of the order parameter R/L for various parameter values of $L/a$. Recombination of the two charges of the ion pair (R → 0) after ion unbinding from their respective binding sites in the membrane transporter leads to a lowering of the Gibbs energy barrier to 0 in such a paired charge configuration.

Fig. 6

Electrostatic Gibbs energy of an ion pair with inter-ionic separation R placed inside a membrane phase ${(\varepsilon }_{\mathrm{m}}=20)$ surrounded by water of dielectric ${\varepsilon }_{\mathrm{w}}$ $\left({\varepsilon }_{\mathrm{m}}\ll {\varepsilon }_{\mathrm{w}}\right)$. Calculations of the Gibbs energy landscapes are shown as a function of the order parameter R/L for various parameter values of $L/a$ (from top to bottom) of 3.5 (gray), 3.0 (orange), and 2.5 (blue). Note that the electrostatic Gibbs energy shows a maxima at intermediate values of R/L, as tabulated in Table 1. Recombination of the two charges of the ion pair (R → 0) after ion unbinding from their respective binding sites in the membrane transporter leads to a lowering of the Gibbs energy barrier to 0 in such a paired charge configuration. In this mode of ion translocation, the two ions move in a coupled way and bind specifically in correlated states at a finite distance of separation, R, before their unbinding from their sites occurs

The electrostatic Gibbs energy involving an ion pair (Fig. 6) can be regarded as an interesting example in physics of a frustrated system with bistability that shows a maxima at intermediate values of R/L. The system reveals two values of the order parameter, R/L, with the same minimum value of Gibbs energy (Fig. 6). Thus, the net ΔG_KTW tends to a value of ~0 at a finite separation distance of H⁺ and A^- owing to almost complete Gibbs energy compensation. A second stable state with a minimum in total Gibbs energy is obtained for formation of a chemical bond (i.e., for R → 0) to yield the charge neutral substrate HA. In between these two stable states, there exists a barrier in the Gibbs energy landscapes (Fig. 6) with a high Gibbs energy state of H^- – A^- at a definite distance of separation. The maxima in ΔG_KTW for the ion pair system occurs at

$${R}_{\mathrm{max}}=\frac{L}{{\left({\varepsilon }_{\mathrm{m}}\left[\frac{1}{{\varepsilon }_{\mathrm{m}}}-\frac{1}{{\varepsilon }_{\mathrm{w}}}\right]\right)}^\frac{1}{3}}$$

(23)

or, for ${\varepsilon }_{\mathrm{m}}\ll {\varepsilon }_{\mathrm{w}}$ at ${R}_{\mathrm{max}}\approx L$. For ${\varepsilon }_{\mathrm{m}}\ll {\varepsilon }_{\mathrm{w}}$, this yields the maximum Gibbs energy change (in kJ mol⁻¹)

$${\Delta G}_{\mathrm{KTW},\mathrm{ max}}=\frac{694}{{\varepsilon }_{\mathrm{m}}}\left[\frac{2}{a}-\frac{3}{L}\right]$$

(24)

These values are tabulated in Table 1. The difference between the maximum value of the Gibbs energy barrier and the ground state Gibbs energy (Table 1; Figs. 6 and 7) is available to perform useful mechanical work in a membrane transporter such as the F_OF₁-ATP synthase. This works out over a 120° catalytic cycle of ATP synthesis to a stored energy in a nonequilibrium state of an enzyme subunit — or in a local region/mechanical degree of freedom of the system — corresponding to ~60 kJ mol⁻¹, a quantum of energy that is sufficient to synthesize one mole of ATP [47, 50, 54, 62].

Fig. 7

Comparison with experimental data of the calculated Gibbs energy landscapes for ion pairs upon their transfer from water into the membrane transporter. The ΔG of the various components and the total ΔG are calculated by the KTW electrostatic theory. The sum of the desolvation Gibbs energy of the two species of ions (horizontal blue line), the Gibbs energy change of charge-charge interactions between positive and negative ions (black curve), the Gibbs energy change of dipole solvation (orange curve), and the net ΔG change (thick ocher curve) are plotted. Calculations are shown as a function of R for ${\varepsilon }_{\mathrm{m}}=20$ and $a=2$ Å for a 1-nm diameter of an individual c-subunit constituting the c-oligomer in the membrane-bound F_O portion of a single molecule of the F_OF₁-ATP synthase. The reference Gibbs energy of the ions far apart in water is taken as the boundary condition, which asymptotes to 0 beyond the right edge of the figure. Experimental measurements of the total Gibbs energy change for interactions in proteins and in water are plotted, and mean and SD values are shown. Estimates are based on NMR ensemble data for long-range ion pairs (open square) [77]; heme groups and surface charges in cytochrome c (open circle) [78]; ion pairs in hemoglobin, lysozyme, and cytochrome c (filled circle) [79,80,81]; and dicarboxylic acids in water (filled triangle) [38]

The results shown in Fig. 6 constitute a classic example of a first-order phase transition in physics with a free energy barrier separating two values of the order parameter with the same minimum value of Gibbs energy. The timescale at which the order parameter moves abruptly between the two values has proved difficult to define for many physical systems. However, for membrane transporters, the timescale at which the order parameter transitions in a discrete step between the two stable states has been calculated for the first time by solving the dynamical equations of the system (see Sect. 3.8).

