Multistate Kinetic Model of the Sodium-Potassium ATPase

Abstract

The Na,K-ATPase is a complex membrane protein that exploits the hydrolysis of ATP as a source of chemical energy to actively transport K⁺ and Na⁺ ions against their electrochemical potential gradient across the cellular membrane. The function of this ATP-driven ion pump is broadly explained by the schematic Post-Albers alternating-access mechanism. Accordingly, the free energy gained from the phosphorylation/ dephosphorylation processes, where the $\gamma$-phosphate of ATP is transferred to a conserved Asp located on a cytoplasmic domain of the protein, is used by the enzyme to interconvert between two main conformational states. As a result of experimentally determined structures at atomic resolution, a detailed Post-Albers transport cycle of the Na,K-ATPase can comprise more than 20 conformational states of the system. This presents a great opportunity to formulate a detailed multi-state kinetic framework model of the transport cycle of the Na,K-ATPase, displaying the thermodynamic and biophysical constraints under which it must operate. Particular attention is given to the effect of coupling to the membrane potential via the incremental displacement charges for the microscopic steps of the transport cycle. On the basis of the multi-state kinetic framework, a simplified continuous model of the transport cycle based on the Smoluchowski equation is formulated and its consequences on the kinetic efficiency of the turnover rate are explored. These considerations lead to conjecture that the free energy of the microstates of the Na,K-ATPase is optimized for achieving a fast turnover rate when the membrane is depolarized.

Graphical Abstract

Introduction

P-type ATPases, named after an obligatory phosphorylated intermediate, are complex membrane proteins that exploit the hydrolysis of adenosine triphosphate (ATP) as a source of chemical energy to actively transport substrates against their electrochemical potential gradient across the cellular membrane. The best-kinetically characterized member of this large family is the Na,K-ATPase (or sodium-potassium pump).^1–3 For each ATP molecule, the enzyme functions according to the overall reaction,⁴

Scheme 1.

where the hydrolysis of one ATP molecule into a molecule of ADP and an inorganic phosphate Pi is accompanied by the translocation of two K⁺ ions into the cell (o→i) and three Na⁺ ions outside the cell (i→o) for a net charge movement of one elementary positive charge toward the extracellular side for each cycle. The function of this ATP-driven ion pump is broadly explained by the schematic Post-Albers alternating-access mechanism,^5,6 whereby the free energy gained from the phosphorylation/dephosphorylation process is used by the enzyme (E) to interconvert between two main conformational states: E1, open to the cytoplasm, and E2, cd /open to the opposite side of the membrane. These two conformations, referred to as E1 and E2, alternate between having a high affinity for sodium and a high affinity for potassium, respectively. During the transport cycle, the $\gamma$-phosphate of ATP is transferred to a conserved Asp located in the cytoplasmic P domain of the protein.

One may briefly summarize the transport cycle the Na,K-ATPase in terms of a few key steps.⁷ Starting in a state facing the cytoplasmic side (E1), the pump binds 3 Na⁺ ions and one ATP molecule with high affinity, leading to the occluded state (3Na⁺)-E1-ATP. Autophosphorylation of a conserved Asp in the P domain forms the state (3Na⁺)-E1P-ADP, which converts to the intermediate E2P-ADP (3Na⁺) followed by a large conformational change prompting the opening of the outer gate E2P-3Na⁺ that renders the ion binding sites accessible to the extracellular solution. The 3 Na⁺ ions are then released on the extracellular side and 2 external K⁺ enter the binding sites, promoting dephosphorylation to form the state E2-Pi-2K⁺, transiting via an occluded state E2(2K⁺)to the state E1–2 K⁺, causing the release of the 2 K⁺ ions on the cytoplasmic side (E1).

Initiated by the landmark work from the groups of Nissen and Toyoshima,^8,9 there is now an abundance of high-resolution X-ray structures displaying several of the intermediate states along the transport cycle, making it possible to begin to understand the mechanism of the Na,K-ATPase at an unprecedented level of atomic detail.¹⁰

As depicted schematically in Figure 1, a basic model accounting for all the structural and functional information about the Na,K-ATPase might easily comprise 20 to 25 structural micro-states. However, despite the wealth of data, our understanding of the mechanism of the transport cycle remains incomplete.^18–28

Figure 1:

A schematic model of the Na,K-ATPase comprising 20 distinct states was developed based on structural studies. Atomistic models for these 20 states were derived from experimental structural data under various conditions, as well as through homology modeling using the Squalus acanthias pump as the target sequence. The elusive intermediate structure (Na₃)-E1-P’·ADP was modeled starting from the occluded state and using steered molecular dynamics to rearrange the intracellular domain.^11,12 The following PDB structures can be used for modeling the microstates of the pump: 7ddf, 8k1l, 2zxe, 7y45, 7y46, 8d3v, and 8jbk.^10,13–17

For example, although the overall pump cycle obviously is driven by a free energy difference of 7.3 kcal/mol from ATP hydrolysis under physiological conditions,²⁹ this does not really inform us about the free energy balance of all the specific steps along the transport cycle. Microscopically all the steps are reversible; these P-type ion pumps can synthesize ATP when coupled to reverse ion transport,³⁰ but whether any particular microscopic step along the transport cycle ought to be down-hill or up-hill energetically is not immediately apparent. Furthermore, crucial for understanding the function of the pump is an understanding of all the electrogenic charge increments from the different microscopic steps.^26,31–40

Molecular dynamics (MD) simulations based on atomic models can help supplement some of the missing information about the mechanism of P-type ATPase, e.g., with regards to the selectivity of the ion binding sites,^41–44 the electrogenic charge increments,³⁸ protonation states of ionizable residues,^45,46 intermediates states,^12,47,48 transition pathways,⁴⁹ and regulation by other proteins.^50,51 Nonetheless, some form of mechanistic model of the transport cycle serving as a general framework is needed to make best use of the information provided by MD simulations. An appealing idea is to construct a detailed kinetic model from all the available the micro-states revealed by the high resolution structural studies. Kinetic models have long played a critical role to understand the mechanism membrane transport and help interpret experimental data.^23,52,53

Traditionally, people sought to construct kinetic models with as few states and parameters as possible in view of the limited information from experiment. Progressively, kinetic models have served as a framework to organize the results from MD simulations.^54–58 In this context, limiting the number of states is less critical and it is preferable to identify clear microscopic transitions.⁵⁹ Now, in an era dominated by atomic structures and MD simulations, kinetic models can be exploited to help integrate knowledge across multiple scales,^60,61 from both experiments and computations.

Our objective is to formulate a detailed multiscale kinetic framework model of the Na,K-ATPase with all known conditions based chemical and physical constraints that could serve as a “central hub” to combine experimental and computational results. Following directly the available structural and functional information, our current model comprises 20 microstates. The model, which can be used as a framework in MD future studies, enables a broad discussion of the general conditions affecting the transport efficacy of the pump cycle as a function of the known physical constraints on molecular machines.^62–69

Theoretical Developments

Multistate Kinetic model

Continuous-time Markov chain with discrete states provides a powerful framework to construct models of complex biomolecular systems. For the sake of simplicity, we consider a closed cycle comprising $N$ states with a simple one-to-one connectivity between adjacent states. A forward transition is $i\to i+1$ and a backward transition is $i\to i-1$. The population of the $i$-th state, represented as $P_i$, obeys a classic master equation,⁷⁰

$${\dot{P}}_i(t)=P_{i-1}(t)k_{i-1,i}+P_{i+1}(t)k_{i+1,i}-P_i(t)\left(k_{i,i+1}+k_{i,i-1}\right)$$ (1)

While it is written as a unimolecular first-order differential equation, one must keep in mind that the concentrations of all substrates (ions, ATP, ADP, etc) are incorporated implicitly in the transition rates. Eq. (1) might be written in matrix form, $\dot{\mathbf{P}}(t)=\mathbf{P}(t)\mathbf{K}$, where the non-diagonal terms of the rate matrix $\mathbf{K}$ are the forward $i\to j$ transition rates $k_{ij}$ of the model, while the diagonal term corresponds to the negative sum of all the outgoing transition rates from the state $i$, i.e., $K_{ii}=-\sum _{j\ne i}{k}_{ij}$. Because $\sum _j{K}_{ij}=0$ by construction, the total probability is conserved for all time, $\sum _{i=1}^N{\dot{P}}_i(t)=0$. Under equilibrium conditions, transition rates between all pairs of states are constrained through state-to-state microscopic detailed balance,

$$P_i^{\text{eq}}{k}_{i,i+1}^{\text{eq}}=P_{i+1}^{\text{eq}}{k}_{i+1,i}^{\text{eq}}$$ (2)

