FÃ¶rster Resonance Energy Transfer between Fluorescent Proteins

Article pubs.acs.org/JPCC

Förster Resonance Energy Transfer between Fluorescent Proteins: Efficient Transition Charge-Based Study Hirotaka Kitoh-Nishioka,*,†,§ Daisuke Yokogawa,†,‡ and Stephan Irle*,†,‡ †

Department of Chemistry, Graduate School of Science and ‡WPI-Research Initiative-Institute of Transformative Bio-Molecules, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan S Supporting Information *

ABSTRACT: Toward a better understanding of the Förster resonance energy transfer (FRET) utilized in genetically encoded biosensors we theoretically examined the excitonic coupling between cyan fluorescent protein (CFP) and yellow FP (YFP) with time-dependent density functional theory (TD-DFT). Going beyond the dipole−dipole (dd) approximation in the original Förster theory, we adopted a transition charge from the electrostatic potential (TrESP) method that approximates the excitonic coupling as classical Coulomb interaction between the transition charges derived from the transition density for each FP fluorophore. From the TD-DFT calculations with embedded point charges for the trajectory generated by classical molecular dynamics (MD) simulations we found that the thermal fluctuation of the fluorophore geometry in FP and the protein electrostatic interactions do not significantly affect the Coulomb interaction between the FP pairs. The TrESP calculations utilizing the Poisson equation indicate that the screening and local field effects by solvent dielectric environment reduce the Coulomb interaction by an almost constant factor of 0.51. Based on these results, we developed a more efficient Frozen-TrESP method that calculates the structure-dependent Coulomb interaction using the reference transition charges preliminarily determined for the isolated fluorophore in the gas phase and confirmed its validity for the evaluation of the Coulomb interaction in the thermally fluctuating CFP-YFP dimer. Finally, we demonstrated the usefulness of the Frozen-TrESP to examine the dependence of the Coulomb interaction on the alignment of YFP with respect to CFP and provide the list of the reference transition charges for the other representative fluorophores of various FPs, which offers guidance on the optimal design of the FRET-based biosensors.

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INTRODUCTION Green fluorescent protein (GFP), discovered in the jellyfish Aequorea victoria by Shimomura et al.,1 plays an essential role in live-cell imaging.2−6 Due to the success in the genetic cloning of GFP7 and the subsequent success in its expression and maturation in E. coli. and C. elegans,8 researchers have extensively used the GFP, its mutants,2 and GFP-like proteins cloned from other marine organisms9−11 as genetically encoded fluorescent labels of the target proteins to elucidate the intracellar molecular activities in real time and space, namely, gene expression, protein localization, protein−protein interaction, and so on. These fluorescent proteins (FPs) fold to form common 11-stranded β-barrel structures without the help of chaperones after their gene expressions. The FPs autocatalytically form their chromophores (that are as well as their fluorophores) from the three amino acids located at the positions corresponding to 65−67 in the wild-type GFP sequence in the presence of an oxygen molecule. Through the introduction of one or a few mutations at the 65−67 positions and/or the amino acids in their vicinity, the GFP attains remarkably different spectral properties such as emission color, brightness, maturation rates, photostability, and so on.2−6 Therefore, many theoretical studies employing quantum chemical (QC) methods with a wide range of computational costs and accuracy have been conducted to understand how the © XXXX American Chemical Society

GFP mutants and GFP-like proteins tune their spectral properties at the atomic scale.12−27 Among the applications of FPs to live-cell imaging, a genetically encoded biosensor based on Förster resonance energy transfer (FRET) between FPs is one of the most powerful tools to monitor the dynamics of intracellular enzyme activities with high spatial and temporal resolution.28−31 Intramolecular FRET-based biosensors, such as the famous Ca2+ indicator, “cameleon”,32 usually consist of a sensor domain sandwiched between two differently colored FPs. When the target small molecule or protein (namely, analyte) binds to or modifies the sensor domain, the biosensor undergoes conformation changes. The conformation changes modulate the distance and orientation between the two FPs, which are detected by a change of the FRET signal by fluorescent microscopes. However, many biosensors exhibit only a small change in FRET signal that is often buried in the background noise of the systems. To improve the dynamic range of the FRET signal change, adjustment of the linkers between the domains33 and modification of the orientation between the FPs by replacing the FPs with the corresponding circularly permuted FPs34 are routine.28,35 However, for now, these Received: January 26, 2017 Published: January 30, 2017 A

DOI: 10.1021/acs.jpcc.7b00833 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

direction of GFP determined only by the π-conjugated part of its fluorophore in the gas phase is in reasonable agreement with the experimental one72−74 and is insensitive to nonconjugated side chains or the protein environment. On the basis of these results, the authors made a list of the TDM directions of the other FPs by the same manner in which the TD-DFT calculations are performed only for the π-conjugated part of the respective fluorophore in the gas phase. Because they observed that the calculated TDMs almost lie along the long axis of the π-conjugated part of the fluorophore for each FP, they suggested that the same rule is applicable to newly developed FPs in the future. If the dd approximation was always valid for estimating the Coulomb interaction between FPs, the list of the TDM directions of FPs would assist the design of new biosensors through the adjustment of the relative orientation between the FPs. Generally, it is believed that the dd approximation is valid if the donor and acceptor fluorophore distance is large enough compared to the molecular size of the fluorophore. However, recent theoretical studies69,70 have shown that the error arising from the dd approximation for Coulomb interaction emerges in the long distance regions at the specific relative orientation. Moreover, when considering the effect of the structural dynamics of the system, Speelman et al.71 have found that the dd approximation based on the TDMs mapped onto the configuration space generated by molecular dynamics (MD) simulations significantly overestimates the FRET rate among the organic dyes attached to oligonucleotides or Lysozyme, compared to the LR approach. Therefore, we have conducted and present here the first theoretical study beyond the dd approximation on the Coulomb interaction in the FRET reaction between FPs. The system employed in this study is a cyan-fluorescence emitting mutant of GFP (CFP) and yellow-fluorescence emitting mutant of GFP (YFP) heterodimer, which is a typical FP pair used in FRET-based biosensor.28−31 Since the GFP and its mutants tend to form a dimer, many biosensors utilize the FRET reaction associated with the GFP dimerization propensity.35,75−79 Studying the FRET reaction in FP dimers is also important to understand the ultrafast fluorescent depolarization80 and anomalous negative fluorescence74 observed in YFP. To consider the balance between the accuracy and the computational resources, we employed the TrESP method;43 for example, the TrESP method with TD-DFT calculations can approximate Coulomb interaction of the EET in xanthorhodopsin obtained from the more advanced method with highly accurate QC calculations, TDFI with the symmetry-adapted cluster CI (SAC−CI).52 We determined the TDMs and transition charges for the evaluation of the Coulomb interaction in the CFP−YFP heterodimer by using the TD-DFT calculations performed for the π-conjugated part of the fluorophore of each CFP and YFP, by following the previous paper.36 We investigated the effect of the thermal fluctuation of the systems on the Coulomb interaction by performing the TDDFT calculations for the MD trajectory of the explicitly solvated CFP-YFP dimer. To consider the effects of electrostatic interaction of the protein environment, the TD-DFT calculations were performed under the molecular mechanics (MM) electrostatic field of protein and solvent. Moreover, we took account of the effect of the optical polarizability of the protein and solvent on the Coulomb interaction by combining the TrESP method with the Poisson equation involving the homogeneous dielectric continuum media.46−49

attempts are conducted by trial-and-error. Therefore, understanding of the FRET reaction in the biosensor at the atomic scale with QC methods should enable the improvement of the dynamical range of the FRET signal by assisting in the design of novel high-performance biosensors. However, in contrast to the studies on the spectral properties of FP,12−27 there is no QCbased theoretical study on the FRET between FPs in biosensor, except for ref 36. The FRET rate constant kFRET is expressed by the original Förster theory37 based on Fermi’s golden rule with the Condon assumption as follows37−39 kFRET =

2π 2 |V | ℏ

∫ dEIA(E)LD(E)

(1)

where the second term ∫ dEIA(E)LD(E) represents the spectral overlap between donor fluorescence and acceptor absorption and corresponds to the Franck−Condon factor weighted by nuclear density of states. The leading term of eq 1, V, is the excitonic coupling, calculated from the Coulomb interaction between one-electron transition densities of the donor (D) and acceptor (A) molecules. The Coulomb interaction remarkably depends on the distance and relative orientation between the D and A molecules. In FRET-based bioimaging, the conformational changes in the biosensors caused by the presence or absence of the analyte lead to the changes in the FRET rate through the Coulomb interaction. Therefore, understanding of the structure dependence of the Coulomb interaction between FPs is of importance for the optimal design of highperformance FRET-based biosensors. The Coulomb interaction in eq 1 is usually approximated as a dipole−dipole (dd) interaction between the transition dipole moments (TDMs) of the D and A molecules because the TDM is the first-order term of a multipole expansion of the transition density.37−39 In addition to the dd approximation, a random relative orientation of the TDMs is also often assumed, which results in the inverse sixth power of the distance dependence of the FRET rate and, therefore, offers the FRET reporter as a spectroscopic molecular ruler in the bioimaging.40 However, these FRET approximations for the Coulomb interaction are questionable, especially for instance in the case of understanding the mechanism of excitation energy transfer (EET) in the early process of photosynthesis. So far, various theoretical methods based on QC calculations beyond the dd approximations have been developed, including the transition density cube,41 transition charge from Mulliken population analysis,42 transition charge from the electrostatic potential (TrESP),43−49 the method based on analytical integrations of atomic orbitals, 50,51 transition density fragment interaction (TDFI),52−54 the linear response (LR) approach,55 LR approach with the polarizable continuum model (PCM),56−60 fragment excitation difference,51 and so on.61,62 In addition to the Förster’s Coulomb interaction, some of them51,54−62 can take account of Dexter’s exchange interaction and the overlap of the D and A orbitals that significantly contribute to the short-range EET. In contrast to electron transfer reactions,63−66 the bridge-mediated superexchange process negligibly contributes to the singlet EET.67,68 Whereas these theoretical methods have been used for the photosynthesis studies,41,43,45−48,52,58−60 the applications of them to the organic dyes for FRET reporter have been limited.69−71 Recently, Ansbacher et al.36 have systematically studied the TDMs of many kinds of FPs by using time-dependent density functional theory (TD-DFT). They have shown that the TDM B


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Figure 1. Workflow of the Frozen-TrESP and TrESP methods.

