LH1–RC light-harvesting photocycle under realistic light–matter conditions

Light–matter interactions are ubiquitous in nature and essential to life on earth. For example, biological light-harvesting is the dominant contribution to energy sources of all known life forms and activities. In such cases, pigment–protein complexes (PPCs) often serve as basic units in the first steps of natural light-harvesting where individual pigment molecules, such as chlorophyll or bacteriochlorophyll (BChl), surrounded by a protein scaffolding, absorb light and pass the excitation energy [excitation energy transfer (EET)] down to the reaction center (RC).^1–4 Similarly, the cis–trans isomerization of retinal initiated by photon capture in rhodopsin constitutes the first step of dim light vision.^5,6

EET in photosynthesis is associated with photosynthetic units, minimal functional elements with multiple chlorophyll molecules that are capable of collecting the photon energy required for photochemistry machinery.^3,7 Experiments conducted under low light intensity confirm that the photosynthetic units are composed of antenna and RCs, where EET funnels the energy from the former to the latter. As early as the 1930s,⁸ it was suggested that EET could be wavelike under natural circumstances, a topic that continues to be of interest to this day.^9–22 

Recently, pulsed laser studies on light-harvesting PPCs have shown coherent beatings of spectroscopic signals under ambient temperature, which have been interpreted as quantum coherence among electronic or vibronic states in the systems.^17–19,21 It was suggested that certain optimal configurations in these natural systems protect nontrivial quantum mechanical effects and that quantum coherence could serve as a potential factor for increasing the rate of EET, the efficiency of energy conversion, and the robustness against environmental fluctuations in natural photosynthesis.^23,24 This work stimulated an investigation of the possibility of nontrivial quantum mechanical phenomena functioning in biological systems.^25–27 

An important misconception associated with these results stems from the associating coherent effects initiated via pulsed laser excitations with the mechanism of biological systems operating under natural solar illumination. The latter is best characterized as a thermal distribution of photons emitted by the Sun. There are two major differences in the nature of radiation in natural incoherent light from that of pulsed lasers: the duration is far longer and the coherence characteristics of the sources are dramatically different. Indeed, the time scales of natural incoherent excitation far exceed the time scales involved in the dynamics of these molecular systems.^28–31 Thus, consideration of light–matter interaction under solar radiation, and the examination of possible nontrivial quantum effects, must be made in the non-equilibrium steady state limit.^32–34 

Here, we focus on characterizing the EET rate within a minimally functional PPC, the LH1–RC complex in purple light-harvesting bacteria. Quantum master equations are used to model both the dynamics and the steady state of the reduced density matrix of the system, where the influence of both the solar radiation and the phonon environment is treated perturbatively. To complete the system photocycle, we include both the charge separation process, which converts the excited RC into the charge-separated (CS) state, and the recovery process, which returns the RC back to the initial open state.³⁵ Once the system is in the excited state, it could return to the ground state by charge separation, nonradiative recombination, recovery channel, or fluorescence emission.³⁶ At steady state, the ratio between the rates of these competing processes will be seen to be dependent on the intensity of the incident radiation. At lower intensities, where the incoming radiation is the rate-limiting process, charge separation/recovery is dominant, whereas at higher intensities, the opposite (emission or non-radiative recombination) is the case. Correspondingly, the steady state EET rate scales linearly with the incident light intensity at low intensities and is bounded at higher intensities.

To model some features of the pulsed laser experiments, we also examine the time-dependent dynamics of light-harvesting under incoherent excitation with sudden turn-on of light–matter interaction. Although the light is incoherent, initial coherences are generated by the sudden turn-on. In this case, we find transient regimes that show EET rates that differ from their steady state values. As in pulsed laser excitation, we take the initial condition to be that where only the bright states of the LH1 are excited. Strong population oscillations between the LH1 and the RC are found on the sub-picosecond time scale, consistent with the pulsed laser experiments. The steady state and sudden turn-on results are compared and shown to differ dramatically.

