Environment Assisted Quantum Transport in Assisted Quantum Transport in Organic Molecules Gabor Vattay, a and Istvan Csabai, a One of the new discoveries in quantum biology is the role of Environment Assisted Quantum Transport (ENAQT) in excitonic ...

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Environment Assisted Quantum Transport in Organic MoleculesGabor Vattay,a and Istvan Csabai,aOne of the new discoveries in quantum biology is the role of Environment Assisted Quantum Transport (ENAQT) in excitonictransport processes. In disordered quantum systems transport is most efficient when the environment just destroys quantuminterferences responsible for localization, but the coupling does not drive the system to fully classical thermal diffusion yet. Thispoised realm between the pure quantum and the semi-classical domains has not been considered in other biological transportprocesses, such as charge transport through organic molecules. Binding in receptor-ligand complexes is assumed to be static aselectrons are assumed to be not able to cross the ligand molecule. We show that ENAQT makes cross ligand transport possibleand efficient between certain atoms opening the way for the reorganization of the charge distribution on the receptor when theligand molecule docks. This new effect can potentially change our understanding how receptors work. We demonstrate roomtemperature ENAQT on the caffeine molecule.1 IntroductionThere is overwhelming evidence that quantum coherenceplays an essential role in exciton transport in photosynthe-sis14. One of the fundamental quantum effects there is theEnvironmentally Assisted Quantum Transport57 (ENAQT).It applies to partially coherent quantum transport in disorderedsystems. At low temperatures transport is dominated by quan-tum walk of excitons over the excitable sites forming a net-work. While a classical walker diffuses away from its initialposition liket in time via taking random turns, a quantumwalker takes a quantum superposition of amplitudes of alter-native paths. In a strongly disordered system the interferenceis destructive and the walker becomes stuck or localized8.At medium temperatures coupling to the environment partiallydestroys quantum interference and the walker is free to moveand diffuse. Then at very high temperatures decoherence be-comes very distractive and the exciton gets frozen due to theZeno effect6. As a result, transport is most efficient at mediumtemperatures or at medium level of decoherence and much lessefficient at low or high temperatures. Transport efficiency isthe highest and mean transport time is the shortest at mediumtemperatures. In photosynthetic systems parameters are suchthat this optimum is near room temperature (290K).The conditions for ENAQT look quite generic and one cansuspect that it can occur in a wide range quantum processesin biology. Yet, the presence of ENAQT has only been estab-lished in the exciton transport of light harvesting systems. Inthis paper we demonstrate that indeed ENAQT can playa major role in electron transport through organic moleculesat ambient temperatures. Accordingly in certain cases our un-derstanding of charge transport in biology needs revision.a Department of Physics of Complex Systems, Eotvos University, Pazmany P.s. 1/A, H-1117 Budapest, Hungary. Fax: +3613722866; Tel: +36308502614;E-mail: vattay@elte.hu 0.001 0.01 0.1 1 10 100-40 -30 -20 -10 0 10 20 30 40 50 60energy spacing (eV)energy (eV)Caffeine level spacings25meV1.4 eVFig. 1 Caffeine electronic energy level spacings En+1En as afunction of the energy En are indicated by red circles and connectedwith green line on a semi-logarithmic plot. Horizontal linesrepresent the mean level spacing at = 1.4eV and the roomtemperature at kBT = 0.025eV . Energies are in the35.3907eV +57.7272eV range, the HOMO is atEHOMO =11.907eV and the LUMO is at ELUMO =8.6776eV .Calculation of N = 66 electronic levels has been carried out with theExtended Huckel method.Charge transport in biological molecules has been studiedextensively with a wide range of quantum chemical meth-ods9. These methods cover fully quantum, mixed quantum-classical, semi-classical, and fully classical approaches. How-ever, none of these cover the parameter range of the valid-ity of ENAQT. In general, electronic levels are broadened dueto