Atomistic Modeling of TripletâTriplet Energy-Transfer Rates from Drug

Article pubs.acs.org/JPCC

Atomistic Modeling of Triplet−Triplet Energy-Transfer Rates from Drug (S)‑Propranolol to (R)‑Cinacalcet in Human α1‑Acid Glycoprotein Yubing Si,†,∥ Baocheng Yang,†,∥ Haimei Qin,§ Jinyun Yuan,† Shuaiwei Wang,† Houyang Chen,*,‡ and Yi Zhao*,§ †

Institute of Nanostructured Functional Materials, Huanghe Science and Technology College, Zhengzhou, Henan 450006, China Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, New York 14260-4200, United States § State Key Laboratory for Physical Chemistry of Solid Surfaces, Collaborative Innovation Center of Chemistry for Energy Materials, Fujian Provincial Key Lab of Theoretical and Computational Chemistry, and College of Chemistry and Chemical Engineering, Xiamen University, Xiamen 361005, China ‡

S Supporting Information *

ABSTRACT: Triplet−triplet energy transfer (TTET) is one of the potential approaches to detect drug−drug interactions in protein environments. Here, quantum mechanism/molecular mechanics (QM/ MM), molecular dynamics (MD), and rate theories are employed to quantitatively predict the TTET rates from a drug (S)-propranolol (PPN) to (R)-cinacalcet (CIN) within the cavity of human α1-acid glycoprotein. The results indicate that the TTET rates from the PPN to CIN can be described by a Marcus-type theory for electron transfer. As the drugs are set into the protein cavity, the total reorganization energy is enhanced from 0.796 to 0.870 eV, and the thermal motions of drugs and protein cause dramatic electronic coupling fluctuations. However, the fluctuation effect on TTET rates can be efficiently considered by a thermally averaged electronic coupling, which is confirmed by the rate calculations obviously incorporating the non-Condon effect. Furthermore, Fermi’s golden rule predicts the consistent TTET rates with experimental ones, demonstrating the importance of nuclear tunneling effect.

1. INTRODUCTION Triplet−triplet energy transfer (TTET) is involved in many biological and chemical processes.1−3 Its concept is that the triplet-state energy of donor, commonly produced by a fast intersystem crossing from a singlet excited-state, is transferred into the triplet state of acceptor. Compared to singlet−singlet energy transfer, TTET has several distinctive behaviors. TTET is sensitive to donor−acceptor distances, and it only occurs when the donor and acceptor have a significant orbital overlap because of a two-electron exchange process involved. Since the intrinsic triplet-state lifetime of donor is on a microsecond time scale, TTET can be designed to detect molecular conformational dynamics, such as in protein folding.4 In addition, the triplet state is 3-fold degenerate, implying that 75% of created excitons would be phosphorescent rather than luminescent. For this reason, TTET has been utilized in phosphorescence-based light-emitting materials and low-energy up-conversion systems.5−9 However, the triplet states also have a negative effect on human and nature;10−12 for instance, the triplets of chlorophylls can sensitize oxygen and result in a highly oxidative singlet oxygen, which is especially harmful to the photosynthetic apparatus of plants. Therefore, it becomes © XXXX American Chemical Society

important to reveal the detailed TTET mechanism from molecular geometries and electronic structures for optimally using TTET and avoiding its demerits. In this paper, theoretical calculations are employed to investigate the TTET mechanism and quantitatively predict the TTET rate from a drug (S)-propranolol (PPN) donor to (R)cinacalcet (CIN) acceptor in human α1-acid glycoproteins (HAAG). When PPN and CIN are used together, their interaction may lead to side effects in patients, such as anxiety, diarrhea, and asthenia.13 Experimentally, Nuin et al.14,15 have suggested the use of the TTET rate for the detection of the PPN−CIN interaction in protein cavities, and they found that the TTET rate is very fast (>5 × 107s−1) because the donor and acceptor have a contact interaction in a common protein binding site. The excited-state dynamics of PPN and CIN encapsulated within a mixed micelle microenvironment has been further revealed by time-resolved fluorescence and laser flash photolysis techniques.16 These investigations are very Received: December 24, 2014 Revised: March 26, 2015

