Plasmonic bio-sensing for the Fenna-Matthews-Olson complex

We study theoretically the bio-sensing capabilities of metal nanowire surface plasmons. As a specific example, we couple the nanowire to specific sites (bacteriochlorophyll) of the Fenna-Matthews-Olson (FMO) photosynthetic pigment protein complex. In this hybrid system, we find that when certain sites of the FMO complex are subject to either the suppression of inter-site transitions or are entirely disconnected from the complex, the resulting variations in the excitation transfer rates through the complex can be monitored through the corresponding changes in the scattering spectra of the incident nanowire surface plasmons. We also find that these changes can be further enhanced by changing the ratio of plasmon-site couplings. The change of the Fano lineshape in the scattering spectra further reveals that “site 5” in the FMO complex plays a distinct role from other sites. Our results provide a feasible way, using single photons, to detect mutation-induced, or bleaching-induced, local defects or modifications of the FMO complex, and allows access to both the local and global properties of the excitation transfer in such systems.

challenging, the recent breakthroughs on coupling SPs to J-aggregates 19,20 and Photosystem I trimer complexes 18 suggests the hybrid devices we study here may be feasible in the future.
Like all other species, the in vivo FMO complex can experience both mutation-induced and bleaching-induced local defects or non-functional sites, leading to blocked excitation transferring pathways. This has been demonstrated in recent experiments 34,35 . These effects may be observed in the changes in excitation dynamics, the efficiency of excitation arrival at the reaction center 36 , or the spectra of the photoluminescence 18,19 . Still lacking however is both a means to prepare the complex with a single localized excitation, and a means to measure populations of local sites of the complex. Here we examine how the SP can be used as both a single photon source 22 and a detector. Through the scattering of the incident SP due to the couplings between the SPs and sites 1 and 6 of the FMO complex, the changes in the transmission spectra can indicate the presence of pathway-inhibition or missing sites. This provides an alternative method to detect mutation-induced defects or non-functional sites, and suggests further applications for such hybrid systems.

Results
We model a single FMO monomer as a network of N = 7 sites (see Fig. 1), which can be described by a general Hamiltonian where the state |n〉 represents an excitation at site n (n ∈ 1, … , 7), ε n is the site energy of chromophore n, and ′ J n n , is the excitonic coupling between the n-th and n ′ -th sites. For simplicity, here the recently discovered 37 eighth site has been omitted because results of molecular dynamics simulations 37 suggest that this site plays a minimal role in the processes we are interested in.
In the bacterial photosynthesis, the excitation from the light-harvesting antenna enters the FMO complex at sites 1 or 6 and then transfers from one site to another. When the excitation gets to the site 3, it hops irreversibly to the reaction center. In the regime that the excitonic coupling ′ J n n , is large compared with the reorganization chromophores. The incident surface plasmon can either be scattered or absorbed, due to the couplings g 1 and g 6 to the sites 1 and 6, respectively. The absorbed excitation then transfers from one chromophore to another, when it arrives at site 3, it can irreversibly transport to the reaction center. energy, the electron-nuclear coupling can be treated perturbatively 3 , and the dynamics of the system can be governed by the quantum Liouville equation. The strong coupling between the excitonic dynamics and the FMO environment, as well as the structure of that environment, will quantitatively affect the excitation transfer 38,39 .
Here, however, we wish to focus on the interplay between the excitation transfer and the SP scattering; so we focus on the simplest possible model of such environmental effects. This allows us to clearly identify in what way defects in the FMO complex affect the SP scattering. A full investigation of the environment influence will be considered in future work. Also, it should be noted that excitonic fluorescence relaxation is not included in the Liouville equation. This is because its time scale (~1 ns) is much longer compared with that of the excitation transfer from site 3 to the reaction center (~1 ps), the typical excitation transfer time across the complex, and the dephasing 40 (~100 fs), such that this relaxation process can then be omitted for simplicity.
The Surface plasmon and FMO Hybrid system. As shown in Fig. 1, we consider a metal nanowire that is placed close to the FMO complex, as a substitute for the light-harvesting antenna found in vivo. A SP with energy = E v k k g  incident from the left end of the wire can be strongly coupled 18,41,42 to both sites 1 and 6. Here, v g and k are the group velocity and wave vector of the incident SP, respectively, and v g is set to be unity throughout this paper. The incident SP can then be scattered by the two sites, due to these strong SP-site couplings. Alternatively, it can be absorbed by these two sites and be dissipated due to loss in the FMO complex. where H sp stands for the energy of the SPs with † a k being the creation operator of the k-mode SP, H sp-site denotes the interaction between the SPs and the sites of the FMO complex. In H sp-site , d is the separation between site 1 and 6, while σ + 1(6) denotes the raising operator for site 1 (6). Here, the coupling strength g 1(6) between the SPs and site 1 (6) is assumed to be independent of k under the Markovian approximation 43,45 . The strong decay rate into SPs

