Conductance of a potassium channel in the limit of zero membrane potential

Abstract

Potassium (K⁺) channels have been studied computationally since the late 1990s, yet a complete quantitative description of their permeation properties has not been achieved. One important metric to validate the computational models is to compare the calculated channel conductance with experiment. One approach to calculate the channel conductance is to carry out nonequilibrium molecular dynamics (MD) simulations in the presence of a membrane potential to determine the mean ionic current as a function of voltage. However, because MD trajectories are of limited length, it is often necessary to apply a fairly large membrane potential to determine the mean current from a statistically meaningful number of ion permeation events. For this reason, a quantitative estimate of the channel conductance at small, physiologically-relevant membrane potentials is lacking. Using Green-Kubo linear response theory, we formulate an alternative approach to estimate the conductance of the MthK K⁺ channel by relying on MD simulations under equilibrium conditions. The channel conductance at 400 mM KCl estimated from equilibrium MD with the nonpolarizable AMBER force field and polarizable Drude force field is on the order of 5–9 pS—about 30–60 times less than the experimental value. For both AMBER and Drude, the ion conduction process occurs according to a hard-knock mechanism, via direct ion-ion contacts with no water molecules between the ions.

INTRODUCTION

The large observed conductance of potassium (K⁺) channels is typically taken as an indication that the ion conduction process takes place near the diffusion limit. The first high resolution crystallographic structure of the KcsA channel revealed specific cation binding sites along a narrow pore lined by backbone carbonyl oxygens (1), and confirmed the multi-ion nature of the conduction process proposed by Hodgkin and Keynes in 1955 (2). This breakthrough enabled the quantitative characterization of the K⁺ channel function through computational studies. The first simulations sought to understand the multi-ion conduction process by calculating free energy landscapes (3, 4). Subsequent studies expanded on these efforts to determine its conductance by simulating the channel in the presence of an applied membrane potential (5–10). Determining transport properties from nonequilibrium simulations is a powerful strategy. However, because MD trajectories are necessarily limited in length, a relatively large membrane potential is applied to produce enough permeation events to accurately compute the mean ionic current. The large potential can be problematic because the current-voltage (IV) relationship of present MD models display strong non-linearities (5–7). The smallest membrane potential considered in computational studies of K⁺ channels is ±50 mV (6, 10), while electrophysiological experiments are often carried out with a membrane potential that is as small as ±25 mV (11). However, as the membrane potential is decreased to realistic values the statistical uncertainty on the average current increases rapidly due to the lack of permeation events. Most of the information about permeation events in simulations is generated with a fairly large membrane potential.

Because the ion conduction mechanism observed in simulations with an explicit membrane potential may not be representative of the process under physiological conditions, it is of interest to seek different computational approaches to estimate the channel conductance. This is possible using Green-Kubo linear response theory, which relies on equilibrium statistical mechanics (12). Assuming the familiar constitutive relation between the electric current and the potential difference across the membrane, $I=g{V}_{\mathrm{m}\mathrm{p}}$, we derive an expression for the electric conductance $g$ of an ion channel valid at small $V_{\mathrm{m}\mathrm{p}}$. We then apply this approach to study the conductance of the prokaryotic Ca²⁺-gated MthK K⁺ ion channel. Two popular nonpolarizable force fields, CHARMM and AMBER, are considered, together with the Drude polarizable force field. The present results indicate that the calculated conductances are considerably less than the experimental value, by a factor of 30 (AMBER) to 60 (Drude). We conclude that the atomic force fields currently used in MD simulations do not provide a quantitatively accurate representation of the ion conduction process in K⁺ channels.

THEORETICAL DEVELOPMENTS

We consider a system comprising an ion channel embedded in a membrane surrounded by aqueous ion solutions. At equilibrium, the classical dynamics of the system is governed by the reference unperturbed Hamiltonian ${\mathscr{H}}_0$. We assume that external electrodes impose a potential $V_{\mathrm{m}\mathrm{p}}$ across the membrane. In the context of a membrane system extending in the $xy$ direction submitted to periodic boundary conditions $\left(L_x,L_y,L_z\right)$, the energetic coupling to the external membrane potential is expressed as, $Q_{\text{dis}}{V}_{\mathrm{m}\mathrm{p}}$, where

$$Q_{\mathrm{d}\mathrm{i}\mathrm{s}}(t)=\sum _i{q}_i\left(\frac{z_i(t)}{L_z}\right)$$ (1)

is the displacement charge (13). Where the sum is over all the charges in the system, i.e., ions, water, channel, and lipids. Thus, any atomic displacement in the microscopic system causes an opposite variation of the external charge $Q_{\text{ext}}=-Q_{\text{dis}}$ detected through the electrical circuit (13).

