Protonic Capacitor: Elucidating the biological significance of mitochondrial cristae formation

For decades, it was not entirely clear why mitochondria develop cristae? The work employing the transmembrane-electrostatic proton localization theory reported here has now provided a clear answer to this fundamental question. Surprisingly, the transmembrane-electrostatically localized proton concentration at a curved mitochondrial crista tip can be significantly higher than that at the relatively flat membrane plane regions where the proton-pumping respiratory supercomplexes are situated. The biological significance for mitochondrial cristae has now, for the first time, been elucidated at a protonic bioenergetics level: 1) The formation of cristae creates more mitochondrial inner membrane surface area and thus more protonic capacitance for transmembrane-electrostatically localized proton energy storage; and 2) The geometric effect of a mitochondrial crista enhances the transmembrane-electrostatically localized proton density to the crista tip where the ATP synthase can readily utilize the localized proton density to drive ATP synthesis.

www.nature.com/scientificreports www.nature.com/scientificreports/ We have also experimentally demonstrated the creation of a transmembrane electrostatically localized proton layer at the water-membrane interface in a biomimetic anode water-membrane-water cathode system 34 .
Our work 29,31,39 has also explained that the Gouy-Chapman theory 48 and the Debye length concept are not applicable to determining the thickness of localized excess proton layer. The reason is that the Debye length equations are defined for charge-balanced solutions including 1:1 electrolyte solutions such as NaCl solution, but not for the special localized excess protons 49 . Furthermore, the mitochondrial inner membrane is an insulator, which is very different from an electrode that the "electric double layer" and Gouy-Chapman theory 48 may apply. Consequently, the transmembrane electrostatically localized excess proton layer at the liquid-membrane interface likely is a special monolayer (with a thickness about 1 nm), but definitely not an "electric double layer" that would be predicted by the Gouy-Chapman theory 48 at a charged electrode-liquid interface. This conclusion as a localized excess proton monolayer at the liquid-membrane interface is also in line with the known "electric double layer" phenomenon in that the excess proton layer may be regarded as an extension from the secondary (proton) layer of the anode's "electric double layer" around the liquid water body surface as reported in our previous experimental study 39 .
Cardiolipin (1,3-bis(sn-3′-phosphatidyl)-sn-glycerol) was once believed to play a role in the "localized proton microcircuits" 38,50 . If its phosphate-associated hydrophilic head group indeed serves as a proton "trap" or a protonic conduction pathway at the liquid-membrane interface along the membrane surface, as previously proposed  10 ). The mitochondrial outer membrane surrounds the inner membrane which separates the inter-membrane space from the matrix. The inner membrane is differentiated into the inner boundary membrane and the cristae. The two regions of the inner membrane are continuous at the crista junctions. The cristae extend more or less deeply into the matrix. (B): Segmented surface of a mitochondrion showing position of ATP synthase dimers (yellow spheres) to cristae membrane (blue) as revealed by electron cryotomography (adapted from Fig. 1 of Davies et al. 11 ). (C and D): ATP synthase distribution in mitochondrial cristae showing rows of ATP synthase dimers along the highly curved ridges of disk-shaped (C) or tubular (D) cristae vesicles as observed through electron cryotomography (adapted from Fig. 4 of Davies et al. 12 ).
www.nature.com/scientificreports www.nature.com/scientificreports/ by Haines and Dencher 51 , that would certainly expose its "anhydrous" protons to the bulk aqueous phase. Therefore, it is not realistic for such cardiolipin "localized proton microcircuits" to function. As we can expect from the transmembrane electrostatic proton localization theory 27-31 that cardiolipin is not required for the formation of protonic capacitor 28,29,39 , cardiolipin is indeed dispensable for oxidative phosphorylation as recently confirmed by an independent experimental study 52 . We have recently also clarified that the protons and ions of any permanent "electric double layer" formed as a result of membrane surface's fixed-charges such as the negatively-charged phosphate groups of any phospholipids including cardiolipin that permanently attract protons and ions are irrelevant to the membrane potential ψ ∆ in proton-coupling bioenergetics 29 . This is also well in line with the conclusion from our previous study 53 : "membrane surface-fixed-charges-attracted protons are not relevant to the protonic motive force" that drives ATP synthesis.