3.5 Comparison of theoretically predicted landscape with experimental data on proteins

A major goal of this work is to calculate of the Gibbs energy landscape upon moving an ion pair from water to the low dielectric medium of a membrane transporter and compare them with experimental ensemble NMR data. The ΔG of the various contributions for such a transfer and the total ΔG have been calculated from KTW theory based on Eq. (22) and plotted in Fig. 7. The effective radius of H₃O⁺ is estimated using the Born formula [58] for ${\varepsilon }_{\mathrm{m}}\ll {\varepsilon }_{\mathrm{w}}$ as $a=-694/{\Delta G}_{\mathrm{sol}}$, where ΔG_sol is the solvation Gibbs energy of H₃O⁺ in water, which is taken as –418 kJ mol⁻¹. Calculations are shown as a function of R for ${\varepsilon }_{\mathrm{m}}=20$, $a$ = 2 Å , and 1 nm effective size of an individual c-subunit [28, 30] that constitutes the c-ring in the membrane-bound F_O portion of ATP synthase (Fig. 7). The reference Gibbs energy of the ions far apart in water is taken as 0. Experimental estimates of the total Gibbs energy change for interactions in various proteins and water are shown, and mean and SD are also given. The calculated values can thereby be readily compared with experimental data on ion pairs, and reasonably good agreement between theory and experiment is obtained (Fig. 7).

3.6 Thermodynamic interpretation of continuum electrostatic Gibbs energy theories

A thermodynamic interpretation can also be provided for the Gibbs energy calculations in Sects. 3.4 and 3.5 and the results shown in Figs. 5–7. It can also help elucidate if pairing of ions of opposite charge can assist ion permeation and translocation in membrane transporters. Let the two ions be labeled H⁺ and A^– and let the aqueous and apolar membrane phases be given the symbols w and m, respectively.

We define ${\Delta G}_{{\mathrm{H}}^{+},\mathrm{ w}\to \mathrm{m}}$ and ${\Delta G}_{{\mathrm{A}}^{-},\mathrm{ w}\to \mathrm{m}}$ as the Gibbs energies of transfer of the individual protons and anions from water into the membrane phase. Furthermore, $\left({\Delta G}_{{\mathrm{H}}^{+}{\mathrm{A}}^{-}, \mathrm{m}}-{\Delta G}_{{\mathrm{H}}^{+}{\mathrm{A}}^{-},\mathrm{ w}}\right)=\Delta {\Delta G}_{{\mathrm{H}}^{+}{\mathrm{A}}^{-},\mathrm{ w}\to \mathrm{m}}$ is defined as the change in interaction/binding Gibbs energy upon transfer of the ion pair from aqueous medium, w, into the membrane phase, m. With these definitions, the Gibbs energy change for transfer of the ion pair from phase w to phase m, ${\Delta G}_{{H}^{+}{A}^{-},\mathrm{ w}\to \mathrm{m}}$ is given by the general thermodynamic equation

$${\Delta G}_{{\mathrm{H}}^{+}{\mathrm{A}}^{-}, \mathrm{w}\to \mathrm{m}}=\left({\Delta G}_{{\mathrm{H}}^{+}{\mathrm{A}}^{-},\mathrm{ m}}-{\Delta G}_{{\mathrm{H}}^{+}{\mathrm{A}}^{-},\mathrm{ w}}\right)+{\Delta G}_{{\mathrm{H}}^{+},\mathrm{ w}\to \mathrm{m}}+{\Delta G}_{{\mathrm{A}}^{-},\mathrm{ w}\to \mathrm{m}}$$

(25)

Equation (25) enables interpretation of various continuum electrostatics Gibbs energy transfer theories in terms of chemical thermodynamics. Thus, ΔG(R)_K in Eq. (4) of Sect. 2.1 that contains the leading n = 0 and n = 1 order terms of Kirkwood’s theory (Eq. (1); Sect. 2.1) can now be interpreted as an estimate of the stabilizing change in interaction Gibbs energy, ${\Delta\Delta G}_{{H}^{+}{A}^{-},\mathrm{ w}\to \mathrm{m}}$ upon transfer of the ion pair from a polar phase, w, into an apolar membrane phase, m. These favorable interactions are interpreted electrostatically by the Kirkwood theory as a sum of charge–charge and Onsager dipole solvation energies and are quantified in Figs. 5–7. The destabilizing ${\Delta G}_{{\mathrm{H}}^{+},\mathrm{ w}\to \mathrm{m}}+{\Delta G}_{{\mathrm{A}}^{-},\mathrm{ w}\to \mathrm{m}}$ terms in Eq. (25) are satisfactorily taken into account by Warshel’s important extension of Kirkwood-Tanford theory in order to consider the finite size of the charges along with the appropriate inclusion of the charge self-energies. These destabilizing components are given by the second term on the RHS of Eq. (5) or the first term on the RHS of Eq. (10) within KTW theory.

The above has several thermodynamic consequences. For example, transfer of a single ion such as H⁺ from phase w to phase m entails a desolvation Gibbs energy penalty ${\Delta G}_{{H}^{+},\mathrm{ w}\to \mathrm{m}}$ that is quantified in KTW theory by the charge self-energy $\frac{694}{a}\left[\frac{1}{{\varepsilon }_{\mathrm{m}}}-\frac{1}{{\varepsilon }_{\mathrm{w}}}\right]$ (kJ mol⁻¹). Models of energy coupling, such as the chemiosmotic theory, that fail to include key self-energy terms and ignore fundamental charge desolvation effects cannot provide an adequate description of ion translocation through the F_OF₁-ATP synthase and other membrane transporters.

Finally, other semi-macroscopic or microscopic electrostatic theories can also be interpreted within the general thermodynamic framework given by Eq. (25), as shown here for the KTW theory. Thus, with progress, electrostatic Gibbs energy theories can be embellished and further sophisticated to help provide ever-increasing and more detailed insights into the contributions and effects of the various stabilizing and destabilizing interactions in enzymes, transporters, and other biological molecular machines.