A

DOI: 10.1021/jp512844q J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

2. NUMERICAL TECHNIQUES 2.1. Electronic Structure Calculations. The singlet- and triplet-state geometries of drugs PPN and CIN are optimized by using DFT at a B3LYP/6-31G* level. A long-rangecorrected functional ωB97X-D38 with a larger basis set 6-311+ +G** is also employed in geometric optimizations for the purpose of comparison. The CDFT module with B3LYP/631G* in Q-Chem software package39 is adopted to construct diabatic states 3PPN−CIN and PPN−3CIN. The initial arrangement of PPN−CIN in HAAG is taken from ref 15, and Becke weights40 are used in the constrained population analysis to guarantee that triplet spin densities are located on PPN and CIN, respectively. The same technique for localization has been successfully used in our previous work.36,37,41 The geometrical fluctuation of PPN/CIN/HAAG is demonstrated by MD simulations under the NVE ensemble protocol at 300 K in the Gromacs 4.5.5 package.42 The unitedatom GROMOS96 43A1 force field is employed with a cutoff of 15 Å for nonbonded interactions. The total simulation time is 30 ns with a 1.0 fs time step. The electronic couplings are then calculated at 1000 snapshots of geometries extracted from the trajectory in last 10 ns. In the concrete implementation, the QM/MM calculations within an ONIOM formalism are performed to incorporate the HAAG effect on states 3PPN− CIN and PPN−3CIN. The system is partitioned into two layers: a “QM layer” including the PPN (or CIN) molecule treated at the DFT/B3LYP/6-31G* level and a “MM layer” including the rest of the residues described by the AMBER parm99 force field.43 The electronic embedding scheme is used in the QM/MM calculations to incorporate the partial charges of the MM region into the QM Hamiltonian. 2.2. TTET Rate Expression. Under a weak coupling limit, the TTET rate from the donor state 3PPN−CIN to acceptor state PPN−3CIN can be described by the Fermi Golden rule (FGR). However, the thermal motions of PPN/CIN/HAAG cause a strong coupling fluctuation, and this fluctuation generally entangles with energy-transfer dynamics. In this case, the corresponding TTET rate in a FGR scheme is given by32

helpful in understanding the relationship between the molecular arrangements in protein environment and energytransfer yield. Essentially, the TTET from PPN to CIN can be considered as a nonadiabatic reaction from the diabatic state 3PPN−CIN to the state PPN−3CIN. The TTET rate is thus controlled by not only the coupling between two diabatic states (drug−drug interaction) but also the driving force and molecular geometric rearrangements. At this point, TTET is very similar to an electron-transfer process. However, the obvious difference between most electron transfer and TTET is in geometric rearrangements during reactions. In the electron-transfer process, the effect of geometric arrangements is described by the reorganization energy because molecular vibrational frequencies in the two diabatic states are very similar. Alternatively, the molecular spin-flip in TTET may result in different vibrational frequencies on the initial and final states. In this case, the definition of single reorganization energy is not suitable, and the nuclear rearrangement effect is commonly estimated by the spectral overlap between the emission of donor and the absorption of acceptor. In the paper, therefore, we will first demonstrate whether the definition of single reorganization energy in electron transfer is applicable or not to the TTET in the present systems. In concrete calculations, the constrained density functional theory (CDFT)17,18 is adopted to construct the initial and finial diabatic states, and then both the four-point model19 and the technique of sum over modespecific reorganization energies20−23 are used to demonstrate the linear response theory. We will show that the TTET in PPN to CIN can be well described by a Marcus-type rate theory24,25 for electron transfer. As drugs are set in a HAAG cavity, the TTET process becomes more complex. The straightforward cavity effect is to restrict two drugs in a small space and cause molecular geometrical deformations compared to their geometries in vacuo. The previous density functional theory (DFT) calculations15 have confirmed this phenomenon and found that the formation of preassociated complexes is a major reason for fast energy transfer. However, it is not clear how the protein environment affects the reorganization energy. Here, the environmental effects in the drugs/protein interfacial region are treated by a quantum mechanics/molecular mechanics (QM/MM) scheme.26−29 Succeedingly, the four-point model19 is used for the calculation of reorganization energy to involve in the protein effect. The other important effect of the cavity is to cause the strong fluctuation of donor−acceptor distance because of the thermal motions of protein, and this fluctuation may result in the fluctuations of both molecular energies and electronic couplings.28,30,31 In particular, the coupling fluctuation, the so-called dynamic disorder or non-Condon effect, may significantly affect the TTET rate. For instance, the nonCondon effect can dramatically change the Marcus parabolic shape with respect to the driving force.32 In the present work, we will further demonstrate the non-Condon effect on TTET rate from the averaged coupling33−37 along the trajectory obtained by molecular dynamics (MD) simulations and the couplings with respect to the mode-specific vibrational coordinates.20−22 The paper is arranged as follows. In section 2, we briefly summarize rate theories and electronic structure methods. Section 3 presents the results and discussion. The concluding remarks are given in section 4.