the coupling to the environment. The width of the levels is proportional to the level of decoherence, which is thenultimately determined by the temperature of the environment kBT . Quantum description is relevant, when the broad-ening is much smaller than the spacing between the consec-utive energy levels En+1En and the mean level spac-16 | 1arXiv:1503.00178v1 [cond-mat.mes-hall] 28 Feb 2015ing = En+1En is much greater than the temperature kBT . The semiclassical and classical or mixed descrip-tion is relevant in the opposite limit, when kBT and manyenergy levels are involved in the process. The most efficientENAQT sets in when the environmental temperature is com-parable with the main level spacing kBT or at least thelevel broadening is comparable10 with the spacings of somelevels En+1 En kBT . In small and medium sized(less than 500 Da) organic molecules the mean level spac-ing of electronic levels is on the electron volt scale, wich isabout two orders of magnitude larger than the room temper-ature (about 25 milli-electron volts). So, at first sight wewould expect that EQNAT is relevant only at temperaturesT = 1eV/kB 12000K, which is twice of the temperature ofthe surface of Sun. However, there is a second possibility:In Fig. 1 we show the nearest neighbor spacings of the en-ergy levels of caffeine. The mean level spacing is = 1.4eV ,which is much larger than the energy scale of the decoherencekBT = 0.025eV . However, there are several level spacings inthe spectrum, which are close to 0.025eV indicating that whileglobal ENAQT extending for the entire molecule is not possi-ble, there might be local pockets on the molecule, where itplays an important role. Next we continue with showing justthat.2 Transport in ligand-receptor complexesOne of the most significant building blocks of biochemicalprocesses in the cell is the docking of signaling moleculesin receptors. A typical situation is shown in Fig. 2, whereadenosine is docked in the A2A adenosine receptor. Besideshydrophobic interactions binding the ligand molecule to thereceptor electrostatically, there are also hydrogen bonds con-necting the two systems. The hydrogen bridge makes possi-ble the exchange of electrons between the two systems. Thereceptor consists of amino acids and these can have very dif-ferent charging situations. Arginine, histidine and lysine havepositive side chains while glutamic acid and asparatic acid hasnegative side chains. The rest of the amino acids is neutralor hydrophobic. An electron from a negatively charged partof the receptor protein can jump via the hydrogen bridge tothe neighboring atom of the ligand molecule. If the ligandmolecule would be a good conductor the electron could walktrough the molecule and reach a positively charged part ofthe receptor trough and other hydrogen bridge. The ligandmolecule would act as a molecular wire, a lightning rod. Asa result, the charge distribution on the receptor protein wouldchange suddenly upon the docking of the ligand molecule andcould spark sudden motion of the parts of the protein, dueto the change of the equilibrium of the electrostatic forces.Our understanding before ENAQT ruled out such transportof electrons through the ligand molecule. Quantum calcula-Fig. 2 Schematic picture of Adenosine in complex with its A2Areceptor. Legend: black dashed lines - hydrogen bonds; green solidlines - hydrophobic interactions; green dashed lines - Pi-Pi, Pi-cationinteractions. This figure has been downloaded from RCSB ProteinData Bank and has been created by the Pose View software1114.tions show that, unless tunneling plays a role in some specialcases, ligand molecules act as insulators. Next, we show thatENAQT changes this picture and provides a mechanism forthe easy and effective transport of electrons between certainatoms in the ligand molecule.3 Transport modelWe can develop a model for the description of the electrontransfer process of the previous section. For the description ofthe molecule only the n electrons in the N atomic orbitals areconsidered. The molecular orbitals j are linear combinationsof the atomic orbitals j =Ni=1C jrr and j,r = 1, ...,N, (1)where r are the valence atomic orbitals 2S,2Px,2Py and 2Pzfor carbon atoms and hetero atoms and 1S for hydrogen atoms.The molecular orbitals can be determined from the overlapmatrix Srs = r | s and Hamiltonian Hrs = r | He f f | smatrices via the generalized eigenequationHC = ESC, (2)where H, S and C are square matrices containing the elementsof Hrs, Srs and C jr respectively. The raw molecular orbitalscan be orthonormalized via the Lowdin transformation15. The2 | 16coefficients in the Lowdin basis can be introduced via C =S1/2C and the transformed Hamiltonian H = S1/2HS1/2satisfies the eigenequationHC = EC, (3)where the eigenenergies E remain unchanged. Next, we willdrop the prime signs and use this basis as default.We assume that the electron is coming trough the H-bridgeor via other mechanisms and initially enters one of the atomicorbits of the molecule. Initially the n electrons are placed inpairs on the lowest molecular orbits with opposite spins andoccupy about the half of the orbitals. The incoming elec-tron occupies one of the atomic orbitals. In a pure quan-tum description the initial wave function is a n + 1 dimen-sional Slater determinant of the n molecular orbitals and thesingle initial electron on the atomic orbital with wave func-tion (t = 0). This wave function is localized on the atomicorbital and it is proportional with the initial atomic orbital(0) = I , where is a normalization constant. The ini-tial atomic orbital can be expanded in terms of all molecularorbitals I = j[C1]I j j, where C1 is the inverse of thecoefficient matrix. The function I is not orthogonal to then occupied molecular orbitals and only the expansion coef-ficients corresponding to unoccupied molecular orbitals willcontribute to the norm of the n+1 electron wave function. Wecan split into two orthogonal parts = o +u,where o is spanned over the occupied and u over the unoc-cupied orbitals of the molecule wit n electrons. The normal-ization condition of the Slater determinant then involves theunoccupied part only | u |2= 1, which yields the followingnormalization condition2 junoccupied[C1]2I j = 1. (4)Consequently the norm| |2= 2 = 1 junoccupied [C1]2I jis larger than unity. This reflects the fact that in this descrip-tion the incoming electron creates both electronic states in theunoccupied sector and hole states in the occupied sector.The time evolution in this model is very simple. The molec-ular orbitals are eigenfunctions of the Hamiltonian and remainunchanged except the stationary phase factors eiE jt/h. Thetime evolution of the incoming electron is governed by theSchrodinger equationih t = H. (5)The time evolution is then a Slater determinant again, com-posed of eiE jt/h j for the occupied orbitals and (t).The reduced density matrix of this wave function containsthe occupied states and the unoccupied part u. In atomicorbital basis =|u u |+j| j j | .This is a very useful expression from the point of view of gen-eralization. The first term represents the evolution of the in-coming electron and the second term represents the rest of theelectrons frozen in the Fermi sea. The incoming electron even-tually leaves the system. Just like in case of light harvestingsystems this can be modeled by adding an imaginary sink termto the HamiltonianH = H i | F F |,where is the rate of the leaking out of the electron via aH-bond coupled to one of the final atomic orbital F . Oncethe excess electron is leaked out the reduced density matrixreduces to the frozen Fermi sea contribution = j| j j | .In our non-interacting electron approximation the leakingout of the electron is fully described by the non-unitarySchrodinger equationih t = H, (6)involving now the anti-Hermitian part ih | F F | de-scribing leaking. The density matrix corresponding to thewave function can be groupped into four sectors| |=|u u |+ |o u |+ |u o |+ |o o |,and only the (u,u) sector has physical meaning. The evolutionof density matrix is described by the von Neumann equationih t=[H,]= [H,] i{H1,} , (7)where H1 = h | F F | is the leaking term and {,} denotesthe anti-commutator.We can now take into account the effect of coupling ofthe electrons to the environment. This can come from manyfactors including the phonon vibrations of the molecule andalso very crude interactions with water molecules and othersources of fluctuating electrostatic forces in the complicatedbiological environment of a cell. Since all the sources ofphase breaking and dissipation cannot be accounted for wecan treat the system