3.7 Molecular interpretation and biological implications

Figure 7 illustrates that the net ΔG_KTW tends to a value of ∼0 at a finite separation distance of H⁺ and A^– (~10 Å under the conditions in Fig. 7) owing to complete Gibbs energy compensation due to ion-ion coupling by local fields. In principle, this localized coupling within membrane transporters arises physically due to interaction between the diffuse electrical layers around the positive and negative space-charge regions (Sect. 3.1). More rigorously, if $\lambda$ is the inter-ionic distance between the bound positive and negative charges, and ${\lambda }_{\mathrm{D}}$ is the Debye length in the aqueous electrolyte of the half-access channels of the membrane transporter, we should have, for coupling

$$\lambda \sim R \approx L<{\lambda }_{\mathrm{D}}$$

(26)

For a univalent 10-mM electrolyte at 25 °C, the Debye length, ${\lambda }_{D}$ measures 3 nm. Hence, as seen, for example, from Fig. 7, the inequality given by Eq. (26) is satisfied for access channels within membrane transporters. Thus, an analysis of system scale sizes helps shed light on molecular mechanism [63].

On the other hand, in the electrolyte solution/interface with the overall membrane, e.g., periplasm/cytoplasm, or matrix/inner membrane in mitochondria, an analysis of scaling in the mesoscopic system [63] shows that $\lambda >{\lambda }_{\mathrm{D}}$. Hence, for the membrane as a whole, the electrical field in the surrounding aqueous phases does not persist beyond length scale ${\lambda }_{D}$, and hence, there can be no overlap of electrical layers of the charges in the solution. Thus, these aqueous spaces are electrically quasi-neutral.

The principle can be formulated that in order to create conditions for overlap of electrical layers (and thereby local coupling), i.e., ${\lambda <\lambda }_{\mathrm{D}}$, a localized confined region or space within the membrane is necessary, as for example, around an access channel. This differs from the conventional capacitance involving the entire membrane leaflet separating two electrolyte solutions on either side of it, generally shown as charged. The charges are considered to be located either in the bulk aqueous media (as in chemiosmosis) [27, 64] or at the membrane surface [26, 65].

Moreover, as explained in detail in Sect. 3.4, a second stable state with a minimum in total Gibbs energy is obtained for formation of a chemical bond (i.e., for R → 0) to yield the charge neutral substrate HA. In between these two stable states, there exists a high Gibbs energy state of H⁺ – A^– on the enzyme, i.e., [E−H⁺A^-]^*, at a definite distance of separation, R (see Eqs. (23) and (24); Table 1; Figs. 6 and 7).

Thus, at the molecular level, oppositely charged ions translocate from a bulk aqueous phase along their concentration gradients sequentially and in a coupled way and bind to their respective binding sites in the membrane of a single enzyme/transporter (i.e., cotransporter or exchanger) molecule, and local fields are created in the membrane transporter around the ion-binding sites. Charge separation of the ion pair thus occurs and the bound H⁺ – A^– charge pair on the enzyme/transporter can be regarded as an “high energy intermediate.” This high-energy state of the enzyme is subsequently discharged and utilized for the performance of useful work by ion-protein interactions. During this energy transduction process, the Gibbs energy maximum is lowered by the interaction of the bound charges with the protein in the membrane (ion-protein interactions), e.g., at the sharp a–c interface/junction in the membrane-bound F_O portion of ATP synthase [30, 34, 49, 53]. This interaction has conformational characteristics and leads to rotation in the membrane-bound F_O portion of ATP synthase by means of a mechanoelectrochemical process that is transmitted to the extra-membrane F₁-portion of ATP synthase, as explained in detail by Nath’s torsional mechanism of energy transduction and ATP synthesis [30, 34, 49, 52,53,54,55, 66, 67]. After unbinding from their respective binding sites in the membrane, the H⁺ and A^– recombine to form neutral substrate HA, which is the second state of low ΔG_KTW, and thereafter the HA molecule leaves the enzyme. In summary, the separation and recombination of ion pairs constitute the elementary events in biological energy transduction and storage in membrane transporters.

The above has major implications for molecular mechanisms of biological energy transduction and points to a central role for interactions between the ions and the enzyme/transporter system. The bound ions help to generate “an energized state” of the enzyme/transporter in local membrane domains. The enzyme/transporter facilitates the separation of the positive and negative charges of an ion pair to definite distances by lowering the Gibbs energy required for this process and enables thereafter their recombination to form the uncharged substrate molecule. Since this is a reversible process, the enzyme can also catalyze the reverse hydrolysis reaction in other biological processes and act as an hydrolase, e.g., as F₁-ATPase.

As desired by one of the reviewers, the above molecular explanation offered for enzyme-catalyzed reactions is compared to that for an ordinary chemical reaction. In the conventional view of catalysis, the reaction progresses from a set of chemical bonds in the reactants to another, different sets of bonds in the products, and the role of the catalyst is to lower the activation Gibbs energy of the reaction. The activated state may contain separated charges; however, their concentrations are considered to be negligible. In our mechanism, the enzyme not only alters reaction rates but it also alters the type of product. For example, the mechanism shows that chemical bonds need not be broken to be replaced by a different set of bonds, but can also be replaced by stable charge-separated species (or vice versa).

The above discussion has clarified that in biological energy transduction, the ability of a membrane-bound enzyme/transporter to remove a positive and a negative charge from a neutral substrate, stabilize them, for example, by binding to specific binding sites, and thereby, separate charges at definite distances are of paramount importance in bioenergetics. The separated charges can interact with each other and with the protein, causing conformational changes within the designed protein structure and medium without dissipation of energy. These possibilities of a network of interactions of the separated charged pairs are not generally available for ordinary chemical reactions.