We assume that $V_{\mathrm{m}\mathrm{p}}$ is suddenly switched on at $t=0$. This introduces the time-dependent perturbation, $\delta \mathscr{H}(t)=-Q_{\text{dis}}{V}_{\mathrm{m}\mathrm{p}}\mathrm{\Theta }(t)$, where $\mathrm{\Theta }(t)$ is a Heaviside step-function (equal to 1 when $t>0$ and 0 otherwise), yielding the total Hamiltonian $\mathscr{H}={\mathscr{H}}_0+\delta \mathscr{H}(t)$. The time-dependent perturbation elicits a nonequilibrium transient response in the system, which in time relaxes to an average steady state electric current $I=⟨{\dot{Q}}_{\text{ext}}⟩$. Using Green-Kubo linear response theory (12), the steady state current in the limit where the membrane potential $V_{\mathrm{m}\mathrm{p}}$ is very small can be expressed as $I=g{V}_{\mathrm{m}\mathrm{p}}$, where the conductance $g$ is (14),

$$g=\frac{1}{k_{\mathrm{B}}T}{\int }_0^{\mathrm{\infty }}{⟨{\dot{Q}}_{\mathrm{d}\mathrm{i}\mathrm{s}}\left(t^{\prime }\right){\dot{Q}}_{\mathrm{d}\mathrm{i}\mathrm{s}}(0)⟩}_{0}d{t}^{\prime }$$ (2)

where $⟨\dots {⟩}_0$ is an average based on the reference unperturbed Hamiltonian ${\mathscr{H}}_0$. Linear response expresses the conductance $g$ in terms of a time-correlation function of the current calculated from the fluctuations of the system at equilibrium. Eq. (2) can be recognized as a form of Nyquist’s theorem (15) adapted here to the periodic boundary conditions of the MD simulation.

This is the essential methodological development allowing the determination of the conductance of the channel from equilibrium simulations in the absence of an applied membrane potential. Following a method from Helfand (16), it is advantageous to re-write Eq. (2) as,

$$g=\underset{t\to \mathrm{\infty }}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{1}{2k_{\mathrm{B}}T}{\dot{C}}_{\mathrm{M}\mathrm{S}\mathrm{D}\mathrm{C}}(t)$$ (3)

where the dot indicates the time derivative of $C_{\text{MSDC}}(t)$, the mean-square displacement charge (MSDC) time-correlation function,

$$C_{\mathrm{M}\mathrm{S}\mathrm{D}\mathrm{C}}(t)={⟨{‖Q_{\mathrm{d}\mathrm{i}\mathrm{s}}(t)-Q_{\mathrm{d}\mathrm{i}\mathrm{s}}(0)‖}^2⟩}_0$$ (4)

This type of transformation is familiar in the calculation of the self-diffusion for a particle in solution (12, 16), which can either be expressed as an integral of the velocity-velocity time-correlation function similar to Eq. (2), or as a limiting slope of the mean squared displacement similar to Eq. (3). The fluctuations in $Q_{\text{dis}}(t)$ reflect spontaneous forward or backward ion conduction events occurring along unperturbed equilibrium MD simulations, and the channel conductance can be determined from the integral of the time-correlation function of the displacement current via Eq. (2), or as the limiting time derivative of $C_{\mathrm{M}\mathrm{S}\mathrm{D}\mathrm{C}}(t)$ via Eq. (3).

SIMULATIONS

The simulation protocol of the MthK channel in a lipid membrane was described previously (17). The model is based on the open-conductive X-ray structure of the MthK channel, PDB ID 3LDC (18). Briefly, the system comprises the channel embedded in a solvated bilayer at 400 mM KCl. Three force fields were considered: the additive nonpolarizable CHARMM C36 (19–23) and AMBER ff14SB (23–27), and the polarizable Drude2023 (28–32). The MD based on the AMBER force field was done with the program PMEMD of the AMBER20 suite (33, 34), and those based on the CHARMM and Drude force fields were done with OpenMM 7.7 (35). Simulations were carried out under NPT conditions at 320 K with periodic boundary conditions. A timestep of 2 fs was used for CHARMM and AMBER, and a timestep of 1 fs was used for Drude. A dual-Langevin integrator was used for Drude with a friction coefficient set to 20 ps⁻¹ for the Drude oscillators, while 1 ps⁻¹ for the Drude-atom pairs to maintain the temperature at 320 K for real atoms while the Drude oscillators were kept at 1 K (28). With the Drude force field, the conformation of the selectivity filter is unstable. To prevent large excursions away from the experimental structure, a flat-bottom restraining harmonic potential was applied to the backbone. In the restrained simulation, the average RMSD fluctuations of the backbone of the selectivity filter are on the order of 0.5 Å, a value that corresponds to the magnitude of the natural fluctuations of the filter according to the unrestrained AMBER and CHARMM simulations. Thus, we are confident that the restrained Drude simulations provide a meaningful picture of the occupancy of the selectivity filter in the context of a filter fluctuating around this average structure. In addition, the intracellular gate was kept open at the level of Pro19 and Phe97 $\mathrm{C}\alpha$ distances between adjacent and diagonally opposing subunits with harmonic distance restraints. For CHARMM and Drude, the MthK channel was kept from drifting with a center-of-mass harmonic restraint on Ile84 $\mathrm{C}\alpha$ atoms. All systems were equilibrated for more than 2 μs. For the AMBER and CHARMM force fields, the systems were simulated each for an additional 3 μs without dihedral restraints. For the Drude force field, five replicas of the equilibrated system were simulated for 600 ns with dihedral restraints, yielding a total of 3 μs. In aggregate, the present MD simulations represent a total sampling time of 9 μs after equilibration of the systems.

