Thermodynamically consistent determination of free energies and rates in kinetic cycle models

Kinetic and thermodynamic models of biological systems are commonly used to connect microscopic features to system function in a bottom-up multiscale approach. The parameters of such models-free energy differences for equilibrium properties and in general rates for equilibrium and out-of-equilibrium observables-have to be measured by different experiments or calculated from multiple computer simulations. All such parameters necessarily come with uncertainties so that when they are naively combined in a full model of the process of interest, they will generally violate fundamental statistical mechanical equalities, namely detailed balance and an equality of forward/backward rate products in cycles due to Hill. If left uncorrected, such models can produce arbitrary outputs that are physically inconsistent. Here, we develop a maximum likelihood approach (named multibind) based on the so-called potential graph to combine kinetic or thermodynamic measurements to yield state-resolved models that are thermodynamically consistent while being most consistent with the provided data and their uncertainties. We demonstrate the approach with two theoretical models, a generic two-proton binding site and a simplified model of a sodium/proton antiporter. We also describe an algorithm to use the multibind approach to solve the inverse problem of determining microscopic quantities from macroscopic measurements and, as an example, we predict the microscopic p K a values and protonation states of a small organic molecule from 1D NMR data. The multibind approach is applicable to any thermodynamic or kinetic model that describes a system as transitions between well-defined states with associated free energy differences or rates between these states. A Python package multibind, which implements the approach described here, is made publicly available under the MIT Open Source license.

Thermodynamically consistent determination of free energies and rates in kinetic cycle models

Kinetic and thermodynamic models of biological systems are commonly used to connect microscopic features to system function in a bottom-up multiscale approach. The parameters of such models-free energy differences for equilibrium properties and in general rates for equilibrium and out-of-equilibrium observables-have to be measured by different experiments or calculated from multiple computer simulations. All such parameters necessarily come with uncertainties so that when they are naively combined in a full model of the process of interest, they will generally violate fundamental statistical mechanical equalities, namely detailed balance and an equality of forward/backward rate products in cycles due to T. Hill. If left uncorrected, such models can produce arbitrary outputs that are physically inconsistent. Here we develop a maximum likelihood approach (named multibind ) based on the so-called potential graph to combine kinetic or thermodynamic measurements to yield state resolved models that are thermodynamically consistent while being most consistent with the provided data and their uncertainties. We demonstrate the approach with two theoretical models, a generic two-proton binding site and a simplified model of a sodium/proton antiporter. We also describe an algorithm to use the multibind approach to solve the inverse problem of determining microscopic quantities from macroscopic measurements and as an example we predict the microscopic p K a s and protonation states of a small organic molecule from 1D NMR data. The multibind approach is applicable to any thermodynamic or kinetic model that describes a system as transitions between well-defined states with associated free energy differences or rates between these states. A Python package multibind , which implements the approach described here, is made publicly available under the MIT Open Source license.

WHY IT MATTERS

The increase in computational efficiency and rapid advances in methodology for quantitative free energy and rate calculations has allowed for the construction of increasingly complex thermodynamic or kinetic "bottom-up" models of chemical and biological processes. These multi-scale models serve as a framework for analyzing aspects of cellular function in terms of microscopic, molecular properties and provide an opportunity to connect molecular mechanisms to cellular function. The underlying model parameters-free energy differences or rates-are constrained by thermodynamic identities over cycles of states but these identities are not necessarily obeyed during model construction, thus potentially leading to inconsistent models. We address these inconsistencies through the use of a maximum likelihood approach for free energies and rates to adjust the model parameters in such a way that they are maximally consistent with the input parameters and exactly fulfill the thermodynamic cycle constraints. This approach enables formulation of thermodynamically consistent multi-scale models from simulated or experimental measurements.

