Kinetic network modeling with molecular simulation inputs: A proton-coupled phosphate symporter

Abstract

Phosphate, an essential metabolite involved in numerous cellular functions, is taken up by proton-coupled phosphate transporters of plants and fungi within the major facilitator family. Similar phosphate transporters have been identified across a diverse range of biological entities, including various protozoan parasites linked to human diseases, breast cancer cells with increased phosphate requirements, and osteoclast-like cells engaged in bone resorption. Prior studies have proposed an overview of the functional cycle of a proton-driven phosphate transporter (PiPT), yet a comprehensive understanding of the proposed reaction pathways necessitates a closer examination of each elementary reaction step within an overall kinetic framework. In this work, we leverage kinetic network modeling in conjunction with a “bottom-up” molecular dynamics approach to show how such an approach can characterize the proton-phosphate co-transport behavior of PiPT under different pH and phosphate concentration conditions. In turn, this allows us to reveal the prevailing reaction pathway within a high-affinity phosphate transporter under different experimental conditions and to uncover the molecular origin of the optimal pH condition of this transporter.

Significance

In this work, multiscale reactive molecular dynamics, hybrid quantum mechanics/molecular mechanics, and classical molecular dynamics, including enhanced free energy sampling, are integrated to construct a bottom-up kinetic model for a proton-coupled phosphate transporter. This framework provides an expansive view of the potential reaction pathways, illuminating the transitions from phosphate-bound outward-facing states to phosphate-released inward-facing states. Through this comprehensive approach, not only can the influence of molecular interactions on reaction rates be investigated, but also the optimal pH observed for the transporter’s functional activity can be revealed. This research also has significant implications for transporters overall by providing a framework for such complex biomolecular processes.

Introduction

Inorganic phosphate (Pi) is essential for all forms of life, given its widespread involvement in numerous metabolic processes. Cells import Pi using phosphate transporters, leveraging the energy from proton or sodium electrochemical gradients across the membrane (1). Increasing attention has been extended to proton-coupled phosphate transporters, due to their critical roles in protozoan parasites (2,3), breast cancer (4), and human bone absorption (5). Transporters in the phosphate-proton symporter (PHS) family, a family within the major facilitator superfamily (MFS), utilize a positive proton gradient across the plasma membranes to drive phosphate import for plant cells and fungi. Examining a transporter from the PHS family can provide significant insight into the molecular mechanisms of how substrate transport is linked with proton transport (PT) during its functional cycle.

The Piriformospora indica phosphate transporter (PiPT), purified from a eukaryotic fungus, holds the unique distinction of being the only transporter in the PHS family with an available crystal structure (6). PiPT shares high homology with all transporters inside the PHS family (6,7), including the Saccharomyces cerevisiae yeast phosphate transporter Pho84. The resolved structure (6) captures PiPT in a phosphate-bound, “inward-facing occluded” (OC) conformational state and confirms that PiPT, sharing the quasi-twofold symmetry feature of MFS transporters, has two homologous domains (N and carboxyl domains) with six transmembrane helices each (Fig. 1). Combined evidence from mutagenesis studies (8,9,10) of the highly analogous Pho84 transporter and structural data (6) from the phosphate-binding site strongly suggest a pivotal role for the aspartate 324 (D324) residue in both the phosphate-release and proton-coupling processes.

Figure 1.

The representative occluded structure of PiPT with phosphate bound at the binding site. (A) The occluded structure. (B) The zoom-in box at the bottom left shows the phosphate-binding site with the proposed proton exit tunnel containing residues D45 and D149. (C) Illustration of the occluded and the inward open (IO) state. To see this figure in color, go online.

Our previous computational work, complemented by mutagenesis studies of a key residue, D45, inside the proton exit tunnel, suggested a probable proton exit path following its departure from D324—a trigger for phosphate release from the binding site into the cytosol (7). This was explored via hybrid quantum mechanics/molecular mechanics (QM/MM) molecular dynamics (MD) combined with an enhanced free energy sampling approach to study the PT event from D324 to D45. Drawing on insight from classical MD simulations and the free energy profile calculation of the PT event, our earlier work suggested that D45, together with D324, is the key to proton-coupled phosphate transport. D45 was not only proposed to receive the transported proton from D324 but also to trigger the initial D324 proton release with its deprotonation event in the occluded state.

