Noise Assisted Excitation Energy Transfer in a Linear Toy Model of a Selectivity Filter Backbone Strand
aa r X i v : . [ c ond - m a t . d i s - nn ] D ec Noise Assisted Excitation Energy Transfer in a Linear Toy Model of a Selectivity Filter BackboneStrand
Hassan Bassereh, Vahid Salari, and Farhad Shahbazi Department of Physics, Isfahan University of Technology, Isfahan 84156-83111, Iran (Dated: August 29, 2018)In this paper, we investigate the effect of noise and disorder on the efficiency of excitation energy transfer(EET) in a N = 5 sites linear chain with ”static” dipole-dipole couplings. In fact, here, the disordered chain isa toy model for one strand of the selectivity filter backbone in ion channels. It is recently discussed that thepresence of quantum coherence in the selectivity filter is possible and can play a role in mediating ion-conductionand ion-selectivity in the selectivity filter. The question is ”how a quantum coherence can be effective in suchstructures while the environment of the channel is dephasing (i.e. noisy)?” Basically, we expect that the presenceof the noise should have a destructive effect in the quantum transport. In fact, we show that such expectation isvalid for ordered chains. However, our results indicate that introducing the dephasing in the disordered chainsleads to the weakening of the localization effects, arising from the randomness, and then increases the efficiencyof quantum energy transfer. Thus, the presence of noise is crucial for the enhancement of EET efficiencyin disordered chains. We also show that the contribution of both classical and quantum mechanical effects arerequired to improve the speed of energy transfer along the chain. Our analysis may help for better understandingof fast and efficient functioning of the selectivity filters in ion channels PACS numbers: 03.65.Yz
I. INTRODUCTION
Energy or charge transfer is one of the most important phe-nomena in physical and biological systems. Life-enablingtransport phenomena in the molecular mechanism of bio-logical systems occur at scales ranging from atoms to largemacro-molecular structures. Charge transfer through DNA or charge and energy transfer processes in photosyntheticstructures are good examples in this context. Recently, ithas been put forward the idea that quantum mechanics mighthave positive effect on the efficiency of energy or chargetransfer in living systems. One of the most important ef-fects of quantum mechanics in biological systems has beenevidenced in the Fenna-Matthews-Olson (FMO) complexes which is observed by experimental methods via ultrafast spec-troscopy .Natural systems, inevitably, suffer from various types ofnoise with internal and external sources, which perturb the in-trinsic dynamics of the real systems. Noise exists everywhereespecially in the hot, wet and complex structures of biologicalsystems. As a general expectation, noise is a destructive fac-tor for any system, but in some living systems it is vice versa .Here, we would like to study excitation energy transfer in a toymodel of a strand of a protein (i.e. a linear harmonic chain)in presence of noise via a quantum mechanical approach. It isknown that in many ion-channel proteins, flow of ions throughthe pore is governed by a gate, comprised by a so-called selec-tivity filter that can be activated by electrical, chemical, light,thermal and/or mechanical interactions. The selectivity filteris believed to be responsible for the selection and fast conduc-tion of particular ions across the membrane of an excitablecell. A selectivity filter is composed of four P-loop chains (P-loop or phosphate-binding loop is a motif in proteins that isassociated with phosphate binding), and each strand is com-posed of the sequences of TGVYG amino acids [T(Threonine,Thr75), V(Valine, Val76), G(Glycine, Gly77), Y(Tyrosine, FIG. 1. P-loop strand of a selectivity filter: the amino acid molecularchains play the main role in conducting the ions. In the structure ofthe selectivity filter N-C=O is an amide group and C=O is a carbonylgroup. Here we have considered each peptide unit H-N-C=O as onemolecule. The P-loop strand is in the form of a chain of peptide units.
