Quantum Model for the Selectivity Filter in K+ Ion Channel

A. A. Cifuentes and F. L. SemiaoCentro de Ciencias Naturais e Humanas, Universidade Federal do ABC, 09210-170, Santo Andre, Sao Paulo, Brazil

In this work, we present a quantum transport model for the selectivity filter in the KcsA potassium ion channel.This model is fully consistent with the fact that two conduction pathways are involved in the translocation ofions thorough the filter, and we show that the presence of a second path may actually bring advantages for thefilter as a result of quantum interference. To highlight interferences and resonances in the model, we considerthe selectivity filter to be driven by a controlled time-dependent external field which changes the free energyscenario and consequently the conduction of the ions. In particular, we demonstrate that the two-pathwayconduction mechanism is more advantageous for the filter when dephasing in the transient configurations islower than in the main configurations. As a matter of fact, K+ ions in the main configurations are highlycoordinated by oxygen atoms of the filter backbone and this increases noise. Moreover, we also show that, fora wide range of driving frequencies, the two-pathway conduction used by the filter leads indeed to higher ioniccurrents when compared with the single path model.

PACS numbers: 03.65.Yz, 87.15.A-, 05.60.Gg,

I. INTRODUCTION

. Quantum biology is an emerging field of research whichaims at investigating the possibility of a functional role forquantum mechanics or coherent quantum effects in biologicalsystems. Notably, much of the work made in the last yearshave been related to transport, being the Fenna-Matthews-Olson (FMO) complex an important example [1]. Thispigment-protein complex is present in green sulfur bacteriaand its function is to channel excitons from the chlorosomeantenna complex, where light is harvested, to the reaction cen-ter which execute the primary energy conversion reactions ofphotosynthesis. Much of the attention given to this complex,and to quantum biology, in general, arose from the experimen-tal observation of long-lived oscillatory features using ultra-fast 2-D spectroscopy [2]. Such oscillations were interpretedas evidence for the presence of long-lived electronic coher-ence, something not trivial given the complexity of these opensystems. After these experimental observations, many effortshave been made to explain the origin of those coherences [3],and also to investigate their possible relation with entangle-ment and other quantum related effects [4].

Due to its particular features and efficiency, ion channelsconstitute another biological system where non trivial quan-tum effects may appear and be functional [5]. These channelsare transmembrane proteins and they have an important rolein the production of electric signals in biological systems [6].Their structure gives rise to a selectivity filter which is a verynarrow channel which catalyses the dehydration, transfer, andrehydration of the ions in a very efficient way, achieving a fluxof about 108 ions per second [7]. Throughly crystallographicstudies and free energy computer simulations showed, morethan ten years ago, that ion translocation in the filter involvestransitions between two main states, and that these transitionsoccur through two physically distinct pathways of conduction[8, 9]. These pathways involve either two or three K+ ionsoccupying the selectivity filter.

In this work, we consider a quantum model with two con-duction pathways in accordance with experimental results andsimulations in potassium channels [8, 9]. Consequently, the

transport in this system constitutes a two-path problem wherequantum superposition effects coming from the competingpaths of conduction play a decisive role. Since we treat herethe ionic current in the filter, the results predicted in this workcan in principle be experimentally accessed with physiolog-ical techniques [5]. This requires the filter to be driven by aperiodic time dependent electric field which is also included inour analyses. In the following, we present the basic elementsof the model and the inclusion of the driving field. We thenstudy the ionic conduction or current, highlighting the pos-sible advantages of having two and not just one conductionpath. In particular, we study the role of having non uniformdephasing in the topology, given that this is the most likelyphysical picture in the selectivity filter.

II. MULTIPLE-PATH CONDUCTION MODEL

In the systematically studied KcsA potassium channelfrom soil bacteria Streptomyces lividians, whose structure isvery well known [10], K+ ions loose their hydrating watermolecules to enter the selectivity filter and carbonyl oxygenatoms in its backbone replace the water molecules. This al-lows the formation of a series of coordination shells throughwhich the K+ ions can move. Qualitatively distinguishableconfigurations of ions and water molecules in the selectivityfilter correspond to different configurations which will be rep-resented here as two-level systems. To be more specific, if asite k is populated i.e., in state |1〉k, this configuration is ac-tive. Otherwise, if it is in state |0〉k, this configuration is notactive or not participating in the ionic conduction. In a classi-cal hoping mechanism, we would never find superpositions inone configuration (being and not being used) or entanglementbetween different configurations (different sites). Such quan-tum coherent events lead to resonances in the ionic currentwhich might be measured directly using physiological tech-niques [5].

