# Quantum Dynamics and Non-Local Effects Behind Ion Transition States during Permeation in Membrane Channel Proteins

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## Abstract

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^{+}ions in the highly conserved KcsA selectivity filter motive of voltage gated ion channels. We first show that the de Broglie wavelength of thermal ions is not much smaller than the periodic structure of Coulomb potentials in the nano-pore model of the selectivity filter. This implies that an ion may no longer be viewed to be at one exact position at a given time but can better be described by a quantum mechanical wave function. Based on first principle methods, we demonstrate solutions of a non-linear Schrödinger model that provide insight into the role of short-lived (~1 ps) coherent ion transition states and attribute an important role to subsequent decoherence and the associated quantum to classical transition for permeating ions. It is found that short coherences are not just beneficial but also necessary to explain the fast-directed permeation of ions through the potential barriers of the filter. Certain aspects of quantum dynamics and non-local effects appear to be indispensable to resolve the discrepancy between potential barrier height, as reported from classical thermodynamics, and experimentally observed transition rates of ions through channel proteins.

## 1. Introduction

^{+}channel by MacKinnon et al. [3]. It turned out that the critical domain of the protein that can combine fast transduction close to the diffusion limit with selective preference for the intrinsic ion species is provided by the narrow selectivity filter (SF) of the protein [4]. In particular, an evolutionary highly conserved sequence of amino acids, the TVGYG (Thr75, Val76, Gly77, Tyr78, Gly79) motive lining the filter region, allows for an inward orientation of backbone carbonyls with oxygen-bound lone pair electrons interacting with the positively charged alkali ions (see Figure 1). This delicate arrangement involving glycine (Gly79, Gly77) residues serving as “surrogate D-amino acids” [5] can offer a unique “interaction topology”, mimicking the ions’ hydration shells prior to entering the filter pore. The interactions are realized by short-range attractive (filter atoms) and repulsive (between ion) Coulombic forces [2,6].

^{+}channel proteins. In these studies, we suggested evidence that at least two important features behind ion permeation and gating dynamics can follow naturally if quantum properties are inserted into the underlying equations of motion. First, inserting quantum interference terms into the canonical version of action potential (AP) initiation can reproduce the fast onset characteristic of APs as seen in experimental recordings of cortical neurons [13]. Second, we have demonstrated evidence for different quantum oscillatory effects within the filter’s atomic environment, which discriminate intrinsic (e.g., K

^{+}) from extrinsic (e.g., Na

^{+}) filter occupations in K

^{+}-type channels [14].

## 2. Methods

^{+}ion during the transition from site S4 to site S3 in the selectivity filter of the KcsA channel (Figure 1). Therefore, the simulation included the carbonyl groups of Thr74, Thr75, and Val76, as shown in Figure 1 (right). The backbone carbons were positioned at the widely used coordinates of Guidoni and Garofoli [17,18]. The carbonyl oxygen–carbon bond was set initially to 0.123 nm bond length and, in the force-free, unperturbed situation, pointed straight to the central axis (the z-axis of the coordinate system). Oxygen atoms were allowed to oscillate within horizontal and vertical bending modes, excluding stretching. The effective spring constant and the damping factor of this oscillation were adjusted to the values of typical thermal frequencies in the range of a few THz and to the expected dissipation of vibrational energy into the protein backbone structure after a few oscillation periods. Although considered in the implementation of the program, the short time interval in the present study did not necessitate setting thermal random kicks from backbone atoms to carbonyl atoms (Table A1). The degrees of freedom for the motion of a single K

^{+}ion were constrained to the central z-axis of the selectivity filter. This allowed us to implement all calculations on a quad-core computer within a reasonable processing time and does not influence or restrict the conclusions to be drawn from the present results.