Nonetheless, several questions and uncertainties persist. For instance, is our presumption concerning the initial proton-release step—where a proton can depart from D45 and reach the cytosolic bulk—entirely accurate? Additionally, does the phosphate remain bound at the binding site while the proton is being transported to D45? Given the multiple potential protonation forms of phosphate binding in the OC state, several reaction pathways could be established from the OC state to an “inward open” (IO) state. So which pathway is the dominant one, and how does it respond to varying pH and phosphate concentration conditions? Arguably, the most challenging question to address is: what makes pH 4.5 the optimal condition for the phosphate uptake activity of PiPT? Our aim in this work is therefore to answer all of these questions with a kinetic network model derived from “bottom-up” MD calculations combined with enhanced free energy sampling and multiscale reactive MD (MS-RMD) modeling (11,12).

Materials and methods

Molecular dynamics

Classical MD simulations were run starting with the PiPT crystal structure (Protein Data Bank ID PDB: 4j05) embedded in a pre-equilibrated and solvated 1,2-dimyristoyl-sn-glycero-3-phosphocholine lipid bilayer with approximately 150 mM NaCl to simulate the biological environment. Models were built and equilibrated using a standard protocol in CHARMM-GUI (13), and simulations were performed in GROMACS (14) version 2019.4 with the Velocity Verlet integrator in the isothermal-isobaric (constant particle number, pressure, temperature) ensemble using a semi-isotropic Parrinello-Rahman barostat (10) at 1 atm and a velocity rescale thermostat set to 310 K. The assigned protonation state of each system was based on the previously published observation for an equilibrated OC or IO state. The simulation protocol can be found in previous literature (7).

Multiscale reactive molecular dynamics

Starting from the equilibrated monobasic and dibasic phosphate-bound occluded structures in classical MD, the MS-RMD simulations were performed to simulate the explicit proton-release process through the proton exit tunnel. Both Asp residues, D45 and D149, were modeled as potential protonation sites. All interactions were described by the CHARMM36 force field with CMAP correction with the developed Asp MS-RMD model (11,12). Simulations were performed with the LAMMPS (15) MD package coupled with the RAPTOR code (16) to enable the PT reactions and processes. The RMD was integrated with time step set to 1 fs in the canonical (constant particle number, volume, temperature) ensemble using a Nose-Hoover chain thermostat at 310 K and with an orthogonal box of previously equilibrated dimension (74.239 Å, 74.239 Å, 112.739 Å) for monobasic phosphate OC structure and (75.485 Å, 75.485 Å, 109.501 Å) for dibasic phosphate OC structure. Electrostatic interactions were calculated using the particle-particle particle-mesh method (17) with the precision of 10⁻⁴ in force. The Lennard-Jones and Coulombic interactions were set to zero at a 10.0 Å radius and utilized a switching function starting at 8.0 Å to smooth the truncation.

Umbrella sampling

A total of over 40 umbrella windows were established for the one-dimensional umbrella sampling (18) of the PT process in the OC states with a spacing of 0.5 Å in the reaction coordinate value. Each umbrella window was run for at least 1 ns, and the total umbrella sampling simulation time exceeded 40 ns. To restrain the lateral diffusion of the hydrated proton when completely dissociated from the protein (19), a harmonic potential was introduced to the reaction coordinate $r_{\perp }$ defined as the radial distance of the transported proton to the D149 carboxylate atoms. The force constant of the wall was set to 5 kcal/mol/Ų, and the restraining potential was switched on once $r_{\perp }$≥10 Å in umbrella windows where $z_{\mathrm{H}}$ ≥ 6 Å. The reaction coordinate $z_{\mathrm{H}}$ to describe the proton cytosolic release is defined as $z_{\mathrm{H}}=z_{\mathrm{D}149}-z_{{\mathrm{H}}^{+}}$, where $z_{{\mathrm{H}}^{+}}$ and $z_{\mathrm{D}149}$ are the z-axial coordinates of the transported proton and the carboxylate group, respectively.