Ion translocation in the KcsA consists of K+ ions that pro-ceed along the pore axis of the selectivity filter in a single fash-ion with water molecules intercalating them. This gives rise

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to two main configurations, usually denoted as 1-3 and 2-4,representing the position of a pair of K+ ions in the selectivityfilter as depicted in Figure 1. The smaller these numbers are,the closer to the extracellular side the ions are. For the sake ofsimplicity in notation, we will denote here 2-4 by s (source)and 1-3 by d (drain), indicating that we will assume that 2-4is in part driven incoherently by collisions of intracellular K+

ions with the carbonyl oxygens in the entrance of the channel,and 1-3 can decay incoherently to another configuration cul-minating with an ion leaving the cell. These processes can bedescribed by Lindblad superoperators in the form [11]

Ls(ρ) = Γs (Cs + 1)(−

σ−sσ

+s , ρ

+ 2σ+

s ρσ−s), (1)

Ld(ρ) = ΓdCd

(−

σ+

dσ−d , ρ

+ 2σ−dρσ

+d

), (2)

where Ls(ρ) (Ld(ρ)) causes incoherent pump (decay) in thesource (drain), Γs (Γd) is the incoherent pump (decay) rate forthe source (drain), and Cs (Cd) is ionic concentration at thesource (drain). Hereafter, ?, ρ denotes the anticommutator?, ρ ≡ ?ρ + ρ?.

FIG. 1: In the selectivity filter, ions move outwards the cell in thepresence of negative oxygen atoms (red segments) of the carbonylgroups in the lateral backbones. Just two of the four backbones areshown. The concerted motion involves alternating potassium ionsK+ and water molecules H2O (not shown). S 1 − S 4 are binding siteswhich are numbered according to convention that numbers decreasewhen the ion proceeds from intra- to extracellular space.

There are two optimal pathways connecting s and d [8, 9],and they will be denoted here by numbers 1 and 2, with no riskof confusing them with the main configurations since theseare now denoted as s and d. Figure 2 depicts this two-pathnetwork on the left panel. On the right panel, we presenta linear single-path chain which will serve as a benchmarkto test for a possible quantum advantage of the two pathwayconduction. Along pathway 1, an ion first approaches the in-tracellular entrance to the selectivity filter, and then pushesthe two ions in the filter, causing the outermost ion to leavethe channel into the extracellular side. This is known asconcentration-dependent path [8]. In the second optimal path-way, the concentration-independent path, the two ions in thefilter move first, leaving a gap in the selectivity filter whichwill attract an incoming ion from intracellular space. All mi-croscopic elementary steps involved in these transitions canoccur reversibly [9]. For this reason, we represent the situa-tion by a hopping term in the Hamiltonian (~ = 1),

Hhop = c(σ+sσ−1 + σ−sσ

+1 ) + c(σ+

1σ−d + σ−1σ

+d )

+βc(σ+sσ−2 + σ−sσ

+2 ) + βc(σ+

2σ−d + σ−2σ

+d ), (3)

where c is a hopping rate. For the two (single) pathway topol-ogy we take β = 1 (β = 0). Of course, these configurationchanges do not take place at the same rate, but we kept themequal in this treatment for the sake of simplicity. Our maingoal is to discuss the general aspects of the problem.

ds

1

2

1s d

FIG. 2: (left) Two-path topology where the source s is incoherentlydriven and excitations hop thorough the network until it is incoher-ently dissipated through the drain d. (right) Single-path topology. Allsites are subjected to dephasing (wavy arrows). The selectivity filterin potassium ion channels uses a two-path configuration for changingbetween s and d.