_{1}, r

_{2}and charges q

_{1}, q

_{2}, including a repulsion term with a characteristic distance r

_{cut}(the distance where the electron shells start to overlap), and ε

_{0}the vacuum dielectric constant, is as follows:

^{+}ions carry unit charge and carbonyl-bound C atoms carry partial charges, usually set to +0.38 units. We assigned two-point charges to oxygen atoms; one at the center of the atom, and the second one representing the effective charge center of the lone pair electrons coordinating the K

^{+}ions and/or water dipoles. The partial charge relocation between the lone pairs and the central O positions was chosen as one of the dynamic variables that determines the depth of the ion-trapping potential (Table A1).

^{+}ion particle waves and was obtained from a non-linear Schrödinger equation (NLSE) (see Equation (2)), with an initial Gaussian wave packet set to an adjustable width and an adjustable mean ion velocity along the z-axis of the filter. The range of these settings is given in Appendix A in Table A1. It is assumed that the wave packet experiences a potential at every instant of time t, which depends on the position of all other particles at this time. Together with the potential term in Equation (1), this can be described by the following NLSE:

_{k}denotes the mass of the K

^{+}ion and r

_{k}its position vector along the z-axis of the filter. Due to the tetrameric lining of the observed motive (Figure 1), summation over the potential term runs over 12 backbone atomic positions for the carbon, oxygen, and lone pair centers within the Thr74, Thr75, and Val76 lining amino acids shown in Figure 1 (right). As the atomic positions r

_{i}change in time and are influenced by the position of other atoms, as well as by the probability distribution of the K

^{+}ion, the situation entails non-linearity in the Schrödinger equation (see also Appendix B). In the above Equation (2), this functional dependence of r

_{i}on ψ is explicitly indicated. The linear gradient “g” expresses the transmembrane electric potential and z

_{k}the z-component of the vector r

_{k}. The parameters that determine the shape and scaling of interaction potentials (i.e., the geometrical embedding) were adjusted to previous models of the KcsA channel [8,11]. This implied initial values for ${r}_{cut}$ of 0.13 nm, with the charge separation distance of the lone pair electrons from an oxygen center being 1.4 times the radius of the oxygen atom, leading to an average of 0.0825 nm from the oxygen atom’s center. The partial charges of an oxygen atom were split to contain a fraction of 30% at central locations and 70% in the lone pair charge point location (Table A1).

^{+}ion to the z-axis and does not include an expansion of the wave packet perpendicular to this axis. This restriction was necessary to keep the computational time within reasonable limits. In addition, the narrow extension and the symmetry of the filter lining in the pore cause sideways forces to mostly cancel each other along the filter’s z-axis. For the further formal description, this implies that the position vector ${r}_{K}$ of the K

^{+}ion is essentially given by its z-component ${z}_{K}$, because its x- and y-components are always zero.

^{+}wave. At this stage, we assume the backbone C atoms to be rigid (see Section 4) but allow for two bending modes of O atoms while keeping the CO distance constant. The differential dz along the z-axis of the wave packet is then found to exert a differential force $d{\overrightarrow{f}}_{i}$ as follows:

^{+}on this O atom is then obtained by integrating over the range defined along z (additional forces acting on this O atom are a restoring force and a decelerating force, as well as attraction/repulsion of the surrounding C and O atoms, see Appendix C.) This will subsequently change the locations r

_{i}of the O atoms and thereby the potential term in the SE acting back on the evolution of the wave packet. The effect of Equation (3) introduces a non-linearity into the SE as shown in Equation (2). The resulting non-linear Schrödinger equation (NLSE) is formally similar but causally different from the description of Bose–Einstein condensation (BEC) at ultra-cold temperatures [21]. This is because under BEC conditions, the probability distribution of the wave function enters into the Hamiltonian itself, while in our case it is the effect integrated over time, as shown in Appendix B. We solved Equation (2) together with Equation (3) in very small-time steps by the Crank-Nicolson method [15] to keep track of the QM phase factor but sampled the positional changes of the O atoms in larger time steps (values given in Appendix A, additional explanations on the NLSE derivation in Appendix B).