A total of over 60 umbrella windows was established for the one-dimensional umbrella sampling of the phosphate process in the IO states with a spacing of 0.5 Å in the reaction coordinate value of $z_{{\mathrm{P}}_{\mathrm{i}}}$. The reaction coordinate $z_{{\mathrm{P}}_{\mathrm{i}}}$, defined as $z_{{\mathrm{P}}_{\mathrm{i}}}=z_{{\text{PO}}_4}-z_{\text{Prot}}$, was used to characterize the phosphate cytosolic release process. Here, $z_{{\text{PO}}_4}$ is the z-axial coordinate value of the center of mass of the non-hydrogen atoms in the corresponding protonation form of the phosphate ion, and $z_{\text{Prot}}$ is the z-coordinate value of the geometric center of the transmembrane backbone of PiPT. Each umbrella window was run for at least 5 ns, and the total simulation time exceeded 300 ns. To restrain the lateral diffusion of the phosphate when completely dissociated from the protein, a harmonic potential was introduced to the reaction coordinate $r_{\perp }$, defined as the radial distance of the phosphate to the center of PiPT transmembrane backbone atoms. The force constant of the wall was set to 5 kcal/mol/Ų, and the restraining potential was switched on once $r_{\perp }$≥10 Å in umbrella windows where $z_{\mathrm{H}}$ ≥ 20 Å.

The weighted histogram analysis method (20) was employed to combine umbrella sampling trajectories and calculate the potential of mean force (PMF; also described herein as free energy profile) for both PT and phosphate-release process. PMF error bars were obtained by partitioning the trajectories of all umbrella windows into six equally sized blocks and calculating the standard deviation using the last five blocks.

Results and discussion

Analysis of the protein transport from D45 to cytosol in the first transition step

To calculate the free energy profile of the PT process from D45 to cytosol, MS-RMD simulations combined with an enhanced sampling approach was employed. The starting two OC states had both D45 and D324 protonated, with the only difference between them being the protonation form of the bound phosphate at the binding site. The reaction coordinate $z_{\mathrm{H}}$ to describe the proton cytosolic release is defined as $z_{\mathrm{H}}=z_{\mathrm{D}149}-z_{{\mathrm{H}}^{+}}$, where $z_{{\mathrm{H}}^{+}}$ and $z_{\mathrm{D}149}$ are the z-axial coordinates of the transported proton and the carboxylate group, respectively. Here, D45 and D149 are the only two acidic residues located inside the proton exit tunnel (Fig. 1). The excess proton residing on D45 corresponds to the free energy minimum at around -5Å and as the proton is being transported toward the cytosolic pore, the value of reaction coordinate ${\mathrm{z}}_{\mathrm{H}}$ increases (Fig. 2, A and B). The free energy profile (i.e., the PMF) reaches a plateau around 7Å where the transported proton reaches the bulk (Fig. 3, A and C). As noted in materials and methods, a cylindrical restraint was applied once the excess proton reached the less-restrictive bulk region to ensure a well-defined one-dimensional (1D) PMF (19). The rate constant of each process was estimated based on suitably modified transition state theory (21). The calculated rate constants for PT from D45 to the bulk with a mono-/dibasic phosphate bound at the binding site are $\left(6.1\pm 5.4\right)\times {10}^6{\mathrm{s}}^{-1}$ and $\left(7.4\pm 5.1\right)\times {10}^5{\mathrm{s}}^{-1}$, respectively. The pK_a of the residue D45 with mono- and dibasic phosphate bound in the OC states were calculated to be $5.2\pm 0.5$ and $5.7\pm 0.4$, respectively based on the expression (11,12,22)

