Like we previously mentioned, NHEs enable extracellular Na⁺ to cross an energy activation barrier constituted by the hydrophobic bulk of membrane lipids and to flow according to its thermodynamic gradient between the outside and inside of cells. At steady-state, the shape of Na⁺ dose response curves for most of the NHEs can be approximated as a hyperbolic saturation function (Pedersen and Counillon 2019). Classically the Michaelis-Menten equation for a saturation curve with extracellular sodium can be written as V = V m a x [ N a e + ] K m + [ N a e + ] (1) Where Km is classically referred from undergraduate textbook biology as the “affinity” for Na⁺. Because there is a transport mechanism, external Na⁺ must bind, be translocated and then released upon proton exchange in the other direction. As NHE1 function is dictated by the thermodynamic gradients, the direction of transport is also fully reversible upon the respective Na⁺ gradients on each side of the membrane. Consequently, we can write the following kinetic scheme in which every step can operate in each direction.

where the symbol (e) represents extracellular and (i) intracellular. For the sake of simplicity, k(h) and k(-h) represent apparent transport rates for H⁺ that are too complex to be developed here mathematically as they contain the cooperative binding and transport mechanism for proton (Lacroix et al., 2004).

Such a simplified mechanism was symbolically encoded in the Maxima computer algebra system (https://maxima.sourceforge.io), thus allowing to express each intermediate as a function of [E_o] (the total quantity of exchanger), [ N a i + ] and [ N a e + ] , and finally express and simplify the net rate of [ N a i + ] evolution that corresponds here to the transporters steady state velocity ( V = d [ N a i + ] / d t ) . A tight monitoring of the symbolic results let us group the constants in a meaningful way, giving. V = [ E o ] Λ i n [ N a e + ] − Λ o u t [ N a i + ] α + β [ N a i + ] + γ [ N a e + ] + δ [ N a i + ] [ N a e + ] (2) with Λ i n = k 1 k 2 k 3 k ( h ) Λ o u t = k − 1 k − 2 k − 3 k ( − h ) and α = [ ( k 2 + k − 1 ) k 3 + k − 1 k − 2 ] ( k ( − h ) + k ( h ) ) β = k − 3 [ ( k − 2 + k 2 + k − 1 ) k ( − h ) + k − 1 k − 2 ] γ = k 1 [ ( k 3 + k − 2 + k 2 ) k ( h ) + k 2 k 3 ] δ = ( k − 2 + k 2 ) k 1 k − 3

Eq. 2 may look rather complex at first but has several interesting built-in features. The numerator is nicely symmetrical with respect to the kinetic constants, to external Na⁺ _e and internal Na⁺ _i. The minus sign shows that the transport direction depends on the respective Na⁺ concentrations. It also shows how a rise in intracellular Na⁺ concentration will slow down the inward Na⁺/H⁺ exchange.

Eq. 2 shape is also very close to the Michaelis-Menten Eq. 1 as it can be rearranged into Eq. 3 below. V = [ E o ] ( Λ i n ( γ + δ [ N a i + ] ) ) [ N a e + ] − ( Λ o u t ( γ + δ [ N a i + ] ) ) [ N a i + ] α + β [ N a i + ] γ + δ [ N a i + ] + [ N a e + ] (3)

If we consider that the intracellular Na⁺ _i concentration is small enough compared to the extracellular Na⁺ _e, we could then simplify Eq. 3 by neglecting Na⁺ _i into Eq. 4 below. V = ( Λ i n ( γ ) ) [ E o ] [ N a e + ] α γ + [ N a e + ] (4) Which is the Michaelis-Menten Eq. 1 for Na⁺ _e where V m a x = Λ i n γ [ E o ] and K m = α γ .

This implies that when a mutation causes a change in Km, it is risky to assign this to a change in extracellular Na⁺ binding, because its effect could also originate from a change in the off constant (k₃) for intracellular sodium or more subtly by a change in the k₂ and or k_-2 kinetic constants of the ion translocation itself. Such intricate effects are in principle impossible to tease out from steady state measurements because as it can be seen in such equations, the apparent constants have a complex shape and there are more unknown constants than measurable values.