In patch clamping [12], it is possible to subject the ionsin the channel to constant and time-dependent potentials dueto applied electric fields [5]. This changes the free energyscenario which rules the translocation of ions in the filter [8].Following [5], we will consider the field to be engineered sothat the configurations follow

Hext = (Ω0 + Ω1 cosωt)[σ+sσ−s + 2(σ+

1σ−1 + βσ+

2σ−2 )

+3σ+dσ−d ], (4)

where Ω0 and Ω1 are essentially the amplitudes of the dc andac parts of the field, respectively, and ω the angular frequencyof the ac part. Similar changes in the free energy scenario oc-cur due to long-range coupling mechanisms in response to aperturbation at a large distance in the protein. This is whathappens, for instance, in ion pumps due to the binding ofAdenosine triphosphate (ATP). The full Hamiltonian for theperiodically driven selectivity filter is then H = Hhop + Hext.

Interaction with the environmental degrees of freedom, es-pecially vibrations of the carbonyl groups of the selectivityfilter backbone [3], surely induces dephasing noise. In thisfirst treatment of the problem, we will assume the simplestmodel where this noise is local and memoryless, i.e., we willuse the following Lindblad superoperator

Ldeph(ρ) =∑

i=s,d,1,2

γi(−

σ+

i σ−i , ρ

+ 2σ+

i σ−i ρσ

+i σ−i), (5)

where γi is a time independent positive dephasing rate. Usu-ally, the drain excitation probability or population is used toquantify transport efficiency in coupled quantum systems [13–16]. Here, we are interested in the time average of this quan-tity or current I which reads [5]

I = limT→∞

1T

∫ T

02 Γdρd(t) dt, (6)

where ρd(t) is the reduced state of the drain which is obtainedby tracing out all other sites.

3

Both, the concentration dependent and independent paths,appear in the experiments and simulations as pathways con-necting site s to site d, and then contribute to ionic conduction[8, 9]. In the quantum regime, these different paths may com-pete leading to interference effects in the observable currentI. We now present our findings about the main trends fol-lowed by this current. Although we are treating transport inthe context of a biological system, it is worthwhile noticingthat coupled two-level systems also appears in a great varietyof physical scenarios including, for example, quantum dots[15] and superconducting qubits [17]. Consequently, the re-sults presented here might be of value for quantum technolo-gies using qubits, where our results might even be promptlysimulated and experimentally observed.

III. RESULTS

In order to investigate a possible quantum advantage of hav-ing two conduction paths linking s and d, we compare the cur-rent I produced with the topologies shown in Figure 2. In allplots, we will be using Ω0 = 256ω0, c = 8ω0, Γs = Γd = ω0,and Cs = Cd = 1 [5], where ω0 = 108s−1 sets a suitable scale.

We first analyze the role of dephasing in the transport, es-pecially in the conduction pathways embodied by sites 1 and2. It is well known that in the selectivity filter, the config-urations s and d have K+ ions residing near the center of abox formed by eight carbonyl oxygens, while in the interme-diate sites 1 and 2 the coordination is reduced to six oxygenatoms, with just four of them provided by the carbonyl groupsof the backbone [8]. The coordination is then dropped to half,and it is expected that by decreasing coordination, dephasingalso decreases. One reason is that when coupling an ion toless carbonyl oxygen atoms of the protein scaffold, vibrationinduced dephasing should be lower. It is then interesting tosee whether having less dephasing in the intermediate config-urations helps conduction. Moreover, would the two pathwayconfiguration, chosen by nature, be more advantageous than asingle pathway in this case?

In order to investigate this point, we compare both topolo-gies considering fixed dephasing in sites s and d (γs = γd =

γ = 0.4ω0), and varying dephasing γ in 1 and 2 (γ1 = γ2 = γ).The current I(γ) for this configuration is presented in Figure3, where we subtracted the current I(γ) which corresponds tothe case of invariant dephasing. Interestingly, it is indeed moreadvantageous to adopt a two pathway topology when passingfrom a high coordination state s to another d, if configurationsof reduced coordination are used in between (γ < γ). At firstsight, it might seem that an advantage of I(γ)− I(γ) ≈ 0.01ω0is not very big, but this impression is false. Actually, givingthat ω0 = 108s−1, this advantage would correspond to an aug-mentation of about 106 ions/s which, for cellular standards, isnot negligible.