More recently, employing the Lee transmembrane electrostatic proton localization theory 27,28,30,31 with a set of new equations for pmf that takes electrostatically localized protons into account, we have now achieved much better understanding of membrane potential ψ ∆ as the voltage difference contributed by the localized surface charge density at the liquid-membrane interface as in "a transmembrane electrostatically localized protons/ cations-membrane-anions capacitor" 29 .
In this article, to answer the fundamental questions on the biological significance of mitochondrial cristae formation, the transmembrane electrostatic proton localization theory [28][29][30][31] is now further employed in calculating the numbers of electrostatically localized protons at the liquid-membrane interface per mitochondrion. Furthermore, the geometric effect of cristae on enhancing localized proton density at the cristae tips is analyzed using an ellipsoidal-shaped mitochondrial crista as a protonic capacitor in accordance with the Lee transmembrane electrostatic proton localization theory [27][28][29][30][31] . This work enables much better understanding of mitochondrial cristae ( Fig. 1) in relation to localized proton coupling bioenergetics. The biological significance of mitochondrial cristae formation in regarding to why the dimers of F 1 F o -ATP synthase are distributed primarily at the mitochondrial cristae tips whereas the proton-pumping "respiratory supercomplexes" are situated at the planar membrane regions is now elucidated with protonic bioenergetics.

transmembrane electrostatic proton Localization theory and calculations
The transmembrane electrostatic proton localization theory. This novel proton-coupling bioenergetics theory [27][28][29][30][31]54 is built on the fundamental understanding that a biologically relevant water liquid phase can act as a protonic conductor. This new understanding is well in line with the well-established knowledge that protons can quickly transfer among water molecules by the "hops and turns" mechanism [55][56][57] . That is, liquid water can serve as a protonic conductor because protons using the "hops and turns" mechanism can translocate quickly among water molecules. Notably, from the negative charge ("proton hole") perspective, hydroxyl anions ("proton holes") translocate among the water molecules in the opposite direction of protonic conduction. Consequently, excess protons (positive charges) in an aqueous liquid on one side of a membrane will repel each other to become electrostatically localized along the membrane surface, attracting an equal number of excess hydroxyl anions (negative charges) to the other side of the membrane and thus resulting in a "protonic capacitor structure" (Fig. 2).
As previously reported [27][28][29] , this protonic capacitor may be mathematically justified employing the Gauss' Law and the knowledge that the electric field E is zero inside a protonic conductor. Although the Gauss' Law has recently been applied to bacteria 28 and a simplified mitochondrion 29 without cristae, it has never been done to a mitochondrion with a crista or cristae before. Therefore, Fig. 2 represents the application of the Gauss' Law to a mitochondrion with a crista for the first time. This application and its result are not entirely obvious. However, it can be justified according to the knowledge that the mitochondrial crista liquid is connected through the crista junction with the liquid in the mitochondrial intermembrane space (Fig. 2).
As described in refs. 58 and 29 , Gauss' Law is known to relate the net charge Q within a volume to the flux of electric field lines through the closed surface surrounding the volume by the following integral equation where ε o is the electric permittivity constant; E is the electric field inside a protonic conductor; and dS is a differential surface element. In this equation (Eq. 1), the small circle on the integral sign indicates that the integration is performed over the closed surface. So, let's now consider a series of integration applications using the Gauss' Law equation (Eq. 1), where a small volume near the center of the mitochondrial matrix liquid is gradually increased until it is just inside the matrix liquid surface as indicated by a polar coordinate r in Fig. 2. Since the electric field E is zero everywhere in a conductive liquid body, the left side of Eq. 1 vanishes in each case. Consequently, the right side of the equation (Eq. 1) must also vanish, which means that no net charge (Q = 0) is within the volume of the mitochondrial matrix liquid phase; the excess hydroxyl anions in this case must therefore be at the matrix liquid-body surface, i.e. at the liquid-membrane interface along the matrix (n) side of mitochondrial inner membrane surface.