Another important point which deserves attention concerns the inhibitory constants of the NHE acylguanidine inhibitors such as amiloride or cariporide, because they compete with extracellular sodium. Reciprocally Na⁺ also makes competition with the inhibitors and as a consequence, the Michaelis Menten rate equation for transport turns into: V = V m a x [ N a e + ] K m ( 1 + [ I ] K i ) + [ N a e + ]

Here [I] is the inhibitor concentration, K _i the inhibitor dissociation constant (a true Kd on this occasion) and where Vmax and Km can have the previous complex expressions. The following facts have to be stressed: 1) a measured K0.5 value cannot be assigned to a Ki value because of the presence of Na ⁺ _e , 2) published work reports different K0.5 for the same inhibitor depending on the extracellular cation concentrations used for the measurements, and 3) in case a mutation changes the dose response curve of an inhibitor, the Km for Na ⁺ has to be always measured and a data analysis has to be performed to calculate the actual K _i value if Km is found changed. Taken together, these considerations show that transport data can yield the K _i values for inhibitors when properly treated. However, the situation is problematic if not unusable for Na⁺ binding and transport as it is impossible to predict whether a particular position in an NHE is involved in the direct coordination or in the transport of Na⁺ or both. In this respect, it is amazing that most of the residues and positions identified by structure-function studies fall in places that belong to these categories in NHE1 3D structure as the structural complexity and organization now offers insight that could not be revealed by placing the identified crucial amino acids within the previous topological models. Indeed, a constellation of side chains atoms is remarkably in the right place to directly interact with the cariporide structure. As this inhibitor contains a guanidine moiety with a structure similar to a partly hydrated Na⁺, the NHE1 structure also reveals how this cation could sit in its external binding pocket (Figure 5). Those correspond to 1) Leu 163 and Phe162 in the remarkable VFFLFLLPPII TM3 sequence (Counillon et al., 1993) (Counillon et al., 1997) (Touret et al., 2001) that makes a very beautiful π−stack with the inhibitors’ aromatic ring (2.6 Å distance), 2) TM8 Glu346 that is at less than 2.8 Å from the guanidine group (Khadilkar et al., 2001; Noël et al., 2003). Of note a recent article by Fliegel’s group highlighted Leu468 of TM11 that is also in a very close position to the hydrophobic 5- substituents of cariporide, thereby providing a molecular basis on the mechanism by which an increased hydrophobicity of these inhibitors’ groups can decrease their Ki values by two orders of magnitude (Li et al., 2021). Finally, the absence of side chain of Gly352 that lies underneath the inhibitor appears to be a steric effect as an amino acid with a large side-chain would collide with the inhibitor (Khadilkar et al., 2001).

In a recent work, Jinadasa et al. (2015) used cysteine substitution and MTS accessibility to map amino acids involved in the interaction with ethylisopropylamiloride, a molecule close to cariporide. This revealed interesting candidates as shown in Figure 5C, clearly accessible to externally-applied MTS, but more likely to exert some distance effect on inhibitor or Na⁺ interaction.

Apart from these residues that are obviously in direct interaction with the competitive inhibitor and very likely with the partially hydrated transported Na⁺, mutations at some other positions appear to have indirect distant effects. As explained above in the kinetic analysis of NHE1 transport, mutations that modify Km or Ki values can have conformational effects that impact kinetic steps instead of binding. When far away but in the same plane as cariporide they likely exert such conformational domino effects on the structure, such as TM11 His 473, Met 476 (Li et al., 2021) or TM2/EL2 Gly 148; and Phe155, or His 349 (Orlowski and Kandasamy, 1996; Khadilkar et al., 2001). Similarly, we described mutations of amino acids in the TM3 sequence, that affected the Km and apparent inhibition constants of competitors by such distance conformational effects. Those were in particular Gly 174 (Counillon et al., 1997), Ile 169 and 170, the mutation of which were able to revert the Phe162Ser mutation (Touret et al., 2001). From the structure, those are indeed far from the external binding site, thereby confirming the difference between binding and kinetic effects.

Positive cooperativity demarks from classical Michaelis-Menten saturation as the hyperbolic shape of the dose-response curve is bent to yield a S shape curve, termed sigmoid. Being steeper than a classical hyperbolic, the cooperative response often constitutes a molecular switch, the setpoint of which can be modified by the interaction with different allosteric modulators. Hence, most enzymes or transporters possessing a cooperative behavior have been selected by evolution at strategic positions to activate or block critical biochemical pathways. As protons are one of the most, if not the most relevant ions in physiology, evolution has firmly incorporated cooperativity in NHEs kinetics. Technically, it must be noted that using pH as a variable can be misleading at first glance, because it is the cologarithm of the actual free H⁺ ion concentration that is the relevant parameter, with sub millimolar concentrations between 10^–7 and 10^–8 mol/L in the cytosol in physiological conditions. This is particularly important for graphical representations as the Log scale that is embedded in pH representations can be confusing when analyzing possible cooperativity.