By virtue of the closed transport cycle, the last steps comprise transitions connecting state $N$ and state 1, i.e., $k_{N,N+1}^{\text{eq}}\equiv k_{N,1}^{\text{eq}}$, and $k_{N+1,N}^{\text{eq}}\equiv k_{1,N}^{\text{eq}}$. Constraining the rates of all adjacent states along the kinetic cycle of Eq. (1) implies that

$$\prod _{i=1}^N\frac{k_{i,i+1}^{\text{eq}}}{k_{i+1,i}^{\text{eq}}}=\frac{P_2^{\text{eq}}}{P_1^{\text{eq}}}\frac{P_3^{\text{eq}}}{P_2^{\text{eq}}}\cdots \frac{P_N^{\text{eq}}}{P_{N-1}^{\text{eq}}}\frac{P_{N+1}^{\text{eq}}}{P_N^{\text{eq}}}=1$$ (3)

where $P_{N+1}^{\text{eq}}\equiv P_1^{\text{eq}}$. So far, we only considered a simple cycle of $N$ states allowing transitions between adjacent states. More complicated kinetic schemes, including parallel branches added to the closed cycle, can also be treated within the same theoretical framework. In this case, satisfying state-to-state microscopic detailed balance Eq. (2) for all pairs of state $i$ and $j$ is sufficient to prohibit the existence of any loops of probability current that are inconsistent with thermodynamic equilibrium. This imposes a robust constraint ensuring that the underlying structure of a multi-state kinetic model is physically sound.

Association and dissociation of various substrates

While the kinetic transport cycle is constructed from apparent pseudo-unimolecular steps, the concentration dependence of the biomolecular association events is implicitly incorporated into the transition rate constants $k_{i,j}$. For example, the kinetic transition rate for a forward step corresponding to the association of a substrate “S” with the enzyme, E+S→E⋅S is a bimolecular process, while the reverse step E⋅S→E+S corresponds to a true unimolecular concentration-independent process. The time-derivative of the probability of the bound and unbound states is,

$${\dot{P}}_{\text{E}\cdot \text{S}}=k_{\text{ass}}^{\prime }\left[\text{S}\right]P_{\text{E}}-k_{\text{diss}}^{\prime }{P}_{\text{E}\cdot \text{S}}$$ (4)

$${\dot{P}}_{\text{E}}=-k_{\text{ass}}^{\prime }\left[\text{S}\right]P_{\text{E}}+k_{\text{diss}}^{\prime }{P}_{\text{E}\cdot \text{S}}$$ (5)

where $k_{\text{ass}}^{\prime }$ and $k_{\text{diss}}^{\prime }$ are the the bimolecular association transition rate and the unimolecular dissociation rate, respectively. Setting the time derivative to zero yields the equilibrium dissociation constant of the substrate,

$$K_{\text{d}}=\frac{k_{\text{diss}}^{\prime }}{k_{\text{ass}}^{\prime }}$$ (6)

To display the consequences of all substrates on the kinetic cycle, it is convenient to re-write the master equation with the pseudo-unimolecular transition rates $k_{ij}$ as $\left[\text{s}\right]k_{ij}^{\prime }$ to display all the concentration dependences explicitly. Regardless of the details of the kinetic scheme, obligatory steps corresponding to bimolecular associations and unimolecular dissociation must occur along the pump cycle with respect to ATP, ADP, and Pi, as well as the Na⁺ and K⁺ ions both inside (i) and outside (o) the cell. There is the association of one ATP, three ${\text{Na}}_{\text{i}}^{+}$ and two ${\text{K}}_{\text{o}}^{+}$ in the forward direction of the cycle, while there is association of one ADP and Pi, three ${\text{Na}}_{\text{o}}^{+}$ and two ${\text{K}}_{\text{i}}^{+}$ in the backward direction. This implies that at equilibrium, we have

$$\left(\frac{[\text{ATP}{]}_{\text{eq}}}{\left[\text{ADP}{]}_{\text{eq}}\right[\text{Pi}{]}_{\text{eq}}}\right){\left(\frac{{\left[{\text{Na}}_{\text{i}}^{+}\right]}_{\text{eq}}}{{\left[{\text{Na}}_{\text{o}}^{+}\right]}_{\text{eq}}}\right)}^3{\left(\frac{{\left[{\text{K}}_{\text{o}}^{+}\right]}_{\text{eq}}}{{\left[{\text{K}}_{\text{i}}^{+}\right]}_{\text{eq}}}\right)}^2\prod _{i=1}^N\frac{k_{i,i+1}^{\prime }}{k_{i+1,i}^{\prime }}=1$$ (7)

with the understanding that $k_{N,N+1}^{\prime }\equiv k_{N,1}^{\prime }$, and $k_{N+1,N}^{\prime }\equiv k_{1,N}^{\prime }$, resulting from the closed transport cycle. One may note that any other species involved in the pump cycle but is not transported across the membrane, such as the transient binding and dissociation of H⁺, does not appear in the overal equilibrium condition. Under thermodynamic equilibrium conditions, each of the 3 sub-reactions, is independently at equilibrium, thus,

$$\begin{gathered} \frac{[\text{ATP}{]}_{\text{eq}}}{\left[\text{ADP}{]}_{\text{eq}}\right[\text{Pi}{]}_{\text{eq}}}=K_{\text{ATP}}^{\text{eq}} \\ \frac{{\left[{\text{Na}}_{\text{i}}^{+}\right]}_{\text{eq}}}{{\left[{\text{Na}}_{\text{o}}^{+}\right]}_{\text{eq}}}=e^{-e{V}_{\text{mp}}/k_{\text{B}}T} \\ \frac{{\left[{\text{K}}_{\text{i}}^{+}\right]}_{\text{eq}}}{{\left[{\text{K}}_{\text{o}}^{+}\right]}_{\text{eq}}}=e^{-e{V}_{\text{mp}}/k_{\text{B}}T} \end{gathered}$$ (8)

where $K_{\text{ATP}}^{\text{eq}}=4.8\times {10}^{-6}{\text{M}}^{-1}$ is the equilibrium constant for ATP hydrolysis, $e$ is the elementary charge, and $V_{\text{mp}}$ is the membrane potential defined relative to the extracellular side, $V_{\text{mp}}=V_{\text{i}}-V_{\text{o}}$. Defining the standard free energy for ATP hydrolysis as $\Delta G_{\text{ATP}}^{(\circ )}=-k_{\text{B}}T\text{ln}{K}_{\text{ATP}}^{\text{eq}}=+7.3\text{kcal}/\text{mol}\left(6.94\times {10}^{-21}\right.\text{joule})$, one can write,

$$\begin{gathered} K_{\text{ATP}}^{\text{eq}}{e}^{(-3+2)e{V}_{\text{mp}}/k_{\text{B}}T}\prod _{i=1}^N\frac{k_{i,i+1}^{\prime }}{k_{i+1,i}^{\prime }}=1 \\ \prod _{i=1}^N\frac{k_{i,i+1}^{\prime }}{k_{i+1,i}^{\prime }}=\frac{1}{K_{\text{ATP}}^{\text{eq}}}\frac{1}{e^{(-3+2)e{V}_{\text{mp}}/k_{\text{B}}T}} \\ =\frac{1}{e^{-\Delta G_{\text{ATP}}^{(\circ )}/k_{\text{B}}T}}\frac{1}{e^{-e{V}_{\text{mp}}/k_{\text{B}}T}} \\ =e^{-\left[-\Delta G_{\text{ATP}}^{(\circ )}-e{V}_{\text{mp}}\right]/k_{\text{B}}T} \\ =e^{-\Delta G_{\text{tot}}^{(\circ )}/k_{\text{B}}T} \end{gathered}$$ (9)

where $\Delta G_{\text{tot}}^{(\circ )}=-\Delta G_{\text{ATP}}^{(\circ )}-e{V}_{\text{mp}}$ is the total electrochemical free energy change per pump cycle, corresponding to the hydrolysis of one ATP (−7.3kcal/mol) and the movement of one net elementary charge from inside to outside the cell experiencing a change in potential of $V_{\text{o}}-V_{\text{i}}=-V_{\text{mp}}$. If we think of $\Delta G_{\text{tot}}^{(\text{o})}$ as the total free energy of the final state minus the total free energy of the initial state, then we have $G_{\left(\text{no}\text{ATP}\text{molecule}\right)}-G_{\left(\text{one}\text{ATP}\text{molecule}\right)}=0-\Delta G_{\text{ATP}}^{(\circ )}$, and $G_{(3\text{ions}\text{out}+2\text{ions}\text{in})}-G_{(3\text{ions}\text{in}+2\text{ions}\text{out})}=\left(3e{V}_{\text{o}}+2e{V}_{\text{i}}\right)-\left(3e{V}_{\text{i}}+2e{V}_{\text{o}}\right)=e\left(V_{\text{o}}-V_{\text{i}}\right)=-e{V}_{\text{mp}}$.