Similarly, considering the proton-conductive aqueous liquid phase at the mitochondrial intermembrane space including cristae space, the electric field E = 0 is true at anywhere within the bulk aqueous liquid phase of both the intermembrane space and the crista space that are connected through a crista junction (Fig. 2). Employing the Gauss' Law integral equation to a series of volumes enclosing the entire mitochondrion and reducing them to be just outside its inner membrane surface (including its cristae membrane surface) as illustrated by a polar coordinate R in Fig. 2, the surface integrals of Eq. 1 in this case also equal to zero so that no net excess charge is found. Since the previous application of Gauss' Law equation (Eq. 1) in the matrix liquid already informed us that the excess hydroxyl anions are on the inner membrane surface at the matrix (n) side, the positive charges (excess www.nature.com/scientificreports www.nature.com/scientificreports/ protons) must be at the membrane-liquid interface along the membrane surface at the intermembrane/cristae space (p) side (including cristae junctions and cristae liquid space side). As a result, the localized positive charges (excess protons) at the membrane-liquid interface on the intermembrane/cristae space (p) side precisely balances the excess hydroxyl anions on the matrix (n) side, making the total net charge of the entire mitochondrion zero. This mathematically justifies that a mitochondrion with cristae is also a protonic membrane capacitor.
New protonic motive force equation with protonic capacitor. The transmembrane electrostatic proton localization theory [27][28][29][30][31]53 and the successful experimental demonstration of a protonic capacitor using biomimetic "water-membrane-water" systems 34,39,54,59 clearly indicate that the textbook Mitchellian chemiosmotic theory is not entirely correct because the Mitchellian delocalized proton coupling view treats protons pretty much like typical solutes in bulk liquid phases and thus fails to consider the existence of localized proton coupling. Consequently, the textbook Mitchellian pmf equation quite fatally misses a major pmf contribution from the electrostatically localized protons at the liquid-membrane interface that is rightly at the ATP synthase protonic inlet mouth. Therefore, the textbook Mitchellian pmf equation must now be revised. It is now quite clear that the population density of localized excess protons is immediately related to the probability for protons to be available at the protonic mouth of ATP synthase, in addition to that implicated from the bulk liquid phase pH. Therefore, based on the new bioenergetics understanding, the author 29,40 has recently rewritten the protonic motive force (pmf) equation for ATP synthesis as where ψ ∆ represents the transmembrane electrical potential difference (positive (p)-phase minus negative (n)-phase defined as in ref. 60 ); R is the gas constant; T is the absolute temperature; F is Faraday's constant;       + H pB is the p-side bulk liquid phase proton concentration in the intermembrane space including cristae space; + H [ ] nB is the proton concentration in mitochondrial matrix (n) bulk liquid phase; and + H [ ] L is the effective concentration of electrostatically localized protons at the liquid-membrane interface along the positive (p) side of the mitochondrial inner membrane.
As pointed out previously 29,40 , the first two terms of Eq. 2 represent the Mitchellian bulk phase-to-bulk phase proton electrochemical potential gradient for which we now call it as the "classic" pmf; whereas the last term represents the "local" pmf from the electrostatically localized protons.

Figure 2.
Cross section for a mitochondrion with crista: transmembrane electrostatic proton localization (protonic capacitor) model for a mitochondrion with crista illustrating how excess protons (H + ) and hydroxyl ions (OH -) could be electrostatically localized at the water-membrane interfaces along the two sides of the mitochondrial inner membrane with crista formation before proton-cation exchange as it would be in a theoretically pure water-membrane-water system. www.nature.com/scientificreports www.nature.com/scientificreports/ For an idealized protonic capacitor, as described previously 29,40 , the ideal localized proton concentration + H [ ] L 0 at the water-membrane interface on the p-side can be expressed as a function of the transmembrane electrical potential difference Δψ in the following equation where C S / represents the specific membrane capacitance per unit surface area; and l is the thickness of the localized proton layer.