Based on our findings, we have proceeded to develop a more efficient approach within the framework of the TrESP method; we preliminary derived the transition charges for the isolated fluorophores of CFP and YFP optimized in the gas phase and then fixed them during the MD simulations. If the transition charges of the FP fluorophore of interest are prepared, this approach requires no QC calculation for the evaluation of the Coulomb interaction, leading to a remarkable reduction of computation cost. We call this approach Frozen-TrESP, which was applied early on to the studies on the EET dynamics in multichromophoric complexes.44,45,49 The Frozen-TrESP method ignores the polarizability of the transition charge due to the thermal fluctuations of the fluorophores and the electrostatic effects of the protein and solvent environment. Nevertheless, we found that there is a reasonable correlation between the Coulomb interactions from the TrESP and Frozen-TrESP methods. From these results, we confirm that the Frozen-TrESP approach is valid to examine the dependence of the Coulomb interaction on the alignment of YFP with respect to CFP in CFP−YFP heterodimers toward the optimal design of FRET-based biosensor. We also present a list of the transition charges obtained for the representative fluorophore models found in various FPs to apply the Frozen-TrESP method to FRET-based biosensors consisting of other FP pairs in future studies.

eg where ρeg D and ρA are the transition densities of the D and A molecules, respectively. Transition charges from the ESP (TrESP) method43,52 approximate the Coulomb interaction in eq 2 as the classical Coulomb interaction between atom-centered “transition” charges of D and A molecules

V

V dd =

|R (D) − R(A) i j |

(3)

(D) (A) 1 κμD (R )μA (R ) 4π ϵ0 R3

(4)

where R is a center-to-center distance between the D and A molecule; ϵ0 is the vacuum permittivity constant; and μD and μA are the amplitudes of the TDMs of D and A molecules, respectively. κ is an orientation factor expressed as

ρDeg *(r1)ρAeg (r2) |r1 − r2|

∑∑

qiTrESP(R(D))qjTrESP(R(A))

where the R(D/A) represents a set of the atomic coordinates of i the D/A molecules. Here the D/A transition charges, qTrESP ’s, i are determined by a fit of the three-dimensional (3D) “pseudoeg ” electrostatic potential (ESP) created from ρeg D /ρA . This procedure is similar to that for determining conventional ESP charges;81 however, in the latter procedure the 3D ESP is created from an electron density rather than the “transition” density. It should be noted that the net determined “transition” charge is always zero, ∑i qiTrESP = 0, because of the orthogonality between ground and excited states. The original Förster theory37 approximates the Coulomb interaction in eq 2 as a dipole−dipole (dd) interaction between transition dipole moments (TDMs) of D and A molecules

THEORY TrESP and dd Interaction Methods. The leading term of eq 1 is the Coulomb interaction between one-electron transition densities of the donor (D) and acceptor (A) molecules, expressed as follows38,39

∫ dr1 ∫ dr2

=

i∈D j∈A

■

V=

TrESP

κ = μD̂ ·μÂ − 3(μD̂ ·R̂ )(μÂ ·R̂ ) (2)

(5)

where the hat represents a unit vector. C


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The Journal of Physical Chemistry C The TrESP method can approximate the TDM as follows TrESP (D/A) μD/A (R )=

∑ qiTrESP(R(D/A))R (D/A) i i

(6)

The Coulomb interaction calculated from the dd method, eq 4, with the μTrESP, eq 6, is denoted by Vdd from TrESP. Dielectric Medium Effects: Poisson-TrESP and Screening Factors for dd Interaction. When considering the dielectric medium effects on the Coulomb interaction with the transition charges, we used the Poisson-TrESP method;46−49 the pseudo-ESP arising from the transition charges of the D molecule, ϕD(r), is first numerically solved by using the following Poisson equation ∇·[ϵopt(r)∇ϕD(r))] = −4π ∑ qiTrESP(R(D))δ(r − R (D) i ) i∈D

(7)

where ϵopt is the optical dielectric constant; ϵopt(r) is set to 1 where the r points into the fluorophore-model cavity consisting of the spheres with CHARMM van der Waals radii82 and is set to 1.9683 where the r points to the position of the protein or solvent. Then, the Coulomb interaction between D and A molecules is determined by V Poisson ‐ TrESP =

TrESP (A) (R ) ∑ ϕD(R(A) j )qj j∈A

(8)

Figure 2. (a) YFP fluorophore model (acceptor molecule), HBDI−, (b) CFP fluorophore model (donor molecule), and (c) the CFP−YFP heterodimer solvated in a water box with counterions.

The adaptive Poisson−Boltzmann Solver (APBS) version 1.484 was used with the Poisson-TrESP method. When considering the dielectric medium effects on the Coulomb interaction with the TDMs, we used the following two models: one is the Förster point-charge model used in the original Förster theory,37 which involves the screening factor of 1/ϵopt against eq 4, and the other is the Onsager dipole model,55 which involves the screening factor of 3/(2ϵopt + 1) against eq 4. More Efficient Methods: Frozen-TrESP and Mappeddd. To aim for more efficient evaluation of the Coulomb interaction in the protein undergoing the thermal fluctuation and/or a large conformational change within the framework of the TrESP method, we set the coordinate-dependent transition charges, qiTrESP(R(D/A)(t)), to the corresponding reference values, qTrESP (R(D/A) ), obtained for the gas-phase optimized D i 0 and A molecule. We call this the Frozen-TrESP method, which can avoid the QC calculation for each MD snapshot except for a single one to obtain qTrESP (R(D/A) ). The qTrESP (R(D/A) ) has the i 0 i 0 same analogy to the ESP charge in nonpolarizable force-field parameters used for conventional MD simulations.85 The condensed flowchart of the Frozen-TrESP method is shown in Figure 1. We also tested the dd method with the reference TDMs obtained for the gas-phase optimized D and A molecule, where the two TDMs are mapped onto each MD snapshot so as to lay the TDM along the long-axis, X, of the corresponding D/A molecule. We call this the ”Mapped-dd” method.36,71

structure of the YFP fluorophore model, the anionic form of phydroxybenzylideneimidazolinone (pHBDI−), which is the same as the model fluorophore of the GFP in the anionic form. The YFP fluorophore model is regarded as the acceptor (A) molecule in the FRET from CFP to YFP. Figure 2(b) shows the chemical structure of the CFP fluorophore model, in which the phenolic ring of pHBDI− is replaced with an indole ring. The CFP fluorophore model is regarded as the donor (D) molecule in the FRET from CFP to YFP. In this study, the direction of the TDM with respect to the corresponding fluorophore model is defined by two different angles: one is an angle with respect to the carbonyl bond of the imidazolinone ring, ω, as shown in Figures 2(a) and (b). The other is an angle with respect to the longest axis of D or A molecules, θ, where the axis, X-axis schematically drawn in Figures 2(a) and (b), is defined as the eigenvector involving the largest eigenvalues of the nuclear quadrupole of the D or A molecule.36 In the mapped-dd method, the D and A TDM vectors, μD and μA, are aligned to the D and A X-axes, respectively, for each MD configuration. Preparation of Fluorescent Protein Heterodimers. The CFP-YFP heterodimer used for MD simulations and the subsequent calculations of the Coulomb interaction between them were prepared by the following procedures:36,74 the coordinates of the wild-type GFP homodimer were first taken from the X-ray crystallographic data in the Protein Data Bank (PDB), code 1GFL.86 After that, the coordinates of YFP were taken from the data in PDB code 1YFP.87 Chain A in 1YFP, including crystal water molecules inside the protein, were then aligned with chain A in 1GFL to minimize the root-meansquare deviation (RMSD) between the coordinates of their backbone heavy atoms by using VMD.88 Next, the coordinates of CFP, including crystal water molecules inside the protein,

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COMPUTATIONAL DETAILS CFP and YFP Fluorophore Models: Donor and Acceptor Molecules. The π-conjugated part of the fluorophore of each FP is treated as the corresponding FP fluorophore model. The side chains of the π-conjugated parts of the fluorophores, linking to the protein, were truncated and capped with a methyl group. Figure 2(a) shows the chemical D