It is important to note that the rates reported in this paper are the transfer rates, or currents, defined by $r=d[P]/dt$⁠, where $[P]$ is the product concentration (RC population). This differs from the rates commonly reported in the light-harvesting literature, which are rate constants that assume the first order kinetics, that is, $k$ as in $r=k⋅[A]$⁠, where $[A]$ is the concentration of the reactant (LH1 population). The resultant $k$ is typically in the range of a few tens of ps for inter-PPC EET processes. A proper comparison of these two “rates” would require justification of the assumption of the first order kinetics (which we do not use) as well as a means of accounting for the (steady state) population inhomogeneity among the 32 sites in the LH1 participating in the EET. As such, our reported transfer rates are devoid of any such approximations.

We consider an LH1 antenna surrounding an RC (Fig. 1), which serves as a minimally functioning light-harvesting PPC. The RC consists of four BChl’s, with two denoted as the special pair. Other cofactors such as the carotenoids, the bacteriopheophytins, and the ubiquinones, as well as the effect of static disorder, are neglected since we assume that the results are not qualitatively affected. A complete photocycle of a functional LH1–RC complex starts with photon absorption from the LH1–RC ground state to the LH1–RC excited state manifold. Excitation energy on the LH1 transfers to the RC, where charge separation takes place at the special pair. Once the charge-separated (CS) state is reached, the RC is considered to be in the closed state that does not absorb additional excitation energy.^36,38–41 This closed state persists until charge transfer dynamics involving quinones carries the charges away from the RC and restores the RC to the open state, completing the photocycle. All processes are included in our simulation, detailed below.

Owing to the weak coupling between the relevant optical transitions in the pigment molecules and the solar radiation, only the zero- and one-particle manifold of the LH1–RC system are considered. In addition, the RC can either be in the open state, where its four constituent BChl pigments participate in both absorption of light and excited manifold dynamics, or in the closed state where the RC pigments cannot absorb light. The relevant orthonormal set of electronic states in this context are

Here, subscript o denotes an open state of the RC, i.e., an RC that can be excited. Subscript c denotes a closed state of the RC, i.e., an RC that, because it is charge-separated, cannot be excited. Only the $S0$ to $S1$ transition, the Qy transition, of the BChl molecule is considered. The indexing of the 36 BChl molecules follows the convention adopted in the PDB entry 3wmm, with $n=1$ to 4 being the ones in the RC (numbers 1 and 4 are the special pair) and $n=5$ to 36 being the ones in the LH1 complex.³⁷ Intercommunication among these manifolds is introduced below.

Figure 1 provides a schematic of the states and the processes involved in the photocycle. Starting from the overall ground state $|0o$ at the bottom left, light–matter interaction described by $Lrad$ (wavy lines) excites the system into its first-excited state manifold $|no,n∈{LH1,RC}$ through photon absorption or conversely de-excites the system through spontaneous and stimulated photon emission. The non-radiative recombination $Lnr$ (not shown) also de-excites the system in a similar fashion. The EET dynamics of interest happens entirely within this manifold (dashed curve). The excited state of the RC $|no,n∈{RC}$⁠, undergoes the charge separation process $Lcs$⁠, thereby closing the RC with all other molecules in the ground state $|0c$⁠. In the closed RC manifold, on the right-hand side of Figure 1, the light–matter interaction continues to excite the LH1 component of the ground state $|0c$ to the excited state manifold $|nc,n∈{LH1}$⁠. Finally, the closed RC manifold is opened up by the recovery process $Lto$⁠, and the full photocycle within this minimal model is completed. We describe the Liouville operators in detail below.

The Hamiltonian for the excitonic system is then (throughout this paper $ℏ=1$⁠)

where $ωn$ is the Qy transition frequency of site $n$ and $Jnm$ is the excitonic coupling between sites $n$ and $m$⁠. Values for these parameters are obtained from quantum chemical calculations based on the crystal structure reported in the literature and are provided in Tables I and II. Note that $Hs$ is the block-diagonal, with the open RC manifold uncoupled from the closed RC and the ground manifold uncoupled from the excited states. We also assume that the state of the RC does not affect the on-site energies or the excitonic couplings among active pigments. As such, actions of the creation and annihilation operators on the relevant states are summarized as follows:

The excitonic system is coupled to both a phonon environment under ambient temperature, representing the protein scaffolding surrounding the BChls and their relevant vibrational modes, and the thermal photon radiation from the Sun. In addition, the exciton dynamics in the LH1 with the closed RC is also taken into account. The dynamics of the system is described by the reduced system density matrix $ρ^$ with the equation-of-motion

where $Lph$ and $Lrad$ are Liouville operators describing the system couplings to the phonon and radiation baths, respectively. $Lnr$ describes the non-radiative recombination. $Lcs$ and $Lto$ account for the formation of the separated charges (closing the RC) and its recovery back to the ground state (opening the RC).