In real biological systems such as mitochondria, the non-proton cations + M pB i (for examples, Na + , K + , and Mg 2+ ) of the bulk liquid p-phase in the intermembrane space including cristae space may exchange with the localized protons at the liquid-membrane interface and thereby reduce the localized proton concentration. As explained in our recent study 29 , for any given number (n) of different non-proton cation species such as Na + , K + , and Mg 2+ present in the bulk aqueous p-phase of the intermembrane space including cristae space, the localized proton concentration after exchanging with all of the cation species at the equilibrium state can be expressed as i is the concentration of any given non-proton cation in the bulk liquid p-phase of the intermembrane space including cristae space; and K Pi is the equilibrium constant for each of the cation species to exchange with the localized protons. As pointed out previously 29,40 , the denominator in Eq. 4 is the reduction factor owing to the effect of cation exchange with the localized protons at the liquid-membrane interface along the inner membrane surface at the intermembrane/cristae space (p) side (Fig. 2). The total contribution of the non-proton cations to this reduction factor surprisingly is a product (multiplication not summation) of the reduction factors associated with each of the non-proton cation species       + M pB i in the bulk liquid p-phase such as Na + , K + , and Mg 2+ at the intermembrane space including the cristae space.
Newly formulated membrane potential equations with protonic capacitor. Another deficiency of the classic Mitchellian chemiosmotic theory is that it does not clearly explain the biophysical origin of membrane potential ∆ψ. Mitchell and his coworker (1969) once stated " Δψ can be equated to the Donnan potential" 61 which seems vaguely regarding membrane potential Δψ as "a delocalized parameter" somehow contributed by delocalized protons and ions in the two bulk liquid phases 60 . As shown in our recent publication 29 and illustrated here with the mitochondrial proton-electrostatics localization model (Fig. 2), the localized excess protons and hydroxyl ions along the two sides of the membrane may directly contribute to the trans-membrane potential difference ∆ψ; In contrast, the Mitchellian "delocalized protons" that can be measured by the bulk liquid phase pH have little to do with the membrane potential Δψ. Consequently, the membrane potential (difference) Δψ created by the localized proton concentration + H [ ] L 0 before cation exchange with the localized protons is now expressed as in the following protonic capacitor-based Δψ equation reported previously 29 : L 0 where C S / is the specific membrane capacitance per unit surface area, l is the thickness of the localized proton layer, and F is the Faraday constant.
From the protonic capacitor-based membrane potential Δψ equation (Eq. 5) as reported in our recent study 29 , we can now clearly understand that the membrane potential (difference) Δψ is a function of the ideal localized proton concentration + H [ ] L 0 . This also elucidates the physical origin of membrane potential Δψ as a function of the localized excess proton concentration at the liquid-membrane interface under the idealized condition.
In actual biological systems such as animal mitochondria, as discussed previously 29 , the cation-proton exchange will reduce the localized proton concentration from  By combining Eqs. 5 and 6, the membrane potential difference Δψ at the cation-proton exchange equilibrium may be expressed as i is the concentration for each of the localized non-proton cations such as sodium and potassium cations in cation-proton exchange equilibrium at a liquid-membrane interface. Therefore, as concluded previously 29 , the membrane potential difference Δψ can also be expressed as at the liquid-membrane interface along the p-side surface of the mitochondrial inner membrane as expressed in Eq. 9. Therefore, the membrane potential Δψ in mitochondria can now be understood as a localized protonic/cationic membrane capacitor behavior 29 .