Membrane potential and incremental displacement charges

One transport cycle results in the movement of one elementary charge from inside to outside the cell. This is coupled to the membrane potential by the energy contribution $-e{V}_{\text{mp}}$. More broadly, all the transition rate $k_{i,j}^{\prime }$ along the pump cycle may be coupled to the membrane potential $V_{\text{mp}}$ via electrogenic charge increments.^23,31 Electrostatically, these correspond to a displacement charge in the field arising from the membrane potential.^71,72 In the literature on voltage-activated channels, these are commonly called the “gating charge”.^73–75 In the context of the pump, we shall refer to those as “incremental displacement charges”.³⁸

In principle, any conformational change of a membrane protein that results in a displacement of either charged residues or the solvent-protein dielectric interface is coupled to the transmembrane potential.^71,72 In that sense, the incremental displacement charge the states do not only reflect the movements of the transported Na⁺ and K⁺ ions, but the displacement of any atom (charged or uncharged) that is part of the system (protein, solvent, ions, and lipids). An important factor is how the various steps associated with molecular motion and/or movement of ions could be coupled to the transmembrane potential $V_{\text{mp}}$. Let us represent the transition from state $i$ to state $i+1$ from the point of view of the dynamics along the reaction coordinate $x$. The total potential of mean force (PMF) along $x$ in the presence of a membrane potential is,

$$W\left(x;V_{\text{mp}}\right)=W_0(x)-𝒬(x)V_{\text{mp}}$$ (10)

where $W_0(x)$ is the PMF in the absence of any membrane potential $V_{\text{mp}}$, and $𝒬(x)$ is the fractional displacement charge associated with the reaction coordinate $x$ associated with the transition pathway between state $i$ and state $i+1$ along the transport cycle.⁴⁹ The function $𝒬(x)$ is an equilibrium quantity that can be calculated using all-atom MD simulations.⁷² Assuming that the membrane extends in the $xy$ plane, the displacement charge of the membrane protein system fixed at $x$ along the reaction coordinate is given by the average,⁷²

$$𝒬(x)={⟨\sum _i{q}_i\left(\frac{z_i}{L_z}\right)⟩}_{(x)}$$ (11)

where $q_i$ and $z_i$ are the charge and position of the $i$-th atom, and $L_z$ is the length of the periodic simulation box in the $z$ direction. The subscript $x$ on the bracket denotes an average constrained along the reaction coordinates between two adjacent conformational or functional states included in the transport cycle (Figure 1). It is important to note that the sum over $i$ runs over all the atoms in the simulation system, including the protein, lipids, solvent, and ions.⁷² The charge-voltage coupling in Eq. (10) is written with the sign convention that a movement of a positive charge from the intracellular side to the extracellular side yields a decrease in the PMF. Formally, the transition-dependent effective charge $𝒬(x)$ represents the coupling of the system to the transmembrane potential along the reaction coordinate. The coupling of charge movements to $V_{\text{mp}}$ alters the relative stability of stable states, and the free energy barrier separating them. As illustrated schematically in Figure 2, the activation barries for the forward $i\to i+1$ and backward $i+1\to i$ transitions, as well as the relative free energy of the two states $i$ and $i+1$ are affected by the membrane potential $V_{\text{mp}}$ via the coupling $𝒬(x)$. The non-linear shape of $𝒬(x)$ in Figure 2 is meant to reflect the fact that the displacement charge does not necessarily vary monotonically along the reaction coordinate between state $i$ and state $i+1$.^38,75–78 In a simple transition rate picture we assume that the rate of the forward $i\to i+1$ transition display the simple dependence $k_{i,i+1}\propto \text{exp}\left[-\Delta W_{ij}^{†}/k_{\text{B}}T\right]$, where $\Delta W_{i,i+1}^{†}$ is the voltage-dependent free energy barrier,

$$\Delta W_{i,i+1}^{†}\left(V_{\text{mp}}\right)=\left(W_0\left(x_{i,i+1}^{†}\right)-W_0\left(x_i\right)\right)-\left(𝒬\left(x_{i,i+1}^{†}\right)-𝒬\left(x_i\right)\right)V_{\text{mp}}$$ (12)

Similar relations are assumed for the backward transition $i+1\to i$. The voltage-dependent forward $i\to i+1$ transition rate is assumed to have the simple exponential form with an activation free energy, $k_{i,i+1}\propto \text{exp}\left[-\Delta W_{i,i+1}^{†}/k_{\text{B}}T\right]$. By virtue of the exponential form, the features related to the transition state drops out and the quantity $-k_{\text{B}}T\text{ln}\left(k_{i,i+1}/k_{i+1,i}\right)$, state-related quantities like free energy and displacement charge of the states $i$ and $i+1$ can be determined.

Figure 2:

Transition rates and coupling; to a membrane potential. (Top) Schematic representation of the PMF governing the $i\to j$ and $j\to i$ transitions. The PMF in the absence (solid lene) and in the presence (dashed line) of the membrane potential is shown, with $W(x;V_{\text{mp}})=W_0(x)—𝒬(x)V_{\text{mp}}$. (Bottom) The quantity $𝒬(x)$ is the effective charge which provides the coupling of the system to the membrane potential $V_{\text{mp}}$ along the reaction coordinate x along the transition pathway between adjacent states of the transport cycle, as defined by Eq. (11). Increasing $𝒬(x)$ is associated with the forward movement of a positive charge in the x direction toward the extracellular side.

It is helpful to develop an intuitive understanding of the impact of the incremental charge on the transition rates. For instance, if a transition $i$ to $i+1$ involves the movement of a positive charge outward along the $x$ coordinate and the membrane potential $V_{\text{mp}}$ is positive, one then expects that the forward transition rate $i\to i+1$ is accelerated when increasing $V_{\text{mp}}$. This intuitive argument implies that the forward movement of a positive charge in the $x$ direction toward the extracellular side yields a positive charge increment, i.e., $𝒬\left(x_{i,i+1}^{†}\right)>𝒬\left(x_i\right)>0$. The situation is illustrated schematically in Figure 2. The forward transition rate is accelerated by the membrane potential because the free energy barrier for the forward transition (top panel) decreases as the displacement charge $𝒬(x)$ increases (bottom panel). This sign arises from the fact that the transition moves the charge away from the intracellular region, which is held at the membrane potential $V_{\text{mp}}$, toward the extracellular solution at 0 mV.

Transposing these considerations to the kinetic model of the transport cycle, we express the forward voltage-dependent transition rate as,

$$\begin{gathered} k_{i,i+1}^{\prime }\left(V_{\text{mp}}\right)\propto e^{-\Delta W_{i,i+1}^{†}\left(V_{\text{mp}}\right)/k_{\text{B}}T} \\ =k_{i,i+1}^{\prime }(0)e^{\Delta {𝒬}_{i,i+1}^{†}{V}_{\text{mp}}/k_{\text{B}}T} \end{gathered}$$ (13)

and the backward voltage-dependent transition rate as,

$$\begin{gathered} k_{i+1,i}^{\prime }\left(V_{\text{mp}}\right)\propto e^{-\Delta W_{i+1,i}^{†}\left(V_{\text{mp}}\right)/k_{\text{B}}T} \\ =k_{i+1,i}^{\prime }(0)e^{\Delta {𝒬}_{i+1,i}^{†}{V}_{\text{mp}}/k_{\text{B}}T} \end{gathered}$$ (14)

where $k_{i,i+1}^{\prime }(0)$ and $k_{i+1,i}^{\prime }(0)$ are the transition rates in the absence of transmembrane potential, and $\Delta {𝒬}_{i,i+1}^{†}=𝒬\left(x_{i,i+1}^{†}\right)-𝒬\left(x_i\right)$ and $\Delta {𝒬}_{i+1,i}^{†}=𝒬\left(x_{i+1,i}^{†}\right)-𝒬\left(x_{i+1}\right)$ are the effective incremental displacement charge movement associated with the forward and backward transitions, respectively. Importantly, the ratio of forward an backward transition rates obeys,