Transmission coefficient
Transmission coefficient  6 . The black-solid curves represent the transmission spectra of the incident surface plasmon under normal conditions (all the normal excitonic couplings remain). For this figure, we set the dephasing rate γ dp = 77 cm −1 , the rate from site 3 to the reaction center γ s = 5.3 cm −1 , the SP-site couplings are set to be g 1 = g 6 = 10 cm −1 , and the Ohmic loss rate γ ol is set to be 1/20 γ sp .
Scientific RepoRts | 7:39720 | DOI: 10.1038/srep39720 then takes the form 24 , and the values we choose for g 1 and g 6 are consistent with a recent analysis of nanowire SPs coupled to J-aggregates [18][19][20] .
When the SP is propagating on the surface of the metal nanowire, it inevitably suffers from dissipation, such as Ohmic loss, that can be described by a Markovian channel. Thus, here we include the Markovian channels for irreversible excitation transfer from site 3 to the reaction center, as well as the Ohmic loss, by introducing non-Hermitian terms in the total Hamiltonian. Moreover, since we are interested in time-independent solutions of the photon state, the effect of quantum jumps 46 on the dynamics is neglected, such that the phonon dephasing leads to population reduction in our model. We therefore simply include the phonon dephasing as a non-Hermitian term. While this introduces extra population loss, it allows us to include in a simple way the broadening effect of dephasing on the SP spectra. The site energy of the FMO complex in Eq. (1) is then modified as where γ n are mostly zero, except for the rates referring to the Markovian channels, γ 1 = γ 6 = γ dp + γ ol and γ 3 = γ s . Here, the Ohmic loss rate γ ol is set to be γ ol = 20 −1 γ sp 21,24,47 . Note that the separation between site 1 and 6 is about 1-2 nm 48 , which is much smaller than the wavelength of the incident SP. Therefore, one can set the separation d = 0. Similarly, since the separation is small compared with the distance (~30 nm) that SPs propagate during the dephasing time of scattering 49,50 , we neglect plasmon dephasing. Note that we only include γ dp on sites 1 and 6, and neglect dephasing for other sites, because the broadening of these sites, which couple to the SPs, typically dominates the scattering spectra 51,52 .
The energy eigenstate of the hybrid system with an energy matching the incident SP, E k = v g k, can be written as 24,43 : sp

Transmission coefficient
Transmission coefficient where |g, 0 sp 〉 describes that the FMO complex is in the ground state with no SPs, and |n, 0 sp 〉 stands for that the excitation is in the site n, while ξ n is the probability amplitude that the site n absorbs the excitation. We also assume that the SP field is incident from the left of the waveguide, the scattering amplitudes φ † x ( ) k R , therefore take the form, Here, t and r are the transmission and reflection amplitudes, respectively, and θ(x) is the unit step function. The total Hamiltonian HT can be further transformed 43 by Fourier transformation into a real-space representation, ps, and the dephasing rate γ dp , proportional to the temperature 40 , is chosen to be 77 cm −1 .

Transmission coefficient
Transmission coefficient . For this figure, we set the dephasing rate γ dp = 77 cm −1 , the rate from the site 3 to the reaction center γ s = 5.3 cm −1 , while the SP-site couplings are set to be g 1 = 10 cm −1 , g 6 = 0.1 cm −1 , such that g 1 /g 6 = 100, and the Ohmic loss rate γ ol is set to be 1/20 γ sp .
Scientific RepoRts | 7:39720 | DOI: 10.1038/srep39720 Detecting defects from changes in the scattering spectra. It has been demonstrated experimentally 34,35 that each chromophore in the FMO complex can be decoupled from its nearest site due to rotations of the chromophore or mutation-induced local defects. This can lead to damage of the chromophores, such as blocked energy pathways or entirely non-functional sites. Here we first calculate the transmission spectra of the incident SP field in the presence of defects in the FMO complex, and compare them to the case without any defects. As can be seen in Fig. 2, the black-solid curve shows the transmission spectrum of the incident SP field for the normal FMO complex. The dips in the spectrum correspond to the the eigenenergies of the hybrid system, i.e., the dips occur when the sites resonant with the SP reflect the incident field 24 . The reason why the dips do not reach zero is because here we have included dephasing and the irreversible excitation transfer from site 3 to the reaction center. These dissipative channels decrease the amplitude of the dips. As seen in Fig. 2, the red-dashed curves show the transmission spectra when certain excitonic couplings are inhibited. The differences between the red-dashed and black-solid curves can indicate the occurrence of a blocked transfer pathway. However, the spectral differences in Fig. 2(a) and (d) are larger than those in Fig. 2(b) and (c). This is because in the normal case these sites are strongly coupled, and thus when suppressed the inhibition strongly affects not only the eigenenergies, but also the quantum coherence 53 in the site basis. On the other hand, the transmission spectra in Fig. 2 show typical Fano lineshapes 54 stemming from the interference between discrete (the sites) and continuous channels (the SPs). As the inhibition of the relatively strong coupling J 5,6 , between sites 5 and 6 leads to the enhancement of the coherence between other the sites of the FMO complex, the Fano resonance is strongly altered [ Fig. 2(d)]. This result is consistent with our previous work 55 on the dynamics of the excitation transfer in FMO complex. Similarly, mutation-induced defects or local environment modification 34,35 can also lead to the disconnection of the chromophores from the FMO complex entirely, rather than just suppression of certain couplings. In Fig. 3, we plot the transmission spectra for the cases where certain sites with relatively large excitonic couplings are removed entirely. The differences between the black-solid (the normal situation) and red-dashed curves can again indicate this drastic alteration of the complex. One interesting question can be raised here, can one distinguish the following two situations: (i) inhibiting a strong excitonic coupling, and (ii) totally removing one of the sites which contains the very excitonic coupling entirely? The answer is yes. By comparing Figs 2 and 3, the inhibition of the transfer pathway only results in changes of the transmission spectra, but the missing site case also leads to vanishing peaks/dips, indicating the removal of an eigenstate of the system. Note that in plotting Figs 2 and 3, we assume the SP-site couplings to be symmetric, i.e., g 1 = g 6 = 10 cm −1 . In the following section, we will discuss the effects of asymmetric couplings.