k=

1 ℏ2

+∞

∫−∞

dtC DA(t )C B(t )

(1)

Here, CDA(t) represents the standard FGR expression (Condon approximation), and it has the form C DA(t ) =

1 Tr[e−βHDeiHDt / ℏe−iHAt / ℏ] QD

(2)

where β = 1/kbT, HD, and HA are the nuclear Hamiltonians for donor and acceptor states, respectively, and QD is the partition function of donor state. CB(t) comes from the coupling fluctuation, and it is described by C B(t ) =

1 Tr[e−βHBeiHBt / ℏHDA(Q )e−iHBt / ℏHDA(Q )] QB

(3)

Here, HB represents the Hamiltonian of intermolecular modes, and the coupling HDA(Q) explicitly depends on the coordinates Q of intermolecular modes. QB is the partition function of intermolecular or bridge modes. In numerical simulations, the time propagate operator exp(±iHt/ℏ) has to be calculated. Fortunately, under the assumption that HD and HA are described by a collection of B


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Figure 1. Chemical structures of PPN (left) and CIN (right).

fluctuation and energy transfer are entangled, the rate must be calculated from eq 1. Commonly, HDA(Q) has a linear dependence on intermolecular mode coordinates, i.e., HDA = VDA(1 + αQ), where VDA represents a constant coupling and α is a constant. For the systems involved in bridges, HDA has an exponential expression HDA(Q) = VDAe−αQ, coming from the McConnell superexchange mechanism.47 In the simulations for realistic organic molecules, it is found that the HDA may also 2 have a Gaussian form HDA(Q) = VDAe−αQ . Fortunately, CB(t) have the analytical expressions for all above coupling forms.32 To obtain the rate prediction, the driving force, reorganization energy, and electronic coupling as well as its coordinate dependence are required. The detailed numerical techniques can be found in reference46 and they are outlined in the following sections.

harmonic oscillators with the same frequencies and different equilibrium positions, eq 2 has an analytical expression44 ⎛i C DA(t ) = exp⎜⎜ ΔGt − ⎝ℏ

∑ Si[(2ni + 1) − nie−iω t i

i

⎞ − (ni + 1)eiωit ]⎟⎟ ⎠

(4)

where ΔG is the driving force, ni = 1/(eβωi − 1), β = 1/kbT, T = temperature, Si = 1/2ωid2i is the Huang−Rhys factor, and ωi and di are the frequency and shift for the i-th mode, respectively. The multimodes can be further mapped into an effective mode with a frequency given by45 ω2 =