$$\begin{gathered} \frac{k_{i,i+1}^{\prime }\left(V_{\text{mp}}\right)}{k_{i+1,i}^{\prime }\left(V_{\text{mp}}\right)}=\frac{k_{i,i+1}^{\prime }(0)}{k_{i+1,i}^{\prime }(0)}{e}^{\left[\Delta {𝒬}_{i,i+1}^{†}-\Delta {𝒬}_{i+1,i}^{†}\right]V_{\text{mp}}/k_{\text{B}}T} \\ =\frac{k_{i,i+1}^{\prime }(0)}{k_{i+1,i}^{\prime }(0)}{e}^{\left[𝒬\left(x_{i+1}\right)-𝒬\left(x_i\right)\right]V_{\text{mp}}/k_{\text{B}}T} \end{gathered}$$ (15)

depends on the difference of the displacement charge between the stable states $i$ and $i+1,\Delta {𝒬}_{i+1,i}={𝒬}_{i+1}-{𝒬}_i$. The value of the displacement charge, $𝒬\left(x_{i,i+1}^{†}\right)$, at the transition state drops out in the difference between stable states. From this observation, it is useful to define the cumulative displacement charge profile for all the states along the transport cycle,

$$\begin{gathered} {𝒬}_1=0 \\ {𝒬}_2=\left(\Delta {𝒬}_{\mathrm{1,2}}^{†}-\Delta {𝒬}_{\mathrm{2,1}}^{†}\right)=\Delta {𝒬}_{\mathrm{2,1}} \\ \dots \\ {𝒬}_j=\sum _{i=1}^{j-1}\left(\Delta {𝒬}_{i,i+1}^{†}-\Delta {𝒬}_{i+1,i}^{†}\right)=\sum _{i=1}^{j=1}\Delta {𝒬}_{i+1,i} \\ \dots \\ {𝒬}_{N+1}=\sum _{i=1}^N\left(\Delta {𝒬}_{i,i+1}^{†}-\Delta {𝒬}_{i+1,i}^{†}\right)=\sum _{i=1}^N\Delta {𝒬}_{i+1,i}=e \end{gathered}$$ (16)

For most transitions, such as the movement of a charged moiety along a narrow pore, the membrane potential is expected to have opposite effects on the forward and backward rates. In other words, the voltage that accelerates the forward rate will generally slow down the backward, and vice versa. Furthermore, the incremental charges $\Delta Q_{i,i+1}^{†}$ and $\Delta Q_{i+1,i}^{†}$ for the forward and backward transition rates are two independent parameters constrained only by the total charge difference over the transport cycle. If one imagines, as illustrated in Figure 2, that the displacement charge $𝒬(x)$ is a continuous function varying along a reaction coordinates $x$, then a reasonable expectation is that the constraint $𝒬\left(x_i\right)<𝒬\left(x_{ij}^{†}\right)<𝒬\left(x_j\right)$ should be respected. This constraint is often expressed in term an electrogenic splitting factor ${\lambda }_i$ such that $\Delta Q_{i,i+1}^{†}={\lambda }_i\left({𝒬}_{i+1}-{𝒬}_i\right)$, and $\Delta Q_{i+1,i}^{†}=\left(1-{\lambda }_i\right)\left({𝒬}_{i+1}-{𝒬}_i\right)$. This is similar to the concept of $\Phi$-values based on Brønsted analysis,^59,79 whereby the relation of the forward and backward $\Delta Q_{i,i+1}^{†}$ and $\Delta Q_{i,i+1}^{†}$ to the cumulative charge displacement $𝒬\left(x_i\right)$ and $𝒬\left(x_{i+1}\right)$ reflects the position of the transition state relative to the stable states.

Importantly, because the displacement charge ${𝒬}_i$ are well-defined state-dependent equilibrium quantities, their calculation does not require the construction of transition pathway between neighboring metastable states along the transport cycle.⁴⁹ In that sense, all the ${𝒬}_i$ can be calculated using Eq. (11) as a straightforward averages from simulations in which the system is restrained to the given state $i$. In contrast, the incremental values $\Delta Q_{i,i+1}^{†}$ and $\Delta Q_{i+1,i}^{†}$ for the forward and backward transitions are much more difficult to obtain because the average of Eq. (11) must be evaluated at the transition state $x_{i,i+1}^{†}$ between the conformations $x_i$ and $x_{i+1}$. A physical transition pathway between known stable conformations can be determined, for example using using the string method, although this represents a considerable undertaking.⁴⁹ Therefore, an appealing intermediate strategy may be to determine the displacement charges ${𝒬}_i$ from explicit simulations of the states, and assuming that the incremental values for the forward and backward transitions can be constructed using some reasonable empirical approximation about the splitting factor ${\lambda }_i$. However, one must consider the physical implications of these splitting factors carefully. For example, it may be tempting as a simplification to postulate a symmetric distribution of the incremental charge for the forward and backward transitions, with ${\lambda }_i=1/2$. While this may be a reasonable assumption in some cases, it is important to keep in mind that strong asymmetries are also possible. For example, the equilibrium dissociation constant of an ion evolving on the intracellular side may depend strongly on the membrane potential,

$$\begin{gathered} K_{\text{d}}\left(V_{\text{mp}}\right)=\frac{k_{\text{diss}}}{k_{\text{ass}}} \\ =K_{\text{d}}(0)e^{-\Delta {𝒬}_{\text{b}}{V}_{\text{mp}}/k_{\text{B}}T} \end{gathered}$$ (17)

where $\Delta {𝒬}_{\text{b}}={𝒬}_{\text{bound}}-{𝒬}_{\text{unboud}}$ is the effective charge gained by the binding process, e.g., the binding becomes stronger ($K_{\text{d}}$ decreases) if $\Delta {𝒬}_{\text{b}}$ and $V_{\text{mp}}$ are both positive. However, while the dissociate rate $k_{\text{diss}}$ could be strongly sensitive to the membrane potential, the diffusion-limited bimolecular association rate $k_{\text{ass}}$ could plausibly be fairly insensitive to the membrane potential.³⁸

Physical considerations and simplifying assumptions regarding the incremental charges for the transitions can help reducing the number of free parameters in multi-state kinetic models in the absence of data from experiments. The importance of this aspect to modeling the transport cycle is highlighted by the fact that different splittings of the electrogenic charge increments of an elementary transition could affect how the turnover rate responds to changes in the transmembrane potential.^62,69,80,81 Ultimately, calculations based on all-atom MD simulations^75–77 may be the only approach to gain detailed information about incremental displacement charges in the microscopic transitions underlying the pump cycle.³⁸

Free energy of the states along the transport cycle

The functional performance of the pump depends on the concentrations of the various substrates as well as on the membrane potential. Considering the transport cycle under general conditions from the pseudo-unimolecular first-order transition rates,

$$\begin{gathered} \prod _{i=1}^N\frac{k_{i,i+1}}{k_{i+1,i}}=\left(\frac{\left[\text{ATP}\right]}{\left[\text{ADP}\right]\left[\text{Pi}\right]}\right){\left(\frac{\left[{\text{Na}}_{\text{i}}^{+}\right]}{\left[{\text{Na}}_{\text{o}}^{+}\right]}\right)}^3{\left(\frac{\left[{\text{K}}_{\text{o}}^{+}\right]}{\left[{\text{K}}_{\text{i}}^{+}\right]}\right)}^2\prod _{i=1}^N\frac{k_{i,i+1}^{\prime }\left(V_{\text{mp}}\right)}{k_{i+1,i}^{\prime }\left(V_{\text{mp}}\right)} \\ =e^{-\left(-k_{\text{B}}T\text{ln}\left(\frac{\left[\text{ATP}\right]}{\left[\text{ADP}\right]\left[\text{Pi}\right]}\right)-3k_{\text{B}}T\text{ln}\left(\frac{\left[{\text{Na}}_{\text{i}}^{+}\right]}{\left[{\text{Na}}_{\text{o}}^{+}\right]}\right)-2k_{\text{B}}T\text{ln}\left(\frac{\left[{\text{K}}_{\text{o}}^{+}\right]}{\left[{\text{K}}_{\text{i}}^{+}\right]}\right)\right)/k_{\text{B}}T}{e}^{-\Delta G_{\text{tot}}^{(\text{o})}/k_{\text{B}}T} \\ =e^{-\Delta G_{\text{tot}}/k_{\text{B}}T} \end{gathered}$$ (18)