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The Journal of Physical Chemistry C were taken from the data in PDB code 1OXD89 and were then aligned with chain B in 1GFL in the same manner as those of YFP. Missing hydrogen atoms in the X-ray crystallographic data were added to the constructed CFP−YFP heterodimer by using the “Psfgen” plugin of VMD.88 When hydrogen atoms were added, titratable residues, aspartic acid (Asp), histidine (His), glutamic acid (Glu), lysine (Lys), and arginine (Arg), were assumed to take their standard protonation states (i.e., Asp−, Glu−, His, Lys+, and Arg+) except for Glu222 of YFP; we set Glu222 of YFP to neutral because its protonation makes possible the formation of the hydrogen bond between Glu222 and the YFP fluorophore.87 Hydrogen atoms were also added to the His residues by following ref 90; the δ nitrogens of His25, His148, His181, His199, and His217 were protonated, and on the other hand, the ϵ nitrogens of His77, His81, His139, and His169 were protonated for both YFP and CFP. The total charge of the CFP−YFP dimer constructed at this state was −12. The structure of the CFP−YFP heterodimer was then solvated in a cubic (100 × 100 × 100 Å) water box by using the “Solvate” plugin of VMD.88 Counter ions (83Na+ and 71Cl−), leading to 0.15 mol/L salt concentration, were added for the neutralization of the whole system by using the “Autoionize” plugin of VMD.88 Figure 2(c) illustrates the final solvated CFP−YFP heterodimer structure which was used as the starting geometry for the MD simulations. Molecular Dynamics (MD) Simulations. Molecular dynamics (MD) simulations were performed using the NAMD2 program91 with the CHARMM22 force field82 including the CMAP correction.92 The TIP3P93 force field was used for the water molecules. The nonstandard CHARMM force field parameters for the YFP fluorophore were taken from ref 90. The CFP fluorophore contains an indole ring instead of a phenolic ring, compared to the YFP fluorophore. Therefore, in this study, the standard CHARMM force fields for the indole ring of the tryptophan (Trp) residue were applied to that of the CFP fluorophore.18,26 The force field parameters for the torsion around the two bridging bonds between the imidazolinone and indole rings were taken from ref 18. For the imidazolinone ring of the CFP fluorophore, we used the same force field parameters as those of the YFP fluorophore. For the MD simulations, the SHAKE method was used, and the integration time step was set to 2 fs. Particle Mesh Ewald (PME) was used for the long-range electrostatic interactions with a cutoff radius of 12 Å. By using the Langevin piston method with a decay period of 100 fs and a damping time 50 fs, we gradually heated the overall system to 310 K with the rate of 1 K per 1 ps, equilibrated the system during the following 10 ns, and then collected the structures every 10 ps over the following 5 ns under NPT conditions at 1 atm. These 501 snapshot geometries were used for subsequent QC calculations. Quantum Chemical Calculations. In this study, we calculated the transition charges and TDMs of the D and A molecules whose structures were obtained from the gas-phase QM optimization and the MD snapshots for the solvated CFP−YFP heterodimer. All ground-state (GS) geometry optimizations of the gas-phase D and A molecules were performed under Cs symmetry restriction with the Gaussian 09 Revision C.01 suite of programs.94 For the D molecule, the excited-state geometry optimization in the gas phase was also performed under Cs symmetry restriction at the TD-CAMB3LYP95 level of theory with the Gaussian 09. All vertical

excited-state calculations at TD-DFT and configurationinteraction singlet (CIS) levels of theory for obtaining transition charges and TDMs were performed using the GAMESS program.96 The excited states contributing to the FRET reaction of interest correspond to the lowest singlet ones with large oscillator strength. In order to address the electrostatic effects of protein and solvent, we performed TDDFT calculations with/without point-charge embedding for each D and A molecule structure taken from the MD snapshots; in the former calculations, we employed quantum mechanical/molecular mechanical (QM/MM) methods at the TD-CAM-B3LYP95/CHARMM82 level of theory; in the latter calculations, we term the obtained results no embedding (NE) ones. In the CHARMM point-charge embedding calculations, solvation water molecules and counterions 10 Å or more from the protein were deleted in MM region for each MD snapshot. The atomic point charges of the QM/MM border amino acids were refined to have integer charges, as shown in Figure S1 in the Supporting Information. We modified the GAMESS program96 so that fitting the atom-centered transition charges of the pseudo-ESP for the TrESP method can be performed by making use of the routines intrinsically implemented in the GAMESS program for computing the ESP charges. For all QM calculations, the Pople’s-type split-valence double-ζ plus polarization and diffuse-function basis sets, 6-31+G(d,p), were used.

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RESULTS Gas-Phase Donor and Acceptor Molecules. The D and A molecules shown in Figure 2 were first optimized in the gas phase under the Cs symmetry restriction. We confirmed that the geometry optimizations under no symmetry restriction provide almost planar Cs symmetry geometries for both the D and A molecules. We employed three density exchangecorrelation (XC) functionals, B3LYP,97−99 M06-2X,100 and CAM-B3LYP,95 and also performed MP2 optimizations. Figures 3(a) and (b) compare the bond lengths of the obtained structures of the YFP and CFP fluorophore models, respectively. Figures 2(a) and (b) number the heavy atoms in the π-conjugated parts of the A and D molecules, respectively. Figure 3 shows that the GS optimized geometry does not significantly depend on the employed methods. Therefore, the structures from the B3LYP/6-31+G(d,p) optimization are used for the following gas-phase excited state calculations at the GS optimized structure. (Those Cartesian coordinates used are listed in Tables S1 and S2 in the Supporting Information.) We performed the TD-DFT calculations with several XC functionals, namely, hybrid PBE0,101 B3LYP,97−99 BHandHLYP,97 meta-hybrid M06-2X,100 and range-separated CAMB3LYP,95 LC-BOP,102 and LC-BLYP,102 and also performed CI-singlet (CIS) excited state calculations. Tables 1 and 2 list the vertical excitation energies, VEE, oscillator strengths, OS, the amplitude of TDM, |μ|, and TDM directions, ω and θ, obtained for the A and D molecules, respectively. Since the TDB3LYP calculation produces S1 state with the small OS of 1.2 × 10−4 for the A molecule, we considered the excitation from the GS to S2 state only for the case. The TD-DFT and CIS calculations for both the D and A molecules show that the “HOMO → LUMO” single electron transition dominantly contributes to the considering excitation. The lambda diagnostic quantities103 obtained for the A and D molecules with the TD-DFT calculations are 0.70−0.74 and 0.76−0.78, respectively, which indicates that the degree of spatial overlap E


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Table 2. Same for Table 1 but Obtained for the CFP Fluorophore Model (Donor Molecule) method TD-PBE0 TD-B3LYP TD-BHHLYP TD-M06-2X TD-CAM-B3LYP TD-LC-BOP TD-LC-BLYP CIS TD-CAM-B3LYP at S1-opt geometrya

VEE

OS

|μ|

ω

θ

3.382 3.301 3.628 3.524 3.542 3.609 3.594 4.262 3.147

0.627 0.597 0.700 0.676 0.685 0.700 0.695 0.934 0.685

2.750 2.717 2.807 2.798 2.810 2.813 2.809 2.990 2.980

76.4 76.5 75.6 75.9 75.8 75.5 75.5 73.7 76.6

11.8 11.7 12.6 12.3 12.4 12.7 12.7 14.6 10.8

a

Gas-phase S1-state optimization with the TD-CAM-B3LYP/6-31+G(d,p) calculations under Cs symmetry condition

reported in the previous theoretical studies.22−27,36 On the other hand, the calculated ω of about 72° is close to the experimental data (75° ± 4°) observed in CD3OD solvent,72 as listed in Table 1. The ω of ca. 76° and θ of ca. 12° obtained for the D molecule are slightly larger than those (ω of ca. 72° and θ of ca. 4°) for the A molecule. The resultant small θ values indicate that the TDM almost lies along the long-axis X, consistent with the previous observations.36 When the FRET occurs from CFP to YFP, the structures of the D molecule in CFP are fully relaxed on the S1 potential energy surface. To approximately estimate the relaxation effect, we also performed the S1 geometry optimization for the gasphase D molecule under Cs symmetry restriction at the TDCAM-B3LYP/6-31+G(d,p) level of theory. The VEE, OS, |μ|, ω, and θ at the obtained relaxed S1-state geometry were listed in Table 2. Comparing the TD-CAM-B3LYP results at the GS optimized geometry with those at the S1 optimized geometry, we can see that S1 geometry relaxation yields a red shift of 0.4 eV in VEE. There are no large differences in the other quantities between the results obtained at the GS and S1 optimized geometries, as shown in Table 2; the S1 geometry relaxation slightly decreased the θ by ca. 1.6° and, on the other hand, slightly increased the ω by ca. 0.8° because the geometry relaxation also changed the direction of the long-axis X. In Figures 4(a) and (b), we plot the transition charges calculated with TD-DFT and CIS calculations for the atoms of π-conjugated parts of the A and D molecules, respectively. The atom numbering is provided in Figures 2(a) and (b). Figure

Figure 3. Bond lengths of (a) YFP and (b) CFP fluorophore models optimized with several QC levels in the gas phase. Average values obtained from 5 ns MD simulations were also plotted. (Error bars represent standard deviation from the mean values.)

between occupied and virtual orbitals involved in the considering excitation is large and implies that the chargetransfer character is small. Tables 1 and 2 show that the CIS calculations give the VEE, OS, and |μ| larger than those from TD-DFT calculations for both the D and A molecules. Table 1 shows that the TD-DFT and CIS calculations overestimate the VEEs for the gas-phase pHBDI− with respect to the experimental data (2.59 eV,104 2.75 eV,105 and 2.84 eV106). Similar overestimations with TD-DFT calculations have been

Table 1. Calculated Vertical Excitation Energies, VEE (in eV), Oscillator Strength, OS, the Amplitude of TDM, |μ| (in au), and TDM Directions, ω and θ (in deg), for the YFP Fluorophore Model (Acceptor Molecule), HBDI−, Optimized with B3LYP/631+G(d,p) in the Gas Phasea method

VEE

OS

|μ|

ω

θ

TD-PBE0 TD-B3LYP TD-BHHLYP TD-M06-2X TD-CAM-B3LYP TD-LC-BOP TD-LC-BLYP CIS exp.