Tyr78), G(Glycine, Gly79)] linked by peptide units H-N-C=O(see Fig. 1). We consider a simplified model for one of theabove chains in the selectivity filter in absence of ions, inwhich the system is made up of sites. Each site, being equiv-alent to a peptide unit linked to an amino acid, is considered asan electric dipole, so the interaction energy between each twosites is assumed to be dipole-dipole type following the inversecube law . Asadian et al considered a rather abstract system,a linear chain of N molecules, and studied the effect of me-chanical vibrations of the system on the energy transfer acrossthe chain. They demonstrated that the collaborative interplaybetween the quantum-coherent excitation and the mechanicalmotion of the molecules would enhance the excitation energytransfer through the linear chain. The general issue is that anexcitation is injected to the first site and after time t , the trans- E x c i t a t i o n sink Disordered State E x c i t a t i o n sink Ordered State
FIG. 2. (Color online) Schematic representation of excitation energytransfer through (top) an ordered chain in which the distances be-tween sites are equal and (bottom) a disordered chain with randomdistances between the molecules. ferred energy to an additional site (i.e. sink) is measured ( e.g.sink can be located after G79 (or G) in Fig. 1). A schematicrepresentation of this energy transfer is illustrated in Fig. 2.We also investigate the effect of noise and disorder on the en-ergy transfer through the structure.Paper is organised as the following. In Sec. II the model isdescribed and quantum transport equations in terms of Lind-blad operators are introduced. Excitation energy transportcalculations, in absence and presence of noise, are given inSec. III. In Sec. IV, we present an alternative approach to findthe contribution of classical dynamics in the efficiency of en-ergy transfer. The results are summarised in Sec. V.
II. STRUCTURE AND TRANSPORT MODEL
Since there are five main peptide units in the P-loop struc-ture, we take N = 5 . However, we are interested in measur-ing the amount of energy transported to the end of the P-loopstrand. As a result, an extra site so-called ’sink’ will be in-troduced at the end of the loop . So, we consider a linearchain of N = 5 sites, composed of two level molecules. Eachmolecule can be stimulated to its excited state and the excita-tion can be transferred to the other sites due to dipole-dipolecouplings between the molecules. For simplicity, we consideronly the interaction between nearest neighbour sites. Thus,the Hamiltonian of the system for a single excitation can bewritten in tight-binding form H = N X n =1 ε | n ih n | + N − X n =1 J n ( | n ih n + 1 | + | n + 1 ih n | ) , (1)where | n i denotes the excited state in the n -th site, and ε isthe excitation energy for a single molecule. The dipole-dipolecoupling between the sites n and n + 1 has the form J n = ˜ J d n , (2) in which ˜ J contains the dipole moment and is set to unity( ˜ J = 1 ), d n,n +1 is the equilibrium distance between twoneighbourings. Since we consider the excitation in an openquantum system, therefore energy dissipation due to interac-tion with the environment is inevitable. Markov approxima-tion formulation of the energy loss due to dissipation, can bedone by using the following Lindblad super-operator : L diss ρ = N X n =1 γ n [2 σ − n ρσ + n − (cid:8) σ + n σ − n , ρ (cid:9) ] , (3)where ρ is the total density matrix of the system, σ + ( σ − ) is the creation (annihilation) operator of the excitation at site n , γ n is the dissipation rate at each site, and (cid:8) A, B (cid:9) = AB + BA . In order to measure how much of the excita-tion energy is transferred along the chain (and not lost dueto dissipation), we introduce an additional site, the sink , rep-resenting the final ( N + 1 )-th trapping site that resembles areaction center, in a photosynthesis system . The sink is pop-ulated via irreversible decay of excitation from the last site, N . This approach is formally implemented by adding the fol-lowing Lindblad operator to the master equation : L sink ρ = γ sink [2 σ + N +1 σ − N ρσ + N σ − N +1 − (cid:8) σ + N σ − N +1 σ + N +1 σ − N , ρ (cid:9) ] , (4)in which γ sink denotes the absorption rate of the sink, forwhich we select the typical values γ sink = 0 . in this work.Later, we will discuss the dependence of the results on γ sink .In order to calculate the efficiency of energy transfer givenby the asymptotic population of the sink, it is necessary tointegrate the following master equation ( ~ = 1 ): ∂ρ∂t = i [ ρ, H ] + L diss ρ + L sink ρ. (5)Here, an excitation is injected at the first site, i. e. ρ (0) = | ih | and we are interested in finding how much of the ex-citation can be transferred to the sink, so we have to find P sink ( t ) = h N + 1 | ρ ( t ) | N + 1 i , which is called Sink pop-ulation (at time t ). In this paper, we use the python packageQUTIP to numerically solve the Lindblad master equations.In our numerical simulations, all energies, time-scales, andrates will be expressed in units of ε , and thereby we consider ε = 1 . III. EET IN ORDERED AND DISORDERED CHAINS
