Modeling light-driven proton pumps in artiﬁcial photosyntheticreaction centers
Pulak Kumar Ghosh,1 Anatoly Yu. Smirnov,1,2 and Franco Nori1,2,a�
1Advanced Science Institute, The Institute of Physical and Chemical Research (RIKEN), Wako-shi,Saitama 351-0198, Japan2Department of Physics, Center for Theoretical Physics, University of Michigan, Ann Arbor,Michigan 48109-1040, USA
�Received 2 April 2009; accepted 12 June 2009; published online 17 July 2009�
We study a model of a light-induced proton pump in artiﬁcial reaction centers. The model containsa molecular triad with four electron states �i.e., one donor state, two photosensitive group states, andone acceptor state� as well as a molecular shuttle having one electron and one proton-binding sites.The shuttle diffuses between the sides of the membrane and translocates protons energeticallyuphill: from the negative side to the positive side of the membrane, harnessing for this purpose theenergy of the electron-charge separation produced by light. Using the methods of quantum transporttheory we calculate the range of light intensity and transmembrane potentials that maximize both thelight-induced proton current and the energy transduction efﬁciency. We also study the effect oftemperature on proton pumping. The light-induced proton pump in our model gives a quantum yieldof proton translocation of about 55%. Thus, our results explain previous experiments on theseartiﬁcial photosynthetic reaction centers. © 2009 American Institute of Physics.�DOI: 10.1063/1.3170939�
I. INTRODUCTION
It would be desirable to create an artiﬁcial system thatexploits the basic principles of natural photosynthesis in or-der to produce energy in an usable form.1–7 Indeed, naturalphotosynthetic structures efﬁciently convert the energy oflight into chemical form.7,8
The overall energy transduction process in plant photo-synthesis occurs through a number of strongly coupled suc-cessive stages �see, e.g., Refs. 1, 2, and 8�. In the ﬁrst step,the light of the appropriate wavelength is absorbed by a lightharvesting complex. The second step involves the conversionof electronic excitation energy to redox potential in the formof the long-lived transmembrane charge separation via mul-tistep electron transfer processes. The ﬁrst two steps involvethree constituents: �a� light-absorbing pigments, �b� an elec-tron acceptor, and �c� an electron donor. In the third step, theenergy stored in the electron subsystem is used for energeti-cally uphill proton pumping, which generates the proton mo-tive force across the membrane.The study of natural photosynthesis has inspired re-searchers to perform the photoinduced energy transductionprocesses in the laboratory.1–7,9–12 A convenient approach tophotosynthesis in artiﬁcial reaction centers is to use syntheticpigments, electron acceptors, and electron donors that arevery similar in molecular structure to natural pigments �e.g.,chlorophylls, carotenoids, and quinones�. In this direction,the experimental model proposed in Refs. 9 and 10 providesa paradigm for the conversion of light energy to a protonpotential gradient. These seminal works9,10 have motivatedresearch in the design and synthesis of new artiﬁcial photo-
synthetic systems13–15 �i.e., light-harvesting antennas and re-action centers� and triggered considerable experimental16–20
and theoretical21–24 activities to investigate more sophisti-cated and more efﬁcient mechanisms for the conversion oflight energy.The transformation of light energy into the electrochemi-cal gradient of protons across the membrane can be quanti-tatively characterized by the quantum yield �or quantum ef-ﬁciency� � of proton translocation. This parameter is deﬁnedas the total number of translocated protons divided by thenumber of photons absorbed by the triad.9 A quantum yieldof the order of 0.4% has been measured in Ref. 9. A muchhigher quantum efﬁciency ��7% for the conversion ofphotons into adenosine triphosphate �ATP� molecules wasfound in Ref. 10. As argued in Ref. 10, the actual quantumyield of ATP formation could be of the order of 15% if wetake into account the real rate of light absorbance, which is�50%. Near four protons are necessary for the synthesis of asingle ATP molecule. This means that the real quantum yield� of proton translocation measured in Ref. 10 can be about60%. The total thermodynamic �or power-conversion� efﬁ-ciency � of the light-to-ATP conversion process is estimatedin Ref. 10 as ��4%.In the present paper, using methods from quantum trans-port theory,25–28 we analyze the photoinduced electron andproton transfer in a molecular triad inserted into a liposomalmembrane, which contains a single molecular shuttle. Wecalculate the photon-to-proton quantum yield ��55% �andthe thermodynamic efﬁciency ��6.3%� for the resonanttunneling conditions when the reorganization energy � of theelectron transitions matches the detuning � between the elec-tron energy levels: ���.We note that due to a small optimal value of the reorga-a�Electronic mail: fnori@riken.jp.
THE JOURNAL OF CHEMICAL PHYSICS 131, 035102 �2009�
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nization energy ���400 meV� the charged recombinationprocess in the triad is described by the inverted region of theMarcus formula.29–31 This further enhances the performanceof the system. Our results explain the experiments made inRef. 10 using artiﬁcial photosynthetic centers. The obtainedpower-conversion efﬁciency corresponds to the highest value��6.5% achieved recently with polymer solar cells.32 It isexpected that the proton current and the efﬁciency shouldincrease with increasing the number of the shuttles in themembrane.This article is organized as follows. In Sec. II �see alsothe Appendix� we introduce the basis set for the system andwrite the Hamiltonian of the problem. In Sec. III, we presentthe master equation for the density matrix coupled to theLangevin equation describing the diffusive motion of theshuttle in the lipid bilayer. In Sec. IV, we numerically solvethese equations and analyze the light-induced proton pump-ing process. In Sec. V we summarize our results.