3.104 3.055 3.231 3.096 3.123 3.084 3.075 3.751 2.59b, 2.75c, 2.84d

1.016 0.982 1.123 1.062 1.080 1.097 1.091 1.547

3.655 3.623 3.766 3.741 3.756 3.810 3.805 4.104

72.4 72.5 72.0 71.8 71.9 71.6 71.6 71.3 75 ± 4e

4.46 4.34 4.77 5.02 4.92 5.23 5.25 5.52

a

The 6-31+G(d,p) basis sets were used. bTaken from ref 104. cTaken from ref 105. dExperimental data taken from ref 106 were obtained from the extrapolation from a Kamlet−Taft fit of absorption peaks recorded in several solvents. eExperimental data taken from ref 72 were obtained in CD3OD solvent. F


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of interest, such as TDM and transition charge, and TD-B3LYP calculations have a possibility of producing the wrong S1 state owing to the underestimation of the charge-transfer state. We tested the TrESP method for a face-to-face eclipsed homodimer of D molecules, which possesses the Cs symmetry, as shown in the inset of Figure S2 in the Supporting Information. The internal coordinates of each D monomer are identical with the GS optimized one described above. For the symmetric system, the reference excitonic coupling can be obtained from one-half of the energy splitting between S1 and S2 states within the framework of the two-state approximation.63,64 Figure S2 in the Supporting Information plots the distance dependence of the excitonic couplings from the exact energy splitting, TrESP, and dd-approximation methods. Figure S2 in the Supporting Information shows that the TrESP method can sufficiently reproduce the reference excitonic coupling even at the short distance of 5 Å, compared to the dd-approximation. Molecular Dynamics Results. To reproduce the realistic molecular dynamics of FPs in the course of the FRET reaction in biosensors, we performed the MD simulations for the solvated CFP-YFP heterodimer. Figure S3 in the Supporting Information plots the RMSD from the starting structure during the entire MD run. Although the CFP and YFP form the heterodimer via the weak nonbonded interactions between the hydrophobic patches on the their surfaces (dissociation constant of 110 μM107), the dimerization was stable during the MD run; RMSD fluctuates around about 2.5 Å during the last 5 ns MD run. We collected the structures each 10 ps over the last 5 ns MD run and used the collected 501 snapshots for the following analyses. The resultant average value of R is 25.1 Å with a small standard deviation of 0.334 Å, which indicates that the CFP−YFP heterodimerization keeps the D and A molecules close to each other during the MD run. In Figures 3(a) and (b), we plot the mean bond lengths for the MD structures of the A and D molecules, respectively. The standard deviations form the average values are represented by the corresponding error bars in Figure 3. Figure 3(a) shows that the mean bond lengths for the A molecule well agree with the corresponding bond lengths from the gas-phase QC optimizations, except for around the bridging bond region (C1−C2, C7−C1, C1−C8, and C8−C10). Figure 3(b) shows that the mean bond lengths for the D molecule from the MD simulations deviate from the corresponding bond lengths from the gas-phase QC optimizations in the indole-ring region. To address the planarity of each fluorophore during the MD simulations, we focus on the bonding and dihedral angles of the bridging bonds. The τ represents the dihedral angles between the atoms C1−C8−C10−N11 for the A molecule and the atoms C1−C10−C12−N13 for the D molecule, respectively. The ψ represents the dihedral angles between the atoms C7− C1−C8−C10 for the A molecule and the atoms C9−C1− C10−C12 for the D molecule, respectively. The α represents the bonding angles between the atoms C1−C8−C10 for the A molecule and the atoms C1−C10−C12 for the D molecule, respectively. The MD simulations produced the average τ, ⟨τ⟩,

Figure 4. Transition charges from the TrESP methods, qTrESP , from i TD-DFT and CIS calculations for the gas-phase optimized (a) YFP and (b) CFP fluorophore models. The 6-31+G(d,p) basis sets were used.

4(a) shows that the transition charges for the A molecule do not depend on the employed XC functionals. On the other hand, Figure 4(a) shows that the CIS calculation overestimates the transition charges on the imidazolinone-ring part (from C10 to C12) with respect to the TD-DFT calculations. Moreover, the transition charges on the phenolic ring part (from C1 to C6) from the CIS calculation deviate from those from the TD-DFT calculations, as shown in Figure 4(a). Similarly, Figure 4(b) shows that many of the transition charges for the D molecule do not depend on the employed XC functionals. However, the TD-DFT calculations with PBE0 and B3LYP functionals exhibit the difference of the transition charges on the bridge part (C1 and from C9 to C12), compared to those with the other XC functionals. Figure 4(b) also shows that the CIS calculation considerably overestimates the transition charges on the bridge part with respect to the TD-DFT calculations. Hereafter, we only use the CAM-B3LYP density functionals because Figure 4 and Tables 1 and 2 show that the TD-CAMB3LYP calculations produce reasonable results of the quantities

Table 3. Average Values of Geometrical Parameters Obtained from the 501 MD Snapshotsa acceptor donor a

τ

ψ

α

R

3.69° (6.69°) 3.24° (6.00°)

3.29° (7.61°) 2.39° (8.05°)

134.6° (2.245°) 129.0° (2.316°)

25.07 Å (0.3343 Å)

The values in parentheses represents the standard deviation. G


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The Journal of Physical Chemistry C of 3.69° with a standard deviation σ(τ) of 6.69° and the average ψ, ⟨ψ⟩ of 3.29°, with a standard deviation σ(ψ) of 7.61° for the A molecule. For the D molecule, the MD simulations produced the ⟨τ⟩ of 3.24° with the σ(τ) of 6.00° and the ⟨ψ⟩ of 2.39° with the σ(ψ) of 8.05°. We obtained the ⟨α⟩’s of 134.6° and 129.0° with the σ(α)’s of 2.245° and 2.316°, for the A and D molecules, respectively. Table 3 lists the obtained statistical values of the geometry parameters. The resultant small ⟨τ⟩, ⟨ψ⟩, σ(τ), and σ(ψ) indicate that the protein environments keep the D and A molecules planar during the thermal fluctuations. Vertical Excitation Calculations of Donor and Acceptor Molecules in Protein. The VEEs and TDMs calculated for D and A molecules in protein are presented in this subsection. To address the effects of thermal fluctuation of the fluorophore geometry, we performed the TD-CAM-B3LYP/6-31+G(d,p) calculations for each fluorophore model whose structure was taken from the 501 MD snapshots of the solvated CFP-YFP dimer. In addition, to address the effects of the electrostatic effects of protein and solvent, we compare the CHARMM point-charge embedding (QM/MM) results with no embedding (NE) ones. We produced the histogram for the distributions of the obtained values of VEE in the range from 320 to 440 nm (in the range from 2.82 to 3.87 eV). To make a correlation between the histogram and absorption spectrum arising from the S0-to-S1 electronic transition, we used the oscillator strength obtained for each VEE calculation as the weighting factor for the corresponding VEE contribution to the histogram. Note that the vibrational contributions to the spectrum are ignored. The results are plotted in Figure 5: the red solid and blue

for the CFP obtained form the QM/MM and NE calculations is 12.4 and 11.0 nm, respectively. The YFP absorption spectrum from the QM/MM calculations exhibited a blue-shift with respect to that from the NE calculations, as shown in Figure 5; the average VEEs from the QM/MM and NE calculations were 398.4 nm (3.11 eV) and 403.8 nm (3.07 eV), respectively. In contrast, the CFP absorption spectrum from the QM/MM calculations exhibited a red shift with respect to that from the NE calculations, as shown in Figure 5; the average VEEs from the QM/MM and NE calculations were 358.0 nm (3.46 eV) and 352.3 nm (3.52 eV), respectively. Table 4 lists the statistical quantities of VEE. Next we show the results of the TDMs of D and A molecules in the protein environment. Table 4 lists the average TDM amplitude |μ| and directions ω and θ taken over the MD snapshots. By comparing the NE results with QM/MM ones listed in Table 4, we can see that the protein electrostatic interactions have little impact on the TDMs for both the D and A molecules, for example, that protein electrostatic interactions reduce the average μ values by only ca. 1−2% for both the D and A molecules. The ω obtained for the A molecule in protein from the MD trajectory were 70.8 ± 3.87° (QM/MM) and 71.6 ± 3.93° (NE), which is in good agreement with 71.8° obtained for the gas-phase A molecule. Among various FPs, the TDM direction of GFP has been experimentally determined.73,74,108,109 Since the fluorophores of GFP and YFP have a common π-conjugated part modeled as pHBDI, the TDM direction of GFP73,74 is often used as that of YFP.74,78,80 We found that the ω obtained for YFP is close to the 67.4 ± 4° observed for GFP.73,74 With respect to the other definition of the TDM direction, the θ obtained for the A molecule in protein from the MD trajectory were 5.52 ± 0.999° (QM/ MM) and 4.49 ± 1.06° (NE), which is in good agreement with 4.49° obtained for the gas-phase A molecule and indicates that the TDM almost lies along the long-axis X of the A molecular geometry during the MD run. The θ obtained for the D molecule in protein from the MD trajectory were 11.3 ± 1.32° (QM/MM) and 11.8 ± 1.35° (NE), which is in good agreement with 12.4° obtained for the gas-phase A molecule and indicates that the TDM roughly lies along the long-axis X of D molecular geometry during the MD run. In the QM/MM calculations, the counterions and water molecules were removed when they were 10 Å away from the protein, similar to the previous QM/MM-based studies on FPs.17,21,26 Since some counterions move in and out of the 10 Å radius, the elimination of them from the QM/MM calculations might lead to extra fluctuations in the VEEs along the MD trajectory. To address the possible artificial effect arising form the elimination, we performed the additional QM/ MM calculations with the different radii, 6, 8, and 12 Å, for the first 200 ps long trajectory. The calculated VEEs are plotted in Figure S4 in Supporting Information, showing that the elimination by using the 10 Å radius provides the reasonably converged QM/MM results for this system. Coulomb Interaction Calculations. Based on the transition charges and TDMs of the D and A molecules obtained from the TD-CAM-B3LYP/6-31+G(d,p) calculations for the 501 MD snapshots, we measured the Coulomb interactions in the solvated CFP−YFP heterodimer. In Figure 6, we plot the time dependence of R, the orientation factor κ from eq 5, and the absolute Coulomb interactions from the TrESP and dd-interaction methods with the no embedding (NE) calculations. As shown in Figure 6(a),

Figure 5. Absorption spectra based on the histogram of the vertical excitation energy distribution calculated for the last 5 ns MD simulations of the CFP−YFP heterodimer. Absorption spectra for YFP calculated with TD-CAM-B3LYP/6-31+G(d,p) with/without CHARMM point-charge embedding are represented by solid red and dashed blue lines, respectively. Absorption spectra for CFP calculated with TD-CAM-B3LYP/6-31+G(d,p) with/without CHARMM pointcharge embedding are represented by solid green and dashed purple lines, respectively.

dashed lines represent the histograms obtained for the YFP fluorophore from the QM/MM and NE calculations, respectively, and the green solid and purple dashed lines represent the histograms obtained for the CFP fluorophore from the QM/MM and NE calculations, respectively. The bin width of the histogram was set to 4 nm. Figure 5 shows that the broadness of the YFP spectrum caused by the thermal fluctuation of the fluorophore geometry is slightly larger than that of the CFP spectrum; the standard deviation of VEEs for the YFP obtained from the QM/MM and NE calculations is 8.87 and 8.85 nm, respectively; the standard deviation of VEEs H


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The Journal of Physical Chemistry C Table 4. Average Values of VEE, OS, |μ| (in au), ω (in deg), and θ (in deg) Taken over the 501 MD Snapshotsa method YFP

NE QM/MM

CFP

exp. NE QM/MM exp.