statistically and use the phenomenological16 | 3approach and add the Lindblad operator to the von Neumannequationih t=[H,]+L (). (8)The most general Lindblad operator16 in our case can be writ-ten in terms of the projection operators of atomic orbitalsAr =| r r | asL = rsLrs (2ArAsAsArAsAr) ,where Lrs is a positive definite covariance matrix of the noiseat different atomic sites. We can get Lrs from detailed models.The crudest approximation is when we assume a completelyuncorrelated external noise and set Lrs = h rs, where is thestrength of the decoherence. Its detailed form is model de-pendent, but it is in the order of thermal energy and without adetailed knowledge of the system can be set to = kBT 17.Now the steps of the full calculations can be summarizedlike this. First we identify the initial I and final F atomicorbitals, where the electron enters and exits the system. Weinitialize the density matrix with the initial wave function =2 | I I |. The trace of this density matrix is Tr = 2and contains both occupied and unoccupied states. We evolvethis density matrix according to (8). The physically relevantreduced density matrix is given by (u,u) the projection of thedensity matrix for the unoccupied sector and we use this forthe calculation of physical quantities.4 Transit time between atomsThe next step is to calculate physical quantities characterizingthe electron transport between various atoms. In our modelwe have to characterize the event of a single electron passingtrough the system. Once the electron enters the system it canleave the system only trough the exit. Unless some symmetryconsideration forbids it the electron will leave the system withprobability 1, so we have to find another characteristics. Thenext quantity is the average time an electron needs to getfrom the entrance to the exit. This depends on two factors inour model: the details of the molecule and the value of . Itis hard to set the value of without detailed models or mea-surements of the system, therefore we have to find its valuefrom other considerations. Since the typical energy range inthe system is the mean level spacing we can expect that therate of leaking is in the order of the time /h. Once thescale of the escape rate is set we determine the average timeneeded from an initial atom to reach the destination FI andalso the time an electron would need to leave the system if itwould be placed immediately to the final site FF . The dif-ference FI FF is a better characteristics of the transport asthe pure pure pumping time is transformed out and mainly theinter-atomic travel time enters.The average transit time can be calculated from the evolu-tion of the reduced density matrix. The outflowing probabilitycurrent at the exit site F at time t is given by dP(t)= JF(t)dt =2dt F | (t) | F and the average transit time is the aver-age = 2 h 0t F | (t) | Fdt. (9)This integral can be calculated analytically using the solutionof the evolution equation. Eqation (8) can be cast into the formrs t= ih pqRrspqpq. (10)We can re-index this equation and introduce a new index re-placing the pairs of indexes J = (r,s) and J = (p,q). With thenew indexing (10) readsJ t= ih JRJJpq. (11)The solution of this is(t) = exp( iRth)(0).Since the initial density matrix has only one nonzero elementJI = (I, I) the integral in (9) yields the matrix element =2 h2[R2]JF ,JI, (12)where JF = (F,F). Next, we show the results of the calcula-tion for a molecule with biological relevance.5 CaffeineIn order to show ENAQT in organic molecules we should picka biologically relevant example which is also computationallyfeasible. Our choice is caffeine as it is a sufficiently smallmolecule with 24 atoms (see in Table 1). Using the ExtendedHuckel method18 implemented in the YAeHMOP19 softwarepackage. There are N = 66 valence atomic orbitals. The com-putational task involves the repeated calculation of the inverseof R, which is an (N N) (N N) = N2N2 matrix, or43564356 in case of caffeine. The rapid growth of the com-putation time with N2 is the strongest numerical limitation inthe problem. In the calculation the escape rate = h/1eV hasbeen used. There are 2145 possible pairs of initial and finalatomic orbitals. In Figure 3 we collected the most interestingresults. These are the transit times of electrons ending on the2S orbital of