II. MODEL
We use a slightly modiﬁed version of the well-acceptedmodel already presented, e.g., in Refs. 9 and 10. In thismodel the reaction center is a molecular triad containing anelectron donor and an electron acceptor both linked to a pho-tosensitive porphyrin group �shown in Fig. 1�. The triad mol-ecule �D-BC-A� is inside the bilayer of a liposome. The lipidbilayer also contains freely diffusing 2,5 diphenylbenzo-quinones acting as proton shuttles. The molecular triad ab-sorbing a photon establishes a negative charge near the outersurface and a positive charge near the inner surface of theliposome by generating charge separated species D+-BC-A−.The freely diffusing quinone shuttle translocates an electron-proton pair across the membrane and neutralizes the molecu-lar triads.In Fig. 2 we schematically illustrate the process of light-induced proton pumping in liposomes by artiﬁcial photosyn-thetic reaction centers.9,10 The transmembrane proton pump-ing requires a symmetric arrangement of the molecular triad�length of �8 nm� inside the bilayer and with a speciﬁcdirection: with the acceptor �A� site toward the outer mem-
brane of the liposome �the negative �N� side of the mem-brane� and with the donor �D� toward the inside of the lipo-some �the positive �P� side of the membrane�.9,10
The energy diagrams of the electron and proton sites areshown in Figs. 3 and 4. There are two electrons in the sys-tem, one of which is initially on the D site and another elec-tron is on the lower energy level B. The quinone molecularshuttle has one electron state S �denoted by S instead of Se�and one proton state Q �denoted here by Q instead of Sp�.Thus, S denotes the shuttle electron state and Q denotes theshuttle proton state.The overall process leading to the proton translocationfrom the N-reservoir with a lower proton potential �N to theP-reservoir with a higher electrochemical potential �P canbe considered as a sequence of eight stages �most of whichare shown in Fig. 2�.
• Step I: The photosensitive moiety of the molecular triadabsorbs light and an electron goes from the ground stateB to the excited state C �see Fig. 3�b��.
• Step II: The unstable excited state C transfers the elec-tron to the acceptor A, producing an unstable charge-separated intermediate species D-BC+-A−.
• Step III: The unstable intermediate charge-separated
FIG. 1. The top ﬁgure presents the triad �donor D, photosensitive part B,C,and acceptor A� and the shuttle S �Refs. 9 and 10�. These are enclosed bycolor circles, which are schematically shown in the bottom ﬁgure. The tetra-arylporphyrin group acts as a photosensitive moiety �B,C� �inside the greencircle in the top structure�. This is connected to both a naphthoquinonemoiety fused to a norbornene system with a carboxylic acid group �whichacts as an electron acceptor �A�� and to a carotenoid polyene �which acts asan electron donor �D��. 2,5-diphenylbenzoquinone is the proton shuttle �S�denoted by a pink hollow circle in the structure and by a solid pink circle inthe cartoon.
FIG. 2. Schematic diagram of the light-induced proton pump across the lipidbilayer in a liposomic membrane. A molecular triad D-BC-C is symmetri-cally inserted in the lipid bilayer. The different stages in the proton pumpingprocess are here denoted by �a�, �b�, �c�, �d�, �e�, and �f�. The two bluishvertical rectangles on both sides schematically represent two proton reser-voirs with electrochemical potentials �P and �N. These two proton reser-voirs correspond to the aqueous phases inside and outside of the liposome,respectively. The shuttle molecule S is shown as a pink-colored oval and theprotonated neutral shuttle is shown as a yellow oval. This shuttle freelydiffuses in �d� �the black scribbled curves represent the thermal stochasticmotion of the shuttle� across the membrane to transport a proton from thelower proton potential �N to the higher proton potential �P side of themembrane, where ��P−�N� denotes the total potential difference betweenthe two reservoirs.
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species is rapidly rearranged to a relatively stablecharge-separated form �D+-BC-A−� by the thermal elec-tron transfer from the state D to the state B+ having alower energy than the state D �Fig. 3�b��.
• Step IV: The shuttle in the position near the N-side ofthe membrane accepts an electron from A− and becomesnegatively charged.
• Step V: The shuttle molecule receives a proton from theN-reservoir and becomes neutralized �Fig. 3�c� rightpanel�.
• Step VI: The neutral shuttle slowly diffuses through thelipid bilayer and carries the electron and the proton tothe P-side of the membrane and to the D-site �stage �d�in Fig. 2�.
• Step VII: The shuttle gives away the electron to thepositively charged site D+ �stage �e� in Fig. 2 and Fig.4�e��.
• Step VIII: The shuttle is deprotonated by donating theproton to the P-reservoir �Fig. 4�f��.
This sequence of eight steps describes the photoinducedelectron transfer that generates the intramembrane redox po-tential, which in turn drives the energetically uphill vectorial
translocation of protons by the shuttle. Electrons in the statei �=D,B,C,A,S� and protons in the state Q are characterizedby the corresponding Fermi operators ai+,ai and bQ+ ,bQ withthe electron population operator ni and the proton populationnQ. We assume that each electron or proton state can beoccupied by a single electron or a single proton. Spin degreesof freedom are neglected. The proton site on the shuttle de-noted by Q can be populated from the N-reservoir providedthe shuttle is within the transition length LQ from the N-sideof the membrane. The protonated shuttle located within thetransition �or tunneling� range from the P-side of the mem-brane can donate its proton to the P-reservoir. Protons in thereservoirs are described by the Fermi operators dk�+ ,dk,�,where �=N,P, and k is an additional parameter which hasthemeaningofawavevectorincondensedmatterphysics.25–28 The number of protons in the reservoirs is de-termined by the operator �kNk� with Nk�=dk�+ dk�.
A. Hamiltonian
The Hamiltonian of the electron-proton system
H = H0 + Hdir + Htr + HB�1�
has a term H0 related to the energies Ei of the electron eigen-states �i=D,B,C,A,S� and to the energy �Q of a proton onthe shuttle
FIG. 3. Energy diagram depicting the energy levels of states involved in anartiﬁcial photosynthetic reaction center before the diffusion of the shuttle tothe P-reservoir. �a�–�c� correspond to the stages �a�, �b�, and �c� in Fig. 2.The left and right panels represent electron and proton energy levels, respec-tively. The abbreviations D, B, C, A, and S are the same as used in the textand in Fig. 1. Also, xD and xA represent the spatial coordinates of sites D andA, respectively. The thick brown arrows denote the path the electrons followin this energy diagram, generating charge separation in �b� and shuttlecharging and protonation in �c�. Initially, light excites an electron from B toC and eventually to A, making it A−. Afterward, in �b�, the donor D loses anelectron, thus becoming D+, and that electron moves to BC. Later on, theshuttle S in �c� receives the electron from A.