Prokaryotic Na+/H+ Exchangers—Transport Mechanism and Essential Residues

Na⁺/H⁺ exchangers are essential for Na⁺ and pH homeostasis in all organisms. Human Na⁺/H⁺ exchangers are of high medical interest, and insights into their structure and function are aided by the investigation of prokaryotic homologues. Most prokaryotic Na⁺/H⁺ exchangers belong to either the Cation/Proton Antiporter (CPA) superfamily, the Ion Transport (IT) superfamily, or the Na⁺-translocating Mrp transporter superfamily. Several structures have been solved so far for CPA and Mrp members, but none for the IT members. NhaA from E. coli has served as the prototype of Na⁺/H⁺ exchangers due to the high amount of structural and functional data available. Recent structures from other CPA exchangers, together with diverse functional information, have allowed elucidation of some common working principles shared by Na⁺/H⁺ exchangers from different families, such as the type of residues involved in the substrate binding and even a simple mechanism sufficient to explain the pH regulation in the CPA and IT superfamilies. Here, we review several aspects of prokaryotic Na⁺/H⁺ exchanger structure and function, discussing the similarities and differences between different transporters, with a focus on the CPA and IT exchangers. We also discuss the proposed transport mechanisms for Na⁺/H⁺ exchangers that explain their highly pH-regulated activity profile.

General principles of secondary active transporter function

Transport of ions and small molecules across the cell membrane against electrochemical gradients is catalyzed by integral membrane proteins that use a source of free energy to drive the energetically uphill flux of the transported substrate. Secondary active transporters couple the spontaneous influx of a "driving" ion such as Na+ or H+ to the flux of the substrate. The thermodynamics of such cyclical non-equilibrium systems are well understood, and recent work has focused on the molecular mechanism of secondary active transport. The fact that these transporters change their conformation between an inward-facing and outward-facing conformation in a cyclical fashion, called the alternating access model, is broadly recognized as the molecular framework in which to describe transporter function. However, only with the advent of high resolution crystal structures and detailed computer simulations, it has become possible to recognize common molecular-level principles between disparate transporter families. Inverted repeat symmetry in secondary active transporters has shed light onto how protein structures can encode a bi-stable two-state system. Based on structural data, three broad classes of alternating access transitions have been described as rocker-switch, rocking-bundle, and elevator mechanisms. More detailed analysis indicates that transporters can be understood as gated pores with at least two coupled gates. These gates are not just a convenient cartoon element to illustrate a putative mechanism but map to distinct parts of the transporter protein. Enumerating all distinct gate states naturally includes occluded states in the alternating access picture and also suggests what kind of protein conformations might be observable. By connecting the possible conformational states and ion/substrate bound states in a kinetic model, a unified picture emerges in which the symporter, antiporter, and uniporter functions are extremes in a continuum of functionality. As usual with biological systems, few principles and rules are absolute and exceptions are discussed as well as how biological complexity may be integrated in quantitative kinetic models that may provide a bridge from the structure to function.

Functional Characterization of SLC Transporters Using Solid Supported Membranes

Here, we present a protocol for the functional characterization of the H⁺-coupled human peptide transporter PepT1 and sufficient notes to transfer the protocol to the Na⁺-coupled sugar transporter SGLT1, the organic cation transporter OCT2, the Na⁺/Ca²⁺ exchanger NCX, and the neuronal glutamate transporter EAAT3.The assay was developed for the commercially available SURFE²R N1 instrument (Nanion Technologies GmbH) which applies solid supported membrane (SSM)-based electrophysiology. This technique is widely used for the functional characterization of membrane transporters with more than 100 different transporters characterized so far. The technique is cost-effective, easy to use, and capable of high-throughput measurements.SSM-based electrophysiology utilizes SSM-coated gold sensors to physically adsorb membrane vesicles containing the protein of interest. A fast solution exchange provides the substrate and activates transport. For the measurement of PepT1 activity, we applied a peptide concentration jump to activate H⁺/peptide symport. Proton influx charges the sensor. A capacitive current is measured reflecting the transport activity of PepT1 . Multiple measurements on the same sensor allow for comparison of transport activity under different conditions. Here, we determine EC₅₀ for PepT1-mediated glycylglycine transport and perform an inhibition experiment using the specific peptide inhibitor Lys[Z(NO₂)]-Val.