Taking the log and substituting $\Delta G_{\text{tot}}^{(\circ )}=-\Delta G_{\text{ATP}}^{(\circ )}-e{V}_{\text{mp}}$ yields the total free energy change over one pump cycle,

$$\begin{gathered} \Delta G_{\text{tot}}=-\Delta G_{\text{ATP}}^{(\circ )}-k_{\text{B}}T\text{ln}\left(\frac{\left[\text{ATP}\right]}{\left[\text{ADP}\right]\left[\text{Pi}\right]}\right) \\ -3k_{\text{B}}T\text{ln}\left(\frac{\left[{\text{Na}}_{\text{i}}^{+}\right]}{\left[{\text{Na}}_{\text{o}}^{+}\right]}\right)-2k_{\text{B}}T\text{ln}\left(\frac{\left[{\text{K}}_{\text{o}}^{+}\right]}{\left[{\text{K}}_{\text{i}}^{+}\right]}\right)-e{V}_{\text{mp}} \end{gathered}$$ (19)

If $\Delta G_{\text{tot}}=0$, then the total free energy change over the pump cycle and the system is at equilibrium and there is no transport.

It is customary to represent the activity of the Na-K ATPase as a closed loop with a series of discrete states (Figure 1).²³ Such a picture is intuitively appealing because the protein is expected to repeatedly return to the same state after each transport cycle. However, while this representation as a cycle captures the sequence of protein states, it is partly misleading and incomplete because it does not explicitly display how the total free energy of the system go down by $\Delta G_{\text{tot}}$ after each transport cycle. One may note that $\Delta G_{\text{tot}}$ is not a true state function one assumes a bath held at fixed conditions. Strictly speaking, the state of the entire system cannot be mapped onto a periodic loop. While $\Delta G_{\text{tot}}$ is sometimes called “free energy dissipation” in the literature, this appellation can be confusing it is not dissipated into heat. For this reason, Wagoner and Dill refer to $\Delta G_{\text{tot}}$ as the change in “basic free energy”, a term previously used by Terrell Hill⁸² to indicate that it is similar to an equilibrium free energy but is corrected for nonequilibrium effects (the term was labeled $\Delta {\mu }_{\text{net}}$ or $\Delta {\mu }_{\text{diss}}$ in their papers.^63,64

An alternative representation is produced by “unwrapping” the transport cycle. In this representation, which bears some similarities to the unwrapped coordinates from MD simulations with periodic boundary conditions, the system is pictured as an infinite line of states, and the total free energy goes downhill by $\Delta G_{\text{tot}}$ for each transport cycle. In this regard, it is useful to define the free energy profile for all the states along the transport cycle,

$$G_i=G_i^{(0)}-{𝒬}_i{V}_{\text{mp}}$$ (20)

where ${𝒬}_i$ is the displacement charge of the $i$-th step from Eq. (17), and $G_i^{(0)}$ is constructed from the free energy increment of the $i$-th step in the absence of membrane potential, $G_{i+1}^{(0)}-G_i^{(0)}=-k_{\text{B}}T\text{ln}\left(k_{i,i+1}^{\prime }(0)/k_{i+1,i}^{\prime }(0)\right)$. The microscopic state of the pump and the transition rates for states $i=1,N$ are repeated periodically as an infinite sequence of states. However, the total free energy by itself is not a periodic series, $\left\{\dots ,G_{i-2},G_{i-1},G_i,G_{i+1},G_{i+2},\dots \right\}$ with $-\infty <i<\infty$, reflecting the global state of the entire system. Likewise, the total displacement charge of the system is also not a periodic series, $\left\{\dots ,{𝒬}_{i-2},{𝒬}_{i-1},{𝒬}_i,{𝒬}_{i+1},{𝒬}_{i+2},\dots \right\}$, reflecting the global electrostatic state of the entire system. By construction, we have that $G_{i+N}=G_i+\Delta G_{\text{tot}}$, and ${𝒬}_{i+N}={𝒬}_i+e$. The kinetic evolution of the ion pump can be pictured as the hopping transitions of the system along this infinite line of states.

It is of interest to note the similarities of the present framework and the treatment of motor proteins that move stochastically along a linear molecular track.^83–85 In particular, the incremental displacement charge $\Delta Q_{i,j}^{†}$ that biases the forward/backward rates in the transport cycle of ion pump are analogous to the load-distribution factors the molecular motor literature.^85,85 While the pump is performing the electrostatic work $V_{\text{mp}}\Delta {𝒬}_{i+1,i}$ in the $i\to i+1$ step, a molecular motor moving along the $x$ axis against an external force $F$ is performing incremental amount of work $F\Delta x_{i+1,i}$.^83,84,86 The total mechanical work per cycle performed by a molecular motor, $W=F\left(\sum _{i=1}^N\Delta x_{i+1,i}\right)$, is equivalent to the work generated from the electrogenic steps, $e{V}_{\text{mp}}$. The terminology often used for models of motors proteins, which relates to the magnitude of the load-distribution and splitting factors, are Brownian ratchet or power stroke mechanisms. However, some differences are worth pointing out. In the case of processing motors, it is often assumed for simplicity that the spatial increments at each step are identical.^83,84,86 In contrast, it is clear that the displacement charges ${𝒬}_i$ along the transport cycle of the pump does not progress monotonically from 0 to one elementary charge $e$ because the inward transport of 2 K⁺ ions and outward transport of 3Na⁺ described in Scheme 1 takes place sequentially (see Figure 3).^4,7,27

Figure 3:

Results for the 6 state model of the Na,K-ATPase after MMC optimization. (Top) Total free energy profile (blue line) for the stable states $(G_i)$ and the free energy barriers between them $\left(G_i+\Delta G_{i,i+1}^{†}\right)$ at −60 mV. Also shown (red line) is the associated cumulative displacement charge for the stable states $\left({𝒬}_i\right)$ and the transition region between them $\left({𝒬}_i+\Delta {𝒬}_{i,i+1}^{†}\right)$. Curves from 400 models sampled by MMC are shown. The free energy barriers are associated with the forward rates as, $\Delta G_{i,i+1}^{†}=-k_{\text{B}}T\text{ln}\left(k_{i,i+1}/k_0\right)$, assuming a time scale $k_0=1/\text{ps}$. In the model, the cumulative displacement charges were fixed to $\left\{{𝒬}_1=0,{𝒬}_2=-1,{𝒬}_3=-2,{𝒬}_4=-1,{𝒬}_4=0,{𝒬}_6=1\right\}$, corresponding to the inward transport of 2 K⁺ (states 1–3) and outward transport of 3 Na⁺ (states 3–6). By periodicity, state $1+N=7$ corresponds to state 1 after one transport cycle. The forward activation displacement charges were expressed as $\Delta {𝒬}_{i,i+i}^{†}={\lambda }_i\left({𝒬}_{i+1}-{𝒬}_i\right)$, where ${\lambda }_i$ is the electrogenic splitting factor. The model’s parameter were refined using MMC under the constraint that the turnover rate is 4/s at −60mV,52/sec at 0mV,55/s at 30 mV, and 54/s at 50 mV to mimic the electrophysiological data from Nakao and Gadsby.^21,22 The optimization methodology is similar to that of Anandakrishnan and Zuckerman.⁶⁸ A total of 18 parameters were optimized: (1) the non-electrogenic contribution to the free energy of the states, (2) the forward free energy barriers, (3) the electrogenic splitting factor ${\lambda }_i$. Physiological conditions were assumed for the ions ATP, yielding a $\Delta \mu =12.0\text{kcal}/\text{mol}$. The total free energy drop over one transport cycle is $\Delta G_{\text{tot}}=-2.1\text{kcal}/\text{mol}$. (Bottom) Turnover rate calculated from the models using Eq. (24).