The above concepts are illustrated by means of a Gibbs energy vs. progress of reaction diagram in Fig. 8. Shown therein is the Gibbs energy as a function of the separation of a positive and negative charge. The minimum in Gibbs energy at r=0 in the medium corresponds to a chemical bond, while the minimum at a separation r_s corresponds to separated charges. Ideally, the Gibbs energy minimum of these two stable states should be equal (Fig. 8). If the separated charges occur in aqueous solution on one side of the membrane, then we can assign a reference value of 0 to the Gibbs energy, as shown by the bold curve. If the neutral undissociated species HA at r=0 in the medium of the transporter diffuses outward and exits to an aqueous medium on the other side of the membrane, it will exist as H⁺A^- in the aqueous phase on the other side of the membrane and still possess the reference value of 0 Gibbs energy. If, on the right-hand side of the diagram, the charges could be separated further, beyond r_s, then the electrostatic Gibbs energy of the system will increase further (dashed curve in Fig. 8). However, there is no logical reason, once the enzyme medium recognizes two separated charges, why it will facilitate further charge separation because the Coulombic attraction between the two charges will resist larger separation. The creation of a transient, finite-time, high-energy charged pair intermediate on the enzyme, [E−H⁺A^-]^*, and its destruction is key to the elusive process of energy transduction. The difference between the value of the electrostatic Gibbs energy of [E–H⁺A^-]^* at the maximum of the barrier and the reference Gibbs energy at the minima (Fig. 8) is transduced to useful mechanical work by the system.

Fig. 8

Gibbs energy as a function of reaction progress/distance of separation of a positive and negative charge for an enzyme-catalyzed reaction. r = 0 corresponds to a chemical bond, while the minimum at a separation r_s corresponds to separated charges in solution. The essential high-energy charged pair intermediate on the enzyme is represented by [E–H⁺A^-]^*. The difference between the value of the electrostatic Gibbs energy of [E–H⁺A^-]^* at the maximum of the barrier and the reference energy at the minima (bold curve) is transduced to useful mechanical work by the system. If the charges are separated beyond distance r_s, then the electrostatic Gibbs energy of the system increases further (dashed curve). However, Coulomb attraction between the two charges will resist separation >r_s

It should be stressed that the above model of biological energy transduction is quite different from the usual model in which the membrane is essentially considered a solid dielectric of packed lipids with a dielectric constant of ~5 that separates electrically conducting media/water on either side across which ion translocation occurs. Looking at our space charge model in a macroscopic way, energy storage is due not to the dielectric between two aqueous media/conductors but to a gradient of immobile charges of opposite sign within a dielectric matrix in which mobile ions are distributed throughout in accordance with the Poisson equation. In the double lattice, the junction of bound positive and bound negative charges gives rise to a transition region which is depleted of mobile ions and therefore possesses high space charge density, almost equal to the fixed charge concentration. In contrast, regions that are distant from the lattice junction have a small space charge density.

As explained above, bound charges uncompensated by mobile counterions lead to strong space charge regions. The electrical field in the depletion layer, which has a thickness of 1-2 nm under physiological conditions, is very high. For a fixed charge concentration of 100 mM, and a dielectric constant between 10 to 20, the (transient) local field for an abrupt fixed charge density profile would be on the order of 10⁶Vcm^-1, even when no electrical potential is imposed across the membrane. Thus, in the presence of steep transition regions of bound charges of opposite sign, as in the double lattice (Sect. 3.1) when two lattices of opposite sign are juxtaposed, the system displays conservative properties, and electrostatic energy storage can readily occur by perturbation of the space charge. In the absence of such localized space charge layers, homogeneous aqueous electrolyte solutions would be primarily resistive in nature.

Finally, it ought to be emphasized that Mitchell’s chemiosmotic theory, being a bulk phase to bulk phase theory according to which only external aqueous media are assumed to be energized, and only the delocalized electrical potentials, φ arising from such energization are considered [27, 64], cannot offer any meaningful insights into such local molecular coupling processes occurring within membrane transporters. On the other hand, Nath’s torsional mechanism of energy transduction and ATP synthesis and Nath’s two-ion theory of energy coupling considers events occurring inside the energy-transducing membrane and can therefore greatly contribute to understand and quantitatively model realistic intramembrane molecular energy transduction processes (Sect. 3.8).

3.8 Calculation of the timescale of movement between two stable states and quantification of the rotary dynamics of F_OF₁-ATP synthase

The timescale at which the enzyme/transporter moves between two stable states of a frustrated system can be estimated by simulation of the dynamics of torque generation in the F_OF₁-ATP synthase as a model example. In Nath’s two-ion theory of energy coupling, a dynamically electrogenic but overall electroneutral ion transport involving membrane permeable anions (such as succinate) and protons has been postulated. In order to utilize the energy of the anion gradient, the anion flows along its concentration gradient to its conserved binding site on the a-subunit of the membrane-bound F_O portion and creates a local Δψ. This local potential is the signal for proton translocation along its concentration gradient, and the proton binds to its conserved binding site on the c-subunit of F_O. Thus, torque generation is a result of change in electrostatic potential, Δ(Δψ) brought about by the binding of proton moving along its concentration gradient to its binding site on the c-subunit that lies within the electrostatic potential field of the bound anion on the a-subunit stator at the abrupt a–c interface/junction within F_O.

