Article pubs.acs.org/JPCB
A Theoretical Investigation into the Effects of Temperature on Spatiotemporal Dynamics of EET in the FMO Complex Colm G. Gillis and Garth A. Jones* School of Chemistry, University of East Anglia, Norwich Research Park, Norwich, Norfolk NR4 7TJ, United Kingdom S Supporting Information *
ABSTRACT: Methodologies are presented in which population dynamics are evolved in the exciton basis and spatiotemporal movement of excitations is subsequently obtained by projection to the site basis. Fluctuations of system eigenstates are explicitly included through vibrations of the chromophores, which are parametrized by ab initio calculations. Two limiting cases of dynamics are considered, namely, the incoherent regime, where state populations correspond to ensembles of classical Landau−Zener (LZ) trajectories, and the coherent regime, where the density matrix is propagated by the quantum Liouville equation (QLE). For QLE simulations, population dynamics show that bacteriochlorophyll a1 and a2 effectively act as a single unit at 77 K but as independent chromophores at 300 K. Population beatings for the lower energy exciton states are considerably slower at physiological temperatures, thus assisting transfer to the sink. Results from LZ trajectories indicate that, within the classical picture, higher temperatures result in a lower probability of the exciton reaching the sink. A broadening of the excitonic spectrum at high temperature alters the pathways of the excitons in the LZ formalism and also increases the possibility of trapping. This study supports the view that a coherent mechanism may assist EET at physiological temperatures since the trapping of excitations in intermediate energy sites is prevented. Furthermore, delocalized vibrations (i.e., superpositions of independent oscillators) are found to assist energy transfer at short times.
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INTRODUCTION Photosynthesis is the umbrella term for the complex chain of events that transpire when organisms such as higher plants or algae convert sunlight into biochemical products. During the primary step of the photosynthetic process, absorbed quanta of light energy undergo transfer to reaction centersthe step widely referred to as electronic energy transfer (EET).1−10 This stage of biophysical transduction within pigment−protein complexes (PPCs) is of particular interest to researchers because quantum yields often exceed 0.9.2−4,7,11−13 Consequently, a detailed understanding of the high efficiency occurring in these natural solar harvesting systems is important within the context of humankind’s search for renewable sources of energy.14,15 Since the 1990s, signatures of quantum behavior have been observed in several species where photosynthesis is used to drive cellular processes.16 Time-resolved measurements of EET within PPCs have revealed ensemble distributions of excitonic populations that may be described by quantum-mechanical probabilities.10,12,17−23 Efficiencies of such systems can be understood outside a purely classical framework.24−26 Such conclusions are controversial, nonetheless.10,26 Previous models of EET in the FMO complex have been developed in order to describe interactions between system and bath. These can be used to avoid the limitations of Redfield27 or Förster28 theory29−32 or to quantify the back-reaction of system on environment,33 for example. These formalisms account for system−bath interactions by modeling trajectories as non-Markovian processes. © 2015 American Chemical Society
Ab initio dynamics (i.e., atomistic dynamics) studies allow one to explicitly incorporate intramolecular vibrations whose time-dependent fluctuations can be incorporated into system Hamiltonians via QM/MM simulations or theoretical models.22,33−37 In many of these models, coherent oscillation of the exciton populations are often suppressed, for example in ref 34. However, one study has reported an atomistic model that has reproduced oscillatory features of physiological EET in the FMO complex. Also, it was shown that under certain temperature regimes there is a need to account for vibrational modes in energy transfer.22 The motivation for this work is in developing a model of EET that can be used to investigate the role of temperature on population dynamics of exciton states and spatial pathways in FMO. This is achieved by introducing energy fluctuations through a normal mode sampling procedure that incorporates intramolecular vibrational modes of the chromophores. The normal modes of the chromophores are perturbed through thermal sampling of the environmental modes via an analytical spectral density. The procedure reported here involves initializing the trajectories by incorporating vibrations into the individual chromophores (giving rise to fluctuating S0 → S1 energy gaps) and then diagonalizing the system Hamiltonian to evolve the population dynamics in the exciton basis.38−40 We note a study by Wong et al.41 where it is stated that “an Received: September 9, 2014 Revised: February 20, 2015 Published: February 23, 2015 4165
DOI: 10.1021/jp509103e J. Phys. Chem. B 2015, 119, 4165−4174
Article
The Journal of Physical Chemistry B adiabatic description of energy transfer is most appropriate” for systems in the strong coupling limit (i.e., when cross peaks occur in experiments). Further, by employing the procedure described in this study, one can gain insight into the role that delocalized vibrations play in modulating the ensuing population transfer between exciton states. Analysis of the exciton transfer pathways between sites is then achieved by projecting back to the site basis by calculating overlap integrals between the site and exciton density of states, as a postprocessing procedure (i.e., it is not part of the dynamical routines). The methodology for generating site fluctuations at specific temperatures is of ad hoc character and is designed specifically so as to allow one to investigate population dynamics in the Fenna−Matthews−Olson (FMO) system.22,42,43 In this work exciton state 7 is initially populated, with the excitation delocalized over several sites. Other studies have explored the possibility of an initial superposition of site populations.34,44 For propagation of exciton populations, two limiting approaches are used. The first is a quasi-classical method based on the Landau−Zener (LZ) condition that approximates transition probabilities between states at an avoided crossing. The second approach involves direct propagation of the quantum Liouville equations (QLE) of motion. These methodologies can be thought of as models that simulate two limiting cases of exciton transport: fully incoherent and fully coherent transport, respectively. In both models, intramolecular vibrations of the BChl a chromophores are explicitly accounted for in simulations by parametrization of quasi-classical trajectories, as outlined in the Methodology section. Environmental modes are also incorporated into simulations by a thermal sampling procedure. The two models are used to compare and evaluate different regimes of EET. This amounts to a decomposition of two limiting cases, which cannot easily be resolved on short time scales. A numerical analysis of EET in both regimes is valuable for providing insight into the coherent and incoherent limits.
