Quantum Model for a Periodically Driven Selectivity Filter in K + Ion Channel
QQuantum Model for a Periodically Driven Selectivity Filter in K + Ion Channel
A. A. Cifuentes and F. L. Semi˜ao
Centro de Ciˆencias Naturais e Humanas, Universidade Federal do ABC, 09210-170, Santo Andr´e, S˜ao 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 dephasing rates and driving frequencies, the two-pathway conduction used by the filter leadsindeed to higher ionic currents 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 e ff ects 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 e ff ortshave been made to explain the origin of those coherences [3],and also to investigate their possible relation with entangle-ment and other quantum related e ff ects [4].Due to its particular features and e ffi ciency, ion channelsconstitute another biological system where non trivial quan-tum e ff ects 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 e ffi cient way, achieving a fluxof about 10 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. Details about the experimentand the simulations demonstrating that ion translocation in thechannel unmistakably follows two distinct paths can be foundin Methods section of [8]. 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, thetransport in this system constitutes a two-path problem wherequantum superposition e ff ects 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. MODELING CONFIGURATIONS AND PATHS IN THECHANNEL
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 di ff erent configurations which will be rep-resented here as two-level systems. To be more specific, if asite k is populated i.e., in state | (cid:105) k , this configuration is ac-tive. Otherwise, if it is in state | (cid:105) 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 di ff erent configurations (di ff erent sites). Such quan-tum coherent events lead to resonances in the ionic currentwhich might be measured directly using physiological tech- a r X i v : . [ phy s i c s . b i o - ph ] M a r 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 riseto two main configurations, usually denoted as and ,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 by s (source)and by d (drain), indicating that we will assume that isin part driven incoherently by interactions of intracellular K + ions with the carbonyl oxygens in the entrance of the channel,and 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] L s ( ρ ) = Γ s (cid:0) − (cid:8) σ − s σ + s , ρ (cid:9) + σ + s ρσ − s (cid:1) , (1) L d ( ρ ) = Γ d (cid:16) − (cid:110) σ + d σ − d , ρ (cid:111) + σ − d ρσ + d (cid:17) , (2)where L s ( ρ ) ( L d ( ρ )) causes incoherent pump (decay) in thesource (drain), Γ s ( Γ d ) is the incoherent pump (decay) ratefor the source (drain), and σ + k ( σ − k ) is the two-level raisingand lowering operator which create (destroy) excitations insite k = s , d . Hereafter, { (cid:63), ρ } denotes the anticommutator { (cid:63), ρ } ≡ (cid:63) ρ + ρ (cid:63) and ρ is the density operator for the foursites. 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 H O (not shown). S − S 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 as concentration-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 which will 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 ( (cid:126) = H hop = c ( σ + s σ − + σ − s σ + ) + c ( σ + σ − d + σ − σ + d ) + β c ( σ + s σ − + σ − s σ + ) + β c ( σ + σ − d + σ − σ + d ) , (3)where c is a hopping rate and σ + k ( σ − k ) with k = , k . For the two (single)pathway topology we take β = β = ds
12 1 s 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 . III. DRIVING THE CHANNEL
The last section, and especially Hamiltonian (3), refers tothe channel under natural conditions in the cell membrane. Inthis case, there are only small electric fields due to concen-trations of di ff erent ions inside and outside the cell and alsocharged residues of aminoacids forming the filter. Now, wewill consider a particular technique which allows one to probeindividual channels in the membrane and to subject them todi ff erent electric fields and chemical environments. This tech-nique, called patch clamping [12], enables one to subject theions in the channel to constant and time-dependent potentialsdue to applied electric fields [5]. Consequently, this changesthe free energy scenario which rules the translocation of ionsin the filter [8]. Following [5], we will consider the field to beengineered so that the configurations follow H ext = ( Ω + Ω cos ω t )[ σ + s σ − s + σ + σ − + βσ + σ − ) + σ + d σ − d ] , (4)where Ω and Ω 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 the periodically driven selectiv-ity filter is then H = H hop + H ext . And this does not refer tothe channel under natural condition but rather to the channelbeing probed in the patch clamping setup. Interaction withthe environmental degrees of freedom, especially vibrationsof the carbonyl groups of the selectivity filter backbone [3],surely induces dephasing noise. In this first treatment of theproblem, we will assume the simplest model where this noiseis local and memoryless, i.e., we will use the following Lind-blad superoperator L deph ( ρ ) = (cid:88) i = s , d , , γ i (cid:0) − (cid:8) σ + i σ − i , ρ (cid:9) + σ + i σ − i ρσ + i σ − i (cid:1) , (5)where γ i is a time independent positive dephasing rate. Usu-ally, the drain excitation probability or population is used toquantify transport e ffi ciency 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 = lim T →∞ T (cid:90) T Γ d ρ d ( t ) dt , (6)where ρ d ( t ) is the reduced state of the drain which is obtainedby tracing out all other sites.