Calculations on total cation exchange reduction factor and electrostatically localized proton concentration. As reported previously 29 , the total cation exchange reduction factor as shown in the denominator of Eq. 4 was calculated from the concentrations of the major cations       + M pB i , their cation-proton exchange equilibrium constants K Pi and the p-side bulk liquid phase proton concentration       + H pB in the mitochondrial intermembrane space and cristae space. Briefly, the concentrations of the major cations 10 mM Na + , 128 mM K + , and 1.0 mM Mg 2+ in the mitochondria were calculated from the composition of the reaction medium reported in ref. 62 . The cation-proton exchange equilibrium constants K Pi for Na + and K + were 5.07 ×10 −8 and 6.93 ×10 −8 , respectively, as determined previously by experimental measurements 39 . The K Pi value of 5.42 ×10 −6 for Mg 2+ was determined from the latest experimental data of divalent cation Mg ++ exchange with electrostatically localized protons at a membrane-liquid interface in the Lee laboratory through a study similar to that reported in ref. 39 .
Chinopoulos et al. 62 elegantly revealed that the bulk-phase pH difference across the mitochondrial inner membrane actually is near zero: the "∆pH max is only ~0.11". This observation, as discussed previously 29 , is well corroborated with our recent biomimetic experimental demonstration that, before and after the membrane is energized with excess protons at the anode side of the membrane and excess hydroxyl anions at the cathode side of the membrane, the bulk-phase liquid pH in the anode liquid chamber remained about the same as that in the cathode chamber liquid 34,39,54,59 . By this corroboration, therefore, it is well justified to use the experimental data from Chinopoulos et al. 62 in regarding to the mitochondria reaction medium pH 7.25 (pH pB ) and matrix liquid pH 7.35 (pH nB ) for our study here as well. Accordingly, the total cation exchange reduction factor (the denominator of Eq. 4) was calculated to be 1.29, indicating a relatively minor role in modulating the electrostatically localized surface proton concentration by the cation-proton exchange process in mitochondria. That is, the electrostatically localized proton concentration + H [ ] L at the cation-proton exchange equilibrium state represent about 78% of the total localized surface charge density in animal mitochondria.
For a mitochondrial membrane potential Δψ in a range from 50 to 200 mV, as reported previously 29  L and mitochondrial inner membrane surface area (S), the amounts of localized protons per mitochondrion can now be calculated for mitochondria with and without cristae using Eqs. 10 and 11 described as follows.

Results and Discussions
Formation of cristae creating more mitochondrial inner membrane surface area and protonic capacitance. From the protonic membrane capacitor of Fig. 2, it is quite apparent that mitochondrial cristae formation creates more mitochondrial inner membrane surface area (S), which immediately contributes to its membrane capacitance (C). Therefore, its bioenergetics significance can now be better understood with the Lee model [28][29][30][31]54 of transmembrane electrostatically localized protons as shown in the membrane potential (Δψ) equation (Eq. 5), which shows that Δψ exists precisely because of the excess cations (including H + ) and the excess anions (such as OH − ) charge layers localized on the two sides of the membrane in a protons-membrane-anions capacitor structure (Fig. 2). This also explains the origin of Δψ and its relationship with the concentration of electrostatically localized protons + H [ ] L as expressed in Eqs. 3 and 4. www.nature.com/scientificreports www.nature.com/scientificreports/ From the membrane potential Δψ equation (Eq. 9), it is now also quite clear that the amount of transmembrane electrostatically localized charges (C · Δψ) per mitochondrion may be calculated from mitochondrial inner membrane surface area (S) and localized proton and cation concentrations + ∑ From here, we can also understand, the amount of transmembrane electrostatically localized charges per mitochondrion is contributed by both the mitochondrial inner membrane capacitance (C) and membrane potential (Δψ). The mitochondrial inner membrane capacitance (C) is intimately related to the membrane area (S). Therefore, when a mitochondrion creates more inner membrane surface area (S) through cristae formation, its capacitance (C) increases and thus enhances its ability to store transmembrane electrostatically localized charges (energy). This also explains part of the biological significance in mitochondrial cristae formation.  and hydroxyl ions (OH -) could be electrostatically localized at the water-membrane interfaces along the two sides of the mitochondrial inner membrane before proton-cation exchange as it would be in a theoretically pure water-membrane-water system. Note, this cross section can be considered as a special result from the tri-axial (a, b, and c) protonic conducting ellipsoidal crista equation (Eq. 12) in 3-dimensional x, y, and z coordinates for its middle cross section (where the z coordinate is zero).