Mathematically, sigmoidal equations rates for enzymes or transporters are different from the logistic equation often used to fit sigmoids and whatever the mechanism, can be written as a fraction of two polynomials for the substrate. This stems from the fact that proteins displaying cooperativity possess more than one binding site for their substrate, leading to multiple equilibria. This means that the reaction order is superior to one for the substrate, and this translates in exponents superior to one for its concentration in the kinetic equations. Another important feature is that those binding sites are neither identical nor independent, meaning that cooperative proteins exist in different conformational states bearing differences in substrate binding or transport. Beyond these general features it is challenging to precisely decipher the underlying mechanisms for any cooperative behavior, because a near infinity of polynomial fractions can fit the same sigmoid. As each equation corresponds to a distinct possible process, it means that it is in principle conceivable to hypothesize an infinite number of mechanisms that would all fit, provided that the constants (that can be fairly complex as explained previously) and exponents are well adjusted. Hence to discriminate between the potential models, extensive information about mutagenesis and/or structure is needed in addition to kinetics and dose responses. Because the required formalism may go beyond the purpose of this review, the interested reader is encouraged to consult advanced enzyme kinetics textbooks such as Bisswanger (2008).

The above-mentioned findings have led to interesting questions concerning the allosteric regulation of NHEs by intracellular H⁺. The first main mechanism proposed after the discovery of the NHEs cooperativity, was that of a monomer with a “proton sensing site” that would allosterically regulate the affinity of the transport site. This led many colleagues to try to identify such a sensor by mutating candidate amino acids, mostly histidines, the pKa of which allows them in principle to bind/unbind H⁺ around pH7. However, without entering in mathematical details, a cooperativity model resulting from a mechanism involving a simple binding of an alternative H⁺ in a non-transport site would unsatisfactorily fit the data due to the reciprocal dependency of the transport and sensor sites (Lacroix et al., 2004). Considering the NHEs dimeric structure and a combination of mutagenesis and kinetic analyses we proposed a mechanism in which the two protomers, strongly interlocked within a symmetrical dimer, would oscillate between a low and a high affinity state for H⁺ in a concerted manner (Monod et al., 1965). Any change that would affect the balance between these two forms, from covalent modification of the protein to its interaction with different molecules or ions, would change the sigmoidal shape and therefore provide H⁺ sensing, without the need for an additional binding site. This is largely mediated through the C-terminal region of NHE1 that contains multiple regulatory sites for signaling pathways. Depending on its interactions with lipids, ATP, proteins and on the balance between its multiple phosphorylations and dephosphorylations (Hendus-Altenburger et al., 2016; Hendus-Altenburger et al., 2019), this region can modify NHE1 cooperativity for protons. In particular, the structure shows that the Pro571-Ser591 short helical antiparallel dimer could be instrumental in this allosteric coupling.

Such regulatory mechanism provides the selective advantage to operate as a sensitive coincidence detector for infinite combinations of stimuli, an action that would be impossible to achieve by direct modification of a proton regulator site.