VEE

OS

|μ|

ω

θ

403.7 nm, 3.07 eV (8.849 nm) 398.4 nm, 3.11 eV (8.866 nm) 514 nm, 2.41 eVb 352.3 nm, 3.52 eV (10.96 nm) 357.5 nm, 3.46 eV (12.43 nm) 434 nm, 2.85 eVd

1.041

3.720 (0.0506) 3.680 (0.0522)

71.6 (3.93) 70.8 (3.87) 67 ± 4c 76.2 (3.94) 76.7 (3.89)

4.49 (1.06) 5.22 (0.999)

1.033

0.641 0.614

2.726 (0.0978) 2.685 (0.123)

11.8 (1.35) 11.3 (1.32)

a

The values in parentheses represent the standard deviation. bRef 87. cExperimental data taken from refs 73 and 74 were obtained for neutral GFP. d Ref 89.

Figure 6. Time dependence of (a) D and A center-to-center distance R, (b) orientation factor in dd method κ, and (c) absolute Coulomb interactions obtained from TrESP and dd methods with no embedding (NE) TD-CAM-B3LYP/6-31+G(d,p) calculations for the 501 MD snapshots.

Figure 7. Plots of |VTrESP(NE)| vs |Vdd(NE)| by the red cross symbol and | VTrESP(NE)| vs |Vdd from TrESP(NE)| by the green open square symbol. The solid red and dashed green lines represent the least-squares fitting lines for the relations between |VTrESP(NE)| and |Vdd(NE)| and between | V TrESP(NE) | and |V dd from TrESP(NE) |, respectively. (b) Plot of | VTrESP(QM/MM)| vs |VTrESP(NE)|. The solid line represents the leastsquares fitting line.

the average value of κ2 associated with the |Vdd(NE)| is 2.37 with a small standard deviation of 0.147, which considerably differs from the random-orientation assumed κ2 of 2/3. Figure 6(b) shows that the resultant R fluctuates around its mean value of 25.1 Å with a standard deviation of 0.334 Å, as described in Section 4.2. By comparing the time course of R with that of the Coulomb interaction in Figures 6(b) and (c), we can see a notable correlation between R and the Coulomb interactions. (We will plot a correlation diagram between R and the Coulomb interaction including the protein/solvent dielectricscreening effect later.) Figure 6, therefore, indicates that the fluctuation of the Coulomb interaction is largely attributed to the R fluctuation rather than the κ fluctuation. Figure 6(c) also shows a good correlation between the Coulomb interactions from the TrESP method and dd method. To see the correlation more clearly, in Figure 7(a), we plot a correlation diagram of | VTrESP(NE)| vs |Vdd(NE)| with the red cross symbol and that of |

VTrESP(NE)| vs |Vdd from TrESP(NE)| with the green open square, respectively. Here |VTrESP(NE)| is the absolute Coulomb interaction obtained from the TrESP method, eq 3, with NE calculations. The |Vdd(NE)| is the absolute Coulomb interaction obtained from the dd method, eq 4, with the TDMs of the D and A molecule based on the NE results. The Coulomb interaction, |Vdd from TrESP(NE)|, is similar to |Vdd(NE)|, but where the TDMs for the dd method are obtained from the transition charges with eq 6. Figure 7(a) shows that the value of | VTrESP(NE)| distributes in the narrow region from 28 to 41 cm−1. We see that the |Vdd from TrESP(NE)| agrees well with the |Vdd(NE)|. The obtained correlation coefficient, η, between |VTrESP(NE)| and I


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The Journal of Physical Chemistry C |Vdd(NE)| is 0.982, indicating a remarkably good correlation between them. We plot the least-squares fitting line for the relation between |VTrESP(NE)| and |Vdd(NE)| in Figure 7(a) by a solid red line. The slope of the obtained fitting line is about 0.89, which indicates that |VTrESP(NE)| and |Vdd(NE)| approach each other as their values are decreased. Next, we assess the electrostatic effects of protein and solvent on the Coulomb interactions. In Figure 7(b), we plot a correlation diagram of |VTrESP(NE)| vs |VTrESP(QM/MM)|, where | VTrESP(QM/MM)| is the absolute Coulomb interaction calculated with the TrESP method based on the transition charges from the charge embedding QC (namely, QM/MM) calculations. In Figure 7(b), we also plot the least-squares fitting line for the relation between |VTrESP(NE)| and |VTrESP(QM/MM)| by a solid line. The obtained slope of the fitting line is about 0.97, and the correlation coefficient is about 0.896. Therefore, we conclude that the electrostatic interactions between the D/A molecules and protein/solvent environment make little impact on the Coulomb interactions of the considering FRET reaction. Next, we address the effect of the electronic polarizability of protein and solvent on the Coulomb interactions by using the NE TD-CAM-B3LYP results. We calculated the Coulomb interactions from the Poisson-TrESP method, eq 8, and the dd method, eq 4, corrected by the screening factor of 1/ϵopt (Förster point-charge model37) or 3/(2ϵopt + 1) (Onsager dipole model55), where the optical dielectric constant ϵOpt is set to 1.96.83 Figure 8 shows the distance dependence of obtained |

Finally, we applied the Frozen-TrESP and Mapped-dd methods to the calculation of the Coulomb interaction. In the Frozen-TrESP method shown in Figure 1, a single set of the reference transition charges {qTrESP (R0)} obtained at the TDi CAM-B3LYP/6-31+G(d,p) level of theory for the gas-phase optimized D and A molecules were used for all the calculations of the Coulomb interactions with the TrESP method. Similarly, in the Mapped-dd calculations, the TDMs obtained for the gasphase optimized D and A molecules were aligned along the long axes of the D and A molecule, respectively, at each MD snapshot, and then the Coulomb interactions were obtained from the dd method with the mapped TDMs. To assess the effect of the fluctuation of the transition charges arising from the molecular thermal fluctuation on the Coulomb interactions, we plot a correlation diagram of |VTrESP(NE)| vs |VFrozen‑TrESP| by a red cross symbol in Figure 9. The solid red line in Figure 9 is a

Figure 9. Correlation between absolute Coulomb interactions from the no embedding TrESP, |VTrESP(NE)|, and those from the no embedding X method, |VX(NE)|, where X = Frozen-TrESP were represented by the red cross and X = Mapped-dd were represented by the open square.

fitting line for the relation between them. We can see that there is a reasonable correlation between |VTrESP(NE) | and | VFrozen‑TrESP|, and the correlation coefficient between them was calculated to be about 0.860. Figure 9 shows that the Frozen-TrESP method slightly overestimates the Coulomb interactions; the average |VFrozen−TrESP| value was calculated to be 36.1 cm−1, which is larger than the average |VTrESP(NE)| value of 34.1 cm−1. In Figure 9, we also plot a correlation diagram of | VTrESP(NE)| vs |VMapped‑dd| by green open squares. The dashed green line in Figure 9 is a fitting line for the relation between them. The calculated correlation coefficient between |VTrESP(NE)| and |VMapped‑dd| is about 0.746, which is smaller than that between |VTrESP(NE)| and |VFrozen‑TrESP|. We obtained the average |VMapped‑dd| value of 31.4 cm−1, which is smaller than the average |VTrESP(NE)| value of 34.1 cm−1. The obtained high correlation between the |VTrESP(NE)| and |VFrozen‑TrESP| strongly supports the applicability of the Frozen-TrESP method to the study on the FRET in the CFP-YFP heterodimer.

Figure 8. (a) Correlation between center-to-center distance, R, and the absolute Coulomb interactions calculated in dielectric medium (ϵOpt = 1.96) for the 501 MD snapshots at every 10 ps. Transition charges and TDMs were obtained from the TD-CAM-B3LYP/631+G(d,p) calculations with no point-charge embedding (NE). The Coulomb interactions from the Poisson-TrESP method, dd with the 3/(2ϵOpt − 1) factor, and dd with 1/ϵOpt factor are plotted by the red cross, green square, and blue asterisk symbols, respectively.

VPoisson−TrESP(NE)| and |Vdd(NE)| corrected by the screening factor. We see that the dd method corrected by the 3/(2ϵopt + 1) screening factor well reproduces |VPoisson−TrESP(NE)| values. With the Poisson-TrESP method, the screening and local field effects by the dielectric environment can be evaluated as a scaling factor, f = |VPoisson−TrESP)/VTrESP|. The resultant mean value of f is 0.570 with a small standard deviation of 0.00207, which is close to the screening factor, 3/(2ϵopt + 1) = 0.610, of the Onsager dipole model. We observed no dependence of f on the D−A distance R and on the relative orientation κ2 for the considered MD configurations.