1 λ

∑ ωi2λi

3. RESULTS AND DISCUSSION 3.1. Electronic Structures of PPN/CIN. Starting from the chemical structures of PPN and CIN, shown in Figure 1, we optimize their singlet and triplet ground states at a DFT/ B3LYP/6-31G* level. It is found that the spin densities in the triplet states of PPN and CIN are located in naphthalene units, manifesting that the TTET should occur dominantly between naphthalene units. The optimized geometric parameters of naphthalene units are thus listed in Table 1, together with those of the singlet state from an X-ray measurement.48 Interestingly, the bond lengths and angles in both singlet and triplet states of PPN are very close to those in CIN, which may be explained by the rigid structure of naphthalene units. The singlet-state parameters in PPN are consistent with the experimental data48 and are also agreement with previous theoretical results.49 However, compared to the ground states, the bond lengths of naphthalene units of PPN and CIN show the interval variation for the triplet states. This may be caused by the redistribution of the π delocalization electrons, which made the C−C/CC bonds decrease or increase regularly. For the same reason, the bond angles do not change too much. Although the DFT/B3LYP calculations correctly predict the geometries of singlet ground states, it is not clear whether the method can also give out reasonable triplet-state geometries. From previous works,36−38,41,50 it is expected that the DFT/ B3LYP and DFT/ωB97XD may be good choices for tripletstate calculations. These functionals with two different basis sets are thus used to calculate the phosphorescence spectra of PPN and CIN in vacuo, and the results are listed in Table 2. It clearly shows that the DFT/B3LYP and DFT/ωB97XD predict similar phosphorescence spectra and the 6-31G* basis set is accurate enough for the present calculations because a more accurate basis set 6-311++G** does not improve energy gaps too much. For the comparison with experimental spectra

(5)

i

with a mode-specific reorganization λi = and total reorganization energy λ = ∑ i λ i . Under the Condon 2 ), eq 1 has a simpler approximation (C B (t) = H DA 32,44 expression 1

k=

/2ω2i d2i

p /2 2 ⎛ 2πHDA n + 1⎞ ⎟ ⎜ exp[−S(2n + 1)]Ip ℏ2ω ⎝ n ⎠

[2S(n(n + 1))1/2 ]

(6)

where S = λ/ω is the Huang−Rhys factor, p = ΔG/ℏω, and Ip is the modified Bessel function. At a high temperature (ℏω ≪ kBT), eq 6 can be further approximated to the well-known Marcus formula46 k=

⎡ (ΔG + λ)2 ⎤ 1 2 π HDA exp⎢ − ⎥ 4λkBT ⎦ ℏ λkBT ⎣

(7)

However, the Condon approximation may not be satisfied in the present system because of the strong coupling fluctuation. The conventional treatment is to use the averaged coupling ⟨H2DA⟩over possible intermolecular distances, which hints that the coupling fluctuation time is much slower than the energy transfer time. In this case, the time evolution of coupling eiHBt/ℏHDA(Q)e−iHBt/ℏ can be replaced by its zero-time operator HDA(Q), resulting in C B(t ) ≈ C B(0) =

1 2 Tr[e−βHBHDA(Q )2 ] = ⟨HDA ⟩ QB

(8)

The rate is then calculated from eq 6 or eq 7 with the replacement of H2DA by its average value. Equation 8 is also named as the static disorder of coupling. As the non-Condon effect, or dynamic disorder begins to play a role, i.e., the C


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DFT techniques.15 To construct triplet diabatic states, we use the QM/MM protocol within a two-layer ONIOM model. In the first-level layer, the PPN (or CIN) molecule is treated at a DFT/B3LYP/6-31G* level, and the second-level layer including the CIN (or PPN) and HAAG is calculated at a MM level. Parts C and D of Figure 2 show the spin density distributions of optimized triplet states of 3PPN/CIN/HAAG and PPN/3CIN/HAAG, respectively. These distributions are similar to those in vacuo, manifesting that the protein environment does not contaminate spins in PPN and CIN, although they have a strong interaction in the ground states. The driving force of TTET is then calculated, and the obtained value is about −0.007 eV, smaller than that in vacuo (−0.02 eV). 3.2. Reorganization Energy. One of methods for calculating reorganization energy is the four-point model originally proposed for electron transfer.19,21,52 In the implementation, the system can be either split into the neutral triplet donor and singlet acceptor parts, or taken as a unit. In both cases, the reorganization energy is calculated from

Table 1. Bond Lengths (Å) and Bond Angles (deg) of PPN and CIN from the Optimized Geometries (at B3LYP/631G* Level) and from X-ray Measurements48(Mo Kα, λ = 0.71069 Å, Monochromator 2θ = 12.16°) parameters