Mutation of two key aspartate residues alters stoichiometry of the NhaB Na+/H+ exchanger from Klebsiella pneumoniae

Bacterial NhaB Na⁺/H⁺ exchangers belonging to the Ion Transporter superfamily are poorly characterized in contrast to Na⁺/H⁺ exchangers of the Cation Proton Antiporter superfamily which have NhaA from Escherichia coli as a prominent member. For a more detailed understanding of the intricacies of the exchanger’s transport mechanism, mutational studies are essential. Therefore, we mutated two protonatable residues present in the putative transmembrane region of NhaB from Klebsiella pneumoniae (KpNhaB), which could serve as substrate binding sites, Asp146 and Asp404, to either glutamate or alanine and analyzed transport function and stability of the mutants using electrophysiological and fluorimetric techniques. While mutation of either Asp residue to Glu only had slight to moderate effects on the transport activity of the exchanger, the mutations D404A and D146A, in particular, had more profound effects on the transport function. Furthermore, a double mutant, D146A/D404A, exhibited a remarkable behavior at alkaline pH, where recorded electrical currents changed polarity, showing steady-state transport with a stoichiometry of H⁺:Na⁺ < 1, as opposed to the H⁺:Na⁺ > 1 stoichiometry of the WT. Thus, we showed that Asp146 and Asp404 are part of the substrate binding site(s) of KpNhaB and engineered a Na⁺/H⁺ exchanger with a variable stoichiometry.

Replacement of Lys-300 with a glutamine in the NhaA Na+/H+ antiporter of Escherichia coli yields a functional electrogenic transporter

Much of the research on Na+/H+ exchange has been done in prokaryotic models, mainly on the NhaA Na+/H+-exchanger from Escherichia coli (EcNhaA). Two conserved aspartate residues, Asp-163 and Asp-164, are essential for transport and are candidates for possible binding sites for the two H+ that are exchanged for one Na+ to make the overall transport process electrogenic. More recently, a proposed mechanism of transport for EcNhaA has suggested direct binding of one of the transported H+ to the conserved Lys-300 residue, a salt bridge partner of Asp-163. This contention is supported by a study reporting that substitution of the equivalent residue, Lys-305, of a related Na+/H+ antiporter, NapA from Thermus thermophilus, renders the transporter electroneutral. In this work, we sought to establish whether the Lys-300 residue and its partner Asp-163 are essential for the electrogenicity of EcNhaA. To that end, we replaced Lys-300 with Gln, either alone or together with the simultaneous substitution of Asp-163 with Asn, and characterized these transporter variants in electrophysiological experiments combined with H+ transport measurements and stability analysis. We found that K300Q EcNhaA can still support electrogenic Na+/H+ antiport in EcNhaA, but has reduced thermal stability. A parallel electrophysiological investigation of the K305Q variant of TtNapA revealed that it is also electrogenic. Furthermore, replacement of both salt bridge partners in the ion-binding site of EcNhaA produced an electrogenic variant (D163N/K300Q). Our findings indicate that alternative mechanisms sustain EcNhaA activity in the absence of canonical ion-binding residues and that the conserved lysines confer structural stability.

Coupling efficiency of secondary active transporters

Secondary active transporters are fundamental to a myriad of biological processes. They use the electrochemical gradient of one solute to drive transport of another solute against its concentration gradient. Central to this mechanism is that the transport of one does not occur in the absence of the other. However, like in most of biology, imperfections in the coupling mechanism exist and we argue that these are innocuous and may even be beneficial for the cell. We discuss the energetics and kinetics of alternating-access in secondary transport and focus on the mechanistic aspects of imperfect coupling that give rise to leak pathways. Additionally, inspection of available transporter structures gives valuable insight into coupling mechanics, and we review literature where proteins have been altered to change their coupling efficiency.