^{2}different starting positions. The positions were generated equidistantly within three times the full 1/e-width of the initial quantum wave packet and weighted with the probability density of this packet. At each initial location 10

^{2}, particles were set into motion with velocities, again sampled equidistantly from within three times the full 1/e-width of the initial Gaussian momentum distribution of the QM wave packet.

## 3. Results

#### 3.1. Classical versus QM Motion

^{+}ion is coordinated at a specific site (e.g., S4) and oscillates within this site due to its thermal energy. In the QM version, the wave packet is placed at the minimum of the potential of this site (at z = 0.15 nm) and assigned a mean velocity corresponding to a kinetic energy sufficiently below the potential barrier to the next site. For the examples shown, we have chosen v

_{0}= 300 m/s. As can be seen from Figure 2, under identical initial conditions at time t = 0, the temporal behavior of a single classical ion (left) is similar to the behavior of the QM wave packet, with the same frequency (around 900 GHz) and amplitudes. This similarity becomes even more striking in a comparison of the single wave packet with a classical ensemble of ions computed for 10

^{4}particles under an identical initial position and velocity distribution as in the QM version (Figure 2, right).

^{+}ion, the QM wave shows a non-vanishing probability that the ion could make a transition to site S3 in the filter (as seen around <1 ps after onset and in more detail in Figure 4, bottom). This observation is particularly interesting as it occurs within just one picosecond (i.e., well within the expected decoherence time due to thermal noise from protein backbone atoms transmitted to carbonyl atoms).

#### 3.2. Transition Behavior: Classical versus QM Evolution

^{+}ion approaches the values needed to cross the potential barrier between these two sites), the QM wave packet and the classical ensemble start to behave quite differently. Figure 3 captures an example for a velocity of v

_{0}= 900 m/s of the ion. At the boundary to S3, the classical ensemble is found to split into roughly ½ (Figure 3, middle), whereas the QM behavior shows that the wave packet manages to cross the barrier with almost all of its location probability (Figure 3, right). Following these initial characteristics, there are also subsequent differences: The splitting in the classical picture remains unchanged during the observed time interval. Ions that have crossed to S3 remain in S3, and ions that did not cross remain in S4.

^{+}ion passes, shown in Figure 5 (left). The deviation depicted in this figure is permanently lower for the QM K

^{+}ion as compared with the classical ensemble of ions.

#### 3.3. Transition Behavior over a Range of Ion Velocities

^{+}ion in S3 was calculated (Figure 4) and the resulting curves were corrected by weights obtained from a Boltzmann distribution (which provides the probabilities for each of the initial mean velocities to occur at a temperature of 310 K). The sum of individual probabilities finally provides a Boltzmann-weighted distribution for the probability to find the QM ion at site S3 at a given instant of time (Figure 5b). We have chosen the same procedure for the weights of classical probabilities.

^{+}ion in S3, setting out from a starting location at S4. As mentioned, the probabilities were Boltzmann weighted for 310 K and summed over velocities from 100 to 900 m/s in steps of 100 m/s.

^{2}(Equation (1)) and these distances can be expected to have a lower bound around ∆r ≥ 0.6nm, the effect on S4–S3 transitions from more distant oxygens can be expected to be small (in addition, the expected intermittent water dipoles will exert a damping effect on these forces). We provide further comments on this situation in Section 4.

## 4. Discussion

^{+}ion between two transition sites in the nano-pore selectivity filter motive of the KcsA channel from two different perspectives: A quantum mechanical simulation implemented by a non-linear version of the Schrödinger equation and a corresponding classical ensemble behavior under identical initial and interaction terms. The non-linear Schrödinger model (NLSE) integrates the solution of the wave equation into its interaction potentials, with all surrounding charges modulating the probability distribution of the wave at a given instance of time. This offers a kind of recursive approach, taking account of mutual interactions of confining Coulomb forces and the QM wave equation, a situation that seems more realistic than the calculation of potentials of mean force (PMF) from classical MD at the atomic scale. The methods were implemented in Java. An executable version is available upon request, as well as a source citation agreement (this requires prior installation of the Java Runtime Environment).