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 di ff erent paths may com-pete leading to interference e ff ects in the observable current I .We now present our findings about the main trends followedby this current. In particular, we investigate a possible quan-tum advantage of having two conduction paths linking s and d by comparing the current I produced with the topologiesshown in Figure 2. Although we are treating transport in thecontext of a biological system, it is worthwhile noticing thatcoupled two-level systems also appears in a great variety ofphysical scenarios including, for example, quantum dots [15]and superconducting qubits [17]. Consequently, the resultspresented here might be of value for quantum technologiesusing qubits, where our results might even be promptly simu-lated and experimentally observed [18]. IV. ESTIMATION OF PARAMETERS ANDDECOHERENCE IN THE SYSTEM
In this section, we provide a more detailed physical discus-sion about the choice of parameters used in the simulationspresented later on in this work. In particular, we try to pre-dict the order of magnitude of the decoherence rates in orderto compare it with estimated frequencies of the system. Itis important to remark that our work is based on an e ff ectivemodel . The same is done in recent descriptions of the FMO[13, 14]. In this case, nonlinear spectroscopy techniques givesdirect information on coupling constants and frequencies ap-pearing in an e ff ective description based on the occupation oftwo-level sites. For the ion channels, however, there is not yetan experimental technique which provides information on theparameters used in e ff ective models. Consequently, we haveto estimate the order of magnitude of the parameters involved in our model using indirect available experimental data. Thisis of course not the most suitable scenario to do predictions,but we think that this is not a reason to prevent serious inves-tigations which pave the way for advances and motivate theproposition of new experimental techniques to verify or fal-sify the findings of the models.The physical constants should be chosen as to fullfill theexpected (and measured) current which is transported by thechannel under regular conditions, i.e., 10 ions / sec [7]. Toachieve this, we basically follow the reasoning presented in[5]. From (3) and (4), it follows that Ω is the energy dif-ference between the configurations and c is the hopping ratebetween them. Let us consider the case c << Ω . In or-der to active a transfer rate of 10 ions / sec, perturbation the-ory tell us that we must have c / Ω ≈ ions / sec. On theother hand, this time dependent transport model presents res-onances when Ω = n ω with n integer and ω the frequency ofthe drive which appears in (4). We will set the system to worknear theses resonances. By defining the constant ω = s − ,these conditions are fulfilled by choosing c ≈ ω and n >> c = ω and Ω = ω . The incoherent pump and disposal of configura-tions given by Γ s and Γ d , respectively, depend on the specificconditions under which the experiment will be set. It is naturalto think them as a monotonic function of the concentration ofions inside and outside the cell, as supposed in [5]. Numericalsimulations show that variations of these constants only limitthe total current but not its dependence on other parameters.So, it does no harm to fix these constants to be the same orderas the rate 10 ions / sec, i.e, Γ s = ω and Γ d = ω . Finally,one have some freedom to set Ω (the drive amplitude) at anydesired value because it is an externally controlled parameter.Since the behavior of the current as a function of Ω consistsof a sequence of maxima and minima [5], we will set this am-plitude such that one has the first minimum of current. This isachieved with Omega = . ω .Giving the complexity of the system, it is not easy to an-ticipate the decoherence rate. For this reason, in the firstsimulation shown in Figure 3, we vary the dephasing rate γ over a wide range. Consequently, we can draw conclusionsin regimes such as pure quantum transport (small decoher-ence) and highly classical transport (massive decoherence).In what regime precisely the channel works is a question tobe answered experimentally such as happened to some photo-syntetic complexes which were shown to keep track of somequantumness due to partial preservation of quantum coher-ences [2]. However, it is interesting to see that it is possi-ble, through simplified assumptions, to provide a rough or-der of magnitude of the dephasing rate. Again, we follow thereasoning originally shown in [5]. One can assume that thedominant form of noise comes from the stretch mode of thecarbonyl groups in the ion channel. This naturally changesthe width of the wells forming the trap sites, causing the fre-quency or energy of the stable configurations to fluctuate. Thesimplified model considering just one trap site and one modeof stretch is then described by the Hamiltonian H t , CO = ω t b † b + ω CO a † a + λ b † b ( a † + a ) , (7)where ω t is the frequency of ion in the trapping well and ω CO is the frequency of the stretch mode. If one consider the trans-port of ions as a sequence of tunelling events through barriersseparating the wells, it is not hard to show that the frequencyof motion in each well must be around ω t ≈ s − in or-der to obtain a tunneling rate of 10 ions / sec, as shown in[5]. Concerning the mode, given its typical high frequency ω CO ≈ s − , only the ground state is appreciably popu-lated in room temperature and the corresponding mean squaredeviation of the position of the oxygen atoms in the carbonylgroups is of the order of 0.02Å. One can then attempt to nu-merically find the order of magnitude of the fluctuations in ω t induced by oscillations of amplitude 0.02Å. This fluctuationturns out to be about one order of magnitude smaller than ω t [5]. Consequently, we can roughly consider λ = ω t /