PPN

R(1,2) R(2,3) R(3,4) R(4,4A) R(4A,5) R(5,6) R(6,7) R(7,8) R(8,8A) R(8A,1) R(8A,4A) A(1,2,3) A(2,3,4) A(3,4,4A) A(4A,5,6) A(5,6,7) A(6,7,8) A(7,8,8A)

1.382 1.418 1.374 1.423 1.422 1.378 1.416 1.379 1.420 1.435 1.432 119.9 121.3 120.1 121.1 120.2 120.3 120.6

3

CIN

X-ray (PPN)48

1.452 1.361 1.440 1.412 1.411 1.434 1.366 1.441 1.409 1.432 1.454 121.4 120.0 121.1 121.6 119.5 120.1 121.9

1.371 1.444 1.364 1.423 1.419 1.369 1.418 1.383 1.416 1.435 1.448 116.9 121.1 122.1 121.9 120.4 120.0 120.5

3

PPN

CIN

1.445 1.362 1.445 1.412 1.413 1.431 1.369 1.440 1.401 1.432 1.448 119.6 120.2 121.5 121.2 119.8 120.4 120.8

1.381 1.415 1.374 1.421 1.422 1.376 1.415 1.379 1.425 1.438 1.438 122.0 120.3 120.2 121.2 119.7 120.6 121.5

λ = λ1 + λ 2 = [E(3Dfc) − E(3Dopt )] + [E(1A fc) − E(1Aopt )]

(9)

where opt and fc represent the optimized ground states and Franck−Condon excited states, respectively, and the spin multiplicity is labeled as the superscript. Two techniques are used to calculate the reorganization energy, and the obtained values are 0.798 and 0.792 eV for the CIN and PPN treated separately and together, respectively. The consistent results manifest that the interaction of triplet states between the donor and acceptor molecules are not strong. Although the four-point method is easy to implement, it cannot predict the mode-specific reorganization energies which are important to understand the energy transfer process in detail. The reorganization energy is alteratively calculated from20−23

shown in Table 2, which is measured in ethanol matrix,14 the phosphorescence spectra are further calculated in a solvent environment with use of a polarizable continuum model (PCM), and the static dielectric constant of ethanol (ϵ = 24.85) is used. The results confirm that the DFT/B3LYP/6-31G* is also suitable for triplet-state calculations. In the subsequent calculations, this method is used to calculate electronic structure parameters. As the PPN and CIN are put together, one needs to define the spin-localized (diabatic) states 3PPN−CIN and PPN−3CIN required in rate calculations. However, the DFT/B3LYP/631G* meets problems because it commonly predicts delocalized distribution of spin density on both PPN and CIN. Here, the CDFT calculation is adopted to constrain the total spin either on PPN or CIN. From the optimized diabatic states of 3PPN−CIN and PPN−3CIN by CDFT calculations, shown in parts A and B of Figure 2, it is found that the spin densities are indeed localized on the PPN and CIN (the Cartesian coordinates and spin density distributions on these states can be found in Tables S1 and S2 in the Supporting Information). Furthermore, the driving force of −0.02 eV is predicted from the two diabatic states, which is consistent with an experimental estimation.15 To investigate the HAAG effect on the driving force, the CDFT/B3LYP/6-31G* still meets a challenge because the HAAG contains 2868 atoms.51 The conventional QM/MM methods should be suitable toward this goal. In fact, Nuin et al. have already calculated the ground-state attractive energy (−14 kJ/mol) between the PPN and CIN and the energy (−46 kJ/ mol) between the HAAG and two drugs by Monte Carlo and

λ=

∑ λi = ∑ i

k

1 2 ωi ΔQ i2 2

(10)

where ΔQi represents the normal-mode coordinate shift between the donor and acceptor states and ωi represents the frequency for the i-th mode. In the numerical simulations, we first make the normal-mode analysis in both optimized donor and acceptor states in the same Cartesian coordinates. The shift Qi of the i-th mode between two optimized geometries is then calculated by matrix transformation techniques.33,46,53−57 The obtained total reorganization energy is 0.796 eV. This result is quite well consistent with that from the four-point method. Therefore, the TTET between PPN and CIN satisfies the linear response theory, and the rate theories for electron transfer are applicable to the TTET. Figure 3 shows mode-specific reorganization energies with respect to mode frequencies. It is interesting to note there are

Table 2. Phosphorescence Spectra (kcal/mol) of PPN and CIN 6-31G* PPN CIN

6-311++G**

6-31G*(PCM)

B3LYP

ωB97XD

B3LYP

ωB97XD

B3LYP

ωB97XD

expt14

52.95 52.17

52.53 52.34

51.86 51.57

52.35 51.80

61.57 61.75

64.70 64.67

61.4 60.9

D


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Figure 2. Spin densities of the 3PPN−CIN and PPN−3CIN in vacuo (A, B) and in protein (C, D), respectively.