■

DISCUSSION We first discuss the applicability of the dd approximation to the estimation of the Coulomb interaction for the FRET in the CFP−YFP heterodimer. As shown in Figure S3 in the Supporting Information, we observed that the CFP−YFP dimer was stable during the entire MD run. Besides, the J


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question for the study on FRET reaction.110 So far, Scholes, Curutchet, and co-workers58,59 have studied the solvent screening effect on the Coulomb interactions among pigments in photosynthetic proteins, namely, Photosystem II, Lightharvesting complex II, Phycoerythrin 545, and Phycocyanin 645, by using the LR approach with PCM. They have observed that the solvent screening effect attenuates exponentially at donor−acceptor distances less than about 20 Å and does not apparently depend on the DA distances at the distances larger than 20 Å.58,59 On the other hand, Adolphs, Renger, and coworkers46−48 have studied the solvent screening effect on the Coulomb interactions among pigments in the other photosynthetic proteins, namely, Photosystem I and FMO, by using the Poisson-TrESP method. They have observed no systematic dependence on the DA distance.46−48 In this study, we analyzed the solvent screening and reaction field contributions to the Coulomb interactions in the thermally fluctuating CFP−YFP heterodimer by using the Poisson-TrESP method. As a result, we obtained the almost constant local field correction/ screening factor f = |VPoisson‑TrESP)/|VTrESP| of 0.570, for the conformation space generated by the MD. Note that the screening factor of the Onsager dipole model,55 3/(2ϵopt + 1) = 0.610, slightly overestimates the resultant f of 0.570, and the dd method slightly underestimates the Coulomb interactions from the TrESP method. Since the errors cancel each other out, the dd-interaction corrected by the Onsager dipole-model screening factor well approximates the Coulomb interactions from the Poisson-TrESP method, as shown in Figure 8. Now we discuss the applicability of the Frozen-TrESP method to the estimation of the Coulomb interaction between FPs, which is one of the main purposes of this study. As shown in Figure 1, in the Frozen-TrESP method, we should preliminarily determine a single set of the reference transition charges of fluorophore from one single-point excited-state calculation performed for its gas-phase optimized structure. We confirmed that the reference transition charges do not significantly depend on the employed DFT XC functional for both CFP and YFP, as shown in Figure 4. Once the reference transition charges are prepared for the D and A molecules, the Frozen-TrESP method can calculate the Coulomb interaction at any configuration through the classical Coulomb interaction among the reference transition charges. By using the GAMESS code locally modified for the TrESP approach, the transition charges of YFP and CFP at each MD snapshot were obtained in around 12 and 18 min on 8 cores with dual intel E5-2650 Xeon CPUs (2.0 GHz) at the TD-CAM-B3LYP/6-31+G(d,p) level of theory, respectively. This result indicates that the FrozenTrESP calculations along the long MD trajectory offer much better computational cost performance than the TrESP ones. On the other hand, the Frozen-TrESP method ignores the effect of the electronic polarization of the transition charges arising from the structural dynamics of the fluorophore geometry and the protein electrostatic interactions. Nevertheless, as shown in Figure 9, the Frozen-TrESP method reasonably reproduces the Coulomb interactions from the TrESP method involving the many TD-DFT calculations for the MD trajectory. To clarify the behavior of the electronic polarization, we calculate the average transition charges and the standard deviations from the average values. Figure 10 plots the average transition charges taken over the NE TD-DFT results for the 501 MD snapshots and those taken over the QM/MM results for the MD snapshots by red open square and closed green square symbols, respectively. The corresponding

geometrical parameters obtained from the MD simulations, listed in Table 3, indicate that the protein environment packs the fluorophore tightly inside the FP and keeps it planar, which is consistent with the previous MD studies.18,90 As a result, the DA distance R and relative orientation factor κ2 moderately fluctuate around the corresponding values at the MD starting structure, as shown in Figure 6(a),(b). Previous QM studies69,70 have shown that such restricted motions around the specific DA orientation have a possibility for the ddinteraction method to fail to approximate the Coulomb interaction even at the long R compared to the D and A molecular sizes. Therefore, we examined the errors arising form the dd approximation by the correlation diagram of the TrESP results, |VTrESP(NE)|, vs the dd-approximation ones, |Vdd(NE)|, in Figure 7(a). Although the dd-approximation underestimates the Coulomb interaction compared to the TrESP method, we obtained a good linear relationship between |VTrESP(NE)| and | Vdd(NE)| with the correlation coefficient η = 0.98. Consequently, the CFP−YFP heterodimer restricts the mutual motion between their fluorophores around the specific orientation, κ = 2.37, which ensures that the dd interaction passably approximates the Coulomb interaction in the dimer. The dd approximation probably holds true for the FRET in other FPpair dimers because we can expect that the dimerizations via the common surface hydrophobic patch35,79 produce the κ-value close to 2.37. Therefore, our results support the previous analyses of the FRET in the FP dimer by using the dd interaction method.74,78,80 However, its applicability to the FRET signal change in practical FP-based biosensors is still unclear because the κ should drastically vary in association with the analyte-induced conformation change. We will discuss this problem by the Frozen-TrESP method in the following subsection. Next we discuss the effect of the protein electrostatic interactions on the Coulomb interaction. Previous theoretical studies12−27 have shown that the electrostatic interactions between fluorophore and protein/solvent are of importance for explaining the spectral properties2−6 of the FPs. In this study, to take into account the effect of the protein electrostatic interactions, we performed the TD-DFT calculation within the CHARMM point-charge embedding scheme, QM/MM. Figure 5 shows that the protein electrostatic interactions lead to a small blue-shift of the YFP absorption spectrum and a small red-shift of the CFP absorption spectrum, which consists of the previous theoretical studies for YFP26 and CFP,18 respectively. Such contradicting contribution of the protein electrostatic interactions is probably due to the difference in the protonation state of Glu222 for the MD simulations and QM/MM calculations. By comparing the QM/MM results with the NE ones listed in Table 4, we can see that those electrostatic interactions have a low impact on the amplitudes and directions of the TDMs for both CFP and YFP, which supports the expectation by Ansbacher et al.36 that the TDMs of all FPs are insensitive to the protein environment. Similarly, by comparing the NE TrESP results, |VTrESP(NE)|, with the QM/MM ones, | VTrESP(QM/MM)|, in Figure 7(b), we found that those electrostatic interactions reduce the Coulomb interaction by only 3% and a good linear correlation η = 0.90 between them for the configuration space by MD. We, therefore, conclude that the effects of the protein/solvent electrostatic interactions on the Coulomb interaction between the FP-pair are negligible. The effect of the electronic polarizability of the fluorophore environment on the Coulomb interaction is an intriguing K


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transition charges. Note that we used poor force field parameters for the indole-ring region of the D molecule in the MD simulations. The poor force field makes the MD snapshot structures in the indole-ring region deviate from the gas-phase QM-optimized one, as shown in Figure 3(b). The structural deviation causes the differences between the reference transition charges and the average transiting charges of the MD simulations in the indole-ring region (C1−C9), as shown in Figure 10(b). Therefore, we can expect that the reparameterization of the force field for the D molecule achieves further improvement of the correlation between | VFrozen‑TrESP| and |VTrESP|. The donor transition charges and TDMs used for the estimation of the Coulomb interaction should be calculated at the structures of the emitting state, namely, the relaxed structures on the S1 potential surface of the D molecule. However, in this study, we have assumed that the donor transition charges and TDMs determined for the gas-phase GS optimized and GS MD structures of the D molecule approximate the corresponding ones for the emitting-state structures and have used them for the estimation of the Coulomb interaction in the CFP−YFP heterodimer. Previous theoretical studies13−17 have succeeded in reproducing the fluorescence energies of FPs by performing QM/MM excitedstate geometry optimizations; these theoretical studies13,17 have shown that the protein environment keeps the fluorophore structure on the S1 potential surface planar and avoids the fluorescence quenching occurring through the conical intersections12 near the twisted structures associated with the geometry parameters, τ and φ. To roughly approximate the emitting-state structure of the D molecule in CFP, we performed the S1 geometry optimization for the gas-phase D molecule under Cs symmetry restriction at the TD-CAMB3LYP/6-31+G(d,p) level of theory. In Figure 10(b), the transition charges calculated from a single TD-CAM-B3LYP/631+G(d,p) calculation at the gas-phase S1-optimized structure of the D molecule are represented by closed black triangle symbols. Figure 10(b) shows that the donor transition charges for the gas-phase S1-optimized structure are in fairly good agreement with those for the gas-phase GS-optimized structure, which supports our assumption for the donor transition charges. (The corresponding numerical values are listed in Table S4 in the Supporting Information.) In the following subsection, we will use these donor transition charges with the Frozen-TrESP method to examine the dependence of the Coulomb interaction on the alignment of YFP with respect to CFP. Next we discuss the effect of the size of the employed QM region for the calculation of the Coulomb interaction between FPs. Following the study by Ansbacher et al.,36 we have assigned only the π-conjugated part of the fluorophore to the QM region for both the CFP and YFP, as shown in Figure 2. To explain the fine color tuning mechanisms of FPs, previous theoretical studies13−17,19−21,25,27 have considered the extended QM regions including the amino acids and water molecules in the vicinity of the fluorophore. Recently, these extensions of the QM regions25,27 have succeeded in accounting for the bathochromic shift in GFP; Kaila et al.25 have shown that the red-shifts of the absorption spectra caused by the protein environment are mainly attributed to the protein electrostatic effects and steric effects. If the charge transfer transitions from the fluorophore to the amino acids or water molecules occur in the photon absorption and emission of FPs, the transition

Figure 10. Average transition charges, qTrESP , taken over the noi embedding (NE) TD-CAM-B3LYP/6-31+G(d,p) results for the 501 MD snapshots (MD (NE), red open square) and the corresponding QM/MM results (MD (QM/MM), green closed square) obtained for the (a) acceptor (YFP fluorophore) and (b) donor (CFP fluorophore) molecules. Error bars represent standard deviation from the mean values. The reference transition charges used in the Frozen-TrESP method are plotted by the blue closed circle. For the D molecule, the reference transition charges obtained in the S1-optimized structure in the gas phase were also plotted by a black closed triangle.