Here, both the exciton–phonon and exciton–photon interactions are treated perturbatively using Redfield theory, where the relevant Liouvillians in the system energy eigenbasis can be written as⁴² 

with $x={ph,rad}$⁠, and the relaxation tensor $Rijkl(x)$ is given by

We assume that the interactions of the individual BChl Qy transition with the photon or phonon environment are uncorrelated. This leads to the following expressions for the damping matrices $Γijkl(x)$⁠:

where $Aijkl(x)$ is the system part of the system–bath interaction Hamiltonian, $J(x)(ω)$ is the bath spectral density, $n¯BE(ω,β)=(eβω−1)−1$ is the Bose–Einstein distribution at inverse temperature $β$⁠, and $β(ph)−1=300$ K and $β(rad)−1=5700$ K. For our purpose,

where in the phonon case we assume independent baths for each of the BChl molecules and $n$ runs over all sites and in the radiation case, $α$ is the polarization. $μ$ represents the transition dipole moment operator. The spectral densities are

That is, we adopt an Ohmic bath of reorganization energy $λ$ with a Lorentzian cutoff $ωc$ for the phonon bath^43,44 and the black body radiation spectral density for the photon bath.^28,32 Both the phonon and the photon baths couple only within the open or the closed RC manifolds, i.e., $Aij,x(x)=Aji,x(x)=0$ for $|i∈{|0c,|nc}$ and $|j∈{|0o,|no}$⁠.

The full operational cycle of the LH1–RC complex includes charge separation at the special pair and the subsequent electron transfer from a cytochrome c back to the oxidized RC, restoring it back to the open state for the next exciton to be processed. Together with non-radiative recombination that returns an excited site back to the ground state, these processes are modeled using Lindblad superoperators^33,45

and the corresponding population draining and dephasing terms. We denote $y={o,c}$⁠, and $knr$⁠, $kcs$⁠, and $kto$ are rate constants, obtained from the experiment, for the non-radiative recombination and charge separation and recovery processes of the RC.³⁹ The sum in Eq. (15) is over the special pair. Other possible quenching processes including intersystem crossing are not included for simplicity.

In modeling the system–radiation interaction, and in contrast to the system–phonon interaction, the system does not evolve under the full strength of the photon source (the Sun). Rather, the incident light is drastically attenuated due to the ambient environment of the bacterial culture as compared to the surface of the Sun. To account for this effect, the Einstein B coefficient for the system–radiation coupling is multiplied by an attenuation factor $C$⁠. Specifically, in Eq. (9), for the photon bath, the Bose–Einstein coefficient is replaced by

where $ωij$ is the energy gap between states $|i$ and $|j$⁠. That is, all stimulated exciton–photon events are attenuated by a factor of $C$⁠. Indeed, among the various fluctuations involved in natural light-harvesting, such as temperature and nutrient concentrations, light intensity is by far the most variable to which photosynthetic organisms are subjected on a relatively short biological time scale.

The effect of $C$ and its biological relevance is discussed in Sec. III A. We obtained these results by substituting Eq. (18a) into Eq. (9) (for all $ωij$⁠) in defining the photon absorption Liouville superoperator $Labs$⁠. This will be utilized in Sec. II B in incorporating the photon flux into the system. Similarly, a photon emission superoperator $Lemi$ can be defined using Eq. (18b), which can be useful in calculating the fluorescence emission quantum yield. Studies in varying $C$ are discussed below and will be seen to provide interesting insights into the overall energy transfer process. That is, although one may anticipate a simple linear relationship between the operation of the LH1–RC system and values of $C$⁠, since the light absorption is in the perturbative regime, this is not the case. Rather, the role of successive ongoing processes and coupling to the proton bath will be seen to result in a rich dependence on $C$⁠. Note that a single exciton manifold is still expected to be sufficient up to $C≤10−2$⁠, as shown below.