$$\mathrm{p}{K}_{\mathrm{a}}=\log \left(c_0{S}_u{\int }_{\text{Reactant}}\mathrm{d}{z}_{\mathrm{H}}{e}^{-\beta \left(W\left(z_{\mathrm{H}}\right)-W\left(+\infty \right)\right)}\right)\text{,}$$ (1)

where $c_0$ is the standard state concentration (1 M) expressed in the number density of value $1/1660{\mathrm{\AA }}^3,W\left(z_{\mathrm{H}}\right)$ is the 1D PMF value, and $W\left(+\infty \right)$ is the value when $z_{\mathrm{H}}$ is sufficiently large such that the PMF profile reaches a plateau. The $S_u$ term corrects for the effect of the introduced cylindrical constraint on the PT in the horizontal plane to its motion and can be expressed as (19)

$$S_u={\int }_0^{\infty }\mathrm{d}{r}_{\perp }2\pi r_{\perp }{e}^{-\beta U_{\text{res}}\left(r_{\perp }\right)}\text{,}$$ (2)

where $r_{\perp }$ is the radial component of the distance between the transported proton and the protein.

Figure 2.

Representative molecular configurations for proton and phosphate release in PiPT (with monobasic phosphate as the reference). (A and B) The PT process from D45 to the cytosol with respect to the proton initially residing on D45 (a reaction coordinate of value ≈ −5 Å in Fig. 3A) and then with the proton released into the cytosolic bulk (reaction coordinate of value >7Å in Fig. 3A). The transporting proton is circled. (C and D) The phosphate-release process with respect to phosphate residing at the binding site (reaction coordinate of value ≈8 Å in Fig. 3B) and phosphate released into the cytosolic bulk (reaction coordinate of value >20 Å in Fig. 3B). The transported phosphate is circled. To see this figure in color, go online.

Figure 3.

PiPT proton and phosphate-release free energy profiles (PMFs). (A and C) PMFs of the PT event from D45 to the cytosol for the phosphate-bound OC states where D324 is protonated. (B and D) PMF of the phosphate-release event from binding site to cytosol at the two proposed IO states (D324 deprotonated for both cases; D45 is deprotonated for monobasic phosphate release, and D45 is protonated for dibasic phosphate release). The forward reaction rate constants were calculated based on transition state theory and the equilibrium constant. Error bars in the PMFs were obtained by partitioning trajectories into six equally sized blocks and calculating the standard deviations of the last five blocks.

The backward reaction rate constant can be obtained from the usual relationship between rate constants and the equilibrium constant (K_a) as follows:

$$K_{\mathrm{a}}=\frac{{\left[{\mathrm{H}}^{+}\right]}_{\mathrm{i}}\left[\mathrm{D}{45}^{-}\right]}{\left[\mathrm{D}45\mathrm{H}\right]}=\frac{k_{\mathrm{f}}}{k_{\mathrm{b}}}\text{,}$$ (3)

where $k_{\mathrm{f}}$ is the forward rate constant and $k_{\mathrm{b}}$ is the backward rate constant. The first-order backward reaction rate constant was estimated to be the product of the second-order backward rate constant and the cytosolic proton concentration. The free energy barrier for the excess proton to leave D45 in the OC state is about 9–10 kcal/mol, and the forward reaction timescale was estimated to be on the order of 100 ns.

Analysis of the phosphate release with classical MD

In our previous work (7), two IO states were found by assigning protonation states to the PiPT protein and equilibrating the system with classical MD, where phosphates then were released to the cytosol within tens of nanoseconds. We inferred based on the classical MD using the assigned protonation states in this previous study that for the cytosolic release of a dibasic phosphate, directly after the PT process from D324 to D45, the dibasic phosphate will leave the binding site and be released into the cytosol. In contrast to the dibasic phosphate, a monobasic phosphate—due to its smaller repulsive force with the negatively charged deprotonated D324 residue—cannot leave the binding site unless D45 transports the proton to the cytosol and becomes deprotonated again.