www.nature.com/scientificreports www.nature.com/scientificreports/ Similarly, when the data of inner membrane surface area (S), the localized proton layer thickness (l), and localized proton concentration + H [ ] L are available, the amounts of transmembrane electrostatically localized protons per mitochondrion may be calculated for mitochondria with and without cristae according to the following equation:

A L
where N A is the Avogadro constant (6.02205 × 10 23 mol −1 ). Table 1 lists the numbers of transmembrane electrostatically localized protons per mitochondrion (in a size about 1500 nm × 300 nm × 300 nm) with and without cristae calculated as a function of transmembrane potential Δψ using Eqs. 3, 4 and 11 under the given reaction medium pH 7.25 (pH pB ), mitochondria matrix pH 7.35 (pH nB ) and taking cation-proton exchange into account. According to a 3-dimensional mitochondrial cristae model of an intermyofibrillar mitochondrion with a size about 1500 nm × 300 nm × 300 nm 63 , the surface area of mitochondrial inner membrane is 5.76 × 10 6 nm 2 , which is about three times larger than the surface area of the mitochondrial outer membrane (1.98 × 10 6 nm 2 ). Using the inner membrane surface area of 5.76 × 10 6 nm 2 for a typical mitochondrion with cristae, the numbers of electrostatically localized protons per mitochondrion were calculated through Eq. 11 to be in a range from 1.84 × 10 4 to 7.36 × 10 4 depending on the membrane potential Δψ in a range from 50 to 200 mV (Table 1 and Fig. 3). On the other hand, assuming 1.98 × 10 6 nm 2 as the inner membrane surface area for a mitochondrion without any cristae, the numbers of electrostatically localized protons per mitochondrion were calculated to be in a range from 6.32 × 10 3 to 2.53 × 10 4 depending on the membrane potential Δψ (Table 1 and Fig. 3).

Transmembrane electrostatically localized protons at the liquid-membrane interface per mitochondrion calculated as a function of membrane potential.
By comparing the numbers of electrostatically localized protons for mitochondria with and without cristae as presented in Fig. 3, it is now quite clear that a mitochondrion with cristae has as many as nearly 3-times more electrostatically localized protons which can be used to drive ATP synthesis than a mitochondrion without any cristae. Therefore, we can now understand that mitochondrial cristae formation may favor the energy transduction process through transmembrane electrostatically localized protons for ATP synthesis. That is, the present study with this finding (Table 1 and Fig. 3) may represent a significant step forward with the protonic bioenergetics as a complementary development to the early empirical understanding such as "the surface area of cristae positively correlates with the amount of ATP produced by oxidative phosphorylation in various tissues" as stated by Zick et al. in a review article 1 .
Consequently, the data in Fig. 3 indicate that, at least, one of the biological significances in the formation of cristae is to create more mitochondrial inner membrane surface area and thus more protonic capacitance for  64 , localized proton layer thickness (l) of 1 nm as discussed in ref. 28,34 , exchange reduction factor of 1.29 as calculated previously 29 , and inner membrane surface area of 5.76 ×10 6 nm 2 for the mitochondrion with cristae and 1.98 ×10 6 nm 2 for the mitochondrion without any cristae.
www.nature.com/scientificreports www.nature.com/scientificreports/ localized proton energy storage. This may represent a type of quantitative (optional) improvement that is biologically advantageous, but not necessarily an absolute requirement for all other organisms. It may also explain why seem only mitochondria have cristae formed so extensively; while some bacteria have an intracytoplasmic membrane structure similar to cristae and others may or may not have such a structure 65 .