To connect to the actual solar power sampled on the Earth’s surface, we proceed to make a simple estimation of the numerical value of $C$⁠. Taking the solar irradiance at 900 nm to be⁴⁶ 900 W/$m2$ and the cross section of an LH1–RC complex to be 100 Ų, the rate of photons incident on a single LH1–RC under direct sunlight at the surface of the Earth is about $∼$4000 $s−1$ per complex. In addition, the light intensities are attenuated at their respective habitats (deep stratified lakes, seawater, sewage, or waste lagoons).⁴⁷ Here, we take the data reported by Kirk⁴⁸ where the measured quantum irradiance a few meters below the surface of a lightly polluted lake drops two to three orders of magnitude compared to the surface value. Provided that all photons are absorbed by the light-harvesting antenna of the bacterium under consideration, an estimated upper bound for the minimal system LH1–RC complex’s photon conversion rate is about 0.1$%$ of the surface solar intensity (⁠$∼4s−1$⁠) corresponding to $C≈10−8$ in the current model.

The rate of change in the population of the RC is

where

are the rate of photon absorption, emission, and non-radiative recombination of RC and the charge separated rate, respectively, and

is the population transfer rate between the RC and the LH1 complexes in units of quanta per unit time. The EET rate between the two complexes is then approximated by multiplying $rLH1→RC$ by $ω¯RC=14∑n∈RCωn$⁠, the average energy gap of the RC pigments. We assume the weak bath limit such that Eq. (20) does not contain cross terms between the interactions with the phonon (⁠$Lph$⁠) and photon baths (⁠$Lrad$⁠). We note that certain spurious terms in Eq. (25), proportional to the system–bath coupling strength, arise if a secular approximation is invoked, as was demonstrated by Ishizaki et al.⁴² This is not the case since a full nonsecular treatment of the baths is employed here.

Our concern is with the EET processes under realistic biological conditions, with the incident solar radiation characterized by incoherent thermal photon states. The overall process occurs in the steady state limit. This is the case since the turn-on time of solar radiation, e.g., sunrise or the passage of a cloud, is many orders of magnitude longer than the typical time scale of the molecular processes involving electronic excited states. Hence, the relevant LH1 to RC transfer rate is obtained in the steady state by taking $ρ=ρss$ in Eq. (25), where $ρss$ satisfies $∂ρss/∂t=0$⁠.

Another important metric describing the light-harvesting apparatus is the quantum yield $η$⁠, defined as the charge separation rate $rcs$ divided by the rate of photon absorption $rabs$⁠,

Note that since no other excited state population quenching is included, we have the following equality:

That is, the quantum yields of charge separation, fluorescence emission, and non-radiative recombination add up to unity.

The discussion below focuses on the EET rate from the antenna LH1 to the RC, where the former is defined by the 32 BChl molecules that surround the RC and the latter by four BChl molecules in the center of the complex (Fig. 1). The center two BChl molecules (the first and the fourth BChl in the 3wmm PDB file convention) are referred to as the special pair, where charge separation occurs. Quantum chemical calculations are employed to calculate the system Hamiltonian [Eq. (1)], while other relevant parameters for the phonon bath, $kcs$ and $kto$⁠, are taken from the literature.^43,44 The numerical values of all parameters are provided in Tables I and II.

Consider then that the steady state EET rate from the LH1 to the RC is obtained by solving for the null space of Eq. (6) and collecting null vectors with finite overlap with the initial condition $ρ^=|0o0o|$⁠, i.e., where all population is in the ground state. In all cases, only a single null vector was found. This procedure provides a direct route to obtaining the steady state for cases involving a complete cycle—i.e., production of the product with associated return to the initial state. For cases where interest is in observables whose time scale is shorter than a full cycle, alternative methods, such as those developed in Ref. 49, must be used.