Here, we employed classical MD simulations combined with enhanced free energy sampling to analyze the three cases mentioned above (D45H [D324]⁻ H₂PO₄⁻, [D45]⁻ [D324]⁻ H₂PO₄⁻, and D45H [D324]⁻ [HPO₄]²⁻) with the 1D PMF calculations. The reaction coordinate $z_{{\mathrm{P}}_{\mathrm{i}}}$, defined as $z_{{\mathrm{P}}_{\mathrm{i}}}=z_{{\text{PO}}_4}-z_{\text{Prot}}$, was used to characterize the phosphate cytosolic release process. Here, $z_{{\text{PO}}_4}$ is the z-axial coordinate value of the center of mass of the non-hydrogen atoms in the corresponding protonation form of the phosphate ion, and $z_{\text{Prot}}$ is the z-coordinate value of the geometric center of the transmembrane backbone of PiPT.

The free energy basin in the 1D PMFs with $z_{{\mathrm{P}}_{\mathrm{i}}}$ ranging from 3 Å to 11 Å in all 1D PMFs corresponds to the phosphate binding at the binding site (Fig. 2 C), and the free energy basin with $z_{{\mathrm{P}}_{\mathrm{i}}}$ value at around 13 Å corresponds to the phosphate being in close contact with the positively charged residue H451 (Fig. 3, B and D; Fig. S1). The phosphate reaches the cytosolic bulk water once it has fully departed from the attractive residue H451 with $z_{{\mathrm{P}}_{\mathrm{i}}}$ value greater than 18 Å (Fig. 2 D), and the PMF profile reaches a plateau after 25 Å for the monobasic phosphate case and after 20 Å for a dibasic phosphate case. The forward rate constants were calculated based on transition state theory for the three phosphate cytosolic release cases, while the equilibrium constant of the phosphate-release process $\mathrm{p}{K}_{\mathrm{d}}$ was calculated using the procedure as described earlier for the PT process from D45 to the bulk. The timescale of phosphate release for these three cases can be estimated as the inverse value of the forward reaction rate constant calculated from the 1D PMFs. Both the dibasic phosphate-release timescale with protonated D45 and monobasic phosphate-release timescale with deprotonated D45 are less than 10 ns (Fig. 3, B and D), while the monobasic phosphate-release timescale with protonated D45 exceeds 10 μs (Fig. S1). These calculated results match our previously published observation (7) that only the first two cases allow for phosphate release.

Bottom-up kinetic network model

With the calculated PMFs available for each elementary step of the kinetic model (i.e., calculated bottom-up), the first-order reaction forward and backward rate constants were calculated based on transition state theory (21,23) and the detailed balanced condition. We explored the impact of three different factors—extracellular pH (pH_o), cytosolic phosphate concentration (Pi_i), and extracellular phosphate concentration (Pi_o)—on the network reaction flux from the OC to the IO state with the cytosolic pH (pH_i) fixed to a value of pH 7.0. The two different OC-state populations were determined by equilibrating the kinetic network model under the extracellular pH and Pi concentration. The three different IO-state populations were determined by equilibrating the kinetic network model under the cytosolic pH and Pi concentration. The populations of the rest of the states were then calculated by solving the transition rate matrix with the fixed reactant and product populations. See the supporting material for more details of these calculations.

Fig. 4 shows the complete kinetic network model for the entire functional cycle of PiPT with an assumption of fast equilibrium between an outward open phosphate-uncaptured state (not shown) and the phosphate-bound OC states. The states A1 and A2 correspond to the previously mentioned OC states, while D1 and D2 correspond to the phosphate-released IO states. The first stage of the PT process from D45 to the cytosol corresponds to elementary reactions from A1 to B1 and from A2 to B2. The second stage of the PT process from D324 to D45 via the titration of both forms of phosphate corresponds to the elementary reactions from B1 to C1, from B2 to C1′, and from B2 to C2. These reaction rates were calculated in our previous work by QM/MM (7) and are shown in Table S1. The third stage of the phosphate-release process involves elementary steps from C1′ to D1, from C1 to D2, and from C2 to D2 (see Fig. 4). The related PT from D45 to the cytosol after the D324 deprotonation event necessitates the monobasic phosphate release as indicated by reaction from C1 to C1′. This PT reaction is assumed to share the same rate constant as the PT reaction from A2 to B2, given that a monobasic phosphate with deprotonated D324 (C1) occupies the same net charge as the one with a dibasic phosphate and a protonated D324 (state A2).