CatSper channels are regulated by protein kinase A

Mammalian sperm must undergo capacitation as a preparation for entering into hyperactivated motility, undergoing the acrosome reaction, and acquiring fertilizing ability. One of the initial capacitation events occurs when sperm encounter an elevated HCO₃ ^- concentration. This anion activates the atypical adenylyl cyclase Adcy10, increases intracellular cAMP, and stimulates protein kinase A (PKA). Moreover, an increase in intracellular Ca²⁺ concentration ([Ca²⁺] _i ) is essential for sperm capacitation. Although a cross-talk between cAMP-dependent pathways and Ca²⁺ clearly plays an essential role in sperm capacitation, the connection between these signaling events is incompletely understood. Here, using three different approaches, we found that CatSper, the main sperm Ca²⁺ channel characterized to date, is up-regulated by a cAMP-dependent activation of PKA in mouse sperm. First, HCO₃ ^- and the PKA-activating permeable compound 8-Br-cAMP induced an increase in [Ca²⁺] _i , which was blocked by the PKA peptide inhibitor PKI, and H89, another PKA inhibitor, also abrogated the 8-Br-cAMP response. Second, HCO₃ ^- increased the membrane depolarization induced upon divalent cation removal by promoting influx of monovalent cations through CatSper channels, which was inhibited by PKI, H89, and the CatSper blocker HC-056456. Third, electrophysiological patch clamp, whole-cell recordings revealed that CatSper activity is up-regulated by HCO₃ ^- and by direct cAMP injection through the patch-clamp pipette. The activation by HCO₃ ^- and cAMP was also blocked by PKI, H89, Rp-cAMPS, and HC-056456, and electrophysiological recordings in sperm from CatSper-KO mice confirmed CatSper's role in these activation modes. Our results strongly suggest that PKA-dependent phosphorylation regulates [Ca²⁺] _i homeostasis by activating CatSper channel complexes.

Dopamine reduces cell surface Na+/H+ exchanger-3 protein by decreasing NHE3 exocytosis and cell membrane recycling

The intrarenal autocrine-paracrine dopamine (DA) system mediates a significant fraction of the natriuresis in response to a salt load. DA inhibits a number of Na⁺ transporters to effect sodium excretion, including the proximal tubule Na⁺/H⁺ exchanger-3 (NHE3). DA represent a single hormone that regulates NHE3 at multiple levels, including translation, degradation, endocytosis, and protein phosphorylation. Because cell surface NHE3 protein is determined by the balance between exocytotic insertion and endocytotic retrieval, we examined whether DA acutely affects the rate of NHE3 exocytosis in a cell culture model. DA inhibited NHE3 exocytosis at a dose-dependent manner with a half maximal around 10^-6 M. The DA effect on NHE3 exocytosis was blocked by inhibition of protein kinase A and by brefeldin A, which inhibits endoplasmic reticulum-to-Golgi transport. NHE3 directly interacts with the ε-subunit of coatomer protein based on yeast-two-hybrid and coimmunoprecipitation. Because NHE3 has been shown to be recycled back to the cell membrane after endocytosis, we measured NHE3 recycling using a biochemical reinsertion assay and showed that reinsertion of NHE3 back to the membrane is also inhibited by DA. In conclusion, among the many mechanisms by which DA reduces apical membrane NHE3 and induces proximal tubule natriuresis, one additional mechanism is inhibition of exocytotic insertion and reinsertion of NHE3 in the apical cell surface.

Competition is the basis of the transport mechanism of the NhaB Na+/H+ exchanger from Klebsiella pneumoniae