A great deal can be learned about the function of LH1–RC by considering variations with the incident light intensity. Figure 2(a) shows the calculated LH1 to RC transfer rate (dashed line) [see Eq. (25)] and the photon absorption rate (solid line) $rabs$ as functions of $C$⁠. While the absorption rate depends linearly on $C$ throughout, the transfer rate scales linearly at lower $C$ values and then plateaus at a value close to the RC recovery rate³⁹ (dashed line) when $C∼10−6$⁠. This is referred to as the transition from active to saturated photosynthesis^36,39 as the charge transfer rate fails to keep up with an increase in the photon absorption rate. Hence, $kto$ provides that rate-determining step. In Fig. 2(b), the steady state populations of the four manifolds, open/closed, ground/excited states, as well as the quantum yield, are shown as functions of $C$⁠. While the dependence of the quantum yield on $C$ reflects that of the rate and the open ground state population, the $C$ dependence of the closed ground state (dotted line) clearly demonstrates that the saturated regime can be further divided into two: $10−6<C≤10−2$⁠, where the population is predominantly in the closed ground state, and (the physically unrealistic regime) $10−2<C≤1$⁠, where significant closed excited state population is recorded. The filled circle at the right-hand side of Fig. 2(b) marks the projected thermal population of the closed ground state $P0c=⟨0c|ρ^th,rad|0c⟩$ at $T=Trad=5800$ K, which is a lower bound to the closed ground state population under the current single-exciton assumption.

It is helpful to define the rate of photon absorption in units of normalized single BChl absorption rate $r0$⁠,

where $P0c(β)=1eβE0−1$ with $β=1/T$ is the closed ground state thermal population, assuming that the LH1 is decoupled from the RC. Taking the transition energy $E0=900$ nm and the transition dipole moment $μ0∼15.6$ Debye, we have $r0∼1.4⋅107$$s−1$⁠. Assuming that $rabs$ is proportional to $NabsC$⁠, we can define the number of absorbers $Nabs$ as

Figure 2(c) shows the dependence of $Nabs(C)$⁠. In agreement with the observations in Fig. 2(b), here the three regimes can be clearly identified: First, in the active photosynthesis regime, the system is predominantly in the open ground state with all 36 BChls of the LH1–RC complex contributing to photon absorption. Second, in the first saturated regime $10−6<C≤10−2$⁠, only the 32 BChls are available for photon absorption. Finally, in the second saturated regime $10−2<C≤1$⁠, the system is in the steady state at such a high photon intensity, approaching the thermal state at the temperature of the surface of the Sun. Here, the number of absorbers continues to shrink due to the considerable excited state populations.

While the overall excited state population remains negligible until the second saturated regime (⁠$≈$0.3% at $C=10−2$⁠), the results presented here in the high light intensity regime could be inaccurate. For example, we caution that the numerical values of $Nabs$ reported in this regime can be unreliable since many-particle processes such as exciton–exciton scattering and annihilation, not included in the current model, play increasingly important roles.⁵⁰ In addition, we note that the current analysis excludes the consideration of higher energy electronic transitions as well as the contributions from pigment molecules other than BChl. Hence, the values of photon rate absorption obtained here serve as lower bounds for real systems where there are other pigment molecules and higher electronic transitions.

The transition from active photosynthesis under a weak illumination condition to saturated photosynthesis under a strong light condition³⁸ is consistent with the rate model employed by Fassioli et al.³⁹ who modeled the EET dynamics on native photosynthetic membranes of Rhodospirillum photometricum. Our single LH1–RC complex can be interpreted as an extreme limit of a functioning photosynthetic membrane under a high-light growing condition, since in that case, the membrane consists exclusively of LH1–RCs, with the difference that intercommunication between different LH1–RC complexes is excluded in the current model.

Under the solar intensity in the natural habitat of purple bacteria, we estimate that the effective $C$ factor is between $10−7$ and $10−6$⁠, near the critical value where the rate-determining step undergoes a transition from photon absorption to RC recovery. In this active regime of $C<10−6$⁠, the rate-determining step is the photon absorption, since the majority of the population is in the ground state and the RC is open and available. On the other hand, in the saturated regimes in Fig. 2, the RC recovery becomes the rate-determining step, and the excess excited state population drains back to the ground state through radiative decay. Note that non-radiative quenching pathways included in this treatment can also contribute to this loss of excited state population.