In this section we study the quantum excitation energytransfer in the static chains in which the distances betweensites are time independent. The ordered static chain is a chainin which the distances between sites are equal and the cou-plings between sites are frequency-independent and don’t varywith time. A disordered static chain is the same as orderedstatic chain but the distances between sites are chosen ran-domly (see Fig. 2).The coupling between neighbouring sites is expressed asthe following γ diss P s i n k ( t →∞ ) DisorderedOrdered
FIG. 3. (Color online) The effect of dissipation rate, γ diss , on thesystem efficiency for ordered (blue (dark) line) and disordered (red(grey) line) chains for γ sink = 0 . . J n = ˜ J [ d n,n +1 ] , (6)Where ˜ J is the dipole moment and d n,n +1 is the equilib-rium dimensionless distance between the sites n and n + 1 ,which is taken to be unity for ordered chain. For the dis-ordered chain, we use the relative ratios of distances basedon the real data obtained from the distances between aminoacids in one strand of the backbone of the selectivity filter ofKcsA ion channels. In the mentioned structure, the distancesbetween neighbouring sites are, r , = 2 . A ◦ , r , =3 . A ◦ , r , = 2 . A ◦ and r , = 4 . A ◦ , therefore di-viding the distances over the minimum one . A ◦ , gives usthe relative distances as d , = 1 , d , = 1 . , d , = 1 and d , = 1 . . Energy population in the sink is calculatedusing the method explained in the Sec. II. As a matter of fact,each molecule feels a different environment, hence the localdissipation rates γ n depend on the sites, nevertheless for thesake of simplicity and due to so far unavailable experimentaldata, we will later assume them to be uniform γ n = γ diss .In order to pick a typical value for dissipation rate, we fixed γ sink = 0 . in Eq. 5 and calculated the sink population interms of γ diss . The results represented in Fig. 3, show therapid decaying of sink population with dissipation rate coef-ficient in both disordered and ordered static chains. Since weare interested in system with efficient energy transfer, hencein the rest of the paper we select a small value γ diss = 0 . for the dissipation rate.With such a selection for dissipation, the sink populationare plotted in Fig. 4versus time for ordered and disorderedchains. It can be seen that population for disordered chainis clearly less than the disordered one, which is a result of t P s i n k Ordered chainDisordered chain
FIG. 4. (Color online)Temporal variation of sink population forstatic ordered (blue (black) solid line) and disordered chain (red(grey) solid line). the Anderson localization of the eigenstates of the tight-binding Hamiltonian in a one dimensional disordered chain.In fact, the randomness in the disordered chain plays the roleof scattering centres for the Bloch states corresponding to theordered chain. The constructive interference, resulting fromsuccessive coherent back scatterings of the Bloch states bysuch scattering centres, would lead to the localization of thesestates within a region characterised by the localization length.Decomposing the initial excitation into the eigenstates of theoriginal Hamiltonian 1, the modes with higher energy havesmaller localization length and are more probable to becomelocalized. The localised modes spend much time inside thechain and then leak out to the environment because of dissipa-tion, hence loss their ability to corporate in the energy transferinto the sink. To explicitly show the localization effect on theenergy transfer, we plot the time dependence energy densityin the first two sites in the left-(a) and right-(a) panels of Fig. 5for ordered and disordered chains, respectively. The temporalbehaviour of energy populations in the other sites are similarto the the first two sites and for the sake of clarification we donot show them. It can be clearly seen that the energy popula-tion in the sites and persists much longer in the disorderedchain than the ones in the ordered chain.Now we proceed to investigate the effect of noise on thequantum energy transfer along the chain. Real systems areopen and interact with environment, so it is reasonable