As the drugs are surrounded by HAAG, the so-called outer reorganization energy caused by environment should be considered. How to calculate the total reorganization energy incorporating the outer contribution is still an uneasy task. It is noted that a functional model analysis has been proposed for this purpose.58 Fortunately, the four-point model can be straightforwardly applied in the two-layer ONIOM simulations, and the obtained result is 0.87 eV, about 0.10 eV larger than the inner reorganization energy (in vacuo). The enhancement of total reorganization energy may be caused by both the geometric distortion of drugs and the polar field of protein. It is seen from Figure 2C,D that the naphthalene chromophores in PPN and CIN become nearly parallel in the protein comparing with their geometries in vacuo, but the individual bond strengths and angles of PPN and CIN do not change too much. This may partially explain why the total reorganization energy is only slightly higher than that in vacuo. On the other hand, the PCM model can be used to investigate the effect of charged and polar groups in protein on the reorganization energy. Several previous works59−63 have already mapped the realistic protein environment into a PCM model. With use of the dielectric constant value of 4 for HAAG,59,62,64,65 the total reorganization energy is further calculated from the four-point model under the nonequilibrium PCM,37 and the obtained result is only 0.81 eV, close to the inner reorganization energy. The detailed analysis shows that TTET does not involve in a major redistribution of the charges between the PPN and CIN molecules, and the dipole moments of two diabatic states may be unchanged.66,67 In the present calculations, the dipole moments are 5.97 debye and 5.48 debye for 3DA and D3A, respectively, and they are very close to each other. Therefore, the corresponding solvent reorganization energy should be very small for HAAG (0.1 eV or less68,69). It is thus expected that the PCM can correctly account for the polarization effect of protein on the reorganization energy, but it cannot consider the geometric distortion effect of drugs caused by protein environment. 3.3. Coupling between States 3 PPN−CIN and PPN−3CIN. At a given molecular geometry, the coupling

Figure 3. Reorganization energies with respect to the mode frequencies in vacuo.

five vibrational modes whose λi are greater than 200 cm−1. They cumulatively account for the most component of reorganization energy, and the rest of the 253 vibrational modes make a slight of contribution. In particular, the three modes with frequencies of 1407.7, 1633.1, and 1659.7 cm−1, corresponding to the C− C/CC bond vibrations in the naphthalene chromophore group, have very large reorganization energies compared to other modes. Although the high-frequency modes (>1750 cm−1) from the C−H stretching motions have a small contribution to the reorganization energy, their cumulation may affect the effective frequency of TTET. Indeed, the calculated effective frequencies from eq 5 are 1339.1 and 1392.9 cm−1 with use of the frequencies in the states 3PPN−CIN and PPN−3CIN, respectively. These quite large frequencies should have a significant quantum effect on the rate even at a room temperature. Furthermore, the consistent effective frequencies manifest that a single reorganization energy is well-defined and Marcus-type rate theory for electron transfer can be used to predict the TTET rate in a weak electronic coupling limit. E


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The Journal of Physical Chemistry C Table 3. Dependence of the Parameters of Coupling on the Intermolecular Mode Coordinates in Vacuo

a

mode

1

2

3

4

5

6

7

8

freq (cm−1) α typea

11.03 −0.0020 linear

50.85 8.3 × 10−6 Gauss

55.41 4.1 × 10−6 Gauss

60.25 0.0024 expo

67.98 1.7 × 10−6 Gauss

171.92 0.0014 linear

185.36 −0.0022 expo

190.01 −0.0018 expo

Linear, Gauss, and expo represent linear, Gaussian, and exponential dependencies, respectively.