In real systems, the organism adapts to the light intensity by adjusting the relative stoichiometry between the peripheral antenna LH2 and the LH1–RC complexes.^51,52 For example, in Rsp. photometricum, the LH2/LH1 ratio is $4∼5$ for bacteria grown under low light intensity (10 W/$m2$⁠). This ratio corresponds to $Nabs≈150$ for such systems, taking the number of BChls to be 27 for an LH2 complex. In this case, we predict, by redoing the computation for this larger system, that the active to saturated transition would occur at $C≈10−7$⁠. We note that these considerations ignore the spectral distribution of the attenuated photons and absorption by other functional pigments in the light-harvesting membrane.

The existence of the transition between the first and the second saturated photosynthesis regime (⁠$C=10−2$⁠) in Fig. 2, albeit in a strong intensity regime, is worth noting as well. In both of the saturated regimes, the charge-separation rate and therefore the LH1 to RC transfer rate are bounded at the maximal value. However, in the first saturated regime (⁠$10−6<C≤10−2$⁠), the overall excited state population is negligible since the dissipation channels (in this case, spontaneous emission and the non-radiative recombination) dominate over the photon absorption. This implies that the system, while saturated, is not exposed to excited state chemistries and the resulting potential photodamages. In contrast, in the second saturated regime, where significant excited state population exists due to the high photon influx, additional photoprotection mechanisms might be necessary,⁵³ which is out of the scope of the current discussion.

The dependence of the transfer rate on the means of preparation of the donor population is not surprising.⁵⁴ Qualitatively, if the donor–acceptor transfer rate is faster than the time it takes to populate the donor, as is the case in natural light-harvesting, then the transfer rate is limited by the donor pumping rate. Provided that the pumping rate is also slower than the charge separation and the recovery processes, a steady state can be reached with the former as the rate-determining step. On the other hand, if the incident light intensity, which is proportional to the $C$ value in our model, is high enough such that the pumping rate is larger than any of the succeeding processes, the EET rate would be bounded by the slowest one among them. Further discussion on the dynamical details of the establishment of the steady state under the sudden turn-on condition is presented in Sec. III B.

The above discussion applies to the case of natural incoherent light. However, additional aspects of the behavior of LH1–RC can be obtained from alternative excitation scenarios. Consider now sudden turn-on of incoherent light for various values of $C$⁠. Figure 3 shows the sequence of resultant quasi-stationary states by displaying the temporal evolution of the LH1–RC transfer rate for several $C$ values. Starting from the bottom curve, where $C=10−9$⁠, the overall rate-determining step is the rate of photon absorption, and the system develops into the steady state at $1∼10$ ns, the lifetime of BChl Qy state. With an increase in $C$⁠, the transfer rate scales linearly as long as all subsequent processes (charge separation and RC recovery) are faster than the rate of photon absorption. This continues until the RC recovery becomes the next rate-determining step, and the dynamics undergoes a transition to a new steady state. The EET rate at this stage plateaus at a value fixed by the RC recovery rate, although the maximum transient rate continues to increase with an increase in $C$⁠.

In the current setup under conditions of the natural environment, the light–matter interaction is treated as that of a thermal radiative bath, and the upper bound to the LH1–RC transfer rate is determined by the thermal distribution established within the open RC manifold, weighted by the oscillator’s strength of each state within the LH1. However, it is readily conceivable that in the case of pulsed laser excitation, where a significant excited state population can be established on a femtosecond time scale or shorter, the transient transfer rate would continue to increase until the transfer itself becomes the rate-determining step. This corresponds to the premise of the generalized Förster resonance energy transfer (FRET) theory, where the donor is assumed to be excited electronically and thermally relaxed vibrationally. The transfer rate obtained under such circumstances is an upper bound to the transient quasi-stationary states and is very different than those under the normal operation of the light-harvesting apparatus in biology.