numerical values are listed in Tables S3 and S4 in the Supporting Information. For comparison, Figures 10(a) and (b) again plot the reference transition charges of A and D molecules by closed blue circle symbols, respectively, which are identical with those plotted by blue lines in Figures 4(a) and (b), respectively. Figure 10 shows that the reference transition charge is in fairly good agreement with the average transition charge on most atom i’th positions except for those on the indole ring (C1−C9) of the D molecule. This agreement is attributed to the fact that the average geometries of D and A molecules taken over the MD snapshots conform with the corresponding GS optimized geometries, respectively, as shown in Figure 3. In Figure 10, the standard deviations of the transition charges from the average values are also represented by the corresponding error bars. The corresponding numerical values are listed in Tables S3 and S4 in the Supporting Information. We observed that the fluctuations of the transition charges of the atoms on the phenolic and indole rings are larger than those on the imidazolinone ring, and the protein electrostatic interactions slightly enhance the fluctuations, as shown in Figure 10. However, there is a good linear correlation between |VFrozen−TrESP| and |VTrESP| with a correlation coefficient η = 0.860 even in the existence of the fluctuations of the L


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The Journal of Physical Chemistry C charges and TDMs calculated from QC calculations would depend on the size of the QM region. However, there is no theoretical study showing the significant contributions of the charge transfer transitions to the absorption and emission properties of FPs, except for the artifact one from the fluorophore to the QM border.20 It has been believed that the π−π stacking interactions between the fluorophore and Try203 in YFP cause the red shift of YFP spectral properties with respect to the anionic GFP ones.2,19,87 The extension of the QM regions including Tyr20321,26 or a polarizable embedding TD-DFT approach26 cannot account for the redshift of YFP because of the difficulty in describing the π−π stacking interactions associated with the local polarizability. Since there is a close similarity in the chemical structures of the fluorophores between GFP and YFP and the π−π stacking interaction probably has a small influence on the TDMs, the TDM direction experimentally determined for the GFP has been used for describing the TDM direction of YFP.74,78,80 On the basis of these results, we assumed that the QM region including only the π-conjugated part of the fluorophore is sufficient to obtain the transition charges and TDM for estimating the Coulomb interaction between FPs. CFP−YFP Alignment Dependence of Coulomb Interactions. One of the main purposes of this study is to develop an efficient method to estimate the Coulomb interaction for the FRET between FPs in genetically encoded biosensor. When a biosensor works, the presence or absence of the targets causes a large conformational change of the sensing domains of the biosensor, leading to the substantial modification of the FRET efficiency observed spectroscopically.28−31 The candidate method, Frozen-TrESP, therefore, should be able to properly describe the variation of the Coulomb interaction caused by the large conformational change of the biosensor. So far, we have demonstrated the usefulness of Frozen-TrESP in estimating the Coulomb interactions for the FRET in the CFP−YFP dimer undergoing thermal fluctuation around the stable dimerization pose not involving the sensing and linker domains; therefore, the FRET reaction that we have presented so far corresponds to the one before or after the large conformational change in the biosensors utilizing the GFP dimerization propensity.75−77 There are difficulties in the atomistic simulations of the biosensors undergoing the large conformational change because they demand huge computational time and cost, and no entire crystal structures of the FRET-based biosensor are available as far as we know. To imitate one of the possible large conformational changes of the FRET-based biosensors as avoiding the difficulties, we rotated YFP with respect to CFP in the CFP−YFP heterodimer and then analyzed the dependence of the Coulomb interactions calculated with the Frozen-TrESP and Mapped-dd methods on the CFP−YFP alignment. This kind of analysis is of importance for optimally adjusting the dimerization manner of the FRETbased biosensors by directly introducing the binding sites at the surface of each of the GFP variants78 or by conducting the point mutations at the dimerization interface.35,79 First, we again adopted the CFP−YFP heterodimer structure used as the starting one for the MD simulations. Therefore, the internal coordinates of the heavy atoms of CFP and YFP are identical with the corresponding X-ray crystallographic data, 1OXD89 and 1YFP,87 respectively. As shown in Figure 11, the rotating axis was set to the longest-axis, XYFP, of YFP that was defined by the eigenvector of the nuclear quadrupole moment of YFP with the highest eigenvalue and lies on the center of nuclear charge

Figure 11. Variations of the orientation factor, κ2, and the absolute Coulomb interactions with Frozen-TrESP and Mapped-dd methods for the CFP−YFP heterodimer when CFP is fixed and YFP is virtually rotated along it is long-axis XYFP. The VFrozen‑TrESP values with the reference transition charges obtained for the gas-phase S1-optimized D molecule are also plotted by green broken lines denoted FrozenTrESP(S1).

of YFP.36 When performing the rigid-body rotation of YFP, we fixed the position of CFP. For the Frozen-TrESP and Mappeddd method, the reference transition charges and the amplitude of TDMs are derived from the TD-CAM-B3LYP/6-31+G(d,p)//B3LYP/6-31+G(d,p) results for the gas-phase D and A molecules. Figure 11 plots the obtained κ2 and absolute Coulomb interaction, |V|, as a function of the rotation angle around XYFP at each 5° rotation. We see that the resultant |V|-values exhibit that their maxima and minima appear at each 90° rotation: two maxima at about 15°−20° and 200°−210° and two minima at about 105° and 290°−300°. Those maxima and minima correspond to those of the resultant κ2 values. As shown in Figure 11, the dependence of the |VMapped−dd| values (red solid line) on the rotation angle are in good agreement with that of the |VFrozen‑TrESP| values (red dashed line). On the other hand, we see that the |VMapped‑dd| value somewhat deviates from the | VFrozen‑TrESP| value around the first peak position where we obtained the largest |VFrozen‑TrESP| of 47.0 cm−1 and the largest | VMapped‑dd| of 34.7 cm−1, respectively. Because the FRET rate can be estimated as kFRET ∝ |V|2 from the Fermi’s rate formula, eq 1, the Mapped-dd method reduces the FRET rate by about 45% compared to the Frozen-TrESP one. It is therefore prefereable to use the Frozen-TrESP method for the finetuning of the FRET-based biosensor. M


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Table 5. Chemical Structures and Reference Transition Charges for the Representative Fluorophore Models of Various FPs Listed in the Supporting Informationa,b chemical structure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure

S5A S5B S5C 2(b) S5D S5E 2(a) S5F S5G S5H S5I S5J S5K S5L S5M S5N S5O

relevant FP name BFPF(Y66F GFP) BFP, EBFP2 mBlueberry ECFP, CyPet, Cerulean GFP GoldFP YFP, EGFP, Venus, Citrine mHoneydew ZFP538 mOrange mKO DsRed, mCherry, mPlum asFP595-like asFP595-like Kaede PSmOrange TagBFP

qFrozen‑TrESP i

VEE

OS

|μ|

θ

Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table

3.76 3.89 3.34 3.54 3.70 3.25 3.12 3.05 2.73 2.63 2.61 2.48 2.68 2.72 2.34 2.46 3.12

0.662 0.586 0.681 0.685 0.771 0.55 1.08 0.733 1.04 1.03 1.05 1.09 0.966 1.02 0.919 1.00 0.538

2.68 2.48 2.88 2.81 2.92 2.63 3.76 3.13 3.95 3.99 4.05 4.24 3.83 3.92 4.01 4.08 2.65

5.08 1.52 16.8 12.4 2.77 14.4 4.92 14.8 12.9 12.6 12.1 2.88 1.81 0.410 7.12 4.327 1.465

S5 S6 S7 S2 S8 S9 S1 S10 S11 S12 S13 S14 S15 S16 S17 S18 S19

The obtained vertical excitation energy VEE (in eV), oscillator strength OS, TDM amplitude μ (in au), and TDM direction θ (in deg) are also listed. bResults obtained from the TD-CAM-B3LYP/6-31+G(d,p)//B3LYP/6-31+G(d,p) calculation in the gas phase.

a

are in good agreement with VFrozen‑TrESP(S1). From these results, we conclude that the Frozen-TrESP method with the reference transition charges for the gas-phase GS optimized structure properly describes the variation of the Coulomb interaction for the FRET in the biosensor. List of Reference Transition Charges of Other Fluorescent Proteins. Although the CFP−YFP pair is still the most practical one for FRET-based biosensor, other FP pairs, including recently optimized GFP-like (red) FPs, also have been used and proposed (see the recommended FP pairs and their spectral properties governing the FRET efficiency in refs 4 and 5). Since the red FPs lack the dimerization propensity and possess their broad absorption spectra with the lower quantum yields, the FRET biosensors with the red FPs tend to be suffering from the low FRET efficiency.6,29,35 Nevertheless, the utilization of the red FP possessing longer absorption and emission wavelength has a potential to expand the applicability of the FRET-based biosensor. For the design of FRET-based biosensor from structural information, Ansbacher et al.36 have made a comprehensive list of the TDM directions of commonly used FPs. In line with their strategy, to make use of the Frozen-TrESP method for a variety of FP pairs, we here make a list of the reference transition charges of the representative fluorophore models found in various FPs. We use the same procedure that we used for obtaining the reference transition charges of CFP and YFP as follows; the fluorophore of each FP is modeled as its πconjugated part, and the side chain is truncated and capped with a methyl group. After that, the gas-phase geometry optimization is performed at the B3LYP/6-31+G(d,p) level of theory for each fluorophore model, and then a single-point TDCAM-B3LYP/6-31+G(d,p) calculation is performed. We only consider the fluorophores in their cis-conformation forms because its trans isomer often exhibits no fluoroscence aside from a couple of exceptions.5,11 The chemical structures of the FP fluorophore model, the FP name represented by the fluorophore model, its gas-phase optimized geometry, and the obtained transition charges are summarized in Table 5. We also list the obtained VEE, OS, TDM amplitude |μ|, and TDM