between the donor state |3D A⟩ and acceptor state |D 3A⟩ represents the overlap of their wave functions. For nonorthogonal wave functions, the coupling is given by18,35

modified NW-chem package,70 dramatically fluctuate with respect to the time of MD trajectory, and the maximum coupling corresponds to the parallel geometries of naphthalene chromophores in the PPN and CIN where the orbital wave function overlap has a maximum value. The averaged coupling is then calculated by

1

Hda =

Vda − 2 (Ed + Ea)Sda 2 1 − Sda

(11)

where Ed and Ea are the diabatic energies of the donor and acceptor states, respectively, and Vda and Sda are defined by Vda = ⟨3DA|Ĥ |D3A⟩ and Sda = ⟨ 3DA|D3A⟩. Since TTET involves in the exchange of two electrons, the coupling should strongly depend on geometrical changes. Indeed, the couplings obtained from eq 11 at the optimized 3 DA and D3A geometries have an obvious difference, and they are 0.22 and 0.18 meV, respectively. It is thus expected that the non-Condon effect may play an important role in the TTET rate. To investigate the detailed geometric effect on coupling, succeedingly, the dynamic disorder of coupling on the TTET rate, we calculate the coupling dependence of mode coordinate by changing the interesting mode coordinate with the other mode coordinates hold at equilibrium positions. The results show that the couplings have dramatic changes along the coordinates of several intermolecular modes, and these modes have little contribution to the reorganization energy. Therefore, the rate eq 1 can be used to investigate the dynamic disorder. Although those mode-coordinate dependencies of coupling have different shapes, they can be fitted by linear, exponential, or Gaussian functions, as shown in Table 3. More details regarding the intermolecular vibrational modes can be seen in Figure 1S of the Supporting Information. Alternatively, the average effect of dynamic disorder can be calculated from MD simulations. The 1000 ground-state geometries of PPN/CIN/HAAG along the trajectory are selected for the calculation of averaged electronic coupling. As shown in Figure 4, the couplings, obtained from the

Vrms =

⟨V 2⟩ =

1 n

n

∑ ⟨Vi2⟩ 1

(12)

where Vi is the electronic coupling at the i-th geometry. This coupling is suitable to estimate the TTET rate as the coupling fluctuation is considered as static disorder, i.e., the coupling fluctuation time is much slower than the energy transfer time. The obtained coupling from Figure 4 is 1.70 meV, about 10 times larger than those at the 3DA and D3A stable geometries, manifesting that the fluctuation helps energy transfer. 3.4. TTET Rate. With use of the obtained driving force, reorganization energy and coupling, the TTET rate from the PPN to CIN can be readily calculated from rate theories. It is noted that 4V/λ is 0.009, much smaller than 1. Therefore, FGR or Marcus theory is very suitable to estimate the rates.71 First, we focus on the TTET rate in vacuo where the driving force and reorganization are −0.02 and 0.796 eV, respectively. Although the protein environment is not incorporated, the drug molecules PPN and CIN have an attractive interaction and their intermolecular modes fluctuate around their equilibrium positions, leading to the non-Condon effect. Therefore, the FGR-type expression of eq 1 is used to calculate the rate to incorporate the dynamic disorder. For the purpose of comparison, the rate under the static disorder approximation is also calculated by eq 6 with the average coupling of 1.7 meV from MD simulation. At 300 K, the TTET rates are 3.7 × 109 s−1 and 2.6 × 109 s−1 for the dynamic and static disorder, respectively. Although the dynamic disorder slightly enhances the rate, both of the rates have the same order of magnitude. The insignificant behavior of dynamic disorder can be explained by the slow motions of intermolecular modes because their frequencies are typically smaller than 200 cm −1 . To demonstrate the nuclear tunneling contribution, we also calculate the rate from Marcus formula (eq 7) with use of the averaged coupling, and the predicted rate is about 5.3 × 107 s−1, 48 times smaller than that from the FGR. As mentioned above, the effective frequency is 1339.1 cm−1. Definitely, the nuclear tunneling effect on the TTET rate is extremely important. As the PPN and CIN are constrained in the HAAG cavity, the dominant cavity effect is to change the reorganization energy. Since the outer reorganization energy is contributed from the polarization and molecular rotational motions, the corresponding frequency of solvent mode is quite small. Therefore, the solvent mode should have little effect on the effective frequency, but it indeed changes the shift of two parabolic potentials for TTET. The rate can be thus calculated from eq 6 with the effective frequency of 1339.1 cm−1, but with