Transmission coefficient
Transmission coefficient E k (1/cm) Figure 5. Enhancing changes by asymmetric couplings in transmission spectra for site 5. The red-dashed curves represent the transmission spectra of the incident surface plasmon when (a) excitonic couplings J 5,6 is inhibited or (b) when site 5 is disconnected entirely from the FMO complex. The black-solid curves represent the transmission spectra of the incident surface plasmon under normal conditions (all the excitonic couplings remain). For this figure, we set the dephasing rate γ dp = 77 cm −1 , the rate from the site 3 to the reaction center γ s = 5.3 cm −1 , while the SP-site couplings are set to be g 1 = 0.1 cm −1 , g 6 = 10 cm −1 , such that g 1 /g 6 = 0.01, and the Ohmic loss rate γ ol is set to be 1/20 γ sp .
Scientific RepoRts | 7:39720 | DOI: 10.1038/srep39720 Discussion So far we have shown that local defects or missing sites can be detected via changes in the transmission spectra of the incident SP. However, if these changes in the transmission spectra can be enhanced, then the observations of such an effect in experimental realizations would become more feasible. We have observed that the transmission spectra contain Fano lineshapes, thus we can predict that a discrepancy between the discrete (the FMO sites) and continuous (the SPs) channels will strongly affect the behavior of such Fano resonances [56][57][58] . Larger changes in the transmission spectra can then be achieved by making these two channels more disparate. In Fig. 4, we show the transmission spectra when inhibiting J 1,2 [ Fig. 4(a)] and entirely removing the site 2 [ Fig. 4(b)] with asymmetric SP-site couplings g 1 /g 6 = 100 (g 1 = 10 cm −1 , g 6 = 0.1 cm −1 ). The changes in the transmission spectra are enhanced compared to those in Figs 2(a) and 3(a), respectively. This is because when the damage (pathway-inhibition or missing site) occurs to site 2, the coherence between other sites in the FMO complex increases 55 , and the large ratio g 1 /g 6 reduces the communication between discrete and continuous channels. As a result, the two channels become more disparate, leading to an enhancement in the transmission spectra.
Similar results can also be obtained as shown in Fig. 5 by either inhibiting J 5,6 [ Fig. 5(a)] or completely removing site 5 [ Fig. 5(b)]. Correspondingly, the asymmetric couplings here are chosen as inverse of the previous example g 1 /g 6 = 0.01 (g 1 = 0.1 cm −1 , g 6 = 10 cm −1 ). Again we see that the changes in the transmission spectra are enhanced compared with those in Figs 2(b) and 3(b). More importantly, in Fig. 5, we can see that when site 5 is disconnected from the FMO complex, the sharp Fano lineshape on the right-hand side of Fig. 5 is smeared out. Since J 1,2 (− 104.1 cm −1 ) is larger than J 5,6 (89.7 cm −1 ), we would expect this effect to be obvious in Fig. 4, but it is not. This reveals that site 5 is very distinct from other sites, consistent with our previous work 55 .
Since the FMO protein complex is in reality a trimer, the validity of the results obtained by studying the monomer alone shall be addressed here. Typically studies have shown that the subunits of the trimer act as separate transport channels 37 from the antenna to the reaction center. On the other hand, the site-couplings of a monomer were found to be increased because of the other monomers in the trimer system 59 . This suggests that studying individual FMO monomers is sufficient to get a qualitative understanding 60 , but that in a full trimer system the precise monomer eigenenergies may be altered.
In summary, we have shown that by observing the scattering of an incident SP, coupled to sites 1 and 6 of the FMO complex, both the inhibition of the excitonic coupling (excitation-transferring pathway) and entirely missing sites can be detected. We also show that by making the discrete (sites) and the continuous (SPs) channels more disparate, it is possible to further enhance the changes in the transmission spectra of the incident SP. These results provide an alternative method to detect the mutation-induced local defects or the non-functional sites in the FMO complex. We expect that future work will reveal even more uses for such hybrid bio-sensors, including probing the role of quantum coherence and environmental effects in photosynthetic complexes.