Geometric effect of a crista enhancing transmembrane-electrostatically localized proton density at the crista tip. As shown in Fig. 1, a mitochondrial crista may be considered as an ellipsoidal-shaped 3D-structure 66-69 which can be described by a tri-axial (a, b, and c) ellipsoid equation (Eq. 12) in 3-dimensional x, y, and z coordinates. As illustrated in Fig. 4 for a middle cross section of an ellipsoidal-shaped mitochondrial crista, the complexes I, III and IV that constitute the "respiratory supercomplexes" are situated at the relatively flat membrane regions, utilizing NADH as the source of electrons and molecular oxygen (O2) as the terminal electron acceptor and pumping protons across the inner membrane from the mitochondrial matrix into the crista liquid space. The excess protons in the crista liquid space created by the redox-driven proton pumps distribute themselves electrostatically to the liquid-membrane interface at the crista liquid side of the inner membrane. The ATP synthase as its dimer located at the crista tip (ridge) utilizes the transmembrane-electrostatically localized protons there to drive ATP synthesis from ADP and Pi. We now understand, the feature that ATP synthase as its dimer located at the crista tip (ridge) has an advantage in relation to protonic bioenergetics, since the localized proton concentration there is higher than that at the relatively flat membrane regions as shown in the following analysis.
As shown in Fig. 4, the electric field E = 0 in both mitochondrial matrix and crista indicates that both the matrix and crista liquids can function as protonic conductors in accordance with the Lee transmembrane-electrostatic electrostatic proton localization theory [27][28][29][30][31] . In recognizing the mitochondrial crista liquid as protonic conductor, based on the mathematic solution of conducting ellipsoid and circular disk electrostatics developed by Prof. Kirk McDonald of Princeton University 70 , we can now understand that the surface protonic charge density σ on a tri-axial (a, b, and c) protonic conducting ellipsoidal crista in 3-dimensional x, y, and z coordinates where Q is the total charge of electrostatically localized protons; and a, b and c are the ellipsoid axial lengths with respect to the 3-dimensional x, y, and z coordinates. At the crista tip where the ATP synthase dimer is located, the coordinates are: x = a, y = 0, and z = 0 (a, 0, 0). Therefore, the localized protonic charge density at the crista tip is Consequently, the ratio of the localized protonic charge density at the crista tip to that at the crista flat region is  Fig. 4, is bioenergetically advantageous. That is, at the cristae ridges, the ATP synthases represent a sink (users) for protons, while the proton pumps of the electron transport chain are in the relatively flat membrane plane regions of cristae. Electrostatically guiding the localized protons from their source to the protonic sink at the ATP synthase, the cristae may work as protonic conduits with the liquid-membrane interface along the membrane surface as a transmembrane electrostatically localized protonic conduction pathway (Fig. 4) that enable efficient ATP production.
This finding (Fig. 4, Eq. 18, and Table 2) is remarkably in line with an independent study 11,71 that "indicate a 3.5-fold increase in surface charge density at the apex, compared with the flat sides" and "the higher surface density of protons in the curved membrane regions would result in a local pH difference of ~0.5 units" for a mitochondrial crista with an axial ratio of 10:1. However, as shown in Table 2 is 10:1, translating to a local pH difference of 1 unit, which is twice as high as the "~0.5 units" reported previously 71 for a mitochondrial crista with an axial ratio of 10:1. Furthermore, based on our protonic capacitor model (illustrated in Figs. 2 and 4, and expressed in Eqs. [3][4][5][6][7][8][9], the excess protons are dynamically held at the liquid-membrane interface by the transmembrane-electrostatic proton localization effect 28,29 without requiring any of the putative interfacial barrier model 72,73 that the "local proton gradient in the cristae space" model proposed by Davies et al. 11 and presented in a subsequent review article 10 seems have to invoke; Consistently, our experimentally demonstrated transmembrane-electrostatic excess proton localization at a liquid-membrane  Table 2. Geometric effect of cristae on transmembrane electrostatically localized proton density at the cristae tips (ridges) in relation to the axial ratios (a b / ) associated with the ellipsoidal cristae length and width.