^{+}wave packet. We presently implement these fluctuations by repetitions of K

^{+}evolutions during time-varying random fluctuations of the surrounding O and C atoms. We expect, however, that the main findings reported here about the difference between a classical ensemble and the K

^{+}wave packet will remain largely resistant to these thermal vibrations. The reason is that the main effect found here is due to the spatial dispersion of the QM wave packet, which in turn dynamically spreads the interacting force directions of the surrounding and coordinating charges. In the pure classical case, these forces would permanently be directed towards the center of the moving ion. It is just this distinction that allows for what we called “quantum sneaking” in the discussion above.

^{+}ions in the filter model around one or a few pico-seconds at warm temperatures.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Charge of the K^{+} ion | +1 q_{0} (q_{0 …} unit charge) |

Charge of the carbon of a CO-group | +0.38 q_{0} |

Charge of the oxygen of a CO-group | −0.38 q_{0} |

Distance C–O of a CO-group | 0.123 nm |

z-coordinate of the CO carbons at Thr74 | 0 nm (by definition) |

z-coordinate of the CO carbons at Thr75 | 0.30 nm |

z-coordinate of the CO carbons at Val76 | 0.62 nm |

Distance of the CO carbon atoms from axis of selectivity filter (z-axis) | 0.38 nm |

Stiffness of bending of the O atom around the C atom in a CO-group | 30°/k_{B}T |

Damping constant of rotational vibrations of O atoms | 1 × 10^{−13} kg/m |

Positions of oxygen atoms at t = 0 | equilibrium positions ^{1} |

Velocity of oxygen atoms at t = 0 | 0 |

Distance of lone pairs charge from the center of the O atom | 1.4 r_{0} (=0.0825 nm) |

Percentage of O partial charge in lone pairs | 70% |

r_{cut} | 0.13 nm |

Thermal random kicks from backbone to carbonyls | None |

Linear potential drop along the axis of the selectivity filter | −100 mV/nm |

Initial position of K^{+} ion | 0.15 nm ^{2} |

Initial mean velocity of K^{+} ion wavepacket or classical ensemble | varied between 100 m/s and 1200 m/s |

Full width of wavepacket (1/e-width) | 0.05 nm ^{3} |

Time step for the classical calculations with Verlet algorithm | 1 fs |

Time step for the quantum mechanical calculations with Crank–Nicolson algorithm | 0.003 fs |

Time step for sampling positional changes of O atoms due to the K^{+} force | 6 fs |

^{1}C–O perpendicular to axis of selectivity filter and pointing to this axis.

^{2}This is approximately the middle of site S4.

^{3}This width entails a velocity spread (1/e-full width) of ±65 m/s. Making the wave packet much narrower would give velocity spreads on the order of the thermal mean velocity of a K

^{+}ion. Making it much wider would bring it beyond the width of the ground state of the harmonic oscillator to which a site potential can be approximated.

## Appendix B. How the Schrödinger Equation Becomes Nonlinear

^{+}ion has an apparently linear form:

^{+}ion, which depends on the quantum mechanical probability distribution of the K

^{+}ion at that moment of time. This latter force introduces a non-linear aspect into the Schrödinger equation:

^{+}ion and ${\overrightarrow{F}}_{other}$ subsumes all other forces acting on the atom.

^{+}ion, ${r}_{K}$, is always on the z-axis, the integration extends over the whole range of the wave packet. For the present “cut-off” in our model, the wave packet has appreciable values only between the limits of ${z}_{min}=-0.2\text{}\mathrm{nm}$ and ${z}_{max}=0.8\text{}\mathrm{nm}$. From the above equations, one can see that the position of an O atom at a given moment is determined by the entire history of the spatial spread of the wave packet of the K

^{+}ion. As this position determines the potential to which the K

^{+}ion is exposed at that instant of time, it introduces the non-linear term into the Schrödinger equation as given by Equation (2).