Free energy balance under physiological conditions

Following the notation of Wagoner and Dill, the total free energy change over one pump cycle can be written as, $\Delta G_{\text{tot}}=-\Delta \mu +W$, where

$$\Delta \mu =\Delta G_{\text{ATP}}^{(\circ )}+k_{\text{B}}T\text{ln}\left(\frac{\left[\text{ATP}\right]}{\left[\text{ADP}\right]\left[\text{Pi}\right]}\right)$$ (21)

is the free energy that is taken from the cellular ATP and put into the system, and

$$W=3k_{\text{B}}T\text{ln}\left(\frac{\left[{\text{Na}}_{\text{o}}^{+}\right]}{\left[{\text{Na}}_{\text{i}}^{+}\right]}\right)+2k_{\text{B}}T\text{ln}\left(\frac{\left[{\text{K}}_{\text{i}}^{+}\right]}{\left[{\text{K}}_{\text{o}}^{+}\right]}\right)-e{V}_{\text{mp}}$$ (22)

is the work performed by a single pump on ions moving across the membrane under the potential $V_{\text{mp}}$. Under physiological conditions for a resting cell, with $\left[{\text{Na}}_{\text{o}}^{+}\right]=143\text{mM},\left[{\text{Na}}_{\text{i}}^{+}\right]=14\text{mM},\left[{\text{K}}_{\text{i}}^{+}\right]=157\text{mM},\left[{\text{K}}_{\text{o}}^{+}\right]=4\text{mM}$, and $V_{\text{mp}}=-60\text{mV}$, one has, $\Delta \mu =12\text{kcal}/\text{mol}$ and $W=9.9\text{kcal}/\text{mol}$. This yields a free energy driving force of $\Delta G_{\text{tot}}=-2.1\text{kcal}/\text{mol}$. Wagoner and Dill defined the thermodynamic efficiency as $\eta =W/\Delta \mu$, which is 0.82 for the conditions considered here. Analyzing a large body of functional data, Wagoner and Dill showed that Na,K-ATPases operate at high thermodynamic efficiency compared to other molecular machines.⁶³ With these physiological concentrations, $W$ becomes negative when the membrane potential is greater than 370 mV. Above this large (physiologically irrelevant) value, the transport cycle carries on but the system is effectively running downhill in free energy while consuming ATP, and the pump is no longer performing any positive work as defined by Eq. (22). As defined, the thermodynamic efficiency $\eta$ is necessarily smaller than 1 because $\Delta G_{\text{tot}}$ must be negative to yield a steady-state forward turnover rate. Therefore, $W$ cannot be larger than $\Delta \mu$. For example, when the pump is working under physiological conditions, transporting K⁺ ions toward the high intracellular concentration and Na⁺ ions toward the high extracellular concentration against a negative membrane potential, then $\Delta G_{\text{tot}}$ is a small negative number, and most of the $\Delta \mu$ is spent to yield a small turnover.

However, this analysis focused on thermodynamic efficiency leaves out consideration about the magnitude of the steady-state turnover rate that the pump is able to deliver for a given value of $\Delta G_{\text{tot}}$. It seems intuitively appealing that the pump is optimally adapted through evolution to maximize its turnover rate under the predominant physiological conditions. A fast rate is not necessarily advantageous for all biological systems. For example, cellular signaling, vision, hearing and neuronal processing, may be designed to integrate input information from a wide range of sources over a period of time before triggering a cascade of events. However, in active membrane transport, once the value of $\Delta G_{\text{tot}}$ is prescribed by the physiology of the cell, there is no obvious advantage to have a slower turnover rate. This observation begs the question of what is the optimal landscapes of microstates under these conditions. This issue is explored in the next section.

Turnover efficiency

Active transport only requires that the overall free energy change, $\Delta G_{\text{tot}}$, be negative. In practice, however, the magnitude of the turnover rate of the pump can be greatly affected by microscopic details of the transport cycle. It seems reasonable to imagine that there should be an evolutionary advantage for a transport protein like the Na,K-ATPase to achieve maximum flux under normal physiological conditions.^67,68 That is, to have the largest possible steady-state flux $J$ for a given free energy $\Delta G_{\text{tot}}$. Motivating this assertion is the observation that there is no perceived biological advantage in spending ATP on an active transport activity that yields an unnecessarily slow turnover rate. However, understanding how a functionally optimized pump with maximum kinetic efficiency as a result of evolutionary pressure should translate into specific microscopic features of the Na,K-ATPase transport cycle is not obvious. At the simplest level, the steady state flux of a kinetic model with $N$ equivalent states and a free energy drop of $\Delta G_{\text{tot}}/N$ per step is,

$$\begin{gathered} J=k_{f}P-k_{b}P \\ =k_{f}P\left(1-\frac{k_b}{k_f}\right) \\ =\frac{k_f}{N}\left(1-e^{\Delta G_{\text{tot}}/N{k}_{\text{B}}T}\right) \end{gathered}$$ (23)

where $P=1/N$ for all states by normalization. When the free energy shift $\Delta G_{\text{tot}}$ is negative, the resulting steady-state flux is positive. If the number of states is large compared to the free energy drop $\Delta G_{\text{tot}}$ then

$$J\approx -\frac{k_f}{N^2}\frac{\Delta G_{\text{tot}}}{k_{\text{B}}T}$$

Under these conditions, the flux $J$ increases directly in proportion with $\Delta G_{\text{tot}}$.

Given these observations, it is of interest to clarify how the evolutionary pressure aimed at optimizing the kinetic efficiency of the pump may affect the free energy profile of the transport cycle. For more complex sequential models of the transport cycle with no jumps and no branching, the steady state flux can be written as,^66–68,87

$$J=\left(1-e^{\Delta G_{\text{tot}}/k_{\text{B}}T}\right){\left[\sum _{i=1}^N\sum _{j=0}^{N-1}\frac{1}{k_{i,i+1}}{e}^{-\left(G_{i-j}-G_i\right)/k_{\text{B}}T}\right]}^{-1}$$ (24)

with a periodic free energy profile, $G_{i\pm N}=G_i\pm \Delta G_{\text{tot}}$. Assuming that $k_{i,i+1}=k_f$, it can be verified that the steady state flux of Eq. (23) is recovered from Eq. (24) with a linear free energy profile $G_i=(i-1)\Delta G_{\text{tot}}/N$. Eq. (24) may be written in a different form by using the Kramers-Smoluchowski expression for the transition rate (see Appendix A),

$$\begin{gathered} J=\left(1-e^{\Delta G_{\text{tot}}/k_{\text{B}}T}\right){\left[\sum _{i=1}^N\sum _{j=0}^{N-1}ab{e}^{\left(G_i^{†}-G_{i-j}\right)/k_{\text{B}}T}\right]}^{-1} \\ =\left(1-e^{\Delta G_{\text{tot}}/k_{\text{B}}T}\right){\left[\sum _{i=1}^N\sum _{j=0}^{N-1}ab{e}^{\Delta G_{i,i-j}^{†}/k_{\text{B}}T}\right]}^{-1} \end{gathered}$$ (25)

where $a$ and $b$ are factors related to the second derivative of $G(x)$ at the minima and maximum. Extensions to sequential periodic kinetic models with jumping, branching and deaths process are also been considered.⁸⁶ Eq. (25) shows that the turnover rate is controlled by a sum of $N^2$ possible activation free energies $\Delta G_{i,i-j}^{†}$, with the wells always to the left of the barriers in the periodic free energy profile. Hence, maximizing the kinetic efficiency requires lowering the height of the dominant free energy maxima, while raising the deepest minima (on the left of the maxima). In other words, to progressively reduce the largest deviations along free energy profile while keeping the overall drop of $\Delta G_{\text{tot}}$ per cycle. Once the dominant extrema are mutated away, progress toward a kinetically more efficient pump shall require changes to the overall free energy profile in a more subtle way. If evolutionary pressure succeeds to reduce all the free energy barriers to similar values, then the largest possible steady-state flux $J$ for a given free energy drop $\Delta G_{\text{tot}}$ is obtained with a linear free energy profile. In effect, any small deviation $\delta G_i$ from the linear free energy landscape actually yields a smaller steady-state flux,

$$\begin{gathered} J_{\delta G}=\left(1-e^{\Delta G_{\text{tot}}/k_{\text{B}}T}\right){\left[\sum _{j=0}^{N-1}\sum _{i=1}^N\frac{1}{k_f}{e}^{-\left[-j\Delta G_{\text{tot}}/N+\delta G_{i-j}-\delta G_i\right]/k_{\text{B}}T}\right]}^{-1} \\ =k_f\left(1-e^{\Delta G_{\text{tot}}/k_{\text{B}}T}\right){\left[\sum _{j=0}^{N-1}{e}^{j\Delta G_{\text{tot}}/N{k}_{\text{B}}T}\sum _{i=1}^N{e}^{-\left[\delta G_{i-j}-\delta G_i\right]/k_{\text{B}}T}\right]}^{-1} \\ =k_f\left(1-e^{\Delta G_{\text{tot}}/k_{\text{B}}T}\right){\left[\sum _{j=0}^{N-1}{e}^{j\Delta G_{\text{tot}}/N{k}_{\text{B}}T}N{B}_j\right]}^{-1} \end{gathered}$$ (26)

