Bubble gating in biological ion channels: A density functional theory study

We study the behavior of a waterlike liquid inside the gate of a biological ion channel following the basic geometry of the well studied potassium channel KcsA. We calculate the three-dimensional density distribution $\rho \left(\mathbf{r}\right)$ of the liquid within the framework of classical density functional theory and observe the formation of a low density region (bubble) when the gate is narrow. This observation corresponds to a finite-size form of capillary evaporation and supports the so-called bubble-gate theory. From the density profile we also compute the energy landscape of the gate and the energy required to change the gate from a closed (narrow) to an open (wide) state and vice versa.

Figure 1

A cut through the model geometries of the ion channel considered in our three-dimensional DFT calculations. Both geometries are cylindrically symmetric around the long ($z$) axis. All measures, except for the lower radius $R_2$, are fixed. (a) The channel connects two bulk systems (model I). (b) The channel is closed by a hemispherical cavity (model II) that leads to the selectivity filter, which is not considered in our paper.

Figure 2

(a) The density profiles $\rho (r,z)$ and (b) $\rho (r=0,z)$ of a waterlike fluid in a narrow closed channel with $R_2=0.2\mathrm{nm}$ considered in model I. Close to the narrowest part of the gate a bubble forms, which can be seen in both representations of the density profile by low densities. This bubble hinders ions from passing through the gate and hence through the channel. Note that the color code of the density scale is not linear.

Figure 3

(a) The density profiles $\rho (r,z)$ and (b) $\rho (r=0,z)$ of a waterlike fluid in a wide open channel with $R_2=0.6\mathrm{nm}$ considered within model I. Although the density inside the channel is slightly below its bulk value, we think that ions still can pass through the open gate and the channel. Note that the color code of the density scale is not linear.

Figure 4

(a) The density profiles $\rho (r,z)$ and (b) $\rho (r=0,z)$ of a waterlike fluid in a narrow closed channel with $R_2=0.2\mathrm{nm}$ considered in model II(b). Close to the narrowest part of the gate, similar to model I, a bubble forms, which can be seen in both representations of the density profile by low densities. This bubble hinders ions from passing through the gate and hence through the channel. Note that the color code of the density scale is not linear.

Figure 5

(a) The density profiles $\rho (r,z)$ and (b) $\rho (r=0,z)$ of a waterlike fluid in a wide open channel with $R_2=0.6\mathrm{nm}$ considered within model II(b). Although the density inside the channel is slightly below its bulk value and a bit below the corresponding density in a wide channel treated within model I, we think that ions still can pass through the open gate and the channel. Note that the color code of the density scale is not linear.

Figure 6

The grand potentials ${\mathrm{\Omega }}_{\mathrm{gate}}\left(R_2\right)$ from our DFT calculations for the cone-only gate (model I, full line) and the gate models with lids and hydrophobic [model II(a), dashed line] and hydrophilic [model II(b), dotted line] bottoms as function of the gate radius $R_2$. The energy landscape shows, in all cases considered here, two minima for the open and the closed states, which are separated by energy barriers of roughly $2k_{B}T$. The zero point of each data set is chosen such that a comparison of the heights of the energy barriers is possible.