Figure 1. Pigments of the FMO monomer of species Chlorobaculum tepidum. Conventional numbering is used.
distribution function of the ground state quantum harmonic oscillator. P0(x) =
⎛ mωx 2 ⎞ ⎛ mω ⎞1/2 ⎜ ⎟ exp⎜ − ⎟ ⎝ πℏ ⎠ ℏ ⎠ ⎝
(1) 47
For each trajectory, a Box−Muller algorithm was used to sample displacements along the each of the normal mode coordinates. Velocities were then calculated from the resulting kinetic energy (i.e., EKIN = EZPE − EPOT). Phases of normal modes were assigned randomly by means of a uniform deviate. Ab initio trajectories were then run in the electronic ground state using the Car−Parrinello type atom centered density matrix propagation (ADMP) molecular dynamics model at the B3LYP/3-21G level.46,48−51 Because of computational limitations, trajectories were run at the B3LYP/3-21G level. Although the 3-21G basis set is not ideal, it nevertheless allows explicit inclusion of molecular vibrations. Vibrational frequencies at this level of theory agreed well with those calculated using the 6-31G (d, p) basis set (see Supporting Information). At each point of the ADMP trajectories, the S0 → S1 energy gap, ΔE(t), corresponding to the Qy site energy for the chromophores was calculated using Zerner’s intermediate neglect of differential orbitals (ZIndo).46 Twenty such trajectories were run for 1 ps with different initial conditions to give a set of ΔE(t) vs t profiles for the BChl a monomers. FFTs were performed using a Lomb periodogram,52 for each of these data sets, to give a distribution of vibrational frequencies that are most strongly coupled to the S0 → S1 energy gap. These FFTs revealed that generally only a small set of intramolecular vibrational modes, typically 10−20 with a few high-intensity modes, significantly perturb the S0 → S1 energy gap. These FFTs formed the basis of the following analytical procedure that incorporated intramolecular vibrational modes explicitly into the simulations. (While it would be ideal to explicitly incorporate ab initio trajectories into the exciton dynamics methodology directly, 7000 ab initio trajectories
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METHODOLOGY Hamiltonian. Simulations assume a single exciton manifold comprising linear combinations of Qy transition (S0 → S1) states for each of the seven chromophores. Numerical values for energies and couplings were taken from the empirically derived site Hamiltonian reported by the Engel group (reproduced in the Supporting Information), and fluctuations of site energies are introduced, as described below.45 Exciton energy levels were then calculated on-the-fly by diagonalization of the site Hamiltonian. It is important to emphasize that while the site Hamiltonian is used for the incorporation of fluctuations, the resulting exciton Hamiltonian is used in the population dynamics. It is the transformation from site to exciton Hamiltonian that gives rise to delocalized vibrations. The structure of FMO can be seen in Figure 1. Site Fluctuations. Incorporation of intramolecular vibrations of the chromophores was achieved by analyzing FFTs of the energy gap fluctuations vs time for a set of individual ab initio trajectories on the BChl a monomer in the gas phase as detailed below. The ab initio trajectories were run using the Gaussian09 package.46 These trajectories were initialized by assigning zero-point energy to each of the normal vibrational modes (excluding the high-frequency stretching modes associated with hydrogen atoms). Initial canonical positions of atoms were assigned randomly using the probability 4166
DOI: 10.1021/jp509103e J. Phys. Chem. B 2015, 119, 4165−4174
Article
The Journal of Physical Chemistry B would be required for a 1000 trajectory ensemble of FMO and therefore beyond reasonable computational feasibility.) Fluctuations of the Hamiltonian site energies are introduced via the equations ⎛ DWi ⎞ ⎟ Ei(t ) = ⟨Ei⟩ + εi(t )⎜ ⎝ σ ⎠ N
εi(t ) =
(2a) N
∑ an |ei(ωn + φn)t |2 − ⟨∑ an |ei(ωn + φn)t |2 ⟩ n=1
n=1
(2b)
where ⟨Ei⟩ is the equilibrium value of the site energy for the ith chromophore (i.e., the site energy corresponding to a diagonal element in the Hamiltonian of ref 45). Terms in eq 2b account for intramolecular vibrations through the ab initio trajectories, described above, where an is the randomly assigned weighting of the nth normal mode, ωn is the frequency of the nth normal mode, and φn is a random phase selected from the environmental spectral density (see later in this section). Given that Qy site energies of the FMO complex are spread over a range of less than 500 cm−1, intramolecular vibrational modes of chromophores were selected from modes of magnitude