In the case of pulsed laser studies of generic donor–acceptor EET dynamics, the donor is prepared in an excited state on a time scale much shorter than the interaction between the donor and the acceptor. For the current LH1–RC model, we can simulate this by starting the dynamics with the pure state of the brightest LH1 singly excited state and setting $C=0$ [see Eq. (18)]. The results of this computation are shown in Fig. 4.

As shown in Fig. 4(b), the sub-picosecond time scale shows strong oscillations of populations between the LH1 and the RC, consistent with an initial condition that is not an eigenstate of the full system under weak damping with the environment. Thus, the EET rate oscillates in sign in the $ps−1$ range, a result in agreement with the experimentally measured EET rate of $∼0.1ps−1$⁠.⁵⁵ The rate drops to a few $μ$s on the $10∼100$ ps scale with all oscillations damped and vanishes at longer times where all the population is in the CS state. These results support our general observation regarding the correlation between the observed EET rate and the character of the excitation that prepares the state. It also serves as the extreme case where the donor LH1 excited state manifold is instantaneously populated, which is far from the case in the natural environment of the light-harvesting PPC.

The crucial dependence of system dynamics on the nature of the incident light is not restricted to issues in biological light-harvesting, as discussed elsewhere (see Ref. 54). An effort to understand the complete picture of physical/biological processes involving light–matter interactions is indisputably assisted by the development and application of the pulsed laser techniques. This effort indeed benefits enormously from the technological developments of increasingly shorter and more controllable laser pulses. However, as these results and others make clear,⁵⁴ there is a dramatic distinction between phenomena probed with the ever-shortening laser pulses and the behavior of systems in their normal steady state operating environments. Rather, the pulsed laser results provide vital input into steady state modeling.

Here, excitation energy transfer (EET) in the LH1–RC complex in purple bacteria is examined under steady state incoherent light–matter interaction conditions that are representative of the EET in the natural bacterial habitat. The intensity of the incident radiation is controlled by linearly scaling the photon occupation number throughout the blackbody spectrum. The transfer rate is shown to be bounded from above by the rate of recovery of the reaction center (RC) from the charge-separated closed state. This occurs on a ms time scale, while the quantum yield is predicted to be near unity within the range of light intensities in the ambient environment. Differences between the steady state results and those obtained under pulsed laser excitation are seen to be related via a sequence of quasi-stationary states where, at each step, the rate is determined by distinct physical processes that are the slowest on the corresponding time scales.

In the context of donor–acceptor energy transfer, this variation of rates correlates with the manner of preparation of the donor population. For example, the dynamics where the LH1 is instantaneously prepared in its excited state manifold was calculated and showed an EET rate comparable to those measured by the pulsed laser experiments. Such rates are seen to play a rather secondary role in the natural EET process.

Our results should motivate experimental studies of LH1–RC dynamics excited by natural incoherent light, and the low probability of absorption in such cases suggests studies that use averaging over pulsed laser results, an approach that we introduced elsewhere⁵⁶ and discussed by Kassal et al.³⁴ At present, the relevant experimental studies to which our results may be compared are those of Timpmann et al.³⁶ Although they used CW laser excitation, they measured the steady state fluorescence quantum yield as a function of excitation intensity in native photosynthetic membranes of Rhodobacter sphaeriodes.⁵⁷ Their experiments utilized membranes grown under different illumination levels and, therefore, involved various systems with a variety of LH2/LH1 ratios. Given our focus on the LH1, our study corresponds to the limit where the LH2/LH1 ratio approaches zero, i.e., to a membrane grown under very strong light conditions. In that case, our results agree qualitatively with the experiment. In particular, they found that the fluorescence emission (charge separation) quantum yield is increased (decreased) in the saturated regime compared to that in the active regime, in accordance with Eq. (28). This result can be understood as fluorescence emission being the dominant pathway in quenching the excited state population in the saturated regime, a dominance that fades away when more RCs are in the open state, capable of carrying out charge separation.

This work was partially supported by the U.S. Air Force Office of Scientific Research under Contract No. FA9550-17-1-0310 and by the NSERC, Canada.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.