A charge geometry at the a–c interface containing bound ion pair and consensus conserved residues for modeling torque generation in the F_O portion of ATP synthase [30] is shown in Fig. 9. The model uses two mutually non-colinear half-access channels, whose existence had been predicted by us ([28, 53]; for a review, see especially pp. 88–93 and Fig. 12 of [30]), several years before structural studies on the ATP synthase by X-ray and cryo-EM techniques revealed their presence [68,69,70]. Thus, entry and exit of protons through the corresponding half-channels are separated by the time it takes the c-rotor to rotate by 15° (for twelve c-subunits in the c-ring). It should be noted that the torsional mechanism works equally effectively for other values of the number of c-subunits constituting the c-oligomer, as shown earlier [54].

Fig. 9

Charge geometry for torque generation and unidirectional rotation of the c-ring with respect to the a-subunit stator in the membrane-bound F_O portion of the F_OF₁-ATP synthase transporter. The conserved Arg-210 and His-245 amino acid residues (Escherichia coli numbering) on the a-stator are depicted by the + symbols. Dicarboxylic acid anion, e.g., succinate (dashed line), translocates along its concentration gradient and binds to its conserved a-subunit site [47, 49,50,51, 62, 63, 82], indicated by the minus symbol, creating a local $\Delta \psi$. H⁺ ion (bold line) subsequently translocates along its concentration gradient through the two mutually non-colinear half-access channels shown in the figure and binds to and unbinds from its conserved Glu/Asp-61 binding site on the c-subunit lying within the electrostatic field of the bound succinate anion [30]. These principles are similar to those uncovered recently from pioneering X-ray structural studies on coupled Na⁺–succinate cotransporters of the DASS family [83,84,85]. The change in local electrical potential, $\Delta \Delta \psi$ due to the binding and unbinding events and subsequent ion pair recombination causes electrostatic interaction between the charges on the a- and c-subunits that leads to mechanical rotation, and ultimately to torsional energy storage in the γ-subunit of F₁ to be utilized thereafter for formation of ATP, as detailed in Nath’s torsional mechanism of energy transduction and ATP synthesis [30, 34, 37, 49, 52,53,54,55, 66]

In the charge geometry of Fig. 9, l₂₃ is the distance between the two negatively charged c-rotor residues and d the channel distance between the rotor and the stator, or more precisely, the distance in space separating the residues harboring the positive and negative charges, which can be smaller than the channel thickness. We represent the vertical offset between the leading negatively charged rotor residue and the upper positively charged a-stator residue by r₁₂.The vertical offset between the trailing negatively charged rotor residue and the lower positively charged stator residue is represented by r₃₄, while the horizontal offset is denoted by l₃₄. θ is the angle subtended by the rotating c-subunit at any instant of time with the center of the c-ring. θ′ is the angle swept by the imaginary line joining the trailing rotor residue and the upper stator residue at any instant of time with respect to the local equilibrium position.

The dynamics of the system can be obtained by simulating in two 15° sub-steps the equation of motion

$$\zeta \frac{{d\theta }^{^{\prime}}}{dt}={\tau }_{\mathrm{m},\mathrm{d}}-{\tau }_{\mathrm{m},\mathrm{r}}$$

(27)

with the initial condition

$${\theta }^{^{\prime}}=0\ \mathrm{at}\ t=0$$

(28)

In Eq. (27), τ_m,d is the instantaneous driving electrostatic motor torque, and τ_m,r is the resisting electrostatic motor torque due to the charge geometry (Fig. 9). The solution of the equation of motion yields the angular displacement–time dynamics which is shown in Fig. 10. The parameters used in the simulation are given in the legend to Fig. 10 (see also ref. [30]). A major difference between the previous simulation [30] and this one (Fig. 10) is that at that time, the existence of 15° sub-steps in the rotatory motion of the c-rotor had not been known to occur. Moreover, the intermediate mechanical drive between c-ring rotation and rotation and twist of the γ-subunit of F₁ was not simulated at that time, which is now modeled as rotation θ″ about the axis of the c-subunits arranged in parallel in the helix ring constituting the c-oligomer and twist of the C-terminal c-subunit α-helices of the c-ring [30, 53]. A timescale of ~0.66 ms for a (15 + 15) = 30° step involving bistability is found, leading to a rotational frequency for the enzyme of ~130 Hz (Fig. 10).

Fig. 10

Dynamics of angular rotation in two discrete 15° elementary substeps for c-rings with 12 c-subunits. The dynamics are quantified by solution of the equations of motion for the F_OF₁-ATP synthase enzyme molecule. The values of the parameters used in the simulation are as follows: $\upzeta =2.5\times {10}^{-25}$ kg m² s⁻¹; d = 0.25 × 10^-9 m; l₂₃ = 1.3 × 10^-9 m; r₁₂ = 0.65 × 10^-9 m; r₃₄ = 0.65 × 10^-9 m; l₃₄ = 0.65 × 10^-9 m; ${\varepsilon }_{\mathrm{m}}=4$. Each 30° step takes approximately 0.66 ms, and one ATP molecule is made per 120° rotation. This leads to a rotational frequency of ~130 Hz, in accord with the experimental value for maximal rate of ATP synthesis of 400 ATP ${\mathrm{s}}^{-1}{{\mathrm{ F}}_{\mathrm{O}}{\mathrm{F}}_{1}}^{-1}$ by the bacterial ATP synthase [86]

The salient details of the simulation, how the parameter values were selected, and other differences from a prior simulation [30] are provided next, in compliance with the second reviewer’s suggestion. Since we are dealing with a quantum of energy required to synthesize a molecule of ATP of >25kT, random Brownian forces on the order of ~1 kT were neglected in the Langevin equations of motion. The moment of inertia, I, of the c- and γ-subunits was calculated as $\sum_{i}\left({m}_{i}{r}_{i}^{2}\right)$ for all atoms of the c- and γ-subunits based on extant X-ray structures [69, 71]. This resulted in an I value of < 4.5 × 10^-37 kg m² for the subunits. Hence, the inertial terms in the previous simulation [30] could be neglected, given that each term was >15 orders of magnitude smaller than the other terms in the dynamical equations of motion. We also verified numerically that the inertial terms did not make any perceptible contribution at the microsecond to millisecond timescale of the simulations. This therefore led to a simplified dynamical equation with focus on the electrostatic and mechanical aspects of the energy transduction process.