www.nature.com/scientificreports www.nature.com/scientificreports/ interface 39 also showed "no support" for the putative interfacial barrier model 72,73 that anyhow is not really required to explain the localized proton coupling bioenergetics. These results all clearly support the suggestion that a position of the ATP synthase at the apex of mitochondrial cristae (as observed in Fig. 1C,D) is optimal for ATP production. This finding (Fig. 4, Eq. 18, and Table 2) is also well corroborated by another independent experimental observation 33 with bioengineered fluorescent protein as pH indicator that elegantly showed "lateral pH gradient between OXPHOS complex IV and F o F 1 ATP-synthase in folded mitochondrial membranes". Therefore, the data in Table 2 indicate another biological significance for cristae formation: the geometric effect of a mitochondrial crista electrostatically enhancing localized proton density to the crista tip where the ATP synthase can readily utilize the localized proton density to drive ATP synthesis. This represents another type of quantitative (optional) improvement that may be evolutionally advantageous, but not necessarily a universal feature to all organisms. This again can explain why seem only mitochondria have cristae formed so extensively; while some bacteria have an intracytoplasmic membrane structure similar to cristae but others may or may not have it 65 . conclusion This study employing the Lee transmembrane electrostatic proton localization theory [27][28][29][30][31] has now shown that the localized proton associated pH ( Log H ([ ] ) L tip 10 0 − + ) at a crista tip (or ridge) can be significantly lower (by as much as "─ 1.00" pH unit) than that (− +

Log H ([ ] )
L flat 10 0 ) at the flat plane region within the same crista (Fig. 4, Eq. 18, and Table 2). Therefore, we can now understand, the formation of cristae with ATP synthase dimer rows located at the cristae ridges (tips) where the localized proton concentration ( + at the relatively flat membrane plane regions where the redox-driven proton-pumping "respiratory supercomplexes" are situated is bioenergetically advantageous. That is, at the cristae ridges, the ATP synthase represents a sink for transmembrane electrostatically localized protons at the liquid-membrane interface, while the redox-driven proton pumps are in the relatively flat membrane plane regions of the cristae. Electrostatically guiding the localized protons to the ATP synthase at the cristae ridges, the liquid-membrane interface along the mitochondrial inner membrane surface may serve as a transmembrane electrostatically localized protonic conduction pathway (Fig. 4) for effective ATP synthesis without requiring any of the putative interfacial barrier 72 .
Furthermore, as shown in Fig. 2 and Table 1, the formation of cristae creates more mitochondrial inner membrane surface area and thus protonic capacitance, which enhances its ability to store transmembrane electrostatically localized charges (energy). For a typical mitochondrion with cristae at a membrane potential Δψ in a range from 50 to 200 mV, the numbers of transmembrane electrostatically localized protons per mitochondrion were calculated to be in a range from 1.84 × 10 4 to 7.36 × 10 4 , which is 3 times more than that of a mitochondrion without cristae (Fig. 3). These results can now better explain the biological significance for mitochondrial cristae formation in relation to protonic bioenergetics.
Therefore, this study has now elucidated the pivotal architectural features of cristae membranes in relation to a curved protonic capacitor in accordance with the Lee transmembrane-electrostatic proton localization theory 27-31 : (1) the formation of mitochondrial cristae (Figs. 1 and 2) may be bioenergetically advantageous since it creates an extended membrane surface area not only for the accumulation of oxidative phosphorylation enzymes (respiratory chain complexes I to IV; F 1 F o -ATP synthase) but also for protonic capacitance to store electrostatically localized charges (energy); and (2) the shaping of mitochondrial cristae as a curved membrane micro-compartment ( Fig. 1C and Fig. 4) may enhance protonic bioenergetics coupling for ATP production through the use of the proton-pumping "respiratory supercomplexes" (complexes I, III and V) at the relatively flat membrane plane regions and the ATP synthase enzymes at the cristae tips (ridges) where the transmembrane electrostatically localized proton concentration ( +

Data availability
All data generated or analyzed during this study are included in this article and the associated supporting information (xls efile showing the detailed calculations of the transmembrane electrostatically localized protons per mitochondrion) available online at the journal website.