to as-sume that the environment is dephasing which means it isnoisy and has a destructive effect on quantum coherence inthe system. The dephasing term can be incorporated to themaster equation, using the following Lindblad form L deph ρ = N X n =1 γ deph [2 σ + n σ − n ρσ + n σ − n − (cid:8) σ + n σ − n , ρ (cid:9) ] (7) P o pu l a t i o n (a) sinksite 2site 1 t P o pu l a t i o n (b) sinksite 2site 1 P o pu l a t i o n (a) sinksite 2site 1 t P o pu l a t i o n (b) sinksite 2site 1 FIG. 5. (Color online) Left) Temporal variation of energy population of the sites , and the sink in ordered chain for (a) absence and (b)presence of dephasing with γ depth = 0 . . Right) Temporal variation of energy population of the sites , and the sink in disordered chain for(a) absence and (b) presence of dephasing with γ depth = 0 . γ deph P s i n k ( t →∞ ) OrderedDisordered
FIG. 6. (Color online) Long time sink population of static ordered(blue (black) solid line) and disordered (red (grey) solid line) chainsversus dephasing rate coefficient. where γ deph is the dephasing rate coefficient. In general, oneexpects decreasing population after adding dephasing term.Fig. 6 illustrates variation of the sink population, as t → ∞ ,versus dephasing rate for the ordered and disordered chains.As we expected, dephasing (i.e. noise) decreases population(i.e. efficiency) of the ordered chain. However, somethingdifferent is observed for the static disordered chain, it is seenthat the sink population does not behave monotonically ver-sus dephasing rate, in such a way that noise is able to increasethe efficiency of energy transfer and reach it to a maximumefficiency for γ depth ∼ . . The reason for noise-assisted en-ergy transfer in the disordered chain is the weakening of thelocalization, as a result of dephasing which reduces the con- structive coherent back-scatterings responsible for such a lo-calization, hence leading to the enhancement of the efficiencyof energy transfer. The panels (b) in Fig. 5 evidently representthe demolishing effect of dephasing on the constructive backscattering in both ordered and disordered chains. As can beseen from the left-(b) panel of Fig 5, the destruction of Ander-son localization in the disordered chain leads to remarkableenhancement in the amount of the sink population, and alsothe speed of energy transfer.In order to gain insight into the effect of possible values ofthe absorption rates of the sink, γ sink , and dephasing rates, γ deph on the energy transfer efficiency, we illustrate the den-sity plots of sink population for the ordered (Fig. 7, left) anddisordered (Fig. 7, right) chains. This figure shows there isno region in the parameter space, γ sink − γ deph , for the noise-assisted energy transfer in the ordered chain, while we see thatsuch a phenomenon occurs in the disordered chain in the re-gion with γ sink . . . IV. HYBRID QUANTUM-CLASSICAL DYNAMICS
We have demonstrated that the collaborative interplay be-tween the quantum-coherent excitation and the mechanicalmotion of the molecules would enhance the excitation en-ergy transfer through the linear chain. It is already concludedthat a closer look at the involved dimensions and energeticsof the process reveals that the underlying mechanism for iontransmission and selectivity might be not entirely classical .Recently it was demonstrated theoretically that, based on thetime and energetic scales involved in the selectivity filter, theion selectivity and transport cannot be entirely a classical pro-cess but involves quantum coherence . In this section wepursue an alternative approach to evaluate the contribution ofthe classical and quantum mechanical effects in the efficiencyof energy transfer across the chain. For this purpose, we useKossakowski-Lindblad master equation in the following γ deph γ s i n k γ deph γ s i n k FIG. 7. (Color online) The density plots of sink population for the disordered chain (left) and ordered chain (right) with γ diss = 0 . . form: ∂ρ∂t = − (1 − η ) i [ H, ρ ] + η X ij [ L ij ρL † ij − / (cid:8) L † ij L ij , ρ (cid:9) ] , (8)where η is a measure for the classical contribution of the en-ergy transfer. If η = 0 we have a pure quantum energy transferand if η = 1 the energy transfer is fully calssical , and for theother values of η (between 0 and 1) we have a joint cooper-ation between classical and quantum mechanical effects. Inthe above equation, L ij = T ij | i ih j | , where | i i denotes thereal space basis state and we select T ij = J /d ij , in which J = 1 and d ij is selected as the real normalized distance be-tween the sites i and j( amino acids in the p-loop strand) in thedisordered chain and d ij = 1 for the ordered chain.Here we neglect the dissipation and again add L sink to the η t s Quantum Classic