Figure 4.

Bottom-up kinetic network model. The double-sided arrows represent the elementary reactions calculated. The gray solid circles highlight the mono- or dibasic phosphate bound at the binding site. The dominant reaction pathway is colored blue under physiological conditions, pH_i = 7.0, cytosolic Pi concentration 1e−4, 1e−5, and 1e−6 M, and cytosolic Pi concentration 1e−5 and 1e−6 M. To see this figure in color, go online.

D45N mutant being unable to release phosphate is confirmed

An additional elementary reaction that could be possible during the functional cycle is the PT from D324 to the bound dibasic phosphate at the OC state (A2 → C1) where D45 stays in its protonated, neutral charged form. We assume that the influence of the charge form of D45 on this PT reaction is rather small and that the rate constants can be well approximated by the reaction from B2 to C1′ where D45 is negatively charged. This possible elementary reaction provides a pathway for transitioning from the holo-OC state to the apo-IO state while D45 maintains its protonated form, eliminating the need for titration. This closely mimics the D45N-mutant scenario, suggesting a possible pathway in the D45N mutant for the phosphate-release process and raises a question about the experimental mutagenesis result (7) that indicates an inability of the D45N mutant to release phosphate into the cytosol. However, our detailed reaction flux analysis of the kinetic network shows that this specific D45N-associated reaction pathway always has a negligible contribution to the total reaction flux toward the phosphate-released IO state, consistent with the experimental result where $K_{\mathrm{m}}$ of the D45N mutant is $6.7\pm 5.3\mu \mathrm{M}$ and $V_{\max }$ is $25.3\pm 6.0\text{pmol}{\min }^{-1}{\mathrm{A}}_{600}^{-1}$. The binding affinity of phosphate to the D45N mutant is seen by the difference of the mutant $K_{\mathrm{m}}$ value compared to the wild-type one ($90.4\pm 10.7\mu \mathrm{M}$), while the absence of phosphate release for D45N is suggested by the same range of $V_{\max }$ between the D45N mutant and the vector-only control $(V_{\max }$ is $11.8\pm 4.9\text{pmol}{\min }^{-1}{\mathrm{A}}_{600}^{-1}$).

Proton and phosphate transport coupling in the reaction flux

With the assumption of cytosolic pH being a constant value of pH 7.0, we explored the impact of proton gradient and phosphate gradient across the membrane on the reaction flux of transforming from the OC to the IO state. The proton gradient was explicitly expressed in $\Delta \text{pH}$, defined as $\Delta \text{pH}={\text{pH}}_{\mathrm{i}}-\mathrm{p}{\mathrm{H}}_{\mathrm{o}}$. The extracellular pH condition ranged from 2.0 to 7.5 in increments of 0.5. The phosphate gradient was explored with cytosolic phosphate in abundant, adequate, and low-level conditions with Pi_i values of ${10}^{-4},{10}^{-5},{\text{and}10}^{-6}\mathrm{M}$ separately, and the extracellular phosphate was set to medium and low-level conditions with Pi_o values of ${10}^{-5}\text{and}{10}^{-6}\mathrm{M}$, respectively.