where $\left(\beta =1/k_{\text{B}}T\right)$,

$$\begin{gathered} B_j=\frac{1}{N}\sum _{i=1}^N{e}^{-\beta \left[\delta G_{i-j}-\delta G_i\right]} \\ \approx \frac{1}{N}\sum _{i=1}^N\left(1-\beta \left[\delta G_{i-j}-\delta G_i\right]+{\beta }^2{\left[\delta G_{i-j}-\delta G_i\right]}^2+\dots \right) \\ \approx \left(1-\beta [\overline{\delta G}-\overline{\delta G}]+2{\beta }^2\left[\overline{\delta G^2}-\overline{\delta G_{i-j}\delta G_i}\right]+\dots \right) \\ \approx \left(1+2{\beta }^2\left[\overline{\delta G^2}-\overline{\delta G_{i-j}\delta G_i}\right]+\dots \right) \\ \ge 1 \end{gathered}$$

where the overline symbol indicates an average over all the states $i=1$ to $N$. The second order term, $\overline{\delta G^2}-\overline{\delta G_{i-j}\delta G_i}$, is positive definite. It follows that $J_{\delta G}\le J_{\delta G\to 0}$ because $B_j\ge 1$. Furthermore, Metropolis Monte Carlo (MMC) simulations seeking to optimize the free energy profile for a given distribution of forward transition rates $k_{i,i+1}$ using Eq. (24) show that a linear free energy profile with small deviations from the order of half the magnitude of the variations in the forward activation free energies yields the maximum turnover rate. These observations are consistent with known optimization principles for enzymes and molecular machines.^63,69,88 Therefore, as long as the heights of the free energy barriers are not too different, the evolutionary advantage of the linear free energy profile to achieved the maximum kinetic efficiency appears to be robust.

While this general discussion provides some guidelines to understand the kinetic efficiency of a simple model, it leaves out key features of the Na,K-ATPase. In particular, its turnover rate is very sensitive to the membrane potential because it is transporting charged species. Using the electrophysiological data from Nakao and Gadsby on the Na,K-ATPase from guinea pig ventricules,^21,22 the turnover rate of the pump rises almost linearly from 13.5/s at −125 mV up to 49.7/s at −25 mV, to saturate to about 55/s above +40 mV. Based on experimental data, the pump appears to be operating near maximum kinetic efficiency over a wide range of membrane potential, departing significantly from the resting state of the cell. At the simplest level, the global charge balance (one net positive elementary charge exported per cycle) is directly responsible for the more negative $\Delta G_{\text{tot}}$ at depolarized membrane potentials that is driving the pump cycle faster. However, when considering the turnover rate, the situation is complex because two K⁺ ions must be transported from the extracellular to the intracellular side, against a positive $V_{\text{mp}}$ when the membrane is depolarized.

In effect, any of the microscopic steps along the transport cycle could be affected in different ways by the membrane potential due to the incremental charge in accord with Eqs. (13) and (14). Therefore, despite the global charge balance causing a shift in the free energy profile with a more negative $\Delta G_{\text{tot}}$, it is not a certainty that an increase in $V_{\text{mp}}$ should necessarily translate in a faster turnover rate. If all the ions were bound and released all at once, in a very concerted manner, one could rationalize how the total charge displacement could be distributed over the microstates. However, this is not how the Na,K-ATPase operates. By virtue of the alternating access mechanism, the protein binds the 2 K⁺ on the extracellular side and release them on the intracellular side, and then binds the 3 Na⁺ on the intracellular side and release them on the extracellular side, all in separate steps. This implies that it is the combined optimization of several aspects of the transport cycle that results in the apparent linear increase in the turnover rate over a range of more than 100 mV.^21,22 For instance, the kinetic efficiency of the pump can also be affected by how the incremental step of free energy are allocated and splitted between the forward and backward transition rates $k_{i,i+1}$ and $k_{i+1,i}$.^62,69,80,81 Splitting factors^62,69,80 may be especially important regarding the impact of the forward $(\Delta G_{i,i+1}^{†})$ and backward $(\Delta G_{i+1,i}^{†})$ incremental displacement charges on the voltage-dependence of the turnover rate of the pump.

To illustrate the impact of the functional constraint on the microscopic parameters we consider a minimalist model of the transport cycle comprising only 6 states. Over the first two steps (state 1–3) the cumulative displacement charge goes from 0 to −2, to account for the inward movement of 2 K⁺ ions, and over the last 3 steps (state 4–6) the cumulative displacement charge goes from −2 to +1, to account for the outward movement of 3 Na⁺ ions. The cummulative displacement charge is $\left\{{𝒬}_1=0,{𝒬}_2=-1,{𝒬}_3=-2,{𝒬}_4=-1,{𝒬}_4=0,{𝒬}_6=1\right\}$. We then sought to optimize the non-electrogenic contribution to the free energy of the states, the non-electrogenic contribution to the free energy barriers for the forward transitinon, and the electrogenic splitting factor ${\lambda }_i$ for the forward transition rate. A MMC algorithm was used to sample the parameter space, similar to the method of Anandakrishnan and Zuckerman.⁶⁸ In total, 18 parameters were optimized to roughly match the electrophysiological data from Nakao and Gadsby.^21,22 The uniqueness of the fit is not of particular importance here, as the exercise serves mainly to show that even a simple model is sufficiently flexible to optimally match a target function. The results are shown in Figure 3.

The turnover rate rises from 4/s at −60 mV to reach a plateau of about 54–55/s at 50 mV, to roughly match experimental data.^21,22 Over this interval, the thermodynamic efficiency, $\eta =W/\Delta \mu$, varies from 0.9 at −100 mV to about 0.6 at 50 mV. This is consistent with the analysis of Wagoner and Dill who showed that Na,K-ATPases are able to maintain a high turnover rate while operating at high thermodynamic efficiency.⁶³ It is interesting to note that several values of the cumulative displacement charge in transition regions $({𝒬}_i+\Delta {𝒬}_{i,i+1}^{†})$ are sampled by the MMC algorithm. These variations suggest that the electrogenic splitting factor ${\lambda }_i$ does not appear to play a crucial role for the 1–2, 2–3, 3–4, and 4–5 transitions in the optimized models. The exception is the electrogenic factor for the 5–6 transition, corresponding to the release of the last Na⁺ ion on the extracellular side, which display almost no variations with ${\lambda }_5=0$. The 6–7 transition is trivially not electrogenic because it is not associated with a change in the cumulative displacement charge in the model. Further tests show that the parameters of the 6-state model can be optimized to match the experimental turnover rate,^21,22 even under markedly different conditions. For instance, the model can accommodate an electro-neutral pump that imports 2 K⁺ and exports 2 Na⁺. More strikingly, it can also represent a pump that imports 3 K⁺ and exports 2 Na⁺, yielding a net transported charge of $-e$ per cycle. These findings suggest that the multi-state kinetic model possesses sufficient flexibility to replicate a broad range of functional behaviors. Admittedly, these simplified models are, by design, intended to present straw man arguments. Nonetheless, the results show that it is possible to optimize the microscopic parameters to enforce an increase of the turnover rate over a wide range of membrane potential while maintaining a constraint on the displacement charge of the microscopic states.

The enhanced efficiency of Na,K-ATPase under depolarized conditions likely provides a functional advantage because rapidly restoring physiological conditions should generally be of critical importance. For example, the concentration of Na⁺ in dendrites and spines can increase dramatically during intense neuronal activity, estimates as high as 100 mM have been proposed,⁸⁹ and restoring intracellular Na⁺ concentration is mainly attributed to the activity of the $\alpha 3$ pump isoform.⁹⁰ There are other examples where temporal physiological events may have shaped the evolution of the pump.¹¹ Furthermore, it has been established that some cardiac Na,K-ATPase isozymes are only activated appreciably upon depolarization, recruited into action for extra pumping during the long-lasting cardiac action potential where most of the Na⁺ entry occurs.⁹¹ The proposition that the pump is not only optimized for maximum turnover under resting conditions, but also for achieving a fast pumping rate over a range of membrane potential is a conjecture, but with profound physiological implications.