Na⁺/H⁺ exchange is essential for survival of all organisms, having a role in the regulation of the intracellular Na⁺ concentration, pH and cell volume. Furthermore, Na⁺/H⁺ exchangers were shown to be involved in the virulence of the bacterium Yersinia pestis, indicating they might be potential targets for novel antibiotic treatments. The model system for Na⁺/H⁺ exchangers is the NhaA transporter from Escherichia coli, EcNhaA. Therefore, the general transport mechanism of NhaA exchangers is currently well characterized. However, much less is known about NhaB exchangers, with only a limited number of studies available. The pathogen Klebsiella pneumoniae, which is a major source of nosocomial infection, possesses three electrogenic Na⁺/H⁺ exchangers, KpNhaA1, KpNhaA2 and KpNhaB, none of which have been previously investigated. Our aim in this study was to functionally characterize KpNhaB using solid supported membrane-based electrophysiology as the main investigation technique, and thus provide the first electrophysiological investigation of an NhaB Na⁺/H⁺ exchanger. We found that NhaB can be described by the same competition-based mechanism that was shown to be valid for electrogenic NhaA and NapA, and for electroneutral NhaP Na⁺/H⁺ exchangers. For comparison we also characterized the activity of KpNhaA1 and KpNhaA2 and found that the three exchangers have complementary activity profiles, which is likely a survival advantage for K. pneumoniae when faced with environments of different salinity and pH. This underlines their importance as potential antibiotic drug targets.

Lysine 300 is essential for stability but not for electrogenic transport of the Escherichia coli NhaA Na+/H+ antiporter

Na+/H+ antiporters are located in the cytoplasmic and intracellular membranes and play crucial roles in regulating intracellular pH, Na+, and volume. The NhaA antiporter of Escherichia coli is the best studied member of the Na+/H+ exchanger family and a model system for all related Na+/H+ exchangers, including eukaryotic representatives. Several amino acid residues are important for the transport activity of NhaA, including Lys-300, a residue that has recently been proposed to carry one of the two H+ ions that NhaA exchanges for one Na+ ion during one transport cycle. Here, we sought to characterize the effects of mutating Lys-300 of NhaA to amino acid residues containing side chains of different polarity and length (i.e. Ala, Arg, Cys, His, Glu, and Leu) on transporter stability and function. Salt resistance assays, acridine-orange fluorescence dequenching, solid supported membrane-based electrophysiology, and differential scanning fluorometry were used to characterize Na+ and H+ transport, charge translocation, and thermal stability of the different variants. These studies revealed that NhaA could still perform electrogenic Na+/H+ exchange even in the absence of a protonatable residue at the Lys-300 position. However, all mutants displayed lower thermal stability and reduced ion transport activity compared with the wild-type enzyme, indicating the critical importance of Lys-300 for optimal NhaA structural stability and function. On the basis of these experimental data, we propose a tentative mechanism integrating the functional and structural role of Lys-300.

NHA2 is expressed in distal nephron and regulated by dietary sodium

Increased renal reabsorption of sodium is a significant risk factor in hypertension. An established clinical marker for essential hypertension is elevated sodium lithium countertransport (SLC) activity. NHA2 is a newly identified Na⁺(Li⁺)/H⁺ antiporter with potential genetic links to hypertension, which has been shown to mediate SLC activity and H⁺-coupled Na⁺(Li⁺) efflux in kidney-derived MDCK cells. To evaluate a putative role in sodium homeostasis, we determined the effect of dietary salt on NHA2. In murine kidney sections, NHA2 localized apically to distal convoluted (both DCT1 and 2) and connecting tubules, partially overlapping in distribution with V-ATPase, AQP2, and NCC1 transporters. Mice fed a diet high in sodium chloride showed elevated transcripts and expression of NHA2 protein. We propose a model in which NHA2 plays a dual role in salt reabsorption or secretion, depending on the coupling ion (sodium or protons). The identified novel regulation of Na⁺/H⁺ antiporter in the kidney suggests new roles in salt homeostasis and disease.

Electrogenic Cation Binding in the Electroneutral Na+/H+ Antiporter of Pyrococcus abyssi

