  Photosynthetic organisms rely on a network of light-harvesting
protein-pigment complexes to efficiently absorb sunlight and transfer
excitation energy to reaction center proteins for charge separation. In
photosynthetic purple bacteria, these complexes are embedded in the cell
membrane, where lipid composition affects their clustering and inter-complex
energy transfer. However, the lipid bilayer's impact on intra-complex
excitation dynamics is less understood. Recent experiments compared
photo-excitation dynamics in detergent-isolated light harvesting complex 2
(LH2) to LH2 embedded in membrane discs mimicking the biological environment,
revealing differences in spectra and intra-complex energy transfer rates. We
use available quantum chemical and spectroscopy data to develop a complementary
theoretical study on the excitonic structure and intra-complex energy transfer
kinetics of the LH2 from photosynthetic purple bacteria Rhodoblastus
acidophilus in two conditions: LH2 in a membrane environment and
detergent-isolated LH2. Dark excitonic states crucial for B800-B850 energy
transfer within LH2 are found to be more delocalised in the membrane model.
Using non-perturbative and generalised F\"orster calculations, it is shown that
the increased quantum delocalisation leads to a B800 to B850 transfer rate 30%
faster than in the detergent-isolated complex, consistent with experimental
results. We identify the main energy transfer pathways in each environment and
show how differences in the B800 to B850 transfer rate stem from changes in
LH2's electronic properties when embedded in the membrane. By considering
quasi-static variations of electronic excitation energies in LH2, we show that
the broadening of the B800 to B850 transfer rate distribution is affected by
lipid composition. We argue that the variation in broadening could indicate a
speed-accuracy trade-off, common in biological systems.