FIG. 4. Energy levels involved in an artiﬁcial photosynthetic reaction center.This ﬁgure is similar to Fig. 3, but now the energy proﬁle corresponds to thestage after the shuttle diffuses to the P-reservoir. Here �d�–�f� correspond tothe stages �d�, �e�, and �f� in Fig. 2. The left and right panels representproton and electron energy levels, respectively. The thick brown arrowsdenote the path followed by the �e� electron and �f� proton. In �d�, an elec-tron on the shuttle S moves to the donor site D, neutralizing it in �e�. Thiselectron transition in the right panels increases the proton energy of theshuttle, as shown in the left panels ��d� and �e��. The proton ﬁnally leavesthe shuttle in the left panel of �f�.
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H0 = �iEini + �QnQ + uDB�1 − nD��1 − nB − nC�
− uDA�1 − nD�nA − uBA�1 − nB − nC�nA − uSQnSnQ.
We include here the electrostatic interaction between theelectron sites uDB,uDA,uBA and the Coulomb attraction uSQbetween the electron and proton sites on the shuttle. It isassumed that the empty donor state D �with nD=0� as well asthe empty photosensitive groups B and C �nB+nC=0� havepositive charges and uDB=uDC, uCA=uBA.The term
Hdir = − �DBaD† aB − �ACaA†aC − �DS�x�aD† aS − �AS�x�aA†aS
− F�t�aB†aC + H.c.�3�
describes the tunneling of electrons between the sites D-B,C-A, A-S, and D-S with the corresponding amplitudes �ii�.Notice that the tunneling elements �DS�x� and �AS�x� dependon the shuttle position x. The Hamiltonian Hdir is also re-sponsible for the electron transitions between the states Band C induced by the electromagnetic ﬁeld �light� F�t�=F0 exp�i�0t� with a frequency �0 and an amplitude F0.Proton transitions between the shuttle �site Q� and the N- andP-proton reservoirs are governed by the Hamiltonian
Htr = − �k�Tk��x�dk�† bQ − �k�Tk�� �x�bQ† dk��4�
with the position-dependent coefﬁcients Tk��x�. We havechosen the following form of Tk��x�:
TkN�x� = TkN��x − �xN − LQ��,
TkP�x� = TkP��xP + LQ − x�,
where ��x� is the Heaviside step function and the parameterLQ deﬁnes the proton loading range of the shuttle.
B. Interaction with the environment
To take into consideration the effect of a dissipative en-vironment we consider the well-known system-reservoirmodel,29,30,33 where the medium surrounding the active sitesis represented by a system of harmonic oscillators with theHamiltonian
HB = �j �pj2
2mj+ mj�j2
2 �xj + 1
2�ixjini�
2�,�5�
where xj,pj are the positions and momenta of the oscillatorswith effective masses mj and frequencies �j. The parametersxji determine the strengths of the coupling between the elec-tron subsystem and the environment. The system of indepen-dent oscillators is conveniently characterized by the spectralfunctions Jii���� deﬁned by
Jii���� = �j
mj�j3�xji − xji��2
2��� − �j�,�6�
so that the reorganization energy �ii� related to the i→i�transition has the form
�ii� =�0
� Jii���� = �j
mj�j2�xji − xji��2
With the unitary transformation Uˆ =�iUˆ i, where
Uˆ i = exp�i2�jpjxjini�,�8�
we can transform the Hamiltonian H to the form H�=U†HU becoming �after dropping the prime�
H = H0 − �ii��ii�e�i/2���i−�i��ai�† ai − F�t�e−�i/2���B−�C�aB†aC
− F��t�aC†aBe�i/2���B−�C� − �k�Tk��x�dk�† bQ
− �k�Tk�� �x�bQ† dk� + �j �pj2
2mj+ mj�j2xj2
2 �,�9�
where �=N,P and the tunneling coefﬁcients �ii�� =�i�i takenonzero values only for transitions between the sites D andB, A and C, A and S, as well as D and S. The stochastic phaseoperator �i is given by
�i = 1
��jpjxji.�10�
The result of this transformation follows from the fact thatfor an arbitrary function ��xj�, the operator Uˆ produces ashift in the oscillator positions
Uˆ †��xj�Uˆ = ��xj + 1
2�ixjini�.�11�
This transformation also results in the phase factors for theelectron amplitudes �see Eq. �9��.The basis sets composed of the electron-proton eigen-states and their corresponding energy eigenvalues are pre-sented in the Appendix. Thus, the reader is encouraged toread this short Appendix before proceeding further.
