^{1,a)}

### Abstract

Photosynthesis, the process by which energy from sunlight drives cellular metabolism, relies on a unique organization of light-harvesting and reaction center complexes. Recently, the organization of light-harvesting LH2 complexes and dimeric reaction center-light-harvesting I-PufX core complexes in membranes of purple non-sulfur bacteria was revealed by atomic force microscopy [S. Bahatyrova *et al.*, Nature (London)430, 1058 (2004)]. Here, we discuss optimal exciton transfer in a biomimetic system closely modeled on the structure of LH2 and its organization within the membrane using a Markovian quantum model with dissipation and trapping added phenomenologically. In a deliberate manner, we neglect the high level detail of the bacterial light-harvesting complex and its interaction with the phonon bath in order to elucidate a set of design principles that may be incorporated in artificial pigment-scaffold constructs in a supramolecular assembly. We show that our scheme reproduces many of the most salient features found in their natural counterpart and may be largely explained by simple electrostatic considerations. Most importantly, we show that quantum effects act primarily to enforce robustness with respect to spatial and spectral disorder between and within complexes. The implications of such an arrangement are discussed in the context of biomimetic photosynthetic analogs capable of transferring energy efficiently across tens to hundreds of nanometers.

The author would like to thank Mark Ratner for proofreading the manuscript and providing helpful feedback.

I. INTRODUCTION

II. THEORY

III. RESULTS AND DISCUSSION

IV. CONCLUSION

## Figures

Light-harvesting apparatus in purple bacteria. Illustration of spherical chromatophore vesicle from *Rh. sphaeroides* showing organization of light-harvesting complexes, LH2, and light-harvesting-reaction center complex (LH1-RC). Architecture and arrangement of constituent chromophores based on AFM images. Reprinted with permission from M. Sener, J. Olsen, C. Hunter, K. Schulten, Proc. Natl. Acad. Sci. U.S.A., Vol. 104, Page 15723, 2007.

Light-harvesting apparatus in purple bacteria. Illustration of spherical chromatophore vesicle from *Rh. sphaeroides* showing organization of light-harvesting complexes, LH2, and light-harvesting-reaction center complex (LH1-RC). Architecture and arrangement of constituent chromophores based on AFM images. Reprinted with permission from M. Sener, J. Olsen, C. Hunter, K. Schulten, Proc. Natl. Acad. Sci. U.S.A., Vol. 104, Page 15723, 2007.

Dipole moment orientation used in these models is shown in top right. Optimal number of elements per ring as found by a genetic algorithm. Average trapping time as a function of the number of elements, *N*, in each ring. The diameter of each ring is 5 nm. The top row of images shows the strength of electrostatic coupling (lines between sites). Dashed lines correspond to >5 cm^{−1} and <10 cm^{−1}, dotted lines to >10 cm^{−1} and <20 cm^{−1}, and thick lines to >20 cm^{−1}. Transition dipole at each site is normal to the plane. Color of connecting lines indicates the dephasing rate between two sites. Color of circles at each site corresponds to their energies. The bottom row of images shows the average residence time at each site, indicated by the color of the circles. The average trapping time is the sum of the average residence times at each site. The donor and trap states are labeled.

Dipole moment orientation used in these models is shown in top right. Optimal number of elements per ring as found by a genetic algorithm. Average trapping time as a function of the number of elements, *N*, in each ring. The diameter of each ring is 5 nm. The top row of images shows the strength of electrostatic coupling (lines between sites). Dashed lines correspond to >5 cm^{−1} and <10 cm^{−1}, dotted lines to >10 cm^{−1} and <20 cm^{−1}, and thick lines to >20 cm^{−1}. Transition dipole at each site is normal to the plane. Color of connecting lines indicates the dephasing rate between two sites. Color of circles at each site corresponds to their energies. The bottom row of images shows the average residence time at each site, indicated by the color of the circles. The average trapping time is the sum of the average residence times at each site. The donor and trap states are labeled.

Chain of rings versus chain of individual chromophores. Comparison of the disorder between rings (left) and between chromophores (right) in a linear arrangement. Total distance from donor to trap is approximately the same in each case (∼32 nm). Δr is the maximum, random displacement in both the x and y direction.

Chain of rings versus chain of individual chromophores. Comparison of the disorder between rings (left) and between chromophores (right) in a linear arrangement. Total distance from donor to trap is approximately the same in each case (∼32 nm). Δr is the maximum, random displacement in both the x and y direction.

Rings are robust to spatial disorder. Plot of average trapping time versus static disorder between rings (red) and single sites (blue) from Figure 3. Error bars for ring arrays are based on five runs through the optimization code. Error bars for array of single sites based on 20 runs through the optimization code. Outliers larger than 5*σ* were removed from the analysis. Single site arrays were fit to third-order polynomial, while ring arrays were fit to a least squares regression line. Inset shows a narrower window of trapping times (maximum of 400 ps). Arrows on the left of the inset indicate the mean residence time with no spatial disorder between rings. The green bar indicates approximate excited-state lifetime of bacteriochlorophyll *a*, which represents the upper limit of relaxation of the sites back to the ground state. The linear array of sites performs better than the linear array of rings with no spatial disorder, but is significantly less robust to imperfections in positioning.