Conclusion

Detailed multi-state kinetic models of the Na,K-ATPase provide a rich framework to achieve an integration of functional, structural, and computational data across scales, from the atom to the physiology, thus enabling us to address a number of questions to help elucidate the fundamental rules governing ATP-driven pumps.²³ Analogous formulations of transport cycles based on kinetic models have been constructed for motor proteins,^{83–85,85,85} and other biological systems.^62,67,68 There are many similarities, for example, with respect to the role of the membrane potential and incremental displacement charges in the pump versus the load-distribution and the incremental displacement of the motor along its track. In the case of molecular motors, fascinating observations were obtained on the basis of fluctuation analysis considering the relation between the mean velocity and the mean-square deviation of the displacement.^92,93 However, despite some early efforts,^94–96 an equivalent analysis has not, so far, been pursued in the case of the Na,K-ATPase. Further work along those lines would be of interest.

It ought to be possible to refine the parameters of the model for the complete transport cycle by exploiting all available function data.^{3,7,18–22,24–27} However, computational methods with MD simulations based on atomic models will be needed to complement the information about the incremental charges of all the microscopic steps.^{38,72,75–77} While the voltage-dependence of the transport cycle has long been established,^{31,34,97–99} only a few microscopic steps have, so far, been amenable to experimental investigation to provide directly information about the incremental charge associated with ion binding on the extracellular side.^34–36,38 This type of analysis shall make it possible to identify the functionally rate-limiting steps in the Post-Albers transport cycle,^100,101 examine the global optimization of the pump for achieving high speed and efficiency,^62–65 and examine the various factors affecting protein adaptation to different environments.^10,102

The present multi-state kinetic model was formulated as a single transport cycle with a unique prescribed sequence of states to incorporate only the main Post-Albers mechanism, constructed. However, there is accumulating evidence that the pump operates in a complex manner that involves parallel branches, slippage, and proton leakage.²⁶ Although they are not normally transported by the pump, protons are critically needed to set the selectivity of the K⁺ binding sites in the occluded E2(2K⁺)state.^43,44,103 In the complete absence of Na⁺, the Na,K-ATPase leaks protons,¹⁰⁴ and mutations linked to neurological disease suggest the presence of a proton-leak pathway.¹⁰⁵ More recent findings have shown that the pump can conduct protons through the Na⁺ binding pathway, not only in the absence of free ions, but under physiological concentrations of Na⁺ and K⁺.¹⁰⁴ These findings show that transport mechanism of P-type ATPase pumps is dynamic.

Appendix A: Smoluchowski diffusion transport cycle

We consider a system evolving along the continous coordinate $x$ in the free energy profile $G(x)$ according to a diffusive Smoluchowski equation. The profile is periodic over the interval $L$ with an additional free energy offset after each period, i.e., $G(x+L)=G(x)+\Delta G_{\text{tot}}$. It has been shown that the steady state flux $J$ is,¹⁰⁶

$$J=\left(1-e^{\Delta G_{\text{tot}}/k_{\text{B}}T}\right){\left[{\int }_0^{L}d{x}^{\prime }{\int }_0^{L}dx\frac{1}{D}{e}^{\left[G\left(x+x^{\prime }\right)-G(x)\right]/k_{\text{B}}T}\right]}^{-1}$$ (A.1)

The coefficient $D$ represents the natural diffusion along the continuous reaction coordinate $x$. There is a steady state flux $J$ the $+x$ direction if $\Delta G_{\text{tot}}$ is negative. The steady state flux is small if the double integral in the denominator is large. Then the question becomes to identify the dominant contributions to this double integral. To identify discrete states and transition rates, we assume that the free energy profile $G(x)$ comprises $N$ minima and maxima, $G_i$ and $G_i^{†}$ for $0<x<L$. In the spirit of a saddle point approximation to the integral, we find the list of maxima at $x+x^{\prime }$ that are on the right of the minima at $x$, such that

$$J=\left(1-e^{\Delta G_{\text{tot}}/k_{\text{B}}T}\right){\left[\sum _{j=0}^{N-1}\sum _{i=1}^N{a}_i{b}_{i+j}{e}^{\left[G\left(x_{i+j}^{†}\right)-G\left(x_i\right)\right]/k_{\text{B}}T}\right]}^{-1}$$ (A.2)

where $a_i=\sqrt{2\pi k_{\text{B}}T/G^{″}\left(x_i\right)}$, and $a_{i+j}=D\sqrt{2\pi k_{\text{B}}T/\Vert \left(G^{″}\left(x_{i+j}^{†}\right)\Vert \right.}$. In this expression, the index $i$ runs from 1 to $N$ over all the minima $x_i\left(x_1,x_2,\dots ,x_N\right)$, while the index $j$ runs from 0 to $N-1$ over all the maxima $x_{i+j}^{†}$ that are on the right of the $i$-th minimum $\left(x_i^{†},x_{i+1}^{†},\dots ,x_{i+N-1}^{†}\right)$. Although this assumption is not essential, it is assumed that the second derivative is the same for all wells and all barriers, $a_i=a$ and $b_i=b$, for the sake of simplicity. To relate this expression to Eq. (24), we represent the forward transition rate according to Kramers-Smoluchowski theory,^107,108

$$k_{i,i+1}=\frac{1}{ab}{e}^{-\left[G\left(x_i^{†}\right)-G\left(x_i\right)\right]/k_{\text{B}}T}$$ (A.3)

Substituting $k_{i,i+1}$ into the expression for the steady state flux, we get,

$$\begin{gathered} J=\left(1-e^{\Delta G_{\text{tot}}/k_{\text{B}}T}\right){\left[\sum _{j=0}^{N-1}\sum _{i=1}^N\frac{ab}{e^{-\left[G\left(x_i^{†}\right)-G\left(x_i\right)\right]/k_{\text{B}}T}}{e}^{-\left[G\left(x_{i-j}\right)-G\left(x_i\right)\right]/k_{\text{B}}T}\right]}^{-1} \\ =\left(1-e^{\Delta G_{\text{tot}}/k_{\text{B}}T}\right){\left[\sum _{j=0}^{N-1}\sum _{i=1}^{N}ab{e}^{\left[G\left(x_i^{†}\right)-G\left(x_{i-j}\right)\right]/k_{\text{B}}T}\right]}^{-1} \end{gathered}$$ (A.4)

In this expression, the index $i$ runs over the maxima from 1 to $N\left(x_1^{†},x_2^{†},\dots ,x_N^{†}\right)$, and the index $i-j$ runs over the minima that are on the left of the $i$-th maxima, from $i$ to $i-N+1\left(x_{i-0},x_{i-1},\dots ,x_{i-N+1}\right)$. The analysis shows that Eq.(24) and Eq.(A.1) are equivalent. Similarly, it is also possible to relate these results to the flux expression from a textbook by Läuger (equation 2.21 on page 35)²³

$$\begin{gathered} J=\left(e^{-\Delta G_{\text{tot}}/k_{\text{B}}T}-1\right){\left[\sum _{j=1}^N\sum _{i=1}^N\frac{1}{k_{i,i+1}}{e}^{-\left[G_{i+j}-G_i\right]/k_{\text{B}}T}\right]}^{-1} \\ =\left(e^{-\Delta G_{\text{tot}}/k_{\text{B}}T}-1\right)\frac{e^{\Delta G_{\text{tot}}/k_{\text{B}}T}}{e^{\Delta G_{\text{tot}}/k_{\text{B}}T}}{\left[\sum _{j=1}^N\sum _{i=1}^N\frac{1}{k_{i,i+1}}{e}^{-\left[G_{i+j}-G_i\right]/k_{\text{B}}T}\right]}^{-1} \\ =\left(1-e^{\Delta G_{\text{tot}}/k_{\text{B}}T}\right){\left[\sum _{j=1}^N\sum _{i=1}^N\frac{ab}{e^{-\left[G_i^{†}-G_i\right]/k_{\text{B}}T}}{e}^{-\left[G_{i+j-N}-G_i\right]/k_{\text{B}}T}\right]}^{-1} \\ =\left(1-e^{\Delta G_{\text{tot}}/k_{\text{B}}T}\right){\left[\sum _{j=1}^N\sum _{i=1}^{N}ab{e}^{\left[G_i^{†}-G_{i+j-N}\right]/k_{\text{B}}T}\right]}^{-1} \end{gathered}$$ (A.5)

In the expression, the index $i$ runs over the maxima, and the index $i+j-N$ runs over the minima that are on the left of the maxima with $G_{i+j-N}=G_{i+j}-\Delta G_{\text{tot}}$.