RESULTS

The time-series of the external charge displacement $Q_{\text{ext}}(t)$ defined from Eq. (1) and the MSDC time correlation function defined from Eq. (4) are shown in Figure 1 on the left and right for AMBER and Drude, respectively. In the case of the AMBER force field, several forward and backward discrete ion conduction events can be distinguished in the time-series shown in Figure 1A. In the case of the Drude force field, only three of the five 600 ns trajectories shown in Figure 1D display spontaneous forward/backward K⁺ permeations. No ion permeation events occurred during the 3 μs simulation based on the CHARMM force field (not shown).

Figure 1:

Ion conduction through the MthK channel examined from equilibrium MD simulations based on the AMBER and Drude force fields. In A and D the displacement charges $Q_{\text{ext}}(t)$ moving through the external circuit defined from Eq. (1) is shown. In B and E, the correlation function $C_{\text{MSDC}}(t)$ defined from Eq. (4) is shown. For AMBER (B), the conductance was determined from the slope between the lagtime range of 0.1 and 100 ns, which is 9 ± 4 pS. For Drude (E), the conductance was determined from the slope between the lagtime range of 40 and 80 ns, which is 5 ± 5 pS. In C and F a characteristic event of the ion conduction process is shown. The snapshots in C and F were taken from A and D on the left and right of the red vertical lines in C and F, respectively. The ions have been colored and labeled for clarity. More information about the transitions is given in Table 1.

For both the AMBER and Drude force fields, the MSDC correlation function has an extremely fast component at short times, followed by a slower linear component at longer times. The slower component arises from the fluctuating ion conduction events through the channel. The fast component arises from the charge fluctuations of a planar capacitor (13), with a relaxation time $\tau =RC=L{C}^{\prime }/\sigma$, where $C^{\prime }=C/A$ is the membrane capacitance per unit area $C^{\prime }=1\mu \mathrm{F}/{\mathrm{c}\mathrm{m}}^2$, and $R=L/\sigma A$ is the resistance of the ionic solution ($L$ is the length of the solution of conductivity $\sigma$ and the area $A$ drops out of the problem). The conductivity of a 400 mM KCl aqueous solution is on the order of 50 mS/cm, and the length $L=80\mathrm{\AA }$, yielding a relaxation time of ~10 ps consistent with Figure 1.

For AMBER the channel conductance, $g$, estimated from the slope of the MSDC time-correlation function (Figure 1B) in the interval from 0.1 to 100 ns is 9 ± 4 pS. Assuming that the permeation events follow a Poisson process, the conductance could be estimated as $g=N{q}^2/\left(2\tau k_{\mathrm{B}}T\right)$, where $N$ is the number of independent events over a period $\tau$ (14). Visual inspection of $Q_{\text{dis}}(t)$ (Figure 1A) suggests that about 5–7 spontaneous conduction events in the forward or backward direction took place over the 3 μs trajectory, yielding an estimated conductance of 5–7 pS. For the Drude force field, the channel conductance estimated from the slope of the MSDC time-correlation function (Figure 1B) in the interval 40–80 ns is 5 ± 5 pS. A value of 4–5 pS would be obtained assuming that the permeation events follow a Poisson process with 4–5 independent conduction events (14). For the CHARMM force field, it is not possible to obtain an estimate from linear-response theory because no spontaneous ion permeation events occurred in the simulation. Assuming a Poisson process with a correlation-time of 0.5 μs, the absence of permeation events during a 3 μs trajectory could be indicative that the conductance of this force field is less than 0.8 pS.