Next we discuss the effect of the electronic polarizability of the protein and solvent on the variation of the Coulomb interaction along the rotation. Figure 11 also plots the |V|values in a dielectric media with ϵOpt = 1.96 by using the Poisson-Frozen-TrESP method (blue solid line) and Mappeddd corrected by the screening factor, 3/(2ϵOpt + 1) (blue dotted line). Figure 11 shows that the Onsager dipole model for the dd method well approximates the screening effect obtained from the |VPoisson‑Frozen‑TrESP| with respect to |VFrozen‑TrESP|, similar to Figure 8. We observed that a local field correction/screening factor, f = |VPoisson‑Frozen‑TrESP/VFrozen‑TrESP|, has its mean value of 0.573 and is almost independent of the rotational angle except for the |VFrozen‑TrESP| minimum position, which is close to the screening factor 3/(2ϵOpt + 1) = 0.610 for the dd method. From this result, the Onsager dipole model can properly take into account the dielectric effects on the Coulomb interactions from the dd method for CFP−YFP dimerizing conformation. As mentioned before, the donor transition charges and TDMs should be evaluated at the relaxed geometry on the S1 potential surface of the D molecule. However, the V calculations we have presented so far are based on the assumption that the transition charges and TDMs obtained for the gas-phase GS optimized and GS MD structures of the D molecule can approximate those for the corresponding S1 relaxed structures. Because the transition charges for the gasphase S1-optimized structure are not much different from those for the gas-phase GS-optimized structure, as shown in Figure 10, we expect that the assumption is probably correct. To roughly estimate the possible errors arising from the assumption, we applied the Frozen-TrESP method with the reference transition charges for the gas-phase S1-optimized structure to the V-calculations depending on the CFP−YFP alignment. Because the internal coordinates of the D molecule are taken from the X-ray crystallographic data,89 VFrozen‑TrESP(S1) includes the S1 relaxed effect on the D transition charges but ignores the S1 relaxed effect on D coordinates. In Figure 11, the resultant Coulomb interactions, VFrozen‑TrESP(S1), are plotted by the green broken line. We see that the VFrozen‑TrESP with the reference D transition charges for the GS-optimized structure N


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The Journal of Physical Chemistry C direction θ for each FP in Table 5. By using eq 6 with the reference transition charges and the geometry summarized in Table 5, one can easily reconstruct the corresponding TDM. Table 5 shows that the extension of the π-conjugated part decreases the VEEs, roughly corresponding to the order of color wavelength implied by the relevant FP name. The resultant θ ranges from 0.4° to 17°, indicating that the TDM almost lies along the long axis of the corresponding fluorophore model regardless of the size of the π-conjugated part. Although this finding is consistent with the previous one36 and supports the applicability of the Mapped-dd method, the increase in D/A sizes generally tends to increase the transition multipole contributions to the Coulomb interaction, and therefore the errors arising from the dd-approximation are probably increased. Modeling each FP fluorophore as its π-conjugated part makes it possible to apply a single set of the transition charges to multiple FPs; for example, the reference transition charges determined for YFP can be used for EGFP, Venus, Citrine, and so on. Hence, note that Table 5 lists the limited FP names represented by a single fluorophore model and is useful for the newly developed FP whose fluorophore possesses the identical π-conjugated part.

the calculations of the Coulomb interaction with the PoissonTrESP method to take account of the screening and local field effects by the protein/solvent dielectric environment with optical dielectric constant ϵOpt = 1.96.83 We obtained an almost constant screening factor f = 0.51 for the MD snapshots, which is close to the Onsager dipole-moment screening factor55 3/ (2ϵOpt + 1) = 0.61 applied to the dd-interaction, as shown in Figure 8. To develop a lower computational-cost approach within the TrESP method, we have tested the Frozen-TrESP method with the reference transition charges preliminary determined for the gas-phase D and A molecules. To clarify the advantage of the Frozen-TrESP method, we have also tested the Mapped-dd method36,71 where the reference TDM preliminary determined for the gas-phase D/A molecule is mapped onto the MD configuration space so as to lay the reference TDM along the long axis of the D/A molecule. Although ignoring the electronic polarizability of the transition charges, the Frozen-TrESP method reasonably reproduces the Coulomb interactions from the TrESP method involving every excited-state calculation for the MD trajectory; we obtained a good linear correlation η = 0.860 between |V Frozen‑TrESP| and |V TrESP(NE) |, which is substantially larger than η = 0.746 between |VMapped‑dd| and | VTrESP(NE)|, as shown in Figure 9. Finally, we have examined the dependence of the Coulomb interaction on the alignment of YFP with respect to CFP using the Frozen-TrESP and Mapped-dd methods. We found that the resultant Coulomb interaction exhibits two maxima at about 15°−20° and 200°−210° and two minima at about 105° and 290°−300° against the rotational angle around XYFP, as shown in Figure 11. Such information assists the design of future sensitive biosensors involving a large change in the FRET signal arising from conformational change of the sensor domain or GFP dimerization propensity. We found that the Mapped-dd method underestimates the FRET rate kFRET ∝ |V|2 by a factor of 0.65 at the large peak positions 15°−20°. Hence, it is preferable to use the Frozen-TrESP method for the fine-tuning of the FRET-based biosensor. In this study, we have demonstrated the usefulness of the Frozen-TrESP method with the MD trajectory and the rigidbody rotation based on the X-ray structures for the evaluation of the structure-dependent Coulomb interaction just for the CFP−YFP heterodimer. Note that all FPs exhibit a number of similarities in the global stricture (11-stranded β-barrel), fluorophore environment, location of the fluorophore in the β-barrel, and the chemical structure of the fluorophore.6 Therefore, we can expect that the Frozen-TrESP method is widely applicable to the estimations of the Coulomb interactions for various FP pairs, not just for the CFP−YFP pair. This prospect prompts us to make a list of the reference transition charges for various FP fluorophores, as summarized in Table 5. Because of the lack of the GFP dimerization propensity, no GFP-like protein FRET pairs with large FRET efficiency have been found so far.35 Making use of the GFP-like proteins with longer absorption and emission wavelength is desirable to expand the applicability of the FRET-based biosensor. As the fluorescent wavelength increases toward red color, the π-conjugated part of the GFP-like protein fluorophore is extended,4−6 and as a result, the errors arising from the dd-approximation are probably increased. Studies for this case using the list of the reference transition charges will be reported in the future.

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SUMMARY AND CONCLUSIONS We have examined the Coulomb interaction of the FRET reaction from CFP to YFP using the TrESP and ddapproximation methods at the TD-DFT level. The transition charges and TDM derived from the transition density for each D and A molecule are used in the TrESP method, eq 3, and ddinteraction method, eq 4, respectively. Hence, the performance of several hybrid and long-range corrected XC functionals was first assessed on the evaluations of the transition charges and TDMs of the gas-phase CFP and YFP fluorophores, namely, D and A molecules, respectively. After confirming the low dependence of the resultant transition charges and TDMs on the tested XC functionals, as shown in Figure 4 and listed in Tables 1 and 2, we entirely adopted the CAM-B3LYP functional for the estimation of the Coulomb interaction. The structure of the CFP−YFP heterodimer was modeled after the X-ray structure of the wild-type GFP homodimer. The thermal fluctuation of the solvated CFP−YFP heterodimer at room temperature was reproduced by the MD simulations. We have employed TD-CAM-B3LYP/6-31+G(d,p) calculations within the QM/MM and NE schemes for each D and A molecular geometry taken from the MD trajectory. As shown in Figure 7(b), we found that there is a satisfactory linear correlation η = 0.90 between the Coulomb interaction obtained from the TrESP method with the QM/MM and NE calculations. These results indicate that the thermal fluctuations of the D and A molecular geometries in protein and the protein/solvent electrostatic interactions do not significantly affect the Coulomb interaction between CFP and YFP. The MD simulations indicate that the CFP−YFP dimerization was stable, and the protein environment packs the fluorophore tightly inside the FPs, resulting in the moderate fluctuation of the DA distance around R = 25.1 Å and the relative orientation between D and A TDMs around κ = 2.37. Figures 6 and 7(a) show that these restricted mutual motions of D and A molecules lead to the validity of the dd-approximation for the estimation of the Coulomb interaction in the CFP−YFP heterodimer, which supports the previous dd-based analyses36,74,78,80 of the FRET in FP dimers. We have performed O


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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b00833. Atomic point charges defined at the QM/MM border, D and A geometries obtained from the gas-phase B3LYP/631+G(d,p) optimizations, D and A reference transition charges from the TD-CAM-B3LYP/6-31+G(d,p) calculations plotted in Figures 4 and 10, the excitonic couplings calculated for the face-to-face eclipsed homodimer of D molecules, RMSDs from the starting structure, the QM/MM convergence tests with respect to the size of the MM region, D and A van der Waals radii used for the Poisson-TrESP method, average transition charges and standard deviations plotted in Figure 10, and the reference transition charges of the representative fluorophore models found in various FPs as summarized in Table 5 (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Phone: +81 52 747 6397. Fax: +81 52 747 63977. ORCID

Stephan Irle: 0000-0003-4995-4991 Present Address §

Center for Computational Sciences, University of Tsukuba, 1− 1−1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS MD simulations were performed at Research Center for Computational Science of the Institute for Molecular Science (IMS) in Okazaki, Japan. All calculations of the Coulomb interactions were supported by Collaborative Research Program for Young Scientists of ACCMS and IIMC, Kyoto University (H.K.-H.). S.I. acknowledges partial support by a CREST grant from JST.

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FÃ¶rster Resonance Energy Transfer between Fluorescent Proteins

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