Computation of Förster Resonance Energy Transfer in Lipid Bilayer Membranes

Calculations of Förster Resonance Energy Transfer (FRET) often neglect the influence of different chromophore orientations or changes in the spectral overlap. In this work, we present two computational approaches to estimate the energy transfer rate between chromophores embedded in lipid bilayer membranes. In the first approach, we assess the transition dipole moments and the spectral overlap by means of quantum chemical calculations in implicit solvation, and we investigate the alignment and distance between the chromophores in classical molecular dynamics simulations. In the second, all properties are evaluated integrally with hybrid quantum mechanical/molecular mechanics (QM/MM) calculations. Both approaches come with advantages and drawbacks, and despite the fact that they do not agree quantitatively, they provide complementary insights on the different factors that influence the FRET rate. We hope that these models can be used as a basis to optimize energy transfers in nonisotropic media.


Contents
References S11 S2 S1 Additional computational details S1.1 Generation of initial geometries Long aliphatic chains can adopt numerous different conformations, rendering it insufficient to merely compute the properties of one conformer. 1-3 Therefore, structures representative of the thermally-accessible conformational ensemble for PDI-C4 and Ru-bpyC9 are generated using the conformer-rotamer ensemble sampling tool (CREST). 4,5 In CREST, the generated structures are optimized using the geometry, frequency, non-covalent extended tight binding model of version 2 (GFN2-xTB). 6 For the sake of computational efficiency, the structure crossing step within the standard CREST workflow, which generates new conformers as difference structures between two already existing conformers, 7 is omitted for the Ru-bpyC9 calculation. Where noted, solvent effects are included implicitly using the analytical linearized Poisson-Boltzmann model. 4,5 Usually, it is sufficient to consider all conformers within an energy range of 3 kcal/mol with respect to the global minimum as -according to the Boltzmann distribution for an idealized, non-degenerate two level system -the population of a conformer that is 3 kcal/mol less stable compared to the lowest energy conformer, would be well below 1%. However, to compensate for inaccuracies in the semiempirical tight-binding GFN2-xTB method on which the conformers are optimized and energies are computed within the CREST workflow, all conformers within an energy window of 7 kcal/mol are included in the initial steps of the ensemble generation.
To reduce the vast ensembles generated with CREST (>38,000 conformers for Ru-bpyC9, structures is reduced to a final 100 for PDI-C4, and 92 for Ru-bpyC9, respectively. Prior to computing the excited states for these representative structures, the geometries are reoptimized utilizing the more accurate B3LYP 9,13 /def2-SVP setup described below.

S1.2 Molecular dynamics simulations
All MD simulations are performed using the program packages Amber20 and AmberTools21. The minimized system is subsequently heated using the Langevin thermostat at a collision frequency of 1.0 ps −1 to 100 K in 2,500 time steps (5 ps), followed by a heating to 300 K in 50,000 steps (100 ps). The first heating is performed at a pressure relaxation time of 1 ps, which is increased to 2 ps for the second heating phase only. Prior to production, the system is equilibrated by running 10 ps of simulation time (5,000,000 time steps).
The analysis of the simulation trajectories is performed using the program CPPTRAJ. 25 Visualization is done using the molecular viewer VMD. 26 The distance to the center of the membrane is computed as the center-of-mass distance between the core atoms of the chromophores, i.e., all non-hydrogen atoms excluding the aliphatic tails, and the C 18 atoms of S5 the DOPG lipids. The C 18 atoms are suitable for defining the center of the membrane, as they are the final carbon atoms in the hydrophobic tail, and thus can be expected to be furthest away from the membrane surface. To ensure that this corresponds to the minimum distance between the chromophore and the membrane center and not some artificially elongated distance misrepresented by lateral movement of the ligand within the membrane, the chromophore is imaged into the center of the periodic box for each frame during the analysis.
In all systems, the membrane is assembled and remained in the xy-plane, so that the angle to the surface of the membrane is evaluated as the angle w.r.t. the xy-plane.

S2.1 Synthesis of chromophores
The Ru-bpyC9 complex was synthesized according to a previously published procedure. 27 The synthesis of PDI-C4 was adapted from a similar compound and is reported below. 28 S2.1.1 Synthesis of N,N'-di(butylenedimethylamine)-3,4,9,10-perylenediimide In 20 mL of isobutanol PTCDA (0.1 g, 0.25 mmol, 1 eq.) and (4-aminobutyl) dimethylamines (0.14 mL, 1.02 mmol, 4 eq.) were combined, stirred and heated at 90 • C for 24 h under argon atmosphere. After the mixture was cooled to room temperature, the crude product was collected by filtration. To remove unreacted PTCDA, to the mixture was added 5% aqueous NaOH solution and stirred at 90 • C for 30 minutes. The product was separated from solvent by filtration and washed with water and ethanol. The red solid powder was then dried under vacuum to give 0.14 g (95% yield) product.  Figure S1: HOMO and LUMO of PDI-C4 on ground state and S 1 excited state geometry.
As visible from the orbitals, the S 0 →S 1 transition is of ππ * character.

S3.2 Ru-bpyC9
The transitions from S 0 to S 5 , S 6 , S 7 , and S 8 are mainly composed of combinations of the highest three occupied molecular orbitals, to the lowest three unoccupied MOs. These are shown below. In the HOMOs, electron density is mostly localized on the metal center, while