III. TIME EVOLUTION OF DENSITY MATRIX
A. Master equations
To describe the time evolution of the diagonal elementsof the density matrix ��m�, we write the Heisenberg equationfor the operators �m with the subsequent averaging over theenvironment ﬂuctuations and over the states of the protonreservoirs,
��˙m� = − �i��m,Hdir�−� − �i��m,Htr�−�.�12�
The protons in the reservoirs ��=N,P� are characterizedby the Fermi distributions,
F��Ek�� =�exp�Ek� − ��
T � + 1�
−1�13�
with the temperature T �kB=1�. The electrochemical poten-tials �N and �P correspond to the negative �N� and positive�P� proton reservoirs, respectively. The proton motive force���� across the membrane is given by
035102-4Ghosh, Smirnov, and NoriJ. Chem. Phys. 131, 035102 �2009�
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�� = �P − �N = V − 2.3RT
F��pH�,�14�
where R and F are the gas constant and Faraday constant,respectively, and V is the transmembrane voltage gradient.Hereafter we change �� by changing the pH of the solutionby �pH.The contribution of the transitions between the shuttleand the proton reservoirs to the time evolution of the densitymatrix is described by the second term in the right hand sideof Eq. �12�, which can be calculated with methods of quan-tum transport theory25,26
�i��m,Htr�−� = �n��nmtr �x���m� − �mntr �x���n���15�
with the relaxation matrix
�mntr �x� = �����x���bQ,mn�2�1 − F���nm��
+ �bQ,nm�2F���mn��.�16�
Here we introduce the frequency-independent coefﬁcients
���x� = 2��k�Tk��x��2��� − Ek��,�17�
which determine the transition rates between the shuttle stateQ and the sides of the membrane �N- and P-reservoirs�. No-tice that these coefﬁcients are functions of the shuttle posi-tion x.The transitions between the electron levels are describedby the Hamiltonian Hdir, which can be written as
Hdir = − �mnAmn�m,n − �mn�n,mAmn†�18�
with the functions
Amn = QDB�aB†aD�mn + QCA�aA†aC�mn + QSA�aA†aS�mn
+ QSD�aD† aS�mn + QCB�aB†aC�mn,�19�
which are deﬁned as superpositions of the heat-bath opera-tors
Qii� = �i�i exp��i/2���i − �i���
= �i�i exp��i/2��jpj�t��xji − xji����20�
forthepairsoftheelectronsites�ii��=�DB�,�CA�,�SA�,�SD�, whereas for the pair �CB� we have
QCB = F0 exp�i�0t�exp��i/2��jpj�t��xjC − xjB��.�21�
In the case of a high-enough temperature of the bath,30 thecumulant functions of the unperturbed operators Qii�
�0� are de-termined by the relations
�Qii��0��t�,Qii��0�†�t��� = ��i�i�2e−i�ii��t−t��e−�ii�T�t − t��2,
�Qii��0�†�t�,Qii��0��t�� = ��i�i�2ei�ii��t−t��e−�ii�T�t − t��2.
The contribution of the electron transitions to Eq. �12� isdetermined by the term
�− i��m,Hdir�−� = i�n�Amn�mn − Anm�nm� + H.c.�23�
Within the theory of open quantum systems developed inRef. 28, the correlation function �Amn�mn� is proportional tothe density matrix elements of the system ��m� with coefﬁ-cientsdeﬁnedbytheunperturbedcorrelators�Amn�0��t�,Amn�0�†�t��� of the bath operators
�Amn�t��mn�t�� = i� dt1��t − t1�ei�mn�t−t1�
���Amn�0��t�,Amn�0�†�t1����m�t��
− �Amn�0�†�t1�,Amn�0��t����n�t���,�24�
�Amn�0��t�,Amn�0�†�t1�� = �QCB�0��t�,QCB�0�†�t1����aB†aC�mn�2
+ �QDB�0��t�,QDB�0�†�t1����aB†aD�mn�2
+ �QCA�0��t�,QCA�0�†�t1����aA†aC�mn�2
+ �QSA�0��t�,QSA�0�†�t1����aA†aS�mn�2
+ �QSD�0��t�,QSD�0�†�t1����aD† aS�mn�2�25�
and the reverse expression can be obtained for the correlator�Amn�0�†�t1�,Amn�0��t��. The formula �24� is valid in the case ofweak tunneling and weak driving force F0. The effects ofquantum coherence are also neglected here.Finally, we derive the master equation for the densitymatrix of the system
��˙m� + �n�nm�x���m� = �n�mn�x���n��26�
with the total relaxation matrix
�mn�x� = �mntr �x� + ��DB�mn + ��CA�mn + ��SA�mn + ��SD�mn
+ ��CB�mn�27�
containing the contribution of proton transitions to and fromthe shuttle �mntr �x�, together with the Marcus rate ��CB�mndescribing the light-induced electron transfer between thesites B and C,
��BC�mn = �F0�2� �
�BCT��aB†aC�mn�2
�exp�− ��mn + �0 + �BC�2
4�BCT�
+ �F0�2� �
�BCT��aB†aC�nm�2
�exp�− ��mn − �0 + �BC�2
4�BCT�,�28�
as well as the rates related to the electron transfers betweenthe pairs of sites �ii��=�DB�,�CA�,�AS�, and �DS�,
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��ii��mn = ��i�i�2� �
�ii�T���ai�† ai�mn�2 + ��ai�† ai�nm�2�
�exp�−��mn + �ii��2
4�ii�T �.�29�
We note that the tunneling coefﬁcients �AS and �DS dependon the shuttle position x.
B. Equation of motion for the shuttle
We assume that the shuttle moves along the linear mo-lecular triad �Fig. 1� and this motion can be described by theoverdamped Langevin equation for the shuttle position x,
�dragdxdt = − dU�x�
dx+ ��t�.�30�
Here �drag is the drag coefﬁcient of the shuttle in the lipidmembrane and the thermal ﬂuctuation of the medium is mod-eled by a zero-mean delta-correlated Gaussian ﬂuctuationforce ��t�, ���t��=0,
���t���t��� = 2�dragT��t − t��,�31�
where T is the temperature of the medium �kB=1�. The dif-fusion of the shuttle is determined by the diffusion coefﬁ-cient Ds=T/�drag. The potential U�x� in Eq. �30� is respon-sible for the spatial conﬁnement of the hydrophobic shuttle�quinone� inside the lipid membrane.
IV. RESULTS AND DISCUSSIONS
To analyze the light-induced proton pumping processquantitatively, we use the standard Heun’s algorithm to nu-merically solve the 20 coupled master equations �26� alongwith the equation �30� for the shuttle. For initial conditionswe have assumed that at t=0, �1,1=1, and the other elementsof the density matrix are zero �this corresponds to one elec-tron on site D and another electron on site B with no elec-trons and no protons on the shuttle�. We also assume that att=0 the shuttle is located nearby the acceptor �A�: x�t=0�=xA�xN. Throughout our simulation we focus on the long-term asymptotic regime, where the effects due to the inﬂu-ence of transient processes have been smoothed out. Thetime-homogeneous statistical properties are obtained in thelong-time limit after the temporal and ensemble averagingare performed.The efﬁciency �quantum yield� of the proton pumpingdevice is deﬁned by the formula
� = number of protons pumped
number of photons absorbed.