A strong support for this mechanism came from the identification of distinct mutations that locked NHE1 in non-cooperative conformation and/or the existence of NHEs with lost cooperativity for protons. A similar low affinity non cooperative NHE1 could be obtained respectively through the mutation of conserved Arginines 327 (Lacroix et al., 2004) and 440 (Wakabayashi et al., 2003a; Wakabayashi et al., 2003b), Serine 375 and Tyrosine 381 (Hisamitsu et al., 2006). Interestingly, total (Lacroix et al., 2004) or partial NHE1 C-terminal truncation, such as the above-mentioned Met 561-Ala575 sequence, yielded the same low affinity non-cooperative NHE1 (visible when plotted as a function of H⁺ instead of pH in Hisamitsu et al., 2004). Interestingly, Arg 327 is not conserved in the non-cooperative NHE7 that displays a high affinity for intracellular H⁺ (Milosavljevic et al., 2014). As arginine’s pKa is 12.5, those amino acids are not good candidates for direct H⁺ binding and release at physiological pH values. In contrast, bearing a positive charged group at the end of a flexible arm could be extremely useful either to couple to other amino acids or act as a short probes for the electrostatics of their environment. Similarly, the Ser 375 and Tyr 381 are not deprotonable at physiological values but may form hydrogen bond and exert a conformational role. Considering the effects of all these mutations, we can predict that these residues must be placed at strategic positions within the dimer. Indeed, the analysis of the NHE1 structure in different conditions reveals a critical position for the two Arg327 that are situated right at the interface between the two protomers (Figure 6A). In the pH6.5 structure (Figure 6A), these two arginines point the positive extremity of their side chains towards the cytosol, and could work exactly as the previously discussed electrostatic probes. Even more strikingly, Arg 327 is in very close vicinity of Ser 375 and Tyr 381, the two critical residues previously mapped in the dimer interface (see above) by the Wakabayashi group. In the 6.5 structure, the main backbone carbonyl of Arg 327, and the hydrogens of Ser 375 and Tyr 381 side chain hydroxyl groups are at optimal distances (∼3 Å) to hydrogen bond. Ser 375 substitution into a cysteine led to a low affinity of NHE1 for protons and this had been interpreted as a possible effect of disulfide bond crosslinking that would block the structure. However, Figure 4B shows that these two side chains are pointing in an opposite direction making disulfide bond formation unlikely. Interestingly, the cariporide-bound structure shows a totally different configuration in which Arg 327 is not in a flexible loop. Indeed, its main chain carbonyl is engaged in an alpha helix, with no possibility to interact with the previous amino acids, its side chain being 6–8 Å apart from the Ser 375 hydroxyl. Moreover, in this conformation, the Tyr 381 side chain is now totally opposite to the dimer interface. Taken together, the input from the structural information highlights the importance of the previously discovered amino acids in a symmetrical dimer. In addition, the conformational changes resulting in the exquisite sensitivity of NHE1 for protons (Lacroix et al., 2004) are uncovered. Arg 440 mutations were also identified for giving a very similar phenotype as those of Arg 327 while Asp 448, Gly 455, and 456 mutations resulted in an enhanced sensitivity to protons. These positions in the structure are far away from the dimer interface. Gly 455 and 456 are in the TM11 non-helical region (Figure 7). Arg 440 and Asp 448 are present in a perfect pocket constituted by helices TM4, 11, 12, and the CHP binding region (Figure 7). Interestingly, mutations that affect NHE1 activity or its response to intracellular pH have been identified between Leu 432 and Lys 443 by the Fliegel’s group (Wong et al., 2019).

It is important to stress that all the mutations that decrease the H⁺ sensing for intracellular H⁺ involve amino acids that cannot be protonated/deprotonated around the NHE1 setpoint for intracellular H⁺, nearby physiological pH. In contrast, they are all at interfacial positions, either at the dimer interface, or at the boundary with the regulatory C-terminal loop or at hinge sequences of NHE1. Another important point is that all the Cryo-EM structures correspond to symmetrical dimers, and not to different conformations within the same dimer. Taken in aggregate, the above summarized studies provide a strong accumulation of results in favor of a concerted cooperative mechanism for proton sensing.

As stated in the introduction, the structural determination of NHE1 and NHE9 have provided a long-awaited and decisive breakthrough in the understanding of the Na⁺/H⁺ exchange world. The aim of our short review in the context of this dedicated issue was to put in a historical perspective some of the main findings that shaped the NHE1 knowledge for many years, namely the topology, dimeric structure, biochemical analysis and key mutations. Learning from the past and considering the advances already made in the field, it is obvious that further progress is contingent on combining structural information with data from functional measurements with the adequate methods that will have to use more sophisticated mathematics and modeling. Future progress will also come from setting up more resolutive measurement methods such as extremely fast presteady state kinetics.

Finally, while the unraveling of the tridimensional structure of the Na⁺/H⁺ exchanger is clearly a spectacular leap forward in the field, it is gratifying for the pioneers in this research area that most of their vision on their favored molecule has stood the test of time. In the meantime, it is fascinating to see how results that could appear rather abstract are now highlighted in a very visual and even aesthetic perspective in those structures. Times are changing -so are models but for both the former and the current Na⁺/H⁺ exchanger scientists “beauty is in the eyes of the beholder” (Aronson et al., 1982; Wakabayashi et al., 2003b).