The parameters of the simulation were selected based on available high-resolution structural information on F_O. Given the diameter of the c-ring of 5 nm [68, 69], the distance between two carboxylate binding sites on adjacent c-subunits (l₂₃) could be estimated to measure 1.3 nm. The distances r₁₂, r₃₄, and l₃₄ were taken as one-half of the value of l₂₃. We view this choice of distance parameters as an improvement over values in the previous simulation [30]. Not only does this choice obey symmetry principles and is aesthetically elegant, but it also ensures that local equilibrium of the c-rotor–a-stator system is reached at regular angular intervals of 360°/n, or every 30° for n = 12 [28, 30], which is also stable to rotational Brownian fluctuations. The value of the frictional coefficient is in accord with drag studies in the fields of biopolymers and molecular motors [72,73,74].

The differential equation (Eq. (27)) was solved numerically by the Runge–Kutta fourth-order method. The relationship between θ′ and θ was derived trigonometrically for both θ =  0–15° and θ =  15–30° sub-steps. Numerical simulation of several test cases revealed that a linear electrostatic motor torque vs. θ′ or θ was obtained in all cases. With respect to θ′, the relationship showed a perfect straight line (R² = 1.0). It is remarkable that a simple, perfectly linear relationship results from such a complex charge geometry (Fig. 9) involving trigonometric functions and polynomial distance formulae [30]. What is the underlying reason for this interesting observation arising from the simulations, which offers tantalizing clues to the molecular mechanism of ATP synthesis? The simulation of the system dynamics (Fig. 10) offers several novel insights into biological energy transduction. The driving electrostatic motor torque, τ_m,d, the resisting motor torque, τ_m,r, due to the charge geometry, and their difference which gives the net motor torque, τ_m,net (Fig. 11), are determined by electrostatic interactions between the charges on a-stator–c-rotor upon ion pair binding/unbinding. τ_m,net is found to decrease linearly as a function of both θ′ and θ over a 15° interval, from 75.8 to 0 pN-nm for the 0–15° sub-step and from 138 to 0 pN-nm for the 15–30° sub-step (Fig. 11). Since the electrostatic forces are time-variant, τ_m,net and its components also vary with time. The underlying reason for a linear torque–angle relationship lies in the need to balance the conservative force between the charges by elastic forces. This necessitates the presence of the intermediate drive of the c-subunits in the c-ring of F_O that twist and store torsional energy before transferring their torsional energy to the γ-subunit to enable ATP synthesis [30, 52,53,54].

Fig. 11

Net electrostatic motor torque, τ_m,net, and its components τ_m,d and τ_m,r as a function of rotary angle, θ, over one-fourth of the conformational cycle in the F_O portion of ATP synthase

Given the new understanding, it is of great importance to model the mechanics of c-subunit deformation in F_O. First, an estimate of the linear spring constant of the c-subunit C-terminal helix is made from the theory of elasticity. The formula for elastic energy storage, U, is given by

$$U=\frac{\pi {R}^{4}E}{8L\left(1+\sigma \right)}{\theta }^{^{\prime\prime} 2}$$

(29)

where R and L are the radius and length of the α-helix, respectively, E is the Young’s modulus, and σ the Poisson’s ratio. From X-ray and cryo-EM data on densely packed c-rings in F_O [68, 69], as also from a statistical analysis of α-helices in protein databases [75], R = 0.7 nm. This dimension is also in consonance with the value of l₂₃ = 1.3 nm employed in the simulations, which should approximately measure 2R. With L = 0.7 nm, E = 1.8 × 10⁹ N m⁻², and σ = 0.4, this yields k″ = 4.8 × 10^-20 kg m² s⁻². Hence k″_eq for the (n – 2) = (12 – 2) = 10 twisting c-subunits arranged in parallel in the c-ring works out to be 4.8 × 10^-19 kg m² s⁻². For the 0–15° sub-step with an initial τ_m,net = 75.8 pN-nm (Fig. 11), θ″ = 75.8 × 10-21 / 4.8 × 10-19 = 0.1566 radians, implying a torsional energy storage $\frac{1}{2}{k}_{eq}^{^{\prime\prime} }{\theta }^{^{\prime\prime} 2}$ in the c-ring helices of 3.5 kJ mol⁻¹. Similarly for the 15–30° sub-step with an initial τ_m,net= 138 pN-nm (Fig. 11), θ″ = 138 × 10^-21 / 4.8 × 10^-19 = 0.285 radians, leading to energy storage $\frac{1}{2}{k}_{\mathrm{eq}}^{^{\prime\prime} }{\theta }^{^{\prime\prime} 2}$ in the c-ring helices of 11.8 kJ mol⁻¹ as twist energy. Hence, the total energy stored as twist in the c-ring for a 30° rotation equals 15.3 kJ mol⁻¹. For 120° rotation corresponding to synthesis of one ATP molecule, this provides a stored energy of 15.3 × 4 = 61.2 kJ mol⁻¹, which adequately meets the energy requirements for synthesis. The maximum angle of rotation θ″ about the axis of a single c-subunit, if it were alone, would be 2.85 × 57.3 = 160°, in agreement with the value determined experimentally using NMR spectroscopy by comparison of the solution structure of an unprotonated c-subunit with the structure of the protonated c-subunit [76].