The photon absorption rate �B→C is approximately equal tothe rate of light-induced transitions from the state B to thestate C. Thus we assume
�B→C,�32�
where Ip is the proton current �the number of protons Nptranslocated across the membrane per unit of time�.
A. Diffusive motion of the shuttle in the lipid bilayer
In Fig. 5 we present the diffusive motion �see Fig. 5�a��of the shuttle in the lipid bilayer together with the time de-pendencies of the electron and proton populations of theshuttle �Fig. 5�b�� complemented by the time evolution of thenumber of pumped protons �Fig. 5�c��. We assume that thereorganization energies for the thermal electron transfers arelow enough to provide a high performance of the system
� � �DB � �AC � �AS � �DS � 400 meV.
This value of � is quite common for porphyrin-quinone dy-ads having a lower limit �the internal reorganization energy�of the order of 0.3 eV.34 Even smaller reorganization energies���230 meV� have been measured for the porphyrin-
FIG. 5. �a� Stochastic motion of the shuttle with time. The horizontal blackdashed lines denote the borders of the membrane xN=40 Å, xP=−40 Å. Viathis diffusion the shuttle transports protons and electrons through the mem-brane. �b� Variation in the electron and proton population on the shuttle.Note that the proton density �red curve� and the electron density �blackcurve� mostly coincide in �b�. �c� Number of protons pumped vs time. Themain parameters used here are the light intensity I=0.138 mW cm−2, tem-perature T=298 K, and the chemical potentials �P=110 meV and �N=−110 meV. The light intensity I corresponds to the photosensitiveBC-group with a dipole moment ��e��1 nm, where e is the electroncharge.
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fullerene dyads.35 The initial stages of electron transfer inbacterial reaction centers36 are also characterized by a lowreorganization energy ��70–300 meV depending on theenvironment. This is due to the fact that the bacteriochloro-phyll molecules �and the molecules of porphyrin involved inour molecular triad� contain highly delocalized �-electronsystems. In Sec. IV B, we also analyze how sensitive theresults are to changes in the values of �.Electrochemical measurements1 show that the energy ofthe carotene�D�-porphyrin�BC�-quinone�A� molecular triadsweeps from the value of �1.9 eV �the ﬁrst excited state ofthe porphyrin, D-B1C-A� to the energy of �1.4 eV related tothe intermediate state D-BC+-A−, and, ﬁnally, to the energyof �1.1 eV of the charge-separated state D+-BC-A−. We as-sume here that the energy of the ﬁrst excited state of theporphyrin EC−EB is 1908 meV, which corresponds to a pho-ton wavelength of 650 nm as used in experiments.9,10 Wehave taken the energy gap between the sites C and A to beapproximately equal to the reorganization energy �EC−EA���=400 meV. This gap is about the energy difference be-tween the D-B1C-A and D-BC+-A− states.The energies of the electron sites S and A are compa-rable, �EA−ES��300 meV, due to a structural similarity ofthe quinone shuttle �S� and quinone moiety of the moleculartriad. The protonation of the shuttle leads to the lowering ofthe electron energy on site S due to the electron-proton Cou-lomb attraction,9 uSQ�360 meV. The other Coulomb inter-action terms are chosen as uDB=uBA=120 meV and uDA=60 meV. These values correspond to the electrostatic inter-action of two charges located at distances of 4 and 8 nm,respectively �in a medium with a dielectric constant of �3�.Furthermore, we assume that ED−EB=400 meV and �Q=200 meV. We have chosen �Q such that for the above men-tioned parameters, the device works well at the transmem-brane potential difference of �200 mV.We choose �P=110 meV, �N=−110 meV, the resonanttunneling rates �/�=15 ns−1, �/�=1.5 ns−1, and the reor-ganization energy for the light-induced electron transfer�BC�80 meV. The majority of parameters in our model arededuced from experimental data. The rates of electron trans-fer reactions are given by
�C→A � �D→B � 26 �s−1,
�A→S � �S→A � 20 �s−1.
Therefore, the loading and unloading time scales of theshuttle are about 0.05 �s. The shuttle has enough time to beloaded and unloaded with electrons and protons when it en-ters the loading/unloading domain with a size of about theelectron tunneling length Ltun�0.5 nm and the proton tran-sition length LQ�0.2 nm. Figures 5�a� and 5�b� show a timesynchronization between the spatial motion of the shuttle andthe time variations in the shuttle populations.It follows from Fig. 5�c� that in 1 ms the shuttle per-forms near 16 trips and translocates ten protons through themembrane,providedthatthelightintensityI=0.133 mW cm−2. We assume that the diffusion coefﬁcientDs is of the order of 2 nm2 �s−1 �Ref. 37� and the dipolemoment of the BC moiety is about �e��1 nm, where e is the
electron charge. The number of photons absorbed in 1 ms is�18. Thus, the approximate quantum yield � of the pump-ing process is �55%. For this parameters, the diffusive mo-tion of the shuttle is the slow and rate-limiting step of thepumping process.
B. Robustness of the model
To show a tolerance of the system to variations in pa-rameters we explore here the parameter space of our model.Keeping ﬁxed the energy difference between the sites B andC, we calculate and plot �see Fig. 6� the pumping efﬁciency� �photon-to-proton quantum yield� as a function of the re-organization energy � and the energy gap �,
� � �DB � �AC � �AS � �DS,
� = EC − EA = ES − ED,
between the energy levels EC and EA, and between the levelsES and ED. Figures 6�a�–6�c� correspond to the different val-ues of detuning between the acceptor energy level EA and theelectron energy level on the shuttle ES: EA−ES=100 meV�Fig. 6�a��; EA−ES=300 meV �Fig. 6�b��; and EA−ES=500 meV �Fig. 6�c��. These plots clearly demonstrate theexistence of quite wide areas in the plane �−�, where thepump performs with maximum efﬁciency. For the detuning
FIG. 6. Contour plots presenting the variations in the quantum efﬁciency �with the reorganization energy � and with the energy gap �, where �=EC−EA=ES−ED.TheparametersusedherearelightintensityI=0.138 mW cm−2, temperature T=298 K, and chemical potentials �P=110 meV and �N=−110 meV. The detunings take the following values:�a� EA−ES=100 meV, �b� EA−ES=300 meV, and �c� EA−ES=500 meV.