In general, for a giving value of the driving field frequencyω, models such as the one considered here present resonanceswhen varying the driving field amplitude Ω1 [5]. In order togain more information about the advantages of having one ortwo conduction pathways in the filter, we now study the global

γ /ω0

0 10 20 30 40∼

0.00

0.02

0.04

0.06

0.08

0.10

0.12

-0.02

I(γ)

/ω0 - I

(γ)/ω

0∼

∼

I(γ)

/ω0 - I

(γ)/ω

0

γ /ω0

∼

0.0 0.2 0.4 0.6 0.8 1.0-0.02

-0.01

0.00

0.01

0.02

∼

FIG. 3: Current I(γ) obtained by fixing dephasing γ = 0.4ω0 in sitess and d and varying γ which is the dephasing in sites 1 and 2. Wesubtracted I(γ) which is the current for equal dephasing in all sites.Solid line corresponds to the single-pathway topology and dashedline to the two-pathway. We used ω = 4ω0 and Ω1 = 284.92ω0.

maximum of current Imax(Ω1) as a function of ω. The resultis shown in Figure 4. It is now clear that for most cases thetwo-path topology can offer advantages with and without de-phasing i.e., the range of ω for which the two-path supersedesthe single path is quite wide. The results are actually quiteconvincing in favor of the two-path topology. Let us take thecase ω = 10ω0, for instance. The current with no dephasingusing the two-path topology is more than twice the currentobserved in the single-path topology. This is a clear evidenceof constructive quantum interference arising from competingconduction paths. Even in the presence of dephasing, the ad-vantage of the two-path topology at ω = 10ω0 is much morepronounced than advantage found with the single-path forsmall ω. Therefore, having two competing paths of conduc-tion gives in general advantages for the filter, and this mighthave actually been used by channel to help it operate under areal noisy environment. As a final comment, it is interestingto see that about ω = 12ω0, dephasing helps conduction in thetwo-path topology. This is a phenomenon called dephasing-assisted transport in the literature [13, 14]. The same happensfor the single-path topology for ω bigger than about 9ω0.

IV. CONCLUSIONS

In this work, we presented a simple quantum model whichtakes into account the most significant features of the KcsApotassium channel. In particular, we included the fact that thissystem employs two pathways of conduction. From a quan-tum point of view, this could be a big advantage given thatpossible constructive interference effects can play a role. Wethen studied the role played by a second pathway of conduc-tion, and we found that there might be indeed some advantagefor the filter to have it. This advantage appears in both theclosed system dynamics, which is certainly not the case in realion channels, and in the open system scenario where system

4

2 8 6 4 10 12ω/ω0

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.1

0.2

Imax(Ω

1)/ω

0

FIG. 4: Behavior of the global maximum of current Imax(ΩI) as afunction of ω. Squares refer to the case with no dephasing and tri-angles to dephasing with γs = γd = 0.4ω0 and γ1 = γ2 = 0.1ω0.Empty shapes in dashed lines refer to the two-path topology andfilled shapes in solid lines refer to the linear single-path topology.

functions. However, it is important to remark that, from the

experimental side, it is still necessary to wait for advances todiscover whether or not this system operates in this moderatenoisy regime where the two-path topology confers advantagesover the single-path. In other words, it is still an open questionwhether or not quantum coherence is present in this interest-ing biological system. On the other hand, given that quantumtransport is a very important topic for modern technologies,our results may still find applications in a great variety of cou-pled quantum systems such as arrays of quantum dots, trappedions, and other systems alike.

Acknowledgments

A.A.C. acknowledges Fundacao de Amparo a Pesquisa doEstado de Sao Paulo (FAPESP) Grant No. 2012/12624-6,Brazil. F.L.S. acknowledge participation as member of theBrazilian National Institute of Science and Technology ofQuantum Information (INCT/IQ). F.L.S. also acknowledgespartial support from CNPq under grant 308948/2011 − 4,Brazil.

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