Furthermore, from the standpoint of τ_m,net, taking the area under the curve of the τ_m,net–θ relationship obtained from the simulation (Fig. 11), the total torsional energy storage per 30° rotation works out to be (9.9 x 10^-21 + 18.1 × 10^-21) 28 pN-nm, or (5.9 + 10.8) = 16.7 kJ mol⁻¹, which corresponds to ~ 67 kJ mol⁻¹ for one catalytic cycle of 120° (Fig. 10). Hence, the above analysis shows that the physical, chemical, and energetic requirements of ATP synthesis are met adequately, that constraints arising from experimental data are satisfied, and that excellent agreement is obtained with all available experimental information by the results of the simulation (Figs. 10 and 11).

The present simulation also explains why the previous simulation [30] worked satisfactorily, although it did not incorporate the “wheels-within-wheels” detailed molecular mechanism of twisting of the c-subunits in the c-ring during ATP synthesis, first proposed back in 2002 ([30, 53]; see esp. pp. 90–93 and Fig. 12 in [30]) and modeled in this work. Neglecting inertia, the equation of motion of the previous simulation [30] can be recast as

$$\zeta \frac{{d\theta }^{^{\prime}}}{dt}={\tau }_{\mathrm{m},\mathrm{d}}-k\theta$$

(30)

In Eq. (30), as in the prior simulation [30], k is the torsional spring constant of the γ-subunit in F₁. However, there is no direct interaction between the c-rotor in F_O and the γ-subunit in F₁ — they each interact directly only with the common intermediary of the c-subunit α-helices of the c-rotor in F_O. But since we have a resisting motor torque, τ_m,r, that is linear with angular displacement, as shown in Fig. 11, the kθ in Eq. (30) inadvertently modeled, or rather mimicked, τ_m,r, as readily seen by comparing Eq. (27) with Eq. (30). Hence, phenomenologically, the right results were obtained earlier, although the actual physical situation was not fully realized, and the detailed molecular mechanism within F_O was not incorporated in the previous simulation [30]. The present simulation takes us a step deeper in our understanding of the process of energy storage and transduction in ATP synthase, and in molecular motors in general.

The above discussion has clarified the important point that if θ″ was identical to θ (or θ′), then as θ increases from 0 to 15 degrees during c-ring rotation (Fig. 10), kθ in the linear springs of the c-subunit α-helices can only increase, implying that there will only be energy storage as twist, but never any energy release. Hence, for efficient release of stored energy, there is the absolute requirement of an intermediate drive between the γ-subunit and the electrostatic torque at the a–c interface, e.g., as twist of c-helices by rotation, θ″, about their own axis. Thus, we are led to the conclusion that in response to the initial impulse of the net electrostatic torque, τ_m,net, θ″ also increases very fast and the c-subunit springs of the c-ring twist until their ${k}_{\mathrm{eq}}^{^{\prime\prime} }{\theta }^{^{\prime\prime} }$ balances τ_m,net. However, τ_m,net is a continually decreasing function of θ as θ progresses over a 15° angular interval.

Thus, once ${k}_{\mathrm{eq}}^{^{\prime\prime} }{\theta }^{^{\prime\prime} }$ balances τ_m,net close to θ  ∼0°, since τ_m,net itself is a decreasing torque due to the charge geometry (i.e., due to non-mechanical reasons), the θ″ will start decreasing, i.e., untwisting occurs, and the ${k}_{\mathrm{eq}}^{^{\prime\prime} }{\theta }^{^{\prime\prime} }$ will always dynamically balance the τ_m,net over the entire range of θ during the release process. Since τ_m,net does not decrease to 0 instantaneously, ${{k}_{\mathrm{eq}}^{^{\prime\prime} }\theta }^{^{\prime\prime} }$ also does not disappear instantaneously but follows the torque. Furthermore, this untwisting energy of the c-subunit helices is off-loaded to the γ-subunit. The energy stored in γ as torsional energy works out to be ~60 kJ mol⁻¹ over a 120° interval in the simulation (Figs. 10 and 11), which is thermodynamically competent to synthesize one mole of ATP [47, 50, 54, 62].

In summary, the twisting of the c-helices is very fast in response to the motor torque but the release of torsional energy is slow and is distributed over most of the 15° rotation cycle in F_O during ATP synthesis. Thus, the applied torsional spring force balances the inertial conservative electrostatic force during the energy release process. This leads to a most efficient transduction of energy in ATP synthesis by ATP synthase.

3.9 A meeting ground for physics, chemistry, and biology

This work has presented the first analysis of ion translocation and energy transduction in membrane transporters from a Gibbs energy landscape perspective. The dynamic regulation of the transport of ions and other polar metabolites through a lipid barrier represents a vital biological attribute of all cells. The complex membrane system requires a study of its structure, architecture, connectivity, interaction, and function that is reflective of life processes, a central topic of biologists. Yet analysis of the biological system requires physics, e.g., by electrostatic theories that quantify Gibbs energy landscapes, phase-transition in frustrated systems, and elucidation of system dynamics by Langevin, Fokker–Planck, or similar equations of motion. These properly constitute the subject matter of physics. However, the translocation of ions through membrane transporters and the ion-protein interactions that are crucial to biological energy transduction traditionally belong to the domain of physical chemistry, as do the thermodynamics of these processes. In fact, even the concept of Gibbs energy itself may be regarded as a mainstay of chemists. In this work, concepts from the above fields have been integrated from an energy landscape perspective. It is shown by physical analysis that membrane-bound enzymes, transporters, and molecular machines such as the F_OF₁-ATP synthase can be treated as a microcosm in which many fundamental concepts from physics, chemistry, and biology are reflected.