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EA−ES=100 meV�Fig.6�a��thepumpingefﬁciencyreaches its maximum at ��48% in the region of parameters�in meV�: 270���500 and 400���700. In this region,the energy gaps between the redox sites are close to thereorganization energy, which results in higher site-to-sitetunneling rates and, consequently, in a high pumpingefﬁciency.Thehigherpumpingefﬁciency��55%canbeachieved at the detuning EA−ES=300 meV �Fig. 6�b��. Inthis case the parameter � can be tuned in such a way that theenergy gaps between all relevant electron sites are equal tothe reorganization energy,
�EC − EA� � �EA − ES� � �ES − uSQ − ED�
� �ED − EB� � �.
We recall that a shuttle populated with a proton has the elec-tron energy ES−uSQ, which differs from the initial value ESby the charging energy uSQ�360 meV. Summing all theabove-mentioned detunings and taking into account the en-ergy difference EC−EB=1908 meV between the opticallyactive levels B and C, we estimate the optimum values of thereorganization energy � and the detuning �,
� � � � �EC − EB − uSQ�/4 = 387 meV.
The maximum of the efﬁciency in Fig. 6�b� is observed at����400 meV, which is very close to our estimations.For a larger energy gap EA−ES=500 meV �see Fig. 6�c��,the proton pumping efﬁciency � decreases and the region ofthe optimum parameters shrinks as compared to Fig. 6�b�.
C. Effects of the resonant tunneling rates
The ﬁne tuning of tunneling couplings between activeelectron sites is feasible in some nanostructures. This tuningcan be implemented by changing the site-to-site distance aswell as by varying the height of the potential barriers �see,e.g., Ref. 38�. Artiﬁcial photosynthetic systems such as themolecular triads also allow to engineer desirable tunnelingand electrostatic properties of the structures1 with the goal toachieve the highest possible efﬁciency. As in colloidalnanocrystals,38 this can be done by inserting additional mo-lecular bridges between the side centers D, A, and the pho-tosensitive part BC utilizing the exponential dependence ofelectron tunneling rates on the distance.5
In Fig. 7 we illustrate the variation in the proton pump-ing efﬁciency � as a function of the resonant tunneling rate��/�� for different values of the reorganization energy ��Figs. 7�a� and 7�b�� and the energy gap � �Figs. 7�c� and7�d��. The detuning EA−ES is ﬁxed to the value of 300 meVfor all plots in Fig. 7.In Fig. 7�a� we plot four curves ���� for the followingset of reorganization energies: �=100, 130, 200, 400 meVand for a detuning �=400 meV. In Fig. 7�b� the efﬁciencies���� are plotted for the reorganization energies: �=500,800, 1000, 1200 meV for the same detuning �. Similar de-pendencies ���� are depicted in Fig. 7�c� for �=100, 130,200, 400 meV and in Fig. 7�d� for �=500, 600, 700, 800meV. In both, Figs. 7�c� and 7�d�, the reorganization energy� is equal to 400 meV.
It follows from Fig. 7 that initially the proton pumpingefﬁciency rapidly increases with increasing � followed by itssaturation for higher values of the resonant tunneling rate.The saturation limit depends on the reorganization energy �as well as on the energy gap �. For the optimum values of �and �:����400 meV, the pumping efﬁciency is sufﬁ-ciently high, ��55%, even for moderate tunneling rates�/��5 ns−1.
D. Effects of Coulomb interactions
In Fig. 8 we plot the efﬁciency � versus the dielectricconstant � of the medium to explore the effects of the Cou-lomb couplings uDB, uBA, and uDA on the performance of theproton pump. The electrostatic interactions between the pho-tosensitive parts B and C and the donor uDB between the sitesB and C and the acceptor uBA and between the donor and the
FIG. 7. Proton pumping quantum efﬁciency � vs resonant tunneling rate �at different reorganization energies � shown in �a� and �b� and for differentdetunings � shown in �c� and �d�. Note that � here represents �/� since weset �=1. We use the following parameters: I=0.138 mW cm−2, T=298 K,�P=110 meV, �N=−110 meV, and the energy gap �EA−ES�=300 meV.Panels �a� and �b� are plotted at ﬁxed �=400 meV, whereas in �c� and �d�the reorganization energy is ﬁxed with �=400 meV.
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acceptor uDA are inversely proportional to the dielectric con-stant � and to the distance between the relevant sites. Forexample, we have
uDB =e2
4��0�rDB,
where rDB characterizes the spatial separation of the sites Dand B and �0 is the vacuum permittivity. The Coulomb inter-actions between the sites D and B and between the sites Band A �with rDB=rBA=4 nm� are decreased from 360 to 36meV when the dielectric constant � scans the range from 1 to10 �see Fig. 8�a��. We note that in our model rDA=8 nm, sothat uDA=uDB/2. Figure 8�b� shows the efﬁciencies ���� fordifferent values of the detuning �:�=100, 200, 400, 600meV for �=400 meV. Moreover, in Fig. 8�c� we plot the
efﬁciencies ���� for ﬁxed detuning �=400 meV and for �=100, 200, 400, 600 meV.It should be emphasized that near the optimum workingpoint �at ����400 meV� the pump operates with the highefﬁciency ��55%, which practically does not depend onthe dielectric properties of the medium.