Figure 4. Fluctuation of the coupling in HAAG. F


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electronic couplings are sensitive to conformational changes, the rates can be correctly estimated from an averaged coupling strength because the non-Condon dynamic effect dominantly comes from low-frequency intermolecular vibrational modes below 200 cm−1, leading to a slow coupling fluctuation. The rate calculations further reveal that the Marcus-type mechanism for electron transfer is applicable, but quantum rate theories should be adopted because of a strong nuclear tunneling effect. Indeed, the rate from Fermi’s golden rule is 2 orders of magnitude greater than that from Marcus formula even at a room temperature. Finally, we point out that the dominant effect of HAAG cavity is to enhance the total reorganization energy and lead to slow TTET rates comparing to those in vacuo. The procedures used in this work may be applicable for the investigation of similar complexes in other biological systems.

the total reorganization energy of 0.870 eV. The estimated rate is 1.6 × 109 s−1, 1.6 times smaller than the one in vacuo. However, the rate from Marcus formula is 2.3 × 107 s−1, nearly two orders smaller than that from FGR. It now becomes obvious that the HAAG has a contribution to the TTET rate from both the coupling fluctuation and reorganization energy, and the former prefers to enhance the rate whereas the latter lowers the rate. The overall effect makes the rate lower. In spite of this negative effect, the TTET rate in HAAG cavity is still quite large, and it has to be measured by ultrafast experiments. For instance, a recent experiment using a nanosecond laser pulse cannot resolve such a fast TTET process, and the estimated rate is higher than 5.0 × 107 s−1.15 The present result should be further confirmed by the experiments with picosecond laser pulses. Figure 5 displays the temperature dependence of rates with and without the HAAG cavity from Marcus and FGR formulas.

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ASSOCIATED CONTENT

S Supporting Information *

Cartesian coordinates and spin density distribution of 3DA and D3A as well as the intermolecular vibrational modes with large non-Condon factors. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: hchen23@buffalo.edu. *E-mail: [email protected]. Author Contributions ∥

These authors contributed equally to this work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Professor Miguel A. Miranda and Dr. Weiwei Zhang as well as Dr. Pengfei Li for their helpful discussions. This work is supported by the Leading Talents for Zhengzhou Science and Technology Bureau (Grant No. 131PLJRC649) and the program for University Innovation Talents of Science and Technology in Henan Province (Grant No. 2012HASTIT036) as well as the NSFC support (Grant Nos. 21206049 and 51472102). Y.Z. thanks the NSFC support (Grant Nos. 21133007, 21073146).

Figure 5. Temperature dependence of the TTET rates.

As expected, the HAAG effect lowers the rates about 1 order of magnitude compared to those in vacuo. In the regime of temperatures smaller than 1000K, FGR always predicts larger rates than Marcus formula. We conclude that although the Marcus-type mechanism for electron transfer can be used to describe the TTET for the present system, quantum rate expressions are required because of a large nuclear tunneling effect.

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4. CONCLUDING REMARKS The triplet−triplet energy transfer (TTET) rate between two drugs PPN and CIN in a HAAG cavity is theoretically investigated by using QM/MM calculations, MD simulations, and rate theories. The key parameters controlling the TTET rate, the driven force, electronic coupling, and reorganization energy have been determined from the diabatic states constructed by a constrained density functional approach. The electronic structure calculations show that only several high-frequency intramolecular vibrational modes are prevalent in the reorganization energy, leading to an important nuclear tunneling effect. Moreover, those high-frequency vibrational modes come from the C−C/CC bonds in the naphthalene chromophore group. It is thus expected that other drugs based on the naphthalene core (such as nabumetone and naproxen), a functional group for intraprotein drug−drug interactions, should have a similar energy-transfer behavior. Although the

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