4 Conclusions

A large number of conclusions on ion translocation and dynamics of membrane transporters were arrived at in this study based on a novel analytical framework of Gibbs energy landscapes.

• 1. Current models of ion translocation through membrane transporters do not take into account the self-energy of the relevant charges, which is a serious lacuna of these models.

• 2. The macroscopic Kirkwood–Tanford–Warshel (KTW) electrostatic theory, originally developed for spherical membranous phases, has been extended and shown to be applicable to planar membranes also. The KTW theory properly takes into account the self-energies of the ions, has the ability to accurately evaluate the electrostatic Gibbs energy of a system of charges, and possesses great conceptual power. It provides a simple model based on continuum electrostatic considerations that offers valuable insights into complex bioenergetic and membrane transport processes, without performing detailed or exact calculations or all-atom molecular dynamics simulations. The theory can be readily extended, and the reliability of calculations further enhanced, by use of semi-macroscopic or microscopic electrostatic theories as further experience is gained by applying them to many biological systems and processes.

• 3. The calculated Gibbs energy landscape for a single species of translocated ion versus translocation of a proton–anion charge pair through a membrane transporter show qualitatively very different characteristics. An important feature is that the Gibbs energy asymptotes to a large positive (destabilizing) value for translocation of a single ion species, owing to the high self-energy of the process. Thus, the energy barrier cannot be lowered to 0 for a charge configuration involving only a single type of ion.

• 4. On the other hand, for a charge pair, the adverse ΔG_desolvation is almost completely compensated by the sum of the electrostatic Gibbs energy of the charge–charge interactions and the solvation energy of the ion pair. In essence, between two stable low Gibbs energy states, corresponding to a long-range ion pair and chemical bond formation of an uncharged substrate HA, a maxima in ΔG_KTW (R) is found at an intermediate value of the interionic distance, R, corresponding to an [E–H⁺A^-]^* “high-energy charge-pair intermediate” on the enzyme. The difference in Gibbs energy between the maxima and the low Gibbs energy state is transduced and stored in local domains of the macromolecule and can be used thereafter for the performance of useful external work.

• 5. The electrostatic Gibbs energy involving an ion pair can therefore be regarded as another illustration of a frustrated system involving bistability that shows a maxima at intermediate values of the order parameter. Hence, the physics of first-order phase transitions can be applied to gain further valuable insights into the process.

• 6. The theoretically calculated Gibbs energy landscape for ion pairs has been compared using experimental data from NMR spectroscopy for a number of globular proteins, and good agreement between theory and experiment has been shown.

• 7. Proton-only and all single-ion theories of ATP synthesis, such as the chemiosmotic theory, are inadequate to understand energy transduction and storage in bioenergetic processes.

• 8. By including the key roles of both proton and anion that translocate in a sequential, coupled way to form an enzyme-bound H⁺ – A^– charge pair in half-access channels of membrane transporters such as the F_OF₁-ATP synthase, Nath’s torsional mechanism of energy transduction and ATP synthesis and Nath’s two-ion theory of energy coupling offer novel insights and lead to a superior understanding of energy transduction at the molecular level.

• 9. The net ΔG_KTW tends to a value of ∼0 at a finite separation distance of H⁺ and  A^– (~10 Å) due to complete Gibbs energy compensation arising from ion-ion coupling by local fields. The profiles of the local electrical field and the local electrical potential have been shown to be amenable to analytical derivation by direct solution of Poisson’s equation. The localized coupling within membrane transporters has been concluded to be caused physically by interaction between the electrical layers around the positive and negative space-charge regions. The necessary condition for coupling has been shown to be $\lambda \sim R<{\lambda }_{D}$. Thus, an analysis of system scale sizes has been shown to shed light on molecular mechanism in ATP synthase. These fundamental physical concepts should be applicable to other mesoscopic systems also.

• 10. The timescale at which the mechanochemical enzyme/transporter system moves between the two stable states of the frustrated system has been shown to be estimated by simulation of the dynamics of torque generation. This works out to measure approximately two-thirds of a millisecond for ATP synthesis by the F_OF₁-ATP synthase.

• 11. Quantification of the dynamics of the enzyme/transporter system has been shown to offer a wealth of novel mechanistic insights and help infer conditions for efficient transduction of energy during ATP synthesis by the F_OF₁-ATP synthase. These include the necessity of an intermediate mechanical drive of the c-subunits in the c-ring of F_O that twists, stores elastic energy, and subsequently untwists and transfers the stored energy to the central γ-subunit as torsional energy that is used thereafter for ATP synthesis in F₁. The energy storage in the c-subunits due to ion binding/unbinding and ion-protein interactions in F_O is almost instantaneous and thus “discrete,” while the energy release from the c-subunits to the γ-subunit is slow, “continuous,” and distributed throughout the ~15° rotation cycle of the c-rotor during the process of ATP synthesis. These features of balanced energy release have been shown to lead to the most efficient energy transfer in ATP synthase.

• 12. The unifying analytical framework developed here for Gibbs energy landscapes in membrane-bound enzymes, membrane transporters, and biological molecular machines such as the F_OF₁-ATP synthase have been shown to provide a meeting ground for physics, chemistry, and biology. They can also serve as a reliable guide for the design and fabrication of efficient mechanochemical devices in nanoscience and nanotechnology applications [34, 53].

• 13. The unified framework of Gibbs energy landscapes has been shown to provide the essential features of biological energy coupling, transduction, and ATP synthesis by membrane transporters that should not be overlooked by further work in this important field of contemporary interdisciplinary research. In summary, the revolutionary impact of energy landscapes to fundamental biological processes stands accentuated.