E. Effect of light intensity
In Fig. 9 we plot the proton current as a function of thelight intensity for different values of the temperature. At zerolight intensity the proton current is zero. Initially, with in-creasing light intensity, the proton current also increases lin-early and then saturates around 0.2 mW cm−2. This satura-
FIG. 8. �a� Coulomb energies uDA and uDB vs the dielectric constant � of themedium. �b� Proton pumping efﬁciency � vs dielectric constant � for dif-ferent values of � and for �=400 meV. �c� The pumping efﬁciency � as afunction of the dielectric constant � for different reorganization energies �and at the ﬁxed detuning �=400 meV. The other parameters are the same asin Fig. 7: I=0.138 mW cm−2, T=298 K, �P=110 meV, �N=−110 meV,and EA−ES=300 meV.
FIG. 9. �a� Proton current vs light intensity I for different temperatures at�N=−110 meV and �P=110 meV. Notice that the proton current isroughly linear for small intensities of light but it saturates with higher lightintensity. In this saturation region, the proton current is larger with highertemperatures. �c� The standard deviation �p of the number Np of pumpedprotons as a function of the light intensity I for different temperatures. �b�The pumping quantum efﬁciency � decreases with light intensity for alltemperatures shown.
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tion is probably caused by the slow diffusion of the shuttleinside the lipid membrane. A similar intensity of saturation��0.1 mW/cm2� has been observed in experiments.10
In a warm environment, the shuttle moves faster andcarries more protons. To do this, the system should absorbmore photons so that at high temperatures a full saturationtakes place at higher light intensities. We note that the lowsaturationlimitobtainedaboveandmeasuredintheexperiment10 with the carotene-porphyrin-quinone triads isfarbelowtheaverageintensityofsolarlight,I�30 mW/cm2.6 This fact points to the relative inefﬁciencyof the energy-conversion process available at normal day-light conditions. An ideal highly efﬁcient photosynthetic sys-tem should not have any saturation limits for the standarddaylight intensity of light.It is evident from Fig. 5 that the number of protons Nptranslocated across the membrane ﬂuctuates in time. To esti-mate these ﬂuctuations we calculate the standard deviation
�p = ��Np2� − �Np�2,�33�
which characterizes the magnitude of its shot noise. The de-pendence of the noise level �p on the light intensity I isshowninFig.9�b�.ForanintensityoflightI�0.14 mW2/cm2 �when �10 protons are translocatedacross the membrane and the efﬁciency ��55% is sufﬁ-ciently high� the uncertainty �p in the number of pumpedprotons is about 1.3. In Fig. 9�c� we demonstrate that theefﬁciency � of the light-induced pumping decreases mono-tonically with increasing light intensity. At low light intensi-ties, a relatively small number of photons are absorbed perunit time. Thus, a higher fraction of the absorbed photons isused for the uphill pumping of the protons.
F. Effect of temperature
Figure 10 shows the effects of temperature on the pump-ing current and on the efﬁciency of the photosynthetic devicefor different values of the light intensity. The temperatureeffects appear in the light-induced proton pumping dynamicsthrough two factors. �i� The electron transfer rates, includingthe loading and unloading rates of the shuttle, increase withincreasing temperature. �ii� The diffusion coefﬁcient of theshuttle increases with temperature. Because of this, theshuttle can perform a higher number of trips to translocateprotons at higher temperatures. Here the electron transferreactions are not rate-limiting ones. The diffusive trips of theshuttle from the N terminal to the P terminal dominate thetransfer rate. Therefore, the increase in the efﬁciency and thepumping current with temperature is due to the increase inthe number of diffusive trips of the shuttle. A temperatureincrease from 200 to 400 K results in an increase of about afactor of 2 in the diffusion constant. It is expected that theproton current should increase at the same rate. However, ourcalculated ratio is about 1.5 �see Fig. 10�a��. This is probablydue to the fact that at high temperatures the shuttle has notenough time to be completely loaded with electrons and pro-tons near the acceptor site A and the N-side of the membrane�and unloaded near the donor site D and the P-side of themembrane�. A similar enhancement of the pumping current
and the efﬁciency with temperature can be useful for photo-synthetic microorganisms to compensate a leakage of pro-tons caused by the high-temperature increase in the mem-brane permeability.39 The simple physical features whichcome into play in our model are also important for the cre-ation of thermostable artiﬁcial photosynthetic devices efﬁ-ciently converting energy of light into electrical and chemi-cal energy in a wide range of temperatures and lightintensities.
G. Effect of the electrochemical potential gradienton the proton current
It follows from Eq. �14� that the difference ��=�P−�N between the electrochemical potentials of P- andN-proton reservoirs can be changed by changing the pH lev-els of the solutions inside and outside of the liposome. Indoing so, one unit change in pH corresponds to �59 meVvariation in the transmembrane proton gradient �� �at stan-dard conditions�. To demonstrate the effect of the pH levelson the performance of the pump, in Fig. 11 we plot thedependencies of the proton current on the electrochemicalpotential �P of the positive side of the membrane at threedifferent values of the N-side potential: �N=−110, �140,and �200 meV. The proton current saturates when the P-sidepotential is sufﬁciently low, �P�160 meV, and goes to zero
FIG. 10. �a� Proton current vs temperature for different values of the lightintensity I. �b� Pumping efﬁciency � vs temperature. Here, the electro-chemical gradient ��=220 meV ��P=110 meV and �N=−110 meV�.
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at �P�200 meV. At this condition, the potential of theP-side exceeds the energy �Q=200 meV of the proton on theshuttle �P��Q, so that the proton cannot be translocated tothe P-reservoir. On the other hand, the shuttle cannot beloaded with a proton at the N-side of the membrane if theelectrochemical potential �N is below the energy �Q−uSQ=−160 meV of the proton on the shuttle populated with asingle electron: �N�−160 meV. This is the reason why thelast curve in Fig. 11 �taken at �N=−200 meV� goes far be-low the other two curves �plotted for �N�−160 meV�.