The reaction flux was calculated as the sum of the elementary net flux from all states directly transferred into each product state (see supporting material for further details). If the calculated overall reaction flux yielded a negative value, which implies the system would transition from the IO state to the OC state under a given circumstance, then such data points were not plotted. As shown in Fig. 5, only a positive pH gradient or a positive Pi gradient necessitates the phosphate uptake by PiPT, or else the phosphate will be transported in a reverse fashion from the cytosol to extracellular bulk. As depicted by the blue, orange, and red lines in the Fig. 5, phosphate can be transported against its electrochemical gradient when there is a positive pH gradient. This demonstrates the ability of PiPT to facilitate Pi accumulation under conditions of low Pi levels. As indicated by the purple line in the plot, in situations where there is a positive phosphate gradient from the extracellular bulk to the cytosol, protons can be transported even against the electrochemical gradient. Hence, the process of proton-coupled phosphate co-transport is demonstrated to be bidirectional; phosphate transport can be facilitated by a positive proton gradient and, conversely, PT can be driven by a positive phosphate gradient. The latter scenario, however, is seldom encountered, as the concentration of extracellular inorganic phosphate is typically less than or equal to the cytosolic concentration.

Figure 5.

Computed reaction flux toward the IO states under various pH and phosphate concentration conditions, with a consistent cytosolic pH of 7. Pi_i and Pi_o represent the cytosolic and extracellular phosphate concentrations, respectively, expressed in molarity (M). To see this figure in color, go online.

Dominant reaction pathway and rate-limiting step

Since multiple reaction pathways exist for the transformation from phosphate-bound OC states to phosphate-released IO states, we applied transition path analysis (24) and determined the dominant reaction pathway for each given condition as the one that has the largest kinetic bottleneck (further detailed in the supporting material). Note that the approach we used here to determine the dominant pathway does not depend on the sequence of edge search and is different from the method in previous work by Swanson and co-workers (25,26) where the dominant pathway was determined from a greedy search algorithm (27) assuming that the pathway that contributes the most to the total reaction flux of the reaction network is always equal to the one that has the largest elementary net flux value for the sequential edge search starting with the given product state D1 or D2.

The dominant reaction pathway for all physiological conditions in Fig. 5 is marked in blue in Fig. 4. The transition from the OC state to the IO phosphate-released state begins with the transport of a proton from D45 to the bulk (state A2 to B2), while a dibasic phosphate remains bound in the OC-state structure. Following this, the proton from D324 is directly transported to the bound dibasic phosphate (state B2 to C1′). Instead of continuously passing the proton from phosphate to D45 (state B2 to C2 then to D2), the phosphate, carrying the transported proton, departs from the binding site immediately (state C1′ to D1).

Since the population ratio between the monobasic and dibasic phosphate-bound OC states only changes with the extracellular pH, as a result the A2 state populates more than the A1 state (see Fig. 4 for these states) when $\Delta \text{pH}\le 1.0$ and the A1 state dominates over A2 when $\Delta \text{pH}>1.0$. This means that for all conditions seen in Fig. 5 with $\Delta \text{pH}>1.0$, phosphate should be dominantly in the monobasic form; however, the phosphate must transform into the dibasic form for the OC state to start the dominant reaction flux toward a phosphate-release state. This observation can be explained by the large reaction rate of transition from C1 to A2 and the proton-shared process between D324 and phosphate in the OC state.

The rate-limiting step of the reaction network was found by perturbing each edge (elementary reaction connecting two states) of the dominant reaction pathway and identifying the one that most impacts the reaction net flux toward state Ds. The rate-limiting step under all circumstances is always the proton-release step from D45 to cytosolic bulk water from the OC state in the functional cycle. This rate-limiting step reveals the strong coupling to the proton gradient driving the functional cycle under physiological conditions. This observation is also consistent with our previous mutagenesis result (7) that the D45N mutant, once losing its ability to transport a proton with D45, cannot release phosphate but keeps it bound at the binding site.

Optimal pH condition and its molecular origins

The molecular underpinnings of the optimal extracellular pH condition for phosphate uptake in PiPT, found to be 4.5 in its close homolog Pho84 (7,28), are important to consider. Our bottom-up kinetic network model has revealed that the optimal pH_o depends on the external phosphate concentration. With extracellular phosphate concentration spanning a range from $1\mu \mathrm{M}$ to $10\mu \mathrm{M}$, the optimal pH increases from pH 3.5 to pH 4.0. This is seen in Fig. 5 where the reaction flux profiles shown by the blue, green, and purple lines, with an external phosphate concentration of $10\mu \mathrm{M}$, all reach a maximum value at $\Delta \text{pH}=3.0$, while the other three with an external phosphate concentration of $1\mu \mathrm{M}$ reach their peaks at $\Delta \text{pH}=3.5$. Our calculation with an external phosphate concentration of $0.1\text{mM}$ renders an optimal pH of value 4.5 (Fig. S2). In that case, the pH 4.5 condition of the Pho84 experimental study was examined in detail, and a $0.1\text{mM}$ phosphate concentration was established in the sample preparation step.

