Living on the edge: light-harvesting efficiency and photoprotection in the core of green sulfur bacteria

Photosynthetic green sulfur bacteria are able to survive under extreme low light conditions. Nevertheless, the light-harvesting efficiencies reported so far, in particular for Fenna–Matthews–Olson (FMO) protein-reaction center complex (RCC) supercomplexes, are much lower than for photosystems of other species. Here, we approach this problem with a structure-based theory. Compelling evidence for a light-harvesting efficiency around 95% is presented for native (anaerobic) conditions that can drop down to 47% when the FMO protein is switched into a photoprotective mode in the presence of molecular oxygen. Light-harvesting bottlenecks are found between the FMO protein and the RCC, and the antenna of the RCC and its reaction center (RC) with forward energy transfer time constants of 39 ps and 23 ps, respectively. The latter time constant removes an ambiguity in the interpretation of time-resolved spectra of RCC probing primary charge transfer and provides strong evidence for a transfer-to-the trap limited kinetics of excited states. Different factors influencing the light-harvesting efficiency are investigated. A fast primary electron transfer in the RC is found to be more important for a high efficiency than the site energy funnel in the FMO protein, quantum effects of nuclear motion, or variations in the mutual orientation between the FMO protein and the RCC.


.1 Intradomain excitation energy transfer: Redfield theory
For the relaxation of excitons within a domain of strongly coupled pigments, we employ Redfield theory. The relaxation rate constant between the exciton states |M a and |N a in domain a is given as 1 the transition frequency ω KaLa = ω Ka − ω La between exciton states |K a and |L a , and the spectral density J(ω) = ξ g 2 ξ δ(ω − ω ξ ) that contains the local coupling constants (eq 1 of the main text), which are assumed to be independent of the site index m a . Note that we have investigated this approximation in a recent normal mode analysis of the spectral density of FMO trimers 2 and found that although the local coupling constants can vary up to a factor of two, these variations have practically no influence on energy transfer.

S1.1.2 Interdomain excitation energy transfer: Generalized Förster theory
The excitation energy transfer between different domains is described by generalized Förster theory. [3][4][5][6][7] Assuming uncorrelated fluctuations of site energies in different domains, as investigated by a recent normal mode analysis of the spectral density, 2 the rate constant in generalized Förster theory k GF Ma→N b between states |M a and |N b of domain a and b, respectively, is given in eq 6 of the main text. It contains the overlap integral between the normalized absorption lineshape function D N b (ω) of the acceptor exciton state and the normalized fluorescence lineshape function D Ma (ω) of the donor exciton state, which are given as 8 of exciton state |M a , the n(ω) from eq S2, and the spectral density J(ω) of the site energy fluctuations. The exciton relaxation-induced inverse dephasing time τ −1 Ma is given as and the shifted transition frequencies are defined as Here, ω Kc is the vertical excitation frequency of exciton state |K c in domain c as given in eq 3 of the main text and is the reorganization energy due to the diagonal elements of the exciton-vibrational coupling of this exciton state, reading with the inverse participation ratio of exciton state |K c (eq S6) and the reorganization energy E where ℘ denotes the principal part of the integral.
Treating the motion of the nuclei classically, the semi-classical rate constant k scGF Ma→N b is obtained as 2 where T is the temperature,ω Ma,N b is given in eq S8, V MaN b is the interdomain exciton coupling (eq 5), and it in the present limit of uncorrelated site energy fluctuations with the E Kc λ in eq S10.

S1.1.3 Exciton population dynamics and initial conditions
The master equations eq 7 of the main text can be written in matrix form as d dt P (t) = −ÂP (t), with the kinetic matrixÂ. This system of ordinary differential equations has the solution where c i and λ i are the eigenvectors and eigenvalues of the matrixÂ, respectively. The constants d i are obtained from the initial conditions P (0) = i d i c i and P (0) denotes the initial population of the FMO-RCC. For the initial population of the system, we assume incoherent excitation energy transfer from the baseplate to the FMO protein and no direct energy transfers from the baseplate to the RCC. The initial population of an exciton state in the FMO protein is assumed to be proportional to the rate constant for the transfer from the baseplate 9 P (a) where V bMa is the electronic coupling between the lowest exciton state in the baseplate and the M th exciton state in domain a in the FMO protein. Taking into account that the transition dipole moment of the lowest energy exciton state of the baseplate is oriented in the plane of the baseplate 10 and, therefore, normal to the symmetry axis e FMO of the FMO protein, the excitonic coupling V bMa is estimated to be Here, θ Ma is the angle between e FMO and the transition dipole moment µ Ma , R 0a is the vertical distance between the center of pigment 8 in domain a and the baseplate. ∆R Ma takes into account the delocalization of the exciton state |M a , it reads where R ma is the center of pigment m in domain a. The sum in eq S16 takes into account only those pigments m a with an exciton coefficient that fulfills the condition c (Ma) ma > 0.3. We normalize the initial population, that is,