Characteristic events of the ion conduction process associated with discrete jumps in the displacement charge (Figure 1A and B) are shown in Figure 1C and F (taken from the time-series at the time indicated by the vertical red line in panels A and B). Table 1 lists the multi-ion configurations in the five binding sites (S₄ to S₀) occurring within a narrow time intervale of 2 ns during the ion conducting transitions displayed by the charge displacement. It is notable that 66% and 70% of all transient configurations occurring during the ion conducting events (Figure 1A and D) show two K⁺ ions occupying simultaneously the sites S₃ and S₂. These configurations, which are correlated with the sudden jumps in $Q_{\text{dis}}(t)$ (Figure 1A and B), might be associated with particular rate limiting steps in the ion conduction process, though further analysis would be required to confirm this possibility. In both cases, the ion events occur via configurations without any water molecules between the permeating ions and frequent direct ion-ion contacts, according to the single-file hard-knock mechanism (7).

Table 1:

Configurations observed during the conducting transitions^a

Sites					Population
S4	S3	S2	S1	S0	AMBER	Drude
-	K+	K+	W	W	0.40	-
K+	K+	-	K+	W	0.16	-
W	K+	K+	-	K+	0.14	0.54
-	K+	K+	-	K+	0.12	0.16
-	K+	-	K+	W	0.02	0.05

^a“W” = water molecule; “-” = empty site.

The measured single channel conductance of the MthK channel with an ionic concentration of 200 mM KCl is ~150 pS (36). The experimental IV curve is linear at membrane potentials within ±50 mV (the smallest applied membrane potential in those experiments). To maximize the number of conduction events, the present MD simulations were carried out with an ionic concentration of 400 mM KCl. If one assumes that the system is far from ion saturation, the estimated conductance at this concentration should be on the order of 300 pS. This is consistent with previous analysis of electrophysiological data on the KcsA channel (11, 37). By comparison, the conductance of the KcsA channel is ~100 pS at 200 mM KCl, indicating that MthK has an even larger conductance. The variation of K⁺ channel conductance as a function of ionic concentration follows roughly a first-order saturation with a very large equilibrium dissociation constant on the order of 600–800 mM (11, 37). For KcsA, the maximum channel conductance at saturating concentration was estimated to be on the order of 550 pS and 360 pS for outward and inward ions flux respectively (37). The estimates from the MD simulations are underestimated by a factor of about 30 for the additive AMBER force field and by a factor of 60 for the polarizable Drude force field. The disagreement with experimental data is considerable, suggesting that the models are of limited quantitative accuracy. One possible interpretation is that the activation free energy barrier of some putative rate limiting step in the ion conduction process is over-estimated by about 2 to 2.5 kcal/mol. Further analysis would be needed to confirm the existence and identity of such a rate limiting step.

CONCLUSION

Using Green-Kubo linear-response theory, we have estimated the conductance of the MthK channel in the formal limit of a vanishing small transmembrane potential. Two additive nonpolarizable force fields (CHARMM and AMBER) as well as one polarizable force field (Drude) were considered. Of the three force field considered, only AMBER and Drude display a quantifiable conductance. No ion conduction events occurred during the trajectories generated with the CHARMM C36 force field, preventing an estimate from linear-response theory. Comparing with experimental measurements, it is clear that the estimates from the MD simulations are underestimated by a factor of 30–60. The error represents a significant discrepancy with experimental data. The atomic force field models are not quantitatively accurate representation of the ion conduction process in K⁺ channels. Interestingly, the conductance estimated from the polarizable force field is smaller than that of the nonpolarizable AMBER force field. This undermines the argument that the main shortcoming of additive nonpolarizable force fields can be corrected by scaling the ion’s charge by the inverse of the square-root of the high-frequency dielectric constant ${ϵ}_{\mathrm{\infty }}$ according to the electrostatic continuum correction (ECC) (10, 38, 39).

Both the AMBER and Drude force fields display a single-file ion conduction mechanism that can be qualified as hard-knock (7), in the sense that there are direct contacts between adjacent K⁺ ions and no intervening water molecules between them along the narrow pore. However, these results must be interpreted with caution because the present atomic models are quantitatively incorrect, yielding a calculated conductance that is much less than the experimental value. It is possible that the mechanism of ion conduction observed during these MD trajectories is affected by the inaccuracies of the force fields. The issue remains unresolved. Although some observations can be reconciled with a hard-knock mechanism (40, 41), a number of experimental results support the presence of water molecules with the K⁺ ions in the narrow pore (40, 42–45). While MD simulations can be of quantitative accuracy for some problems, the inherent complexities of ion conduction through K⁺ channels pose unique challenges that underscore the need for continued refinement of atomic force fields (17).

SIGNIFICANCE.

Equilibrium molecular dynamics simulations with different force fields are used to estimate the conductance of a potassium channel within the framework of linear-response theory. The estimates are around 5–9 pS, which is about 30 to 60 times less than the experimental value. The simulations also provide information about the ion conduction process.

DATA AND CODE AVAILABILITY

Computer files are available upon request.