DisorderOrder
FIG. 8. (Color online) Saturation time versus the coefficient ofclassical part of energy transfer dynamics above Eq. 8 and calculate the saturation time t s ( the timewhich all of the first injected energy is transferred to the sink)versus parameter η . As shown in Fig. 8, t s is not much sensi-tive to η in the ordered chain, however in the disordered chainwe observe that t s rapidly drops by increasing the classicalpart of the energy transfer dynamics. The optimal stationarytime is obtained for η ∼ . , meaning that the maximum effi-ciency in the speed of energy transfer is acquired by approxi-mately equal contribution of classical and quantum mechani-cal effects. V. DISCUSSION AND CONCLUSION
Ion channels are proteins in the membranes of excitablecells that cooperate for the onset and propagation of electri-cal signals across membranes by providing a highly selectiveconduction of charges bound to ions through a channel likestructure. The selectivity filter is a part of the protein forminga narrow tunnel inside the ion channel which is responsiblefor the selection process and fast conduction of ions acrossthe membrane . Certainly, the magnitude of the thermal fluc-tuations of the backbone atoms forming the selectivity filteris large relative to the small size difference between Na andK, raising fundamental questions about the mechanism thatgives rise to ion selectivity. This suggests that the traditionalexplanation of ionic selectivity should be reexamined . Re-cently, Vaziri et al. and Ganim et al. proposed the presenceof quantum coherence in K + ion-channels, in their backboneamide groups that can play a role in mediating ion-conductionand ion-selectivity in the selectivity filter. In summary, weanalysed quantum excitation energy transfer (EET) in a linearchain composed of five sites, as a toy model for one strand ofselectivity filter backbone in ion channels. The inter-site sepa-rations is adjusted to be proportional to the distances betweenpeptide units in the selectivity filter. EET is studied both inabsence and presence of dephasing noise. Comparison of theresult with those obtained in an ordered chain, indicates thatdisorder in such systems has always a destructive role in EET,because of the Anderson localization effect. When dephasingis introduced, it is found that noise has destructive effect inEET in ordered chains. Nevertheless, for disordered chainsit is shown that the noise is able to significantly increase theefficiency of energy transfer across the chain either in amount and speed. The main message of this work is the significanceof dephasing in the efficiency of quantum energy transfer indisordered chains. The living systems are mostly disorderedand in some cases are very efficient, so it is possible in suchsystems that noise has a key role for efficient energy transferthrough their structures, e.g. selectivity filter backbone. B. Giese, J. Amaudrut, A.-K. Kohler, M. Spormann, and S. Wes-sely, Nature ,318 (2001). T. Brixner, J. Stenger, H. M. Vaswani, M. Cho, R. E. Blankenship,and G. R. Fleming, Nature , 625 (2005). G. S. Engel, T. R. Calhoun, E. L. Read, T.-K. Ahn, T. Mancal,Y.-C. Cheng, R. E. Blankenship,and G. R. Fleming, Nature ,782 (2007). H. Lee, Y.-C. Cheng, and G. R. Fleming, Science , 1462(2007). A. Ishizaki and G. Fleming, Proc. Nat. Acad. Sci. ,17255(2009). R. E. Fenna and B. W. Matthews, Nature , 573 (1975);J.Adolphs and T. Renger, Biophys. J. , 2778 (2006). G. S. Engel, T. R. Calhoun, E. L. Read, T. -K. Ahn, T. Mancal,Y. -C. Cheng, R. E. Blankenship, and G. R. Fleming, Nature ,782 (2007); G. Panitchayangkoon, D. Hayes, K. A.Fransted, J. R.Caram, E. Harel, J. Wen, R. E. Blankenship, G.S. Engel, Proc.Natl. Acad. Sci. USA ,12766 (2010). K. B. Plenio and S.F. Huelga , New J. Phys. , 113019 (2008) Ali Asadian, Markus Tiersch, Gian Giacomo Guerreschi, Jian-ming Cai, Sandu Popescu and Hans J Briegel, New J. Phys. ,075019 (2010) T. Allen, S. Kuyucak, S. Chung, Biophysical Journal , 2502(1999). J. Johansson, P. Nation, and F. Nori, Comp. Phys.Comm. ,1234 (2013). P.W. Anderson, Phys. Rev. , 1492 (1958); E. Abrahams, P.W.Anderson, D. C. Licciardello, and T.V. Ramakrishnan, Phys. Rev.Lett. , 673 (1979). Vaziri A, Plenio MB, New J. Phys., , 085001(2011). Ganim Z, Tokmakoff A,Vaziri A, New J. Phys., , 113030(2011). B. Roux and K. Schulten, Structure , 1343 (2004). J. Summhammer, V. Salari, G. Bernroider, J. Integ. Neurosci., (2), 123-135, (2012). F. Caruso, Universally optimal noisy quantum walks on complexnetworks, New J. Phys. , 055015, (2014). Ingarden, Roman S.; Kossakowski, A.; Ohya, M. ”