Having successfully reproduced the optimal pH with the reaction flux calculated from the bottom-up kinetic network model, the next step was to take advantage of this bottom-up kinetic network model and use it to explore the molecular origin of the optimal pH condition. Using $\mathrm{p}{\mathrm{H}}_{\mathrm{i}}=7.0,{\text{Pi}}_{\mathrm{i}}={10}^{-5}\mathrm{M},{\text{and}\text{Pi}}_{\mathrm{o}}={10}^{-4}\mathrm{M}$ with the different $\mathrm{p}{\mathrm{H}}_{\mathrm{o}}$ values ranging from 3.5 to 7.5 at a 0.5 pH unit interval as an example, the net flux that directly produces the IO states goes up and then down as the extracellular pH increases its value. The concentration of states A2, B2, and C1′ along the dominant reaction pathway all peak concurrently at the same pH value (Table S2). The A2-state population was solved solely based on the extracellular pH and phosphate condition according to the calculation scheme described earlier. As $\mathrm{p}{\mathrm{H}}_{\mathrm{o}}$ increases, the equilibrium between the mono- and dibasic phosphate increases the concentration of dibasic phosphate (A2 state), and the equilibrium between the protonated and deprotonated D45 suppresses the concentration of a protonated D45 (A2 state). Thus, pH 4.5 becomes the “sweet spot” for the balance of these two opposing driving forces and becomes the (optimal) pH to reach the reaction flux maximum. The optimal pH corresponds to the optimal balance between the phosphate protonation state and D45 protonation state.

Conclusions

We have employed a kinetic network model enhanced by bottom-up MD simulations to unravel the mechanistic behavior of (in this case as a key example) the PiPT symporter. These calculations involve multiscale reactive MD combined with enhanced free energy sampling to characterize the elementary PT reactions. Furthermore, classical MD simulations were performed to explore various phosphate-release processes, again by utilizing enhanced free energy sampling to explore the timescale of the final phosphate-release process. Each elementary reaction step in the kinetic network model was described by the calculated reaction free energy profile (PMF), and the associated forward and backward reaction rate constants were determined from transition state theory. We note that we omitted the outward-facing (OF) state from our kinetic modeling, effectively assuming that the equilibrium concentrations of the OC state closely approximate its steady-state concentrations. This assumption is based on our previous observation whereby the OC-to-OF transition was found to be fast.

Utilizing this bottom-up kinetic network framework, we were able to then construct a comprehensive picture of the myriad potential reaction pathways facilitating the transition from the phosphate-bound OC states to the phosphate-released IO states. This holistic approach also allowed us to examine the impact of environmental factors on the overall reaction flux, such as extreme pH and phosphate concentration. We discovered that the PiPT transporter can adaptively transport the various forms of phosphate under differing conditions; however, it does so by utilizing the same dominant reaction pathway under all conditions studied. Interestingly, according to the dominant reaction pathway of the full kinetic framework, the deprotonation event at D324 is directly succeeded by a joint transport of proton and phosphate via the newly opened cytosolic gate rather than a separate PT through the previously identified proton exit tunnel. Our kinetic network model revealed the co-transport of proton and phosphate to be bidirectional, which can be driven by either a positive phosphate gradient or a positive proton gradient across the membrane. The kinetic model also reproduces the optimal pH conditions in the experimental system, which arises from control of the equilibrium between the D45 and phosphate protonation states.

Author contributions

This research was conceived, carried out, and written through the contributions of all authors.

Editor: Thomas DeCoursey.