Na⁺/H⁺ antiporters in the CPA1 branch of the cation proton antiporter family drive the electroneutral exchange of H⁺ against Na⁺ ions and ensure pH homeostasis in eukaryotic and prokaryotic organisms. Although their transport cycle is overall electroneutral, specific partial reactions are electrogenic. Here, we present an electrophysiological study of the PaNhaP Na⁺/H⁺ antiporter from Pyrococcus abyssi reconstituted into liposomes. Positive transient currents were recorded upon addition of Na⁺ to PaNhaP proteoliposomes, indicating a reaction where positive charge is rapidly displaced into the proteoliposomes with a rate constant of k >200 s^-1 We attribute the recorded currents to an electrogenic reaction that includes Na⁺ binding and possibly occlusion. Subsequently, positive charge is transported out of the cell associated with H⁺ binding, so that the overall reaction is electroneutral. We show that the differences in pH profile and Na⁺ affinity of PaNhaP and the related MjNhaP1 from Methanocaldococcus jannaschii can be attributed to an additional negatively charged glutamate residue in PaNhaP. The results are discussed in the context of the physiological function of PaNhaP and other microbial Na⁺/H⁺ exchangers. We propose that both, electroneutral and electrogenic Na⁺/H⁺ antiporters, represent a carefully tuned self-regulatory system, which drives the cytoplasmic pH back to neutral after any deviation.

Simulating the Function of the MjNhaP1 Transporter

The structures of transport proteins have been steadily revealed in the last few decades, and yet the conversion of this information into molecular-level understanding of their function is still lagging behind. In this study, we try to elucidate how the action of the archaeal sodium/proton antiporter MjNhaP1 depends on its structure-energy relationship. To this end, we calculate the binding energies of its substrates and evaluate the conformational change barrier, focusing on the rotation of the catalytic residue D161. We find that sodium ions and protons compete against a common binding site and that the accessibility of this binding site is restricted to either the inside or outside of the cell. We suggest that the rotation of D161 χ1 angle correlates with the conformational change and is energetically unfavorable when D161 does not bind any substrate. This restriction ensures coupling between the sodium ions and the protons, allowing MjNhaP1 and probably other similar transporters to exchange substrates with minimal leak. Using Monte Carlo simulations we demonstrate the feasibility of our model. Overall we present a complete picture that reproduces the electroneutral (at 1:1 substrate ratio) and coupled transport activity of MjNhaP1 including the energetic basis for the criteria provided by Jardetzky half a century ago.

Mechanism of pH-dependent activation of the sodium-proton antiporter NhaA

Escherichia coli NhaA is a prototype sodium-proton antiporter, which has been extensively characterized by X-ray crystallography, biochemical and biophysical experiments. However, the identities of proton carriers and details of pH-regulated mechanism remain controversial. Here we report constant pH molecular dynamics data, which reveal that NhaA activation involves a net charge switch of a pH sensor at the entrance of the cytoplasmic funnel and opening of a hydrophobic gate at the end of the funnel. The latter is triggered by charging of Asp164, the first proton carrier. The second proton carrier Lys300 forms a salt bridge with Asp163 in the inactive state, and releases a proton when a sodium ion binds Asp163. These data reconcile current models and illustrate the power of state-of-the-art molecular dynamics simulations in providing atomic details of proton-coupled transport across membrane which is challenging to elucidate by experimental techniques.

The pH dependence of the activity of Escherichia coli main sodium-proton antiporter NhaA is still not fully understood. Here, the authors use continuous constant pH molecular dynamics simulations to identify NhaA proton carrier residues and elucidate its gating and ion transport processes.

Na(+)/H(+) antiporter (NHE1) and lactate/H(+) symporters (MCTs) in pH homeostasis and cancer metabolism

The Na(+)/H(+)-exchanger NHE1 and the monocarboxylate transporters MCT1 and MCT4 are crucial for intracellular pH regulation, particularly under active metabolism. NHE1, a reversible antiporter, uses the energy provided by the Na(+) gradient to expel H(+) ions generated in the cytosol. The reversible H(+)/lactate(-) symporters MCT1 and 4 cotransport lactate and proton, leading to the net extrusion of lactic acid in glycolytic tumors. In the first two sections of this article we review important features and remaining questions on the structure, biochemical function and cellular roles of these transporters. We then use a fully-coupled mathematical model to simulate their relative contribution to pH regulation in response to lactate production, as it occurs in highly hypoxic and glycolytic tumor cells. This article is part of a Special Issue entitled: Mitochondrial Channels edited by Pierre Sonveaux, Pierre Maechler and Jean-Claude Martinou.