Rings are robust to spatial disorder. Plot of average trapping time versus static disorder between rings (red) and single sites (blue) from Figure 3. Error bars for ring arrays are based on five runs through the optimization code. Error bars for array of single sites based on 20 runs through the optimization code. Outliers larger than 5*σ* were removed from the analysis. Single site arrays were fit to third-order polynomial, while ring arrays were fit to a least squares regression line. Inset shows a narrower window of trapping times (maximum of 400 ps). Arrows on the left of the inset indicate the mean residence time with no spatial disorder between rings. The green bar indicates approximate excited-state lifetime of bacteriochlorophyll *a*, which represents the upper limit of relaxation of the sites back to the ground state. The linear array of sites performs better than the linear array of rings with no spatial disorder, but is significantly less robust to imperfections in positioning.

Grid-like array of chromophores is highly inefficient. Excitonic transfer through a grid-like arrangement of sites after optimization. Left: Colors of circles represent site energies. Color of lines represent dephasing rate. Right: Color represent mean residence time at each site. A significant amount of time is “wasted” at sites that do not directly link the donor to acceptor, i.e., sites 7, 13, and 19.

Grid-like array of chromophores is highly inefficient. Excitonic transfer through a grid-like arrangement of sites after optimization. Left: Colors of circles represent site energies. Color of lines represent dephasing rate. Right: Color represent mean residence time at each site. A significant amount of time is “wasted” at sites that do not directly link the donor to acceptor, i.e., sites 7, 13, and 19.

Two-dimensional packing is more robust to spatial disorder than linear chains. Electrostatic coupling between staggered (left) versus linear (right) in the presence of spatial disorder between rings for a 7-ring system. Thin lines indicate weak coupling (>5 cm^{−1} and <10 cm^{−1}), medium lines indicate intermediate coupling (>10 cm^{−1} and <20 cm^{−1}), and thick lines indicates strong coupling (>20 cm^{−1}). Staggered arrangement maintains non-negligible coupling strength and hence a path from donor to acceptor through multiple, neighboring rings. Linear arrangement more easily forms breaks, which may effectively block energy transfer across large distances. Bottom right: spectrum calculated by diagonalizing the system Hamiltonian. Energy spans approximately 200 cm^{−1} in each case—a major role of coupling is to break site degeneracy and broaden the spectrum for efficient absorption of solar flux.

Two-dimensional packing is more robust to spatial disorder than linear chains. Electrostatic coupling between staggered (left) versus linear (right) in the presence of spatial disorder between rings for a 7-ring system. Thin lines indicate weak coupling (>5 cm^{−1} and <10 cm^{−1}), medium lines indicate intermediate coupling (>10 cm^{−1} and <20 cm^{−1}), and thick lines indicates strong coupling (>20 cm^{−1}). Staggered arrangement maintains non-negligible coupling strength and hence a path from donor to acceptor through multiple, neighboring rings. Linear arrangement more easily forms breaks, which may effectively block energy transfer across large distances. Bottom right: spectrum calculated by diagonalizing the system Hamiltonian. Energy spans approximately 200 cm^{−1} in each case—a major role of coupling is to break site degeneracy and broaden the spectrum for efficient absorption of solar flux.

Near-optimal tapping is achieved without fine-tuning the system and bath. Left: Exciton transfer optimization achieved by a genetic algorithm to minimize the average trapping time as a function of the mutual dephasing between sites, trapping rate, and site energies. In the case of three rings in this arrangement, the optimal trapping time was found to be 63 ps. Right: Keeping the dephasing between sites constant and the site energies identical, the optimal trapping time is found to be ∼70 ps. This indicates that fine-tuning of the system and the system-bath interactions is not necessary to achieve near-optimal transfer efficiency.

Near-optimal tapping is achieved without fine-tuning the system and bath. Left: Exciton transfer optimization achieved by a genetic algorithm to minimize the average trapping time as a function of the mutual dephasing between sites, trapping rate, and site energies. In the case of three rings in this arrangement, the optimal trapping time was found to be 63 ps. Right: Keeping the dephasing between sites constant and the site energies identical, the optimal trapping time is found to be ∼70 ps. This indicates that fine-tuning of the system and the system-bath interactions is not necessary to achieve near-optimal transfer efficiency.

Quantum transport is robust to static energetic disorder. Comparison of classical and quantum transport in the absence (a) and presence (b) of static energetic disorder—difference in energies at each site. (a) When the site energies are degenerate, classical transport predicts a shorter trapping time—near zero line width. Quantum transport predicts a slower trapping time by about a factor of three, but at a modest value of the dephasing rate. (b) When the site energies are non-degenerate, classical transport undergoes a dramatic shift in the optimal line width. For the quantum case, changes in the optimal dephasing rate and trapping rate are negligible. In this case, the quantum transport is faster and significantly more robust to changes in site energies.

Quantum transport is robust to static energetic disorder. Comparison of classical and quantum transport in the absence (a) and presence (b) of static energetic disorder—difference in energies at each site. (a) When the site energies are degenerate, classical transport predicts a shorter trapping time—near zero line width. Quantum transport predicts a slower trapping time by about a factor of three, but at a modest value of the dephasing rate. (b) When the site energies are non-degenerate, classical transport undergoes a dramatic shift in the optimal line width. For the quantum case, changes in the optimal dephasing rate and trapping rate are negligible. In this case, the quantum transport is faster and significantly more robust to changes in site energies.

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