S1.2 Estimation of the intrinsic quenching time constant
If the FMO protein is under oxidative stress, the measured fluorescence rate constant is k f l = 60 ps −1 . 11 The inverse rate constant is calculated as where f M is the Boltzmann factor, k M →g is the transition rate from the M th excited state into the ground state,  For each pigment m a , the Poisson equation

S1.3 Parameters
is solved numerically using MEAD, 12 revealing the ESP of the transition density of pigment m a in a heterogeneous dielectric environment. The optical dielectric constant ε(r) = 1 in the cavity and ε(r) = 2 in the protein/solvent environment. Using the electrostatic potential φ m (r), the excitonic coupling V ma,n b between pigment m in domain a and pigment n in domain b is given as is the electrostatic potential of the transition charges of pigment m a at the position of the Ith atom in pigment n b , and f RF is the reaction field factor that takes into account an enhancement of the transition density of the chromophores by the polarization of the environment. 13 The excitonic couplings obtained for f RF = 1.15 can be found in the accompanying file "exc coupling.dat".

S1.3.2 Site energies: The Charge Density Coupling (CDC) method
In the CDC-method 14 , the charge density of the ground and excited state of the pigments is approximated by atomic partial charges. These atomic partial charges are obtained from a fit of the ab-initio ESP, as described below. The site energy E ma of pigment m in domain a is obtained as E ma = E 0 + ∆E ma , where E 0 is a reference energy and ∆E ma is the site energy shift. The site energy shift in the CDC method is then given as 14 where the sum over I runs over all partial charges of pigment m a , the sum over J runs over all ground state background charges q J (0, 0), and q Jth atomic partial charge of the background, respectively, and ε eff is an effective dielectric constant, which takes into account screening and polarization effects.

S1.3.3 Quantum chemical calculations
The ab-initio charge density of the ground (S 0 ) and the first excited (S 1 ) states of BChl a and Chl a were obtained with (time-dependent) density functional theory ((TD)DFT) using Q-Chem 20 . The geometries of the isolated BChl a and Chl a pigments were optimized with DFT using the B3LYP exchange-correlation (XC) functional and a 6-31G(d,p) basis set. Based on this geometry, the ground and excited state charge densities and the transition densities between those two states were calculated with (TD)DFT using the CAM-B3LYP XC-functional and a 6-31G(d,p) basis set. The fit of the ESP of the ground and excited state charge densities and the transition densities was performed with CHELP-BOW 21 . The resulting atomic partial charges q k (0) and q k (1) of the ground and excited state, respectively, are given in Table S3  The latter has been obtained from fluorescence line narrowing spectra of the B777 complex 8 and later applied to many different PPCs 7,17,23-25 , including the FMO protein 17 . This spectral density reads with the normalized ( dωJ 0 (ω) = 1) function : Atomic partial charges of the ground (q k (0)) and excited (q k (1)) state of Chl a in units of the elementary charge e, obtained with (time-dependent) density functional theory using the CAM-B3LYP exchange-correlation functional and a 6-31G(d.p) basis set. The labeling of the heavy atoms is according to the protein data bank file 6M32.  and the Huang-Rhys factor S. The parameters for J 0 (ω) are s 1 = 0.8, s 2 = 0.5, ω 1 = 0.069 meV, ω 2 = 0.24 meV.
The Huang-Rhys factor S can be obtained from the temperature dependence of optical spectra. For the FMO protein, we obtained 17 S = 0.42. We will assume the same value for the pigments in the RCC.

S1.4 Linear Optical Spectra
The linear absorption OD(ω) is given as where . . . dis denotes an average over static disorder in site energies. The linear absorption of domain a is given where µ ma is the local transition dipole moment of pigment m a . The lineshape function D Ma (ω) for the optical excitation of the M a th exciton state is described in eq S3. Further details can be found in the original Ref. 8 .
The linear dichroism (LD) spectrum 26,27 is obtained by replacing the square of the transition dipole moment |µ Ma | 2 in eq S25 with the expression |µ Ma | 2 1 − 3 cos 2 (Θ Ma ) , where Θ Ma denotes the angle between the transition dipole moment µ Ma and the membrane normal vector. The linear dichroism spectrum of a domain is then given The circular dichroism (CD) spectrum reads where the rotational strength R Ma is given as The distance vector R ma,na = R ma − R na connects the centers of pigments m a and n a . Figure S1: Optical spectra calculated for alternative set of site energies. Left: Same as Figure 2 of the main text, but using the site energies from CDC calculations (Table S1). Right: Same as Figure 3 of the main text, but using the site energies from CDC calculations (Table S2) (Table S1) Figure S5: Correlation between different sets of site energies for the RCC. Correlation between the site energies of the BChl a pigments of the RCC obtained from a fit of optical spectra using a genetic algorithm and the site energies calculated with the CDC method based on the cryo-EM structural model 19 (Table S2).  Figure 4 of the main text (no quenching), but assuming that a hypothetical second FMO protein is bound to the homo-dimeric RCC at a symmetric position with respect to the first. Please note that the populations of FMO1 and FMO2 are practically identical. The same is true for RCC-Ant1 and RCC-Ant2