## Appendix C. Restoring and Dissipating Forces on Carbonyl O Atoms

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**Figure 1.**A section through the tetrameric KcsA filter motive, showing a sketch of two transmembrane helices for binding sites S4–S1, with two ions and two waters molecules (left). On the right, a window (insert) for atomic locations of the filter lining during the passage of a K

^{+}ion (green) from S4 to S3 is sketched. The carbon atoms (brown) of the carbonyl groups are situated at the corners of a square (including all four backbone strands). The charge (blue) of oxygen atoms (in red) is partly contained in the center of the atom and partly within a point location slightly outside the oxygen. As these charges are drawn towards the central axis, they represent the effective charge center of the lone pair electrons (shown in blue). The size of atoms and the K

^{+}ion on the right are drawn to scale approximately.

**Figure 2.**Single ions and the classical ensemble: Comparison of the evolution (within 3 ps time) of a single classical K

^{+}ion (

**left**, blue curve) with initial velocity of 300 m/s at the minimum of site S4 with a quantum mechanical wave packet of minimum uncertainty of this ion (

**middle**). The red lines are the z-coordinates of the carbonyl oxygen atoms. Middle: Probability density from a quantum mechanical (QM) calculation along the z-axis of the wave packet as a function of time (intensity of blue reflects higher probability densities). The initial full width of this density is 0.05 nm (at 1/e).

**Right**: Classical probability density of finding an ion from the ensemble of 10

^{4}ions at the given z-coordinate as a function of time.

**Figure 3.**Transition behavior between S4 and S3 (

**left**insert) for a classical ensemble (

**middle**) and the simulated QM wave packet (

**right**), with shades of black and blue coding normalized probability densities for location and time. Red lines (

**right**) are again the z-coordinates of carbonyl oxygens. Note: whereas the classical ensemble splits after around 0.8 ps (

**middle**), the QM distribution goes beyond the barrier to S3 almost completely (

**right**).

**Figure 4.**Time-dependent probabilities to find an ion in S3, when the ion was implanted into S4 with different mean onset velocities (900 m/s blue top for the QM wave packet, black for the classical ensemble) and at 300 m/s for the QM wave packet (with some probability <0.1 to cross over to S4). At this initial velocity of 300 m/s, the classical particles do not cross to S3. Note: most classical particles with 900 m/s are in S3 after 0.5 ps but eventually about 45% return to S4 due to oxygen charge derived forces (the spring that returns these ions to equilibrium positions with vibrations around 3 THz, see Figure 2). The QM wave exhibits a small but remaining probability (<10%) of returning to S4.

**Figure 5.**(

**a**) Mean deviation of Tyr75 carbonyl oxygens from their equilibrium positions, while a K

^{+}is moving past from location S4 to S3 (in nm). The classical particles are in black, QM wave packets in blue; (

**b**) Probability for a K

^{+}ion to be found in S3, setting out from S4 with mean velocities between 100 m/s and 900 m/s, weighted according a Boltzmann velocity distribution at 310 K (blue QM, black classical).

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**MDPI and ACS Style**

Summhammer, J.; Sulyok, G.; Bernroider, G.
Quantum Dynamics and Non-Local Effects Behind Ion Transition States during Permeation in Membrane Channel Proteins. *Entropy* **2018**, *20*, 558.
https://doi.org/10.3390/e20080558

**AMA Style**

Summhammer J, Sulyok G, Bernroider G.
Quantum Dynamics and Non-Local Effects Behind Ion Transition States during Permeation in Membrane Channel Proteins. *Entropy*. 2018; 20(8):558.
https://doi.org/10.3390/e20080558

**Chicago/Turabian Style**

Summhammer, Johann, Georg Sulyok, and Gustav Bernroider.
2018. "Quantum Dynamics and Non-Local Effects Behind Ion Transition States during Permeation in Membrane Channel Proteins" *Entropy* 20, no. 8: 558.
https://doi.org/10.3390/e20080558