V. CONCLUSIONS
We have analyzed a simple model for light-induced pro-ton pumps in artiﬁcial photosynthetic systems. This modelhas ﬁve electron sites �four sites �D,B,C,A� for the triadmolecule and one site for the shuttle �S�� and one proton-binding site on the shuttle �Q�. The shuttle exhibits diffusivemotion in the lipid bilayer, so that the electron and protonpopulations of the shuttle depend on the shuttle position.Based on the methods of quantum transport theory we havederived and solved numerically a system of master equationsfor electron and proton state probabilities evolving in timetogether with the Langevin equation for the position of theshuttle. This allows us to calculate the proton current and thepumping efﬁciency of the system and determine their depen-dence on the intensity of light, temperature, and electro-chemical potential gradient.For a reasonable set of parameters closely related to theexperimental setup, we demonstrate that this photosyntheticdevice can translocate protons against an electrochemicalgradient of the order of 220 meV with the efﬁciency �photon-to-proton quantum yield� which exceeds 55%. Our resultsexplain the experiments on artiﬁcial photosynthetic reactioncenters.10 We predict that both the proton current and thepumping efﬁciency grow linearly with temperature due tothe related increase in the number of diffusive trips of theshuttle. We also show that the pumping current increases
linearly with the light intensity and saturates at the experi-mentally observed limit, which is lower than the averageintensity of solar light.
ACKNOWLEDGMENTS
We acknowledge the partial support from the NationalSecurity Agency �NSA�, Laboratory for Physical Sciences�LPS�, Army Research Ofﬁce �ARO�, and National ScienceFoundation �NSF� Grant No. EIA-0130383. We also ac-knowledge the RIKEN Super Combined Cluster System forcomputational facilities.
APPENDIX: BASIS SETS: ELECTRON-PROTONEIGENSTATES AND ENERGY EIGENVALUES
The electron-proton system studied here with no leadscan be characterized by the 20 basis states of the Hamil-tonian H0,
�1� = aD† aB†�0�;�11� = aD† aB†bQ† �0�,
�2� = aD† aC†�0�;�12� = aD† aC†bQ† �0�,
�3� = aD† aA†�0�;�13� = aD† aA†bQ† �0�,
�4� = aB†aC†�0�;�14� = aB†aC†bQ† �0�,
�5� = aB†aA†�0�;�15� = aB†aA†bQ† �0�,�A1��6� = aC†aA†�0�;�16� = aC†aA†bQ† �0�,
�7� = aD† aS†�0�;�17� = aD† aS†bQ† �0�,
�8� = aB†aS†�0�;�18� = aB†aS†bQ† �0�,
�9� = aC†aS†�0�;�19� = aC†aS†bQ† �0�,
�10� = aA†aS†�0�;�20� = aA†aS†bQ† �0�.
Here, �0� represents the vacuum state when all electron andproton sites are empty. The state �1�=aD† aB†�0� corresponds tothe case when one electron is located on site D, one on siteB, and so on. The state �11�=aD† aB†bQ† �0� indicates that inaddition to two electrons on sites D and B, there is also aproton on the shuttle. States �1� to �10� describe the shuttlewith no protons, whereas states �11� to �20� are related to theshuttle populated with a single proton.An arbitrary operator A of the combined electron-protonsystem can be expressed in terms of the basis Heisenbergmatrices �m,n=�m��n�,
A = �m,nAmn�m,n,
where m and n label the basis states m,n=1, ... ,20. Thediagonal operator is denoted as �m��m,m. Thus the electronpopulation operators �nD,nB,nC,nA,nS� can be represented inthe form
nD = �1 + �2 + �3 + �7 + �11 + �12 + �13 + �17,
FIG. 11. Proton pumping current vs electrochemical potential �P of thepositive side �P-reservoir� of the membrane for different values of the po-tential �N of the negative side �N-reservoir� for the light intensity I=0.132 mW cm−2 and temperature T=298 K.
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nB = �1 + �4 + �5 + �8 + �11 + �14 + �15 + �18,
nC = �2 + �4 + �6 + �9 + �12 + �14 + �16 + �19,�A2�
nA = �3 + �5 + �6 + �10 + �13 + �15 + �16 + �20,
nS = �7 + �8 + �9 + �10 + �17 + �18 + �19 + �20,
and for the operator of the proton population of the shuttlewe obtain
nQ = �11 + �12 + �13 + �14 + �15 + �16 + �17 + �18 + �19
+ �20.�A3�
Using the eigenfunctions �see Eq. �A1��, we can rewrite theHamiltonian H0 in a simple diagonal form
H0 = �m=1
20�m�m�A4�
with the following energy spectrum:
�1 = ED + EB;�11 = �1 + �Q
�2 = ED + EC;�12 = �2 + �Q,
�3 = ED + EA − uBA;�13 = �3 + �Q,
�4 = EB + EC − uDB;�14 = �4 + �Q,
�5 = EB + EA − uDA;�15 = �5 + �Q,�A5��6 = EC + EA − uDA;�16 = �6 + �Q,
�7 = ED + ES;�17 = �7 + �Q − uSQ,
�8 = EB + ES;�18 = �8 + �Q − uSQ,
�9 = EC + ES;�19 = �9 + �Q − uSQ,
�10 = EA + ES + uDB − uBA − uDA;�20 = �10 + �Q − uSQ.
The terms ai†ai� describing the direct tunneling between allpossible coupled sites i and i� are given by the expressions
aB†aD = �4,2 + �5,3 + �8,7 + �14,12 + �15,13 + �18,17,
aA†aC = �3,2 + �5,4 + �10,9 + �13,12 + �15,14 + �20,19,
aB†aC = �1,2 + �5,6 + �8,9 + �11,12 + �15,16 + �18,19,�A6�
aA†aS = �3,7 + �5,8 + �6,9 + �13,17 + �15,18 + �16,19,
aD† aS = − �1,8 − �2,9 − �3,10 − �11,18 − �12,19 − �13,20.
It should be noted that the operator Hdir in Eq. �3� is nondi-agonal. The proton operator bQ can also be expressed in asimilar form,
bQ = �mnbQ,mn�m,n.�A7�
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