10 in (7).In the scope of this simplified model, an initial super-position state of the ion such as ( | (cid:105) + | (cid:105) ) / √ t d ≈ ω − t = / e . This suggests a decoherence rate about γ ≈ / t d = s − . Therefore, in this crude estimation, oneobtains that the dephasing rate γ is approximatelly one or-der of magnitude stronger than ω . But this is not enough todiscard the analyses of regimes where γ is slightly bigger orsmaller than ω . In fact, we will show in next section that thetwo-path topology is more e ffi cient than the single path oneeven for high dephasings. As said before, it is likely that onlyan experiment will be able to precisely determine γ . In [5], itis proposed the use of physiological techniques to experimen-tally estimate γ from measurements of current. V. SIMULATIONS
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]. Since coupling to the stretch mode ofthe carbonyl groups is expected to be the main cause of de-coherence, we expect that the intermediate sites will possesslower decoherence rates. It is then interesting to see whetherhaving less dephasing in the intermediate configurations helpsconduction.In order to investigate this point, we compare both topolo-gies considering fixed dephasing in sites s and d ( γ s = γ d = γ = . ω ), and varying the dephasing ˜ γ in 1 and 2 ( γ = γ = ˜ γ ). The current I (˜ γ ) for this configuration is presented in Fig-ure 3, where we subtracted the current I ( γ ) which correspondsto the case of invariant dephasing. Interestingly enough, it isclear that the two pathway topology benefits from the passingthrough configurations of reduced dephasing (˜ γ < γ = . ω )which is consistent with the fact that the intermediate sites areless coordinated.It is import to remark that Figure 3 does not allow us to de- γ /ω ∼ I ( γ ) / ω - I ( γ ) / ω ∼ ∼ I ( γ ) / ω - I ( γ ) / ω γ /ω ∼ ∼ FIG. 3: Current I (˜ γ ) obtained by fixing dephasing γ = . ω in sites s 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 = ω and Ω = . ω . cide which topology is best suited for transport for a givenvalue of γ . It only allows us to evaluate the advantage ofhaving intermediate sites of lower decoherence rates than thebinding sites. In figure 4, we address this point by fixing theration ˜ γ/γ and varying γ . We show the behavior of the cur-rent in the cases where decoherence in the intermediate sitesis lower or stronger than in the binding sites. It is now quiteclear that, in respect to di ff erent decoherence regimes, thetwo-pathway topology is more likely to bring transport ad-vantages to the channel compared to the linear topology.In general, for a giving value of the driving field frequency ω , models such as the one considered here present resonanceswhen varying the driving field amplitude Ω [5]. In order togain more information about the advantages of having one ortwo conduction pathways in the filter, we now study the globalmaximum of current I max ( Ω ) as a function of ω . The resultis shown in Figure 5. It is now clear that for most cases thetwo-path topology can o ff er 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 = ω , 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 = ω 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 = ω , 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 9 ω . γ /ω I( γ ) / ω γ /ω I( γ ) / ω ]FIG. 4: (Above) Current I ( γ ) obtained by fixing ˜ γ/γ = . γ which is the dephasing in sites s and d . The quantity ˜ γ isthe dephasing in the intermediate configurations 1 and 2. Solid linecorresponds to the single-pathway topology and dashed line to thetwo-pathway. (Below) The same for ˜ γ/γ =
3. Solid line correspondsto the single-pathway topology and dashed line to the two-pathway.We used ω = ω and Ω = . ω .
2 8 6 4 10 12 ω / ω I m a x ( Ω ) / ω FIG. 5: Behavior of the global maximum of current I max ( Ω ) as afunction of ω . Squares refer to the case with no dephasing and tri-angles to dephasing with γ s = γ d = . ω and γ = γ = . ω .Empty shapes in dashed lines refer to the two-path topology andfilled shapes in solid lines refer to the linear single-path topology. VI. FINAL REMARKS
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 e ff ects 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 advan-tage for the filter to have it. This advantage appears in boththe closed system dynamics, which is certainly not the casein real ion channels, and in the open system scenario wheresystem functions. It is important to remark that, from the ex-perimental 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 interestingbiological system. However, the measure of the current usingthe scheme proposed in [5] would be enough to decide on thevalidity of the model. 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 Fundac¸ ˜ao deAmparo a Pesquisa do Estado de S˜ao Paulo (FAPESP) GrantNo. 2012 / / IQ). F.L.S.also acknowledges partial support from CNPq under grant308948 / −
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