oxygen no. 6 in Table 1. The blue curve showsthe transit time between this orbital and itself. This is the mini-mum time FF needed for an electron to leak out from this site.This value is constant for low dephasing and starts increasing4 | 160.0001 0.01 1 100 10000Dephasing (eV) 300 K = 0.025 eV1100100001e+06Charge transfer time (arbitrary units)Fig. 3 Transit times between atomic orbitals in the caffeinemolecule at escape rate = h/1eV . Dephasing rate in units of eVis shown on the horizontal axis. Curves show transition timesbetween the N = 66 atomic orbitals and the 2S orbital of oxygenatom no. 6 in Table 1. The blue curve corresponds to the transit timebetween 2S orbital of oxygen no. 6 and itself. The green curvesshow the transit time between the 2S orbital of oxygen no. 6 and theorbits 2Px,2Py and 2Pz of oxygen no. 6. The red curve shows thetransition time between the 2S orbital of oxygen no. 11 and 2S ofoxygen no. 6. The black curves show the rest of the transit timecurves.in the classical limit of large dephasing. The green curvesshow the leaking time from the same atom but the electron isinitially placed on the three 2P orbitals of the atom. In thiscase we can observe a shallow minimum in the = 110eVrange. The black curves represent the transitions between allthe atomic orbitals and the 2S orbital of oxygen no. 6, except2S of oxygen no. 11, which is shown in red. The black and thegreen curves show the sign of ENAQT and are large both forsmall and large with a minimum in the middle. The mini-mum corresponds again to about 110eV in accordance withthe expectation based on the mean level spacing 1.4eV , whichis 2-3 orders of magnitude larger than room temperature. Thered curve also has a minimum in this range. However, a sec-ond shallow minimum is also developed around 0.01eV and atroom temperature 0.025eV the transition time is still close tothis minimum. This second optimum of ENAQT signals fasttransport between the 2S orbits of the two oxygen atoms.The transition time between the two oxygen atoms is just10 times as large as the minimum leaking time set by thescale of . This should be contrasted with the quantum limit,where the transition times are 1000 to 100000 times largerthan the minimum leaking time. A pure quantum calculationwould find the caffeine molecule strongly insulating betweenthe atoms and electron transport would be practically impos-sible.Table 1 Index and position of atoms in the caffeine moleculeIndex Atom x y z1 N 1.047000 -0.000000 -1.3120002 C -0.208000 -0.000000 -1.7900003 C 2.176000 0.000000 -2.2460004 C 1.285000 -0.001000 0.0160005 N -1.276000 -0.000000 -0.9710006 O -0.384000 0.000000 -2.9930007 C -2.629000 -0.000000 -1.5330008 C -1.098000 -0.000000 0.4020009 C 0.193000 0.005000 0.91100010 N -1.934000 -0.000000 1.44400011 O 2.428000 -0.000000 0.43700012 N 0.068000 -0.000000 2.28600013 C -1.251000 -0.000000 2.56000014 C 1.161000 -0.000000 3.26100015 H 1.800000 0.001000 -3.26900016 H 2.783000 0.890000 -2.08200017 H 2.783000 -0.889000 -2.08300018 H -2.570000 -0.000000 -2.62200019 H -3.162000 -0.890000 -1.19800020 H -3.162000 0.889000 -1.19800021 H -1.679000 0.000000 3.55200022 H 1.432000 -1.028000 3.50300023 H 2.024000 0.513000 2.83900024 H 0.839000 0.513000 4.1670006 ConclusionsWe investigated the possibility of ENAQT in organicmolecules at room temperature. We worked out a frame-work which makes it possible to treat this problem in first ap-proximation. We showed that global ENAQT is not presentin these molecules in general as the optimal temperature isseveral orders of magnitude higher than the room tempera-ture. However, room temperature ENAQT can be observedin the electron transport between certain atoms in the caffeinemolecule. This indicates that in ligand-receptor systems theligand molecule can act as a molecular wire connecting differ-ently charged areas of the receptor protein and can contributeto their response for the docking. This can have a signifi-cant importance in understanding signaling and drug actionin cells.The authors thank D. Salahub and S. Kauffman for help,encouragement and numerous discussions over the last threeyears.16 | 5References1 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, 2007, 446, 782786.2 E. Collini, C. Y. Wong, K. E. Wilk, P. M. 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